Abstract:-The generalized integral transform technique (GITT)is employed to obtain a hybrid numericalanalytical solution of natural convection in a cavity with volumetric heat generation. The hybrid nature of this approach allows for the establishment of benchmark results in the solution of non-linear partial differential equation systems, including the coupled set of heat and fluid flow equations that govern the steady natural convection problem under consideration. Through performing the GITT, the resulting transform… Show more
“…The GITT has been applied to the solution of heat and fluid flow problems under the Navier–Stokes formulation in both primitive variables (Lima et al , 2007; Curi et al , 2014; Souza et al , 2016) and streamfunction-only (Perez-Guerrero and Cotta, 1992; Perez-Guerrero et al , 2000; Silva et al , 2010; An et al , 2013; Matt et al , 2017; Pontes et al , 2018; Fu et al , 2018) formulations. The latter approach has been preferred due to the enhanced convergence, the elimination of the pressure field, and automatic satisfaction of the continuity equation granted by the definition of the streamfunction, though restricted to two-dimensional fluid flow problems.…”
Purpose The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation.
Design/methodology/approach The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme.
Findings A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification.
Originality/value A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.
“…The GITT has been applied to the solution of heat and fluid flow problems under the Navier–Stokes formulation in both primitive variables (Lima et al , 2007; Curi et al , 2014; Souza et al , 2016) and streamfunction-only (Perez-Guerrero and Cotta, 1992; Perez-Guerrero et al , 2000; Silva et al , 2010; An et al , 2013; Matt et al , 2017; Pontes et al , 2018; Fu et al , 2018) formulations. The latter approach has been preferred due to the enhanced convergence, the elimination of the pressure field, and automatic satisfaction of the continuity equation granted by the definition of the streamfunction, though restricted to two-dimensional fluid flow problems.…”
Purpose The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation.
Design/methodology/approach The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme.
Findings A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification.
Originality/value A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.
“…In the last decades, the hybrid analytical-numerical computational methodology (GITT) has been recognised as a powerful computational tool to explain the heat transfer and diffusion-convection phenomena (Nogueira and Cotta, 1990;Pontedeiro et al, 2008;Monteiro et al, 2009;Knupp et al, 2014Knupp et al, , 2015aKnupp et al, , 2015bCotta et al, 2016) and for the solutions of Navier-Stokes equations (Lima et al, 1997;Pereira et al, 1998Pereira et al, , 2000de Lima et al, 2007;Paz et al, 2007;Silva et al, 2010). In the convection behaviour analysis, the GITT technique has been approved as an accurate hybrid analytical-numerical methodology (Leal et al, 1999(Leal et al, , 2000Alves et al, 2002;Neto et al, 2006;Monteiro et al, 2010;An et al, 2013). Leal et al (1999Leal et al ( , 2000 used the GITT methodology to investigate the steady and transient convection behaviours in a steady and transient laminar flow in a square cavity, respectively.…”
Section: Introductionmentioning
confidence: 99%
“…Later, Neto et al (2006) performed a study to investigate the effect of the aspect ratio on the natural convection using a three-dimensional model. Recently, the natural convection in enclosure with the uniformly distributed heat generation was investigated using the GITT technique (An et al, 2013).…”
Purpose
The purpose of this study is to propose the generalised integral transform technique to investigate the natural convection behaviour in a vertical cylinder under different boundary conditions, adiabatic and isothermal walls and various aspect ratios.
Design/methodology/approach
GITT was used to investigate the steady-state natural convection behaviour in a vertical cylinder with internal uniformed heat generation. The governing equations of natural convection were transferred to a set of ordinary differential equations by using the GITT methodology. The coefficients of the ODEs were determined by the integration of the eigenfunction of the auxiliary eigenvalue problems in the present natural convection problem. The ordinary differential equations were solved numerically by using the DBVPFD subroutine from the IMSL numerical library. The convergence was achieved reasonably by using low truncation orders.
Findings
GITT is a powerful computational tool to explain the convection phenomena in the cylindrical cavity. The convergence analysis shows that the hybrid analytical–numerical technique (GITT) has a good convergence performance in relatively low truncation orders in the stream-function and temperature fields. The effect of the Rayleigh number and aspect ratio on the natural convection behaviour under adiabatic and isothermal boundary conditions has been discussed in detail.
Originality/value
The present hybrid analytical–numerical methodology can be extended to solve various convection problems with more involved nonlinearities. It exhibits potential application to solve the convection problem in the nuclear, oil and gas industries.
“…Even though it has been investigated for more than a century, the RB problem has been the subject of a large number of studies in recent years, involving, in particular, modifications of the classical RB system. For example, the problem of natural and mixed convection in enclosures is very important in the study of the cooling of electronic devices and thermal comfort (Fontana et al, 2015;An et al, 2013;Mariani and Coelho, 2007).…”
-Natural convection in superimposed layers of fluids heated from below is commonly observed in many industrial and natural situations, such as crystal growth, co-extrusion processes and atmospheric flow. The stability analysis of this system reveals a complex dynamic behavior, including the potential multiplicity of stationary states and occurrence of periodic regimes. In this study, a linear stability analysis (LSA) was performed to determine the onset of natural convection as a function of imposed boundary conditions, geometrical configuration and specific perturbations. To investigate the effects of the non-linear terms neglected in LSA, a direct simulation of the full nonlinear problem was performed using computational fluid dynamics (CFD) techniques. The numerical simulation results show an excellent agreement with the LSA results near the onset of convection and an increase in the deviation as the Rayleigh number increases above the critical value.
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