Heat transfer in turbulent pipe flow based on a new mixing length model

A new model for the heat transfer in turbulent pipe flow is presented based on a modified form of the mixing length theory developed by Cebeci [1] for boundary layer flow problems. The model predicts the velocity and temperature distributions and the Nusselt number for fluids with low, medium and high Prandtl numbers (Pr=.02 to 15) and fits the available experimental data very accurately for values of Reynolds number exceeding 10⁴. Expressions for the eddy conductivity and for the turbulent Prandtl number are presented and shown to be dependent upon the Reynolds number, the Prandtl number, and the distance from the tube wall.     

Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface

The mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent fluid is analyzed using similarity solution technique. Wall temperature and stretching velocity are assumed to have specific exponential function forms. The influence of buoyancy along with viscous dissipation on the convective transport in the boundary layer region is analyzed in both aiding and opposing flow situations. The flow is governed by the mixed convection parameter Gr/Re². The velocity and temperature inside the boundary layer are observed to be influenced by the parameters like Prandtl number Pr, Gebhart number Gb. Significant changes are observed in non-dimensional skin friction and heat transfer coefficients due to viscous dissipation in the medium. The flow and temperature distributions inside the boundary layer are analyzed and the results for non-dimensional skin friction and heat transfer coefficients are discussed through computer generated plots.     

Numerical simulation of heat transfer in a pipe with non-homogeneous thermal boundary conditions

Direct numerical simulations of heat transfer in a fully-developed turbulent pipe flow with circumferentially-varying thermal boundary conditions are reported. Three cases have been considered for friction Reynolds number in the range 180–360 and Prandtl number in the range 0.7–4. The temperature statistics under these heating conditions are characterized. Eddy diffusivities and turbulent Prandtl numbers for radial and circumferential directions are evaluated and compared to the values predicted by simple models. It is found that the usual assumptions made in these models provide reasonable predictions far from the wall and that corrections to the models are needed near the wall.     

Effect of Prandtl number on the linear stability of compressible Couette flow

Accurate prediction of laminar to turbulent transition in high-speed flows is a challenging task. Compressibility, and the resultant large variations in the transport properties can affect the transition process significantly. In this paper, we study the influence of Prandtl number, the ratio of momentum to heat diffusivity, in Couette flows at high Mach numbers. It is a part of an ongoing research programme to isolate and understand the transport property effects on the stability of high-speed flows. As a first step, we neglect the high-temperature effects and vary the Prandtl number in the range 0.9 to 0.2, by changing the relative magnitudes of viscosity and conductivity. A temporal linear stability analysis shows that the variation of phase speed with Prandtl number leads to synchronization between two acoustic modes, with peaks in growth rate at the synchronization points. Two types of branching patterns are observed depending on the Prandtl number, and the branch type determines which of the two modes is destabilized and which one is stabilized due to synchronization. Further, the mode shapes are either retained as earlier or interchanged between the two acoustic modes depending on the branching pattern. The stability diagrams for varying Mach and Reynolds numbers show a destabilizing role of decreasing the Prandtl number, both in terms of increased disturbance growth rates, and of larger regions of instability in the parameter space. It also results in a significant reduction in the critical Reynolds number of the flow, especially at high Mach numbers with wall cooling.     

Development of a turbulent near-wall temperature model and its application to channel flow

A numerical study of fluid flow and heat transfer in a two-dimensional channel under fully developed turbulent conditions is reported. A computer program which is capable of treating both forced and natural convection problems under turbulent conditions has been developed. The code uses the high-Reynolds-number form of the two equation turbulent model(k-?) in which a turbulent kinetic energy near-wall model is incorporated in order to accurately represent the behavior of the flow near the wall, particularly in the viscous sublayer where the turbulent Reynolds number is small. A near-wall temperature model has been developed and incorporated into the energy equation to allow accurate prediction of the temperature distribution near the wall and, therefore, accurate calculation of heat transfer coefficients. The sensitivity of the prediction of flow and heat transfer to variations in the coefficients used in the turbulence model is investigated. The predictions of the model are compared to available experimental and theoretical results; good agreement is obtained. The inclusion of the near-wall temperature model has further improved the predictions of the temperature profile and heat transfer coefficient. The results indicate that the turbulent kinetic energy Prandtl number should be a function of Reynolds number.     

