Pressure boundary conditions, simpleFoam, plate near a wall

RyArPa · October 21, 2019, 12:37

Hi all,

I am trying to solve steady, laminar, shear flow about a plate near a wall. The aim is to understand how separation and flow reversal occur as viscosity changes and the gap between the plate and wall decreases.

I am using simpleFoam, however cannot work out which BCs to use. I have searched the internet and literature high and low and cannot get to a suitable answer.

- Here I am not sure that the zeroGradient condition holds because nu is non-zero/relatively large and with the plate inclined towards, and close to the wall, the usual arguments of du/dx = 0 etc do not follow.

As a result I feel as though the way forward is to have dp/dn = nu d2v/dn2 on the lower wall and plate boundaries. Is there a way of achieving this in OF and simpleFoam? or should I use a different solver?

The other BCs are:
- Plate: U = zeroGradient; p = ???
- Wall: U = U_w (positive or negaitve_; p = ???
- Inlet: U = U_w + y; p = zeroGradient
- Outlet: U = zeroGradient; p = 0
- Upper: U = U_W + y_max; p = zeroGradient

Any help or advice is appreciated!

Carlo_P · October 22, 2019, 08:30

Hey,
I corrected your BCs:

-Plate: U = zeroGradient; p = zeroGradient
- Wall: U = U_w (positive or negaitve_; p = zeroGradient
- Inlet: U = U_w + y; p = zeroGradient
- Outlet: U = zeroGradient; p = 0
- Upper: U = U_W + y_max; p = zeroGradient

but..I have a question: how you can calculate the separation using a laminar flow?

RyArPa · October 22, 2019, 08:57

Thanks Carlo_P for your time and response.

My supervisor and I are not convinced that the zeroGradient condition holds in this case though due to the presence of the plate and the viscousity of the fluid. I have read the other threads on here and I am not sure the justification there hold here for either the wall or the plate. Could you explain why the zeroGradient would be the correct choice?

As for separation, why would laminar flow be an issue? In theory it is admissible in a laminar boundary layer, and we do see the typical characteristics with flow reversal about the plate, eddies forming below the plate and in the wake, and adverse pressure gradients in the cases we've run.

RyArPa · October 23, 2019, 05:08

In addition, out of interest - is there a way to set a boundary condition of dp/dn = nu d^2v/dn^2 in OF?

cryabroad · October 23, 2019, 05:19

I think it would be better if you put up a figure that shows the domain the boundaries. I'm kind of interested in this, is the plate parallel to the wall and you decrease the distance between them to see what are the effects?

zeroGradient for pressure is pretty straightforward, if you have a pressure gradient near a wall boundary, there will be flow penetrating it, which is physically not possible. You mentioned something about very large fluid viscosity, but that should have nothing to do with pressure. I see that you are using zeroGradient for velocity of the plate, is the plate moving? Anyway, a simple figure would help a lot here.

RyArPa · October 23, 2019, 05:51

Thanks for your interest and advice cryabroad - that is helpful, and it is a fascinating problem when you get into it!

Here is a figure.

- The plate is inclined to the wall, with varying inclination for different scenarios.
- We are in a frame of reference fixed on the plate, so it is static but the wall is moving relative to the plate in either an upstream or downstream direction (giving a positive or negative wall velocity) such that the inlet velocity profile is: U = Uw + y, with the wall at y = 0, and the plate some varying distance from the wall.

The plate U BC was an error - it is no-slip. Sorry.

As for zeroGradient pressure, my understanding was that a pressure gradient near the wall would be equivalent to flow penetrating it, however, from the equations we keep landing at: dp/dn = nu d^2v/dn^2 as the correct condition on the wall... I can't quite resolve this.

cryabroad · October 24, 2019, 05:41

Okay now I get the point. I think the problem may come from the reference frame you are working with. I guess you have to add certain terms in the equation to make the reference frame work, and that's probably why you end up with additional terms. Can you share the derivation of the dp/dn term you mentioned? I think this is where the confusion is. I still think the pressure boundary condition should be zeroGradient, because there should be no additional acceleration term (pressure gradient is basically acceleration) if things are moving at constant speeds in your original problem (before you change the reference frame).

Additionally, do check if OpenFOAM has built-in moving reference frame (MRF) modules. I think it does, at least I know Fluent has it. Such built-in modules should be designed in a way that the user doesn't need to worry about things related to MRF, in that case the pressure boundary condition should definitely be zeroGradient on the wall.

