Application of Variational Iteration and Homotopy-Perturbation Methods to Nonlinear Heat Transfer Equations with Variable Coefficients

Since analytical perturbation methods depend on a small parameter and finding this small parameter is difficult, two powerful analytical methods are introduced to be applied to solve nonlinear heat transfer problems in this article. One is He's variational iteration method ( VIM ) and the other is the homotopy-perturbation method ( HPM ). The VIM is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. The HPM converts a difficult problem into a simple problem which can be easily solved. In this article, the VIM is used to solve some nonlinear heat transfer equations with variable heat transfer coefficient. The results are then compared with those obtained by the HPM and the exact solution.     

Investigations on several compact ADI methods for the 2D time fractional diffusion equation

ABSTRACT

In this article, based on the superconvergent approximation for fractional derivative and the Riemann-Liouville fractional integral, several compact alternating direction implicit (ADI) methods are investigated for solving the 2D time fractional diffusion equation with subdiffusion α ∈ (0, 1). All these methods are second-order-accurate in time and fourth-order-accurate in space, which are independent of the values of anomalous diffusion exponent α. The unconditional stability of the first two methods is discussed by the Fourier analysis method. Numerical examples are computed to justify the theoretical results, and comparisons are made among these methods.     

A comparative study of four low-Reynolds-number k-ε turbulence models for periodic fully developed duct flow and heat transfer

ABSTRACT

In this study, streamwise-periodic fully developed turbulent flow and heat transfer in a duct is investigated numerically. The governing equations are solved by using the finite-control-volume method together with nonuniform staggered grids. The velocity and pressure terms of the momentum equations are solved by the SIMPLE algorithm. A cyclic tri-diagonal matrix algorithm (TDMA) is applied in order to increase the convergence rate of the numerical solution. Four versions of the low-Reynolds-number k-ε model are used in the analysis: Launder-Sharma (1974), Lam-Bremhorst (1981), Chien (1982), and Abe-Kondoh-Nagano (1994). The results obtained using the models tested are analyzed comparatively against some experimental results given in the literature. It is discussed that all the models tested failed in the separated region just behind the ribs, where the turbulent stresses are underpredicted. The local Nusselt numbers are overpredicted by all the models considered. However, the Abe-Kondoh-Nagano low-Re k-ε model presents more realistic heat transfer predictions.     

Numerical Simulation of Non-Darcy Forced Convection through a Channel with Nonuniform Heat Flux in an Open Cavity Using Nanofluid

This article aims to numerically investigate forced convection heat transfer phenomena in a two-dimensional horizontal channel having an open cavity with porous medium. A nonuniform heat flux is considered to be located on the bottom surface of the cavity. Three different heating modes are considered at this wall. The rest of the surfaces are taken to be perfectly adiabatic. The physical domain is filled with water based nanofluid containing TiO₂ naparticles. The fluid enters from left and exits from right with initial velocity U _i and temperature T _i . Governing equations are discretized using the penalty finite element method. The simulation is carried out for a range of Prandtl number Pr(=5.2–12.2) and solid volume fraction φ (=0%–15%). Results are presented in the form of streamlines, isothermal lines, local and average Nusselt number, average temperature of the fluid, horizontal and vertical velocities at mid-height of the channel, and mean velocity field for various Pr and φ. It is found that increasing Pr and φ cause the enhancement of the heat transfer rate.     

Lattice Boltzmann Simulation of Natural Convection in an Inclined Square Cavity with Spatial Temperature Variation

In this paper, natural convection heat transfer in an inclined square cavity filled with pure air (Pr = 0.71) was numerically analyzed with the lattice Boltzmann method. The heat source element is symmetrically embedded over the center of the bottom wall, and its temperature varies sinusoidally along the length. The top and the rest part of the bottom wall are adiabatic while the sidewalls are fixed at a low temperature. The influences of heat source length, inclination angle, and Rayleigh number (Ra) on flow and heat transfer were investigated. The Nusselt number (Nu) distributions on the heat source surface, the streamlines, and the isotherms were presented. The results show that the inclination angle and heat source length have a significant impact on the flow and temperature fields and the heat transfer performance at high Rayleigh numbers. In addition, the average Nu firstly increases with γ and reaches a local maximum at around γ = 45°, then decreases with increasing γ and reaches minimum at γ = 180° in the cavity with ? = 0.4. Similar behaviors are observed for ? = 0.2 at Ra = 10⁴. Moreover, nonuniform heating produces a significant different type of average Nu and two local minimum average Nu values are observed at around γ = 45° and γ = 180° for Ra = 10⁵ in the cavity with ? = 0.2.     

Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number: Part I. Theory,resolutions

ABSTRACT

Rigorously space filtering the thermal, multispecies Navier–Stokes (NS) conservation principle partial differential equation (PDE) system embeds a priori undefined tensor and vector quadruples. Large eddy simulation (LES) computational fluid dynamics algorithm resolutions replace the tensor quadruple with a single tensor then secures closure through “physics-based” modeling, assuming the velocity field is turbulent, i.e., the Reynolds number (Re) is large. In complete distinction, a totally analytical closure is derived for the rigorously generated tensor/vector quadruples, achieved totally absent any modeling component or Re assumption. For Gaussian filter of uniform measure δ, derived analytical filtered Navier–Stokes (aFNS) theory PDE system state variable is significance scaled O(1; δ²; δ³) through classic fluid mechanics perturbation theory. That uniform measure δ filter penetrates domain boundaries requires O(1) resolved scale PDE system inclusion of boundary commutation error (BCE) integrals, (unfiltered) NS state variable extension in the sense of distributions, and domain enlargement to encompass all surfaces with Dirichlet boundary condition (DBC) specification. Theory-derived O(δ²) resolved–unresolved scale interaction PDE system, also the O(1) system, is rendered bounded domain, well posed through a priori identification of O(1; δ²) state variable nonhomogeneous DBCs. BCE and DBC resolution algorithm derivations use O(δ⁴) approximate deconvolution (AD) differential definition Galerkin weak forms. Theory analytically derived unresolved scale O(δ³) state variable annihilates discretization-induced O(h²) dispersion error at unresolved scale threshold δ, h the mesh measure. Net is an analytical theory closing rigorously space-filtered NS exhibiting potential for first principles prediction of viscous laminar–turbulent transition, separation, and relaminarization.     

A Novel Alpha Gradient Smoothing Method (αGSM) for Fluid Problems

In this article, a novel alpha gradient smoothing method (αGSM) based on the strong form of governing equations for fluid problems is presented. The basic principle of αGSM is that the spatial derivatives at a location of interest are approximated by the gradient smoothing operation. The main difference among the piecewise-constant gradient smoothing method (PC-GSM), piecewise-linear gradient smoothing method (PL-GSM), and αGSM is the selection of smoothing function. In the αGSM, the α value controls the contribution of the PC-GSM and PL-GSM. The αGSM is also verified by the solving the Poisson equation. The proposed αGSM has been tested for one benchmark example. All the numerical results demonstrate that the αGSM is remarkably accurate, robust, and stable. Finally, the αGSM has been applied to analyze the flow characteristic in the diseased artery in terms of stenosis.     

Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number: Part II. Resolution algorithms,validations

ABSTRACT

For uniform measure δ Gaussian filter, Part I derives the totally analytical aFNS theory closing rigorously space filtered Navier–Stokes (NS) partial differential equation (PDE) system absent a Reynolds number (Re) assumption. aFNS theory state variable is scaled O(1; δ²; δ³) via classic fluid mechanics perturbation theory which also identifies the O(δ²) elliptic PDE system. Filter penetration of domain bounding surfaces requires O(1) PDE system inclusion of boundary commutation error (BCE) integrals. For the O(1; δ²), PDE system to be bounded domain well-posed requires derivation of domain encompassing nonhomogeneous Dirichlet boundary conditions (DBC). Resolution of BCE and DBC requirements is theorized via O(δ⁴) approximate deconvolution (AD) Galerkin differential definition weak form algorithms. Amenable to any space-time discretization, detailed is aFNS theory insertion in the optimal Galerkin weak form CFD algorithm, finite element linear tensor product basis implemented. Coupled Galerkin CFD/BCE/DBC code a posteriori data reported herein validate theory resolution algorithms including accuracy/convergence assessments.     

