**A Methodology Based on Experimental Investigation of a DBD-Plasma Actuated Cylinder Wake for Flow Control**

Kelly Cohen1, Selin Aradag2, Stefan Siegel3, Jurgen Seidel3 and Tom McLaughlin3 *1University of Cincinnati, Ohio, 2TOBB University of Economics and Technology 3US Air Force Academy, Colorado 1,3USA 2Turkey* 

## **1. Introduction**

116 Low Reynolds Number Aerodynamics and Transition

Paksoy, A.; Apacoglu, B. & Aradag, S. (2010). Analysis of Flow over a Circular Cylinder by

Paksoy, A. & Aradag, S. (2011). Prediction of Lid-Driven Cavity Flow Characteristics Using

Samarasinghe S. (2006). *Neural Networks for Applied Sciences and Engineering, From* 

Sanghi, S. & Hasan, N. (2011). Proper Orthogonal Decomposition and Its Applications. *Asia-*

Siegel, S.; Cohen, K.; Seidel, J.; Aradag, S. & McLaughlin, T. (2008). Low Dimensional Model

Seidel, J.; Cohen, K.; Aradag, S.; Siegel, S. & McLaughlin, T. (2007). Reduced Order

Sirovich, L. (1987). Turbulence and the Dynamics of Coherent Structures, Part I: Coherent Structures. *Quarterly Applied Mathematics*, Vol. 45, No. 3, pp. 561-571 Smith, T.R.; Moehlis, J. & Holmes P. (2005). Low Dimensional Models for Turbulent Plane

Travin, A.; Shur, M.; Strelets, M. & Spalart, P. (1999). Detached-Eddy Simulations Past a Circular Cylinder Flow. *Turbulence and Combustion*, Vol. 63, pp. 293- 313 Unal, M. & Rockwell, D. (2002). On Vortex Shedding from a Cylinder, Part 1: The Initial

Wissink, J.G. & Rodi, W. (2008). Numerical Study of the Near Wake of a Circular Cylinder.

Xie G.; Sunden B.; Wang Q. & Tang L. (2009). Performance Predictions of Laminar and

Zhang, L.; Akiyama, M.; Huang, K.; Sugiyama H. & Ninomiya, N. (1996). Estimation of Flow

Turbulent Heat Transfer and Fluid Flow of Heat Exchangers Having Large TubeRow by Artificial Neural Networks. *International Journal of Heat and Mass* 

Patterns by Applying Artificial Neural Networks. *Information Intelligence and* 

*AIAA Fluid Dynamics Conference and Exhibit*, Miami, Florida, June 2007 Sen., M.; Bhaganagar, K. & Juttijudata, V. (2007). Application of Proper Orthogonal

*Engineering Systems Design and Analysis*, Istanbul, Turkey, July 2010

*Computational Heat and Mass Transfer*, Istanbul, Turkey, July 2011

*Pacific Journal of Chemical Engineering*, Vol. 6, pp. 120-128

with Rough Walls. *Journal of Turbulence*, Vol. 8, No. 41

Instability. *Journal of Fluid Mechanics*, Vol. 190, pp. 491-512

*Transfer*, Vol. 52, pp. 2484-2497

*Systems*, Vol. 4, pp. 1358-1363

*International Journal of Heat and Fluid Flow*, Vol. 29, pp. 1060-1070

Francis Group

June 2008

110.

CFD and Reduced Order Modeling, *Proceedings of ASME 10th Biennial Conference on* 

an Artificial Neural Network Based Methodology Combined With CFD and Proper Orthogonal Decomposition, *Proceedings of the 7th International Conference on* 

*Fundamentals to Complex Pattern Recognition*. Auerbach Publications Taylor and

Development Using Double Proper Orthogonal Decomposition and System Identification, *Proceedings of the 4th Flow Control Conference*, Seattle, Washington,

Modeling of a Turbulent Three Dimensional Cylinder Wake, *Proceedings of the 37th* 

Decomposition (POD) to Investigate a Turbulent Boundary Layer in a Channel

Couette Flow in a Minimal Flow Unit. *Journal of Fluid Mechanics*, Vol. 538, pp. 71-

The main purpose of flow control is to improve the mission performance of air vehicles. Flow control can either be passive or active and active flow control is further characterized by open-loop or closed-loop techniques. Gad-el-Hak (1996) provides an insight into the advances in the field of flow control. Research of closed-loop flow control methods has increased over the past two decades. Cattafesta et al (2003) provide a useful classification of active flow control.

Before proceeding into the details of modeling and control, it is imperative to appreciate the reasons as to why closed-loop control is of importance and the main advantages associated with its application to flow control problems. It is advantageous to opt for closed-loop flow control for the following reasons:


Several applications of closed-loop control have been reported in literature, namely, specific areas of interest include flow-induced cavity resonance. (Cattafesta et al, 2003, Samimy et al, 2003), vectoring control of a turbulent jet (Rapoport et al, 2003), separation control of the NACA-4412 Airfoil (Glauser, 2004) and control of vortex shedding in circular cylinder wakes (Gerhard et al, 2003, Gillies, 1995). The ability to control the wake of a bluff body could be used to reduce drag, increase mixing and heat transfer, and vibration reduction.

We can consider the cylinder wake problem. In a two-dimensional cylinder wake, selfexcited oscillations in the form of periodic shedding of vortices referred to as the von Kármán Vortex Street. Shedding of counter-rotating vortices is observed in the wake of a two-dimensional cylinder above a critical Reynolds number (Re ~ 47, non-dimensionalized with respect to free stream speed and cylinder diameter). An effective way of suppressing the self-excited flow oscillations, without making changes to the geometry or introducing vast amounts of energy, is by the incorporation of active closed-loop flow control (Gillies, 1995). A closed-loop flow control system is comprised of a controller that introduces a perturbation into the flow, via a set of actuators, to obtain desired performance. Furthermore, the controller acts upon information provided by a set of sensors. During the past years, the closed-loop flow control program research effort at the United States Air Force Academy (USAFA) focused on developing a suite of low-dimensional flow control tools based on the low Reynolds numbers (Re ~ 100-300) cylinder wake benchmark (Cohen et al, 2003, Cohen et al, 2004, Cohen et al, 2005, Cohen et al, 2006a, Siegel et al, 2003a) Several computations and experiments were also performed for the cylinder wake at high Reynolds numbers (Re=20000) (Aradag, 2009, Aradag et al, 2010)

Energy is introduced into the flow via actuators and the flow field in the wake of a cylinder may be influenced using several different forcing techniques with the wake response being similar for different types of forcing (Gillies, 1998) The following forcing methods have been employed: external acoustic excitation of the wake, longitudinal, lateral or rotational vibration of the cylinder, and alternate blowing and suction at the separation points (Gillies, 1998). Work at USAFA has shown that the Dielectric Barrier Discharge (DBD) plasma actuator (Munska and McLaughlin, 2005) is an effective means of forcing at higher frequencies without mechanical movement. This relatively simple actuation device is composed of two thin electrodes separated by a dielectric barrier. When an AC voltage is applied to the electrodes, a plasma discharge propagates from the edge of the exposed electrode over the insulated electrode. The emergence of this plasma is accompanied by a coupling of directed momentum into the surrounding air as the plasma propagates over the buried electrode during each oscillation forcing cycle (Enloe et al, 2004). This momentum can effectively alter a moving flow or generate flow in the direction of plasma propagation, as several application-based papers have shown (List et al, 2003, Asghar and Jumper, 2003, Bevan et al, 2003). The non-mechanical nature of the plasma actuator makes it ideal for high Re flow control applications. Its high fundamental operating frequency suggests it can be effective over a very wide bandwidth (by fluid time scale standards). This enables operation over a much broader range of frequencies than mechanical actuators. It has no moving parts, and has no resonant frequency. Munska and McLaughlin (2005) established that plasma actuators can achieve vortex shedding lock-in and span-wise coherence over a range of forcing conditions. They employed a cylinder arrangement similar to that of Asghar and Jumper (2003), with electrodes at ±90º and Re up to 88x103, and used a similar amplitudemodulated forcing scheme.Low-dimensional modeling is a vital building block when it comes to realizing a structured model-based closed-loop flow control strategy. For control purposes, a practical procedure is needed to represent the velocity field, governed by the Navier Stokes partial differential equations, by separating space and time. A common method used to substantially reduce the order of the model is Proper Orthogonal Decomposition (POD). This method is an optimal approach in that it will capture a larger amount of the flow energy in the fewest modes of any decomposition of the flow. The two

We can consider the cylinder wake problem. In a two-dimensional cylinder wake, selfexcited oscillations in the form of periodic shedding of vortices referred to as the von Kármán Vortex Street. Shedding of counter-rotating vortices is observed in the wake of a two-dimensional cylinder above a critical Reynolds number (Re ~ 47, non-dimensionalized with respect to free stream speed and cylinder diameter). An effective way of suppressing the self-excited flow oscillations, without making changes to the geometry or introducing vast amounts of energy, is by the incorporation of active closed-loop flow control (Gillies, 1995). A closed-loop flow control system is comprised of a controller that introduces a perturbation into the flow, via a set of actuators, to obtain desired performance. Furthermore, the controller acts upon information provided by a set of sensors. During the past years, the closed-loop flow control program research effort at the United States Air Force Academy (USAFA) focused on developing a suite of low-dimensional flow control tools based on the low Reynolds numbers (Re ~ 100-300) cylinder wake benchmark (Cohen et al, 2003, Cohen et al, 2004, Cohen et al, 2005, Cohen et al, 2006a, Siegel et al, 2003a) Several computations and experiments were also performed for the cylinder wake at high

Energy is introduced into the flow via actuators and the flow field in the wake of a cylinder may be influenced using several different forcing techniques with the wake response being similar for different types of forcing (Gillies, 1998) The following forcing methods have been employed: external acoustic excitation of the wake, longitudinal, lateral or rotational vibration of the cylinder, and alternate blowing and suction at the separation points (Gillies, 1998). Work at USAFA has shown that the Dielectric Barrier Discharge (DBD) plasma actuator (Munska and McLaughlin, 2005) is an effective means of forcing at higher frequencies without mechanical movement. This relatively simple actuation device is composed of two thin electrodes separated by a dielectric barrier. When an AC voltage is applied to the electrodes, a plasma discharge propagates from the edge of the exposed electrode over the insulated electrode. The emergence of this plasma is accompanied by a coupling of directed momentum into the surrounding air as the plasma propagates over the buried electrode during each oscillation forcing cycle (Enloe et al, 2004). This momentum can effectively alter a moving flow or generate flow in the direction of plasma propagation, as several application-based papers have shown (List et al, 2003, Asghar and Jumper, 2003, Bevan et al, 2003). The non-mechanical nature of the plasma actuator makes it ideal for high Re flow control applications. Its high fundamental operating frequency suggests it can be effective over a very wide bandwidth (by fluid time scale standards). This enables operation over a much broader range of frequencies than mechanical actuators. It has no moving parts, and has no resonant frequency. Munska and McLaughlin (2005) established that plasma actuators can achieve vortex shedding lock-in and span-wise coherence over a range of forcing conditions. They employed a cylinder arrangement similar to that of Asghar and Jumper (2003), with electrodes at ±90º and Re up to 88x103, and used a similar amplitudemodulated forcing scheme.Low-dimensional modeling is a vital building block when it comes to realizing a structured model-based closed-loop flow control strategy. For control purposes, a practical procedure is needed to represent the velocity field, governed by the Navier Stokes partial differential equations, by separating space and time. A common method used to substantially reduce the order of the model is Proper Orthogonal Decomposition (POD). This method is an optimal approach in that it will capture a larger amount of the flow energy in the fewest modes of any decomposition of the flow. The two

Reynolds numbers (Re=20000) (Aradag, 2009, Aradag et al, 2010)

dimensional POD method was used to identify the characteristic features, or modes, of a cylinder wake as demonstrated by Gillies (1998) and Gerhard et al (2003).

The major building blocks of the structured approach presented here are comprised of a reduced-order POD model, a state estimator and a controller. The desired POD model contains an adequate number of modes to enable accurate modeling of the temporal and spatial characteristics of the large scale coherent structures inherent in the flow in order to model the dynamics of the flow. A Galerkin projection may be used to derive a set of reduced order ordinary differential equations by projecting the Navier-Stokes equations on to the modes (Holmes et al, 1996). Further details of the POD method may be found in Holmes et al (1996). A common approach referred to as the method of "snapshots" introduced by Sirovich (1987) is employed to generate the basis functions of the POD spatial modes from flow-field information obtained using either experiments or numerical simulations. This approach to controlling the global wake behavior behind a circular cylinder was effectively employed by Gillies (1998) and Noack et al (2004) and is also the approach followed in the current research effort.

For practical applications, it is important to estimate the state of the flow, i.e. the relevant POD time coefficients, using body mounted sensors. The advantages of body mounted sensors are:


Pressure sensors, mounted on the surface have been used on a back-ward facing ramp by Taylor and Glauser (2004) and by Glauser et al (2004) on a NACA 4412 airfoil. Recent efforts have successfully demonstrated estimation of the time coefficients of the POD model for a "D" shaped cylinder for laminar flow at low Reynolds numbers (Cohen et al, 2004, Stalnov et al, 2005). The body mounted sensors may measure skin friction or surface pressures, as is done in this effort. The intention of the proposed strategy is that the measurements, provided by a certain configuration of body mounted pressure sensors placed on the model surface, are processed by an estimator to provide the real-time estimates of the POD time coefficients that are used to close the feedback loop. The estimation scheme is to behave as a modal filter that "combs out" the higher modes. The main aim of this approach is to thereby circumvent the destabilizing effects of observation "spillover". The estimation scheme may be based on the linear stochastic estimation procedure introduced by Adrian et al (1977) or a quadratic stochastic estimation proposed by Murray and Ukeiley (2002) as well as by Ausseur et al (2006).

This chapter is organized as follows: The following section provides the main objective of this chapter. The basic approach to feedback flow control for turbulent wake flows is presented in Section III. A wind tunnel experiment of a plasma actuated cylinder wake, at a Reynolds number of 20,000, is described in Section IV. Preliminary experimental results using POD and a Neural Network based estimator and a subsequent discussion are presented in Section V. Finally, the conclusions of this research effort and recommendations for future work are summarized in Section VI.

## **2. Aims and concerns**

Technological advances in sensors, actuators, on-board computational capability, modeling and control sciences have offered a possibility of seriously considering closed-loop flow control for practical applications. The main strategies to closed-loop control are a modelindependent, full-order optimal control approach based on the Navier-Stokes equations and a reduced order model strategy. This effort emphasizes the methodology based on the lowdimensional, proper orthogonal decomposition method applied to the problem concerning the suppression of the von Kármán vortex-street in the wake of a circular cylinder. Focus is on the validity of the low-dimensional model, selection of the important modes that need representation, incorporation of ensembles of snapshots that reflect vital transient phenomena, selection of sensor placement and number, and linear stochastic estimation for mapping of sensor data onto modal information. Furthermore, additional issues surveyed include observability, controllability and stability of the closed-loop systems based on lowdimensional models. Case studies based on computational and experimental studies on the cylinder wake benchmark are presented to illuminate some of the important issues.

