Simulation of anisotropic turbulence transport
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Turbulence in engineering problems, in meteorology, or in plasma physics
is mostly anisotropic. For instance the buoyancy and Coriolis forces in
the ocean surpress motion in vertical direction, making the flow two-dimensional
of nature. A typical property of 2D-turbulence is the self organization
of the flow, resulting in large eddies (see Figure 1).
Figure 1 - Vorticity plots, showing the self organization of 2D
turbulence, for the time t = 1 and t = 50 with Reynolds number Re = 3000
Tracer transport in anisotropic turbulence is characterized by a large
scale separation between the size of the turbulent eddies and the filaments
of the tracer material. For solving the tracer concentration even a higher
resolution is required than for the turbulent flow field. Transport can
be studied more efficiently by applying the LDC technique on a problem with
a local dye distribution (see Figure 2).
Figure 2 - Tracer transport of an initial local dye
distribution in the flow field of Figure 1 for t > 50.
Figure 3 - Coarse and fine grid to solve a transport problem with
a high activity in a small part of the domain using LDC
Local Defect Correction (LDC)
Local Defect Correction (LDC) is a method that can be used to efficiently
solve problems whose solutions exhibit large variations in a small part
of the domain. LDC iteratively combines the solution over a global coarse
grid and the solution over one or more local fine grids ; the fine grids
are located where the high activity occurs (see Figure 3).
Figure 5 - Finite differences solution of a 1D Possion equation
improved using LDC
The global grid provides artificial boundary conditions
for the fine grid interface whereas the (more accurate) local solutions
provide an error estimate for the coarse grid discretization: the defect
(see Figure 4). The main advantages of LDC are saving CPU time, since the
grid is fine only where it needs to be, and preserving a simple data structure.
Figure 4 - Scheme of LDC
LDC: an example of application
Figure 5 shows the Finite Differences approximation of a 1D Poisson
equation, whose exact solution is a tanh function, treated with LDC.
LDC considerably improves the initial approximation after a single iteration.
LDC seems to be a promising tool to solve transport problems
in anisotropic turbulent flows.
 S.R. Maassen, PhD thesis, TUE, 2000.
 M.J.H. Anthonissen, PhD thesis, TUE, 2001.