**1. Introduction**

30 Will-be-set-by-IN-TECH

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Viscoelastic liquids with very small amounts of polymer/surfactant additives can, as well known since B.A. Toms' observation in 1948, provide substantial reductions in frictional drag of wall-bounded turbulence relative to the corresponding Newtonian fluid flow. Friction reductions of up to 80% compared to the pure water flow can be occasionally achieved with smooth channel/pipe flow of viscoelastic surfactant solution [11, 54]. This friction-reducing effect, referred to as turbulent drag reduction (DR) or Toms effect, has been identified as an efficient technology for a large variety of applications, e.g. oil pipelines [25] and heating/cooling systems for buildings [43], because of major benefits in reducing energy consumption.

It has been known that long, high-molecular-weight, flexible polymers or rod-like micelle networks of surfactant are particularly efficient turbulence suppressor, so that those solutions lead to different turbulent states both qualitatively and quantitatively, resulting in dramatic DRs. One of promising additives, which may allow their solutions to induce DR, is a cationic surfactant such as "cetyltrimethyl ammonium chloride (CTAC)" under appropriate conditions of surfactant chemical structure, concentration, counter-ion, and temperature to form micellar networks in the surfactant solution. Those resulting micro-structures give rise to viscoelasticity in the liquid solution. The properties and characteristics of the viscoelastic fluids measured even in simple shear or extensional flows are known to exhibit appreciably different from those of the pure solvent. From a phenomenological perspective, their turbulent flow is also peculiar as is characterized by extremely elongated streaky structures with less bursting events. Therefore, the viscoelastic turbulence has attracted much attention of researchers during past 60 years. Intensive analytical, experimental, and numerical works have been well documented and many comprehensive reviews are available dealing with this topic: [cf., 18, 19, 26, 35, 51, , and others].

Although the mechanism of DR is still imperfectly understood, but some physical insights have emerged. In particular, with the aid of recent advanced supercomputers, direct

©2012 Tsukahara and Kawaguchi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Viscoelasticity 34 Viscoelasticity – From Theory to Biological Applications Turbulent Flow of Viscoelastic Fluid Through Complicated Geometry <sup>3</sup>

numerical simulations (DNSs) of viscoelastic fluid as well as the Newtonian fluid have been increasingly performed [e.g., 1, 7, 17, 41, 44]. Some progresses in the model of DR and in the understanding of modulated turbulent structures have been made by L'vov et al. [20] and Roy et al. [39]. Later, Kim et al. [16] carried out DNS to examine interactions between the coherent structures and the fluid viscoelasticity. They reported a dependency of the vortex-strength threshold for the auto-generation of new hairpin vortices in the buffer layer on the viscoelasticity. Most of DNS studies in the literature are performed on flows over smooth wall surface and other simple flow configurations, such as channel flow, boundary layer, isotropic turbulence, and shear-driven turbulence.

Configuration Author(s) Method Expansion ratio

Tsurumi et al. [50] Exp. 1:2

Castro & Pinho [3] Exp. 1:1.54 Escudier & Smith [8] Exp. 1:1.54 Poole & Escudier [32, 33] Exp. 1:2, 1:4 Oliveira [28] Sim. 1:2 Manica & De Bortoli [21] Sim. 1:3 Dales et al. [5] Exp. 1:1.5 Poole et al. [34] Sim. 1:3

> 0.128 0.096 0.064 0.032 0 -0.032 -0.064 -0.096 -0.128

Turbulent Flow of Viscoelastic Fluid Through Complicated Geometry 35

0.128 0.096 0.064 0.032 0 -0.032 -0.064 -0.096 -0.128

Orifice Present Sim. 1:2

Sudden expansion Pak et al. [29] Exp. 1:2, 3:8

Backward-facing step Poole & Escudier [31] Exp. 1:1.43

simulation.

the swirling strength *λ*ci*ω<sup>z</sup>*

direction is from left to right. Cited from [50].

