HIGHER ORDER THEORY ACROSS A THREE DIMENSIONAL AXIALLY-SYMMETRICAL CURVED SHOCK NEAR THE STAGNATION POINT

The next order conditions across a three-dimensional curved shock near stagnationpoint have been established,including the effects of heat conduction,viscosity and the shockstructure.These shock conditions involve the local shock curvature in addition to its localinclination.Explicit results have been obtained for the correctional formulations in themass flux across the shock,the stagnation enthalpy,the tangential component of velocityand the normal component of momentum flux.     

On the frame dependence of constitutive equations. I. Heat transfer through a rarefied gas between two rotating cylinders

To verify the principle of material frame indifference a numerical calculation of the heat flux field in a rotating gas has been carried out based on the kinetic equation over wide ranges of gas rarefaction and angular velocity. It has been confirmed that a radial gradient of the temperature causes a tangential heat flux. Also, it has been found that the radial heat flux is affected by the rotation.On temporary leave from Department of Physics, Urals State University, 620083 Ekaterinburg, Russia     

Physical criterion study on forward stagnation point heat flux CFD computations at hypersonic speeds

In order to evaluate uncertainties in computational fluid dynamics （CFD） computations of the stagnation point heat flux, a physical criterion is developed. Based on a quasi-one-dimensional hypothesis along the stagnation line, a new stagnation flow model is applied to obtain the governing equations of the flow near the stagnation point at hypersonic speeds. From the above equations, the compatibility relations are given at the stagnation point and along the stagnation line, which consist of the physical criterion for checking the accuracy in the stagnation point heat flux computations. The verification of the criterion is made with various numerical results.     

Turbulence enhancement of stagnation point heat transfer on a body of revolution

Enhancement, by free stream turbulence, of convective heat transfer to the stagnation region of a hemispherical-nosed cylinder has been studied. Increases in heat transfer were found to depend primarily on the Reynolds number and turbulence intensity of the free stream, experimental results being most successfully correlated on a NuRe^?0.5 versus TuRe^0.5 basis. Flow visualization studies have demonstrated the validity of a phenomenological model of the enhancement process, predictions of this theory showing good agreement with experimental results. The effect of free stream turbulence on the stagnation point velocity gradient has also been evaluated.     

Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

Comparisons and applications of numerical simulation methods for predicting aerodynamic heating around complex configurations

Numerical simulation methods of aerodynamic heating were compared by considering the inuence of numerical schemes and turbulence models,and attempting to investigate the applicability of numerical simulation methods on predicting heat flux in engineering applications. For some typical cases provided with detailed experimental data,four spatial schemes and four turbulence models were adopted to calculate surface heat flux. By analyzing and comparing,some inuencing regularities of numerical schemes and turbulence models on calculating heat flux had been acquired. It is clear that AUSM+-up scheme with rapid compressibilitymodified high Reynolds number k-ω model should be appropriate for calculating heat flux. The numerical methods selected as preference above were applied to calculate the heat flux of a 3-D complex geometry in high speed turbulent flows. The results indicated that numerical simulation can capture the complex flow phenomena and reveal the mechanism of aerodynamic heating. Especially,the numerical result of the heat flux at the stagnation point of the wedge was well in agreement with the prediction of Kemp-Riddel formula,and the surface heat flux distribution was consistent with experiment results,which implied that numerical simulation can be introduced to predict heat flux in engineering applications.     

Influence of turbulence intensity and free stream velocity oscillations on stagnation point heat transfer

A model is proposed for the momentum eddy diffusivity induced by free stream turbulence intensity and integral length scale. The eddy diffusivity model is applied to the stagnation point of a cylinder situated in a steady uniform crossflow in the presence of free stream turbulence. A numerical solution of the governing steady-state momentum and energy equations with the proposed eddy diffusivity model yielded results for the skin friction coefficient and the Nusselt number. Agreement between the numerical predictions of this work and experimental data is very good. The experimental data concerning the unsteady stagnation point heat transfer under the combined influence of free stream velocity oscillations and turbulence intensity have been successfully correlated by means of a new turbulence parameter     

Convective heat-transfer rate distributions over a missile shaped body flying at hypersonic speeds

Experiments are carried out with air as the test gas to obtain the surface convective heating rate on a missile shaped body flying at hypersonic speeds. The effect of fins on the surface heating rates of missile frustum is also investigated. The tests are performed in a hypersonic shock tunnel at stagnation enthalpy of 2 MJ/kg and zero degree angle of attack. The experiments are conducted at flow Mach number of 5.75 and 8 with an effective test time of 1 ms. The measured stagnation-point heat-transfer data compares well with the theoretical value estimated using Fay and Riddell expression. The measured heat-transfer rate with fin configuration is slightly higher than that of model without fin. The normalized values of experimentally measured heat transfer rate and Stanton number compare well with the numerically estimated results.     

Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field

The steady MHD mixed convection flow of a viscoelastic fluid in the vicinity of two-dimensional stagnation point with magnetic field has been investigated under the assumption that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary layer theory is used to simplify the equations of motion, induced magnetic field and energy which results in three coupled non-linear ordinary differential equations which are well-posed. These equations have been solved by using finite difference method. The results indicate the reduction in the surface velocity gradient, surface heat transfer and displacement thickness with the increase in the elasticity number. These trends are opposite to those reported in the literature for a second-grade fluid. The surface velocity gradient and heat transfer are enhanced by the magnetic and buoyancy parameters. The surface heat transfer increases with the Prandtl number, but the surface velocity gradient decreases.     

Flow measurement in an impinging jet cell with three-segment microelectrocles

Three-dimensional flow close to the stagnation point of an impinging jet cell was studied by means of three-segment electrodiffusion probes. The total current and the directional characteristics providing the wall velocity gradient and the flow direction respectively were measured at different distances from the stagnation point. The linear dependence of the wall velocity gradient on this distance was confirmed in the stagnation zone. The normal velocity component was evaluated from the flatness of the directional characteristics. The values of the normal velocity component were rather higher than those corresponding to the linear change of the velocity gradient and the dimensionless ratio of the probe radius to distance. The deviation appearing on the normal velocity component was explained by geometrical imperfections of the probes due to the technology of fabrication.This work was supported by the grant no. 101(9511421 awarded by the Grant Agency of the Czech Republic.     

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     

高超声速高焓条件下的内嵌式温敏漆测量方法

热流密度点测量结果并不能完全反映详细的热流分布特征, 尤其是针对热流梯度较大、热流分布复杂的区域, 需要热流密度场测量技术以获取全场精细的热流分布特征. 应用温敏漆测量热流密度场的方法得到了广泛应用, 但实验条件来流总温较低, 与真实飞行环境存在明显差异, 真实飞行条件下的辐射效应严重限制了温敏漆技术的应用. 针对高超声速高焓条件下缺乏热流密度场测量方法的难题, 提出了内嵌式温敏漆测量方法, 基本思想是利用温敏漆测量内壁面温度的变化历程结合热传导反问题的求解确定热流密度. 本文详细介绍了内嵌式温敏漆测量方法的测量原理、测量系统构成、数据处理方法、设计原则及该测量方法的优势. 针对高超声速风洞实验中常见的阶跃、线性和局部突变等热流密度分布进行了数值验证, 验证了内嵌式温敏漆测量方法的可行性, 并分析了风洞实验温度测量精度及噪声对测量结果的影响. 内嵌式温敏漆测量方法可用于测量高超声速真实飞行环境下细致的气动热特征, 扩展了温敏漆测量方法的应用范围, 解决了高超声速高焓条件下缺乏热流密度场测量方法的难题.      

Momentum and heat transfer in two-phase bubble flow—I. Theory

A theory is proposed which describes the transfer process of momentum and heat in a two-phase bubble flow in channels. The eddy diffusivity to express the turbulent structure of the liquid phase is subdivided into the two components, one for the inherent wall turbulence independent of bubble agitation and the other for the additional turbulence caused by bubbles. On the basis of the theory, the velocity profile and the frictional pressure gradient for a given flow can be predicted when its void fraction profile is known. Furthermore, when a uniform heat flux is added to the system, its temperature distribution and heat transfer coefficient can be determined. A method for the numerical calculation of these parameters is also presented.     

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.     

Heat transfer at and near the stagnation point in turbulent flow over bodies

Results are presented of experimental investigations of heat transfer in the neighborhood of the stagnation point in flow of a turbulent gas over bodies. It is assumed that the outer flow is capable of rendering the boundary layer turbulent over the whole body surface, i.e., the hypothesis is invoked that there is a turbulent stagnation point. Using the method of integral relations [1] and the flat plate heat-transfer law, transformed in such a way as to satisfy the heat-transfer conditions at the stagnation point, simple formulas have been obtained for calculating the heat flux.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–181, July–August, 1975.     

