Roel Verstappen

Symmetry-preserving discretization of turbulent flow

We propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators, i.e., the convective operator is represented by a skew-symmetric coefficient matrix and the diffusive operator is approximated by a symmetric, positive-definite matrix. Mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner. We give it a concrete form by noting that a symmetry-preserving discretization of the Navier-Stokes equations is stable on any grid, and conserves the total mass, momentum and kinetic energy (for the latter the physical dissipation is to be turned off, of coarse). Being stable on any grid, the choice of the grid may be based on the required accuracy solely, and the main question becomes: how accurate is a symmetry-preserving discretization? Its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies. The comparison is carried out for a Reynolds number of 5600, which is based on the channel width and the mean bulk velocity (based on the channel half-width and wall shear velocity the Reynolds number becomes 180). The comparison shows that with a fourth-order, symmetry-preserving method a 64 × 64 × 32 grid suffices to perform an accurate numerical simulation.

Proper orthogonal decomposition and low-dimensional models for driven cavity flows

A proper orthogonal decomposition (POD) of the flow in a square lid-driven cavity at Re=22,000 is computed to educe the coherent structures in this flow and to construct a low-dimensional model for driven cavity flows. Among all linear decompositions, the POD is the most efficient in the sense that it captures the largest possible amount of kinetic energy (for any given number of modes). The first 80 POD modes of the driven cavity flow are computed from 700 snapshots that are taken from a direct numerical simulation (DNS). The first 80 spatial POD modes capture (on average) 95% of the fluctuating kinetic energy. From the snapshots a motion picture of the coherent structures is made by projecting the Navier–Stokes equation on a space spanned by the first 80 spatial POD modes. We have evaluated how well the dynamics of this 80-dimensional model mimics the dynamics given by the Navier–Stokes equations. The results can be summarized as follows. A closure model is needed to integrate the 80-dimensional system ...

Spectro-consistent discretization of Navier-Stokes: a challenge to RANS and LES

In this paper, we discuss the results of a fourth-order, spectro-consistent discretization of the incompressible Navier-Stokes equations. In such an approach the discretization of a (skew-)symmetric operator is given by a (skew-)symmetric matrix. Numerical experiments with spectro-consistent discretizations and traditional methods are presented for a one-dimensional convection-diffusion equation. LES and RANS are challenged by giving a number of examples for which a fourth-order, spectro-consistent discretization of the Navier-Stokes equations without any turbulence model yields better (or at least equally good) results as large-eddy simulations or RANS computations, whereas the grids are comparable. The examples are taken from a number of recent workshops on complex turbulent flows.

Direct numerical simulation of turbulence at lower costs

Direct Numerical Simulation (DNS) is the most accurate, but also the most expensive, way of computing turbulent flow. To cut the costs of DNS we consider a family of second-order, explicit one-leg time-integration methods and look for the method with the best linear stability properties. It turns out that this method requires about two times less computational effort than Adams–Bashforth. Next, we discuss a fourth-order finite-volume method that is constructed as the Richardson extrapolate of a classical second-order method. We compare the results of this fourth-order method and the underlying second-order method for a DNS of the flow in a cubical driven cavity at Re= 104. Experimental results are available for comparison. For this example, the fourth-order results are clearly superior to the second-order results, whereas their computational effort is about twenty times less. With the improved simulation method, a DNS of a turbulent flow in a cubical lid-driven flow at Re = 50,000 and a DNS of a turbulent flow past a square cylinder at Re = 22,000 are performed.

Symmetry-preserving discretization of Navier-Stokes equations on collocated unstructured grids

A fully-conservative discretization is presented in this paper. The same principles followed by Verstappen and Veldman (2003) 3] are generalized for unstructured meshes. Here, a collocated-mesh scheme is preferred over a staggered one due to its simpler form for such meshes. The basic idea behind this approach remains the same: mimicking the crucial symmetry properties of the underlying differential operators, i.e., the convective operator is approximated by a skew-symmetric matrix and the diffusive operator by a symmetric, positive-definite matrix. A novel approach to eliminate the checkerboard spurious modes without introducing any non-physical dissipation is proposed. To do so, a fully-conservative regularization of the convective term is used. The supraconvergence of the method is numerically showed and the treatment of boundary conditions is discussed. Finally, the new discretization method is successfully tested for a buoyancy-driven turbulent flow in a differentially heated cavity.

