Transient Energy Growth Research Articles

By-pass transition to airfoil flutter by transient growth due to gust impulse

We present an experimental study which shows that the mechanism known as transient growth of energy, can cause flutter instability of a nonlinearly flexible airfoil at a wind velocity below the linear critical flutter velocity. A flap mounted upstream a flexible airfoil in a wind tunnel generates a single gust which triggers the plunge and pitch oscillations. This gust is characterized using two-component hot-wire anemometry. For the first time experimental evidence is provided to confirm the theoretical scenario of a by-pass transition to flutter by transient growth. From an engineering point of view, transient growth might explain also the premature structural fatigue encountered in structures subject to wind.

Three-dimensional instabilities and transient growth of a counter-rotating vortex pair

This paper investigates the three-dimensional instabilities and the transient growth of perturbations on a counter-rotating vortex pair. The two dimensional base flow is obtained by a direct numerical simulation initialized by two Lamb–Oseen vortices that quickly adjust to a flow with elliptic vortices. In the present study, the Reynolds number, ReΓ=Γ/ν, with Γ the circulation of one vortex and ν the kinematic viscosity, is taken large enough for the quasi steady assumption to be valid. Both the direct linearized Navier–Stokes equation and its adjoint are solved numerically and used to investigate transient and long time dynamics. The transient dynamics is led by different regions of the flow, depending on the optimal time considered. At very short times compared to the advection time of the dipole, the dynamics is concentrated on the points of maximal strain of the base flow, located at the periphery of the vortex core. At intermediate times, depending on the symmetry of the perturbation, one of the hyperbolic stagnation points provides the optimal amplification by stretching of the perturbation vorticity as in the classical hyperbolic instability. The growth of both short time and intermediate time transient perturbations are non- or weakly dependent of the axial wavenumber whereas the long time behavior strongly selects narrow bands of wavenumbers. We show that, for all unstable spanwise wavenumbers, the transient dynamics last until the nondimensional time t=2, during which the dipole has traveled twice the separation distance between vortices b. During that time, all the wavenumbers exhibit a transient growth of energy by a factor of 50, for the Reynolds number ReΓ=2000. For time larger than t=2, energy starts growing at a rate given by the standard temporal stability theory. For all wavenumbers and two Reynolds numbers, ReΓ=2000 and ReΓ=105, different instability branches have been computed using a high resolution Krylov method. At large Reynolds number, the computed Crow and elliptic instability branches are in excellent agreement with the inviscid theory [S. C. Crow, AIAA J. 8, 2172 (1970); S. Le Dizes and F. Laporte, J. Fluid Mech. 471, 120 (2002)] and numerical analysis [D. Sipp and L. Jacquin, Phys. Fluids 15, 1861 (2003)]. A novel oscillatory elliptic instability involving Kelvin waves with azimuthal wavenumbers m=0 and |m|=2, that was missed in previous numerical analysis [D. Sipp and L. Jacquin, Phys. Fluids 15, 1861 (2003)] is found. For the stationary elliptic instability, we show that viscous effect may be estimated using the large Reynolds number direct and adjoint eigenmodes. This asymptotically exact estimate of the viscous damping of elliptic instability mode agrees with our direct numerical computation of instability branches at moderate Reynolds number and demonstrates that formula proposed by Le Dizes and Laporte [J. Fluid Mech. 471, 120 (2002)] strongly over estimated the viscous correction.

Global transient dynamics of three-dimensional hydrodynamical disturbances in a thin viscous accretion disk

Thin viscous Keplerian accretion disks are considered asymptotically stable, even though they can show significant dynamic activity on short time scales. In this paper the dynamics of nonaxisymmetric hydrodynamical disturbances of disks are investigated analytically, building upon the steady state three-dimensional structure and evolution of axisymmetric perturbations explored in previous work. Assuming a polytropic equation of state, solutions are found by means of an asymptotic expansion in the small parameter measuring the ratio of the disk thickness to characteristic radius. In-depth analysis shows that every perturbation that disturbs the radial velocity induces significant transient growth in the (acoustic) energy of the evolving disturbance. This effect is most evident in the density and vertical velocity. The transient growth observed is tied to the nonseparable nature of the solutions where, in particular, pattern evolution is controlled by a similarity variable composed of the radial coordinate and time. This leads to growing winding perturbations that display successive radial peaks and troughs. We argue that these transient nonaxisymmetric structures may precipitate secondary instabilities which, consequently, may be a critical element for a new alternative picture of turbulence arousal in nonmagnetized astrophysical disks.

