Numerical study of plane couette flow in a rotating framework

Summary A numerical study of laminar plane Couette flow subjected to a steady spanwise rotation is conducted. The full nonlinear Navier-Stokes equations in a steadily rotating framework are solved by a finite difference method for a long, large-aspect-ratio rectangular channel where the outer wall moves at a constant velocity. In this manner, nonlinear and wall end effects which are present in any real laboratory Couette flow experiment are taken into account. The computations demonstrate the existence of a roll instability, at intermediate rotation rates, when the Reynolds number exceeds a critical value of 42. The associated secondary flow in the form of longitudinal rolls is shown to have a severe distortional effect on the primary axial velocity in the interior of the channel which is not linear like its counterpart in an inertial framework. Comparisons are made with previously conducted linear stability analyses as well as with other analogous numerical and experimental studies.With 15 Figures     

The stability of low-amplitude three-dimensional disturbances in a nonequilibrium compressible boundary layer

The stability of Tollmien-Schlichting waves propagating at an angle to the main flow in a nonequilibrium compressible supersonic boundary layer is investigated within the linear theory of hydrodynamic stability. The dependences of the critical Reynolds number on the degree of disequilibrium and on the Mach number of undisturbed flow are found at different angles of wave propagation. It is demonstrated that the critical Reynolds number in a nonequilibrium medium may decrease appreciably with increasing degree of disequilibrium, which results in the reduction of the characteristic length of the linear region of transition to turbulence.     

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

The effect of inertia on the Yih–Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr–Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.     

Axisymmetric flow of two fluids in a pulsating pipe

The motion of a viscous thread surrounded by an annular viscous layer inside a pulsating cylindrical pipe whose radius is a periodic function of time is investigated. At zero Reynolds number, a stagnation-point-type solution may be written down in closed form. A Floquet linear stability analysis for Stokes flow reveals the pulsations either decrease or increase the growth rate of longwave disturbances depending on the initial radius of the thread. For a moderate-sized initial thread radius, increasing the amplitude of the pulsations decreases the critical wavenumber for instability to below the classical Rayleigh threshold. Increasing the viscosity contrast, so that the fluid in the annular layer becomes more viscous than the fluid in the thread, tends to decrease the growth rate of disturbances. In the second part of the paper, the basic stagnation-point-type flow at arbitrary Reynolds number is computed using a numerical method on the assumption that the interface is a circular cylinder at all times. During the motion, either the thread radius tends to increase and the thickness of the annular layer decreases, or else the thread tends to thin and the thickness of the annular layer increases, depending upon the initial conditions and the parameter values. For a judicious choice of initial condition, a time-periodic exact solution of the Navier–Stokes equations is identified.     

The influence of nonparallelism in channel flow stability

Summary Considered below are the high Reynolds number flows of an incompressible fluid, and their nonparallel stability properties, in certain plane channels whose widths vary slowly in the streamwise direction. The first approximation to the steady-state flow is governed by the classical boundary layer equations but with the pressure unknown. Solutions, some including separation and reattachment, to this are obtained numerically for a range of the parameters involved, and the stability of the resulting flows is considered using fixed frequency disturbances and taking into account the nonparallel nature of the basic flow by use of a W.K.B. method. The calculations yield critical Reynolds numbers, below which all the disturbances decay downstream, and for various Reynolds numbers above the critical values the growths of the small disturbances are calculated. The results are specialized to a particular class of channels which are straight far upstream and far downstream but vary in between. So the predictions should be much more easily amenable to experimental investigation and comparisons than those of the more idealized, diverging channel, flow problem studied by Eagles & Weissman [1] and the only other basic flow seriously studied from the viewpoint of nonparallel flow stability theory, the Blasius boundary layer. The results also represent the first application of both quasiparallel and slightly non-parallel stability theory to channels involving slowly varying but finite changes in width.     

