Thanks for your videos on CFD. Just for this video I think at minute 27:53 you should say that for the interpolation of the pressure at the w and e faces is Pw=(PW+PP)/2 and PE=(PP+PE)/2 and not Pw=(PW-PP)/2 and PE=(PP-PE)/2.
Thanks a lot for covering this topic in your talks, your content and delivery style helps me understand difficult concepts quite easily. Looking forward to remaining two part.
This was the topic I always had some problems with its understanding. I really appreciate your clear and detailed explanations. By the way, I am looking forward to watching the video on the rest of LES method. Thanks an ocean for your great videos.
4:45 I am not really sure why ideal gas law can not be used to compute the pressure field. The reasoning you give is that the change in density is small (are you referring to spatial change in density or temporal change in density or both?). Regardless, why would the numerical value of change in density make the ideal gas law inapplicable P =rho*r*T, for a particular flow regime? If the change in density is small or even zero, why would it matter in a numerical algorithm? I don't see a division by zero in the equations. I just can't seem to understand this reasoning, from a mathematical point of view. Could you please clarify?
@@fluidmechanics101 Thanks! I see your point and that is kind of where I am getting at too - the numerical algorithm has to be general for all fluids. We should never be using ideal gas law, except for ideal gases. We have the unknowns ( ho, ho u, ho v, ho w, p) but only four equations (1 continuity and 3 momentum). What is the most general equation that relates pressure-density-momenta and applies to all fluids (gas, air, water) and all flow regimes (compressible or incompressible)?
Generally in a pressure based solver, this relationship can be anything of the form: rho = function (variables). Rho is calculated separately which leaves the unknowns in the momentum equation as u, v and w (leaving rho as a constant). Pressure is then calculated from a pressure correction equation. With this approach we can account for all constitutive relationships by leaving them as a variable for the user to define
Am I correct when assuming, if you would have selected an upwind discretization on the face velocities (Uf) the chequerboard oscillations would not occur: even if you do use central difference for pressure at the face?
I haven't come across people using upwind for the face velocities before. I think it may be because we already know the velocities at the cell centroids, so we are doing an explicit calculation of the face velocities (Rhie Chow is not used to discretise the momentum equations, it is used afterwards so we don't need an implicit scheme here). But your point is still valid, I am not sure why they haven't tried upwind
Hi Aidan, I wanted to know more about the formulation of matrix ap & an from final discretised Momentum Eq [Eq-19], 21:24 . Have you gone through this on CFD Fundamentals course - 3 on udemy? I would appreciate any sources.
I haven't been through the coefficients for the full Navier Stokes system yet. It's quite difficult! Fundamentals 3 on Udemy is still dealing with the heat conduction equation but in 3D on an unstructured mesh. I am hoping to get the full NS system sorted soon and then I can start teaching that one as well. I have mainly been using the famous book by Patankar but I will check out that book. Thanks for the reference ☺️
@Fluid Mechanics 101 Thank you for the video. I encoded a 2d laminar cfd solver myself, and at the very beginnig I had hard times defining the right staggered and scalar grids and also where to place the phsical domain in between those grids. Could you please record a video about it as well?
Yes, this is the tricky part of writing the staggered grid solvers. Have you checked your pressure correction equation and interpolation of velocities and pressures onto the right grids? This is typically the difficult bit and I haven't go around to right a video for this yet
@@fluidmechanics101Eventually I could manage that by placing the physical domain not to the edges but to just a half way inside of the scalar /main/grids. So the unit cells’ boundaries of staggered grids perfectly matched with the physical domain boundaries. It was tricky at first but then when applying the pressure correction to the scalar cell I appreciated how it was easily adapted since no interpolation needed as u also said. The weird part in the staggered grid was that I needed to solve momentum equation over u- and v- control volumes individually while I solved the pressure correction equation ( mass conservation) only one time over the scalar control volume. And indices was also tricky as you indicate because for any (i,j) location there are three different cell centers and as I named them (I,J) for scalar cell center; (I,j) for u- cell location of backward staggered of (I,J) cell; (i,J) for v- cell center of backward staggered of (I,J) cell. And the sad part is that my own code solves the equations slower than ANSYS Fluent :) Fluent solves the equations faster and converges much earlier. I guess I should apply the TDMA method for the solution of momentum equations instead of Under relaxed Gauss Seidel. I have also some suspicion about if the pressure correction equation converges in the iteration step. I use SOR method for the pressure correction. I still need to go further and dig into works and books :)
Thank you for your all amazing videos. Would you please describe more about the flows at moderate mach numbers-flows, somehow between the incompressible and compressible ranges. This caregory may happen through unstable flow regimes in tirbomachined i.e. axial fans or axial compressors near the stall conditions. Regards
Thank you very much for the video and as usual you always come with amazing content. Could you please explain how we treat the boundary conditions for both staggered grid and riew chow interpolation. Mainly the pressure BC.
