In this study, a new coupled surface shape design (SSD) methodology named direct design method is presented for the solution of problems containing different types of convection heat transfer in which a specific distribution of either heat flux or temperature is given instead of the shape of a boundary. In the proposed method, the governing equation, without using any mathematical transformation for the physical domains, is manipulated so that the grid generation, solving fluid flow, and heat transfer as well as shape updating can all be carried out simultaneously. Five different inverse shape design problems containing different types of convection heat transfer are solved by the proposed method. All the problems are also solved using the ball-spine algorithm (BSA), which is a recently developed de-coupled algorithm, for the sake of comparison. In all problems, the effects of using different under-relaxation parameters are investigated and the capability of both approaches is compared with each other. The results show that the proposed coupled method can solve the problems better than the BSA in the sense that the direct design method converges sooner than the BSA when the same under-relaxation parameter is used for both methods. Also, it is shown that the computational cost of solving a SSD problem using the direct design method is slightly greater than solving an analysis problem.
Issue Section:
Forced Convection
References
1.
Gosselin
, L.
, Tye-Gingras
, M.
, and Mathieu-Potvin
, F.
, 2009
, “Review of Utilization of Genetic Algorithms in Heat Transfer Problems
,” Int. J. Heat Mass Transfer
, 52
(9–10
), pp. 2169
–2188
.2.
Huang
, C. H.
, and Wang
, S. P.
, 1999
, “A Three-Dimensional Inverse Heat Conduction Problem in Estimating Surface Heat Flux by Conjugate Gradient Method
,” Int. J. Heat Mass Transfer
, 42
(18
), pp. 3387
–3403
.3.
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
, 37
(1
), pp. 69
–83
.4.
Lan
, C. H.
, Cheng
, C. H.
, and Wu
, C. Y.
, 2001
, “Shape Design for Heat Conduction Problems Using Curvilinear Grid Generation Conjugate Method and Redistribution Method
,” Numer. Heat Transfer, Part A
, 39
(5
), pp. 487
–510
.5.
Zhou
, J.
, Zhang
, Y.
, Chen
, J. K.
, and Feng
, Z. C.
, 2010
, “Inverse Estimation of Surface Heating Condition in a Three-Dimensional Object Using Conjugate Gradient Method
,” Int. J. Heat Mass Transfer
, 53
(13–14
), pp. 2643
–2654
.6.
Huang
, C.
, and Lee
, C.
, 2015
, “An Inverse Problem to Estimate Simultaneously Six Internal Heat Fluxes for a Square Combustion Chamber
,” Int. J. Therm. Sci.
, 88
, pp. 59
–76
.7.
Huang
, C. H.
, and Chen
, W. C.
, 2000
, “A Three-Dimensional Inverse Forced Convection Problem in Estimating Surface Heat Flux by Conjugate Gradient Method
,” Int. J. Heat Mass Transfer
, 43
(17
), pp. 3171
–3181
.8.
Cheng
, C. H.
, and Chang
, M. H.
, 2003
, “Shape Design for a Cylinder With Uniform Temperature Distribution on the Outer Surface by Inverse Heat Transfer Method
,” Int. J. Heat Mass Transfer
, 46
(1
), pp. 101
–111
.9.
Katamine
, E.
, Kawase
, Y.
, and Azegami
, H.
, 2008
, “Shape Optimization of Thermal Forced Convection Fields
,” Heat Transfer-Asian Res.
, 37
(5
), pp. 313
–328
.10.
Park
, H. M.
, and Chung
, O. Y.
, 1999
, “An Inverse Natural Convection Problem of Estimating the Strength of a Heat Source
,” Int. J. Heat Mass Transfer
, 42
(23
), pp. 4259
–4273
.11.
Prud'homme
, M.
, and Hung Nguyen
, T.
, 2001
, “Solution of Inverse Free Convection Problems by Conjugate Gradient Method: Effects of Rayleigh Number
,” Int. J. Heat Mass Transfer
, 44
(11
), pp. 2011
–2027
.12.
Park
, H. M.
, and Shin
, H. J.
, 2003
, “Shape Identification for Natural Convection Problems Using the Adjoint Variable Method
,” J. Comput. Phys.
