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Old   November 15, 2008, 15:05
Default Inflow-Outflow
  #1
Rajesh
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I was wondering if someone could throw some light on inflow-outflow problem in a CFD study. I understand that fluid motion is caused by a pressure gradient. What is the driving force in case of an inflow-outflow problem if not the pressure gradient ?

Thanks n regards Rajesh
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Old   November 16, 2008, 16:02
Default Re: Inflow-Outflow
  #2
Ahmed
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Look the Navier's Stokes equation, move the pressure term to the left hand side and read it. It is the combined effect of convection and pressure gradient that cause the motion of fluids, Hope you get the picture, and Good luck.
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Old   November 16, 2008, 18:08
Default Re: Inflow-Outflow
  #3
Jonas Holdeman
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It is well-known (though mostly to mathematicians) that the NSE can be decomposed into solenoidal and irrotational parts by the Helmholtz decomposition. The solenoidal part does not (and cannot) contain the (irrotational) pressure-gradient term. In this case, the governing equation for the velocity cannot be pressure-driven, but something else must take the place of the pressure gradient. In 2D, that something is the boundary condition on the stream function, that is, the difference in the stream function across the inlet or the outlet, which determines the net flow. So where does the stream function come from? Any divergence-free field can be written as the curl of a stream function (or vector potential in 3D). So the divergence-free velocity is inevitably tied to a stream function, even if the stream function does not appear in the primative variable formulation. Of course other formulations such as the stream function-vorticity method eliminate the pressure gradient, while explicitly introducing the stream function. In his book, Phil Gresho remarks that any method that tries to use divergence-free bases seems to result in something that looks like a stream function.

So how does the pressure-driven formulation arise? Given the velocity field, the pressure can be found using something that always looks like the presssure-poisson equation. One can solve the coupled velocity and pressure equations simultaneously without BC on the stream function, in which case the pressure is a control variable, with a velocity field that is consistent with the pressure boundary conditions.
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Old   November 17, 2008, 09:55
Default Re: Inflow-Outflow
  #4
otd
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Is this true for compressible flows?
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Old   November 17, 2008, 16:01
Default Re: Inflow-Outflow
  #5
Jonas Holdeman
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otd asks: Is this true for compressible flows?

No. Compressible flows are not, in general, divergence-free, but rather satisfy the more-general compressibility condition. Compressible fluids may exhibit "incompressible flow," flowing as if they were incompressible at low Mach number (v < .3 Mach). Pressure equilibrates at the speed of sound (sound waves are pressure waves). So if the flow is slow enough (much less than the speed of sound), pressure stays in equilibrium on time scales of interest.
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Old   November 18, 2008, 10:00
Default Re: Inflow-Outflow
  #6
Rajesh
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Thanks all. In particular, Ahmed's reply is closer to what I am looking at. So, in absence of pressure gradient, it's the convection that's making the fluid flow. And, this is called inflow - outflow problem. right ? And, do we need a cause, as such, for Convection? Like, what's causing convection ?
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Old   November 18, 2008, 16:02
Default Re: Inflow-Outflow
  #7
Ahmed
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definitley you can ask that question, but you will end up in a vicious circle of what causes what etc. Let me give you a simple example, go to the nearest river, throw away a small branch and observe what will happen. the water stream will carry the branch away, right, this is convection, now you can start the what questions, what causes the water to flow Gravity what causes gravity or what is that gravity, etc...you see what I mean Good Luck
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Old   November 18, 2008, 17:03
Default Re: Inflow-Outflow
  #8
Rajesh
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Thanks Ahmed. Actually my senior gave me a CFD code to go through. In that code, only velocity profile is specified at the inlet. Now, physically, we understand that fluid flows, but how would we ensure the flow in the computational domain numerically ? I thought a pressure gradient needs be defined, but the answer I got was that it is an inflow-outflow problem. I had asked my first question in this context.
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Old   November 19, 2008, 10:46
Default Re: Inflow-Outflow
  #9
Jonas Holdeman
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If you have specified the inlet profile and hence the net flow, you have specified the stream function across the inlet. As I stated earlier, this is all that is necessary to develop a solution without reference to a pressure.
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Old   November 19, 2008, 15:24
Default Re: Inflow-Outflow
  #10
Paolo Lampitella
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Actually, as also stated by ahmed, i think the question is a little more subtle. Obviously we are talking about divergence free velocity fields.

Numerically speaking you're right, there's no need to specify the pressure (at least not its absolute value), neither in a primitive variable approach (for example in the fractional step method of Chorin), more than ever in a vorticity-streamfunction approach.

Anyway there is a sound connection between them, both satisfying an elliptic equation and both being responsible for satisfying the cinematic condition:

dU/dx + dV/dy + dW/dz = 0

that is a divergence free velocity field.

