Abstract
The simulation of turbulent flows by means of computational fluid dynamics is highly challenging. The costs of an accurate direct numerical simulation (DNS) are usually too high, and engineers typically resort to cheaper coarse-grained models of the flow, such as large-eddy simulation (LES). To be suitable for the computation of turbulence, methods should not numerically dissipate the turbulent flow structures. Therefore, energy-conserving discretizations are investigated, which do not dissipate energy and are inherently stable because the discrete convective terms cannot spuriously generate kinetic energy. They have been known for incompressible flow, but the development of such methods for compressible flow is more recent. This paper will focus on the latter: LES and DNS for turbulent subsonic flow. A new theoretical framework for the analysis of energy conservation in compressible flow is proposed, in a mathematical notation of square-root variables, inner products, and differential operator symmetries. As a result, the discrete equations exactly conserve not only the primary variables (mass, momentum and energy), but also the convective terms preserve (secondary) discrete kinetic and internal energy. Numerical experiments confirm that simulations are stable without the addition of artificial dissipation. Next, minimum-dissipation eddy-viscosity models are reviewed, which try to minimize the dissipation needed for preventing sub-grid scales from polluting the numerical solution. A new model suitable for anisotropic grids is proposed: the anisotropic minimum-dissipation model. This model appropriately switches off for laminar and transitional flow, and is consistent with the exact sub-filter tensor on anisotropic grids. The methods and models are first assessed on several academic test cases: channel flow, homogeneous decaying turbulence and the temporal mixing layer. As a practical application, accurate simulations of the transitional flow over a delta wing have been performed.
Highlights
Reduction of the aerodynamic drag of aircraft is a formidable task, because viscous friction forces are subject to the chaotic process of turbulence, which engineers would like to better understand
In contrast with the accurate results obtained in underresolved simulations of channel flow, the under-resolved simulation of decaying grid turbulence with the symmetrypreserving discretization disagrees with the experimental measurements
The initial energy decay is considerably smaller than in the experiment, which leads to over-prediction of the total kinetic energy at the second and third measurements station
Summary
Reduction of the aerodynamic drag of aircraft is a formidable task, because viscous friction forces are subject to the chaotic process of turbulence, which engineers would like to better understand. The momentum equation for compressible flow is [6] ∂ ∂t (ρ ) + ∇ ⋅ (ρ ⊗ ) + ∇p = ∇ ⋅ σ, (1). Note that throughout the paper, the symbol ⋅ will denote an inner product in R3 with corresponding norm | ⋅ |3. The common physical explanation of turbulence is that it is a cascade of progressively smaller and more complex flow structures. The driving force of the turbulent cascade is the non-linear convective term ∇ ⋅ (ρ ⊗ ). This term models the transfer of momentum to smaller scales (energy cascade) and conserves both momentum and kinetic energy, i.e. it only redistributes kinetic energy over the scales of motion
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