Effects of constant turbulent eddy viscosity assumption on gradient-based design optimization

Abstract

A feasibility study is carried out by investigating the effects of a usual assumption of constant turbulent eddy viscosity on the aerodynamic design using an adjoint variable method, one of most efficient gradient-based optimization techniques. Accurate steady and unsteady flow analyses are followed by the aerodynamic sensitivity analysis for the Navier-Stokes equations coupled with two-equation turbulence models. The capability of the present sensitivity analysis code to treat complex geometry using the chimera overlaid grid is also demonstrated by analyzing the flow over multielement airfoil. Like the mean flow equations, the turbulence model equations are also hand-differentiated to accurately calculate the sensitivity derivatives of flow quantities in turbulent viscous flows. With two-equation turbulence it is observed that the constant turbulent eddy viscosity assumption in the adjoint variable method could lead to inaccurate results in several test cases of transonic airfoil with strong shock and multielement airfoil at high angle of attack. Especially for the flow over high-lift airfoil close to stall, both flow analysis and sensitivity analysis are performed in an unsteady, time-accurate manner using the dual time stepping method. In addition, the effects of constant eddy viscosity assumption on single and multielement airfoil design optimization are carefully investigated in various design examples. Introduction As computational power advances, design optimization tools using computational fluid dynamics (CFD) have played a more important role in aerodynamic design. Among advanced optimization techniques, gradient-based optimization method has been widely used and even applied to multidisciplinary design optimization (MDO). In general, a gradient-based design optimization requires two steps. The first is to obtain the search direction that defines how the design variables will be changed for design improvement. The second is, so called one-dimensional search, to determine how far the design variables will move in this direction. This basic process is repeated until it approaches to the optimum. In one-dimensional search, an accurate and efficient flow solver is indispensable for the computation of pressure distribution and aerodynamic load coefficients such as lift, drag and pitching moment which are used in an objective function to be either minimized or maximized. Especially for the high-lift design optimization, it was * Postdoctorial Researcher, Institute of Advanced Machinery and Design, AIAA Member. T Assistant Professor, Dep't of Aerospace Engineering, AIAA Member. 1 Professor, Dep't of Aerospace Engineering, AIAA Senior Member. Copyright © 2002 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. previously noted that the unsteady, time-accurate computation is definitely required for the accurate computation of the flow over a high-lift device at a higher angle of attack close to stall angle where massive flow separation might occur. In determining the search direction, the gradients of design variables were traditionally calculated by the finitedifference method. It is, however, too expensive to compute the flow field iteratively with the incremented values of a design variable for complex two-dimensional or three-dimensional problems. In addition, this method is so sensitive to the step size of a design variable that it sometimes provides inaccurate signs or sensitivity derivatives. 4 Therefore, more robust techniques have been proposed using direct differentiation methods and adjoint variable methods. Compared with the direct differentiation methods, the adjoint variable methods are more advantageous for their capability to compute the gradients of the objective function and constraints when the number of design variables is much larger than that of the objective function and constraints. The adjoint variable methods adopt the formulation of the gradient in either a discrete or a continuous approach. In the discrete approach, which is used in the present work, the discretized governing equations are differentiated with respect to design variables whereas the adjoint equations are first differentiated and American Institute of Aeronautics and Astronautics Paper 2002-0262 (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. then discretized in the continuous approach.' It is necessary to incorporate the effect of turbulence in differentiating the governing equations to treat high Reynolds number flows more accurately. It is, however, very difficult to fully hand-differentiate the governing equations including the viscous terms and turbulence terms. Thus, some software tools such as automatic differentiation' ' n are used for the Navier-Stokes equations with a turbulence model. However, this approach is generally less efficient, in terms of computing time and computer memory, than hand-differentiation codes.' In the present work, the Navier-Stokes equations coupled with two-equation turbulence models are fully differentiated by human hand. Among most popular two-equation turbulence models, the k-co SST model' 14 is mainly used and then compared with the original k — co model' 16 and the standard k-s model. Like the mean flow equations, the turbulence model equations are also hand-differentiated to accurately compute the sensitivity derivatives of flow quantities with respect to design variables in turbulent flows. A usual assumption of constant turbulent eddy viscosity (CTEV, hereafter) in the adjoint variable methods has been previously used in aerodynamic design optimizations '. Recently, the accuracy of this assumption in sensitivity analysis for turbulent flows was reported using two-equation turbulence models on chimera overlaid grids and a one-equation turbulence model on unstructured grids in Refs. 2 and 4, respectively. In the present study, the feasibility of constant eddy viscosity assumption is carefully studied by comparing the sensitivity gradients using the CSAAV code, and the effects of this assumption on single and multielement airfoil design optimization are further investigated in various design examples: subsonic and transonic designs for drag minimization and lift maximization, and high-lift designs for lift to drag ratio maximization and even C/max maximization. Numerical Background Flow Analysis The Compressible Flow Analysis Navier-Stokes (CFANS2D/3D) solver, which has been well verified in many applications'', is used for the computation of turbulent viscous flows over single and multielement airfoils. The governing equations are the twodimensional, unsteady, compressible Navier-Stokes equations coupled with two-equation turbulence models: the k-co SST model', the k-co model', and the standard k-s model. The governing equations are transformed in generalized coordinates and are solved with a finite-volume method. Using a backward Euler implicit method, the governing equations adopting the dual time stepping method are discretized in time and linearized in delta form as: