Spanwise Wavenumber Research Articles

A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator

A novel reduced-order model for time-varying nonlinear flows arising from a resolvent decomposition based on the time-mean flow is proposed. The inputs required for the model are the mean-flow field and a small set of velocity time-series data obtained at isolated measurement points, which are used to fix relevant frequencies, amplitudes and phases of a limited number of resolvent modes that, together with the mean flow, constitute the reduced-order model. The technique is applied to derive a model for the unsteady three-dimensional flow in a lid-driven cavity at a Reynolds number of 1200 that is based on the two-dimensional mean flow, three resolvent modes selected at the most active spanwise wavenumber, and either one or two velocity probe signals. The least-squares full-field error of the reconstructed velocity obtained using the model and two point velocity probes is of the order of 5 % of the lid velocity, and the dynamical behaviour of the reconstructed flow is qualitatively similar to that of the complete flow.

An asymptotic study of instability to Langmuir circulation in shallow layers

The rôle of boundary conditions applied to perturbation equations describing the instability of wavy shear flows to counter-rotating vortical structures aligned with the flow in shallow water layers is considered. In the ocean context the structures are described by the Craik-Leibovich equations and are known as Langmuir circulation (LC); they arise via an instability mechanism known as CL2, which requires the presence of shear and differential drift of the same sense and that a threshold Rayleigh number be exceeded. While Neumann or stress-free boundary conditions applied at the free surface in deep water act to yield a physically plausible spanwise wavenumber l at onset, that is not the case in shallow layers, where non-zero l at onset is ensured only when mixed boundary conditions are applied. This paper questions why that is so and goes on to determine how onset is affected by distributions of and . In order to proceed analytically, results are computed by a small-l asymptotic approximation for which an algorithm is generated enabling computation to, essentially, any order. It is found that the extra stress induced by the perturbed motion, which is reflected in mixed but not Neumann boundary conditions, acts to suppress the growth rate of the instability over all wavenumbers, thereby restricting instability to non-zero l; further details are seen to depend on the distributions of and . Specifically, while the critical wavenumber is not overly sensitive to precise distributions of and , the critical Rayleigh number is sensitive. Minor variations in and can also lead to multiple rather than single layers of LC and even contrive to ensure non-zero l at onset with Neumann boundary conditions. Finally, although crafted as an asymptotic approximation for , the results concur closely with numerical calculations at much larger l.

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

We perform BiGlobal linear stability analysis on flow past a NACA0012 airfoil at 16° angle of attack and Reynolds number ranging from 400 to 1000. The steady-state two-dimensional base flows are computed using a well-tested finite difference code in combination with the selective frequency damping method. The base flow is characterized by two asymmetric recirculation bubbles downstream of the airfoil whose streamwise extent and the maximum reverse flow velocity increase with the Reynolds number. The stability analysis of the flow past the airfoil is carried out under very small spanwise wavenumber β = 10−4 to approximate the two-dimensional perturbation, and medium and large spanwise wavenumbers (β = 1–8) to account for the three-dimensional perturbation. Numerical results reveal that under small spanwise wavenumber, there are at most two oscillatory unstable modes corresponding to the near wake and far wake instabilities; the growth rate and frequency of the perturbation agree well with the two-dimensional direct numerical simulation results under all Reynolds numbers. For a larger spanwise wavenumber β = 1, there is only one oscillatory unstable mode associated with the wake instability at Re = 400 and 600, while at Re = 800 and 1000 there are two oscillatory unstable modes for the near wake and far wake instabilities, and one stationary unstable mode for the monotonically growing perturbation within the recirculation bubble via the centrifugal instability mechanism. All the unstable modes are weakened or even suppressed as the spanwise wavenumber further increases, among which the stationary mode persists until β = 4.

Receptivity coefficients at excitation of cross-flow waves due to scattering of free-stream vortices on surface vibrations

This paper is devoted to an experimental investigation of receptivity of a laminar swept-wing boundary layer due to scattering of free-stream vortices on localized (in the streamwise direction) surface vibrations. The experiments were conducted under completely controlled disturbance conditions by means of a hot-wire anemometer on a model of a swept wing with a sweep angle of 25°. Both the free-stream vortices and the surface vibrations were generated by disturbance sources; their frequency–wavenumber spectra were measured thoroughly. The free-stream vorticity vectors were directed perpendicular to the incident-flow velocity vector and parallel to the swept-wing-model surface. The linearity of the receptivity mechanism under investigation (in a sense that the corresponding receptivity coefficients are independent of the disturbances amplitudes) has been checked carefully. The main goal of this experiment was to estimate the vibration-vortex receptivity coefficients as functions of the disturbance frequency, spanwise wavenumber and vortex offset parameter. This goal has been attained. Being defined in Fourier space, the obtained receptivity coefficients are independent of the specific surface vibration shape and can be used for verification of various receptivity theories.

