Siva Nadarajah

A COMPARISON OF THE CONTINUOUS AND DISCRETE ADJOINT APPROACH TO AUTOMATIC AERODYNAMIC OPTIMIZATION

This paper compares the continuous and discrete adjoint-based automatic aerodynamic optimization. The objective is to study the trade-off between the complexity of the discretization of the adjoint equation for both the continuous and discrete approach, the accuracy of the resulting estimate of the gradient, and its impact on the computational cost to approach an optimum solution. First, this paper presents complete formulations and discretization of the Euler equations, the continuous adjoint equation and its counterpart the discrete adjoint equation. The differences between the continuous and discrete boundary conditions are also explored. Second, the results demonstrate two-dimensional inverse pressure design and drag minimization problems as well as the accuracy of the sensitivity derivatives obtained from continuous and discrete adjoint-based equations compared to finite-difference gradients.

Optimum Shape Design for Unsteady Flows with Time-Accurate Continuous and Discrete Adjoint Methods

This paper presents an adjoint method for the optimal control of unsteady flows. The goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time-accurate method. First, this paper presents the complete formulation of the time-dependent optimal design problem. Second, we present the time-accurate unsteady continuous and discrete adjoint equations. Third, we present results that demonstrate the application of the theory to a two-dimensional oscillating airfoil. The results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization.

Effect of Shape Parameterization on Aerodynamic Shape Optimization

This paper studies the effect of shape parameterization on automatic aerodynamic shape optimization. Four shape parameterization techniques are studied: mesh points, Hicks-Henne bump functions, B-spline curves and the PARSEC method. The objective is to compare them based on accuracy and impact on the computational cost. First, this paper presents the complete formulation of the optimal design problem for the Navier-Stokes equations. Second, the implementation of these surface representation methods is explored. Finally, results are presented for inverse and drag minimization problems.

OPTIMAL CONTROL OF UNSTEADY FLOWS USING A TIME ACCURATE METHOD

This paper presents an adjoint method for the optimal control of unsteady flows. The goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time accurate method. First, this paper presents the complete formulation of the time dependent optimal design problem. Second, we present the time accurate unsteady continuous and discrete adjoint equations. Third, we present results that demonstrate the application of the theory to a two-dimensional oscillating airfoil. The results are compared to a multipoint approach to illustrate the added benefit of performing full unsteady optimization.

Optimum Shape Design for Unsteady Three-Dimensional Viscous Flows Using a Nonlinear Frequency-Domain Method

Thispaperpresentsanadjointmethodfortheoptimumshapedesignofunsteadythree-dimensionalviscous flows. The goal is to develop a set of discrete unsteady adjoint equations and the corresponding boundary condition for the nonlinear frequency-domain method. First, this paper presents the complete formulation of the time-dependent optimal design problem. Second, we present the nonlinear frequency-domain adjoint equations for threedimensional viscous transonic flows. Third, we present results that demonstrate the application of the theory to a three-dimensional wing.

SURVEY OF SHAPE PARAMETERIZATION TECHNIQUES AND ITS EFFECT ON THREE-DIMENSIONAL AERODYNAMIC SHAPE OPTIMIZATION

This paper applies three shape parameterization techniques to few design cases typical in aerospace studies in order to quantify the effect of shape parameterization on automatic aerodynamic shape optimization. The three methods in study are mesh points, B-spline surfaces and the Class function / Shape function Transformation (CST). Given the extensive prevalence of CFD in aerodynamic design, it is of interest to study the efficiency of parameterization approaches and compare them in terms of accuracy and performance aspects of the shape optimization process. Initially, the complete formulation of the optimal design problem for the Euler equations is presented. Then, the implementation of these surface representation methods is explored and finally, results are presented for three-dimensional inverse design and drag minimization problems.

SONIC BOOM REDUCTION USING AN ADJOINT METHOD FOR WING-BODY CONFIGURATIONS IN SUPERSONIC FLOW

This paper presents an adjoint method for the calculation of remote sensitivities in supersonic flow. The goal is to develop a set of adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the formulation and discretization of the adjoint equations. The special treatment of the adjoint boundary condition to obtain remote sensitivities is also discussed. Second, we present results that demonstrate the application of the theory to a three-dimensional remote inverse design problem using a low sweep biconvex wing and a supersonic business jet wing-body configuration.

A stable interface element scheme for the p-adaptive lifting collocation penalty formulation

This paper presents a procedure for adaptive polynomial refinement in the context of the lifting collocation penalty (LCP) formulation. The LCP scheme is a high-order unstructured discretization method unifying the discontinuous Galerkin, spectral volume, and spectral difference schemes in single differential formulation. Due to the differential nature of the scheme, the treatment of inter-cell fluxes for spatially varying polynomial approximations is not straightforward. Specially designed elements are proposed to tackle non-conforming polynomial approximations. These elements are constructed such that a conforming interface between polynomial approximations of different degrees is recovered. The stability and conservation properties of the scheme are analyzed and various inviscid compressible flow calculations are performed to demonstrate the potential of the proposed approach.

Detached-Eddy Simulation of a Wing Tip Vortex at Dynamic Stall Conditions

The behavior of the tip vortex behind a square NACA0015 wing was numerically investigated. The problems studied include the stationary and the oscillating wings at static and dynamic stall conditions. Reynolds-averaged Navier-Stokes and detached-eddy simulation schemes were implemented. Vortex structures predicted by Reynolds-averaged Navier-Stokes were mainly diffused while detached-eddy simulation was able to produce qualitatively and quantitatively better results as compared to the experimental data. The breakup of the tip vortex, which started at the end of the upstroke and continued to the middle of the downstroke over an oscillation cycle, was observed in detached-eddy simulation data.

Implicit Nonlinear Frequency-Domain Spectral-Difference Scheme for Periodic Euler Flow

This paper combines a nonlinear frequency-domain scheme with a high-order spectral-difference discretization for the two-dimensional unsteady Euler equations. An implicit lower/upper symmetric Gauss-Seidel method is introduced to solve the nonlinear frequency-domain equations. High-order accuracy and solution acceleration due to the implicit treatment are numerically verified on the vortex advection and subsonic airfoil test cases. Finally, the performance of this implicit high-order scheme on a fully compact stencil for periodic flows is evaluated on a pitching-airfoil problem.