Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 103
Gradient
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0 5 10 15 20 25
(a)
Variable
Gradient
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0 5 10 15 20 25
(b)
Variable
initial
optimal
Fig. 4.8 Total pressure losses minimization in a 2D compressor cascade (a) convergence
history of the objective function gradient values and (b) gradient values for the initial and
optimal cascade airfoils
y
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
0 0.2 0.4 0.6 0.8 1
(a)
x
initial
optimal
y
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
(b)
x
initial
optimal
Fig. 4.9 Total pressure losses minimization in a 2D compressor cascade (a) initial and
optimal set of Bézier control points and (b) the corresponding cascade airfoil contours
corrected gradient values “push” both suction and pressure side control points
away from each other.
Improvements in aerodynamic characteristics are shown in Figs. 4.10 and
4.11 where the initial and optimal pressure and friction coefficients are shown.
A close-up view of the flow developed close to the trailing edge over the suction
side is shown. Both figures show that separation on the optimal blade airfoil
is kept minimum or even disappears. The static pressure plateau and the
negative friction coefficient values are significantly reduced. The improvement
of separation is also shown in Fig. 4.12 where the Mach number distribution
is shown for the initial and optimal configuration.
Similar results can be obtained using the entropy generation instead of
the total pressure loss as objective function [52, 53]. These are omitted in the
interest of space.