`Mathematical Problems in EngineeringVolume 2014 (2014), Article ID 405202, 9 pageshttp://dx.doi.org/10.1155/2014/405202`
Research Article

## A Nonlinear k- Turbulence Model Applicable to High Pressure Gradient and Large Curvature Flow

1School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
2Beijing Institute of Control Engineering, Beijing 100190, China
3State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China

Received 22 September 2013; Revised 4 December 2013; Accepted 30 December 2013; Published 17 February 2014

Copyright © 2014 Xiyao Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Most of the RANS turbulence models solve the Reynolds stress by linear hypothesis with isotropic model. They can not capture all kinds of vortexes in the turbomachineries. In this paper, an improved nonlinear k-ε turbulence model is proposed, which is modified from the RNG k-ε turbulence model and Wilcox's k-ω turbulence model. The Reynolds stresses are solved by nonlinear methods. The nonlinear k-ε turbulence model can calculate the near wall region without the use of wall functions. The improved nonlinear k-ε turbulence model is used to simulate the flow field in a curved rectangular duct. The results based on the improved nonlinear k-ε turbulence model agree well with the experimental results. The calculation results prove that the nonlinear k-ε turbulence model is available for high pressure gradient flows and large curvature flows, and it can be used to capture complex vortexes in a turbomachinery.

#### 1. Introduction

Boussinesq hypothesis is a first-order-closure model of turbulence which assumes an explicit algebraic relationship between Reynolds stresses and mean-velocity gradients. The Boussinesq hypothesis is a popular linear eddy-viscosity model, which is used in most Reynolds averaged Navies-Stocks equations to predict the sheer stress. The model and model are the most common turbulence models [1], which are a two-equation model including two extra transport equations to represent the turbulent properties of the flow. The model and model do not perform well in cases of large adverse pressure gradients [2]. The use of a formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sublayer [3]. The SST model can be used as a Low-Re turbulence model without any extra damping functions. The SST formulation also switches to model in the free stream and thereby avoids the common problem that the model is too sensitive to the inlet free-stream turbulence properties [4]. However, practical engineering flows exhibit complex mean strain associated, for example, with high pressure gradients, separation, impingement, streamline curvature, and swirl. The linear hypothesis is not suitable to capture the flow of anisotropy.

One way to model the anisotropy is to use the Reynolds stress transport equations. Craft and Launder [5] discussed different Reynolds stress turbulence models and find out that higher order pressure strain models which consider the stress redistribution near walls can predict the lateral spreading rate well. Lübcke et al. [6] used another way to simulate the anisotropy. They presented an explicit Reynolds-stress closure which offers a physically sound extension of the most prominent linear Boussinesq viscosity models with modest computational effort. Pettersson Reif [7] also presented a promising explicit Reynolds stress closure. Ji et al. [8, 9] studied unsteady turbulence flows around a twisted hydrofoil using partially averaged Navier-Stokes (PANS) methods. The subfilter scale stress in PANS methods was a positive semidefinite tensor exhibiting much of the same properties as Reynolds stress, and it was a nonlinear stress model.

Durbin [10] introduced the elliptic relaxation approach within the framework of the linear eddy-viscosity formulation, the so-called model. The nonlinear turbulence model differs from the family of conventional nonlinear eddy-viscosity models. In this approach, elliptic wall effects are accounted for indirectly through the solution of modified Helmholtz equation [1113]. The turbulence model is a seven-equation model, which consumes more computational time and resources. Nonlinear turbulence models often arise from scientific research, modelling of nonlinear phenomena, and optimal control of complex systems [14, 15]. Nonlinear turbulence model can be also used in the study of heat transfer [16] and phase-change process [17]. The nonlinear solutions are more accurate in the solution of complex phenomena.

In this paper, an improved nonlinear turbulence model was proposed, which was modified from RNG turbulence model and Wilcox’s turbulence model. The Reynolds stresses were solved by nonlinear methods which were proposed by Ehrhard [18]. The nonlinear turbulence model was used to simulate the flow in a curved rectangular duct, and the results of the internal flow were compared with experimental data.