边界层流中当地感受性过程的数值研究

The research on local receptivity in the boundary layer is very important for the prediction and control of the laminar-turbulent transition, and especially the study on the formation mechanism of 3D Tollmien-Schlichting (T-S) waves is meaningful in theory. The high-order high-resolution non-uniform compact finite difference schemes were utilized to study the local receptivity under the interaction of free-stream turbulence and 2D localized wall roughness. The numerical results verify the exsistance of the local receptivity under the interaction of free-stream turbulence and 2D localized wall roughness, and the streamwise vorticity forms and gets stronger downstream as the excited 3D T-S wave packets evolve in the streamwise direction. The numerical results also show that the propagation direction of the excited 3D T-S wave packets is influenced by the propagation direction of the free-stream turbulence, and the propagation speed is close to 1/3 of the free-stream velocity; the wave-length conversion mechanism only changes streamwise wave number α, whereas spanwise wave number β keeps unchanged. In addition, the relation between the free-stream amplitude and incident angle, the localized wall roughness height and length, and the local receptivity, is confirmed. The in-depth research on this subject is helpful for further understanding of the laminar-turbulent transition and tubulence formation.

Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flow modifications

The question of optimal spanwise-periodic modification for the stabilisation of spanwise-invariant flows is addressed. A second-order sensitivity analysis is conducted for the linear temporal stability of parallel flows $U_{0}$ subject to small-amplitude spanwise-periodic modification ${\it\epsilon}U_{1},{\it\epsilon}\ll 1$. It is known that spanwise-periodic flow modifications have a quadratic effect on stability properties, i.e. the first-order eigenvalue variation is zero, hence the need for a second-order analysis. A second-order sensitivity operator is computed from a one-dimensional calculation, which allows one to predict how eigenvalues are affected by any flow modification $U_{1}$, without actually solving for modified eigenvalues and eigenmodes. Comparisons with full two-dimensional stability calculations in a plane channel flow and in a mixing layer show excellent agreement. Next, optimisation is performed on the second-order sensitivity operator: for each eigenmode streamwise wavenumber ${\it\alpha}_{0}$ and base flow modification spanwise wavenumber ${\it\beta}$, the most stabilising/destabilising profiles $U_{1}$ are computed, together with lower/upper bounds for the variation in leading eigenvalue. These bounds increase like ${\it\beta}^{-2}$ as ${\it\beta}$ goes to zero, thus yielding a large stabilising potential. However, three-dimensional modes with wavenumbers ${\it\beta}_{0}=\pm {\it\beta}$, $\pm {\it\beta}/2$ are destabilised, and therefore larger control wavenumbers should be preferred. The most stabilising $U_{1}$ optimised for the most unstable streamwise wavenumber ${\it\alpha}_{0,max}$ has a stabilising effect on modes with other ${\it\alpha}_{0}$ values too. Finally, the potential of transient growth to amplify perturbations and stabilise the flow is assessed with a combined optimisation. Assuming a separation of time scales between the fast unstable mode and the slow transient evolution of the optimal perturbations, combined optimal perturbations that achieve the best balance between transient linear amplification and stabilisation of the nominal shear flow are determined. In the mixing layer with ${\it\beta}\leqslant 1.5$, these combined optimal perturbations appear similar to transient growth-only optimal perturbations, and achieve a more efficient overall stabilisation than optimal spanwise-periodic and spanwise-invariant modifications computed for stabilisation only. These results are consistent with the efficiency of streak-based control strategies.