## **3. Research methods**

## **3.1 Closed loop control methodology**

Based on the research effort at the USAF Academy over the past years, a methodology for approaching closed-loop flow control has been developed. This approach has been applied to control of laminar bluff body wakes at low Reynolds numbers (Re~50-180). In this work, this methodology is extended to higher Reynolds number turbulent wakes (Re~20,000). A schematic representation of the setup is presented in Figure 1.

Fig. 1. Methodology for Closed-Loop Flow Control.

The following is a more detailed look into each of the six steps:

a. Identification of the "Lock-In" Region

120 Low Reynolds Number Aerodynamics and Transition

Technological advances in sensors, actuators, on-board computational capability, modeling and control sciences have offered a possibility of seriously considering closed-loop flow control for practical applications. The main strategies to closed-loop control are a modelindependent, full-order optimal control approach based on the Navier-Stokes equations and a reduced order model strategy. This effort emphasizes the methodology based on the lowdimensional, proper orthogonal decomposition method applied to the problem concerning the suppression of the von Kármán vortex-street in the wake of a circular cylinder. Focus is on the validity of the low-dimensional model, selection of the important modes that need representation, incorporation of ensembles of snapshots that reflect vital transient phenomena, selection of sensor placement and number, and linear stochastic estimation for mapping of sensor data onto modal information. Furthermore, additional issues surveyed include observability, controllability and stability of the closed-loop systems based on lowdimensional models. Case studies based on computational and experimental studies on the

cylinder wake benchmark are presented to illuminate some of the important issues.

Based on the research effort at the USAF Academy over the past years, a methodology for approaching closed-loop flow control has been developed. This approach has been applied to control of laminar bluff body wakes at low Reynolds numbers (Re~50-180). In this work, this methodology is extended to higher Reynolds number turbulent wakes (Re~20,000). A

**2. Aims and concerns** 

**3. Research methods** 

**3.1 Closed loop control methodology** 

schematic representation of the setup is presented in Figure 1.

Fig. 1. Methodology for Closed-Loop Flow Control.

In order to obtain a meaningful low-order representation of the flow, it is imperative that the behavior of the flow be constrained so that it can be characterized using a relatively small number of parameters. A good example that illustrates this feature is the "lock-in" envelope of a cylinder wake. The cylinder wake flow can be forced in an open loop fashion using sinusoidal displacement of the cylinder with a given amplitude and frequency. Koopmann (1967) investigated the response of the flow to this type of forcing in a wind tunnel experiment. He found a region around the natural vortex shedding frequency where he could achieve "lock-in", which is characterized by the wake responding to the forcing by establishing a fixed phase relationship with respect to the forcing. The frequency band around the natural vortex shedding frequency for which lock-in may be achieved is amplitude dependent. In general, the larger the amplitude, the larger the frequency band for which lock-in is possible. However, there exists a minimum threshold amplitude below which the flow will not respond to the forcing any more. In Koopmann's experiment (1967), this amplitude was at 10% peak displacement of the cylinder. Siegel et al (2003a) show that for a circular cylinder, at Reynolds number of 100, a closed-loop controller operating within the "lock-in region" achieves a drag reduction of close to 90% of the vortex-induced drag, and lowers the unsteady lift force by the same amount.

Recently, the dielectric barrier discharge (DBD) plasma electrode has been developed as a flow control actuator, showing the ability to affect flow behavior in a range of applications. McLaughlin et al (2006) applied the DBD plasma actuator to a circular cylinder at Reynolds numbers of up to 3x105. Hot film measurements show that vortex shedding frequency can be driven to the actuator forcing frequency, within the lock-in range, at all Reynolds numbers investigated. The wake forced with plasma actuators exhibits "lock-in" behavior similar to that previously reported by Koopmann using cylinder displacement for forcing (Munska and McLaughlin, 2005).

b. Open-Loop, Transient Excitation using Actuators

Since the intended use of the low dimensional model, based on POD, is feedback flow control, the low dimensional state of the flow field needs to be accurately estimated as input for a controller. This poses the problem of snapshot selection: For the state to which the feedback controller drives the flow, usually no snapshots are available beforehand. We investigated POD bases derived from steady state, transient startup and open loop forced data sets for the two dimensional circular cylinder wake at Re = 100. None of these bases by itself is able to represent all features of the feedback controlled flow field. However, a POD basis derived from a composite snapshot set consisting of both transient startup as well as open loop forced data accurately models the features of the feedback controlled flow. For similar numbers of modes, this POD basis, which can be derived *a* priori, represents the feedback controlled flow as well as a POD model developed from the feedback controlled data a posteriori. These findings have two important implications: Firstly, an accurate POD basis can be developed without iteration from unforced and open loop data. Secondly, it appears that the feedback controlled flow does not leave the subspace spanned by open loop and unforced startup data, which may have important implications for the performance limits of feedback flow control**.** Further details on this approach are presented by Siegel et al (2005a) and Seidel et al (2006).

An important aspect of the developed methodology is to obtain a low-dimensional model that can predict the modal behavior of the flow when subject to various forcing inputs within the lock-in region. The emphasis is on the robustness of the predictive capability of the model. The main aim here is to predict the time histories of the time coefficients of the truncated POD model under the influence of open-loop control within the lock-in region. For the low Reynolds number, circular cylinder wake problem, Cohen et al (2006b) used nine different data sets, as marked in Figure 2, for the open loop forced cases at 10, 15, 20, 25 and 30 percent cylinder displacement. Some of the cases use 5-10% lower or higher frequency at 30% displacement, which is still within the lock-in region. In this example, the 25 percent cylinder displacement sinusoidal forcing serves as design point for model development.

Fig. 2. Model Building within "Lock-In" Envelope for the Circular Cylinder Wake.

#### c. Development of a Low-Dimensional Model (LOM) based on POD

In the developed approach, the main advantage of POD, namely its optimality and thus ability to capture the global behavior of a flow field with a minimum number of modes, is combined with established system identification techniques developed for the modeling of dynamical systems. Over the past decades, the controls community has developed methods to identify the dynamic properties of complex structures based on experimental measurements. These rely on the acquisition of transient measurements based on a known excitation input to the system. So called System Identification methods are then used to develop a dynamical mathematical model that can be used later for design and analysis of an effective control law as well as dynamic observer development. The main emphasis is then to develop an effective system identification technique that captures the dynamics of the time dependent coefficients of the POD modes with respect to transient actuation inputs within the lock-in region.

An important aspect of the developed methodology is to obtain a low-dimensional model that can predict the modal behavior of the flow when subject to various forcing inputs within the lock-in region. The emphasis is on the robustness of the predictive capability of the model. The main aim here is to predict the time histories of the time coefficients of the truncated POD model under the influence of open-loop control within the lock-in region. For the low Reynolds number, circular cylinder wake problem, Cohen et al (2006b) used nine different data sets, as marked in Figure 2, for the open loop forced cases at 10, 15, 20, 25 and 30 percent cylinder displacement. Some of the cases use 5-10% lower or higher frequency at 30% displacement, which is still within the lock-in region. In this example, the 25 percent cylinder

displacement sinusoidal forcing serves as design point for model development.

Fig. 2. Model Building within "Lock-In" Envelope for the Circular Cylinder Wake.

In the developed approach, the main advantage of POD, namely its optimality and thus ability to capture the global behavior of a flow field with a minimum number of modes, is combined with established system identification techniques developed for the modeling of dynamical systems. Over the past decades, the controls community has developed methods to identify the dynamic properties of complex structures based on experimental measurements. These rely on the acquisition of transient measurements based on a known excitation input to the system. So called System Identification methods are then used to develop a dynamical mathematical model that can be used later for design and analysis of an effective control law as well as dynamic observer development. The main emphasis is then to develop an effective system identification technique that captures the dynamics of the time dependent coefficients of the POD modes with respect to transient actuation inputs

c. Development of a Low-Dimensional Model (LOM) based on POD

within the lock-in region.

An important question that needs to be answered is: "What are the desired characteristics most sought after in a low-order, POD based model?" It is imperative to understand that given the complexity of the problem at hand, it may not be possible to address this problem with off-the-shelf methods but instead we propose a unique synthesis of software tools that appear to be promising. The important features are:


The ARX-ANN algorithms used in this effort are a modification of the toolbox developed by Nørgaard et al (2000). After the POD time coefficients were extracted, a basic single hidden layer ANN-ARX architecture was selected. The training set was then developed using Input-Output data obtained from CFD simulations. The model was validated for off-design cases and if the estimation error was unacceptable, then the ANN architecture was modified. This cycle repeated until estimation errors were acceptable for all off-design cases. Cohen et al (2006b) successfully demonstrated this approach for modeling of a cylinder wake at Reynolds numbers of 100.

Most of the modeling effort has been based on the velocity field. The low Reynolds number cylinder wake flow is two dimensional in nature, simplifying the spatial characteristics of the POD modes. However, as the Reynolds number is increased to turbulent regimes or as the geometry of the model becomes more complex, the flow field becomes three dimensional in nature.

a. Sensor Configuration and Estimator Development

A major design challenge lies in finding an appropriate number of sensors and their locations that will best enable the flow estimation. For low-dimensional control schemes to be implemented, a real-time *estimation* of the modes present in the wake is necessary, since it is not possible to measure them directly. Velocity field (Cohen et al, 2006a), surface pressure (Cohen et al, 2004), or surface skin friction (Stalnov et al, 2005) measurements, provided from either simulation or experiment, are used for estimation. This process leads to the state and measurement equations, required for design of the control system. For practical applications it is desirable to reduce the information required for estimation to the minimum. The spatial modes obtained from the POD procedure provide information that can be used to place sensor in locations where modal activity is at its highest. These areas would be the maxima and minima of the spatial modes (Cohen et al, 2006a). Placing sensors at the energetic maxima and minima of each mode is the basic hypothesis of the developed approach and the purpose of the CFD simulation is to design a sensor configuration which is later validated using experiments (Cohen et al, 2004).

The estimation scheme may be based on the linear stochastic estimation procedure introduced by Adrian (1977) a quadratic stochastic estimation (Ausseur et al, 2006) or in the form of an artificial neural network estimator, ANNE (Cohen et al, 2006c). Cohen et al (2006c) compare the effectiveness of the conventional LSE versus the newly proposed ANNE. The development of the procedure was based on CFD simulations of a cylinder at a Reynolds number of 100. Results show that for the estimation of the first four modes, it is seen that for the design condition (no noise) 4 sensors using ANNE provide significantly better results than 4 sensors using LSE. For the estimation of the first four modes, we show that a considerably smaller number of sensors using ANNE estimation provide better results than more sensors using LSE estimation. Furthermore, ANNE displays robust behavior when the signal to noise ratio of the sensors is artificially degraded.

b. Development and Analysis of a Control Law

A simple approach to control the von Kármán Vortex Street behind a two dimensional circular cylinder, based on the proportional feedback of the estimate of *just* the first POD mode was presented by Cohen et al (2003). A stability analysis of this control law was conducted after linearization about the desired equilibrium point and conditions for controllability and asymptotic stability were developed. The control approach, applied to the 4 mode cylinder wake POD model at a Reynolds number of 100 stabilizes the POD based low-dimensional wake model. While the controller uses only the estimated amplitude of the first mode, all four modes are stabilized. This suggests that the higher order modes are caused by a secondary instability. Thus they are suppressed once the primary instability is controlled. This simple control approach was later modified by Siegel et al (2003a) when applying it to a high resolution CFD simulation. An adaptive gain strategy, based on the estimation of the "mean-flow" mode incorporated to tune the phase of a Proportional-

Most of the modeling effort has been based on the velocity field. The low Reynolds number cylinder wake flow is two dimensional in nature, simplifying the spatial characteristics of the POD modes. However, as the Reynolds number is increased to turbulent regimes or as the geometry of the model becomes more complex, the flow field becomes three

A major design challenge lies in finding an appropriate number of sensors and their locations that will best enable the flow estimation. For low-dimensional control schemes to be implemented, a real-time *estimation* of the modes present in the wake is necessary, since it is not possible to measure them directly. Velocity field (Cohen et al, 2006a), surface pressure (Cohen et al, 2004), or surface skin friction (Stalnov et al, 2005) measurements, provided from either simulation or experiment, are used for estimation. This process leads to the state and measurement equations, required for design of the control system. For practical applications it is desirable to reduce the information required for estimation to the minimum. The spatial modes obtained from the POD procedure provide information that can be used to place sensor in locations where modal activity is at its highest. These areas would be the maxima and minima of the spatial modes (Cohen et al, 2006a). Placing sensors at the energetic maxima and minima of each mode is the basic hypothesis of the developed approach and the purpose of the CFD simulation is to design a sensor configuration which

The estimation scheme may be based on the linear stochastic estimation procedure introduced by Adrian (1977) a quadratic stochastic estimation (Ausseur et al, 2006) or in the form of an artificial neural network estimator, ANNE (Cohen et al, 2006c). Cohen et al (2006c) compare the effectiveness of the conventional LSE versus the newly proposed ANNE. The development of the procedure was based on CFD simulations of a cylinder at a Reynolds number of 100. Results show that for the estimation of the first four modes, it is seen that for the design condition (no noise) 4 sensors using ANNE provide significantly better results than 4 sensors using LSE. For the estimation of the first four modes, we show that a considerably smaller number of sensors using ANNE estimation provide better results than more sensors using LSE estimation. Furthermore, ANNE displays robust

A simple approach to control the von Kármán Vortex Street behind a two dimensional circular cylinder, based on the proportional feedback of the estimate of *just* the first POD mode was presented by Cohen et al (2003). A stability analysis of this control law was conducted after linearization about the desired equilibrium point and conditions for controllability and asymptotic stability were developed. The control approach, applied to the 4 mode cylinder wake POD model at a Reynolds number of 100 stabilizes the POD based low-dimensional wake model. While the controller uses only the estimated amplitude of the first mode, all four modes are stabilized. This suggests that the higher order modes are caused by a secondary instability. Thus they are suppressed once the primary instability is controlled. This simple control approach was later modified by Siegel et al (2003a) when applying it to a high resolution CFD simulation. An adaptive gain strategy, based on the estimation of the "mean-flow" mode incorporated to tune the phase of a Proportional-

behavior when the signal to noise ratio of the sensors is artificially degraded.

dimensional in nature.

a. Sensor Configuration and Estimator Development

is later validated using experiments (Cohen et al, 2004).

b. Development and Analysis of a Control Law

Differential (PD) controller was used (Siegel et al, 2003a). The closed loop feedback simulations explore the effect of both fixed phase and variable phase feedback on the wake. While fixed phase feedback is effective in reducing drag and unsteady lift, it fails to stabilize this state once the low drag state has been reached. Variable phase feedback, however, achieves the same drag and unsteady lift reductions while being able to stabilize the flow in the low drag state. In the low drag state, the near wake is entirely steady, while the far wake exhibits vortex shedding at a reduced intensity. We achieved a drag reduction of close to 90% of the vortex-induced drag, and lowered the unsteady lift force by the same amount.

c. Validation of the Closed-Loop Controller

A low-dimensional model allows for controller development and if a more accurate nonlinear model, having more modes that those used for controller development, is employed then the controller features may be analyzed as well. However, as the common saying goes "the taste of the pudding is in the eating", we need to validate the controller effectiveness in experiment. Nevertheless, a high resolution, CFD based truth simulation can provide very important insight into the complexities of feedback flow control. Both of these comprehensive approaches have been used by the USAFA team and the following are some highlights of these studies (Seidel et al, 2006, Siegel et al, 2004).