In both collocated and staggered grids, pressure is calculated in the same cell so you can use the same boundary conditions. If you are using a pressure correction approach (SIMPLE, PISO) then the pressure correction on the wall faces is zero because you know the pressure there. This roughly translates into a neighbour coefficient of zero in the matrix equations (complicated I know)
Thanks a lot, very useful. I have a question. In your video about simple scheme, the continuity just contains the pressure and we can calculate the new pressure from continuity equation, then to calculate the new velocity. But in this video, continuity equation contains pressure and U~. So how can I calculate the pressure?
Very interesting video. I'm dealing with a transient simulation problem where low mach number (0.3) and high mach number flow (14) occur together. I'm simulating with a implicit pressure-based segregated solver and get much better convergence and faster convergence comparing to a implicit density-based coupled solver. I experienced that the pressure based segregated solver deals much better with my kind of simulation when i use fine mesh and low cfl number. Although the solver is not made for such problems. Do you know why the density based coupled solver can not reach the same level of convergence per timestep like the segregated one?
Thank you for this video. Can i ask you to do video about using simple algorithm in supersonic flow. Can this algorithm model shockwaves without oscilations without using artificial diffusion? You are doing very understanding lectures for very complicated things!!!
Aidan, where I can find ap an coefficients for convection part of the Navier Stokes eq.? I have a question how the discretization works in that particular case because there is should be U^2 ? Can't find an answer in lectures or courses
I will be covering this soon. In the meantime you could check out the book by Ferziger and Peric (computational methods for fluid dynamics). They cover ap in there. In short, ap has units of mass flow rate. The mass flow rate is taken from the previous iteration of the solver. Recall that mass flow rate = rhp U A (so this is one of the velocities in the U^2). The other velocity in the U^2 is the unknown in the momentum equation. Hence we are linearising U^2 by lagging one of the velocities Hope that helps for now 👍
@@fluidmechanics101 Thank you! So one U is from previous iteration. Seems clear now to me. The only question is how this lag is justified mathematically. I will be waiting your lectures on this theme. Your videos are interesting to watch and easy to understand 👍 Best CFD material presentation I have ever found
Thanks a lot. I had never really understood chequerboard problem, I knew it's name and never actually "understood" it from book. Also can't using one sided differencing, for pressure gradient, instead of central differencing, solve this issue?
Yes Im sure one sided differencing could stop the oscillations but it might reduce the accuracy of the solution, in the same sort of way that 1st order upwind does for the convection term
First of all, Thanks a lot for your work! It might be a dumb question. But i dont really understand the dimensions of equation 19. On the left hand side we have a velocity (m/s). On the right hand side we have a pressure gradient multiplied by a volume (m^3 * Pa/m = N). Does someone understand this issue?
Yes, the units for all the terms in the momentum equation should have units of N. So pressure gradient times volume does indeed give Pa /m = N. The left hand side is more tricky. The aP terms have units of mass flow rate (kg / s). You can ways check this because aN = mass flow rate through the face if we use the simple case of upwind differencing. Mass flow rate × velocity = momentum, which has units of N 👍
aP and aN will depend on your choice of differencing scheme for convection and diffusion terms. I cover this in my course 'computational fluid dynamics fundamentals' on Udemy in a lot of detail if you are interested. (There is a link in the description of the vid)
Velocity, temperature, pressure, turbulent kinetic energy etc. are all calculated at the centroid of all the cells in the mesh. The value of these variables at positions between the cell centroids can be calculated using linear interpolation (this is used in the finite volume method to discretise the equations into algebraic form). Hence, we know that in the finite volume method, the flow variables all vary linearly across the cell (implicitly)
@@fluidmechanics101 But a linear solution representation is just one choice. My point is, the true solution to the PDE is definitely nonlinear, so I was just baffled by the language you were using which seemed to suggest the exact solution varies linearly - definitely not the case.
