Non-modal Growth Research Articles

Wave interactions in neutrally stable shear layers: Regular and singular modes, and non-modal growth

A recent letter [J. R. Carpenter and A. Guha, “Instability of a smooth shear layer through wave interactions,” Phys. Fluids 31, 081701 (2019)] compared the neutral modes of a smooth two-dimensional shear profile without an inflection point to the modes of its corresponding piecewise-linear profile. The regular mode in the smooth profile was identified as the one least sensitive to the numerical resolution, while the singular modes displayed high sensitivity. Here, we provide a physical interpretation using a wave interaction approach for understanding the structure and behavior of both the regular and singular modes. The regular modes are the interfacial Rossby waves located at the concentrated mean vorticity gradient of the shear profile. In contrast, the singular modes result from a one way phase-locking interaction between singular vorticity disturbances, passively advected by the mean flow at different levels of the profile, and the interfacial Rossby waves. We show that this one way interaction can also lead to a sustained non-modal growth of the interfacial Rossby waves that cannot be captured by standard eigenvalue analysis.

The bypass transition mechanism of the Stokes boundary layer in the intermittently turbulent regime

This numerical study focuses on the coherent structures and bypass transition mechanism of the Stokes boundary layer in the intermittently turbulent regime. In particular, the initial disturbance is produced by a temporary roughness element that is removed immediately after triggering a two-dimensional vortex tube under an inflection-point instability. The present study reveals a complete scenario of self-induced motion of a vortex tube after rollup from the boundary layer. The trajectory of the vortex tube is reasonably described based on the Helmholtz point-vortex equation. The three-dimensional transition of the vortex tube is attributed to the Crow instability, which leads to a sinusoidal disturbance that eventually evolves into a ring-like structure, especially for the weaker vortex. Further investigation demonstrates that three-dimensional or quasi-three-dimensional vortex perturbations in the free stream play a critical role in the boundary layer transition through a bypass mechanism, which is featured by the non-modal and explosive transient growth of the subsequent boundary layer instabilities. This transition scenario is found to be analogous to the oblique transition in the steady boundary layer, both of which are characterised by the formation of streaks, rollup of hairpin-like vortices and burst into turbulent spots. In addition, the streamwise propagation of turbulent spots is discussed in detail. To shed more light on the nature of the intermittently turbulent Stokes boundary layer, a conceptual model is proposed for the periodically self-sustaining mechanism of the turbulent spots based on the present numerical results and experimental evidence reported in the literature.

Mechanism for frustum transition over blunt cones at hypersonic speeds

Numerical and experimental studies have demonstrated laminar–turbulent transition in hypersonic boundary layers over sharp cones via the modal growth of planar Mack-mode instabilities. However, due to the strong reduction in Mack-mode growth at higher nose bluntness values, the mechanisms underlying the observed onset of transition over the cone frustum are currently unknown. Linear non-modal growth analysis has shown that both planar and oblique travelling disturbances that peak within the entropy layer experience appreciable energy amplification for moderate to large nose bluntness. However, due to their weak signature within the boundary-layer region, the route to transition onset via non-modal growth of travelling disturbances remains unclear. Nonlinear parabolized stability equations (NPSE) and direct numerical simulations (DNS) are used to identify a potential mechanism for transition over a 7-degree blunt cone that was tested in the AFRL Mach-6 high-Reynolds-number facility. Specifically, computations are conducted to study the nonlinear development of a pair of oblique, unsteady non-modal disturbances in the regime of moderately blunt nose tips. Excellent agreement was demonstrated between the NPSE and DNS predictions. Results reveal that, even though the linear non-modal disturbances are primarily concentrated outside the boundary layer, their nonlinear interaction can generate stationary streaks that penetrate and amplify within the boundary layer, eventually inducing the onset of transition via the breakdown of these streaks. The results indicate that a pair of oblique, controlled non-modal disturbances can produce transition at the location measured in the experiment when their initial amplitude is chosen to be approximately 0.15 % of the free-stream velocity.

