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Gassan Abdoulaev April 7, 1999 05:02

Pressure BC for NSEs
I am developing a code for the flow in a tube with the prescribed inflow and outflow pressures (actually, the difference between them). The flow is viscous incompressible and is governed by the Navier-Stokes equation for velocity and pressure. The system is intergrated in time with the projection method. The question is about boundary conditions for velocity equation (conv-diff step) and for pressure equation (projection step). Is it enough to impose homogeneous Neumann for velocity at the inflow and outflow boundaries (and no-slip on the walls), and vice versa for pressure: the Dirichlet conditions at the in/outflow and Neumann on the walls. Can anybody share his experience with respect to this problem? Thanks you in advance!

Gassan Abdoulaev

andy April 7, 1999 06:00

Re: Pressure BC for NSEs
Neumann conditions for an inlet velocity is not a physically reasonable boundary condition. If you are lucky, and the Reynolds number low enough, the flow might settle down to the fully developed laminar solution where the integrated wall shear stress balances the imposed pressure drop. This is an obvious solution to the problem posed. It is also the solution to the more constrained problem of a fixed normal inlet flow angle (a more common thing to do) - I would guess it very unlikely that the flow would settle down.

If the flow is turbulent (unsteady), the boundary conditions do not stand a chance. Attention needs to be focussed on ensuring that the pressure field obtained matches the velocity field (for an incompressible flow, the pressure field can be determined directly from the velocity field apart from a constant value).

In order to make further suggestions I would need to know if you are trying to obtain a fully developed solution? If not, the fact I am asking the question is a clear indication that information is missing from the statement of the problem (i.e. boundary conditions are missing).

Gassan Abdoulaev April 8, 1999 04:01

Re: Pressure BC for NSEs
This problems arises from the vascular (blood) flow simulation, where you don't know the inflow velocity profile, but rather the pressure or the flux. For instance, one, can impose the mean pressure on the artificial boundaries (or the total pressure P+0.5*|U|^2) as a function of time. Here I follow the paper of Heywood, Rannacher and Turek. This problem is not well-posed. The boundary conditions can be reformulated as p-d_n*u_n=P_i, d_n*u_t=0 on the artificial boundaries, where d_n is the normal derivative, u_n, u_t are the normal and tangential components of the velocity. The flow is unsteady (pulsatile), but the turbulence is negligible. Under centain conditions P_i is in fact the mean pressure across artificial boundaries. When you decouple the system using the projection method (Chorin, Temam, etc..), the boundary conditions should also be decoupled. The question was about the way to decouple boundary conditions. Thank you!

andy April 8, 1999 11:56

Re: Pressure BC for NSEs
I am not sufficiently familiar with the numerical scheme you site to answer your b.c. question with confidence.

However, as an aside, I was sufficiently intrigued by the inlet condition to modify an LES code and implement the problem (I think) you describe (apart from rectangular shaped blood vessels). The flow remained reasonable for a while (imposed sine wave pressure drop) but then strong uncontrolled transverse velocities developed in the inlet plane (the pressure gradient is held at zero within the plane by the boundary condition) and the solution procedure failed. Although mistakes in the implementation cannot be discounted, I am fairly sure you will have to modify the d_n*u_t=0 boundary conditions by introducing information from upstream and/or allow pressure gradients to develop within the inlet plane.

John C. Chien April 8, 1999 12:44

Re: Pressure BC for NSEs
It is just my personal opinion that for incompressible flow through a tube, the only physical quantity which controls the whole process is the inlet velocity or mass flow rate(integrated). The pressure should be considered as the derived quantities mainly because of the frictional loss. For inviscid flow, you are not going to have the pressure difference between the inlet and the exit station for the constant diameter tube. The story is completely different for compressible flows, where the pressure does play important role because of the equation of state relationship. In this case, you can set the pressure in any way you like at the inlet and the exit. The flow will adjust itself to this condition. This is not the case for incompressible flow in a tube. Inlet and exit planes are 2-D planes, so, specifying pressure distributions at two 2-D planes in the flow field will force the whole velocity filed to seek a configuration such that the frictional force can balance out the pressure force created by the boundary conditions. And if the tube is connected to a larger tube at the inlet, then is the pressure at the inlet plane a constant value or a non-uniform distribution? So, what I am saying is, for incompressible internal flows we normally specify the inlet velocity distribution. We can easily control the inlet velocity ( or mass flow rate) but it is not practical to control both the inlet and the exit pressure distribution.( that is what you are trying to do mathematically)

andy April 8, 1999 14:46

Re: Pressure BC for NSEs
I am not sure I fully understand this posting but disagree with some of it and feel obliged to reply.

