By Roger Peyret, Egon Krause
This publication collects the lecture notes in regards to the IUTAM institution on complex Turbulent stream Computations held at CISM in Udine September 7–11, 1998. The direction was once meant for scientists, engineers and post-graduate scholars attracted to the applying of complicated numerical strategies for simulating turbulent flows. the subject includes heavily attached major topics: modelling and computation, mesh pionts essential to simulate advanced turbulent flow.
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Extra info for Advanced Turbulent Flow Computations
Let us note u; = uN(x;), the relation between the physical space (the space of grid values u;, i = 0, ... , N) and the spectral space (the space of the coefficients uk, k = 0, ... , N) is characterized by the equations u; uk N = L:ukn(x;), i k=O 1 = o, ... ,N 1 L ::-u;Tk(x;), i=O N = 2_ N Ck C; = 0, ... 148) = eN = 2, ck = 1 for 1 :::; k :::; N - 1. Taking into account the fact that = cos (k cos- 1 x) and X; = cos1rijN, we have Tk(x;) = cos (1rikjN). 148) are discrete cosine Fourier expansions and can be evaluated by FFT.
138) yields N = u~(x;) = L u~ ukT~(x;), i k=O = 0, ... +. 150) G Note that calculations of these coefficients can be subject to round-off errors when N is large. Among the various techniques proposed to reduce the effect of round-off errors, 44 R. Peyret this suggested by Bayliss et al.  seems to be the most efficient as experienced by Sabbah . One source of error identified by Bayliss et al.  is the failure of the above differentiation formulas to exactly represent the derivative of a constant, that is to numerically satisfy the identity N L d~~ = 0' i = 0, ...
113) with ~i = x;). Another way to conceive the finite-volume method is to define a mesh x;, so that the basic unknowns are the grid values u; of the solution u(x;) (vertex-centered method). Then, the equations are integrated in a "control volume" constructed around the point x; and the divergence theorem is applied as above. Therefore, it remains to evaluate the mean value u; in the control volume and the fluxes through its boundary in terms of the neighbouring grid values ui+)• j = -J, ... , J'.
Advanced Turbulent Flow Computations by Roger Peyret, Egon Krause