Nonmodal growth of atmospheric disturbances relevant to the East Asian pressure surge in boreal winter

Significant events of East Asian pressure surge are identified based on sea level pressure for boreal winters of the period 1979–2016. These events typically grow from a high pressure anomaly over Northern Europe and the northwestern Russia and attain their peak over Siberia around ten days after the initiation. The structure of the pressure surge at its peak intensity mimics that of the semi-permanent Siberian High. With a two-level quasi-geostrophic (QG) model, a nonmodal instability analysis of the winter climatological flow reveals that one of the most rapidly amplifying “optimal disturbances” with an optimization time of 2 days resembles the structure of the East Asian pressure surge. This optimal disturbance originates from small-scale precursor disturbances over Northern Europe and the Barents Sea. Such initial disturbances grow in magnitude via Siberia towards East Asia, forming a low-level ridge extending to around 50° N. The optimal disturbance successfully captures the temporal and spatial scales and the overall structure of the circulation anomaly associated with the pressure surge, albeit with varying degrees of accuracy of the location of individual trough/ridge of the surge anomaly. Additional experiments suggest that (1) adding damping effect to the two-level QG model extends the low-level ridge further south to around 40° N and (2) the upper level wave train feature associated with pressure surge can be captured even in a barotropic model by optimal disturbances with an optimization time of 4 days.

A Nonmodal Instability Perspective of the Declining Northern Midlatitude Synoptic Variability in Boreal Summer

AbstractThe Pacific–North America–North Atlantic sector in general experienced a dryer and warmer climate in summer during the past 40 years. These changes are partly associated with declining midlatitude synoptic variability in boreal summer, especially over the two ocean basins. A nonmodal instability analysis of the boreal summer background flow is conducted for two periods, 1979–94 and 2000–15, to understand dynamical processes potentially responsible for the observed decline of synoptic variability. The synoptic variability associated with fast, nonmodal growth of atmospheric disturbances shows a decline over northern midlatitudes in the later period, in both a barotropic model and a two-level quasigeostrophic model. These results highlight the importance of the changing summer background flow in contributing to the observed changes in synoptic variability. Also discussed are factors likely associated with background flow changes including sea surface temperature and sea ice change.

Reducing noise from twin supersonic jets using very-low-frequency control

The intense jet noise radiated by closely spaced, twin supersonic hot jets leads to sound-induced structural vibration, fatigue and personnel related operational difficulties. Experimental, theoretical, and computational investigations into the physics and control of jet noise have identified several important sound sources, including wavepackets, screech, Mach wave radiation, and broadband shock associated noise (BBSAN). Reducing the loudest sources of jet noise, without sacrificing propulsive performance, has relied on intuition, parametric survey, or optimal control techniques. With the aim of developing a more general and robust method of jet noise reduction (JNR), we present a physics-based approach that leverages very-low-frequency jet dynamics in order to achieve JNR whilst preserving propulsive performance. Our approach formulates the control problem using the very-low-frequency global modes of the compressible Navier-Stokes operator linearized about the jet mean flow to disrupt the nonmodal transient growth processes. The presentation will showcase uncontrolled and controlled single and twin supersonic hot jets issuing from biconical nozzles, with conditions and geometries motivated by tactical Naval aircraft. The predictions utilize fully resolved numerical simulations whose data inform the control development and which evaluate its performance on the jet exhaust and on its radiated noise.

Investigation of the transient growth in plane jet by non-modal stability analysis

Linear stability analysis is used to characterize the dynamics of a plane jet by incorporating non-modal stability analysis besides classical global temporal stability analysis. It is explained that similar shapes of different global modes are the result of non-normal characteristics of linearized Navier Stokes equations. Optimal initial disturbances and their eigenfunctions together with transient energy growth are obtained for different time horizons and Reynolds numbers of the jet in the linear unstable configuration. These structures are localized at the upstream of the jet nozzle at the boundary layer. The transient growth of the inlet perturbation in limited time bounds is found in the presence of unstable eigenmodes in the flow. The evolution of optimal initial disturbances yields robust energy growth of more than two orders of magnitude. It is found that by increasing Reynolds number in linear unstable flow framework, the non-modal transient growth can marginally surpass the exponential growth in short times.

