Generalized Integral Transform Technique Research Articles

Integral Transform Solution for Natural Convection Within Horizontal Concentric Annular Cavities

Abstract Natural convection inside horizontal concentric annular cavities is dealt with through the generalized integral transform technique (GITT), offering a hybrid numerical-analytical solution of the continuity, Navier–Stokes, and energy equations in cylindrical coordinates. The flow is in steady-state, laminar regime, two-dimensional, buoyancy-induced, and the governing equations are written in the streamfunction-only formulation. Two strategies of integral transformation are adopted to verify the best computational performance, namely, the usual one with eigenvalue problems for both streamfunction and temperature defined in the radial variable, and a novel alternative with eigenvalue problems defined in the azimuthal angular coordinate. First, the eigenfunction expansions convergence behavior is analyzed to critically compare the two integral transform solution strategies. Then, test cases for different aspect ratios and Rayleigh numbers are validated with experimental data from the classical work of Kuehn and Goldstein. A maximum relative deviation of 5% is found comparing the GITT results for the average Nusselt number against the experimental data, while an 8% maximum relative deviation is found comparing against an empirical correlation by the same authors. It is concluded that the GITT solution with the eigenvalue problem in the angular coordinate yields better convergence rates than the more usual eigenfunction expansion in the radial variable. This is due to the originally homogeneous boundary conditions in the angular direction, which do not require filtering for convergence enhancement, as opposed to the required filter in the radial direction that introduces a source term in the filtered equation for the streamfunction.

Buckling analysis of rectangular plates with rotationally restrained edges subjected to linearly varying in-plane loading

This study addresses a comprehensive procedure for the buckling behavior of rectangular plates with rotationally restrained edges. They are subjected to uniaxial in-plane load that varies linearly from compression to bending. The load is applied to opposite edges with variable boundary conditions, ranging from simply supported to clamped. The other edges can also be free, guided, or between them. For the first time, the generalized integral transform technique (GITT) is applied to the buckling equation of such a plate. The integral kernel perfectly satisfies the plate boundary conditions and its constituent terms are limited to the approximate range of ±1 which prevents the numerical instability. The transformed equation is a set of linear algebraic equations which establish an eigenvalue problem. The plate buckling coefficient, and corresponding mode shape (contours of the buckled plate, the number of half-waves in each direction, etc.) are presented according to variations of the edge relative stiffness as well as the plate aspect ratio, the loading shape, and Poisson's ratio. The latter parameter has a significant effect on the buckling behavior when at least one of the edges is free, guided, or between them. This portion is developed for auxetic materials (negative Poisson's ratios), and it was concluded that the minimum buckling load mostly occurs as the Poisson's ratio tends to zero.

INTEGRAL TRANSFORM SOLUTION OF HEAT CONDUCTION IN ANISOTROPIC HETEROGENEOUS MEDIA

The generalized integral transform technique (GITT) is advanced to deal with conduction heat transfer in anisotropic heterogeneous media. A formal solution for exact integral transformation of conduction in anisotropic media is extended to account for heterogeneities expressed as space variable equation coefficients and source terms. The proposed eigenfunction expansion is based on biorthogonal eigenvalue problems, which results in an exact integral transformation for a class of linear problems and in a coupled transformed ordinary differential system for nonlinear situations. An algorithm is proposed for the associated eigenvalue problems, also handled through the GITT, by considering simpler biorthogonal eigenvalue problems of known analytical solution, leading to transformed algebraic eigenvalue problems. A single domain reformulation strategy is adopted to merge the information from multiple regions and materials, either isotropic or anisotropic, into one single diffusion equation. A two-dimensional transient test case is considered that presents an abrupt transition between isotropic and anisotropic materials yielding a marked change in thermal behavior in a defined region of interest formed by the anisotropic inclusion. Convergence behavior of the integral transform solution is illustrated, and the fully converged results are employed as a benchmark to inspect the accuracy of a commercial finite element code for automatically defined mesh refinement levels.