Secondary instabilities of Görtler vortices in high-speed boundary layer flows

Görtler vortices developed in laminar boundary layer experience remarkable changes when the flow is subjected to compressibility effects. In the present study, five $\mathit{Ma}$ numbers, covering incompressible to hypersonic flows, at $\mathit{Ma}=0.015$, 1.5, 3.0, 4.5 and 6.0 are specified to illustrate these effects. Görtler vortices in subsonic and moderate supersonic flows ($\mathit{Ma}=0.015$, 1.5 and 3.0) are governed by the conventional wall-layer mode (mode W). In hypersonic flows ($\mathit{Ma}=4.5$, 6.0), the trapped-layer mode (mode T) becomes dominant. This difference is maintained and intensifies downstream leading to different scenarios of secondary instabilities. The linear and nonlinear development of Görtler vortices which are governed by dominant modal disturbances are investigated with direct marching of the nonlinear parabolic equations. The secondary instabilities of Görtler vortices set in when the resulting streaks are adequately developed. They are studied with Floquet theory at multiple streamwise locations. The secondary perturbations become unstable downstream following the sequence of sinuous mode type I, varicose mode and sinuous mode type II, indicating an increasing threshold amplitude. Onset conditions are determined for these modes. The above three modes can each have the largest growth rate under the right conditions. In the hypersonic cases, the threshold amplitude $A(u)$ is dramatically reduced, showing the significant impact of the thermal streaks. To investigate the parametric effect of the spanwise wavenumber, three global wavenumbers ($B=0.5$, 1.0 and $2.0\times 10^{-3}$) are specified. The relationship between the dominant mode (sinuous or varicose) and the spanwise wavenumber of Görtler vortices found in incompressible flows (Li & Malik, J. Fluid Mech., vol. 297, 1995, pp. 77–100) is shown to be not fully applicable in high-speed cases. The sinuous mode becomes the most dangerous, regardless of the spanwise wavelength when $\mathit{Ma}>3.0$. The subharmonic type can be the most dangerous mode while the detuned type can be neglected, although some of the sub-dominant secondary modes reach their peak growth rates under detuned states.

Resolution effects in compressible, turbulent boundary layer simulations

This paper presents the first systematic study of resolution in compressible, turbulent boundary simulations, and the best-resolved simulation carried out to date. Direct numerical simulation (DNS) and wall-resolved, implicit large-eddy simulation (ILES-NWR) were carried out for a turbulent boundary layer at a Mach number of M=2.3 and maximum momentum thickness Reynolds number of Reθi=2×103. The wall temperature was fixed at a constant value corresponding to the nominal adiabatic wall temperature. The flow was developed spatially from a laminar boundary layer similarity solution specified at the inflow, and transition to turbulence was promoted with an artificial body force trip. The effects of spatial resolution in the range of ILES-NWR, conventional DNS, and very strict DNS were considered. The finest grid in the spatial resolution study consisted of 3.3 × 1010 points, and maintained max(Δx1+,Δx2+,Δx3+)≤1 everywhere. With the resolution at the wall held at (Δx2+)w<1, statistics characterizing large-scale flow features converged for max(Δxi+)≤10, and agreed well with experimental data. Velocity spectra in the outer part of the boundary layer agreed well at low wavenumber for all grids in this range of mesh spacing, and increasing resolution acted to fill out the high-wavenumber end of the spectrum. In all cases, the resolved region of the spectrum agreed well with a well-validated model of the spectrum of isotropic turbulence. Thus, ILES-NWR can be concluded to converge seamlessly to DNS as the spatial resolution is increased. The low-wavenumber aspect of spatial resolution was examined by varying the width of the computational domain, with a fixed level of small-scale resolution. The primary influence of increasing domain width was to capture additional spectral content in low spanwise wavenumbers; other statistics were found to be identical. Further, turbulence statistics were found to be essentially independent of the domain width for values between two and eight times the maximum boundary layer thickness. This work predicts computationally, for the first time, the full velocity spectrum in the outer part of the boundary layer. In doing so, it justifies use of the ILES-NWR approach as long as the maximum grid spacing is less than 10 in inner units, and the domain width is at least two times the maximum boundary layer thickness.

The effect of three-dimensionality on the aerodynamic admittance of thin sections in free stream turbulence

The unsteady behaviour of a thin plate section or aerofoil strip in a turbulent wind field is described by a two-wavenumber extension of the well known single wavenumber Sears function. The second (spanwise) wavenumber is introduced to deal with the three dimensional effects of the turbulence. The present work considers the unsteady forces induced on finite span sections and investigates the influence of the ratio of span to chord (aspect-ratio) of the section on these three dimensional effects. The analysis shows that the influence of the spanwise wavenumber becomes negligible for the aerodynamic admittance of buffeting lift forces for large enough aspect-ratios, such as those typical of modern long-span bridges, thus supporting the use of strip-theory in these cases.

