On Tollmien–Schlichting-like waves in streaky boundary layers

The linear stability of the boundary layer developing on a flat plate in the presence of finite-amplitude, steady and spanwise periodic streamwise streaks is investigated. The streak amplitudes considered here are below the threshold for onset of the inviscid inflectional instability of sinuous perturbations. It is found that, as the amplitude of the streaks is increased, the most unstable viscous waves evolve from two-dimensional Tollmien–Schlichting waves into three-dimensional varicose fundamental modes which compare well with early experimental findings. The analysis of the growth rates of these modes confirms the stabilising effect of the streaks on the viscous instability and that this stabilising effect increases with the streak amplitude. Varicose subharmonic modes are also found to be unstable but they have growth rates which typically are an order of magnitude lower than those of fundamental modes. The perturbation kinetic energy production associated with the spanwise shear of the streaky flow is found to play an essential role in the observed stabilisation. The possible relevance of the streak stabilising role for applications in boundary layer transition delay is discussed.     

Temporal and spatial instability of an inviscid compound jet

This paper examines the linear hydrodynamic stability of an inviscid compound jet. We perform the temporal and the spatial analyses in a unified framework in terms of transforms. The two analyses agree in the limit of large jet velocity. The dispersion equation is explicit in the growth rate, affording an analytical solution. In the temporal analysis, there are two growing modes, stretching and squeezing. Thin film asymptotic expressions provide insight into the instability mechanism. The spatial analysis shows that the compound jet is absolutely unstable for small jet velocities and admits a convectively growing instability for larger velocities. We study the effect of the system parameters on the temporal growth rate and that of the jet velocity on the spatial growth rate. Predictions of both the temporal and the spatial theories compare well with experiment.Dedicated to the memory of Professor Tasos C. Papanastasiou     

Interactions between pairs of oblique waves in a Bickley jet

We consider the weakly nonlinear spatial evolution of a pair of varicose oblique waves and a pair of sinuous oblique waves superimposed on an inviscid Bickley jet, with each wave being slightly amplified on a linear basis. The two pairs are assumed to both be inclined at the same angle to the plane of the jet. A nonlinear critical layer analysis is employed to derive equations governing the evolution of the instability wave amplitudes, which contain a coupling between the modes. These equations are discussed and solved numerically, and it is shown that, as in related work for other flows, these equations may develop a singularity at a finite distance downstream.     

Dynamics of large-scale eddies in turbulent flows

A model equation based on the equipartition of the turbulent dissipation is proposed for describing the dynamics of large-scale eddies in turbulent flows. The equation is reducible to the equation of motion of an inviscid fluid, so that the motion of the large-scale eddies can be described in terms of inviscid fluid dynamics. It is found that the large-scale eddies are always weakened by the background turbulence and their evolution is slowed down compared with the corresponding inviscid motion. In the case of turbulent mixing layer, its linear growth in downstream direction is accounted for by the exponential growth in time of the perturbation in an inviscid plane vortex sheet.     

Phase averaged velocity field in an axisymmetric jet subject to a sudden velocity decrease

An unsteady transient axisymmetric turbulent jet was studied experimentally. The initial flow perturbation consisted of a sudden and large decrease in the ejection velocity. The temporal evolution of the mean and fluctuating unsteady velocity field was measured by using X hot-wire probes. In the jet far field, adaptation of the externally imposed unsteadiness to the local jet time scale is confirmed quantitatively. The main features of the phase averaged velocity field are presented and comments are made about the instantaneous state of the turbulence energetics. Transient mean radial velocities are deduced and an important increase of the instantaneous rate of entraining external fluid into the jet is found. Finally, we show that the pressure effect due to radial impusle terms plays an important role in the propagation of the mean perturbation. The longitudinal adaptation of the perturbation time scale driven by the local jet time scale provides a turbulent flow that is intermediate to quasi-static flows and rapidly distorted flows.We wish to thank Professor H. Fiedler and Professor M. Wolfshtein for their helpful comments about this work. We have benefited greatly from discussions with Dr. H.J. Nuglisch, Professor E.K. Longmire and Dr. A. Sevrain and of the technical support of G. Couteau and J.F. Alquier.     

