Thermal Wave Front Research Articles

Dynamic response of two-dimensional generalized thermoelastic coupling problem subjected to a moving heat source

Based on the Lord and Shulman generalized thermoelasticity theory with one relaxation time, an isotropic semi-infinite plate subjected to a moving heat source has been studied by employing the finite element method directly in time domain, whose distributions of nondimensional temperature, displacement and stress are illustrated graphically. The results show that the present method is an effective and exact numerical one for solving the thermoelastic coupling problem and is capable of overcoming the defects of traditional integrated transformation and inverse integrated transformation methods. At the same time, the temperature step of the thermal wave front is obtained exactly in contrast with conventional numerical transformation methods.

Study on Generalized Thermoelastic Problem of Semi-infinite Plate Heated Locally by the Pulse Laser

A generalized thermoelastic coupled problem for the semi‐ infinite plane induced by pulsed laser heating locally is studied by adopting L‐S generalized thermoelasticity. In order to avoid the loss of the precision in general integral transformation method, the finite element control equations are solved directly in the time domain. Temperatures and displacements distributions are obtained and represented graphically under pulsed laser heating in the semi‐infinite plane. The results show that the maximum temperature on the structure always locates near the thermal wave front and reduces gradually alone with time evolution.

Analysis of non-Fourier conduction and volumetric radiation in a concentric spherical shell using lattice Boltzmann method and finite volume method

Application of the lattice Boltzmann method (LBM) has been extended to formulate and solve the energy equation of a non-Fourier conduction and radiation heat transfer problem in a concentric spherical shell. The enclosed conducting-radiating medium is absorbing, emitting and scattering. The non-Fourier conduction effect is induced by thermally perturbing one of the boundaries and incorporating the finite propagation speed of the thermal wave front in Fourier’s law of heat conduction. The volumetric radiative information needed in the energy equation has been computed using the finite volume method (FVM). To establish the accuracy of the LBM approach, with volumetric radiative information obtained from the FVM, the energy equation is also solved using the FVM. Effects of extinction coefficient, scattering albedo, conduction–radiation parameter, emissivity, radius ratio and the magnitude of thermal perturbations are studied on transient temperature distributions. Effects of the aforesaid parameters on the steady-state conduction, radiation and total energy flow rates are also studied. Steady-state LBM and FVM results are compared. In all the cases, the LBM results compare exceedingly well with the FVM results, and LBM has a faster convergence than the FVM.

Generalized thermoelastic solutions for the axisymmetric plane strain problem

Thermal disturbance propagates in the elastic medium with a finite speed during the transient heat transfer, which leads to the thermal-mechanical behavior of materials being significantly different than that of conventional heat transfer. The axisymmetric plane strain problem is investigated in this paper, the generalized thermoelastic solutions for different generalized thermoelasticity are derived by means of the Laplace transform and its limit theorem and the asymptotic formulas of Bessel functions. A case of an infinite axisymmetric medium with a cylindrical hole under thermal shock is analyzed. It is pointed out that each of physical field has a phased distribution when the propagation of thermal disturbance with a finite speed, and both the temperature and stresses have jumps at the location of thermal elastic wavefront and thermal wavefront. The L-S and G-N theories predict similar patterns for the distributions of displacement, temperature and stresses in the transient thermal shock problem, but for G-L theory a Dirac-delta-type result for the stresses is predicted and the displacement is discontinuous at both the wavefronts.

Analysis of thermal effect and beam wavefront properties for LD-pumped Q-switch Nd:YAG laser

Thermal effects and beam wavefront properties of a diode-pumped Q-switch Nd:YAG laser have been investigated. The influence of thermal lens effect on laser output mode and beam quality is discussed by means of transmission matrix optics. The functional relations between the thermal lens focal length and the beam spot size for a plano-concave stable resonator have been calculated numerically. Further experiment on the output beam quality of a laser diode pumped acousto-optic Q-switched laser has been also performed by using a Hartmann–Shack (H–S) wavefront sensor and Zernike mode reconstruction theory. In this way, the PV and RMS values of the wavefront aberration, the first 35 orders Zernike aberrations, can be acquired, and point spread function (PSF) of the beam and the distribution of circle energy can also be obtained by calculation, so the output beam quality can be known fully. The experimental results show that the wavefront aberration of laser is mainly includes the defocus Z3, the low-order astigmatism Z4 and Z5, the spherical aberration Z10, etc. because of the thermal effect of gain medium, which will lay an experimental foundation for solid-state laser optimum design.