3. Seidel et al (2006) conduct high resolution, three-dimensional feedback controlled simulations of the wake behind a circular cylinder. In the current simulations, a threedimensional sensor array was placed in the wake to estimate the flow state based on two dimensional POD Modes, which were applied at multiple span-wise locations. An LSE algorithm was used to map sensor readings to the temporal coefficients of the POD modes. The simulations were aimed at investigating the efficacy of three dimensional flow sensing to improve feedback control. Because the control input had only one degree of freedom (1 DOF), the mode amplitudes had to be combined into one actuator signal. Starting from an idealized, highly two-dimensional open loop case, the threedimensional feedback controlled simulations show that, independent of the number and location of the sensor planes, control is initially successful for the whole span-wise extent. For approximately two seconds or ten vortex shedding cycles, the controller is able to significantly reduce the vortex shedding, resulting in a reduction of the drag coefficient of more than ten percent.

#### **3.2 Experimental set-up**

All tests were conducted in the USAFA Low Speed Wind Tunnel (LSWT). This tunnel has a 3 ft x 3 ft test section with a usable velocity range from 16 ft/s to 115 ft/s. A 3.5 in diameter, D, PVC cylinder spanned the entire height of the test section. Plasma actuators were placed along the span at the ±90° marks based on previous work done by List et al (2003) indicating this as the best position. The actuators consisted of two strips of copper tape, one buried beneath the dialectic barrier and one on top. Computer controlled voltage was amplified and transformers were used to significantly increase the magnitude to 11kV. The plasma formed atop the Teflon tape over the area of the buried electrode. Five layers of Teflon dielectric tape were used, as shown effective through McLaughlin et al (2006). In this case however, the Teflon tape was only used on the front side of the cylinder to make room for the sensors on the back half (Figure 3).

Fig. 3. Top view of cylinder set-up.

3. Seidel et al (2006) conduct high resolution, three-dimensional feedback controlled simulations of the wake behind a circular cylinder. In the current simulations, a threedimensional sensor array was placed in the wake to estimate the flow state based on two dimensional POD Modes, which were applied at multiple span-wise locations. An LSE algorithm was used to map sensor readings to the temporal coefficients of the POD modes. The simulations were aimed at investigating the efficacy of three dimensional flow sensing to improve feedback control. Because the control input had only one degree of freedom (1 DOF), the mode amplitudes had to be combined into one actuator signal. Starting from an idealized, highly two-dimensional open loop case, the threedimensional feedback controlled simulations show that, independent of the number and location of the sensor planes, control is initially successful for the whole span-wise extent. For approximately two seconds or ten vortex shedding cycles, the controller is able to significantly reduce the vortex shedding, resulting in a reduction of the drag

All tests were conducted in the USAFA Low Speed Wind Tunnel (LSWT). This tunnel has a 3 ft x 3 ft test section with a usable velocity range from 16 ft/s to 115 ft/s. A 3.5 in diameter, D, PVC cylinder spanned the entire height of the test section. Plasma actuators were placed along the span at the ±90° marks based on previous work done by List et al (2003) indicating this as the best position. The actuators consisted of two strips of copper tape, one buried beneath the dialectic barrier and one on top. Computer controlled voltage was amplified and transformers were used to significantly increase the magnitude to 11kV. The plasma formed atop the Teflon tape over the area of the buried electrode. Five layers of Teflon dielectric tape were used, as shown effective through McLaughlin et al (2006). In this case however, the Teflon tape was only used on the front side of the cylinder to make room for

coefficient of more than ten percent.

the sensors on the back half (Figure 3).

Fig. 3. Top view of cylinder set-up.

**3.2 Experimental set-up** 

A panel was cut from the downstream side of the cylinder for sensor placement. Sixteen pressure ports consisting of four rows of four ports were placed into this panel and a Scanivalve pressure multiplexer was fixed inside the cylinder with tubing connected to each of the sixteen ports (Figure 4).

Fig. 4. Scanivalve pressure multiplexer and pressure ports in cylinder.

The location of the pressure ports was determined by doing hot film testing across the back side cylinder 1/8" behind the cylinder wall. The plasma actuators were operated at the natural shedding frequency to ensure lock-in and provide adequate flow control. Before data was collected, flow visualization was conducted to see the flow characteristics and ensure the plasma was effective in forcing the flow. Hot film anemometers were also used to validate the theoretical values for frequency downstream of the cylinder. The hot film anemometers were used to gather preliminary data very near the surface of the cylinder. The data collected was used to enable a preliminary guess in choosing pressure port locations for identifying certain flow characteristics. Figure 5 shows the tunnel set-up of the cylinder with pressure ports and the hot films positioned in the wake.

The Validyne pressure sensor was used in conjunction with a Scanivalve pressure multiplexer unit to cycle through the 16 different pressure ports. These ports were drilled into the removable rear section of the cylinder. The locations of the ports can be found in Figures 3-4. The pressure sensor has a pressure range of ±0.03 psid, an analog output of ±10Vdc, and accuracy of 0.25%. To use both the Scanivalve pressure multiplexer and Validyne sensor together, the central transducer of the Scanivalve pressure multiplexer was removed and replaced with a "dummy" plug which simply makes the Scanivalve pressure multiplexer a switching mechanism for the separate pressure ports. A period of 60 seconds was required between each pressure reading in order to ensure that the flow had "settled" after each Scanivalve pressure multiplexer switch. The remote placement of the sensor eliminated EMI issues because it was physically separated from the plasma actuators so that

Fig. 5. Hot film anemometers and pressure ports.

id was not subject to any interference. To ensure the data acquired was not contaminated by the remote set-up, the characteristics of the plumbing were examined. For the sensor to output reliable data, the natural frequency of the plumbing system must be at least five times that of the largest frequency to be measured according to the documentation included with the sensor. The natural frequency of the system was found using the equation

$$\alpha\_n = \frac{c}{L\sqrt{\frac{1}{2} + \frac{Q}{aL}}},\tag{1}$$

where *ωn* is the natural frequency, *c* is the speed of sound (1089.2 ft/s), *L* is the length of tubing (2.5 ft), *Q* is the transducer cavity volume (2.03E-5 ft3), and *a* is the cross sectional area of the tubing (2.13053E-5 ft2). This yielded ωn=73.87 Hz, which was well within the specified range since the maximum frequency measured was 9.1 Hz at Reynolds number of 20,000. This gave around 2-3% amplification of pressure waveform.

#### **4. Results**

The Validyne sensor that was connected through the Scanivalve pressure multiplexer to the pressure ports provided the surface mounted measurements required for flow state estimation. The collection of wake mounted hot film measurements and the pressure readings at each port was acquired at a sampling rate of 1 kHz. This ensured that the comparative studies could be adequately analyzed. The fundamental frequency, associated with the von Kármán vortex shedding frequency is very distinctly identified. The frequency content of the data, pertaining to the von Kármán vortex shedding frequency, from the surface mounted pressure measurements perfectly correlates with that of the hot film anemometers. For the unforced flow, it can be seen that both sensors are picking up the

id was not subject to any interference. To ensure the data acquired was not contaminated by the remote set-up, the characteristics of the plumbing were examined. For the sensor to output reliable data, the natural frequency of the plumbing system must be at least five times that of the largest frequency to be measured according to the documentation included

> 1 2

where *ωn* is the natural frequency, *c* is the speed of sound (1089.2 ft/s), *L* is the length of tubing (2.5 ft), *Q* is the transducer cavity volume (2.03E-5 ft3), and *a* is the cross sectional area of the tubing (2.13053E-5 ft2). This yielded ωn=73.87 Hz, which was well within the specified range since the maximum frequency measured was 9.1 Hz at Reynolds number of

The Validyne sensor that was connected through the Scanivalve pressure multiplexer to the pressure ports provided the surface mounted measurements required for flow state estimation. The collection of wake mounted hot film measurements and the pressure readings at each port was acquired at a sampling rate of 1 kHz. This ensured that the comparative studies could be adequately analyzed. The fundamental frequency, associated with the von Kármán vortex shedding frequency is very distinctly identified. The frequency content of the data, pertaining to the von Kármán vortex shedding frequency, from the surface mounted pressure measurements perfectly correlates with that of the hot film anemometers. For the unforced flow, it can be seen that both sensors are picking up the

*c <sup>Q</sup> <sup>L</sup> aL*

, (1)

with the sensor. The natural frequency of the system was found using the equation

*n*

20,000. This gave around 2-3% amplification of pressure waveform.

Fig. 5. Hot film anemometers and pressure ports.

**4. Results** 

exact same shedding frequency of 9.1 Hz. Again, with plasma forcing within the lock-in regime, the same data are taken and using a fast Fourier transformation, the fundamental frequency is found to be very distinct. During the DBD plasma forcing, the flow's shedding frequency gets locked into the plasma actuation, which was set to a frequency of 8.8 Hz. The velocity measured by the hot films in the wake at 1.5-2.5 diameters downstream was 3-5% greater than the velocity set for the tunnel which was expected and within the range of the calculated blockage error. Since the area of the test section is reduced by the relatively large model (blockage ratio of 9.7%), the flow's velocity was increased while the resulting natural shedding frequency was also increased. Furthermore, the shedding frequency of the Re=20k flow was increased from 8.7 Hz to 9.1 Hz.

Feasible real time estimation and control of the cylinder wake may be effectively realized by reducing the model complexity of the cylinder wake using POD techniques. POD, a nonlinear model reduction approach is also referred to in the literature as the Karhunen-Loeve expansion (Holmes et al, 1996). The truncated POD model will contain an adequate number of modes to enable modeling of the temporal and spatial characteristics of the large-scale coherent structures inherent in the flow. Since a pressure multiplexer is used to collect data from the 16 pressure ports, it is imperative to synchronize the time histories of the pressure measurements before any meaningful analysis of the results can be made. For this purpose, the hot film velocity measurements are used to initiate all pressure signals based on the very distinctive fundamental frequency. While this approach is inaccurate, it does provide some interesting insight into the applicability of surface mounted pressure sensors for low-order modeling of the cylinder wake at Re~20,000. In order to examine the robustness of this procedure, the POD procedure was applied to 4 snapshot sets each containing 1601, 2601, 3601 and 4601 snapshots for both plasma off and plasma on cases. The resulting Eigenvalues, without and with the mean flow mode, are presented in Tables 1 and 2 respectively. It can be seen that the Eigen-value distribution is relatively insensitive to the number of snap-shots. Also, the spatial modes for plasma-off, as shown in Figure 6 (1601 snap-shots), and for plasma off (4601 snapshots), as shown in Figure 7, are fairly similar. The temporal coefficients were also found to be of a similar nature as will be discussed later. Additionally, it can be seen in Tables 1-2 that as the plasma is turned on, the intensity of the Eigen-values of modes one and two (von Kármán modes) is increased while the mean mode as well as the higher mode amplitudes are reduced.


Table 1. POD – Eigen-values of Surface Pressure @ Re~20K (after extraction of the mean).



Fig. 6. First two POD Spatial Periodic Modes (1601 Snap-Shots) – Plasma Off.

Plasma On [%]

> 96.73 1.52 1.05 0.10 0.09 0.07 0.07 0.06 0.05

Table 2. POD – Eigen-values of Surface Pressure @ Re~20K (after inclusion of the mean).

Fig. 6. First two POD Spatial Periodic Modes (1601 Snap-Shots) – Plasma Off.

1601 Snapshots 2601 Snapshots 3601 Snapshots 4601 Snapshots

Plasma Off [%]

> 97.36 0.69 0.48 0.31 0.21 0.17 0.13 0.11 0.10

Plasma On [%]

> 96.74 1.47 1.09 0.09 0.08 0.07 0.07 0.06 0.06

Plasma Off [%]

> 97.37 0.60 0.41 0.34 0.20 0.18 0.16 0.13 0.10

Plasma On [%]

> 96.76 1.44 1.11 0.08 0.08 0.07 0.06 0.06 0.05

Mode

Plasma Off [%]

> 97.38 0.94 0.51 0.28 0.17 0.11 0.10 0.09 0.08

Plasma On [%]

> 96.74 1.51 1.06 0.11 0.10 0.08 0.07 0.06 0.05

Plasma Off [%]

> 97.25 0.78 0.56 0.28 0.23 0.14 0.12 0.11 0.09

Fig. 7. First two POD Spatial Periodic Modes (4601 Snap-Shots) – Plasma On at 8.8 Hz, 11 KVolt, Position of Sensor is marked with .

The spatial modes obtained from the POD procedure provide information concerning the location of areas where modal activity is at its highest. These energetic areas are the maxima and minima of the spatial modes (Cohen et al, 2006a). In this effort, 5 of the surface mounted pressure sensors, which are positioned at the energetic maxima and minima of each of the von Kármán modes, are used to provide an estimate of the POD time coefficients.

Now that the sensor configuration is determined an Artificial Neural Network Estimator (ANNE) is developed for the real-time mapping of pressure measurements onto POD time coefficients. The main features of ANNE, as described in a flow-chart and schematically in Figures 8-9, are as follows:


Fig. 8. Mapping of Body Mounted Pressure Measurements to POD time coefficients.

Fig. 9. Basic Architecture of ANNE.

**Weighting Matrices** 

132 Low Reynolds Number Aerodynamics and Transition

Fig. 8. Mapping of Body Mounted Pressure Measurements to POD time coefficients.

Fig. 9. Basic Architecture of ANNE.


The estimations provided by ANNE for the 3 mode model is given in Figure 10 for the training data and in Figure 11 for the validation data. These preliminary results appear to be promising. However, one must be reminded that the main aim in this exercise is to obtain an insight for the application of the d low-dimensional suite of tools, which were primarily developed for low Reynolds laminar bluff body wakes, to higher Reynolds number turbulent wakes.

Fig. 10. Predictions based on ANNE (Training Data).

Fig. 11. Predictions based on ANNE (Validation Data).

## **5. Conclusions**

In this chapter, we present a potentially promising approach for closed-loop flow control of a turbulent wake behind a circular cylinder at higher Reynolds numbers (Re ~ 20,000), with the ultimate goal being closed-loop flow control of the cylinder wake using DBD plasma actuators. The proposed methodology for approaching closed-loop flow control is based on the research effort at the USAF Academy over the past five years. This approach has been developed with a focus on control of laminar bluff body wakes at low Reynolds numbers (Re~50-180). The approach consists of six steps, namely: Identification of the "lock-in" region; open-loop, transient excitation using actuators; development of a lowdimensional model based on POD; sensor configuration and estimator development; development and analysis of a control law; and finally validation of the closed-loop controller.

Experimental results using plasma actuation and surface mounted pressure sensors for a circular cylinder at Reynolds number of 20,000 show that the fundamental frequency, which is paramount for feedback, is distinctly and accurately picked up by the surface mounted pressure measurements. Surface mounted pressure measurements seem to be useful for feedback of plasma forced cylinder wake at Reynolds number of 20000. Based on these experimental results, it appears that the low dimensional approach and tools developed by USAFA/DFAN for low Reynolds number (Re~100) (Sensor placement and number strategy; and ANNE estimation of the POD temporal coefficients based on surface mounted pressure sensors) feedback flow control is applicable to much higher Reynolds number (Re~20,000).

