Gas Near a Wall: Shortened Mean Free Path,Reduced Viscosity,and the Manifestation of the Knudsen Layer in the Navier–Stokes Solution of a Shear Flow

For the gas near a solid planar wall, we propose a scaling formula for the mean free path of a molecule as a function of the distance from the wall, under the assumption of a uniform distribution of the incident directions of the molecular free flight. We subsequently impose the same scaling onto the viscosity of the gas near the wall and compute the Navier–Stokes solution of the velocity of a shear flow parallel to the wall. Under the simplifying assumption of constant temperature of the gas, the velocity profile becomes an explicit nonlinear function of the distance from the wall and exhibits a Knudsen boundary layer near the wall. To verify the validity of the obtained formula, we perform the Direct Simulation Monte Carlo computations for the shear flow of argon and nitrogen at normal density and temperature. We find excellent agreement between our velocity approximation and the computed DSMC velocity profiles both within the Knudsen boundary layer and away from it.     

Asymptotic study of turbulent Poiseuille flow with wall transpiration

An incompressible, pressure–driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary–value problem for a second–order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. A limiting transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one–parameter family of curves. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Turbulent boundary layer on a plate with transpiration: Dynamical and thermal problems

Scaling laws are established for the profiles of mean velocity and temperature, Reynolds-stress components, turbulent heat flux and mean-square temperature fluctuation, skin friction and wall heat transfer in the turbulent boundary layer on a flat plate with transpiration. In the case of blowing, the velocity and temperature distributions represented in scaling variables outside the viscous sublayer have universal forms known from experimental data for flow over an impermeable flat plate. The turbulent shearing stress and heat flux also can be represented in terms of these two functions. In the case of suction, the mean quantities are described by one-parameter families of curves. Universal skin-friction and heat-transfer laws provide a basis for representation of the skin-friction and heat-flux distributions corresponding to different Reynolds numbers and transpiration velocities in terms of universal functions of one variable. The results are obtained without invoking any special closure hypotheses. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The heat transfer problem in the wall region of a turbulent boundary layer

The problem of heat transfer in the wall region of a turbulent boundary layer has been investigated. The resonant triad in the theory of hydrodynamic stability was used to obtain the velocity field induced by the coherent structures in the wall region of the turbulent boundary layer, while the small scale turbulence was represented by a simple model. By such a new approach of modeling, the 3-D temperature field is calculated, the mean temperature profile in the wall region and the Nusselt number characterizing the heat flux, which was found to be in good agreement with the experimental observations are obtained. The instantaneous temperature field had streaky structures, thus offering a mechanism for their generation found in numerical simulations. Project supported by the National Natural Science Foundation of China (Grant No. 19132011).     

Hybrid modeling of convective laminar flow in a permeable tube associated with the cross-flow process

The confined flows in tubes with permeable surfaces are associated to tangential filtration processes (microfiltration or ultrafiltration). The complexity of the phenomena do not allow for the development of exact analytical solutions, however, approximate solutions are of great interest for the calculation of the transmembrane outflow and estimate of the concentration polarization phenomenon. In the present work, the generalized integral transform technique (GITT) was employed in solving the laminar and permanent flow in permeable tubes of Newtonian and incompressible fluid. The mathematical formulation employed the parabolic differential equation of chemical species conservation (convective–diffusive equation). The velocity profiles for the entrance region flow, which are found in the connective terms of the equation, were assessed by solutions obtained from literature. The velocity at the permeable wall was considered uniform, with the concentration at the tube wall regarded as variable with an axial position. A computational methodology using global error control was applied to determine the concentration in the wall and concentration boundary layer thickness. The results obtained for the local transmembrane flux and the concentration boundary layer thickness were compared against others in literature.     

Similarity solutions for mixed convection boundary layer flow over a permeable horizontal flat plate

The effects of suction and injection on steady laminar mixed convection boundary layer flow over a permeable horizontal flat plate in a viscous and incompressible fluid is investigated in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction and injection parameter f₀, the constant exponent n of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using a finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the reduced local Nusselt number, and the velocity and temperature profiles are obtained for various values of the parameters considered. Dual solutions are found to exist for the opposing flow.     