Design of poiseuille flow controllers using the method of inequalities

This paper investigates the use of the method of inequalities (MoI) to design output-feedback compensators for the problem of the control of instabilities in a laminar plane Poiseuille flow. In common with many flows, the dynamics of streamwise vortices in plane Poiseuille flow are very non-normal. Consequently, small perturbations grow rapidly with a large transient that may trigger nonlinearities and lead to turbulence even though such perturbations would, in a linear flow model, eventually decay. Such a system can be described as a conditionally linear system. The sensitivity is measured using the maximum transient energy growth, which is widely used in the fluid dynamics community. The paper considers two approaches. In the first approach, the MoI is used to design low-order proportional and proportional-integral (PI) controllers. In the second one, the MoI is combined with McFarlane and Glover’s H ∞ loop-shaping design procedure in a mixed-optimization approach.

Temporal behaviour of global perturbations in compressible axisymmetric flows with free boundaries

AbstractThe dynamics of small global perturbations in the form of a linear combination of a finite number of non‐axisymmetric eigenmodes is studied in the two‐dimensional approximation. The background flow is assumed to be an axisymmetric perfect fluid with adiabatic index γ = 5/3 rotating with a power law angular velocity distribution Γ ∝ r–q , 1.5 < q < 2.0, confined by free boundaries in the radial direction. The substantial transient growth of acoustic energy of optimized perturbations is discovered. An optimal energy growth G is calculated numerically for a variety of parameters. Its value depends essentially on the perturbation azimuthal wavenumber m and increases for higher values of m. The closer the rotation profile to the Keplerian law, the larger growth factors can be obtained but over a longer time. The highest acoustic energy increase found numerically is of order ∼102 over ∼6 typical Keplerian periods. Slow neutral eigenmodes with corotation radius beyond the outer boundary mostly contribute to the transient growth. The revealed linear temporal behaviour of perturbations may play an important role in angular momentum transfer in toroidal flows near compact relativistic objects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

Characterizing energy growth during combustion instabilities: Singularvalues or eigenvalues?

Non-normality in thermoacoustic systems has received attention recently. It has been shown that in a non-normal but classically linearly stable system, there can be significant transient energy growth of small perturbations before their eventual decay. This growth occurs in the absence of non-linear effects. This phenomenon can be explained by the non-normality of the governing linear operator or the non-orthogonality of the eigenvectors of the system. In this paper, we study various aspects of this transient energy growth for a general combustor, with localized heat release approximated by the popular n– τ model. Galerkin technique is used to simplify the governing acoustic equations in a duct in the presence of a localized heat source with appropriate boundary conditions. Singularvalue decomposition (SVD) is used to compute the transient energy growth. SVD is used as a tool to obtain the maximum possible energy amplification and the optimal initial conditions required for this amplification. The necessary and sufficient conditions for no energy growth are discussed. A parametric analysis is performed to highlight the effect of system parameters on the maximum transient growth rate and to obtain regions of stability of the thermoacoustic system.

Minimizing transient energy growth in plane Poiseuille flow

The feedback control of laminar plane Poiseuille flow is considered. In common with many flows, the dynamics of plane Poiseuille flow is very non-normal. Consequently, small perturbations grow rapidly with a large transient that may trigger non-linearities and lead to turbulence, even though such perturbations would, in a linear flow, eventually decay. This sensitivity can be measured using the maximum transient energy growth. The linearized flow equations are discretized using spectral methods and then considered at one wave-number pair in order to obtain a model of the flow dynamics in a form suitable for advanced control design. State feedback controllers that minimize an upper bound on the maximum transient energy growth are obtained by the repeated solution of a set of linear matrix inequalities. The controllers are tested using a full Navier—Stokes solver, and the transient energy response magnitudes are significantly reduced compared with the uncontrolled case.

