[jacktat7]

Hello ,
I was wondering when it is allowed to prescribe a perssure BC at the inlet in a (cardiovascular) CFD problem. I am making a model of a couple of vessels, and want to have it pressure-driven. However, someone said that it is not always physically correct due to the energy balance not always being fulfilled or something. However, I know that it can be correct but I want to know what needs to be met / when is it OK to prescribe pressure at inlet, does anyone have some scientific references?

Cheers,
Jack

[LuckyTran]

If you are talking specifically about static pressure, it's okay when the inlet is supersonic.

We use inlet total/stagnation pressure as a BC all the time.

[jacktat7]

I do not think it's super sonic, it's blood through a vessel. So the pressure will also vary since it will replicate pressures created by the heart pumping.

[andy_]

A common combination is a specified total pressure inlet condition and static pressure outlet condition. The mass flow rate through the solution region will then adjust until the pressure loss balances the inlet dynamic pressure.

[LuckyTran]

Btw I hope we are talking about compressible fluids. For incompressible fluids, pressure has even less meaning and it gets wonky.

If you want to fix the flowrate AND the inlet static pressure, this is possible only when the inlet is supersonic.

You can fix the inlet and outlet static pressure if you are willing to accept whatever flowrate corresponds to this combination. However, there are two possible solutions to this pair. One subsonic and potentially one supersonic. Actually the supersonic one might not even be possible if the static pressure at the outlet is being imposed as a hard constraint. But since we are talking about compressible fluids, well now you have to be super careful how you apply the BC's for the energy equation.

Just to be clear, we are talking about hard clamping the inlet static pressure (that's how I interpret imposing it as a BC). There are workarounds of course (by iterating other things) to achieve a targeted static pressure.

[andy_]

Originally Posted by LuckyTran

Btw I hope we are talking about compressible fluids. For incompressible fluids, pressure has even less meaning and it gets wonky.

Blood typically flows at pretty low Mach numbers and so would normally be modelled as an incompressible fluid. Specifying an upstream total pressure and downstream static pressure is common when there is something like a pump upstream driving fluid through a system and venting to something like atmosphere. Of course what is labelled a "total pressure" or "static pressure" boundary condition in CFD software will involve more than a single value since boundary conditions will need to be supplied for the full set of equations solved. But the user input is typically an inlet total pressure value/profile and outlet static pressure value.

There are of course other combinations of specified quantities that work (the absolute value of static pressure is usually irrelevant in incompressible flows). The key to success is likely ensuring the mass flow adjusts to ensure a force balance where the pressure drop across the solution region balances the shear on the walls.

[FMDenaro]

If the vessel is modelled as elastic then the blood and vessel have to be considered as a compressible medium.
If the vessel is rigid the blood is an incompressible fluid.

[andy_]

Originally Posted by FMDenaro

If the vessel is modelled as elastic then the blood and vessel have to be considered as a compressible medium.

Why? Is an incompressible fluid and a linear elastic solid a compressible medium?

[FMDenaro]

Originally Posted by andy_

Why? Is an incompressible fluid and a linear elastic solid a compressible medium?

Because there is a difference in the sound velocity through the blood and that through the pulsatile vessel. Therefore it is the global system to be compressible.

[jacktat7]

Yes I was modeling it as an incompressible fluid, and I think total pressure at inlet and static at outlet is good combination. I am still wondering if any of you know of a good piece of documentation / literature which shows WHY and WHEN this is possible / allowed. (this due to the fact that I need to change a moddel which has flowrate and make it pressure driven, so I want to be able to show in my paper that this also is physically correct.)

[FMDenaro]

Originally Posted by jacktat7

Yes I was modeling it as an incompressible fluid, and I think total pressure at inlet and static at outlet is good combination. I am still wondering if any of you know of a good piece of documentation / literature which shows WHY and WHEN this is possible / allowed. (this due to the fact that I need to change a moddel which has flowrate and make it pressure driven, so I want to be able to show in my paper that this also is physically correct.)

The issue is that in the incompressible flow model, the "pressure" has no physical meaning and does not enter by means of the absolute values. To the pressure field any added function of time produces the same velocity field

[jacktat7]

do they normally perscribe traction vector i think right? t = sigma * n

[andy_]

Originally Posted by jacktat7

Yes I was modeling it as an incompressible fluid, and I think total pressure at inlet and static at outlet is good combination. I am still wondering if any of you know of a good piece of documentation / literature which shows WHY and WHEN this is possible / allowed. (this due to the fact that I need to change a moddel which has flowrate and make it pressure driven, so I want to be able to show in my paper that this also is physically correct.)

Can't help with references but the most reliable one is likely to be the boundary condition documentation for you code. It is physically correct because a force balance is physically correct (energy is not normally solved directly in an incompressible code).

As mentioned earlier the physics is that the flow rate adjusts to raise or lower the pressure drop down the tube that is required to overcome the wall stresses. The details of what balance is strictly enforced follows from the details of the differencing in the momentum equations and, if present, the form of pressure smoothing.

My vaguely recalled conclusions from many decades ago in the 90s when playing with what to pin and what to float from time step to time step when driving the flow in an incompressible LES code was that the differences were reasonably unimportant if the solution domain was of sufficient size. Had the flow settled to a steady laminar one there would have been no differences simply a different way of expressing the same boundary condition. However, I wasn't imposing a constant pressure at exit which would be incompatible with eddies leaving the solution domain so this may not quite match your situation.

