[HectorRedal]

I am searching for a reference that explains the type of boundary conditions to use in open flow.
Is mandatory to fix the pressure at the exit when the velocity is fixed at the entrance? Is equivalent to use the gradient of presure at the exit instead of a fixed value for the pressure?

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[FMDenaro]

Originally Posted by HectorRedal

I am searching for a reference that explains the type of boundary conditions to use in open flow.
Is mandatory to fix the pressure at the exit when the velocity is fixed at the entrance? Is equivalent to use the gradient of presure at the exit instead of a fixed value for the pressure?

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Are you considering the incompressible or compressible flow model?
For open flow do you mean an outflow in a open enviroment?

[HectorRedal]

Originally Posted by FMDenaro

Are you considering the incompressible or compressible flow model?
For open flow do you mean an outflow in a open enviroment?

I am subsonic considering incompressible flow. But I am also interested in knowing what happens with boundary conditions in the case of compressible flow model, just for future use.

When referring to open flow, I mean open environment (no walls). I am referring to symmetric boundary conditions in the top and bottom part of the domain and a spedified subsonic inlet velocity.

Up to my knowledge, since the flow is subsonic, at the inlet one of the variables (either v or p) are set but the other one should be allowed to float.
In the exit, the same should be applied. If I need to determine the the velocity field, then the p set has to be set. But, can be used the dp / dx instead?

[FMDenaro]

Subsonic is not automatically also incompressible... if you choose the incompressible model then you assume M=0 (due to the infinite sound velocity) and the divergence-free constraint substitutes the continuity equation. In such case, either velocity or pressure (that is however not a theromodinamic function) can be prescribed.
As you wrote, for compressible subsonic flows, the characteristic line analysis shows that one boundary condition at the inlet can not be fixed but comes from the interior. Viceversa, at outlet you can fix only one boundary condition.
Symmetric conditions, periodic conditions are of mathematical nature and does not depend on the regime of the flow.

You can find many papers about BC.s, I suggest to read this hystorical paper
https://www.researchgate.net/publica..._viscous_flows

[HectorRedal]

Originally Posted by FMDenaro

Subsonic is not automatically also incompressible... if you choose the incompressible model then you assume M=0 (due to the infinite sound velocity) and the divergence-free constraint substitutes the continuity equation. In such case, either velocity or pressure (that is however not a theromodinamic function) can be prescribed.
As you wrote, for compressible subsonic flows, the characteristic line analysis shows that one boundary condition at the inlet can not be fixed but comes from the interior. Viceversa, at outlet you can fix only one boundary condition.
Symmetric conditions, periodic conditions are of mathematical nature and does not depend on the regime of the flow.

You can find many papers about BC.s, I suggest to read this hystorical paper
https://www.researchgate.net/publica..._viscous_flows

I agree with you that subsonic does not mean incompressible. I wrongly specified the requirement. I was referring to incompressible. As far as I know, when choosing M < 0.3, it can be assumed that the flow is incompressible. May I be wrong? I don't think so.
Thanks for the reference. Unfortunately, the subscription I have do not allow me to read papers in Journal of Computational Physics before 1995. I will for a paper that references this article.

[FMDenaro]

Originally Posted by HectorRedal

I agree with you that subsonic does not mean incompressible. I wrongly specified the requirement. I was referring to incompressible. As far as I know, when choosing M < 0.3, it can be assumed that the flow is incompressible. May I be wrong? I don't think so.
Thanks for the reference. Unfortunately, the subscription I have do not allow me to read papers in Journal of Computational Physics before 1995. I will for a paper that references this article.

the link I posted is for a free copy on researchgate, you can read...

When you consider a compressible flow model, even at low Mach number, the equations are mathematically different from those written for a pure incompressible flow model. As a consequence, the BC.s are different.

[HectorRedal]

Originally Posted by FMDenaro

the link I posted is for a free copy on researchgate, you can read...

When you consider a compressible flow model, even at low Mach number, the equations are mathematically different from those written for a pure incompressible flow model. As a consequence, the BC.s are different.

Thanks. I forgot to check if it can be downloaded freely from the website.