单声道超声水表测量特性分段校正方法的研究

当管道内流体处于不同雷诺数测量条件时,超声水表的线平均流速v_L与面平均流速v_S之间存在着显著的非线性。根据管道内被测流体介质流动分布状态不同,提出了一种分段流量测量特性校正新方法,在其临界处设立校正分界点,当层流时,采用常系数校正;湍流与过渡流时,分别采用拟合直线方程校正。经实验验证,该方法是可行的。     

Stability of Bödewadt flow

The stability of the flow produced over an infinite stationary plane in a fluid rotating with uniform angular velocity at an infinite distance from the plane is considered. The basic flow is an exact solution of the Navier-Stokes equations making it amenable to theoretical study. An asymptotic investigation is presented in the limit of large Reynolds number. It is shown that the stationary spiral instabilities observed experimentally can be described by a linear inviscid stability analysis. The prediction obtained for the wave angle of the disturbances is found to agree well with the available experimental and numerical results.     

微管道内湍流转捩的实验研究

研究旨在确定微管道内流动从层流到湍流转捩的临界雷诺数。利用微观粒子图像测速技术(Micro-PIV)研究了去离子水在内径为230μm的圆形截面玻璃微管道内的流场结构,得到了从层流到充分发展湍流各流动状态下的轴向平均速度分布和湍流度分布,实验雷诺数为1020~3145,同时研究了微管道内的流动阻力特性。平均速度场和脉动速度场的实验结果表明微管道内从层流到湍流的转捩发生在Re=1800~1900左右,与流动阻力的测量结果一致,与宏观流动比较,并未发现微管道内的流动转捩有明显提前。实验结果还显示,当Re>2700时,微管道内的平均流速分布和相对湍流度分布呈现典型的充分发展湍流状态特征。     

Experiments on vertical slender flexible cylinders clamped at both ends and subjected to axial flow

Three series of experiments were conducted on vertical clamped-clamped cylinders in order to observe experimentally the dynamical behaviour of the system, and the results are compared with theoretical predictions. In the first series of experiments, the downstream end of the clamped-clamped cylinder was free to slide axially, while in the second, the downstream end was fixed; the influence of externally applied axial compression was also studied in this series of experiments. The third series of experiments was similar to the second, except that a considerably more slender, hollow cylinder was used. In these experiments, the cylinder lost stability by divergence at a sufficiently high flow velocity and the amplitude of buckling increased thereafter. At higher flow velocities, the cylinder lost stability by flutter (attainable only in the third series of experiments), confirming experimentally the existence of a post-divergence oscillatory instability, which was previously predicted by both linear and nonlinear theory. Good quantitative agreement is obtained between theory and experiment for the amplitude of buckling, and for the critical flow velocities.     

Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows

The failure of normal-mode linear stability analysis to predict a transition Reynolds number (Retr) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re. The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re, are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al. indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Retr of approximately 2000 associated with the bifurcation between two deviation solutions.     

Stability of internally mixing flow

Hydrodynamic stability of viscous shear flow resulting from the mixing of two parallel streams of fluid along the centre line of a channel is investigated. A linear stability analysis is employed and eigenvalues are extracted using a variational principle combined with a Rayleigh-Ritz procedure. Velocity profiles for the base flow are obtained from a numerical solution of the laminar mixing problem. Results show a sharp drop in critical Reynolds number as the streams start to mix. However, the flow is generally stable at the mean flow Reynolds number investigated here. These results have practical implications for enhancing mixing and heat or mass transfer in thermal systems of small characteristic dimensions, as found in microelectronic equipment packages.     

Transition to Turbulence in Pipe Flow

We review the results of recent experimental investigations into transition to turbulence in fluid flow through a circular straight pipe, at room temperature. The stability of Hagen–Poiseuille flow was investigated using impulsive perturbations by either injecting or sucking small amounts of fluid through holes in the wall of the pipe. The evolution of the induced patches of disturbed flow were observed using flow visualization and laser Doppler velocimetry. The principle result obtained was a finite amplitude stability curve where the critical amplitude of the disturbance required to cause transition is found to be inversely proportional to the Reynolds number. Estimates for the lower threshold value of Reynolds number which is required to sustain turbulence were also measured.     

