Why incompressible fluid flow is advantageous in numerical computation

In summary: Hi, In summary, the problem with not including pressure in continuity equation is that spurious pressure values can be generated. Treating the fluid as incompressible can lead to round-off problems in the numerical solution, and also doesn't have any significant effect on the flow.
  • #1
Zoli
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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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  • #2
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.
 
  • #3
Zoli said:
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
 
  • #4
Chestermiller said:
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.
 
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  • #5
AlephZero said:
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
 
  • #6
Chestermiller said:
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
 
  • #7
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.
 
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  • #8
Zoli said:
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)

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

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)
 
  • #9
bigfooted said:
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)

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

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.
 
  • #10
Zoli said:
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
 
  • #11
bigfooted said:
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.
 

1. Why is incompressible fluid flow advantageous in numerical computation?

Incompressible fluid flow is advantageous in numerical computation because it simplifies the equations used to model fluid flow. In incompressible flow, the fluid density remains constant, which eliminates the need to solve for the density variable in the equations. This reduces the complexity of the equations and makes them easier to solve numerically.

2. How does incompressibility affect the accuracy of numerical simulations?

Incompressibility is a key assumption in many numerical simulation techniques, and it greatly improves the accuracy of the results. By assuming that the fluid is incompressible, the equations used to model the flow become simpler and easier to solve, resulting in more accurate numerical solutions.

3. What are the limitations of using incompressible flow in numerical computations?

One limitation of using incompressible flow in numerical computations is that it can only be applied to fluids with low Mach numbers, meaning that the fluid velocity is much lower than the speed of sound. In addition, incompressibility assumes that the fluid is inviscid, meaning that it has no internal friction. This can limit the accuracy of the results in situations where viscous effects are significant.

4. How does the assumption of incompressibility impact the computational cost of simulations?

The assumption of incompressibility can significantly reduce the computational cost of simulations. Since the equations used to model incompressible flow are simpler, they require less computational resources to solve. This allows for faster simulation times and the ability to simulate larger and more complex systems.

5. Are there any real-world applications where incompressible flow is not a valid assumption?

There are many real-world applications where incompressible flow is not a valid assumption. For example, in high-speed aerodynamics and supersonic flow, the assumption of incompressibility breaks down due to the compressibility of air at these speeds. In these situations, compressible flow equations must be used instead.

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