Sinan Eyi

Two-point transonic airfoil design using optimization for improved off-design performance

A two-point, aerodynamic design method is presented that improves the aerodynamic performance of transonic airfoils over a range of the flight envelope. It couples an Euler flow solver and a numerical optimization tool. The major limitation of single-point design is the poor off-design performance. Two-point design is used to extend the optimized performance range over more of the desired flight envelope. The method is applied to several transonic flow design points, and the results are compared to single-point design results. The secondary design points are chosen by varying the Mach number and the angle of attack. The two-point designs perform better than the single-point design over the design-point range.

Aerodynamic design via optimization

An aerodynamic design optimization method is presented that generates an airfoil, producing a specified surface pressure distribution at a transonic speed. The design procedure is based on the coupled Euler and boundary-layer technology to include the rotational viscous physics which characterizes transonic flows. A leastsquare optimization technique is used to minimize pressure discrepancies between the target and designed airfoils. The method is demonstrated with several examples at transonic speeds. The design optimization process converges quickly, that makes the method attractive for practical engineering applications. I. Introduction I N recent years, computational fluid dynamics (CFD) has become a valuable engineering tool in the aircraft industry. CFD plays a complementary role, not a replacement, to experiments in practical design communities. Rubbert1 showed some good examples of the use of CFD and experiment, in combination, for transonic design. A major strength of CFD is the ability to produce detailed insights into complex flow phenomena. The process of decomposition and parameterization can help identify the cause of weak aerodynamic performance, and the microscopic understanding of the flow can lead to improved design. Continuing advances in computer hardware and simulation techniques provide an unprecedented opportunity for CFD. Now simulations of more complete configurations with more complex physics can be performed at an affordable cost. Accuracy and reliability of the computation have been continuously improved. The use of high-level flow models and large-size refined grids enables one to analyze flows with complicated structures and various length scales. Compared to the remarkable advances in analysis capability, however, relatively few advances have been made in design technology. Conventional design practices, therefore, often depend on analysis methods through iterative cut-and-try approaches. A unique advantage of CFD is the capability of inverse design. Inverse design directly determines the airfoil geometry that produces the pressure distribution specified by a designer. Many existing inverse design methods are based on the potential flow assumption due to its simplicity. Volpe and Melnik2 employed an inverse design method using the nonlinear full potential formulation. Bauer and colleagues3 used the hodograph method that solves the full potential equation in the hodograph plane where the equations are linear. The potential flow model, however, cannot properly represent transonic features such as embedded shock waves and shock-boundarylayer interactions. An accurate analytic capability is a prerequisite for a successful design, because the quality of the design depends on the quality of the method used to predict the flowfield. Several inverse design methods were demonstrated using the Euler formulations by Giles and Drela,4 and Mani.5 Instead of achieving the prescribed pressure distribution, some design methods use a constrained optimization process

Transonic Airfoil Design by Constrained Optimization

A N aerodynamic design method is developed which cou- ples flow analysis and numerical optimization to find an airfoil shape with improved aerodynamic performance. The flow analysis code is based on the coupled Euler and bound- ary-layer equations in order to include the rotational, viscous physics of transonic flows. The numerical optimization pro- cess searches for the best feasible design for the specified design objective and design constraints. The method is dem- onstrated with several examples at transonic flow conditions. Contents The optimization process is performed with a commercially available constrained optimization tool.3 The sensitivity of the flow to the perturbation is calculated by finite differences. The effectiveness and efficiency of the design process are influenced by many factors: the number and the shape of the base functions, the number and the tolerance of the con- straints, the flow model and the grid used for flow analyses, and the flight condition at the design point. Design Demonstration The objective of the present design is to produce minimum drag at a specified transonic flight condition. Inequality con- straints are imposed on lift, pitching moment, and cross-sec- tional area of the optimized airfoil. The lift and the area of the optimized airfoil should not be smaller than those of the original airfoil, and the pitching moment should not increase in absolute value. Also imposed are side constraints which limit the magnitude of the design variables. Side constraints are important because a large geometry change can cause boundary-layer separation leading to a termination of the flow solver.

