Subgrid Stress Model Study

6.1. Turbulent Channel Flow 41

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0.0 0.2 0.4 0.6 0.8 1.0 1.2

< u > /U 50 / 20

50 / 10 25 / 20 25 / 10

DNS

10

⁰

10

¹

10

²

y

⁺

0 5 10 15 20 25

< u > /u

τ

Figure 6.2: Mean streamwise velocity for different span- and streamwise mesh resolutions. In outer coordinates (left), in inner coordinates (right).

Figure 6.3 shows the non-zero components of the Reynolds stress tensor.

The streamwise component gets over-predicted on all meshes besides the finest one. The coarsest mesh gives the largest over-prediction which is not surprising.

The behaviour for the wall-normal and spanwise components is similar.

The meshes with ∆z

⁺

= 10 perform well, whereas using a coarser spanwise resolution leads to a larger under-prediction. The turbulent shear stress is very well predicted using all meshes.

It is possible to conclude that decreasing ∆x

⁺

while keeping the same

∆z

⁺

has very little effect on all the Reynolds stresses, with the exception of the streamwise component. There, refining from ∆x

⁺

= 50 to 25, while keeping ∆z

⁺

= 10 gives a big boost in accuracy. On the other hand, a similar refinement but with ∆z

⁺

kept at 20 does not lead to an equally large improvement.

The overall conclusion of the study is that using ∆x

⁺

= 25, ∆z

⁺

= 10

along with the used cell-size distribution in the wall normal direction leads to

very accurate results for both first- and second-order statistics. If coarsening

is necessary, it is better to coarsen ∆x

⁺

rather then ∆z

⁺

, as the latter has a

larger effect on the accuracy in the inner layer. However, a small ∆x

⁺

does

lead to a better prediction in the outer layer for the mean velocity profile.

direc-0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0 2 4 6 8 10 12

< u

0

u

0

> /u

2 τ

50 / 20 50 / 10 25 / 20 25 / 10

DNS

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0.0 0.2 0.4 0.6 0.8 1.0 1.2

< v

0

v

0

> /u

2 τ

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0.0 0.5 1.0 1.5 2.0

< w

0

w

0

> /u

2 τ

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

1.0 0.8 0.6 0.4 0.2 0.0

< u

0

v

0

> /u

2 τ

Figure 6.3: The components of the Reynolds stress tensor for different span-and streamwise mesh resolutions.

tions respectively. In the wall-normal direction the distribution (6.1) will be used. This rather coarse mesh is chosen so that the SGS modelling has a significant effect on the results. In the previous section it was shown that us-ing no SGS model with this resolution leads to a profile which over-predicts the mean velocity in the inner layer and under-predicts it in the core of the channel. Possibly, an SGS model can improve on that result.

The following models are considered in the study (see section 2.2.2 for definitions).

• ILES, i.e. no SGS model at all.

• The Smagorinsky model, see (2.11).

• The one-equation model, see (2.9).

6.1. Turbulent Channel Flow 43

• The one-equation model with near-wall damping using the van Driest damping function (2.12).

• A dynamic version of the one-equation model.

It is important to note that the models were not tuned in any way with respect to the values of the constants that they depend on. Default values suggested in OpenFOAM were used.

Figure 6.4 shows the mean streamwise velocity profiles obtained with the considered models. The difference in the results is more easily observed in the inner coordinates. The ILES and the dynamic model perform very similarly to each other. The one-equation and the Smagorinsky model also perform equally and significantly worse than the other models. However, the large discrepancy seen in the plot in inner coordinates is mostly due to the over-predicted value of the friction velocity. The shape of the profile is not that significantly different from those produced by ILES and the dynamic models. Applying a damping function improves the performance of the one-equation model significantly. It is hard to judge whether it performs better or worse than ILES, the overall error in the profile is of similar magnitude, but its distribution is different.

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0.0 0.2 0.4 0.6 0.8 1.0 1.2

< u > /U ILES

Smag.

One-eq.

One-eq., damped Dyn. one-eq.

DNS

10

⁰

10

¹

10

²

y

⁺

0 5 10 15 20 25

< u > /u

τ

Figure 6.4: Mean streamwise velocity for different SGS models. In outer coordinates (left), in inner coordinates (right).

Figure 6.5 shows the computed non-zero components of the Reynolds stress tensor. The observed trends are mostly similar to that exhibited by the mean velocity profile. The dynamic model performs similar to the ILES.

The Smagorinsky and one-equation models perform significantly worse.

In-terestingly, the one-equation model with damping only differs in

perform-ance from the undamped version in the accuracy of the hu

⁰

u

⁰

i profile. There,

the damping slightly improves the result; in fact, the model gives the best prediction of this component among all the considered models.

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0 2 4 6 8 10 12

< u

0

u

0

> /u

2 τ

ILES Smag.

One-eq.

One-eq., damped Dyn. one-eq.

DNS

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0.0 0.2 0.4 0.6 0.8 1.0 1.2

< v

0

v

0

> /u

2 τ

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

0.0 0.5 1.0 1.5 2.0

< w

0

w

0

> /u

2 τ

0.0 0.2 0.4 0.6 0.8 1.0 y/δ

1.0 0.5 0.0 0.5 1.0

< u

0

v

0

> /u

2 τ

Figure 6.5: The components of the Reynolds stress tensor for different SGS models.

The conclusion that can be drawn from this study is that the considered SGS models, at least as they are implemented in OpenFOAM, do not offer significant improvement over not using any model at all and relying on numerical dissipation. Only the dynamic one-equation model performed on par with ILES, but no improvement that would justify the associated increase in simulation time was observed.

While this study is too limited to draw any general conclusions regarding

SGS modelling, it can serve as a guideline for users of OpenFOAM and other

solvers employing similar numerical techniques. It shows that simply not

using an SGS model can be a justifiable decision and a possibility that is

worth investigating.

In document T M InflowGenerationforScale-ResolvingSimulationsofTurbulentBoundaryLayers (Page 51-55)