Mechanism of unsteady aerodynamic heating with sudden change in surface temperature

The characteristics and mechanism of unsteady aerodynamic heating of a transient hypersonic boundary layer caused by a sudden change in surface temperature are studied. The complete time history of wall heat flux is presented with both analytical and numerical approaches. With the analytical method, the unsteady compressible boundary layer equation is solved. In the neighborhood of the initial and final steady states, the transient responses can be expressed with a steady-state solution plus a perturbation series. By combining these two solutions, a complete solution in the entire time domain is achieved. In the region in which the analytical approach is applicable, numerical results are in good agreement with the analytical results, showing reliability of the methods. The result shows two distinct features of the unsteady response. In a short period just after a sudden increase in the wall temperature, the direction of the wall heat flux is reverted, and a new inflexion near the wall occurs in the profile of the thermal boundary layer. This is a typical unsteady characteristic. However, these unsteady responses only exist in a very short period in hypersonic flows, meaning that, in a long-term aerodynamic heating process considering only unsteady surface temperature, the unsteady characteristics of the flow can be ignored, and the traditional quasi-steady aerodynamic heating prediction methods are still valid.     

Controlled random search technique for estimation of convective heat transfer coefficient

This paper is concerned with a method for solving inverse heat conduction problem. The method is based on the controlled random search (CRS) technique in conjunction with modified Newton–Raphson method. The random search procedure does not need the computation of derivative of the function to be evaluated. Therefore, it is independent of the calculation of the sensitivity coefficient for nonlinear parameter estimation. The algorithm does not depend on the future-temperature information and can predict convective heat transfer coefficient with random errors in the input temperature data. The technique is first validated against an analytical solution of heat conduction equation for a typical rocket nozzle. Comparison with an earlier analysis of inverse heat conduction problem of a similar experiment shows that the present method provides solutions, which are fully consistent with the earlier results. Once validated, the technique is used to investigate another estimation of heat transfer coefficient for an experiment of short duration, high heating rate, and employing indepth temperature measurement. The CRS procedure, in conjunction with modified Newton–Raphson method, is quite useful in estimating the value of the convective heat-transfer coefficient from the measured transient temperature data on the outer surface or imbedded thermocouple inside the rocket nozzle. Some practical examples are illustrated, which demonstrate the stability and accuracy of the method to predict the surface heat flux.     

Application of a second-moment closure model to simulate the turbulent dispersion from an elevated line source

In this study, a second-moment closure model and an algebraic stress and flux model are used to simulate the turbulent dispersion from an elevated line source located in a turbulent boundary layer. In the modeling of the Reynolds stress equation, several models for the triple velocity correlation are examined by comparing their predicting results with the direct numerical simulation data. Predicted mean and fluctuating correlation quantities of velocity and temperature by the second-moment closure model and the algebraic model are compared with the experimental data. From the comparisons, it is found that the second-moment closure is capable of reproducing the experimental results satisfactorily. However, predictions of the algebraic stress and flux model are quite poor. Received on 12 June 1998     

Quasiequilibrium Knudsen boundary layer on a nonisothermal porous body

The state of a gas near a permeable nonisothermal body with ultrathin pores, that is, pores in which the motion of molecules is not accompanied by intermolecular collisions, is studied. A boundary layer of a new type, namely, the quasiequilibrium Knudsen boundary layer on the porous body surface, is investigated. It is formed on condition that within the body there is a heat flux, even when the body is in an equilibrium gas. The statistical methods for solving the Boltzmann equation are used to determine the temperature and pressure jumps across the Knudsen layer near thin perforated and porous membranes.     