Localized disturbances in parallel shear flows

The development of localized disturbances in parallel shear flows is reviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to bring out the key asymptotic results in an explicit form. In this context, the distinctive differences between the wave-packet associated with the asymptotic behavior of eigenmodes and the non-dispersive (inviscid) continuous spectrum is highlighted. The largest growth is found for three-dimensional disturbances and occurs in the normal vorticity component. It is due to an algebraic instability associated with the lift-up effect. Comparison is also made between the analytical results and some numerical calculations.Next the viscous case is treated, where the complete solution to the initial value problem is presented for bounded flows using eigenfunction expansions. The asymptotic, wave-packet type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum transient growth possible to obtain from this mechanism is also presented, the so called optimal growth.Lastly the application of the dynamics of three dimensional disturbances in modeling of coherent structures in turbulent flows is discussed.     

On the inviscid acoustic-mode instability of supersonic shear flows

The same methods used previously to study acoustic-mode instability in supersonic boundary layers are applied to free shear layers, and new calculations are made for boundary layers with cooling and suction. The objective is to obtain additional information about acoustic-mode instability, and to find what features of the instability are common to boundary layers and free shear flows. Acoustic modes exist whenever there is an embedded region of locally supersonic flow relative to the phase speed of the instability wave. Consequently, they can be found in boundary layers, wakes, and jets, but not in mixing layers unless the flow is confined. In this first part of a two-part paper, attention is directed principally to two-dimensional waves. The linear, inviscid stability theory is used to calculate spatial amplification rates at Mach number 3 for the sinuous and varicose modes of a single wake flow and a single jet flow, each made up of the same mixing-layer profile plus a central region of uniform flow. Along with sequences of sinuous and varicose unstable modes clearly identifiable as acoustic modes, both of these flows, unlike the boundary layer, have a lowest sinuous mode that is the most unstable. The unstable modes include both subsonic and radiating disturbances with large amplification rates. The latter phenomenon is also found for highly cooled boundary layers with suction. In these boundary layers, suction is generally stabilizing for nonradiating acoustic disturbances, but destabilizing for radiating disturbances.The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Support from the Aerodynamics Division of the Office of Aeronautics and Exploration Technology is gratefully acknowledged. A preliminary version of this paper was presented at the Fourth Symposium on Numerical and Physical Aspects of Aerodynamic Flows, California State University, Long Beach, CA, 16–19 January 1989.     

Generation of internal waves by a turbulent jet in a stratified fluid

The process of generation of internal waves by an initially cylindrical, turbulent jet with a Gaussian profile of the average horizontal velocity component in a fluid with stable linear density stratification is investigated by direct numerical simulation. It is shown that on time intervals Nt < 30, where N is the buoyancy frequency, the vertical velocity pulsations collapse, which is accompanied by the generation of internal waves whose spatial period is close to the wavelength of the spiral mode of jet instability in a homogeneous fluid. The wave dynamics and kinematics can be satisfactorily described by the linear theory for a pulsed source and their parameters are in good agreement with the parameters of the “coherent” internal waves generated by a stratified wake in a laboratory experiment. At large times the wave generation ceases and the variations of the fluid density are localized in the neighborhood of the centers of large-scale vortices formed in the horizontal plane in the neighborhood of the jet.     

