Non-modal Growth Research Articles

Experimental investigation on the steady and unsteady disturbances in a flat plate boundary layer

Recent experiments have shown that miniature vortex generators (MVGs) are coveted devices to stabilize unsteady disturbances in flat plate boundary layers and to delay the onset of turbulence by modulating the base flow in the spanwise direction. The spanwise modulation is a result from the non-modal transient growth of steady and spanwise periodic streamwise vortices being generated by the MVGs. The present experimental investigation aims at studying the transient growth of non-modal disturbances induced by a spanwise periodic array of MVGs and its stabilizing effect on non-linear unsteady disturbances in the boundary layer originating from planar Tollmien-Schlichting (TS) waves. Measurements consist of cross-stream planes at different downstream locations in the boundary layer and a spatio-temporal analysis of different modes of the disturbances is carried out. In the streaky boundary layer generated by the MVGs the fundamental spanwise mode, with the same wavelength as the MVG pairs in the array, and its first harmonic, both undergo transient growth whereas the higher harmonics decay immediately downstream of the array. In the unstable region formed in the wake of the MVG blades, i.e., just downstream of the array, a wide range of spanwise modes contributes to an initial growth in the energy of unsteady disturbances. Similar behavior is observed upstream of branch II position of the neutral stability curve where the unsteady disturbances undergo a second energy growth in the unstable region. It is shown that the spatial gradients of the base flow in the wall-normal and spanwise directions are contributing to the amplification and attenuation of the TS wave disturbances, respectively, in the streaky boundary layer.

An inviscid modal interpretation of the ‘lift-up’ effect

AbstractIn this paper, we give a modal interpretation of the lift-up effect, one of two well-known mechanisms that lead to an algebraic instability in parallel shearing flows, the other being the Orr mechanism. To this end, we first obtain the two families of continuous spectrum modes that make up the complete spectrum for a non-inflectional velocity profile. One of these families consists of modified versions of the vortex-sheet eigenmodes originally found by Case (Phys. Fluids, vol. 3, 1960, pp. 143–148) for plane Couette flow, while the second family consists of singular jet modes first found by Sazonov (Izv. Acad. Nauk SSSR Atmos. Ocean. Phys., vol. 32, 1996, pp. 21–28), again for Couette flow. The two families are used to construct the modal superposition for an arbitrary three-dimensional distribution of vorticity at the initial instant. The so-called non-modal growth that underlies the lift-up effect is associated with an initial condition consisting of rolls, aligned with the streamwise direction, and with a spanwise modulation (that is, a modulation along the vorticity direction of the base-state shearing flow). This growth is shown to arise from an appropriate superposition of the aforementioned continuous spectrum mode families. The modal superposition is then generalized to an inflectional velocity profile by including additional discrete modes associated with the inflection points. Finally, the non-trivial connection between an inviscid eigenmode and the viscous eigenmodes for large but finite Reynolds number, and the relation between the corresponding modal superpositions, is highlighted.

Non-modal disturbances growth in a viscous mixing layer flow

The non-modal transient growth of disturbances in an isothermal viscous mixing layer flow is studied for Reynolds numbers varying from 100 up to 5000 at different streamwise and spanwise wavenumbers. It is found that the largest non-modal growth takes place at the wavenumbers for which the mixing layer flow is stable. In linearly unstable configurations, the non-modal growth can only slightly exceed the exponential growth at short times. Contrarily to the fastest exponential growth, which is two-dimensional, the most profound non-modal growth is attained by oblique three–dimensional waves propagating at an angle with respect to the base flow. By comparing the results of several mathematical approaches, it is concluded that within the considered mixing layer model with a tanh base velocity profile, the non-modal optimal disturbances growth is governed only by eigenvectors that are decaying far from the mixing zone. Finally, a full three-dimensional DNS with optimally perturbed base flow confirms the presence of the structures determined by the transient growth analysis. The time evolution of optimal perturbations is presented. It exhibits a growth and a decay of flow structures that sometimes become similar to those observed at the late stages of the time evolution of the Kelvin–Helmholtz billows. It is shown that non-modal optimal disturbances yield a strong mixing without a transition to turbulence.

