Prehistory of Instability in a Hypersonic Boundary Layer

The initial phase of hypersonic boundary-layer transition comprising excitation of boundary-layer modes and their downstream evolution from receptivity regions to the unstable region (instability prehistory problem) is considered. The disturbance spectrum reveals the following features: (1) the first and second modes are synchronized with acoustic waves near the leading edge; (2) further downstream, the first mode is synchronized with entropy and vorticity waves; (3) near the lower neutral branch of the Mack second mode, the first mode is synchronized with the second mode. Disturbance behavior in Regions (2) and (3) is studied using the multiple-mode method accounting for interaction between modes due to mean-flow nonparallel effects. Analysis of the disturbance behavior in Region 3) provides the intermodal exchange rule coupling input and output amplitudes of the first and second modes. It is shown that Region (3) includes branch points at which disturbance group velocity and amplitude are singular. These singularities can cause difficulties in stability analyses. In Region (2), vorticity/entropy waves are partially swallowed by the boundary layer. They may effectively generate the Mack second mode near its lower neutral branch. Received 17 July 2000 and accepted 23 March 2001     

高超声速边界层中模态转化的数值研究

在高超声速边界层中,第一模态和第二模态是与转捩有关的两个主要不稳定模态.除了不稳定模态,还存在一类稳定模态,其相速度在前缘接近快声波的相速度称为快模态.在感受性过程中,这类模态对激发边界层中不稳定模态起着很重要的作用.前缘感受性理论解释了边界层外扰动激发边界层中第一模态波的机理.针对高超声速平板边界层,利用相似性解剖面作为基本流,采用线性稳定性理论和直接数值模拟的方法研究了快模态和慢模态的稳定性行为.研究发现模态转化的位置与马赫数有关.根据线性稳定性理论的结果定义了临界频率.当扰动频率高于临界频率,第一模态与第二模态同支;而当扰动频率低于临界频率,第一模态与第二模态的共轭模态同支.借助稳定性方程的伴随方程分析了直接数值模拟的结果.直接数值模拟结果表明不论上游是快模态还是慢模态,当它们经过第二模态的不稳定区,它们都会演化成第二模态. 这可用模态在非平行流中传播的特征来解释.      

On the görtler vortex instability mechanism at hypersonic speeds

The linear instability of the hypersonic boundary layer on a curved wall is considered. As a starting point the viscosity of the fluid is taken to be a linear function of temperature and real-gas effects are ignored. It is shown that the flow is susceptible to Görtler vortices and that they are trapped in the logarithmically thin adjustment layer in which the temperature of the basic flow changes rapidly to its free stream value. The vortices decay exponentially in both directions away from this layer and are most unstable when their wavelength is comparable with the depth of the adjustment layer. The nonuniqueness of the neutral stability curve associated with incompressible Görtler vortices is shown to disappear at high Mach numbers if the appropriate fast streamwise dependence of the instability is built into the disturbance flow structure. It is shown that in the hypersonic limit wall-cooling affects the leading-order term in the expansion of the Görtler number which is independent of the wavenumber and which is due to the curvature of the basic state but it has a negligible effect on the other terms which are dependent on the wavenumber.This research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the first author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. This work was also partially supported by NASA under Grant NASA 18107 and by USAF under Grant AFOSR89-0042.     

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.     

Stability of Flow with Viscous Dissipation in a Horizontal Porous Layer with an Open Boundary Having a Prescribed Temperature Gradient

A buoyancy-induced stationary flow with viscous dissipation in a horizontal porous layer is investigated. The lower boundary surface is impermeable and subject to a uniform heat flux. The upper open boundary has a prescribed, linearly varying, temperature distribution. The buoyancy-induced basic velocity profile is parallel and non-uniform. The linear stability of this basic solution is analysed numerically by solving the disturbance equations for oblique rolls arbitrarily oriented with respect to the basic velocity field. The onset conditions of thermal instability are governed by the Rayleigh number associated with the prescribed wall heat flux at the lower boundary, by the horizontal Rayleigh number associated with the imposed temperature gradient on the upper open boundary, and by the Gebhart number associated with the effect of viscous dissipation. The critical value of the Rayleigh number for the onset of the thermal instability is evaluated as a function of the horizontal Rayleigh number and of the Gebhart number. It is shown that the longitudinal rolls, having axis parallel to the basic velocity, are the most unstable in all the cases examined. Moreover, the imposed horizontal temperature gradient tends to stabilise the basic flow, while the viscous dissipation turns out to have a destabilising effect.     

