William H. Cabot

A dynamic subgrid‐scale eddy viscosity model

One major drawback of the eddy viscosity subgrid‐scale stress models used in large‐eddy simulations is their inability to represent correctly with a single universal constant different turbulent fields in rotating or sheared flows, near solid walls, or in transitional regimes. In the present work a new eddy viscosity model is presented which alleviates many of these drawbacks. The model coefficient is computed dynamically as the calculation progresses rather than input a priori. The model is based on an algebraic identity between the subgrid‐scale stresses at two different filtered levels and the resolved turbulent stresses. The subgrid‐scale stresses obtained using the proposed model vanish in laminar flow and at a solid boundary, and have the correct asymptotic behavior in the near‐wall region of a turbulent boundary layer. The results of large‐eddy simulations of transitional and turbulent channel flow that use the proposed model are in good agreement with the direct simulation data.

A dynamic subgrid-scale model for compressible turbulence and scalar transport

The dynamic subgrid‐scale (SGS) model of Germano et al. [Phys. Fluids A 3, 1760 (1991)] is generalized for the large eddy simulation (LES) of compressible flows and transport of a scalar. The model was applied to the LES of decaying isotropic turbulence, and the results are in excellent agreement with experimental data and direct numerical simulations. The expression for the SGS turbulent Prandtl number was evaluated using direct numerical simulation (DNS) data in isotropic turbulence, homogeneous shear flow, and turbulent channel flow. The qualitative behavior of the model for turbulent Prandtl number and its dependence on molecular Prandtl number, direction of scalar gradient, and distance from the wall are in accordance with the total turbulent Prandtl number from the DNS data.

A Lagrangian dynamic subgrid- scale model of turbulence

The dynamic model for large-eddy simulation of turbulence samples information from the resolved velocity field in order to optimize subgrid-scale model coefficients. When the method is used in conjunction with the Smagorinsky eddy-viscosity model, and the sampling process is formulated in a spatially local fashion, the resulting coefficient field is highly variable and contains a significant fraction of negative values. Negative eddy viscosity leads to computational instability and as a result the model is always augmented with a stabilization mechanism. In most applications the model is stabilized by averaging the relevant equations over directions of statistical homogeneity. While this approach is effective, and is consistent with the statistical basis underlying the eddy-viscosity model, it is not applicable to complex-geometry inhomogeneous flows. Existing local formulations, intended for inhomogeneous flows, are most commonly stabilized by artificially constraining the coefficient to be positive. In this paper we introduce a new dynamic model formulation, that combines advantages of the statistical and local approaches. We propose to accumulate the required averages over flow pathlines rather than over directions of statistical homogeneity. This procedure allows the application of the dynamic model with averaging to in-homogeneous flows in complex geometries. We analyse direct numerical simulation data to document the effects of such averaging on the Smagorinsky coefficient. The characteristic Lagrangian time scale over which the averaging is performed is chosen based on measurements of the relevant Lagrangian autocorrelation functions, and on the requirement that the model be purely dissipative, guaranteeing numerical stability when coupled with the Smagorinsky model. The formulation is tested in forced and decaying isotropic turbulence and in fully developed and transitional channel flow. In homogeneous flows, the results are similar to those of the volume-averaged dynamic model, while in channel flow, the predictions are slightly superior to those of the spatially (planar) averaged dynamic model. The relationship between the model and vortical structures in isotropic turbulence, as well as ejection events in channel flow, is investigated. Computational overhead is kept small (about 10% above the CPU requirements of the spatially averaged dynamic model) by using an approximate scheme to advance the Lagrangian tracking through first-order Euler time integration and linear interpolation in space.

Subgrid-scale backscatter in turbulent and transitional flows

Most subgrid‐scale (SGS) models for large‐eddy simulations (LES) are absolutely dissipative (that is, they remove energy from the large scales at each point in the physical space). The actual SGS stresses, however, may transfer energy to the large scales (backscatter) at a given location. Recent work on the LES of transitional flows [Piomelli et al., Phys. Fluids A 2, 257 (1990)] has shown that failure to account for this phenomenon can cause inaccurate prediction of the growth of the perturbations. Direct numerical simulations of transitional and turbulent channel flow and compressible isotropic turbulence are used to study the backscatter phenomenon. In all flows considered roughly 50% of the grid points were experiencing backscatter when a Fourier cutoff filter was used. The backscatter fraction was less with a Gaussian filter, and intermediate with a box filter in physical space. Moreover, the backscatter and forward scatter contributions to the SGS dissipation were comparable, and each was often much...

Approximate Wall Boundary Conditions in the Large-Eddy Simulation of High Reynolds Number Flow

The near-wall regions of high Reynolds numbers turbulent flows must be modelled to treat many practical engineering and aeronautical applications. In this review we examine results from simulations of both attached and separated flows on coarse grids in which the near-wall regions are not resolved and are instead represented by approximate wall boundary conditions. The simulations use the dynamic Smagorinsky subgrid-scale model and a second-order finite-difference method. Typical results are found to be mixed, with acceptable results found in many cases in the core of the flow far from the walls, provided there is adequate numerical resolution, but with poorer results generally found near the wall. Deficiencies in this approach are caused in part by both inaccuracies in subgrid-scale modelling and numerical errors in the low-order finite-difference method on coarse near-wall grids, which should be taken into account when constructing models and performing large-eddy simulation on coarse grids. A promising new method for developing wall models from optimal control theory is also discussed.

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor-Green vortex, Shu-Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

The mixing transition in Rayleigh-Taylor instability

A large-eddy simulation technique is described for computing Rayleigh–Taylor instability. The method is based on high-wavenumber-preserving subgrid-scale models, combined with high-resolution numerical methods. The technique is verified to match linear stability theory and validated against direct numerical simulation data. The method is used to simulate Rayleigh–Taylor instability at a grid resolution of

Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation

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High-Reynolds number Rayleigh–Taylor turbulence

. The growth rate is found to depend on the mixing rate. A mixing transition is observed in the flow, during which an inertial range begins to form in the velocity spectrum and the rate of growth of the mixing zone is temporarily reduced. By measuring growth of the layer in units of dominant initial wavelength, criteria are established for reaching the hypothetical self-similar state of the mixing layer. A relation is obtained between the rate of growth of the mixing layer and the net mass flux through the plane associated with the initial location of the interface. A mix-dependent Atwood number is defined, which correlates well with the entrainment rate, suggesting that internal mixing reduces the layers growth rate.

Progress in understanding turbulent mixing induced by Rayleigh–Taylor and Richtmyer–Meshkov instabilities

The cost of large eddy simulation (LES) in the near-wall region of attached turbulent boundary layers scales as the square of the friction Reynolds number, thus limiting LES to moderate Reynolds numbers. Wall stress boundary conditions are frequently used to alleviate this resolution requirement, but commonly used models are shown to perform poorly at high Reynolds numbers even in turbulent channel flow. Techniques from optimal control theory are used to find wall stresses that yield much better results in turbulent channel flow at high Reynolds numbers than existing models even on extremely coarse grids. In this approach, a suboptimal control strategy is used in which the objective is to force the outer LES towards a desired solution by using the wall stress boundary conditions as control. The suboptimal wall stresses are not necessarily physical, rather they are whatever is necessary to overcome the numerical and modeling errors present in the near-wall region to yield the correct mean velocity profile....