Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary

Analytical solutions play important roles in the understanding of fluid dynamics and heat transfer related problems. Some analytical solutions for incompressible steady/unsteady 2-D problems have been obtained in literature, but only a few of those are found under heat transfer conditions (which brings more complexities into the problem). This paper is focused on the analytical solutions to the basic problem of incompressible unsteady 2-D laminar flows with heat transfer. By using the traveling wave method, fluid dynamic governing equations are developed based on classical Navier–Stokes equations and can be reduced to ordinary differential equations, which provide reliable explanations to the 2-D fluid flows. In this study, a set of analytical solutions to incompressible unsteady 2-D laminar flows with heat transfer are obtained. The results show that both the velocity field and the temperature field take an exponential function form, or a polynomial function form, when traveling wave kind solution is assumed and compared in such fluid flow systems. In addition to heat transfer problem, the effects of boundary input parameters and their categorization and generalization of field forming or field evolutions are also obtained in this study. The current results are also compared with the results of Cai et al. (R. X. Cai, N. Zhang. International Journal of Heat and Mass Transfer, 2002, 45: 2623-2627) and others using different methods. It is found that the current method can cover the results and will also extend the fluid dynamic model into a much wider parameter ranges (and flow situations).     

Optimum wall thickness ratio based on the minimization of entropy generation in a viscous flow between parallel plates

The influence of the geometrical and physical parameters on entropy generation for a viscous flow between infinite parallel walls of finite thickness is studied by solving the momentum and energy conservation equations. The conjugate heat transfer problem in the fluid and solid walls is solved analytically using thermal boundary conditions of the third kind at the outer surfaces of the walls and continuity of temperature and heat flux across the fluid–wall interfaces. Analytic solutions for the velocity and temperature fields in the fluid and walls are used to calculate the local and global entropy generation rate. Conditions under which this quantity is minimized are determined for certain suitable combination of geometrical and physical parameters of the system. Special attention has been given to the effect of the wall thickness on the entropy generation rate. It is found that the global entropy generation reaches a minimum for specific values of the wall thickness ratio, when the other parameters are fixed.     

Convective Heat Transfer of an Air Bubble in Water With Variable Thermophysical Properties

A numerical study of convective heat transfer of an air bubble in water with variable thermophysical properties is considered. Two-dimensional simulations of multifluid flows with heat transfer include the Navier–Stokes, energy, and volume of fluid (VOF) advection equations. The solver computes the flow field and temperature by solving the systems of Navier–Stokes equations and the energy equation using the finite–volume method with the SIMPLE algorithm and tracks the position of interface between two fluids with different fluid properties by the VOF method with piecewise linear interface construction technique. Empirical correlations in terms of temperature for thermophysical properties are considered in the simulations. The convective heat transfer model is assessed with a benchmark problem of cooling of water and compared with previous literature data showing good agreement. Finally, a numerical study of the effect of the bubble diameter in the range from 2 mm to 3 mm on heat transfer is performed.     

The boundary element method applied to forced convection heat problems

An integral equation formulation for steady flow of a viscous fluid is presented based on the boundary element method. The continuity, Navier–Stokes and energy equations are used for calculation of the flow and temperature fields. The governing differential equations, in terms of primitive variables, are derived using velocity–pressure–temperature parameters. The calculation of fundamental solutions and solutions tensor is shown. Applications to simple flow cases, such as driven cavity, forward facing step, deep cavity and channel are presented. Convergence difficulties are indicated, which have limited the applications to flows of low Reynolds numbers.     

Simplified model and lattice Boltzmann algorithm for microscale electro-osmotic flows and heat transfer

The extremely small length scale of the electric double layer (EDL) of electro-osmotic flows (EOF) in a microchannel makes it difficult to simulate such flows and associated thermal behaviors. A feasible solution to this problem is to neglect the details in the thin EDL and replace its effects on the bulk flow and heat transfer with effective velocity-slip and temperature-jump boundary conditions outside the EDL. In this paper, by carrying out a scale analysis on the fluid flow and heat transfer in the thin EDL, we analytically obtain the velocity and the temperature at the interface between the EDL and the bulk flow region. The Navier–Stokes equations and the conservation equation of energy, along with the interfacial velocity and temperature as the velocity-slip and temperature-jump boundary conditions, form a simple model for the electro-osmotic flows with thermal effects in a microchannel with a thin EDL. We use the double distribution function lattice Boltzmann algorithm to solve this model and found that numerical results are in good agreement with those by the conventional complete model with inclusion of the EDL, particularly for the cases when channel size is about 400 times larger than the Debye length. Moreover, we found that the present model can substantially reduce the computational time by four to five times of that using the conventional complete model. Therefore, the simplified model proposed in this work is an efficient tool for simulating electro-osmosis-based microfluidic systems.     

