## Publications

84 results found

Wu X, Qin F, 2024, Excitation and evolution of radiating modes in supersonic boundary layers. Part 1. Fundamental resonance with impinging sound waves, *Journal of Fluid Mechanics*, ISSN: 0022-1120

Wu X, Qin F, 2024, Excitation and evolution of radiating modes in supersonic boundary layers. Part 2. Back effect of spontaneously radiated Mach waves, *Journal of Fluid Mechanics*, ISSN: 0022-1120

Xu J, Liu J, Zhang Z,
et al., 2023, Spatial-temporal transformation for primary and secondary instabilities in weakly non-parallel shear flows, *Journal of Fluid Mechanics*, Vol: 959, ISSN: 0022-1120

When studying instability of weakly non-parallel flows, it is often desirable to convert temporal growth rates of unstable modes, which can readily be computed, to physically more relevant spatial growth rates. This has been performed using the well-known Gaster's transformation for primary instability and Herbert's transformation for the secondary instability of a saturated primary mode. The issue of temporal–spatial transformation is revisited in the present paper to clarify/rectify the ambiguity/misunderstanding that appears to exist in the literature. A temporal mode and its spatial counterpart may be related by sharing either the real frequency or wavenumber, and the respective transformations between their growth rates are obtained by a simpler consistent derivation than the original one. These transformations, which consist of first- and second-order versions, are valid under conditions less restrictive than those for Gaster's and Herbert's transformations, and reduce to the latter under additional conditions, which are not always satisfied in practice. The transformations are applied to inviscid Rayleigh instability of a mixing layer and a jet, secondary instability of a streaky flow as well as general detuned secondary instability (including subharmonic and fundamental resonances) of primary Mack modes in a supersonic boundary layer. Comparison of the transformed growth rates with the directly calculated spatial growth rates shows that the transformations derived in this paper outperform Gaster's and Herbert's transformations consistently. The first-order transformation is accurate when the growth rates are small or moderate, while the second-order transformations are sufficiently accurate across the entire instability bands, and thus stand as a useful tool for obtaining spatial instability characteristics via temporal stability analysis.

Zhang Z, Wu X, 2023, Generation of sound waves by nonlinearly evolving ring-mode coherent structures on a turbulent subsonic circular jet: a comparative study of two mechanisms, *ACTA MECHANICA SINICA*, Vol: 39, ISSN: 0567-7718

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Zhang Z, Wu X, 2023, A unified theory for the envelope radiation of ring-mode coherent structures in the very-near-nozzle and developed regions of a circular jet, *PHYSICS OF FLUIDS*, Vol: 35, ISSN: 1070-6631

Zhu K, Wu X, 2022, Effects of spanwise-periodic surface heating on supersonic boundary-layer instability, *Journal of Fluid Mechanics*, Vol: 940, ISSN: 0022-1120

The effects of streamwise-elongated, spanwise-periodic surface heating on a supersonic boundary-layer instability are investigated under the assumption of high Reynolds number. Our focus is on the lower-branch viscous instability and so the spanwise spacing of the elements is chosen to be of O(Re−3/8L) , the wavelength of the latter, where Re is the Reynolds number based on L , the distance from the leading edge to the centre of the elements. The streamwise length is assumed to be much longer in order to simplify the mathematical description. Starting with classical triple-deck theory, the equations governing the heating-induced streaky flow are derived by appropriate rescaling. When Chapman's viscosity law is adopted, a similarity solution is found. The stability of the streaky flow, which is of a bi-global nature, is shown to be governed by a novel triple-deck structure characterised by fully compressible dynamics in the lower deck. Through asymptotic analysis, the bi-global stability is reduced to a one-dimensional eigenvalue problem, which involves only the spanwise-dependent wall temperature and wall shear. The instability modes may be viewed as a continuation of oncoming first Mack modes, but might also be considered as a new kind since they exhibit two distinctive features: strong temperature perturbation near the wall and spontaneous radiation of an acoustic wave to the far field, neither of which is shared by first Mack modes. Numerical calculations, performed for two simple patterns of spanwise-periodic heating elements, demonstrate their stabilising/destabiling effects on modes with different frequencies and spanwise wavelengths.

Zhang Z, Wu X, 2022, Nonlinear evolution and low-frequency acoustic radiation of ring-mode coherent structures on subsonic turbulent circular jets, *Journal of Fluid Mechanics*, Vol: 940, ISSN: 0022-1120

