Second-mode Instability 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.

Nonlinear Nonmodal Analysis of Hypersonic Flow over Blunt Cones

The linear amplification of modal disturbances that lead to boundary-layer transition in two-dimensional/axisymmetric hypersonic configurations is strongly reduced by the presence of a blunt nose tip, and the mechanisms underlying the low Mack’s second-mode N-factor values at the observed onset of transition over the cone frustum are currently unknown. Linear nonmodal analysis has shown that both planar and oblique traveling disturbances that peak within the entropy-layer experience appreciable energy amplification for moderate to large nose-tip bluntness. The present study extends the previous linear analysis by including the nonlinear effects. Specifically, the harmonic Navier–Stokes equations (HNSE) are solved with a fully implicit formulation and the Newton–Raphson method. The increased number of degrees of freedom for the nonlinear system presents difficulties for solution strategies based on direct solution of the linearized system. Such difficulties are overcome by using the generalized minimal residual method (GMRES) iterative method with a preconditioner corresponding to a simplified Jacobian without the cross-derivative terms. The HNSE solver is verified by comparing with nonlinear parabolized stability equation results for the nonlinear evolution of planar waves in an incompressible Blasius boundary layer and in a Mach 6 flow over a blunt cone. Finally, nonlinear nonmodal results are presented for planar traveling disturbances over a blunt cone configuration with reduced transition N factor as measured in wind-tunnel experiments. The nonmodal analysis demonstrates that entropy-layer disturbances generated close to the nose tip can seed the amplification of higher-frequency Mack’s second-mode instabilities farther downstream and hence can lead to a reduction in the transition N factor.

Implementation Of Optical Diagnostics For Study Of A Non-Canonical Hypersonic Geometry

The study of shock/boundary-layer interactions on a non-canonical hypersonic geometry involves physical complexities not found on the canonical investigations that dominate the literature. The present case examines a cone-slice-ramp, which combines an upstream asymmetric flow expansion with a downstream three-dimensional compression ramp. Optical diagnostics of the off-body flowfield are a necessary complement to surface instrumentation including high-frequency pressure sensors and Temperature Sensitive Paint (TSP). Focused Laser Differential Interferometry (FLDI) measured the flowfield distribution of second-mode instability waves responsible for transition. Filtered Rayleigh Scattering (FRS) was used for flow visualization of the separation region and unsteady shear layer. Velocimetry through the separation shear layer was provided by Femtosecond Laser Electronic Excitation Tagging (FLEET) whereas Coherent Anti-Stokes Raman Scattering (CARS) measured the thermal profiles through multiple flow features generated by the interaction. These flowfield diagnostics detail the alterations in the shock/boundary-layer interaction as the flow state moves from laminar to transitional to turbulent conditions, revealing behaviors absent from canonical flows.

Skin-friction measurements using oil-film interferometry in hypersonic transitional boundary layer flows

Skin-friction measurements in hypersonic transitional boundary layer (HTBL) flows on a flared cone at Mach 6 at zero angle of attack are obtained using oil-film interferometry (OFI). In addition, the amplitude evolution of instability waves is investigated using fast-response pressure sensors and Rayleigh-scattering flow visualization (RSFV). The connection between the skin-friction coefficient and near-wall dynamics during the whole transition process is illustrative and the underlying mechanism of the generation of the wall skin friction is revealed. There exist two peaks of skin-friction rise along the streamwise direction of the model. The first one is at the transitional region (denoted as FS) which is caused by the second-mode instability. The second one is at the region where the transition is completed (denoted as FT). The drastic growth of skin friction near the end of transition is mainly attributed to the rapid boundary-layer breakdown into small scales. It is found that the strong shear in the near-wall region in the quiet zone where dominant hairpin-shaped vortices evolved from the second modes can also cause a local skin-friction rise (denoted as FQ). Further investigations show that in the region between FS and FQ, there is a slight increase in the skin friction, which is associated with the growing low-frequency waves. The skin friction coefficient value decreases slowly after FT but does not reach the expected turbulent theoretical value by the end of the test domain, indicating that the development towards the completely turbulent boundary layer appears to be a gradual process.

