Free-stream Vortical Disturbances Research Articles

Excitation and stability of nonlinear compressible Görtler vortices and streaks induced by free-stream vortical disturbances

We study the generation, nonlinear development and secondary instability of unsteady Görtler vortices and streaks in compressible boundary layers exposed to free-stream vortical disturbances and evolving over concave, flat and convex walls. The formation and evolution of the disturbances are governed by the compressible nonlinear boundary-region equations, supplemented by initial and boundary conditions that characterise the impact of the free-stream disturbances on the boundary layer. Computations are performed for parameters typical of flows over high-pressure turbine blades, where the Görtler number, a measure of the curvature effects, and the disturbance Reynolds number, a measure of the nonlinear effects, are order-one quantities. At moderate intensities of the free-stream disturbances, increasing the Görtler number renders the boundary layer more unstable, while increasing the Mach number or the frequency stabilises the flow. As the free-stream disturbances become more intense, vortices over concave surfaces no longer develop into the characteristic mushroom-shaped structures, while the flow over convex surfaces is destabilised. An occurrence map identifies Görtler vortices or streaks for different levels of free-stream disturbances and Görtler numbers. Our calculations capture well the experimental measurements of the enhanced skin friction and wall-heat transfer over turbine-blade pressure surfaces. The time-averaged wall-heat transfer modulations, termed hot fingers, are elongated in the streamwise direction and their spanwise wavelength is half of the characteristic wavelength of the free-stream disturbances. Nonlinearly saturated disturbances are unstable to secondary high-frequency modes, whose growth rate increases with the Görtler number. A new varicose even mode is reported, which may promote transition to turbulence at the stem of nonlinear streaks.

Influence of spanwise wall vibration on non-modal perturbations subject to freestream vortical disturbances in hypersonic boundary layers

In this paper, we study the effect of lateral wall vibrations on the excitation and evolution of non-modal perturbations in hypersonic boundary layers subject to low-frequency freestream vortical disturbances (FSVDs). A novel, high-efficiency numerical approach, combining the harmonic weakly nonlinear Navier–Stokes and nonlinear parabolised stability equation approaches, is developed, which is sufficient to accommodate both the rapid distortion of the perturbation in the leading-edge vicinity and the nonlinear development of finite-amplitude high-order harmonics in the downstream region. The boundary-layer response to low-frequency FSVDs shows a longitudinal streaky structure, for which the temperature perturbation shows much greater magnitude than the streamwise velocity perturbation. The lateral vibration induces a Stokes layer solution for the spanwise velocity perturbation, which interacts with the FSVD-induced perturbations and leads to a suppression of the non-modal perturbation and an enhancement of the downstream modal perturbation. The new perturbations excited by the FSVD–vibration interaction strengthen as the vibration intensifies, and they could become comparable with the FSVD-induced perturbations in downstream locations at a high vibration intensity, indicating a remarkable modification of the streaky structure and its instability property. Secondary instability (SI) analyses based on the streaky base flow indicate that the vibration could enhance or suppress the SI modes, depending on their initial phases over the vibration period. Overall, the average effect is that the low-frequency and high-frequency SI modes are stabilised and destabilised by the vibration, respectively. Since the high-frequency SI modes undergo higher amplifications, the subsequent bypass transition is likely to be promoted by relatively strong vibrations.

Nonlinear evolution of vortical disturbances entrained in the entrance region of a circular pipe

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$ -fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

Global receptivity analysis: physically realizable input–output analysis

In the context of transition analysis, linear input–output analysis determines the worst-case disturbances to a laminar base flow based on a generic right-hand-side volumetric/boundary forcing term. The worst-case forcing is not physically realizable, and, to our knowledge, a generic framework for posing physically realizable worst-case disturbance problems is lacking. In natural receptivity analysis, disturbances are forced by matching (typically local) solutions within the boundary layer to outer solutions consisting of free-stream vortical, entropic and acoustic disturbances. We pose a scattering formalism to restrict the input forcing to a set of realizable disturbances associated with plane-wave solutions of the outer problem. The formulation is validated by comparing with direct numerical simulations of a Mach 4.5 flat-plate boundary layer. We show that the method provides insight into transition mechanisms by identifying those linear combinations of plane-wave disturbances that maximize energy amplification over a range of frequencies. We also discuss how the framework can be extended to accommodate scattering from shocks and in shock layers for supersonic flow.

Entrainment and growth of vortical disturbances in the channel-entrance region

The entrainment of free-stream unsteady three-dimensional vortical disturbances in the entry region of a channel is studied via matched asymptotic expansions and by numerical means. The interest is in flows at Reynolds numbers where experimental studies have documented the occurrence of intense transient growth, despite the flow being stable according to classical stability analysis. The analytical description of the vortical perturbations at the channel mouth reveals how the oncoming disturbances penetrate into the wall-attached shear layers and amplify downstream. The effects of the channel confinement, the streamwise pressure gradient and the viscous/inviscid interplay between the oncoming disturbances and the boundary-layer perturbations are discussed. The composite perturbation velocity profiles are employed as initial conditions for the unsteady boundary-region perturbation equations. At a short distance from the channel mouth, the disturbance flow is mostly confined within the shear layers and assumes the form of streamwise-elongated streaks, while farther downstream the viscous disturbances permeate the whole channel although the base flow is still mostly inviscid in the core. Symmetrical disturbances exhibit a more significant growth than anti-symmetrical disturbances, the latter maintaining a nearly constant amplitude for several channel heights downstream before growing transiently, a unique feature not reported in open boundary layers. The disturbances are more intense as the frequency decreases or the bulk Reynolds number increases. We compute the spanwise wavelengths that cause the most intense downstream growth and the threshold wall-normal wavelengths below which the perturbations are damped through viscous dissipation.

Elevated low-frequency free-stream vortical disturbances eliminate boundary-layer separation

Abstract

Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability

Abstract

Neutral stability curves of compressible Görtler flow generated by low-frequency free-stream vortical disturbances

Pre-transitional compressible boundary layers perturbed by low-frequency free-stream vortical disturbances and flowing over plates with streamwise-concave curvature are studied via matched asymptotic expansions and numerically. The Mach number, the Görtler number and the frequency of the free-stream disturbance are varied to obtain the neutral stability curves, i.e. curves in the space of the parameters that distinguish spatially growing from spatially decaying perturbations. The receptivity approach is used to calculate the evolution of Klebanoff modes, highly oblique Tollmien–Schlichting waves influenced by the concave curvature of the wall, and Görtler vortices. The Klebanoff modes always evolve from the leading edge, the Görtler vortices dominate when the influence of the curvature becomes significant and the Tollmien–Schlichting waves may precede the Görtler vortices for moderate Görtler numbers. For relatively high frequencies the triple-deck formalism allows us to confirm the numerical result of the negligible influence of the curvature on the Tollmien–Schlichting waves when the Görtler number is an order-one quantity. Experimental data for compressible Görtler flows are mapped onto our neutral-curve graphs and earlier theoretical results are compared with our predictions.

Compressible unsteady Görtler vortices subject to free-stream vortical disturbances

The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave walls are studied numerically and through asymptotic methods. We employ an asymptotic framework based on the limit of high Görtler number, the scaled parameter defining the centrifugal effects; we use an eigenvalue formulation where the free-stream forcing is neglected; and we solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions that synthesize the influence of the free-stream vortical flow. Near the leading edge, the boundary-layer perturbations develop as thermal Klebanoff modes and, when centrifugal effects become influential, these modes turn into thermal Görtler vortices, i.e. streamwise rolls characterized by intense velocity and temperature perturbations. The high-Görtler-number asymptotic analysis reveals the condition for which the Görtler vortices start to grow. The Mach number is destabilizing when the spanwise diffusion is negligible and stabilizing when the boundary-layer thickness is comparable with the spanwise wavelength of the vortices. When the Görtler number is large, the theoretical analysis also shows that the vortices move towards the wall as the Mach number increases. These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behaviour further downstream than the velocity perturbations. A matched-asymptotic composite profile, found by combining the inviscid core solution and the near-wall viscous solution, agrees well with the receptivity profile sufficiently downstream and at high Görtler number. The Görtler vortices tend to move towards the boundary-layer core when the flow is more stable, i.e. as the frequency or the Mach number increase, or when the curvature decreases. As a consequence, a region of unperturbed flow is generated near the wall. We also find that the streamwise length scale of the boundary-layer perturbations is always smaller than the free-stream streamwise wavelength. During the initial development of the vortices, only the receptivity calculations are accurate. At streamwise locations where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed with sufficient accuracy by the eigenvalue analysis, although the correct amplitude and evolution of the Görtler vortices can only be determined by the receptivity calculations. It is further proved that the eigenvalue predictions of the growth rate and wavenumber worsen as the Mach number increases as these quantities show a dependence on the wall-normal direction. We conclude by qualitatively comparing our results with the direct numerical simulations available in the literature.