Of course. The true solution is non linear. The CFD solution is 'a piecewise linear approximation' that improves its accuracy as the mesh is refined. Sorry if this was confusing!
@@fluidmechanics101 The vast majority of your viewers are without background in numerical analysis so perhaps it would be beneficial to articulate that more often. Your content is great though, thanks a lot.
![[CFD] Rhie & Chow Interpolation (Part 2): Staggered Grids](http://i.ytimg.com/vi/UuZF0Zou5II/mqdefault.jpg)
![[CFD] Rhie & Chow Interpolation (Part 2): Staggered Grids](/img/tr.png)
![[CFD] Rhie & Chow Interpolation (Part 3): Deriving the Correction!](http://i.ytimg.com/vi/PmEUiUB8ETk/mqdefault.jpg)
![[CFD] Pressure-based Coupled Solver (Part 1)](/img/n.gif)
Thanks for your videos on CFD. Just for this video I think at minute 27:53 you should say that for the interpolation of the pressure at the w and e faces is Pw=(PW+PP)/2 and PE=(PP+PE)/2 and not Pw=(PW-PP)/2 and PE=(PP-PE)/2.
Agreed! My mistake 😅
This is the greatest lecture on Rhie-Chow type scheme I have ever came across. Thank you.
Thanks a lot for always bringing the quality videos here for free.
Thanks a lot for covering this topic in your talks, your content and delivery style helps me understand difficult concepts quite easily. Looking forward to remaining two part.
Thanks for the video! Both your explanation and line of reasoning are quite enlightening!
This was the topic I always had some problems with its understanding. I really appreciate your clear and detailed explanations. By the way, I am looking forward to watching the video on the rest of LES method. Thanks an ocean for your great videos.
Best video on chequerboard oscillations
4:45 I am not really sure why ideal gas law can not be used to compute the pressure field. The reasoning you give is that the change in density is small (are you referring to spatial change in density or temporal change in density or both?). Regardless, why would the numerical value of change in density make the ideal gas law inapplicable P =rho*r*T, for a particular flow regime? If the change in density is small or even zero, why would it matter in a numerical algorithm? I don't see a division by zero in the equations. I just can't seem to understand this reasoning, from a mathematical point of view. Could you please clarify?
You can't always rely on the ideal gas law because .... what if your fluid is liquid water? The algorithm needs to be general for all fluids
@@fluidmechanics101 Thanks! I see your point and that is kind of where I am getting at too - the numerical algorithm has to be general for all fluids. We should never be using ideal gas law, except for ideal gases. We have the unknowns (
ho,
ho u,
ho v,
ho w, p) but only four equations (1 continuity and 3 momentum). What is the most general equation that relates pressure-density-momenta and applies to all fluids (gas, air, water) and all flow regimes (compressible or incompressible)?
Generally in a pressure based solver, this relationship can be anything of the form: rho = function (variables). Rho is calculated separately which leaves the unknowns in the momentum equation as u, v and w (leaving rho as a constant). Pressure is then calculated from a pressure correction equation. With this approach we can account for all constitutive relationships by leaving them as a variable for the user to define
Very nice explaination, very usefull for me to understand the setting in some CFD software....
great work, thank you Aidan!
Very informative video. Thanks man
Am I correct when assuming, if you would have selected an upwind discretization on the face velocities (Uf) the chequerboard oscillations would not occur: even if you do use central difference for pressure at the face?
