On buoyancy in disperse two-phase flow and its impact on well-posedness of two-fluid models

Abstract

The Maxey-Riley-Gatignol equation for the flow around a sphere at low particle Reynolds number tells us that the fluid-particle interaction force decomposes into a contribution from the local flow disturbance caused by the particle's boundary -- consisting of the drag, virtual-mass, and history forces, and their Faxén corrections -- and another contribution from the stress of the background flow, termed generalized-buoyancy force. There is also a consensus that, for general disperse two-phase flow, the interfacial force density, coupling the average fluid phase and dispersed-phase momentum balances, decomposes in a likewise manner. However, there has been a long-standing controversy about the physical closure separating the generalized-buoyancy from the interfacial force density, especially whether or not pseudo-stresses, such as the Reynolds stress, should be attributed to the background flow. Furthermore, most existing propositions for this closure involve small-particle approximations. Here, we show that all existing buoyancy closures are inconsistent with particle-resolving numerical simulations and/or at least one of two simple thought experiments designed to determine the roles of pseudo-stresses and small-particle approximations. We then derive the unique consistent closure. It requires no approximation and implies that all stresses and pseudo-stresses in the average fluid phase momentum balance, except the Reynolds stress, fully contribute to the background flow responsible for buoyancy. Remarkably, it exhibits a low-pass filter property, attenuating buoyancy at short wavelengths, that prevents it from causing Hadamard instabilities, constituting a first-principle-based solution to the long-standing ill-posedness problem of two-fluid models. When employing the derived closure, even simplistic two-fluid models are linearly well-posed.

Figures (10)

• Figure 1: Snapshot of a particle-resolving simulation of statistically steady, uniform sediment transport driven by viscous flow.
• Figure 2: Vertical profiles of (a) the dispersed-phase volume fraction $\beta_\mathrm{s}$ and (b) the fluid-phase-averaged streamwise flow velocity $u_{\mathrm{f}x}$ and dispersed-phase-averaged streamwise flow velocity $u_{\mathrm{s}x}$, obtained from a particle-resolving simulation of statistically steady, uniform viscous sediment transport.
• Figure 3: (a) Fluid phase shear stress contributions obtained from a particle-resolving simulation of statistically steady, uniform viscous sediment transport. (b) The sum of all contributions results in the total fluid phase shear stress $\sigma^\mathrm{f}_{zx}+\sigma_{\mathrm{Re}zx}^\mathrm{f}=\beta_\mathrm{f}\langle\sigma_{zx}\rangle^\mathrm{f}+\sigma_{\mathrm{s}zx}^\mathrm{f}+\sigma_{\mathrm{Re}zx}^\mathrm{f}$.
• Figure 4: (a) Fluid phase normal stress contributions obtained from a particle-resolving simulation of statistically steady, uniform viscous sediment transport. (b) The sum of all contributions results in the total fluid phase normal stress $\sigma^\mathrm{f}_{zz}+\sigma_{\mathrm{Re}zz}^\mathrm{f}=\beta_\mathrm{f}\langle\sigma_{zz}\rangle^\mathrm{f}+\sigma_{\mathrm{s}zz}^\mathrm{f}+\sigma_{\mathrm{Re}zz}^\mathrm{f}$, which exhibits a nearly hydrostatic profile, $\sigma^\mathrm{f}_{zz}+\sigma_{\mathrm{Re}zz}^\mathrm{f}\approx\rho_\mathrm{f}(H-z)g_z$.
• Figure 5: (a) Test of the buoyancy closures by Zhangetal07b, (\ref{['BuoyancyZhang']}), and RevilBaudardChauchat13, (\ref{['BuoyancyChauchat']}), against data from a particle-resolving simulation of statistically steady, uniform viscous sediment transport. If lift forces are not significant, as indicated by $\beta_\mathrm{s}\langle f_{\mathrm{h}z}\rangle^\mathrm{s}/(-\beta_\mathrm{s}\rho_\mathrm{f}g_z)\approx1$ in (b), one expects $\beta_\mathrm{s}\langle f_{\mathrm{B}z}\rangle^\mathrm{s}/(\beta_\mathrm{s}\langle f_{\mathrm{h}z}\rangle^\mathrm{s})\approx1$ in (a). This is, indeed, the case for the closure by RevilBaudardChauchat13, but not for the closure by Zhangetal07b. The peaks around $z=4$ likely have a numerical origin due to insufficient data for statistical averaging when $\beta_\mathrm{s}$ is very small (figure \ref{['ConcentrationVelocityProfiles']}(a)).