In this investigation, the flexible string algorithm (FSA), used before for inverse design of subsonic and supersonic ducts in compressible flows with and without normal shock, is developed and applied for inverse design of 2D incompressible viscous internal flow with and without separation. In the proposed method, the duct wall shape is changed under an algorithm based on deformation of a virtual flexible string in flow. At each modification step, the difference between current and target wall pressure distributions is applied to the string. The method is an iterative inverse design method and utilizes the analysis code for the flow field solution as a black-box. Some validation test cases and design examples are presented here, which show the robustness and flexibility of the method in handling complex geometries. In cases with separated flow pressure distribution, a unique solution for inverse design problem does not exist. The design algorithm is a physical and quick converging approach and can efficiently utilize commercial flow analysis software.

1.
Stanitz
,
J. D.
, 1953, “
Design of Two-Dimensional Channels With Prescribed Velocity Distributions Along the Duct Walls
,” Lewis Flight Propulsion Laboratory Technical Report No. 1115.
2.
Stanitz
,
J. D.
, 1980, “
General Design Method for Three-Dimensional, Potential Flow Fields, I—Theory
,” NASA Technical Report No. 3288.
3.
Stanitz
,
J. D.
, 1987, “
A Review of Certain Inverse Methods for the Design of Ducts With 2- or 3-Dimensional Potential Flows
,”
Proceedings of the Second International Conference on Inverse Design Concepts and Optimization in Engineering Sciences (ICIDES-II)
,
G. S.
Dulikravich
, ed., Pennsylvania State University, University Park, PA, Oct. 26–28.
4.
Zannetti
,
L.
, 1986, “
A Natural Formulation for the Solution of Two-Dimensional or Axis-Symmetric Inverse Problems
,”
Int. J. Numer. Methods Eng.
0029-5981,
22
, pp.
451
463
.
5.
Ashrafizadeh
,
A.
,
Raithby
,
G. D.
, and
Stubley
,
G. D.
, 2003, “
Direct Design of Ducts
,”
ASME J. Fluids Eng.
0098-2202,
125
, pp.
158
165
.
6.
Ghadak
,
F.
, 2005, “
A Direct Design Method Based on the Laplace and Euler Equations With Application to Internal Subsonic and Supersonic Flows
,” Ph.D. thesis, Sharif University of Technology, Iran.
7.
Cheng
,
C. -H.
, and
Wu
,
C. -Y.
, 2000, “
An Approach Combining Body Fitted Grid Generation and Conjugate Gradient Methods for Shape Design in Heat Conduction Problems
,”
Numer. Heat Transfer, Part B
1040-7790,
37
(
1
), pp.
69
83
.
8.
Jameson
,
A.
, 1994, “
Optimum Aerodynamic Design Via Boundary Control
,” Optimum Design Methods for Aerodynamics, AGARD Report No. R-803, pp.
3.1
3.33
.
9.
Kim
,
J. S.
, and
Park
,
W. G.
, 2000, “
Optimized Inverse Design Method for Pump Impeller
,”
Mech. Res. Commun.
0093-6413,
27
(
4
), pp.
465
473
.
10.
Dedoussis
,
V.
,
Chaviaropoulos
,
P.
, and
Papailiou
,
K. D.
, 1993, “
Rotational Compressible Inverse Design Method for Two-Dimensional Internal Flow Configurations
,”
AIAA J.
0001-1452,
31
(
3
), pp.
551
558
.
11.
Demeulenaere
,
A.
, and
van den Braembussche
,
R.
, 1998, “
Three-Dimensional Inverse Method for Turbomachinery Blading Design
,”
ASME J. Turbomach.
0889-504X,
120
(
2
), pp.
247
255
.
12.
de Vito
,
L.
, and
Van den Braembussche
,
R.
, 2003, “
A Novel Two-Dimensional Viscous Inverse Design Method for Turbomachinery Blading
,”
ASME J. Turbomach.
0889-504X,
125
, pp.
310
316
.
13.
Demeulenaere
,
A.
,
Leonard
,
O.
, and
Van den Braembussche
,
R.
, 1997, “
A Two-Dimensional Navier–Stokes Inverse Solver for Compressor and Turbine Blade Design
,”
Proc. Inst. Mech. Eng., Part A
0957-6509,
211
, pp.
299
307
.
14.
Henne
,
P. A.
, 1980, “
An Inverse Transonic Wing Design Method
,” AIAA Paper No. 80-0330.
15.
Volpe
,
G.
, 1989, “
Inverse Design of Airfoil Contours: Constraints, Numerical Method Applications
,” Computational methods for Aerodynamic design (Inverse) and Optimization, AGARD CP-463, Paper No. 4.
16.
Barger
,
R. L.
and
Brooks
,
C. W.
, 1974, “
A Streamline Curvature Method for Design of Supercritical and Subcritical Airfoils
,” NASA Report No. TN D-7770.
17.
Garabedian
,
P.
, and
McFadden
,
G.
, 1982, “
Design of Supercritical Swept Wings
,”
AIAA J.
0001-1452,
30
(
3
), pp.
444
446
.
18.
Malone
,
J.
,
Vadyak
,
J.