DNS,LES and RANS of turbulent heat transfer in boundary layer with suddenly changing wall thermal conditions

The objectives of this study are to investigate a thermal field in a turbulent boundary layer with suddenly changing wall thermal conditions by means of direct numerical simulation (DNS), and to evaluate predictions of a turbulence model in such a thermal field, in which DNS of spatially developing boundary layers with heat transfer can be conducted using the generation of turbulent inflow data as a method. In this study, two types of wall thermal condition are investigated using DNS and predicted by large eddy simulation (LES) and Reynolds-averaged Navier–Stokes equation simulation (RANS). In the first case, the velocity boundary layer only develops in the entrance of simulation, and the flat plate is heated from the halfway point, i.e., the adiabatic wall condition is adopted in the entrance, and the entrance region of thermal field in turbulence is simulated. Then, the thermal boundary layer develops along a constant temperature wall followed by adiabatic wall. In the second case, velocity and thermal boundary layers simultaneously develop, and the wall thermal condition is changed from a constant temperature to an adiabatic wall in the downstream region. DNS results clearly show the statistics and structure of turbulent heat transfer in a constant temperature wall followed by an adiabatic wall. In the first case, the entrance region of thermal field in turbulence can be also observed. Thus, both the development and the entrance regions in thermal fields can be explored, and the effects upstream of the thermal field on the adiabatic region are investigated. On the other hand, evaluations of predictions by LES and RANS are conducted using DNS results. The predictions of both LES and RANS almost agree with the DNS results in both cases, but the predicted temperature variances near the wall by RANS give different results as compared with DNS. This is because the dissipation rate of temperature variance is difficult to predict by the present RANS, which is found by the evaluation using DNS results.     

Experimental characterization of non-premixed turbulent jet propane flames

This paper reports an experimental study conducted on turbulent jet propane flames aiming at further understanding of turbulent structure in non-premixed slow-chemistry combustion systems. Measurements of mean and fluctuating velocity and temperature fields, mean concentration of major chemical species, correlation between velocity and temperature fluctuations, and dissipation of temperature fluctuations are reported in a turbulent round jet non-premixed propane flame, Re=20 400 and 37 600, issuing vertically in still air. The experimental conditions were designed to provide a complete definition of the upstream boundary conditions in the measurement domain for the purpose of validating computational models. The measured data depicts useful flow field information for describing turbulent non-premixed slow-chemistry flames. Velocity–temperature correlation measurements show turbulent heat fluxes tended to be restricted to the mixing layer where large temperature gradients occurred. Observations of non-gradient diffusion of heat at x/D=10 were verified. Temperature fluctuation dissipation, χ, showed the highest values in the shear layer, where the variance of temperature fluctuations was maximum and combustion occurred. The isotropy between the temperature dissipation in the radial and tangential directions was confirmed. By contrast, the observed anisotropy between axial and radial directions of dissipation suggests the influence of large structures in the entrainment shear layer on the production of temperature fluctuations in the flame region. The value of the normalized scalar dissipation at the stoichiometric mixture fraction surface, χ_st, was calculated, and ranges between 2 and 4 s⁻¹. The measured data were used to estimate the budgets in the balance equations for turbulent kinetic energy, Reynolds shear stresses, turbulent heat flux and temperature variance, quantifying the mechanisms involved in the generation of turbulence as well as in the transport of the temperature.     

Comparison of different kinetic models for the heat transfer problem

The steady-state heat transfer problem between two parallel plates is investigated using kinetic models of the Boltzmann equation: BGK, S-model and ES-model. The discrete velocity method is used to determine the values of physical parameters: density, bulk velocity, temperature and heat flux. The obtained results are compared with the analytical expressions and some experimental data.     

