Compressible and immiscible fluids with arbitrary density ratio

Abstract

For the water-air system, the bulk density ratio is as high as about 1000; no model can fully tackle such a high density ratio system. In the Navier-Stokes and Euler equations, the density $ρ$ within the water-air interface is assumed to be a constant based on the Boussinesq approximation namely $ρ(\mathrm{d} \mathbf u/\mathrm{d} t)$, which does not account for the true momentum evolution $\mathrm{d} (ρ\mathbf u)/\mathrm{d} t$ ($\mathbf u$-fluid velocity). Here, we present an alternative theory for the density evolution equations of immiscible fluids in computational fluid dynamics, differing from the concept of Navier-Stokes and Euler equations. Our derivation is built upon the physical principle of energy minimization from the aspect of thermodynamics. The present results provide a generalization of Bernoulli's principle for energy conservation and a general formulation for the sound speed. The present model can be applied for immiscible fluids with arbitrarily high density ratios, thereby, opening a new window for computational fluid dynamics both for compressible and incompressible fluids.

Figures (3)

• Figure 1: Sketch for the validity of the convection flux in a water-air system. The water-air interface consists of water (blue) and gas (orange) molecules. Within the water-air interface, the density is not uniform. For example, when measuring the convection flux across the reference line (green), the density at the left and right hand side of the reference line is different.
• Figure 2: The volume concentration within the volume element $V$. (a) Conventional definition. The volume element $V$ is divided into the volume of air ($v_a$) and the volume of water ($v_w$), plus the excess volume $v_e$. In this definition, the densities $\rho_a$ and $\rho_w$ are constants. (b) Current concept. The volume element $V$ is divided into the volume of air ($v_a^\prime$) and the volume of water ($v_w^\prime$). We have distributed the excess volume into the volumes of $v_a^\prime$ and $v_w^\prime$. By this way, the densities $\rho_a^\prime$ and $\rho_w^\prime$ are affected by the local excess volume and no more constants.
• Figure 3: Analytical solution of the equation system, Eq. \ref{['eq:94']}-Eq. \ref{['eq:98-00']} in 2D for a moving flat water-air interface in a tube: (a) Volume concentration $\phi(x)$. (b) Pressure $p(x)$ and fluid velocity $u(x)$, with $u_0=0.025$ and $a=0.33$. (c) Density $\rho(x)/\rho_w$, where $\rho_w$ is the density of bulk water. Square symbols represent DFT results from Refs. pezzotti20182dcreazzo2024water; the dashed line shows a linear interpolation of the density. (d) Sum of the pressure energy and kinetic energy, $p+\rho u^2$.