## **6. Acknowledgments**

134 Low Reynolds Number Aerodynamics and Transition

In this chapter, we present a potentially promising approach for closed-loop flow control of a turbulent wake behind a circular cylinder at higher Reynolds numbers (Re ~ 20,000), with the ultimate goal being closed-loop flow control of the cylinder wake using DBD plasma actuators. The proposed methodology for approaching closed-loop flow control is based on the research effort at the USAF Academy over the past five years. This approach has been developed with a focus on control of laminar bluff body wakes at low Reynolds numbers (Re~50-180). The approach consists of six steps, namely: Identification of the "lock-in" region; open-loop, transient excitation using actuators; development of a lowdimensional model based on POD; sensor configuration and estimator development; development and analysis of a control law; and finally validation of the closed-loop

Experimental results using plasma actuation and surface mounted pressure sensors for a circular cylinder at Reynolds number of 20,000 show that the fundamental frequency,

Fig. 11. Predictions based on ANNE (Validation Data).

**5. Conclusions** 

controller.

The authors would like to acknowledge funding by the Air Force Office of Scientific Research, LtCol Sharon Heise, Program Manager. The authors would like to thank cadets Brandon Snyder, Joshua Lewis, and Assistant Professor Christopher Seaver for preparing the experimental aspects of this effort and collecting the experimental data in the wind tunnel. The contributions of SSgt Mary Church in set-up, tear-down, and everything in between, Ken Ostasiewski and his help with the Scanivalve pressure multiplexer, and many other facets of the wind tunnel testing, and Jeff Falkenstine with the manufacturing of the model were vital to the success of the work. We would also like to thank Mr. Tim Hayden for the assistance in experimentation. The authors appreciate the discussions with Dr. Young-Sug Shin on system identification using Artificial Neural Networks. This work has been presented at an AIAA conference and published in the conference proceedings. It has no copyright protection in US since it is considered as US government work.

## **7. References**


Cattafesta Iii, L.N., Williams, D.R., Rowley, C.W., and Alvi, F.S. (2003). Review of Active

Cohen K., Siegel S., Mclaughlin T., and Gillies E. (2003). Feedback Control of a Cylinder

Cohen, K., Siegel, S., Wetlesen, D., Cameron, J., and Sick, A. (2004). Effective Sensor

Cohen, K., D., Siegel, S., Luchtenburg, M., and Mclaughlin, T., And Seifert, A. (2004). Sensor

Cohen, K., Siegel S., Mclaughlin T., Gillies E., and Myatt, J. (2005). Closed-loop approaches

Cohen, K., Siegel S., and Mclaughlin T. (2006a). A Heuristic Approach to Effective Sensor

Cohen, K., Siegel, S., Seidel, J., and Mclaughlin, T. (2006b). System Identification of a Low

Cohen, K., Siegel, S., Seidel, J., and Mclaughlin, T. (2006c). Neural Network Estimator

Cybenko, G.V. (1989). Approximation by Superpositions of a Sigmoidal function,

Enloe, C.L., Mclaughlin, T.E., Vandyken, R.D., Kachner, K.D., Jumper, E.J., and Corke, T.C.

Gad-El-Hak, M. (1996). Modern Developments in Flow Control", Applied Mechanics

Gerhard, J., Pastoor, M., King, R., Noack, B.R., Dillmann, A., Morzynski, M. and Tadmor, G.

Gillies, E. A. (1995). Low-Dimensional Characterization and Control of Non-Linear Wake

Gillies, E. A. (1998). Low-dimensional Control of the Circular Cylinder Wake, Journal of

Glauser, M., Young, M., Higuchi, H., Tinney, C.E., and Carlson, H. (2004). POD Based

Haykin, S. (1999). Neural Networks – A comprehensive foundation, Second Edition,

Dimensional Model of a Cylinder Wake, AIAA-2006-1411.

Mathematics of Control, Signals and Systems, 2, 303-314.

Plasma Morphology, AIAA Journal, 42 (3).

Reviews, 49, 365–379.

Scotland, U.K.

0574.

Models, AIAA Paper2003-4262.

Fluid Mechanics, 371, 157-178.

Prentice Hall, New Jersey, USA.

Wake Low-Dimensional Model, AIAA Journal, 41 (8).

2003.

1857-1880.

2004-2523.

2006.

3491.

and Fluids, 34 (8) , 927-949.

Control of Flow-Induced Cavity Resonance, AIAA Paper 2003-3567,

Placements for the Estimation of Proper Orthogonal Decomposition Mode Coefficients in von Kármán Vortex Stree, Journal of Vibration and Control, 10 (12),

Placement for Closed-loop Flow control of a 'D' Shaped Cylinder Wake, AIAA-

to control of a wake flow modeled by the Ginzburg-Landau equation, Computers

Placement for Modeling of a Cylinder Wake, Computers and Fluids, 35 (1), 103-120,

for Low-Dimensional Modeling of a Cylinder Wake, AIAA-2006-

(2004). Mechanisms and Responses of a Single Dielectric Barrier Plasma Actuator:

(2003). Model-based Control of Vortex Shedding using Low-dimensional Galerkin

Flows, Ph.D. Dissertation, Faculty of Engineering, Univ. Of Glasgow, Glasgow,

Experimental Flow Control on a NACA-4412 Airfoil (Invited), AIAA Paper 2004-


## **Thermal Perturbations in Supersonic Transition**

Hong Yan *Northwestern Polytechnical University P.R.China*

#### **1. Introduction**

138 Low Reynolds Number Aerodynamics and Transition

Stalnov, O., Palei, V., Fono, I., Cohen, K., and Seifert, A. (2005). Experimental Validation of Sensor Placement for Control of a "D" Shaped Cylinder Wake, AIAA 2005-5260. Taylor, J.A. and Glauser, M.N. (2004). Towards Practical Flow Sensing and Control via POD

345.

and LSE Based Low-Dimensional Tools, Journal of Fluids Engineering, 126, 337-

In recent years, there has been considerable interest in the study of bump-based methods to modulate the stability of boundary layers (Breuer & Haritonidis, 1990; Breuer & Landahl, 1990; Fischer & Choudhari, 2004; Gaster et al., 1994; Joslin & Grosch, 1995; Rizzetta & Visbal, 2006; Tumin & Reshotko, 2005; White et al., 2005; Worner et al., 2003). These studies are mostly focused on the incompressible regime and have revealed several interesting aspects of bump modulated flows. Surface roughness can influence the location of laminar-turbulent transition by two potential mechanisms. First, they can convert external large-scale disturbances into small-scale boundary layer perturbations, and become possible sources of receptivity. Second, they may generate new disturbances to stabilize or destabilize the boundary layer. Breuer and Haritonidis (Breuer & Haritonidis, 1990) and Breuer and Landahl (Breuer & Landahl, 1990) performed numerical and experimental simulations to study the transient growth of localized weak and strong disturbances in a laminar boundary layer. They demonstrated that the three-dimensionality in the evolution of localized disturbances may be seen at any stage of the transition process and is not necessarily confined to the nonlinear regime of the flow development. For weak disturbances, the initial evolution of the disturbances resulted in the rapid formation of an inclined shear layer, which was in good agreement with inviscid calculations. For strong disturbances, however, transient growth gives rise to distinct nonlinear effects, and it was found that resulting perturbation depends primarily on the initial distribution of vertical velocity. Gaster *et al.*(Gaster et al., 1994) reported measurements on the velocity field created by a shallow oscillating bump in a boundary layer. They found that the disturbance was entirely confined to the boundary layer, and the spanwise profile of the disturbance field near the bump differed dramatically from that far downstream. Joslin and Grosch (Joslin & Grosch, 1995) performed a Direct Numerical Simulation (DNS) to duplicate the experimental results by Gaster *et al.* (Gaster et al., 1994). Far downstream, the bump generated a pair of counter-rotating streamwise vortices just above the wall and on either side of the bump location, which significantly affected the near-wall flow structures. Worner *et al.* (Worner et al., 2003) numerically studied the effect of a localized hump on Tollmien-Schlichting waves traveling across it in a two-dimensional laminar boundary layer. They observed that the destabilization by a localized hump was much stronger when its height was increased as opposed to its width. Further, a rounded shape was less destabilizing than a rectangular shape.

Researchers have also studied the effect of surface roughness on transient growth. White *et al.*(White et al., 2005) described experiments to explore the receptivity of transient disturbances to surface roughness. The initial disturbances were generated by a spanwise-periodic array of roughness elements. The results indicated that the streamwise flow was decelerated near the protuberances, but that farther downstream the streamwise flow included both accelerated and decelerated regions. Some of the disturbances produced by the spanwise roughness array underwent a period of transient growth. Fischer and Choudhari (Fischer & Choudhari, 2004) presented a numerical study to examine the roughness-induced transient growth in a laminar boundary layer. The results showed that the ratio of roughness size relative to array spacing was a primary control variable in roughness-induced transient growth. Tumin and Reshotko (Tumin & Reshotko, 2005) solved the receptivity of boundary layer flow to a three dimensional hump with the help of an expansion of the linearized solution of the Navier-Stokes equations into the biorthogonal eigenfunction system. They observed that two counter-rotating streamwise vortices behind the hump entrained the high-speed fluid towards the surface boundary layer. Rizzetta and Visbal (Rizzetta & Visbal, 2006) used DNS to study the effect of an array of spanwise periodic cylindrical roughness elements on flow instability. A pair of co-rotating horseshoe vortices was observed, which did not influence the transition process, while the breakdown of an unstable shear layer formed above the element surface played a strong role in the initiation of transition.

Although the effect of physical bumps on flow instabilities has been studied extensively, far fewer studies have explored the impact of thermal bumps. A thermal bump may be particularly effective at supersonic and hypersonic speeds. One approach to introduce the bump is through an electromagnetic discharge in which motion is induced by collisional momentum transfer from charged to neutral particles through the action of a Lorentz force (Adelgren et al., 2005; Enloe et al., 2004; Leonov et al., 2001; Roth et al., 2000; Shang, 2002; Shang et al., 2005). Another approach is through a high-frequency electric discharge (Samimy et al., 2007). Joule heating is a natural outcome of such interactions, and is assumed to be the primary influence of the notional electric discharge plasma employed here to influence flow stability.

For supersonic and hypersonic flows, heat injection for control have considered numerous mechanisms, including DC discharges (Shang et al., 2005), microwave discharges (Leonov et al., 2001) and lasers (Adelgren et al., 2005). Recently however, Samimy *et al.* (Samimy et al., 2007) have employed Localized Arc Filament Plasma Actuators in a fundamentally unsteady manner to influence flow stability. The technique consists of an arc filament initiated between electrodes embedded on the surface to generate rapid (on the time scale of a few microseconds) local heating. Samimy *et al.* (Samimy et al., 2007) employed this method in the control of high speed and high Reynolds number jets. The results showed that forcing the jet with *m* = ±1 mode at the preferred column mode frequency provided the maximum mixing enhancement, while significantly reducing the jet potential core length and increasing the jet centerline velocity decay rate beyond the end of the potential core.

Yan *et al.* (Yan et al., 2007; 2008) studied the steady heating effect on a Mach 1.5 laminar boundary layer. Far downstream of the heating, a series of counter-rotating streamwise vortex pairs were observed above the wall on the each side of the heating element. This implies that the main mechanism of the thermal bump displays some degree of similarity to that of the physical bump. This finding motivates the further study on thermal bumps since they have several advantages over physical bumps. These include the ability to switch on and off on-demand, and to pulse at any desired frequency combination. Yan and Gaitonde (Yan & Gaitonde, 2010) studied both the steady and pulsed thermal rectangular bumps in supersonic boundary layer. For the steady bump, the velocity fluctuation profile across the span bore some similarity to the physical bump in an overall sense. The disturbance decayed 2 Will-be-set-by-IN-TECH

flow was decelerated near the protuberances, but that farther downstream the streamwise flow included both accelerated and decelerated regions. Some of the disturbances produced by the spanwise roughness array underwent a period of transient growth. Fischer and Choudhari (Fischer & Choudhari, 2004) presented a numerical study to examine the roughness-induced transient growth in a laminar boundary layer. The results showed that the ratio of roughness size relative to array spacing was a primary control variable in roughness-induced transient growth. Tumin and Reshotko (Tumin & Reshotko, 2005) solved the receptivity of boundary layer flow to a three dimensional hump with the help of an expansion of the linearized solution of the Navier-Stokes equations into the biorthogonal eigenfunction system. They observed that two counter-rotating streamwise vortices behind the hump entrained the high-speed fluid towards the surface boundary layer. Rizzetta and Visbal (Rizzetta & Visbal, 2006) used DNS to study the effect of an array of spanwise periodic cylindrical roughness elements on flow instability. A pair of co-rotating horseshoe vortices was observed, which did not influence the transition process, while the breakdown of an unstable shear layer formed above the element surface played a strong role in the initiation of

Although the effect of physical bumps on flow instabilities has been studied extensively, far fewer studies have explored the impact of thermal bumps. A thermal bump may be particularly effective at supersonic and hypersonic speeds. One approach to introduce the bump is through an electromagnetic discharge in which motion is induced by collisional momentum transfer from charged to neutral particles through the action of a Lorentz force (Adelgren et al., 2005; Enloe et al., 2004; Leonov et al., 2001; Roth et al., 2000; Shang, 2002; Shang et al., 2005). Another approach is through a high-frequency electric discharge (Samimy et al., 2007). Joule heating is a natural outcome of such interactions, and is assumed to be the primary influence of the notional electric discharge plasma employed here to influence flow

For supersonic and hypersonic flows, heat injection for control have considered numerous mechanisms, including DC discharges (Shang et al., 2005), microwave discharges (Leonov et al., 2001) and lasers (Adelgren et al., 2005). Recently however, Samimy *et al.* (Samimy et al., 2007) have employed Localized Arc Filament Plasma Actuators in a fundamentally unsteady manner to influence flow stability. The technique consists of an arc filament initiated between electrodes embedded on the surface to generate rapid (on the time scale of a few microseconds) local heating. Samimy *et al.* (Samimy et al., 2007) employed this method in the control of high speed and high Reynolds number jets. The results showed that forcing the jet with *m* = ±1 mode at the preferred column mode frequency provided the maximum mixing enhancement, while significantly reducing the jet potential core length and increasing the jet

Yan *et al.* (Yan et al., 2007; 2008) studied the steady heating effect on a Mach 1.5 laminar boundary layer. Far downstream of the heating, a series of counter-rotating streamwise vortex pairs were observed above the wall on the each side of the heating element. This implies that the main mechanism of the thermal bump displays some degree of similarity to that of the physical bump. This finding motivates the further study on thermal bumps since they have several advantages over physical bumps. These include the ability to switch on and off on-demand, and to pulse at any desired frequency combination. Yan and Gaitonde (Yan & Gaitonde, 2010) studied both the steady and pulsed thermal rectangular bumps in supersonic boundary layer. For the steady bump, the velocity fluctuation profile across the span bore some similarity to the physical bump in an overall sense. The disturbance decayed

centerline velocity decay rate beyond the end of the potential core.

transition.

stability.

downstream, suggesting that the linear stability theory applies. For pulsed heating, non-linear dynamic vortex interactions caused disturbances to grow dramatically downstream. Yan and Gaitonde (Yan & Gaitonde, 2011) assessed the effect of the geometry of the thermal bump and the pulsing properties. It was shown that the rectangular element was more effective than the circular counterpart. The smaller width of the rectangular element produced higher disturbance energy, while the full-span heating indicated delayed growth of the disturbances. The disturbance energy increased with the initial temperature variation, and the lower frequency produced lesser disturbance energy.