Influence of thermophysical and acoustic properties on temperature elevation at muscle/bone interface induced by ultrasound hyperthermia

Therapeutic ultrasound heating is a very common technique in the treatment of muscular and joints injuries. However, insufficient or excessive heating delivered to living tissues can be either innocuous or hazardous to the patient. The present work deals with the heat transfer process in living tissues, with particular concern on the behavior at the muscle-bone interface, when ultrasound irradiation is applied. Based on the Generalized Integral Transform Technique (GITT), a hybrid numerical-analytical solution of the Pennes’ equation using the open-source UNIT code (“UNified Integral Transforms”) is proposed, considering different boundary conditions. The influence of physical properties and parameters, such as beam incidence angle, acoustic attenuation coefficients for refracted longitudinal and shear waves in the bone, and blood perfusion in the muscle, were more closely analyzed. Numerical results of temperature profiles showing variations of up to 2 °C for different situations are presented, illustrating a marked dependence of heating patterns on the perfusion rates as well as on the effect of shear waves. The conclusions are that such type of more detailed analysis is essential for the accurate identification of physical properties in living tissues and phantoms and can contribute to the treatment planning for therapeutic ultrasound.

Conduction-radiation heat transfer in a gray semi-transparent medium: combining integral transforms and discrete ordinate method

This work presents a novel approach to solve the unsteady one-dimensional coupled heat conduction-radiation problem for a gray semi-transparent, absorbing, emitting, and diffusing medium, non-linearly anisotropic, bounded by two gray parallel-plane surfaces that are diffuse in emission, and diffuse and specular in reflection. A combination of the Discrete Ordinate Method (DOM), to discretize the angular coordinate of the Radiative Transfer Equation (RTE), and the Generalized Integral Transform Technique (GITT), to solve the energy equation, is proposed as solution methodology. Results are presented in terms of the temperature field, incident radiation, hemispherical reflectance, and conductive, radiative and total heat fluxes. The versatility and the accuracy of the methodology are verified/ validated with solutions of the literature for different phase functions, conduction-radiation coupling parameters, and albedo, emphasizing the effects of these parameters, as well as, the characteristics of the present methodology for different kinds of quadrature rules. The results show that the combination DOM/GITT proved to be adequate as a tool for solving the governing equations of the coupled conduction-radiation problem.

Vibrational Responses of an Ultra-Large Cold-Water Pipe for Ocean Thermal Energy Conversion: A Numerical Approach

The transportation of seawater on a grand scale via an ultra-large cold-water pipe situated within the context of ocean thermal energy conversion (OTEC) floating installations inherently presents challenges associated with instability and potential malfunction in the face of demanding operational circumstances. This study endeavors to augment the stability and security of cold-water pipe (CWP) operations by scrutinizing their vibrational attributes across diverse boundary configurations. Initially, we invoke Euler–Bernoulli beam theory to forge the analytical framework and proffer a semi-analytical resolution by utilizing the generalized integral transform technique (GITT). Subsequently, we authenticate the convergence and precision of our proposed approach through comparative analysis with extant theories. Our findings underscore the conspicuous influence of boundary conditions on the convergence of transverse displacement. The influence of internal flow on the transverse displacement and the natural frequency manifests substantial variability under different boundary conditions. Significantly, an escalation in the internal flow velocity triggers a concomitant reduction in the natural frequency, ultimately culminating in instability once the critical velocity threshold is reached. Additionally, the reliance of the transverse displacement and the natural frequency on the clump weight at the bottom is markedly pronounced. Our discoveries propose that pipe stability can be ameliorated by adjusting the clump weight at the bottom. Furthermore, the novel insights obtained through our proposed approach can significantly aid in the early-stage design and analysis of CWP.

