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#21 |
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Lucky
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You can pull the dh out of the time derivative when you apply the temporal discretization. Presenting it this way just obfuscates how, what, and when things are being discretized. You are calling it an "approximation" when really you have invoked a discretization. Just keep it neat.
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#22 |
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Filippo Maria Denaro
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After some further reading of your notes I still do not understand why you do not put directly Eqs.(1) in divergence form and integrate them over Eulerian FVs to solve. That is drho/dt + d/dx(u*rho)=0 du/dt + d/dx(u^2/2-D*du/dx)=0 and then drho_bar/dt + (1/h)[(u*rho)|x+h/2-(u*rho)|x-h/2]=0 du_bar/dt + (1/h)[(u^2/2-D*du/dx)|x+h/2-(u^2/2-D*du/dx)|x-h/2]=0 with the second order approximation rho_bar=rho and u_bar=u. Do you really need to convert in a p-system like?? Why? |
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#23 | |
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Matthew
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nu_t=u_h (nu*u)_t= ((D/nu)*u_h+u^2/2)_h These equations scream out for the finite volume method. The more general equations can be put into these form, so I want to solve the easier equations so I want to understand how to solve these, and the finite volume method seems to be the best method. Thanks to your explanation, I understand the method a lot better, but I need to understand how to apply the boundary conditions correctly. At the momen t, I'm a little confused about how it's done. |
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#24 | |
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Filippo Maria Denaro
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The original is complicated? Are you kidding?? |
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#25 |
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Lucky
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For canonical BCs you simply insert them at the level of (20) & (21) directly as face fluxes on the boundary faces. Then you reconstruct missing values according to your interpolation scheme. Hence it's really important that you write out, entirely, the FVM system without any "approximations" and only apply the discretizations, term by term, when and where they are needed.
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#26 |
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Matthew
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They appear more complicated than first meets the eye. I tied to solve them using the method of lines and matlab's ode solvers, but this failed. I tried to simply do an Euler method, but this failed as well, as I got unphysical results.
I'm at a loss, but this seems to be the best way forward so far. |
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#27 | |
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Matthew
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#28 | |
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Filippo Maria Denaro
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You have the 1d Burgers equations, the continuity equation can be solved later. |
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#29 | |
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Lucky
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Yes I understand you have boundary conditions just like every other person that has ever worked with a boundary value probem before you. BCs are exactly the fluxes you need to apply to (20) and (21), if not, then it is not a BC. At the face where you have u you apply u and at the face where you have u_h, the you apply u_h. You would encounter this exact same issue if you solve advection equation, diffusion equation, wave equation, etc. |
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#30 | |
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Matthew
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You're missing the idea I think, it's not a viscous Burger's equation, it's linked to the other one. The separate when you linearise them, but that's a simple set of equations to solve the issue becomes when they're coupled. |
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#31 | |
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Matthew
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#32 |
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Lucky
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@Filippo, what mat calls complicated is the linearized Cauchy Stress equations not shown here. The motivation is they would like to solve this Berger's equation to learn how to solve slightly more complicated stress transport equation. We are all in agreement that what is shown, is a very simple equation.
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#33 | |
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Lucky
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2) You don't have enough constraints to fully define your boundary and need to go determine what additional boundary conditions you need. Consider in navier-stokes for example that we have no-slip at the wall which is u=0 at y=0. This leaves free v, w, and du, dv, and dw, just what you are asking. Yet another example is heat equation where you apply a heat flux dT/dx=a, this leaves T to be whatever it needs to be. |
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#34 | |
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Matthew
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#35 | |
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Lucky
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It's much simpler than the linearized navier-stokes which itself is much simpler than navier-stokes. At the same time, the fully Cauchy-Stress momentum transport equation itself is a more generalized form of the navier-stokes. I wouldn't say navier-stokes is difficult to solve, thousands of people do it daily. In your case however, there are some gaps in the know-how that you need to address. |
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#36 | |
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Filippo Maria Denaro
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My opinion, as a CFD teacher, is that so far he lost time … |
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#37 | |
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Filippo Maria Denaro
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Are you able to apply in conservative form on the single Burgers equation first using periodic BCs? |
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#38 |
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Matthew
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I can solve the viscous Burgers equation using backwards Euler and Newton raphson method.
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#39 | |
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Filippo Maria Denaro
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Try first the explicit first order accurate upwind method in conservative form for the single viscous Burgers equation. Define a flux(i) on each face and store the convective and diffusive fluxes. Then the update of the velocity and the Bcs |
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#40 | |
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Matthew
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