Suppression of vortex-shedding noise via derivative ...

PHYSICS OF FLUIDS

VOLUME 16, NUMBER 10

OCTOBER 2004

Suppression of vortex-shedding noise via derivative-free shape optimization Alison L. Marsden Department of Mechnical Engineering, Stanford University, Stanford, California 94305

Meng Wang Center for Turbulence Research, Stanford University, Stanford, California 94305

J. E. Dennis, Jr. Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005

Parviz Moin Department of Mechanical Engineering, Stanford University, Stanford, California 94305

(Received 9 April 2004; accepted 11 June 2004; published online 8 September 2004) In this Letter we describe the application of a derivative-free optimization technique, the surrogate management framework (SMF), for designing the shape of an airfoil trailing edge which minimizes the noise of vortex shedding. Constraints on lift and drag are enforced within SMF using a filter. Several optimal shapes have been identified for the case of laminar vortex shedding with reasonable computational cost using several shape parameters, and results show a significant reduction in acoustic power. Physical mechanisms for noise reduction are discussed. © 2004 American Institute of Physics. [DOI: 10.1063/1.1786551] The model airfoil geometry, shown in Fig. 1, is a shortened version of that used in experiments of Blake.4 The airfoil chord is 10 times its thickness, and the right half of the surface is allowed to deform. The flow is from left to right, at a chord Reynolds number of Re= 10 000. We begin by defining a cost function, which is the quantity we aim to minimize through optimization. For unsteady laminar flow past an airfoil at low Mach number, the acoustic wavelength associated with the vortex shedding is typically long relative to the airfoil chord. Noise generation from an acoustically compact surface can be expressed using Curle’s extension to the Lighthill theory,5

Optimization of unsteady flow problems, such as aeroacoustic problems, are difficult using standard gradient-based methods. Although adjoint solvers provide an efficient method of computing gradients for problems with many design variables,1 they present data storage issues for unsteady flows and their implementation is flow-solver dependent. Derivative-free optimization methods offer a viable alternative, but are often dismissed because of computational expense. In this Letter, we illustrate the use of a derivative-free technique, the surrogate management framework (SMF), to optimize the trailing edge of an airfoil for noise reduction in unsteady laminar flow. The method is shown to be effective and computationally efficient. The SMF method, developed by Booker et al.,2 relies on convergence theory of pattern search methods. It increases the efficiency of pattern search methods for computationally expensive problems that may have little or no gradient information by making use of surrogate functions which approximate the true function.2 The majority of optimization is performed on the inexpensive surrogate function rather than on the expensive cost function. A novel adaptation of the SMF method with constraints has been validated for optimal aeroacoustic shape design by Marsden et al.3 The purpose of this Letter is to bring these methods to the attention of the fluid mechanics community by demonstrating their success on a constrained airfoil trailing-edge shape design problem. The results presented here extend the previous work3 by allowing both the upper and lower surfaces of the trailing edge to deform with constraints on lift and drag. We demonstrate the robustness and efficiency of the SMF method, and present trailing-edge designs which achieve as much as 70% reduction in vortexshedding noise. Physical mechanisms for noise reduction are discussed. 1070-6631/2004/16(10)/L83/4/$22.00

␳⬇

M 3 xi ⳵ 4␲ 兩x兩2 ⳵ t

冕

n j pij共y,t − M兩x兩兲d2y,

共1兲

S

where ␳ is the acoustic density at far field position x, pij = p␦ij − ␶ij is the compressive stress tensor, composed of pressure and viscous wall stress, n j is the direction cosine of the outward normal to the airfoil surface S, M is the free stream Mach number, and y is the source field position vector. All variables have been made dimensionless, with airfoil chord C as the length scale, free stream velocity U⬁ as velocity scale, and C / U⬁ as the time scale. The density and pressure are normalized by their ambient values. From Eq. (1), a cost function directly proportional to the radiated acoustic power per unit span can be derived: J=

冉冕 ⳵

⳵t

S

冊冉冕 2

n j p1j共y,t兲d2y

+

⳵

⳵t

S

n j p2j共y,t兲d2y

冊

2

. 共2兲

The noise field is of dipole type, containing contributions from the unsteady lift (second term) and drag (first term). L83

© 2004 American Institute of Physics

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FIG. 1. Model airfoil shape and control points. Squares: control points for optimization of upper surface. Circles: control points used for optimization of upper and lower airfoil surfaces.