, 186
(1
), pp. 198
–211
.13.
Park
, H. M.
, and Yoon
, T. Y.
, 2000
, “Solution of the Inverse Radiation Problem Using a Conjugate Gradient Method
,” Int. J. Heat Mass Transfer
, 43
(10
), pp. 1767
–1776
.14.
Hosseini Sarvari
, S. M.
, and Mansouri
, S. H.
, 2004
, “Inverse Design for Radiative Heat Source in Two-Dimensional Participating Media
,” Numer. Heat Transfer, Part B
, 46
(3
), pp. 283
–300
.15.
Salinas
, C. T.
, 2010
, “Inverse Radiation Analysis in Two-Dimensional Gray Media Using the Discrete Ordinates Method With a Multidimensional Scheme
,” Int. J. Therm. Sci.
, 49
(2
), pp. 302
–310
.16.
Demeulenaere
, A.
, and Van Den Braembussche
, R. A.
, 1998
, “Three-Dimensional Inverse Method for Turbomachinery Blading Design
,” ASME J. Turbomach.
, 120
(2
), pp. 247
–255
.17.
De Vito
, L.
, Van den Braembussche
, R. A.
, and Deconinck
, H.
, 2003
, “A Novel Two-Dimensional Viscous Inverse Design Method for Turbomachinery Blading
,” ASME J. Turbomach.
, 125
(2
), pp. 310
–316
.18.
Barger
, R. L.
, and Brooks
, C. W.
, 1974
, “A Streamline Curvature Method for Design of Supercritical and Subcritical Airfoils
,” NASA Langley Research Center, Hampton, VA, Technical Report No. NASA TN D-7770
.https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19740024303.pdf19.
Garabedian
, P.
, and McFadden
, G.
, 1982
, “Design of Supercritical Swept Wings
,” AIAA J.
, 20
(2
), pp. 289
–291
.20.
Malone
, J.
, Vadyak
, J.
, and Sankar
, L. N.
, 1987
, “Inverse Aerodynamic Design Method for Aircraft Components
,” J. Aircr.
, 24
(1
), pp. 8
–9
.21.
Safari
, M.
, Nili-Ahmadabadi
, M.
, Ghaei
, A.
, and Shirani
, E.
, 2014
, “Inverse Design in Subsonic and Transonic External Flow Regimes Using Elastic Surface Algorithm
,” Comput. Fluids
, 102
, pp. 41
–51
.22.
Nikfar
, M.
, Ashrafizadeh
, A.
, and Mayeli
, P.
, 2015
, “Inverse Shape Design Via a New Physical-Based Iterative Solution Strategy
,” Inverse Probl. Sci. Eng.
, 23
(7
), pp. 1138
–1162
.23.
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.
, 17
(8
), pp. 1037
–1057
.24.
Nili-Ahmadabadi
, M.
, Hajilouy
, A.
, Ghadak
, F.
, and Durali
, M.
, 2010
, “A Novel 2-D Incompressible Viscous Inverse Design Method for Internal Flows Using Flexible String Algorithm
,” ASME J. Fluids Eng.
, 132
(3
), p. 031401
.25.
Nili-Ahmadabadi
, M.
, Ghadak
, F.
, and Mohammadi
, M.
, 2013
, “Subsonic and Transonic Airfoil Inverse Design Via Ball-Spine Algorithm
,” Comput. Fluids
, 84
, pp. 87
–96
.26.
Vaghefi
, N. S.
, Nili-Ahmadabadi
, M.
, and Roshani
, M. R.
, 2012
, “Optimal Design of Axe-Symmetric Diffuser Via Genetic Algorithm and Ball-Spine Inverse Design Method
,” ASME
Paper No. IMECE2012-87924.27.
Madadi
, A.
, Kermani
, M. J.
, and Nili-Ahmadabadi
, M.
, 2011
, “Application of an Inverse Design Method to Meet a Target Pressure in Axial-Flow Compressors
,” ASME
Paper No. GT2011-46091.28.
Madadi
, A.
, Kermani
, M. J.
, and Nili-Ahmadabadi
, M.
