# Finite Volume Method

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 April 9, 2016, 14:18 Finite Volume Method #1 Member   Join Date: Apr 2016 Posts: 30 Rep Power: 9 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 #2 Senior Member   Michael Prinkey Join Date: Mar 2009 Location: Pittsburgh PA Posts: 363 Rep Power: 24 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 cell-averaged 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. First-order 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 plain-old Talyor series expansions. Errors increase as you make FV larger. The rate of increase is governed by the order of the method. First-order schemes scale error linearly. Second-order sceheme scale error quadratically, etc.

April 11, 2016, 04:41
#3
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Filippo Maria Denaro
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Quote:
 Originally Posted by shanvach 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?

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 #4 Member   sajad Join Date: Apr 2014 Location: Iran Posts: 46 Rep Power: 11 Does upwind scheme cause problem for Euler equation?

April 11, 2016, 04:54
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Filippo Maria Denaro
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Quote:
 Originally Posted by sajad6 Does upwind scheme cause problem for Euler equation?

For Euler equations the upwind discretization is used, often for the characteristic-based variables. You can find details in the book of Leveque

May 21, 2016, 02:28
#6
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LUQILIN
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Quote:
 Originally Posted by shanvach 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?
1. It depends what problem you are dealing with. Generally 1st order scheme is too dissipate. But you should try it first to totally understand its meaning. Just use the center value of element then you can get a 2nd order scheme. Do a accuracy test after you finish your code and make sure 2nd order is achieved.
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.

 Tags discretization, function, fvm, node, size