Thermal Modeling and Analysis of a Thermal Barrier Coating Structure Using Non-Fourier Heat Conduction

This paper presents a numerical solution of the hyperbolic heat conduction equation in a thermal barrier coating (TBC) structure under an imposed heat flux on the exterior of the TBC. The non-Fourier heat conduction equation is used to model the heat conduction in the TBC system that predicts the heat flux and the temperature distribution. This study presents a more realistic approach to evaluate in-service performance of thin layers of TBCs typically found in hot sections of land based and aircraft gas turbine engines. In such ultrafast heat conduction systems, the orders of magnitude of the time and space dimensions are extremely short which renders the traditional Fourier conduction law, with its implicit assumption of infinite speed of thermal propagation, inaccurate. There is, therefore, the need for an advanced modeling approach for the thermal transport phenomenon taking place in microscale systems. A hyperbolic heat conduction model can be used to predict accurately the transient temperature distribution of thermal barrier structures of turbine blades. The hyperbolic heat conduction equations are solved numerically using a new numerical scheme codenamed the mean value finite volume method (MVFVM). The numerical method yields minimal numerical dissipation and dispersion errors and captures the discontinuities such as the thermal wave front in the solution with reliable accuracy. Compared with some traditional numerical methods, the MVFVM method provides the ability to model the behavior of the single phase lag thermal wave following its reflection from domain boundary surfaces. In addition, parametric studies of properties of the substrate on the temperature and the heat flux distributions in the TBC revealed that relaxation time of the substrate material, unlike the thermal diffusivity and thermal conductivity has very little effect on the transient thermal response in the TBC. The study further showed that for thin film structures subject to short time durations of heat flux, the hyperbolic model yields more realistic results than the parabolic model.

Lattice Boltzmann Method Applied to the Solution of Energy Equation of a Radiation and Non-Fourier Heat Conduction Problem

This article concerns the application of the lattice Boltzmann method (LBM) to solve the energy equation of a combined radiation and non-Fourier conduction heat transfer problem. The finite propagation speed of the thermal wave front is accounted by non-Fourier heat conduction equation. The governing energy equation is solved using the LBM. The finite-volume method (FVM) is used to compute the radiative information. The formulation is validated by taking test cases in 1-D planar absorbing, emitting, and scattering medium whose west boundary experiences a sudden rise in temperature, or, with adiabatic boundaries, the medium is subjected to a sudden localized energy source. Results are analyzed for the various values of parameters like the extinction coefficient, the scattering albedo, the conduction-radiation parameter, etc., on temperature distributions in the medium. Radiation has been found to help in facilitating faster distribution of energy in the medium. Unlike Fourier conduction, wave fronts have been found to reflect from the boundaries. The LBM-FVM combination has been found to provide accurate results.

Hyperbolic Thermoelastic Analysis due to Pulsed Heat Input by Numerical Simulation

Thermo-mechanical behavior in a rod subjected to a pulsed heat input was investigated by numerical simulation using the hyperbolic thermo-elasticity theory derived from the thermal dynamics in the present paper. Unlike the classical thermo-elastic theory with the parabolic energy equation and the hyperbolic motion equation, temperature response and thermal stress due to the temperature change exhibit significant wavy characteristics in the hyperbolic thermo-elasticity theory which is based on the non-Fourier heat conduction. The whole region of the rod is split into the heat disturbed region and the heat undisturbed region by the thermal wave front which is determined by the propagating velocity of the heat wave. The heat wave and elastic wave travel in the body at a finite velocity and reflect at the end of the rod. Thermal shock due to the discontinuous jump in thermal condition, and the reflection of thermal stress at the end terminate of the rod are significant during the heating process.

Simulation of ballistic and non-Fourier thermal transport in ultra-fast laser heating

In this work, the lattice Boltzmann method (LBM) is developed to simulate pico- and femtosecond laser heating of silicon. The temperature fields calculated by the LBM are compared with those obtained from the parabolic heat conduction equation (PHCE) and the hyperbolic heat conduction equation (HHCE). Although the HHCE overcomes the dilemma of infinite thermal propagation speed of the PHCE, it cannot be applied to length scales comparable to the mean free path of energy carriers because of the breakdown of continuum approaches under severe nonequilibrium conditions. The LBM, considering both effects, can be used in both short temporal and spatial scales. From the results of the LBM, it is found that the speed of thermal wave at the ballistic limit is equal to the speed of sound, instead of the value predicted by the HHCE, which is valid only in the diffuse limit. It is also demonstrated that the traditional way of calculating heat flux using the temperature gradient gives rise to physically unreasonable results at the thermal wave front, while the LBM has no such drawback.