Analysis of transient laminar forced convective heat transfer in parallel-plate channels with time-varying thermal boundary conditions of the inlet and wall

The problem of transient laminar forced convective heat transfer in parallel-plate channels with time-varying thermal boundary conditions of the inlet and wall is thoroughly investigated in this study. By employing the methodologies of generalized integral transform technique (GITT) and Duhamel's theorem, we successfully derive an analytical solution for this problem. To validate the accuracy of our analytical findings, a comparison is made with the corresponding numerical solution, demonstrating a remarkable level of agreement between the two approaches. In order to obtain the general patterns of heat transfer, we employ a decoupling technique to separate the thermal response in the case of harmonic temperature boundary conditions of the inlet and wall. This approach enables us to uncover the intricate relationship between the weight coefficient of heat input and the two-dimensional spatial location of the channel, as well as the dimensionless angular frequency of heat input. Furthermore, we present the corresponding dimensionless cut-off angular frequency of heat input at various locations within the channel. To gain further insights, we conduct an in-depth analysis of the effects of integration time or distance on the dimensionless optimum angular frequency corresponding to the dimensionless maximum average heat flux.

Changes in groundwater flow in an unconfined aquifer adjacent to a river under surface recharge

ABSTRACT The groundwater level in an unconfined aquifer near a river is strongly affected by rainfall and river level. The linearized Boussinesq equation has been used in numerous studies to analytically investigate the changes in groundwater level. A perturbation method was used to determine the non-linear term of the Boussinesq equation. The generalized integral transform technique (GITT) was subsequently used to obtain a semi-analytical solution. We compared this solution with a linearized analytical and non-linear numerical solution and investigated the influence of each parameter that affects the number of convergent terms in analytical solutions. To estimate the recharge volume, two methods, the Horton infiltration formula and a method in which the fraction of rainfall intensity is considered, were employed. Subsequently, the simulated groundwater levels were compared with the field data. Finally, the recharge type, namely unimodal, with and without the Horton infiltration formula was used to model the groundwater flow.

Semi-analytical solution of single-well push–pull test under transient flow conditions

The single-well push–pull (SWPP) test has been extensively used in parameter estimation models. Numerous analytical solutions to the problem are available, all of which consider solute transport in a non-uniform flow field from steady radial flow created by the SWPP test. However, the non-uniform flow velocity is based on the Thiem equation and varies in a spatial manner, rather than a temporal manner. No analytical solution has been previously described for the solute transport equation of SWPP under fully transient flow. In this study, the generalized integral transform technique (GITT) was used to develop a new semi-analytical solution for solute transport in the SWPP test under fully transient flow. Four phases of the SWPP test were included: injection, chaser, rest, and extraction. With the proposed solution, the differences between a transient flow SWPP model solution and a piecewise steady-state flow SWPP model solution were non-negligible; such differences increased with decreases in the dimensionless parameter related to aquifer flow properties, which is proportional to transmissivity, aquifer thickness, and porosity, but inversely proportional to the storage coefficient and pumping rate. Additionally, long tails of BTCs are characteristic of type curves under the transient flow model. The long tails of BTCs under transient flow could result in overestimation of dispersivity if the SWPP model uses piecewise steady-state flow, rather than transient flow, for parameter estimation. The proposed semi-analytical solution provides a useful tool for curve-fitting during parameter estimation when the effects of transient flow are significant. The proposed solution can also serve as a performance comparison for testing numerical solutions to the SWPP test.

Integral Transforms with Single Domain Formulation for Transient Three-Dimensional Conjugated Heat Transfer

An analysis based on integral transforms is undertaken for transient three-dimensional conjugated conduction-convection heat transfer, with a focus on mini or micro channels-based devices. The numerical-analytical approach Generalized Integral Transform Technique (GITT) is combined with a single domain reformulation, providing accurate, robust, and cost-effective simulations for determining temperature distributions within the domain. The fluid and solid subdomains are represented as one single region, while the integral transformation is carried out using a three-dimensional eigenvalue problem encompassing the thermophysical properties and velocity field abrupt spatial variations. The steady state problem solution is employed as a filter and solved through an integral transformation based on the corresponding two-dimensional eigenvalue problem defined for the channel cross section. The transformed ordinary differential systems for both the steady and homogeneous transient problems are handled analytically, requiring only the numerical solution of the associated matrix eigensystem analysis. Converged numerical results for dimensionless temperature distributions are then critically compared with a finite element solution and a previously proposed GITT solution that implements a partial transformation scheme under a pseudo-transient formulation, both for the conjugated problem with thermally developing laminar flow in a rectangular channel.

