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.     

Effects of mesh resolution on hypersonic heating prediction

Aeroheating prediction is a challenging and critical problem for the design and optimization of hypersonic vehicles. One challenge is that the solution of the Navier-Stokes equations strongly depends on the computational mesh. In this letter, the effect of mesh resolution on heat flux prediction is studied. It is found that mesh-independent solutions can be obtained using fine mesh, whose accuracy is confirmed by results from kinetic particle simulation. It is analyzed that mesh-induced numerical error comes mainly from the flux calculation in the boundary layer whereas the temperature gradient on the surface can be evaluated using a wall function. Numerical schemes having strong capability of boundary layer capture are therefore recommended for hypersonic heating prediction.     

Transient laminar forced convection with arbitrary variation in the wall heat flux

 The aim of the work is to present a detailed numerical study of the transient forced laminar convection flow over a flat plate, when thermal conditions are due to arbitrary wall heat flux variations in space. The energy governing equation is modelled using the Karman–Pohlhausen integral approach in the wide range of Prandtl numbers. The influence of both the thermal problem nature (transient heating and/or cooling processes) and the wall flux function on the resulting mathematical expressions is evidenced and the thermal boundary layer thickness behaviour is discussed. In addition, a particular attention has been focused on both the change in sign of the flux and the duration of the transient heating and cooling. Detailed thermal responses and convective heat coefficient evolutions due to the change of wall conditions are presented. Received on 14 April 2000 / Published online: 29 November 2001     

Thermal Analysis of Flow in a Porous Medium Over a Permeable Stretching Wall

This work presents a similarity solution for boundary layer flow through a porous medium over a stretching porous wall. Two considered wall boundary conditions are power-law distribution of either wall temperature or heat flux which are general enough to cover the isothermal and isoflux cases. In addition to momentum, both first and second laws of thermodynamics analyses of the problem are investigated. Independent numerical simulations are also performed for verification of the proposed analytical solution. The results, from the two independent approaches, are found to be in complete agreement. A comprehensive parametric study is presented and it is shown that heat transfer and entropy generation rates increase with Reynolds number, Prandtl number, and suction to the surface.     

Unsteady magnetohydrodynamic stagnation point flow — closed-form analytical solutions

This paper investigates the unsteady stagnation point flow and heat transfer of magnetohydrodynamic(MHD) fluids over a moving permeable flat surface. The unsteady Navier-Stokes(NS) equations are transformed into a similarity nonlinear ordinary differential equation, and a closed form solution is obtained for the unsteadiness parameter of 2. The boundary layer energy equation is transformed into a similarity equation,and is solved for a constant wall temperature and a time-dependent uniform wall heat flux case. The solution domain, velocity, and temperature profiles are calculated for different combinations of parameters including the Prandtl number, mass transfer parameter, wall moving parameter, and magnetic parameter. Two solution branches are obtained for certain combinations of the controlling parameters, and a stability analysis demonstrates that the lower solution branch is not stable. The present solutions provide an exact solution to the entire unsteady MHD NS equations, which can be used for validating the numerical code of computational fluid dynamics.     

振动激发对高超声速气动力/热影响

随着飞行马赫数的不断提高,空气的高温气体效应越来越明显,对高超声速飞行器的气动力/热特性产生重要影响.高温气体效应对气动力/热的影响机理复杂,影响参数众多,迄今为止国内外尚未完全研究清楚.发生高温气体效应时,多个非线性物理过程耦合在一起,地面试验和数值模拟无法将这些过程解耦,无法给出关键物理机理.为了解决这一问题,文章提出一种理论分析与数值模拟相结合的两步渐进新方法:先通过牛顿迭代法得到发生振动激发过程的斜激波无黏解;再将该无黏解的结果作为边界条件,求解边界层的黏性解.利用该方法研究了振动激发过程对二维斜劈的气动力/热特性的影响规律.研究结果表明,振动激发过程对斜激波后的温度、密度、马赫数、雷诺数和斜激波角影响较大,而对压力和速度影响较小.斜激波波后的无黏流动与边界层流动是耦合在一起的.发生振动激发后,斜激波波后雷诺数的增大会导致边界层厚度减小,结合多个物理量的变化,如速度增大和温度减小,共同对边界层内的摩擦阻力和气动热产生影响.对比完全气体的结果发现,振动激发使壁面摩阻升高,而使壁面热流降低.分别通过影响激波层和边界层,振动激发对摩阻的影响是弱耦合的,而对热流的影响则是强耦合的.     

Stagnation point heat transfer from laminar,high temperature methane flames

Combustion of methane-rich fuels frequently provides forced convective heating in industry, and the ability to predict the rate of heat transfer from such flames to solid surfaces is often desirable. Mathematical modelling of stagnation point heat flux has been achieved by numerical solution of the boundary layer equations, and by an analytical equation modified to include the effects of chemical reaction in the free stream flow and to allow for the enhancement in heat flux caused by the diffusion and exothermic recombination of reactive species in the boundary layer surrounding the heat receiving body. Predictions from these models have been compared with experimental data obtained in high temperature methane flames of various equivalence ratios. Within the equilibrium region of these flames, predictions from the modified analytical equation based on total Lewis numbers equal to and greater than one form a tight envelope around the experimental results, and hence provide a relatively simple method of predicting heat flux. Numerical solutions tend to slightly underestimate predictions from the analytical equation and experimental data, although agreement with the alternative prediction method increases with the surface temperature of the heat receiving body     

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.     

