Nonlinear evolution of secondary instabilities in boundary-layer transition

Methods of nonlinear stability theory are applied to analyze the evolution of disturbances in the three-dimensional stage immediately preceding the breakdown of a laminar boundary layer. A perturbation scheme is used to solve the nonlinear equations and to develop a dynamical model for the interaction of primary and secondary instabilities. The first step solves for the two-dimensional primary wave in the absence of secondary disturbances. Once this finite-amplitude wave is calculated, it is decomposed into a basic-flow component and an interaction component. The basic-flow component acts as a parametric excitation for the three-dimensional secondary wave, while the interaction component captures the resonance between the secondary and primary waves. Results are presented in two principal forms: amplitude growth curves and velocity profiles. Our results agree with experimental data and the few available results of transition simulations and, moreover, reveal the origin of the observed phenomena. The method described establishes the basis for physical transition criteria in a given disturbance environment.This work has been supported by the Air Force Office of Scientific Research under Contract F46920-87-K-0005 and Grant AFOSR-88-0186 (TH) and by an ONT Postdoctoral Fellowship (JDC).     

Three-dimensional waves in a boundary layer

As is known [1], two-dimensional waves develop in the boundary layer and then become three-dimensional waves with increase of the Reynolds number R. Since Squire [2] has shown that the linear growth of three-dimensional waves is more intense than that of the two-dimensional, it is natural that the behavior of three-dimensional waves in the boundary layer is explained by nonlinear intersection [3], However, Gaster [4] has noted that although disturbances which increase with time are usually considered, experimentally we observe disturbances which grow in space. (Squire's proof does not extend to this case.) It has been shown that the spatially growing disturbances cannot explain the occurrence of the three-dimensional waves (in the linear formulation).The author wishes to thank his scientific advisor G. I. Petrov and also A. A. Zaitsev for valuable discussions of the study.     

Nonlinear development of a wave in a boundary layer

In recent years definite progress has been achieved in the construction of theoretical models of nonlinear wave processes which lead to a transition from laminar to turbulent flow [1, 2]. At the same time, there is a shortage of actual experimental material, especially for flows in a boundary layer. Fairly thorough experimental studies have been carried out only on the initial stage of the development of disturbances in a boundary layer, which is satisfactorily describable by the linear theory of hydrodynamic stability. In evaluating the theoretical models of subsequent stages of the transition, investigators have been forced to turn chiefly to much earlier experiments carried out by the United States National Bureau of Standards [3, 4], in which the main attention was concentrated on the three-dimensional structure of the transition region. The present investigation was undertaken for the purpose of obtaining detailed data on the structure of the flow in the transition region when there is disturbance in the laminar boundary layer of a two-dimensional wave. In order to make the two-dimensional nonlinear effects stand out more clearly, the amplitude of the wave was specified to be fairly large from the very outset. In contrast to earlier investigations, the main attention was centered on the study of the spectral composition of the disturbance field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 49–58, May–June, 1977.     

Interaction between an unsteady pressure disturbance and a hypersonic boundary layer

The development of disturbances in a hypersonic boundary layer on a cooled surface is investigated in the case in which the characteristic velocity of disturbance propagation is small but greater than the flow velocity in the wall region of the three-layer disturbed zone with interaction. The nonlinear boundary value problem formulated involves a single similarity parameter that characterizes the contribution made by the main, on average either subsonic or supersonic, region of the boundary layer to the generation of the pressure disturbance. In the linear approximation, an analytical solution and an algebraic dispersion equation are derived. It is shown that only waves exponential in time and in the streamwise coordinate can propagate downstream when themain region of the undisturbed boundary layer is subsonic on average.     

Characteristics of an unstably localized disturbance in a compressed boundary layer

Experiments have demonstrated [1] that the transition of streamline-type flow into turbulent flow in a boundary layer occurs as a result of the formation and development of turbulent spots apparently arising from small natural disturbances. A study of the nonlinear evolution and interaction of localized disturbances requires knowledge of their characteristics to a linear approximation [2]. In the current work, results are presented of calculations of such characteristics for the first two unstable modes in a supersonic boundary layer on a two-dimensional plate (M = 4.5, T_w = 4.44).Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 50–53, January–February, 1976.     