Réponse a un signal thermique sinusoïdal dans le cas d'un écoulement laminaire a plan directeur

A theoretical and numerical study of laminar forced convection in both parallel-plate channels, subjected to a sinusoidally varying inlet temperature, is presented. The plate's thermal response is coupled to the fluid via the congugation conditions at the interface (boundary condition of the fourth kind). A thermal diffusion in the plate and a boundary condition that accounts for external convection are considered. The temperature amplitude and phase lag are determined as a function of four parameters. The results are compared at the classical hypothesis of isothermal plates used by several authors. Exactness of the numerical scheme solutions are established by comparison with exact solutions for slug flows.     

A Fully Coupled Navier-Stokes Solver for Fluid Flow at All Speeds

This article deals with the formulation and testing of a newly developed, fully coupled, pressure-based algorithm for the solution of fluid flow at all speeds. The new algorithm is an extension into compressible flows of a fully coupled algorithm developed by the authors for laminar incompressible flows. The implicit velocity–pressure–density coupling is resolved by deriving a pressure equation following a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique. The coefficients of the momentum and continuity equations are assembled into one matrix and solved simultaneously, with their convergence accelerated via an algebraic multigrid method. The performance of the coupled solver is assessed by solving a number of two-dimensional problems in the subsonic, transsonic, supersonic, and hypersonic regimes over several grid systems of increasing sizes. For a desired level of convergence, results for each problem are presented in the form of convergence history plots, tabulated values of the maximum number of required iterations, the total CPU time, and the CPU time per control volume.     

Boundary conditions at a fluid–porous interface for a convective heat transfer problem: Analysis of the jump relations

This paper presents a two-step up-scaling approach to determine the jump relations that must be imposed at the interface between a homogeneous porous domain and a free domain. We study convective heat transfer at such an interface under the assumption of local thermal equilibrium. The two-step approach has the capability of providing closed jump relations depending on intrinsic characteristic of the interface. In addition, from the resulting jump relations, it is possible to determine a particular interface location where the condition of continuity are sufficient. Thus, the use of jump or continuity conditions depend only on the interface location inside the fluid–porous transition region.     

ADVANCED SIMULATION OF OPTICAL FIBER DRAWING PROCESS

Precise modeling of the optical fiber drawing process is extremely important for identifying optimum drawing conditions in a furnace that would produce high-quality fibers at a low cost. In this study, a numerical approach for detailed simulation of thermal transport in an optical fiber drawing furnace is developed by implementing a primitive variable computational fluid dynamics algorithm. To accurately simulate the underlying fluid dynamics, the complete Navier–Stokes equations are solved for both glass and external gas, which are coupled by the conjugate boundary conditions at the interface. The governing equations are discretized by a finite volume approach and the solution algorithm for the discretized equations is based on the semi-implicit method for the pressure linked equations revised (SIMPLER) method instead of the traditional streamfunction approach. Radiative heat transfer is the most dominant mode of heat transfer during optical fiber drawing and it is modeled by the finite volume method. The gas-preform interface is treated as a Fresnel surface instead of a diffuse surface. To validate the present numerical approach and to examine the effects of the Fresnel interface condition on the preform temperature, a benchmark optical fiber drawing problem containing a prescribed neck-down profile is investigated with various fiber drawing speeds, furnace wall temperatures, and preform diameters. For the diffuse interface, the present prediction in temperature is found to match the available other solution very well. For the Fresnel interface, the present prediction is usually higher in comparison with that of the same approach with the diffuse interface. However, the temperature difference between two interfaces is found to be small, implying that the error caused by the diffuse assumption may be not significant.     

Magnetized mixed convection second‐grade fluid flow adjacent to a lubricated vertical surface

The aspects of magnetized mixed convection in second‐grade fluid flow near stagnant point induced by vertical wall are reported. The fluid is impinging orthogonally on the power law lubricant surface. The convective surface temperature and concentration distribution have been assumed. Both the lubricant and the base fluid are governed by the partial differential mathematical expressions. The velocity of the second‐grade fluid and the lubricant are supposed to be continuous at interface. To get the solution of defined nonlinear problem, an implicit numerical technique namely Keller‐Box scheme is applied. The influential constraints are visualized by plotting graphs on velocity, concentration, and thermal profiles. The results of skin‐friction factors, and mass and heat transport rates for both opposing and assisting flows are tabulated and evaluated. The obtained results are validated through available data for limiting conditions.     

Boundary conditions at a planar fluid–porous interface for a Poiseuille flow

The velocity boundary condition that must be imposed at an interface between a porous medium and a free fluid is investigated. A heterogeneous transition zone characterized by rapidly varying properties is introduced between the two homogeneous porous and free fluid regions. The problem is solved using the method of matched asymptotic expansions and boundary conditions between the two homogeneous regions are obtained. The continuity of the velocity is recovered and a jump in the stress built using the viscosity (and not the effective viscosity) appears. This result also provides an explicit dependence of the stress jump coefficient to the internal structure of the transition zone and its sensitivity to this microstructure is recovered.     