RyArPa · October 24, 2019, 13:45

[Duplicated post]

RyArPa · October 24, 2019, 15:08

Thanks that its helpful advice - I will look into it!

As for the derivation of the BC for pressure (working for the wall, for ease, where the normal is in the

-axis), I am solving the continuity and momentum equations in 2D:

(1)

(2) $uu_x + vu_y = -p_x + \nu (u_{xx}+u_{yy})$
(3) $uv_x + vv_y = -p_y + \nu (v_{xx}+v_{yy})$

The velocity BCs on the wall are:

, which is constant for

(such that $u_x = u_{xx}=0$ ); and

which is constant in

(but not necessarily in

)

Hence,

such that by (1):

Plugging all of this into (3) we get: $p_y = \nu v_{yy} \neq 0$ (although

and

, the overall profile of

is unknown, hence, for example, there could be quadratic contributions such that the above all holds - in fact such contributions are likely given the locality of the plate to the wall and the effect on the fluid flow as a result of its presence).

As a result this makes the BC: $p_y = \nu v_{yy}$ .

This all leaves me with three questions that I am pondering/looking in to... (which I will continue to mull over!!!):
(1) does the above mean that the zeroGradient condition is inappropriate for pressure?
(2) if so, what alternatives/methods are there available in OF to resolve this?
(3) if it is appropriate, why? are there nuances in OF's implementation of the SIMPLE algorithm that I have not considered or perhaps the mathematics above incorrect?

cryabroad · November 4, 2019, 02:43

Sorry for my late response, got caught up in some projects.

I went through your derivation, and I'm not sure if the second inequality in this expression holds, $p_y = \nu v_{yy} \neq 0$ . We don't know what the gradient of wall normal velocity is before we solve the problem, why the inequality? The correct way of doing this is to assume

and go from there.

For boundary value problems, one can only specify one type of boundary condition for one variable at one boundary. In this case,

at the wall is sufficient for

, to state $v_{yy} \neq 0$ at the wall is an overconstraint I think. For example, just from the above expression,

can be a linear function with respect to the wall distance and satisfies the constraint of

at the wall. In such case, $p_y = \nu v_{yy} = 0$ . Anyway, I think searching for MRF in OpenFOAM should be a good option for your case.

RyArPa · November 20, 2019, 05:30

Quote:

Originally Posted by cryabroad

I went through your derivation, and I'm not sure if the second inequality in this expression holds, $p_y = \nu v_{yy} \neq 0$ . We don't know what the gradient of wall normal velocity is before we solve the problem, why the inequality? The correct way of doing this is to assume

and go from there.

For boundary value problems, one can only specify one type of boundary condition for one variable at one boundary. In this case,

at the wall is sufficient for

, to state $v_{yy} \neq 0$ at the wall is an overconstraint I think.

Hi cryabroad, sorry for the delay (I have been away on leave).

One slight confusion, it's my fault for not being clearer - I am not imposing $v_{yy} \neq 0$ , rather I probably should have said $v_{yy} = C$ , where

is some constant that may or may not be 0. Hence, $v_{yy}$ is calculated at the boundary, for which

is found rather than set. If anything, forcing

over-constrains the problem in my mind?

cryabroad · November 23, 2019, 06:46

I think we both agree that the first equality holds, which is $p_y = \nu v_{yy}$ . The problem is how do we go from here.

As I mentioned, for such a boundary value problem, to state anything associated with $v_{yy}$ when we already establish the fact that

is not right, and it does over-constrain the problem. Even if you are assuming $v_{yy} = some constant$ , that is not right in my mind.

However, in fluid mechanics pressure and velocity are coupled. When you specify a boundary condition for pressure/velocity, it implicitly constrains the velocity/pressure (in one form or the other) at the same boudary, which looks like you are over-constraining the problem. In other words, it is not that the $v_{yy}=0$ condition is forced by us, it is forced by continuity. Again, I do think that for an accelerating or decelerating system this is NOT the case, and more derivations have to be done to figure out the correct boundary conditions. But for the system you are considering (which is at constant speed), I think it should be just as simple as a normal zero pressure gradient condition.

RyArPa · November 27, 2019, 07:06

Thanks for your time on this - I think I understand what you are saying now and it is becoming clearer in my mind. I still have a reservation though.

