Development of the direct stress solution technique for three-dimensional hydrodynamic models using finite elements

Velocity varies rapidly near sheared boundaries. Therefore in many practical fluid problems it can be inefficient to solve discrete equations with velocity as the dependent variable. Conversely, shear stress varies slowly near sheared boundaries, suggesting that it may be well suited for use as the dependent variable in discrete equations. This paper describes a formulation of the internal mode equations for a three-dimensional hydrodynamic model using shear stress as the dependent variable. The resulting direct stress solution (DSS), coupled with a spatial discretization using linear finite elements, yields a system matrix that can be set up and solved with the efficiency of a banded matrix with bandwidth 8. If the eddy viscosity distribution is assumed to be piecewise linear over the depth (with an arbitrary number of time-varying segments), the recovery of velocity from stress can be easily accomplished in closed form, thereby avoiding any difficulty resulting from the logarithmic singularity in the velocity profile that occurs at a boundary. Results from tidal and wind-driven test cases with realistic boundary layers are used to demonstrate the accuracy and computational advantages of a DSS formulation versus a standard velocity-based formulation.     

Symmetries, Invariance and Scaling-Laws in Inhomogeneous Turbulent Shear Flows

An approach to derive turbulent scaling laws based on symmetry analysis is presented. It unifies a large set of scaling laws for the mean velocity of stationary parallel turbulent shear flows. The approach is derived from the Reynolds averaged Navier–Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. For the plane case the results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile that is found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law is confirmed in both the centre and the near wall regions in both experimental and DNS data of turbulent channel flows. For a non-rotating and a moderately rotating pipe about its axis an algebraic law was found for the axial and the azimuthal velocity near the pipe-axis with both laws having equal scaling exponents. In case of a rapidly rotating pipe, a new logarithmic scaling law for the axial velocity is developed. The key elements of the entire analysis are two scaling symmetries and Galilean invariance. Combining the scaling symmetries leads to the variety of different scaling laws. Galilean invariance is crucial for all of them. It has been demonstrated that two-equation models such as the k– model are not consistent with most of the new turbulent scaling laws.     

Reynolds number effects on the behavior of a non-buoyant round jet

The behavior of a non-buoyant circular water jet discharged from a contraction nozzle was experimentally investigated. In this experiment, the Reynolds number of the jet, based on the mean velocity results obtained by particle image velocimetry (PIV), ranged from 177 to 5,142. From the experimental results, we found that the cross-sectional profile of the axial velocity for a laminar flow near the nozzle did not show a top-hat distribution, whereas the profiles with Reynolds number higher than 437 were almost top-hat. The length of the zone of flow establishment (ZFE) was found to decrease with increasing Reynolds number. The measured centerline velocity decayed more rapidly and, consequently, approached the theoretical equation earlier near the nozzle as the Reynolds number increased. The decay constant for the centerline velocity of the turbulent cases was relatively lower than that discovered in theory. It is assumed that this probably resulted from the use of the contraction nozzle. Verifying the similarity of the lateral velocity profiles demonstrated that the Gaussian curve was properly approximated only for the turbulent jets and not for the laminar or transitional flows. The jet half width seldom grew for the laminar or transitional flows, whereas it grew with increasing axial distance for the turbulent flows. The spreading rates for the turbulent flows gradually decreased with increasing Reynolds number. The normalized turbulence intensity along the jet centerline increased more rapidly with the axial distance as the Reynolds number increased, and tended to the constant values proposed by previous investigators. The Reynolds shear stress levels were also found to increase as the Reynolds number increased for the turbulent jets.     

One-dimensional model for microscale shock tube flow

A one-dimensional model for the numerical simulation of transport effects in small-scale, i.e., low Reynolds number, shock tubes is presented. The conservation equations have been integrated in the lateral directions and three-dimensional effects have been introduced as carefully controlled sources of mass, momentum and energy, into the axial conservation equations. The unsteady flow of gas behind the shock wave is reduced to a quasi-steady flow by choosing a coordinate system attached to the shock. The boundary layer problem is thereby reduced to a laminar solution, similar to the Blasius solution, with the exception that the wall velocity can be nonzero. The resulting one-dimensional equations are then solved numerically using a two-step Lax-Wendroff/ MacCormack scheme with flux correction transport. For validation purposes, comparisons are performed against previously published shock structure and low Reynolds number shock tube experiments; good agreement is observed. The model has been used to predict the performance of a 10μm shock tube and the result of this simulation shows the possibility of shock wave disappearance at lower pressure ratios for a micro-scale shock tube.      