A new numerical technique of electric field determination within dielectric materials plate and cable using the TSM method

In the frame of the thermal step method (TSM) used to characterize the space charge in dielectric materials, we present an original numerical technique for determining the electric field distribution in the bulk of a dielectric plate or cable. The first stage of our technique is the application of the finite element method (FEM) in order to find instantaneous distributions of temperature profiles. The calculation of the electric field distribution is based on these obtained profiles. During the stage of the determination, our mathematical treatment is based on the TSM charge q(t) in order to avoid numerical instabilities on the temperature derivatives. Therefore, we have transformed the integral equation for the TSM current I(t) used in previous works to an integral equation for the TSM charge q(t). The control of the instantaneous propagation of the thermal wave front, produced by submitting one face of the dielectric to a thermal step, and the application of Simpson's method to the integral equation of the TSM charge q(t), allow to determine the electric field distribution. The results of the electric field distribution are validated with those obtained in previous works. A good agreement and an improvement near the dielectric thickness boundaries are observed on these results. The numerical space charge density within the material is obtained by numerical derivation of the field according to Poisson's equation.

Characteristic surfaces in gas flows

It is shown that weak discontinuities of three types may occur in flows of an inviscid heat-conducting gas: on sonic lines, on contact surfaces and on a thermal wave front. Weak discontinuities may occur in flows of a viscous non-heat-conducting gas, but only of one type — contact weak discontinuities.

Several-temperature systems: extended irreversible thermodynamics and thermal wavefront propagation

We develop an extension of local-equilibrium thermodynamics for systems with components at different temperatures. The description includes heat conduction, diffusion, a chemical reaction and heat losses. The ensuing equations are applied to the problem of thermal wavefront propagation. There is satisfactory agreement between the predicted and the experimental values for the speed of combustion flames over cellulosic fuels. Heat losses cause a decrease of the front speed. This generalizes Zeldovich's result. For radiative losses, the effect can easily be about 15%. When the finite speed of thermal signals is taken into account, an additional lowering of the thermal wavefront speed results.

Two-dimensional nonlinear heat conduction wave in a layer-inhomogeneous medium and the characteristics of heat transfer in laser thermonuclear fusion targets

An analytical solution is obtained to the problem of propagation of a 2-D nonlinear heat conduction wave from a cylindrical energy source, which acts in a planar layer of a material surrounded by a medium with different mass density and degree of ionisation. A theoretical justification is given of several interesting phenomena of 2-D thermal wave propagation through an inhomogeneous medium. These phenomena are related to the difference between the thermal wave velocities in the media with different thermal diffusivities. When the mass density in a layer experiencing the action of an energy source exceeds the density of the surrounding medium, the thermal wave front is shown to glide along the layer boundaries with a spatial velocity exceeding the velocity of the wave inside the layer. Moreover, there is a possibility of 'themal flow' of a layer across the boundaries between the layer and the surrounding medium in front of a thermal wave propagating inside the layer. The problems of heat transfer in multilayer targets for laser thermonuclear fusion are considered as an application.

PROPAGATION AND REFLECTION OF THERMAL WAVES IN A RECTANGULAR PLATE

The wave nature of heat propagation in a two-dimensional rectangular plate with an instantaneous thermal disturbance released in an arbitrary position is investigated by solving the hyperbolic heat conduction equation. The exact analytical solutions are developed for the temperature field and heat flux using the Green's function technique to deal with two limiting boundary conditions, the constant wall temperature and the adiabatic condition, around the region. The disturbance gives rise to a severe thermal wave front, which differs completely from that obtained through one-dimensional analysis, traveling through the medium at a finite speed with a sharp peak at the leading edge. The significant findings in these results are that a negative trailer is generated and follows behind the wave front. In addition, the magnitude of the front is significantly attenuated from the side adjacent to the trailer because the increasing area available to it for diffusion, and decays exponentially along its path of travel, since the thermal energy is dissipated in the wake of the moving wave front. The results also reveal that different boundary conditions strongly influence the reflection of a thermal wave front from the exterior surfaces and the reflection and interaction among thermal waves are more complicated than those found through one-dimensional analysis.