Actually this connection can also be shown to be (in indicial notation):

d2P/dx_ix_i = - S_ij * S_ij + 1/2 * d2SF_i/dx_jx_j d2SF_i/dx_kx_k

with:

S_ij = 1/2 (dU_i/dx_j + dU_j/dx_i)

SF_i = i component of the 3D StreamFunction

P = Pressure

d2()/dx_ix_i = sum of second derivatives

Also, it can be shown that the pressure acts like a lagrange multiplier enforcing the divergence free condition. And, for 3D problems you have just 1 equation, which is no more true for the streamfunction (they becomes 3).

Anyway, going to the crude physics, the cause of a fluid motion is a force. You can have surface forces or volume forces. Now let specify this for inflow outflow problems.

Consider a long pipe open at both the endings as your domain. You can have an inflow-outflow flow in 3 ways (maybe more):

1) Creating, with something like a pump, a pressure gradient, between the inflow and the outflow, which will drive the flow. This can be seen as both kind of forces (consequence of the divergence theorem):

a)Surface - if considering the pressure on inflow and outflow faces

b)Volume - if considering the pressure gradient acting in any point of your domain, like a volume force

2) Using any kind of volume force, like gravity (which can actually be put in a divergence form so becoming like the previous case), or anything else can't be put in the form of a potential function

3) Using your pipe like a missile and launching it through the air. In this case, the cause of the velocity field inside the pipe is a combination of the external velocity given to it and the viscous forces acting on the internal surface of it (the external ones being of not so much interest in this case). Even in this case there is a pressure gradient inside the pipe but obviously it's not a cause of motion but a consequence of the pipe motion through the air (A combination of pressure increase in front of the pipe, which has to equilibrate the viscous effects inside the pipe, and an incomplete recover behind it, still due to viscous effects, actually a separation of the flow).

Whatever approach you choose the pressure has still the same strong meaning while in a 3D case like the 3rd one i just can't figure out what the streamfunction is supposed to be.
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Old   November 20, 2008, 10:33
Default Re: Inflow-Outflow
  #11
Jonas Holdeman
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If one uses divergence-free basis functions for the velocity, any linear combination of them is divergence-free. Then there is no Lagrange multiplier needed to enforce continuity. The pressure is that consistent with the flow, and invoking the Lagrange multiplier - pressure connection doesn't make sense. As I said, the NSE can be split into orthogonal solenoidal (for v) and irrotational (for P) parts. Likewise any body forces also split into orthogonal parts. The velocity is influenced only by the solenoidal forces, while the pressure is influenced only by the velocity and irrotational forces (such as gravity gradient). An example of solenoidal driving body forces would be the electromagnetic propulsive force produced by crossed electric and magnetic fields in a conducting fluid.

The stream function in 2D is just one component of a vector potential function. In 3D, it is the line integral of the vector potential around the inlet or outlet or anywhere along the pipe confining the flow that replaces the pressure as mathematically determining the flow.

Realize that all fluids we experience are compressible, and incompressibility is an ideal for which we have no direct experience (the closest thing in nature to incompressibility is probably the core of a neutron star, and we certainly have no experience with that). Incompressibility is inconsistent with the accepted principle of relativity, but it is still useful, though we must be careful using "common sense" (experience) to make arguments about it.
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Old   November 20, 2008, 18:38
Default Re: Inflow-Outflow
  #12
Paolo Lampitella
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Actually, i don't think i have mentioned any kind of fluid, which of course are more or less all compressible.

We were talking about divergence-free velocity fields. Also, in my knowledge, the streamfunction SF (more precisely the vector potential function for the general 3D case) being defined as (the only one i know):

rho * U = curl(SF)

such that the mass flux through a surface, due to the Stokes theorem, becomes the line integral of SF all around the boundary of the surface, always give a divergence-free velocity field (if considered unsteady) due to the properties of the symmetric-antisymmetric tensor product so...we are definitely talking about divergence-free velocity fields.

That said, if i remember about the helmoltz-hodge decomposition, it says that any vector field U defined in a bounded domain D, both with certain (very general) costraints about regularity, can be split in two parts, a curl-free one and a divergence-free one. Also this decomposition can be shown to exist, be unique and orthogonal.

So we can argue that a divergence-free velocity field can be composed in:

U = grad(phi) + curl(PSI)

Moreover the theorem also states that you can impose just one boundary condition, on PSI or phi, to have a well posed problem.

So, as you can see, i agree on the fact that all you need to solve a problem is a boundary condition on PSI and you have a well defined problem.

Now, one question is why this should invalidate the "pressure view"? And why the pressure - lagrange multiplier connection is invalidated by this?

