Transient Energy Growth Research Articles

Optimal disturbances in round submerged jets

In this paper, optimal growth analysis of spatial disturbances in round submerged jets is performed. Optimal energy growth is studied for various Reynolds numbers Re and frequencies ω. Different velocity profiles proposed by Michalke [“Survey on jet instability theory,” Prog. Aerosp. Sci. 21, 159–199 (1984)] are investigated by varying the shear layer momentum thickness δ, while the jet radius R is considered fixed (as characteristic length). In the case of stationary disturbances, there are no amplified eigenmodes, so eventually the energy of such disturbances decays downstream. Such disturbances are characterized by optimal energy growth as a function of streamwise coordinate z. At certain location downstream zmax this value reaches its maximum Gmax and then decays, although not monotonically, unlike the result of the temporary setting [Jimenez-Gonzalez et al., “Modal and non-modal evolution of perturbations for parallel round jets,” Phys. Fluids 27, 044105 (2015); Jimenez-Gonzalez and Brancher, “Transient energy growth of optimal streaks in parallel round jets,” Phys. Fluids 29, 114101 (2017)]. The well-known scaling Gmax∝Re2 and zmax∝Re is confirmed for stationary disturbances. Conversely, for nonstationary disturbances there is a range of frequencies in which an amplified mode is present. In this case, optimal perturbations far downstream consist only of the amplified mode. The main effect of non-modality here is “the energy pumping” of the amplified mode, so the disturbances are characterized by the “energy ratio” G/Em, i.e., the ratio of the energy of an arbitrary perturbation to the energy of the amplified mode. The influence of the shear layer momentum thickness on optimal disturbances is studied; increasing the shear layer momentum thickness of the base flow “smears” the area occupied by the optimal disturbance, although it does not dramatically affect energy gain values, slightly decreasing maximum energy as shear layer momentum thickness increases. Spatial oscillations in the energy of stationary optimal disturbances in jet flows are discovered. The spectrum structure examination shows that the frequency of these oscillations corresponds to the frequencies of the two least damped discrete eigenmodes.

Non-modal analysis of transient growth in a liquid annular jet surrounded by gas flow

Transient energy growth is a common mathematical concept in many fluid flow systems, and it has been widely investigated in recent years using non-modal analysis. Non-modal analysis can characterize the short-term energy amplification of perturbations, which is influenced by the Reynolds number, the Weber number, and the initial conditions such as the wavenumber. In gas–liquid coaxial nozzles, annular jets are often produced, and their breakup process is influenced by transient energy growth. However, research in this area has been limited so far. This paper for the first time investigates the transient energy growth of an annular liquid jet in static gas and validates it using a modified annular jet model. In the derivation process, the gas–liquid interfaces inside and outside the annular liquid film are taken into account. It has been found that there exists an optimal initial condition for a certain Reynolds number and a Weber number. The increase in the Reynolds number and ratio of inner and outer radius of the annular jet can maximize the transient growth under a specific initial wavenumber, while the increase in gas/liquid density ratio and the Weber number will minimize the transient growth. It is also found that transient energy growth is caused by the displacement of the free boundary.

Linear stability analysis of the viscoelastic Navier–Stokes–Voigt fluid model through Brinkman porous media: Modal and non-modal approaches

The linear stability analysis of a viscoelastic Navier-Stokes-Voigt fluid flow, or the Kelvin-Voigt fluid of zero order in a Brinkman porous medium, is investigated using both modal and non-modal analysis. The numerical solution is obtained using the Chebyshev collocation method. The combined effects of the medium's porosity, represented by the porous parameter, the fluid viscosity, represented by the ratio of effective viscosity to fluid viscosity, and the fluid elasticity, represented by the Kelvin-Voigt parameter are investigated using both modal and non-modal analysis. The modal analysis describes the long-term behavior of the system, obtained through plotting the eigenspectrum, eigenfunctions, growth rate curves, neutral stability curves, and streamline plots, along with accurate values of critical triplets. In non-modal analysis, the pseudospectrum of the Orr-Sommerfeld operator, transient energy growth curves, and regions of stability, instability, and potential instability are depicted. The results obtained from modal analysis indicate that the porous parameter, Kelvin-Voigt parameter, and the ratio of effective viscosity to fluid viscosity act as stabilizing agents. However, using non-modal analysis, it is observed that while the porous parameter and the ratio of effective viscosity to fluid viscosity act as stabilizing agents, the Kelvin-Voigt parameter acts as a destabilizing agent over shorter periods.

Nonmodal stability analysis of Poiseuille flow through a porous medium

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like O(Re2) for a fixed porous material, where Re is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.

Modal and non-modal stability for Hagen–Poiseuille flow with non-ideal fluid

Modal and non-modal stability analyses are applied to Hagen–Poiseuille flow with a non-ideal fluid. The non-ideal fluid is defined as a fluid close to its vapor–liquid critical point. In this region, properties of the fluid deviate significantly from the assumptions of the ideal gas model. In this paper, the specific example of CO2 near the critical point is taken as a non-ideal fluid. We studied fluids at supercritical pressure and different wall temperatures so that the centerline temperatures can be lower, equal, and higher than the pseudo-critical temperature. Flow instability is characterized by the Reynolds number, and the product of the Prandtl and Eckert numbers. In modal stability analysis, we observe that there is no unstable mode in Hagen–Poiseuille flow with a non-ideal fluid. Regarding the growth rate, as the axial wavenumber increases, another mode becomes the least stable. The non-modal theory is employed to investigate the optimal response to harmonic external force and transient energy growth. The influence of axial and azimuthal wave numbers, Prandtl and Eckert numbers, and thermodynamic states are also taken into account. In this study, we identify an generalized inflection point in the transcritical base profile, causing the transcritical state to be the most unstable. In non-modal instability, we observe that the optimal response mainly occurs at time invariant axisymmetric disturbance. This suggests that the axisymmetric disturbance could potentially initiate the transition to turbulence.