The lift-up effect: The linear mechanism behind transition and turbulence in shear flows

The formation and amplification of streamwise velocity perturbations induced by cross-stream disturbances is ubiquitous in shear flows. This disturbance growth mechanism, so neatly identified by Ellingsen and Palm in 1975, is a key process in transition to turbulence and self-sustained turbulence. In this review, we first present the original derivation and early studies and then discuss the non-modal growth of streaks, the result of the lift-up process, in transitional and turbulent shear flows. In the second part, the effects on the lift-up process of additives in the fluid and of a second phase are discussed and new results presented with emphasis on particle-laden shear flows. For all cases considered, we see the lift-up process to be a very robust process, always present as a first step in subcritical transition.

Double diffusive natural convection of CO2 in a brine saturated geothermal reservoir: Study of non-modal growth of perturbations and heterogeneity effects

In a recent study (Islam et al., 2013) double diffusive natural convection of CO2 in brine saturated homogeneous permeable geothermal reservoir was investigated. Here we have shown the study of heterogeneous reservoirs. Firstly, we present the non-modal stability theory to analyze time-dependent double diffusive convection. Linear stability analysis is performed and the maximum amplification of perturbations optimized over the entire space of initial perturbations is examined. These provide us onset time of convection (or the critical time) and the structure of the most-amplified perturbations. Secondly, we investigate numerical simulation of an anisotropic and layered reservoir with permeability variations, which is impervious from the sides, and is open to CO2 at the top. We show the propagation of CO2 plumes over long period of time by analyzing different combination of problem parameters: solutal Rayleigh number (100≤Ras≤10,000), the buoyancy ratio (2≤N≤100), Dykstra-Parsons coefficient (0≤Vdp≤0.85) and a fixed Lewis number. How permeability variation in vertical (kz) to horizontal (kx) direction affects the CO2 dissolution is discussed for different ratios of kz/kx (0.1≤kz/kx≤0.8). Results of different cavity aspect ratios (0.5≤A≤2) are also presented. It is found that heterogeneous porous reservoir renders better mass transfer of CO2 than the homogeneous one. CO2 dissolution is enhanced with increasing both Ras and Vdp. N, which represents geothermal effect, has minor impact on overall dissolution process. The results have implications in enhanced oil recovery and CO2 based geothermal systems.

The temporal evolution of the kinetic drift-Alfven instability of plasma shear flow

The linear non-modal kinetic theory of the kinetic drift-Alfven instability, in the presence of shearing plasma flow, exhibits temporal non-modal amplitude growth with time-increasing growth rate. The drift-Alfven instability develops when the evolving frequencies of the density-gradient-driven drift wave and the kinetic Alfven wave become comparable and mode coupling increases significantly. Because the development of the instability depends on the plasma density gradient, drift-Alfven turbulence may be invoked as a possible turbulence driver in the edge pedestal, limiting the pedestal density gradients during the edge localized modes crash. In contrast to the case of shear-modified electrostatic drift-wave turbulence, the scattering of plasma ions by the shear-modified electromagnetic drift-Alven turbulence does not only suppress the turbulence, but can be responsible for transient increase in the instability's growth rate. The instability, i.e., turbulence, stabilizes when the drift and Alfven wave frequencies eventually become different enough that effective coupling between the two modes ends. So, perpendicular-flow shear can cause the growth rate to increase, decrease, become zero, and become negative as a result of shear's time-changing modification to the frequencies of the drift and Alfven waves.

