J. Cent. South Univ. Technol. (2008) 15(s1): 243-246 

DOI: 10.1007/s11771-008-355-6 

Drag reduction by linear viscosity model in turbulent channel flow of polymer solution

WU Gui-fen(吴桂芬)¹, LI Chang-feng(李昌烽)¹, HUANG Dong-sheng(黄东升)²,

ZHAO Zuo-guang(赵作广)¹, FENG Xiao-dong(冯晓东)¹, WANG Rui(王 瑞)¹

(1. School of Energy and Power Engineering, Jiangsu University, Zhenjiang 212013, China;

2. Shanghai Maple Automobile Co. Ltd, Shanghai 201501, China)

Abstract: A further numerical study of the theory that the drag reduction in the turbulence is related to the viscosity profile growing linearly with the distance from the wall was performed. The constant viscosity in the Navier-Stokes equations was replaced using this viscosity model. Some drag reduction characteristics were shown comparing with Virk’s phenomenology. The mean velocity and Reynolds stress profiles are consistent with the experimental and direct numerical simulation results. A drag reduction level of 45% was obtained. It is reasonable for this linear viscosity model to explain the mechanism of turbulence drag reduction in some aspects.

Key words: turbulence; drag reduction; viscosity profile; polymers; velocity; Reynolds stress

1 Introduction

Turbulent flows of dilute polymeric solutions play a central role in many engineering applications, such as turbulence reduction, fire fighting and agricultural spraying^[1-4]. It is well known that the addition of small amounts of soluble high relative molecular mass polymers to inertia-dominated, wall bounded flows gives rise to a reduction of turbulent drag. Specifically, it was observed experimentally that polymer concentration of O(100) ppm is sufficient to reduce drag up to 70%. This phenomenon has stimulated tremendous research effort in the past several decades. In spite of a large amount of experimental and situational data, the fundamental mechanism remained under debate for a long time^[4-6]. After adding the polymer into the turbulence, the polymer will interact with the vortex in the turbulence and tends to be stretched, thus increasing the extensional viscosity and the bulk viscosity greatly. WANG and DOU^[7] found that the bulk viscosity increased 2 to 8 times in the drag reduced flows. Then a contradiction appears that the drag should increase owning to the increased viscosity but the drag reduction occurs indeed. So there must exist a mechanism that compensates for the increased viscosity.

In wall-bounded turbulence, the drag is caused by momentum dissipation at the walls. For Newtonian flows (in which the kinematical viscosity is a constant) the momentum flux is mainly dominated by the so-called Reynolds stress, leading to the von Karman log law for the mean velocity profile^[5]. However, with polymers, the drag reduction entails a change in the von Karman log law such that a much higher mean velocity is achieved. In particular, for high concentrations of polymers, a regime of maximum drag reduction is attained (the ‘‘MDR asymptote’’), independent of the chemical identity of the polymer^[2]. For a fixed rate flow generated by the constant pressure gradient or the mass flux, reducing the Reynolds stress can reduce the momentum dissipation. In a recent theory of drag reduction in wall turbulence^[8], it was proposed that the polymer stretching gives rise to a self-consistent effective viscosity that increases linearly with the distance from the wall. Such a linear viscosity profile could reduce the Reynolds stress (i.e. the momentum flux to the wall) more than increase the viscous drag, then the result is drag reduction. The aim of this paper is to substantiate this mechanism for drag reduction on the basis of numerical simulations and to show further drag reduction characteristics from this model.

2 Virk’s phenomenology for polymer drag reduction

In the turbulent channel flow of a polymer solution, for the viscous sublayer, because its turbulence dissipation rate is the smallest in the entitle flow, the velocity profile maintains the same as the Newtonian flow^[9]. For y⁺＜11.6,

U⁺=y⁺                                           (1)

According to Virk’s mean flow model^[2] for polymer drag reduction, in the presence of polymer the velocity profile in the log region is shifted upwards by an amount of S⁺ with no change of slope compared with the Newtonian flow. That is,

U⁺=2.5ln y⁺+5.5+S⁺                              (2)

But for the buffer layer, the difference cannot be ignored in drag reduction flow (see Fig.1). In fact, after adding the polymer, the buffer layer will grow up towards the central line of the channel (i.e. the buffer is thickened by the polymer). If possible, its maximum limit is to the core, and then the maximum drag reduction occurs. According to Virk’s asymptote, for the buffer region,

U⁺=11.7ln y⁺-17.0                                (3)

where the mean stream-wise velocity U⁺ is normalized in units of the turbulent velocity scale, , and the distance normal to the wall y⁺ is measured in units of the turbulent length scale, where τ_w is the wall shear stress, and ρ is the fluid density.

Fig.1 Mean velocity profiles vs distance from wall

3 Governing equations of drag reduction induced by viscosity profile

The mechanism for the polymer to induce drag reduction in the turbulence can be expressed in the modified Navier-Stokes equation^[5-10] in consideration of the polymer stress:

              (4)

where  v₀ is the kinematic viscosity of the carrier fluid; τ_p is the extra stress tensor due to the polymer; U is the bulk velocity and is the pressure gradient. For the interaction of the polymer and the vortex of the turbulence, the vortex will stretch the polymer thus increasing the fluid viscosity. Then the momentum balance equation in the reduced flow can be summed up:

                 (5)

where  is the viscous stress; is the Reynolds stress; is the shear stress due to the polymer in the turbulence, and h is the half height of the channel. If the viscous stress and the polymer stress are put together, Eqn.(5) can be written as:

                      (6)

where  v_e can be called as an effective viscosity. This effective viscosity in the viscous sublayer remains unchanged as the Newtonian flow, and above the viscous sublayer it increases and grows linearly with the distance from the wall towards the centre of the channel^[8]:

v_e=v₀  (y≤y_v)                             (7a)

v_e/v₀=1+D(y-y_v)/h  (y＞y_v)                      (7b)

where  D is a coefficient which dominates the amount of the drag reduction and is the function of the polymer identity, concentration and flow conditions.

