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We present our approach of adaptive finite elements for laminar
compressible flows. The approach is validated by
the heat-driven cavity benchmark of Paillere and Le Quere. We are
able to determine the Nusselt number up to 5 digits on a single
workstation.
Furthermore, we present first results of an extension of that
benchmark to the three dimensional case. Here the obtained accuracy
without using a parallel machine is expected to be
worse. We comment on local mesh refinement in this 3D case.
An automatic adaptive hybrid Cartesian grid generator is presented together with
solutions of flows around complicated three-dimensional geometries.
The primary computational grid is a Cartesian grid, which
is generated based on an octree-data structure.
A secondary grid may be added for resolving the boundary layer region of
viscous flow around the solid body. It consists of quasi triangular-prismatic
cells generated by marching the body surface triangulation outward.
The unstructured flow solver TAU from the german DLR (Deutsches Zentrum fuer Luft-
und Raumfahrt e.V.) is applied for the compressible
Euler and Navier Stokes equations.
The automatic generation of coarse grids for multigrid smoothing bases on the hierarchical
octree data structure for the Cartesian grid. Whereas for the quasi-prismatic
grid, coarsening profits by its semi-structure.
The hybrid Cartesian grid is improved due to manifold
adaptation features, which are optimized in order to avoid any user interaction.
Therefore, the final grid depends only on the particular geometry and flow structure
(self-organized grid).
The AMR algorithm of Berger and Oliger is the most efficient adaptive method for
hyperbolic conservation laws on blockstructured grids. Instead of refining single
cells a multi-level hierarchy of recursively embedded subgrids is constructed.
Data structures remain regular and the adaptive algorithm only requires an
implementation of a time-explicit finite-volume on a single rectangular grid.
The design of an object-oriented framework in C++ for the Berger-Oliger-method
in arbitrary space dimensions on distributed memory computers (e.g. PC-Clusters)
will be presented. On top of a base level of hierarchical data structures that
capsulate all parallelization details, the mere AMR-solver and its specific
components are formulated. Standard interfaces allow the definition of applications
by simple Fortran functions without any knowledge of AMR. Typical benchmarks for
inviscid Euler equations show a computational performance similar to existing
single processor Fortran codes.
The ability of the parallel AMR method to attack challenging problems in acceptable
real time will be demonstrated by numerical simulations of detonation phenomena in
all space dimensions. In these computations the simplified ZND-model or detailed
reactions mechanisms are coupled to the multi-component Euler equations by a
fractional step method.
The entire package consisting of the parallel AMR-program, examples utilizing
standard flux-vector- and flux-difference-splitting discretizations in one up to
three space dimensions and appropriate visualization tools will be accessible as
public domain software by the end of the year and might be valuable for parts of
the audience.
Numerical simulations of relativistic matter flows in the presence of
strong gravitational fields are a highly complex problem. They involve
the solution of coupled evolution equations for hydrodynamic (matter)
and spacetime (metric) quantities.
We present results from a code which simulates gravitational collapse
of rotating stellar cores in general relativity. The hydrodynamic
equations are formulated in a hyperbolic flux-conservative form and
solved using a high-resolution shock-capturing method. To simplify the
complexity of the metric field equations of general relativity, we use
the CFC approximation; in this scheme, the Einstein equations for the
spacetime reduce to a set of 5 coupled elliptic equations.
Whereas the numerical integration of the hyperbolic system of
hydrodynamic equations is straight-forward and computationally
inexpensive, the metric equations demand a more complicated and
computationally expensive solution scheme. We show how the large
elliptic system arising from the discretized metric equations can be
solved efficiently on a vector computer architecture.
Shallow flows are very common in nature. Examples of hydraulic or environmental engineering interest are wide
rivers, lakes and coastal regions. The shallowness limits the size of the three-dimensional turbulent eddies, but
there is room for coherent two-dimensional structures with length-scales orders of magnitude greater than the depth.
For practical geophysical flows three-dimensional simulations are mostly to expensive. Therefore depth-integrated
models are introduced to obtain more cost-efficient simulations. With statistical turbulence models it is very
difficult to handle these complex turbulent flow structures adequately, especially when large horizontal eddies
dominate the momentum exchange. Using LES makes it possible to resolve these leaving only the small scales to be
modelled.
For two-dimensional depth-averaged LES (DA-LES) the continuity and Navier-Stokes equations are depth-integrated and
filtered. The bottom shear stresses and the total stress which results from combining the depth integration and the
filtering need to be modeled.
The shallow water flow around a cylinder has been investigated by the author using 3D-LES and DA-LES. The LES
results have been verified by comparison with experiments, performed by Carmer and Jirka at the authors institute.