#### 2. Model Development

In Wilcox’s turbulence model, the turbulence kinetic energy transport equation is given by the following equation: and the turbulence specific dissipation rate equation is The relationship between turbulence dissipation and turbulence specific dissipation rate is It can be calculated that Then, the improved equation for turbulence dissipation can be derived: where is the diffusion term, represents the generation of turbulent dissipation due to the mean velocity gradients, and denotes the damping term for turbulent dissipation rate.

For turbulence model, the equation for turbulence kinetic energy is given by the following equation: where the kinematic eddy viscosity The closure coefficients are as follows: The turbulence kinetic energy equation (5) and turbulence dissipation equation (9) form the improved turbulence model.

The combination of RNG turbulence model and Wilcox’s turbulence model does not need to make use of wall functions, because it is valid up to solid walls.

The Reynolds stress anisotropy tensor which was proposed by Ehrhard [18] was combined with the improved turbulence model. Ehrhard developed a nonlinear turbulence model which was calibrated by the use of simple two-dimensional shear and complex flows with recirculation zones. The final form of the nonlinear turbulence model is given by the nonlinear solution shown as follows.

The Reynolds stress can be calculated by The nonlinear solution of Reynolds stress is where, The closure coefficients are as follows: where is the turbulence velocity scale and   is the turbulence time scale.

#### 3. Test Case

To prove that the nonlinear turbulence model is available for capturing complex flow, a three-dimensional (3-D) curved rectangular duct was studied and the results of the internal flow were compared with experimental data. The nonlinear turbulence model was performed by Fluent software, using user define function (UDF). The computational results of the nonlinear turbulence model were also compared with the results performed by RNG turbulence model and SST turbulence model. Experimental data of the internal flow in the curved rectangular duct were performed by Yakinthos et al. [19].

The structure of the curved rectangular duct was shown in Figure 1.  m, and the radius of the inner circle was 0.608 m. Velocity at the inlet of the tunnel was  m/s, and the Reynolds number was . Mesh was generated by ICEM software, which was shown in Figure 2. Grid dependency is studied by two kinds of number of meshes, and the results of pressure distribution on the local convex surface are shown in Figure 3. It can be seen that the result based on 4.3 million cells is the same as 6.8 million cells. All the following calculations are performed by the mesh with 6.8 million cells. on the surface in the computational domain is less than 2.

Figure 1: A curved rectangular duct.
Figure 2: Mesh of the rectangular duct.
Figure 3: Grid dependency studies (Pa).

The transverse velocity (perpendicular to the measuring line and along the flow direction), axial velocity (perpendicular to the measuring line and flow direction), and longitudinal velocity (along the measuring line) on the measuring curve were calculated.

Velocity inlet and pressure outlet were used. No slip boundary conditions were used to solve the near wall region. Couple method was used to correct the pressure during the calculation. Second-order upwind scheme was used to resolve the Navies-Stocks equations. Convergence was determined by the residual error less than 0.0001.

#### 4. Results and Discussions

##### 4.1. Velocity Distribution on the Measuring Line

Figure 4 shows results of the transverse velocity on the measuring line. The result of the nonlinear turbulence model is in good agreement with experimental result. The transverse velocity of experimental result on the measuring line has a hump characteristic, and it can be captured by the nonlinear turbulence model. The results of RNG model and SST model could not obtain the nonlinear phenomena.

Figure 4: Transverse velocity.

The longitudinal velocity on the measuring line is shown in Figure 5. The result of RNG model and SST model is smaller than the result of nonlinear turbulence model at the region of dominant flow. Longitudinal velocity on the measuring line agrees well with the experiment result. The result of longitudinal velocity by nonlinear turbulence model can also capture the nonlinear phenomena when .

Figure 5: Longitudinal velocity.