Modal versus energy stability analysis of kinematic dynamos in cylindrical configurations

The kinematic dynamo problem is solved in a cylindrical geometry using Galerkin expansions of the magnetic field components. The difference with the modal Galerkin analysis [L. Marié et al., Phys. Fluids 18, 017102 (2006)] concerns the weighting functions which here belong to the same set as the trial functions. The new procedure allows to determine the magnetic Reynolds number RmE for energy growth. Lower bounds on the value of RmE are derived for magnetic modes of azimuthal wavenumber m. Using a variational principle, more accurate values of RmE are obtained in the case of helical flows. It is found that the threshold value for the axisymmetric magnetic mode m=0 is slightly higher than its value for the antisymmetric mode m=1. Although excluded by Cowling’s theorem the mode m=0 exhibits transient energy growth and could play a role in the nonlinear equilibration of cylindrical dynamos.

Convective instability and transient growth in flow over a backward-facing step

Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components.

Linear stability, transient energy growth, and the role of viscosity stratification in compressible plane Couette flow

Linear stability and the nonmodal transient energy growth in compressible plane Couette flow are investigated for two prototype mean flows: (a) the uniform shear flow with constant viscosity, and (b) the nonuniform shear flow with stratified viscosity. Both mean flows are linearly unstable for a range of supersonic Mach numbers (M). For a given M , the critical Reynolds number (Re) is significantly smaller for the uniform shear flow than its nonuniform shear counterpart; for a given Re, the dominant instability (over all streamwise wave numbers, alpha ) of each mean flow belongs to different modes for a range of supersonic M . An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean flow to perturbations. It is shown that the energy transfer from mean flow occurs close to the moving top wall for "mode I" instability, whereas it occurs in the bulk of the flow domain for "mode II." For the nonmodal transient growth analysis, it is shown that the maximum temporal amplification of perturbation energy, G(max), and the corresponding time scale are significantly larger for the uniform shear case compared to those for its nonuniform counterpart. For alpha=0 , the linear stability operator can be partitioned into L ~ L+Re(2) L(p), and the Re-dependent operator L(p) is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: G(t/Re) ~ Re(2). In contrast, the dominance of L(p) is responsible for the invalidity of this scaling law in nonuniform shear flow. An inviscid reduced model, based on Ellingsen-Palm-type solution, has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and nonmodal instability, it is shown that the viscosity stratification of the underlying mean flow would lead to a delayed transition in compressible Couette flow.

