Predictive simulations for problems with solution non-uniqueness

We present our main conclusions regarding the simulation of turbulent mixing, with a summary of previous results and the inclusion of new evidence in support of these conclusions. Our main conclusions are: 1. Turbulent simulations in the Large Eddy Simulation (LES) regime are inherently non-unique, and require experimental validation before they can be used for scientific or engineering purposes. 2. The level of non-uniqueness is mesh dependent, decreasing to zero as the mesh is refined. The rate of decrease is slow and governed by Kolmogorov exponents when in a scaling regime. 3. Simulation uncertainty is greatly reduced by the use of subgrid scale (SGS) models. For mixing problems (in the case of small time scales or small diffusion parameters), tracking of discontinuities or steep gradients is also essentiAl. 4. Simulation codes have implicitly defined subgrid scale terms within them; each new code or code revision needs its own validation study. For the same reason, higher order subgrid terms, or even summing the missing parts of the Kolmogorov spectrum to reconstruct more exact subgrid terms is not a panacea, and validation (adjustment) of subgrid models is still needed. Within our own validation experiments, we find that low order SGS terms, second order differencing and front tracking (to control excess diffusion) result, basically, in a perfect fit to experimental data. That is, no further tuning is required for this algorithm. 5. Experimental validation (essential) can be performed at a high but experimentally feasible Reynolds number (Re), followed by an extrapolation to Reynolds numbers needed for engineering design or scientific studies. 6. Convergence of the cumulative distribution functions (CDFs) is important for reactive chemistry coupled to turbulent mixing. We provide a mathematical framework for this convergence, and show that it does occur, with slow rates influenced by Kolmogorov exponents. Engineering simulations are often regarded as interpolative, not predictive, in their dependence on experimental data. Our conclusions both support this view and indicate a mitigation strategy based on mesh refinement and the appropriate choice of numerical methods.