Redistribution of Energy in Dense Matter Irradiated by ShortIntense Laser Pulses

The absorption and redistribution of the energy of short intenselaser pulses in solid targets are studied by means of 1-D Lagrangiancomputer code. It is shown the considerable influence of the plasmaexpansion and ionization phenomena on the laser energy absorption andheating of a target. Nonlocal heat conductivity is taken into accountby means of the Epperlein–Short model. It is demonstrated thatthe nonlocality of heat transport does not change considerably thelaser energy deposition into the target and plasma temperature, butleads to preheating of the plasma ahead of the thermal wave front andproduces a more concave profile of the heating wave.

Flashlamp-Pumped ND:Glass Amplifiers for the National Ignition Facility

This paper describes the design and performance of flashlamp-pumped, Nd:glass. Brewster-angle slab amplifiers intended to be deployed in the National Ignition Facility (NIF). To verify performance, we tested a full-size, three-slab-long, NIF prototype amplifier, which we believe to be the largest flashlamp-pumped Nd:glass amplifier ever assembled. Like the NIF amplifier design, this prototype amplifier had eight 40-cm-square apertures combined in a four-aperture-high by two-aperture-wide matrix. Specially-shaped reflectors, anti-reflective coatings on the blastshields, and preionized flashlamps were used to increase storage efficiency. Cooling gas was flowed over the flashlamps to remove waste pump heat and to accelerate thermal wavefront recovery.The protoytpe gain results are consistent with model predictions and provide high confidence in the final engineering design of the NIF amplifiers. Although the dimensions, internal positions, and shapes of the components in the NIF amplifiers will be slightly different from the prototype, these differences are small and should produce only slight differences in amplifier performance.

A GENERALIZED THERMO-VISCO-ELASTIC PROBLEM OF AN INFINITELY EXTENDED THIN PLATE CONTAINING A CIRCULAR HOLE

This article is concerned with the investigation of a two-dimensional thermo-visco-elastic problem of an infinitely extended thin plate containing a circular hole. The material is a viscoelastic solid of Kelvin-Voigt type. The formulation of the problem is based on the generalized theory of thermoelasticity proposed by Lord and Shulman. The inner boundary of the hole in the plate is stressfree and is subjected to a step rise in temperature. Laplace transform technique is applied to obtain the short-time approximations for the distribution of displacement, temperature, and stresses. It is observed that the deformation and the radial stress are continuous at the thermal wave front, while the temperature and the hoop stress suffer discontinuities at the thermal wave front. The discontinuities at the wave front of various fields have also been analyzed. Using PC the numerical values of the displacement, temperature, and stresses at different points are computed numerically and the results are displayed graphically for an appropriate material for illustration.

An analytical study on the fast-transient process in small scales

The energy equation accounting for the microstructural interaction effect in the fast-transient process displays a new-type of differential equation in engineering science. In addition to a wave term reflecting the fast-transient effect, the equation contains a mixed-derivative term reflecting the microscopic interaction effect. Employing the method of Laplace transform, the present work derives the analytical expressions for temperature by performing the Bromwich contour integrations. The contour integral for the classical thermal wave equation is also derived for comparison to that obtained earlier under very special conditions. Special features in this work include the direct extraction of the location of the thermal wavefront from the Laplace transform solution by the method of partial expansions.

Thermal wave behavior in high-speed penetration

Temperature rise in the local area adjacent to the nose of a penetrator is studied in an uncoupled approach. The high-rate of change in temperature in penetration is modelled by the thermal wave model. For steel penetrating into an aluminum target, the cavity-expansion model is used to estimate the particle velocity and the radial stress at the wall. Both effects of heat generation by high-speed friction and intensified plasticity are included. The thermal wavefront in relation to the particle velocity is specially emphasized. Depending on the penetration speed, it has been found that the local temperature rise in the aluminum target ranges from several hundred to a thousand degrees over the ambient. The material boundary layer in which such a temperature rise occurred is on the order of microns.

On the global behaviour of weak discontinuities in radiation gasdynamics

The nature of weak gasdynamic discontinuities has been studied by considering the interaction between the time dependent gasdynamic and the radiation fields. The effects of radiative transfer have been investigated using a quasi-equilibrium and quasi-isotropic hypothesis of a differential approximation which is valid over the entire optical depth range from the transparent limit to the optically thick limit. It is shown that the existence of the time dependent radiation field gives rise to a radiation induced wave which has a negligibly small effect on the non-relativistic flow properties of the gasdynamic field. The effect of the thermal radiation and initial wave front curvature on the non-linear breaking of weak discontinuities are discussed.