this link will help you i think.

http://guava.physics.uiuc.edu/~nigel/courses/563/Essays_2010/PDF/hegg1.pdf

Dear Deewarak Sharma.

This is a question that I have been working on for 10-15 years in the early days of my research career, when I used to work with several groups from France, the US and Japan focused on the dynamics of near-critical fluids.

The question you raise here calls for a different answer depending on wether you need to compute the bulk temperature in the linear or in the nonlinear regime of the fluid evolution, and also depending upon the time scale that you are interested in observing. I have written a short recap of different cases of interest below, but should you wish to get a more specific answer for YOUR particular configuration, feel free to contact me at [email protected] .

As you may know, the Piston Effect is a thermo-acoustic coupling taking place in ALL fluids, but one that is particularly dominant in the near proximity to the Critical Point. You can therefore observe this phenomenon either on the short acoustic time-scale (if you are willing to get a direct glimpse at the sound waves created by the local heat input), or, alternately, you can be merely interested in checking the spatially averaged bulk temperature on longer time-scales (i.e. the nearly homogeneous temperature observed far from the heated walls).

1. Let me first address the simpler, linear case, where you need to predict DT_Bulk(t), the spatially homogeneous temperature elevation on the Piston-Effect time scale. Here, "linear case" means that the temperature evolution is considered to be small enough that the average fluid's properties are not changed during the thermal process. In such a case, a simple and efficient means to calculate DT_Bulk(t) for all heating profiles is through a formula of the type:

      DT_Bulk(t) = DT_Wall(t) * F_PE(t/tpe)                         (1)  ,

where DT_Wall(t) is the time evolution of the temperature increase prescribed at the heated wall, F_PE() is a specific transfer function that depends upon the global geometry of the fluid container, tpe is the typical time-scale of the Piston Effect, and " * * denotes a convolution product.

Classically, tpe is defined as:   tpe = td / (gamma - 1)2                (2)  ,

where gamma is the ratio of Cp/Cv and td is the typical heat diffusion time of the fluid (td = L2/D, with D the heat diffusivity, and L a typical length of the fluid container _ more about this later).

This formula will be helpful if heating of the container is known under the form of a given wall temperature evolution (namely here, T_Wall(t)).

2. There are many cases however where what is known is the heat flux Phi_Wall(t) at the heated wall rather than its temperature T_wall(t). In such cases, formula (1) becomes, quite naturally:

      DT_Bulk(t) = Phi_Wall(t) * G_PE(t/tpe)                         (3)  ,

where G_PE() is another specific transfer function, also dependent upon the geometry of the container, and different form F_PE() above.

Note: So far, I have not explained how to calculate the typical transfer functions (or, to speak more properly, the impulse response functions) F_PE() and G_PE().

The way you can derive these two functions is a little technical, and it is briefly explained in an early paper of ours, Garabos et al., PRE 57 (1998), and earlier in my PhD thesis (Carles, PhD Thesis 1995). When application is made of expressions (1) and (3) to the particular heating conditions chosen in Onuki et al., PRE 41 (1990), their particular former results are recovered.

3. As discussed above, another possibility is to explore the fluid's response to heating on the acoustic time-scale, so as to observe the time and space evolution of the Piston-Effect acoustic waves. This was first addressed in Zappoli and Carles, EJM 14 (1995) and Zappoli and Carles, Physica D 89 (1996) for van der Waals fluids, and later generalized to real fluids in Carles, Phys. Fluids 10 (1998).

The way to calculate the bulk temperature in that case is very similar to the previous cases, with however two important differences: first, the bulk temperature is no longer spatially homogeneous (waves travel back and forth in the container), and second, the appropriate time scale is no longer tpe but ta, the acoustic time-scale:

             ta = L / c                (4) ,

with L a typical length of the container and c the sound velocity in the fluid. The best way to estimate L in the case of complex geometries is derived in Carles, PhD Thesis (1995) and Garrabos et al., PRE 57 (1998).

Formulas (1) and (3) therefore become, respectively:

          DT_Bulk(t, x) = DT_Wall(t) * Fa_PE(t/ta, x/L)                         (5)  ,

          DT_Bulk(t, x) = Phi_Wall(t) * Ga_PE(t/ta, x/L)                         (6) .

Note the x-dependence of DT_Bulk on the acoustic time scale.

4. Finally, the full nonlinear case remains to be discussed. When the temperature evolution in the fluid is large enough to impart significant local property changes in the fluid, the linear "transfer functions" approach that I described above fails. Two options remain, then.

First, the "brute force" option, consisting in solving the full compressible Navier-Stokes system for a near-critical fluid. This was the approach followed in most of Zappoli's work, starting from Zappoli et al., PRA 41 (1990) and later. The pros of this method are obvious: with the right equation of state, pretty much anything can be calculated. The cons are equally obvious: Even simple cases and geometry require a very involved numerical work and lengthy CPU times.

For "moderately" non-linear heating curves, a more efficient approach exists: It consists in solving numerically the partial derivative equations proposed by Onuki in Onuki et al., PRA 41 (1990), expression (12). This expression, when "re-read" in the framework of numerical analysis, happens to fit exactly the classical Acoustic Filtering Method commonly used in compressible fluid dynamics and aeroacoustics.

I hope that this answer gives you enough informations and references to solve your problem. Once again, feel free to write a message to me at [email protected], should you want to discuss your particular problem.

Best

Pierre Carles

@Eldar Mansimov,

I read the note in the link you provided, and although it is reasonably complete with respect to the earliest published works on the subject in the 90s, most of the important references later on are missing.

But what perplexed me more is the fact that I could recognize several of the figures from articles and reports that I know (some, that I wrote myself !), and which are reproduced here without any credit (let alone authorization).

I understand that this may be merely an unpublished note written as a tutorial for colleagues or students, but even so, it sets a very bad example of how to properly trace and source all used information.

Would you happen to know where that note come from ? There is no reference on it that I can see or recognize.

Thank you,

Best,

Pierre

@Pierre Carles 

Hello and thanks for your kind reply. I could understand most of the stuff but need to go through the articles you have mentioned.

I will go through these and get back to you in case any further guidance is required.

Thanks

Deewakar

Welcome !