Integral Transform Benchmarks of Diffusion, Convection–Diffusion, and Conjugated Problems in Complex Domains

  • Renato M. CottaEmail author
  • Diego C. Knupp
  • João N. N. Quaresma
  • Kleber M. Lisboa
  • Carolina P. Naveira-Cotta
  • José Luiz Z. Zotin
  • Helder K. Miyagawa


The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numerical–analytical approach for linear or nonlinear diffusive and convective–diffusive partial differential formulations, including an important class of conjugated problems in heat transfer and fluid flow analyses. This chapter focus is on the handling of irregular regions and heterogeneous domains, as a tribute to Prof. D. B. Spalding, who stimulated this research direction in a private communication with the first author, back in 1994. First, formal solutions for nonlinear diffusion and convection–diffusion formulations are reviewed, including the alternatives of adopting nonlinear and/or convective eigenvalue problems, either on total or partial transformation schemes. Next, the GITT itself is formalized in the solution of linear and nonlinear eigenvalue problems, including the direct integral transformation of problems defined in irregular domains, based on simpler auxiliary eigenvalue problems written for the same geometry. Then, a single domain reformulation strategy is discussed, which accounts for heterogeneities on either physical properties or geometrical forms, by rewriting the different media transitions as space variable equation coefficients and source terms. The two complementary strategies are then illustrated through representative examples in convection and conjugated conduction–convection problems, confirming the excellent convergence characteristics of the proposed eigenfunction expansions, toward the establishment of sets of benchmark reference results. The present hybrid solutions are also co-verified against results from purely numerical general-purpose CFD codes.


Hybrid methods Integral transforms GITT Irregular domains Heterogeneous media Conjugated problems Eigenvalue problems 



The authors are grateful for the financial support offered by the Brazilian Government agencies CNPq (projects no. 401237/2014-1 and no. 207750/2015-7), CAPES-INMETRO, and FAPERJ. RMC is also grateful to the Leverhulme Trust for the Visiting Professorship (VP1-2017-028) and to the kind hospitality of the Department of Mechanical Engineering, University College London (UCL), UK, along 2018.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Renato M. Cotta
    • 1
    • 2
    Email author
  • Diego C. Knupp
    • 3
  • João N. N. Quaresma
    • 4
  • Kleber M. Lisboa
    • 1
  • Carolina P. Naveira-Cotta
    • 1
  • José Luiz Z. Zotin
    • 5
  • Helder K. Miyagawa
    • 4
  1. 1.Mechanical Engineering Department POLI & COPPE, CTFederal University of Rio de Janeiro, UFRJRio de JaneiroBrazil
  2. 2.General Directorate of Nuclear and Technological DevelopmentDGDNTM, Brazilian NavyRio de JaneiroBrazil
  3. 3.Mechanical Engineering Department, Instituto Politécnico, IPRJ/UERJState University of Rio de JaneiroNova FriburgoBrazil
  4. 4.Chemical Engineering Department and Graduate ProgramNatural Resource Engineering in the Amazon, Institute of Technology, University of Pará, UFPABelémBrazil
  5. 5.Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, CEFET/RJItaguaíBrazil

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