You say:

Quote:

Originally Posted by cryabroad

As I mentioned, for such a boundary value problem, to state anything associated with $v_{yy}$ when we already establish the fact that

is not right, and it does over-constrain the problem. Even if you are assuming $v_{yy} = some constant$ , that is not right in my mind.

To clarify: I am not looking to say that $v_{yy}$ is definitely some constant and thus imposed that as a BC alongside a pressure BC and another velocity BC, since this would over constrain the problem.

Rather, $v_{yy}$ could be constant, or zero, or something else, we don't know. Hence I am looking for a pressure BC such that $p_{y} = \nu v_{yy}$ on the boundary.

I think is mathematically valid since with a zeroGradient BC we have $p_{y} = 0 = \nu v_{yy}$ and are imposing a value on

through the pressure coupling. Which is what I would like to do here, but for a potentially non-zero value.

October 21, 2019, 12:37	Pressure boundary conditions, simpleFoam, plate near a wall	#1
RyArPa New Member Ryan Join Date: Oct 2019 Posts: 8 Rep Power: 7	Hi all, I am trying to solve steady, laminar, shear flow about a plate near a wall. The aim is to understand how separation and flow reversal occur as viscosity changes and the gap between the plate and wall decreases. I am using simpleFoam, however cannot work out which BCs to use. I have searched the internet and literature high and low and cannot get to a suitable answer. - Here I am not sure that the zeroGradient condition holds because nu is non-zero/relatively large and with the plate inclined towards, and close to the wall, the usual arguments of du/dx = 0 etc do not follow. As a result I feel as though the way forward is to have dp/dn = nu d2v/dn2 on the lower wall and plate boundaries. Is there a way of achieving this in OF and simpleFoam? or should I use a different solver? The other BCs are: - Plate: U = zeroGradient; p = ??? - Wall: U = U_w (positive or negaitve_; p = ??? - Inlet: U = U_w + y; p = zeroGradient - Outlet: U = zeroGradient; p = 0 - Upper: U = U_W + y_max; p = zeroGradient Any help or advice is appreciated!

October 22, 2019, 08:57		#3
RyArPa New Member Ryan Join Date: Oct 2019 Posts: 8 Rep Power: 7	Thanks Carlo_P for your time and response. My supervisor and I are not convinced that the zeroGradient condition holds in this case though due to the presence of the plate and the viscousity of the fluid. I have read the other threads on here and I am not sure the justification there hold here for either the wall or the plate. Could you explain why the zeroGradient would be the correct choice? As for separation, why would laminar flow be an issue? In theory it is admissible in a laminar boundary layer, and we do see the typical characteristics with flow reversal about the plate, eddies forming below the plate and in the wake, and adverse pressure gradients in the cases we've run. Last edited by RyArPa; October 22, 2019 at 12:44. Reason: Spelling error

October 24, 2019, 13:45		#8
RyArPa New Member Ryan Join Date: Oct 2019 Posts: 8 Rep Power: 7	[Duplicated post] Last edited by RyArPa; October 25, 2019 at 05:20. Reason: Accidental duplication of posts - see below

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October 22, 2019, 08:30		#2
Carlo_P Senior Member Carlo_P Join Date: May 2019 Location: Italy Posts: 176 Rep Power: 8	Hey, I corrected your BCs: -Plate: U = zeroGradient; p = zeroGradient - Wall: U = U_w (positive or negaitve_; p = zeroGradient - Inlet: U = U_w + y; p = zeroGradient - Outlet: U = zeroGradient; p = 0 - Upper: U = U_W + y_max; p = zeroGradient but..I have a question: how you can calculate the separation using a laminar flow?

October 23, 2019, 05:08		#4
RyArPa New Member Ryan Join Date: Oct 2019 Posts: 8 Rep Power: 7	In addition, out of interest - is there a way to set a boundary condition of dp/dn = nu d^2v/dn^2 in OF?

October 23, 2019, 05:19		#5
cryabroad Senior Member Ruiyan Chen Join Date: Jul 2016 Location: Hangzhou, China Posts: 162 Rep Power: 10	I think it would be better if you put up a figure that shows the domain the boundaries. I'm kind of interested in this, is the plate parallel to the wall and you decrease the distance between them to see what are the effects? zeroGradient for pressure is pretty straightforward, if you have a pressure gradient near a wall boundary, there will be flow penetrating it, which is physically not possible. You mentioned something about very large fluid viscosity, but that should have nothing to do with pressure. I see that you are using zeroGradient for velocity of the plate, is the plate moving? Anyway, a simple figure would help a lot here.