Regarding the compressible / incompressible model, I'd like to comment that I am trying to simulate a incompressible flow: An incompressible fluid past a circular cylinder in an open domain.
For simulating simulating the incompressiblity I am using a sound speed equal to 10000 m/s. The specified fluid inlet velocity is 1 m/s (M=1e-4).
Taking into account your comments, do you still think that my simulation is not correct since the compressiblity effects cannot be totally ruled out?
Based on your comments, I should use BC for the compressiblity case instead of BC for the incompressibility case, right?

[FMDenaro]

Originally Posted by HectorRedal

Thanks. I forgot to check if it can be downloaded freely from the website.

Regarding the compressible / incompressible model, I'd like to comment that I am trying to simulate a incompressible flow: An incompressible fluid past a circular cylinder in an open domain.
For simulating simulating the incompressiblity I am using a sound speed equal to 10000 m/s. The specified fluid inlet velocity is 1 m/s (M=1e-4).
Taking into account your comments, do you still think that my simulation is not correct since the compressiblity effects cannot be totally ruled out?
Based on your comments, I should use BC for the compressiblity case instead of BC for the incompressibility case, right?

Running a code implemented for solving compressible flows in the M->0 limit is not simple due to the stiffness of the problem. When dp/drho ->Inf you have strong fluctuations and the time step have to be so small as to descrive the sound velocity propagation. For such reason, the low-Mach formulation is mandatory. Alternatively, a pre-conditioning technique is used.
Otherwise, you need to rewrite your code in terms of the incompressible form of the NS equations.
I strongly suggest to read carefully the theory of these formulations.

[HectorRedal]

Originally Posted by FMDenaro

Running a code implemented for solving compressible flows in the M->0 limit is not simple due to the stiffness of the problem. When dp/drho ->Inf you have strong fluctuations and the time step have to be so small as to descrive the sound velocity propagation. For such reason, the low-Mach formulation is mandatory. Alternatively, a pre-conditioning technique is used.
Otherwise, you need to rewrite your code in terms of the incompressible form of the NS equations.
I strongly suggest to read carefully the theory of these formulations.

The scheme I am using can be used both for compressible and for incompressible flows. When used semi-implicitly (explicitly for momentum conservation equation and implicitly for mass conservation equation), the time step in each iteration does not depend on the value of M.
That the reason why I am using this scheme for solving incompressible flow.

I tried to remove the M term from the implementation, since in the limit M=0, but it only worked in the lid-driven cavity flow problem. When used in a flow through a channel two parallel walls, it does not converge at all. So, that's the reason why I am still using M but with a low value.

[FMDenaro]

Originally Posted by HectorRedal

The scheme I am using can be used both for compressible and for incompressible flows. When used semi-implicitly (explicitly for momentum conservation equation and implicitly for mass conservation equation), the time step in each iteration does not depend on the value of M.
That the reason why I am using this scheme for solving incompressible flow.

I tried to remove the M term from the implementation, since in the limit M=0, but it only worked in the lid-driven cavity flow problem. When used in a flow through a channel two parallel walls, it does not converge at all. So, that's the reason why I am still using M but with a low value.

I don't know how your code works, therefore I can't be sure but I think that when you solve incompressible, that means low-Mach flow not the incompressible formulation of the NS (M=0). The BC.s for these two formulations have differences due to the different mathematical character.

[HectorRedal]

Originally Posted by FMDenaro

I don't know how your code works, therefore I can't be sure but I think that when you solve incompressible, that means low-Mach flow not the incompressible formulation of the NS (M=0). The BC.s for these two formulations have differences due to the different mathematical character.

As commented before (previous response), I tried to remove the term M (since M=0) from the pressure equation. After this change, I understand that the algorithm can work as an incompressible solver.
In this case, for outer boundaries, I can fix the pressure at the exit. But, when solving for the compressible flow (M!=0), the pressure cannot be fixed at the exit (Both characteristics lines go out of the domain in question, in contrast with incompressible flow, where one characteristic line goes out of the domain, whereas the other characteristic line goes inside the domain).

This is my understanding, but maybe I am wrong.

[FMDenaro]

You re wrong ...
At M=0 the system is no longer hyperbolic

[HectorRedal]

Originally Posted by FMDenaro

You re wrong ...
At M=0 the system is no longer hyperbolic

Thanks for the feedback.

