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Research Papers: Gas Turbines: Turbomachinery

Adjoint Method for Shape Optimization in Real-Gas Flow Applications

[+] Author and Article Information
M. Pini

Assistant Professor
Propulsion and Power,
Aerospace Engineering Faculty,
Delft University of Technology,
Kluyverweg 1,
Delft 2629 HS, The Netherlands
e-mail: m.pini@tudelft.nl

G. Persico

Assistant Professor
Laboratorio di Fluidodinamica delle Macchine,
Dipartimento di Energia,
Politecnico di Milano,
via Lambruschini, 4,
Milano 20156, Italy
e-mail: giacomo.persico@polimi.it

D. Pasquale

Dipartimento di Ingegneria
Meccanica e Industriale,
Universitá degli studi di Brescia,
via Branze, 38,
Brescia 25123, Italy
e-mail: david.pasquale@ing.unibs.it

S. Rebay

Associate Professor
Dipartimento di Ingegneria
Meccanica e Industriale,
Universitá degli studi di Brescia,
via Branze, 38,
Brescia 25123, Italy
e-mail: stefano.rebay@ing.unibs.it

Contributed by the Turbomachinery Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received May 24, 2014; final manuscript received June 26, 2014; published online October 14, 2014. Editor: David Wisler.

J. Eng. Gas Turbines Power 137(3), 032604 (Oct 14, 2014) (13 pages) Paper No: GTP-14-1245; doi: 10.1115/1.4028495 History: Received May 24, 2014; Revised June 26, 2014

An adjoint-based shape optimization approach for supersonic turbine cascades is proposed for application to organic Rankine cycle (ORC) turbines. The algorithm is based on an inviscid discrete adjoint method and encompasses a fast look-up table (LuT) approach to accurately deal with real-gas flows. The turbine geometry is defined by adopting state-of-the-art parameterization techniques (NURBS), enabling to handle both global and local control of the shape of interest. A preconditioned steepest descent method has been chosen as gradient-based optimization algorithm to efficiently search for the nearest minimum. The potential of the optimization approach is first verified by application on the redesign of an existing converging–diverging turbine nozzle operating in thermodynamic regions characterized by relevant real-gas effects. A significant efficiency improvement and a more uniform flow at the blade outlet section are achieved, with expected beneficial effects on the aerodynamics of the downstream rotor. The optimized configuration is also assessed by means of high-fidelity turbulent simulations, which point out the capability of the present inviscid approach in optimizing supersonic turbine cascades with very limited computational burdens. Finally, the newly developed real-gas adjoint method is compared against adjoints based on ideal equations of state on the same design problem. Results show that the performance gain obtained by a fully real-gas optimization strategy is by far higher than that achieved with simplified approaches in case of ORC turbines. This proves the relevance of including accurate thermodynamic models in all steps of ORC turbine design.

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References

Figures

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Fig. 1

Left: thermodynamic region covered by a LuT for the siloxane MDM in the T−s diagram. Blue dots located on the vapor saturation line represent the support points of the grid. Right: thermodynamic grid of 10,000 elements built within the tabulated region. Notice the high distortion of the mesh, as it is originally generated using different independent variables.

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Fig. 2

Fully automated adjoint-based optimization algorithm. The aerodynamic gradient refers to the sensitivities with respect to all grid points, the active gradient is the vector of derivatives against surface grid points, whereas the surface gradient indicates the sensitivities with respect to NURBS control points. Red labels denote the names of the numerical solvers.

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Fig. 3

Profile contour and control points distribution of the converging–diverging blade. Red circles indicate the design variables, while blue ones are kept fixed during optimization.

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Fig. 4

Left: predicted Mach flow-field using the baseline and optimized cascade. Right: comparison of blade loading between the baseline and optimized cascade.

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Fig. 5

Sketch of the nonsmoothed (a) and smoothed and normally projected (b) gradient vectors with respect to blade surface mesh nodes. The gradient components are properly scaled to be better visualized. Notice that flow uniformity, i.e., the cost functional, highly depends on the deflection imparted by the rear suction profile.

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Fig. 6

Convergence history of the design process

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Fig. 7

Left: predicted Mach number distribution of the baseline cascade. Right: predicted Mach number distribution of the optimized cascade.

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Fig. 8

Left: isentropic Mach number distribution along the blade surface of the baseline and optimized configurations. Right: predicted spanwise Mach number distribution at outflow boundary.

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Fig. 9

Left: predicted isentropic Mach number of the baseline blade surface for inviscid and turbulent flows. Right: predicted isentropic Mach number of the optimized blade surface for inviscid and turbulent flows.

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Fig. 10

Left: predicted inviscid fishtail wave pattern of the baseline configuration. Right: predicted turbulent fishtail shock pattern of the baseline configuration.

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Fig. 11

Left: predicted inviscid fishtail wave pattern of the optimized configuration. Right: predicted turbulent fishtail shock pattern of the optimized configuration.

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Fig. 12

Mach flowfield obtained through the three optimized blades (PIG, PVdW, and LuT)

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Fig. 13

Left: predicted isentropic Mach number distribution along the blade profile computed through the Span–Wagner EoS for the three optimized blades (PIG, PVdW, and LuT). Notice that curvilinear abscissa s/smax is used to plot the trends in place of the nondimensional streamwise coordinate. Right: area ratio (Ar = a/o) for the optimized PIG and LuT cascades. Notice that Aropt = ArLuT>ArPIG and correspondingly results that ArLuT>ArPVdW leading to the formation of an over-expanded freejet outside the bladed channel.

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Fig. 14

Outlet Mach number pitchwise distribution obtained with the three optimized blades (PIG, PVdW, and LuT)

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