Evolution Of Instability Waves Research Articles

Interaction of second-mode instability and distributed roughness elements in the hypersonic boundary layer over a sharp cone

The ablation of the thermal protection system (TPS) on the aircraft surface generates distributed roughness elements (DRE), which is expected to affect the hypersonic boundary-layer instability seriously. This work presents the experimental investigation of hypersonic boundary-layer instability waves on a 7° half-angle sharp cone with DRE allocated downstream of synchronization point at 0° angle of attack using surface flush-mounted pressure sensors, focused laser differential interferometer (FLDI), infrared thermography and schlieren technique. The experimental results showed that when the DRE (lower than the local boundary-layer height) was located downstream of the synchronization point, the Mack second mode still dominate the boundary-layer instability, and the transition onset was not significantly affected by DRE at various Reynolds numbers. Based on further analysis in the region downstream of roughness, the DRE didn't influence the nonlinear interaction among instability waves materially, although it promoted the growth of instability waves compared with the smooth case. However, the nonlinear interactions were decaying adjacent to the DRE, as the Mack second-mode instability waves cannot penetrate into the separation bubble because of the flow separation induced by the DRE. The FLDI measurement along the wall-normal direction revealed the second-mode instability waves were suppressed close to the wall by the complex roughness surface at the DRE region, and also confirmed the existence of second-mode instability waves at these streamwise locations. The evolution of instability waves followed the linear stability theory (LST), despite the existence of complex wall surfaces. This work enriches the understanding of the effect of DRE on hypersonic boundary-layer transition and demonstrates the spatial characteristics of instability waves above the complex surface.

Experimental investigation of hypersonic boundary layer instability with wavy wall located downstream of synchronization point

The wavy wall surface has been reported to be effective in suppressing the growth of second mode instability waves in hypersonic boundary layer, while the allocation of wavy wall relative to the synchronization point requires further study. Upon this work experimental investigation of hypersonic boundary layer instability with wavy wall located downstream of synchronization point based on 7° half-angle sharp cone at zero angle of attack was carried out in a hypersonic noisy wind tunnel. The evolution of instability waves with wavy wall surface is measured using both surface-mounted pressure fluctuation sensors and focused laser differential interferometer (FLDI). The results reveal that the growth of instability waves has been enhanced significantly when the wavy wall region is allocated downstream of the synchronization point when the wavy wall height is slightly smaller than the local boundary layer thickness. Additionally, second mode instability waves are observed at both crests and troughs within the wavy wall region based on the FLDI measurement and moreover, the high-frequency instability waves measured in the separation bubbles of troughs are speculated to be the transmission of second mode instability waves from the shear layer, rather than the local generation of second mode instability.

Experimental study of bluntness effects on hypersonic boundary-layer transition over a slender cone using surface mounted pressure sensors

In this work, we studied the bluntness effect on the hypersonic boundary-layer transition over a slender cone at Mach 6 with interchangeable tips in a noisy Ludwieg tube tunnel before the so-called “transition reversal” phenomenon occurs. The evolution of instability waves is characterized using surface flush-mounted pressure sensors deployed along the streamwise direction within unit Reynolds number from 4E+ 6/m ≤ Reunit ≤ 10E+ 6/m, and the bluntness of the cone nose ranges from 0.1 mm to 5 mm. Power spectral density (PSD) of pressure fluctuation indicates that small nose bluntness (ReR ≤ 2000) has little influence on the evolution of instability waves along the hypersonic boundary-layer, whereas with a moderate nose size (2000 ≤ ReR ≤ 5000), the hypersonic boundary layer transition is delayed monotonically as the nose radius increases before the boundary-layer turns into fully laminar without instability waves. The delaying effect can be attributed to the increased entropy-layer swallowing distance with a large tip radius. Instability wave characterization reveals that the second mode instability wave plays a dominant role before the transition reversal happens. The quadratic phase locking of second mode instabilities can be identified by bispectral analysis, and it attenuates as the nose tip radius increases.