I haven't come across people using upwind for the face velocities before. I think it may be because we already know the velocities at the cell centroids, so we are doing an explicit calculation of the face velocities (Rhie Chow is not used to discretise the momentum equations, it is used afterwards so we don't need an implicit scheme here). But your point is still valid, I am not sure why they haven't tried upwind
Hi Aidan, I wanted to know more about the formulation of matrix ap & an from final discretised Momentum Eq [Eq-19], 21:24 . Have you gone through this on CFD Fundamentals course - 3 on udemy? I would appreciate any sources.
Found this in a book called:
Computational Fluid Dynamics for Incompressible Flows - Roy Chowdhury
I haven't been through the coefficients for the full Navier Stokes system yet. It's quite difficult! Fundamentals 3 on Udemy is still dealing with the heat conduction equation but in 3D on an unstructured mesh. I am hoping to get the full NS system sorted soon and then I can start teaching that one as well. I have mainly been using the famous book by Patankar but I will check out that book. Thanks for the reference ☺️
Same question. Did you found a source with explanation?
Yes! If you see my other comment I have left an explanation there
@Fluid Mechanics 101 Thank you for the video. I encoded a 2d laminar cfd solver myself, and at the very beginnig I had hard times defining the right staggered and scalar grids and also where to place the phsical domain in between those grids. Could you please record a video about it as well?
Yes, this is the tricky part of writing the staggered grid solvers. Have you checked your pressure correction equation and interpolation of velocities and pressures onto the right grids? This is typically the difficult bit and I haven't go around to right a video for this yet
@@fluidmechanics101Eventually I could manage that by placing the physical domain not to the edges but to just a half way inside of the scalar /main/grids. So the unit cells’ boundaries of staggered grids perfectly matched with the physical domain boundaries. It was tricky at first but then when applying the pressure correction to the scalar cell I appreciated how it was easily adapted since no interpolation needed as u also said. The weird part in the staggered grid was that I needed to solve momentum equation over u- and v- control volumes individually while I solved the pressure correction equation ( mass conservation) only one time over the scalar control volume.
And indices was also tricky as you indicate because for any (i,j) location there are three different cell centers and as I named them (I,J) for scalar cell center; (I,j) for u- cell location of backward staggered of (I,J) cell; (i,J) for v- cell center of backward staggered of (I,J) cell.
And the sad part is that my own code solves the equations slower than ANSYS Fluent :) Fluent solves the equations faster and converges much earlier. I guess I should apply the TDMA method for the solution of momentum equations instead of Under relaxed Gauss Seidel. I have also some suspicion about if the pressure correction equation converges in the iteration step. I use SOR method for the pressure correction. I still need to go further and dig into works and books :)
❤Thanks to your detailing introduction!
However, the eq. 26 has typo for interpolation; but it’s not big deal for your brilliant work!🎉
Thank you for your all amazing videos. Would you please describe more about the flows at moderate mach numbers-flows, somehow between the incompressible and compressible ranges.
This caregory may happen through unstable flow regimes in tirbomachined i.e. axial fans or axial compressors near the stall conditions.
Regards
Thank you very much for the video and as usual you always come with amazing content. Could you please explain how we treat the boundary conditions for both staggered grid and riew chow interpolation. Mainly the pressure BC.
In both collocated and staggered grids, pressure is calculated in the same cell so you can use the same boundary conditions. If you are using a pressure correction approach (SIMPLE, PISO) then the pressure correction on the wall faces is zero because you know the pressure there. This roughly translates into a neighbour coefficient of zero in the matrix equations (complicated I know)
Thank you for the detailed explanation.
Thanks a lot, very useful. I have a question. In your video about simple scheme, the continuity just contains the pressure and we can calculate the new pressure from continuity equation, then to calculate the new velocity. But in this video, continuity equation contains pressure and U~. So how can I calculate the pressure?
This is quite a tricky question. I am going to do a full derivation soon with all the terms, as it can be quite confusing!
Thank you for this video！Can you make video to explain PRESTO！discretization method for pressure？
Is this any better then the interpolation using H/a - grad(p)/a ? Can I make rhie and chow using H operator?
Very interesting video. I'm dealing with a transient simulation problem where low mach number (0.3) and high mach number flow (14) occur together. I'm simulating with a implicit pressure-based segregated solver and get much better convergence and faster convergence comparing to a implicit density-based coupled solver. I experienced that the pressure based segregated solver deals much better with my kind of simulation when i use fine mesh and low cfl number. Although the solver is not made for such problems. Do you know why the density based coupled solver can not reach the same level of convergence per timestep like the segregated one?