同轴圆柱间滑移旋转流动问题的同伦解析解

从理论上研究了具有延伸柱面的同轴圆柱间滑移流动问题.通过引入适当的相似变换将控制方程组转化为一类非线性边值问题, 利用同伦分析方法首次获得问题的近似解析解, 分析讨论滑移参数、雷诺数和内外筒半径比对流动的影响.结果表明: 滑移边界参数对剪切应力有较大的影响, 滑移边界小的流体对壁面和边界层内流场施加了更大的剪切力; 增大雷诺数Re, 能增大内筒的壁面剪切力和扭矩; 内外筒半径比对流动结构也有较大的影响, 内筒半径固定, 增大外筒半径能够减少内筒的纵向剪切力.      

Stability of viscoelastic shear flows in the limit of high Weissenberg and Reynolds numbers

We consider a limit of the upper convected Maxwell model where both the Weissenberg and Reynolds numbers are large. The limiting equations have a status analogous to that of the Euler equations for the high Reynolds number limit. These equations admit parallel shear flows with an arbitrary profile of velocity and normal stress. We consider the stability of these flows. An extension of Howard’s semicircle theorem can be used to show that the flow is stabilized if elastic effects are sufficiently strong. We also show how to analyze the long wave limit in a fashion similar to the inviscid case.     

Experiments in Turbulent Pipe Flow with Polymer Additives at Maximum Drag Reduction

In this paper we report on (two-component) LDV experiments in a fully developed turbulent pipe flow with a drag-reducing polymer (partially hydrolyzed polyacrylamide) dissolved in water. The Reynolds number based on the mean velocity, the pipe diameter and the local viscosity at the wall is approximately 10000. We have used polymer solutions with three different concentrations which have been chosen such that maximum drag reduction occurs. The amount of drag reduction found is 60–70%. Our experimental results are compared with results obtained with water and with a very dilute solution which exhibits only a small amount of drag reduction. We have focused on the observation of turbulence statistics (mean velocities and turbulence intensities) and on the various contributions to the total shear stress. The latter consists of a turbulent, a solvent (viscous) and a polymeric part. The polymers are found to contribute significantly to the total stress. With respect to the mean velocity profile we find a thickening of the buffer layer and an increase in the slope of the logarithmic profile. With respect to the turbulence statistics we find for the streamwise velocity fluctuations an increase of the root mean square at low polymer concentration but a return to values comparable to those for water at higher concentrations. The root mean square of the normal velocity fluctuations shows a strong decrease. Also the Reynolds (turbulent) shear stress and the correlation coefficient between the stream wise and the normal components are drastically reduced over the entire pipe diameter. In all cases the Reynolds stress stays definitely non-zero at maximum drag reduction. The consequence of the drop of the Reynolds stress is a large polymer stress, which can be 60% of the total stress. The kinetic-energy balance of the mean flow shows a large transfer of energy directly to the polymers instead of the route by turbulence. The kinetic energy of the turbulence suggests a possibly negative polymeric dissipation of turbulent energy. This revised version was published online in July 2006 with corrections to the Cover Date.     

A unified theory of turbulent flow

Summary A general theory of turbulent flow is applied to incompressible plane Couette flow. It is found that a unique formulation is not obtained because of a singularity in the equations and problems relating to the boundary conditions. Solutions are obtained for several different assumptions. The characteristic feature is a square root velocity profile for high Reynolds numbers. The logarithmic law is obtained as a divergent approximation. There are discrepancies in the available experimental data; one set agreeing with the square root form, and a second set with the logarithmic form.     

Velocity distribution of flow with submerged flexible vegetations based on mixing-length approach

By choosing a PVC slice to simulate flexible vegetation, we carried out experiments in an open channel with submerged flexible vegetation. A 3D acoustic Doppler velocimeter （micro ADV） was used to measure local flow velocities and Reynolds stress. The results show that hydraulic characteristics in non-vegetation and vegetation layers are totally different. In a region above the vegetation, Reynolds stress distribution is linear, and the measured velocity profile is a classical logarithmic one. Based on the concept of new-riverbed, the river compression parameter representing the impact of vegetation on river is given, and a new assumption of mixing length expression is made. The formula for time-averaged velocity derived from the expression requires less parameters and simple calculation, and is useful in applications.     