Relaminarization effects in hypersonic flow on a three-dimensional expansion–compression geometry

This experimental work explores the flow field around a three-dimensional expansion–compression geometry on a slender cone at Mach 8 using high-frequency pressure sensors, high-frame-rate schlieren, temperature-sensitive paint, shear-stress measurements and oil-flow visualizations. The $7^\circ$ cone geometry has a hyperbolic slice which acts as an expansion corner and suppresses the disturbances present in the upstream boundary layer. Downstream of the cone-slice corner, high-frequency boundary-layer disturbances attenuate in all cases. Under laminar conditions, second-mode instabilities from the cone diminish and lower-frequency second-mode waves develop on the slice at a frequency commensurate with the increased boundary layer thickness. For fully turbulent cases, the boundary layer over the slice shows evidence of a two-layered nature with a turbulent outer region and a near-wall region with strong attenuation of high-frequency disturbances and reappearance of lower-frequency instability waves. When a downstream compression ramp is added to the slice, the expanded boundary layer shows enhanced susceptibility to separation such that separation is observed at a $10^{\circ }$ deflection, which is smaller than expected for turbulent conditions. For a $30^{\circ }$ ramp, boundary-layer separation occurs further upstream where the heat flux contours show a decrease in heating that is characteristic of a transitional separation. These results demonstrate the effect of relaminarization caused by an upstream expansion on a subsequent shock-wave/boundary-layer interaction.

Response of second-mode instability to backward-facing steps in a high-speed flow

Stability in a Mach 4.5 boundary layer over backward-facing steps (BFSs) is investigated using numerical methods. Two types of cases are considered with different laminar inflow conditions, imposed with single-frequency or broadband-frequency modes, respectively. Compared with the typical K-type transition over a flat plate, the boundary layer transition initiated by 90 kHz-frequency second mode appears to follow the same pattern but with a noticeable delay over the step. A larger step height leads to a better inhibition of the downstream Λ-vortices and thus a later transition, providing the step height is smaller than the local boundary layer thickness. Moreover, both the frequency weighted power spectral density and the root mean square of the streamwise velocity indicate the presence of Kelvin–Helmholtz (K–H) instability when the step height is equivalent to the thickness of the nearby boundary layer. There may exist an optimal step height for suppressing single-frequency (90 kHz) mode without exciting significant K–H modes. Similar to the previous studies on roughness, BFS can act as an amplifier for the low-frequency second modes and a suppressor for the high-frequency second modes. The critical frequency is equal to that of the unstable mode whose synchronization point is exactly located at the step corner. Additionally, the correction effects of the step induce the change of the phase speed of the fast mode, which correspondingly results in the movement of the synchronization point. Generally, the BFS is not able to completely alleviate the transition initiated by the broadband-frequency second modes but can still delay the boundary layer transition in a certain degree by suppressing the high-frequency unstable waves.

Wall-Normal Focused Laser Differential Interferometry

A new approach for measuring boundary-layer disturbances with focused laser differential interferometry (FLDI) for planar models is presented. By integrating a glass window into a flat plate, the optical axis was aligned normal to the model surface, and the focal plane was set inside the boundary layer. By determining the extent of the sensitive volume along the optical axis and calculating the analytical transfer function of the setup, the implications of the FLDI properties on the measured data are analyzed. Measurements performed on a flat plate at Mach 6 are used to demonstrate the effects of the laminar–turbulent transition on the spectral distribution of the power density and to explicitly verify the detectability of the expected second-mode instabilities. Advantages and disadvantages of the proposed setup compared to the conventional one are discussed.

The Experiments and Stability Analysis of Hypersonic Boundary Layer Transition on a Flat Plate

Experimental and linear stability theory (LST) investigation of boundary layer transition on a flat plate was conducted with a flow of Mach number 5. The temperature distributions and second-mode disturbances on the flat plate surface at different unit Reynolds number (Reunit) values were captured by infrared thermography and PCB technology, respectively, which revealed the transition location of the flat-plate boundary layer. The PCB sensors successfully captured the second-mode disturbances within the boundary layer initially at a frequency of about 100 kHz, with a gradually expanding frequency range as the distance travelled downstream increased. The evolution characteristics of the second-mode instabilities were also investigated by LST and obtained for the second mode, ranging from 100 to 250 kHz. The amplitude amplification factor (N-factor) of the second-mode instabilities was calculated by the eN method. The N-factor of the transition location in the wind tunnel experiment predicted by LST is about 0.98 and 1.25 for Reunit = 6.38 × 106 and 8.20 × 106, respectively.

