# Direct Numerical Simulation of Turbulent Flow

### General description of DNS research

The project aims to use direct numerical simulations (DNS) and their mathematical analysis as a tool for turbulence research. Its long term goal concentrates around enlarging our understanding of turbulence and transition in physical terms (e.g. energy transport, coherent structures) and in mathematical terms (e.g. bifurcation scenarios, proper orthogonal decomposition, attractor structure). Ultimately it should lead to improved turbulence models for industrial simulation methods based on LES or Reynolds-averaged Navier-Stokes. Our first steps on the long and challenging road towards this goal have concentrated on two aspects: (i) the development of numerical methods for DNS; and (ii) the analysis of DNS data.

(i) Development of numerical methods for DNS Numerically reliable solutions of the unsteady, incompressible Navier-Stokes equations are to be computed using grids that are sufficiently fine to resolve all scales of motion. In view of the computational complexity of DNS our first concern has been to push the algorithmic efficiency as far as we can get. Hereto we have concentrated on higher-order space-discretization methods, long-term time-integration methods and sparse-matrix solvers. As a result, since the start of our project in 1990 the efficiency of our numerical tools has increased by two orders of magnitude. In conjunction with the progress made in computer hardware, 3D DNS at the lower end of the industrial Reynolds numbers range has now become within reach. (PhD thesis Wissink 1995)

(ii) Analysis of DNS data The mathematical analysis of the DNS results is performed by using tools from non-linear dynamical systems theory, such as time-series analysis and proper orthogonal decomposition (POD). In the latter technique, from the unsteady DNS flow field a set of basis functions is extracted, which is optimal in the sense that any other decomposition (e.g. in Fourier modes) captures less kinetic energy. The Galerkin projection of the Navier-Stokes equations on the space spanned by the basis functions yields a low-dimensional set of ordinary differential equations which mimics the dynamical behaviour of the real flow. With this computationally `cheap' system bifurcation analyses have been performed, which shed more light on the mechanism of laminar-turbulent transition. (PhD thesis Cazemier 1997)

### Comparison between DNS, LES and experiment

At a Reynolds number of 10,000 the turbulent flow in a driven cavity with spanwise aspect ratio 0.5 has been studied by means of DNS, LES and experiment. The figure shows a comparison of the Reynolds stress component u'v' along the horizontal center line of the cross-sectional 'symmetry' plane.

The DNS results have been obtained on a 50x50x50 grid using fourth-order space discretization (Verstappen, 1995)

The LES results have been obtained on a 64x64x32 grid using a dynamic mixed model (Zang, Street and Koseff, 1993).

### Higher-order vs. lower-order discretization

As an example of the power of higher-order discretization, we present a comparison between a second-order approach and our fourth-order approach for flow in a 3D driven cavity at Re=10000. Experimental results are available for comparison (Prasad and Koseff, 1989). The figure below shows the vertical velocity along the central axis of the cavity as obtained from a second-order calculation on a 100x100x100 grid, a fourth-order calculation on a 50x50x50 grid and the experimental results. In this example the fourth-order results are clearly superior to the second-order results, whereas their computational effort is more than an order of magnitude less. Essential to this succes is the way in which the discretization of the convective terms copes with the stretching of the computational grid.

### DNS versus experiment for external flow

In a recent workshop on LES and DNS that was held in Grenoble (16-18 sept. 1996) the simulation of an external flow around a square cylinder at Re=22,000 was at stand. Below we show a close-up of the instantaneous flow near the cylinder (as visualized by Wim de Leeuw at CWI).

Fig. 3: Mean stream-wise velocity for flow past a square cylinder at Re=22,000: DNS versus experiment. The continuous lines correspond to the DNS; the experimental data is depicted by the dots.

### DNS versus RaNS: flow past an array of surface mounted onbstacles

Next we show an example of a turbulent flow that has been the subject of the 6th ERCOFTAC Workshop on Refined Flow Modelling held in Delft in June 1997. It concerns the fully developed flow in a channel where an array of 25x10 cubes is placed regularly at the bottom. The Reynolds number based on the height of the channel and the bulk velocity is 13,000. Experimental data for this flow are available (Meinders 1997). A periodic part of the flow domain has been covered by a slightly stretched 100x100x100 grid. The first mesh point is spaced 0.006 units away from a wall; a cube is represented by 40 grid points in each direction.