Airfoil flow receptivity to simulated tonal noise emissions

On airfoils at low to moderate Reynolds numbers, trailing edge sound can have a feedback effect on the development of instability waves resulting in strongly periodic vortex shedding and tonal noise. This complex problem is decoupled in this study to investigate the flow response to deterministic noise emissions. The flow receptivity and response to tonal noise are examined using controlled steady and transient acoustic excitation. Experiments are performed on a NACA 0018 airfoil at a Reynolds number of 1.15×105 and an angle of attack of 5°, with a separation bubble on the suction side. Controlled acoustic perturbations are introduced via a loudspeaker. A combination of mean and time-resolved surface pressure and simultaneous high-speed particle image velocimetry measurements provides insight into the response of the bubble to simulated acoustic forcing. Amplification of shear layer perturbations in the fore portion of the separation bubble is observed only after some time delay from the onset of acoustic forcing. This places the receptivity region within the boundary layer upstream of the separation bubble, extending from the leading edge to just downstream of the suction peak. The acoustically excited perturbations in the receptivity region attain higher amplitudes than natural disturbances and are amplified significantly in the separation bubble, leading to a transient reduction in the separation bubble size. Results elucidate the ensuing transient response of the separation bubble, illustrating the flow dynamics expected during the initiation of the acoustic feedback loop during natural tonal noise emissions on an airfoil.

Nonlinear Instability Characterization of Hypersonic Laminar Boundary Layer

This study presents the experimental characterization of instability waves upon a hypersonic laminar boundary layer of a flared-cone model using a nonintrusive optical diagnostic [that is, a focused laser differential interferometer (FLDI)], with special emphasis on the nonlinear behavior of instability waves. To begin with, the application of FLDI technique on hypersonic instability waves was validated with a surface-mounted fast response pressure sensor. Subsequently, the development of instability waves along both streamwise and wall-normal directions was comprehensively investigated by traversing the focus of the FLDI through the hypersonic laminar boundary layer. Power spectral density analysis shows the coexistence of various instability waves up to 1.4 MHz, and the corresponding variation of each instability wave was described. Bicoherence analysis of FLDI measurement identifies different types of nonlinear interaction through a phase-locked mechanism, and the nonlinear evolution along the streamwise and wall-normal directions was elaborated on.

Evolution and transition mechanisms of internal swirling flows with tangential entry

The characteristics and transition mechanisms of different states of swirling flow in a cylindrical chamber have been numerically investigated using the Galerkin finite element method. The effects of the Reynolds number and swirl level were examined, and a unified theory connecting different flow states was established. The development of each flow state is considered as a result of the interaction and competition between basic mechanisms: (1) the centrifugal effect, which drives an axisymmetric central recirculation zone (CRZ); (2) flow instabilities, which develop at the free shear layer and the central solid-body rotating flow; (3) the bouncing and restoring effects of the injected flow, which facilitate the convergence of flow on the centerline and the formation of bubble-type vortex breakdown; and (4) the damping effect of the end-induced flow, which suppresses the development of the instability waves. The results show that the CRZ, together with the free shear layer on its surface, composes the basic structure of swirling flow. The development of instability waves produces a number of discrete vortex cores enclosing the CRZ. The azimuthal wave number is primarily determined by the injection angle. Generally, the wave number is smaller at a higher injection angle, due to the reduction of the perimeter of the free shear layer. At the same time, the increase in the Reynolds number facilitates the growth of the wave number. The end-induced flow tends to reduce the wave number near the head end and causes a change in wave number from the head end to the downstream region. Spiral-type vortex breakdown can be considered as a limiting case at a high injection angle, with a wave number equal to 0 near the head end and equal to 1 downstream. At lower Reynolds numbers, the bouncing and restoring effect of the injected flow generates bubble-type vortex breakdown.