Cooled Vortex Street Generated by Cooling a Cylinder at Low Reynolds Number: Effects of Reynolds Number Difference on Cooled Wake and Cooled Vortex Street

First, the dominant action in the cooled wakes in mercury and water at the Reynolds number Re = 22 and 44 is discussed. Next, the cooled wake at Re = 22 is numerically simulated and is compared with the previous results at Re = 44. Finally, effects of Re on the cooled wake and the cooled vortex street are elucidated, and are found to be extremely powerful as follows.

• 1. The dominant action can be determined at different fluids and different Re. Here, the vorticity and the temperature are relating with each other.

• a. “Table of diffusion intensity order” is invented. By obtaining and using this table, the dominant action can be determined automatically.

• b. The kinds of the dominant action are the advection and the diffusion in the vorticity and the temperature.

• c. In mercury and water at Re = 44 and 22, the dominant action is the vorticity diffusion at the low Re, the vorticity advection at the high Re, the temperature diffusion at the low Peclet number Pe, and the temperature advection at the high Pe.

• d. The dominant action in air at Re = 44 is between the dominant action in mercury and water at Re = 44. The dominant action in air at Re = 22 is the same as the dominant action in mercury at Re = 22.

• e. By using the dominant action, the wake variations and item 2 below, i.e., the characteristics of the cooled wake behavior, can be explained.

• 2. When Re is decreased, the following occurs and its cause is elucidated.

• f. In the cooled wake, the Karman vortex street does not occur, but the cooled vortex street with g below occurs.

• g. The vortex spiral size is not changed in mercury but is increased in water.

• h. The following is decreased in mercury but is increased in water. The range of the absolute Richardson number|Ri|generating the cooled vortex street, the spiral degree in the cooled vortex, the critical|Ri| for the symmetric wake onset, and the reciprocal of the temperature wake area.

     

Natural Convection Coupled with Radiation Heat Transfer in Slanted Square and Shallow Enclosures Containing an Isotropic Scattering Medium

A finite-volume method (FVM) is used to address combined heat transfer, natural convection, and volumetric radiation with an isotropic scattering medium, in a tilted shallow enclosure. Numerical simulations are performed in the in-house fluid flow software GTEA. All the results obtained by the present FVM agree very well with the numerical solutions in the references. The effects of various influencing parameters such as the Planck number (0.0001 ≤ Pl ≤ 10), the scattering albedo (0 ≤ ω ≤ 1), the inclination angle (?60° ≤ α ≤ 90°), and aspect ratio (1 ≤ AR ≤ 5) on flow and heat transfer have been numerically studied. For a constant Pl number, flow is slightly intensified at the midplane as the Ra number increases from 10⁶ to 5 × 10⁶. As the scattering albedo increases, the effect of radiation heat transfer decreases on both slanted and horizontal enclosures. In positive tilt angles, the influence of α on heat transfer is quite remarkable. The highest Nu_rad appears at α = 30° (ω = 1)and 0° (ω = 0, 0.5), whereas Nu_rad is maximum at α = ? 15° (ω = 1) and ?45° (ω = 0, 0.5). At α = ?15°, the maximum and minimum values of Nu_rad are presented for ω = 0, AR = 1 and ω = 1, AR = 5.     

Analytical and Level Set Method-Based Numerical Study for Two-Phase Stratified Flow in a Pipe

A nondimensional analytical study for fully developed and three-dimensional numerical study of developing viscosity stratified flow is presented, at various values of viscosity ratio η, inlet area fraction of the less viscous fluid α _1,in , and Reynolds number. With increasing α _1,in , a change in the trend of axial variation in interface–from inlet to the fully developed region–is found at α _1,in  = α _1,in,c . With increasing η, development length is found to asymptotically decrease for α _1,in  = 0.2 and increase for α _1,in  = 0.5 and 0.8. A physical model for flow development is also presented for single- and two-fluid flow. A favorable operating condition to reduce the cost of transportation of more viscous fluids by pipe is proposed.     