October 24, 2019, 05:41		#7
cryabroad Senior Member Ruiyan Chen Join Date: Jul 2016 Location: Hangzhou, China Posts: 162 Rep Power: 10	Okay now I get the point. I think the problem may come from the reference frame you are working with. I guess you have to add certain terms in the equation to make the reference frame work, and that's probably why you end up with additional terms. Can you share the derivation of the dp/dn term you mentioned? I think this is where the confusion is. I still think the pressure boundary condition should be zeroGradient, because there should be no additional acceleration term (pressure gradient is basically acceleration) if things are moving at constant speeds in your original problem (before you change the reference frame). Additionally, do check if OpenFOAM has built-in moving reference frame (MRF) modules. I think it does, at least I know Fluent has it. Such built-in modules should be designed in a way that the user doesn't need to worry about things related to MRF, in that case the pressure boundary condition should definitely be zeroGradient on the wall.

October 24, 2019, 15:08		#9
RyArPa New Member Ryan Join Date: Oct 2019 Posts: 8 Rep Power: 7	Thanks that its helpful advice - I will look into it! As for the derivation of the BC for pressure (working for the wall, for ease, where the normal is in the -axis), I am solving the continuity and momentum equations in 2D: (1) (2) $uu_x + vu_y = -p_x + \nu (u_{xx}+u_{yy})$ (3) $uv_x + vv_y = -p_y + \nu (v_{xx}+v_{yy})$ The velocity BCs on the wall are: , which is constant for (such that $u_x = u_{xx}=0$ ); and which is constant in (but not necessarily in ) Hence, such that by (1): Plugging all of this into (3) we get: $p_y = \nu v_{yy} \neq 0$ (although and , the overall profile of is unknown, hence, for example, there could be quadratic contributions such that the above all holds - in fact such contributions are likely given the locality of the plate to the wall and the effect on the fluid flow as a result of its presence). As a result this makes the BC: $p_y = \nu v_{yy}$ . This all leaves me with three questions that I am pondering/looking in to... (which I will continue to mull over!!!): (1) does the above mean that the zeroGradient condition is inappropriate for pressure? (2) if so, what alternatives/methods are there available in OF to resolve this? (3) if it is appropriate, why? are there nuances in OF's implementation of the SIMPLE algorithm that I have not considered or perhaps the mathematics above incorrect?

November 4, 2019, 02:43		#10
cryabroad Senior Member Ruiyan Chen Join Date: Jul 2016 Location: Hangzhou, China Posts: 162 Rep Power: 10	Sorry for my late response, got caught up in some projects. I went through your derivation, and I'm not sure if the second inequality in this expression holds, $p_y = \nu v_{yy} \neq 0$ . We don't know what the gradient of wall normal velocity is before we solve the problem, why the inequality? The correct way of doing this is to assume and go from there. For boundary value problems, one can only specify one type of boundary condition for one variable at one boundary. In this case, at the wall is sufficient for , to state $v_{yy} \neq 0$ at the wall is an overconstraint I think. For example, just from the above expression, can be a linear function with respect to the wall distance and satisfies the constraint of at the wall. In such case, $p_y = \nu v_{yy} = 0$ . Anyway, I think searching for MRF in OpenFOAM should be a good option for your case.

November 23, 2019, 06:46		#12
cryabroad Senior Member Ruiyan Chen Join Date: Jul 2016 Location: Hangzhou, China Posts: 162 Rep Power: 10	I think we both agree that the first equality holds, which is $p_y = \nu v_{yy}$ . The problem is how do we go from here. As I mentioned, for such a boundary value problem, to state anything associated with $v_{yy}$ when we already establish the fact that is not right, and it does over-constrain the problem. Even if you are assuming $v_{yy} = some constant$ , that is not right in my mind. However, in fluid mechanics pressure and velocity are coupled. When you specify a boundary condition for pressure/velocity, it implicitly constrains the velocity/pressure (in one form or the other) at the same boudary, which looks like you are over-constraining the problem. In other words, it is not that the $v_{yy}=0$ condition is forced by us, it is forced by continuity. Again, I do think that for an accelerating or decelerating system this is NOT the case, and more derivations have to be done to figure out the correct boundary conditions. But for the system you are considering (which is at constant speed), I think it should be just as simple as a normal zero pressure gradient condition.