Integral transform solution of internal flow problems based on Navier–Stokes equations and primitive variables formulation

Abstract

SUMMARYThe generalized integral transform technique (GITT) is employed in the solution of incompressible laminar channel flows as formulated by the steady-state Navier-Stokes and continuity equations under the primitive variables mathematical representation. A hybrid numerical-analytical solution is developed based on eigenfunction expansions in one space co-ordinate and error-controlled numerical solution of the resulting system of coupled ordinary differential equations in the remaining space direction. The approach is illustrated for developing flow between parallel-plates with uniform and irrotational inlet flow condition. The conventional Poisson-type equation for the pressure field with appropriate boundary conditions is also transformed and simultaneously solved with the momentum equation along the longitudinal direction, by considering eigenvalue problems for each of the two potentials, defined in the transversal direction. The transversal velocity component is then explicitly determined from the continuity equation. Numerical results of the longitudinal velocity component and friction factor fields are reported to illustrate the convergence behaviour and user prescribed error control inherent to the proposed hybrid approach. Critical comparisons with previous contributions on the same method that made use of the streamfunction-only formulation are also provided.

On the application of generalized integral transform technique to wind‐induced vibrations on overhead conductors

“…Substituting Equations (29) and (30) into Equation (31) and then using the inverse relation one recovers Equation (9). Before closing this section, it seems convenient to make three important remarks.…”

“…New perspectives have been opened by novel hybrid numericalanalytical approaches that attempt to incorporate the advantages associated with classical analytical approaches, while offering sufficient flexibility for dealing with more than just the mathematical equations, and aiming at providing a feasible alternative to purely numerical schemes. One such hybrid approach is the so-called generalized integral transform technique (GITT) [25][26][27], which extends the classical integral transform analytical approach toward the hybrid treatment of both linear and non-linear diffusion and convection-diffusion problems, and has been successfully 905 applied to a number of significant heat and fluid flow phenomena [28][29][30]. The basic idea behind the GITT approach is to propose eigenfunction expansions for the dependent variables, based on the associated diffusion operator behavior, and to perform the integral transformation of the related partial differential equations.…”

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.…”

Hybrid solution for the flow and heat transfer of a second-grade viscoelastic fluid on a stretching sheet with injection or suction

“…The so-called GITT methodology has been successfully employed in the hybrid numerical-analytical solution of several problems in the field of heat and fluid flow [13,[15][16][17][18][19][20][21][22]. Particularly, for applications involving internal and external flows under either the boundary layer or the full Navier-Stokes formulations, one may cite the works of Pérez Guerrero and Cotta [23][24][25], Machado and Cotta [26], Figueira da Silva and Cotta [27], Lima et al [28], Quaresma and Cotta [29], Cotta and Pimentel [30], Pereira et al [31], Bolivar et al [32], Pérez Guerrero et al [33], de Lima et al [34], Paz et al [35], Naveira et al [36], Silva et al [37], Silva et al [38] and de Souza [39]. Therefore, the present work extends the ideas of the GITT approach for solving the boundary layer equations in the flow of a second-grade viscoelastic fluid over a stretched plate.…”