Axial drive to nonlinear flow between rotating cylinders

Stability of pseudoplastic rotational flow between cylinders in presence of an independent axial component is investigated. The fluid is assumed to follow the Carreau model and mixed boundary conditions are imposed. The conservation of mass and momentum equations give rise to a four-dimensional low-order dynamical system, including additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. In absence of the axial flow, as the pseudoplasticity effects increases, the purely-azimuthal base flow loses its stability to the vortex structure at a lower critical Taylor number. Emergence of the vortices corresponds to the onset of a supercritical bifurcation also present in the flow of a linear fluid. However, unlike the Newtonian case, pseudoplastic Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Existence of an axial flow induced by a pressure gradient appears to further advance each critical point on the bifurcation diagram. In continuation, complete flow field together with viscosity maps is analyzed for different flow scenarios. Through evaluation of the Lyapunov exponent, flow stability and temporal behavior of the system for cases with and without axial flow are brought to attention.     

Hydrodynamic interaction of two neutrally-buoyant smooth spheres suspended in plane Poiseuille flow: the BEM simulations versus the MoR approximations

The hydrodynamic interaction between two particles suspended in shear flows is fundamental to the macroscopic characterization of suspension flows. Although such interaction in quiescent or linear shear flow is well understood, studies on that in a nonlinear shear field are rare. In this study, the hydrodynamic interaction between two neutrally-buoyant smooth spheres moving at negligible Reynolds numbers in an unbounded plane Poiseuille flow has been calculated by three-dimensional boundary element method (BEM) simulations. The BEM results have been compared with the analytical results obtained with the method-of-reflection (MoR) approximations. The BEM simulations have been found to provide satisfactory predictions if the number of elements on the spheres are more than 200, whereas the MoR approximations provide satisfactory predictions only when the minimum separation between the spheres is relatively large although this MoR method has the advantage to easily calculate the hydrodynamic interaction between two spheres freely moving at negligible Reynolds numbers in unbounded quadratic flow by solving ordinary differential equations. Furthermore, it is found that there is a preferential cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of shear in plane Poiseuille flow which does not arise in simple shear flow. This migration is always directed towards low shear regions when the sphere having larger translational velocity approaches the other sphere, and reverses towards high shear regions when the faster sphere leads the other sphere in the plane of shear. There is also a cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of vorticity, but this migration does not have a preferential direction. These migrations are symmetric about the point where the spheres are at the minimum separation, and are only significant when the hydrodynamic interaction of the spheres is strong. These results show that the migration of the center-of-gravity of the sphere-pair can be attributed to the nonlinearity of the shear field, which agrees with the MoR approximations. The hydrodynamic interaction between the two spheres has been quantified under various conditions by the BEM simulations for both identical and disparate spheres.     

Gas–Liquid Interfacial Friction Factor for the Transition from Stratified to Slug Flow

Calculations based on the slug stability model and simplified stratified flow model provide predictions of the critical liquid height and the critical superficial velocities of a stratified flow for the transition to a slug flow in a horizontal pipe. Since slug flow derives from different interfacial waves patterns, previous interfacial waves model in stratified gas–liquid flows brings about the discrepancy between theoretical prediction and experimental data. A partial analysis for this behavior is given, which recognizes that the values of gas–liquid interfacial friction factor at the onset of slug flow have been underestimated, especially at high gas flows, and they should be obtained indirectly from other measured variables. Modified correlations for the interfacial friction factor are presented and better agreement between predicted and measured critical superficial velocities has been obtained.     

临界雷诺数下带人工水线斜拉索气动性能研究

通过风洞试验,在临界雷诺数下,研究带上水线拉索模型的气动性能。测得不同风向角下拉索表面风压分布情况,得到作用在拉索模型上的气动力系数,分析临界雷诺数下拉索的气动稳定性。研究表明:基于准定常假定,在临界雷诺数下,拉索发生风雨激振的可能性较亚临界区低;上水线改变了拉索表面的流体分离并出现分离流再附现象,导致拉索气动力随上水线位置的改变而发生剧烈变化。     