, 2014
, “Application of the Ball-Spine Algorithm to Design Axial-Flow Compressor Blade
,” Sci. Iran.
, 21
(6
), pp. 1981
–1992
.http://scientiairanica.sharif.edu/article_3590.html29.
Madadi
, A.
, Kermani
, M. J.
, and Nili-Ahmadabadi
, M.
, 2014
, “Applying the Ball-Spine Algorithm to the Design of Blunt Leading Edge Airfoils for Axial Flow Compressors
,” J. Mech. Sci. Technol.
, 28
(11
), pp. 4517
–4526
.30.
Madadi
, A.
, Kermani
, M. J.
, and Nili-Ahmadabadi
, M.
, 2014
, “Aerodynamic Design of S-Shaped Diffusers Using Ball–Spine Inverse Design Method
,” ASME J. Eng. Gas Turbines Power
, 136
(12
), p. 122606
.31.
Shumal
, M.
, Nili-Ahmadabadi
, M.
, and Shirani
, E.
, 2016
, “Development of the Ball-Spine Inverse Design Algorithm to Swirling Viscous Flow for Performance Improvement of an Axisymmetric Bend Duct
,” Aerosp. Sci. Technol.
, 52
, pp. 181
–188
.32.
Hoghooghi
, H.
, Nili-Ahmadabadi
, M.
, and Manshadi
, M. D.
, 2016
, “Optimization of a Subsonic Wind Tunnel Nozzle With Low Contraction Ratio Via Ball-Spine Inverse Design Method
,” J. Mech. Sci. Technol.
, 30
(5
), pp. 2059
–2067
.33.
Mayeli
, P.
, Nili-Ahmadabadi
, M.
, and Besharati-Foumani
, H.
, 2016
, “Inverse Shape Design for Heat Conduction Problems Via Ball Spine Algorithm
,” Numer. Heat Transfer, Part B
, 69
(3
), pp. 249
–269
.34.
Mayeli
, P.
, Nili-Ahmadabadi
, M.
, Pirzadeh
, M. R.
, and Rahmani
, P.
, 2017
, “Determination of Desired Geometry by a Novel Extension of Ball Spine Algorithm Inverse Method to Conjugate Heat Transfer Problems
,” Comput. Fluids
, 154
, pp. 390
–406
.https://doi.org/10.1016/j.compfluid.2016.05.02235.
Hesami
, H.
, and Mayeli
, P.
, 2016
, “Development of the Ball-Spine Algorithm for the Shape Optimization of Ducts Containing Nanofluid
,” Numer. Heat Transfer, Part A
, 70
(12
), pp. 1371
–1389
.http://dx.doi.org/10.1080/10407782.2016.124397636.
Stanitz
, J. D.
, 1988
, “A Review of Certain Inverse Methods for the Design of Ducts With 2 or 3-Dimensional Potential Flow
,” ASME Appl. Mech. Rev.
, 41
(6
), pp. 217
–238
.37.
Nelson
, C. D.
, and Yang
, T.
, 1977
, “Design of Branched and Unbranched Axially Symmetrical Ducts With Specified Pressure Distribution
,” AIAA J.
, 15
(9
), pp. 1272
–1277
.38.
Chaviaropoulos
, P.
, Dedoussis
, V.
, and Papailion
, K. D.
, 1995
, “On the 3-D Inverse Potential Target Pressure Problem—Part I: Theoretical Aspects and Method Formulation
,” J. Fluid Mech.
, 282
, pp. 131
–146
.39.
Chaviaropoulos
, P.
, Dedoussis
, V.
, and Papailion
, K. D.
, 1995
, “On the 3-D Inverse Potential Target Pressure Problem—Part II: Numerical Aspects and Application to Duct Design
,” J. Fluid Mech.
, 282
, pp. 147
–162
.https://doi.org/10.1017/S002211209500007340.
Zannetti
, L.
, 1986
, “A Natural Formulation for the Solution of Two–Dimensional or Axis–Symmetric Inverse Problems
,” Int. J. Numer. Methods Eng.
, 22
(2
), pp. 451
–463
.41.
Borges
, J. E.