The cost function J depends on control parameters which each correspond to a deformation point on the airfoil surface. Parameter values must lie strictly within prescribed bounds. Figure 1 shows control points for deforming only the upper surface (squares) and both the upper and lower surfaces (circles). When optimizing the upper surface only, each parameter value is defined as the control point displacement relative to the original shape in the direction normal to the airfoil surface. When optimizing both sides, the parameter values are defined by displacement in the vertical direction, and lower surface deformation is defined by the airfoil thickness relative to the upper surface. In both cases, a Hermite spline connects all the deformation points to the trailing edge point and the left (undeformed) region to give a continuous airfoil surface. To calculate the cost function value for a given shape, a finite-difference code6 is used to solve the time-dependent incompressible two-dimensional Navier–Stokes equations in generalized curvilinear coordinates. The mesh is a C-type mesh consisting of approximately 131 000 cells. Because the flow has unsteady vortex shedding, the cost function is oscillatory. In the optimization procedure, the mean cost function J [cf. (2)] is used, which is obtained by averaging in time after the flow is statistically steady. To make the aeroacoustic shape design problem more realistic, constraints are added to keep lift and drag at desirable levels. Constraints are enforced by using a filter, as added to pattern search methods by Audet and Dennis.7 To construct a filter, we first define a constraint violation function H 艌 0 which indicates how closely the problem constraints are being met. The goal of the optimization is to find solutions which have a small cost function value, together with a small (or zero) value of H. In the trailing-edge problem, H is defined to prevent the lift from decreasing and the drag from increasing, as follows:

冉

H = max 0,

冊冉

冊

L* − L D − D* + max 0, , L* D*

共3兲

where L*共⬇0.1兲 and D*共⬇0.033兲 are the dimensionless lift and drag of the original airfoil, equal to twice the lift and drag coefficients, respectively. The set of points that exactly satisfy H共x兲 = 0 defines the feasible region. Thus, a point x is infeasible if H共x兲 ⬎ 0. An infeasible point x⬘ is considered filtered, or dominated, if there is an infeasible point x belonging to the filter for which H共x兲艋 H共x⬘兲 and J共x兲艋 J共x⬘兲. A filter F, is defined here to be the finite set of nondominated infeasible points plus the best feasible point found so far. For example, the final filter from the trailing-edge optimization with deformation of two sides is shown in Fig. 2, where the cost function value is

FIG. 2. Final filter for optimization with deformation of both sides of the airfoil. The best feasible point is the square, the least infeasible point is the triangle, the filter points are the circles, and filtered points are stars.

plotted vs the constraint violation. The points in the filter are connected with vertical and horizontal lines to form a dividing line between filtered and unfiltered regions. The best feasible point (square) has the lowest cost function value which satisfies the constraints (i.e., where H = 0). The least infeasible point (triangle) is the filter point with the lowest nonzero constraint violation function value. Other points in the filter are marked with circles. We present an outline of the SMF algorithm here, with reference to a detailed description given by Marsden et al.3 Like pattern search methods, SMF requires that all points evaluated are restricted to lie on a mesh in the parameter space. An initial set of data is chosen using Latin hypercube sampling,8 and evaluated for construction of the surrogates, which are Kriging functions in the present work. The SMF algorithm consists of two steps, SEARCH and POLL. The SEARCH step provides means for local and global exploration of the parameter space. In the SEARCH step, optimization is performed on the surrogate in order to predict the location of one or more minimizing points, and the function is evaluated at these points. If a new nondominated point is found (i.e., the filter is improved), the search is considered successful, the surrogate is updated, and another SEARCH step is performed. If the SEARCH fails to find an improved point, a POLL step is performed. Convergence of the SMF algorithm is guaranteed by the POLL step, in which points neighboring the current best point on the mesh are evaluated in a positive spanning set of directions. If the POLL produces an improvement in either the best feasible or least infeasible point, then a SEARCH step is performed on the current mesh. Otherwise, the current best point is a local minimizer of the function on the mesh. For greater accuracy, the mesh may be refined, at which point the algorithm continues with a SEARCH. Convergence is reached when a local minimum on the mesh is found, and the mesh has been refined to the desired accuracy. Each time new data points are evaluated, the surrogate is updated.