This chapter summarizes some of the key findings in thermal perturbation induced supersonic flow transition in our research group. The chapter is organized as follows. The flow configuration, setup and numerical components are described first. The effect of the pulsed heating is then explored in the context of disturbance energy growth, and correlated with linear stability analysis. Subsequently, the phenomenology of the non-linear dynamics between the vortices produced by the pulsed bump and the compressible boundary layer is examined with emphasis on later stages of the boundary layer transition.

### **2. Flow configuration**

A Mach 1.5 flat plate flow is considered with the total temperature and pressure of 325 *K* and 3.7 <sup>×</sup> 105 *Pa*, respectively. The thermal bump is imposed as a surface heating element and centered in the spanwise direction as shown schematically in Fig. 1. For some simulations, the

Fig. 1. Flat plate with thermal bump

nominally two-dimensional case is considered where the bump extends cross the entire span of the plate. Even for this case, the three-dimensional domain is discretized since the primary disturbance growth is three-dimensional. The heating effect is modeled as a time-dependent step surface temperature rise Δ*Tw* with a monochromatic pulsing frequency (*f*) and duty cycle as shown in Fig. 2, where the pulsing time period *Tt* = 1/ *f* . The subscript *w* denotes the value at the wall. For simplicity, it is assumed that Δ*Tw* = *Tw* − *Tw*0, where *Tw* and *Tw*<sup>0</sup> are wall temperature inside and outside of the heating region, respectively, and *Tw*<sup>0</sup> is fixed at the adiabatic temperature (*Tad*) as shown in Fig. 1.

In all perturbed simulations, the heating element is placed immediately upstream of the first neutral point in the stability neutral curve for an adiabatic flat plate boundary layer with the freestream Mach number (*M*∞) of 1.5. The stability diagram, shown in Fig. 3, is obtained from the Langley Stability and Transition Analysis Codes (LASTRAC) (Chang, 2004). LASTRAC performs linear calculations and transition correlation by using the *N*-factor method based on linear stability theory, where the *N* factor is defined by *N* = *<sup>s</sup>*<sup>1</sup> *<sup>s</sup>*<sup>0</sup> *γds*, and *s*<sup>0</sup> is the point at which the disturbance first begins to grow, *s*<sup>1</sup> is the end point of the integration, which may be at upstream or downstream of where transition is correlated and *γ* is the characteristic growth rate of the disturbance. For disturbances at *f* = 100 kHz, the first neutral point is located at the Reynolds number of *Re* = 610 based on the similarity boundary-layer length scale (*η*) defined as <sup>=</sup> <sup>√</sup>*ν*∞*x*/*u*∞, where *<sup>ν</sup>*<sup>∞</sup> and *<sup>u</sup>*<sup>∞</sup> are the freestream kinematic viscosity

Fig. 2. Two time periods of surface temperature rise, *Tt*

Fig. 3. Neutral curve for Mach 1.5 adiabatic flat plate boundary layer

and streamwise velocity, respectively and is shown as the solid rectangle in Fig. 3. The local Reynolds number based on the running distance from the leading edge of the plate1 is defined by *Rex* = *Re*2. Thus, the distance from the leading edge of the plate to the leading edge of the heating element is 7.65 mm (*i.e.* corresponding to *Re* = 610). The nominal spanwise distance between bumps is determined from the most unstable mode, which for the present Mach number is oblique. The N factor profile, shown in Fig. 4 for different spanwise wave lengths (*λ*) at *M*∞=1.5 and *f* = 100 kHz, indicates that *λ* = 2 mm is the most unstable mode. Thus, the nominal distance between two adjacent heating elements is set to 2 mm to excite the most unstable wave. This is accomplished by choosing a spanwise periodic condition on a domain of 2 mm, at the center of which a bump is enforced.

<sup>1</sup> *Rex* grows linearly with *x* and is adopted in all the figures and tables except in the neutral stability curve figure, where *Re* is used instead.

4 Will-be-set-by-IN-TECH

Fig. 2. Two time periods of surface temperature rise, *Tt*

Fig. 3. Neutral curve for Mach 1.5 adiabatic flat plate boundary layer

of 2 mm, at the center of which a bump is enforced.

curve figure, where *Re* is used instead.

and streamwise velocity, respectively and is shown as the solid rectangle in Fig. 3. The local Reynolds number based on the running distance from the leading edge of the plate1 is defined by *Rex* = *Re*2. Thus, the distance from the leading edge of the plate to the leading edge of the heating element is 7.65 mm (*i.e.* corresponding to *Re* = 610). The nominal spanwise distance between bumps is determined from the most unstable mode, which for the present Mach number is oblique. The N factor profile, shown in Fig. 4 for different spanwise wave lengths (*λ*) at *M*∞=1.5 and *f* = 100 kHz, indicates that *λ* = 2 mm is the most unstable mode. Thus, the nominal distance between two adjacent heating elements is set to 2 mm to excite the most unstable wave. This is accomplished by choosing a spanwise periodic condition on a domain

<sup>1</sup> *Rex* grows linearly with *x* and is adopted in all the figures and tables except in the neutral stability

Fig. 4. N factor for different spanwise wavelengths at Mach=1.5 and *f* = 100 kHz

#### **3. Numerical model**

The governing equations are the full compressible 3-D Navier-Stokes equations. The Roe scheme (Roe, 1981) is employed together with the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL)(Van Leer, 1979) to obtain up to nominal third order accuracy in space. Solution monotonicity is imposed with the harmonic limiter described by Van Leer (Van Leer, 1979). Given the stringent time-step-size limitation of explicit schemes, an implicit approximately factored second-order time integration method with a sub-iteration strategy is implemented to reduce computing cost. The time step is fixed at 4.2 <sup>×</sup> <sup>10</sup>−<sup>8</sup> s for all the cases.

The Cartesian coordinate system is adopted with *x*, *y* and *z* in the streamwise, wall-normal and spanwise direction, respectively. The *x* axis is placed through the center of the plate with the origin placed at the leading edge of the plate. The computational domain is *Lx* =38 mm long, *Ly* =20 mm high and *Lz* =2 mm wide for case 1, and *Lx* =76 mm long, *Ly* =51 mm high and *Lz* =2 mm wide for cases 2 and 3. This is determined by taking two factors into consideration. In the streamwise direction, the domain is long enough to capture three-dimensional effects induced by heating and to eliminate the non-physical effects at the outflow boundary. Based on this constraint, the Reynolds number at the trailing edge of the plate is *ReL* <sup>=</sup> 1.80 <sup>×</sup> 106 for case 1, and 3.68 <sup>×</sup> <sup>10</sup><sup>6</sup> for cases 2 and 3. In the wall-normal direction, the domain is high enough to avoid the reflection of leading edge shock onto the surface. The upper boundary is positioned at 86*δ<sup>L</sup>* above the wall for case 1, and 220*δ<sup>L</sup>* for cases 2 and 3, where *δ<sup>L</sup>* is the boundary layer thickness at the trailing edge of the plate. The velocity, pressure and density in the figures shown in Section *Results and analyses* are normalized by *u*∞, *p*<sup>∞</sup> and *ρ*∞, respectively. The vorticity is normalized by *u*∞/*Lx*, where *u*<sup>∞</sup> = 450 m/s and *Lx* =0.038 m.

The grid is refined near the leading edge of the flat plate and near the heating element. Approximately 150 grid points are employed inside the boundary layer at the leading edge of the heating element to resolve the viscous layer and capture the heat release process. Previous results (Yan & Gaitonde, 2008) indicated that this is fine enough to capture the near-field effect of the thermal perturbation. The grid sizes are 477 × 277 × 81 in the *x*, *y* and *z* direction, respectively for case 1, and 854 × 297 × 81 for cases 2 and 3.

For boundary conditions, the stagnation temperature and pressure and Mach number are fixed at the inflow. The no-slip condition with a fixed wall temperature is used on the wall. The pulse is imposed as a sudden jump as shown in Fig. 2. The symmetry condition is enforced at the spanwise boundary to simulate spanwise periodic series of heating elements spaced *Lz* apart in the finite-span bump cases. This boundary condition is also suitable to mimic two-dimensional perturbation in the full-span bump case. First-order extrapolation is applied at the outflow and upper boundaries.

## **4. Results and analyses**

The study is comprised of two parts. The first part studies the effects of the pulsed bump whose properties are listed in case 1 in Table 1. The pulsed bump introduces the disturbance at *<sup>f</sup>* <sup>=</sup> 100 kHz, and is located at *Re*<sup>0</sup> <sup>=</sup> 6102 <sup>=</sup> 0.3721 <sup>×</sup> <sup>10</sup>6, immediately upstream of the first neutral point (*Re*=610) for this particular frequency, where the subscript 0 denotes the streamwise location of the thermal bump. The Reynolds number at the trailing edge of the plate is *ReL* <sup>=</sup> <sup>1341</sup><sup>2</sup> <sup>=</sup> 1.80 <sup>×</sup> 106, which corresponds to the location immediately downstream of the second neutral point (*Re* =1300) as shown in Fig. 3. The rectangular bump is considered with spanwise width (*w*) of 1 mm and its streamwise length (*l*) is arbitrarily set to 0.2 mm.


Table 1. Classification of cases simulated, <sup>Δ</sup>*Tw* <sup>=</sup> 0.76*Tad*, *Re*<sup>0</sup> <sup>=</sup> 0.3721 <sup>×</sup> 106

The second part examines the breakdown process at later stages of flow evolvement. To this end, the plate is extended far downstream of the second neutral point to *Re* = 1918 (*ReL* = 3.68 <sup>×</sup> <sup>10</sup>6) as shown in Fig. 3. Both 3D and 2D thermal bumps are considered. The cases are denoted as cases 2 and 3 in Table 1.

#### **4.1 Effect of pulsed bump**

#### **4.1.1 Unperturbed flow (basic state)**

The basic or unperturbed state is a Mach 1.5 adiabatic flat plate boundary layer with Reynolds number at the trailing edge of the plate of *ReL* <sup>=</sup> 1.80 <sup>×</sup> 106. Figs. 5 and 6 show the streamwise and wall-normal velocity profiles along the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 1.4 <sup>×</sup> 106 at the spanwise center and side of the plate. The *y* coordinate is normalized with the local theoretical boundary layer thickness (*δ*). Both boundary layer thickness and velocity profiles are predicted correctly compared to the compressible boundary layer theory. In particular, the wall-normal velocity, which is of much smaller order *v* ∼ *u*∞/ <sup>√</sup>*Rex*, is captured correctly as well. The fact that the profiles on the center and side of the plate collapse demonstrates flow two-dimensionality as expected.

6 Will-be-set-by-IN-TECH

the heating element to resolve the viscous layer and capture the heat release process. Previous results (Yan & Gaitonde, 2008) indicated that this is fine enough to capture the near-field effect of the thermal perturbation. The grid sizes are 477 × 277 × 81 in the *x*, *y* and *z* direction,

For boundary conditions, the stagnation temperature and pressure and Mach number are fixed at the inflow. The no-slip condition with a fixed wall temperature is used on the wall. The pulse is imposed as a sudden jump as shown in Fig. 2. The symmetry condition is enforced at the spanwise boundary to simulate spanwise periodic series of heating elements spaced *Lz* apart in the finite-span bump cases. This boundary condition is also suitable to mimic two-dimensional perturbation in the full-span bump case. First-order extrapolation is

The study is comprised of two parts. The first part studies the effects of the pulsed bump whose properties are listed in case 1 in Table 1. The pulsed bump introduces the disturbance at *<sup>f</sup>* <sup>=</sup> 100 kHz, and is located at *Re*<sup>0</sup> <sup>=</sup> 6102 <sup>=</sup> 0.3721 <sup>×</sup> <sup>10</sup>6, immediately upstream of the first neutral point (*Re*=610) for this particular frequency, where the subscript 0 denotes the streamwise location of the thermal bump. The Reynolds number at the trailing edge of the plate is *ReL* <sup>=</sup> <sup>1341</sup><sup>2</sup> <sup>=</sup> 1.80 <sup>×</sup> 106, which corresponds to the location immediately downstream of the second neutral point (*Re* =1300) as shown in Fig. 3. The rectangular bump is considered with spanwise width (*w*) of 1 mm and its streamwise length (*l*) is arbitrarily set

> Case *<sup>w</sup>*/*<sup>l</sup>* (mm) *<sup>f</sup>* (kHz) *ReL* <sup>×</sup> <sup>10</sup>−<sup>6</sup> 1/0.2 100 1.80 1/0.2 100 3.68 2/0.2 (full span) 100 3.68

The second part examines the breakdown process at later stages of flow evolvement. To this end, the plate is extended far downstream of the second neutral point to *Re* = 1918 (*ReL* = 3.68 <sup>×</sup> <sup>10</sup>6) as shown in Fig. 3. Both 3D and 2D thermal bumps are considered. The cases are

The basic or unperturbed state is a Mach 1.5 adiabatic flat plate boundary layer with Reynolds number at the trailing edge of the plate of *ReL* <sup>=</sup> 1.80 <sup>×</sup> 106. Figs. 5 and 6 show the streamwise and wall-normal velocity profiles along the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 1.4 <sup>×</sup> 106 at the spanwise center and side of the plate. The *y* coordinate is normalized with the local theoretical boundary layer thickness (*δ*). Both boundary layer thickness and velocity profiles are predicted correctly compared to the compressible boundary layer theory. In particular, the wall-normal velocity,

profiles on the center and side of the plate collapse demonstrates flow two-dimensionality as

<sup>√</sup>*Rex*, is captured correctly as well. The fact that the

Table 1. Classification of cases simulated, <sup>Δ</sup>*Tw* <sup>=</sup> 0.76*Tad*, *Re*<sup>0</sup> <sup>=</sup> 0.3721 <sup>×</sup> 106

respectively for case 1, and 854 × 297 × 81 for cases 2 and 3.

applied at the outflow and upper boundaries.

**4. Results and analyses**

denoted as cases 2 and 3 in Table 1.

**4.1.1 Unperturbed flow (basic state)**

which is of much smaller order *v* ∼ *u*∞/

**4.1 Effect of pulsed bump**

expected.

to 0.2 mm.

Fig. 5. Streamwise velocity in the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 1.4 <sup>×</sup> 106 (basic state for case 1)

Fig. 6. Vertical velocity in the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 1.4 <sup>×</sup> 106 (basic state for case 1)

#### **4.1.2 Perturbed flow by pulsed bump**

A pulsed thermal bump at a frequency of 100 kHz is turned on to introduce the disturbance after the basic state is obtained. Recall that the bump is placed immediately upstream of the first neutral point (where *Re* = 610) for disturbances at frequency of 100 kHz.

