Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement

With discretized particle velocity space, a multi-scale unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the kinetic model in one-dimensional case (J. Comput. Phys., vol. 229 (2010), pp. 7747-7764). For the kinetic equation, to extend a one-dimensional scheme to multidimensional flow is not so straightforward. The major factor is that addition of one dimension in physical space causes the distribution function to become two-dimensional, rather than axially symmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on the Shakhov model in two-dimensional space will be presented. Instead of particle-based modeling for the rarefied flow, such as the direct simulation Monte Carlo (DSMC) method, the philosophical principal underlying the current study is a partial-different- ial-equation (PDE)-based modeling. Since the valid scale of the kinetic equation and the scale of mesh size and time step may be significantly different, the gas evolu- tion in a discretized space is modeled with the help of kinetic equation, instead of directly solving the partial differential equation. Due to the use of both hydrody- namic and kinetic scales flow physics in a gas evolution model at the cell interface, the unified scheme can basically present accurate solution in all flow regimes from the free molecule to the Navier-Stokes solutions. In comparison with the DSMC and Navier-Stokes flow solvers, the current method is much more efficient than DSMC in low speed transition and continuum flow regimes, and it has better capability than NS solver in capturing of non-equilibrium flow physics in the transition and rarefied flow regimes. As a result, the current method can be useful in the flow simulation where both continuum and rarefied flow physics needs to be resolved in a single com- putation. This paper will extensively evaluate the performance of the unified scheme from free molecule to continuum NS solutions, and from low speed micro-flow to high speed non-equilibrium aerodynamics. The test cases clearly demonstrate that the uni- fied scheme is a reliable method for the rarefied flow computations, and the scheme provides an important tool in the study of non-equilibrium flow.

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases

Mechanics of Rarefied Gases

This work concerns the breakup of millimetre-scale liquid droplets in gaseous flow fields that are disturbed from free-stream conditions by the presence of solid obstacles or other drops. A broad range of flow conditions is considered - from subsonic to supersonic, from highly rarefied to ambient pressures, and from fixed cylindrical obstacles to free liquid droplets (as obstacles). The liquid is water or tributyl phosphate, a water-like low-viscosity fluid of very low vapour pressure. We present data on deformation and breakup regimes, and, aided by numerical simulations, we discuss governing mechanisms and the time scaling of these events. Thereby a methodology is demonstrated for conveniently forecasting first-order behaviours in disturbed flow fields more generally. The highly resolved images lend themselves to testing/benchmarking numerical simulations of interfacial flows. These results, along with the experimental capability developed, constitute one of the key building blocks for our overall long-term aim towards predicting ultimate particle-size distributions from such intense aerodynamic interactions involving very large quantities of Newtonian and viscoelastic liquids.

Aerobreakup in disturbed subsonic and supersonic flow fields

A Monte Carlo simulation technique has been used to conduct numerical experiments with a model gas on a digital computer. Drag, heat transfer, and general flowfield information was obtained for cylinders, spheres, and zero incidence flat plates. Speed ratios of 5 and 10 were treated, while the Knudsen number ranged from 0.1 to 30. It was found that drag coefficient results give a poor indication of the onset of collisional effects, since the fall in the streamwise momentum of the molecules striking the body is initially compensated by an increase in the flux of these molecules. For cylinders and spheres with cold surfaces, the drop in momentum may be overcompensated? and the drag coefficient may rise above the free molecule value before falling, with Knudsen number, towards the continuum value. The shear stress on a flat plate was found to rise above the free molecule value for both hot and cold surfaces. Some results were obtained for the pressure distribution on finite length flat plates, and these indicate that, contrary to a common assumption, there is no free molecule region at the leading edge of such plates. Nomenclature d = parameter related to the mean free path e = probability of event I = separation distance n = number flux p = pressure r — random number s = molecular speed ratio t = time tm = time interval u = streamwise velocity component v = molecular speed vr = relative velocity x = probabilistic variable C = counter CD = drag coefficient Kn = Knudsen number N = number density No = total number of molecules Q = heat transfer per unit area per unit time R = gas constant St = modified Stantori number T = temperature Tr = free molecule recovery temperature X = mean free path p = density a- = molecular radius r = shear stress Subscripts oo = freestream value continuum stagnation value surface value leading-edge value free molecule value o = w = le fm =

Aerodynamic properties of some simple bodies in the hypersonic transition regime.

The drag coefficients for cylinders normal to the flow have been determined experimentally at Mach ∼2, Mach ∼4, and Mach ∼6 with Knudsen numbers extending from continuum conditions to free‐molecule‐flow conditions. The results indicate a smooth transition from inviscid values at low Knudsen numbers to free‐molecule‐flow predictions for diffuse reflection at high Knudsen number. Small departure theories which are applicable to the ``near free molecule flow'' regime are compared to the experimental data.

Cylinder Drag in the Transition from Continuum to Free‐Molecule Flow

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Cylinder Drag in the Transition from Continuum to Free‐Molecule Flow