By adapting the triple decomposition of an instantaneous turbulent flow into atime-averaged mean field, large-scale coherent motion and fine-scale random fluctuations,and treating the coherent motion as instability modes on the mean flow, a mathematicaltheory is developed to describe the nonlinear spatial–temporal modulation and acousticradiation of a coherent structure (CS) on a circular jet in the form of a wavepacketconsisting of an axisymmetric (ring) mode and its sideband components. The effect offine-scale turbulence on the CS is characterised via a gradient closure model, and thenon-parallelism due to the axial variation and radial velocity of the mean flow is takeninto account. By employing the matched asymptotic expansion and multi-scale techniques,a strongly nonlinear system is derived, which governs the envelope of the CS and itsvorticity and temperature in the critical layer. Numerical solutions to the evolution systemshow that the theory captures the nonlinear amplitude attenuation and vorticity roll-upas observed in experiments. The large-distance asymptotic properties of the CS allow usto describe and predict its acoustic radiation on the basis of first principles. The CS istrapped within the jet, but its self-interaction generates a temporally and axially modulatedmean-flow distortion, which acts as the emitter to radiate low-frequency sound waves, withthe Reynolds stresses driving this mean-flow distortion being identified unambiguouslyto be the physical source in the present context. An equivalent source in the Lighthilltype of acoustic analogy is also identified. For the present ring-mode CS in the fullydeveloped region of a circular jet, the equivalent source can be determined before theacoustic field is, and the intensity of the radiated sound waves is found to be O(3), wheremeasures the magnitude of the CS. The directivity and spectrum of the acoustic far field are calculated for representative parameters, and the predicted features resem

Xu J, Wu X, 2022, Surface-roughness effects on crossflow instability of swept-wing boundary layers through generalized resonances, *AIAA Journal: devoted to aerospace research and development*, Vol: 60, Pages: 2887-2904, ISSN: 0001-1452

It is known that crossflow instability and transition can be influenced significantly by micrometer-sized surface roughness. A recent study sought to explain such a sensitive effect from the standpoint of a generalized resonant-triad interaction between crossflow instability modes and distributed roughness-induced perturbations. The mechanism was demonstrated for Falkner–Skan–Cooke similarity velocity profiles. In the present paper, we examine its role in destabilizing stationary and travelling crossflow vortices in the boundary layers over the NLF(2)-0415 swept wing, for which experiments found that micrometer-sized roughness caused earlier transition. Our analysis shows that the generalized resonance mechanism operates in a swept-wing boundary layer. Under the assumption that roughness consists of all spectral components, a crossflow mode with a fixed-dimensional frequency and wavelength resonates at each chordwise location with one or more other modes. We derive the amplitude equations for the interacting modes. The calculations show that the resonance is highly effective, especially when the roughness elements are near the leading edge. Importantly, it is found that the wave numbers of the roughness spectra participating in the most effective resonant interactions are very close to those of the right-branch neutral stationary eigenmode. As a result, micrometer-sized distributed roughness generates a perturbation of much larger amplitude, which alters, through the resonant interactions, the local growth rates of the crossflow vortices by an O(1) amount for both stationary and travelling vortices. We also traced the chordwise development of crossflow vortices with fixed-dimensional frequencies and spanwise wave numbers by integrating the local growth rate, or by solving the initial-value problem of the amplitude equations, for roughness without and with chordwise modulation. For roughness with a height of 10 μm, the accumulated effect

Xu J, He J, Wu X, 2022, Triadic Resonance Analysis of Distributed Roughness Effects on Crossflow Instability in Swept-Wing Boundary Layers, 9th IUTAM Symposium on Laminar-Turbulent Transition, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 825-836, ISSN: 1875-3507

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Zhang Z, Wu X, 2022, Nonlinear Evolution of Multiple Helical Modes in the Near-Nozzle Region of Subsonic Circular Jets: A Weakly Nonlinear Critical-Layer Theory, 9th IUTAM Symposium on Laminar-Turbulent Transition, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 137-147, ISSN: 1875-3507

Liu J, Marensi E, Wu X, 2022, Effects of Streaky Structures on the Instability of Supersonic Boundary Layers, 9th IUTAM Symposium on Laminar-Turbulent Transition, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 587-598, ISSN: 1875-3507

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Xu D, Wu X, 2021, Elevated low-frequency free-stream vortical disturbances eliminate boundary-layer separation, *Journal of Fluid Mechanics*, Vol: 920, ISSN: 0022-1120

steady two-dimensional boundary layer subject to an adverse streamwise pressure gradient usually separates. In this paper, we investigate how free-stream vortical disturbances (FSVD) of moderate level prevent the separation in such a boundary layer over a plate or concave wall. The focus is on physically realisable FSVD with sufficiently long wavelength (low frequency) as they have the most significant impact on the boundary layer. The FSVD intensity ϵ is taken to be small but nevertheless strong enough that the streaks or Görtler vortices generated in the boundary layer are fully nonlinear and can alter the mean-flow profile by an order-one amount. The excitation and evolution of streaks and Görtler vortices are governed by the nonlinear unsteady boundary-region equations supplemented by appropriate initial (upstream) and boundary (far-field) conditions, which describe appropriately the action of FSVD on the boundary layer. The flow variables are decomposed into two parts: the steady spanwise-averaged and the unsteady or spanwise-varying components. These two parts are coupled and are computed simultaneously. Numerical results show that the separation is eliminated when the FSVD level exceeds a critical intensity ϵc . It is inferred that the strong nonlinear mean-flow distortion associated with the nonlinear streaks or Görtler vortices prevents the separation. The critical FSVD intensity ϵc depends on the streamwise curvature, the pressure gradient and the frequency of FSVD. The value of ϵc decreases significantly with the Görtler number, indicating that concave curvature inhibits separation. A higher ϵc is required to prevent the separation in the case of stronger adverse pressure gradient. Interestingly, unsteady FSVD with low frequencies are found to be more effective than steady ones in suppressing the separation.