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FIG. 4. Normalized cost function vs number of function evaluations. --Upper surface deformation; —, deformation of both sides. The best feasible solution is marked with the right-most square and the least infeasible solution is marked with a triangle.

FIG. 3. Instantaneous vorticity contours with minimum −25, maximum 25 and 20 contour levels. Figure shows shapes corresponding to original (upper plot), upper surface optimization (second from top), best feasible from twosided optimization (third), and least infeasible from two-sided optimization (fourth).

A novel method of incorporating a penalty function into the SMF method for surrogate building has been developed.3 In this method, the penalty is used only for construction of the surrogate, and does not affect convergence theory of the method. The penalty function Jˆ = Jorig + ␣H is formed, and a surrogate is constructed to approximate the function Jˆ, so that it is used to predict areas of the function which satisfy the constraint. The minimum of the modified surrogate function is then evaluated in the SEARCH step. A systematic method for choosing the penalty parameter ␣ has been developed.3 Shapes resulting from optimization of the upper surface, and both the upper and lower surfaces of the trailing edge using five parameters are shown in Fig. 3, with the original airfoil shape shown at the top. All plots show instantaneous contours of vorticity. The optimized airfoil shape for deformation of the upper surface only is shown in the second plot from the top of Fig. 3. The constraints are effective in keeping the lift at the target value, and the drag for this case has decreased by 9% compared with the original shape. The cost function reduction for this case is 43%, requiring 22 iterations (88 total function evaluations) for convergence. All points in either the SEARCH or POLL step can be evaluated in parallel so that one iteration consists of several simultaneous function evaluations.3 The cost function history for this case is shown by the dashed line in Fig. 4. It is observed from Fig. 3 that

vortices are shed farther away from the trailing edge compared to the original case, resulting in smaller pressure fluctuations on the airfoil surface. In addition, the magnitude of wake vortices is smaller. This explains the reduction in unsteady lift, and therefore of acoustic power. Results for the case of deforming both the upper and lower trailing-edge surfaces are shown in the lower plots in Fig. 3. The third shape from the top results in a 48% cost function reduction, and exactly satisfies the lift and drag constraints (H = 0). This corresponds to the best feasible point in Fig. 2 marked with a square. The lift in this case has fortuitously increased by 229% and the drag has decreased by 4%. The large increase in lift is not surprising since the modified airfoil has a greater effective angle of attack. The optimization required 135 function evaluations in 31 iterations, and the cost function history is shown by the solid line in Fig. 4. The least infeasible point, marked with a triangle in Fig. 2, yields the bottom shape in Fig. 3. Although this shape does not exactly satisfy the drag constraint, it achieves a significantly greater cost function reduction of 70%. The drag has increased by only 1.5% and the lift has increased by 262%. The slight relaxation of the constraint requirement for drag allows for drastic reduction in the cost function. It is clear from this result that the SMF filter method can provide an attractive range of possible solutions, rather than returning a single solution. A comparison of vorticity contours in Fig. 3 suggests that vortex shedding for the two-sided optimal cases is nonperiodic, in contrast to the original and one-sided optimal cases. The vortices shed from the upper and lower surfaces are not always out of phase as in the one-sided case, resulting in more interaction at the trailing edge and generation of smaller vortical structures. This interaction results in cancellation of forces on the airfoil surface, leading to reduced noise. Figure 5 shows the power spectrum of unsteady lift for the four shapes in Fig. 3 as a function of frequency. The spectra for the original and one-sided optimal shapes are


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FIG. 5. Frequency spectrum of lift. —, Original shape, - · · -, constrained upper surface deformation, - · -, best feasible solution with both sides deformed, — — — , least infeasible solution with both sides deformed.

