A PISO-like algorithm to simulate superfluid helium flow with the two-fluid model
Introduction
The physical phenomena of superfluid liquid helium can be macroscopically represented in a general framework best illustrated by Landau’s two fluids model [1], [2]. In this model, He II is seen as though it were a mixture of two different fluids. One of these is a normal viscous fluid, the other is a superfluid and moves with zero viscosity along a solid surface. Besides this particularity, it must be noted that the superfluid flow does not carry entropy. Moreover, the momentum equations include a thermo-mechanical force that occurs when a temperature gradient exists. When the superfluid velocity reaches a certain critical value, some turbulence phenomena arise and the above equations are no longer valid. In such a case, one introduces the so-called Gorter–Mellink mutual friction term, which has been proposed to estimate the interaction between the two components [3], [4].
Only few multi-dimensional codes that solve the complete two-fluid model are reported in the literature. Actually, as pointed out by Kitamura et al. [5], authors have met numerical difficulties, probably due to the fact that the thermo-mechanical and the Gorter–Mellink mutual friction terms are several orders of magnitude larger than the other terms in the superfluid momentum equation. The heat wave propagation intrinsic to superfluidity, the so-called second sound, may also lead to additional instabilities. To overcome these numerical difficulties, authors have either focused on one-dimensional solutions [6], [7] or proposed simplified multi-dimensional models. For instance, Ramadan and Witt [8] have modified the momentum equations as the thermo-mechanical and the mutual friction terms are larger than the others in balancing each other (hence they can be dropped from the equations) to create a heat flux with the «familiar» dependence. Assuming also that these two terms are approximately equal, Kitamura et al. [5], and later Suekane et al. [9] and Pietrowicz and Baudouy [10], proposed a simplified model made of a conventional continuity equation, a modified Navier–Stokes equation for the total velocity and a heat equation for temperature. Tatsumoto et al. [11] proposed a numerical segregated solution to solve superfluid equations. In their algorithm, the total fluid velocity and the pressure fields are first solved, then the superfluid momentum is computed from the resulting pressure field, after which the normal velocity is deduced point-wise. Finally, the temperature equation is solved and the whole sequence iterated for a new time step.
Actually, the discussion regarding the choice of using segregated or coupled algorithms is still an open debate. On one hand, the superfluid equations are strongly coupled, mainly from the thermo-mechanical and the mutual friction terms, and coupled algorithms offer suitable solutions [12], [7]. On the other hand, segregated approaches tend to consume less memory precisely because the entire problem is never assembled. However, in that case, how can one choose the optimal sequence? Why should the normal velocity be deduced from the superfluid velocity as suggested by Tatsumoto et al. [11] and not the opposite? In this paper, we propose an alternative segregated approach to solve superfluid equations. Our algorithm is an extension to the two-fluid model of the Pressure Implicit Operator Splitting (PISO) algorithm by Issa [13]. In our method, a pressure equation is directly derived from the total mass balance and both momentum equations. We called this algorithm Super-PISO. It is at the core of HellFOAM, the superfluid code we have developed using the OpenFOAM® technology.
The paper is organized as follows. In Section 2, we present the superfluid equations and the main assumptions. In Section 3, we introduce the discretization of the superfluid equations and the Super-PISO algorithm. Then in Section 4, we solve numerically the superfluid equations in different configurations to validate and illustrate the potential of HellFOAMcode.
Section snippets
Mathematical model and assumptions
In this section, we introduce the mathematical model on which our numerical study relies. In what follows we denote by subscripts and all quantities related to the normal and superfluid flow respectively. Hence, and are the normal and superfluid densities and their sum is the actual density of He II, These quantities vary according to the temperature: vanishes at the - point where the fluid becomes fully normal, is null at absolute zero. Moreover, as a classical fluid,
Numerical approach to solve the problem
All numerical developments of HellFOAM are performed with OpenFOAM® version 2.3. This code is a C++ library that solves partial differential equations with the method of finite volumes [19], [20]. It handles 3D geometry by default. One of its features is to solve equations using segregated approaches. Since the superfluid problem presents a strong coupling between all its unknown variables (, , , and ), it is important to develop a suitable algorithm. We introduce in this section the
Some simulation results
In this section, we present some simulations we have performed with HellFOAM. The objective is two-fold: (i) a comparison of the numerical results with analytical ones when available, (ii) simulation examples for cases of current research interest. First, steady-state results for a capillary containing He II in both Landau and Gorter–Mellink flow regimes are compared to analytical solutions. Then transient simulation results of forced flow of He II are compared to the experimental data by
Conclusions
We have presented in this paper the first version of HellFOAM, a helium superfluid simulator based on the OpenFOAM® technology. Given the sequential nature of OpenFOAM® algorithms, we had, in order to solve superfluid equations, to develop a new algorithm (called Super-PISO) extending the type of PISO algorithm used in OpenFOAM®.
Using the implemented code, different scenarii were simulated which correspond to situations of current research interest and with significant complexity. Solutions on
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