**Table 1.** Relevant previous studies on viscoelastic turbulent flow: Exp., experiment; Sim., numerical

(a) Water

(b) CTAC, 150 ppm **Figure 1.** Snapshots of flow fields behind the orifice, taken by PIV measurement: vector, (*u*, *v*); contour,

using PIV (particle image velocimetry) [50]. Figure 1 shows the instantaneous velocity vectors in a plane of interest. Also shown is the contour of swirling strength, by which the vortex core can be extracted by plotting iso-surface of *λ*ci > 0, the imaginary part of complex conjugate eigenvalue of velocity-gradient tensor in the two-dimensional plane, and the rotational direction be evaluated by the sign of spanwise rotation *ωz*. As can be seen in the figure, the sudden expansion of the orifice leads to generation of strong separated shear layers just behind the orifice-rib edges. This shear layer enhances turbulence dominantly


As well as smooth turbulent flows in plane channel and pipe, the turbulent flow through complex geometries has both fundamental scientific interest and numerous practical applications: such flows are associated with the chemical, pharmaceutical, food processing, and biomedical engineering, where the analysis and designing for their pipe-flow systems are more difficult than for its Newtonian counterpart. This is mainly because severe limitations in the application of ideal and Newtonian flow theories to these relevant flow problems. Most of the previous work presented in the literature concerning this subject has been done with flows either through sudden expansion or over backward-facing step. The flow even in such relatively simple cases of complex geometries exhibits important features that partain to complex flows containing flow separation, reattachment, and often an extremely high level of turbulence. A better understanding of viscoelastic-fluid behavior and turbulent flow properties of those flows should lead to both the design and the development of hydrodynamically more efficient processes in various pipe-flow systems and to an improved quality control of the final products. Consequently, in situations of both practical and fundamental importance, we have investigated the detailed mechanism and efficiency of DR for viscoelastic turbulent flow through roughened channel, or an orifice flow, that is one of canonical flows involving separation and reattachment. The goal of a series of our works is to better understand the physics of viscoelastic turbulent flow in complicated flow geometry.

The following subsections give a brief introduction to the preceding studies that motivated us to further investigate the viscoelastic turbulent orifice flow and describe the more specific purpose of the study reported in this chapter.

## **1.1. Related studies**

As far as we know there exist no other DNS studies on the viscoelastic turbulent orifice flow than those carried out by authors' group recently. However, there are a few experimental and numerical works on sudden expansion and backward-facing step owing to their geometrical simplicity. Table 1 summarizes several earlier works.

As for the Newtonian fluid, Makino et al. [22, 23] carried out DNSs of the turbulent orifice flow, and investigated also the performance of heat transfer behind the orifice. They reported several differences in turbulent statistics between the orifice flow and other flows of the sudden expansion and the backward-facing step. Recently, the authors' group investigated the viscoelastic fluid in the channel with the same rectangular orifice using DNS [46, 49]. We found phenomenologically that the fluid viscoelasticity affected on various turbulent motions in just downstream of the orifice and attenuated spanwise vortices.

By means of experiments, we confirmed the turbulence suppression in the region behind the orifice and analyzed the flow modulation with respect to the turbulent structures by


2 Viscoelasticity

numerical simulations (DNSs) of viscoelastic fluid as well as the Newtonian fluid have been increasingly performed [e.g., 1, 7, 17, 41, 44]. Some progresses in the model of DR and in the understanding of modulated turbulent structures have been made by L'vov et al. [20] and Roy et al. [39]. Later, Kim et al. [16] carried out DNS to examine interactions between the coherent structures and the fluid viscoelasticity. They reported a dependency of the vortex-strength threshold for the auto-generation of new hairpin vortices in the buffer layer on the viscoelasticity. Most of DNS studies in the literature are performed on flows over smooth wall surface and other simple flow configurations, such as channel flow, boundary

