I've spent an entire day trying to understand Rhie-Chow interpolation from books but to no avail. This set of lectures sealed the deal for me. Thank you Dr. Aidan.
Thank you so much for this series of lectures. You are the first one who explains the Rhie Chow interpolation clearly! Cannot find other lecturers regarding this topic. Finally understand the famous Rhie Chow algorithm! It makes my day!
Thank you for the fantastic series about Rhie and Chow type corrections, really helped a lot understanding what's going on. I'm a little troubled with the use of CFD, because CFD is the full set of numerical methods we use to approximate the Navier-Stokes, among them the finite volume method is one method. There are modern CFD codes utilizing the lattice Boltzmann method or build on finite elements or even use smoothen particle hydrodynamics. Not all CFD methods use the concept of centroids and faces. Apart from that your videos are very instructive and very helpful :)
Probably something like: A.Wimshurst, 'Rhie & Chow Interpolation', Available online: www.youtube.com/..., Accessed 20/06/2022. I think I would go with this, which is a similar format to citing web pages on the internet
In the linear interpolation of equation (1), I think there is a typo. The linear interpolation should be Uf = lx UE + (1-lx) Up. In other words, in linear interpolation, the closest point (P) to the face f must have higher weight in the linear average than the farest point (E)..... Or you can define lx=1-d1/d2, then the written interpolation should be ok. Anyway, all the tutorial is brilliant! Thanks in advance
Related to the Brandon Lobo's comment...I think there is a mistake in Ix definition...Ix should be (distance from f to E)/(distance from P to E) instead of (distance from f to P)/(distance from P to E)...just the lever rule...With the formula that you provide if Ix=0 (f and P coincide) Uf would equal to UE instead to UP. Thanks for your excellent series of videos
Thanks for the lecture. The correction term in the given derivation seems quite tricky since it will be canceled out if a same linear interpolation scheme is used for the first term in the parenthesis in Eq (23), as for the second term. This formula is also prone to the mesh non-orthogonality. Regards.
Yep, the orthogonality issue is very tricky. I would really recommend the paper by Bartholomew et al. (And read it very carefully) if you would like to know more about this one
It is probably too early and I am making a mistake here, but around time 8:15. If face f is very close to P, then distance 1 is very small compared to distance 2. So the ratio of distance1/distance2 (small/big) is going to be small and tending to zero and Uf tends to Up. I agree up to this point. But from your equation for Uf, Uf = lx*Up + (1 - lx)Ue. So for lx tending to zero Uf = Ue. So, the formulation for Uf is then wrong? No biggie just wanted to point it out. EDIT: I just completed watching the video and thanks a lot for the quick tutorial. I have problems with convergence due to a checkerboard effect even though the Rhie-Chow method is implemented. The issue is with the time-step size being really small for high Re flows (wall-resolved LES around an airfoil). I am currently reading into the issue and methods to make it time-step independent.
24:51 Seems like Equation (21) is insufficient to derive Equation (22) from Equation (20). How did you factorise out the d in the third RHS term of Equation (20)? Because bar(d*dp/dx) is not equal to bar(d)*bar(dp/dx).
This is an approximation that is commonly paid. It is not exactly correct. However, it is ok because the Rhie and Chow correction = 0 when the simulation converges, so this approximation only affects the speed of convergence
I'm curious how you do it in boundary conditions. If, in a rectangular mesh, the left, lower ,right and upper are all surrounded by walls, U left = U bottom = U right = 0 and When U up = 1m/s, Can I put that boundary value on the U face as it is?? Without interpolation?
You don't need to use Rhie and Chow on the boundary faces, only the interior faces ☺️ you can use regular extrapolation here (if Nemann BC) or the values are known already (if Dirichlet)
Hey Aidan, I've been following your excellent lectures, I wish there were something nearly as good as your courses during my master's. Unfortunately, I am still struggling with a step in this derivation. When Rhie and Chow's correction is calculated, as you said, velocities at points W, P, and E are knowns from the momentum equation. However, we need the pressure gradient to ensure the velocities satisfy the continuity restriction. So, how are UW, UP, and UE already defined despite the pressure gradient not being yet determined since we do not know the face velocities needed for the continuity (that is pressure) equation?