高超声速全动舵面的热气动弹性研究

根据分层求解原理对考虑舵轴及舵轴与机身间隙影响下的高超声速飞行器全动舵面进行了热气动弹性分析. 采用计算流体力学（CFD）方法求解N-S 方程计算舵面周围的热环境,在该温度分布下根据结构壁面温度计算热流,应用傅里叶（Fourier）定律确定结构热传导过程及其内部温度分布,进而分析结构考虑热应力和温度对材料属性的影响下的模态固有特性,结合基于CFD 技术的当地流活塞理论,在状态空间中对舵面进行了热气动弹性分析. 结果表明,气动加热效应改变了结构的固有频率以及弯扭耦合频率之间的间距,进而改变了结构的颤振速度和颤振频率;随着热传导的进行,结构固有频率和颤振频率先快速减小后基本保持不变,弯扭耦合频率之间的间距和颤振速度则先快速减小后略有上升;舵轴及舵轴与机身间隙的存在对舵面的固有频率、颤振频率、颤振速度都产生了影响,使其最大下降了6%.      

STUDY ON AEROTHERMOELASTICITY OF A HYPERSONIC ALL-MOVABLE CONTROL SURFACE

根据分层求解原理对考虑舵轴及舵轴与机身间隙影响下的高超声速飞行器全动舵面进行了热气动弹性分析. 采用计算流体力学（CFD）方法求解N-S 方程计算舵面周围的热环境,在该温度分布下根据结构壁面温度计算热流,应用傅里叶（Fourier）定律确定结构热传导过程及其内部温度分布,进而分析结构考虑热应力和温度对材料属性的影响下的模态固有特性,结合基于CFD 技术的当地流活塞理论,在状态空间中对舵面进行了热气动弹性分析. 结果表明,气动加热效应改变了结构的固有频率以及弯扭耦合频率之间的间距,进而改变了结构的颤振速度和颤振频率;随着热传导的进行,结构固有频率和颤振频率先快速减小后基本保持不变,弯扭耦合频率之间的间距和颤振速度则先快速减小后略有上升;舵轴及舵轴与机身间隙的存在对舵面的固有频率、颤振频率、颤振速度都产生了影响,使其最大下降了6%.     

近似黎曼解对高超声速气动热计算的影响研究

高超声速流场计算一般采用TVD型格式,这些格式中,大多采用了不同形式的近似黎曼解. 通过分析和数值验证,论述了激波捕捉格式中近似黎曼解的耗散性质,说明其对高超声速热流计算的影响. 数值实验证明,采用低耗散格式可大大提高热流计算精度,降低热流计算对网格的依赖程度,从而获得精确的热流数值解.      

Wave induced thermal boundary layers in a compressible fluid: analysis and numerical simulations

The response of a semi-infinite compressible fluid to a step-wise change in temperature of its boundary is investigated analytically and numerically. Numerical results of the boundary layer structure are compared with Clarke’s analytical solution for a gas with thermal conductivity proportional to temperature. To avoid unwanted numerical dissipation in the numerical analysis, the space-time conservation element and solution element (CESE) method has been adopted to solve the unsteady 1-D Navier-Stokes equations. Good agreement between analytical and numerical results has been found for the development of the thermal boundary layer on a long time scale. Weak shock waves and expansion waves induced by the thermal boundary layer due to its compressibility, are observed in the numerical simulation. Finally, the numerical method has been applied to the reflection of a non-linear expansion wave and to a shock wave from an isothermal wall, thereby illustrating the effect of the boundary layer on the external flow field.     

Unsteady boundary layer and its separation over a heated circular cylinder

A numerical method is developed to solve the coupled unsteady laminar momentum and thermal boundary layers over a circular cylinder impulsively started from rest. The present non-iterative finite difference method, which requires relatively fewer grid points in the reversed flow region than any other method, can easily handle the separating boundary layer flows. The results indicate that the present method has accuracy comparable with the earlier methods, while consuming computer time approximately one order of magnitude less. The present numerical method allowed investigation of the effect of buoyancy parameter on the starting boundary layer. The time-dependent behaviour of the boundary layer is studied in terms of the appearance of the singularity, the distribution of skin friction and wall heat flux, and the wall position of the inflection point of the velocity profile. The transient as well as buoyancy-dependent patterns of the streamlines and isotherms are also studied.     

考虑瞬态效应的承载隔热多功能结构拓扑优化

同时满足承载和隔热要求的多功能结构在高超飞行器热防护结构设计中倍受关注.实际隔热材料通常承载能力弱,而高承载材料隔热性能差,如何在有限空间内协同结构的承载与隔热成为关键问题.高超飞行器气动加热时间有限,存在加热时间短、热荷载变化大的特点.因此结构设计需要考虑时间因素和瞬态效应,而现有稳态传热与承载的多功能协同优化设计模...     