For all the pulsed heating cases, the solution is marched until a statistically stationary state is obtained. This determination is made by monitoring all primitive variables at several points in the domain. Mean statistics are then gathered over numerous cycles until time invariant values are obtained. The instantaneous results presented are those obtained after the time-mean quantities reach invariant values.

Fig. 7 shows the instantaneous streamwise vorticity contours on the wall. Since these values

Fig. 7. Instantaneous *ω<sup>x</sup>* contours on the wall (case 1)

are plotted after the flow reaches an asymptotic state, the vortex pattern is formed by the dynamic vortex interaction from numerous heating pulses. When the bump is pulsed, a complex vortex shedding and dynamic interaction process results in a vortical pattern with the alternating sign in the streamwise direction. These structures are constrained in the spanwise direction, but move away from the surface, which will be shown in the time-mean values. Smaller eddies are observed at about *Rex* <sup>=</sup> 1.25 <sup>×</sup> 106 near the central region and intensified downstream of *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106.

The effect of pulsing on the time-mean streamwise vorticity is shown in Fig. 8. The spanwise-varying streaky structures are formed downstream with concentration in the central

Fig. 8. Time-mean *ω<sup>x</sup>* contours on the wall (case 1)

region and intensified after *Rex* <sup>=</sup> 1.5 <sup>×</sup> <sup>10</sup>6. These results bear some similarity to free shear flow control with tabs. For example, Zaman *et al.* (Zaman et al., 1994) demonstrated with a comprehensive experimental study that the pressure variation induced by the tabs installed on the nozzle wall generated streamwise vorticity, which significantly enhanced the mixing downstream of the nozzle exit.

The vortex interaction and penetration can be seen on the cross sections in Fig. 9. The first cross section (Fig. 9(a)) is cut immediately downstream of the bump, therefore the top pair 8 Will-be-set-by-IN-TECH

For all the pulsed heating cases, the solution is marched until a statistically stationary state is obtained. This determination is made by monitoring all primitive variables at several points in the domain. Mean statistics are then gathered over numerous cycles until time invariant values are obtained. The instantaneous results presented are those obtained after

Fig. 7 shows the instantaneous streamwise vorticity contours on the wall. Since these values

are plotted after the flow reaches an asymptotic state, the vortex pattern is formed by the dynamic vortex interaction from numerous heating pulses. When the bump is pulsed, a complex vortex shedding and dynamic interaction process results in a vortical pattern with the alternating sign in the streamwise direction. These structures are constrained in the spanwise direction, but move away from the surface, which will be shown in the time-mean values. Smaller eddies are observed at about *Rex* <sup>=</sup> 1.25 <sup>×</sup> 106 near the central region and intensified

The effect of pulsing on the time-mean streamwise vorticity is shown in Fig. 8. The spanwise-varying streaky structures are formed downstream with concentration in the central

region and intensified after *Rex* <sup>=</sup> 1.5 <sup>×</sup> <sup>10</sup>6. These results bear some similarity to free shear flow control with tabs. For example, Zaman *et al.* (Zaman et al., 1994) demonstrated with a comprehensive experimental study that the pressure variation induced by the tabs installed on the nozzle wall generated streamwise vorticity, which significantly enhanced the mixing

The vortex interaction and penetration can be seen on the cross sections in Fig. 9. The first cross section (Fig. 9(a)) is cut immediately downstream of the bump, therefore the top pair

the time-mean quantities reach invariant values.

Fig. 7. Instantaneous *ω<sup>x</sup>* contours on the wall (case 1)

Fig. 8. Time-mean *ω<sup>x</sup>* contours on the wall (case 1)

downstream of *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106.

downstream of the nozzle exit.

Fig. 9. Time-mean *ω<sup>x</sup>* on different cross sections (case 1)

of vortices above the wall possesses the same sign as that at the leading edge of the bump shown in Fig. 8 (positive at *z* = 0.5*w* and negative at *z* = −0.5*w*). As they move downstream, they are lifted away from the wall and induce additional vortices near the wall to satisfy the noslip condition as well on the sides where periodic conditions apply. The original pairs form a double pattern as indicated in Fig. 9(b). As they move downstream, vortices are stretched and intensified as shown in Fig. 9(c). At *Rex* <sup>=</sup> 1.25 <sup>×</sup> <sup>10</sup><sup>6</sup> (Fig. 9(d)), the vortex pattern is distorted and the vortices break into smaller eddies. This leads to the complex vortex dynamic interaction downstream at *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 (Fig. 9(e)), which completely distorts the double pattern and results in a vortex trace that appears to move towards the center region. At the last station (Fig. 9(f)), the vorticity is intensified around the center region.

The accumulated effect of the streamwise vorticity distorts the basic state in nonlinear fashion. Fig. 10 shows the streamwise perturbation velocity contours on the downstream cross sections. The quantity plotted is *u*¯ − *ub*, where *u*¯ is the time-mean value of the pulsed case. Please note change in contour levels between Figs. 9 and 10. Immediately downstream of the bump (Fig. 10(a)), a velocity excess region is formed above the wall due to flow expansion. Proceeding downstream, a velocity deficit is generated downstream of the center of the heating element, while an excess is observed on both sides of the bump (Fig. 10(b)). This behavior is similar to the observation in the flow over a shallow bump studied by Joslin and Grosch (Joslin & Grosch, 1995) and the steady heating case discussed earlier. The intensity of the excess region is at the same level as that in the steady heating (compare Fig. 10(a) with Fig. 9(a)). Proceeding downstream, the pulsed bump behaves differently from the steady one. The velocity distortion is amplified as seen in Figs. 10(c) and (d). The velocity excess regions grow in the region near the wall across the entire span of the domain (Figs. 10(e) and (f)) as the streamwise vortices bring the high-momentum fluid from the freestream towards the wall.

The above observations are further explored in Fig. 11, which plots *u*¯ and *u*� = *u*¯ − *ub* along the *y* direction at *z*=0 and *z*=-0.5*w* (*i.e.*, at the spanwise edge of the bump). The intensity of the velocity excess in the near-wall region increases along the downstream and reaches about 20% of *<sup>u</sup>*<sup>∞</sup> at *Rex* <sup>=</sup> 1.75 <sup>×</sup> 106, while in the outer region, a velocity deficit is observed. This results in an inflection point in the mean flow near the centerline (Fig. 11(a)), giving rise to the rapid breakdown observed in Fig. 9. On the edges of the bump, the flow is accelerated cross the entire boundary layer and no inflection points are observed (Fig. 11(b)).

The strength of disturbance energy growth for the compressible flow is measured by the energy norm proposed by Tumin and Reshotko (Tumin & Reshotko, 2001) as

$$E = \int\_0^\infty \vec{q}^T A \vec{q} dy \tag{1}$$

where �*q* and *A* are the perturbation amplitude vector and diagonal matrix, respectively, and are expressed as

$$\vec{q} = (\mu', v', w', \rho', T')^T \tag{2}$$

$$A = \text{diag}\left[\rho\_\prime \rho\_\prime \rho\_\prime T/(\gamma \rho M\_\infty^2), \rho / (\gamma(\gamma - 1) T M\_\infty^2)\right] \tag{3}$$

The first three terms represent the components of the kinetic disturbance energy denoted as *Eu*, *Ev* and *Ew*, respectively and the last two represent the thermodynamic disturbance energy as *E<sup>ρ</sup>* and *ET*. The spanwise-averaged disturbance energy is plotted in Fig. 12. The initial growth rate of the total disturbance energy is small and becomes larger as the disturbances are amplified in the region of 0.9 <sup>×</sup> 106 <sup>&</sup>lt; *Rex* <sup>&</sup>lt; 1.4 <sup>×</sup> 106. The disturbances then saturate and reach finite amplitude shown as a plateau in Fig. 12(a). At this stage, the flow reaches 10 Will-be-set-by-IN-TECH

of vortices above the wall possesses the same sign as that at the leading edge of the bump shown in Fig. 8 (positive at *z* = 0.5*w* and negative at *z* = −0.5*w*). As they move downstream, they are lifted away from the wall and induce additional vortices near the wall to satisfy the noslip condition as well on the sides where periodic conditions apply. The original pairs form a double pattern as indicated in Fig. 9(b). As they move downstream, vortices are stretched and intensified as shown in Fig. 9(c). At *Rex* <sup>=</sup> 1.25 <sup>×</sup> <sup>10</sup><sup>6</sup> (Fig. 9(d)), the vortex pattern is distorted and the vortices break into smaller eddies. This leads to the complex vortex dynamic interaction downstream at *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 (Fig. 9(e)), which completely distorts the double pattern and results in a vortex trace that appears to move towards the center region. At the

The accumulated effect of the streamwise vorticity distorts the basic state in nonlinear fashion. Fig. 10 shows the streamwise perturbation velocity contours on the downstream cross sections. The quantity plotted is *u*¯ − *ub*, where *u*¯ is the time-mean value of the pulsed case. Please note change in contour levels between Figs. 9 and 10. Immediately downstream of the bump (Fig. 10(a)), a velocity excess region is formed above the wall due to flow expansion. Proceeding downstream, a velocity deficit is generated downstream of the center of the heating element, while an excess is observed on both sides of the bump (Fig. 10(b)). This behavior is similar to the observation in the flow over a shallow bump studied by Joslin and Grosch (Joslin & Grosch, 1995) and the steady heating case discussed earlier. The intensity of the excess region is at the same level as that in the steady heating (compare Fig. 10(a) with Fig. 9(a)). Proceeding downstream, the pulsed bump behaves differently from the steady one. The velocity distortion is amplified as seen in Figs. 10(c) and (d). The velocity excess regions grow in the region near the wall across the entire span of the domain (Figs. 10(e) and (f)) as the streamwise vortices bring the high-momentum fluid from the freestream towards the wall.

The above observations are further explored in Fig. 11, which plots *u*¯ and *u*� = *u*¯ − *ub* along the *y* direction at *z*=0 and *z*=-0.5*w* (*i.e.*, at the spanwise edge of the bump). The intensity of the velocity excess in the near-wall region increases along the downstream and reaches about 20% of *<sup>u</sup>*<sup>∞</sup> at *Rex* <sup>=</sup> 1.75 <sup>×</sup> 106, while in the outer region, a velocity deficit is observed. This results in an inflection point in the mean flow near the centerline (Fig. 11(a)), giving rise to the rapid breakdown observed in Fig. 9. On the edges of the bump, the flow is accelerated cross

The strength of disturbance energy growth for the compressible flow is measured by the

where �*q* and *A* are the perturbation amplitude vector and diagonal matrix, respectively, and

The first three terms represent the components of the kinetic disturbance energy denoted as *Eu*, *Ev* and *Ew*, respectively and the last two represent the thermodynamic disturbance energy as *E<sup>ρ</sup>* and *ET*. The spanwise-averaged disturbance energy is plotted in Fig. 12. The initial growth rate of the total disturbance energy is small and becomes larger as the disturbances are amplified in the region of 0.9 <sup>×</sup> 106 <sup>&</sup>lt; *Rex* <sup>&</sup>lt; 1.4 <sup>×</sup> 106. The disturbances then saturate and reach finite amplitude shown as a plateau in Fig. 12(a). At this stage, the flow reaches

<sup>∞</sup>), *<sup>ρ</sup>*/(*γ*(*<sup>γ</sup>* <sup>−</sup> <sup>1</sup>)*TM*<sup>2</sup>

�*q<sup>T</sup> A*�*qdy* (1)

)*<sup>T</sup>* (2)

<sup>∞</sup>)] (3)

 ∞ 0

, *v*� , *w*� , *ρ*� , *T*�

the entire boundary layer and no inflection points are observed (Fig. 11(b)).

energy norm proposed by Tumin and Reshotko (Tumin & Reshotko, 2001) as

are expressed as

*E* =

�*q* = (*u*�

*A* = *diag*[*ρ*, *ρ*, *ρ*, *T*/(*γρM*<sup>2</sup>

last station (Fig. 9(f)), the vorticity is intensified around the center region.

Fig. 10. *u*� = *u*¯ − *ub* on different cross sections (case 1)

a new state which becomes a basic state on which secondary instabilities can grow (Schmid & Henningson, 2001). The new basic state is represented by the appearance of the inflection point at *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 in the left plot of Fig. 11(a). The disturbances grow more rapidly after *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 because the secondary instability susceptible to high frequency disturbances usually grows more rapidly than the primary instabilities. The thermodynamic disturbance energy (*E<sup>ρ</sup>* and *ET*) in Fig. 12 (b) shows a similar trend except for a spike in the vicinity of the thermal bump as expected. However the thermodynamic components are four orders of magnitude lower than the *Eu*, indicating that the primary disturbance quickly develops a vortical nature.

Fig. 11. *u*¯ and *u*� = *u*¯ − *ub* along the *y* direction (case 1)

With pulsed heating, the disturbances grow significantly downstream and the flow becomes highly inflectional. The observation is consistent with the linear stability theory. However the velocity fluctuation reaches 20% of *u*∞ at the downstream, indicating that non-linear growth comes into the play and the assumption that disturbances are infinitesimal is not valid any 12 Will-be-set-by-IN-TECH

a new state which becomes a basic state on which secondary instabilities can grow (Schmid & Henningson, 2001). The new basic state is represented by the appearance of the inflection point at *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 in the left plot of Fig. 11(a). The disturbances grow more rapidly after *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 because the secondary instability susceptible to high frequency disturbances usually grows more rapidly than the primary instabilities. The thermodynamic disturbance energy (*E<sup>ρ</sup>* and *ET*) in Fig. 12 (b) shows a similar trend except for a spike in the vicinity of the thermal bump as expected. However the thermodynamic components are four orders of magnitude lower than the *Eu*, indicating that the primary disturbance quickly develops a

(a) at *z* = 0

(b) at *z* = −0.5*w*

With pulsed heating, the disturbances grow significantly downstream and the flow becomes highly inflectional. The observation is consistent with the linear stability theory. However the velocity fluctuation reaches 20% of *u*∞ at the downstream, indicating that non-linear growth comes into the play and the assumption that disturbances are infinitesimal is not valid any

Fig. 11. *u*¯ and *u*� = *u*¯ − *ub* along the *y* direction (case 1)

vortical nature.

Fig. 12. Spanwise-averaged disturbance energy along the *x* direction (case 1)

more. Thus, the dynamic vortex non-linear interaction plays an important role in the later development of the sustained disturbance growth, and will be discussed in the following section.

#### **4.2 Analyses of breakdown process**

This section explores the phenomenology of the non-linear dynamics between the vortices produced by the bump and the compressible boundary layer. To this end, the domain size is extended in both streamwise and normal directions relative to the case 1, but the spanwise width remains unchanged. The Reynolds number at the end of the plate is *ReL* <sup>=</sup> 3.68 <sup>×</sup> 106. Two cases (cases 2 and 3 in Table 1) are examined; the first considers a three-dimensional perturbation associated with a finite-span thermal bump, and the second is comprised of full-span disturbances. In both cases, the bump is positioned at the same streamwise location as in the case 1 with the same pulsing frequency and magnitude shown in Table 1.