The axial velocity on the measuring line is shown in Figure 6. The maximum point of the axial velocity by the nonlinear model is very close to the result of experimental data, while the results based on RNG model and SST model have large error at this point. The nonlinear characteristic of the axial velocity when can also be captured by the nonlinear model.

Figure 6: Axial velocity.
##### 4.2. Pressure and Velocity Distribution Near the Convex Surface

Pressure distributions near the convex surface of the curved rectangular duct are shown in Figure 7. The results based on the nonlinear model have a large region with negative pressure, while the results by RNG model can only capture a small region with negative pressure. The pressure on the convex surface by SST model does not have negative value. The velocity distribution on the convex surface is shown in Figure 8. Results by RNG model are almost the same with SST model. Results of the nonlinear model are much different form the other two turbulence models. In consideration of the velocity results on the measuring line, the pressure and velocity distribution on the convex surface by the nonlinear model are more close the real flow.

Figure 7: Pressure distribution on the convex surface (Pa).
Figure 8: Velocity distribution near the convex surface (m/s).

Normal velocity distributions near the convex surface are shown in Figure 9. The normal velocity calculated by the nonlinear model has negative value on the surface near the convex surface. The results based on SST model and RNG model could not have this phenomenon. The nonlinear solution of Reynolds stress can predict the complex flow near the wall.

Figure 9: Normal velocity distribution near the convex surface (m/s).
##### 4.3. Limit Streamline Near the Convex Surface

Figure 10 shows the results of limit streamline near the convex surface. Results of RNG model and SST model have flow separation phenomena which can be seen from Figure 10(b). The flow also shows symmetric distribution and no vortex near the convex surface by RNG model and SST model. The results of the limit streamline near the convex surface based on the nonlinear model show great difference from the other two models. Both flow separation and reverse flow can be captured by the nonlinear model. The flow separation at the front of the curved rectangular duct is the same as the other turbulence model, but the results of RNG model and SST model could not predict the reverse flow at the curved region.

Figure 10: Velocity distribution on the convex surface (m/s).
##### 4.4. Reynolds Stress Close to the Convex Surface

One of the Reynolds stresses, which is sheer stress, is shown in Figure 11. The distribution of sheer stress close to the convex surface calculated by the nonlinear model is greatly different from the results of the other two turbulence models. The results of sheer stress based on RNG model and SST model close to the convex surface are all above 0, while the result performed by the nonlinear model has negative value. The range of the value of sheer stress by RNG model and SST model is very small, and it has uniform distribution close to the convex surface. So the nonlinear model can calculate the nonlinear distribution of Reynolds stresses close to the wall. It can be used to predict the complex flow with nonlinear phenomena.

Figure 11: Sheer stress distributions on the convex surface (Pa·s).

#### 5. Conclusions

This paper presents an improved nonlinear turbulence model to include sensitivity to high pressure gradient flow and large curvature flow. The nonlinear turbulence model incorporates the advantage of Wilcox’s model for the prediction of near wall region and the model for the mainstream flow. The improved nonlinear turbulence model improves accuracy by incorporating a nonlinear solution of Reynolds stress, which is realized through the turbulence velocity scale and turbulence time scale. The resulting formulation for the nonlinear solution is relatively simple and satisfies principles of frame invariance and realizability.

Results for the curved rectangular duct indicate that the improved model shows a nontrivial improvement compared with the original model and RNG model. Results of velocity on the measuring line are in great agreement with experimental result. The model is able to capture the flow separation and reverse flow, and it can predict the nonlinear flow caused by flow separation. Due to its simplicity and realizability, this model can provide a method for flow prediction of complex flows with significant effects of streamline curvature.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The research work was funded by China Postdoctoral Science Foundation (no. 2013M531173) and the National Natural Science Foundation of China (nos. 51176168 and 51076077). The project was also supported by National Science and Technology Ministry (ID: 2011BAF03B01).

