Abstract
A computer-aided design (CAD) and adjoint based multipoint optimization of the LS89 high pressure axial turbine vane is presented. The aim is to reduce the entropy generation at both subsonic and transonic flow conditions by means of employing CAD and adjoint based methods during the optimization process. The performance metrics at design and off-design conditions are grouped into a single objective function using equal weights. A steady state Reynolds-Averaged density based Navier-Stokes solver and the one-equation transport Spalart-Allmaras turbulence model are used to predict the losses. The entropy generation is reduced whilst keeping the trailing edge thickness and the axial chord length as manufacturing constraints and the exit flow angle as a flow constraint, which is enforced via the penalty formulation. The resulting unconstrained optimization problem is solved by a L-BFGS-B algorithm. At every optimization iteration a new profile is constructed using B-splines and the grid is rebuilt by elliptic grid generation. The gradients used for the optimization are obtained via a novel approach in which both the CAD kernel and grid generation are differentiated using Algorithmic Differentiation techniques. The sensitivities of the objective function with respect to the grid coordinates are computed by a hand-derived adjoint solver. The off-design performance of the LS89 is significantly improved and the optimal geometry is analyzed in more detail.
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- c :
-
chord length
- \(c_{ax}\) :
-
axial chord length
- g:
-
pitch
- \({J_{1}}\) :
-
term proportional to entropy generation
- \({J_{2}}\) :
-
exit flow angle
- \({J_{MP}}\) :
-
multi point pseudo cost function
- \({\dot{m}}\) :
-
mass flow
- \(M_{ise}\) :
-
isentropic Mach number
- \(M_{ise,2}\) :
-
downstream isentropic Mach number
- \({P_{01}}\) :
-
inlet total pressure
- \({P_{02}}\) :
-
outlet (downstream) total pressure
- \(p_{2}\) :
-
outlet (downstream) static pressure
- \(R_{LE}\) :
-
leading edge radius
- \(R_{TE}\) :
-
trailing edge radius
- t:
-
throat height
- \({t_{PS}^1}\), ..., \({t_{PS}^4}\):
-
PS thickness
- \({t_{SS}^1}\), ..., \({t_{SS}^9}\):
-
SS thickness
- \(\mathbf {X}\) :
-
grid x, y, z coordinates
- \(\mathbf {\varvec{\alpha }}\) :
-
design vector
- \({\beta }_{in}\) :
-
inlet angle
- \({\beta }_{out}\) :
-
outlet angle
- \({\gamma }\) :
-
stagger angle
- \(\frac{dJ}{d\mathbf {\varvec{\alpha }}}\) :
-
performance sensitivity vector
- \(\frac{dJ}{d\mathbf {X}}\) :
-
adjoint sensitivity vector
- \(\frac{d\mathbf {X}}{d\mathbf {\varvec{\alpha }}}\) :
-
grid sensitivity vector
- \({\varphi _{PS}}\) :
-
pressure side trailing edge wedge angle
- \({\varphi _{SS}}\) :
-
suction side trailing edge wedge angle
- \({\sigma }\) :
-
solidity
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Sanchez Torreguitart, I., Verstraete, T., Mueller, L. (2019). CAD and Adjoint Based Multipoint Optimization of an Axial Turbine Profile. In: Andrés-Pérez, E., González, L., Periaux, J., Gauger, N., Quagliarella, D., Giannakoglou, K. (eds) Evolutionary and Deterministic Methods for Design Optimization and Control With Applications to Industrial and Societal Problems. Computational Methods in Applied Sciences, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-89890-2_3
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