, and
Sankar
,
L. N.
, 1987, “
Inverse Aerodynamic Design Method for Aircraft Components
,”
J. Aircr.
0021-8669,
24
(
1
), pp.
8
9
.
19.
Malone
,
J.
,
Vadyak
,
J.
, and
Sankar
,
L. N.
, 1985, “
A Technique for the Inverse Aerodynamic Design of Nacelles and Wing Configurations
,” AIAA Paper No. 85-4096.
20.
Campbell
,
R. L.
, and
Smith
,
L. A.
, 1987, “
A Hybrid Algorithm for Transonic Airfoil and Wing Design
,” AIAA Paper No. 87-2552.
21.
Bell
,
R. A.
, and
Cedar
,
R. D.
, 1991, “
An Inverse Method for the Aerodynamic Design of Three-Dimensional Aircraft Engine Nacelles
,”
Proceedings of the Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences, ICIDES-III
,
G. S.
Dulikravich
, ed., Washington, DC, Oct. 23–25, pp.
405
417
.
22.
Malone
,
J. B.
,
Narramore
,
J. C.
, and
Sankar
,
L. N.
, 1989, “
An Efficient Airfoil Design Method Using the Navier–Stokes Equations
,” Computational methods for Aerodynamic design (Inverse) and Optimization, AGARD CP-463, Paper No. 5, pp.
5.1
5.18
.
23.
Malone
,
J. B.
,
Narramore
,
J. C.
, and
Sankar
,
L. N.
, 1991, “
Airfoil design method using the Navier–Stokes equations
,”
J. Aircr.
0021-8669,
28
(
3
), pp.
216
224
.
24.
Takanashi
,
S.
, 1985, “
Iterative Three-Dimensional Transonic Wing Design Using Integral Equations
,”
J. Aircr.
0021-8669,
22
, pp.
655
660
.
25.
Hirose
,
N.
,
Takanashi
,
S.
, and
Kawai
,
N.
, 1987, “
Transonic Airfoil Design Procedure Utilizing a Navier–Stokes Analysis Code
,”
AIAA J.
0001-1452,
25
(
3
), pp.
353
359
.
26.
Dulikravich
,
G. S.
and
Baker
,
D. P.
, 1999, “
Aerodynamic Shape Inverse Design Using a FOURIER SERIES METHOD
,” AIAA Paper No. 99-0185.
27.
Ashihara
,
K.
and
Goto
,
A.
, 2001, “
Turbomachinery Blade Design Using 3-D Inverse Design Method, CFD and Optimization Algorithm
,” ASME Paper No. GT2001-0358.
28.
Min
,
J.
,
Dang
,
T. Q.
, and
Cave
,
M. J.
, 2009, “
Fully Three-Dimensional Viscous Semi-Inverse Method for Subsonic Mixed-Flow and Radial Impeller Design
,” ASME Paper No. GT2009-59679.
29.
Roidl
,
B.
and
Ghaly
,
W.
, 2009, “
Dual Point Redesign of Axial Turbines Using a Viscous Inverse Design Method
,” ASME Paper No. GT2009-59707.
30.
Daneshkhah
,
K.
, and
Ghaly
,
W.
, 2007, “
Aerodynamic Inverse Design for Viscous Flow in Turbomachinery Blading
,”
J. Propul. Power
0748-4658,
23
(
4
), pp.
814
820
.
31.
Daneshkhah
,
K.
, and
Ghaly
,
W.
, 2006, “
An Inverse Blade Design Method for Subsonic and Transonic Viscous Flow in Compressors and Turbines
,”
Inverse Probl. Sci. Eng.
1741-5977,
14
(
3
), pp.
211
231
.
32.
Roidl
,
B.
and
Ghaly
,
W.
, 2008, “
Redesign of a Low Speed Turbine Stage Using a New Viscous Inverse Design Method
,” ASME Paper No. GT2008-51468.
33.
Nili-Ahmadabadi
,
M.
,
Durali
,
M.
,
Hajilouy
,
A.
, and
Ghadak
,
F.
, 2009, “
Inverse Design of 2D Subsonic Ducts Using Flexible String Algorithm
,”
Inverse Probl. Sci. Eng.
1741-5977,
17
(
8
), pp.
1037
1057
.
34.
Nili-Ahmadabadi
,
M.
,
Hajilouy
,
A.
,
Durali
,
M.
and
Ghadak
,
F.
, 2009, “
Duct Design in Subsonic & Supersonic Flow Regimes With & Without Shock Using Flexible String Algorithm
,” ASME Paper No. GT2009-59744.
35.
Farhanieh
,
B.
, and
Davidson
,
L.
, 1992, “
Numerical Calculation of Navier-Stokes Equations in Biological Ducts
,”
Proceedings of the Sixth Biomechanics Seminar
, Gothenburg, Apr., Vol.
6
, pp.
154
–167.
36.
Sagi
,
C. J.
, and
Johnston
,
J. P.
, 1967, “
The Design of Performance of Two Dimensional, Curved Diffusers
,”
ASME J. Basic Eng.
0021-9223,
89
, pp.
715
731
.
You do not currently have access to this content.