Research Papers

Employing the Time-Domain Unsteady Discrete Adjoint Method for Shape Optimization of Three-Dimensional Multirow Turbomachinery Configurations

[+] Author and Article Information
Georgios Ntanakas

Rolls-Royce Deutschland,
Blankenfelde-Mahlow 15827, Germany;
Parallel CFD & Optimization Unit,
Lab of Thermal Turbomachines,
School of Mechanical Engineering,
National Technical University of Athens,
Athens 15780, Greece
e-mail: gntanak@central.ntua.gr

Marcus Meyer

Rolls-Royce Deutschland,
Blankenfelde-Mahlow 15827, Germany
e-mail: marcus.meyer@rolls-royce.com

Kyriakos C. Giannakoglou

Parallel CFD & Optimization Unit,
Lab of Thermal Turbomachines,
School of Mechanical Engineering,
National Technical University of Athens,
Athens 15780, Greece
e-mail: kgianna@central.ntua.gr

Manuscript received December 3, 2017; final manuscript received June 12, 2018; published online July 26, 2018. Assoc. Editor: Li He.

J. Turbomach 140(8), 081006 (Jul 26, 2018) (11 pages) Paper No: TURBO-17-1229; doi: 10.1115/1.4040564 History: Received December 03, 2017; Revised June 12, 2018

In turbomachinery, the steady adjoint method has been successfully used for the computation of derivatives of various objective functions with respect to design variables in gradient-based optimization. However, the continuous advances in computing power and the accuracy limitations of the steady-state assumption lead toward the transition to unsteady computational fluid dynamics (CFD) computations in the industrial design process. Previous work on unsteady adjoint for turbomachinery applications almost exclusively rely upon frequency-domain methods, for both the flow and adjoint equations. In contrast, in this paper, the development the discrete adjoint to the unsteady Reynolds-averaged Navier–Stokes (URANS) solver for three-dimensional (3D) multirow applications, in the time-domain, is presented. The adjoint equations are derived along with the adjoint to the five-stage Runge–Kutta scheme. Communication between adjacent rows is achieved by the adjoint sliding interface method. An optimization workflow that uses unsteady flow and adjoint solvers is presented and tested in two cases, with objective functions accounting for the transient flow in a turbine vane and the periodic flow in a compressor three-row setup.

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Fig. 1

Simplified two-dimensional sliding interface at a radial (a) and an axial (b) cut (thick dashed line: interior sliding plane (donors), thick line: exterior sliding plane (receivers), arrows: interpolation direction)

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Fig. 2

Schematic of the temporal coarsening technique used for the unsteady adjoint solver

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Fig. 5

Turbine vane: (a) instantaneous pressure ratio values and the L2 norm of the adjoint field. The two vertical dotted lines define the time interval over which the unsteady objective function is defined (from the 750th to the 800th time-step). (b) Typical convergence plot of the flow and adjoint equations in pseudo-time at an arbitrarily selected time-step.

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Fig. 10

Rig250 compressor: inlet and outlet flow condition profiles

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Fig. 3

Turbine vane: geometry and blade-to-blade grid at midspan

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Fig. 4

Turbine vane: inlet and outlet flow condition profiles

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Fig. 6

Turbine vane: (a) comparison of gradients computed using the unsteady adjoint method and finite differences and (b) optimization of the total pressure ratio

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Fig. 7

Turbine vane: baseline (light) versus improved (dark) vane geometries: (a) pressure side and (b) suction side

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Fig. 8

Rig250 compressor: cross section of the compressor and the selected configuration

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Fig. 9

Rig250 compressor: geometry and blade-to-blade grids at midspan

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Fig. 14

Rig250 compressor: value change (%) of objective and constraint functions during the first three optimization steps. Objective function: axial force on rotor blade. Constraints: exit capacity and total pressure coefficient.

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Fig. 15

Rig250 compressor: baseline (gray) versus improved (green) stator blade and rotor blade geometries: (a) stator blade (S2; view from the leading edge) and (b) rotor blade (R3; view from the trailing edge)

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Fig. 16

Rig250 compressor: Baseline versus improved blade's profiles near hub (5% blade's span) and tip (95% blade's span): (a) S2 near tip, (b) S2 near hub, (c) R3 near tip, and (d) R3 near hub

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Fig. 17

Rig250 compressor: gradients computed with the temporal coarsening method compared with the reference gradients. Objective function: axial force on the R3 blade.

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Fig. 11

Rig250 compressor: Steady (mixing interface; (a)) and unsteady (sliding interface; (b)) Spalart–Allmaras variable fields at midspan

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Fig. 12

Rig250 compressor: steady (mixing interface; (a)) and unsteady (sliding interface; (b)) adjoint Spalart–Allmaras variable fields at midspan. Objective function: axial force on the R3 blade.

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Fig. 13

Rig250 compressor: steady (a), mean (b), and instantaneous unsteady (c) sensitivity maps plotted over the pressure side of an S2 blade. Objective function: axial force on the R3 blade.



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