Gas-kinetic BGK Scheme For Viscous Transonic Flows

Transonic aerodynamics plays an important role in the operation of long rangeaircrafts. The cruise speed of most civil airplanes is in the transonic regime. The physi-cal natures associated with transonic flow, including shock wave, discontinuity, shockwave-boundary layer interaction, flow separation, etc., are still not fully understood.Mathematically, the governing equations are nonlinear and hybrid, which show ellip-tic feature in the subsonic region and hyperbolic property in the supersonic parts. Thusit is hard to find analytical solutions to predict the flow patterns and then to guide theairplane designing. Following the pioneering work of Murman and Cole in the early1970s, many numerical methods for transonic aerodynamics have been developed, fromthe small disturbance equation to the full potential equation, the Euler equations, and theNavier-Stokes equations. With the ever-increasing high performance of super-computerand sophisticated software, CFD has been considered as a convenient and useful tool foraerodynamic analysis and designing. However, to simulate high-Reynolds number flowsor around complex configurations flows is still a challenge. The main difculty is not thevast need of computing resources, but the physical modeling and numerical approaches.Instead of solving the Euler and Navier-Stokes equations directly, the gas-kineticBGK scheme(GKS) based on the Boltzmann equation has been developed and attractedmore and more attentions since the early1990s. Many progresses have been obtained onthe validation, analysis and improvement of the GKS during the last two decades. Thereare numerous literatures on these topics. Based on its constructive modeling, the GKShas a sound physical basis and can be employed as a robust and reliable flow solver tosimulate problems from low Mach number flows to hypersonic flows.Nevertheless the GKS has an obvious disadvantage in terms of its weak computa-tional efciency and expensive computational cost, which limits its practical application-s, in particular for multidimensional problems.To overcome this drawback, in this thesis some acceleration techniques, includinglocal time stepping, implicit LU-SGS method, and multigrid strategy, are implementedinto the original GKS and the new scheme is applied to study3D steady transonic viscousflows.Firstly, we check the efectiveness of those acceleration techniques. Implicit LU-SGS method is utilized to adopt larger CFL number than that can be used for explicit time integration. Multigrid has been proved as a powerful acceleration tool for stationaryflows. Take implicit multigrid with3-level V-cycle for example, the speedup is up toabout6, even more. Totally, numerical results show a significant speedup of the schemeto obtain the steady convergence solutions. Otherwise, for various problems, coveringSod shock tube problem,2D lid-driven cavity flow, Mach3step problem, etc., the currentscheme shows good capability on shock-capturing, high resolution and robustness.Secondly, we extend GKS to solve3D high-Reynolds number flow problems. Onthe basis of the simulations of ONERA M6wing and ARA M100wing-body config-uration, we know the extension of the GKS to viscous flows coupled with engineeringturbulence models is available. The code performs well in combination with the incor-porated models. Also, we investigate the influences of the computational grid and theturbulence model.Finally, we develop parallel GKS algorithm on multi-block structured grid andpredict the aerodynamic properties of DLR-F6wing-body configuration and wing-body-nacelle-pylon configuration under the design cruise condition of Ma=0.75. The pres-sure distributions on the wing at selected span-wise positions and on the nacelle out-board/inboard at special measured points show good agreement with the experimentaldata. But the predicted lift is higher than the wind-tunnel data, drag is lower and pitchingmoment is more negative. The predicted forces will be closed to the experimental datawith increased grid density. It is noted that the flow separations occurred on the upperwing surface near the trailing edge and on the lower wing surface near the pylon. Theseparation could be predicted in CFD, but the details with respect to the size, locationand shape are afected by the local grid density and quality, numerical dissipation, tur-bulence models and others. The diferences exist for separation predictions will resultin diferences in the aerodynamic forces. At last we analyze the efciency of parallelcomputing. When the total processes below a threshold, the speedup is very close to theideal status. As the processes increased over the threshold, the increment of speedup istiny. This odd may be caused by load-imbalance and should be researched more in thefuture.