Transport Equation For Temperature Research Articles

Large eddy simulation of particle hydrodynamic characteristics in a dense gas-particle bubbling fluidized bed

A subgrid scale particle kinetic turbulent energy model coupled with a second order moment model is developed to describe the particle hydrodynamic characteristics in dense gas-particle fluidized bed. Anisotropic behaviors of particle dispersions and interactions between gas and particle are fully revealed by means of two-phase Reynolds stresses transport eqs. A four-way coupling strategy for the consideration of particle-particle collisions is modeled by frictional granular temperature transport equation. Simulation results by large eddy simulation are validated by experimental measurement with acceptable agreement. Heterogenous flow structures, anisotropic characteristics of particle dispersions and wider-size distribution behaviors of bubbles are indicated. At the height of 0.2 m, the largest dominant frequency of porosity is 25.0 Hz at lateral 0.375 cm away from center, it is approximately 5.0 times larger than that of lateral 9.375 cm. Compared to 0.45 m, the standard deviations of 82.5 cm/s at lateral region is approximately 1.5 times larger than that of center region.

Flow and thermal fields modeling in jet impingement configurations using a Reynolds stress turbulence closure within the RANS and Sensitized-RANS framework

The flow and thermal field characteristics underlying turbulent convective heat transfer in two reference configurations, representing a jet exiting a planar slot and a circular tube and impinging on a heated wall, are studied computationally. The performance of a conventional Reynolds Stress Model (RSM) and its scale-resolving variant, employed in the context of a time-accurate Reynolds-Averaged Navier–Stokes (RANS) simulation, is investigated in conjunction with different modeling approaches for the turbulent heat flux uj′θ′¯, whose gradient represents the turbulence-induced heat source in the temperature transport equation. In this so-called Sensitized-RANS modeling strategy, employing an improved eddy-resolving instability-sensitive Reynolds Stress Model (IISRSM, Jakirlić and Maduta (2015)), the unresolved ‘sub-scale’ turbulent correlations represent the solutions of a set of corresponding transport equations for the entire stress tensor ui′uj′¯. The turbulent scalar-flux models follow from the classical gradient-diffusion approach with the diffusion-like coefficients formulated as a function of both the scalar eddy-viscosity and the anisotropic Reynolds stress tensor. The latter model expressions also take into account various non-linear relationships in terms of turbulent stress components. The computational results of the fully-developed channel flow, featured only by the mean shear and constant wall heat flux, representing a preliminary investigated flow configuration, agree very well with the corresponding DNS (Direct Numerical Simulation) database for both RSM versions in conjunction with all the turbulent heat flux models adopted. However, the study of the two flow impingement cases with the different structure of the impinging jet and nozzle-to-wall distance variations, characterized by a much more complex flow straining, using the considered conventional Reynolds stress model shows significantly different results for the Reynolds stress and thermal fields compared to the available DNS data, especially in the immediate impact region manifested by a significant overprediction of the turbulence intensity. In contrast, the IISRSM-relevant results for the mean flow and temperature fields, the associated integral characteristics, and the corresponding turbulence quantities are in very good agreement with the reference DNS data, due to the much higher predictive capabilities of this eddy-resolving turbulence model compared to its baseline counterpart.

Modeling the interstitial fluid effect in turbulent disperse slurry flow in a vertical pipe