微槽-吸气组合控制平板边界层多模态稳定性研究

边界层转捩会使高超声速飞行器壁面摩阻和热流显著增加,因此在高超声速飞行器设计过程中往往占据重要地位.针对高超声速飞行器多模态转捩控制问题,提出了微槽道(1 mm)与边界层吸气的组合控制方法,并通过直接数值模拟和线性稳定性理论研究了Ma=4.5平板边界层的稳定性及组合控制效果.边界层在无控状态时,同时存在失稳的第一、二模态波,且二维第二模态波最不稳定;单纯施加微槽道控制时,边界层第二模态波会被抑制但第一模态波会被略微激发.对比而言,采用“微槽-吸气”组合控制后,不仅增强了对第二模态波的抑制效果,而且减弱了第一模态波的激发程度;同时随着吸气强度的增加,第二模态波不稳定区域明显收缩、频率显著增高,而第一模态波则变化不明显.相较于单纯的微槽道,吸气增强了“微槽吸收”与“声波散射”作用,因此中等吸气强度下该组合控制方法对第一和第二模态波的增长率分别实现了12.63%和28.02%的抑制效果.以上结果表明“微槽-吸气”组合控制手段具有适用宽频、布置区域灵活的优点,展现出了一定的多模态控制效果.     

Stability analysis of the onset of vortex shedding for wakes behind flat plates

Above a critical Reynolds number, wake flows behind flat plates become globally unstable, the leading modal instability in this case is known as Kelvin–Helmholtz mechanism. In this article, both local and BiGlobal linear instability analyses are performed numerically to study the onset of the shedding process. Flat plates with different base shapes are considered to assess geometry effects, and the relation between the critical shedding Reynolds number, \(Re_\mathrm{cr}\), and the boundary layer thickness is studied. Three types of base shapes are used: square, triangular and elliptic. It is found that the base shape has a great impact on the growth rate of least stable disturbance mode, thus would influence \(Re_\mathrm{cr}\) greatly, but it has little effect on the vortex shedding frequency. The shedding frequency is determined mainly by boundary layer thickness and has little dependence on the Reynolds number and base shape. We find that for a fixed Reynolds number, increasing boundary layer thickness acted in two ways to modify the global stability characteristics: It increases the length of the absolute unstable region and it makes the flow less locally absolutely unstable in the near-wake region, and these two effects work against each other to destabilize or stabilize the flow.     

Linear stability analysis in compressible, flat-plate boundary-layers

The stability problem of two-dimensional compressible flat-plate boundary layers is handled using the linear stability theory. The stability equations obtained from three-dimensional compressible Navier–Stokes equations are solved simultaneously with two-dimensional mean flow equations, using an efficient shoot-search technique for adiabatic wall condition. In the analysis, a wide range of Mach numbers extending well into the hypersonic range are considered for the mean flow, whereas both two- and three-dimensional disturbances are taken into account for the perturbation flow. All fluid properties, including the Prandtl number, are taken as temperature-dependent. The results of the analysis ascertain the presence of the second mode of instability (Mack mode), in addition to the first mode related to the Tollmien–Schlichting mode present in incompressible flows. The effect of reference temperature on stability characteristics is also studied. The results of the analysis reveal that the stability characteristics remain almost unchanged for the most unstable wave direction for Mach numbers above 4.0. The obtained results are compared with existing numerical and experimental data in the literature, yielding encouraging agreement both qualitatively and quantitatively.      