A lattice Boltzmann model for interphase conjugate heat transfer

A lattice Boltzmann model is proposed with a newly modified equilibrium distribution function for solving the conservation form of the energy equation to treat the interphase conjugate heat transfer problems under both steady state and unsteady state. The temperature and heat flux continuity conditions at the interface can be inherently satisfied without needing any additional treatments, such as iterative computation, correcting procedure for the incoming distribution function, and the complicated calculation procedure for the source term, to account for the interphase conjugate heat transfer. The implementation of the present LB model, therefore, is more straightforward and more efficient than those in most previous models, especially for problems with complex interfaces. The applicability and accuracy of the proposed LB model were evaluated by some benchmark problems including both simple flat interface and complex interface geometry. The results show excellent agreements with analytical solutions or finite volume results, demonstrating that the present model can serve as a promising numerical technique for dealing with fluid flow and heat transfer in complex heterogeneous systems.     

Prediction of the phase change front using the optimization technique

A new method for tracking the solid-liquid interface is proposed. Experiments on solidification of a flowing fluid in a vertical tube were carried out to obtain the wall temperature of the tube as a function of time. Using the experimental data for the tube wall temperature, the governing equation, initial and boundary conditions for the solidified layer are transformed into an equivalent optimization problem. This problem is then discretized using the space-time finite element so that the continuously moving solid-liquid interface can be easily incorporated. This discretized problem is solved by Fibonacci search method to find the position of the transient solid-liquid interface. Solidified-layer thickness versus time curves are obtained from the present theory and they are compared with the curves obtained from the steady-state formula for solidification.     

Dynamical and thermal performance of molten salt pipe during filling process

The dynamical and thermal performances of molten salt pipe during the filling process are numerically investigated using volume of fluid model. The whole filling process has three main stages, or the developing stage with the interface width quickly increasing, the fluctuating stage with the interface width fluctuating and the fully developed stage with stable interface, and then associated interface structures, flow and temperature fields are described in detail. Before the molten salt flows through a certain position, the fluid temperature will jump within a short time, while the wall temperature only linearly increases after that. The heat transport during the filling process is mainly dependent upon the pipe wall and molten salt flow, while natural convection outside can almost be ignored. The dimensionless interface temperature has similar evolution process under different surrounding temperatures, but it will apparently increase with the flow velocity rising. In addition, the pipe will be blocked when the interface temperature drops below the freezing point, so a model of penetration distance is derived by correlating the interface temperature evolution process, and it has a good agreement with available experimental results.     

Flow due to a porous stretching/shrinking sheet with thermal radiation and mass transpiration

This study investigates the behavior of carbon nanotubes (CNT) approaching an unsteady flow of a Newtonian fluid over a stagnation point on a stretching surface employing porous media. It flows when the liquid begins to move with the progression of time. Heat exchange with the environment has an impact on the flow. The implicitly limited component technique is used to solve the nondimensional partial differential equation with an associated boundary layer, which is an unstable system. Analytically, the solutions, as well as the required boundary conditions, are obtained. The effects of mass transpiration, volume fraction, and heat radiation on Newtonian fluid flow through porous media are explored. Single- and multi-walled CNTs are used as well as water, as base fluids in the experiment. The impact of thermal radiation and heat source/sink is shown in the energy equation, which is solved under four different cases: uniform heat flux case, constant wall temperature case, general power-law wall heat flux case, and general power-law wall temperature case. By supplying distinct physical characteristics, a theoretical analysis of the existence and nonexistence of unique and dual solutions may be explored. These physical parameters determine the velocity distribution and temperature distribution. Prescribed surface temperature (PST) and prescribed wall heat flux (PHF) heat transfer solutions can be written using confluent hypergeometric equations, and generic power-law PST and PHF situations can also be expressed using confluent hypergeometric equations. The graphical representations assist in the discussion of the current study's findings.     

Modeling of Heat Transfer Across the Interface in Two-Fluid Flows

A model is presented in this article to deal with heat transfer across the interface separating two immiscible fluids. It is suitable to be incorporated into interface-tracking methods, such as volume-of-fluid (VOF) methods, because a sharp interface is available in these approaches. The temperature at the interface and the heat flux through it are calculated in such a way that the continuity of the two properties at the contact surface is satisfied explicitly. With use of these values, the temperature either at the centroid or on a face of the interface cell can be estimated, which serves as Dirichlet boundary condition for the energy equation. The temperature field is then calculated by solving the energy equations for the two fluids simultaneously in an implicit way. This method is first assessed via testing on two heat conduction problems in which two solids are in contact. Good agreement between numerical solutions and theory is obtained. To demonstrate its capability, it is applied to two kinds of heat transfer problems, one being the collapse of a heated water column in a cavity, and the other the falling of a molten tin droplet in an oil tank. The effect of fluid flow on the heat transfer is clearly illustrated.     

