[Jaydi_21]

I'm working with a finite volume scheme to solve the shallow water equations on a 2d flow. However, I'm having issues understanding the boundary conditions.

So far, I think I understand that the condition at the wall can be either non slip, or slip. I don't know however when the slip condition can be applied. At the same time I know that a boundary can be reflective or transmissive? (I know there are many more possibilities, but for now I'm focused on these cases).

My question is, for a 2d case if I want to put a reflective boundary, then I should fix the normal velocity of the next cell (beyond the domain) equal to the one of the last cell within the domain, right? If that is the case, then what should I fix for the tangential velocity? Because I'm using an explicit scheme, I need a value for the tangential velocity also. Does this value depend either on a non slip or a slip condition? Then the boundary is reflective and non slip? Or just the reflective boundary condition involves fixing also the tnagential velocity to some value?

[praveen]

If you have a first order problem like inviscid flow, you can only say that normal velocity must be zero because the fluid cannot penetrate into a solid wall, so there can be tangential or slip velocity.

For second order (parabolic) problem like navier-stokes, the full velocity must be zero at a stationary wall, which is called no-slip condition.

Reflective boundary is used as a synonym for slip boundary.

For slip boundary: assuming you are using a cell-centered formulation, i.e., your first grid point is above the wall, you can set the velocity in the ghost cell by reflecting the normal component and the tangential component is unchanged. So if is velocity vector at first cell, then you set



where is the normal vector to the boundary.

[Jaydi_21]

Originally Posted by praveen

If you have a first order problem like inviscid flow, you can only say that normal velocity must be zero because the fluid cannot penetrate into a solid wall, so there can be tangential or slip velocity.

For second order (parabolic) problem like navier-stokes, the full velocity must be zero at a stationary wall, which is called no-slip condition.

Reflective boundary is used as a synonym for slip boundary.

For slip boundary: assuming you are using a cell-centered formulation, i.e., your first grid point is above the wall, you can set the velocity in the ghost cell by reflecting the normal component and the tangential component is unchanged. So if is velocity vector at first cell, then you set



where is the normal vector to the boundary.

Should that expression give me that the normal component is just the opposite direction and the tangential one, remains the same?. Let's say (normal) and (tangential) are the components of the velocity vector , then I was setting for the ghost cell (yes, I have a cell-centered formulation):




But I have a cartesian grid and with that expression, I'd only have one value of velocity, is that just for the normal component? And then the tangential, I just simply say it remains unchanged?

[FMDenaro]

Originally Posted by Jaydi_21

Should that expression give me that the normal component is just the opposite direction and the tangential one, remains the same?. Let's say (normal) and (tangential) are the components of the velocity vector , then I was setting for the ghost cell (yes, I have a cell-centered formulation):




But I have a cartesian grid and with that expression, I'd only have one value of velocity, is that just for the normal component? And then the tangential, I just simply say it remains unchanged?

In your original question I see a mix of doubts about physical and numerical BC.s to se at a wall.

1) Physical BC.s: use the no-slip condition for real fluid where viscosity is taken into acccount. The slip condition is used for the simplified inviscid flow model (Euler equations).

2) Numerical B.C.s: if you are looking for a way to set a value in a ghost point, say at a location yh (1d example), having the value at the wall (yw) and in a set of interior nodes (y1, y2, y3,..), you have to prescribe a value that is congruent with the accuracy order of your scheme. If you have a low order scheme, no more than second order, you can use the values at yw and y1 in order to evaluate a linear law and extrapolate the value at yh. If the scheme has higher order of accuracy you need to extrapolate the value at yh using more infos, for example the value at y2 (or, when possible, the condition of the normal derivative at the wall).

[Jaydi_21]

Originally Posted by FMDenaro

In your original question I see a mix of doubts about physical and numerical BC.s to se at a wall.

1) Physical BC.s: use the no-slip condition for real fluid where viscosity is taken into acccount. The slip condition is used for the simplified inviscid flow model (Euler equations).

2) Numerical B.C.s: if you are looking for a way to set a value in a ghost point, say at a location yh (1d example), having the value at the wall (yw) and in a set of interior nodes (y1, y2, y3,..), you have to prescribe a value that is congruent with the accuracy order of your scheme. If you have a low order scheme, no more than second order, you can use the values at yw and y1 in order to evaluate a linear law and extrapolate the value at yh. If the scheme has higher order of accuracy you need to extrapolate the value at yh using more infos, for example the value at y2 (or, when possible, the condition of the normal derivative at the wall).

Let's say that I use the non slip condition as you suggest. Does that imply that the wave will not reflect? I mean, I want the boundary to reflect the incoming waves...I have a first order scheme, so I'm just using linear interpolation.

Because the non slip condition actually set the normal velocity to 0, right? and the tangential one also is 0? But if I want the waves to be reflected, then I need to null normal velocity!

[FMDenaro]

Originally Posted by Jaydi_21

Let's say that I use the non slip condition as you suggest. Does that imply that the wave will not reflect? I mean, I want the boundary to reflect the incoming waves...I have a first order scheme, so I'm just using linear interpolation.

Because the non slip condition actually set the normal velocity to 0, right? and the tangential one also is 0? But if I want the waves to be reflected, then I need to null normal velocity!

Reflection of a wave on a wall is a quite different issue and depends on the type of wave (acoustic, convective, ..). Generally, this aspect is relevant for the Euler equations (inviscid flow model) or for compressible Navier-Stokes equations at very high Reynolds number.
There are many papers about the types of BC.s for such flow problem. One of the relevant paper is from Poinsot and Lele on JCP.

[Jaydi_21]

Originally Posted by FMDenaro

Reflection of a wave on a wall is a quite different issue and depends on the type of wave (acoustic, convective, ..). Generally, this aspect is relevant for the Euler equations (inviscid flow model) or for compressible Navier-Stokes equations at very high Reynolds number.
There are many papers about the types of BC.s for such flow problem. One of the relevant paper is from Poinsot and Lele on JCP.

Ok, I'll read about it because actually, at this moment I'm not interested in seeing reflection. I just wanted to set a condition that says that the water should reflect at that boundary but if it's so complicated, I think I'll use the non slip condition. For the non slip condition, both components (normal and tangential) of the velocity at the ghost cell, are null, right? Or do I look for a value that gives me null velocity at the boundary? If the normal velocity at the first cell is , then at the ghost cell, I should have a velocity, is that correct? And the same for the tangential component, ? That way, I have non null velocities at the ghost cell, but null velocity at the boundary, for both components.

And the last question, for the water depth, is it correct to simply use the same water depth than the first cell within the domain (not interpolating)?