Study of Secondary Instability of a Streaky Boundary Layer Under Spanwise-Localized Free-Stream Vortical Disturbances

Study of Secondary Instability of a Streaky Boundary Layer Under Spanwise-Localized Free-Stream Vortical Disturbances

Receptivity of secondary instability modes in streaky boundary layers

In the presence of relatively high freestream turbulence, laminar-turbulent transition in boundary layers follows a so-called bypass route. This process, in general, is described as follows: low-frequency freestream vortical disturbances (FSVDs) with large amplitude excite streaky disturbances that grow transiently in boundary layers; high-frequency disturbances amplify exponentially due to the secondary-instability regime; streaks break down and the intermittent turbulence spots emerge when the secondary modes reach finite amplitude. However, one process still remains unclear, namely, the mechanism that the secondary instability modes are excited by in the early stage of bypass transition. This paper first verifies the solutions of the boundary-region and edge-layer equations on describing the boundary-layer response to low- and high-frequency FSVDs, respectively. Then, using the above models as the inflow perturbation, the generation of the secondary instability modes is studied by direct numerical simulations. The transformation of Wigner-Ville distribution is employed on analyzing the numerical data, and the formation of the secondary instability modes in the vicinity of the neutral position and time is exhibited in detail. The receptivity process is described as the interaction between the high-frequency FSVDs and the non-parallel streaky disturbance that generates the secondary instability modes when the latter first becomes unstable. A simplified model is established by selecting a steady base flow according to the group velocity of the excited secondary modes, which is confirmed to be an efficient means to determine the receptivity coefficient. Moreover, the nonlinear evolution of the secondary instability modes is simulated, exhibiting the whole process of bypass transition.

Neutral stability curves of low-frequency Görtler flow generated by free-stream vortical disturbances

The neutral curves of the boundary layer Görtler-vortex flow generated by free-stream disturbances, i.e., curves that distinguish the perturbation flow conditions of growth and decay, are computed through a receptivity study for different Görtler numbers, wavelengths, and low frequencies of the free-stream disturbance. The perturbations are defined as Klebanoff modes or strong and weak Görtler vortices, depending on their growth rate. The critical Görtler number below which the inviscid instability due to the curvature never occurs is obtained and the conditions for which only Klebanoff modes exist are thus revealed. A streamwise-dependent receptivity coefficient is defined and we discuss the impact of the receptivity on the $N$-factor approach for transition prediction.

Non-localized boundary layer instabilities resulting from leading edge receptivity at moderate supersonic Mach numbers

This paper uses matched asymptotic expansions to study the non-localized (which we refer to as global) boundary layer instabilities generated by free-stream acoustic and vortical disturbances at moderate supersonic Mach numbers. The vortical disturbances produce an unsteady boundary layer flow that develops into oblique instability waves with a viscous triple-deck structure in the downstream region. The acoustic disturbances (which for reasons given herein are assumed to have obliqueness angles that are close to a certain critical angle) generate slow boundary layer disturbances which eventually develop into oblique stable disturbances with an inviscid triple-deck structure in a region that lies downstream of the viscous triple-deck region. The paper shows that both the vortically generated instabilities and the acoustically generated oblique disturbances ultimately develop into modified Rayleigh-type instabilities (which can either grow or decay) further downstream.

Growth and wall-transpiration control of nonlinear unsteady Görtler vortices forced by free-stream vortical disturbances

The generation, nonlinear evolution, and wall-transpiration control of unsteady Görtler vortices in an incompressible boundary layer over a concave plate is studied theoretically and numerically. Görtler rolls are initiated and driven by free-stream vortical perturbations of which only the low-frequency components are considered because they penetrate the most into the boundary layer. The formation and development of the disturbances are governed by the nonlinear unsteady boundary-region equations with the centrifugal force included. These equations are subject to appropriate initial and outer boundary conditions, which account for the influence of the upstream and free-stream forcing in a rigorous and mutually consistent manner. Numerical solutions show that the stabilizing effect on nonlinearity, which also occurs in flat-plate boundary layers, is significantly enhanced in the presence of centrifugal forces. Sufficiently downstream, the nonlinear vortices excited at different free-stream turbulence intensities Tu saturate at the same level, proving that the initial amplitude of the forcing becomes unimportant. At low Tu, the disturbance exhibits a quasi-exponential growth with the growth rate being intensified for more curved plates and for lower frequencies. At higher Tu, in the typical range of turbomachinery applications, the Görtler vortices do not undergo a modal stage as nonlinearity saturates rapidly, and the wall curvature does not affect the boundary-layer response. Good quantitative agreement with data from direct numerical simulations and experiments is obtained. Steady spanwise-uniform and spanwise-modulated zero-mass-flow-rate wall transpiration is shown to attenuate the growth of the Görtler vortices significantly. A novel modified version of the Fukagata-Iwamoto-Kasagi identity, used for the first time to study a transitional flow, reveals which terms in the streamwise momentum balance are mostly affected by the wall transpiration, thus offering insight into the increased nonlinear growth of the wall-shear stress.

Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances

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 $\pm k_{3}$, 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 $G_{\unicode[STIX]{x1D6EC}}$ (based on the spanwise wavelength $\unicode[STIX]{x1D6EC}$ of the disturbances) is $O(1)$, 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 $G_{\unicode[STIX]{x1D6EC}}\neq 0$. 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 $(0,2)$ (which has zero frequency and wavenumber $2k_{3}$) 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 relevance is delineated. The characteristics of dominant unstable modes, including their frequency ranges and eigenfunctions, are in good agreement with experiments. The secondary instability is intermittent when FSVD are unsteady and of low frequency. However, the intermittence diminishes as the frequency increases. The present theoretical framework, which allows for a detailed and integrated description of the key transition processes, from generation, through linear and nonlinear evolution, to the onset of secondary instability, represents a useful step towards predicting the pre-transitional flow and transition itself of the boundary layer over a blade in turbomachinery.

Boundary-layer disturbances subjected to free-stream turbulence and simulation on bypass transition

The phenomena associated with the entrainment of free-stream turbulence (FST) into boundary-layer flows are relevant for a number of subjects. It has been believed that the continuous spectra of the Orr-Sommerfeld (O-S)/Squire equations describe the entrainment process, and thus they are used to specify the inlet condition in simulation of bypass transition. However, Dong and Wu (Dong, M. and Wu, X. On continuous spectra of the Orr-Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances. Journal of Fluid Mechanics, 732, 616–659 (2013)) pointed out that continuous spectra exhibit several non-physical features due to neglecting the non-parallelism. They further proposed a large-Reynolds-number asymptotic approach, and showed that the non-parallelism is a leading-order effect even for the short-wavelength disturbance, for which the response concentrates in the edge layer. In this paper, the asymptotic solution is verified numerically by studying its evolution in incompressible boundary layers. It is found that the numerical results can be accurately predicted by the asymptotic solution, implying that the latter is adequate for moderate Reynolds numbers. By introducing a series of such solutions as the inflow perturbations, the bypass transition is investigated via the direct numerical simulation (DNS). The transition processes, including the evolution of streaks, the amplification of secondary-instability modes, and the emergence of turbulent spots, agree with the experimental observations.

Response and receptivity of the hypersonic boundary layer past a wedge to free-stream acoustic, vortical and entropy disturbances

This paper analyses the response and receptivity of the hypersonic boundary layer over a wedge to free-stream disturbances including acoustic, vortical and entropy fluctuations. Due to the presence of an attached oblique shock, the boundary layer is known to support viscous instability modes whose eigenfunctions are oscillatory in the far field. These modes acquire a triple-deck structure. Any of three elementary types of disturbance with frequency and wavelength on the triple-deck scales interacts with the shock to generate a slow acoustic perturbation, which is reflected between the shock and the wall. Through this induced acoustic perturbation, vortical and entropy free-stream disturbances drive significant velocity and temperature fluctuations within the boundary layer, which is impossible when the shock is absent. A quasi-resonance was identified, due to which the boundary layer exhibits a strong response to a continuum of high-frequency disturbances within a narrow band of streamwise wavenumbers. Most importantly, in the vicinity of the lower-branch neutral curve the slow acoustic perturbation induced by a disturbance of suitable frequency and wavenumbers is in exact resonance with a neutral eigenmode. As a result, the latter can be generated directly by each of three types of free-stream disturbance without involving any surface roughness element. The amplitude of the instability mode is determined by analysing the disturbance evolution through the resonant region. The fluctuation associated with the eigenmode turns out to be much stronger than the free-stream disturbances due to the resonant nature of excitation, and in the case of acoustic disturbances, to the well-known amplification effect of a strong shock. Moreover, excitation at the neutral position means that the instability mode grows immediately without undergoing any decay, or missing any portion of the unstable region. All these indicate that this new mechanism is particularly efficient. The boundary-layer response and coupling coefficients are calculated for typical values of parameters.

Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers

The fundamental difference between continuous modes of the Orr–Sommerfeld/Squire equations and the entrainment of free-stream vortical disturbances (FSVD) into the boundary layer has been investigated in a recent paper (Dong & Wu, J. Fluid Mech., vol. 732, 2013, pp. 616–659). It was shown there that the non-parallel-flow effect plays a leading-order role in the entrainment, and neglecting it at the 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 characteristic wavelength of the order of the local boundary-layer thickness, is extended in this 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 remains a leading-order effect in the present scaling, which turns out to be more general in that the 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 characterize the entrainment. It is found that when the Prandtl number $\mathit{Pr}<1$, free-stream vortical disturbances of relatively low frequency generate very strong temperature fluctuations within the edge layer, leading to formation of thermal streaks. A composite solution, uniformly valid across the entire boundary layer, is constructed, and it can be used in receptivity studies and as inlet conditions for direct numerical simulations of bypass transition. For compressible boundary layers, continuous spectra of the disturbance equations linearized about a parallel base flow exhibit entanglement between vortical and entropy modes, namely, a vortical mode necessarily induces an entropy disturbance in the free stream and vice versa, and this amounts to a further non-physical behaviour. High Reynolds number asymptotic analysis yields the relations between the amplitudes of entangled modes.

Entrainment of Short-wavelength Free-stream Vortical Disturbances into Boundary Layers

This paper is concerned with how free-stream vortical disturbances (FSVD) enter the boundary layer, a process which we refer to as entrainment. A self-consistent description of this physical process is presented, which shows that the non-parallel-flow effect plays a leading-order role even for disturbances with short wavelength comparable with the boundary-layer thickness or shorter. The entrained disturbances are thus drastically different from continuous modes of Orr-Sommerfeld/Squire equations, which neglect non-parallelism at outset. Indeed neglecting non-parallelism is found to cause several nonphysical features of continuous modes, including ‘Fourier entanglement’, abnormal anisotropy of FSVD, and for compressible boundary layers, entanglement of vortical and entropy modes. Eigenfunctions of continuous modes and their superpositions have been used to specify the inlet conditions in direct numerical simulations of bypass transition and boundary-layer receptivity. The findings of present study indicate that this popular practice is inappropriate, and a remedy is suggested.

Laminar Streaks in Oscillating Boundary Layers

The growth of laminar streaks in a flat-plate boundary layer is investigated in the presence of both spanwise wall oscillations and steady streamwise-dependent spanwise wall forcing. The laminar streaks are forced by free-stream vortical disturbances, which interact with the boundary layer to produce disturbances, which first grow and then decay. The oscillating plate produces a base flow that matches Blasius boundary layer in the streamwise and wall-normal directions, but produces a generalized Stokes layer (GSL) in the spanwise direction. This depends on the Blasius flow and tends to the classical Stokes layer (CSL) in the asymptotic limit of high oscillation frequency or large downstream distance. Spanwise oscillations of the plate and spanwise wall forcing can both reduce the total energy and maximum amplitude of the disturbances. Their relative effects are discussed.