#### References

1. D. C. Wilcox, Turbulence Modeling for CFD, DCW Industries, Los Angeles, Calif, USA, 2nd edition, 2004.
2. J.-L. Yin, D.-Z. Wang, Y.-L. Wu, and D. K. Walters, “A modified k-ε model for computation of flows with large streamline curvature,” Advances in Mechanical Engineering, vol. 2013, Article ID 592420, 10 pages, 2013.
3. X.-L. Tang, Z.-C. Chen, and Y.-L. Wu, “Numerical models for turbulent flows through a centrifugal impeller,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 1, pp. 81–88, 2008.
4. F. R. Menter, “Two-equation eddy-viscosity turbulence models for engineering applications,” AIAA Journal, vol. 32, no. 8, pp. 1598–1605, 1994.
5. T. J. Craft and B. E. Launder, “On the spreading mechanism of the three-dimensional turbulent wall jet,” Journal of Fluid Mechanics, vol. 435, pp. 305–326, 2001.
6. H. M. Lübcke, T. H. Rung, and F. Thiele, “Prediction of the spreading mechanism of 3D turbulent wall jets with explicit Reynolds-stress closures,” International Journal of Heat and Fluid Flow, vol. 24, no. 4, pp. 434–443, 2003.
7. B. A. Pettersson Reif, “A nonlinear constitutive relationship for the v2f model,” in Annual Research Briefs, pp. 267–276, Center for Turbulence Research, 1999.
8. B. Ji, X. W. Luo, Y. L. Wu, X. X. Peng, and Y. L. Duan, “Numerical analysis of unsteady cavitating turbulent flow and shedding horse-shoe vortex structure around a twisted hydrofoil,” International Journal of Multiphase Flow, vol. 51, pp. 33–43, 2013.
9. B. Ji, X. W. Luo, Y. L. Wu, and H. Y. Xu, “Unsteady cavitating flow around a hydrofoil simulated using partially-averaged Navier-Stokes model,” Chinese Physics Letters, vol. 29, no. 7, Article ID 076401, 2012.
10. P. Durbin, “Separated flow computations with the k-ε-v2 model,” AIAA Journal, vol. 33, no. 4, pp. 659–664, 1995.
11. D. R. Laurence, J. C. Uribe, and S. V. Utyuzhnikov, “A robust formulation of the v2-f model,” Flow, Turbulence and Combustion, vol. 73, no. 3-4, pp. 169–185, 2005.
12. M. Popovac and K. Hanjalic, “Compound wall treatment for RANS computation of complex turbulent flows and heat transfer,” Flow, Turbulence and Combustion, vol. 78, no. 2, pp. 177–202, 2007.
13. K. Duraisamyy and G. Iaccarino, “Curvature correction and application of the v2-f turbulence model to tip vortex flows,” in Annual Research Briefs, pp. 157–168, Center for Turbulence Research, 2005.
14. X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013.
15. Y. H. Wu, L. D. Liu, B. Wiwatanapataphee, and S. Y. Lai, “Nonlinear functional analysis of boundary value problems: novel theory, methods, and applications,” Abstract and Applied Analysis, vol. 2013, Article ID 158358, 3 pages, 2013.
16. Y. Wu and B. Wiwatanapataphee, “Modelling of turbulent flow and multi-phase heat transfer under electromagnetic force,” Discrete and Continuous Dynamical Systems B, vol. 8, no. 3, pp. 695–706, 2007.
17. B. Wiwatanapataphee, Y. Wu, J. Archapitak, P. F. Siew, and B. Unyong, “A numerical study of the turbulent flow of molten steel in a domain with a phase-change boundary,” Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 307–319, 2004.
18. J. Ehrhard, Untersuching linearer und nichtlinearer wirbelviskositatsmodelle zur berechnung turbulenter stomungen um gebaude [Ph.D. thesis], VDI, Düsseldorf, Germany.
19. K. Yakinthos, Z. Vlahostergios, and A. Goulas, “Modeling the flow in a 90° rectangular duct using one Reynolds-stress and two eddy-viscosity models,” International Journal of Heat and Fluid Flow, vol. 29, no. 1, pp. 35–47, 2008.