Effects of base flow modifications on noise amplifications: flow past a backward-facing step

Amplifications of flow past a backward-facing step with respect to optimal inflow and initial perturbations are investigated at Reynolds number 500. Two mechanisms of receptivity to inflow noise are identified: the bubble-induced inflectional point instability and the misalignment effect downstream of the secondary bubble. Further development of the misalignment results in decay of perturbations from $x=28$ onwards (the step is located at $x=0$), as has been observed in previous non-normality studies (Blackburn et al., J. Fluid Mech., vol. 603, 2008, pp. 271–304), and eventually limits the receptivity. The receptivity is found to be maximized at an inflow perturbation frequency of ${\it\omega}=0.50$ and a spanwise wavenumber of ${\it\beta}=0$, where the inflow noise takes full advantage of both mechanisms and is amplified over two orders of magnitude in terms of the velocity magnitude. In direct numerical simulations (DNS) of the flow perturbed by optimal or random inflow noise, vortex shedding, flapping of bubbles, three-dimensionality and turbulence are observed in succession as the magnitude of the inflow noise increases. Similar features of linear and nonlinear receptivity are observed at higher Reynolds numbers. The Strouhal number of the bubble flapping is 0.08, at which the receptivity to inflow noise reaches a maximum. This Strouhal number is close to reported values extracted from DNS or large eddy simulations (LES) at larger Reynolds numbers (Le et al., J. Fluid Mech., vol. 330, 1997, pp. 349–374; Kaiktsis et al., J. Fluid Mech., vol. 321, 1996, pp. 157–187; Métais, New Trends in Turbulence, 2001, Springer; Wee et al., Phys. Fluids, vol. 16, 2004, pp. 3361–3373). Methods to further clarify the mechanisms of receptivity and to suppress the noise amplifications by modifying the base flow using a linearly optimal body force are proposed. It is observed that the mechanisms of optimal noise amplification are fully revealed by the distribution of the base flow modification, which weakens the bubble instabilities and misalignment effects and subsequently reduces the receptivity significantly. Comparing the base flow modifications with respect to amplifications of inflow and initial perturbations, it is found that the maximum receptivity to initial perturbations is highly correlated with the receptivity to inflow noise at the optimal frequency ${\it\omega}=0.50$, and the correlation reduces as the inflow frequency deviates from this optimal value.

Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

Spatio-temporal spectra in the logarithmic layer of wall turbulence: large-eddy simulations and simple models

Motivated by the need to characterize the spatio-temporal structure of turbulence in wall-bounded flows, we study wavenumber–frequency spectra of the streamwise velocity component based on large-eddy simulation (LES) data. The LES data are used to measure spectra as a function of the two wall-parallel wavenumbers and the frequency in the equilibrium (logarithmic) layer. We then reformulate one of the simplest models that is able to reproduce the observations: the random sweeping model with a Gaussian large-scale fluctuating velocity and with additional mean flow. Comparison with LES data shows that the model captures the observed temporal decorrelation, which is related to the Doppler broadening of frequencies. We furthermore introduce a parameterization for the entire wavenumber–frequency spectrum $E_{11}(k_{1},k_{2},{\it\omega};z)$, where $k_{1}$, $k_{2}$ are the streamwise and spanwise wavenumbers, ${\it\omega}$ is the frequency and $z$ is the distance to the wall. The results are found to be in good agreement with LES data.

Three-dimensional flow past a rotating cylinder

AbstractThree-dimensional computations are carried out for a spinning cylinder placed in a uniform flow. The non-dimensional rotation rate is varied in the range $0.0\leqslant {\it\alpha}\leqslant 5.0$. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in primitive variables formulation. Linear stability analysis of the steady state shows the existence of several new unstable three-dimensional modes for $200\leqslant \mathit{Re}\leqslant 350$ and $4.0\leqslant {\it\alpha}\leqslant 5.0$. The curves of neutral stability of these modes are presented in the $\mathit{Re}{-}{\it\alpha}$ parameter space. For the flow at $\mathit{Re}=200$ and rotation rate in the ranges $0.0\leqslant {\it\alpha}\leqslant 1.91$ and $4.34\leqslant {\it\alpha}\leqslant 4.7$, the vortex shedding, earlier reported in two dimensions and commonly referred to as parallel shedding, can also exist as oblique shedding. In this mode of shedding, the vortices are inclined to the axis of the cylinder. In fact, parallel shedding is a special case of oblique shedding. It is found that the span of the cylinder plays a significant role in the time evolution of the flow. Of all the unstable eigenmodes, with varied spanwise wavenumber, only the ones whose integral number of wavelengths fit the span length of the cylinder are selected to grow. For the flow at $\mathit{Re}=200$, two steady states exist for $4.8\leqslant {\it\alpha}\leqslant 5.0$. While one of them is associated with unstable eigenmodes, the other is stable to all infinitesimal perturbations. In this regime, irrespective of the initial conditions, the fully developed flow is steady and devoid of any instabilities.