A numerical study was conducted to assess the performance of a Two-Fluid Model (TFM) developed for gas-solid flow in predicting liquid-solid flow, as well as a TFM specifically modified for liquid-solid flow, which considers the effect of the interstitial fluid. Both cases use the Kinetic Theory of Granular Flow (KTGF) for prediction of the solid phase flow features. Some of the physics involved in gas-solid and liquid-solid flows is intuitively different, and some model terms that can be neglected in a gas-solid formulation turn out to be important for liquid-solid flow. For instance, the much larger density and viscosity of the liquid compared to a gas suggest that the solid particle fluctuations and collisions are reduced. Three intermediate model formulations were developed to assess the relevance of the interstitial terms. The intermediate model formulations, together with the two base case models, were assessed based on their predictions for the case of fully developed, turbulent, steady flow of a liquid solid mixture in a vertical pipe. The main differences in the model formulations pertain to the transport equations for the streamwise liquid momentum, granular temperature and turbulence kinetic energy. The predictions considered such flow features as: the velocity profiles of both the liquid and solid phases, the solids volume fraction profile and budgets of the transport equations for the granular temperature and turbulence kinetic energy. The results obtained were used to identify the most significant model terms. These include modified formulations for the solids viscosity and granular temperature conductive coefficient to include the effects of the interstitial fluid. The single most important term was the model for the long-range particle fluctuations through the fluid, which played a dominant role in the balance of the turbulence kinetic energy and granular temperature transport equations. The present analysis demonstrates that this term should be reconfigured as a sink term in the granular temperature equation and a source term in the turbulence kinetic energy equation. With this modification, the numerical predictions were much closer to the experimental data, especially in terms of the solids volume fraction profile.

Bouncing drop impingement on heated hydrophobic surfaces

We extend a diffuse interface phase-field method for two-phase flow simulations so as to include interfacial heat transfer and the thermal Marangoni effect. The set of governing equations hold non-standard terms, which originally stem from the underlying variational consideration of the total free energy of the two-phase system. It consists of the coupled Cahn-Hilliard Navier-Stokes equations with a temperature-dependent mixing energy term and a temperature transport equation implemented in the OpenFOAM framework. The underlying solver phaseFieldFoam is validated for the test cases of thermocapillary convection in a two-fluid-layer and thermocapillary migration of a drop.In the main part of the paper, hydrodynamics and heat transfer of a droplet impinging on a heated hydrophobic surface with subsequent bouncing are studied in detail. By comprehensive simulations, the effects of impact velocity, droplet diameter (in mm range) and substrate wettability (contact angle) are investigated. The numerical results for spreading ratio, wall contact time and cooling effectiveness are found to compare well with experiments indicating that the developed code is well suited for heat transfer simulation in two-phase flow. For the maximum spreading ratio, a new generalizing correlation is proposed which models the kinetic energy and energy losses at maximum spreading by two coefficients, which are determined from simulation results. A new correlation is also proposed for the contact time, which takes into account surface wettability. For the time evolution of drop mean temperature, a crossover is observed when comparing simulations with constant and temperature-dependent surface tension, indicating that the inclusion of Marangoni effects increases both, the heat transfer between drop and wall during spreading and the heat transfer between drop and surrounding air after rebound.

Uniform global convergence of non-isentropic Euler–Maxwell systems with dissipation

The purpose of this paper is to investigate uniformly global smooth solutions to the Cauchy problem for a non-isentropic Euler–Maxwell system with velocity dissipation and small physical parameters. As the parameters go to zero, we prove that the quasi-neutral limit of the non-isentropic Euler–Maxwell system is the e-MHD system and the linear transport equation for temperature, while its non-relativistic quasi-neutral limit is the incompressible Euler equation and the linear transport equation for temperature. The uniform energy estimates are achieved through controlling the divergence and the curl of the velocity, the electric and magnetic fields.

Turbulent Inflow Generation Through Buoyancy Perturbations with Colored Noise

An inflow generation method for large-eddy simulations of wall-bounded turbulent flows is presented. The method builds upon the work of Muñoz-Esparza et al. (“A Stochastic Perturbation Method to Generate Inflow Turbulence in Large-Eddy Simulation Models: Application to Neutrally Stratified Atmospheric Boundary Layers,” Physics of Fluids, Vol. 27, No. 3, 2015, Paper 035102). In the present adaptation of the original idea, colored noise is generated in the temperature field by adding white noise as source term perturbations in the filtered form of the temperature transport equation. Perturbation boxes introduced within a buffer zone near the inlet encompass multiple computational cells to induce a volumetric buoyancy effect. The amplitudes of the source term perturbations are determined by assigning a bulk Richardson number based on the size of the perturbation box and the height-dependent velocity profile. The temperature field with colored noise is then coupled to the incompressible flowfield within a buffer zone only via the Boussinesq approximation. A turbulent-channel flow and a backward-facing step with a spatially evolving boundary layer upstream of the step are simulated to assess the predictive performance of the inflow generation technique. The results are in good agreement with direct numerical simulation benchmark data, both for the first- and second-order statistics of turbulence. It is also found that the same set of parameters works well in both cases studied.