Unified Integral Transforms and Non-Classical Eigenvalue Problems in Heat and Mass Transfer

Abstract The generalized integral transform technique (GITT) is reviewed as a computational–analytical methodology in linear and nonlinear convection–diffusion problems, based on eigenfunction expansions extracted from characteristic differential operators, coefficients, and boundary conditions present in the original partial differential problem formulation. Here, the emphasis is on the employment of nonclassical eigenvalue problems as the expansion basis, which do not fall into the more usual framework of Sturm–Liouville problems. The goal is to enable or improve the eigenfunction expansions convergence, by incorporating more information from the original operators into the chosen eigenvalue problem, while requiring the handling of such a more involved expansion base. In this concern, the proposed differential eigenvalue problem can itself be handled by the GITT, leading to an algebraic eigensystem analysis. Different classes of nonclassical eigenvalue problems are then reviewed and associated with typical applications in heat and mass transfer. Representative test cases are then chosen to illustrate the extended methodology and demonstrate the convergence rates attainable by this enhanced hybrid solution path.

Heat transfer analysis of compressible laminar flow in a parallel-plates channel via integral transforms

The present work analyzes a coupled nonlinear mathematical model for heat transfer in compressible laminar flow within a parallel plates channel. The governing partial differential equations are obtained considering the conservation of mass, momentum and energy, based on a two-dimensional steady laminar flow of an ideal gas with constant physical properties and taking into account viscous dissipation. Two kinds of boundary conditions at the wall were considered, namely, prescribed arbitrary temperature or heat flux longitudinal distributions. The mathematical model was solved using the hybrid numerical-analytical method known as the Generalized Integral Transform Technique (GITT). A convergence analysis of the proposed eigenfunction expansions for velocity and temperature fields was undertaken, indicating that fairly low truncation orders are required for the accurate representation of the potentials. The proposed hybrid solution was also critically compared with the Mathematica NDSolve finite difference-based routine and additional theoretical and experimental results in the literature, with good agreement in all cases considered.

Bending of variable thickness rectangular thin plates resting on a double-parameter foundation: integral transform solution

PurposeThe purpose of this study is to propose a generalized integral transform technique (GITT) to investigate the bending behavior of rectangular thin plates with linearly varying thickness resting on a double-parameter foundation.Design/methodology/approachThe bending of plates with linearly varying thickness resting on a double-parameter foundation is analyzed by using the GITT for six combinations of clamped, simply-supported and free boundary conditions under linearly varying loads. The governing equation of plate bending is integral transformed in the uniform-thickness direction, resulting in a linear system of ordinary differential equations in the varying thickness direction that is solved by a fourth-order finite difference method. Parametric studies are performed to investigate the effects of boundary conditions, foundation coefficients and geometric parameters of variable thickness plates on the bending behavior.FindingsThe proposed hybrid analytical-numerical solution is validated against a fourth-order finite difference solution of the original partial differential equation, as well as available results in the literature for some particular cases. The results show that the foundation coefficients and the aspect ratio b/a (width in the y direction to height of plate in the x direction) have significant effects on the deflection of rectangular plates.Originality/valueThe present GITT method can be applied for bending problems of rectangular thin plates with arbitrary thickness variation along one direction under different combinations of loading and boundary conditions.

