Yong Zhao, Zhi Zong, Li Zouand Tianlin Wang

1. Transportation Equipment and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China

2. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

3. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

4. School of Naval Architecture, Dalian University of Technology, Dalian 116024, China

Turbulence Model Investigations on the
Boundary Layer Flow with Adverse Pressure Gradients

Yong Zhao1,2, Zhi Zong3,4*, Li Zou3,4and Tianlin Wang1

1. Transportation Equipment and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China

2. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

3. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

4. School of Naval Architecture, Dalian University of Technology, Dalian 116024, China

In this paper, a numerical study of flow in the turbulence boundary layer with adverse and pressure gradients (APGs) is conducted by using Reynolds-averaged Navier-Stokes (RANS) equations. This research chooses six typical turbulence models, which are critical to the computing precision, and to evaluating the issue of APGs. Local frictional resistance coefficient is compared between numerical and experimental results. The same comparisons of dimensionless averaged velocity profiles are also performed. It is found that results generated by Wilcox (2006) k-ω are most close to the experimental data. Meanwhile, turbulent quantities such as turbulent kinetic energy and Reynolds-stress are also studied.

adverse pressure gradient; turbulent boundary layer; turbulence models; local frictional resistance coefficient; RANS; Reynolds-stress

1 Introduction1

The flow pasting curved surface often induces an adverse pressure gradient (APG). This phenomenon offen happens in many engineering applications, such as ships, turbo machinery blades, wind turbines, and the trailing edges of airfoils. The APG plays a significant role in such engineering devices. The flow could become unstable if a turbulent boundary layer flow encounters large APG and, consequently, the flow would separate from the surface (Lee and Sung, 2008; Spalart and Watmuff, 1993; Tulapurkara et al., 2001; Liu et al., 2002; Knopp et al., 2014). Many important features of APG flows are quite well understood. For instance, as the magnitude of APG increases, the mean velocity profile develops a large wake region and the turbulent kinetic energy decreases in the near-wall region. Therefore, it is really difficult to predict flow with large APG by using common industrial methods such as RANS (Samuel and Joubert, 1974).Most results are acquired either from experiments (Spalart and Watmuff, 1993; Tulapurkara et al., 2001; Liu et al., 2002; Knopp et al., 2014) , direct numerical simulation (DNS) (Coleman and Kim, 2003) or large eddy simulation (LES) (Inoue et al.,2013). However, DNS and LES are still limited to simulating relative simple flows with low Reynolds number and experiments are expensive and time-consuming. Hence, it is practically useful to simulate boundary layer flow with APG. However, the precision is intensively impacted by the selection of turbulence model.

For the purpose of highlighting the effect APG, only flow passing a flat plate is considered. Six typical models are selected for the research in order to find out which one is optimal. The six typical models are respectively applied for different turbulence model types： Wilcox (2006) stress-ω model for complete equations type, Wilcox (2006) k-ω model and Launder-Sharma k-ε model for two-equation type, Spalart-Allmaras model for one-equation type, Johnson-King model for 1/2-equation type, and Baldwin-Lomax model for zero-equation type in this paper.

2 Governing equations and turbulence models

2.1 Governing equations

The governing equations for incompressible and steady boundary layer flow in a two-dimensional space by RANS are stated as follows (Schlichting, 2003).

where U, V are streamwise： the normal averaged velocity components ρ, P are fluid density and pressure, and ν, νT are molecular and turbulent eddy kinetic viscosities, respectively. The boundary of the solid wall is under theno-slip condition, i.e.

Next the used six turbulence models’ formulas and the corresponding boundary conditions will be introduced.

2.2 Wilcox (2006) stress-ω model

which yield

with

For the closure coefficients and related functions in the above equations, refer to Wilcox (2006). There is no specific far field boundary condition for this model. For the Reynolds stresses and length scale at solid walls,

2.3 Wilcox (2006) k-ω model

The k-ω model was firstly created by Saffman. Wilcox has continually refined and improved the model during the past three decades and demonstrated its accuracy for a wide range of turbulent flows (Wilcox, 2006). In Wilcox (2006) k-ω model, the turbulent eddy viscosity is represented by

Turbulence kinetic energy equation is given as

and specific dissipation rate is

For closure coefficients and auxiliary relations, refer to Wilcox (2006). For solid wall, the boundary condition is specified as

where d is the distance from the nearest grid to the wall.

