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April 9, 2016, 14:18 
Finite Volume Method

#1 
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Hi,
Could someone help with a little problem that i have regarding how FVM is applied in CFD. After discretization,we assume a general function that can be first order upwind,second order upwind etc.How to decide which kind of function to take? And also what are the effects of larger cell size in FVM? 

April 10, 2016, 08:43 

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Michael Prinkey
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First order upwind assumes that the solution is piecewise constant over the cells...the value at the each face is the same as value at the centroid. Second order upwind assumes a linear expansion with the centroid value being the cellaveraged value and the linear terms as the gradient of the function dotted with the vector connecting the cell centroid to the test location:
phi(x) = phi_c + grad(phi) . (x  x_c) Formal order of accuracy is important, but really, the use of higher order schemes, especially for advection/convection terms is governed by artificial diffusion. Firstorder upwind schemes are very diffusive, so you lose details very quickly in practical simulations. Errors scale in FVM as they do in finite difference, finite element, or even plainold Talyor series expansions. Errors increase as you make FV larger. The rate of increase is governed by the order of the method. Firstorder schemes scale error linearly. Secondorder sceheme scale error quadratically, etc. 

April 11, 2016, 04:41 

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Filippo Maria Denaro
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Quote:
The FV is based on the integral form of the equation. The function you are referring is the numerical flux function. Upwind (first, second or higher) flux has intrinsically numerical viscosity (mprinkey already wrote) while central flux discretizations have not. What you use depends on the flow case and formulation. DNS/LES use discretization without numerical viscosity conversely RANS use also upwind. 

April 11, 2016, 04:48 
upwind scheme for Euler equation

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sajad
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Does upwind scheme cause problem for Euler equation?


April 11, 2016, 04:54 

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Filippo Maria Denaro
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May 21, 2016, 02:28 

#6  
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LUQILIN
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Quote:
What you say about upwind is the way to compute flux at cell interface when discrete convention terms. 2. You can test the effects of cell size yourself. But if time integration is an explicit scheme, the cell size is not independent, i.e., associated with time steps. Check CFL number and ensure that your scheme is stable. 

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discretization, function, fvm, node, size 
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