Central recirculation zones and instability waves in internal swirling flows with an annular entry

The characteristics of the central recirculation zone and the induced instability waves of a swirling flow in a cylindrical chamber with a slip head end have been numerically investigated using the Galerkin finite element method. The effects of Reynolds number as well as swirl level adjusted by the injection angle were examined systematically. The results indicate that at a high swirl level the flow is characterized by an axisymmetric central recirculation zone (CRZ). The fluid in the CRZ takes on a solid-body rotation driven by the outer main flow through a free shear layer. Both the solid-body rotating central flow and the free shear layer provide the potential for the development of instability waves. When the injection angle increases beyond a critical value, the basic axisymmetric flow loses stability, and instability waves develop. In the range of Reynolds numbers considered in this study, three kinds of instability were identified: inertial waves in the central flow, and azimuthal and longitudinal Kelvin-Helmholtz waves in the free shear layer. These three types of waves interact with each other and mix together. The mode selection of the azimuthal waves depends strongly on the injection angle, through the perimeter of the free shear layer. Compared with the injection angle, the Reynolds number plays a minor role in mode selection. The flow topologies and characteristics of different flow states are analyzed in detail, and the dependence of flow states on the injection angle and Reynolds number is summarized. Finally, a linear analysis of azimuthal instabilities is carried out; it confirms the mode selection mechanisms demonstrated by the numerical simulation.

Experimental study of second-mode instability growth and breakdown in a hypersonic boundary layer using high-speed schlieren visualization

Visualization experiments are performed to investigate the development of instability waves within the boundary layer on a slender cone under high Mach number conditions. The experimental facility is a reflected-shock wind tunnel, allowing both low (Mach-8 flight equivalent) and high-enthalpy conditions to be simulated. Second-mode instability waves are visualized using a high-speed schlieren set-up, with pulse bursting of the light source allowing the propagation speed of the wavepackets to be unambiguously resolved. This, in combination with wavelength information derived from the images, enables the calculation of the disturbance frequencies. At the lower-enthalpy conditions, we concentrate on the late laminar and transitional regions of the flow. General characteristics are revealed through time-resolved and ensemble-averaged spectra on both smooth and porous ceramic surfaces of the cone. Analysis of the development of individual wavepackets is then performed. It is found that the wavepacket structures evolve from a ‘rope-like’ appearance to become more interwoven as the disturbance nears breakdown. The wall-normal disturbance distributions of both the fundamental and first harmonic, which initially have local maxima at the wall and near $y/{\it\delta}=0.7$–0.75, exhibit an increase in signal energy close to the boundary-layer edge during this evolution. The structure angle of the disturbances also undergoes subtle changes as the wavepacket develops prior to breakdown. Experiments are also performed at high-enthalpy ($h_{0}\approx 12~\text{MJ}~\text{kg}^{-1}$) conditions in the laminar regime, and the visualization technique is shown to be capable of resolving wavepacket propagation speeds and frequencies at such conditions. The visualizations reveal a somewhat different wall-normal distribution to the low-enthalpy case, with the disturbance energy concentrated much more towards the wall. This is attributed to the highly cooled nature of the wall at high enthalpy.

Experimental investigation of sound generation by a protuberance in a laminar boundary layer

Sound radiation from a two-dimensional protuberance glued on the wall in a laminar boundary layer was investigated experimentally at low Mach numbers. When the protuberance was as high as the boundary-layer thickness, a feedback-loop mechanism set in between protuberance-generated sound and Tollmien-Schlichting (T-S) waves generated by the leading-edge receptivity to the upstream-propagating sound. Although occurrence of a separation bubble immediately upstream of the protuberance played important roles in the evolution of instability waves into vortices interacting with the protuberance, the frequency of tonal vortex sound was determined by the selective amplification of T-S waves in the linear instability stage upstream of the separation bubble and was not affected by the instability of the separation bubble.

