**2.1 eN Method**

*Boundary Layer Flows - Theory, Applications and Numerical Methods*

universal and depends on the type of flow.

Reynolds Averaged Navier-Stokes (RANS) solvers are widely available for numerically predicting fully turbulent part of flow fields by frequent use of, for instance, one- or two-equation turbulence closure models. However, none of these models are adequate to handle flows with significant transition effects due to the lack of practical transition modeling. Menter et al. [1] state that some of the main requirements for pragmatic transition modeling are the following: calibrated prediction of the onset and length of transition, allow inclusion of different mechanisms, allow local formulation, and allow a robust integration with background turbulence

Nevertheless, transition modeling as applied to CFD methods has followed certain line of evolution covering a range of methods starting from simple linear stability methods such as the eN method [2, 3] to almost or fully predictive methods such as LES and DNS that are very costly for engineering applications [1]. The eN method is the lowest level transition model based on linear stability theory. This method has found quite wide application in numerical boundary layer methods [4], but translating this into RANS methods has proven quite demanding as it requires a high-resolution boundary layer code that must work hand in hand with the RANS method. Also, this method is also dependent on the empirical factor-n that is not

Following the eN method, a better level of complexity that is compatible with the CFD methods is the low Reynolds number turbulence models [5]. Yet, they do not reflect real flow physics and lack the true predictive capability. These methods take advantage of the fortuitous ability of the wall damping terms mimicking some of the effects of transition. Next in the line of increasing complexity comes the class of the so-called correlation-based transition models [1]. These models are based on the fundamental approach of blending the laminar and the turbulent regions of the flow field by introducing intermittency equations to the turbulence equations. In this line, based on the boundary layer methods, there are three similar examples of intermittency equation approach that was introduced by Dhawan and Narasimha [6], Steelant and Dick [7], and Cho and Chung [8]. First, Dhawan and Narasimha [6] used a generalized form of intermittency distribution function in order to combine the laminar and the turbulent flow regions. Second, Steelant and Dick [7] proposed an intermittency equation that behaves like an experimental correlation. Third, Cho and Chung [8] introduced the k-ε-γ model which was formulated by an additional transport equation-γ to the well-known k-ε turbulence model. Finally, Suzen and Huang [9] significantly improved intermittency equation approach for flow transition prediction by combining the last two methods with a model that simulates transition in both streamwise and cross-stream directions. However, these models all rely on nonlocal flow data, and it was difficult to embed these models into practical CFD codes. These models require calculating the momentum thickness Reynolds number-Reθ, which is an integral parameter, and comparing it with a critical momentum thickness Reynolds number. For this reason, these early models are "nonlocal" methods that require exhausting search algorithms for flows with

After the success of the "nonlocal" transition models that use intermittency transport equations including experimental correlations, a range of new methods [10, 11] has been developed, called as the local correlation-based transition models (LCTM) by Menter et al. [1] that are compatible with the modern CFD codes. This compatibility has been achieved by the experimental observation that a locally calculated parameter called as the vorticity Reynolds number (Rev) is proportional to the momentum thickness Reynolds number (Reθ) in a Blasius boundary layer. This observation is also shown to be quite effective for a wide class of flow types with moderate pressure gradients. This is due to the fact that the relative error between

**102**

complex geometries.

models.

The well-known eN method is based on the linear stability theory [25], and it is developed by assuming that the flow is two-dimensional and steady, the boundary layer is thin and the level of disturbances in the flow region is initially very low. In this method, the Orr-Sommerfeld eigenvalue equations are solved by using the previously obtained velocity profiles over a surface in order to calculate the local instability amplification rates of the most unstable waves for each profile. By taking the integral of those rates after a certain point where the flow first becomes unstable along each streamline, an amplification factor is calculated. Transition is said to occur when the value of the amplification factor exceeds a threshold N value. Typical values of N vary between 7 and 9.

#### **2.2 Low Reynolds number turbulence models**

In the low Reynolds number turbulence models, the wall damping functions are modified in order to capture the transition effects [5]. To be able to predict the transition onset, these models depend on the diffusion of the turbulence from freestream into the boundary layer and its interaction with the source terms of the turbulence models. For this reason, these models are more suitable for bypass transition flows. Nonetheless, due to the similarities between a developing laminar boundary layer and a viscous sublayer, their success is thought to be coincidental, and thus these modes are mostly unreliable. These models also lack sensitivity to adverse pressure gradients and convergence problems arise for separation-induced transition cases.

### **2.3 Intermittency equation transition models**

It has been known from experiment that turbulence has an intermittent character with large fluctuations in flow variables like velocity, pressure, etc. Based on this observation, transition to turbulence has been tried to be modeled using the so-called intermittency function. One-, two- or three-equation partial differential equations have been derived to include the intermittency equation as one of the equations of the complete equation set including relevant experimental calibrations that mimic the actual physical behavior. To this end, "nonlocal" [7–9] and "local" [1, 10, 11] correlation transition models have been proposed. In the following, a systematic line of progress is presented that reveals the evolution of such models.