Flat-plate hypersonic boundary-layer ?ow instability and transition prediction considering air dissociation

The effects of air dissociation on ?at-plate hypersonic boundary-layer ?ow instability and transition prediction are studied. The air dissociation reactions are assumed to be in the chemical equilibrium. Based on the ?at-plate boundary layer, the ?ow stability is analyzed for the Mach numbers from 8 to 15. The results reveal that the consideration of air dissociation leads to a decrease in the unstable region of the ?rst-mode wave and an increase in the maximum growth rate of the second mode. High frequencies appear earlier in the third mode than in the perfect gas model, and the unstable region moves to a lower frequency region. When the Mach number increases, the second-mode wave dominates the transition process, and the third-mode wave has little effect on the transition. Moreover, when the Mach number increases from 8 to 12, the N-factor envelope becomes higher, and the transition is promoted. However, when the Mach number exceeds 12, the N-factor envelope becomes lower, and the transition is delayed. The N-factor envelope decreases gradually with the increase in the altitude or Mach number.     

Second mode unstable disturbance measurement of hypersonic boundary layer on cone by wavelet transform

Experimental investigation of hypersonic boundary layer instability on a cone is performed at Mach number 6 in a hypersonic wind tunnel.Time series signals of instantaneous fluctuating surface-thermal-flux are measured by Pt-thin-film thermocouple temperature sensors mounted at 28 stations on the cone surface in the streamwise direction to investigate the development of the unstable disturbance.Wavelet transform is employed as a mathematical tool to obtain the multi-scale characteristics of fluctuating surfacethermal-flux both in the temporal and spectrum space.The conditional sampling algorithm using wavelet coefficient as an index is put forward to extract the unstable disturbanceThe generic waveform for the second mode unstable disturbance is obtained by a phase-averaging technique.The development of the unstable disturbance in the streamwise direction is assessed both in the temporal and spectrum space.Our study shows that the local unstable disturbance detection method based on wavelet transformation offers an alternative powerful tool in studying the hypersonic unstable mode of laminar-turbulent transition.It is demonstrated that,at hypersonic speeds,the dominant flow instability is the second mode,which governs the course of laminar-turbulent transition of sharp cone boundary layer.     

Continuous Mode Transition in High-speed Boundary-layers

Mode interaction is studied via direct numerical simulations of a Mach 4.5 boundary layer with discrete and continuous modes imposed at the inflow. An approximate decoupling procedure is developed to create separate vortical, acoustic and entropic continuous mode components. Oblique horizontal vorticity modes induce boundary layer disturbances that grow with downstream distance, similarly to their incompressible counterpart. One salient difference is that a low frequency vorticity mode, alone, is found to induce transition by spawning two-dimensional, unstable discrete modes. The discrete modes are non-linearly excited at high harmonics of the inlet perturbation. Adding a Mack second mode, in addition to the vorticity mode, causes even earlier transition, suggesting that, in supersonic flow, unstable discrete modes play a crucial role in breakdown of boundary-layer streaks.     

Vortex-speed selection within the boundary-layer flow over a rotating sphere placed in an enforced axial flow

This paper furthers existing work into the instability mechanisms within the boundary-layer flow over a rotating sphere through the study of amplification rates within the convectively-unstable region. The onset of convective instability is associated with the experimentally observed onset of spiral vortices reported in the literature. Axial flow is found to stabilize the boundary layer by both delaying the onset of convective instability at all latitudes and also by significantly reducing the spatial amplification rates. We find that the type II (streamline curvature) mode becomes increasingly amplified with respect to the type I (crossflow) mode and is therefore likely to be selected in practice for sufficiently high axial flow rates. Furthermore, in experiments where special care is taken to remove all surface roughness, we predict that vortices will rotate at around 75% of the local surface speed. This is consistent with the experimental observations of Kobayashi & Arai who note a speed of around 76% under particular experimental conditions. These predictions are entirely consistent with related work on the rotating-disk and cone boundary layers.     