Regularized thermal lattice Boltzmann method for natural convection with large temperature differences

A new thermal lattice Boltzmann (LB) method is proposed for the simulation of natural convection with large temperature differences and high Rayleigh number. A regularization procedure is developed on LB equation with a third order expansion of equilibrium distribution functions, in which a temperature term is involved to recover the equation of state for perfect gas. A hybrid approach is presented to couple mass conservation equation, momentum conservation equations and temperature evolution equation. A simple and robust non-conservative form of temperature transport equation is adopted and solved by the finite volume method. A comparison study between classical Double Distribution Function (DDF) model and the hybrid finite volume model with different integration schemes is presented to demonstrate both consistency and accuracy of hybrid models. The proposed model is assessed by simulating several test cases, namely the two-dimensional non-Boussinesq natural convection in a square cavity with large horizontal temperature differences and two unsteady natural convection flows in a tall enclosure at high Rayleigh number. The present method can accurately predict both the steady and unsteady non-Boussinesq convection flows with significant heat transfer. For unsteady natural convection, oscillations with chaotic feature can be well captured in large temperature gradient conditions.

Numerical Study of Discharged Heat Water Effect on Aquatic Environment From Coastal Thermal Power Plant by Using Two Water Discharged Pipes

The article presents a numerical study of the thermal load on the aquatic environment by using two water discharge pipes under various operational capacities of thermal power plant. It is solved by the Navier-Stokes and temperature transport equations for an incompressible fluid in a stratified medium. The aim of this article is to improve the existing water discharge system for reduce the heat load on the reservoir-cooler of the thermal power plants operation (Ekibastuz SDPP-1). In this article, thermal pollution using only two water discharge pipes, using the existing one and building only one additional in the eastern part of the reservoir-cooler is numerically simulated. The numerical method is based on the projection method, which was approximated by the finite volume method. The obtained numerical results of three-dimensional stratified turbulent flow for two water discharge pipes under various operational capacities of the thermal power plant were compared with experimental data and with the numerical results for one water discharge pipe.

Numerical Modeling of Disperse Materials Process in a Continuous-Flow Plasma Reactor

The paper presents a numerical simulation of the propagation of the direct-flow temperature plasma reactor, which is solved by the compressible Navier–Stokes equations, numerical algorithm based on SIMPLE algorithm that are approximated by finite volume method. In the numerical solution of the equation system can be divided into four stages. The first stage the transfer of momentum carried out only by convection and diffusion. The intermediate velocity field is solved by the solution of the differential velocity gradient equation, the Green-Gauss Cell Based scheme is used. The second stage for the pressure field, PRESTO numerical scheme is applied. In the third step it is assumed that the transfer is carried out only by the pressure gradient. The fourth step of the equation is solved for the temperature transport equation as well as the momentum equations by the Green-Gauss Cell Based scheme is used. The algorithm is parallelized on high-performance systems. With this numerical algorithm numerical results of temperature distribution in a continuous-flow plasma reactor was obtained. Numerical modeling allows us to give a more precise description of the processes that have been identified or studied theoretically by laboratory methods, and can reveal new physical phenomena processes that are not yet available, seen in experimental studies. Simulation results show that the constructed numerical model provides the necessary accuracy and stability, which should accurately describe the process during the time interval.

Numerical Study of the Discharged Heat Water Effect on the Aquatic Environment from Thermal Power Plant by using Two Water Discharged Pipes

Abstract The paper presents a numerical study of the discharged heat water effect on the aquatic environment from the thermal power plant by using two water discharged pipes. It is solved by the Navier–Stokes and temperature transport equations for an incompressible fluid in a stratified medium. The aim of this study is to improve the existing water discharge system to reduce the heat load on the reservoir-cooler of the thermal power plants operation (Ekibastuz SDPP-1). In this study thermal pollution to the reservoir-cooler using only two water discharged pipes as so using the existing one and building only one additional in the eastern part of the reservoir-cooler is numerically simulated. The numerical method is based on the projection method which was approximated by the finite volume method. The numerical solution of the equation system is divided into four stages. The algorithm is parallelized on a high-performance computer. The obtained numerical results of three-dimensional stratified turbulent flow for two water discharged pipes of the thermal power plant were compared with experimental data and with numerical results for one water discharged pipe. General thermal load in the reservoir-cooler decreases comparing one water discharged pipe and revealed qualitatively and quantitatively approximately the basic laws of hydrothermal processes occurring in the reservoir-cooler can be seen that from numerical simulations where two water discharged pipes were used.

CFD modeling the hydrodynamics of binary particle mixture in pseudo-2D bubbling fluidized bed: Effect of model parameters

The hydrodynamics of binary coal-sand mixture in a pseudo-2D rectangular bubbling fluidized bed (0.385m×0.005m×0.128m) was simulated using the multi-fluid Eulerian model incorporating the kinetic theory of granular flow. Parametric studies of the boundary wall condition, particle-particle restitution coefficient, friction packing limit, as well as transport equation for granular temperature were performed to investigate their influences on the predicted mixing/segregation behavior. The CFD simulation results demonstrated that the predicted mixing behavior was closely related to the expression for granular temperature transport equation and specularity coefficient. When the full transport equation for granular temperature was adopted, the predicted mixing degree decreased with the increase of specularity coefficient. And the best agreement between simulation results and experimental data was achieved when specularity coefficient was equal to 1.0. Nevertheless, when the algebraic transport equation for granular temperature was adopted, the system was always predicted in well-mixing rather than segregation state. Under the full transport equation for granular temperature and the no slip boundary wall condition, the predicted mixing degree decreased with the increase of the particle-particle restitution coefficient and frictional packing limit. The supplementary simulations indicated that for the considered gas-solid system there exist a critical bed thickness larger than which the system was in well-mixing state and the simulation results were independent from the investigated parameters. The hydrodynamic analysis indicated that the reduction of bubble size and the solid axial movement could be the mechanism responsible for the occurrence of axial segregation.

Linear stability and energetics of rotating radial horizontal convection

The effect of rotation on horizontal convection in a cylindrical enclosure is investigated numerically. The thermal forcing is applied radially on the bottom boundary from the coincident axes of rotation and geometric symmetry of the enclosure. First, a spectral element method is used to obtain axisymmetric basic flow solutions to the time-dependent incompressible Navier–Stokes equations coupled via a Boussinesq approximation to a thermal transport equation for temperature. Solutions are obtained primarily at Rayleigh number $\mathit{Ra}=10^{9}$ and rotation parameters up to $Q=60$ (where $Q$ is a non-dimensional ratio between thermal boundary layer thickness and Ekman layer depth) at a fixed Prandtl number $\mathit{Pr}=6.14$ representative of water and enclosure height-to-radius ratio $H/R=0.4$. The axisymmetric solutions are consistently steady state at these parameters, and transition from a regime unaffected by rotation to an intermediate regime occurs at $Q\approx 1$ in which variation in thermal boundary layer thickness and Nusselt number are shown to be governed by a scaling proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an increase in $Q$ sees the flow accumulate available potential energy and more strongly satisfy an inviscid change in potential energy criterion for baroclinic instability. At the strongest $Q$ the flow is dominated by rotation, accumulation of available potential energy ceases and horizontal convection is suppressed. A linear stability analysis reveals several instability mode branches, with dominant wavenumbers typically scaling with $Q$. Analysis of contributing terms of an azimuthally averaged perturbation kinetic energy equation applied to instability eigenmodes reveals that energy production by shear in the axisymmetric mean flow is negligible relative to that produced by conversion of available potential energy from the mean flow. An evolution equation for the quantity that facilitates this exchange, the vertical advective buoyancy flux, reveals that a baroclinic instability mechanism dominates over $5\lesssim Q\lesssim 30$, whereas stronger and weaker rotations are destabilised by vertical thermal gradients in the mean flow.