If you do not know the mass flow or inlet velocity profile you cannot prescribe it whatever is normally done. Something else has to be tried.

So long as the problem is correctly formulated, I cannot see that it matters whether you accelerate the flow by prescribing a varying inlet velocity, specifing some pressure relationship between inlet and exit, specifying the wall shear stress or applying body forces. However, if there is something putting time varying work into the fluid upstream I know which one is likely to be easiest to formulate in the axial direction. My interest was in the inlet boundary conditions in the transverse directions. For a 2D flow, du/dx=0 would force dv/dy=0 via continuity because v=0 on the wall and there would be no problem. This is not the case in 3D where dvdy = -dwdz = anything from continuity. At low Reynolds numbers, viscous effects propagating upstream would (probably) prevent things getting out of hand (although the meaning of the simulation would be questionable) but at higher Reynolds numbers the controlling mechanism is information propagating from upstream (like at the exit which behaves itself) and this is absent in the problem formulation. In order to proceed further, one would need to know more about what is going on upstream so that a parameterized relationship could be formulated for the inlet boundary condition. Prescribing v and w, assuming a parameterised profile shape or prescribing a parameterised flow angle are all likely to work. So long as the flow is not evolving strongly in the axial direction, the assumption of a constant pressure across the tube would seem reasonable.

Gassan Abdoulaev April 9, 1999 04:38

Re: Pressure BC for NSEs
Certainly, it would be better to have the inlet velocity prescribed, but unfortunately it's practically impossible to give a reasonable velocity distribution beforehand, whereas the (mean) pressure can be measured or estimated. Besides, you don't know in advance, where the flow goes in or where it goes out. Well, for a single tube it's obvious, but if you've got a more complex system of connected tubes with three or more in/outlets, everything would depend on the relative pressure values on artificial boundaries (cross-sections). The pressure can be assumed constant at the section plane.

The alternative formulation is to prescribe the net flux, then the mean pressure values would be unknowns.

John C. Chien April 9, 1999 12:04

Re: Pressure BC for NSEs
(1). this is just an exercise. (2). there is a long tube, and you apply (specify) the inlet pressure at station-1 and apply the exit pressure at station-2. (3). then someone decided to apply the velocity boundary condition upstream of the station-1, and set it equal to zero ( just like a wall, or to seal off the inlet). he also decided to specify the velocity condition downstream of the exit and set it equal to zero. (4). the question is: what is the flow field between the station-1 and the station-2 ? ( especially, when you don't know what is happening upstream of station-1 and downstream of station-2.)

John C. Chien April 9, 1999 12:29

Re: Pressure BC for NSEs
by the way, station-1 and station-2 are somewhere on the tube and the computational domain is the tube between station-1 and station-2.

John C. Chien April 9, 1999 13:01

Re: Pressure BC for NSEs
(1). to make the Friday's exercise short, the answer is: to specify the pressure difference only tells the iterations loop when to stop, it does not produce a solution.(2). In reality, at each iteration, the inlet velocity is specified ( or changed) or re-specified, and the flow field is then solved. If the result is not consistent with the desired pressure difference condition, then the inlet velocity is readjusted and a new case ( iteration) computed. (3). In this case the pressure difference is acting like a source term to modulate the viscous effect. (4). When this is no viscous effect, there is no pressure difference, but still you can keep the flow moving by specifying the inlet velocity.

Gassan Abdoulaev April 12, 1999 10:21

Re: Pressure BC for NSEs
I've mentioned earlier, that fixing pressure only is not enough for the problem to be well posed. There is no point to discuss. The boundary conditions should be modified in a known way (see Heywood, Rannacher, Turek). The original question was how those conditions are applied within the framework of the projection time stepping scheme, when the problems for the velocity and pressure are decoupled. This question is addressed particularly for those who have any experience concerning the problem.

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