dominated by a single peak at the shedding frequency, whereas the spectra for the two-sided optimal shapes are more broadband albeit with distinct spectral peaks. In general, reduction in acoustic power can be caused by a reduced amplitude or frequency of lift and drag oscillations [see Eq. (2)]. Compared to the spectrum for the original shape 共−兲, the spectrum for the one-sided optimal case 共- · · -兲 shows a lower peak amplitude, but the same shedding frequency, indicating that the former is the cause for the observed noise reduction. The contribution from the unsteady drag in all cases was found to be small. The spectrum for the best feasible case with optimization of both sides 共- · -兲 shows a slight reduction in magnitude but a large reduction in the dominant shedding frequency. The spectrum for the least infeasible case 共---兲 confirms reductions in both magnitude and frequency compared with the original. These results all agree with qualitative observations of the flow field in Fig. 3. Calculation of the far-field noise spectra confirms that not only the total acoustic power, but also the peak amplitude, has been reduced by the optimized shapes. Theoretical analysis of trailing-edge noise for the Blake airfoil geometry in turbulent flow is presented by Howe.9 Although Howe’s work is an analysis of turbulent trailingedge flow, it is worth noting that a decrease in trailing-edge noise with an increase in trailing-edge angle was predicted. This result agrees qualitatively with the blunt shape found by

the optimization method in the case where the upper surface was deformed. In conclusion, application of the SMF method to trailing-edge optimization has demonstrated significant reduction in acoustic power, as well as several interesting and previously unexpected airfoil shapes. The SMF method is robust and cost effective for several design parameters, and the filter method is effective in enforcing constraints on airfoil lift and drag. In this work, both sides of the airfoil were deformed, extending the previous work,3 where the method was validated by deforming the upper surface. Use of the SMF method for time-dependent fluid dynamics problems avoids significant difficulties with the addition of constraints, implementation and data storage that arise with adjoint solvers. The methodology described in this paper is not restricted to the laminar flow problem considered here, but can be applied to a wide range of fluid flow problems including complex geometries and turbulence. The portability of the method would easily allow it to be coupled to turbulent flow solvers based on LES or unsteady RANS for high Reynolds number flows. This work was supported by ONR under Grant No. N00014-01-1-0423 with Ronald Joslin as program manager. Computer time was provided by the DoD’s HPCMP through NRL-DC and ARL/MSRC. J.E.D. was supported by the AFOSR, CSRI, the IMA, the NSF, and the Ordway Endowment at the University of Minnesota. 1

A. Jameson, L. Martinelli, and N. A. Pierce, “Optimum aerodynamic design using the Navier–Stokes equations,” Theor. Comput. Fluid Dyn. 10, 213 (1998). 2 A. J. Booker, J. E. Dennis, Jr., P. D. Frank, D. B. Serafini, V. Torczon, and M. W. Trosset, “A rigorous framework for optimization of expensive functions by surrogates,” Struct. Optim. 17, 1 (1999). 3 A. L. Marsden, M. Wang, J. E. Dennis, Jr., and P. Moin, “Optimal aero acoustic shape design using the surrogate management framework,” Optimization and Engineering 5, 235 (2004). 4 W. K. Blake, “A statistical description of pressure and velocity fields at the trailing edge of a flat strut,” David Taylor Naval Ship Research and Development Center Report No. 4241 Bethesda, MD, 1975. 5 N. Curle, “The influence of solid boundary upon aerodynamic sound,” Proc. R. Soc. London, Ser. A 231, 505 (1955). 6 M. Wang and P. Moin, “Computation of trailing-edge flow and noise using large-eddy simulation,” AIAA J. 38, 2201 (2000). 7 C. Audet and J. E. Dennis, Jr., “A pattern search filter method for nonlinear programming without derivatives,” SIAM J. Optim. 14, 980 (2004). 8 M. D. McKay, W. J. Conover, and R. J. Beckman, “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,” Technometrics 21, 239 (1979). 9 M. S. Howe, “The influence of surface rounding on trailing edge noise,” J. Sound Vib. 126, 503 (1988).