2.4 Launder-Sharma k-ε model

As a widely used model type, k-epsilon type has many versions. This study selected the typical model for calculation, namely Launder-Sharma (LS) version version (Launder and Sharma, 1974), which is also called the “standard” k-ε model. Turbulent eddy viscosity is given by

The turbulent kinetic energy equation is

and dissipation rate equation is

where the auxiliary relations are as follows：

and the closure coefficients are listed bellow：

There is no specific far field boundary condition recommended for this model. On solid wall, the boundary conditions are

and

where n is the normal direction along the wall.

2.5 Spalart-Allmaras model

In the Spalart-Allmaras (SA) model (Spalart and Allmaras, 1994), turbulent eddy viscosity is given by

where the auxiliary relations are as follows：

and the closure coefficients are listed bellow：

The boundary condition on the wall is

2.6 Johnson-King and Baldwin-Lomax models

Without introducing unknown quantity, both Johnson-King (JK) and Baldwin-Lomax (BL) models simulate the turbulent viscosity μt=ρνtdirectly.

The differences between the two models are based on the auxiliary functions μT0,μTiand the related coefficients among them. Details can be found in Johnson and King (1984) and Granville (1987), respectively. These two models do not introduce additional quantity and no boundary condition is supplemented.

3 APG flow and numerical parameters

For the purpose of validating the numerical results, the same flow parameters as in the experiment conducted by Samuel and Joubert (1974) were selected： the velocity of free stream U∞=25 m/s, temperature T=293 K, density ρ=1.21 kg/m3, and the Reynolds number based on unit length ReL=U∞·1/ν=1.7×106, where ν is the kinematic viscosity coefficient. Samuel and Joubert measured the local frictional resistance coefficient Cfand dimensionless velocity U+, which were defined as follows：

On the basis of laminar flow, the codes for the above models are supplemented to calculate the adverse pressure gradient flow. An iterative algorithm is used to solve the equations based on the order of momentum equations, the continuous equation and the turbulence model equations. A threshold value setting as 10-4for streamwise averaged velocity’s relative error between the two consecutive iterative results can be used for checking numerical convergence. In the discrete equations, the second-order upwind difference scheme is used for the convection items, and for the others the second-order central difference scheme is used. The uniform incoming velocity is set for inlet boundary condition, zero-gradient in stream-wise direction is set for outlet boundary condition and zero-gradient in the normal direction is set for upper boundary condition. The no-slip wall and pressure distribution are specified for the wall condition. The calculated zone is 3.4 m long in streamwise direction and the height in normal direction is 0.14 m with nodes number of 151×51 for all calculations.

Fig. 1 Pressure distribution along the wall

4 Numerical results and discussion

Firstly, the local frictional coefficient Cfcalculated by the six turbulence models and the experimental results (Samuel and Joubert, 1974) are compared in Fig. 2. For a clear comparison, numerical results are only given in the measured region, i.e. 1.05-3.4 m. As shown in Fig. 2(a), the standard k-ε model overestimates resistance as the distance increases in the streamwise. In contrast, Wilcox stress-ω model underestimates the result, but the difference may be kept relatively small at about 5 %-15 % from the measured value in the whole range. Wilcox (2006) k-ω model predicts the numerical result of Cf,deviating between 3 %-5 % from the experimental value in the whole range. Local frictional coefficient predicted by the SA, BL, JK models and the measured ones are compared in Fig. 2(b). The BL model coincides with the results in the former part and underestimates the results in the latter part about 20 %-30 % lower than the experimental value. The SA model’s performance is poor, which overestimates the results much greater than the experimental value in the whole range. JK model can predict the matching result in the middle plate part, but underestimate the result in the former and latter parts about 10 %-40 % lower than the experimental value. By comparing numerical local frictional coefficient with the measured value, Wilcox (2006) k-ω model predicts the most closely results with the experimental value among the six selected models.