Liu Y, Dong M, Wu X, 2021, Receptivity of inviscid modes in supersonic boundary layers to wall perturbations, *Journal of Engineering Mathematics*, Vol: 128, ISSN: 0022-0833

The present paper investigates the receptivity of inviscid first and second modes in a supersonic boundary layer to time-periodic wall disturbances in the form of local blowing/suction, streamwise velocity perturbation and temperature perturbation, all introduced via a small forcing slot on the flat plate. The receptivity is studied using direct numerical simulations (DNS), finite- and high-Reynolds-number approaches, which complement each other. The finite-Reynolds-number formulation predicts the receptivity as accurately as DNS, but does not give much insight to the detailed excitation process, nor can it explain the significantly weaker receptivity efficiency of the streamwise velocity and temperature perturbations relative to the blowing/suction. In order to shed light on these issues, an asymptotic analysis was performed in the limit of large Reynolds number. It shows that the receptivity to all three forms of wall perturbations is reduced to the same mathematical form: the Rayleigh equation subject to an equivalent suction/blowing velocity, which can be expressed explicitly in terms of the physical wall perturbations. Estimates of the magnitude of the excited eigenmode can be made a priori for each case. Furthermore, the receptivity efficiencies for the streamwise velocity and temperature perturbations are quantitatively related to that for the blowing/suction by simple ratios, which are of O(R−1/2) and have simple expressions, where R is the Reynolds number based on the boundary-layer thickness at the centre of the forcing slot. The simple leading-order asymptotic theory predicts the instability and receptivity characteristics accurately for sufficiently large Reynolds numbers (about 104), but appreciable error exists for moderate Reynolds numbers. An improved asymptotic theory is developed by using the appropriate impedance condition that accounts for the O(R−1/2) transverse velocity induced by the viscous motion in the Stokes layer adjacent to t

Xu D, Liu J, Wu X, 2020, Gortler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability, *Journal of Fluid Mechanics*, Vol: 900, ISSN: 0022-1120

This paper investigates streaks and Görtler vortices in a boundary layer over a flat or concave wall in a contracting or expanding stream, which provides a favourable or adverse pressure gradient, respectively. We consider first the excitation of streaks and Görtler vortices by free stream vortical disturbances (FSVD), and their nonlinear evolution. The focus is on FSVD with sufficiently long wavelength, to which the boundary layer is most receptive. The formulation is directed at the general case where the Görtler number GΛ (based on the spanwise length scale Λ of FSVD) is of order one, and the FSVD is strong enough that the induced vortices acquire an O(1) streamwise velocity in the region where the boundary layer thickness becomes comparable with Λ, and the vortices are governed by the nonlinear boundary region equations (NBRE). An important effect of a pressure gradient is that the oncoming FSVD are distorted by the non-uniform inviscid flow outside the boundary layer through convection and stretching. This process is accounted for by using the rapid distortion theory. The impact of the distorting FSVD is analysed to provide the appropriate initial and boundary conditions, which form, along with the NBRE, the appropriate initial boundary value problem describing the excitation and nonlinear evolution of the vortices. Numerical results show that an adverse/favourable pressure gradient cause the Görtler vortices to saturate earlier/later, but at a lower/higher amplitude than that in the case of zero-pressure-gradient. On the other hand, for the same pressure gradient and at low levels of FSVD, the vortices saturate earlier and at a higher amplitude as GΛ increases. Raising FSVD intensity reduces the effects of the pressure gradient and curvature. At a high FSVD level of 14 %, the curvature has no impact on the vortices, while the pressure gradient only influences the saturation intensity. The unsteadiness o

Dong M, Liu Y, Wu X, 2020, Receptivity of inviscid modes in supersonic boundary layers due to scattering of free-stream sound by localised wall roughness, *Journal of Fluid Mechanics*, Vol: 896, ISSN: 0022-1120