Numerical study of laminar-turbulent transition in particle-laden channel flow

We present direct numerical simulations of subcritical transition to turbulence in a particle-laden channel flow, with particles assumed rigid, spherical, and heavier than the fluid. The equations describing the fluid flow are solved with an Eulerian mesh, whereas those describing the particle dynamics are solved by Lagrangian tracking. Two-way coupling between fluid and particles is modeled with Stokes drag. The numerical code is first validated against previous results from linear stability: the nonmodal growth of streamwise vortices resulting in streamwise streaks is still the most efficient mechanism for linear disturbance amplification at subcritical conditions as for the case of a single phase fluid. To analyze the full nonlinear transition, we examine two scenarios well studied in the literature: (1) transition initiated by streamwise independent counter-rotating streamwise vortices and one three-dimensional mode and (2) oblique transition, initiated by the nonlinear interaction of two symmetric oblique waves. The threshold energy for transition is computed, and it is demonstrated that for both scenarios the transition may be facilitated by the presence of particles at low number density. This is due to the fact that particles may introduce in the system detrimental disturbances of length scales not initially present. At higher concentrations, conversely, we note an increase of the disturbance energy needed for transition. The threshold energy for the oblique scenario shows a more significant increase in the presence of particles, by a factor about four. Interestingly, for the streamwise-vortex scenario the time at which transition occurs increases with the particle volume fraction when considering disturbances of equal initial energy. These results are explained by considering the reduced amplification of oblique modes in the two-phase flow. The results from these two classical scenarios indicate that, although linear stability analysis shows hardly any effect on optimal growth, particles do influence secondary instabilities and streak breakdown. These effects can be responsible of the reduced drag observed in turbulent channel flow laden with heavy particles.

Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow

AbstractStreamwise rolls and accompanying streamwise streaks are ubiquitous in wall-bounded shear flows, both in natural settings, such as the atmospheric boundary layer, as well as in controlled settings, such as laboratory experiments and numerical simulations. The streamwise roll and streak structure has been associated with both transition from the laminar to the turbulent state and with maintenance of the turbulent state. This close association of the streamwise roll and streak structure with the transition to and maintenance of turbulence in wall-bounded shear flow has engendered intense theoretical interest in the dynamics of this structure. In this work, stochastic structural stability theory (SSST) is applied to the problem of understanding the dynamics of the streamwise roll and streak structure. The method of analysis used in SSST comprises a stochastic turbulence model (STM) for the dynamics of perturbations from the streamwise-averaged flow coupled to the associated streamwise-averaged flow dynamics. The result is an autonomous, deterministic, nonlinear dynamical system for evolving a second-order statistical mean approximation of the turbulent state. SSST analysis reveals a robust interaction between streamwise roll and streak structures and turbulent perturbations in which the perturbations are systematically organized through their interaction with the streak to produce Reynolds stresses that coherently force the associated streamwise roll structure. If a critical value of perturbation turbulence intensity is exceeded, this feedback results in modal instability of the combined streamwise roll/streak and associated turbulence complex in the SSST system. In this instability, the perturbations producing the destabilizing Reynolds stresses are predicted by the STM to take the form of oblique structures, which is consistent with observations. In the SSST system this instability exists together with the transient growth process. These processes cooperate in determining the structure of growing streamwise roll and streak. For this reason, comparison of SSST predictions with experiments requires accounting for both the amplitude and structure of initial perturbations as well as the influence of the SSST instability. Over a range of supercritical turbulence intensities in Couette flow, this instability equilibrates to form finite amplitude time-independent streamwise roll and streak structures. At sufficiently high levels of forcing of the perturbation field, equilibration of the streamwise roll and streak structure does not occur and the flow transitions to a time-dependent state. This time-dependent state is self-sustaining in the sense that it persists when the forcing is removed. Moreover, this self-sustaining state rapidly evolves toward a minimal representation of wall-bounded shear flow turbulence in which the dynamics is limited to interaction of the streamwise-averaged flow with a perturbation structure at one streamwise wavenumber. In this minimal realization of the self-sustaining process, the time-dependent streamwise roll and streak structure is maintained by perturbation Reynolds stresses, just as is the case of the time-independent streamwise roll and streak equilibria. However, the perturbation field is maintained not by exogenously forced turbulence, but rather by an endogenous and essentially non-modal parametric growth process that is inherent to time-dependent dynamical systems.

Spatial optimal growth in three-dimensional compressible boundary layers

AbstractThis paper represents a continuation of the work by Tempelmann et al. (J. Fluid Mech., vol. 646, 2010b, pp. 5–37) on spatial optimal growth in incompressible boundary layers over swept flat plates. We present an extension of the methodology to compressible flow. Also, we account for curvature effects. Spatial optimal growth is studied for boundary layers over both flat and curved swept plates with adiabatic and cooled walls. We find that optimal growth increases for higher Mach numbers. In general, extensive non-modal growth is observed for all boundary layer cases even in subcritical regions, i.e. where the flow is stable with respect to modal crossflow disturbances. Wall cooling, despite stabilizing crossflow modes, destabilizes disturbances of non-modal nature. Curvature acts similarly on modal as well as non-modal disturbances. Convex walls have a stabilizing effect on the boundary layer whereas concave walls have a destabilizing effect. The physical mechanisms of optimal growth in all studied boundary layers are found to be similar to those identified for incompressible flat-plate boundary layers.