Numerical Evaluation of Radiation Extinction Coefficient Using Fractal Geometry for Vegetation Modeling

This article focuses on the accurate determination of the radiation extinction coefficient (β) for a vegetal set like a tree in the frame of forest fires. Usually, this coefficient is calculated using the De Mestre relationship, not taking into account the leaf orientation and position. In order to evaluate their role, a realistic vegetal structure is numerically created using LAI data, De Wit's models, and IFS geometry. A ray-tracing method is used to simulate the radiative transfer inside the numerical tree. The results show that the leaf distribution and position cannot be neglected in β determination. The use of well-known correlations are not sufficient to predict β: over-estimations of up to 100% have been observed.     

Numerical Study of Heat Transfer Around the Small Scale Airfoil Using Various Turbulence Models

A numerical investigation of low-Reynolds number flows with thermal effect around the MAV airfoils using various turbulence models, including an algebraic Baldwin-Lomax model, Spalart-Allmaras one equation, and two equation (k-ω and SST-k-ω) turbulence models, is presented. First, the thermal effect on the aerodynamic efficiency is studied for flow around a rectangular MAV wing, based on the NACA0012 airfoil section at low-aspect ratio (AR = 2) and an angle of attack equal to 0°. Second, details of the thermal effect are limited to the two-dimensional NACA0012 airfoil with chord length of 3.81 cm. This study shows that the improvement of aerodynamic efficiency (increase lift and reduce drag) is achieved by the generation of a temperature difference between extrados and intrados of the airfoil (by cooling the upper surface and heating the lower surface). The numerical results obtained with various turbulence models are in good agreement with experiment data, except the k-ω turbulence model. These results are performed with the CFD-FASTRAN code, using the fully implicit scheme for time integration and the upwind Roe flux difference splitting scheme for space discretization augmented by a high order Osher-Chakravarthy limiter.     

DIRECT-SIMULATION MONTE CARLO MODELING ON TWO-DIMENSIONAL RAYLEIGH-BÉNARD INSTABILITIES OF RAREFIED GAS

ABSTRACT

This study investigates numerically the convective instabilities of rarefied gas in two-dimensional enclosures by the direct-simulation Monte Carlo (DSMC) method. A simulation using an enclosure with length-to-height aspect ratio AS = 2.016 is conducted to validate the value of the critical Rayleigh number on the onset of convection. Enclosures of AS = 2 and 4 are chosen to explore the influences of the Knudsen number Kn and initial wall temperatures on the flow. The results indicate that the different initial conditions may cause multiple solutions for the convection rolls and the Boussinesq approximation is no longer valid for rarefied gases of high Kn.     

Turbulence Models for Fluid Flow and Heat Transfer Between Cross-Corrugated Plates

Three-dimensional numerical predictions of fluid flow and heat transfer between cross-corrugated plates were obtained for the same geometry and grid using eight turbulence models, i.e., LBKE, SKE, RKE, RNGKE, RSM, KW, SST and LES, for the purpose of model performance evaluation. The average Colburn factor j, friction factor f, and local Nusselt number distribution were presented and compared with available experimental data. The velocity, temperature, and turbulent viscosity ratio distributions were recorded and discussed. It has been found that all models can predict practically satisfactory j and acceptable f within the current Reynolds number range. LBKE and SST provide the best overall agreement with experimental data and thus are highly recommended for application. Taking into account its robustness and economy, SKE with enhanced wall treatment is also recommended for CC channel flow and heat transfer simulation. In addition, near wall treatment approach seems to be significant for the current wall-bounded flow simulation. Since some models predict similar j and f values but very different velocity and temperature distributions, it seems not quite sufficient to only compare the overall heat transfer and pressure drop data between simulation and experiment for comprehensive model evaluations.     