The present paper investigates the receptivity of inviscid first and second modes in super/hypersonic boundary layers due to the interaction between a weak free-stream acoustic wave and a small isolated surface roughness element. The large-Reynolds-number asymptotic analysis reveals the detailed processes of the excitation. The distortion of the acoustic signature by the curved wall contributes to the leading-order receptivity, producing an eigenmode of O(Eh) amplitude, where E≪1 is the magnitude of the acoustic wave and h≪1 the roughness height normalised by the local boundary-layer thickness δ . The interactions between the roughness-induced mean-flow distortion and the acoustic signature contribute to the second-order receptivity, which is of O(EhR−1/3) with R≫1 being the Reynolds number based on δ . Interestingly, the leading-order receptivity is equivalent to a canonic receptivity problem, the excitation by time-periodic blowing and suction through a local slot on the wall, and the effective periodic outflux velocity forced from the underneath Stokes layer can be determined explicitly in terms of the roughness shape function. This equivalence holds when h=O(R−1/3) , for which the roughness-induced mean-flow distortion becomes nonlinear. A systematic parametric study is carried out for the excitation of the first and second modes by both fast and slow free-stream acoustic waves, and the dependence of the receptivity efficiency on the relevant parameters is provided. Interestingly, the second-order receptivity could become dominant (e.g. in the case of slow acoustic waves with low frequencies and small incident angles), but the present mathematical theory remains valid. In order to check the accuracy of the asymptotic predictions, we have carried out direct numerical simulations (DNS) and also extended the existing finite-Reynolds-number theory to the supersonic regime. The asymptotic solutions agree with the results given by

Katai C, Wu X, 2020, Effects of streamwise elongated and spanwise periodic surface roughness elements on boundary-layer instability, *Journal of Fluid Mechanics*, Vol: 899, Pages: 1-47, ISSN: 0022-1120

We investigate the impact on the boundary-layer stability of spanwise periodic, stream-wise elongated surface roughness elements. Our interest is in their effects on the so-called lower-branch Tollmien–Schlichting modes, and so the spanwise spacing of the elements is taken to be comparable with the spanwise wavelength of the latter, which is ofO(R−3/8L), whereLis the dimensional length from the leading edge of the flat plate to the surface roughness, and R is the Reynolds number based onL. The streamwise length is much longer, consistent with experimental setup. The roughness height is chosen such that the wall shear is altered byO(1). From the generic triple-deck theory for three-dimensional roughness elements with both the streamwise and spanwise length scales being ofO(R−3/8L), we derived the relevant governing equations by appropriate rescaling. The resulting equations are nonlinear but parabolic because the pressure gradient in the streamwise direction is negligible while in the spanwise direction is completely determined by the roughness shape. Appropriate upstream, boundary and13matching conditions are derived for the problem. Due to the parabolicity, the equations are solved efficiently using a marching method to obtain the streaky flow. The instability of the streaky flow is shown to be controlled by the spanwise dependent (periodic) wall shear. Two- and weakly three-dimensional lower-frequency modes are found to be stabilised by the streaks, confirming previous experimental findings, while stronger three-dimensional and higher-frequency modes are destabilised. Among the three roughness shapes considered, the roughness elements in the form of hemispherical cap is found to be most effective for a given height. A resonant subharmonic interaction was found to occur for modes with spanwise wavelength twice that of the roughness elements.

Mikhaylov K, Wu X, 2020, Nonlinear evolution of interacting sinuous and varicose modes in plane wakes and jets: quasi-periodic structures, *Physics of Fluids*, Vol: 32, ISSN: 1070-6631

A plane wake or jet supports sinuous and varicose instability modes. The nonlinear interaction between them following their linear development was described previously by Leib and Goldstein [“Nonlinear interaction between the sinuous and varicose instability modes in a plane wake,” Phys. Fluids A 1, 513–521 (1989)] using the strongly nonlinear non-equilibrium critical-layer approach in the case of the Bickley jet for which the frequencies of the sinuous and varicose modes have an integer ratio of 2. This paper develops the theory for general profiles where the frequencies of the sinuous and varicose modes are non-commensurable. The disturbance is quasi-periodic in time and space and must be expressed as a function of two phase variables. Using matched asymptotic expansions simultaneously with the multi-scale method, we derived a set of coupled evolution equations governing the development of the amplitudes and critical-layer vorticities of these modes. The evolution system is solved for the base-flow profiles mimicking those in experiments. The sinuous mode suppresses the varicose mode but also causes the latter to saturate in a highly oscillatory manner. The varicose mode inhibits the sinuous mode initially. However, in the later stage, it lends the sinuous mode a significantly higher saturating amplitude. For a wide range of initial modal compositions and Reynolds numbers, the ratio of the varicose mode amplitude to that of the sinuous mode eventually tends to an almost constant value in the range of 0.4–0.6, in line with the experimental measurement. Due to the self and mutual interactions, the vorticities roll up to form vortices, which are non-symmetric in the transverse direction and quasi-periodic in the streamwise direction as well as in time. With such an increased complexity, the vortices resemble those observed in experiments. The nonlinear interactions of the sinuous and varicose modes in the critical layer generate all harmonics

Liu Y, Dong M, Wu X, 2020, Generation of first Mack modes in supersonic boundary layers by slow acoustic waves interacting with streamwise isolated wall roughness, *Journal of Fluid Mechanics*, Vol: 888, ISSN: 0022-1120