In the usual Newtonian flow with a constant viscosity, the drag monotonically increases with the viscosity. It looks paradoxical for the possibility of drag reduction by increasing the viscosity, which is the essential difference from previous viscous drag reduction theory^[5]. Therefore, the crucial test and aim of this paper is to introduce such a linear viscosity profile into the Navier-Stokes equations instead of Eqn.(4) as the governing equation, and see whether the drag reduction can be observed together with statistical aspects.

4 Numerical simulations and analysis

Instead of simulating the viscoelastic equations, the Navier-Stokes equations were simulated with a space dependent viscosity profile which remains a constant for the sublayer as the same as the Newtonian flow, and then grows linearly towards the center. All numerical simulations were performed in a computational domain size of 40h×2h×5h in x, y, z-directions respectively with a spatial discretization of 200×80×28 with periodic boundary conditions in z span-wise directions and no slip conditions on the walls that were separated by 2h in y wall-normal directions. To capture the velocity profile in the near wall region, the enhanced wall functions were used and the distance of the first grid from the wall was 0.275 in wall units. The same mass flux in x direction and Newtonian initial conditions were used with the mean Reynolds number Re_m=U_m2h/v₀=  10 000 (based on the mean velocity cross the channel) in all the runs.

4.1 Rate of drag reduction

Through changing the slope of the viscosity profile (the value of D), the simulations were performed from onset, LDR (low drag reduction) to HDR (high drag reduction) regimes. Since a constant mass flux is applied to all simulations, the decreased value of the wall shear stress  is a manifestation of the drag reduction (DR) for all the runs, which can be expressed as:

                                        (10)

where   is the wall shear stress in the Newtonian flow;  is the wall shear stress induced by the viscosity profile in the polymeric solutions. The wall shear stress and the DR are shown in Table 1.

Table 1 Numerical simulation parameters for all runs

From Table 1 and Fig.2, it is clearly illustrated that the linear viscosity profile model can really induce the turbulence drag reduction. It can be seen that up to the slope D=4, the drag reduction almost cannot be observed. However, after D＞4, DR increases greatly with increasing to D=16. Then drag reduction increases slowly, and drag reduction rate saturates to DR=45.4% at D=25. There is still some gap to the value of maximal

Fig.2 Drag reduction rate vs slope of linear viscosity profile

drag reduction asymptote (about 80%), which implies that the slope D should also be a function of flow conditions (such as the mean Reynolds number), and the linear viscosity profile model can only give partial explanation of turbulence drag reduction mechanism induced by polymers.

4.2 Mean velocity profile

The mean velocity profiles of drag reduction cases induced by the linear viscosity profiles with selected slope D are shown in Fig.3. It can be seen that in drag reduced flows, all mean velocity profiles in the viscous sublayer collapse on the linear distribution U⁺=y⁺. The upward shift of the logarithmic profile can be interpreted as the thickening of the buffer layer^[3].

Fig.3 Mean streamwise velocity profiles of Newtonian flow and reduced drag flow induced by viscosity profiles

4.3 Reynolds stress and viscous stress

The Reynolds stress and viscous stress normalized by wall shear stress in the different runs with the same slope of Fig.3 are shown in Fig.4 and Fig.5. It can be seen that the Reynolds stress is significantly reduced

Fig.4 Profile of decreased Reynolds stress induced by linear viscosity profiles

Fig.5 Increased viscous stress profile induced by increased viscosity profiles

with increasing DR, and overwhelms the increased viscous stress. This evidence demonstrates that the great decrease of Reynolds stress can compensate for the increased drag induced by the increased viscosity, and then the drag reduction occurs. The behaviors of stresses are consistent with the experimental observations^[11] and direct numerical simulation results^[12-14].

5 Conclusions

1) Numerical simulations were performed using the viscosity profile growing linearly with the distance from the wall, instead of simulating the viscoelastic equations directly. It was shown that this linear viscosity profile model could really reduce the drag of the turbulent flow.

2) Drag reduction onset, low drag reduction regimes and mediate drag reduction were obtained through increasing the slope of the linear viscosity profile. Ultimately the drag reduction rate saturates to DR=45.4% at D=25, lower than the maximal drag reduction asymptote (about 80%). The behaviors of objects like the Reynolds stress and the viscous stress are in close correspondence with the experimental results and simulation results at the same drag reduction level.

3) It is reasonable for this linear viscosity profile model to partially explain the mechanism for drag reduction induced by the polymer in the turbulence. Further study is needed to go on more depth and more details.

References

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(Edited by YUAN Sai-qian)

Foundation item: Project(10672069) supported by the National Natural Science Foundation of China

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: LI Chang-feng; Tel: +86-13775378215; E-mail: cfli@ujs.edu.cn