Energy budgets in two-phase flows

[Andrés Castillo-Castellanos]

Hello, 
I have a question that is half physics, half numerics. I'm interested in the Rayleigh-Taylor instability between two non-miscible fluids and we are doing simulations using the vof method. 

From the energetics viewpoint, this system starts at rest (zero kinetic energy) and has "infinite" potential energy. As the RTI develops, the potential energy drops and one part becomes kinetic energy, other part becomes elastic energy due to the stretching of the interface and the rest is dissipated. So, one expects that 

-ΔPE  = ΔKE + ΔEE + int_0^t (viscous dissipation rate) dt 

where Δ is the change between t=0 and t. When we do this, there is a missing part. I think that part of this missing energy is the part that is spent mixing the non-miscible fluids. My hunch is that f[] has more and more mixed cells because of the stretching of the interface, and this creates a mixing of sorts. This would mean that the unexplained energy is proportional to the interface area, like the elastic energy, which seems to be the case.

I was just surprised to see, that this part is about 5% to 10% of the energy budget and I was wondering if there is something that l’m missing here. Any opinions or remarks are welcomed. 

Cheers,   

[Wojciech Aniszewski]

Well, that type of flow classes into them two-phase turbulent flows where the production
is not due to overwhelming injected KE (otherwise known as an atomising jet) but mixing
which happens at multiple scales. This opens the Pandora's Box of Nonresolved Enstrlophy,
which is 1:1 correlated with interface fragmentation and proportional to interface area as
you noted.

In the times of Yore, even before first iPhone premiered and people lived in caves sitting
along the fire from unread newspapers, there were works that already prophetized these issues,
such as the seminal (for me) work by S.Vincent's group: 10.1016/j.compfluid.2007.02.017 wherein
the terms im discussing are a priori measured. Some naive folk (i.e. yours trurly) then went on to model these terms
as LES-closures with varying luck (dx.doi.org/10.1016/j.jcp.2016.09.03). Mind you this normally wasn't RTI but "1/4 of it", as also visible
in pretty pictures here: Computers and Fluids 176 (2018) 245–259 (same group).

The contributions have been shown to be even above 10% (already in the 2007 Vincent's paper).

[tangent] This also, AFAIK partially motivated the 'manifold death' methods (Chirco et al) since, in asymmetric RTI the main 'enstrophy generating event' is a sheet breakup which tends to happen for numerical reasons, so there was an impression it would be better to control it by choosing a breakup threshold scale. [end of tangent]

I think SP has, among examples in the basilisk.fr website, the homogenous isotropic turbulence generated by the ABC forcing. That, IMHO, is a somewhat ""better" " (quadruple quotes purposefully) tool to analyse the enrgy budget than RTI, mainly because you avoid the 'atomisation in a box' scenario.

Cordialement
Voitek

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( (••) ) Wojciech (Vôitek) ANISZEWSKI
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[Stephane Zaleski]

Dear Wojtek

You are allowed to cite your own papers on exactly that topic

%0 Journal Article
%T A phase inversion benchmark for multiscale multiphase flows
%A Estivalezes, J-L
%A Aniszewski, Wojciech
%A Auguste, Franck
%A Ling, Yue
%A Osmar, Ludovic
%A Caltagirone, J-P
%A Chirco, Leonardo
%A Pedrono, Annaïg
%A Popinet, Stéphane
%A Berlemont, Alain
%J Journal of Computational Physics
%V 450
%P 110810
%@ 0021-9991
%D 2022
%I Elsevier



cheers

Stéphane

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[Andrés Castillo-Castellanos]

Hi, 

Thanks for the feedback and for the references. By the way, these are some images from the configuration we are considering. The mini-energy budget is on the bottom left corner, while the pdf of droplet size obtained using tag.h is on the bottom right corner. The red part is the energy that is unnaccounted for, while in the pdf the vertical lines indicate (from left to right) delta x, capillary lenght and the lenght of an initial perturbation. The leftmost peak on the pdf is hard to explain, but there are two main suspects: it could be the cell size influencing the droplet distribution, or it could be a Kolmogorov-ish scale, since below this scale, droplets would stop fragmenting. I do have a question concerning droplet size, since we are getting many droplets with characteristic lenghts below delta_x (i remove them from the image below), but I wanted to know if this is "normal" for this type of simulations. 

Cheers, 

[Daniel Fuster]

Hi Andres

Just to let you know that we are about to submit a paper on the Rayleigh Taylor instability in 2D. We did some energy budgets, but we did not include them in the paper. For instance, I do not understand why you say that the potential energy is infinity at t=0. Anyway, you can find a draft here

https://hal.science/hal-05553377

I may update it soon... :)

best

Daniel

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[Andrés Castillo-Castellanos]

Hi Daniel, 

Thanks for the pre-print! I'm very interested. 

By "infinite" i just meant very large. This is more or less true in our since we are interested in unconfined cases, i.e. where the height of the domain H is larger than the mixing zone and the width W remains larger than the size of the flow structures. Even if this value is finite, what matters, i think, is the drop in potential energy.