Linear stability analysis of supersonic axisymmetric jets

Stabilities of supersonic jets are examined with different velocities, momentum thicknesses, and core temperatures. Amplification rates of instability waves at inlet are evaluated by linear stability theory (LST). It is found that increased velocity and core temperature would increase amplification rates substantially and such influence varies for different azimuthal wavenumbers. The most unstable modes in thin momentum thickness cases usually have higher frequencies and azimuthal wavenumbers. Mode switching is observed for low azimuthal wavenumbers, but it appears merely in high velocity cases. In addition, the results provided by linear parabolized stability equations show that the mean-flow divergence affects the spatial evolution of instability waves greatly. The most amplified instability waves globally are sometimes found to be different from that given by LST.

Instability waves in a subsonic round jet detected using a near-field phased microphone array

We propose a diagnostic technique to detect instability waves in a subsonic round jet using a phased microphone array. The detection algorithm is analogous to the beam-forming technique, which is typically used with a far-field microphone array to localize noise sources. By replacing the reference solutions used in the conventional beam-forming with eigenfunctions from linear stability analysis, the amplitudes of instability waves in the axisymmetric and first two azimuthal modes are inferred. Experimental measurements with particle image velocimetry and a database from direct numerical simulation are incorporated to design a conical array that is placed just outside the mixing layer near the nozzle exit. The proposed diagnostic technique is tested in experiments by checking for consistency of the radial decay, streamwise evolution and phase correlation of hydrodynamic pressure. The results demonstrate that in a statistical sense, the pressure field is consistent with instability waves evolving in the turbulent mean flow from the nozzle exit to the end of the potential core, particularly near the most amplified frequency of each azimuthal mode. We apply this technique to study the effects of jet Mach number and temperature ratio on the azimuthal mode balance and evolution of instability waves. We also compare the results from the beam-forming algorithm with the proper orthogonal decomposition and discuss some implications for jet noise.

Nonlinear sinusoidal and varicose instability in a boundary layer

It is well known [1] that the laminar‐turbulent transition at a low turbulence level of the free flow is associated with the development of instability waves, the so-called Tollmien‐Schlichting waves. When a twodimensional Tollmien‐Schlichting wave reaches a certain amplitude at the nonlinear stage of its development, it undergoes three-dimensional distortion and, as a result, characteristic three-dimensional Λ structures arise [1]. Owing to certain features of the appearance and development of these structures, they are not only typical for the classical laminar‐turbulent transition, but are also inevitable attributes of a transition to more complex flows, e.g., flows modulated with longitudinal streaky structures, such as Hertler vortices, transverseflow vortices on sliding wings, etc., as well as flows in the viscous sublayer of a turbulent boundary layer. In these cases, they arise in particular due to the secondary high-frequency instability of such flows and may be manifested not only as Λ structures, but also in the form of horseshoe vortices ( Ω structures), hairpin vortices, etc. A characteristic feature of the development of such structures, e.g., on a sliding wing, is the disappearance of one of the counter-rotating vortices due to the transverse flow, whereas the development of a classical Λ structure can be observed on a straight wing [1]. The high-frequency secondary instability of transition and turbulent near-wall flows in the presence of streaky structures is often attributed to so-called sinusoidal and varicose instability. Both instability modes were investigated under controlled conditions at the linear and initial stages of nonlinear development. When the transverse size of the streaky structure was larger than the thickness of the shear layer, growth of varicose instability was observed. At the same time, when the transverse size of the streaky structure was comparable to or smaller than the thickness of the shear layer, it became more instable with respect to antisymmetric (sinusoidal) modes than to symmetric (varicose) modes. The experiment reported in [2] clearly shows that the growth of the symmetric mode leads to the formation of hairpin vortices, which are a pair of counterrotating longitudinal vortices that are connected by a head, i.e., a Λ vortex, while an antisymmetric mode is developed to a train of quasi-longitudinal vortices with alternating-sign vorticity. Unfortunately, the experiments reported in [2] concerned only the initial stage of the nonlinear development of disturbances, and spatial resolution was insufficiently high to reveal the structure of the flow in more detail. In this paper, we report on our experimental investigations of the nonlinear stage of the varicose and sinusoidal instability of the streaky structure in the Blasius boundary layer. In contrast to the experiment reported in [2], the study is more detailed (thermal anemometer measurements of the longitudinal velocity component and velocity pulsations in space ( xyz ) at 5 × 10 4 points) in order to reveal the features of the dynamics of the appearance, development, and internal structure of coherent formations up to the later stages of their nonlinear development. The experiments were carried out under controlled conditions in a low-turbulent wind tunnel. A plane plate 14 mm in thickness, 1000 mm in width, and 2000 mm in length was placed in parallel in the operation part of the tunnel. The streaky structure was generated by means of a cylindrical roughness element, which had a height of 1.1 mm and a diameter of 5.8 mm and was placed in the center of the plate at a distance of x 0 = 438 mm from the fore. The velocity of the flow was equal to U ∞ = 7.8 m/s, and the turbulence level was no higher than 0.04%. In the absence of the roughness element, the laminar boundary layer was developed without any waves and the velocity profile was close to the Blasius profile. A roughness-element height of h = 1.1 mm is close to the thickness of the displacement of