Vortex-induced vibrations of a flexible cylinder at large inclination angle.

The free vibrations of a flexible circular cylinder inclined at 80° within a uniform current are investigated by means of direct numerical simulation, at Reynolds number 500 based on the body diameter and inflow velocity. In spite of the large inclination angle, the cylinder exhibits regular in-line and cross-flow vibrations excited by the flow through the lock-in mechanism, i.e. synchronization of body motion and vortex formation. A profound reconfiguration of the wake is observed compared with the stationary body case. The vortex-induced vibrations are found to occur under parallel, but also oblique vortex shedding where the spanwise wavenumbers of the wake and structural response coincide. The shedding angle and frequency increase with the spanwise wavenumber. The cylinder vibrations and fluid forces present a persistent spanwise asymmetry which relates to the asymmetry of the local current relative to the body axis, owing to its in-line bending. In particular, the asymmetrical trend of flow-body energy transfer results in a monotonic orientation of the structural waves. Clockwise and counter-clockwise figure eight orbits of the body alternate along the span, but the latter are found to be more favourable to structure excitation. Additional simulations at normal incidence highlight a dramatic deviation from the independence principle, which states that the system behaviour is essentially driven by the normal component of the inflow velocity.

Nonlinear Mode Interactions in the Wake of a Medium Height Roughness Element

The present study is devoted to nonlinear (resonant) mode interactions in the wake of a medium height, cylindrical roughness element, which is placed in a laminar, airfoil boundary layer. The roughness element causes a considerable mean flow distortion in the near wake centerline region and two counter-rotating vortex pairs arise in the outer spanwise domain. In the streamwise evolution the mean flow stabilizes and a spanwise uniform flow is recovered in the far wake. Upstream of the roughness controlled, low-amplitude Tollmien-Schlichting (TS) wave modes are excited in the upper branch unstable region. The interference of the excited, 2D fundamental modes with the roughness results in the excitation of oblique modes in a broad spanwise wave number range. Nonlinear interactions in-between the (oblique) fundamental modes in the destabilized near wake lead to the generation of primary, low-frequency modes at the difference frequencies of the fundamental modes, before the fundamental modes can recover linear stability characteristics in the far wake. In contrast, the primary, subharmonic-type modes experience a nonlinear growth, which is a result of frequency and spanwise wave number detuned resonant mode interactions with the 2D fundamental modes. The resonant growth of the primary interaction modes initiates a symmetrization of the spectrum in the low-frequency, subharmonic range. That is, symmetric subharmonic-type (secondary) modes are resonantly amplified in the far wake. Therefore, although the fundamental modes recover linear stability characteristics with the stabilization of the mean flow, resonant mode interactions in the low-frequency, subharmonic range initiate the onset of boundary layer transition in the far wake of the medium height roughness element.

Non-Parallel-Flow Effects on Stationary Crossflow Vortices at Their Genesis

The linear stability of a laminar boundary-layer flow over a swept wing is studied for a stationary perturbation with spanwise wavenumber ▪, where β is non-dimensionalised by the chordwise distance L to the leading edge, and Re is the Reynolds number based on L This scaling corresponds to the lower-branch regime of the neutral curve, and hence represents the genesis stage of stationary crossflow vortices. We show that in this stage or regime, the non-parallelism of the flow plays a leading order role in the growth of the instability. Non-parallel-flow effects consist of two parts: the streamwise variation of the base-flow profiles, and the distortion of the eigenfunction. These two contributions to the growth rate are found to be of the same order of magnitude, and both are closely related to the 4th-order vertical derivatives of the base-flow profile on the surface of the wing. The implication of the present finding on the description of stationary vortices using the Orr-Sommerfeld equation is discussed.