Great question. This is a difficult one to answer concisely in the comment section, so I will try my best! The continuity equation states that: Sum rhof( Uf dot nf) Af = 0 (Where f is a subscript to indicate a quantity at the face centre, nf is the unit normal vector and Af is the face area) Substitute in the Rhie and Chow correction: Sum rhof ( pressure gradient dot nf) Af - Sum rhof (interpolated pressure gradient dot nf) Af = Sum rhof (interpolated velocity at face centre dot nf) Af This can be rearranged into the pressure equation. The velocity on the right hand side (the interpolated solution of the momentum equation) becomes the source term, and the left hand side (the solution) is the pressure field which satisfies the Rhie and Chow correction and would correct the velocity field so that it satisfies the continuity equation. So we are solving for a pressure field which corrects the velocity field AND satisfies the Rhie and Chow correction. There are actually many different velocity fields that (for a given cell) satisfy the continuity equation, so we need to pick the right one. The Rhie and Chow correction allows us to do this by constraining the choice This is probably the best explanation I can give in the comments section 😅 i would really recommend checking out my 'SIMPLE algorithm course' where I do the full derivation in lots of detail. It should help clear up any remaining questions you might have.
@Fluid Mechanics 101 I've seen the SIMPLE video aleeady, another great lesson. So, the conclusion I draw from your explanation is that actually, the point W, P, and E's velocities are not the correct ones already, is that right? These knowns satisfy the momentum balance but do not have any commitment with the continuity restriction at this stage. If it is correct, then Rhie and Chow are inplemented in the correction part of a prediction-correction algorithm, reaching 0 at the convergence. If the above conclusion is right, so we can substitute interpolation schemes like Upwind for the Rhie and Chow's correction. Thank you tremendously for your attention and answers.
Both are types of interpolation to calculate the velocity at the face centre from the centroids on either side of the face. Distance based is the same as standard linear interpolation. Rhie and Chow includes an additional correction on the interpolation to try and reduce /avoid chequerboard oscillations
Just a small clarification. Because of all this notation, I lost the bigger picture. I want to take the example of the SIMPLE algorithm. There we solve momentum equation first (so we have velocity at the centroids). Now we wish to enforce continuity to get the pressure correction. To do this we need fluxes for which we require velocity at the cell faces. This is where the checkerboard oscillations occur. Is this correct? So can I make a generic statement and say that checkerboard oscillations are a consequence of the non-linearity of the equation?
Also, since the checkerboard effect is a consequence of discretisation (?), won't it (to an extent) improve with refinement (at least until mesh convergence is achieved)?
Hello, in answer to your questions: 1) your description of the SIMPLE Algorithm is correct 2) The oscillations occur because of the 'pressure-velocity coupling' that is inherent in the N-S equations when they are solved in a discrete form, not the non linearity. The nonlinearity arises from the advection term in the N-S equations, which is unrelated to the pressure gradient. It is probably more correct to say that the pressure in a given cell can become uncoupled from the pressure in the cell of its immediate neighbours, depending on the interpolation adopted in the numerical algorithm. One way of correcting this is to correct the interpolation of the face mass flow rate carefully, so that strong neighbour-neigbour coupling is introduced
With regards to your second question, the decoupling happens between immediate neighbour cells. If you refine the grid, the oscillations still occur between neighbour cells, the cells are just closer together now!! This is a very hard topic to understand, I hope this makes some sense
![[CFD] Rhie & Chow Interpolation (Part 1): Chequerboard Oscillations](http://i.ytimg.com/vi/yqZ59Xn_aF8/mqdefault.jpg)
![[CFD] Rhie & Chow Interpolation (Part 1): Chequerboard Oscillations](/img/tr.png)
![[CFD] Relaxation in CFD (Part 1) - Explicit Relaxation, Under-Relaxation Factor](http://i.ytimg.com/vi/GSsv2ncNJN8/mqdefault.jpg)
![[CFD] Eddy Viscosity Models for RANS and LES](http://i.ytimg.com/vi/SVYXNICeNWA/mqdefault.jpg)
I've spent an entire day trying to understand Rhie-Chow interpolation from books but to no avail. This set of lectures sealed the deal for me. Thank you Dr. Aidan.
Thank you so much for this series of lectures. You are the first one who explains the Rhie Chow interpolation clearly! Cannot find other lecturers regarding this topic. Finally understand the famous Rhie Chow algorithm! It makes my day!