Linear quadratic control of plane Poiseuille flow–the transient behaviour

This paper describes the design of optimal linear quadratic controllers for single wavenumber-pair periodic 2-D disturbances in plane Poiseuille flow, and subsequent verification using a finite-volume full Navier–Stokes solver, at both linear and non-linear levels of initial conditions selected to produce the largest linear transient energy growth. For linear magnitude initial conditions, open and closed-loop finite-volume solver results agree well with a linear simulation. Transient energy growth is an important performance measure in fluid flow problems. The controllers reduce the transient energy growth, and the non-linear effects are generally seen to keep energy levels below the scaled linear values, although they do cause instability in one simulation. Comparatively large local quantities of transpiration fluid are required. The modes responsible for the transient energy growth are identified. Modes are shown not to become significantly more orthogonal by the application of control. The synthesis of state estimators is shown to require higher levels of discretiation than the synthesis of state-feedback controllers. A simple tuning of the estimator weights is presented with improved convergence over uniform weights from zero initial estimates. Nomenclature Greek symbols α streamwise (x) wave number, cycles per 2π distance β spanwise (z) wave number, cycles per 2π distance ε(t) synchronic transient energy bound at time t synchronic error energy bound at time t ζ eigenvalue in synchronic transient energy bound eigensystem η(x,y,z,t) wall-normal vorticity perturbation η Fourier coefficient at wavenumber pair α,β θ diachronic transient energy bound θ Error diachronic error energy bound θ Est estimated energy bound ith eigenvalue diagonal eigenvalue matrix μ dynamic viscosity ρ fluid density ith singular value of A (A) spectral norm or largest singular value of A modal amplitude vector, χ0 initial χ, at time t = 0 Ψ matrix of right eigenvectors ψ i ith right eigenvector frequency Roman symbols system matrix input matrix output matrix multiplying co-efficient for nth Chebyshev polynomial c i amplitude of mode i E(t) transient energy, , at time t E Est(t) estimated transient energy, , at time t E Error (t) error energy , at time t E 0 E of worst open-loop perturbation of , at t = 0 (2.26 × 10− 9) E pair,bound upper bound on mode pair energy growth h channel wall separation identity matrix j state feedback gain matrix estimator gain matrix N highest Chebyshev polynomial degree used, final collocation point index P pressure P b steady base flow pressure p pressure perturbation state variable weighting (energy) matrix control weighting matrix R Reynolds number r control weight multiplier s measurement noise weight multiplier invertible matrix for conversion between state variables and , excludes next-to-wall velocities and wall vorticities t time x,y,z streamwise, wall-normal and spanwise co-ordinates flow velocity vector steady base flow velocity U cl Ub at centreline velocity perturbation vector u,v,w Fourier coefficients at wavenumber pair α,β control vector measurement noise power spectral density process noise power spectral density state variable vector state estimates vector estimate error vector, which generates θ which generates θ Error transformed to values at collocation points state variables transformed to , thus measurement vector y n y at nth Chebyshev–Gauss–Lobatto collocation point

Disturbance growth in two-fluid channel flow: The role of capillarity

This work examines the role of capillarity in the non-modal linear stability properties of three-dimensional disturbances in sheared two-layer flow of immiscible fluids of similar density. Capillarity reduces the transient growth of energy that occurs due to the non-normality of the linear stability problem according to a scaling of peak energy with We 1/2 over a wide range of Weber number, viscosity ratio and wavenumber. More importantly, the participation of capillary modes in non-modal growth leads to oscillatory energy growth and to larger disturbance growth rates, features that are confirmed by computing the numerical range and numerical abscissa of the non-normal disturbance evolution operator. Examination of energy components and disturbance structure reveals that early rapid growth and subsequent oscillations are due to the coupling of streamwise vortices – the two-fluid analog of lift-up – to the displaced interface.

On the Minimization of Maximum Transient Energy Growth

The problem of minimizing the maximum transient energy growth is considered. This problem has importance in some fluid flow control problems and other classes of nonlinear systems. Conditions for the existence of static controllers that ensure strict dissipativity of the transient energy are established and an explicit parametrization of all such controllers is provided. It also is shown that by means of a Q-parametrization, the problem of minimizing the maximum transient energy growth can be posed as a convex optimization problem that can be solved by means of a Ritz approximation of the free parameter. By considering the transient energy growth at an appropriate sequence of discrete time points, the minimal maximum transient energy growth problem can be posed as a semidefinite program. The theoretical developments are demonstrated on a numerical example.

Optimal disturbances in swept Hiemenz flow

The initial perturbation with the largest transient energy growth is computed in the context of the swept leading-edge boundary layer. The highest energy amplification is found for perturbations which are homogeneous in the spanwise z-direction, although on shorter time scales the most amplified disturbances have a finite spanwise wavenumber. In both cases the production term associated with the shear of the spanwise velocity is responsible for the energy amplification in the perturbation energy equation. A connection is made with the amplification mechanism exhibited by optimal perturbations in streaky boundary layers (Hoepffner et al. J. Fluid Mech. vol. 537, 2005, p.91) and the results are compared to the optimal Görtler–Hämmerlin disturbances computed by Guégan et al. (J. Fluid Mech. vol. 566, 2006, p. 11).