Turbulent thermal boundary layer on a plate. Reynolds analogy and heat transfer law over the entire range of prandtl numbers

Arational asymptotic theory is proposed,which describes the turbulent dynamic and thermal boundary layer on a flat plate under zero pressure gradient. The fact that the flow depends on a finite number of governing parameters makes it possible to formulate algebraic closure conditions relating the turbulent shear stress and heat flux with the gradients of the averaged velocity and temperature. As a result of constructing an exact asymptotic solution of the boundary layer equations, the known laws of the wall for velocity and temperature, the velocity and temperature defect laws, and the expressions for the skin friction coefficient, Stanton number, and Reynolds analogy factor are obtained. The latter makes it possible to give two new formulations of the temperature defect law, one of which is identical to the velocity defect law and contains neither the Stanton number nor the turbulent Prandtl number, and the second formulation does not contain the skin friction coefficient. The heat transfer law is first obtained in the form of a universal functional relationship between three parameters: the Stanton number, the Reynolds number, and the molecular Prandtl number. The conclusions of the theory agree well with the known experimental data.     

Heat transport mechanisms of low Mach number turbulent channel flow with spanwise wall oscillation

Large eddy simulation （LES） of low Mach num- ber compressible turbulent channel flow with spanwise wall oscillation （SWO） is carried out. The flow field is analyzed with emphases laid on the heat transport as well as its rela- tion with momentum transport. When turbulent coherent structures are suppressed by SWO, the turbulent transports are significantly changed, however the momentum and heat transports change in the same manner, which gives the evi- dence of inherently consistent transport mechanisms between momentum and heat in turbulent boundary layers. The Reynolds analogies of all the flow cases are quite good, which confirms again the fact that the transport mechanisms of momentum and heat are consistent, which gives theoreti- cal support for controlling the wall heat flux control by using the drag reducing techniques.     

Inflow boundary condition for DNS of turbulent boundary layers on supersonic blunt cones

For direct numerical simulation (DNS) of turbulent boundary layers, gen- eration of an appropriate inflow condition needs to be considered. This paper proposes a method, with which the inflow condition for spatial-mode DNS of turbulent boundary layers on supersonic blunt cones with different Mach numbers, Reynolds numbers and wall temperature conditions can be generated. This is based only on a given instant flow field obtained by a temporal-mode DNS of a turbulent boundary layer on a flat plate. Effectiveness of the method is shown in three typical examples by comparing the results with those obtained by other methods.     

Heat transfer in magnetohydrodynamic laminar radial channel flow

Heat transfer in the steady axisymetrical laminar source flow of an incompressible electrically conducting fluid between two parallel disks in the presence of a transverse applied magnetic field is analyzed. The energy equation is solved numerically for the temperature distribution, where both Joulean and viscous heating are included. Both local and average Nusselt numbers for the case of constant wall temperature are evaluated. For fluids of moderate and high Prandtl numbers, Nusselt number is seen to be a strong function of both Hartmann number and a heat generation parameter together with a modified Peclet number. However, for fluids of small Prandtl number, Joulean heating and viscous dissipation can be neglected without appreciable error.     