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

This study analyses the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. The analysis is performed using a frequency-domain low-order network model as well as a time-domain spatially discretised model. Anti-correlated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. The time-domain analysis results in a delay-differential equation and allows the investigation of non-normal behaviour and its consequences. Pseudospectral analysis reveals that energy amplification is crucially linked to an interplay between acoustic and ITA modes. While higher non-orthogonality between two modes is correlated with peaks in modal sensitivity, transient energy growth does not necessarily involve the most sensitive modes. In particular, growth estimates based on the Kreiss constant demonstrate that transient amplification relies critically on the proximity of the non-normal modes to the imaginary axis. The time scale for transient amplification is identified as the flame time delay, which is further corroborated by determining the optimal initial conditions responsible for the bulk of the non-modal energy growth. The flame is identified as an active and dominant contributor to energy gain. The frequency of the optimal perturbation matches the acoustic time scale, once more confirming an interplay between acoustic and ITA structures. Flame-based amplification factors of two to five are found, which are significant when feeding into the acoustic dynamics and eventually triggering nonlinear limit-cycle behaviour.

Passive Drag Reduction via Bionic Hull Coatings

The resistance of ships is often dominated by friction between the hull and the water. This study explores possibilities of reducing skin-friction drag in a way inspired by dolphins. These possess a soft skin believed to diminish drag by delaying the transition from laminar to turbulent flow. The underlying mechanism builds on a stabilization of the laminar boundary layer by the compliant surface. To transfer this mechanism to ship hulls, coatings similar to dolphin skin have been designed numerically, made from polymeric materials, and tested in a water tunnel. For the best coating, a drag reduction by almost 3% has been predicted in the boundary layer along the hull model of a small search-and-rescue vessel. The trends of the numerical predictions have been confirmed in the experiments. 1. Overview Dolphins are able to sustain a swimming speed of about 9 m/sec across long distances (Carpenter et al. 2000). This fascinating endurance has stimulated the hypothesis that the pliable dolphin skin interacts with the surrounding flow so as to maintain low-drag laminar flow via a delay of transition to turbulence (Gad-el-Hak 1996). Although this hypothesis is under debate (Fish & Lauder 2006), laboratory experiments have indeed demonstrated that soft coatings delay laminar-turbulent transition (Gaster 1987). According to boundary-layer linear stability theory (LST), compliant surfaces are able to mitigate the amplification of Tollmien-Schlichting (TS) waves, the forerunners of transition in low-disturbance environments (Carpenter & Garrad 1985). However, wall compliance may also promote transition in that the flow may excite instability waves borne by the soft coating material. These waves are known as fluid-induced surface instabilities (FISI) (Carpenter & Garrad 1985) or travelling-wave flutter (Gad-el-Hak 1996). FISI waves may in turn via nonlinear effects develop into quasi-steady surface corrugations ("divergence") (Carpenter & Garrad 1986). This process is associated with an absolute instability mechanism (Tsigklifis & Lucey 2015) that rapidly triggers transition via the bypass route. The boundary-layer instability over a compliant coating is hence more complex than that over a rigid surface owing to the presence of two wave-bearing media. Compliant-wall boundary layers also support non-modal transient growth inducing bypass transition (Tsigklifis & Lucey 2015).