The other question is which view is the best one? I mean, there are some reasons because everyone uses the primitive variable approach. One being the fact that the (u,v,w,p) equations are easier to solve and with an immediate meaning (what should i do with one component of the vector potential function?); the second being the fact that nearly always the pressure is needed to compute forces.

Actually, having the velocity provided on the domain boundaries is a well posed problem also without considering the streamfunction, but it doesn't mean it is an explanation of why the fluid has that velocity profile on the boundaries. More than ever you can specify the velocity at the turbulent exit of a channel with a bluff body inside it...i mean, it is part of the solution or at least it should be to have physical sense. To be more specific, if at some station along a pipe (which until now we have called inflow) we ha ve a specified velocity profile this doesn't mean that upstrem of that station nothing exists. Probably there will be some other station which has influenced the profile specified at the inflow and so on... as stated by ahmed, a river doesn't flows because someone has specified a velocity profile but because of gravity.
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Old   November 21, 2008, 12:09
Default Re: Inflow-Outflow
  #13
Jonas Holdeman
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I am not an expert on functional analysis so I cannot construct mathematical proofs of existence and uniqueness, though much of this was done several decades ago. Even the mathematicians are cautious about physical interpretations. One distinguished mathematician remarked to me many years ago that the decomposition of the NSE equations was just a mathematical trick to prove existence and uniqueness (in the mathematical sense). I am just a retired physicist working alone using (mostly continuous) divergence-free finite elements to compute incompressible flow. This approach was attempted in the mid-eighties and largely abandoned as impractical because no one was able to devise pointwise (strong) divergence-free elements. One has to be careful with the terminology here, because the term "divergence-free" has been applied mostly in the weak sense, or even to describe discontinuous elements giving fields which are not divergence-free at element boundaries.

The trick, it turns out, is to use Hermite elements for the stream function, where the derivative degrees-of-freedom are the components of the curl, i.e. velocities. One takes the curl of the stream function element to get the divergence-free velocity elements. If the elements are sufficiently continuous (vanishing on the appropriate parts of the element boundary), then it doesn't matter if you include the pressure gradient, the velocity functions will be orthogonal to the pressure gradient and the pressure will be decoupled from the velocity and not appear in the assembled equations. You don't need the projection operators found in the mathematical theory, orthogonality is sufficient. BUT, all workable divergence-free elements that I have found (and no one else has ever published any) result from taking the curl of a suitable Hermite (stream function) element, and so inescapably involve the stream function. Then one has to rethink how to redefine all the classical problems in a way that does not involve the pressure.

So why use this approach? Because the problem is now solving a simple partial differential equation rather than a saddle point problem -- no Lagrange multipliers to worry about and no operator splitting and LBB condition to satisfy and no algebraic differential equations to solve. And the stream function, often used for visualizing flow with contour plots comes right out of the solution, not as a post-processing step.

A curious result is that many flow-through problems, which is the topic of this discussion thread, can be solved with remarkably little input data. The backwards-facing step can be solved without specifying the inlet profile and without any condition on the outflow. One specifies the stream function difference across the inlet and one gets flow results which match the benchmark solutions. If the inlet channel is long enough (and it doesn't need to be very long), the inlet flow automatically becomes parabolic. One can truncate the mesh right through the recirculation bubbles and one still gets a flow which is very close to the results in a longer channel.

Another problem I solve is flow in a (2D) room (perhaps with boxes on the floor) driven by a ceiling fan. I simply specify the stream function difference across the fan tips. To do this with the pressure, one would have to specify a discontinuous pressure and the pressure difference across the fan. If I go down to Home Depot and look at the specifications on ceiling fans, they will give me the flow (say in cubic feet per minute), but not the pressure difference.

Then given the velocity field, I calculate the pressure and vorticity, and the results agree with benchmark problems, and are perhaps benchmark results themselves. Given the efficacy of this method, I don't think it unreasonable to assert, as I have done, that the problems can be formulated with stream function boundary conditions.

With regard to the Helmholtz-Hodge-Whoever decomposition, most of the classical flow problems for which there are solutions can also be solved by orthogonal decomposition. The decomposition U=grad(phi)+curl(PSI) which Paolo states is not quite correct and the actual statement avoids this ambiguity. There is the possibility of an additional term which is a solution of a Laplace equation and is both divergence-free and irrotational. The trick in solving classical solvable problems (which are mostly Stokes problems) involves the partitioning of the "Laplace" term between the divergence-free and gradient parts to satisfy the problem conditions.

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Old   November 21, 2008, 12:44
Default Re: Inflow-Outflow
  #14
momentum_waves
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A more simple way to look at the issue is based on Newton's 2nd law -> sum of forces = mass x total acceleration.