This paper investigates the receptivity of a supersonic boundary layer to slow acoustic waves whose characteristic frequency and wavelength are on the triple-deck scales, and the phase speed is thus asymptotically small. Acoustic waves on these scales are of special importance as they have the interesting property that a perturbation with a magnitude of O(ϵu) in the free stream generates much larger, O(εuR1/8) , velocity fluctuations inside the boundary layer, where R is the Reynolds number based on the distance to the leading edge. Their interaction with streamwise localized roughness elements, leading to stronger receptivity, is studied using triple-deck theory and direct numerical simulations (DNS). The receptivity coefficient, defined as the ratio of the streamwise-velocity amplitude of the instability mode excited to that of the incident free-stream acoustic wave, serves to characterize receptivity efficiency. Its dependence on the roughness width, the Reynolds number R , the free-stream Mach number M and the incident angle of the acoustic wave is examined. The theoretical predictions, obtained assuming R≫1 , are found to be in quantitative agreement with the DNS results at moderate values of R when the roughness elements are located near the lower branch of the instability. The receptivity is sensitive to the incident angle (or the phase speed) of the acoustic wave, being highly effective within a small range of angles close to cos−1(1/M) and π+cos−1(1/M) for downstream and upstream propagating sound waves, respectively. The amplitude of the instability mode excited is proportional to the streamwise-velocity amplitude of the acoustic signature inside the boundary layer, and scales with the roughness height h∗ as (h∗/δ∗)R1/4 , where δ∗ is the boundary-layer thickness.

Zhang Z, Wu X, 2020, Nonlinear evolution and acoustic radiation of coherent structures in subsonic turbulent free shear layers, *Journal of Fluid Mechanics*, Vol: 884, Pages: A10-1-A10-68, ISSN: 0022-1120

Large-scale coherent structures are present in compressible free shear flows, where they are known to be a main source of aerodynamic noise. Previous studies showed that these structures may be treated as instability waves or wavepackets supported by the underlying turbulent mean flow. By adopting this viewpoint in the framework of triple decomposition of the instantaneous flow into the mean field, coherent motion and small-scale turbulence, a strongly nonlinear dynamical model was constructed to describe the formation and development of coherent structures in incompressible turbulent free shear layers (Wu & Zhuang, J. Fluid Mech., vol. 787, 2016, pp. 396–439). That model is now extended to compressible flows, for which the coherent structures are extracted through a density-weighted (Favre) phase average. The nonlinear non-equilibrium critical-layer theory for instability waves in a laminar compressible mixing layer is adapted to analyse coherent structures in its turbulent counterpart. The strong non-parallelism associated with the fast spreading of the turbulent mean flow is taken into account and found to be significant. The model also accounts for the effect of fine-scale turbulence on coherent structures via a gradient type of closure model which now allows for a phase lag between the phase-averaged small-scale Reynolds stresses and the strain rates of coherent structures. The analysis results in an evolution system comprising of an amplitude equation, the critical-layer temperature and vorticity equations along with the appropriate initial and boundary conditions. The physical processes of acoustic radiation from the coherent structures are described by examining the far-field asymptote of the hydrodynamic fluctuations. We demonstrate that the nonlinearly generated slowly breathing mean-flow distortion radiates low-frequency sound waves. The true physical sources are identified. Equivalent sources in a Lighthill type of acoustic analogy context also

Wu X, Zhang Z, 2019, First-principle description of acoustic radiation of shear flows, *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 377, Pages: 1-18, ISSN: 1364-503X

As a methodology complementary to acoustic analogy, the asymptotic approach to aeroacoustics seeks to predict aerodynamical noise on the basis of first principles by probing into the physical processes of acoustic radiation. The present paper highlights the principal ideas and recent developments of this approach, which have shed light on some of the fundamental issues in sound generation in shear flows. The theoretical work on sound wave emission by nonlinearly modulated wavepackets of supersonic and subsonic instability modes in free shear flows identifies the respective physical sources or emitters. A wavepacket of supersonic modes is itself an efficient emitter, radiating directly intensive sound in the form of a Mach wave beam, the frequencies of which are in the same band as those of the modes in the packet. By contrast, a wavepacket of subsonic modes radiates very weak sound directly. However, the nonlinear self-interaction of such a wavepacket generates a slowly modulated mean-flow distortion, which then emits sound waves with low frequencies and long wavelengths on the scale of the wavepacket envelope. In both cases, the acoustic waves emitted to the far field are explicitly expressed in terms of the amplitude function of the wavepacket. The asymptotic approach has also been applied to analyse generation of sound waves in wall-bounded shear flows on the triple-deck scale. Several subtleties have been found. The near-field approximation has to be worked out to a sufficiently higher order in order just to calculate the far-field sound at leading order. The back action of the radiated sound on the flow in the viscous sublayer and the main shear layer is accounted for by an impedance coefficient. This effect is of higher order in the subsonic regime, but becomes a leading order in the transonic and supersonic regimes.

