# Why incompressible fluid flow is advantageous in numerical computation

Hi,

When we want to solve the Navier-Stokes equations coupled with the conservation of mass for incompressible fluids using the primitive-variable approach, we have to face to the problem that the equation for the continuity equation does not contain the pressure which leads to spurious pressure values. Moreover the compressible fluid assumption characterizes the fluid properties better. Then why don't we solve the compressible Navier-Stokes during numerical simulations?

Thanks,
Zoli

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mathman
It depends on the physical sutiation. Flow through liquid is incompressible. Flow through gas (such as air) can be handled as incompressible until the speed gets near the speed of sound.

Chestermiller
Mentor
Hi,

When we want to solve the Navier-Stokes equations coupled with the conservation of mass for incompressible fluids using the primitive-variable approach, we have to face to the problem that the equation for the continuity equation does not contain the pressure which leads to spurious pressure values. Moreover the compressible fluid assumption characterizes the fluid properties better. Then why don't we solve the compressible Navier-Stokes during numerical simulations?

Thanks,
Zoli
Since when does treating the fluid as incompressible lead to spurious pressure values? I've never heard of this before, and I've been doing fluid mechanics for many decades. Treating an nearly incompressible fluid as compressible, however, can lead to round-off problems in the numerical solution. Also, if you treat the fluid as incompressible, then there is a piece of data that you don't need to have to solve the problem, namely, the bulk modulus. Personally, I think it is much easier to solve fluid mechanics problems in which the fluid is treated as incompressible.

Chet

AlephZero
Homework Helper
Since when does treating the fluid as incompressible lead to spurious pressure values? I've never heard of this before, and I've been doing fluid mechanics for many decades.
Well, there is a sort-of-strawman argument that if you do it wrong, you can get spurious oscillations. The fix is to do it right, of course. See "colocated grids" in http://www3.nd.edu/~gtryggva/CFD-Course/2011-Lecture-22.pdf

To answer the OP's question, including the compressibility is not "better" if
(1) you have no interest in it, and
(2) it doesn't have any significant effect on the flow.

• 1 person
Chestermiller
Mentor
Well, there is a sort-of-strawman argument that if you do it wrong, you can get spurious oscillations. The fix is to do it right, of course. See "colocated grids" in http://www3.nd.edu/~gtryggva/CFD-Course/2011-Lecture-22.pdf

To answer the OP's question, including the compressibility is not "better" if
(1) you have no interest in it, and
(2) it doesn't have any significant effect on the flow.
Excellent points.

Chet

Since when does treating the fluid as incompressible lead to spurious pressure values? I've never heard of this before, and I've been doing fluid mechanics for many decades. Treating an nearly incompressible fluid as compressible, however, can lead to round-off problems in the numerical solution. Also, if you treat the fluid as incompressible, then there is a piece of data that you don't need to have to solve the problem, namely, the bulk modulus. Personally, I think it is much easier to solve fluid mechanics problems in which the fluid is treated as incompressible.

Chet
Spurious pressure values come up when we use the non-staggered approximation for the velocity and pressure. However programming spectral methods (as I deal with the spectral collocation of the Stokes-equations) is hard for staggered-grids except the case when we apply homogeneous boundary conditions.

Zoli

Gold Member
Also for, compressible flows, the momentum and continuity equations are in germ real no longer decoupled from the energy equation, so you've effectively increased the dimensionality of the problem.

• 1 person
bigfooted
Gold Member
Spurious pressure values come up when we use the non-staggered approximation for the velocity and pressure. However programming spectral methods (as I deal with the spectral collocation of the Stokes-equations) is hard for staggered-grids except the case when we apply homogeneous boundary conditions.

Zoli
This spurious pressure problem is as far as I know a problem when using central difference type schemes for the incompressible NS equations (skipping nodes in the discretization), because checkerboard/sawtooth distributions for the velocity and pressure then are valid solutions. I thought it suffices to use different orders for the discretization of pressure and velocity.