Stability of Compressible Mixing Layers Modified by Wakes and Jets

Binary free shear layers are frequently used as a model to study the stability of flows relevant in propulsion devices. The stability behaviour of binary, compressible mixing layer have been investigated and reported in the literature by many research groups and detailed description of the stability characteristics and flow topology have been published. The stability of more complex free shear layers, such as mixing layers modified by jets and wakes, has received less attention and a smaller number of references report on the stability characteristics of such flows. The availability of advanced numerical techniques as well as computational power nowadays allow the study of complex flows by direct numerical simulation. The present investigation is an extension of previous work on the stability of complex free shear flows where besides growth rates, wave numbers and frequencies, the flow topology resulting from the interaction of mixing layers, jets and wakes are investigated. The results show that, depending on the jet or wake strength relative to the mixing layer fast and slow stream velocities, the characteristic Kelvin-Helmholtz vortices may be enhanced and a more complex vortex distribution topology results. These vorticity distributions have a significant effect on the mixture between different species on the upper and lower stream and the mixing layer to jet species ratio. The effect of three-dimensional disturbances with spanwise wavenumber β * 0 are also considered and the most amplified modes are determined for a low convective mach number condition. Different combinations of hydrogen and oxygen on the mixing layer and jets are considered. For that study the base flow is determined from a parabolic, variable properties boundary layer code. The results show that oblique disturbances are more stable than two-dimensional disturbances but with different strength depending on the particular binary free shear layer considered.

Effect of Freestream Turbulence on Roughness-induced Crossflow Instability

The effect of freestream turbulence on generation of crossflow disturbances over swept wings is investigated through direct nu- merical simulations. The set up follows the experiments performed by Downs et al. (2012). In these experiments the authors use ASU(67)-0315 wing geometry which promotes growth of crossflow disturbances. Distributed roughness elements are locally placed near the leading edge with a given spanwise wavenumber to excite the corresponding stationary crossflow vortices. In present study, we partially reproduce the isotropic homogenous freestream turbulence through direct numerical simulations using freestream spectrum data from the experiments. The generated freestream fields are then applied as the inflow boundary condition for direct numerical simulation of the wing. The distributed roughness elements are modelled through wing surface deformation and placed near the leading edge to trigger the stationary crossflow disturbances. The effects of the generated freestream turbulence on the initial amplitudes and growth of the boundary layer perturbations are then studied.

Study of the Discrete Spectrum in a Mach 4.5 Görtler Flow

Evolution of the discrete spectrum in the M a = 4.5 boundary layer is studied with LST and PSE approaches. Both two-dimensional (2-D) and three-dimensional (3-D) disturbances are considered with streamwise curvature effects. The concave curvature shows a destabilizing effect on the 2-D second/third mode when the fast mode (mode F(1), mode F(2)...) synchronizes with the slow mode (mode S). The spectrum branching in the synchronization between the mode F(2) and mode S is also observed. The increase in the spanwise wavenumber(3-D disturbances), on the other hand, suppresses the synchronization between mode F and mode S and reduces the growth rate of the unstable mode. With regard to the 3-D disturbances subjecting to the concave curvature, the mode S originating from the slow acoustic wave amounts to the unsteady Gortler mode while the quasi-steady Gortler mode emanates from the continuous spectrum of the vorticity/entropy wave.

Nonlinear interaction mechanisms of disturbances in supersonic flat-plate boundary layers

Due to the complexity of compressible flows, nonlinear hydrodynamic stability theories in supersonic boundary layers are not sufficient. In order to reveal the nonlinear interaction mechanisms of the rapidly amplified 3-D disturbances in supersonic boundary layers at high Mach numbers, the nonlinear evolutions of different disturbances in flat-plate boundary layers at Mach number 4.5, 6 and 8 are analyzed by numerical simulations. It can be concluded that the 3-D disturbances are amplified rapidly when the amplitude of the 2-D disturbance reaches a certain level. The most rapidly amplified 3-D disturbances are Klebanoff type (K-type) disturbances which have the same frequency as the 2-D disturbance. Among these K-type 3-D disturbances, the disturbances located at the junction of upper branch and lower branch of the neutral curve are amplified higher. Through analyzing the relationship between the amplification rate and the spanwise wavenumber of the 3-D disturbances at different evolution stages, the mechanism of the spanwise wavenumber selectivity of K-type 3-D disturbances in the presence of a finite amplitude 2-D disturbance is explained.