Fig. 2 Comparison of local frictional coefficients along the plate among the models’ numerical results and experimental value

In Fig. 3, the dimensionless averaged velocity profiles U+～y+computed by the six models are compared with the measured one at the position of x=3.4 m. As shown in Fig. 3(a), both the stress-ω model and the k-ω model can predict the matching results with the experimental value. The k-ε model underestimates the values very much in the whole range. Fig. 3(b) shows the numerical results predicted by SA, BL, JK models with the measured value. SA model underestimates the velocity in the region where y+＞1000 and BL model overestimates the value in the middle region where 30＜ y+＜1000. JK model’s performance is poor, which underestimates the velocity in the whole range.

Wilcox (2006) k-ω model performs the best based on the comprehensive comparison. Each model has its own advantage in particular applications. However, making the decisions in which turbulence model is the best for the intended type of applications is a difficult task. Some reasons of failure and success of each turbulent model are given below. As suggested by Yorke and Coleman (2004), two-equation model is generally superior to one/zero-equation model for the AGP flow, such as SA, BL, JK model. This is the critical aspect in the study. The main reason for the inaccuracy in the Launder-Sharma k-ε formation of νTis excessive dissipation. The k-ε model uses the gradient diffusion hypothesis to relate the Reynolds stresses to the mean velocity gradients and the turbulent viscosity and performs poorly for complex flows involving adverse pressure gradient, separation, and strong streamline curvature (Menter, 1997). Using a blending function to combine the original k-ω model for near-wall and the standard k-ε model away from walls, Wilcox (2006) k-ω eliminates the standard k-ω model’s strong sensitivity under free stream conditions without sacrificing near-wall performance.

Fig. 3 Comparison of computed and measured dimensionless averaged stream-wise velocity profiles at x=3.4m

In the following, the numerical results are calculated by the Wilcox (2006) k-ω model. Turbulent kinetic energy k and Reynolds stress -are associated with the turbulence developmental level, which are shown in Fig. 4. Fig. 4(a) is the contour lines of turbulent kinetic energy, showing that the turbulent kinetic energy decreases in the near-wall region. It is found that the larger value part is located in the middle part of the boundary layer, which indicates energy containing eddy in the boundary layer is developing outward from the boundary wall. Another important turbulence statistical quantity, the contour lines of Reynolds stress -are shown in Fig.4 (b), whose distribution is similar as turbulent kinetic energy, indicating that the flow would become unstable in the following development (Rahgozar and Maciel, 2012).

Fig. 4 Contour lines of turbulent kinetic energy

Fig. 5 Contour lines of Reynolds stress

By investigating the turbulent characteristics such as turbulent kinetic energy and Reynolds stress, it is found that the most noticeable feature of boundary layer’s structure lies in its multi-layered structure. The most energetic flow is located in the middle part of the boundary layer (Lu and Li, 2002).

5 Conclusions

In the presented study, a series of numerical analysis is performed to investigate the effect of APG on turbulent boundary layer by using RANS. Six typical models were selected as such： one for complete equations type, two for two-equation type, one for one-equation, one for 1/2-equation and one for algebraic type, respectively. The flow parameters of the numerical test in this study are the same as those of the experiment conducted by Samuel-Joubert. By a comprehensive comparison, Wilcox (2006) k-ω model is considered to be the best candidate for numerical simulation of APG flow. Turbulent quantities, i.e. turbulent kinetic energy and Reynolds stress, are obtained by using the k-ω model. It is found that the multi-layer structure is developed obviously in the boundary layer. Another observation is that the most energetic turbulent flow is located in the middle layer caused by the increasing pressure’s retarding effect along the wall. Other combined effects, such as Reynolds number and rough surface (Pailhas et al., 2008), are to be included in the future research.

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1671-9433(2015)02-0170-05

10.1007/s11804-015-1303-0

Received date： 2014-05-01.

Accepted date： 2014-12-16.

Foundation item： Supported by the National Natural Science Foundation of China (Nos.51309040, 51379033, 51209027, 51309025), Open Research Fund of State Key Laboratory of Ocean Engineering (Shanghai Jiao Tong University) (Grant No.1402), and Fundamental Research Fund for the Central Universities (DMU3132015089).

*Corresponding author Email： zongzhi@dlut.edu.cn

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015