Thank you for this video. Can i ask you to do video about using simple algorithm in supersonic flow. Can this algorithm model shockwaves without oscilations without using artificial diffusion? You are doing very understanding lectures for very complicated things!!!
It can do supersonic flow but you are probably better off using a density based solver or coupled solver instead 👍
Nice talk
Thanks for this wonderful video. I have confusions how do you derive from equation 18 to equation 19?
Aidan, where I can find ap an coefficients for convection part of the Navier Stokes eq.? I have a question how the discretization works in that particular case because there is should be U^2 ? Can't find an answer in lectures or courses
I will be covering this soon. In the meantime you could check out the book by Ferziger and Peric (computational methods for fluid dynamics). They cover ap in there.
In short, ap has units of mass flow rate. The mass flow rate is taken from the previous iteration of the solver. Recall that mass flow rate = rhp U A (so this is one of the velocities in the U^2). The other velocity in the U^2 is the unknown in the momentum equation. Hence we are linearising U^2 by lagging one of the velocities
Hope that helps for now 👍
@@fluidmechanics101 Thank you! So one U is from previous iteration. Seems clear now to me. The only question is how this lag is justified mathematically. I will be waiting your lectures on this theme. Your videos are interesting to watch and easy to understand 👍 Best CFD material presentation I have ever found
Thanks a lot. I had never really understood chequerboard problem, I knew it's name and never actually "understood" it from book. Also can't using one sided differencing, for pressure gradient, instead of central differencing, solve this issue?
Yes Im sure one sided differencing could stop the oscillations but it might reduce the accuracy of the solution, in the same sort of way that 1st order upwind does for the convection term
@@fluidmechanics101 thanks for the reply
First of all, Thanks a lot for your work!
It might be a dumb question. But i dont really understand the dimensions of equation 19. On the left hand side we have a velocity (m/s). On the right hand side we have a pressure gradient multiplied by a volume (m^3 * Pa/m = N). Does someone understand this issue?
Also, i don't know exactly how to get ap and an in eq.19 or 20, If you already handle it, could you tell me how?
Yes, the units for all the terms in the momentum equation should have units of N. So pressure gradient times volume does indeed give Pa /m = N. The left hand side is more tricky. The aP terms have units of mass flow rate (kg / s). You can ways check this because aN = mass flow rate through the face if we use the simple case of upwind differencing. Mass flow rate × velocity = momentum, which has units of N 👍
aP and aN will depend on your choice of differencing scheme for convection and diffusion terms. I cover this in my course 'computational fluid dynamics fundamentals' on Udemy in a lot of detail if you are interested. (There is a link in the description of the vid)
@@fluidmechanics101 Thanks a lot!
@@Muuui What exactly is your issue?
You ever though about making videos about inviscid flux schemes (FVS/FDS/etc)?
What are you talking about that the flow variables vary linearly across a cell?
Velocity, temperature, pressure, turbulent kinetic energy etc. are all calculated at the centroid of all the cells in the mesh. The value of these variables at positions between the cell centroids can be calculated using linear interpolation (this is used in the finite volume method to discretise the equations into algebraic form). Hence, we know that in the finite volume method, the flow variables all vary linearly across the cell (implicitly)
@@fluidmechanics101 But a linear solution representation is just one choice. My point is, the true solution to the PDE is definitely nonlinear, so I was just baffled by the language you were using which seemed to suggest the exact solution varies linearly - definitely not the case.
Of course. The true solution is non linear. The CFD solution is 'a piecewise linear approximation' that improves its accuracy as the mesh is refined. Sorry if this was confusing!
@@fluidmechanics101 The vast majority of your viewers are without background in numerical analysis so perhaps it would be beneficial to articulate that more often. Your content is great though, thanks a lot.
Good point. Thank you for the feedback. I will bear that in mind in future
thank you
thank you thank you thank you ...
wow~ thanks alot
Thank you so much
这英国口音听着太舒服了
👍