Stratified flow over a backward‐facing step: hybrid solution by integral transforms

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

“…The integral transform analysis of fluid flow problems governed by the Navier-Stokes equations has required the proposition of new eigenfunction expansions, other than those normally employed in diffusion or convection-diffusion problems, directly derived from the general Sturm-Liouville eigenvalue problem. Along the years, in the present methodological context, the Navier-Stokes equations have been mostly dealt with in the streamfunction-only formulation [25,[35][36][37][38][39][40][41][42][43][44][45][46], and less frequently in the primitive variables formulation [47,48]. In two-dimensional problems, the streamfunction formulation offers the advantages of automatically satisfying the continuity equation and eliminating the pressure field.…”

“…However, the extension of this concept to three-dimensional flows, leading to vector and scalar potentials, has been shown to be less advantageous when dealt with by the same hybrid approach [49]. Nevertheless, the integral transform method under the two-dimensional streamfunction formulation has been applied to various classes of problems, including cavity and channel flows, rectangular and cylindrical geometries, regular and irregular domains, laminar and turbulent flows, steady and transient states, natural and forced convection, as well as on magnetohydrodynamics [25,[35][36][37][38][39][40][41][42][43][44][45][46]. The integral transformation of the streamfunction formulation is the first one to be here reviewed, for both steady and transient state situations, in light of its popularity among the contributions that employed this hybrid approach so far.…”

Unified Hybrid Theoretical Analysis of Nonlinear Convective Heat Transfer

“…In this case the emphasis is placed on extending the classical integral transform method making it sufficiently flexible to handle problems that are not a priori transformable, such as in the case of problems with nonlinear coefficients in either the equation or the boundary conditions [8][9][10][11][12][13][14][15]. Various classes of nonlinear problems in heat and fluid flow have been handled by the GITT, and among them convective heat transfer problems formulated by either the boundary layer of full Navier-Stokes formulations, for cavity, duct or external flows, here reviewed just for the internal flow situation of closer interest to the application to be later considered [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Nevertheless, only in a few situations [23][24] the full nonlinear nature of these equations have been dealt with, including not only the usual nonlinear terms that derive from the convective formulation, but also those due to the variable physical properties, especially in their dependence with temperature.…”