Self-sustained amplification of disturbances in pipe Poiseuille flow

The nonlinear development of disturbances in pipe Poiseuille flow is studied with a low-dimensional model. The basic system from which the model is derived governs disturbances closely related to the radial velocity and radial vorticity disturbances. The analysis is restricted to the interaction of the two first harmonics of streamwise elongated disturbances since they are the most transiently amplified ones in linear theory. In the resulting dynamical system a nonlinear feedback from the normal vorticity disturbance (which is transiently amplified according to linear theory) to the radial velocity disturbance is present. Above a threshold of the initial amplitude, the feedback leads to a self-sustained amplification of the disturbances continuing for all times.     

Secondary instability modes generated by a Tollmien-Schlichting wave scattering from a bump

The plane finite-amplitude Tollmien-Schlichting wave interaction with a three-dimensional bump on a wall is considered for plane channel flow. The scattering of this wave leads to the production of unsteady three-dimensional disturbances which transform into growing secondary instability modes. The generation of such modes is studied assuming the three-dimensional disturbances to be small in comparison with the primary plane instability wave. The solution predicts that secondary disturbance amplification takes place only within a narrow wedge downstream of the bump. The qualitative comparison of results with experiments on turbulent wedge origination at an isolated roughness in a boundary layer is presented.     

Studies on nonlinear stability of three-dimensional H-type disturbance

The three-dimensional H-type nonlinear evolution process for the problem of boundary layer stability is studied by using a newly developed method called parabolic stability equations (PSE). The key initial conditions for sub-harmonic disturbances are obtained by means of the secondaryinstability theory. The initial solutions of two-dimensional harmonic waves are expressed in Landau expansions. The numerical techniques developed in this paper, including the higher order spectrum method and the more effective algebraic mapping for dealing with the problem of an infinite region, increase the numerical accuracy and the rate of convergence greatly. With the predictor-corrector approach in the marching procedure, the normalization, which is very important for PSE method, is satisfied and the stability of the numerical calculation can be assured. The effects of different pressure gradients, including the favorable and adverse pressure gradients of the basic flow, on the “H-type“ evolution are studied in detail. The results of the three-dimensional nonlinear “H-type“ evolution are given accurately and show good agreement with the data of the experiment and the results of the DNS from the curves of the amplitude variation, disturbance velocity profile and the evolution of velocity.     

Resonance properties of two-dimensional wave disturbances in a boundary layer

The nonlinear development of disturbances of the traveling wave type in the boundary layer on a flat plate is examined. The investigation is restricted to two-dimensional disturbances periodic with respect to the longitudinal space coordinate and evolving in time. Attention is concentrated on the interactions of two waves of finite amplitude with multiple wave numbers. The problem is solved by numerically integrating the Navier-Stokes equations for an incompressible fluid. The pseudospectral method used in the calculations is an extension to the multidimensional case of a method previously developed by the authors [1, 2] in connection with the study of nonlinear wave processes in one-dimensional systems. Its use makes it possible to obtain reliable results even at very large amplitudes of the velocity perturbations (up to 20% of the free-stream velocity). The time dependence of the amplitudes of the disturbances and their phase velocities is determined. It is shown that for a fairly large amplitude of the harmonic and a particular choice of wave number and Reynolds number the interacting waves are synchronized. In this case the amplitude of the subharmonic grows strongly and quickly reaches a value comparable with that for the harmonic. As distinct from the resonance effects reported in [3, 4], which are typical only of the three-dimensional problem, the effect described is essentially two-dimensional.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–44, March–April, 1990.     

Three-wave resonance interaction of disturbances in a boundary layer

The three-frequency resonance of Tolman-Schlichting waves, one of which propagates along the stream while the other two propagate at adjacent angles to it, is investigated as a function of the spectrum and initial intensity in incompressible flows of the boundary-layer type within the framework of a weakly nonlinear theory. In the parallel-flow approximation such an interaction leads to the formation of unstable self-oscillations. The spatial evolution of the associated disturbances is studied with allowance for the self-similar deformation of the velocity profile of the main flow. It is shown that such development can lead to a sharp amplification of the oscillations, primarily of those propagating at an angle to the flow. The role of the effects under consideration in the transitional process and the connection with experimental data are discussed. As experiments [1, 2] show, in the process of a transition from a laminar boundary layer to a turbulent region, well described by the linear theory of hydrodynamic stability, there first comes a section of the excitation of harmonics of a Tolman-Schlichting wave, the appearance of three-dimensional structures, and a rapid growth in the intensity of low-frequency oscillations. There is no doubt that in this section the phenomena are dependent on the nonlinear character of the development with disturbances. The resonance interaction of wave triads can play an important role in this. For small enough amplitudes such an interaction is described by a first-order theory [3, 4], and in the general case the nonlinear effects associated with them should occur sooner than others. The importance of resonance triads for the explanation of the development of three-dimensional structures in a layer and the generation of intense pulsations has already been emphasized in [5, 6]. The clarification of the properties of the evolution of resonantly interacting disturbances therefore is important for an understanding of this transitional process.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 78–84, September–October, 1978.The authors thank V. Ya. Levchenko for a discussion of the work.     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

