One-Shot Pseudo-Time-Stepping Method for Aerodynamic Shape Optimization Using the Navier-Stokes Equations

Introduction

In all the applications of pseudo-time-stepping method mentioned in Chapters 7- 11, the state equations have been the Euler equations. In this chapter we extend the method to viscous compressible flow modeled by the Reynolds Averaged Navier- Stokes equations together with algebraic turbulence model of Baldwin and Lomax. While inviscid formulations are useful for the design in transonic cruise conditions, inclusion of viscous effects are essential for optimal design encompassing off-design conditions and high-lift configurations. The computational complexity in viscous design is at least an order of magnitude greater than that in inviscid design since the number of mesh points are to be increased by a factor of two or more to resolve the boundary layer. The convergence of the Navier-Stokes solver is much more slower than the Euler solver due to discrete stiffness and directional decoupling arising from the highly stretched boundary layer cells. Since we use inaccurate state and costate solutions, and hence inaccurate gradients, in our one-shot pseudotime- stepping method, therefore slow convergence of the Navier-Stokes (forward and adjoint) solver may affect the convergence of this method. We investigate that numerically in this chapter.