Investigation of a hypersonic viscous shock layer on rotating axisymmetric bodies in the presence of blowing

The hypersonic flow of a laminar stream of viscous compressible gas past blunt axisyrametric bodies rotating about the longitudinal axis is considered. It is assumed that gas blows from the surface of the body. The solution of the problem is obtained by a finite-difference method in a wide range of Reynolds numbers and blowing and rotation parameters. Some results of the calculations characterizing the effect of the rotation on the velocity and temperature profiles across the shock layer, on the friction and heat transfer coefficients, and the shock wave separation are given for the neighborhood of the stagnation point. For large Reynolds numbers and strong blowing an analytic solution of the problem is found in an approximation of two inviscid layers separated by a contact surface. The calculations are made for the flow past a sphere and a paraboloid and it is shown that in the presence of rotation the maximum of the heat flux is shifted from the stagnation point onto the side surface of the body. The dependence of the pressure distribution, the heat flux, and the friction coefficient is investigated for cases of constant and variable blowing over the contour of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–114, January–February, 1986.     

Mixed Convection Stagnation-Point Flow Over a Vertical Plate with Prescribed Heat Flux Embedded in a Porous Medium: Brinkman-Extended Darcy Formulation

This article considers the problem of mixed convection stagnation-point flow towards a vertical plate embedded in a porous medium with prescribed surface heat flux. It is assumed that the free stream velocity and the surface heat flux vary linearly from the stagnation point. Using a similarity transformation, the governing system of partial differential equations is transformed into a system of ordinary differential equations, before being solved numerically by a finite-difference method. The features of the flow and the heat transfer characteristics are analyzed and discussed. It is found that dual solutions exist for both buoyancy assisting and opposing flows.     

Modeling Fluid Flow in Fractured Porous Media with the Interfacial Conditions Between Porous Medium and Fracture

One of the most popular models that has been applied to predict the fluid velocity inside the fracture with impermeable walls is the cubic law. It highlights that the mean flux along the fracture is proportional to the cubic of fracture aperture. However, for a fractured porous medium, the normal and tangential interface conditions between the fracture and porous matrix can change the velocity profile inside the fracture. In this paper, a correction factor is introduced for flow equation along the fracture by imposing the continuity of normal and tangential components of velocity at the interface between the fracture and porous matrix. As a result, the mean velocity inside the fracture depends not only on the fracture aperture, but also on a set of non-dimensional numbers, including the matrix porosity, the ratio of intrinsic permeability of fracture to that of matrix, the wall Reynolds number, and the ratio of normal velocity on one wall to the other. Finally, the introduced correction factor is employed within the extended finite element method, which is widely used for numerical simulation of fluid flow within the fractured porous media. Several numerical results are presented for the fluid flow through a specimen containing single fracture, in order to investigate the deviation from the cubic law in different case studies.

     

高超声速压缩拐角峰值热流位置预测模型研究

旨在通过建立高超声速压缩拐角流动的再附点附近的模型理论来揭示峰值热流的产生机理以及如何可靠预测其位置.首先,分析再附点附近流动特征,提出了基于可压缩斜驻点流动模型来近似该处的局部流动.其次,近似求解了该流动模型,发现边界层厚度存在极小值的特征,从而揭示了再附点后峰值热流的产生机理,并由此得到了再附点和峰值热流点间距离的半解析半数值估算方法.最后,对此结果作了数值验证,并做了讨论.      

Investigation of heat flux at the forward critical point of bodies placed in a high-temperature subsonic stream

We present the results of measurements of the total heat flux at the forward critical point of axially symmetric models placed in a high-temperature subsonic stream of air and nitrogen in an apparatus which uses a high-frequency inductive discharge [1] to heat the gas. The heat fluxes were measured for cylindrical models whose forward part had one of three possible shapes: a hemisphere, a hemisphere with a blunt nose, or a flat circular end-face. A water-cooled calorimeter sensor was set up at the forward critical point of the model; the calorimeter sensor was made of different materials, so that it was possible to estimate the radiant and convective components of the total heat flux and determine the effect of the sensor material on the heat flux measured. The convective component of the heat flux was compared to a calculated value obtained by Fay and Riddell's formula. The heat-flux values found for two shapes of models were used in determining the effective radius of streamline flow for a model with a plane end-face.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 133–141, September–October, 1973.The authors are grateful to N. I. Nesterov for his help with the work.