[FMDenaro]

You should consider that M=0 is a model for an incompressible fluid that leads to parabolic/elliptic equations.
On the other hand using the compressible formulation for M->0 still retains the hyperbolic character of the equations. However, u+c and u-c slopes for the characteristic lines tends to +Inf and -Inf.

[HectorRedal]

Originally Posted by FMDenaro

You should consider that M=0 is a model for an incompressible fluid that leads to parabolic/elliptic equations.
On the other hand using the compressible formulation for M->0 still retains the hyperbolic character of the equations. However, u+c and u-c slopes for the characteristic lines tends to +Inf and -Inf.

I have taken a look at the Fletcher's book that explains the boundary conditions for incompressible flows, and according to this reference the way to impose boundary condition for outflow is specifying the traction value at that boundary:
For the normal traction -> t_n = -p + nu * du/dx
For the trangential traction -> t_t = nu * dv/dx

My question is: What is supposed to be the value of the traction at the exit? Can it be assumed that the traction is zero (t_n = 0)?
In this case, then p = nu * du/dx, isn't it?

[FMDenaro]

Originally Posted by HectorRedal

I have taken a look at the Fletcher's book that explains the boundary conditions for incompressible flows, and according to this reference the way to impose boundary condition for outflow is specifying the traction value at that boundary:
For the normal traction -> t_n = -p + nu * du/dx
For the trangential traction -> t_t = nu * dv/dx

My question is: What is supposed to be the value of the traction at the exit? Can it be assumed that the traction is zero (t_n = 0)?
In this case, then p = nu * du/dx, isn't it?

Again, what equations are you considering? If you are considering only the momentum equation with the divergence-free constraint the BC.s for either velocity or pressure must fulfill some specific requirement for the mass conservation.
The BC.s you wrote cannot be used at the outflow

[HectorRedal]

Originally Posted by FMDenaro

Again, what equations are you considering? If you are considering only the momentum equation with the divergence-free constraint the BC.s for either velocity or pressure must fulfill some specific requirement for the mass conservation.
The BC.s you wrote cannot be used at the outflow

I am considering the equations for the incompressible limit, that is M= 0.
I am not only considering the equation for the momentum.
I am solving a three step algorithm: (1) prediction, (2) solving for pressure and (3) correction.
In the 3rd step is the step to enforce the boundary condition after having obtained the real value for pressure.
According to Fletcher these are the B.C to use. What is the reason why you state that these B.C. cannot be used? It's just for my understanding.

[FMDenaro]

At M=0 (not low..), the system reduces to

Div v=0

a + Grad p = a*

being a the divergence-free local acceleration. The pressure BC ( velocity is zero on a non-permeable no-slip wall) is expressed in terms of the Neumann condition

dp/dn = n.(a* -a)

Note that such BC must fulfill the constraint that the surface integral of the momentum reduces to

Int [S] dp/dn dS = Int [S] n.a* dS


In case of low-Mach problem, you can introduce a linear expansion around the basic state and produce a modified elliptic problem.

From a different point of view, you can use the compressible form of the equations with density, momentum and energy but introducing a pre-conditioning technique. For example Fluent uses such approach.

In conclusion, only once you have specified the mathematical equations, their classification will provide the correct set of equations.

[FMDenaro]

Gives a look here https://pdfs.semanticscholar.org/836...2ec302afa2.pdf

but you can find more modern papers

[HectorRedal]

Originally Posted by FMDenaro

Gives a look here https://pdfs.semanticscholar.org/836...2ec302afa2.pdf

but you can find more modern papers

Hi Filippo,

Thanks for the reference and answer.
In a previous response, you have stated to me that in the incompressiblity limit (M=0), the NS equations are not longer hyperbolic.
The kind of problem I am solving is the Viscous Incompressible Flow.
As far as I know, this kind of problem is hyperbolic in time and eliptical in space. So, if I am trying to solve a transient problem, the kind of BCs for the outflow needed are for hyperbolic problems. Taken as input the paper you have provided, this mean I can use at the exit (du/dx = dv/dx = 0).