被均匀流缓慢调制的有限水深毛细重力波

非线性的存在会产生高次谐波,这些谐波又反作用于原来的低次谐波,使波幅发生缓慢变化,从而产生缓慢调制现象.这里从考虑均匀流作用下的毛细重力水波基本方程出发,在不可压缩、无旋、无黏条件假设下,使用多重尺度分析方法推导出了在均匀流影响下有限深水毛细重力波振幅所满足的非线性Schr¨odinger方程(NLSE).分析了NLSE解的调制不稳定性.给出了毛细重力波调制不稳定的条件和钟型孤立波产生的条件.分析了无量纲最大不稳定增长率随无量纲水深和表面张力的变化趋势.同时给出了无量纲不稳定增长率随无量纲微扰动波数变化的曲线,呈现出了先增大后减小的趋势.最后指出均匀顺流减小了无量纲不稳定增长率及最大增长率,逆流增大了它们.由表面张力作用产生的毛细波及重力与表面张力共同作用产生的毛细重力波,与流的相互作用可以改变海表粗糙度和海洋上层流场结构,进而影响海气界面动量、热量及水汽的交换.了解海表这些短波动力机制,对卫星遥感的精确测量、海气相互作用的研究及海气耦合模式的改进等有重要意义.     

气流作用下同轴带电射流的不稳定性研究

通过对气体驱动同轴电流动聚焦的实验模型进行简化,开展了电场力和惯性力共同作用下同轴带电射流的不稳定性理论研究.在流动为无黏、不可压缩、无旋的假设下,建立了三层流体带电射流物理模型并得到了扰动在时间域内发展演化的解析形式色散关系,利用正则模方法求解色散方程发现了流动的不稳定模态,进而分析了主要控制参数对不稳定模态的影响.结果表明,在参考状态下轴对称模态的最不稳定增长率最大,因此轴对称扰动控制整个流场.外层气流速度越高,气体惯性力越大,射流的界面越容易失稳.内外层液-液同轴射流之间的速度差越大,射流越不稳定.表面张力对射流不稳定性起到促进作用.轴向电场对射流不稳定性具有双重影响:当加载电场强度较小时,射流不稳定性被抑制;当施加电压大于某一临界值时,轴向电场会促进射流失稳.临界电压的大小与界面上自由电荷密度和射流表面扰动发展关系密切.这些结果与已有的实验现象吻合,能够对实验的过程控制提供理论指导.     

Transient behavior of an axisymmetric turbulent jet

The prediction of the response of unsteady flows submitted to external excitation is a real challenge for the optimization of industrial processes. As the jet flow is a very basic turbulent flow related to mixing and entrainment phenomena via turbulent structure dynamic, we investigate the transient behavior of an axisymmetric jet submitted to a large and sudden decrease of its ejection velocity. The non stationary flow evolution is studied experimentally. Measurements along the jet axis based on pure ensemble averaging show clearly the convective motion of the perturbation and the adaptation of the local interaction to the local jet time scale. A transverse investigation in the non stationary region show that the mean flow and its turbulence is deeply affected during the local velocity decrease.     

Instability of a viscoelastic liquid jet with axisymmetric and asymmetric disturbances

The temporal instability behavior of a viscoelastic liquid jet in the wind-induced regime with axisymmetric and asymmetric disturbances moving in an inviscid gaseous environment is investigated theoretically. The corresponding dispersion relation between the wave growth rate and the wavenumber is derived. The linear instability analysis shows that viscoelastic liquid jets are more unstable than their Newtonian counterparts, and less unstable than their inviscid counterparts, for both axisymmetric and asymmetric disturbances, respectively. The instability behavior of viscoelastic jets is influenced by the interaction of liquid viscosity and elasticity, in which the viscosity tends to dampen the instability, whereas the elasticity results in an enhancement of instability. Relatively, the effect of the ratio of deformation retardation to stress relaxation time on the instability of viscoelastic jets is weak. It is found that the liquid Weber number is a key measure that controls the viscoelastic jet instability behavior. At small Weber number, the axisymmetric disturbance dominates the instability of viscoelastic jets, i.e., the growth rate of an axisymmetric disturbance exceeds that of asymmetric disturbances. When the Weber number increases, both the growth rate and the instability range of disturbances increase drastically. The asymptotic analysis shows that at large Weber number, more asymmetric disturbance modes become unstable, and the growth rate of each asymmetric disturbance mode approaches that of the axisymmetric disturbance. Therefore, the asymmetric disturbances are more dangerous than that of axisymmetric disturbances for a viscoelastic jet at large Weber numbers. Similar to the liquid Weber number, the ratio of gas to liquid density is another key measure that affects the viscoelastic jet instability behavior substantially.     