Fig. 4: Side-view of a sub-channel unit showing an instantaneous flow field in a plane through the centre of the cubes; the flow is directed from left to right (left-hand picture). Large structures of recirculating flow behind the obstacles are not present in any of the snapshots of this flow. They can only be observed if the flow is averaged over a long period in time. The right-hand picture shows the average streamwise velocity (continuous lines) in comparison with experiment (dots).

Mean velocity profiles and Reynolds stresses have been computed. Figure 4 shows a comparison with experimental data in the vertical symmetry plane parallel to the streamwise direction that bisects the cubes. The sampling time of data for the computation of the first- and second-order statistics of the flow started after a transitional period of 100 (nondimensional) time units. The time-averages were computed over 200 time units; samples were taken each time step. The entire computation (including start-up and sampling time) took about 100 hours on one vector-processor of a CRAY C90.

Figure 5: the fluctuating streamwise velocity $\overline{u'u'}$ (solid line), demonstrating a quite good agreement with experiment (dots). In particular, the sharp peak just on top of the cube is accurately resolved.

In order to challenge RaNS computations, we have also performed a simulation at a 60x60x60 grid. On this grid, a cube is represented by 30 grid points in each direction. The first grid point of the 60^3 grid lies approximately twice as far from the wall than the first grid point of the 100^3 grid. The coarse grid simulation takes about one tenth of the CPU-time of the fine grid simulation, among other things because the time step can be increased by a factor of about two.
There were four groups who presented results of their RaNS computations of the flow in the channel with surface-mounted cubes at the workshop (Hanjali\'c and Obi 1997). Here, we restrict ourselves to the RaNS computation (of Dr. S. Jansson from the Department of Fluid Mechanics of Vattenfall Entvecklung AB in Sweden) that agreed the best with the available experimental data. It may be noted that there were no results of large-eddy simulations submitted to this workshop, nor have there been any reported elsewhere.
The best RaNS result was based on a second-order, cell-centered, finite-volume method. The QUICK scheme was used for the velocities and the turbulent quantities were integrated by means of a second-order accurate scheme with a Van Leer limiter. Periodic boundary conditions were applied in the streamwise direction; the period was taken equal to four cube lengths. Symmetry conditions were applied in the spanwise directions. The spanwise boundaries were taken two cube lengths apart. Dirichlet conditions were applied at the solid walls, except for the dissipation rate $\epsilon$: the normal derivative of $\epsilon$ was put to zero at solid walls. The turbulence model consisted of a two-layer eddy-viscosity combined with a standard $k-\epsilon$ model. The transport equation for the dissipation rate $\epsilon$ was not solved in near-wall regions, but instead it was computed explicitly from a predicted length scale. This RaNS computation was performed on a stretched, orthogonal grid of 67x72x57 points in the streamwise, the normal and the spanwise direction, respectively. Except for the spanwise direction, the grid spacing of this RaNS computation is slightly finer than that of our 60^3 simulation. In the spanwise direction, the average resolution of the RANS computation is about two times finer than the average resolution of the 60^3 grid, due to the fact that the RaNS computation uses symmetry conditions and therefore can restrict its spanwise computational domain to two cube lengths, while in our simulation periodic boundary conditions have been applied in the spanwise direction with a period of four cube lengths.

On a corresponding grid, the mean velocities computed from the Reynolds averaged Navier-Stokes equations agree less with the experimental data than the results of the symmetry-preserving simulation. The velocity profiles of the RaNS computation are much too smooth. In addition, the maxima of the velocities are located in the symmetry-plane between two cubes, which is in distinct disagreement with the experimental data. Finally, it may be observed that the convergence of the symmetry-preserving simulation upon grid refinement is plain: the results on the 100^3 grid are closer to the measurements than those of the 60^3 grid.

Fig. 6: Comparison of the mean streamwise velocity at half cube height. The flow comes from below; the horizontal corresponds to the spanwise direction. The dashed vertical lines are lines of symmetry; their distance is two cube lengths. The lowermost profiles are located at 0.3 cube lengths before the front of the cube, the middlemost profiles at 0.3 cube lengths after the front an the uppermost profiles at one cube length after the middlemost. The velocity scale is shown for the uppermost profiles only. The experimental data is taken from Meinders (1997); the RaNS data is taken from Hanjali'c and Obi (1997).

### Local grid refinement

In the project Direct Numerical Simulation of Complex Turbulent Flow, sponsored by NWO Computational Science, the symmetry-preserving discretization method is combined with local grid refinement.
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