(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)

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).

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.

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).

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Department of Mathematics

University of Groningen