Aerothermodynamic shape optimization of hypersonic blunt bodies

The aim of this study is to develop a reliable and efficient design tool that can be used in hypersonic flows. The flow analysis is based on the axisymmetric Euler/Navier–Stokes and finite-rate chemical reaction equations. The equations are coupled simultaneously and solved implicitly using Newtons method. The Jacobian matrix is evaluated analytically. A gradient-based numerical optimization is used. The adjoint method is utilized for sensitivity calculations. The objective of the design is to generate a hypersonic blunt geometry that produces the minimum drag with low aerodynamic heating. Bezier curves are used for geometry parameterization. The performances of the design optimization method are demonstrated for different hypersonic flow conditions.

Convergence Error and Higher-Order Sensitivity Estimations

differential equation with respect to the design variables. The accuracies of the convergence error and higher-order sensitivity estimation methods are verified using Laplace, Euler, and Navier–Stokes equations. The developed methods are used to improve the accuracy of the finite-difference sensitivity calculations in iteratively solved problems.Aboundonthenormvalueofthe finite-differencesensitivityerrorinthestatevariablesisminimizedwith respect to the finite-difference step size. The optimum finite-difference step size is formulated as a function of the norm values of both convergenceerror andhigher-order sensitivities. Thesensitivities calculated with the analytical and the finite-difference methods are compared. The performance of the proposed methods on the convergence of inverse design optimization is evaluated.

Design Optimization in Hypersonic Flows

The objective of this study is to develop a reliabl e and efficient design tool that can be used in hyp ersonic flows. The flow analysis is based on axisymmetric Euler an d the finite rate chemical reaction equations. Thes e coupled equations are solved by using Newton’s method. The analytical method is used to calculate Jacobian mat rices. Sensitivities are evaluated by using the adjoint me thod. The performance of the optimization method is demonstrated in hypersonic flow.

Performances of Numerical and Analytical Jacobians in Flow and Sensitivity Analysis

The effects of flux Jacobian evaluation on flow and sensitivity analysis are studied. A cell centered finite volume method with various upwinding schemes is used. A Newton’s method is applied for flow solution, and the resulting spa rse matrix is solved by LU factorization. Flux Jacobians are evaluated both numerically and analytically. The sources of the error in numerical Jacobian calculation are studied. The optimum finite difference perturbation magnitude that minimizes the error is searched. The effects of error numerical Jacobians on the convergence of flow solver are studied. The sen sitivities of the flow variables are evaluated by direct-differentiation method. The Jac obian matrix which is constructed in the flow solution is also used in sensitivity calculati on. The influence of errors in numerical Jacobians on the accuracy of sensitivities is analy zed. Results showed that, the error in Jacobians significantly affects the convergence of flow analysis and accuracy of sensitivities. Approximately the same optimum perturbation magnitude enables the most accurate numerical flux Jacobian and sensitivity calculation s.

INVERSE AIRFOIL DESIGN USING THE NAVIER STOKES EQUATIONS

The feasibility of the use of the Navier-Stokes equations in aerodynamic design is examined. The Navier-Stokes equations can include the rotational viscous physics at transonic speeds, and hence are expected to produce more reliable designs. The target pressure is specified and a least-squares optimization is used to minimize the pressure discrepancy between the target and designed airfoils. The sensitivity analysis which determines the response of the flow to a geometry perturbation is performed based on a finite-difference evaluation. The performance of the design method is evaluated with various design practices.

Comparison of Newton and Newton-GMRES Methods for Three Dimensional Supersonic Nozzle Design

The aim of this study is to implement Newton-GMRES method into supersonic flows as an alternative to Newton method which is a reliable and efficient solver. The flow analysis is based on the three dimensional Euler equations. The objective of this study is compare the performances of Newton and Newton-GMRES methods in a 3-D nozzle design. The Jacobian matrix needed for Newton method is evaluated analytically. Newton-GMRES method is a matrix free solution technique hence it does not require the Jacobian matrix evaluation. A gradient based method is used for numerical optimization. Sensitivities are evaluated by using adjoint method. Design optimization is conducted by Newton’s method.

Performance Evaluation of the Numerical Flux Jacobians in Flow Solution and Aerodynamic Design Optimization

A direct sparse matrix solver is utilized for the flow solution and the analytical sensitivity analysis. The effects of the accuracy of the numerical Jacobians on the accuracy of sensitivity analysis and on the performance of the Newton’s method flow solver are analyzed in detail. The gradient based aerodynamic design optimization is employed to demonstrate those effects.