Transient growth and rotor–stator interaction noise in mean swirling duct flow

The phenomenon of spatial transient growth in swirling duct flow with application to aeroengine duct acoustics is investigated. In a typical aeroengine a region of swirling flow exists between the rotor and the downstream stator. The possibility for optimal transient growth of disturbance energy, and of a norm related to the acoustic power radiated back upstream from the stator is shown. Significant transient growth of both energy and noise norms can be observed over length scales relevant to aeroengine size, and over a wide range of parameter values such as the degree of mean swirl, azimuthal wavenumber and frequency. The optimal disturbances are found to have significant three dimensionality. The growth in energy results from increases in the azimuthal and radial components of the disturbance velocity field as it propagates downstream, whilst increases in the azimuthal velocity bring about non-modal growth of the noise measure.

Transient magnetic energy growth in spherical stationary flows

The non-normality associated with the induction equation may lead to subcritical growth of magnetic field structures even if all linear eigenmodes decay. We compute the magnitude of these transient effects for a collection of predominantly axisymmetric stationary spherical flows and find without exception the dominance of axisymmetric field growth above all other symmetries. The transient growth is robust under small flow perturbations and can be understood by simple physical mechanisms: either field line shearing or stretching. Magnetic energy amplification of is possible at magnetic Reynolds numbers of , and such effects could therefore lead the system from a principally non-magnetic state into one where the magnetic field plays a significant role in the dynamics.

Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

Nonmodal transient growth studies and an estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The steady mean flow is characterized by a nonuniform shear rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The maximum amplification of perturbation energy over time, Gmax, is found to increase with increasing Reynolds number Re, but decreases with increasing Mach number M. More specifically, the optimal energy amplification Gopt (the supremum of Gmax over both the streamwise and spanwise wave numbers) is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wave number, αopt, is nonzero at M=0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G(t∕Re)∼Re2, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with an increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow (Ė1) in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of Ė1 for the present mean flow.

Optimal Growth of Antarctic Circumpolar Waves

Abstract Generalized stability theory is applied to a simple dynamical model of interannual ocean–atmosphere variability in the southern midlatitudes to determine the perturbations that create the most rapid growth of energy in the system. The model is composed of a barotropic quasigeostrophic atmosphere coupled to a 1.5-layer quasigeostrophic ocean, each linearized about a zonally invariant mean state, and with atmospheric and ocean surface temperature obeying a simple heat balance. Eigenanalysis of the system reveals modes of interannual variability that resemble the so-called Antarctic Circumpolar Wave (ACW), consistent with an earlier analytical study of the system. The optimal excitation of these modes relative to an energy norm is found to be a perturbation almost entirely restricted to the ocean momentum field and is shown to resemble strongly the optimal perturbations in energy for the system. Over interannual time scales most rapid growth is seen in zonal wavenumbers 4–6, despite the fact that the least-damped eigenmodes of the system are of a lower zonal wavenumber. The rapid transient growth in energy occurs by extracting perturbation energy from the mean state through advection of the mean meridional oceanic temperature gradient. This transient growth of high-zonal-wavenumber modes dominates the model’s variability when it is forced by noise that is white in space or time. A dominant low-zonal-wavenumber response, consistent with the observed and modeled ACW, occurs only when the forcing is red in space or time, with decorrelation scales greater than 3 yr or 10 000 km. It is concluded that, if the ACW is a coupled mode analogous to that supported in this simple model, then it is excited by other large-scale phenomena such as ENSO rather than by sources of higher-frequency forcing.

Experimental evidence of transient growth of energy before airfoil flutter

Abstract This paper presents experimental evidence of the transient growth of energy for the coupled-mode flutter of an airfoil. The phenomenon occurs even in linearly stable dynamical systems. Its application is new in the context of fluid–structure interactions where only theoretical and numerical studies of transient growth exist. The experimental set-up allows an NACA 0015 airfoil to oscillate in rotational and vertical degrees of freedom when it is subjected to air-flow. Measurements consist of time series of the two motions obtained by laser displacement sensors. Structural parameters are first estimated without air-flow. The transient evolution of energy is measured, and amplification is observed for a given set of initial conditions. Our experiments agree well with numerical simulations based on unsteady airfoil theory.