Nonmodal Growth of Traveling Waves on Blunt Cones at Hypersonic Speeds

The existing database of transition measurements in hypersonic ground facilities has established that, as the nose-tip bluntness is increased, the onset of boundary-layer transition over a circular cone at zero angle of attack shifts downstream. However, this downstream movement in transition slows down and eventually reverses at sufficiently large values of the nose Reynolds number. Because modal amplification is too weak to initiate the measured frustum transition at moderate to large bluntness values, nonmodal growth is investigated as the potential basis for a physics-based model. The present analysis investigates the nonmodal growth of traveling disturbances initiated within the nose-tip vicinity. Results show that both planar and oblique traveling disturbances that peak within the entropy layer experience appreciable energy amplification for successively higher disturbance frequencies as the nose bluntness is increased. For relatively small bluntness values, the initial nonmodal growth is followed by a partial decay that is overcome by an eventual, modal growth as Mack-mode waves. For larger bluntness values, the Mack-mode waves are not amplified anywhere upstream of the experimentally measured transition location, but the traveling waves still undergo a significant amount of nonmodal growth. These findings do not provide a definitive link between optimal growth and the onset of frustum transition on moderately blunt cones, but they are qualitatively consistent with the experimental observations of an initial peak near the nose tip followed by a decay and growth in disturbance energy, as well as the appearance of inclined structures above the boundary-layer edge.

3D global optimal forcing and response of the supersonic boundary layer

3D optimal forcing and response of a 2D supersonic boundary layer are obtained by computing the largest singular value and the associated singular vectors of the global resolvent matrix. This approach allows to take into account both convective-type and component-type non-normalities responsible for the non-modal growth of perturbations in noise selective amplifier flows. It is moreover a fully non-parallel approach that does not require any particular assumptions on the baseflow. The numerical method is based on the explicit calculation of the Jacobian matrix proposed by Mettot et al. [1] for 2D perturbations. This strategy uses the numerical residual of the compressible Navier-Stokes equations imported from a finite-volume solver that is then linearised employing a finite difference method. Extension to 3D perturbations, which are expanded into modes of wave number, is here proposed by decomposing the Jacobian matrix according to the direction of the derivatives contained in its coefficients. Validation is performed on a Blasius boundary layer and a supersonic boundary layer, in comparison respectively to global and local results. Application of the method to a boundary layer at M=4.5 recovers three regions of receptivity in the frequency-transverse wave number space. Finally, the energy growth of each optimal response is studied and discussed.

Stability of slip channel flow revisited

In this work, we revisit the temporal stability of slip channel flow. Lauga and Cossu [“A note on the stability of slip channel flows,” Phys. Fluids 17, 088106 (2005)] and Min and Kim [“Effects of hydrophobic surface on stability and transition,” Phys. Fluids 17, 108106 (2005)] have investigated both modal stability and non-normality of slip channel flow and concluded that the velocity slip greatly suppresses linear instability and only modestly affects the non-normality. Here, we study the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did. We find that, with a sufficiently large slip, both streamwise and spanwise slip trigger three-dimensional leading instabilities. Overall, the critical Reynolds number is only slightly increased by streamwise slip, whereas it can be greatly decreased by spanwise slip. Streamwise slip suppresses the nonmodal transient growth, whereas the spanwise slip enlarges the nonmodal growth, although it does not affect the base flow. Interestingly, as the spanwise slip length increases, the optimal perturbations exhibit flow structures different from the well-known streamwise rolls. However, in the presence of equal slip in both directions, the three-dimensional leading instabilities disappear and the flow is greatly stabilized. The results suggest that earlier instability and larger transient growth can be triggered by introducing anisotropy in the velocity slip.

Numerical Investigation of Roughness Effects on Transition on Spherical Capsules

To address the hitherto unknown mechanism of boundary-layer transition on blunt reentry capsules, the role of roughness-induced disturbance growth on a spherical-section forebody is assessed via optimal transient growth theory and direct numerical simulations (DNS). Optimal transient-growth studies have been performed for the blunt capsule experiments at Mach 5.9 in the Hypersonic Ludwieg tube at the Technische Universität Braunschweig (HLB), which included measurements behind a patch of controlled, distributed micron-sized surface roughness. Transient-growth results for the HLB capsule indicate similar trends as the corresponding numerical data for a Mach 6 experiment in the Actively Controlled Expansion (ACE) facility of the Texas A&M University (TAMU) at a lower Reynolds number. Both configurations indicate a similar dependence on surface temperature ratio and, more important, rather low values of maximum energy gain. DNS are performed for the conditions of the HLB experiment to understand the generation of stationary disturbances by the roughness patch and the accompanying evolution of unsteady perturbations. However, no evidence of either modal or nonmodal disturbance growth in the wake of the roughness patch is found in the DNS data; thus, the physical mechanism underlying the observed onset of transition still remains unknown.