, 2007
, “Computational Method for the Design of Ducts
,” Comput. Fluids
, 36
(2
), pp. 480
–483
.42.
Ashrafizadeh
, A.
, Raithby
, G. D.
, and Stubley
, G. D.
, 2002
, “Direct Design of Shape
,” Numer. Heat Transfer, Part B
, 41
(6
), pp. 501
–520
.43.
Ashrafizadeh
, A.
, Raithby
, G. D.
, and Stubley
, G. D.
, 2003
, “Direct Design of Ducts
,” ASME J. Fluids Eng.
, 125
(1
), pp. 158
–165
.44.
Ashrafizadeh
, A.
, Raithby
, G. D.
, and Stubley
, G. D.
, 2004
, “Direct Design of Airfoil Shape With a Prescribed Surface Pressure
,” Numer. Heat Transfer, Part B
, 46
(6
), pp. 505
–527
.45.
Ashrafizadeh
, A.
, and Raithby
, G. D.
, 2006
, “Direct Design Solution of the Elliptic Grid Generation Equations
,” Numer. Heat Transfer, Part B
, 50
(3
), pp. 217
–230
.46.
Taiebi-Rahni
, M.
, Ghadak
, F.
, and Ashrafizadeh
, A.
, 2008
, “A Direct Design Approach Using the Euler Equation
,” Inverse Probl. Sci. Eng.
, 16
(2
), pp. 217
–231
.47.
Ghadak
, F.
, Taiebi-Rahni
, M.
, and Ashrafizadeh
, A.
, 2009
, “Direct Design of Branched Ducts
,” Sci. Iran., Trans. B
, 16
(2
), pp. 111
–120
.http://www.sid.ir/en/VEWSSID/J_pdf/95520092B09.pdf48.
Ashrafizadeh
, A.
, Bavafa
, B. A.
, and Mayeli
, P.
, 2015
, “A New Co-Located Pressure-Based Discretization Method for the Numerical Solution of Incompressible Navier-Stokes Equations
,” Numer. Heat Transfer, Part B
, 67
(6
), pp. 563
–589
.49.
Nikfar
, M.
, and Ashrafizadeh
, A.
, 2016
, “A Coupled Element-Based Finite Volume Method for the Solution of Incompressible Navier-Stokes Equations
,” Numer. Heat Transfer, Part B
, 69
(5
), pp. 447
–472
.50.
Ashrafizadeh
, A.
, and Nikfar
, M.
, 2016
, “On the Numerical Solution of Generalized Convection Heat Transfer Problems Via the Method of Proper Closure Equations—Part I: Description of the Method
,” Numer. Heat Transfer, Part A
, 70
(5
), pp. 187
–203
.51.
Ashrafizadeh
, A.
, and Nikfar
, M.
, 2016
, “On the Numerical Solution of Generalized Convection Heat Transfer Problems Via the Method of Proper Closure Equations—Part II: Application to Test Problems
,” Numer. Heat Transfer, Part A
, 70
(5
), pp. 204
–222
.52.
Bejan
, A.
, 2004
, Convection Heat Transfer
, 3rd ed., Wiley
, New York
.53.
Al-Amiri
, A.
, Khanafer
, K.
, Bull
, J.
, and Pop
, I.
, 2007
, “Effect of the Sinusoidal Wavy Bottom Surface on Mixed Convection Heat Transfer in a Lid-Driven Cavity
,” Int. J. Heat Mass Transfer
, 50
(10
), pp. 1771
–1780
.54.
Nikfar
, M.
, and Mahmoodi
, M.
, 2012
, “Meshless Local Petrov-Galerkin Analysis of Free Convection of Nanofluid in a Cavity With Wavy Side Walls
,” Eng. Anal. Boundary Elem.
, 36
(3
), pp. 433
–445
.55.
Arefmanesh
, A.
, and Nikfar
, M.
, 2013
, “Analysis of Natural Convection in a Nanofluid-Filled Triangular Enclosure Induced by Cold and Hot Sources on the Walls Using Stabilised MLPG Method
,” Can. J. Chem. Eng.
, 91
(10
), pp. 1711
–1728
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