Physical consistency of subgrid-scale models for large-eddy simulation of incompressible turbulent flows

We study the construction of subgrid-scale models for large-eddy simulation of incompressible turbulent flows. In particular, we aim to consolidate a systematic approach of constructing subgrid-scale models, based on the idea that it is desirable that subgrid-scale models are consistent with the mathematical and physical properties of the Navier-Stokes equations and the turbulent stresses. To that end, we first discuss in detail the symmetries of the Navier-Stokes equations, and the near-wall scaling behavior, realizability and dissipation properties of the turbulent stresses. We furthermore summarize the requirements that subgrid-scale models have to satisfy in order to preserve these important mathematical and physical properties. In this fashion, a framework of model constraints arises that we apply to analyze the behavior of a number of existing subgrid-scale models that are based on the local velocity gradient. We show that these subgrid-scale models do not satisfy all the desired properties, after w...

Minimum-dissipation models for large-eddy simulation

Minimum-dissipation eddy-viscosity models are a class of sub-filter models for large-eddy simulation that give the minimum eddy dissipation required to dissipate the energy of sub-filter scales. A previously derived minimum-dissipation model is the QR model. This model is based on the invariants of the resolved rate-of-strain tensor and has many desirable properties. It appropriately switches off for laminar and transitional flows, has low computational complexity, and is consistent with the exact sub-filter tensor on isotropic grids. However, the QR model proposed in the literature gives insufficient eddy dissipation. It is demonstrated that this can be corrected by increasing the model constant. The corrected QR model gives good results in simulations of decaying grid turbulence on an isotropic grid. On anisotropic grids the QR model is not consistent with the exact sub-filter tensor and requires an approximation of the filter width. It is demonstrated that the results of the QR model on anisotropic grids are primarily determined by the used filter width approximation, and that no approximation gives satisfactory results in simulations of both a temporal mixing layer and turbulent channel flow. A new minimum-dissipation model for anisotropic grids is proposed. This anisotropic minimum-dissipation (AMD) model generalizes the desirable practical and theoretical properties of the QR model to anisotropic grids and does not require an approximation of the filter width. The AMD model is successfully applied in simulations of decaying grid turbulence on an isotropic grid and in simulations of a temporal mixing layer and turbulent channel flow on anisotropic grids.

Fourth-Order DNS of flow past a square cylinder: First results

In this paper we present some initial results of a fourth-order direct numerical simulation of flow past a square cylinder at Re = 22,000. The flow is identical to the second test case which is considered at this workshop. Mean velocities, the mean Strouhal number, the mean drag coefficient Cd, the mean lift coefficient C1 and the rms fluctuations of Cd and C1 are computed.

Direct Numerical Simulation of Driven Cavity Flows

Direct numerical simulations of 2D driven cavity flows have been performed. The simulations exhibit that the flow converges to a periodically oscillating state at Re=11,000, and reveal that the dynamics is chaotic at Re=22,000. The dimension of the attractor and the Kolmogorov entropy have been computed. Explicit time-integration techniques are discussed.

A symmetry-preserving discretisation and regularisation model for compressible flow with application to turbulent channel flow

Most simulation methods for compressible flow attain numerical stability at the cost of swamping the fine turbulent flow structures by artificial dissipation. This article demonstrates that numerical stability can also be attained by preserving conservation laws at the discrete level. A new mathematical explanation of conservation in compressible flow reveals that many conservation properties of convection are due to the skew-symmetry of the convection operator. By preserving this skew-symmetry at the discrete level, a fourth-order accurate collocated symmetry-preserving discretisation with excellent conservation properties is obtained. Also a new symmetry-preserving regularisation subgrid-scale model is proposed. The proposed techniques are assessed in simulations of compressible turbulent channel flow. The symmetry-preserving discretisation for compressible flow has good stability without artificial dissipation and yields acceptable results already on coarse grids. Regularisation does not consistently improve upon no-model results, but often compares favourably with eddy-viscosity models.