Self-similar solution of the unsteady mixed convection flow in the stagnation point region of a rotating sphere

An analysis is performed to present a new self-similar solution of unsteady mixed convection boundary layer flow in the forward stagnation point region of a rotating sphere where the free stream velocity and the angular velocity of the rotating sphere vary continuously with time. It is shown that a self-similar solution is possible when the free stream velocity varies inversely with time. Both constant wall temperature and constant heat flux conditions have been considered in the present study. The system of ordinary differential equations governing the flow have been solved numerically using an implicit finite difference scheme in combination with a quasilinearization technique. It is observed that the surface shear stresses and the surface heat transfer parameters increase with the acceleration and rotation parameters. For a certain value of the acceleration parameter, the surface shear stress in x-direction vanishes and due to further reduction in the value of the acceleration parameter, reverse flow occurs in the x–component of the velocity profiles. The effect of buoyancy parameter is to increase the surface heat transfer rate for buoyancy assisting flow and to decrease it for buoyancy opposing flow. For a fixed buoyancy force, heating by constant heat flux yields a higher value of surface heat transfer rate than heating by constant wall temperature.     

Thermo-fluid dynamics of the unsteady channel flow

We present an exact analytical representation of the unsteady thermo-fluid dynamic field arising in a two-dimensional channel with parallel walls for a fluid with constant properties. We assume that the axial pressure gradient is an arbitrary function of time that can be expanded in Taylor series; a particular case is the impulsive motion generated by a sudden jump to a constant value; for large time values the flow reaches the well-known steady Poiseuille solution. As boundary conditions for the dynamic field we consider fixed and moving walls (unsteady Couette flow). The assigned temperature on the walls can be an arbitrary function of time. We also consider the coupling of the energy and momentum equations (i.e. Eckert number different from zero). The solution is obtained by series with simple expressions of the coefficients in terms of the error functions. The fundamental physical parameters, such as shear stress, mass flow and heat flux at the wall are obtained in explicit analytical form and discussed by means of their diagrams.     

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.     

Numerical simulation of the ground effect using the boundary element method

As is well known, the lift of a wing passing over the ground becomes larger than that of a wing in a finite air field because of the ground effect. Owing to its special aerodynamic characteristics and applications, the problem of the ground effect has become increasingly common. In this paper some investigations were conducted to calculate the unsteady aerodynamic forces for long and short ground plates by means of boundary element techniques. In order to calculate the pressure variation on a long ground plate, the steady boundary element method was used. However, when using a short ground plate, the boundary element method was modified to treat the unsteady aerodynamic phenomena. Experimental studies were also made for both ground plates to confirm the validity of the numerical results. At low angles of attack the qualitative behaviour of the unsteady aerodynamic pressure on both ground plates was well predicted by the boundary element methods and qualitative agreement is found between the calculated and measured results. © 1997 John Wiley & Sons, Ltd.     

Immersed boundary method for unsteady kinetic model equations

Predicting unsteady flows and aerodynamic forces for large displacement motion of microstructures requires transient solution of Boltzmann equation with moving boundaries. For the inclusion of moving complex boundaries for these problems, three immersed boundary method flux formulations (interpolation, relaxation, and interrelaxation) are presented. These formulations are implemented in a 2‐D finite volume method solver for ellipsoidal‐statistical (ES)‐Bhatnagar‐Gross‐Krook (BGK) equations using unstructured meshes. For the verification, a transient analytical solution for free molecular 1‐D flow is derived, and results are compared with the immersed boundary (IB)‐ES‐BGK methods. In 2‐D, methods are verified with the conformal, non‐moving finite volume method, and it is shown that the interrelaxation flux formulation gives an error less than the interpolation and relaxation methods for a given mesh size. Furthermore, formulations applied to a thermally induced flow for a heated beam near a cold substrate show that interrelaxation formulation gives more accurate solution in terms of heat flux. As a 2‐D unsteady application, IB/ES‐BGK methods are used to determine flow properties and damping forces for impulsive motion of microbeam due to high inertial forces. IB/ES‐BGK methods are compared with Navier–Stokes solution at low Knudsen numbers, and it is shown that velocity slip in the transitional rarefied regime reduces the unsteady damping force. Copyright © 2015 John Wiley & Sons, Ltd.     

低压涡轮内部流动及其气动设计研究进展

随着高空无人飞行器研究的不断升温, 高空低雷诺数条件下动力装置的研究越来越受到人们的重视.结合近年来国内外相关领域的研究工作, 对低雷诺数低压涡轮内部复杂流动机理的研究进展进行了介绍, 包括低雷诺数情况下低压涡轮内部非定常流动的特点, 叶片边界层分离及转捩现象机理, 上游周期性尾迹与下游叶片边界层相互作用机理等. 在此基础上给出了适合低雷诺数条件的低压涡轮气动设计方法:尾迹通过与边界层的相互作用, 能够抑制分离, 进而减小叶型损失, 在气动设计中有效引入非定常效应可以大幅度提高低压涡轮的气动负荷或降低气动损失, 最终达到提高性能的目的;数值及实验结果验证了这种设计方法的有效性.