Discrete Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids

Abstract

An adjoint-based methodology for design optimization of unsteady turbulent flows on dynamic unstructured grids is described. The implementation relies on an existing unsteady three-dimensional unstructured grid solver capable of dynamic mesh simulations and discrete adjoint capabilities previously developed for steady flows. The discrete equations for the primal and adjoint systems are presented for the backward-difference family of time-integration schemes on both static and dynamic grids. The consistency of sensitivity derivatives is established via comparisons with complex-variable computations. The current work is believed to be the first verified implementation of an adjoint-based optimization methodology for the true time-dependent formulation of the Navier-Stokes equations in a practical computational code. Large-scale shape optimizations are demonstrated for turbulent flows over a tiltrotor geometry and a simulated aeroelastic motion of a fighter jet. = cost function component weight 1 = quantity at freestream conditions * = target quantity

Objective function choice for control of a thermocapillary flow using an adjoint-based control strategy

“…Derivations of the adjoint equations can be found in Muldoon (2008), Giles et al (2003), Nielsen (1998). Details of issues involved in deriving and solving the adjoint equations can be found in Jameson (2000, 2001), Nielsen et al (2009, 2004), and McNamara et al (2004, and references therein. Two issues are worth mentioning, however.…”

Computing Stability Derivatives and Their Gradients for Aerodynamic Shape Optimization

“…Unfortunately, computing the stability derivatives with a full time-dependent solution in order to include that information would be extremely expensive. Several authors have examined the use of adjoint methods in time-dependent optimizations, both in two dimensions [33,34,35] and three dimensions [36,37]. While the timedependent adjoint method is certainly an improvement over finite-difference sensitivity methods, it still incurs a high computational cost.…”

Error estimation and adaptation for functional outputs in time-dependent flow problems

“…(33) gives us an expression for the second and final contribution to the computable correction term as:…”

“…In our work, the cost of performing the backward sweep in the adjoint solver is generally similar to the cost of performing the forward integration in time to obtain the analysis solution. However, other researchers have noted that for large-scale unsteady viscous problems with turbulence models, the adjoint solution can in fact cost more than the analysis solution [33,34].…”