Radiative heat flux characteristics of methane flames in oxy-fuel atmospheres

Oxy-fuel combustion is a promising alternative for power generation with CO₂ capture, where the fuel is burned in an atmosphere enriched with oxygen and CO₂ is used as a diluent. This type of combustion is characterised by uncommon characteristics in terms of thermal heat transfer budget as compared to air supported systems. The study presents experimental results of radiative heat flux along the flame axis and radiant fractions of non-premixed jet methane flames developing in oxy-fuel environments with oxygen concentrations ranging from 35% to 70%, as well as in air. The flames investigated have inlet Reynolds numbers from 468 to 2340. The data collected have highlighted the effects of the flame structure and thermo-chemical properties of oxy-fuel combustion on the heat flux radiated by the flames. It was first observed that peak heat flux increases considerably with oxygen concentration. More generally the radiant fraction increases with both increasing Reynolds number in the laminar regime and oxygen concentration. It was found that despite a difference in flame temperature, the radiative characteristics of the flames (heat flux distributions and radiant fraction) in air were similar to those with 35% O₂ in CO₂. The radiative properties of flames in oxy-fuel atmosphere with CO₂ as diluents appear to be dominated by the flame temperature.     

Exact solution of conductive heat transfer in cylindrical composite laminate

This paper presents an exact solution for steady-state conduction heat transfer in cylindrical composite laminates. This laminate is cylindrical shape and in each lamina, fibers have been wound around the cylinder. In this article heat transfer in composite laminates is being investigated, by using separation of variables method and an analytical relation for temperature distribution in these laminates has been obtained under specific boundary conditions. Also Fourier coefficients in each layer obtain by solving set of equations that related to thermal boundary layer conditions at inside and outside of the cylinder also thermal continuity and heat flux continuity between each layer is considered. In this research LU factorization method has been used to solve the set of equations.     

Prediction of periodic boundary layers

A relatively simple, yet efficient and accurate finite difference method is developed for the solution of the unsteady boundary layer equations for both laminar and turbulent flows. The numerical procedure is subjected to rigorous validation tests in the laminar case, comparing its predictions with exact analytical solutions, asymptotic solutions, and/or experimental results. Calculations of periodic laminar boundary layers are performed from low to very high oscillation frequencies, for small and large amplitudes, for zero as well as adverse time-mean pressure gradients, and even in the presence of significant flow reversal. The numerical method is then applied to predict a relatively simple experimental periodic turbulent boundary layer, using two well-known quasi-steady closure models. The predictions are shown to be in good agreement with the measurements, thereby demonstrating the suitability of the present numerical scheme for handling periodic turbulent boundary layers. The method is thus a useful tool for the further development of turbulence models for more complex unsteady flows.     

COUPLING BETWEEN LAMINAR FILM CONDENSATION AND NATURAL CONVECTION ON OPPOSITE SIDES OF A VERTICAL PLATE

Theoretical analyses which incorporate one-dimensional heat conduction along a plate and transverse heat conduction approximations are presented to predict the net heat transfer between laminar film condensation of a saturated vapour on one side of a vertical plate and boundary layer natural convection on the other side. It is assumed that countercurrent boundary layer flows are formed on the two sides. The governing boundary layer equations of this problem and their corresponding boundary conditions are all cast into dimensionless forms by using a non-similarity transformation. Thus the resulting system of equations can be solved by using the local non-similarity method for the boundary layer equations and a finite difference method for the heat conduction equation of the plate. The plate temperature and the heat flux through the plate are repetitively determined until the solutions for each side of the plate match. The predicted results show that the effect of Pr_c is not negligible for larger values of A* (thermal resistance ratio between natural convecti on side and condensing film side) and the approximation of transverse heat conduction overpredicts the plate temperature for lower values of R_t (thermal resistance ratio between plate and condensing film). However, no significant differences are observed between the two different approximations for higher values of R_t. © by 1997 John Wiley & Sons, Ltd.     