Subcritical spatial transition of swept Hiemenz flow

It is known from experimental investigations that the leading-edge boundary layer of a swept wing exhibits transition to turbulence at subcritical Reynolds numbers, i.e. at Reynolds numbers which lie below the critical Reynolds number predicted by linear stability theory. In the present work, we investigate this subcritical transition process by direct numerical simulations of a swept Hiemenz flow in a spatial setting. The laminar base flow is perturbed upstream by a pair of stationary counter-rotating vortex-like disturbances. This perturbation generates high- and low-speed streaks by a non-modal growth mechanism. Further downstream, these streaky structures exhibit a strong instability to secondary perturbations which leads to a breakdown to turbulence.The observed transition mechanism has strong similarities to by-pass transition mechanisms found for two-dimensional boundary layers. It can be shown that transition strongly depends on the amplitude of the primary perturbation as well as on the frequency of the secondary perturbation.

Modal and non-modal stability of particle-laden channel flow

Modal and non-modal linear stability analysis of channel flow with a dilute particle suspension is presented where particles are assumed to be solid, spherical, and heavy. The two-way coupling between particle and fluid flow is therefore modeled by the Stokes drag only. The results are presented as function of the particle relaxation time and mass fraction. First, we consider exponentially growing perturbations and extend previous findings showing the potential for a significant increase of the critical Reynolds number. The largest stabilization is observed when the ratio between the particle relaxation time and the oscillation period of the wave is of order one. By examining the energy budget, we show that this stabilization is due to the increase of the dissipation caused by the Stokes drag. The observed stabilization has led to the hypothesis that dusty flows can be more stable. However, transition to turbulence is most often subcritical in canonical shear flows where non-modal growth mechanisms are responsible for the initial growth of external disturbances. The non-modal analysis of the particle-laden flow, presented here for the first time, reveals that the transient energy growth is, surprisingly, increased by the presence of particles, in proportion to the particle mass fraction. The generation of streamwise streaks via the lift-up mechanism is still the dominant disturbance-growth mechanism in the particle laden flow; the length scales of the most dangerous disturbances are unaffected, while the initial disturbance growth can be delayed. These results are explained in terms of a dimensionless parameter relating the particle relaxation time to the time scale of the instability. The presence of a dilute solid phase therefore may not always work as a flow-control strategy for maintaining the flow as laminar. Despite the stabilizing effect on modal instabilities, non-modal mechanisms are still strong in internal flows seeded with heavy particles. Our results indicate that the initial stages of transition in dilute suspensions of small particles are similar to the stages in a single phase flow.

Spectral and algebraic instabilities in thin Keplerian discs under poloidal and toroidal magnetic fields

Linear instability of two equilibrium configurations with either poloidal (I) or toroidal (II) dominant magnetic field components are studied in thin vertically-isothermal Keplerian discs. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. In both equilibrium configurations the perturbations are decoupled into in-plane and vertical modes. For equilibria of type I those two modes are the Alfv\'en-Coriolis and sound waves, while for equilibria of type II they are the inertia-Coriolis and magnetosonic waves. Exact expressions for the growth rates as well as the number of unstable modes for type I equilibria are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. It is further shown that the axisymmetric MRI is completely suppressed by dominant toroidal magnetic fields (i.e. equilibria of type II). This renders the system prone to either non-axisymmetric MRI or non-modal algebraic growth mechanisms. The algebraic growth mechanism investigated in the present study occurs exclusively due to the rotation shear, generates the inertia-Coriolis driven magnetosonic modes due to non-resonant or resonant coupling that induces, respectively, linear or quadratic temporal growth of the perturbations.