A new basic state (no perturbation) is obtained for the cases with the extended domain. In the absence of imposed perturbations, no tendency is observed towards transition even at the higher Reynolds number. Figs. 13 and 14 show the streamwise and wall-normal velocity profiles along the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 3.5 <sup>×</sup> 106. The comparisons with the compressible theoretical profiles are good and the fact that the profiles on the center and side of the plate collapse demonstrates flow two-dimensionality as expected.

The heating element is turned on after the basic state is obtained. For the finite-span case, a series of counter-rotating streamwise vortices are generated at the edges of the thermal bump by heating induced surface pressure variation as discussed earlier. These vortices shed from their origins when the element is switched off, forming a traveling vortical pattern with an alternating sign in the streamwise direction up to *Rex* <sup>=</sup> 1.25 <sup>×</sup> <sup>10</sup><sup>6</sup> as shown in Fig. 15(a), where the instantaneous streamwise vorticity contours are plotted on the wall. Further downstream, small organized alternating structures appear near the center region at *Rex* = 1.5 <sup>×</sup> <sup>10</sup>6. Up to this point, the perturbed flow structures are similar to case 1 as expected. Subsequently, the vortices are intensified at about *Rex* <sup>=</sup> 2.0 <sup>×</sup> 106 due to vortex stretching

Fig. 13. Streamwise velocity in the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 3.5 <sup>×</sup> 106 (basic state for cases 2 and 3)

Fig. 14. Vertical velocity in the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 3.5 <sup>×</sup> 106 (basic state for cases 2 and 3)

14 Will-be-set-by-IN-TECH

Fig. 13. Streamwise velocity in the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 3.5 <sup>×</sup> 106 (basic state for cases 2 and 3)

Fig. 14. Vertical velocity in the *<sup>y</sup>* direction at *Rex* <sup>=</sup> 3.5 <sup>×</sup> 106 (basic state for cases 2 and 3)

Fig. 15. Instantaneous *ω<sup>x</sup>* contours on the wall (cases 2 and 3)

and interaction, which is described in more detail later. After *Rex* <sup>=</sup> 2.5 <sup>×</sup> 106, the flow tends to relax to a relatively universal stage. The full-span case shows a different development as shown in Fig. 15(b). The counter-rotating vortices are not observed immediately downstream of the bump. Rather, an asymmetric instantaneous vortical pattern is initiated with small successive structures starting at about *Rex* <sup>=</sup> 1.25 <sup>×</sup> 106, which are concentrated on the lower half of the domain. The fact that these small structures occur at the same location for both cases suggests that they are unlikely to be related to the original counter-rotating vortices, and an inherent stability mechanism that stimulates their appearance.

The vortex development is examined in a three-dimensional fashion in Fig. 16 which shows the iso-surface of the non-dimensionalized vorticity magnitude at |*ω*| = 100 colored with

Fig. 16. Iso-surface of instantaneous vorticity magnitude at |*ω*| = 100, *x* : *y* = 1 : 10, *x* : *z* = 1 : 10 (cases 2 and 3)

the distance from the wall. This iso-level is chosen to reveal the near-wall structures. For visualization purpose, the *y* and *z* axes are equally stretched with a ratio to the *x* axis of 10. The same length unit is used for these three axes and Reynolds numbers are only marked for discussion purpose. Thus the structures are closer to the wall and more elongated in the streamwise direction than they appear in the plot. In the finite-span case shown in Fig. 16(a), a sheet of vorticity is generated by the wall shear and rolls up into three rows of hairpin-like vortices across the span at about *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106. The vortices are then slightly lifted away from the wall at about *Rex* <sup>=</sup> 2.0 <sup>×</sup> 106. Correspondingly, the vortices are stretched in the streamwise direction in the wall region, resulting in the intensification of streamwise vorticity as previously shown in Fig. 15(a). The vortices become weaker as the flow relaxes further downstream. As shown in Fig. 16(b), hairpin-like structures are also observed for the full-span case, but they develop in an asymmetric fashion. Similar to the full-span case, vortex stretching in the lifting process induces strong streamwise vorticity in the wall region.

The hairpin structures are better displayed by lowering the iso-levels to |*ω*| = 25 as shown in Fig. 17. The hairpin-like vortices are initiated across the span at about *Rex* <sup>=</sup> 2.0 <sup>×</sup> 106.

Fig. 17. Iso-surface of instantaneous vorticity magnitude at |*ω*| = 25, *x* : *y* = 1 : 10, *x* : *z* = 1 : 10 (cases 2 and 3)

The legs of the hairpin constitute a pair of counter-rotating vortices oriented in the streamwise direction in the wall region. They are mainly comprised of *ω<sup>x</sup>* and can be difficult to discern in the total vorticity iso-surface plot since the spanwise vorticity *ω<sup>z</sup>* is dominant in the boundary layer. On the other hand, the heads, mainly comprised of *ωz*, can be easily identified in the total vorticity variable because they penetrate into the boundary layer about 2.24*δ* and 2.35*δ* at *Rex* <sup>=</sup> 2.5<sup>×</sup> 106 and 3.5<sup>×</sup> 105, respectively, where *<sup>δ</sup>* is the local unperturbed laminar boundary layer thickness. It is also observed that for the full-span case, the hairpin vortices are tilted higher in the boundary layer than in the finite-span case. The fact that hairpin vortices appear in the full-span case confirms that the initial counter-rotating streamwise vortices are not a necessity in generating the hairpin vortices.

The vorticity concentration can be viewed through vorticity deviation from the basic state as shown in Fig. 18. Looking downstream, close examination reveals that the right leg rotates with positive *ω<sup>x</sup>* and the head with negative *ωz*. Three hairpin vortices are annotated on the plot. The legs can be more clearly seen in the iso-surface of *ω<sup>x</sup>* difference in Fig. 19 and the head in the iso-surface of *ω<sup>z</sup>* difference in Fig. 20. Since the value of *ω* changes, the structures appear to be broken, but other values confirm the coherence of the structures. The hairpin vortices are aligned in the streamwise direction, forming a pattern similar to K-type breakdown, which results from fundamental modes (Klebanoff et al., 1962). In addition, they appear to be highly asymmetric for both cases. Robinson (Robinson, 1991) pointed out that in a turbulent boundary layer, the symmetry of vortex was predominantly distorted, yielding structures designated "one-legged hairpins". Fig. 21 shows a hairpin vortex schematically. Low-momentum fluid is lifted away from the wall between the legs while high-momentum fluid from the freestream is brought down to the wall outside the legs.

The above described motion of the hairpin vortices alters the velocity distribution in the wall region. In the finite-span case, the passage of the counter-rotating vortices generates 16 Will-be-set-by-IN-TECH

vorticity as previously shown in Fig. 15(a). The vortices become weaker as the flow relaxes further downstream. As shown in Fig. 16(b), hairpin-like structures are also observed for the full-span case, but they develop in an asymmetric fashion. Similar to the full-span case, vortex stretching in the lifting process induces strong streamwise vorticity in the wall region.

The hairpin structures are better displayed by lowering the iso-levels to |*ω*| = 25 as shown in Fig. 17. The hairpin-like vortices are initiated across the span at about *Rex* <sup>=</sup> 2.0 <sup>×</sup> 106.

(a) finite span (b) full span

The legs of the hairpin constitute a pair of counter-rotating vortices oriented in the streamwise direction in the wall region. They are mainly comprised of *ω<sup>x</sup>* and can be difficult to discern in the total vorticity iso-surface plot since the spanwise vorticity *ω<sup>z</sup>* is dominant in the boundary layer. On the other hand, the heads, mainly comprised of *ωz*, can be easily identified in the total vorticity variable because they penetrate into the boundary layer about 2.24*δ* and 2.35*δ* at *Rex* <sup>=</sup> 2.5<sup>×</sup> 106 and 3.5<sup>×</sup> 105, respectively, where *<sup>δ</sup>* is the local unperturbed laminar boundary layer thickness. It is also observed that for the full-span case, the hairpin vortices are tilted higher in the boundary layer than in the finite-span case. The fact that hairpin vortices appear in the full-span case confirms that the initial counter-rotating streamwise vortices are not a

The vorticity concentration can be viewed through vorticity deviation from the basic state as shown in Fig. 18. Looking downstream, close examination reveals that the right leg rotates with positive *ω<sup>x</sup>* and the head with negative *ωz*. Three hairpin vortices are annotated on the plot. The legs can be more clearly seen in the iso-surface of *ω<sup>x</sup>* difference in Fig. 19 and the head in the iso-surface of *ω<sup>z</sup>* difference in Fig. 20. Since the value of *ω* changes, the structures appear to be broken, but other values confirm the coherence of the structures. The hairpin vortices are aligned in the streamwise direction, forming a pattern similar to K-type breakdown, which results from fundamental modes (Klebanoff et al., 1962). In addition, they appear to be highly asymmetric for both cases. Robinson (Robinson, 1991) pointed out that in a turbulent boundary layer, the symmetry of vortex was predominantly distorted, yielding structures designated "one-legged hairpins". Fig. 21 shows a hairpin vortex schematically. Low-momentum fluid is lifted away from the wall between the legs while high-momentum

The above described motion of the hairpin vortices alters the velocity distribution in the wall region. In the finite-span case, the passage of the counter-rotating vortices generates

fluid from the freestream is brought down to the wall outside the legs.

Fig. 17. Iso-surface of instantaneous vorticity magnitude at |*ω*| = 25, *x* : *y* = 1 : 10,

*x* : *z* = 1 : 10 (cases 2 and 3)

necessity in generating the hairpin vortices.

Fig. 18. Iso-surface of instantaneous vorticity magnitude difference at |*ω*| − (|*ω*|)*<sup>b</sup>* = 25, *x* : *y* = 1 : 10, *x* : *z* = 1 : 10 (cases 2 and 3)

Fig. 19. Iso-surface of instantaneous streamwise vorticity difference at *ω<sup>x</sup>* − (*ωx*)*<sup>b</sup>* = ±15, *x* : *y* = 1 : 10, *x* : *z* = 1 : 10 (cases 2 and 3)

Fig. 20. Iso-surface of instantaneous spanwise vorticity difference *ω<sup>z</sup>* − (*ωz*)*<sup>b</sup>* = 25, *x* : *y* = 1 : 10, *x* : *z* = 1 : 10 (cases 2 and 3)

Fig. 21. Schematic of hairpin vortex in pulsed heating

several streamwise streaks in the center region and a low-speed streak is flanked alternately by high and low-speed streaks as shown in Fig. 22(a), which plots the instantaneous streamwise

Fig. 22. Instantaneous *u* contours for the finite-span case (case 2)

velocity (*u*) contours at the first grid point above the wall (*i.e. y* = *δ*0/200). The central low-speed region is intensified and concentrated towards the center between 1.5 <sup>×</sup> <sup>10</sup><sup>6</sup> <sup>&</sup>lt; *Rex* <sup>&</sup>lt; 2.0 <sup>×</sup> <sup>10</sup>6, resulting in a strong growth of the boundary layer as shown in Fig. 22(b), which depicts the velocity contours on the symmetry plane passing through the center of the domain. At *Rex* <sup>=</sup> 2.0 <sup>×</sup> 106 which is just downstream of the second neutral point, the flow pattern changes dramatically. The low-speed streaks weaken in the wall region and the near-wall low-momentum region becomes thinner as the hairpin vortices pump the low-momentum fluid away from the wall. Strong three-dimensional fluctuations are observed in the upper portion of the boundary layer where the hairpin vortices interact with the high-momentum fluid, leading to the boundary layer distortion.

In the full-span case, the initial spanwise structures are almost two-dimensional in nature. Subsequently, the low-speed streaks are formed at about the location where the hairpin vortices start to appear as shown in Fig. 23(a). This indicates that the low-speed streaks are the footprints of the hairpin vortices. The boundary layer growth is not as strong as that in the finite-span case between 1.5 <sup>×</sup> 106 <sup>&</sup>lt; *Rex* <sup>&</sup>lt; 2.0 <sup>×</sup> 106 (compare Figs. 23(b) and 22(b)). However, downstream of *Rex* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup>6, strong three-dimensional fluctuations are observed, similar to the finite-span case. It suggests that the non-linear disturbance growth becomes dominant and the initial disturbance form becomes less important. This 18 Will-be-set-by-IN-TECH

several streamwise streaks in the center region and a low-speed streak is flanked alternately by high and low-speed streaks as shown in Fig. 22(a), which plots the instantaneous streamwise

(a) at *y* = *δ*0/200 (b) on the center plane

velocity (*u*) contours at the first grid point above the wall (*i.e. y* = *δ*0/200). The central low-speed region is intensified and concentrated towards the center between 1.5 <sup>×</sup> <sup>10</sup><sup>6</sup> <sup>&</sup>lt; *Rex* <sup>&</sup>lt; 2.0 <sup>×</sup> <sup>10</sup>6, resulting in a strong growth of the boundary layer as shown in Fig. 22(b), which depicts the velocity contours on the symmetry plane passing through the center of the domain. At *Rex* <sup>=</sup> 2.0 <sup>×</sup> 106 which is just downstream of the second neutral point, the flow pattern changes dramatically. The low-speed streaks weaken in the wall region and the near-wall low-momentum region becomes thinner as the hairpin vortices pump the low-momentum fluid away from the wall. Strong three-dimensional fluctuations are observed in the upper portion of the boundary layer where the hairpin vortices interact with the

In the full-span case, the initial spanwise structures are almost two-dimensional in nature. Subsequently, the low-speed streaks are formed at about the location where the hairpin vortices start to appear as shown in Fig. 23(a). This indicates that the low-speed streaks are the footprints of the hairpin vortices. The boundary layer growth is not as strong as that in the finite-span case between 1.5 <sup>×</sup> 106 <sup>&</sup>lt; *Rex* <sup>&</sup>lt; 2.0 <sup>×</sup> 106 (compare Figs. 23(b) and 22(b)). However, downstream of *Rex* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup>6, strong three-dimensional fluctuations are observed, similar to the finite-span case. It suggests that the non-linear disturbance growth becomes dominant and the initial disturbance form becomes less important. This

Fig. 21. Schematic of hairpin vortex in pulsed heating

Fig. 22. Instantaneous *u* contours for the finite-span case (case 2)

high-momentum fluid, leading to the boundary layer distortion.

Fig. 23. Instantaneous *u* contours for the full-span case (case 3)

is confirmed in the disturbance energy growth in Fig. 24, which plots the spanwise-averaged

Fig. 24. Spanwise-averaged time-mean total disturbance energy along the *x* direction (cases 2 and 3)

time-mean total disturbance energy for both finite- and full-span cases. The energy growth in the 2-D perturbations is much weaker than that in the 3-D ones near the bump. However, as the non-linear stability mechanism becomes dominant after about *Rex* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup>6, the disturbance energy growth in both cases becomes comparable.

The accumulated effect of high-frequency pulsing is now described by the time-mean quantities. Only the finite-span results are shown unless otherwise specified. The time-mean pressure (*p*¯) contours are shown on the center plane in Fig. 25. A series of expansion waves is formed at about *Rex* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup><sup>6</sup> and propagates outside the boundary layer. This is caused by the strong boundary layer distortion as shown in the time-mean streamwse velocity contours on the center plane in Fig. 26. The momentum thickness at *Rex* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup><sup>6</sup> is increased by a factor of 1.7 compared to that in laminar flow, indicating that the boundary layer is highly energized downstream of *Rex* <sup>=</sup> 2.0 <sup>×</sup> 106 and shows signs of transition to turbulence. The

Fig. 25. *p*¯ contours on the center plane for the finite-span case (case 2)

Fig. 26. *u*¯ contours on the center plane for the finite-span case (case 2)

expansion waves in the downstream location are also partially observed in case 1, in which the outlet boundary is set at *ReL* <sup>=</sup> 1.80 <sup>×</sup> 106.

