
[Sponsors] 
January 4, 2005, 22:41 
Discretisation scheme in CFX1st or 2nd order?

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Can anyone advise me on the discretisation scheme used in CFX? I learn from the theory manual that: advection term: 1 or 2 order, depending on the chosen scheme transient term: 1 or 2 order, depending on the chosen scheme, default=2 order but what about other terms like diffusion etc...Can anyone kind enough to point out for me all the other terms in the eqns & what order do they have in CFX? I'm still not quite sure after reading thru the manual. Thanks,Pete 

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January 5, 2005, 09:41 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi,
Which version of CFX are you using? 

January 5, 2005, 20:22 
Re: Discretisation scheme in CFX1st or 2nd order?

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CFX5.7. Would you mind to let me know the discretisation technique of other terms (like pressure, diffusion) in the governing equations used in CFX & whether it's 1st or 2nd order accurate? I find quite some info about the discretisation scheme of Transient & Convection terms in theory manual but a bit confused about the others. Thanks a lot for your help. Pete


January 6, 2005, 12:05 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi, In the CFX5.7 manual, Solver Theory, Discretisation and Solution Theory, page 234, it's described how the pressure and the diffusion terms are evaluated at the integration points (ip).
The pressure at ip, Pip, is obtained as: Pip=Sum(Nn(ip)*Pn), where Nn(ip) indicates the shape function for node n at ip, and Pn the value of pressure at node n. The summation is over all the nodes of the mesh element. The derivatives for the diffusion terms are also obtained using shape functions: d_phi/d_t=Sum(d_Nn)/d_t(ip)*phin), where d_ indicates a derivative, and phin the value of phi at node n. The summation is also over all nodes of the mesh element. The method for obtaining the kinematic diffusivity (viscosity or thermal conductivity) at the integration points is not described in the documentation, but I suppose it is obtained as the pressure. However I'm not sure of that as Patankar mentions that an harmonic average of the diffusion coefficient is a much better alternative than the arithmetic average (which is equivalent to the use of shape functions). If you want more information about the discretization methods, take a look at the CFXTascFlow Theory Documentation at http://wwwwaterloo.ansys.com/cfxcom...on/default.asp ; and at CFX5 Supplementary Therory at http://wwwwaterloo.ansys.com/cfxcom...oursenotes.htm Regards, Rui 

January 6, 2005, 22:28 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi Rui,
Thanks for your reply. Just one more quick question. When we set blend factor=1 in the advection scheme, is it a 2nd order UDS or CDS? Thanks, Pete 

January 7, 2005, 11:59 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi,
For the blend factor=1, the value of phi at ip, phiip, is given by phiip=phiup+grad(phi)dotR (Discretisation and Solution Theory, page 235). If the value of grad(phi) is obtained with linear shape functions (which as many other things about discretisation is not mentioned in the documentation), then it corresponds to CDS (it is 2nd order accurate). In a paper available at the CXFCommunity site (http://wwwwaterloo.ansys.com/cfxcom...ail.asp?id=329), it's mentioned that "If the blend factor=1, the scheme is a 2nd order upwindbiased scheme". However, for most of the cases you should use the high resolution scheme. By the way, about the transient discretisation, in the CFX5.7 documentation, Discretisation and Solution Theory, the 2 equations at the top of page 231 are the correct ones (rho is inside brackets). The 2 equations at the bottom of page 230 are incorrect. Also on page 230, where is indicated (Eqn. 199), the second one is the correct (rho is inside brackets). This has been corrected in CFX5.7.1 documentation (http://wwwwaterloo.ansys.com/cfxcom...tionTheory.pdf). But if you are dealing with incompressible fluids, it doesn't matter if rho is inside or outside brackets. Regards, Rui 

January 8, 2005, 07:27 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi Rui,
Thanks for the extra details. I only used 2nd order upwindbiased scheme (blend factor =1) because the manual said it's formally 2 order accurate. From my understanding, high resolution scheme doesn't necessary give us 2 order accurate solution. Don't you think the 2nd order upwindbiased scheme should be used whenever possible as the solution will always be 2nd order accurate? Please correct me if I were wrong. Thank you for your great help. 

January 8, 2005, 14:09 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi Pete,
I think you are only partially right. The documentation says: "The choice b = 1 is formally second order accurate, but it may display nonphysical overshoots and undershoots in the solution (see Numerical Dispersion (p. 248))" ; "The High Resolution Scheme computes b locally to be as close to 1 as possible without violating boundedness principles. The high resolution scheme is therefore both accurate and bounded" The high (2nd, 3rd ...) order schemes are known to produce in some cases nonphysical results. If you have access to the book "Numerical Heat Transfer and Fluid Flow" by S. Patankar, take a look at section 5.2.1, where for a simple 1D example, the centraldifference scheme produces unrealistic results. I allways use the highresolution scheme. But I think there isn't the "best scheme", it's case dependent. You may even chose different schemes for different equations. If you chose the highresolution scheme, in CFXPost you can visualize the blend factor used for each equation (for example: Velocity u.Beta). I think you can try the same problem with the highresolution scheme, and with blend factor = 1, and then compare the results. What kind of situation are you trying to model? Transient or steadystate? Regards, Rui 

January 10, 2005, 13:21 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi Rui,
The advection gradient is not obtained by shape functions unless you specifically run a central difference scheme (hidden from the user but used for LES). When you run High Resolution or Blend Factor, the second order correction is is caluclated based on the upwind gradient and is therefore a second order upwind scheme. As for the remaining terms, diffusion terms are calucated based on element shape functions and are thus second order accurate. Pressure is further corrected by appling a fourth order Pressure Correction term to prevent pressurevelocity decoupling (so called Rhie & Chow correction). High Resolution the default scheme and is recommended. It will maintain second order accuracy of gradients, but will reduce the second order correction to zero where there are no gradients to minimize numerical instability (the second order correction is nearly zero in these cases anyway). Also, regardless of what you choose under the general settings, turbulence quantities are advected with a 1st order scheme (advection itself has only a second order effect on turbulence). Hope this helps. Regards, Robin 

January 12, 2005, 13:11 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi,
Thanks again for your help, Robin. You should post in this forum and in the CFXCommunity forum more often. Just one more question: how is obtained the diffusion coefficient (if it varies with temperature, for example) at the integration points? By shape functions? Reagards, Rui 

January 12, 2005, 13:48 
Re: Discretisation scheme in CFX1st or 2nd order?

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Hi Rui,
Fluid properties are evaluated at the node and the Diffusion terms are interpolated to the face centers by shape functions. Regards, Robin P.S. I post whenever I can, usually on the CFX Community Forum. 

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