Layered structure of turbulent plane wall jet

This paper investigates the layered structure of a turbulent plane wall jet at a distance from the nozzle exit. Based on the force balances in the mean momentum equation, the turbulent plane wall jet is divided into three regions: a boundary layer-like region (BLR) adjacent to the wall, a half free jet-like region (HJR) away from the wall, and a plug flow-like region (PFR) in between. In the PFR, the mean streamwise velocity is essentially the maximum velocity, and the simplified mean continuity and mean momentum equations result in a linear variation of the mean wall-normal velocity and Reynolds shear stress. In the HJR, as in a turbulent free jet, a proper scale for the mean wall-normal flow is the mean wall-normal velocity far from the wall and a proper scale for the Reynolds shear stress is the product of the maximum mean streamwise velocity and the velocity scale for the mean wall-normal flow. The BLR region can be divided into four sub-layers, similar to those in a canonical pressure-driven turbulent channel flow or shear-driven turbulent boundary layer flow. Building on the log-law for the mean streamwise velocity in the BLR, a new skin friction law is proposed for a turbulent wall jet. The new prediction agrees well with the correlation of Bradshaw and Gee (1960) over moderate Reynolds numbers, but gives larger skin frictions at higher Reynolds numbers.     

Computation of the two-dimensional viscous incompressible fluid flow with separation at the nlet edge around a cascade

A complex flow consisting of an outer inviscid stream, a dead-water separation domain, and a boundary layer, which interact strongly, is formed in viscous fluid flows with separation at the streamlined profile with high Re numbers. Different jet and vortex models of separation flow are known for an inviscid fluid; numerical, asymptotic, and integral methods [1–3] are used for a viscous fluid. The plane, stationary, turbulent flow through a turbine cascade by a constant-density fluid without and with separation from the inlet edge of the profile and subsequent attachment of the stream to the profile (a short, slender separation domain) is considered in this paper.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 34–44, May–June, 1978.     

Absolute inviscid instability of a ring jet with back-flow and swirl

The absolute instability of a ring jet with back-flow and swirl has been investigated on the basis of the inviscid linearized theory for incompressible flow. An axisymmetric disturbance mode is found to be only convectively unstable. The first azimuthal mode can become absolutely unstable, if the ring jet has a back-flow on the jet axis, and an additional swirl can increase the instability. However, for large swirl the absolute instability is suppressed. A ring jet without back-flow becomes absolutely unstable only in the presence of swirl.     

Linear instability characteristics of incompressible coaxial jets

The present study deals with the local linear instability of axisymmetric coaxial jets with a duct wall separating the two streams. The flow is assumed to be locally parallel, inviscid and incompressible. The objective of the work is to understand how the various parameters describing this flow geometry (i.e. boundary layers thicknesses at the exit, velocity ratio, wall thickness) may influence the instability of the flow and, in particular, the convective/absolute instability transition. A specific family of profiles is chosen for the modelling of the mean undisturbed flow and a spatial stability analysis is performed in order to identify the unstable modes and to assess how they are affected by the wake region behind the wall. An absolutely unstable mode is found, and its characteristics, depending on the velocity ratio and shear layers thicknesses, are determined. Results show that the absolute unstable mode is present only for a limited range of velocity ratios and that the corresponding frequency is almost constant if normalized with the mean velocity and wake thickness. This frequency value and the extension of the range of velocity ratios is similar to those found in the experiments on a similar geometry. Finally, a specific velocity ratio is found that maximizes the region at the jet exit for which an absolute instability behind the wall is present. This may increase the possibility for the onset of a global mode that may sustain the instability of the whole jet, enhancing considerably the mixing and entrainment characteristics between the two streams.     