Topology optimization of Superhydrophobic Surfaces to delay spatially developing modal laminar–turbulent transition

Super-Hydrophobic Surfaces (SHSs) have been shown to reduce skin friction of an overlying fluid as a consequence of gas pockets trapped within the surface’s microstructure. More recently, they have also been shown capable of delaying laminar–turbulent transition. This article investigates the applicability of topology optimization in designing the macroscopic layout of SHSs in a channel that are able to further delay K-type transition in a spatial setting. Unsteady direct numerical simulations are performed to simulate the transition scenario. This is coupled with adjoint–based sensitivity analysis and gradient based optimization. The optimized designs found through this procedure are capable of moving the transition location further downstream compared to a homogeneous counterpart by inhibiting the growth of secondary instability modes. This article provides the first application of topology optimization to a spatially developing transition scenario.

Effects of Nose Bluntness on Hypersonic Transition over Ogive-Cylinder Forebodies

Ogive cylinders are representative designs of hypersonic forebodies and harbor several flow complexities. This work evaluates the effects of nose bluntness of ogive-cylinder forebodies on their laminar, transitional, and turbulent boundary layers (BLs). Laminar simulations indicate that the thickness of BL and entropy layer (EL) increases with nose bluntness, with the latter displaying a stronger sensitivity to nose radius due to the increased strength and curvature of the leading-edge shock system. Linear perturbation analysis identifies the presence of the second mode, the first mode, and EL modes with increasing bluntness. While the former two are prevalent over a significant streamwise extent, EL modes are limited to the upstream region, well within the swallowing length. Using direct numerical simulations, it is found that transition over the sharpest nose design occurs through the modal fundamental resonance route, driven by second-mode instabilities. Due to weakened BL modes, the blunter geometries undergo transition further downstream in the low-amplitude freestream perturbation environment tested in this study. The transition mechanism in the blunter geometries is intermittent in nature and appears to be driven by turbulent spots generated from streamwise vortices reminiscent of Klebanoff modes.

Mixing enhancement of a compressible jet over a convex wall

A highly compressive effect would suppress the mixing of the shear layer in a convex wall jet. The spanwise distributed protrusions at the nozzle lip are employed to achieve mixing enhancement in this study. The mixing characteristics and enhancement mechanisms are numerically investigated by the delayed detached-eddy simulation method based on the two-equation shear-stress transport model. A widely applicable flow spatiotemporal analysis method, called proper orthogonal decomposition (POD), is used to gain further insight into the dynamical behaviours of the flow instability mode. The results reveal that the centrifugal effect maintains and amplifies the initial perturbations induced by the spanwise distributed heterogeneities, resulting in forced streamwise vortices. The instabilities induced by the streamwise vortices significantly increase the growth rate of the jet half-width and the shear layer vorticity thickness. The spanwise wavelength of the streamwise vortices is consistent with the spanwise distributed forced excitation. In addition, the spanwise meandering motion of the streamwise vortices is observed, which is usually associated with the streamwise travelling wave. This is further confirmed by the POD analysis of the spanwise velocity fluctuation in both stream-radial and stream-span sections. Also, the spatial distributions of the POD modes with the highest energy provide information on the secondary instability modes. Both sinuous and varicose types of disturbances are observed in the unforced jet, whereas the forced jet seems to be dominated by the sinuous type instability, which is more easily excited than the varicose type instability. Moreover, the turbulence intensity in the forced jet is also significantly enhanced as expected due to the earlier and stronger streamwise vortices and associated instabilities. The enhanced turbulent characteristics of the highly compressible condition tend to be isotropic, whereas in the unforced jet, it is anisotropic due to the strong compressibility suppressing the spanwise turbulent fluctuations.

