V - Reynolds Stress Model (RSM)

Different from most turbulence models, the Reynolds stress models (RSM) closes the Reynolds-Averaged Navier-Stokes equations by solving additional transport equations for the six independent Reynolds stresses.

Three varieties of the Reynolds Stress Models are available and are known as (i) LRR (Launder-Reece-Rodi), (ii) SSG (Spiezale-Sakar-Gatski) and (iii) Wilcox Stress-.

The LRR model is the best known and tested model based on the dissipation transport equation. The other RSM are mostly variations of LRR differing mainly in terms of modeling the distribution pressure.

These models are available in FLUENT as RSM linear, RSM quadratic and RSM stress-omega.

It is advisable to use the RSM models when dealing with the following types of flow:

Free shear flows with strong anisotropy (e.g. flows in rotating fluids).

Flows with sudden changes in the mean strain rate. Flows where the strain fields are complex. Flows with strong streamline curvature. Secondary flow. Buoyant flow.

Nine case studies were performed as to cover all the model options available in FLUENT. The main characteristics used in the simulations are presented in Table 1. The solver used was the steady pressure –based implicit solver.

Table 1 – Characteristics of the models.

Casestudy

Characteristics

RSM modelNear-Wall treatment

Scheme Order Residuals

1 Linear Standard PISO 2nd 10-42 Linear Scalable PISO 2nd 10-43 Linear Non-Equilibrium PISO 2nd 10-44 Linear Enhanced PISO 2nd 10-45 Quadratic Standard PISO 2nd 10-46 Quadratic Scalable PISO 2nd 10-47 Quadratic Non-Equilibrium PISO 2nd 10-4

k-omega options8 Stress-Omega Low-Re PISO 2nd 10-49 Stress-Omega Shear Flow PISO 2nd 10-4

Cases 1 to 4 and 5-7 were performed with the objective of evaluating the Linear and Quadratic RSM models in conjunction with different wall functions.

Cases 8 and 9 were performed with the objective of evaluating the Stress-Omega model in conjunction with the k-omega options available.

V.1 – Results and Discussion

Figure 1 present a comparison of U-velocity obtained with RSM model with experimental and LES results at x=0.5

Figure 1 – Comparison of U-velocity obtained with RSM model with experimental and LES results at x=0.5

According to the results shown in Figure 1, it can be observed that all of the RSM models presented a similar behaviour, independent of the near-wall treatment in the case of Linear and Quadratic models, or the k-omega option in the case of Stress-Omega model. Essentially, the U-velocity is overpredicted for values of y/H in the ranges of 1.0-1.15 and 1.4-2.0. On the other hand the U-velocity is underpredicted for values of y/H in the range 1.15-1.4.

When comparing the RSM results with the LES results, it can be seen that the same differences encountered when comparing with the experimental results are observed, except in the top wall of the cube, where the values of the U-velovcity obtained with LES are superior to those obtained using RSM.

Figure 2 presents a comparison of U-velocity obtained with RSM model with experimental and LES results at x=1.0

Figure 2 – Comparison of U-velocity obtained with RSM model with experimental and LES results at x=1.0

In a similar fashion to the observed for the results at x=0.5, it can be seen in Figure 2 that U-velocity is overpredicted for values of y/H in the range of 1.4-2.0 and underpredicted for values of y/H in the range of .1.15-1.4. Different from results for x=0.5, there is a better fitting to the experimental data near the walls.

When comparing the RSM results with the LES results, it can be seeh that the RSM data fit well the LES data except for values of y/H in the range of 1.25-1.45.

Figure 3 presents a comparison of U-velocity obtained with RSM model with experimental and LES results at x=1.0.

Figure 3 – Comparison of U-velocity obtained with RSM model with experimental and LES results at x=2.0

Figure 3 shows that all the models overpredict the U-velocity for values of y/H in the range of 1.5-2.0. Linear and Quadratic models overpredict U-velocity for values of y/H in the range of 0.0-1.5, whereas Stress-Omega overpredict in the range of 0.5-1.5.

V.1 – Results and Discussion

In order to have a better understanding of the adequability of the models in the present study, calculation of the errors were made.

The absolute errors between numerical and experimental values of U-velocity for x=0.5, x=1.0 and x=2.0 are presented in Figures 4, 5 and 6, respectively.

Figure 4 – Absolute errors between numerical and experimental values of U-velocity for x=0.5

Figure 5 – Absolute errors between numerical and experimental values of U-velocity for x=1.0

Figure 6 – Absolute errors between numerical and experimental values of U-velocity for x=2.0

Figures 4, 5 and 6 show that the performance of the models are similar in all the cases. It is important to mention the models don’t perform well for values of y/H between 1.2 an 1.4. This may be because of the occurrence of separation and reattachement in this region.

The magnitude of the error obtained in the cases studies can be evaluated by the root mean square (RMS) of the errors. The RMS is calculated using equation (1):

RMSerror=2√ 1n

(|error|)2

RMS error for the nine case studies at x=0.5, 1.0 and 2.0 are presented in Table 2.

Table 2 - RMS for the case studies at x=0.5, 1.0 and 2.0.

Case StudyX=0.5 X=1.0 X=2.0

1 0.2170 0.220 0.22462 0.2203 0.2434 0.25613 0.2069 0.2071 0.21034 0.2123 0.1816 0.23035 0.2662 0.3300 0.38566 0.2741 0.3367 0.41487 0.2641 0.3227 0.37288 0.2801 0.3313 0.40509 0.3041 0.3789 0.4746

Comparing the magnitude of the errors, it can be concluded that the best results were obtained using the Linear RSM model with Non-Equilibrium near wall treatment. The worst results were obtained using Stress-Omega with shear flow as k-omega option.

Figure 6 shows the streamlines obtained using RSM Linear-Non-Equilibrium wall.

Figure 6 – Streamlines obtained using RSM Linear-Non-Equilibrium wall

In Figure 6, it is noticeable the vortices produced by the interaction of the fluid and the corners of the cube.