The collisional drift wave instability in steep density gradient regimes

The collisional drift wave instability in a straight magnetic field configuration is studied within a full-F gyro-fluid model, which relaxes the Oberbeck–Boussinesq (OB) approximation. Accordingly, we focus our study on steep background density gradients. In this regime we report on corrections by factors of order one to the eigenvalue analysis of former OB approximated approaches as well as on spatially localised eigenfunctions, that contrast strongly with their OB approximated equivalent. Remarkably, non-modal phenomena arise for large density inhomogeneities and for all collisionalities. As a result, we find initial decay and non-modal growth of the free energy and radially localised and sheared growth patterns. The latter non-modal effect sustains even in the nonlinear regime in the form of radially localised turbulence or zonal flow amplitudes.

Sustaining turbulence in spectrally stable shear flows – interplay of linear transient growth and nonlinear transverse cascade

We analyze the sustaining mechanism of nonlinear perturbations/turbulence in spectrally stable smooth shear flows. The essence of the sustenance is a subtle interplay of linear transient growth of Fourier harmonics and nonlinear processes. In spectrally stable shear flows, the transient growth of perturbations is strongly anisotropic in spectral (k-)space. This, in turn, leads to anisotropy of nonlinear processes ink-space and, as a result, the main (new) nonlinear process appears to be not a direct/inverse, but rather a transverse/angular redistribution of harmonics in Fourier space referred to as the nonlinear transverse cascade. It is demonstrated that nonlinear state is sustained owing to the interplay of the linear nonmodal growth and the transverse cascade. The possibility of such course of events has been described ink-space byG. Chagelishvili, J.-P. Zahn, A. Tevzadze and J. Lominadze, A&A, 402, 401 (2003)that reliably exemplifies the well-known bypass scenario of subcritical turbulence in spectrally stable shear flows. We present selected results of the simulations performed in different (HD and MHD; 2D and 3D; plane and Keplerian) shear flows to demonstrate the transverse cascade in action.

From three-dimensional to quasi-two-dimensional: transient growth in magnetohydrodynamic duct flows

This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of $5000$ and $15\,000$ and an extensive range of Hartmann numbers $0\leqslant \mathit{Ha}\leqslant 800$ were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes, corresponding to low, moderate and high magnetic field strengths, are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness $R_{H}$ and Reynolds number built upon the Shercliff layer thickness $R_{S}$, respectively. Transition between regimes respectively occurs at $\mathit{Ha}\approx 2$ and no lower than $R_{H}\approx 33.\dot{3}$. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be excellent predictors of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-two-dimensional turbulence at much higher Hartmann numbers than is currently achievable.

Non-normality increases variance of gravity waves trapped in a tilted box

ABSTRACTWe study the prototype problem of internal gravity waves in a square domain tilted with respect to the gravity vector by an angle θ. Only for and a countable number of other angles regular normal modes exist, for all other angles wave attractors and singularities dominate the flow. We show that the linear operator of the governing PDE becomes non-normal for giving rise to non-modal transient growth. This growth depends on the underlying norm: for the variance norm significant growth rates can be found whereas for the energy norm, no growth is possible since, in contrast to shear flows for which the mean flow feeds the perturbations, there is no energy source. We continue by showing that the non-normality of the system matrix is increasing with θ and reaches a maximum for . Moreover, with increasing tilt angle the growth rate is increasing as can be expected from the increasing non-normality of the matrix. Our results imply that at least the most simple wave attractors can be seen as those initial flow fields that gain most of the variance during a given time period. We conclude that in a closed confinement with walls tilted with respect to gravity, internal wave variance can be maintained by transient growth of perturbations without a direct contribution of unstable normal modes.