Large eddy simulation in Code_Saturne of thermal mixing in a T junction with brass walls

Following on from Kuhn et al (2010) we study the capability of large eddy simulation with conjugate heat transfer to predict thermal fluctuations with thermal mixing. Wall functions are used to model the wall heat transfer. Comparison with experimental results show that the temperature variance on the outer skin of the solid is well predicted by the simulation. It is shown that the variance of thermal flux in the fluid closely maps the temperature variance at the outer boundary of the solid. Since the variance of thermal flux is closely related to the dissipation of temperature variance it can be concluded that the dissipation of temperature variance in the fluid is linked to temperature variance in the solid. Analysis of the equation of the temperature variance in the solid confirms this is indeed the case. It is the dissipation of temperature variance in the fluid that characterizes how the temperature variance penetrates the solid. Thus RANS modelling can be used to predict thermal variance in solids provided that there is an accurate model for the dissipation of temperature variance at the wall and an equation for the thermal variance in the solid is solved.     

Analytical study of heat transfer from circular cylinder in liquid metals

In this study the influence of a thin hydrodynamic boundary layer on the heat transfer from a single circular cylinder in liquid metals having low Prandtl number (0.004–0.03) is investigated under isothermal and isoflux boundary conditions. Two separate analytical heat transfer models, viscous and inviscid, are developed to clarify the discrepancy between previous results. For both models, integral approach of the boundary layer analysis is employed to derive closed form expressions for the calculation of the average heat transfer coefficients. For an inviscid model, the energy equation is solved using potential flow velocity only whereas for a viscous model, a fourth-order velocity profile is used in the hydrodynamic boundary layer and potential flow velocity is used outside the boundary layer. The third-order temperature profile is used inside the thermal boundary layer for both models. It is shown that the inviscid model gives higher heat transfer coefficients whereas viscous flow model gives heat transfer results in a fairly good agreement with the previous experimental/numerical results.     

Turbulent boundary layer heat transfer for a constant property particle-laden gas flow

Solution of a turbulent boundary layer for a constant property, particle-laden gas flow is obtained by a differential method. A dimensionless analysis shows importance of an interaction parameter in increasing heat flux. Boundary layer analysis is done in usual manner by transforming partial differential equations and solution is started at the leading edge by Runge-Kutta method. Velocity and temperature profiles at downstream planes for gas and particles are calculated by an implicit finite-difference iterative procedure, and numerical results are compared with available experimental data.     

Thermoelastic contact instability of a functionally graded layer and a homogeneous half-plane

The conductive heat transfer between two elastic bodies in the static contact can cause the system to be unstable due to the interaction between the thermoelastic distortion and pressure-dependent thermal contact resistance. This paper investigates the thermoelastic contact instability of a functionally graded material (FGM) layer and a homogeneous half-plane using the perturbation method. The FGM layer and half-plane are exposed to a uniform heat flux and are pressed together by a uniform pressure. The material properties of the FGM layer vary exponentially along the thickness direction. The characteristic equation governing the thermoelastic stability behavior is obtained to determine the stability boundary. The effects of the gradient index, layer thickness and material combination on the critical heat flux are discussed in detail through a parametric study. Results indicate that the thermoelastic stability behavior can be modified by adjusting the gradient index of the FGM layer.     

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

Analytical solution to the unsteady one-dimensional conduction problem with two time-varying boundary conditions: Duhamel’s theorem and separation of variables

An analytical resolution of the time-dependent one-dimensional heat conduction problem with time-dependent boundary conditions using the method of separation of variables and Duhamel’s theorem is presented. The two boundary conditions used are a time-dependent heat flux at one end and a varying temperature at the other end of the one-dimensional domain. It is put forth because the author found that the prescribed resolution method using separation of variables and Duhamel’s theorem presented in heat conduction textbooks is not directly applicable to problems with more than one time-dependent boundary condition. The analytical method presented in this paper makes use of one of the property of the heat conduction equation: the apparent linearity of the solutions. For that reason, in order to solve a problem with two time-dependent boundary conditions, the author first separates the initial problem into two independent but complementary problems, each with only one time-dependent boundary condition. Doing that, both simpler problems can be solved independently using a prescribed method that is known to work and the final solution can be obtained by joining the two independent solutions from the simpler separated problems. Every step of the resolution method is presented in this paper, along with a numerical validation of the final solution of three test case problems.