Investigation of a Confined Laminar Impinging Jet on a Plate with a Porous Layer Using the Preconditioned Density-Based Algorithm

The preconditioned density-based algorithm and two-domain approach were used to investigate the fluid flow and heat transfer characteristics of a confined laminar impinging jet on a plate covered with porous layer. In the porous zone, the momentum equations were formulated by the Darcy-Brinkman-Forchheimer model; the thermal nonequilibrium model was adopted for the energy equation. At the porous/fluid interface, the applicability and influence of different hydrodynamic and thermal interfacial conditions were analyzed for the problem. The governing equations were solved by the preconditioned density-based finite-volume method, with preconditioning matrix for equations of porous domain adopted, aiming to eliminate the equation stiffness of porous seepage flows. The effects of Reynolds number, porosity, Darcy number, thermal conductivity ratio, Biot number, and porous layer thickness on the flow pattern and local heat transfer performance were studied. Results indicate that the Reynolds number and porosity don't strongly influence the flow pattern of porous channel, while the Darcy number and porous layer thickness have obvious influence on the flow pattern. The heat transfer performance are greatly influenced by the parameters studied.     

Power-optimization of non-ideal energy converters under generalized convective heat transfer law via Hamilton-Jacobi-Bellman theory

A multistage irreversible Carnot heat engine system operating between a finite thermal capacity high-temperature fluid reservoir and an infinite thermal capacity low-temperature environment with generalized convective heat transfer law [q∝m(ΔT)] and the irreversibility of heat resistance and internal dissipation is investigated in this paper. Optimal control theory is applied to derive the continuous Hamilton-Jacobi-Bellman (HJB) equations, which determine optimal fluid temperature configurations for maximum power output under the conditions of fixed duration and fixed initial temperature of the driving fluid. Based on general optimization results, the analytical solution for the case with Newtonian heat transfer law (m=1) is further obtained. Since there are no analytical solutions for the other heat transfer laws (m≠1), the continuous HJB equations are discretized and dynamic programming (DP) algorithm is adopted to obtain complete numerical solutions of the optimization problem, and the relationships among the maximum power output of the system, the process period and the fluid temperature are discussed in detail.     

Entropy generation minimization of a MHD (magnetohydrodynamic) flow in a microchannel

The dissipative processes that arise in a microchannel flow subjected to electromagnetic interactions, as occurs in a MHD (magnetohydrodynamic) micropump, are analyzed. The entropy generation rate is used as a tool for the assessment of the intrinsic irreversibilities present in the microchannel owing to viscous friction, heat flow and electric conduction. The flow in a parallel plate microchannel produced by a Lorentz force created by a transverse magnetic field and an injected electric current is considered assuming a thermally fully developed flow and conducting walls of finite thickness. The conjugate heat transfer problem in the fluid and solid walls is solved analytically using thermal boundary conditions of the third kind at the outer surfaces of the walls and continuity of temperature and heat flux across the fluid-wall interfaces. Velocity, temperature and current density fields in the fluid and walls are used to calculate the global entropy generation rate. Conditions under which this quantity is minimized are determined for specific values of the geometrical and physical parameters of the system. The Nusselt number is also calculated and explored for different conditions. Results can be used to determine optimized conditions that lead to a minimum dissipation consistent with the physical constraints demanded by the microdevice.     

Numerical Simulation of Low-Mach-Number Laminar Mixing and Reacting Flows Using a Dual-Purpose Pressure-Based Algorithm

Benefitting from an analogy between compressible and incompressible governing equations, a novel dual-purpose, pressure-based finite-volume algorithm is suitably extended to simulate laminar mixing and reacting flows in low-Mach-number regimes. In our test cases, the Mach number is as high as 0.00326. Definitely, such low-Mach-number flows cannot be readily solved by either regular density-based solvers or most of their extensions. To examine the accuracy and performance of the extended formulation and algorithm, we simulate two benchmark cases including the mixing natural-convection flow in a square cavity with strong temperature gradients and the premixed reacting flow through annuli with high, sharp density variations. In both cases, the fluid flow is treated as an ideal gas, whose properties vary with temperature variation assuming Sutherland's law. Additionally, we do not take into account the Boussinesq limit in treating highly thermobuoyant flow fields. The current results are validated against other available benchmarks and reliable numerical solutions. Despite using a pressure-based algorithm, the Mach number and density variations are predicted very accurately.