From the 3D-LES results the instantaneous dispersive stresses can be extracted, providing valuable data for
modelling.
We present a posteriori error estimates and adaptive finite volume
schemes
for weakly coupled nonlinear convection-diffusion-reaction system.
The estimates for the error between the exact solution and an implicite
vertex centered upwind finite volume approximation are derived
independent
of the diffusion parameters. The resulting a posteriori error estimate
is used to define a grid adaptive solution algorithm for the finite
volume
scheme. Such systems of convection-diffusion-reaction equations are of
special interest as they model physical motions in which the convective
fluxes dominate the diffusive ones. Such problems occur for example in
fluid
dynamics with high Reynolds numbers, and in density driven or two-phase
flow
problems in porous media. Another rather important application is the
transport of contaminants in subsurface flow models. In this
presentation
we continue our work on a posteriori error estimates for finite volume
approximations of nonlinear conservation laws and convection diffusion
equations, which was started in [1], and [3]. On the other hand we
extend the numerical analysis of weakly coupled systems, as it was
studied
in the hyperbolic framework in [4] and [5]. The presented results will
be published in [6] and [7].
[1] D. Kröner and M. Ohlberger.
A-posteriori error estimates for upwind finite volume schemes for
nonlinear conservation laws in multi dimensions.
Math. Comput., 69:25-39, 2000.
[3] M. Ohlberger.
A posteriori error estimates for vertex centered finite volume
approximations of convection-diffusion-reaction equations.
M2AN Math. Model. Numer. Anal., 35(2):355-387, 2001.
[4] Christian Rohde.
Entropy solutions for weakly coupled hyperbolic systems in several
space dimensions.
Z. Angew. Math. Phys., 49(3):470-499, 1998.
[5] Christian Rohde.
Upwind finite volume schemes for weakly coupled hyperbolic systems
of
conservation laws in 2D.
Numer. Math., 81(1):85-123, 1998.
[6] M. Ohlberger and C. Rohde.
Adaptive finite volume approximations for weakly coupled convection
dominated parabolic systems.
To appear in IMA J. Numer. Anal. (2001).
[7] R. Klöfkorn ,D. Kröner, M. Ohlberger: Local adaptive methods for convection dominated problems. To appear in Internat. J. Numer. Methods Fluids (2001).
The work is focussed towards designing a robust numerical scheme for the
solution of compressible viscous flows on arbitrary complex geometries
in 2D and 3D. The solution to the boundary layer setting for the scalar convection-diffusion equation in 2D is
being considered for the moment. This particular
choice of model problem is motivated by the objective to handle Navier-Stokes
equation in future. The numerical scheme is a cell-centered finite volume
scheme with explicit integration in time.
Accuracy of spatial discretization regardless of grid distortions is a crucial
issue and hence we currently focus our attention on this. A piecewise high-order
polynomial approximates the flow variables in the cell. This process is
referred to as reconstruction. The viscous flux for such flows depends on the
gradient of the solution. Hence, accuracy of reconstruction of gradients influe
nces the accuracy of the viscous discretizations. Design of a scheme which has
a grid independent first-order accuracy demands that the order of the reconstruc
tion polynomial need to be at least two.
An algorithm for the construction of a quadratic polynomial on arbitrary
unstructured grids is developed, which uses a recursive correction approach.
Options available to obtain the reconstruction polynomial and the problems
associated to the corresponding choices in an unstructured grid frame work have
been analyzed. Influence of boundary conditions for these algorithm are being
studied. Consistent quadratic reconstruction regardless of grid-distortions
has been developed.
Our group has investigated a numerical scheme for low Mach number flows and developed
the so called MPV-(Multiple Pressure Variables) code.
This scheme should be applicable for the incompressible, compressible and
higher Mach number flows.
At the starting phase we have implemented the operator splitting method
for the time-integration. But with the Strang-splitting unfortunately one can only achieve
1st-order accuracy in time for incompressible flows, i.e. for M=0.
Therefore we have recently examined some semi-implicit time-integration methods
in order to obtain global higher temporal accuracy independent of Mach number.
In this poster we show and discuss the results of the convergence tests.
In this talk we present a wavelet-based technique for the solution of PDEs.
The method uses anisotropic tensor products of interpolating wavelets as ansatz functions
and finite differences for the discretization of operators.
In this way it combines the advantages of adaptive sparse grid approximation and the flexibility of finite differences.
We show some numerical simulations of two- and three-dimensional shear flows.
The equations of magneto-hydrodynamics (MHD equations) characterize
the behaviour of a conducting, compressible fluid under the influence
of a magnetic field. We consider the so-called `ideal MHD equations',
in which the viscosity and thermal conductivity are neglected and the
electroconductivity is assumed to be infinite.
