Barakos, G.

Colonia, S.

Steijl, R.

English

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Assessment of Implicit Implementation of the AUSM+ Method and the SST Model for Viscous High Speed Flow

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EUCASS 2013Assessment of Flux Evaluation Methods and Turbulence Models forViscous Hypersonic Flows.Simone Colonia ∗, Rene´ Steijl †, George N. Barakos ‡School of Engineering, University of LiverpoolLiverpool, L69 3GHKeywords: Hypersonic, Turbulence, Computational Fluid Dynamics.ABSTRACTEfficient and accurate computation of aerodynamics and thermal environment of hypersonic vehicle isessential in their design and development of hypersonic vehicles. Computational fluid dynamics meth-ods have gain significant prominence in recent years and have been used in hypersonic vehicle design;however, a number of challenges remain, including the prediction of the laminar to turbulent transitionas well as the turbulent flow field development are active areas of research at present. Furthermore,substantial challenges exist in devising accurate and robust numerical schemes for the convective fluxcomputation of Navier-Stokes solvers. Popular approximate Riemann solvers for transonic and super-sonic flow simulation, e.g. the Roe scheme [6], have shown deficiencies, such as over-expansions andoscillations around the sonic line, fig. 1. For this reason, improved schemes have been developed.In recent years the AUSM family has been widely used in hypersonic CFD. The original AUSM schemehas been introduced for the first time by M.-S. Liou in [2] and then improved in [3] to obtain theAUSM+. Recently Liou [4] extended the AUSM-family schemes to solve flows at all speed regimeswith the AUSM+up family of schemes. One of the key points of the AUSM+up scheme is the choice ofthe interface speed of sound.The present work involves a systematic analysis of the Roe and AUSM+up family of schemes for arange of generic hypersonic test cases. The work focuses on two main aspects. The first is the behaviourof these scheme in the laminar and turbulent flow regimes. The second part of the work involves theanalysis of different compressibility corrections to the two-equation k − ω [9], SST [5] turbulencemodel as well as the one-equation Spalart-Allmaras model [8]. Turbulence modelling remains a greatchallenge in the hypersonic flow simulation, as a major source of errors in the prediction of aerodynamicforces and heat transfer. Turbulence models are commonly used for incompressible flows but for highMach flows compressibility corrections are needed. From the review paper of Roy [7] it is clear thata lot of work has been done in order to validate the different turbulence models for simple hypersonicflows but studies are still needed to validate these models for more complex cases such as Edney Typesshock/shock iterations [1] and boundary layer iterations.Figures 2, 3, 4 and 5 show examples of results obtained for an infinite cylinder (R = 1). The re-sults show that for cases with even a week expansion, the use of the entropy satisfying interface speedof sound it is needed. Results are presented with the AUSM+up scheme, for inviscid and laminarflow (Re = 25000), and 4-stage Runge-Kutta method in time with a CFL of 0.25 was used. Fur-ther investigations include more complex flow fields such as Edney types shock/shock interaction andshock/boundary layer interaction and these are discussed a the full paper.∗PhD Candidate - s.colonia@liv.ac.uk†Lecturer - r.steijl@liv.ac.uk‡Professor - g.barakos@liv.ac.ukFigure 1: Infinite cylinder inviscid case. The flow was computed using the Roe scheme, 4-stages Runge-Kutta, CFL = 0.05 and M = 3.Figure 2: Infinite cylinder inviscid case. The flow was computed using the AUSM+up with entropysatisfying a1/2, 4-stages Runge-Kutta, CFL = 0.5 and M = 3.Figure 3: Infinite cylinder laminar case. The flow was computed using the AUSM+up with entropysatisfying a1/2, 4-stages Runge-Kutta, CFL = 0.25 and M = 10. RHO is the dimensionless density( ρρ∞ ).Figure 4: Infinite cylinder laminar case. The flow was computed using the AUSM+up with entropysatisfying a1/2, 4-stages Runge-Kutta, CFL = 0.25 and M = 10. Mach iso-surfaces.XYZdUdX0.00-3.98-7.96-11.94-15.93-19.91-23.89-27.87-31.85-35.83-39.81-43.80-47.78-51.76-55.74-59.72-63.70-67.69-71.67-75.65-79.63-83.61-87.59-91.57-95.56-99.54-103.52-107.50Figure 5: Infinite cylinder laminar case. The flow was computed using the AUSM+up with entropysatisfying a1/2, 4-stages Runge-Kutta, CFL = 0.25 and M = 10.References[1] B. Edney: Anomalous heat transfer and pressure distributions on blunt bodies at hypersonic speedsin the presence of an impinging shock. Aeronautical Research Institute of Sweden Report, 115(1968).[2] M.-S. Liou and C.J. Steffen Jr.: A New Flux Splitting Scheme. Journal of Computational Physics,107 (1993), 23–39.[3] M.-S. Liou: A Sequel to AUSM: AUSM+. Journal of Computational Physics, 129 (1996), 364–382.[4] M.-S. Liou: A Sequel to AUSM, Part II: AUSM+-up for All Speed. Journal of ComputationalPhysics, 214 (2006), 137–170.[5] F.R. Menter: Two-Equation Eddy-Viscosity Turbulence Model for Engineering Applications.AIAA Journal, 32 (1994), 1598–1605.[6] P.L. Roe: Approximate Riemann Solvers, Parameter Vectors and Different Schemes. Journal ofComputational Physics, 43 (1981), 357–372.[7] C.J. Roy and F.G. Blottner: Review and Assessment of Turbulence Models for Hypersonic Flows.Progress in Aerospace Sciences, 42 (2006), 469–530.[8] P.R. Spalart and S.R. Allmaras: A One-Equation Turbulence Model for Aerodynamic Flows. AIAAPaper 90-0439 (1981).[9] D.C. Wilcox: A One-Equation Turbulence Modeling for CFD. DCW Industries, Inc. (1994).

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Assessment of Implicit Implementation of the AUSM+ Method and the SST Model for Viscous High Speed Flow

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