Gonzalez Hernandez C, Wu X, 2019, Receptivity of supersonic boundary layers over smooth and wavy surfaces to impinging slow acoustic waves, *Journal of Fluid Mechanics*, Vol: 872, Pages: 849-888, ISSN: 0022-1120

In this paper, we investigate the receptivity of a supersonic boundary layer toimpinging acoustic waves. Unlike previous studies of acoustic receptivity, wherethe sound waves have phase speeds comparable with or larger than the free-streamvelocity U∞, the acoustic waves here have much slower (O(R−1/8U∞)) phase velocity,and their characteristic wavelength and frequency are of O(R−3/8L) and O(R1/4U∞/L)respectively, compatible with the triple-deck structure, where L is the distance to theleading edge and R the Reynolds number based on L and U∞. A significant feature ofa sound wave on the triple-deck scale is that an O(εs) perturbation in the free streamgenerates much stronger (O(εsR1/8)) velocity fluctuations in the boundary layer. Tworeceptivity mechanisms are considered. The first is new, involving the interaction oftwo such acoustic waves and operating in a boundary layer over a smooth wall. Thesecond involves the interaction between an acoustic wave and the steady perturbationinduced by a wavy wall. The sound–sound, or sound–roughness, interactions generatea forcing in resonance with a neutral Tollmien–Schlichting (T–S) wave. The latter isthus excited near the lower branch of the neutral curve, and subsequently undergoesexponential amplification. The excitation through sound–sound interaction may offera possible explanation for the appearance of instability modes downstream of theirneutral locations as was observed in a supersonic boundary layer over a smoothwall. The triple-deck formalism is adopted to describe impingement and reflection ofthe acoustic waves, and ensuing receptivity, allowing the coupling coefficient to becalculated. The two receptivity processes with the acoustic waves on the triple-deckscale are much more effective compared with those involving usual sound waves,with the coupling coefficient being greater by a factor of O(R1/4) and O(R1/8) inthe sound–sound a

He J, Butler A, Wu X, 2019, Effects of distributed roughness on crossflow instability through generalized resonance mechanisms, *Journal of Fluid Mechanics*, Vol: 858, Pages: 787-831, ISSN: 0022-1120

Experiments have shown that micron-sized distributed surface roughness can significantly promote transition in a three-dimensional boundary layer dominated by crossflow instability. This sensitive effect has not yet been fully explained physically and mathematically. Past studies focused on surface roughness exciting crossflow vortices and/or changing the local stability characteristics. The present paper seeks possible additional mechanisms by investigating the effects of distributed surface roughness on crossflow instability through resonant interactions with eigenmodes. A key observation is that the perturbation induced by roughness with specific wavenumbers can interact with two eigenmodes (travelling and stationary vortices) through triadic resonance, or interact with one eigenmode (stationary vortices) through Bragg scattering. Unlike the usual triadic resonance of neutral, or nearly neutral, eigenmodes, the present triadic resonance can take place among modes with growth rates, provided that these are equal; unlike the usual Bragg scattering involving neutral waves, crossflow stationary vortices can also be unstable. For these amplifying waves, the generalized triadic resonance and Bragg scattering are put forward, and the resulting corrections to the growth rates are derived by a multiple-scale method. The analysis is extended to the case where up to four crossflow vortices interact with each other in the presence of suitable roughness components. The numerical results for Falkner–Skan–Cooke boundary layers show that roughness with a small height (a few percent of the local boundary-layer thickness) can change growth rates substantially (by a more-or-less amount). This sensitive effect is attributed to two facts: (i) the resonant nature of the triadic interaction and Bragg scattering, which makes the correction to the growth rate proportional to the roughness height, and (ii) the wavenumbers of the roughness component required for the resonance

Wu X, 2019, Nonlinear Theories for Shear Flow Instabilities: Physical Insights and Practical Implications, ANNUAL REVIEW OF FLUID MECHANICS, VOL 51, Editors: Davis, Moin, Publisher: ANNUAL REVIEWS, Pages: 451-485

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Butler A, Wu X, 2018, Stationary crossflow vortices near the leading edge of three-dimensional boundary layers: the role of non-parallelism and excitation by surface roughness, *Journal of Fluid Mechanics*, Vol: 845, Pages: 93-140, ISSN: 0022-1120

Non-parallelism, i.e. the effect of the slow variation of the boundary-layer flow in the chordwise and spanwise directions, in general produces a higher-order correction to the growth rate of instability modes. Here we investigate stationary crossflow vortices, which arise due to the instability of the three-dimensional boundary layer over a swept wing, focusing on a region near the leading edge where non-parallelism plays a leading-order role in their development. In this regime, the vortices align themselves with the local wall shear at leading order, and so have a marginally separated triple-deck structure, consisting of the inviscid main boundary layer, an upper deck and a viscous sublayer. We find that the streamwise (and spanwise) variations of both the base flow and the modal shape must be accounted for. An explicit expression for the growth rate is derived that shows a neutral point occurs in this regime, downstream of which non-parallelism has a stabilising effect. Stationary crossflow vortices thus have a viscous and non-parallel genesis near the leading edge. If an ‘effective pressure minimum’ occurs within this region then the growth rate becomes unbounded, and so the previous analysis is regularised within a localised region around it. A new instability is identified. The mode maintains its three-tiered structure, but the pressure perturbation now plays a passive role. Downstream, the instability evolves into a Cowley, Hocking & Tutty (Phys. Fluids, vol. 28, 1985, pp. 441–443) instability associated with a critical layer located in the lower deck. Finally, we consider the receptivity of the flow in the non-parallel regime: generation of stationary crossflow modes by arrays of chordwise-localised, spanwise-periodic surface roughness elements. The flow responds differently to different Fourier spectral content of the roughness, giving the lower deck a two-part structure. We find that roughness elements with sharper edges generate str

