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February 4, 2012, 18:44
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UDS for compressible flow
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Senior Member
Join Date: Jun 2010
Posts: 111
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Hi, Quick question on Upwind differencing scheme in finite volume, for compressible flows, where density and velocity both vary.
I know that when density and velocity are constants, then you would have: ![\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\rho u\frac{\phi_{i}-\phi_{i-1}}{x_{i}-x_{i-1}}](/Forums/vbLatex/img/4dea65fae0b049d861a737315d62d7ce-1.gif "\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\rho u\frac{\phi_{i}-\phi_{i-1}}{x_{i}-x_{i-1}}") if  and ![\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\rho u\frac{\phi_{i+1}-\phi_{i}}{x_{i+1}-x_{i}}](/Forums/vbLatex/img/522315f5a38a2d916dca81fabbf38452-1.gif "\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\rho u\frac{\phi_{i+1}-\phi_{i}}{x_{i+1}-x_{i}}") if  Can someone please tell me what happens when density and velocity are variables. Would I get something like this? ![\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\frac{\left(\rho u\phi\right)_{i}-\left(\rho u\phi\right)_{i-1}}{x_{i}-x_{i-1}}](/Forums/vbLatex/img/d42661931830986523ba1d912f54c936-1.gif "\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\frac{\left(\rho u\phi\right)_{i}-\left(\rho u\phi\right)_{i-1}}{x_{i}-x_{i-1}}") if ![\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\frac{\left(\rho u\phi\right)_{i+1}-\left(\rho u\phi\right)_{i}}{x_{i+1}-x_{i}}](/Forums/vbLatex/img/e13cdb6e06f367c18ce95ec7f45366d7-1.gif "\left[\frac{\partial(\rho u\phi)}{\partial x}\right]_{i}=\frac{\left(\rho u\phi\right)_{i+1}-\left(\rho u\phi\right)_{i}}{x_{i+1}-x_{i}}") if I'm guessing the same principle applies even if density is constant and velocity varies. I didn't manage to find anything useful online. Please help! Thanks! Last edited by Hooman; February 5, 2012 at 04:51. |
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