Turbulent Poiseuille flow with near-critical wall transpiration

An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress with the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary-value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. The case of near-critical transpiration, when the shear stress at the injection wall vanishes, is considered. It is shown that the maximum point on the mean velocity profile lies in a thin sublayer near the suction wall in this case. A formula for the position of the maximum point as a function of the transpiration factor is obtained. The mean velocity profiles near the suction wall are calculated. A friction law for Poiseuille flow with near-critical transpiration is found, which makes it possible to describe the relation between the shear stress at the wall, the Reynolds number, and the transpiration velocity by a single function of one variable. Direct numerical simulation of the flow for some transpiration factors is performed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

壁面加热湍流耗散率标度指数测量的实验研究

对中等雷诺数下壁面常温和壁面加热的平板湍流边界层中速度和温度粗粒化的耗散率结构函数标度指数进行了实验测量.用热线风速仪测量了风洞中壁面常温和加热的平板湍流边界层中不同法向位置的流向速度分量和温度的时间序列信号,研究了由于湍流边界层近壁区域相干结构的存在而导致的非各向同性、非均匀性对湍流耗散率结构函数标度指数的影响,研究发现,中等雷诺数下壁面加热的边界条件和剪切湍流的平均速度梯度对速度和温度耗散率结构函数的标度指数没有影响,均匀各向同性湍流的耗散率结构函数标度指数的层次结构模型对壁面加热平板湍流边界层的速度和温度耗散率结构函数的标度指数也是适用的.     

Asymptotic behaviour of commutation errors and the divergence of the Reynolds stress tensor near the wall in the turbulent channel flow

The derivation of the space averaged Navier–Stokes equations for the large eddy simulation (LES) of turbulent incompressible flows introduces two groups of terms which do not depend only on the space averaged flow field variables: the divergence of the Reynolds stress tensor and commutation errors. Whereas the former is studied intensively in the literature, the latter terms are usually neglected. This note studies the asymptotic behaviour of these terms for the turbulent channel flow at a wall in the case that the commutation errors arise from the application of a non‐uniform box filter. To perform analytical calculations, the unknown flow field is modelled by a wall law (Reichardt law and 1/αth power law) for the mean velocity profile and highly oscillating functions model the turbulent fluctuations. The asymptotics show that near the wall, the commutation errors are at least as important as the divergence of the Reynolds stress tensor. Copyright © 2006 John Wiley & Sons, Ltd.     

Turbulent boundary layer on a plate. Reynolds analogy and new formulations of the temperature defect law

A consistent asymptotic theory describing hydrodynamic and thermal turbulent boundary layers on a flat plate in zero pressure gradient is developed. The fact that the flow depends on a limited number of governing parameters allows us to formulate algebraic closure conditions that relate the turbulent shear stress and turbulent heat flux to mean velocity and temperature gradients. As a result of an exact asymptotic solution of the boundary-layer equations, the known laws of the wall for the velocity and temperature and the velocity and temperature defect laws as well as the expressions for the skin-friction coefficient, Stanton number, and Reynolds-analogy factor are obtained. The latter implies two new formulations for the temperature defect law one of which is completely similar to the velocity defect law and does not contain the Stanton number and the turbulent Prandtl number, and the other does not contain the skin-friction coefficient. A heat-transfer law is obtained that relates only thermal quantities. The theoretical conclusions agree well with experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

湍流边界层底层相干结构的一个理论模型*

本文采用非线性稳定性分析方法,研究了湍流边界层底层相干结干结构的成因.计算得到的增长最快的不稳定波的展向尺度与纵向尺度都与实验相符.这一分析的特点是采用了不同于湍流平均速度剖面的更合理的速度剖面作为稳定性分析的基础,并采用了新的非线性理论.文中结果有助于理解湍流边界层底层相干结构的拟有序现象.     

