## Computational Fluid Dynamics Analysis of a Mixing Layer

This fluid dynamics analysis involved simulating the interactions of two streams of air as they merge behind a splitter plate. The configuration was modeled after experiments performed at Ohio State by Samimy and Elliott. Both streams of fluid were at the same pressure, but the high speed side was moving at Mach 1.5 and the low speed side at Mach 0.5.

### Approaches to Turbulence Modeling in CFD

The purpose of these simulations was to evaluate the performance of various turbulence models when applied to a highly unsteady flow. Three different classes of model were examined.

Most CFD solvers were originally written to solve the Reynolds-Averaged Navier-Stokes (**RANS**) equations. These equations explicitly “average out” the unsteady turbulent fluctuations and use various models to account for their effects on the mean flow.

This is a perfectly legitimate thing to do when seeking a steady state solution, but for a time-accurate fluid dynamics analysis, it may not work as well since the models do not necessarily distinguish between turbulent fluctuations, which should be modeled, and large scale unsteadiness which should be resolved. To overcome this weakness, various techniques have been developed.

Large Eddy Simulation (**LES**) takes a different approach to deriving the governing equations and only tries to model the scales of motion which cannot be resolved on the computational grid. LES requires that the dominant scales of turbulent motion be resolved, however, so the grid requirements can be quite restrictive.

Then there are **Hybrid RANS/LES** models which use a RANS model in regions where the grid is too coarse for an LES. In regions with more resolution, hybrid models switch over to an LES approach. In this way, one attempts to combine the best features of the LES and RANS approaches, but the transition between the two regimes can be tricky.

This study looked at the performance of seven different models covering the spectrum of RANS, hybrid RANS/LES, and pure LES models to see how they would perform in a situation where LES should be applicable. Obviously, we expect RANS models to be at a disadvantage in this situation, but there is a surprising amount of CFD performed where unsteady simulations are performed with an unmodified RANS model, so it bears looking at how these models perform.

### Results

The figure below is a snapshot of the mixing layer at an instant in time. The flow moves from left to right, with the high speed region on top. The figure shows contours of the concentration of a so-called “passive scalar” quantity which has a value of 1.0 in the high-speed flow and 0.0 in the low speed flow. Regions with intermediate values develop as the two streams mix.

The next figure shows the prediction of mixing layer thickness as a function of distance downstream of the splitter plate for the different models. Not too surprisingly, the RANS models underpredict growth rate of the mixing layer. The other models are roughly consistent with each other and the experimental results. The absolute magnitude is not as important for this quantity, because of uncertainties in the experimental conditions which make it difficult to match the initial growth.

The second line plot shows the intensity of the turbulent velocity fluctuations in the vertical direction along a vertical cut through the mixing layer. Here we see that the LES models and the Hybrid models generally do a good job of predicting the unsteadiness, although the older Smagorinsky model is less capable. The RANS models badly underpredict this quantity.

The final plot shows the “skewness” of the streamwise velocity fluctuations in a vertical slice through the mixing layer. Again, the turbulence models designed specifically for unsteady fluid dynamics analysis (i.e. the LES and hybrid RANS/LES models) do an excellent job, even on this higher order statistical quantity.

Note that the non-dimensionalization of the spatial coordinate is such that the mixing layer extends from -0.5 to 0.5. Beyond this region, the grid resolution coarsens rapidly, and one would not expect much from any turbulence model. Nevertheless, the unsteady models appear to capture the experimentally observed behavior of many quantities quite well.

### Conclusions

What we learned from this study was that our Hybrid models worked just as well as the LES models for this mixing layer configuration. This is an important step in the validation of these models, although it is by no means the final word on the subject.

The other important finding was that RANS models are truly inappropriate for use in simulations where one wishes to resolve unsteadiness associated with large-scale turbulence. While one can certainly use them for some unsteady fluid dynamics analysis (e.g. simulations of moving shocks and store drop simulations), they should be used with great care.

As a final note, while not useful for the unsteady CFD analysis, the RANS models were able to capture the mixing layer growth quite well in steady-state mode. Obviously, the turbulent statistics which can be obtained from such a simulation are limited, but depending on your application, the steady-state results may be sufficient. Certainly, running a 2D steady-state RANS simulation requires much less computational horsepower than the full 3-D unsteady fluid dynamics analysis.

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