As well as smooth turbulent flows in plane channel and pipe, the turbulent flow through complex geometries has both fundamental scientific interest and numerous practical applications: such flows are associated with the chemical, pharmaceutical, food processing, and biomedical engineering, where the analysis and designing for their pipe-flow systems are more difficult than for its Newtonian counterpart. This is mainly because severe limitations in the application of ideal and Newtonian flow theories to these relevant flow problems. Most of the previous work presented in the literature concerning this subject has been done with flows either through sudden expansion or over backward-facing step. The flow even in such relatively simple cases of complex geometries exhibits important features that partain to complex flows containing flow separation, reattachment, and often an extremely high level of turbulence. A better understanding of viscoelastic-fluid behavior and turbulent flow properties of those flows should lead to both the design and the development of hydrodynamically more efficient processes in various pipe-flow systems and to an improved quality control of the final products. Consequently, in situations of both practical and fundamental importance, we have investigated the detailed mechanism and efficiency of DR for viscoelastic turbulent flow through roughened channel, or an orifice flow, that is one of canonical flows involving separation and reattachment. The goal of a series of our works is to better understand the physics of viscoelastic turbulent flow in complicated flow geometry. The following subsections give a brief introduction to the preceding studies that motivated us to further investigate the viscoelastic turbulent orifice flow and describe the more specific

As far as we know there exist no other DNS studies on the viscoelastic turbulent orifice flow than those carried out by authors' group recently. However, there are a few experimental and numerical works on sudden expansion and backward-facing step owing to their geometrical

As for the Newtonian fluid, Makino et al. [22, 23] carried out DNSs of the turbulent orifice flow, and investigated also the performance of heat transfer behind the orifice. They reported several differences in turbulent statistics between the orifice flow and other flows of the sudden expansion and the backward-facing step. Recently, the authors' group investigated the viscoelastic fluid in the channel with the same rectangular orifice using DNS [46, 49]. We found phenomenologically that the fluid viscoelasticity affected on various turbulent motions

By means of experiments, we confirmed the turbulence suppression in the region behind the orifice and analyzed the flow modulation with respect to the turbulent structures by

layer, isotropic turbulence, and shear-driven turbulence.

purpose of the study reported in this chapter.

simplicity. Table 1 summarizes several earlier works.

in just downstream of the orifice and attenuated spanwise vortices.

**1.1. Related studies**

**Table 1.** Relevant previous studies on viscoelastic turbulent flow: Exp., experiment; Sim., numerical simulation.

**Figure 1.** Snapshots of flow fields behind the orifice, taken by PIV measurement: vector, (*u*, *v*); contour, the swirling strength *λ*ci*ω<sup>z</sup>* |*ωz*| (positive, anti-clockwise rotation; negative, clockwise). The main flow direction is from left to right. Cited from [50].

using PIV (particle image velocimetry) [50]. Figure 1 shows the instantaneous velocity vectors in a plane of interest. Also shown is the contour of swirling strength, by which the vortex core can be extracted by plotting iso-surface of *λ*ci > 0, the imaginary part of complex conjugate eigenvalue of velocity-gradient tensor in the two-dimensional plane, and the rotational direction be evaluated by the sign of spanwise rotation *ωz*. As can be seen in the figure, the sudden expansion of the orifice leads to generation of strong separated shear layers just behind the orifice-rib edges. This shear layer enhances turbulence dominantly

4 Viscoelasticity 36 Viscoelasticity – From Theory to Biological Applications Turbulent Flow of Viscoelastic Fluid Through Complicated Geometry <sup>5</sup>

**Figure 3.** Configuration of the roughened-channel flow for the simulation, where a sequence of

the heat-transfer augmentation would be deeply related to the turbulence-viscoelasticity

Turbulent Flow of Viscoelastic Fluid Through Complicated Geometry 37

We performed DNSs without any turbulence model but with the Giesekus' viscoelastic-fluid model, valid for a polymer/surfactant solution, which is generally capable of reducing the turbulent frictional drag in a smooth channel. The geometry considered here is periodic

In this section, the equations governing incompressible viscoelastic-fluid flows are presented in their dimensional and non-dimensional forms. Rheological properties relating to a model we employed here to calculate the polymer/surfactant, or the fluid-viscoelasticity,

