Non-modal Disturbances Research Articles

Receptivity of a hypersonic flow over a blunt wedge to a slow acoustic wave

The receptivity of the second mode for Mach 6 flow over a blunt wedge with a nose radius of 0.258 mm is investigated numerically. When a plane slow acoustic wave is imposed on the freestream, the receptivity process of the boundary layer goes through three stages, with the development of an entropy-layer mode, a fast mode, and nonmodal external disturbances, before exciting the second mode. The first two dominant modes can be obtained by linear stability theory in the entropy layer and the boundary layer, respectively, whereas the third is a complicated type of disturbance whose shape function peaks in the entropy layer. The results show that the nonmodal disturbances in the entropy layer play a leading role in the excitation of the second mode, which suggests that the growth of nonmodal disturbances could dominate nonmodal amplification, eventually leading to boundary-layer transition.

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.

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.

Optimal Growth in Hypersonic Boundary Layers

The linear form of the parabolized linear stability equations is used in a variational approach to extend the previous body of results for the optimal, nonmodal disturbance growth in boundary-layer flows. This paper investigates the optimal growth characteristics in the hypersonic Mach number regime without any high-enthalpy effects. The influence of wall cooling is studied, with particular emphasis on the role of the initial disturbance location and the value of the spanwise wave number that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary-layer equations, mean flow solutions based on the full Navier–Stokes equations are used in select cases to help account for the viscous–inviscid interaction near the leading edge of the plate and for the weak shock wave emanating from that region. Using the full Navier–Stokes mean flow is shown to result in further reduction with Mach number in the magnitude of optimal growth relative to the predictions based on the self-similar approximation to the base flow.

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.

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 critical layer in linear-shear boundary layers over acoustic linings

AbstractAcoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as $O(1/ {x}^{4} )$, where $x$ is the distance downstream of the source. For sources within the sheared flow, the contribution from the critical layer is found to consist of a non-modal disturbance of constant amplitude and a disturbance decaying algebraically as $O(1/ {x}^{3} )$. For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.

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.

A ‘receptive’ boundary layer

Receptivity is the process which describes how environmental disturbances (such as gusts, acoustic waves or wall roughness) are filtered by a boundary layer and turned into downstream-growing waves. It is closely related to the identification of initial conditions for the disturbances and requires knowledge of the characteristics of the specific external forcing field. Without such a knowledge, it makes sense to focus on worst case scenarios and search for those initial states which maximize the disturbance amplitude at a given downstream position, and hence to identify upper bounds on growth rates, which will be useful in predicting the transition to turbulence. This philosophical approach has been taken by Tempelmann, Hanifi & Henningson (J. Fluid Mech., 2010, vol. 646, pp. 5–37) in a remarkably complete parametric study of ‘optimal disturbances’ for a model of the flow over a swept wing; they pinpoint the crucial importance both of the spatial variation of the flow and of non-modal disturbances, even when the flow is ‘supercritical’ and hence subject to classical ‘normal mode’ instabilities.

On the growth (and suppression) of very short-scale disturbances in mixed forced–free convection boundary layers

The two-dimensional boundary-layer flow over a cooled/heated flat plate is investigated. A cooled plate (with a free-stream flow and wall temperature distribution which admit similarity solutions) is shown to support non-modal disturbances, which grow algebraically with distance downstream from the leading edge of the plate. In a number of flow regimes, these modes have diminishingly small wavelength, which may be studied in detail using asymptotic analysis.Corresponding non-self-similar solutions are also investigated. It is found that there are important regimes in which if the temperature of the plate varies (in such a way as to break self-similarity), then standard numerical schemes exhibit a breakdown at a finite distance downstream. This breakdown is analysed, and shown to be related to very short-scale disturbance modes, which manifest themselves in the spontaneous formation of an essential singularity at a finite downstream location. We show how these difficulties can be overcome by treating the problem in a quasi-elliptic manner, in particular by prescribing suitable downstream (in addition to upstream) boundary conditions.