Large eddy simulation of vertical turbulent jets under JONSWAP waves

The effect of random waves on vertical plane turbulent jets is studied numerically and the mechanism behind the interaction of the jet and waves is analyzed. The large eddy simulation method is used and the σ-coordinate system is adopted. Turbulence is modeled by a dynamic coherent eddy model. The σ-coordinate transformation is introduced to map the irregular physical domain with a wavy free surface and an uneven bottom onto a regular computational domain. The fractional step method is used to solve the fil...     

Direct numerical simulation of particle-laden plane turbulent wall jet and the influence of Stokes number

The distribution and motion of inertial particles in plane turbulent wall jet are investigated using direct numerical simulation, under the assumption of one-way coupling. To our knowledge, this appears to be the first direct numerical simulation of a particle-laden plane turbulent wall jet. It is shown that, in outer part of the wall jet, the behaviour of particles closely resembles that of a free plane jet. Due to the streamwise decay of particle Stokes number, the particle streaks formed in the near wall region of the wall jet are characterized by their intensity variation, which differs significantly from those in the channel flow. The streamwise growth of the particle velocity half-width is approximately equal to that of the fluid velocity half-width and the maximum velocity of particles decays slower than that of fluid due to inertia. The outer scaling can collapse the mean particle velocity in both the inner and outer region for heavier particles. In the buffer region, the particle–fluid velocity difference can be negative or positive depending on the Stokes number since there are two competing effects, namely the memory effect and turbophoresis. In the viscous region, the larger particles are on average faster than fluid and the velocity difference is found to be self-similar depending on outer Stokes number. The near-wall distribution of velocity difference is significantly correlated with the presence of high-momentum particles which are entrained by vortical structures generated in the outer region of the wall jet. These results are useful for environmental and engineering applications.     

Prediction of turbulent offset jet flows with an assessment of QUICKER scheme

The QUICKER scheme extended for non-uniform rectangular grid systems has been applied to predict the turbulent offset jet flows. Computational results obtained with the QUICKER scheme are compared with those from the skew-upwind and the hybrid schemes. Computational results include the reattachment length, the velocity profile, the axial velocity decay curve, and the shear stress distribution. In the sense of an overall agreement with the experimental data, the QUICKER scheme is found to be superior to the other two schemes. Boundary conditions are carefully set up to account for various flow conditions. Special attention has been given to the set-up of entrainment boundary condition. It is emphasized that the numerical diffusion due to streamline-to-grid skewness far exceeds the turbulent diffusion in offset jet flows; therefore, a numerical scheme that would minimize the numerical diffusion is a prerequisite for a better prediction of the turbulent offset jet flows.     

Delayed die swell

The experiments reported here establish that there is a general critical condition associated with die swell which we call delayed die swell. This condition is defined by a critical speed which is the area-averaged velocity, the extrusion velocity, at the exit of the pipe when the swell is first delayed. The delayed swell ratio and delay distance first increase for larger, post-critical values of the extrusion velocity; then the increases are terminated either by instabilities or by smoothing. The maximum post-critical velocity at the pipe exit was always greater than the shear wave speed measured on the shear-wave-speed meter. The post critical area averaged velocity at the position of maximum swell before termination was always less than the shear wave speed. There were always points in the region of swelling where the ratio of the local velocity to the shear wave speed, the viscoelastic Mach number, was unity. The swelling of the jet is a nonlinear phenomenon which we suggest is finally terminated either by instability or when the variations of the velocity, vorticity and stress field are reduced to zero by the inward propagation of shear waves from the free surface of the jet. This propagation is generated by discontinuous “initial” data along χ in which the prescribed values of velocity at the boundary change from no-slip in the pipe to no-shear in the jet. The measurements raise the possibility that the delay may be associated with a change of type from supercritical to subcritical flow.