The motion, for a fluid, results from the sum of the forces acting on a fluid particle. Forces at work are external body force & (nabla o stress tensor).

Take a look at eqn (1.1) in "The Naver-Stokes Equations - A classification of Flows and Exact Solutions", Drazin P. & Riley N., CUP,(2006).

The constitutive equation for a Newtonian fluid - for stress tensor - then brings us to a relationship in terms of pressure gradient & stress-relationships.

The convection term arises out of the total acceleration term (Lagrangian viewpoint), when viewed from a Eulerian viewpoint. Convection is part of the total acceleration of the 'squishy' particle - the spatial acceleration component. The other component is the temporal (time-based) acceleration term.

So, essentially, the driving forces acting on the fluid particle turn out to be: pressure-gradient (force), shear stress (force) & external body force. The N-S equation is an acceleration formulation & needs to be multiplied by the particle mass to convert into a force balance.

mw...

www.adthermtech.com/wordpress3

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Old   November 22, 2008, 12:15
Default Re: Inflow-Outflow
  #15
Paolo Lampitella
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I had a suspect that we were going to introduce FEM methods.

Even if i really hate this approach for fluids, basically because it seems to me much more a math trick with which matematicians feel comfortables than a pratical way to handle CFD problems, i have full respect of this and there's no way i can say that your method is not a viable approach.

That said, i really don't understand why the projection on a local function basis should be better than any other method and why a mathematical trick (as also defined by you) should let us redefine our understanding of why fluids flows.
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Old   November 22, 2008, 14:41
Default Re: Inflow-Outflow
  #16
Ahmed
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I am just a retired physicist

Jonas, I am a retiring mechanical engineer, so I guess we can talk openly, my experience taught me to use the basic fundamental laws to explain and communicate with other people no matter how qualified they are. Tensor analysis is a very difficult way of explaining especially when talking to undergraduates, the case of the original post. He is happy with the simple physical explanations I gave him.

I do not want to be rude, but honestly speaking, though I am very familiar with the decomposition of tensor fields to rotational and irrotational fields, but I have never used it in my whole life, and though finite element methods have been around longer than the finite volume methods, but today, most commercial and academic programmes use the finite volume method, why?

There is a physical phenomena, explaining it using basic principles or complicated mathematical tools is a matter of personal choice, but in both cases, one has to feel happy that the audience has understood the message.

If you want to know the pressure rise across a fan, contact the manufacturer and they will be happy to provide you with their performance curves. (an Engineer solution) Enjoy your computing
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Old   November 24, 2008, 12:16
Default Re: Inflow-Outflow
  #17
Jonas Holdeman
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Ahmed,

When one finds something that works very well, as is the case with the method I described, then it is useful to understand why it works. A person watching the flow of water quickly develops a recognition that flow does not accumulate in corners and flow goes around objects without having to know about pressure.

In learning mechanics, one first learns to solve problems (a limited set) using kinematics and later dynamics. Kinematics uses conservation laws or rules that have a deeper basis in forces, but kinematics can solve problems where the forces are not known or are very complicated. I view the method using divergence-free finite elements (DFFEM) as a kinematic method (though that term already has use in fluid flow with a different meaning). The "conservation law" in this case is incompressibility. I argue that incompressibility is all one needs to solve the class of flows called incompressible. The case of compressible flow is more complicated and requires the use of dynamics and thus forces including pressure.

So there remains the problem of explaining why the DFFEM works. I have learned one very practical rule in my later life: if you are looking for something and you have looked everywhere it could possibly be, then start loking where it could not possible be. Rephrasing, if you can find no explanation within your belief system, then look for an answer outside your belief system (and many people hold multiple incompatible belief systems). But I am waxing philosophical.

The best quasi-physical explanation I have goes something like this: first realize that pressure waves propogate at infinite speed in a (hypothetical) incompressible fluid. If one follows the elementary process of examining the forces on a small volume of fluid with the intent of applying the law of conservation of momentum, then one might consider the force of pressure. One imagines the pressure as being different on opposite sides of the volume. This leads to a net pressure force on the fluid and a contribution to the acceleration. But this assumes that one actually can create such a force. If the fluid is incompressible, then a pressure difference would be instantly relaxed and vanish. In this view one cannot create the pressure difference that would lead to a pressure gradient in the governing equation. In a sense, this is done all the time when neglecting the propagation of sound in a fluid. One considers averages over a time interval that is large compared with the time for sound to move across the small fluid volume, but small compared with the bulk movement of the fluid.

I admit I am not entirely comfortable with this dynamic explanation, but maybe some day I will be when I expand my belief system a little more. I do think that the strongest argument is the one that this is a kinematic solution (in the sense of classical physics).
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