[mihirmakwana6]

I am trying to simulate a warm bubble(air) that rises in an adiabatic atmosphere using FVM.

The paper that i am reffering to is
http://www.sciencedirect.com/science...21999107005384

The problem is defined in section 3.2) case 2 : Rising thermal bubble
where it is mentioned that they are using no-flux B.C at all boundaries.

Based on section 4.4.2 No-flux B.C, it is clear that no-flux B.C mean that velocity normal to boundary is zero along with gradient of tangential velocity being zero ( i.e free slip B.C at all boundaries)



The kind of equations that i am solving are
1) continuity eqn
2) momentum eqn
3) pressure correction eqn

Thus i need B.C for density, velocity and pressure correction at all boundaries

Now, velocity B.C is obtained based on free slip.

But what about the B.C for density and pressure correction ???


Based on the problem definition, what can be B.C for density and pressure correction ?

Please help.

Thanks in advance.

[FMDenaro]

I am not sure to understand you in what you wrote and now I cannot access the paper, I suppose pressure correction should imply the solving of the continuity equation.
What set of equations are you solving?

[mihirmakwana6]

Originally Posted by FMDenaro

I am not sure to understand you in what you wrote and now I cannot access the paper, I suppose pressure correction should imply the solving of the continuity equation.
What set of equations are you solving?

As atmospheric flows occurs at low mach number, i am using the following low mach number equations ( these are compressible N.S equations which are modified to capture low mach number effects : here pressure is split as base pressure + M^2 * pressure_pertubation i.e p_zero + M^2*p_two)

The continuity and momentum equations are
http://i.imgur.com/2Kr1hxT.jpg

The energy equation is converted to pressure equation and that is in turn converted to a pressure correction( dp(2) ) equation which is
http://i.imgur.com/DvG7sBo.jpg
In the pressure correction equation, terms on RHS are source terms
Also, i have applied time integration to the pressure correction equation.


The snippets from the paper which discuss the problem are
1) section 3.2 : case 2 : Rising thermal bubble
http://i.imgur.com/RUZ307C.jpg

2) section 4.4.2 No-flux B.C
http://i.imgur.com/OIzD8Fp.jpg

In the paper the set of equations used are different from the one that i am using


Also, based on my calculations Reynolds number for the flow is approx 10^7.

[FMDenaro]

the images are not available....

[mihirmakwana6]

Originally Posted by FMDenaro

the images are not available....

i have edited the post. please have a look at it.

[FMDenaro]

from the image of the energy equation, the field p_2 is a solution of an alliptic equation, right? Therefore the Neuman condition are derived from the Grad p_2 term in the momentum under projection along the normal direction to the boundary and using the condition u.n=0

[mihirmakwana6]

Originally Posted by FMDenaro

from the image of the energy equation, the field p_2 is a solution of an alliptic equation, right? Therefore the Neuman condition are derived from the Grad p_2 term in the momentum under projection along the normal direction to the boundary and using the condition u.n=0

yes the field p_2 is obtained by solving for elliptic eqn of dp_2 i.e the pressure correction equation

Due to no-flux B.C , u.n=0 thus grad_dp2 will be zero at the boundaries.
This gives the B.C for pressure correction equation.
but is this condition of grad_dp2 = 0 ,correct for the problem at hand.
i.e can we apply it for this problem

also, we need a B.C for density

[mihirmakwana6]

Originally Posted by mihirmakwana6

yes the field p_2 is obtained by solving for elliptic eqn of dp_2 i.e the pressure correction equation

Due to no-flux B.C , u.n=0 thus grad_dp2 will be zero at the boundaries.
This gives the B.C for pressure correction equation.
but is this condition of grad_dp2 = 0 ,correct for the problem at hand.
i.e can we apply it for this problem

also, we need a B.C for density

i applied the following B.C

1) grad_dp2 = 0 at all 4 boundaries

2) grad_density = 0 at all 4 boundaries ( i am not sure whether it is correct for the thermal bubble problem )

Based on the above B.C's along with free slip B.C for velocity, i ran the simulation.

But the results are not matching

I think there is some problem with the B.C that i am imposing

[FMDenaro]

maybe you can find useful this recent paper and its references

http://eprints.eemcs.utwente.nl/2607...ukhvostova.pdf

[mihirmakwana6]

Originally Posted by FMDenaro

maybe you can find useful this recent paper and its references

http://eprints.eemcs.utwente.nl/2607...ukhvostova.pdf

ok. thankyou sir.
I will have a look at it.

[FMDenaro]

As you can read, they set (1/rho) dp/dn = 0

[mihirmakwana6]

Originally Posted by FMDenaro

As you can read, they set (1/rho) dp/dn = 0

so, in my case it will be (1/rho) grad(dp2) = 0 i.e. grad(dp2) = 0 at all the boundaries.

Now,
Density is obtained by solving continuity equations only for internal points
i.e. I don't have values of density at the boundaries

So, some B.C for density is needed

[FMDenaro]

you can consider to solve the density equation in a FV around the node boundary...otherwise, an approximation such as d rho/dn = 0 (actually, the normal derivative should be expressed by the state equation)

[mihirmakwana6]

Originally Posted by FMDenaro

you can consider to solve the density equation in a FV around the node boundary...otherwise, an approximation such as d rho/dn = 0 (actually, the normal derivative should be expressed by the state equation)

i did implement d rho/dn = 0 at all boundaries but it is not giving the expected result

[FMDenaro]

Originally Posted by mihirmakwana6

i did implement d rho/dn = 0 at all boundaries but it is not giving the expected result

of course that is not a good approximation in specific cases, you can try to solve the density equation at the boundary

[mihirmakwana6]

Originally Posted by FMDenaro

of course that is not a good approximation in specific cases, you can try to solve the density equation at the boundary

is there a paper that discusses solving continuity at boundary ?

[FMDenaro]

Originally Posted by mihirmakwana6

is there a paper that discusses solving continuity at boundary ?

I used that many years ago in a FV formulation on unstructured grid, unfortunaltely I don't remember those references.
However, it is just a local integral formulation for a FV built around the node at the wall.