Convective Heat Transfer from a Thick Hemispherical Plate during Free Liquid Jet Impingement

Convective heat transfer during free liquid jet impingement on a hemispherical solid plate of finite thickness has been examined. The model included the entire fluid region (impinging jet and flow spreading out over the hemispherical surface) and solid plate as a conjugate problem. Solution was done for both isothermal and constant heat flux boundary conditions at the inner surface of the hemispherical plate. Computations were done for jet Reynolds number (Re _j ) ranging from 500 to 2,000, dimensionless nozzle-to-target spacing ratio (β) from 0.75 to 3, and for various dimensionless plate thicknesse-to-nozzle diameter ratios (b/d _n ) from 0.08 to 1.5. Results are presented for local Nusselt number using water (H₂O), flouroinert (FC-77), and oil (MIL-7808) as working fluids, and aluminum, Constantan, copper, silicon, and silver as solid materials. It was observed that plate materials with higher thermal conductivity maintained a more uniform temperature distribution at the solid–fluid interface. A higher Reynolds number increased the Nusselt number over the entire solid–fluid interface.     

Numerical Study of Sphere Drag Coefficient in Turbulent Flow at Low Reynolds Number

The drag coefficient of a sphere immersed in turbulent air flow in the Reynolds number (Re = U _∞ d/ν _∞) range up to 250 and turbulence intensity (u _∞′/U _∞) up to 60% is computed numerically. Reynolds-averaged Navier-Stokes equations (RANS) are solved in Cartesian coordinates by using a blocked-off technique. To our knowledge, the present work is the first to employ the blocked-off technique for flow over a sphere. Closure for the turbulence stress term is accomplished by testing four different turbulence closure models. The main findings of the present investigation are that the laminar numerical data compare well with numerical and experimental published work. However, different turbulence closure models produce different trends in the range of Reynolds number up to Re = 100, and this difference is demarcated by the nonagreement between the turbulent predictions and the “standard” drag coefficient results. However, the results obtained using Menter's SST turbulence model show fair agreement with the well-known sphere “standard” drag over the range of test conditions explored here. Thus, the present results confirm recently published findings, which suggest that the free-stream turbulence intensity does not have a significant effect on the sphere mean drag.     

The Effect of Conjugate Heat Transfer on Film Cooling Effectiveness

The film cooling effectiveness of a two-dimensional gas turbine endwall is compared for the cases of conjugate heat transfer and an adiabatic wall condition using five common turbulence models. The turbulence models employed in this study are: the RNG k–ε model, the realizable k–ε model, the standard k–ω model, the SST k–ω model, and the RSM model. The computed flow field and surface temperature profiles along with the film effectiveness for one and two cooling slots at different injection angles are presented. The results show the strong effect of conjugate heat transfer on the film effectiveness compared to the adiabatic case and also compared to the effectiveness values obtained from analytically solvable models.     

Effect of Inclination on Heat Transfer and Fluid Flow in a Finned Enclosure Filled with a Dielectric Liquid

The problem of two-dimensional natural convection flow of a dielectric fluid in a square inclined enclosure with a fin placed on the hot wall is investigated numerically. The fin thickness and length are 1/10 and 1/2 of the enclosure side, respectively. The Rayleigh number is varied from 10³ to 5 × 10⁵ and the solid to fluid thermal conductivity ratio is fixed at 10³. The enclosure tilt or inclination angle is varied from 0° to 90°. The streamlines and isotherms within the enclosure are produced and the heat transfer is calculated. It is found that for 2.5 × 10⁴ ≤ Ra ≤ 2.5 × 10⁵, the average Nusselt number is maximum when γ = 0° and minimum when γ = 90°. For Ra = 5 × 10⁵, the values of enclosure tilt angle for which the average Nusselt number is maximum or minimum are completely different due to the transition to unsteady state. In this case, the maximum heat transfer is obtained for γ = 60°, while the minimum heat transfer is predicted for γ = 0°. Monomial correlations relating the average Nusselt number with the different values of the Rayleigh number from 10⁴ to 10⁵ are determined for two different angles, γ = 0° and γ = 90°.