Thanks Dr. Aidan, definitely useful and helpful talk as usual, you are doing a great work, we are very grateful, thank you very much.
Thank you for the fantastic series about Rhie and Chow type corrections, really helped a lot understanding what's going on.
I'm a little troubled with the use of CFD, because CFD is the full set of numerical methods we use to approximate the Navier-Stokes, among them the finite volume method is one method. There are modern CFD codes utilizing the lattice Boltzmann method or build on finite elements or even use smoothen particle hydrodynamics. Not all CFD methods use the concept of centroids and faces. Apart from that your videos are very instructive and very helpful :)
Fantastic, thank you Aidan, this whole series on Rhie & Chow interpolation has been very useful for my thesis.
Fantastic! I'm glad it helps
@@fluidmechanics101 If I wanted to cite your videos what would be the best way to go about it?
Probably something like:
A.Wimshurst, 'Rhie & Chow Interpolation', Available online: www.youtube.com/..., Accessed 20/06/2022.
I think I would go with this, which is a similar format to citing web pages on the internet
In the linear interpolation of equation (1), I think there is a typo. The linear interpolation should be Uf = lx UE + (1-lx) Up. In other words, in linear interpolation, the closest point (P) to the face f must have higher weight in the linear average than the farest point (E)..... Or you can define lx=1-d1/d2, then the written interpolation should be ok. Anyway, all the tutorial is brilliant! Thanks in advance
Yes, there was a typo here. The overall approach should still be fine. Well spotted!
excellent and extremely valuable lecture!
Really good, great visual representation
Related to the Brandon Lobo's comment...I think there is a mistake in Ix definition...Ix should be (distance from f to E)/(distance from P to E) instead of (distance from f to P)/(distance from P to E)...just the lever rule...With the formula that you provide if Ix=0 (f and P coincide) Uf would equal to UE instead to UP. Thanks for your excellent series of videos
Yep, I think you are correct 👍
Thanks for the lecture. The correction term in the given derivation seems quite tricky since it will be canceled out if a same linear interpolation scheme is used for the first term in the parenthesis in Eq (23), as for the second term. This formula is also prone to the mesh non-orthogonality. Regards.
Yep, the orthogonality issue is very tricky. I would really recommend the paper by Bartholomew et al. (And read it very carefully) if you would like to know more about this one
Really well explained! Thank you!
Thanks, Dr. Aiden.
Is the correction applicable to interior faces only, or do we have to apply the correction for boundary faces as well?
25:11 I don't think the following holds mathematically, unless some approximation is considered:
(d_f * gradP_f)_bar = d_f_bar * gradP_f_bar (??)
Excellent sir...
Thanks you so much Sir for the efforts. Very good content on the topic.
It is probably too early and I am making a mistake here, but around time 8:15. If face f is very close to P, then distance 1 is very small compared to distance 2. So the ratio of distance1/distance2 (small/big) is going to be small and tending to zero and Uf tends to Up. I agree up to this point. But from your equation for Uf, Uf = lx*Up + (1 - lx)Ue. So for lx tending to zero Uf = Ue. So, the formulation for Uf is then wrong? No biggie just wanted to point it out.
EDIT: I just completed watching the video and thanks a lot for the quick tutorial. I have problems with convergence due to a checkerboard effect even though the Rhie-Chow method is implemented. The issue is with the time-step size being really small for high Re flows (wall-resolved LES around an airfoil). I am currently reading into the issue and methods to make it time-step independent.
Yes, that sounds more like an issue with your time step (and maybe spatial discretisation schemes becoming unbounded) for LES, rather than Rhie Chow
24:51 Seems like Equation (21) is insufficient to derive Equation (22) from Equation (20). How did you factorise out the d in the third RHS term of Equation (20)? Because bar(d*dp/dx) is not equal to bar(d)*bar(dp/dx).
This is an approximation that is commonly paid. It is not exactly correct. However, it is ok because the Rhie and Chow correction = 0 when the simulation converges, so this approximation only affects the speed of convergence
@@fluidmechanics101 I see, thanks for the prompt response!
really good
I'm curious how you do it in boundary conditions.
If, in a rectangular mesh, the left, lower ,right and upper are all surrounded by walls,
U left = U bottom = U right = 0 and
When U up = 1m/s,
Can I put that boundary value on the U face as it is?? Without interpolation?
