A Thermodynamically Consistent Relaxation Model for Turbulent Binary Mixture Undergoing Phase Transition

Abstract

Non-equilibrium phenomena within a turbulent binary mixture undergoing a phase transition are modelled based on the one fluid, homogeneous approach. The set of governing evolution equations is given and the averaged basic balance laws are considered. The continuum under consideration consists of two constituents of which the first is undergoing a phase transition (vaporisation, cavitation, flashing, condensation) during the flow with turbulent transport of mass, heat and momentum. Starting from the Clausius-Duhem inequality for the recoverable specific entropy, several thermodynamical relationships for equilibrium and non-equilibrium fields are developed. The relaxation-retardation constitutive model for turbulent, phasic and diffusive fluxes are described. It is shown that the resulting model uncovers a few known two-phase models of turbulence and also the classical relaxation models as limiting cases.

Neoclassical Navier–Stokes Equations Considering the Gyftopoulos–Beretta Exposition of Thermodynamics

“…The neoclassical statement offers value for constitutive relations. The first is an entropy flux, and the second is a volume flux vector (see Equations (63) and (64)). In addition, connections between the heating and working fluxes as well as with the entropy, volume, and momentum fluxes are not simple (Equations (24) and (25)); they can be extended to the cases where Cimmelli's interstitial heating and interstitial working vectors appear.…”

“…Our results showed that it is possible to adopt the Gyftopoulos-Beretta exposition for a continuum media like a Navier-Stokes fluid. We are now enforcing this exposition to the case of turbulent fluid, where, as we suppose [64], the classical model of Osborne Reynolds needs some thermodynamic reconsiderations. For instance, if one takes a so-called k − ε model as a well-known phenomenological model of turbulent momentum evolution, then the question becomes, "Which role for two scalar evolution equations k and ε is proper?"…”

“…Another type of induced motion stems from the difference of an electric potential ϕ on a surface-this phenomenon is called "electrophoresis" and was discovered by von Smoluchowski in 1916 [62,63] and governed by an electro-mobility coefficient c vϕ . Yet another mobility mechanism is connected with the phase transition change [64,65] and the surface gradient of the phase order parameter x. Let us note that all of the mobilities: pressure, thermal, diffusional, phase, and electrical define only an external mobility force in the Stokes layer [66,67].…”

On the mass and momentum transport in the Navier–Stokes slip layer

“…However, if we suppose that the surface tension c is a surface dependent quantity, for instant, Bilicki and Badur (2003):…”

“…where s-is the Bilicki-Kestin relaxation time (Bilicki and Badur 2003). Next, using the surface mass balance…”