Finite-volume WENO scheme for viscous compressible multicomponent flows

We develop a shock- and interface-capturing numerical method that is suitable for the simulation of multicomponent flows governed by the compressible Navier–Stokes equations. The numerical method is high-order accurate in smooth regions of the flow, discretely conserves the mass of each component, as well as the total momentum and energy, and is oscillation-free, i.e. it does not introduce spurious oscillations at the locations of shockwaves and/or material interfaces. The method is of Godunov-type and utilizes a fifth-order, finite-volume, weighted essentially non-oscillatory (WENO) scheme for the spatial reconstruction and a Harten–Lax–van Leer contact (HLLC) approximate Riemann solver to upwind the fluxes. A third-order total variation diminishing (TVD) Runge–Kutta (RK) algorithm is employed to march the solution in time. The derivation is generalized to three dimensions and nonuniform Cartesian grids. A two-point, fourth-order, Gaussian quadrature rule is utilized to build the spatial averages of the reconstructed variables inside the cells, as well as at cell boundaries. The algorithm is therefore fourth-order accurate in space and third-order accurate in time in smooth regions of the flow. We corroborate the properties of our numerical method by considering several challenging one-, two- and three-dimensional test cases, the most complex of which is the asymmetric collapse of an air bubble submerged in a cylindrical water cavity that is embedded in 10% gelatin

A quasi-conservative discontinuous Galerkin method for multi-component flows using the non-oscillatory kinetic flux

In this paper, a high order quasi-conservative discontinuous Galerkin (DG) method using the non-oscillatory kinetic flux is proposed for the 5-equation model of compressible multi-component flows with Mie-Gr\"uneisen equation of state. The method mainly consists of three steps: firstly, the DG method with the non-oscillatory kinetic flux is used to solve the conservative equations of the model; secondly, inspired by Abgrall's idea, we derive a DG scheme for the volume fraction equation which can avoid the unphysical oscillations near the material interfaces; finally, a multi-resolution WENO limiter and a maximum-principle-satisfying limiter are employed to ensure oscillation-free near the discontinuities, and preserve the physical bounds for the volume fraction, respectively. Numerical tests show that the method can achieve high order for smooth solutions and keep non-oscillatory at discontinuities. Moreover, the velocity and pressure are oscillation-free at the interface and the volume fraction can stay in the interval [0,1].Comment: 41 pages, 70 figure

A random projection method for sharp phase boundaries in lattice Boltzmann simulations

Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting

A Computational Study of the Inertial Collapse of Gas Bubbles Near a Rigid Surface

Cavitation research is essential to a variety of applications ranging from naval hydrodynamics to medicine and energy sciences. Vapor cavities can grow from sub-micron-sized nuclei to millimeter-sized bubbles, and collapse violently in an inertial fashion. This implosion, which concentrates energy into a small volume, can produce high pressures and temperatures, generate strong shock waves, and even emit visible light. One of the main consequences of cavitation is structural damage to neighboring surfaces due to bubble collapse. The propagation of shock and rarefaction waves in a multiphase medium results in a complicated multiscale and multiphysics problem. Laboratory experiments of such flows are challenging due to the wide range of spatial and temporal scales, difficult optical access, and limitations of measurement devices. To better understand these flows, we use highly resolved numerical simulations of the inertial collapse of individual vapor bubbles near a rigid surface. For this purpose, we developed a novel numerical multiphase model combined with high-performance computing techniques to perform accurate and efficient simulations of the three-dimensional compressible Navier-Stokes equations for a binary, gas-liquid system. We present the detailed dynamics of the Rayleigh collapse of a single vapor bubble near a rigid wall for different geometrical configurations and driving pressures. We explain that the presence of a rigid boundary breaks the symmetry of the collapse and hinders the energy concentration. As a result, a liquid re-entrant jet directed toward the wall forms, ultimately giving rise to lower pressure and temperatures produced upon collapse. We characterize the collapse non-sphericity, and show that this quantity, which strongly depends on the initial stand-off distance of the bubble from the wall, significantly affects the overall dynamics. We further show that bubbles initially close to the wall or attached to the surface are responsible not only for the high pressure loads along the wall, but also the elevated temperatures on the solid surface. In fact, for certain soft materials, instantaneous temperatures greater than the melting point may be achieved on the surface, thus confirming that thermal damage is a potential threat to such materials exposed to cavitating flows. Furthermore, the development of scalings for important collapse properties (jet velocity, shock pressure, wall pressures/temperatures), in terms of the initial stand-off distance and driving pressure, not only illustrates universality of non-spherical bubble dynamics but also provides means to predict these phenomena. Since real flows involve many bubbles, we also investigate the inertial collapse of a pair of vapor bubbles near a rigid surface. We explain that the presence of a second bubble in the vicinity of the original (primary) bubble leads to far more complicated dynamics and completely changes the single-bubble scalings. Strong interactions between the bubbles and the boundary drastically increase the collapse non-sphericity and amplify/hinder the pressures and temperatures produced by the collapse. Our simulations show that the re-entrant jets in both bubbles form at distorted angles, and for certain configurations, ``double jetting'', occurs, in which two jets penetrate the primary bubble. The results indicate that bubble-bubble interactions and their effects on collapse dynamics near a wall are non-negligible. Furthermore, given the complexity of even this simple problem and the large number of parameters, the value of extending such high-resolution simulations to develop scalings for the collapse of many bubbles is debatable at the present time; it may be worth considering alternative modeling approaches.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144079/1/alahyari_1.pd