You don't need to use Rhie and Chow on the boundary faces, only the interior faces ☺️ you can use regular extrapolation here (if Nemann BC) or the values are known already (if Dirichlet)
Hey Aidan, I've been following your excellent lectures, I wish there were something nearly as good as your courses during my master's.
Unfortunately, I am still struggling with a step in this derivation. When Rhie and Chow's correction is calculated, as you said, velocities at points W, P, and E are knowns from the momentum equation. However, we need the pressure gradient to ensure the velocities satisfy the continuity restriction.
So, how are UW, UP, and UE already defined despite the pressure gradient not being yet determined since we do not know the face velocities needed for the continuity (that is pressure) equation?
Great question. This is a difficult one to answer concisely in the comment section, so I will try my best!
The continuity equation states that:
Sum rhof( Uf dot nf) Af = 0
(Where f is a subscript to indicate a quantity at the face centre, nf is the unit normal vector and Af is the face area)
Substitute in the Rhie and Chow correction:
Sum rhof ( pressure gradient dot nf) Af - Sum rhof (interpolated pressure gradient dot nf) Af = Sum rhof (interpolated velocity at face centre dot nf) Af
This can be rearranged into the pressure equation. The velocity on the right hand side (the interpolated solution of the momentum equation) becomes the source term, and the left hand side (the solution) is the pressure field which satisfies the Rhie and Chow correction and would correct the velocity field so that it satisfies the continuity equation.
So we are solving for a pressure field which corrects the velocity field AND satisfies the Rhie and Chow correction. There are actually many different velocity fields that (for a given cell) satisfy the continuity equation, so we need to pick the right one. The Rhie and Chow correction allows us to do this by constraining the choice
This is probably the best explanation I can give in the comments section 😅 i would really recommend checking out my 'SIMPLE algorithm course' where I do the full derivation in lots of detail. It should help clear up any remaining questions you might have.
@Fluid Mechanics 101 I've seen the SIMPLE video aleeady, another great lesson.
So, the conclusion I draw from your explanation is that actually, the point W, P, and E's velocities are not the correct ones already, is that right? These knowns satisfy the momentum balance but do not have any commitment with the continuity restriction at this stage.
If it is correct, then Rhie and Chow are inplemented in the correction part of a prediction-correction algorithm, reaching 0 at the convergence.
If the above conclusion is right, so we can substitute interpolation schemes like Upwind for the Rhie and Chow's correction.
Thank you tremendously for your attention and answers.
very interesting and usefull. Thank you very much.
Help full
what is Rhie chow momentum based and distanced based approach?
Both are types of interpolation to calculate the velocity at the face centre from the centroids on either side of the face. Distance based is the same as standard linear interpolation. Rhie and Chow includes an additional correction on the interpolation to try and reduce /avoid chequerboard oscillations
I have a big question: aP in equation 5 can be 0 if the velocity is 0 right? In that case how do you handle it when you do 1/aP
Just a small clarification. Because of all this notation, I lost the bigger picture. I want to take the example of the SIMPLE algorithm. There we solve momentum equation first (so we have velocity at the centroids). Now we wish to enforce continuity to get the pressure correction. To do this we need fluxes for which we require velocity at the cell faces. This is where the checkerboard oscillations occur. Is this correct? So can I make a generic statement and say that checkerboard oscillations are a consequence of the non-linearity of the equation?
Also, since the checkerboard effect is a consequence of discretisation (?), won't it (to an extent) improve with refinement (at least until mesh convergence is achieved)?
Hello, in answer to your questions: 1) your description of the SIMPLE Algorithm is correct 2) The oscillations occur because of the 'pressure-velocity coupling' that is inherent in the N-S equations when they are solved in a discrete form, not the non linearity. The nonlinearity arises from the advection term in the N-S equations, which is unrelated to the pressure gradient. It is probably more correct to say that the pressure in a given cell can become uncoupled from the pressure in the cell of its immediate neighbours, depending on the interpolation adopted in the numerical algorithm. One way of correcting this is to correct the interpolation of the face mass flow rate carefully, so that strong neighbour-neigbour coupling is introduced
With regards to your second question, the decoupling happens between immediate neighbour cells. If you refine the grid, the oscillations still occur between neighbour cells, the cells are just closer together now!! This is a very hard topic to understand, I hope this makes some sense
Amazing.
thanks
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