Instability-wave propagation in boundary-layer flows at subsonic through hypersonic Mach numbers

Direct numerical simulations (DNS) form an important ingredient to physics-based prediction of laminar-turbulent transition in boundary-layer flows, particularly in applications where it is desirable or even essential to model the various stages of transition process in an integrated manner. This paper addresses two building-block issues towards such capability: application to instability-wave propagation in boundary layers over curvilinear surfaces and robust outflow boundary conditions across the speed regime. In particular, detailed comparisons of linear and nonlinear development of instability waves in a range of boundary-layer flows are used to cross-validate a high-order direct numerical simulation algorithm against the approximate but computationally more efficient technique of parabolized stability equations (PSE). Three separate flow configurations are investigated in this study: (i) development of a Tollmien-Schlichting (TS) instability wave over a two-dimensional (2D), symmetric, low-speed airfoil, (ii) both first and second-mode development in a self-similar, flat plate boundary layer at Mach 4.5, and (iii) amplification of first and second modes of Rayleigh instability and a stationary Gortler vortex in the hypersonic, axisymmetric boundary layer over a flared cone. The satisfactory agreement between the DNS and PSE predictions for both amplitudes and mode shapes of the instability waves confirms the overall efficacy of the DNS algorithm, while underscoring the accuracy of predictions based on the PSE approximation.

An experimental study of the nonlinear evolution of instability waves on a flat plate for mach number M=3

An experimental study of the nonlinear evolution of instability waves on a flat plate for mach number M=3

Direct Numerical Simulation of Supersonic Jet Flow

A numerical method is given for direct numerical simulation of the nonlinear evolution of instability waves in supersonic round jets, with spatial discretisation based on high-order compact finite differences. The numerical properties of a class of symmetric and asymmetric schemes are analysed. Implementation for the Navier–Stokes equations in cylindrical polar coordinates is discussed with particular attention given to treatment of the origin to ensure stability and efficiency. Validation of the schemes is carried out by detailed comparison with linear stability theory. The computer code is applied to study the initial stages of nonlinear development of unstable modes in a Mach 3 jet. The modes of instability that are present include strongly unstable axisymmetric acoustic and helical vortical waves, as well as weakly unstable radiating acoustic and vortical modes. Three distinctive wave patterns are observed from the simulations including a cross-hatched internal shock structure. Nonlinear interactions between the vortical and acoustic modes are investigated.

