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Section: Research Program

Computational fluid mechanics: resolving versus modelling small scales of turbulence

A typical continuous solution of the Navier Stokes equations is governed by a spectrum of time and space scales. The broadness of that spectrum is directly controlled by the Reynolds number defined as the ratio between the inertial forces and the viscous forces. This number is quite helpful to determine if the flow is turbulent or not. In the former case, it indicates the range of scales of fluctuations that are present in the flow under study. Typically, for instance for the velocity field, the ratio between the largest scale (the integral length scale) to the smallest one (Kolmogorov scale) scales as $R{e}^{3/4}$ per dimension. In addition, for internal flows, the viscous effects near the solid walls yield a scaling proportional to $Re$ per dimension. The smallest scales may have a certain effect on the largest ones which implies that an accurate framework for the modelling and the computation of such turbulent flows must take into account all these scales of time and space fluctuations. This can be achieved either by solving directly the Navier-Stokes (NS) equations (Direct numerical simulations or DNS) or by first applying to them a filtering operation either in time or space. In the latter cases, the closure of the new terms that appear in the filtered equations due to the presence of the non-linear terms implies the recourse to a turbulence model before discretizing and then solving the set of resulting governing equations. Among these different methodologies, the Reynolds averaged Navier-Stokes (RANS) approach yields a system of equations aimed at describing the mean flow properties. The term mean is referring to an ensemble average which is equivalent to a time average only when the flow is statistically stationnary. In that case, the turbulence model aims at expressing the Reynolds stresses either through the solution of dedicated transport equations (second order modelling) or via the recourse to the concept of turbulent viscosity used to write and ad-hoc relation (linear or not) between the Reynolds stress and the mean velocity gradient tensor. If the filtering operation involves a convolution with a filter function in space of width $\delta$, this corresponds to the large-eddy simulation (LES) approach. The structures of size below $\delta$ are filtered out while the bigger structures are directly resolved. The resulting set of filtered equations is again not closed and calls for a model aimed at providing a suitable expression for the subgrid scale stress tensor.

From a computational point of view, the RANS approach is the less demanding, which explains why historically it has been the workhorse in both the academic and the industrial sectors. Although it has permitted quite a substantive progress in the understanding of various phenomena such as turbulent combustion or heat transfer, its inability to provide a time-dependent information has led to promote in the last decade the recourse to either LES or DNS as well as hybrid methods that combine RANS and LES. By simulating the large scale structures while modelling the smallest ones supposed to be more isotropic, the LES, alone or combined with the most adanced RANS models such as the EB-RSM model [4] proved to be quite a step through that permits to fully take advantage of the increasing power of computers to study complex flow configurations. In the same time, DNS was progressively applied to geometries of increasing complexity (channel flows, jets, turbulent premixed flames), and proved to be a formidable tool that permits (i) to improve our knowledge of turbulent flows and (ii) to test (i.e. validate or invalidate) and improve the numerous modelling hypotheses inherently associated to the RANS and LES approaches. From a numerical point of view, if the steady nature of the RANS equations allows to perform iterative convergence on finer and finer meshes, this is no longer possible for LES or DNS which are time-dependent. It is therefore necessary to develop high accuracy schemes in such frameworks. Considering that the Reynolds number in an engine combustion chamber is significantly larger than 10000, a direct numerical simulation of the whole flow domain is not conceivable on a routine basis but the simulation of generic flows which feature some of the phenomena present in a combustion chamber is accessible considering the recent progresses in High Performance Computing (HPC).