2D flow over backward facing step

Underlying Flow Regime 3-15 © copyright ERCOFTAC 2004

Evaluation

Comparison of CFD calculations with Experiments

The published calculations on the tests cases are summarized in Table 1. It includes the model and the experiment against which the results are compared as well as the recirculation length obtained and the grid size employed (when they were presented).

Reference	Model	Exp.	x_r/h JD (Grid size)	x_r/h DS(Grid size)
Durbin (1995)	k-ε-χ²	JD,DS	6.2 (120x120)	6.2 (120x120)	1
Manceau and Parnaix (1999)	k-ε-χ²	JD	6.2	-	2
Driver and Seegmiller (1985)	k-ε	DS	-	4.3 (42x42)	3
	k-ε*		-	4.6 (42x42)
	ASM		-	4.7 (42x42)
	ASM*		-	5.7 (42x42)
Menter (1994)	k-ω	DS	-	6.4 (120x120)	4
	k-ω SST		-	6.5 (120x120)
	k-ω BSL		-	5.9 (120x120)
	k-ε		-	5.5 (120x120)
Rodi (1991)	k-ε	DS	-	4.9
Rodi (1991)	two layer k-ε	DS	-	5.5
Menter (1996)	k-ε	DS	-	5.5 (120x120)
Menter (1996)	k-ε 1 eq	DS	-	6.1 (120x120)
Apsley and Leschziner (1998)	Linear k-ε	DS	-	5.45 (152x112)
	quadratic k-ε		-	5.55 (152x112)
	cubic k-ε		-	6.2 (152x112)
	cubic k-ε		-	7.0 (152x112)
Akselvoll and Moin (1993)	LES	JD	6.0 (192x48x32)		5
Yoder et al.(1999)	k-ε (NPARC code)	DS	-	5.31 (238 x 185)
	k-ε (WIND code)		-	5.30 (238 x 185)
	k-ε C_μ		-	5.55 (238 x 185)
	SST		-	6.43 (238 x 185)
Parnaix and Durbin (1996)	SSG	JD	6.2 (120x120)	-
Michelassi et al. (1996)	k-ε-χ² version 1	DS	-	6.1 (120x120)
Michelassi et al. (1996)	k-ε-χ² version 2	DS	-	6.5 (120x120)
Hanjalic and Jakirlic (1998)	LRRG+WF+Sl	DS JD	6.38 (90x46)	6.21 (140x110)
Hanjalic and Jakirlic (1998)	HJ	DS JD	6.26(180x100)	6.20 (175x145)
Experimental			6.26	6.28

Table 1: CFD calculations for the chosen test cases. The numbers in the last column stand for:

1: A refinement study is referenced showing that the resolution is appropriate.

2: The results are very similar to those presented in 1, but they were obtained using a finite element code using unstructured meshes.

3: The reattachment length is presented in a figure showing its dependence with the top wall reflection angle. The * stands for modified versions of the models that use a modified equation for P

4: A refinement study was performed using 90x90 and 240x240 grids, obtaining the same results.

5: The experiments used in this reference are a previous version of those performed by JD.

Some calculations present comparisons of the performance including the k-ε model, from which the well-known fact that it underpredicts the reattachment length can be seen. In the work of Rodi (1991) it is shown the reattachment length found using the two layer k-ε model is closer to the experimental value than that obtained using the k-ε model, but is still 14% shorter. In later results presented by Menter (1996), the reattachment length using the k-ε model was 5% short and, remarkably, using the one equation model is within the experimental uncertainty. A significant improvement was found by Driver and Seegmiller (1985) when using a modified P equation by a new production term, but only when using ASM models and not with the k-ε. It is to be noted that the grid employed is coarse. Better predictions of the recirculation length (Menter (1994) and Yolder (1999)) are found when using other two equation models, k-ω and SST. The SST was found to slightly overpredict it. The same tendency to overpredict the reattachment length was found with non linear two equation models in Apsley and Leschziner (1998), although the values are not presented in this study, this can be seen in the friction coefficient distribution. The predictions are much better when using the k-ε-χ² model, or second moment closures. Large eddy simulations provide values that are less than 2% far from DNS.

The k-ε model gives smaller velocities than those observed in the DS experiments in the reverse flow region, a fact that is observed in the skin friction coefficient distribution along the bottom wall (Rodi(1991)). The calculations presented by Driver and Seegmiller (1985) show an improvement in this distribution when using ASM models, but mainly in the recirculation length and not in the C_f negative peak values, so the experimental velocity gradients near the wall are steeper than those calculated. The situation is improved with the two layer k-ε model as reported by Rodi (1991). The C_f peaks are now a little overpredicted. This fact is even more pronounced in the JL k-ε model as shown by Menter (1994) and Menter (1996) were significant improvements have been achieved with the k-ω, the k-ω BSL, the k-ω SST and also with the k-ε 1eq model, using a fine grid in all cases. A similar behaviour is reported by Yoder et al. (1999) (that can be seen at NPARC Alliance) where it is shown that the k-ε models (including k-ε with variable C_μ) not only underpredict the recirculation length but also present some overshoots and undershoots in the friction coefficient. It is mentioned that this behaviour can be reduced by strongly increasing the discretization near the bottom wall. The use of a variable C_μ, tends to reduce the turbulent viscosity within the separation region, and thus reduces the magnitude of skin friction. It is also found a good prediction of the skin friction distribution when using the SST model.

In Figure 4 the skin friction coefficient distribution taken from Durbin (1995) is shown including comparisons to both DS and JD experimental results.