Stability properties of the vertical boundary layers in differentially heated cavities

In the present study, the two-dimensional (2-D) stability properties of the vertical boundary layers in a cavity that is differentially heated over two opposing vertical walls is considered. The study is performed by introducing artificial, controlled perturbations at the base of the vertical boundary layer along the hot cavity wall and by following the evolution of these disturbances. For small initial perturbations, the evolution is governed by linear effects. This method accurately predicts the frequency of the bifurcation, which occurs for (much) larger Rayleigh numbers. Convective instability sets in for Rayleigh numbers much smaller than those at which the absolute instability (i.e., the bifurcation) occurs, and these Rayleigh numbers are in reasonable agreement with those for the boundary layer along a plate. The absolute instability does not result from the first wave which becomes unstable. For small Prandtl numbers (≤ 2), the unstable waves which lead to the absolute instability are shear-driven, and a single frequency is introduced in the flow after the bifurcation. For larger Prandtl numbers, the unstable waves are buoyancy driven and no single-frequency unsteady flow is observed after the bifurcation.     

Stability analysis and transition prediction of hypersonic boundary layer over a blunt cone with small nose bluntness at zero angle of attack

Stability and transition prediction of hypersonic boundary layer on a blunt cone with small nose bluntness at zero angle of attack was investigated. The nose radius of the cone is 0.5 mm; the cone half-angle is 5°, and the Mach number of the oncoming flow is 6. The base flow of the blunt cone was obtained by direct numerical simulation. The linear stability theory was applied for the analysis of the first mode and the second mode unstable waves under both isothermal and adiabatic wall condition, and eN method was used for the prediction of transition location. The N factor was tentatively taken as 10, as no experimentally confirmed value was available. It is found that the wall temperature condition has a great effect on the transition location. For adiabatic wall, transition would take place more rearward than those for isothermal wall. And despite that for high Mach number flows, the maximum amplification rate of the second mode wave is far bigger than the maximum amplification rate of the first mode wave, the transition location of the boundary layer with adiabatic wall is controlled by the growth of first mode unstable waves. The methods employed in this paper are expected to be also applicable to the transition prediction for the three dimensional boundary layers on cones with angle of attack.     

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.     

Viscous and Inviscid Instabilities of Flow Along a Streamwise Corner

Here we consider the stability of flow along a streamwise corner formed by the intersection of two large flat plates held perpendicular to each other. Self-similar solutions for the steady laminar mean flow in the corner region have been obtained by solving the boundary layer equations for zero and nonzero streamwise pressure gradients. The stability of the mean flow is investigated using linear stability analysis. An eigensolver has been developed to solve the resulting linear eigenvalue problem either in a global mode to obtain an approximation to all the dominant eigenmodes or in a local mode to refine a particular eigenmode. The stability results indicate that the entire spectrum of two-dimensional and oblique viscous modes of a two-dimensional Blasius boundary layer is active in the case of a corner layer as well, but away from the cornerline. In a corner region of finite spanwise extent, the continuous spectrum of oblique modes degenerates to a discrete spectrum of modes of increasing spanwise wave number. The effect of the corner on the two-dimensional viscous instability is small and decreases the growth rate. The growth rate of outgoing oblique disturbances is observed to decrease, while the growth rate of incoming oblique disturbances is enhanced by the corner. This asymmetry between the outgoing and incoming viscous modes increases with increasing obliqueness of the disturbance. The instability of a zero pressure gradient corner layer is dominated by the viscous modes; however, an inviscid corner mode is also observed. The critical Reynolds number of the inviscid mode rapidly decreases with even a small adverse streamwise pressure gradient and the inviscid mode becomes the dominant one. Received 17 March 1998 and accepted 28 April 1999     

Thermal Instability in a Horizontal Porous Channel with Horizontal Through Flow and Symmetric Wall Heat Fluxes