Wu X, Xu D, Zhang Y, 2017, Nonlinear evolution and secondary instability of steady and unsteady vortices induced by free-stream vortical disturbances, *Journal of Fluid Mechanics*, Vol: 829, Pages: 681-730, ISSN: 0022-1120

We study the nonlinear development and secondary instability of steady and unsteady Görtler vortices which are excited by free-stream vortical disturbances (FSVD) in a boundary layer over a concave wall. The focus is on low-frequency (long-wavelength) components of FSVD, to which the boundary layer is most receptive. For simplification, FSVD are modelled by a pair of oblique modes with opposite spanwise wavenumbers , and their intensity is strong enough (but still of low level) that the excitation and evolution of Görtler vortices are nonlinear. For the general case that the Görtler number (based on the spanwise wavelength of the disturbances) is , the formation and evolution of Görtler vortices are governed by the nonlinear unsteady boundary-region equations, supplemented by appropriate upstream and far-field boundary conditions, which characterize the impact of FSVD on the boundary layer. This initial-boundary-value problem is solved numerically. FSVD excite steady and unsteady Görtler vortices, which undergo non-modal growth, modal growth and nonlinear saturation for FSVD of moderate intensity. However, for sufficiently strong FSVD the modal stage is bypassed. Nonlinear interactions cause Görtler vortices to saturate, with the saturated amplitude being independent of FSVD intensity when . The predicted modified mean-flow profiles and structure of Görtler vortices are in excellent agreement with several steady experimental measurements. As the frequency increases, the nonlinearly generated harmonic component (which has zero frequency and wavenumber ) becomes larger, and as a result the Görtler vortices appear almost steady. The secondary instability analysis indicates that Görtler vortices become inviscidly unstable in the presence of FSVD with a high enough intensity. Three types of inviscid unstable modes, referred to as sinuous (odd) modes I, II and varicose (even) modes I, are identified, and their rel

Huang Z, Wu X, 2017, A local scattering approach for the effects of abrupt changes on boundary-layer instability and transition: a finite-Reynolds-number formulation for isolated distortions, *Journal of Fluid Mechanics*, Vol: 822, Pages: 444-483, ISSN: 0022-1120

We investigate the influence of abrupt changes on boundary-layer instability and transition. Such changes can take different forms including a local porous wall, suction/injection and surface roughness as well as junctions between rigid and porous walls. They may modify the boundary conditions and/or the mean flow, and their effects on transition have usually been assessed by performing stability analysis for the modified base flow and/or boundary conditions. However, such a conventional local linear stability theory (LST) becomes invalid if the change occurs over a relatively short scale comparable with, or even shorter than, the characteristic wavelength of the instability. In this case, the influence on transition is through scattering with the abrupt change acting as a local scatter, that is, an instability mode propagating through the region of abrupt change is scattered by the strong streamwise inhomogeneity to acquire a different amplitude. A local scattering approach (LSA) should be formulated instead, in which a transmission coefficient, defined as the ratio of the amplitude of the instability wave after the scatter to that before, is introduced to characterize the effect on instability and transition. In the present study, we present a finite-Reynolds-number formulation of LSA for isolated changes including a rigid plate interspersed by a local porous panel and a wall suction through a narrow slot. When the weak non-parallelism of the unperturbed base flow is ignored, the local scattering problem can be cast as an eigenvalue problem, in which the transmission coefficient appears as the eigenvalue. We also improved the method to take into account the non-parallelism of the unperturbed base flow, where it is found that the weak non-parallelism has a rather minor effect. The general formulation is specialized to two-dimensional Tollmien–Schlichting (T–S) waves. The resulting eigenvalue problem is solved, and full direct numerical simulations (DNS)

Marensi E, Ricco P, Wu X, 2017, Nonlinear unsteady streaks engendered by the interaction of free-stream vorticity with a compressible boundary layer, *JOURNAL OF FLUID MECHANICS*, Vol: 817, Pages: 80-121, ISSN: 0022-1120

Qin F, Wu X, 2016, Response and receptivity of the hypersonic boundary layer past awedge to free-stream acoustic, vortical and entropy disturbances, *Journal of Fluid Mechanics*, Vol: 797, Pages: 874-915, ISSN: 0022-1120

