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Salem September 21, 2005 16:50

periodic boundary conditions fro pressure
Dear all,

I am solving NSE using a finite volume method for laminar flow in a 2D channel. The momentum equations has a body force. This body force is due to the electric field (which is coupled with the charge conservation equations and solved before the momentum equations are). NSE were non-dimensionlized in a way the Re is (Reynolds number based in the electric filed applied; i.e. if electric field goes to zero no flow is generetaed). My question is if I apply periodic boundary conditions to the velocities and pressure am I:

1) over-spesfiying the boundary conditions for flow field by applying periodic to the pressure.

2) I understand that periodic boundary condition for pressure means that no pressure gradient in the system and thus no flow. Howevere, can I say that the flow I am generating is solely due to the body force.

agg September 22, 2005 10:29

Re: periodic boundary conditions fro pressure
1) You are not over-specifying the bc by applying periodic bc to the pressure. You are solving a second order pde for presure, and you will need bc for this.

2) Yes, the mean pressure gradient that is driving your flow is generated by the body force.

Salem September 22, 2005 12:25

Re: periodic boundary conditions fro pressure
Thanks for your response.

How it is a second order Pde for pressure?

If it is, then is the peroidic bcs for pressure are:

dp/dx)in=dp/dx)out and Pin=Pout



agg September 22, 2005 15:13

Re: periodic boundary conditions fro pressure
d2p/dx2 + dp2/dy2 + dp2/dz2 = F is a second order Poisson equation for pressure and in 3D you need 2 boundary conditions for each direction.

For periodic, pin = pout

Mani September 22, 2005 15:21

Re: periodic boundary conditions fro pressure
> 2) Yes, the mean pressure gradient that is driving your flow is generated by the body force.

Contradiction. If there is a non-zero "mean pressure gradient" throughout the channel (generated by a body force), how can Pin be equal to Pout?

Isn't it enough to specify the pressure gradient for incompressible flow? (dpdx)in = (dp/dx)out is not a periodic boundary condition for pressure, but a periodic condition on pressure "gradient"! That would be perfectly fine. I am not so sure that the actual pressure condition Pin = Pout is needed, and is problematic anyway (see above).

jasond September 22, 2005 18:57

Re: periodic boundary conditions fro pressure
Well, what you can do is "subtract out" the mean gradient part of the pressure, add that to the body force term, and then the pressure term in the NSE is no longer the real pressure (and can be periodic). In this specific case, I'm not sure that there must neccessarily be a mean gradient in pressure.


agg September 23, 2005 00:21

Re: periodic boundary conditions fro pressure
The body force provides a mean pressure gradient that drives the flow. The pressure however, that appears in the NS equations can be periodic. For example, refer to Kim and Moin JFM Paper on channel flow simulations (1982). In their work, they add a mean pressure gradient (like body force) to their equations, whilst maintaining the pressure gradient term, and using periodic bc for solving pressure Poisson equation.

andy September 23, 2005 06:41

Re: periodic boundary conditions fro pressure
Is your code incompressible or compressible? This will influence what can or cannot be done with the pressure.

If your flow is in a channel then there will be shear at the wall. This must balance what is driving the flow. Usually this is a drop in pressure between the inlet and exit of the channel caused by something external. In your case this could be the electric field or a combination of electric field and pressure drop/rise. What do you know about the pressure at inlet and exit? That is, what is going on outside your solution domain to establish your appropriate boundary conditions?

Salem September 23, 2005 08:44

Re: periodic boundary conditions fro pressure
Thank you all,

My code is incomprissible and steady. The overall balance in the problem can be predifined to be one between the electtric field body force and the fluidic resistance of the channel with no exterel pressure difference maintaned between inlet and outlet. Can I in this case use the following boundary conditions for the pressure (in addition to the periodic ones for the velocities)

1) Pin=Pout

2) dp/dx)in= dp/dx) out


andy September 23, 2005 09:53

Re: periodic boundary conditions fro pressure
Are you using staggered velocity components or are variables stored at the same location on the grid? If the former, you may not need to specify any pressure condition although you will have to a constraint on the velocity to ensure global mass conservation.

The appropriate boundary condition will still depend on what problem you want to specify. What physically is your inlet and outlet? What determines the mass flowing down your channel? Do you want to determine the fully developed profiles in a straight channel? Or do you want to study the development from known inlet conditions?

Without knowing what problem you are trying to setup it is still not possible to answer precisely.

Does condition (1) mean a single constant value over both inlet or exit or that the pressure can vary over the inlet/exit but must do so in the same way (a periodic condition or, at least, part of one).