Elastic design of skid beams in offshore load-out systems

Skid beams are essential components in skidded load-out systems, widely used in offshore operations, and are usually made in plate or box girder forms. They are continued from the jetty to the barge for a relatively long length and are the foundation of portable superstructures to transfer the distributed vertical load to the jetty/barge. Finite element analysis is usually used to verify the strength and stability of the skid beams' components, including flange, web, and stiffener. As FEM usually includes many analyses with a trial-and-error process, this study presents a general and explicit design procedure for skid beams in load-out operations. The main design criteria are the flange's bending capacity as well as the web and stiffener's buckling capacity. The generalized integral transform technique (GITT) is applied to solve the bending equation of the flange and the buckling equations of the web and stiffener with hinged and rigid connections. An algorithm is suggested by which some design curves (concise formulas) are obtained to find the maximum slenderness ratios of the skid beam's components. Accordingly, a step-by-step design procedure is presented, which can be easily used for practical purposes and can be a better replacement for finite element analysis. Comparing the analytical method with FE modeling, there is acceptable agreement and excellent convergence. A realistic design shows that the reduction of skid beam weight is not more than 10% due to rigid connections. In addition, the weight is reduced by increasing the web and decreasing the stiffener aspect ratios.

Applications of generalized integral transform technique in the rapid solidification of metallic metals in the planar flow casting process / Aplicações da técnica de transformação integral generalizada na solidificação rápida de metais metálicos no processo de fundição em fluxo planar

The Planar Flow Casting (PFC) is a single-stage fast solidification technique to produce thin metallic ribbons. The objective of this research is to develop a mathematical model to analyze the phenomenon of heat transfer and phase change during the formation of the puddle, to determine the interface position and velocity and the temperature profile. The applied methodology consists of the use of the energy balance, where the equations of the energy (liquid and solid phases), and the equation of the interface are transformed through the Generalized Integral Transform Technique (GITT) being solved by the NDSolve routine of the Matemathica. This tool was capable to solve problem, can study the fast cooling of metals and obtain ribbons of thickness controlled by the speed of the wheel and for the heat transfer coefficient. Considering that the height of the pool is very small and the process time is very short, many eigenvalues were used to obtain the solution convergence. The results of the temperature distribution along the length of the puddle, the evolution of the solidification front and the interface velocity were compared with existing results in the literature, obtaining good harmony.

Three-dimensional transient flow in heterogeneous reservoirs: Integral transform solution of a generalized point-source problem

Appraisal well testing plays a key role in optimizing a complex field’s exploitation strategy — and increasing complexity often calls for more elaborate reservoir models for the proper interpretation of each test’s results. The classical analytical methods usually require restrictive simplifying assumptions, which limit their usefulness when dealing with reservoir heterogeneity. We seek to solve this issue by applying the hybrid analytical–numerical Generalized Integral Transform Technique (GITT) to the pressure diffusivity equation — and specifically, to the point-source problem extended to a reservoir with arbitrary permeability variation irregularly distributed throughout the 3D domain. The technique is first demonstrated to produce an exact alternative form of the well-known solution to the classical point-source in a homogeneous reservoir. This equivalence is then used to derive a computationally efficient working expression for the generalized point-source in a heterogeneous reservoir. Finally, this building block is applied to construct a uniform flow solution for a limited entry vertical well through spatial superposition — thus demonstrating the usage of the GITT for other wellbore geometries. Synthetic examples are used to show that the obtained expressions are in good agreement with results from a commercial reservoir simulator. In all cases presented, the eigenfunction expansions for both the drawdown pressure and its logarithmic derivative converge to at least four and two significant digits, respectively, within the adopted practical ranges of the series’ truncation orders. It is clear that the pressure expansions have better convergence characteristics, which are also influenced by the time value and by the distance from the position under consideration to the point-source. The resulting novel solution to the point-source provided in the present work is the most general and least restrictive expression presented so far to this single problem in a heterogeneous domain, and it is suitable for pressure transient analysis. • Generalized Integral Transform Technique (GITT) for pressure transient analysis. • Generalization of the point-source solution for a heterogeneous reservoir. • Building block for any wellbore geometry — illustrated with the limited entry well.