307 平板境界層におけるNon-modal成長に対する主流乱れの強さとスケールの影響(OS5-2熱流体の可視化と計測)

307 平板境界層におけるNon-modal成長に対する主流乱れの強さとスケールの影響(OS5-2熱流体の可視化と計測)

Erratum: “Linear study of the nonmodal growth of drift waves in dusty plasmas” [Phys. Plasmas 17, 063703 (2010)

First Page

Modal and Nonmodal Growths of Symmetric Perturbations in Unbounded Domain

Abstract Modal and nonmodal growths of nonhydrostatic symmetric perturbations in an unbounded domain are examined in comparison with their hydrostatic counterparts. It is shown that the modal growth rate is a function of a single internal parameter s, the slope of the cross-band wave pattern. The maximum nonmodal growth of total perturbation energy norm is produced, also as a function of s, by an optimal combination of one geostrophic neutral mode and two paired nongeostrophic growing and decaying (or propagating) modes in the unstable (or stable) region. The hydrostatic approximation inflates the maximum modal growth rate significantly (or boundlessly) as the basic-state Richardson number Ri is small (or → 0) and inflates the maximum nonmodal growth rate significantly (or boundlessly) as |s| is large (or → ∞). Inside the unstable region, the maximum nonmodal growth scaled by the modal growth is a bounded increasing function of growth time τ but reduces to 1 at (Ri, s) = (¼, −2) where the three modes become orthogonal to each other. Outside the unstable region, the maximum nonmodal growth is a periodic function of τ and the maximum growth time τm is bounded between ¼ and ½ of the period of the paired propagating modes. The scaled maximum nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 (the marginal-stability boundary) for any τ if Ri ≤ 1, or at s = −1 ± (1 − Ri−1)1/2 for τ = τm if Ri > 1. When the neutral mode is filtered, the nonmodal growth becomes nongeostrophic and smaller than its counterpart growth constructed by the three modes but still significantly larger than the modal growth in general. The scaled maximum nongeostrophic nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 for any τ if Ri ≤ 1, or at s = −Ri−1/2 for τ = τm if Ri > 1. Normalized inner products between the modes are introduced to measure their nonorthogonality and interpret their constructed nonmodal growths physically.

Linear study of the nonmodal growth of drift waves in dusty plasmas

The main effect of dust on drift wave turbulence is the enhancement of the nonadiabaticity. Previous work found that nonmodal behavior is important in the nonadiabatic regime of the drift wave system. Here, the modal and nonmodal properties of the linear Hasegawa–Wakatani system of dusty plasmas are investigated. The non-normality of the linear evolution operator can lead to enhanced transient growth rates compared to the modal growth rates.

Viscous symmetric stability of circular flows

The linear stability properties of viscous circular flows in a rotating environment are studied with respect to symmetric perturbations. Through the use of an effective energy or Lyapunov functional, we derive sufficient conditions for Lyapunov stability with respect to such perturbations. For circular flows with swirl velocity V(r) we find that Lyapunov stability is determined by the properties of the function ℱ(r) = (2V/r + f)/Q (with f the Coriolis parameter, r the radius and Q the absolute vorticity) instead of the customary Rayleigh discriminant Φ(r) = (2V/r + f)Q. The conditions for stability are valid for flows with non-zero Q everywhere. Further, the flows are presumed stationary, incompressible and velocity perturbations are required to vanish at rigid boundaries. For Lyapunov stable flows an upper bound for the increase of the total perturbation energy due to transient non-modal growth is derived which is valid for any Reynolds number. The theory is applied to Couette flow and the Lamb–Oseen vortex.

Receptivity to free-stream vorticity of flow past a flat plate with elliptic leading edge