Also, solving the incompressible NS equations with a compressible code leads to stability issues. For explicit schemes, the stability criterion on the time step is of the form (e.g. MacCormack scheme for compressible NS)

$\Delta t < \frac{1}{\frac{u}{\Delta x} + \frac{v}{\Delta y} + a*\sqrt(\frac{1}{\Delta x^2}+\frac{1}{\Delta y^2})}$

with a the speed of sound. For incompressible flow, the speed of sound is infinite, so you cannot use your compressible NS code with the McCormack scheme to solve the incompressible NS equations.

Codes that can solve compressible and incompressible NS equations usually use some kind of preconditioning for the low mach number limit, or they switch from one numerical method to the other.

Anyway, staggered grids should not be necessary. there are plenty of methods to circumvent using staggered grids (weak formulations, pseudo-spectral methods)

This spurious pressure problem is as far as I know a problem when using central difference type schemes for the incompressible NS equations (skipping nodes in the discretization), because checkerboard/sawtooth distributions for the velocity and pressure then are valid solutions. I thought it suffices to use different orders for the discretization of pressure and velocity.

Also, solving the incompressible NS equations with a compressible code leads to stability issues. For explicit schemes, the stability criterion on the time step is of the form (e.g. MacCormack scheme for compressible NS)

$\Delta t < \frac{1}{\frac{u}{\Delta x} + \frac{v}{\Delta y} + a*\sqrt(\frac{1}{\Delta x^2}+\frac{1}{\Delta y^2})}$

with a the speed of sound. For incompressible flow, the speed of sound is infinite, so you cannot use your compressible NS code with the McCormack scheme to solve the incompressible NS equations.

Codes that can solve compressible and incompressible NS equations usually use some kind of preconditioning for the low mach number limit, or they switch from one numerical method to the other.

Anyway, staggered grids should not be necessary. there are plenty of methods to circumvent using staggered grids (weak formulations, pseudo-spectral methods)
I do not know the finite difference approach, but I speak about the steady-state spectral or finite element solution of the problem. To prevent the checkerboard pressure distribution, one should use elements that satisfies the Babuska-Brezzi condition.

bigfooted
Gold Member
I do not know the finite difference approach, but I speak about the steady-state spectral or finite element solution of the problem. To prevent the checkerboard pressure distribution, one should use elements that satisfies the Babuska-Brezzi condition.
OK. The inf-sup condition should be satisfied 'almost always' when using normal Galerkin FEM. If you are worried about this, you can switch to e.g. a least-squares formulation and recast your system to first order.

If you try to find a solution to the steady-state problem by solving the transient equation with false time-stepping, then there is a lot of literature on how to get a stable solution. If you use splitting schemes for instance, the inf-sup condition is not required and you can use equal order for velocity and pressure. This is a fast and stable method, and is usually preferred over solving the NS equations without the d/dt term.

To get back to your original point: compressible NS equations are not used to simulate in the incompressible regime because either 1) it takes much longer than solving the incompressible NS equations because of stability requirements, 2) you need a much more complicated numerical method involving preconditioning based on the mach number, 3) it will not give you 'more' information

OK. The inf-sup condition should be satisfied 'almost always' when using normal Galerkin FEM. If you are worried about this, you can switch to e.g. a least-squares formulation and recast your system to first order.

If you try to find a solution to the steady-state problem by solving the transient equation with false time-stepping, then there is a lot of literature on how to get a stable solution. If you use splitting schemes for instance, the inf-sup condition is not required and you can use equal order for velocity and pressure. This is a fast and stable method, and is usually preferred over solving the NS equations without the d/dt term.

To get back to your original point: compressible NS equations are not used to simulate in the incompressible regime because either 1) it takes much longer than solving the incompressible NS equations because of stability requirements, 2) you need a much more complicated numerical method involving preconditioning based on the mach number, 3) it will not give you 'more' information
Thank you, now it is clear.