Subcritical spatial transition of swept Hiemenz flow

It is known from experimental investigations that the leading-edge boundary layer of a swept wing exhibits transition to turbulence at subcritical Reynolds numbers, i.e. at Reynolds numbers which lie below the critical Reynolds number predicted by linear stability theory. In the present work, we investigate this subcritical transition process by direct numerical simulations of a swept Hiemenz flow in a spatial setting. The laminar base flow is perturbed upstream by a pair of stationary counter-rotating vortex-like disturbances. This perturbation generates high- and low-speed streaks by a non-modal growth mechanism. Further downstream, these streaky structures exhibit a strong instability to secondary perturbations which leads to a breakdown to turbulence.The observed transition mechanism has strong similarities to by-pass transition mechanisms found for two-dimensional boundary layers. It can be shown that transition strongly depends on the amplitude of the primary perturbation as well as on the frequency of the secondary perturbation.     

Boundary-layer separation on conical bodies

Some characteristics of the variation in the linear dimensions of the flow separation zones on conical bodies with expanding conical skirts and of variation of the pressure within these zones as a function of variation of the Mach number, Reynolds number, and intensity of the disturbance that causes the boundary layer separation are examined. Experiments were conducted in laminar, transitional, and turbulent flows in flow separation regions. The interaction of viscous and nearly inviscid flows is quite common. This phenomenon occurs in flow past a concave corner, when a compression shock impinges on a boundary layer, and in many other cases. The characteristics of this phenomenon in flow about two-dimensional bodies have been investigated experimentally in [1, 2] and other studies. Attempts have been made to analyze the interaction of compression shocks with the boundary layer theoretically. In “free” separated flows, when the points of separation and reattachment of the boundary layer are not fixed (for example, on a flat plate with a long wedge attached to it), theoretical studies are usually made within the framework of the boundary layer theory with use of the approximate integral methods [3, 4]. In this article we examine some results from studies of free separated flows on conical bodies with conical skirts in laminar, transitional, and turbulent flows (Fig. 1).     

Steady convective motions in a plane horizontal fluid layer with permeable boundaries

In a plane horizontal fluid layer bounded by permeable plane surfaces which are heated to different temperatures and between which transverse flow takes place with uniform velocity, convection occurs at a definite critical Rayieigh number. The study of the disturbance spectrum and the convective stability, made within the framework of linear theory in [1], showed that convective instability in the layer with permeable boundaries, just as in the case of the Rayieigh problem, is associated with the development of monotonie disturbances. It turns out that the transverse motion in the layer leads to a considerable increase of the Rayieigh number. Linear theory does not permit analysis of the development of the disturbances in the supercritical region. Analysis of the developed nonlinear motion can be made only on the basis of the complete nonlinear convection equations.In this investigation we made a numerical study of nonlinear motions in the supercritical region. Calculations were made on a computer via the grid method. Solutions are obtained for the nonlinear equations of motion over a wide range of Rayieigh numbers for different values of the Peclet number, whichdefines the intensity of the transverse motion in the layer.The author wishes to thank E. M. Zhukovitskii for his guidance, and G. Z. Gershuni and E. L. Tarunin for their interest and assistance in the study.     

Influence of nonparallel flow on the development of disturbances in a supersonic boundary layer

The linear theory is used to solve the problem of the development of two-dimensional disturbances in the boundary layer of compressible fluid. In contrast to the stability theory of plane-parallel flows, the present paper takes into account the presence in the boundary layer of transverse (at right angles to the flow direction) motions, the dependence of the averaged flow parameters on the longitudinal coordinate, and also the deformation of the amplitude distribution profile of the disturbances as a function of the longitudinal coordinate. The calculations are made for Mach number M = 4.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–31, March–April, 1980.     