Application of the Spectral Multidomain Method to the Navier-Stokes Equations

An application of the spectral multidomain method to the two-dimensional, time-dependent, incompressible Navier-Stokes equations is presented. The governing equations are discretized on a non-staggered, stretched mesh with a mixed finite difference/Chebyshev method and are integrated by a time-splitting procedure. The accuracy of the multidomain method is demonstrated for two different channel flow configurations, one with a backward-facing step and the other with a rectangular step. The method is then applied to the investigation of the effects of an isolated, two-dimensional roughness element on the spatial development of instability waves in boundary layers.

MECHANISMS OF AIR-ASSISTED LIQUID ATOMIZATION

A well-controlled experimental apparatus was used to investigate the air-assisted liquid drop atomization process. High-magnification, high-speed photography as well as conventional spray field photography was used to study the breakup of a monodisperse stream of drops injected into a transverse high-velocity air stream. The gas velocity profile, liquid drop velocity, and size were determined by laser Doppler velocimetry (LDV) and a phase Doppler particle analyzer (PDPA). The experiments gave information about the microscopic structure of the liquid breakup process, drop breakup regimes, drag coefficients of atomizing drops, breakup drop size distributions, and drop trajectories. At low gas-liquid relative velocities, the microscopic photographs confirmed the existence of bag and stripping breakup regimes. The photographs also revealed the nature of high-speed, "catastrophic," liquid atomization, which appears to be due to the development of instability waves on the liquid surface. A model based on aerodynamic liquid breakup theory was used to study the influences of the drop drag coefficient and the breakup time on the drop trajectory during the drop atomization process. Previously proposed expressions for the drop drag coefficient and the breakup time constant were tested by comparing the calculated trajectories with the measurements.

Evolution of coherent structures in shear flows: A phase dynamics approach

A general amplitude evolution model is proposed to describe the development of instability waves in parallel free shear flows which are invariant under both space translations and Galilean transformations. It is argued that temporal and spatial modulations of finite-amplitude states are strongly coupled with a marginally unstable large-scale horizontal velocity field. As a result, an array of spatially periodic coherent structures is shown to exhibit two distinct types of phase instabilities: an Eckhaus-like modulational instability dominated by two-dimensional disturbances and a fully three-dimensional secondary instability generated by coupling between pattern deformation and advection by the large scale field. Anisotropic phase waves can also travel on the basic array, the global velocity field acting as a restoring force. Applications of these concepts to the phase dynamics of Kelvin-Helmholtz vortices are briefly considered.

Effect of the acoustic regularization of initial perturbations on the development of ordered structures

Mechanisms are considered by which acoustic oscillations influence the structure of subsonic shear flows. Analysis of the experimental data [1–7] confirms the assumption made in [6] that the regularization of initial perturbations, which causes a higher degree of ordering and an increase in the life of vortices formed because of the development of instability waves or interaction of acoustic oscillations with the edge of the nozzle, is one of the mechanisms by which acoustics influences various shear flows. Photographs are given which show the regularizing effect of acoustics on the development of vortices in the wake behind the edge.

Energy integral description of the development of Kelvin-Helmholtz billows

DOI: 10.1111/j.2153-3490.1976.tb00669.xIn this paper we study the development of instability waves in a stratified shear flow which occurs, for instance, in frontal zones and on large scale internal waves where the Richardson number is sufficiently low. The model is based on splitting the flow into the mean flow and instability wave components. The basis for the interaction between the mean flow and the wave is their respective vertically integrated energy flux equations. The wave description is obtained through a shape assumption: the time dependent wave amplitude is determined by its energy equation solved jointly with the mean flow and the vertical shape function is given by the local linear theory. The instability wave kinetic energy development is determined by the balance between energy production from the mean flow and the conversion of fluctuation kinetic to potential energy. From the energy integral considerations, the numerical results show moderately good agreement with observations in our estimate of the lifetime of the wave and the doubling of the thickness of the mean shear layer during this time. The modification of the mean flow velocity and temperature profiles is explained via the effects of the wave generated vertical momentum and heat (or buoyancy) fluxes, respectively. DOI: 10.1111/j.2153-3490.1976.tb00669.x