Accretion disk instability revisited

Accretion disk flow in a local Cartesian (or shearing box) approximation is examined for viscous three-dimensional linear disturbances. Eigenvalue computations predict that the flow is asymptotically stable unless the rotation number falls in the range 0 < Ro < 1/2, in agreement with predictions of inviscid theory; Keplerian flow (Ro = 1/q = 2/3) is accord- ingly stable. Analysis of non-modal disturbances predicts large transient amplification factors, implying that the flow, although asymptotically stable, may be transiently unstable. Strong rotation, including Keplerian, two-dimensionalizes the system: the largest growth factors are found for disturbances which are uniform along the direction of the rotation axis and amplification occurs via the Orr mechanism, as in 2D shear flow. Amplification factors scale as Re 2/3 , implying very strong growth in actual disks. The implications of transient instability of rotating shear on disk turbulence are discussed.

Optimal Control of Nonmodal Disturbances in Boundary Layers

"The optimal control of infinitesimal flow disturbances experiencing the largest transient gain over a fixed time span, commonly termed ""optimal perturbations,"" is undertaken using a variational technique in two- and three-dimensional boundary layer flows. The cost function employed includes various energy metrics which can be weighted according to their perceived importance, simplifying the task of determining which terms are essential for a ""good"" control scheme. In the accelerated boundary layers investigated, disturbance kinetic energy can be typically reduced by about one order of magnitude. However, it seems impossible to suppress perturbations completely over the entire control interval; ""good"" control strategies still permit approximately an order of magnitude growth over the initial energy at some point in the interval. It is shown that the control effort efficiently targets the physical mechanisms behind transient growth."

The effect of barotropic shear on baroclinic instability Part II: The initial value problem

Abstract The effect of barotropic shear on baroclinic instability has been investigated using both a linear quasi-geostrophic β-plane channel model and a multilevel primitive equation model on the sphere when a nonmodal disturbance is used as the initial perturbation condition. The analysis of the initial value problem has demonstrated the existence of a rapid transient growth phase of the most unstable mode. The inclusion of a linear barotropic shear reduces initial rapid transient growth, although at intermediate times the transient growth rates of the sheared cases can be larger than in the unsheared case owing to downgradient eddy momentum fluxes. Certain disturbances can amplify by factors of 4.5–60 times (for the L2 norm), or 3–30 times (for the perturbation amplitude maximum), as large as disturbances based on the linear normal modes. However, linear horizontal shear always reduces the amplification factors. The mechanism is that the shear confines the disturbance meriodionally and therefore limits the energy conversion from the zonal available potential energy to eddy energy. The effect of barotropic shear on the transient growth is not changed much in the presence of either thermal damping or Ekman pumping. Nonmodal integrations of baroclinic wave lifecycles show that the energy level reached by eddies is not very sensitive to the structure of the initial disturbance if the amplitude of the initial disturbance is small. Although in some cases the eddy kinetic energy level reached by the wave integrated from nonmodal disturbance can be 25–150% larger than the normal mode integrations, barotropic shear, characterized by large shear vorticity with small horizontal curvature, always reduces the eddy kinetic energy level reached by the wave, confirming the results of normal mode studies.

Evidence for transient disturbance growth in a 1961 pipe-flow experiment

Evidence is presented for what appears to have been nonmodal disturbance growth in a pipe-flow experiment performed by Kaskel [Technical Report No. 32-138, Jet Propulsion Laboratory, California Institute of Technology, 1961], based on a comparison of the experimental traces of disturbance amplitude with predictions of transient amplitude growth calculated from the underlying linear stability operator. Owing to the geometry of the experimental disturbance generator, modes having azimuthal wavenumbers n=0, 6 and 12 are expected to contribute the bulk of the disturbance energy, and numerically predicted amplitude growth versus downstream distance is in good agreement with the experiment.

Instability of Surface and Upper-Tropospheric Shear Lines

Abstract An analytic linear stability analysis is carried out for a shear line associated with a surface temperature anomaly in uniform potential vorticity, quasigeostrophic flow. Previous studies of this type of flow, albeit with more realistic basic states, have relied on numerical solution. The instability can be interpreted as the result of the interaction of counterpropagating edge waves on the two opposing potential temperature gradients that bound the shear line. The analytic normal mode analysis can easily be extended to investigate nonmodal disturbances. The disturbances defined by maximizing the growth of selected norms over the norms over the normal mode e-folding time generally show similar growth rates to the normal modes. There is weaker scale selectivity and a shift to longer wave-lengths. The enstrophy norm provides an exception to this behavior. This norm is sensitive to small-scale structures and can grow much faster than the large-scale disturbance. Nonlinear integrations show the insta...