Nose-Tip Bluntness Effects on Transition at Hypersonic Speeds

The existing database of transition measurements in hypersonic ground facilities has established that as the nose-tip bluntness is increased the onset of boundary-layer transition over a circular cone at zero angle of attack shifts downstream. However, this trend is reversed at sufficiently large values of the nose-tip Reynolds number, so the transition onset location eventually moves upstream with a further increase in nose-tip bluntness. This transition-reversal phenomenon was the focus of a collaborative investigation under the NATO Science and Technology Organization (STO) Applied Vehicle Technology (AVT)-240 group. The current Paper provides an overview of that effort, which included wind tunnel measurements and theoretical analysis related to modal and nonmodal amplification of boundary-layer disturbances. Because modal amplification is too weak to initiate transition at large bluntness values, transient growth has been investigated as the potential basis for a physics-based model for the transition-reversal phenomenon. Results indicate that stationary disturbances that are initiated within the nose-tip vicinity can undergo relatively significant nonmodal amplification that increases with the nose-tip bluntness. This finding does not provide a definitive link between transient growth and the onset of transition but is qualitatively consistent with the experimental observations that frustum transition during the reversal regime was highly sensitive to wall roughness and, furthermore, was dominated by disturbances originating near the nose tip. The present analysis shows significant nonmodal growth of traveling disturbances that peak within the entropy layer and could also play a role in the transition-reversal phenomenon if those disturbances are realizable in a natural disturbance environment.

Instability wave-streak interactions in a hypersonic boundary layer at flight conditions.

The interaction of stationary streaks undergoing nonmodal growth with modally unstable instability waves in a hypersonic boundary-layer flow is studied using numerical computations. The geometry and flow conditions are selected to match a relevant trajectory location from the ascent phase of the HIFiRE-1 flight experiment; namely, a 7 degree half-angle, circular cone with 2.5 mm nose radius, freestream Mach number equal to 5.30, unit Reynolds number equal to 13.42 m-1, and wall-to-adiabatic temperature ratio of approximately 0.35 over most of the vehicle. This paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite-amplitude streaks, followed by an analysis of the modal instability characteristics of the perturbed, streaky boundary-layer flow. The investigation is performed with stationary direct numerical simulations (DNS) and plane-marching parabolized stability equations (PSE), in conjunction with partial-differential-equation-based planar eigenvalue analysis. The overall effect of streaks is to reduce the peak amplification factors of instability waves, indicating a possible downstream shift in the onset of laminar-turbulent transition. The present study confirms previous findings that the mean flow distorsion of the nonlinear streak perturbation reduces the amplification rates of the Mack-mode instability. More importantly, however, the present results demonstrate that the spanwise varying component of the streak can produce a larger effect on the Mack-mode amplification. The study with selected azimuthal wavenumbers for the stationary streaks reveals that a wavenumber of approximately 1.4 times larger than the optimal wavenumber is more effective in stabilizing the planar Mack-mode instabilities. In the absence of unstable first-mode waves for the present cold-wall condition, transition onset is expected to be delayed until the peak streak amplitude increases to nearly 35 percent of the freestream velocity, when intrinsic instabilities of the boundary-layer streaks begin to dominate the transition process. For streak amplitudes below that limit a significant net stabilization is achieved, yielding a potential transition delay that can exceed 100 percent of the length of the laminar region in the uncontrolled case.

Non-modal stability analysis of stratified two-phase channel flows

The non-modal transient growth of perturbations in horizontal and inclined channel flows of two immiscible fluids is studied. 3D perturbations are examined in order to find the optimal perturbations that attain the maximum amplification of perturbation energy at relatively short times. Definition of the energy norm is extended to account for the gravitational potential energy along with the kinetic energy and the interfacial capillary energy. Contrarily to the fastest exponential growth, which is reached by essentially 2D perturbations, the maximal non-modal energy growth is attained mostly by three-dimensional spanwise perturbations. Significant transient energy growth is found to occur in linearly stable flow configurations, which, similarly to single phase shear flows, may trigger non-linear destabilizing mechanisms within one of the phases. It is shown that the transient energy growth in linearly stable cases can be accompanied by noticeable interface deformations. Therefore, flow pattern transition due to non-modal transient growth and reduction of the range of operational conditions for which stratified-smooth flow remains stable cannot be ruled out.