The linear thermoconvective instability of the basic parallel flow in a plane and horizontal porous channel is investigated. The boundary walls are assumed to be impermeable and subject to symmetric and uniform heat fluxes. The wall heat fluxes produce either a net heating or a net cooling of the fluid saturated porous medium. A horizontal mass flow rate is externally impressed leading to a stationary basic state with a temperature gradient inclined to the vertical. A region of possibly unstable thermal stratification exists either in the lower half-channel (boundary heating), or in the upper half-channel (boundary cooling). The convective instability of the basic flow is governed by the Rayleigh number and by the Péclet number. In the case of boundary heating, the thermal instability arises when the Rayleigh number exceeds its critical value, that depends on the Péclet number. The change of the critical Rayleigh number as a function of the Péclet number is determined numerically for arbitrary normal modes oblique to the basic flow direction. The most dangerous modes are the longitudinal rolls, with a wave vector perpendicular to the basic velocity. There exists a minimum value of the Péclet number, 19.1971, below which no linear instability is detected.     

Effect of transverse stream velocity in an incompressible boundary layer on the stability of the laminar flow regime

The stability of the laminar flow regime in the boundary layer developed on a wall is increased considerably by the relatively slight extraction of fluid from the wall [1–4]. In the theoretical study of this phenomenon, all the investigators known to the present authors have taken into account only the increase in the fullness of the velocity profile in the boundary layer with suction. Computations of the stability characteristics have been made on the assumption that there are no transverse velocities in the laminar boundary layer.We present below an analysis of the stability of the laminar boundary layer in the presence of a constant transverse velocity in the near-wall region (suction). The calculations made show the existence for a given velocity profile in the boundary layer of a relative suction velocity v=v such that with suction velocities greater than v the flow remains stable at all Reynolds numbers, while the method used in the cited references gives a definite finite critical Reynolds number, equal in our notation to the Reynolds number at v=0, for each relative suction velocity.It was found that with suction of fluid from the boundary layer the region of instability has finite dimensions, i.e., there exist lower and upper critical Reynolds numbers. The flow is stable if its Reynolds number is less than the lower, or greater than the upper values of the critical Reynolds number.The instability region diminishes with increase in the relative suction velocity, and at a value of this velocity which is specific for each value of the velocity profile the instability region degenerates into a point-the flow becomes absolutely stable. Thus, with distributed suction it is advisable to increase the relative suction velocity only to a definite magnitude corresponding to disappearance of the instability region. The computational results presented make it possible to estimate this velocity for velocity profiles ranging from a Blasius profile to an asymptotic profile. Specific calculations were made for a family of Wuest profiles, since under actual conditions with suction there always exists a starting segment of the boundary layer [1, 2].     

Development of slow disturbances in a hypersonic boundary layer

The mechanisms of development of slow time-dependent disturbances in the wall region of a hypersonic boundary layer are established and a diagram of the disturbed flow patterns is plotted; the corresponding nonlinear boundary value problem is formulated for each of these regimes. It is shown that the main factors that form the disturbed flow are the gas enthalpy near the body surface, the local viscous-inviscid interaction level, and the type, either subsonic or supersonic, of the boundary layer as a whole. Numerical and analytical solutions are obtained in the linear approximation. It is established that enhancement of the local viscous-inviscid interaction or an increased role for the main supersonic region of the boundary layer makes the disturbed flow by and large “supersonic”: the upstream propagation of the disturbances becomes weaker, while their downstream growth is amplified. Contrariwise, local viscous-inviscid interaction attenuation or an increased role for the main subsonic region of the boundary layer has the opposite effect. Surface cooling favors an increased effect of the main region of the boundary layer while heating favors an increased wall region effect. It is also found that in the regimes considered disturbances travel from the turbulent flow region downstream of the disturbed region under consideration counter to the oncoming flow, which may be of considerable significance in constructing the nonlinear stability theory.     

Instability waves in the wall region of a turbulent boundary layer on a flat plate

Coherent structures and the bursting phenomena in the wall region of a turbulent boundary layer play a very important role in determining the characteristics of the boundary layer. Yet the nature and the origin of the coherent structures are unclear until now. In this paper, nonlinear stability calculations for the wall region of a turbulent boundary layer have been made. It was found that there do exist instability waves which may be responsible for the coherent structures. The project is supported by the National Natural Science Foundation of China.