Is condition (2) constant over the inlet/exit plane?

Salem September 23, 2005 10:48

Re: periodic boundary conditions fro pressure
I am trying to study a flow driven solely due to the electric field body force. In reality this channel is connected to other channel dowensreams and upstreams. So the presure coming from the upstream channel equal to the inlet of my channel and the exit from mu channel equal the inlet at the downstream one. So pressure inelt = pressure exit at my channel no specified value is given. and since the pressure equation using finite volume staggered (TDMA is applied to solve) is second order PDE then I need another boundary condition which is pressure gradient at inlet and exit are equal.


Salem September 23, 2005 10:51

Re: periodic boundary conditions fro pressure
Basically trying to apply periodic pressure boundary conditions as I am doing the same for the velocities.

Mani September 23, 2005 13:31

Re: periodic boundary conditions fro pressure
In that case the body force is replaced by an "equivalent" pressure gradient, added to the equation as a source term. That still doesn't mean there is any real pressure gradient in the flow,... just be careful with your wording.

andy September 23, 2005 17:55

Re: periodic boundary conditions fro pressure
If you apply periodic conditions on the pressure and velocity you will be performing a simulation fairly common with LES codes. Note you must solve the normal velocity component on one of the periodic planes if you are not already doing so.

The flow will accelerate under the action of the body force until it is balanced by the wall shear stress. You should end up with the fully developed velocity profile at all locations along the duct assuming the body force is constant.

jasond September 24, 2005 14:44

Re: periodic boundary conditions fro pressure
No, even if you move the mean gradient to a body force term, there still is a pressure gradient. The "pressure term" just is not the real pressure anymore.


Mani September 26, 2005 13:10

Re: periodic boundary conditions fro pressure
Ok, either way, my point was: you cannot have a quantity uniform with a non-zero gradient. That's a contradiction. So either the gradient is not real, or the pressure is not real, as you say. However, if the pressure term is not real pressure, than the gradient of that term cannot be the real pressure gradient either. The question now is, how do you apply coundary conditions. Using the real pressure, and real pressure gradient or the pseudo pressure? I suppose it doesn't matter, since you have to prescribe the body force anyway.

jasond September 26, 2005 17:05

Re: periodic boundary conditions fro pressure
I think you may be misunderstanding me. If we think of the pressure as being made of two parts, as in

p = p(local) + p(mean gradient),

then we can apply periodic bc's to the "local" part if we wish to. The pressure term in the NSE is then

dp/dxi = dp(local)/dxi + dp(mean gradient)/dxi

For many cases, the second term is a constant (and probably known somehow), and can be thought of as a body force. The bc's applied to the "local" part will play a large role in the success of the simulation. If they are inconsistent with other considerations (force balances, etc.), then bad stuff should happen. The "local" term is not the real pressure, but the gradient in pressure is real.

Glen September 28, 2005 06:46

Re: periodic boundary conditions fro pressure
Hello All,

In the absence of a body foce like an electric field, how is the flow being pushed ? For the LES in channel flows, the mean Pressure gradient is what pushes the flow , but how do we know the 'fixed mean Pr gradient(body force)' priori, as per Moin and Kim. In fact how does one calcualte the same. I am sure that it should be corrected everytime, but the means of calculating the same is not mentioned, nor the terms that is used to calculate. Kindly clarify the same.

Kindly help if someone has any idea,

Thanks in advance,


andy September 28, 2005 08:11

Re: periodic boundary conditions fro pressure
It depends on what you want to simulate.

The most common LES simulation of this type imposes a constant mean shear stress and the corresponding constant mean pressure gradient. The predicted mass flow through the channel will vary with time.

An alternative would be to hold the mass flow constant and iteratively adjust the wall stress and/or pressure drop to maintain an instantaeous force balance.

Neither is "perfect" and other variations are possible. An infinitely long fully developed duct will have all three constant but, of course, will not have periodic conditions (apart from a jump in pressure level between inlet and exit).

agg September 28, 2005 11:24

Re: periodic boundary conditions fro pressure

Here is a reference:

"Numerical methods for direct simulation of turbulent shear flows" - Tran, K.D. and Morchoisne, Y., von Karman Institute for Fluid Dynamics, Lecture Series 1989-03

Here they give a formula to calculate the forcing function as a function of time. In general, at each time step, whatever flow rate is lost, is added back again.

F_n+1 = F_n + 2/dt*(Q_n+1 - Q_0) - 1/dt*(Q_n - Q_0)


Q is the flow rate per unit cross-sectional area

n is current time step

Q_0 is initial value of flow rate per unit cross-sectional area

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