Direct-Forcing Immersed Boundary Method for Mixed Heat Transfer

AbstractA direct-forcing immersed boundary method (DFIB) with both virtual force and heat source is developed here to solve Navier-Stokes and the associated energy transport equations to study some thermal flow problems caused by a moving rigid solid object within. The key point of this novel numerical method is that the solid object, stationary or moving, is first treated as fluid governed by Navier-Stokes equations for velocity and pressure, and by energy transport equation for temperature in every time step. An additional virtual force term is then introduced on the right hand side of momentum equations in the solid object region to make it act exactly as if it were a solid rigid body immersed in the fluid. Likewise, an additional virtual heat source term is applied to the right hand side of energy equation at the solid object region to maintain the solid object at the prescribed temperature all the time. The current method was validated by some benchmark forced and natural convection problems such as a uniform flow past a heated circular cylinder, and a heated circular cylinder inside a square enclosure. We further demonstrated this method by studying a mixed convection problem involving a heated circular cylinder moving inside a square enclosure. Our current method avoids the otherwise requested dynamic grid generation in traditional method and shows great efficiency in the computation of thermal and flow fields caused by fluid-structure interaction.

Mathematical Modeling of the Discharged Heat Water Effect on the Aquatic Environment from Thermal Power Plant

Abstract The paper presents a mathematical model of the thermal load on the aquatic environment under operational capacity 200 MW of thermal power plant. It is solved by the Navier–Stokes and temperature equations for an incompressible fluid in a stratified medium based on numerical method, the splitting method by physical parameters which approximated the finite volume method. The numerical solution of the equation system is divided into four stages. At the first step it is assumed that the momentum transfer is carried out only by convection and diffusion. Intermediate velocity field is solved by the five-step Runge–Kutta method. At the second stage, the pressure field is solved by the intermediate velocity field. Poisson equation for the pressure field is solved by Jacobi method. The third step is assumed that the transfer is carried out only by pressure gradient. The fourth step of the transport equation for temperature is also solved as momentum equations, with five-step Runge–Kutta method. The obtained numerical results of temperature distribution for operational capacity of 200 MW of three-dimensional stratified turbulent flow were compared with experimental data, which revealed qualitatively and quantitatively approximately the basic laws of hydrothermal processes occurring in the reservoir-cooler.

Principal interval decomposition framework for POD reduced‐order modeling of convective Boussinesq flows

SummaryA principal interval decomposition (PID) approach is presented for the reduced‐order modeling of unsteady Boussinesq equations. The PID method optimizes the lengths of the time windows over which proper orthogonal decomposition (POD) is performed and can be highly effective in building reduced‐order models for convective problems. The performance of these POD models with and without using the PID approach is investigated by applying these methods to the unsteady lock‐exchange flow problem. This benchmark problem exhibits a strong shear flow induced by a temperature jump and results in the Kelvin–Helmholtz instability. This problem is considered a challenging benchmark problem for the development of reduced‐order models. The reference solutions are obtained by direct numerical simulations of the vorticity and temperature transport equations using a compact fourth‐order‐accurate scheme. We compare the accuracy of reduced‐order models developed with different numbers of POD basis functions and different numbers of principal intervals. A linear interpolation model is constructed to obtain basis functions when varying physical parameters. The predictive performance of our models is then analyzed over a wide range of Reynolds numbers. It is shown that the PID approach provides a significant improvement in accuracy over the standard Galerkin POD reduced‐order model. This numerical assessment of the PID shows that it may represent a reliable model reduction tool for convection‐dominated, unsteady‐flow problems. Copyright © 2015 John Wiley & Sons, Ltd.

