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Research Papers

Optimum Shape Design for Multirow Turbomachinery Configurations Using a Discrete Adjoint Approach and an Efficient Radial Basis Function Deformation Scheme for Complex Multiblock Grids

[+] Author and Article Information
Benjamin Walther

Department of Mechanical Engineering,
McGill University,
Montreal, QC H3A 0C3, Canada
e-mail: benjamin.walther@mail.mcgill.ca

Siva Nadarajah

Department of Mechanical Engineering,
McGill University,
Montreal, QC H3A 0C3, Canada
e-mail: siva.nadarajah@mcgill.ca

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received December 9, 2014; final manuscript received December 11, 2014; published online January 28, 2015. Editor: Kenneth C. Hall.

J. Turbomach 137(8), 081006 (Aug 01, 2015) (20 pages) Paper No: TURBO-14-1314; doi: 10.1115/1.4029550 History: Received December 09, 2014; Revised December 11, 2014; Online January 28, 2015

This paper proposes a framework for fully automatic gradient-based constrained aerodynamic shape optimization in a multirow turbomachinery environment. The concept of adjoint-based gradient calculation is discussed and the development of the discrete adjoint equations for a turbomachinery Reynolds-averaged Navier–Stokes (RANS) solver, particularly the derivation of flow-consistent adjoint boundary conditions as well as the implementation of a discrete adjoint mixing-plane formulation, are described in detail. A parallelized, automatic grid perturbation scheme utilizing radial basis functions (RBFs), which is accurate and robust as well as able to handle highly resolved complex multiblock turbomachinery grid configurations, is developed and employed to calculate the gradient from the adjoint solution. The adjoint solver is validated by comparing its sensitivities with finite-difference gradients obtained from the flow solver. A sequential quadratic programming (SQP) algorithm is then utilized to determine an improved blade shape based on the gradient information from the objective functional and the constraints. The developed optimization method is used to redesign a single-stage transonic flow compressor in both inviscid and viscous flow. The design objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.

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Figures

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Fig. 1

Periodicity in circumferential coordinate direction

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Fig. 2

Dependency of the boundary states at a mixing-plane interface

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Fig. 3

Boundary states contributing to the adjoint flux of a cell next to a mixing-plane interface

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Fig. 4

Multiblock grid structure for a single-stage transonic compressor

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Fig. 5

Perturbed skeleton after the first stage (left) and perturbed grid after completion of the RBF grid perturbation scheme (right)

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Fig. 6

Computational grid for Darmstadt rotor No. 1 at 50% span and blade surfaces

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Fig. 7

Relative Mach number contours (left) and ψ2-contours (right), inviscid flow, plotted section slice: 50% span

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Fig. 8

Gradient comparison for the functionals entropy generation rate (left) and total pressure ratio (right)

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Fig. 9

Shape modification of rotor (left) and stator (right), inviscid design case, constrained optimization

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Fig. 10

Relative Mach number contours of Darmstadt rotor No. 1 baseline design (left) and redesign (right) at 75% span (top), 50% span (center), and 25% span (bottom), inviscid design case, constrained optimization

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Fig. 11

Relative Mach number distribution on the rotor suction side (top) and rotor pressure side (bottom) of the baseline design (left) and redesign (right), inviscid design case, constrained optimization

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Fig. 12

Normalized objective function and isentropic efficiency (left), change in mass flow rate and total pressure ratio (right), inviscid case

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Fig. 13

Relative Mach number contours of Darmstadt Rotor No. 1 baseline design (left) and redesign (right) at 75% span (top) and 50% span (bottom), viscous design case

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Fig. 14

Isentropic Mach number distribution on rotor suction side (top) and rotor pressure side (bottom) of baseline design (left) and redesign (right), viscous design case

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Fig. 15

Entropy contours baseline design (top) and optimized design (bottom) at the compressor exit plane, viscous design case

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Fig. 16

Normalized objective function and isentropic efficiency (left), change in mass flow rate and total pressure ratio (right), unconstrained (UC) optimization

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Fig. 17

Final grid after optimization (top) and baseline grid before optimization (bottom), 33% span cut, viscous design case

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Fig. 18

Final grid after optimization, 75% span cut, viscous design case

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