A Priori Tests of RANS Models for Turbulent Channel Flows of a Dense Gas

Dense gas effects, encountered in many engineering applications, lead to unconventional variations of the thermodynamic and transport properties in the supersonic flow regime, which in turn are responsible for considerable modifications of turbulent flow behavior with respect to perfect gases. The most striking differences for wall-bounded turbulence are the decoupling of dynamic and thermal effects for gases with high specific heats, the liquid-like behavior of the viscosity and thermal conductivity, which tend to decrease away from the wall, and the increase of density fluctuations in the near wall region. The present work represents a first attempt of quantifying the influence of such dense gas effects on modeling assumptions employed for the closure of the Reynolds-averaged Navier–Stokes equations, with focus on the eddy viscosity and turbulent Prandtl number models. For that purpose, we use recent direct numerical simulation results for supersonic turbulent channel flows of PP11 (a heavy fluorocarbon representative of dense gases) at various bulk Mach and Reynolds numbers to carry out a priori tests of the validity of some currently-used models for the turbulent stresses and heat flux. More specifically, we examine the behavior of the modeled eddy viscosity for some low-Reynolds variants of the \(k-\varepsilon \) model and compare the results with those found for a perfect gas at similar conditions. We also investigate the behavior of the turbulent Prandtl number in dense gas flow and compare the results with the predictions of two well-established turbulent Prandtl number models.     

Prediction of hypersonic boundary layer transition on sharp wedge flow considering variable specific heat

When the air temperature reaches 600 K or higher, vibration is excited. The specific heat is not a constant but a function of temperature. Under this condition, the transition position of hypersonic sharp wedge boundary layer is predicted by using the improved eN method considering variable specific heat. The transition positions with different Mach numbers of oncoming flow, half wedge angles, and wall conditions are computed condition, the nearer to the Mach number The results show that for the same oncoming flow condition and wall transition positions of hypersonic sharp wedge boundary layer move much leading edge than those of the flat plate. The greater the oncoming flow the closer the transition position to the leading edge.     

Viscous dissipation effects on heat transfer in flow over an inclined plate

The effects of viscous dissipation are considered for natural convection flow past a semi-infinite inclined plate with variable surface temperature. Velocity and temperature profiles, skin friction, and rate of heat transfer are obtained. The effects of Grashof and Prandtl numbers, inclination angle, exponent in the wall temperature variation law, and viscous dissipation parameter on the flow are discussed. It is shown that the time required to reach steady states increases with increasing Prandtl number of the fluid. In addition, an increase in the plate temperature due to viscous dissipation was found to lead to a rise in the average skin friction and a decrease in the average Nusselt number.     

Thermo-mechanical modeling of turbulent heat transfer in gas–solid flows including particle collisions

A thermo-mechanical turbulence model is developed and used for predicting heat transfer in a gas–solid flow through a vertical pipe with constant wall heat flux. The new four-way interaction model makes use of the thermal k_θ–τ_θ equations, in addition to the hydrodynamic k–τ transport, and accounts for the particle–particle and particle–wall collisions through a Eulerian/Lagrangian formulation. The simulation results indicate that the level of thermal turbulence intensity and the heat transfer are strongly affected by the particle collisions. Inter-particle collisions attenuate the thermal turbulence intensity near the wall but somewhat amplify the temperature fluctuations in the pipe core region. The hydrodynamic-to-thermal times-scale ratio and the turbulent Prandtl number in the region near the wall increase due to the inter-particle collisions. The results also show that the use of a constant or the single-phase gas turbulent Prandtl number produces error in the thermal eddy diffusivity and thermal turbulent intensity fields. Simulation results also indicate that the inter-particle contact heat conduction during collision has no significant effect in the range of Reynolds number and particle diameter studied.     