蜿蜒边界下平面流的线性流动稳定性

为便于数值分析,蜿蜒河流水动力和演变模型中一般隐性假设二次时均流-二次涡的关系与明渠流时均流-明渠湍流的关系相同,但由于高雷诺数下的DNS算力限制和实验尺度限制,这种隐含假设是否成立目前尚无相关湍流研究来支撑.文章试图通过分析明渠湍流和二次湍流发展初期的研究,侧面揭示其湍流结构的异同.通过对曲线正交坐标系下的平面二维NS方程使用双参数摄动的方法,建立了一种求解蜿蜒边界弱非线性层流的摄动解法,并推导得出一个适用于蜿蜒边界的EOS方程以及其特征值问题的解法.蜿蜒边界下弱非线性层流解为一系列蜿蜒谐波分量的叠加,其中线性部分使得两壁产生流速差,非线性部分随着雷诺数增大呈指数增长.水流的扰动增长率特征谱的第一模态与直道流相似,由3条曲线、4个波段合成,但其长波段和短波段的扰动流场与直道流不同,所有短波段的扰动流速近似于KH涡.蜿蜒边界对内部水流扰动有一定的选择性.偏角幅值越大扰动增长越快;蜿蜒波数的影响则为先增后减,有一个使扰动增长最快的蜿蜒波数.扰动流场由一个典型的TS波和一对波包形式的二次涡叠加而成,波包只有纵向流速分量,包络线由蜿蜒波数控制,波包内是与直道扰动波参数相同的TS波.     

Experiments on localized disturbances in a flat plate boundary layer. Part 1. The receptivity and evolution of a localized free stream disturbance

The receptivity of a laminar boundary layer to free stream disturbances has been experimentally investigated through the introduction of deterministic localized disturbances upstream of a flat plate mounted in a wind tunnel. Hot-wire measurements indicate that the spanwise gradient of the normal velocity component (and hence the streamwise vorticity) plays an essential role in the transfer of disturbance energy into the boundary layer. Inside the laminar boundary layer the disturbances were found to give rise to the formation of longitudinal structures of alternating high and low streamwise velocity. Similar streaky structures exist in laminar boundary layers exposed to free stream turbulence, in which the disturbance amplitude increases in linear proportion to the displacement thickness. In the present study the perturbation amplitude of the streaks was always decaying for the initial amplitudes used, in contrast to the growing fluctuations that are observed in the presence of free stream turbulence. This points out the importance of the continuous influence from the free stream turbulence along the boundary layer edge.     

Algebraic growth in a Blasius boundary layer: Nonlinear optimal disturbances

The three-dimensional, algebraically growing instability of a Blasius boundary layer is studied in the nonlinear regime, employing a nonparallel model based on boundary layer scalings. Adjoint-based optimization is used to determine the “optimal” steady leading-edge excitation that provides the maximum energy growth for a given initial energy. Like in the linear case, the largest transient growth is found for inlet streamwise vortices, that yield streamwise streaks downstream. Two different definitions of growth are employed, providing qualitatively similar results, although the spanwise wavenumbers of optimal growth differ by up to 20% in the two cases. The wavelength of the most amplified optimal disturbance increases with the initial amplitude. For large input amplitudes, significant deformations of the mean velocity field are found; in such cases it is reasonable to expect that nonlinear streaks may break down through a secondary instability.     

On the Secondary Instability of Three-Dimensional Boundary Layers

One of the possible transition scenarios in three-dimensional boundary layers, the saturation of stationary crossflow vortices and their secondary instability to high-frequency disturbances, is studied using the Parabolized Stability Equations (PSE) and Floquet theory. Starting from nonlinear PSE solutions, we investigate the region where a purely stationary crossflow disturbance saturates for its secondary instability characteristics utilizing global and local eigenvalue solvers that are based on the Implicitly Restarted Arnoldi Method and a Newton–Raphson technique, respectively. Results are presented for swept Hiemenz flow and the DLR swept flat plate experiment. The main focuses of this study are on the existence of multiple roots in the eigenvalue spectrum that could explain experimental observations of time-dependent occurrences of an explosive growth of traveling disturbances, on the origin of high-frequency disturbances, as well as on gaining more information about threshold amplitudes of primary disturbances necessary for the growth of secondary disturbances. Received 13 July 1998 and accepted 7 July 2000     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.