Receptivity of the two-dimensional boundary layer on a flat plate with elliptic leading edge is studied by numerical simulation. Vortical perturbations in the oncoming free stream are considered, impinging on two leading edges with different aspect ratio to identify the effect of bluntness. The relevance of the three vorticity components of natural free-stream turbulence is illuminated by considering axial, vertical and spanwise vorticity separately at different angular frequencies. The boundary layer is most receptive to zero-frequency axial vorticity, triggering a streaky pattern of alternating positive and negative streamwise disturbance velocity. This is in line with earlier numerical studies on non-modal growth of elongated structures in the Blasius boundary layer. We find that the effect of leading-edge bluntness is insignificant for axial free-stream vortices alone. On the other hand, vertical free-stream vorticity is also able to excite non-modal instability in particular at zero and low frequencies. This mechanism relies on the generation of streamwise vorticity through stretching and tilting of the vertical vortex columns at the leading edge and is significantly stronger when the leading edge is blunt. It can thus be concluded that the non-modal boundary-layer response to a free-stream turbulence field with three-dimensional vorticity is enhanced in the presence of a blunt leading edge. At high frequencies of the disturbances the boundary layer becomes receptive to spanwise free-stream vorticity, triggering Tollmien–Schlichting (T-S) modes and receptivity increases with leading-edge bluntness. The receptivity coefficients to free-stream vortices are found to be about 15% of those to sound waves reported in the literature. For the boundary layers and free-stream perturbations considered, the amplitude of the T-S waves remains small compared with the low-frequency streak amplitudes.

Spatial optimal growth in three-dimensional boundary layers

A parabolized set of linear equations is derived, which, in combination with the proposed solution procedure, allows for the study of both non-modal and modal disturbance growth in three-dimensional boundary layers. The method is applicable to disturbance waves whose lines of constant phase are closely aligned with the external streamline. Moreover, strongly growing disturbances may fall outside the scope of application. These equations are used in conjunction with a variational approach to compute optimal disturbances in Falkner–Skan–Cooke boundary layers subject to adverse and favourable pressure gradients. The disturbances associated with maximum energy growth initially take the form of streamwise vortices which are tilted against the mean crossflow shear. While travelling downstream these vortical structures rise into an upright position and evolve into bent streaks. The physical mechanism responsible for non-modal growth in three-dimensional boundary layers is therefore identified as a combination of the lift-up effect and the Orr mechanism. Optimal disturbances smoothly evolve into crossflow modes when entering the supercritical domain of the flow. Non-modal growth is thus found to initiate modal instabilities in three-dimensional boundary layers. Optimal growth is first studied for stationary disturbances. Influences of parameters such as sweep angle, spanwise wavenumber and position of inception are studied, and the initial optimal amplification of stationary crossflow modes because of non-modal growth is investigated. Finally, general disturbances are considered, and envelopes yielding the maximum growth at each position are computed. In general, substantial growth is already found upstream of the first neutral point. The computations show that at supercritical conditions, maximum growth of optimal disturbances in accelerated boundary layers can exceed the growth predicted for modal instabilities by several orders of magnitude.

Steady and unsteady Görtler boundary-layer instability on concave wall

The subject of the present combined experimental and theoretical investigation is the steady and unsteady linear Görtler instability. The majority of previous experiments were devoted to the steady Görtler vortices, despite the unsteady ones are also observed in real transitional flows. Moreover, even for the steady Görtler vortices no quantitative agreement between the experimental and theoretical linear-stability characteristics was obtained, especially for disturbance amplification rates. The experimental difficulties were connected, in particular, with a rather poor accuracy of measurements at zero disturbance frequency, a possible influence of nonlinearity, and an admixture of non-modal (transient) growth mechanism. All these difficulties have been overcome in the experimental part of the present study by means of: ( i) tuning-out of the exact zero frequency of Görtler vortices and working, instead, with quasi-steady perturbations of very low frequencies, ( ii) performing measurements at low disturbance amplitudes, and ( iii) minimization and careful estimation of the disturbance-source near-field by means of utilizing a special controlled disturbance source and performing special numerical computations for exact experimental conditions. A detailed study of all linear-stability characteristics for essentially unsteady Görtler vortices was performed in this paper as well. The results are obtained in a range of Görtler numbers 13 ≲ Gö ≲ 17.3, frequency parameters F = 0.56–22.70, and spanwise wavelength parameters Λ = 149–775 (close to the most amplified Görtler modes). Appropriate calculations based on locally-parallel and non-local non-parallel linear-stability theories were performed and compared quantitatively with experimentally obtained linear-stability characteristics. For the first time all stability characteristics measured for steady Görtler vortices (in quasi-steady regimes) are found to agree very well with those calculated for the most amplified first discrete-spectrum mode of the linear Görtler-instability problem. Similar good agreement is obtained for essentially unsteady Görtler vortices. The roles of effects of the base-flow non-parallelism and the disturbance-source near-field are examined.