基于直接数值模拟的可压缩湍流模型评估和改进

对来流Mach数2.25和6的平板边界层湍流进行了直接数值模拟, 并通过与理论、实验及他人计算结果的对比对数值结果进行了验证. 基于直接数值模拟得到的湍流数据库, 对常用的湍流模型进行了先验评估. 评估的湍流模型有k-εvarepsilon模型(包括标准k-εvarepsilon 模型、可实现的k-εvarepsilon模型及低Reynolds数k-εvarepsilon模型)、SA模型及BL模型. 结果显示, 对于Mach2.25的平板边界层, 可实现的k-εvarepsilon 模型及低Reynolds 数k-εvarepsilon模型具有较好的预测能力, 而标准k-εvarepsilon模型预测的湍流黏性系数偏高; SA模型在边界层内层预测准确度较高, 而在外层预测值偏高. 而对于Mach6的平板边界层, k-εvarepsilon模型及SA模型预测的湍流黏性系数均偏高, 尤其是标准k-εvarepsilon模型. 对于Mach6的平板边界层, BL模型低估了内-外层交界位置, 造成湍流黏性系数预测值严重偏低. 作者通过修改模型系数及内-外层交界位置对BL模型进行了修改, 修改后模型预测的湍流黏性系数与DNS给出的值吻合较好.     

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.     

Effects of Mach number on turbulent separation behaviours induced by blunt fin

 An experimental study of the interaction between shock wave and turbulent boundary layer induced by blunt fin has been carried out at M _∞=7.8 using oil flow visualization and simultaneous measurements of fluctuating wall pressure and heat transfer. This paper presents the effects of Mach number on turbulent separation behaviours induced by blunt fin. Received: 21 July 1996/Accepted: 4 February 1998     

Effect of viscous dissipation and heat source on flow and heat transfer of dusty fluid over unsteady stretching sheet

This paper investigates the problem of hydrodynamic boundary layer flow and heat transfer of a dusty fluid over an unsteady stretching surface.The study considers the effects of frictional heating(viscous dissipation) and internal heat generation or absorption.The basic equations governing the flow and heat transfer are reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations.The transformed equations are numerically solved by the Runge-Kutta-Fehlberg-45 order method.An analysis is carried out for two different cases of heating processes,namely,variable wall temperature(VWT) and variable heat flux(VHF).The effects of various physical parameters such as the magnetic parameter,the fluid-particle interaction parameter,the unsteady parameter,the Prandtl number,the Eckert number,the number density of dust particles,and the heat source/sink parameter on velocity and temperature profiles are shown in several plots.The effects of the wall temperature gradient function and the wall temperature function are tabulated and discussed.     

Unsteady flow with heat and mass transfer due to impingement of slot jet on an inclined plate

The unsteady laminar incompressible boundary layer flow due to a two-dimensional slot jet on a flat plate at an angle of attack has been studied. The unsteadiness in the flow field is due to the free stream velocity distribution or wall temperature (concentration) which varies with time. The governing partial differential equations in primitive variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effect of the variation of the free stream velocity distribution with time is found to be more pronounced on the skin friction than on the heat or mass transfer. The Prandtl number and the variation of the wall temperature with time strongly affect the heat transfer. Similarly, the Schmidt number and the variation of the concentration at the wall with time strongly affect the mass transfer. Beyond a certain critical value of the viscous dissipation parameter, the plate gets heated instead of being cooled.     

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.     

Influence of turbulent Prandtl number on heat transfer of a flat plate

In computations involving heat transfer in turbulent flow past bodies it is necessary to assume turbulent Prandtl number distribution across the boundary layer. A review and comparison of results obtained by different authors are given, e.g., in [1–5]. Unfortunately, the existing data are so contradictory that, at present, it does not appear to be possible to establish reliably a function that determines turbulent Prandtl number distribution across the boundary layer. The absence of sufficiently reliable and general results on the distribution of turbulent Prandtl number led to the result that in the majority of studies conducted in earlier years its value was assumed a constant and either close to or equal to one. The effect of turbulent Prandtl number on the intensity of heat transfer from a flat plate is numerically investigated in the present paper. The thermal, turbulent boundary layer equation is integrated for this purpose at different values of turbulent Prandtl number and results are compared with experimental data. Results from [6], where the thermal boundary layer was numerically integrated with Pr_t=1 and compared with experimental data, were used for comparison in the present paper. The same numerical integration procedure as in [6] was used here.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 81–85, July–August, 1984.