Meng-Sing Liou

A sequel to AUSM, Part II: AUSM+-up for all speeds

In this paper, we present ideas and procedure to extend the AUSM-family schemes to solve flows at all speed regimes. To achieve this, we first focus on the theoretical development for the low Mach number limit. Specifically, we employ asymptotic analysis to formally derive proper scalings for the numerical fluxes in the limit of small Mach number. The resulting new scheme is shown to be simple and remarkably improved from previous schemes in robustness and accuracy. The convergence rate is shown to be independent of Mach number in the low Mach number regime up to M~=0.5, and it is also essentially constant in the transonic and supersonic regimes. Contrary to previous findings, the solution remains stable, even if no local preconditioning matrix is included in the time derivative term, albeit a different convergence history may occur. Moreover, the new scheme is demonstrated to be accurate against analytical and experimental results. In summary, the new scheme, named AUSM^+-up, improves over previous versions and eradicates fails found therein.

A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme

In this paper, we propose a new approach to compute compressible multifluid equations. Firstly, a single-pressure compressible multifluid model based on the stratified flow model is proposed. The stratified flow model, which defines different fluids in separated regions, is shown to be amenable to the finite volume method. We can apply the conservation law to each subregion and obtain a set of balance equations. Secondly, the AUSM^+ scheme, which is originally designed for the compressible gas flow, is extended to solve compressible liquid flows. By introducing additional dissipation terms into the numerical flux, the new scheme, called AUSM^+-up, can be applied to both liquid and gas flows. Thirdly, the contribution to the numerical flux due to interactions between different phases is taken into account and solved by the exact Riemann solver. We will show that the proposed approach yields an accurate and robust method for computing compressible multiphase flows involving discontinuities, such as shock waves and fluid interfaces. Several one-dimensional test problems are used to demonstrate the capability of our method, including the Ransoms water faucet problem and the air-water shock tube problem. Finally, several two dimensional problems will show the capability to capture enormous details and complicated wave patterns in flows having large disparities in the fluid density and velocities, such as interactions between water shock wave and air bubble, between air shock wave and water column(s), and underwater explosion.

How to solve compressible multifluid equations: a simple, robust and accurate method

Solving multifluid equations of compressible multiphase flows has proven to be extremely demanding because of some peculiar mathematical properties, such as nonhyperbolicity, nonconservative form, and stiffness due to disparity in fluid properties and flow scales occurring typically. In this paper, we first consider the mathematical issues concerning nonhyperbolicity and nonconservative form. Their effects on the stability and convergence of numerical solutions are the theme of our presentation; we shall present solutions for a range of problems selected to illuminate these numerical issues. To this end, we present a new numerical method that is simple to implement for a general class of fluids and yet is capable of robustly and accurately calculating phenomena involving material and shock discontinuities and interactions between them. Additionally, the paper is completed with a new information for ensuring hyperbolicity under an interfacial pressure representation.

Numerical prediction of interfacial instabilities: Sharp interface method (SIM)

We introduce a sharp interface method (SIM) for the direct numerical simulation of unstable fluid-fluid interfaces. The method is based on the level set approach and the structured adaptive mesh refinement technology, endowed with a corridor of irregular, cut-cell grids that resolve the interfacial region to third-order spatial accuracy. Key in that regard are avoidance of numerical mixing, and a least-squares interpolation method that is supported by irregular datasets distinctly on each side of the interface. Results on test problems show our method to be free of the spurious current problem of the continuous surface force method and to converge, on grid refinement, at near-theoretical rates. Simulations of unstable Rayleigh-Taylor and viscous Kelvin-Helmholtz flows are found to converge at near-theoretical rates to the exact results over a wide range of conditions. Further, we show predictions of neutral-stability maps of the viscous Kelvin-Helmholtz flows (Yih instability), as well as self-selection of the most unstable wave-number in multimode simulations of Rayleigh-Taylor instability. All these results were obtained with a simple seeding of random infinitesimal disturbances of interface-shape, as opposed to seeding by a complete eigenmode. For other than elementary flows the latter would normally not be available, and extremely difficult to obtain if at all. Sample comparisons with our code adapted to mimic typical diffuse interface treatments were not satisfactory for shear-dominated flows. On the other hand the sharp dynamics of our method would appear to be compatible and possibly advantageous to any interfacial flow algorithm in which the interface is represented as a discrete Heaviside function.

Numerical Prediction of Interfacial Instability

“Free”, fluid-fluid interfaces, in the presence of body forces and/or differential velocities are subject to deformation and breakup - processes that are principally responsible for flow regime development, and thus for the macroscopic features in all multi-fluid systems. Body forces normal to an interface can be stabilizing or destabilizing, differential velocities parallel to an interface are always destabilizing, and for development of instability such driving forces must be sufficient to overcome the surface tension force (always stabilizing). Under unstable conditions, early growth of an interfacial disturbance is exponential in time, and the theory for understanding this regime, based on the linearized Navier-Stokes equations, rests on firm grounds. At amplitudes that are a significant fraction of the wavelength, this theory breaks down, non-linear analysis becomes scarcely feasible, and numerical simulation is the key to further progress. In this paper we develop and demonstrate the first reliable approach to this quest.

A nonlinear modeling approach using weighted piecewise series and its applications to predict unsteady flows

To preserve nonlinearity of a full-order system over a range of parameters of interest, we propose an accurate and robust nonlinear modeling approach by assembling a set of piecewise linear local solutions expanded about some sampling states. The work by Rewienski and White 1 on micromachined devices inspired our use of piecewise linear local solutions to study nonlinear unsteady aerodynamics. These local approximations are assembled via nonlinear weights of radial basis functions. The efficacy of the proposed procedure is validated for a two-dimensional airfoil moving with different pitching motions, specifically AGARDs CT2 and CT5 problems 27, in which the flows exhibit different nonlinear behaviors. Furthermore, application of the developed aerodynamic model to a two-dimensional aero-elastic system proves the approach is capable of predicting limit cycle oscillations (LCOs) by using AGARDs CT6 28 as a benchmark test. All results, based on inviscid solutions, confirm that our nonlinear model is stable and accurate, against the full model solutions and measurements, and for predicting not only aerodynamic forces but also detailed flowfields. Moreover, the model is robust for inputs that considerably depart from the base trajectory in form and magnitude. This modeling provides a very efficient way for predicting unsteady flowfields with varying parameters because it needs only a tiny fraction of the cost of a full-order modeling for each new condition-the more cases studied, the more savings rendered. Hence, the present approach is especially useful for parametric studies, such as in the case of design optimization and exploration of flow phenomena.

On the Numerical Prediction of Interfacial Instabilities in Shear Flows (Keynote)

We assess the state of the art in numerical prediction of interfacial instabilities due to shear in layered flows of viscous fluids. Basic ingredients of this assessment include linear stability analysis results for both miscible (diffuse) and immiscible (sharp) interfaces, the physics and resolution requirements of the critical layer, and convergence properties of the relevant numerical schemes. Behaviors of physically and numerically diffuse interfaces are contrasted and the case for a sharp interface treatment for reliable predictions of this class of flows is made.