As pointed out by P. Durbin, the skin friction is larger in JD experiments due to the low Reynolds number of the flow and this is correctly predicted by his k-ε-χ² model, although a larger negative value than the observed one is predicted for the DS case. The main difference with respect to experiments is the extent of the positive skin friction coefficient that is due to the secondary bubble located in the base corner of the step. In Parneix and Durbin's (1996) second moment calculations, the size of this secondary bubble is still smaller than the observed one. Figure 5, taken from Parneix and Durbin (1996), shows the streamlines obtained with the second moment closure model compared to those obtained by DNS.

This aspect was studied in depth by Hanjalic and Jakirlic (1998) for second moment closures. The predicted length of the secondary bubble is approximately 0.5 with the LRRG and SSG models (a little bigger in the first one) as can be seen in the streamline patterns presented. A small improvement is obtained when using the modified P equation.

Figure 5: Stream lines pattern from Parneix and Durbin (1996) a) Second moment closure results b) DNS results

They presented, however, a much better result with their second moment low Reynolds number model, finding the size of the secondary bubble (and even the small third one) very close to that obtained by DNS, as shown in the Table 2:

*Table 2: results taken from Hanjalic and Jakirlic (1998)*
	main bubble	secondary bubble		third bubble
Model	Xr/H	Xr/H	Yr/H	Xr/H
DNS	6.28	1.76	0.8	0.042
HJ Low Re RSM	6.38	1.55	0.66	0.04

This secondary bubble is absent when using k-ε model, and appeared when using two layer models as pointed out by Rodi (1991). In the works of Menter (1994) and Menter (1996) it can be inferred from the C_f distribution. These two equation calculations predicted a small (second) bubble except the k-ε one equation of Menter (1996).

Turbulent kinetic energy and shear stress calculations, carried out using the modified ASM model, presented by Driver and Seegmiller (1985), showed in general a good agreement to experimental values. The increase in the kinetic energy and turbulent stresses after the step and their decrease after the reattachment are correctly predicted. There are however some differences in the backflow region mainly in the position of the peak values of both turbulent shear stress and turbulent kinetic energy. The authors suggest some problem in the convection of kinetic energy as one would expect the energy to be convected along the streamlines. The authors also comment the tendency of the unmodified k-ε and ASM to overpredict the turbulent kinetic energy and shear stresses. The same situation is found by Rodi (1991) where better predictions of turbulent shear stress by two-layer models are shown. Results reported by Yoder et al. (1999) for turbulent kinetic energy profiles and turbulent shear stress show that k-ε model implemented in different codes and with variable C_μ, overpredict both the kinetic energy and the shear stress. The SST model on the other hand, underpredicts these values due to the form of the k-ω model used in the near-wall region. The incorrect peak locations can be appreciated clearly in these figures (available at NPARC Alliance), especially at x/h= 2.5, 5 and 6 for both kinetic energy and shear stress and for all the turbulence models employed. It can be also mentioned that the use of variable C_μ reduces the turbulent kinetic energy and shear stresses mainly in the recirculation region. According to the author this is due to an increase in the turbulent dissipation rate within the recirculation zone because of the variable C_μ.

Figure 6, taken from Durbin (1995) presents turbulent kinetic energy profiles for both DS and JD experiments. As can be seen there is a little overprediction of turbulent kinetic energy profiles in the separated region. The peak value locations are well predicted for the JD case, and a little shifted in the DS case near the reattachment.

Figure 6: Turbulent kinetic energy taken from Durbin (1995) a) Comparison to DS experiments b) Comparison to JD experiments

Large eddy simulations performance is very good in this aspect. Figure 7, taken from Alkselvol and Moin (1993), show the prediction of turbulent shear stress compared to DNS of Le and Moin (1997).

Figure 7 : turbulent shear stress taken from Alkselvol and Moin (1993). Solid lines LES dashed lines DNS a) x/h=2.0 b) x/h=7.0 c) x/h=13 d) x/h=18

A very important feature of this problem is the absence of balance between production and dissipation rate of turbulent kinetic energy in the separated region, because this balance (the local equilibrium) is assumed when using wall functions. Driver and Seegmiller (1985) presented experimental data and calculations of the kinetic energy balance. The computations carried out with the different models overpredict production and dissipation of turbulent kinetic energy, The modified ASM is slightly better in predicting production but it still overestimates dissipation mainly in the shear layer. Figure 8 from Driver and Seegmiller (1985) shows experimental and calculated production and dissipation rate of turbulent kinetic energy.

Figure 8: Kinetic energy budget at two steps upstream the reattachemnt taken from Driver and Seegmiller (1985).

Second moment calculations presented by Hanjalic and Jakirlic (1998) show an excellent agreement between their low Reynolds number model and the DNS results.

The profiles of mean velocities show a general problem of RANS turbulence models. Figure 9 taken from Durbin (1995), shows how these models are unable to predict the recovery of the boundary layer downwards the reattachment. In the case of DS experiment the prediction is worst than in the JD case. Calculations using second moment closures present the same behaviour.

Figure 9: Mean velocity profiles in the stream direction taken from Durbin (1995) a) Comparison to DS experiments b) Comparison to JD experiments

Figure 10, from Hanjalic and Jakirlic (1998) shows the velocity profiles at three stations in the recovery region comparing the results to those of the DNS. It is a very instructive plot showing how far from the log-law (which predicts local equilibrium) the velocity profile is, even at x/h=19 and this is handled quite well by the RSM models.