考虑温粘热效应的多瓦轴承模型及其应用

构筑了轴向解析、周向有限元压力分布的一维变粘度场有限宽轴承模型。在绝热边界条件下,忽略泊肃叶流项对速度的影响,不考虑温度轴向变化并沿油膜厚度方向积分,三维能量方程可降阶为平均温度场只沿周向分布的一维形式,结合滑动轴承非线性油膜力的一维直接解法,能量方程与雷诺方程可分别求解,既考虑了温粘效应对滑动轴承非线性动力学性能的影响,又提供了无需迭代直接确定油膜破裂边界和求解非线性油膜力的快速新方法。作为应用,针对进油槽位于水平两侧的椭圆瓦轴承进行了动力润滑热效应分析,与工程数据比较,计算结果吻合,证明该模型合理,适用于工程上多瓦轴承的分析计算。     

Some Results Based on Near-wall Turbulence

In this paper some results based on the near-wall mean characteristics of a bounded turbulent flow are presented. In the study empirical polynomials and experimental data for an attached wall-bounded flow are used with the objective of studying the time-scales similarities in the very-near-wall region. As a result of this analysis a new parameter to characterize the high to low-Reynolds turbulence transition in the context of turbulence models is proposed. A relation for the Reynolds stress in the buffer region is also proposed, which allows a mean velocity profile through the buffer region to be obtained. This mean velocity profile joins the logarithmic ones at the beginning of the inertial sub-layer and fits appropriately to experimental data. Another result derived from the previous analysis is an expression for the eddy viscosity through the very-near-wall region. Comparison of this expression with those relations used by four known low-Reynolds models reveals that it has a very good performance.     

Semi-analytic solution of the two-dimensional turbulent energy equation in round tubes using an analytic velocity profile and its experimental validation

 A semi-analytic solution of the temperature development of single-phase, turbulent viscous flows inside smooth round tubes is performed. The special feature of the theoretical analysis revolves around two single universal functions of analytic form for the accurate characterization of the turbulent diffusivity of momentum and the turbulent velocity profile in the entire cross-section of a round tube. Using this valuable information that emanates from the analytic solution of the one-dimensional momentum balance equation, the two-dimensional energy balance equation was reformulated into an adjoint system of ordinary differential equations of first–order with constant coefficients. Each equation in the system of differential equations governs the axial variation of the average temperature of a finite volume of fluid of annular shape. Exploiting the linearity of the system of differential equations, an analytic solution of it was obtained via the matrix eigenvalue method with LAPACK, a library of Fortran 77 subroutines for numerical linear algebra. Reliable series have been determined for the axial variation of the two thermal quantities of importance: (a) the time-mean bulk temperature and (b) the local Nusselt number. The semi-analytic nature of the local Nusselt number distribution is advantageous because it may be viewed as an analytic-based correlation equation. Prediction of the local Nusselt numbers for turbulent air flows compare satisfactorily with the comprehensive correlation equations and the abundant experimental data that are accessible from the literature. The air flows are regulated by a wide spectrum of turbulent Reynolds numbers. Received on 4 June 2001 RID="★" ID="★" Current address Mechanical Engineering Dept. The University of Vermont Burlington, VT 05405, USA     

The behavior of viscoelastic squeeze films subject to normal oscillations, including the effect of fluid inertia

The problem of the squeeze film flow of a viscoelastic fluid between parallel, circular disks is analyzed. The upper disk is subject to small, axial oscillations. Lodge's “rubber-like liquid” is used as the viscoelastic fluid model, and fluid inertia forces are included. An exact solution to the equations of motion is obtained involving in-phase and out-of-phase components of velocity field and load, with respect to the plate velocity. Peculiar resonance phenomena in the load amplitude are exhibited at high Deborah number. At certain combinations of Reynolds number and Deborah number, the in-phase and/or out-of-phase velocity field components may attain an unusual circulating type of motion in which the flow reverses direction across the film. In the low Deborah number limit, and in the low Reynolds number limit, the results of this study reduce to those obtained by other workers.     

Viscous dissipation of Rankine vortex profile in zero meridional flow

An analytical solution is given for a time-decay Rankine vortex profile due to viscous effects. The vortex filament is assumed to be isolated, strong, concentrated and having zero-meridional flow (i.e. radial and axial velocities are equal to zero). Zero-meridional renders the governing equations for an unsteady, incompressible and axisymmetric vortex in a simple form. Based on the tangential momentum equation, the spatial-temporal distributions of the swirl velocity are given in terms of Fourier-Bessel series by using separation of variables technique. A general formula is derived by total differentiation of the swirl velocity with respect to time, depicting the viscous dissipation for Oseen and Taylor-like vortex profiles. This analysis is validated by comparison with previous experimental data.     

Asymptotic solutions of the flow of a Johnson-Segalman fluid through a slowly varying pipe using double perturbation strategy

A double perturbation strategy is presented to solve the asymptotic solutions of a Johnson-Segalman (J-S) fluid through a slowly varying pipe. First, a small parameter of the slowly varying angle is taken as the small perturbation parameter, and then the second-order asymptotic solution of the flow of a Newtonian fluid through a slowly varying pipe is obtained in the first perturbation strategy. Second, the viscoelastic parameter is selected as the small perturbation parameter in the second perturbation strategy to solve the asymptotic solution of the flow of a J-S fluid through a slowly varying pipe. Finally, the parameter effects, including the axial distance, the slowly varying angle, and the Reynolds number, on the velocity distributions are analyzed. The results show that the increases in both the axial distance and the slowly varying angle make the axial velocity slow down. However, the radial velocity increases with the slowly varying angle, and decreases with the axial distance. There are two special positions in the distribution curves of the axial velocity and the radial velocity with different Reynolds numbers, and there are different trends on both sides of the special positions. The double perturbation strategy is applicable to such problems with the flow of a non-Newtonian fluid through a slowly varying pipe.     