Instability and transition onset downstream of a laminar separation bubble at Mach 6

Instability measurements of an axisymmetric, laminar separation bubble were made over a sharp cone-cylinder-flare with a$12^{\circ }$flare angle under hypersonic quiet flow. Two distinct instabilities were identified: Mack's second mode (which peaked between 190 and 290 kHz) and the shear-layer instability in the same frequency band as Mack's first mode (observed between 50 and 150 kHz). Both instabilities were measured with surface pressure sensors and were captured with high-speed schlieren. Linear stability analysis results agreed well with these measured instabilities in terms of both peak frequencies and amplification rates. Lower-frequency fluctuations were also noted in the schlieren data. Bicoherence analysis revealed nonlinear phase-locking between the shear-layer and second-mode instabilities. For the first time in axisymmetric, low-disturbance flow, naturally generated intermittent turbulent spots were observed in the reattached boundary layer. These spots appeared to evolve from shear-layer-instability wave packets convecting downstream. This work presents novel experimental evidence of the hypersonic shear-layer instability contributing directly to transition onset for an axisymmetric model.

Optimal two-dimensional roughness for transition delay in high-speed boundary layer

The influence of surface roughness on transition to turbulence in a Mach 4.5 boundary layer is studied using direct numerical simulations. Transition is initiated by the nonlinearly most dangerous inflow disturbance, which causes the earliest possible breakdown on a flat plate for the prescribed inflow energy and Mach number. This disturbance primarily comprises two normal second-mode instability waves and an oblique first mode. When localized roughness is introduced, its shape and location relative to the synchronization points of the inflow waves are confirmed to have a clear impact on the amplification of the second-mode instabilities. The change in modal amplification coincides with the change in the height of the near-wall region where the instability wave speed is supersonic relative to the mean flow; the net effect of a protruding roughness is destabilizing when placed upstream of the synchronization point and stabilizing when placed downstream. Assessment of the effect of the roughness location is followed by an optimization of the roughness height, abruptness and width with the objective of achieving maximum transition delay. The optimization is performed using an ensemble-variational (EnVar) approach, while the location of the roughness is fixed upstream of the synchronization points of the two second-mode waves. The optimal roughness disrupts the phase of the near-wall pressure waves, suppresses the amplification of the primary instability waves and mitigates the nonlinear interactions that lead to breakdown to turbulence. The outcome is a sustained non-turbulent flow throughout the computational domain.

Amplification factor transport transition model for high-speed flows

A fully local amplification factor transport model is developed for high-speed transitional flows. On the basis of the approximate eN envelope method of linear stability theory, two transport equations are established to describe the evolution of amplification factors for first- and second-mode instabilities in high-speed boundary layers. An intermittency factor transport equation is then constructed based on the transported amplification factors and is coupled with Menter’s k−ω shear-stress transport eddy-viscosity turbulence model. The new model is validated through several test cases under different flow conditions, including flat plates, straight and flared cones, and a double ramp configuration. Comparisons with linear stability theory and experimental data demonstrate the ability of the model to predict the transition behavior induced by first- or second-mode instabilities. The model provides a reasonable reflection of the effects of different parameters that influence transitions, including the Mach number, temperature, nose bluntness, and pressure gradient.

Effect of porous coatings on the nonlinear evolution of Mack modes in hypersonic boundary layers

The influence of a porous wall on the nonlinear evolution of Mack modes in a hypersonic boundary layer is studied by solving the nonlinear parabolized stability equations. The fundamental resonance of the second mode is particularly considered. It is found that the porous effect leads to (1) a much stronger mean-flow distortion in an indirect way, (2) a greater suppression of the saturated fundamental mode, and (3) slower amplification rates of the secondary instability modes, which eventually delays the transition onset. Detailed explanations of the three mechanisms are provided.

Energy growth of vortical, acoustic, and entropic components of the second-mode instability in the hypersonic boundary layer

The acoustic, vortical, and entropic (thermal) components of the second-mode instability, regarded as an asymptotic behavior of the free-stream counterparts, were found to interact with each other in a well-defined way. However, the mechanisms of the energy growth of each component and the resulting second mode instability remain to be clarified. The present paper provides a quantitative energy analysis of the key sources responsible for the modal growth in the momentum potential theory framework. The acoustic, vortical, and entropic components are governed by energy source effects and interexchange effects, characterized by explicit transport terms and relationships between the growth rate and the energy source. The thermal-acoustic source, induced by the interaction between the fluctuation pressure and the fluctuation entropy, is revealed to be the most pronounced cause of the second-mode instability in the hypersonic boundary layer. The thermal-acoustic source is further decomposed into the dissipative (viscous) part and the non-dissipative (inviscid) part. The dissipative thermal-acoustic source is dominant near the wall surface and destabilizes the second mode. The non-dissipative thermal-acoustic source destabilizes the second mode significantly at the critical layer, while the dissipative thermal-acoustic source stabilizes the second mode in this region.