This paper analyses the response and receptivity of the hypersonic boundary layerover a wedge to free-stream disturbances including acoustic, vortical and entropyfluctuations. Due to the presence of an attached oblique shock, the boundary layer isknown to support viscous instability modes whose eigenfunctions are oscillatory in thefar field. These modes acquire a triple-deck structure. Any of three elementary typesof disturbances with frequency and wavelength on the triple-deck scales interacts withthe shock to generate a slow acoustic perturbation, which is reflected between theshock and the wall. Through this induced acoustic perturbation, vortical and entropyfree-stream disturbances drive significant velocity and temperature fluctuations withinthe boundary layer, which is impossible when the shock is absent. A quasi-resonancewas identified, due to which the boundary layer exhibits a strong response to a continuumof high-frequency disturbances within a narrow band of streamwise wavenumbers.Most importantly, in the vicinity of the lower-branch neutral curve the slow acousticperturbation induced by a disturbance of suitable frequency and wavenumbers is inexact resonance with a neutral eigen mode. As a result, the latter can be generated directlyby each of three types of free-stream disturbances without involving any surfaceroughness element. The amplitude of the instability mode is determined by analysingthe disturbance evolution through the resonant region. The fluctuation associatedwith the eigen mode turns out to be much stronger than free-stream disturbances dueto the resonant nature of excitation and in the case of acoustic disturbances, to thewell-known amplification effect of a strong shock. Moreover, excitation at the neutralposition means that the instability mode grows immediately without undergoingany decay, or missing any portion of the unstable region. All these indicate that thisnew mechanism is particularly efficient. The boundary-layer response and c

Wu X, Dong M, 2016, Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers, *Journal of Fluid Mechanics*, Vol: 797, Pages: 683-728, ISSN: 0022-1120

The fundamental difference between continuous modes of the Orr-Sommerfeld/Squireequations and the entrainment of free-stream vortical disturbances (FSVD) into theboundary layer has been investigated in a recent paper (Dong & Wu 2013, J. FluidMech.). It was shown there that the non-parallel-flow effect plays a leading-order role inthe entrainment, and neglecting it at outset, as is done in the continuous-mode formulation,leads to non-physical features of ‘Fourier entanglement’ and abnormal anisotropy.The analysis, which was for incompressible boundary layers and for FSVD with a characteristicwavelength of the order of the local boundary-layer thickness, is extended inthis paper to compressible boundary layers and FSVD with even shorter wavelengths,which are comparable with the width of the so-called edge layer. Non-parallelism remainsa leading-order effect in the present scaling, which turns out to be more general in thatthe equations and solutions in the previous paper are recovered in the appropriate limit.Appropriate asymptotic solutions in the main and edge layers are obtained to characterizethe entrainment. It is found that when the Prandtl number Pr < 1, free-streamvortical disturbances of relatively low frequency generate very strong temperature fluctuationswithin the edge layer, leading to formation of thermal streaks. A compositesolution, uniformly valid across the entire boundary layer, is constructed, and it can beused in receptivity studies and as inlet conditions for direct numerical simulations of bypasstransition. For compressible boundary layers, continuous spectra of the disturbanceequations linearised about a parallel base flow exhibit entanglement between vortical andentropy modes, namely, a vortical mode necessarily induces an entropy disturbance inthe free stream and vice versa, and this amounts to a further nonphysical behaviour.High-Reynolds-number asymptotic analysis yields the relations between the amplitudesof entangled mo

Wu X, Dong M, 2016, A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue, *Journal of Fluid Mechanics*, Vol: 794, Pages: 68-108, ISSN: 0022-1120

This paper is concerned with the rather broad issue of the impact of abrupt changes (such as isolated roughness, gaps and local suctions) on boundary-layer transition. To fix the idea, we consider the influence of a two-dimensional localized hump (or indentation) on an oncoming Tollmien-Schlichting (T-S) wave. We show that when the length scale of the former is comparable with the characteristic wavelength of the latter, the key physical mechanism to affect transition is through scattering of T-S waves by the roughness-induced mean-flow distortion. An appropriate mathematical theory, consisting of the boundary-value problem governing the local scattering,is formulated based on triple deck formalism. The transmission co efficient, defined as the ratio of the amplitude of the T-S wave downstream the roughnessto that upstream, serves to characterize the impact on transition. The transmission coefficient appears as the eigenvalue of the discretized boundary-value problem. The latter is solved numerically, and the dependenceof the eigenvalue on the height and width of the roughness and the frequencyof the T-S wave is investigated. For a roughness element without causing separation, the transmission coefficient is found to be about 1:5 for typical frequencies, indicating a moderate but appreciable destabilizing effect. For a roughness causing incipient separation, the transmission coefficient can be as large as O(10), suggesting that immediate transition may take place at the roughness site. A roughness element with a fixed height produces the strongest impact when its width is comparable with the T-S wavelength, in which case the traditional linear stability theory is in valid. The latter how ever holds approximately when the roughness width is sufficiently large. By studying the two-hump case, a criterion when two roughness elements can be regarded as being isolated is suggested. The transmission coefficient can be converted to an equivalent N-factor increment, by makin

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