Higher-dimensional charged shear-free relativistic models with heat flux

We analyse shear-free spherically symmetric relativistic models of gravitating fluids with heat flow and electric charge defined on higher dimensional manifolds. The solution to the Einstein-Maxwell system is governed by the pressure isotropy condition which depends on the spacetime dimension. We study this highly nonlinear partial differential equation using Lie's group theoretic approach. The Lie symmetry generators that leave the equation invariant are determined. We provide exact solutions to the gravitational potentials using the first symmetry admitted by the equation. Our new exact solutions contain the earlier results for the four-dimensional case. Using the other Lie generators, we are able to provide solutions to the gravitational potentials or reduce the order of the master equation to a first order nonlinear differential equation. We derive the temperature transport equation in higher dimensions and find expressions for the causal and Eckart temperatures showing their explicit dependance on the dimension. We analyse a particular solution, obtained via group techniques, to show its physical applicability.

An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation

We propose a semi-Lagrangian numerical algorithm for a time-dependent, anisotropic temperature transport equation in magnetized plasmas in regimes with negligible variation of the magnitude of the magnetic field B along field lines. The approach is based on a formal integral solution of the parallel (i.e., along the magnetic field) transport equation with sources. While this study focuses on a Braginskii (local) heat flux closure, the approach is able to accommodate nonlocal parallel heat flux closures as well. The numerical implementation is based on an operator-split formulation, with two straightforward steps: a perpendicular transport step (including sources), and a Lagrangian (field-line integral) parallel transport step. Algorithmically, the first step is amenable to the use of modern iterative methods, while the second step has a fixed cost per degree of freedom (and is therefore algorithmically scalable). Accuracy-wise, the approach is free from the numerical pollution introduced by the discrete parallel transport term when the perpendicular to parallel transport coefficient ratio χ⊥/χ∥ becomes arbitrarily small, and is shown to capture the correct limiting solution when ϵ=χ⊥L∥2χ∥L⊥2→0 (with L∥, L⊥ the parallel and perpendicular diffusion length scales, respectively). Therefore, the approach is asymptotic-preserving. We demonstrate the performance of the scheme with several numerical experiments with varying magnetic field complexity in two dimensions, including the case of heat transport across a magnetic island in cylindrical geometry in the presence of a large guide field.

Effects of electrode contact condition on electrical dynamic resistance during resistance spot welding

This study systematically investigates the effects of electrical resistance at the workpiece/electrode interface or electrode face on temperature dependent dynamic resistance during resistance spot welding (RSW). To evaluate temperature transport equations of mass, momentum, energy, species and magnetic field intensity in workpieces, the energy and magnetic equations in the electrode are solved. Contact resistances composed of constriction and film resistances are functions of hardness, temperature, electrode force and surface conditions. The results show that dynamic resistance is complicated due to different variations of film and constriction resistances with temperature at not only the faying surface but also the electrode face in the early stage, i.e. shorter than around 3 cycles. Dynamic resistance in this stage is fortunately insignificant to transport processes. When the power is off, dynamic resistance depends on competition between decreased bulk resistance and increased constriction resistance at the electrode face. Decreased constriction resistance at the electrode face reduces dynamic resistance and temperature.

Global well-posedness of the stochastic 2D Boussinesq equations with partial viscosity

This paper deals with the stochastic 2D Boussinesq equations with partial viscosity. This is a coupled system of Navier-Stokes/Euler equations and the transport equation for temperature under additive noise. Global well-posedness result of this system under partial viscosity is proved by using classical energy estimates method.

A finite element model for the ocean circulation driven by wind and atmospheric heat flux

A finite element model for solving ocean circulation forced by winds and atmospheric heat fluxes is presented. It is vertically integrated 1 1/2 layer model which solves the motion and continuity hydrodynamic equations coupled with the advection-diffusion transport equation for temperature. A space-time Petrov-Galerkin formulation is used to minimize the undesirable numerical spurious oscillation effects of unresolved boundary layers solution of classical Galerkin method. The model is employed to simulate the circulation of the eastern South Pacific Ocean represented by a mesh covering the area between the equator and 30oS and between the 70oW to 100oW. Monthly climatological data are used to determine the wind and heat fluxes forcing functions of the model. The model simulates the main features of observed sea surface temperature (SST) pattern during the summer months showing the upwelling along the coastal boundary with currents oriented northwestward and the presence of southward flows and warm water intrusion in offshore side. The calculated SST fields are compared with the mean observed SST showing that the coastal processes and the interaction with the equatorial band are physically better resolved.