Monte Carlo simulation of boundary layer transition

A statistical method for simulating a boundary layer transition flow is proposed as based on experimental data on the kinematics and dynamics of turbulent spots (Emmons spots) on a flat plate placed in an incompressible fluid. The method determines intermittency with allowance for overlapping spots, which makes it possible to determine the forces on the plate surface and the flow field near the transition region if the mean streamwise velocity field in a developed turbulent boundary layer is known as a function of the Reynolds number. In contrast to multiparameter transition models, this approach avoids the use of nonphysical parameter values.     

Boundary layer along the initial length of a flat plate at zero incidence with homogeneous suction

An investigation has been made into the two-dimensional laminar incompressible boundary layer along the initial length of a semi-infinite flat plate at zero incidence with homogeneous suction. The momentum and the kinetic energy integral equations have been numerically integrated with the aid of a singly infinite system of boundary layer velocity profiles. The results obtained are well in agreement with the known exact solution and the process of integration is simpler to be carried out by the use of a monoparametric family of velocity profiles. The method can be used to investigate the boundary layer along a flat plate with arbitrary suction starting either at the leading edge or at some point downstream.     

反射型激波风洞中激波与边界层的相互作用

本文研究了反射型激波风洞中由于非完全反射对激波与壁面边界层相互作用的影响,给出了在反射激波坐标系中计算边界层速度分布、温度分布和马赫数分布的计算方法.算例表明,在计及氮气的平衡真实气体效应的情形下,随着入射激波马赫数M_s的增大,边界层的最小马赫数从壁面处移到边界层内;随着喷管喉道面积的增大,边界层的最小马赫数、反射激波的分叉角α和分叉区后的射流速度均随之减小.计算结果与实验值相比是一致的.     

用区域分解法求不可压N-S方程的差分解

§1.引言 对不可压小粘性流的数值解,[1]和[2]用奇异摄动观点提出了一个区域分解法.从常微分方程(组)的奇异摄动问题出发,解分解为外部解加边界修正解(以下简称为修正解).外部解的边界条件有:给定(原边界条件)、待定(用原边界条件和修正解)和延拓类.修正解的边界条件有:给定(用原边界条件和外部解延拓)渐近(在边界层外缘)和待定     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

湍流边界层外区扰动激发近壁区相干结构的一种机制

应用直接数值模拟的方法,研究了湍流边界层近壁区由于外区扰动的作用而导致相干结构产生的问题．结果表明:在湍流边界层近壁区的上边界存在的速度扰动,可以在近壁区内激发出相干结构,从而完善了单个相干结构的理论模型．     

A Lagrangian simulation of particle deposition in a turbulent boundary layer in the presence of thermophoresis

Knowledge of particle deposition in turbulent flows is often required in engineering situations. Examples include fouling of turbine blades, plate-out in nuclear reactors and soot deposition. Thus it is important for numerical simulations to be able to predict particle deposition. Particle deposition is often principally determined by the forces acting on the particles in the boundary layer. The particle tracking facility in the CFD code uses the eddy lifetime model to simulate turbulent particle dispersion, no specific boundary layer being modelled. The particle tracking code has been modified to include a boundary layer. The non-dimensional yplus, y⁺, distance of the particle from the wall is determined and then values for the fluid velocity, fluctuating fluid velocity and eddy lifetime appropriate for a turbulent boundary layer used. Predictions including the boundary layer have been compared against experimental data for particle deposition in turbulent pipe flow. The results giving much better agreement. Many engineering problems also involve heat transfer and hence temperature gradients. Thermophoresis is a phenomena by which small particles experience a force in the opposite direction to the temperature gradient. Thus particles will tend to deposit on cold walls and be repulsed by hot walls. The effect of thermophoresis on the deposition of particles can be significant. The modifications of the particle tracking facility have been extended to include the effect of thermophoresis. A preliminary test case involving the deposition of particles in a heated pipe has been simulated. Comparison with experimental data from an extensive experimental programme undertaken at ISPRA, known as STORM (Simplified Tests on Resuspension Mechanisms), has been made.     

The Design of axisymmetric ducts for incompressible flow with vorticity.

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function and the function as independent variables where for irrotational flow can be recognized as the velocity potential function, for rotational flow ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences on a uniform mesh is employed. The technique described is capable of tackling the so–called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct wall shapes. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)