The Method of Transport is a Riemann-solver free method for systems of
hyperbolic conservation laws. It is based on decoupling the system
using a multi-dimensional wave model. The decoupled system is
linearized and the single linear transport equations are solved by
backtracking characteristics.
In order to get a second order scheme the error that arises from the
decoupling requires the consideration of some correction terms. These
depend on the equation itself and on the chosen wave model. In the
case of the ideal MHD equations, they are so extremely complicated
that the effort in programming them explicitely is not acceptable. In
particular, this effort would have to be repeated if the wave model
was being changed. We now found a possibility to determine the
correction terms implicitely from some intermediate result of the
previous time step. This will simplify the programming of the scheme
substantially. However, as it turns out, in the case of
discontinuities the correction terms will now have to be limited
somehow.
Two different ideas for this limitation process will be
discussed. First, a heuristic factor obtained from the gradient
limitation (which may for example be done using the WENO limiter) can
be used. Second, the correction terms can be disabled specifically
near the discontinuities (in the most simple case, the positions of
the discontinuities are assumed to be known and hardcoded). Both
methods do not yield a satisfying result.
The standard model of type Ia supernovae describes these astrophysical
objects as thermonuclear explosions of white dwarfs. Observations
show that the combustion must start as a deflagration, which later
accelerates due to turbulence. Possible mechanisms for the development
of turbulence are the Rayleigh-Taylor instability and the
Landau-Darrieus instability. The effect of the latter on the behavior
of the flame on small scales is invesigated using a numerical method
with level set description of the flame front and a PPM hydro
solver. In-cell reconstruction of burnt and unburnt states provide the
treatment of the front as a discontinuity. Objectives of the
investigation are to determine the acceleration of the flame and a
possible development of turbulence ("active turbulent combustion") due to the Landau-Darrieus
instability. These effects eventually could lead to a deflagration-to-detonation transition.
Due to the feedback between turbulence, gas expansion and flame front
dynamics a continous acceleration of premixed flames can occur. This
process occurs, e.g., in large scale gas explosions and astrophysical
nova- and supernova explosions. In the context of flame accelerations
and DDT one is faced with rapidly changing thermodynamic, mean flow and
turbulence conditions. One consequence is that the internal structure of
the propagating combustion front will become inherently time dependent.
In addition, the turbulence
intensities associated with the accelerating flow will increase and grow
rapidly beyond the characteristic burning velocity of a laminar flame.
While turbulence intensities are still low, quasi laminar combustion
takes place in thin ``flamelets''. Turbulent combustion modelling will
in this case aim at a description of the net flame surface area and of
the mean quasi-laminar burning velocity in order to arrive at the net
rate of unburnt gas consumption. If, on the other hand, turbulence
intensities increase dramatically, then the turbulence-induced
strains will locally distort the flamelet structures or even quench them
completely and a more stochastic interaction between reaction, turbulent
transport and diffusion becomes significant. As a consequence in these
regimes, the ``thin-rection-zone regime'' and the ``well-stirred
reactor'' regime, very different effective turbulent combustion models
must be employed. Here we present a new numerical technique
which--given such a set of (turbulent) combustion models--allows us to
consistently represent laminar deflagrations, fast
turbulent deflagrations as well as detonation waves. Supplemented with
suitable DDT criteria, the complete evolution of a DDT process can be
implemented in principle.
In the poster presentation we will summarize the numerical
technique, explain some relevant details of the implementation of different
turbulent combustion models and show some recent results on flame
acoustic interactions as well as on 3D propagating flames.
The equations of MHD build a hyperbolic system which
describe flows of a plasma in interaction with a
magnetic field. The hyperbolic properties of the
system are lively discussed due to the formal
existence of non-regular waves, like compound-waves
and overcompressive shocks.
In this work the Rankine-Hugoniot-conditions of MHD
are analysed and an exact Riemann-solver is
constructed by solving the resulting nonlinear
system with Newtons method numerically. The exact
results of the 1d-Riemann-problems are then used to
investigate the convergence behavior of several
finite-volume-schemes (Roe-type, HLLE, central
schemes).
The work places emphasis on the case of almost-planar
initial conditions. Even though in this regime only
regular solutions are possible, the numerical
solution interferes with the non-regular solution of
the completely planar problem. This leads for all
FV-schemes to a kind of pseudo-convergence: Up to a
certain number of grid-points the numerical solution
seems to converge to a solution with typical non-
regular features. The empirical order of convergence
breaks down, indicating that the error does not
decrease with finer grids. But a strong increase of
grid-points reestablishes convergence, the non-regular
features vanish and all schemes converge to the
regular solution.
These results confirm the stability-considerations
of other authors and may have an impact on the
reliability of higher dimensional calculations which
lack high-resolution grids.