Prior to introducing the equations, let us depict the configuration of the computational domain in Fig. 3. In the three dimensional Cartesian coordinate system, *x*, *y*, and *z* indicate the streamwise, wall-normal, and spanwise directions, respectively. The main flow is driven by the streamwise mean pressure gradient. The flow that we analyzed by DNS was assumed to be fully-developed turbulent flow through an obstructed channel, of height *Ly* = 2*h*, with periodically repeating two-dimensional orifices (i.e., transverse rectangular orifices): namely, in the simulations, the periodic boundary condition was adopted in the *x* direction as well as the *z* direction to allow us to demonstrate an infinite channel and regularly-spaced obstructions Note that, by contrast with the above-mentioned experiment, where transient flows past only one orifice were studied, we numerically investigated the fully-developed flows through a sequence of orifices. As illustrated in Fig. 3, the transverse orifices are placed

The height of each rib is chosen as 0.5*h*—the channel half height is *h*—and thus the expansion ratio of the orifice is 1:2, that is equivalent to the experimental condition but the thickness in the *x* direction is small as 0.1*h*. These present conditions relating to the orifice installation are the same as those studied by Makino et al. [22]. The no-slip boundary condition is used on all

The domain size in the streamwise direction (*Lx* = 12.8*h*) was not sufficiently long that the effects of the orifice on the flow approaching next one could be neglected. The domain size

regularly-spaced, rectangular, orifices is considered.

contribution to the extra-stress tensor are also described.

interaction, and suggest its scenario.

orifices with the 1:2 expansion ratio.

**2. Problem formulation**

**2.1. Flow configuration**

in every *Lx* in the *x* direction.

the wall surfaces including the faces of the orifice.

**Figure 2.** Outline of experimental apparatus with PIV system.

in the water flow, while the viscoelastic flow seems rather calm. Here, the viscoelastic fluid they employed was the CTAC solution with 150 ppm of weight concentration. A schematic of the experimental set up is depicted in Fig. 2. The Reynolds number based on the actual bulk mean velocity passing the orifice were *Re*<sup>m</sup> = 8150 for water and 7840 for the viscoelastic fluid (CTAC solution), which were obtained under the same pumping power. It is interesting to note that the hydrodynamic drag throughout the channel including the orifice is rather increased in the viscoelastic flow despite the presence of turbulence-suppression phenomenon. We conjectured that, in the experiment, any DR did not apparently occur because an increment of the skin friction by an extra shear stress due to viscoelasticity exceeded a decrement of the Reynolds shear stress. It might be difficult to determine the individual contribution of either turbulence, viscosity, or viscoelasticity in such an experimental study. To achieve clearer pictures of the role of viscoelasticity and turbulence modulations affecting on DR, we should re-examine the viscoelastic turbulent orifice flow with emphasis on the viscoelastic force (stress) exerted on the fluctuating flow motion.

#### **1.2. Purpose**

In the present study, we will focus on an instantaneous field of the viscoelastic turbulent flow past the rectangular orifice and discuss mainly the interaction between the turbulent fluid motion and the (polymer/surfactant) additive conformation field, i.e. the balance of the inertia, viscous, and viscoelastic forcing terms in the governing momentum equation. We have made some preliminary studies which have shown that this flow exhibits a change in the augmentation of the local heat transfer dependently on the streamwise distance from the orifice [49]. Therefore, we propose in this chapter that this streamwise variation of 36 Viscoelasticity – From Theory to Biological Applications Turbulent Flow of Viscoelastic Fluid Through Complicated Geometry <sup>5</sup> Turbulent Flow of Viscoelastic Fluid Through Complicated Geometry 37

**Figure 3.** Configuration of the roughened-channel flow for the simulation, where a sequence of regularly-spaced, rectangular, orifices is considered.

the heat-transfer augmentation would be deeply related to the turbulence-viscoelasticity interaction, and suggest its scenario.

We performed DNSs without any turbulence model but with the Giesekus' viscoelastic-fluid model, valid for a polymer/surfactant solution, which is generally capable of reducing the turbulent frictional drag in a smooth channel. The geometry considered here is periodic orifices with the 1:2 expansion ratio.