#### *2.3.1 Models depending on nonlocal flow variables*

### *2.3.1.1 Dhawan and Narasimha model*

Dhawan and Narasimha [6] proposed a scalar intermittency function-γ that would provide some sort of a measure of progression toward a fully turbulent boundary layer. Based on the experimentally measured streamwise intermittency distributions on flat plate boundary layers, for instance, Dhawan and Narasimha [6] introduced the following function for streamwise intermittency profile:

$$\text{reduced the following function for streamwise interactions profile:}$$

$$\text{Y} = \begin{cases} 0 & \text{x } < \text{x}\_t\\ 1.0 - \exp\left[-\frac{\text{(x} - \text{x}\_t)^2 n\sigma}{U}\right] & \text{= 1.0 - \exp\left(-0.41\xi^2\right)} \text{ x } \le \text{ x}\_t \end{cases} \tag{1}$$

In the above function, xt is the known transition onset location, n is the turbulence spot formation rate per unit time per unit distance in the spanwise direction, σ is a turbulence spot propagation parameter, and U is the freestream velocity.

### *2.3.1.2 Cho and Chung model*

Cho and Chung [8] developed the k-ε-γ turbulence model that is not designed for prediction of transitional flows but for free shear flows. In this model, the intermittency effect is incorporated into the conventional k-ε turbulence model with the addition of an intermittency transport equation for the intermittency factor γ. In this model, the turbulent viscosity is defined in terms of k, ε, and γ. The intermittency transport equation is given as:

$$
\mu\_{j} \frac{\partial \chi}{\partial \mathbf{x}\_{j}} = D\_{\mathbf{y}} + \mathbf{S}\_{\mathbf{y}} \tag{2}
$$

**105**

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications*

before, although the model was not designed for transition prediction, the γ intermittency profile for the turbulent-free shear layer flows was quite realistic.

Steelant and Dick [7] developed an intermittency transport model that can be used with the so-called conditioned Navier-Stokes equations. In this model, the intermittency function of Dhawan and Narasimha [6] is first differentiated along the streamline direction, s, and the following intermittency transport equation is obtained:

<sup>∂</sup>*<sup>y</sup>* <sup>=</sup> (1 <sup>−</sup> <sup>γ</sup>)ρ<sup>√</sup>

In the above equation, β(s) is a turbulent spot formation and propagation term,

Suzen and Huang [9] proposed an intermittency transport equation model by mixing the production terms of the Cho and Chung [8] and Steelant and Dick [7] models by means of a new blending function. An extra diffusion-related production term due to Cho and Chung is also added to the resultant equation. This model successfully reproduces experimentally observed streamwise intermittency profiles and demonstrates a realistic profile for the cross-stream direction in the transition region. This model is coupled with the Menter's k-ω SST turbulence model [27] in which the intermittency factor calculated by the Suzen and Huang model is used to scale the eddy viscosity field computed by the turbulence model. This model is successfully tested against several flat plate and low-pressure turbine experiments. However, as mentioned before, this model is not a fully local formulation, and thus it cannot be implemented in straightforward fashion in the modern CFD codes.

Langtry and Menter's formulation of the two-equation γ-Reθ model [11] is one of the most widely used transition models as far as general CFD applications in aeronautics are concerned. This model is formulated in such a way that allows calibrated prediction of transition onset and length that are valid for both the 2-D and 3-D flows. It uses the so-called local variables and thus applicable to any type of grids generated around complex geometries with robust convergence characteristics. As mentioned in the introduction part, this model is based on an important experimental observation that a locally calculated parameter called as the vorticity Reynolds number (Rev) and the momentum thickness Reynolds number (Reθ) where

2.193 and *Rev* <sup>=</sup> <sup>ρ</sup>*dw*

2 \_\_\_\_

*<sup>μ</sup>* Ω (4)

*Revmax*

which is seen in the exponential function part of the Dhawan and Narasimha model. Steelant and Dick tested their model for zero, adverse and favorable pressure gradient flows by using two sets of the so-called conditioned averaged Navier-Stokes equations. Although their model reproduces the intermittency distribution of Dhawan and Narasimha for the streamwise direction, a uniform intermittency distribution in the cross-stream direction is assumed. Yet, this is inconsistent with the experimental observations of, for instance, Klebanoff [26] where a variation of the intermittency in the normal direction by means of an error function formula.

\_\_\_\_\_\_

*u*<sup>2</sup> + *v*<sup>2</sup> β(*s*) (3)

*DOI: http://dx.doi.org/10.5772/intechopen.83520*

*2.3.1.3 Steelant and Dick model*

*<sup>d</sup>* \_\_\_

*2.3.1.4 Suzen and Huang model*

*2.3.2 Models depending on local flow variables*

*2.3.2.1 Langtry and Menter γ-Reθ model*

*Re*<sup>θ</sup> = \_\_\_\_\_\_

∂τ <sup>+</sup> ∂*u* \_\_\_\_\_

*dx* <sup>+</sup> ∂*v* \_\_\_\_\_

where Dγ is the diffusion term and Sγ is the source term. This model is tested for a plane jet, a round jet, a plane far-wake, and a mixing layer case. As mentioned

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications DOI: http://dx.doi.org/10.5772/intechopen.83520*

before, although the model was not designed for transition prediction, the γ intermittency profile for the turbulent-free shear layer flows was quite realistic.