The boundary layer distortion can be assessed by the variation of shape factor *H* obtained from the mean velocity profile as shown in Fig. 27. The shape factor for the basic state, shown for comparison, reaches an asymptotic value of 2.6 as the flow becomes fully-developed laminar (Fig. 27(a)). With heating, the mean flow is strongly distorted, causing the shape factor to oscillate taking values of 2.85 and 1.35 between *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 and 2.0 <sup>×</sup> 106, respectively as shown in Fig. 27(a). A lower shape factor indicates a fuller velocity profile. After *Rex* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup><sup>6</sup> the shape factor decreases rapidly, indicating an increase of the flow momentum in the boundary layer, and starts to level off around *Rex* <sup>=</sup> 3.0 <sup>×</sup> 106. Strong spanwise non-uniformity is observed at *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 and 2.0 <sup>×</sup> <sup>10</sup><sup>6</sup> as shown in Fig. 27(b), while in later stages, only mild distortion is observed and the shape factor reduces to around 1.5, which is close to the turbulent value.

Features of the turbulence statistics are examined through the transformed velocity and Reynolds stresses. Fig. 28 shows the transformed velocity profiles at different downstream locations along the center line (*z*=0) and the side line of the bump (*z*=-0.5*w*). In the viscous sublayer of a compressible turbulent boundary layer where *y*<sup>+</sup> < 5, the turbulent stresses are negligible compared to viscous stress and the velocity near the wall grows linearly with the distance from the wall as *u*<sup>+</sup> = *y*+, where *u*<sup>+</sup> is defined as *uvd*/*uτ*, and *y*<sup>+</sup> as *yuτ*/*νw*. The friction velocity *u<sup>τ</sup>* is defined as *τw*/*ρw*, where *τ<sup>w</sup>* is wall stress. The detailed formulation of the transformed velocity *uvd* may be found in Smits and Dussauge (Smits & Dussauge, 2006). 20 Will-be-set-by-IN-TECH

Fig. 25. *p*¯ contours on the center plane for the finite-span case (case 2)

Fig. 26. *u*¯ contours on the center plane for the finite-span case (case 2)

the outlet boundary is set at *ReL* <sup>=</sup> 1.80 <sup>×</sup> 106.

1.5, which is close to the turbulent value.

expansion waves in the downstream location are also partially observed in case 1, in which

The boundary layer distortion can be assessed by the variation of shape factor *H* obtained from the mean velocity profile as shown in Fig. 27. The shape factor for the basic state, shown for comparison, reaches an asymptotic value of 2.6 as the flow becomes fully-developed laminar (Fig. 27(a)). With heating, the mean flow is strongly distorted, causing the shape factor to oscillate taking values of 2.85 and 1.35 between *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 and 2.0 <sup>×</sup> 106, respectively as shown in Fig. 27(a). A lower shape factor indicates a fuller velocity profile. After *Rex* <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup><sup>6</sup> the shape factor decreases rapidly, indicating an increase of the flow momentum in the boundary layer, and starts to level off around *Rex* <sup>=</sup> 3.0 <sup>×</sup> 106. Strong spanwise non-uniformity is observed at *Rex* <sup>=</sup> 1.5 <sup>×</sup> 106 and 2.0 <sup>×</sup> <sup>10</sup><sup>6</sup> as shown in Fig. 27(b), while in later stages, only mild distortion is observed and the shape factor reduces to around

Features of the turbulence statistics are examined through the transformed velocity and Reynolds stresses. Fig. 28 shows the transformed velocity profiles at different downstream locations along the center line (*z*=0) and the side line of the bump (*z*=-0.5*w*). In the viscous sublayer of a compressible turbulent boundary layer where *y*<sup>+</sup> < 5, the turbulent stresses are negligible compared to viscous stress and the velocity near the wall grows linearly with the distance from the wall as *u*<sup>+</sup> = *y*+, where *u*<sup>+</sup> is defined as *uvd*/*uτ*, and *y*<sup>+</sup> as *yuτ*/*νw*. The friction velocity *u<sup>τ</sup>* is defined as *τw*/*ρw*, where *τ<sup>w</sup>* is wall stress. The detailed formulation of the transformed velocity *uvd* may be found in Smits and Dussauge (Smits & Dussauge, 2006).

Fig. 27. Shape factor for the finite-span case (case 2)

Fig. 28. Transformed velocity for the finite-span case at different Reynolds number (case 2)

Good agreement is found with the theory at different Reynolds numbers at both center and side locations. The turbulent stresses become large between *<sup>y</sup>*<sup>+</sup> <sup>&</sup>gt; 30 and *<sup>y</sup>*/*<sup>δ</sup>* � 1 where the log law holds with *u*<sup>+</sup> = <sup>1</sup> *<sup>κ</sup>* ln(*y*+) + *<sup>C</sup>* with *<sup>κ</sup>* <sup>=</sup> 0.4 and *<sup>C</sup>* <sup>=</sup> 5.1 (Smits & Dussauge, 2006). It is shown in Fig. 28 that the logarithmic region gradually forms with increasing Reynolds number and the velocity slope approaches the log law. However, a large discrepancy remains between the velocity profile at the end of the plate (*Rex* <sup>=</sup> 3.5×106) and the log law, indicating that the perturbed flow has not reached fully-developed turbulence.

The Reynolds stress profiles are shown to further examine the evolution of the flow. Fig. 29 shows the streamwise Reynolds stress (*ρ*¯*u*�*u*� ) and Reynolds shear stress (*ρ*¯*u*�*v*�) normalized by the local wall stress (*τw*) at *Rex* <sup>=</sup> 3.5 <sup>×</sup> 106. Note that the local boundary layer thickness *δ* varies across the span. Experimental and numerical results by other researchers (Johnson

Fig. 29. Reynolds stress for the finite-span case (case 2)

& Rose, 1975; Konrad, 1993; Konrad & Smits, 1998; Muck et al., 1984; Yan et al., 2002; Zheltovodov et al., 1990) are plotted for comparison. The predicted streamwise Reynolds stress presents a similar trend to the experiments and other numerical data. It reaches the peak at about *y* = 0.05*δ*–0.1*δ* and decays rapidly between 0.1*δ* < *y* < 0.3*δ*. A large spanwise variation of the peak value is observed with the value at *z* = 0 being 1.8 times that at *z* = −0.5*w*. The same observation holds for the Reynolds shear stress as shown in Fig. 29(b), which is a main source of turbulence production in the wall-bounded flows. The largely scattered data implies that the flow is still in transitional stage, where the strong non-linear disturbances continue to extract energy from the mean flow to maintain their mobility before the energy redistribution equilibrates and the flow exhibits some features of fully-developed turbulence. This is also consistent with that the mean velocity profile being located above the log law in Fig. 28.

Overall, the effect of the disturbance introduced by thermal bumps is observed to follow classical stability theory in the linear growth region. For the parameters considered, the gross features of transitional flow appear near the second neutral point. These features consist of hairpin vortex structures which are non-staggered and resemble K-type transition. Comparison of 3-D (finite span) with 2-D (full span) perturbations effects indicate that although the near field consequences of the bump are profoundly different, the development further downstream is relatively similar, suggesting a common non-linear mechanism associated with the interaction of the disturbance with the boundary layer vorticity.

#### **5. Concluding remarks**

This chapter explores the stability mechanism of a thermally perturbed Mach 1.5 flat plate boundary layer. With pulsed heating at frequency of 100 kHz immediately upstream of the first neutral point, non-linear dynamic vortex interactions cause disturbances to grow dramatically downstream and the maximum velocity fluctuation reaches about 20% of *u*∞. The inflectional velocity profile makes the flow highly susceptible to the secondary instabilities.

The dynamic vortex interaction at later stages of the boundary layer development is studied by extending the flat plate further downstream. Hairpin structures, considered as one kind of the basic structures in turbulence, are observed and serve to increase the momentum in the wall region. The fact that the hairpin vortices are observed in the full-span case suggests that the initial counter-rotating vortices generated by the finite-span bump might not be directly associated with the formation of hairpin structures. The boundary layer is observed to grow noticeably downstream relative to the unperturbed case. The Reynolds stresses and shape factor profiles suggest that the boundary layer is approaching turbulence, but remains transitional at the end of the computational domain. These results suggest that pulsed heating can be used as an effective mechanism to modulate the supersonic laminar-turbulence transition. One effective way to generate pulsed heating is through plasma actuator where Joule heating and electrode heating are effectively assumed as surface heating.

#### **6. References**

22 Will-be-set-by-IN-TECH

(a) Streamwise stress (b) Shear stress

& Rose, 1975; Konrad, 1993; Konrad & Smits, 1998; Muck et al., 1984; Yan et al., 2002; Zheltovodov et al., 1990) are plotted for comparison. The predicted streamwise Reynolds stress presents a similar trend to the experiments and other numerical data. It reaches the peak at about *y* = 0.05*δ*–0.1*δ* and decays rapidly between 0.1*δ* < *y* < 0.3*δ*. A large spanwise variation of the peak value is observed with the value at *z* = 0 being 1.8 times that at *z* = −0.5*w*. The same observation holds for the Reynolds shear stress as shown in Fig. 29(b), which is a main source of turbulence production in the wall-bounded flows. The largely scattered data implies that the flow is still in transitional stage, where the strong non-linear disturbances continue to extract energy from the mean flow to maintain their mobility before the energy redistribution equilibrates and the flow exhibits some features of fully-developed turbulence. This is also consistent with that the mean velocity profile being located above the

Overall, the effect of the disturbance introduced by thermal bumps is observed to follow classical stability theory in the linear growth region. For the parameters considered, the gross features of transitional flow appear near the second neutral point. These features consist of hairpin vortex structures which are non-staggered and resemble K-type transition. Comparison of 3-D (finite span) with 2-D (full span) perturbations effects indicate that although the near field consequences of the bump are profoundly different, the development further downstream is relatively similar, suggesting a common non-linear mechanism

This chapter explores the stability mechanism of a thermally perturbed Mach 1.5 flat plate boundary layer. With pulsed heating at frequency of 100 kHz immediately upstream of the first neutral point, non-linear dynamic vortex interactions cause disturbances to grow dramatically downstream and the maximum velocity fluctuation reaches about 20% of *u*∞. The inflectional velocity profile makes the flow highly susceptible to the secondary

associated with the interaction of the disturbance with the boundary layer vorticity.

Fig. 29. Reynolds stress for the finite-span case (case 2)

log law in Fig. 28.

**5. Concluding remarks**

instabilities.


24 Will-be-set-by-IN-TECH

162 Low Reynolds Number Aerodynamics and Transition

Roe, P. (1981). Approximate riemann solvers, parameter vectors and difference schemes, *J.*

Roth, J. R., Sherman, D. M. & P, W. S. (2000). Electrohydrodynamic flow control with a glow

Samimy, M., Kim, J.-H., Kastner, J., Adamovich, J. & Utkin, Y. (2007). Active control of

Schmid, P. J. & Henningson, D. S. (2001). *Stability and Transition in Shear Flows*, Springer-Verlag,

Shang, J. S. (2002). Plasma injection for hypersonic blunt body drag reduction, *AIAA J.*

Shang, J. S., Surzhikov, S. T., Kimmel, R., Gaitonde, D., Menart, J. & Hayes, J. (2005).

Smits, A. J. & Dussauge, J.-P. (2006). *Turbulent Shear Layers in Supersonic Flow, second edition*,

Tumin, A. & Reshotko, E. (2001). Spatial theory of optimal disturbances in boundary layers,

Tumin, A. & Reshotko, E. (2005). Receptivity of a boundary-layer flow to a three-dimensional

Van Leer, B. (1979). Towards the ultimate conservative difference scheme. v. a second order

White, E. B., Rice, J. M. & Ergin, F. G. (2005). Receptivity of stationary transient disturbances

Worner, A., Rist, U. & Wagner, S. (2003). Humps/steps influence on stability characteristics of

Yan, H. & Gaitonde, D. (2008). Numerical study on effect of a thermal bump in supersonic

Yan, H. & Gaitonde, D. (2010). Effect of thermally-induced perturbation in supersonic

Yan, H. & Gaitonde, D. (2011). Parametric study of pulsed thermal bumps in supersonic

Yan, H., Gaitonde, D. & Shang, J. (2007). Investigation of localized arc filament plasma actuator

Yan, H., Gaitonde, D. & Shang, J. (2008). The effect of a thermal bump in supersonic flow,

Yan, H., Knight, D. & Zheltovodov, A. A. (2002). Large eddy simulation of supersonic flat plate boundary layer using miles technique, *J. Fluids Eng.* 124(4): 868–875. Zaman, K., Samimy, M. & Reeder, M. F. (1994). Control of an axisymmetric jet using vortex

Zheltovodov, A. A., Trofimov, V. M., Schülein, E. & Yakovlev, V. N. (1990). An experimental

documentation of supersonic turbulent flows in the vicinity of forward- and backward-facing ramps, *Rep No 2030, Institute of Theoretical and Applied Mechanics,*

hump at finite reynolds numbers, *Phys. Fluid* 17(9): 094101.

to surface roughness, *Physics of Fluids* 17: 064109.

flow control, *AIAA Paper 2008-3790*.

generators, *Phys. Fluids* 6(2): 778–793.

*AIAA Paper 2008-1096*.

*USSR Academy of Sciences.*

boundary layer, *Physics of Fluids* 22: 064101.

boundary layer, *Shock Waves* 21(5): 411–423.

in supersonic boundary layer, *AIAA Paper 2007-1234*.

sequel to godunov's method, *J. Computational Physics* 32: 101–136.

two-dimensional laminar boundary layer, *AIAA J.* 41(2): 192–197.

high-speed and high-reynolds-number jets using plasma actuators, *J. Fluid Mech.*

Mechanisms of plasma actuators for hypersonic flow control, *Progress in Aerospace*

*Computational Physics* 43(2): 357–372.

578: 305–330.

New York, NY.

40: 1178–1186.

*Sciences.* 41: 642–668.

Springer, New York, PA.

*Phys. Fluid* 13(7): 2097–2104.

discharge surface plasma, *AIAA J.* 38(7): 1166–1172.

## *Edited by Mustafa Serdar Genc*

This book reports the latest development and trends in the low Re number aerodynamics, transition from laminar to turbulence, unsteady low Reynolds number flows, experimental studies, numerical transition modelling, control of low Re number flows, and MAV wing aerodynamics. The contributors to each chapter are fluid mechanics and aerodynamics scientists and engineers with strong expertise in their respective fields. As a whole, the studies presented here reveal important new directions toward the realization of applications of MAV and wind turbine blades.

Low Reynolds Number Aerodynamics and Transition

Low Reynolds Number

Aerodynamics and Transition

*Edited by Mustafa Serdar Genc*

Photo by yuran-78 / iStock