Attenuation of hypersonic second-mode boundary-layer instability with an ultrasonically absorptive silicon-carbide foam

Attenuation of hypersonic second-mode boundary-layer instability with an ultrasonically absorptive silicon-carbide foam

Effect of Local Thermal Strips on Hypersonic Boundary-Layer Instability

An effect of localized thermal strips on hypersonic boundary-layer instability is investigated using linear stability theory (LST). The linear evolution of Mack's second mode is analyzed for a Mach 4.5 flat-plate boundary-layer with and without the strips. The validity of the LST analysis is assessed through the comparison of the results with implicit large eddy simulations (ILES) for selected cases. Parametric studies on the temperature intensity, length, and location of a single strip are performed to examine the influence of the strip on the second-mode instability. A reversal in the stabilizing and destabilizing effect according to the local streamwise surface pressure gradient induced by the strip is identified. The reversal point of the stabilizing/destabilizing effect is located in the vicinity of the location of the maximum growth rate of the second mode. The influence of three strips in a series arrangement is also analyzed for several combinations. The proper combination of strips, considering the reversal phenomenon, results in more significant stabilizing/destabilizing effects for a disturbance with a specific frequency. The overall effects are also evaluated and analyzed in terms of the envelope of the N-factor curves by carrying out the LST calculations for disturbances with various frequencies.

Hypersonic Boundary-Layer Instability Suppression by Transverse Microgrooves with Machining Flaw

Transverse rectangular microgrooves with local machining flaw are investigated using direct numerical simulations from the viewpoint of suppressing the second-mode instability in a Mach 6 hypersonic boundary layer. Two types of machining flaw are modeled for the local deformation of microgrooves, i.e., chamfered and protruded ones. The effect of locally deformed microgrooves on the boundary-layer stability is studied by superposing the second-mode disturbance to the base flow of the boundary layer. The investigated unstable second mode is 400 kHz, and the microgrooves are installed in its synchronous region. The results show that compression and expansion waves are generated on the surface of the deformed microgrooves, and microscale separation bubbles are formed at the rear edge of the deformed microgrooves. With the increase of local deformation height of microgrooves, the suppression effect of the second-mode disturbance gradually increases. As the deformation height reaches 0.09 mm, the amplitude inhibition rate of locally deformed microgrooves on the second-mode disturbance reaches 44.7%, compared to that of the regular one.

Modal laminar–turbulent transition delay by means of topology optimization of superhydrophobic surfaces

When submerged under a liquid, the microstructure of a SuperHydrophobic Surface (SHS) traps a lubricating layer of gas pockets, which has been seen to reduce the skin friction of the overlying liquid flow in both laminar and turbulent regimes. More recently, spatially homogeneous SHS have also been shown to delay laminar–turbulent transition in channel flows, where transition is triggered by modal mechanisms. In this study, we investigate, by means of topology optimization, whether a spatially inhomogeneous SHS can be designed to further delay transition in channel flows. Unsteady direct numerical simulations are conducted using the spectral element method in a 3D periodic wall-bounded channel. The effect of the SHS is modelled using a partial slip length on the walls, forming a 2D periodic optimization domain. Following a density-based approach, the optimization procedure uses the adjoint-variable method to compute gradients and a checkpointing strategy to reduce storage requirements. This methodology is adapted to optimizing over an ensemble of initial perturbations.This study presents the first application of topology optimization to laminar–turbulent transition. We show that this method can design surfaces that delay transition significantly compared to a homogeneous counterpart, by inhibiting the growth of secondary instability modes. By optimizing over an ensemble of streamwise phase-shifted perturbations, designs have been found with comparable mean transition time and lower variance.