[Gfs-devel] about thermocapillary flow

[Gibbs.Leonid]

Hi, Stephane and all

    

      I plane to develop  the branch of thermocapillary flow into Gerris, in which the surface tension

depends on the temperature that the Marangoni effect drives the flow.

 

      The equations are obtained that a tangential component of surface tension due to temperature

gradient should be added  into the pressure correction step. The CSF model is also adopted here.

 

       Before I start to implement the branch, I think it's better to discuss with you about the numerical

scheme I should select to keep the scheme consistent and conservative

 

      the normal surface tension is expressed as sigma*kappa*(\nabla C) on cell face and integrated into

 u^{\star}. With the Gauss theorem, the surface integral of surface tension can be written as

 

      \Sigma ( sigma*kappa*\nabla C)*n

 

 and (\nabla C*n) is calculated with gfs_face_gradient.

 

     However, in considering the tangential component of surface tension, we should calculate the norm of (\nabla C) and

the inner product of (\nabla C)*(\nabla T) , that means all three directions (3D) of gradient vector on cell face should be calculated.

 

     Suppose the cell face is normal to x-direction, then the x-direction gradient vector is easily obtained by gfs_face_gradient.

The problem lies in how to properly calculate the y- and z-direction gradient vector on cell face.

 

      I've two scheme for this. The first, use the "gfs_center_gradient" to calculate them on cell center, and interpolate

them onto cell face.     The second, interpolate the value of C and T onto cell vertex and then obtain the gradient vector

 using the interpolated vertex value. I'm not sure which one is better and if they are appropriate. Maybe a better scheme should

be adopted because the surface tension simulation is a complex work and sensitive with the numerical technique.

 

      By the way, I'm not sure if anyone has implement this thermalcapillary flow branch, it's just on my working plane and any

suggestion or interest from anyone is appreciated.

 

       Then , Merry Christmas 

 

Cheers

 

Leonid

---

[Stephane Popinet]

Hi Leonid,

> I plane to develop the branch of thermocapillary flow into Gerris, in
> which the surface tension
> depends on the temperature that the Marangoni effect drives the flow.

Yes, that sounds like an excellent project. I am not aware of anybody
working on this at present, although the idea has been "in the air"
for a while.

> I've two scheme for this. The first, use the "gfs_center_gradient" to
> calculate them on cell center, and interpolate
> them onto cell face. The second, interpolate the value of C and T onto
> cell vertex and then obtain the gradient vector
> using the interpolated vertex value. I'm not sure which one is better and
> if they are appropriate. Maybe a better scheme should
> be adopted because the surface tension simulation is a complex work and
> sensitive with the numerical technique.

I don't really have arguments for or against either technique,
although the "vertex interpolation" scheme will lead to a wider
stencil which may lead to smoother results. I think you should have
the option to do either and evaluates their respective merits using
test cases. Also, people probably have already looked at this issue
and there may be some literature on the topic.

Please keep in touch on a regular basis (send small code patches
etc...) as this makes collaboration much easier (than a very large bit
of code in a year's time!) and provides people with an opportunity to
give feedback which may be useful to you.

cheers

Stephane

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[Gibbs.Leonid]

Hi, Stephane

 

       I've read some literatures on this topic and compile a piece of code for this, see it attached.

 

1.  For thermocapillary flow , the surface tension coefficient varies linearly with the temperature

 

       sigma=sigma_0 + sigma_t*(T-T_0)

    

      and the main algorithm I choose is adhered in the attachment.

 

2.   As depicted in the algorithm file, the surface tension in thermocappilary flow falls into  two parts,

 the normal component and  the tangential component.  For the normal component should be balanced

 by pressure gradient, so it's calculated in the correct step of projection method. But the tangential

component of surface tension is balanced by the vicious sheer stress, so in the code, I place this in the

 prediction step as a source term which is computed in the cell center.  I'm not sure if this is really appropriate?

 

3.   The third document I attach is the parameter file and the non-dimensional parameter is set according

to the paper of Nas and Tryggvason (2003, International journal of multiphase flow ).  Then the comparison

 of rise velocity is presented in fig1, it seems promising.

 

*4.  However, if I add  "AdaptGradient {istep=1} {cmax=1.0e-6 minlevel=5 maxlevel=6} T " in the parameter

 file, the  result seems to be incorrect as showed in fig2 that it's smaller than the reference, and through examination

 I found this faulty comes from the tangential component of surface tension. I'm not sure why this happens when

 the mesh is non-uniform. Maybe it's a slight mistake in my code......

 

5.   I've simulated another two cases of low Marangoni number, and the terminal velocity of drop is also in good

 agreement with  the analytic solution. Of cause, they are conducted on uniform meshes.

 

*6.   I'd like to try another way to complete this branch, that is calculating the tangential component of surface tension

together with the normal component in the correct step , but I don't think it can be complied in an independent module

 because the surface tension added in Gerris is written in 'gfs_velocity_face_sources' in Timestep.c and it is

binding with pressure gradient in the function of  'correct_normal_velocity'.  it will be more complicated and

the consistency should be considered since it's calculated on face. According to your advice, I'll try 'vertex interpolation' first.

 

     Wait for your suggestion.

 

Cheers

 

Leonid

---

Gibbs.Leonid