A sharp interface approach for compressible multiphase flow

The numerical simulation of compressible multiphase flow is a difficult task due to the close connection of thermodynamics and fluid dynamics. Our approach is based on a sharp interface treatment. The two phases are coupled by a ghost fluid method to avoid a mixing of the phases. To obtain information about the location of the interface a level-set equation is used. The bulk flows are approximated by a spectral element discontinuous Galerkin scheme for the Navier-Stokes equations that supports a general equation of state. To define values in the ghost cells we solve the multi-phase Riemann problem to determine the phase front velocity and to get the values at the phase interface from both sides. We focus in the solution of the multi-phase Riemann problem itself. We present a first step in a comparison of the Riemann problem solution for the Euler equations with a microscopic solution based on molecular dynamics (MD) solution. The MD simulations are based on the Lennard-Jones model fluid with truncated and shifted potential, for which a highly accurate equation of state is available. This allows a clear connection of both simulations. However, the quite different time and space scales have to be bridged. We can show a perfect coincidence of these solutions for Riemann problems in the supercritical regime without occurring phase transitions. The other class of problems, addressed in the talk, describe expansion waves into vacuum, which allows simpler microscopic simulations. The MD simulation shows a clustering of molecules that is interpreted as a starting of droplet formation initiated by the strong pressure drop. We also show finite volume simulations for this case. The other topic in the talk is the consideration of heat conduction that is important at phase transitions to balance the latent heat. Here, we propose a novel calculation of the numerical flux based on the generalized Riemann problem for advection diffusion problems.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada de la Universidad de Málaga. Departamento de Matemática Aplicada de la Universidad de Málaga. Vicerrectorado de Investigación de la Universidad de Málaga. ICIAM2019 Sociedad Española de Matemática Aplicada ModCompShoc

Comparison of macro- and microscopic solutions of the Riemann problem I. Supercritical shock tube and expansion into vacuum

The Riemann problem is a fundamental concept in the development of numerical methods for the macroscopic flow equations. It allows the resolution of discontinuities in the solution, such as shock waves, and provides a powerful tool for the construction of numerical flux functions. A natural extension of the Riemann problem involves two phases, a liquid and a vapour phase which undergo phase change at the material boundary. For this problem, we aim at a comparison with the macroscopic solution from molecular dynamics simulations. In this work, as a first step, the macroscopic solution of two important Riemann problem scenarios, the supercritical shock tube and the expansion into vacuum, were compared to microscopic solutions produced by molecular dynamics simulations. High fidelity equations of state were used to accurately approximate the material behaviour of the model fluid. The results of both scenarios compare almost perfect with each other. During the vacuum expansion, the fluid obtained a state of non-equilibrium, where the microscopic and macroscopic solutions start to diverge. A limiting case was shown, where liquid droplets appeared in the expansion fan, which was approximated by the macroscopic solution, assuming an undercooled vapour.DFG, 84292822, TRR 75: Tropfendynamische Prozesse unter extremen Umgebungsbedingunge

A Relaxation Model for the Non-Isothermal Navier-Stokes-Korteweg Equations in Confined Domains

The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Secondly, any real gas equation of state is based on a non-convex Helmholtz free energy potential which may cause the eigenvalues of the Jacobian of the first-order fluxes to become imaginary numbers inside the spinodal region. In this work, a thermodynamically consistent relaxation model is presented which is used to approximate the NSK equations. The model is complimented by thermodynamically consistent non-equilibrium boundary conditions which take contact angle effects into account. Due to the relaxation approach, the contribution of the Korteweg tensor in the linear momentum and total energy equations can be reduced to second-order terms which enables a straightforward implementation of contact angle boundary conditions in a numerical scheme. Moreover, the definition of a modified pressure function enables to formulate first-order fluxes which remain strictly hyperbolic in the entire spinodal region. The present work is a generalization of a previously presented parabolic relaxation model for the isothermal NSK equations

Expansion rates of bubble clusters in superheated liquids

[EN] The present work analyses the growth of multiple bubbles in superheated liquid jets by means of direct numerical simulations (DNS). A discontinuous Galerkin approach is used to solve the Euler equations and an adequate interface resolution is ensured by applying finite-volume sub-cells in cells with interfaces. An approximate Riemann solver has been adapted to account for evaporation and provides consistency of all conserved quantities across the interface. The setup mimics conditions typical for orbital manoeuvring systems when liquid oxygen (LOX) is injected into the combustion chamber prior to ignition. The liquid oxygen will then be in a superheated state, bubble nucleation will occur and the growth of the bubbles will determine the break-up of the liquid jet. The expansion rates of bubble groups under such conditions are not known and standard models rely on single bubble assumptions. This is a first DNS study on bubble-bubble interactions in flash boiling sprays and on the effects of these interactions on the growth rates of the individual bubbles. The present simulations resolve a small section of the jet close to the nozzle exit and the growth of bubble groups inside of the jet is analysed. The results suggest that an individual bubble within the group grows more slowly than conventional models for single bubble growth would predict. The reduction in bubble growth can amount to up to 30% and depends on the distances between the bubbles and the number of bubbles within the bubble group. In the present case, the volume expansion of the superheated liquid decreases by approximately 50% if the distance between the bubbles is doubled.The authors acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) as part of the Collaborative Research Center SFB TRR 75 “Droplet dynamics under extreme ambient conditions” held by University of Stuttgart and University of Technology Darmstadt.Dietzel, D.; Hitz, T.; Munz, C.; Kronenburg, A. (2017). Expansion rates of bubble clusters in superheated liquids. En Ilass Europe. 28th european conference on Liquid Atomization and Spray Systems. Editorial Universitat Politècnica de València. 1068-1075. https://doi.org/10.4995/ILASS2017.2017.4714OCS1068107

On the investigation of oblique shock‐wave/turbulent boundary‐layer interactions with a high‐order discontinuous Galerkin method

Shock-wave/turbulent boundary-layer interactions are still a challenge for numerical simulation. The shock capturing needs dissipation to avoid spurious oscillations while turbulence will be falsified by introducing dissipation. Especially, an accurate prediction of quantities such as the skin-friction coefficient inside the interaction area of shock wave and turbulent flow is a critical point. In this article, we investigate a wall-resolved large eddy simulation of oblique shock-wave/turbulent boundary-layer interactions by a high-order discontinuous Galerkin scheme. The high-order scheme handles the turbulent flow very well. The shock-capturing is confined to the near shock region by switching locally to a finite volume second-order TVD scheme on subcells. This strategy is completed with the application of a shock indicator to a filtered flow field. A global spanwise filter is applied to avoid switching on the shock-capturing procedure in regions of under-resolved turbulent structures. We validate our numerical results first at shock-wave/laminar boundary-layer interaction. The main simulation under consideration is a Mach 2 turbulent boundary-layer with an inlet momentum-thickness Reynolds number of 1628, interacting with an oblique shock that deflects the incoming flow by 8°. We employ a reformulated synthetic eddy method at the inlet to avoid the influence of recycling-based turbulence generating schemes on the low-frequency unsteadiness. The anisotropic linear forcing technique is adopted to further reduce the turbulence recovery length. Through the spectral analysis of wall pressure probes, a typical Strouhal number of around 0.03 is observed. We attribute the discrepancies between an experimental scaling law and our computation to the three-dimensional sidewall effects in the experiment. With the assistance of numerical results from this article and other authors, a new scaling law for the spanwise-periodic computations is suggested to quantify the difference between experimental and computed data.China Scholarship CouncilDeutsche ForschungsgemeinschaftProjekt DEA