Renato M. Cotta

Benchmark results in computational heat and fluid flow : the integral transform method

Abstract The integral transform method is reviewed as a benchmark tool in computational heat and fluid flow, with special emphasis on nonlinear problems. The hybrid numerical-analytical nature of this approach collapses the numerical task into one single independent variable, and thus allows for a simple computational procedure with automatic global error control and mild increase in computational effort for multidimensional situations. Various applications on nonlinear diffusion and convection-diffusion are more closely considered, followed by sample results from recent contributions.

HYBRID NUMERICAL/ANALYTICAL APPROACH TO NONLINEAR DIFFUSION PROBLEMS

A class of nonlinear diffusion-type problems is handled through a hybrid method. This method incorporates the ideas in the generalized integral transform technique to reduce the original partial differential equation into a denumerable system of coupled ordinary differential equations. These equations can then be solved through standard numerical techniques, once the system is truncated to a finite order. Sufficient conditions for the convergence of the truncated finite system are then examined. An application is considered that deals with a transient radiative fin problem, which is quite suitable for illustrating the solution methodology and convergence behavior.

Integral transform solutions of transient natural convection in enclosures with variable fluid properties

Abstract This paper is aimed at the application of the Generalized Integral Transform Technique to the transient version of the classical differentially heated square cavity problem, considering both constant and variable fluid properties. The streamfunction-only formulation of the flow equations and the associated energy equation under laminar flow regime are employed in seeking a hybrid numerical–analytical solution to this natural convection problem. The computational procedure is carefully validated and a thorough convergence analysis is undertaken, yielding sets of reference results. The computed transient behavior of the coupled heat and fluid flow phenomena is compared to some previously reported results. The solution for variable fluid properties with partial Boussinesq approximation (density variation in the body force term only) is presented and compared with the constant properties results. Both models are investigated for different values of the Rayleigh number, from 103 to 105, and Prandtl number equal to 0.71.

An inverse problem of parameter estimation for heat and mass transfer in capillary porous media

Abstract This work deals with the solution of an inverse problem of parameter estimation involving heat and mass transfer in capillary porous media, as described by the dimensionless linear Luikov’s equations. The physical problem under picture involves the drying of a moist porous one-dimensional medium. The main objective of this paper is to simultaneously estimate the dimensionless parameters appearing in the formulation of the physical problem by using transient temperature and moisture content measurements taken inside the medium. The inverse problem is solved by using the Levenberg–Marquardt method of minimization of the least-squares norm with simulated measurements.

Thermally developing laminar flow inside rectangular ducts

Abstract Laminar forced convection inside rectangular ducts is analytically studied by extending the generalized integral transform technique, allowing for the solution of convection-diffusion problems with non-separable eigenvalue problems. Reference results are established for quantities of practical interest within the thermal entry region, for a wide range of the axial variable and various aspect ratios. The accuracy of previously reported results from direct numerical approaches is then critically examined, for both the developing and fully developed regions.

Improved One-Dimensional Fin Solutions

The classical problem of determining heat transfer performance of extended surfaces is revisited. An improved one-dimensional formulation of the energy equation is proposed, which takes approximately into account the nonuniformity of temperature distributions across the fin, as opposed to the classical approach, which neglects transversal temperature gradients. The resulting modified expressions are as simple as those obtained from the classical solution, but accuracy on heat transfer rates at the base and average temperature profiles along the fin is significantly improved, as demonstrated by numerical results for rectangular longitudinal fins and cylindrical pins. The present analysis extends the applicability range of the very simple one-dimensional formulation to considerably larger values of Biot number at the fins lateral surface.

Benchmark integral transform results for flow over a backward-facing step

Abstract Benchmark results for flow over a backward-facing step are obtained through the so-called generalized integral transform technique (GITT). This hybrid numerical-analytical approach is employed to handle the steady two-dimensional incompressible Navier-Stokes equations in streamfunction-only formulation. Numerical results with automatic global accuracy control are produced for different values of Reynolds number. Critical comparisons with previously reported experimental results are performed with excellent agreement. Also, a few different purely numerical approaches are validated, from a survey of the recent literature.

Estimation of dimensionless parameters of Luikov's system for heat and mass transfer in capillary porous media

Abstract This work deals with the solution of inverse problems of parameter estimation involving heat and mass transfer in capillary porous media, as described by the linear one-dimensional Luikovs equations. Our main objective is to use the D-optimum criterion to design the experiment with respect to the magnitude of the applied heat flux, heating and final experimental times, as well as the number and locations of sensors. The present parameter estimation problem is solved with Levenberg–Marquardts method of minimization of the ordinary least-squares norm, by using simulated temperature data containing random errors. Moisture content measured data is not considered available for the inverse analysis in order to avoid quite involved measurement techniques. We show that accurate estimates can be obtained for Luikov, Kossovitch and Biot numbers by using only temperature measurements in the inverse analysis. Also, the experimental time can be reduced if the body is heated during part of the total experimental time.

Stability analysis of natural convection in porous cavities through integral transforms

Abstract The onset of convection and chaos related to natural convection inside a porous cavity heated from below is investigated using the generalized integral transform technique (GITT). This eigenfunction expansion approach generates an ordinary differential system that is adequately truncated in order to be handled by linear stability analysis (LSA) as well as in full nonlinear form through the Mathematica software system built-in solvers. Lorenzs system is generated from the transformed equations by using the steady-state solution to scale the potentials. Systems with higher truncation orders are solved in order to obtain more accurate results for the Rayleigh number at onset of convection, and the influence of aspect ratio and Rayleigh number on the cell pattern transition from n to n+2 cells (n=1,3,5,…) is analyzed from both local and average Nusselt number behaviors. The qualitative dependence of the Rayleigh number at onset of chaos on the transient behavior and aspect ratio is presented for a low dimensional system (Lorenz equations) and its convergence behavior for increasing expansion orders is investigated.

Integral transform solutions of diffusion problems with nonlinear equation coefficients

Abstract A hybrid numerical-analytical procedure is described for the accurate and reliable solution of nonlinear diffusion problems due to potential dependent equation coefficients. A sufficiently general formulation of a transient multidimensional problem is first considered, and formal solutions provided in terms of the related transformed potentials, obtained from the numerical solution of a denumerable system of coupled nonlinear ordinary differential equations. An application related to heat conduction with temperature dependent thermal conductivity is then more closely studied, and numerical results presented to illustrate convergence behavior for increasing truncation order of the associated infinite O.D.E. system.