Viscous dissipation of Rankine vortex profile in zero meridional flow

An analytical solution is given for a time-decay Rankine vortex profile due to viscous effects. The vortex filament is assumed to be isolated, strong, concentrated and having zero-meridional flow (i.e. radial and axial velocities are equal to zero). Zero-meridional renders the governing equations for an unsteady, incompressible and axisymmetric vortex in a simple form. Based on the tangential momentum equation, the spatial-temporal distributions of the swirl velocity are given in terms of Fourier-Bessel series by using separation of variables technique. A general formula is derived by total differentiation of the swirl velocity with respect to time, depicting the viscous dissipation for Oseen and Taylor-like vortex profiles. This analysis is validated by comparison with previous experimental data. The English text was polished by Yunming Chen.     

Axisymmetric stagnation flow obliquely impinging on a moving circular cylinder with uniform transpiration

Laminar stagnation flow, axisymmetrically yet obliquely impinging on a moving circular cylinder, is formulated as an exact solution of the Navier–Stokes equations. Axial velocity is time‐dependent, whereas the surface transpiration is uniform and steady. The impinging free stream is steady with a strain rate k?. The governing parameters are the stagnation‐flow Reynolds number Re=k?a²/2ν, and the dimensionless transpiration S=U₀/k?a. An exact solution is obtained by reducing the Navier–Stokes equations to a system of differential equations governed by Reynolds number and the dimensionless wall transpiration rate, S. The system of Boundary Value Problems is then solved by the shooting method and by deploying a finite difference scheme as a semi‐similar solution. The results are presented for velocity similarity functions, axial shear stress and stream functions for a variety of cases. Shear stresses in all cases increase with the increase in Reynolds number and suction rate. The effect of different parameters on the deflection of viscous stagnation circle is also determined. Copyright © 2010 John Wiley & Sons, Ltd.     

Non-Isotropic Length Scales During the Compression Stroke of a Motored Piston Engine

For the case of axial compression the two-point velocity correlation equations of axisymmetric homogeneous turbulence are derived. Appropriate integrations then lead to equations for the components of the Reynolds stress tensor as well as to those for the two independent integral length-scales characterizing axisymmetric homogeneous turbulence. These equations contain a certain number of empirical constants. Values for these constants are taken from the literature, or were adjusted from the present data.The resulting model is validated using data from a motored piston engine. The flow field, which has negligible swirl and tumble, has been measured using particle image velocimetry (PIV). Since turbulence is axisymmetric and homogeneous in the counter region, two-dimensional PIV provides the time history of the axial and radial length-scales. The experimental data are compared with the mathematical model.     

Reynolds stress modelling of rectangular open‐channel flow

A Reynolds stress model for the numerical simulation of uniform 3D turbulent open‐channel flows is described. The finite volume method is used for the numerical solution of the flow equations and transport equations of the Reynolds stress components. The overall solution strategy is the SIMPLER algorithm, and the power‐law scheme is used to discretize the convection and diffusion terms in the governing equations. The developed model is applied to a flow at a Reynolds number of 77000 in a rectangular channel with a width to depth ratio of 2. The simulated mean flow and turbulence structures are compared with measured and computed data from the literature. The computed flow vectors in the plane normal to the streamwise direction show a small vortex, called inner secondary currents, located at the juncture of the sidewall and the free surface as well as the free surface and bottom vortices. This small vortex causes a significant increase in the wall shear stress in the vicinity of the free surface. A budget analysis of the streamwise vorticity is carried out. It is found that both production terms by anisotropy of Reynolds normal stress and by Reynolds shear stress contribute to the generation of secondary currents. Copyright © 2006 John Wiley & Sons, Ltd.     

Fundamentals of physics of fluid flow reconsidered

Recently, two significant defects of the state-of-the-art of theoretical fluid mechanics have been evidenced, namely the ambiguity of the Navier Stokes equations with respect to the viscous stress tensor, and the inconsistency of the Reynolds stress approach with the symmetry of the inertial tensor. Therefore, by means of two classical topics of fluid mechanics, namely the universal velocity profile and the laminar boundary layer, the physics of fluid flow are scrutinized. The validity of the so obtained results is tested at the example of an exact solution of the Navier Stokes equations, namely the diffusion of the circulation of an eddy.