A spacetime framework for aerodynamics of complex motions

  • Imanol Flamarique Ederra

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


A two-dimensional 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 rotor-stator configurations. It avoids methods such as Chimera or overset grids, or even re-meshing, by the use of a finite-volume 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 Lagrangian-Eulerian 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 time-periodic problems.

The finite-difference 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 central-difference scheme in spacetime yields non-physical 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 (central-difference 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 Spalart-Allmaras one-equation turbulence model.

Several two-dimensional unsteady aerodynamics problems have been computed with the different formulations and compared with the central-difference scheme. In particular, the following problems are presented in this work: a one-dimensional periodic piston problem and one with a rapid change of direction; the shock tube problem; a two-dimensional isentropic Euler vortex transport problem; a periodic pitching NACA-0012 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 rotor-stator 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 central-difference stencil leads to non-physical solutions as a consequence of pressure waves travelling backwards in physical-time, as expected. It has also been proved that upwind (e.g. Van Leer, Roe) and hybrid (CSUT = central-difference 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 central-difference one are more noticeable in the case of non-periodic 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 central-difference counterpart. The potential for very efficient time-accurate simulations through the spacetime method has been demonstrated by the use of a variable time-step size across the spatial domain. It is possible to use small time-steps in the neighbourhood of the geometry, where big gradients occur, whilst retaining very large time-steps 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 Lagrangian-Eulerian (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.
Date of Award19 Mar 2019
Original languageEnglish
Awarding Institution
  • University of Bristol
SupervisorThomas C S Rendall (Supervisor) & Ann L Gaitonde (Supervisor)


  • Spacetime
  • Aerodynamics
  • CFD modelling
  • Unsteady
  • Time-accurate
  • Finite-volume
  • Navier-Stokes
  • Euler
  • Spalart-Allmaras
  • Central-difference
  • Upwind
  • Stability analysis
  • Alternative to ALE
  • Complex motions
  • Topological geometry change
  • Relative motions
  • Explicit Runge-Kutta
  • 2D landing case
  • Slat and flap
  • Spoiler
  • Transonic
  • Supersonic
  • Subsonic
  • RANS
  • Stencil
  • Boundary conditions
  • Cell-centered
  • CFL condition
  • Negative Spalart-Allmaras
  • Rotor-stator
  • Sock-tube problem
  • Isentropic Euler vortex
  • NACA 0012
  • Adverse lift
  • Pitching motion

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