MATHEMATICAL MODEL AND ANALYTICAL SOLUTION FOR THE VIBRATION OF INCLINED FLUID-TRANSPORTING SUBMARINE PIPELINES

Due to the complexity of submarine environments, the nature of the dynamic response of free-spanning submarine pipelines, particularly inclined pipelines, is unclear. This paper aims to theoretically analyze the vibration behaviors of inclined fluid-transporting free-spanning submarine pipelines in the deepwater area. The mathematical model for the vibration of inclined fluid-transporting pipelines is established considering the influence of gravity on vibration response, and a non-linear wake oscillator is employed to model the vortex shedding behind the pipeline free span. The partial differential equation system is solved through the generalized integral transform technique (GITT), which is an analytical or semi-analytical method. Parametric analysis are carried out to investigate the effects of the inclination on the dynamic response of fluid-transporting pipelines. It is found that the inclination of the free- spanning pipeline will radically alter the natural frequency of the structure, and consequently the VIV lock-in region. In addition, the slope of the seabed will cause a more significant internal flow effect. The thorough theoretical understanding of inclined fluid-transporting pipelines helps increase the design accuracy for pipelines installed on a seabed with a highly irregular topography.

Analytical solution for thermally developing laminar core-annular flow with two immiscible liquids considering axial diffusion in semi-infinite ducts

Flows involving core-annular immiscible liquids provide a wide range of useful engineering applications, such as transportation of heavy oils and cooling processes. The present article considers an analytical solution for the internal convection heat transfer in hydrodynamically developed and thermally developing laminar core-annular liquid–liquid immiscible flows in circular ducts. The axial diffusion is considered in the model and the problem is investigated for semi-infinite domain under convective boundary condition on duct walls and prescribed inlet flow temperature. The steady-state problem is solved through the generalized integral transform technique (GITT) with a single domain approach, in which, the coupled ODE system arising from the GITT is analytically solved using the matrix exponential method. Moreover, integral balance is used to compute the Nusselt number. The convergence process of the local Nusselt number is shown in different flow regions and the results are presented for multiple flow parameters. The effects of the boundary conditions, flow properties, and other parameters are discussed. An analysis of axial diffusion effects is performed, which confirms that it can interfere with the local Nusselt number, especially for lower Péclet numbers.

Effects of an Axial Magnetic Field on Vortex Breakdown and Fluid Layer

The effects of an axial magnetic field on both the vortex breakdown process and fluid layers development in a cylindrical container filled with a conducting viscous fluid are numerically analyzed by using the Generalized Integral Transform Technique (GITT) with a stream function-only formulation. A temperature gradient is imposed in the axial direction on the swirling flow which is advanced by the rotation of the bottom disk under the stabilizing effect of the external magnetic field. Flows are studied for a range of parameters: the Richardson number, Ri, 0 ≤Ri ≤2.0; and three values of the Prandtl number are investigated, Pr = 0.025 (liquid Mercury), 0.032 ( PbLi 17 alloy), and 0.065 (the molten lithium). Three combinations of aspect ratios (H/R) and Reynolds numbers are compared: (case A: Re=1500, H/R=1.5); (case B: Re=1855, H/R=2.0) and (case C: Re=2400, H/R=2.5). The results reveal that the increase in the values of Hartmann number, Ha suppresses the vortex breakdown in the isothermal case and reduces the number of fluid layers in the layering case. The stability diagram (Hacr–Ri) corresponding to the transition from the multiple fluid layers zone to the one fluid layer zone for increasing Prandtl number is obtained‎.

Prediction of coupled in-line and cross-flow vortex-induced vibrations of fluid-transporting free-spanning submarine pipelines: an integral transform solution

ABSTRACT The Generalised Integral Transform Technique (GITT) was adopted to predict the dynamic behaviours of fluid-transporting free-spanning submarine pipelines, considering both the in-line and cross-flow vibrations. The coupled in-line and cross-flow fluid-structural system was analytically solved by GITT, where wake oscillator models were used to simulate in-line and cross-flow forces acting on the pipeline. The mathematical model and the solution method were validated against experimental results. The convergence behaviour of the solution method was analyzed, parametric studies were conducted, and the influence of internal fluid and external velocities were evaluated. Findings in this paper also demonstrated that GITT, which is a semi-analytical solution, is faster than finite element methods, hence highly suitable for conducting parametric studies for submarine pipelines.