Perhaps a useful way to look at things is how is the inlet mass flow (which will be constant at every location down the tube) is determined. Assuming the flow at exit is well settled so that a zero gradient or constant pressure condition are almost equivalent then a total pressure inlet condition is simply changing the specified inlet velocity until things have settled down. When they have, assuming your initial calculation of the pressure drop was correct, then both solutions will have the same specified inlet velocity. Physically correct doesn't really enter into it (so long as it converges) because the boundary conditions imposed by the code are effectively the same. Probably not exactly but effectively. Looking up how total pressure and static pressure boundary conditions are implemented for incompressible codes should hopefully make this clearer.

[andy_]

Originally Posted by FMDenaro

Because there is a difference in the sound velocity through the blood and that through the pulsatile vessel. Therefore it is the global system to be compressible.

I still cannot see this. We have a soft tube with slow bending waves and a hard almost incompressbile fluid with a fast sound speed. I cannot see why an infinite rather than a fast fluid sound speed is not a reasonable assumption. Perhaps I should add that I have never performed or even thought about such simulations prior to this thread so am happy to be wrong. I am interested though because I hope to soon start a project on a general (e.g. moving boundary) low Mach number code.

[FMDenaro]

Originally Posted by andy_

I still cannot see this. We have a soft tube with slow bending waves and a hard almost incompressbile fluid with a fast sound speed. I cannot see why an infinite rather than a fast fluid sound speed is not a reasonable assumption. Perhaps I should add that I have never performed or even thought about such simulations prior to this thread so am happy to be wrong. I am interested though because I hope to soon start a project on a general (e.g. moving boundary) low Mach number code.

The problem is not different from the water hammer issue where the liquid and the pipe is considered compressible. In pulsatile blood vessel you have to consider 2 characteristic sound velocities. This is the hyperbolic model to use if you want to study the vessel with deformation of the elastic walls due to pressure (for example a simplified 1d model https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4183203/) otherwise the standar incompressible flow model in a pipe is used.

[andy_]

Originally Posted by FMDenaro

The problem is not different from the water hammer issue where the liquid and the pipe is considered compressible. In pulsatile blood vessel you have to consider 2 characteristic sound velocities. This is the hyperbolic model to use if you want to study the vessel with deformation of the elastic walls due to pressure (for example a simplified 1d model https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4183203/) otherwise the standar incompressible flow model in a pipe is used.

Thanks for the clarification.

I have only very briefly skimmed the linked paper and the PhD from Olufsen which describes the original method. The original method was derived for a compliant tube of varying cross section containing an incompressible fluid. In an Appendix this is shown to be equivalent to 1D gas dynamics for a straight tube but using a speed of sound derived from the wall bending wave and not the speed of sound in blood. There would seem to be only one "speed of sound" involved.

I still cannot see a requirement to involve the compressibility of blood but would like to given there seem to be few applications for a low Mach number compressible code.

[FMDenaro]

Originally Posted by andy_

Thanks for the clarification.

I have only very briefly skimmed the linked paper and the PhD from Olufsen which describes the original method. The original method was derived for a compliant tube of varying cross section containing an incompressible fluid. In an Appendix this is shown to be equivalent to 1D gas dynamics for a straight tube but using a speed of sound derived from the wall bending wave and not the speed of sound in blood. There would seem to be only one "speed of sound" involved.

I still cannot see a requirement to involve the compressibility of blood but would like to given there seem to be few applications for a low Mach number compressible code.

It is not a requirement for the variable density of the blood but it is a result of the fluid-structure interaction with a time-varying flow rate. Have a look to Sec. 6 here
https://www.mate.polimi.it/bibliotec...qmox/mox01.pdf

[andy_]

Originally Posted by FMDenaro

It is not a requirement for the variable density of the blood but it is a result of the fluid-structure interaction with a time-varying flow rate. Have a look to Sec. 6 here
https://www.mate.polimi.it/bibliotec...qmox/mox01.pdf

Thanks for further clarification. Again the derivation excludes the compressibility of blood and associated sound waves.

[FMDenaro]

Originally Posted by andy_

Thanks for further clarification. Again the derivation excludes the compressibility of blood and associated sound waves.

We should accord on the fact that herein compressibility is not the property of the blood to have a density variation but the property of pressure waves to have a finite velocity of propagation (a finite Mach number). That is described by the finite characteristic velocity (6.17). See also remark 2 in the text (page 78).

The standard incompressible flow model conversely assumes an infinite sound velocity, the pressure waves spread at infinite velocity.

[LuckyTran]

Originally Posted by jacktat7

Yes I was modeling it as an incompressible fluid, and I think total pressure at inlet and static at outlet is good combination. I am still wondering if any of you know of a good piece of documentation / literature which shows WHY and WHEN this is possible / allowed. (this due to the fact that I need to change a moddel which has flowrate and make it pressure driven, so I want to be able to show in my paper that this also is physically correct.)

Total pressure at inlet and pressure at outlet is always a good combination for CFD in general. The other combinations is a mass-flow inlet and a pressure outlet. There's also a total pressure inlet and mass-flow outlet (which is really a pressure outlet in disguise).

For incompressible fluids (constant density or temperature dependent density), you can still use a total pressure inlet and static pressure outlet. But you can also use a static pressure inlet and static pressure outlet because all that matters is that you have the right driving pressure differential. This proof is trivial. You can just look and see that you only have the gradient of pressure in the governing equations (i.e. continuity and momentum). You have two equations and two unknowns in (the gradient of) pressure and velocity.

The trouble-maker is a static pressure inlet and static pressure outlet for a compressible flow. This one is special for physical reasons.

Well I'm sorry I derailed this thread into the compressible world if it was not at all applicable. I automatically assumed we were in compressible land because cardiovascular flows are more-often-than-not modeled using a compressible approach. For those that are interested in compressible talk, Fillipo has given a nice primer so far and even given the water hammer example. You can't go into the water hammer problem with an infinite propagation speed.