Title:

A spacetime framework for aerodynamics of complex motions

A twodimensional spacetime framework is presented to solve unsteady aerodynamics problems as an alternative to conventional approaches for complex unsteady problems involving large deformations or topological change. Some examples of problems that the spacetime method can cope with seamlessly are store separation, slat and flap deployment, spoiler deflection or rotorstator configurations. It avoids methods such as Chimera or overset grids, or even remeshing, by the use of a finitevolume approach both in space and time. The simulation of unsteady problems of dimension N is effectively done as the simulation of steady problems of dimension N+1. Hence, both the geometry and its motion are defined by a spacetime mesh in an N+1 dimensional space. The use of an arbitrary LagrangianEulerian formulation along with a geometric conservation law are also avoided by the spacetime formulation. Moreover, it is a conservative method both in space and time. Therefore, it is very suitable for the solution of timeperiodic problems. The finitedifference approach used for the time integration in conventional methods based on an ALE formulation uses directionally biased schemes since the solution is only know at previous time levels. In contrast with this, the use of a centraldifference scheme in spacetime yields nonphysical transient solutions as a consequence of pressure waves travelling backwards in time. The search for a more realistic time stencil has led to the formulation of one hybrid (centraldifference in space, upwind in time) and two upwind stencils. Initially, most of the work has been done based on an Euler solver. Then, a RANS formulation has also been implemented with the SpalartAllmaras oneequation turbulence model. Several twodimensional unsteady aerodynamics problems have been computed with the different formulations and compared with the centraldifference scheme. In particular, the following problems are presented in this work: a onedimensional periodic piston problem and one with a rapid change of direction; the shock tube problem; a twodimensional isentropic Euler vortex transport problem; a periodic pitching NACA0012 aerofoil at different flight conditions; a simple flap deflection; a slat and slotted flap deployment; a spoiler deployment; an investigation of adverse lift due to rapidly deploying spoiler; a full landing case with a combination of slat, flap and spoiler deployments along with ground effect; a case where aerofoils fly in opposite directions at subsonic and supersonic speeds; and a rotorstator configuration with infinite relative motions. Moreover, some of the spacetime solutions have been correlated with a couple of analytical solutions and some empirical data. It has been demonstrated that the use of a centraldifference stencil leads to nonphysical solutions as a consequence of pressure waves travelling backwards in physicaltime, as expected. It has also been proved that upwind (e.g. Van Leer, Roe) and hybrid (CSUT = centraldifference in space, upwind in time) stencils yield more representative physical solutions and improve the rate of convergence. The benefits derived from the use of an upwind stencil as opposed to a centraldifference one are more noticeable in the case of nonperiodic problems, especially in fast transients. Unfortunately, upwind stencils are more dissipative and, as implied by the results of the isentropic Euler vortex transport problem, they did not seem to achieve as high a temporal accuracy as the centraldifference counterpart. The potential for very efficient timeaccurate simulations through the spacetime method has been demonstrated by the use of a variable timestep size across the spatial domain. It is possible to use small timesteps in the neighbourhood of the geometry, where big gradients occur, whilst retaining very large timesteps far away in the farfield, where the solution remains almost constant throughout the whole simulation. The versatility and broad applicability of the spacetime method to almost any kind of unsteady problems have been shown by the simulation of a wide range of problems involving complex boundary motions. Large relative motions or topological changes in the geometry are simulated with ease by the use of a spacetime formulation, which avoids the use of an arbitrary LagrangianEulerian (ALE) formulation in combination with a geometric conservation law (GCL). The solver did not need any modifications to cope with any of the problems presented here which proves its potential for highly automated CFD simulations. This could, in turn, speed up the design cycle of industrial applications.