Freckle formation in a solidifying binary alloy

The effect of solutal solidification on the unidirectional solidification of a binary alloy cooled from below is considered. Soon after the onset of convection a mushy region often forms, and is accompanied by vigorous convection in the molten alloy above. In contrast, the fluid in the mush appears unaffected by the neighbouring flow and remains essentially quiescent. This work considers the nonlinear convective stability of the mush and determines a criterion for channelling in the mush to occur. The basic state is a similarity solution, so that a quasi-static approximation must be applied in order to apply conventional stability theory. Moreover, although the model for solidification is relatively simple, an analytical expression for linear stability is not available. Thus the series of equations arising from the nonlinear stability analysis lead to a complicated set of symbolic and numerical calculations. Stable finite-amplitude solutions are found for Rayleigh numbers larger than critical for all values of the chosen superheat. The nonlinear solutions demonstrate the possibility of channelling within the mush dependent upon the strength of convection. These finite-amplitude solutions are extended further by the calculation of a numerical solution of the model equations. The evolution of the stream function and the mass fraction is followed, the onset of convection and freckling can then be deduced from the numerical simulation. The onset of convection in the mush is found in terms of the mush Rayleigh number, and compares favourably with linear stability theory and experimental data. The onset of freckling is also given in terms of a Rayleigh number, but is sensitive to the initial conditions. This appears to explain the large disagreement found in experiments aimed at finding a criterion for freckling.     

The interaction between inner and outer regions of turbulent wall-bounded flow

The nature of the interaction between the inner and outer regions of turbulent wall-bounded flow is examined. Townsend's theory of inactive motion is shown to be a first-order, linear approximation of the effect of the large eddies at the surface that acts as a quasi-inviscid, low-frequency modulation of the shear-stress-bearing motion. This is shown to be a 'strong' asymptotic condition that directly expresses the decoupling of the inner-scale active motion from the outer-scale inactive motion. It is further shown that such a decoupling of the inner and outer vorticity fields near the wall is inappropriate, even at high Reynolds numbers, and that a 'weak' asymptotic condition is required to represent the increasing effect of outer-scale influences as the Reynolds number increases. High Reynolds number data from a fully developed pipe flow and the atmospheric surface layer are used to show that the large-scale motion penetrates to the wall, the inner-outer interaction is not describable as a linear process and the interaction should more generally be accepted as an intrinsically nonlinear one.     

Effect of long undulated bottoms on thin gravity-driven films

Summary We carry out a perturbation analysis for steady gravity-driven film flow over undulations of moderate steepness that are long compared to the film thickness and study the linear stability of the flow in the framework of Floquet analysis. The effect of geometric nonlinearities on the instability becomes relevant for moderate bottom variations. We find that the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom. At higher inclination angles, the theoretical results are in good quantitative agreement with experiment. At inclination angles where the flat part of the undulation is close to being horizontal, the basic solution for the steady flow fails to describe the flow in the flat part, and the linear stability analysis overestimates the critical Reynolds number.     

Separating shear flow past a surface-mounted blunt obstacle

The planar flow of incompressible fluid past a blunt obstacle mounted on a flat (horizontal) fixed solid surface of infinite extent is examined in the presence of an incident linear velocity profile, modelling the fluid behaviour close to a small surface roughness for instance. The motion is taken to be steady and laminar. The obstacle is blunt in the sense that its typical surface slopes are not small, a feature which here always induces flow separation both upstream and downstream of the obstacle. Computations and nonlinear theory are applied, together with comparisons. The direct computations of the Navier-Stokes equations, using for example a higher order upwind-difference scheme, deal with a moderate range of Reynolds numbers up to 200, based on the obstacle height and the incident uniform shear. In addition the accuracy is necessarily limited as the Reynolds number increases. The theory is for large Reynolds numbers and is based on viscous-inviscid reasoning, back-pressure effects from the obstacle and slender-layer separation locally, among other influences. The comparisons nevertheless yield encouragingly close agreement, for the present computed cases of a vertical flap or a rectangular block. This is both quantitatively, in terms of the upstream separation and downstream reattachment positions in particular, and generally, in terms of the separating flow structure, even at the notably moderate Reynolds numbers covered accurately by the computations.