Stability patterns in plane porous Poiseuille flow with uniform vertical cross-flow: A dual approach

In this study, we examine the effects of a porous parameter and a uniform vertical cross-flow on the onset of instability in a plane Poiseuille flow (PPF) subjected to infinitesimal disturbances. The linearized Navier-Stokes (N–S) equations, which govern the stability of the fluid flow system, are addressed using the Chebyshev collocation method (CCM). Our approach includes both modal and non-modal analyses, each providing distinct insights into the system's behavior. The modal analysis involves a linear stability assessment using both non-normalized and normalized basic velocity approaches. Assuming a constant pressure gradient, the first approach aligns with the traditional Poiseuille flow framework, where a uniform pressure difference drives the flow. This simplifies calculations and aids in predicting flow characteristics. The second approach, which varies the pressure gradient to maintain a constant maximum streamwise velocity, offers a different perspective by focusing on a specific flow velocity profile through the porous medium. This variation in the pressure gradient allows us to explore changes in flow behavior and the system's responses under these conditions. This analysis includes examining the eigenspectrum, neutral stability curves, and determining critical triplets. On the non-modal front, we explore the intricacies of the non-normal Orr-Sommerfeld (O–S) operator. This involves studying the pseudospectrum and transient energy growth rate (G(t)) plots for optimal perturbations. Identification of regions of stability, potential instability, and instability is achieved using contour plots of maximum transient energy growth (Gmax). Interestingly, our results reveal that the porous parameter acts as a stabilizing agent, mitigating instability. In contrast, the uniform vertical cross-flow plays a dual role, contributing to both stabilization and destabilization of the system. Additionally, our study highlights the impact of examining basic velocity through both normalized and non-normalized approaches, leading to different stability predictions.

Transient energy growth in channel flow with compliant walls

Transient energy growth in channel flow with compliant walls

Control of linear instabilities in two-dimensional airfoil boundary layers

The present study deals with the control of linear, modal and non-modal growth of perturbations in a trapezoidal wing boundary layer. The asymmetric NACA 2515 airfoil is chosen for the analysis. The boundary layer velocity profiles, at different locations on the airfoil, are obtained using the Falkner–Skan equations. The non-modal linear stability theory is used to obtain the amplification of the initial perturbations, for which the state-space approach is adopted. The control is brought about using a Linear Quadratic Regulator (LQR) controller by the means of addition of fluid momentum in the wall-normal direction from the wing surface. Maximum transient energy growth is calculated across a range of wavenumber pairs, and control is applied to the pair with the highest energy growth. In this numerical analysis, we show that the transient growth is higher towards the trailing edge because of the adverse pressure gradient of the flow. Upon control actuation, the reduction in the transient energy growth of perturbations is achieved up to a maximum of 54.6%.

Outlet boundary condition and mean temperature gradient effects on the minimum acoustics disturbances energy in triggering nonlinear thermoacoustic instability

In this study, we theoretically investigate the impact of outlet boundary conditions and mean temperature gradients on the maximum transient growth rate of acoustical energy and the critical energy required for triggering. Our analysis encompasses open–open and open–closed thermoacoustic systems. The theoretical models developed focus on horizontal ducts with a mean temperature jump over the heat source, employing the modified King's law. By linearizing the unsteady heat release, the nonlinear thermoacoustic equations transform into linearized-delay ones. This approach enables us to predict optimal initial perturbations for linearized-delay and nonlinear systems, corresponding to maximum transient growth rates of acoustic energy over short and long periods, respectively, thus providing insights into critical energy for triggering. We find that a closed outlet leads to higher transient energy growth and a lower critical energy for triggering compared to an open outlet. The increased mean temperature gradient has a “destructive” impact on triggering in open–open systems but a “constructive” effect in open–closed systems. Raising the mean temperature ratio generally increases the critical energy for triggering in the open–open system, whereas it decreases the critical energy in the open–closed system. The critical energy for nonlinear optimal initial perturbations is notably affected by the minimum energy of critical unstable periodic solutions, while the critical energy for linearized-delay optimal initial perturbations is closely tied to the energy level of stable periodic solutions. Due to the transient energy growth rate, the critical energy for nonlinear optimal initial perturbations is significantly lower than that for linearized-delay optimal initial perturbations.

Transient energy growth analysis of flat-plate boundary layer with an oblique and non-uniform wall suction and injection

This paper presents a non-modal bi-global linear stability analysis of an incompressible flat-plate boundary layer under the effects of oblique and non-uniform wall suction and injection. The base flow velocity profile is two-dimensional and fully non-parallel. The flow is laminar at the inflow boundary, and no reverse flow occurs in the flow domain. The Chebyshev spectral collocation method has been used for the discretization of the governing stability equations. The discretized stability equations along with appropriate boundary conditions form an initial value problem (IVP). The transient energy growth (G(t)) and associated optimal perturbations are computed for different oblique angles (θ=30°,60°,90°,120°, and 150°), suction and injection intensities (I=0.5%, 1.5%, and 2.5% of U∞), Reynolds numbers (Re=195,284, and 411), and different spanwise wavenumbers (β=0−2) for uniform and non-uniform profiles of wall suction and injection. The behaviour of the boundary layer is found to be approximately similar but opposite in nature for θ=90° to 180° and θ=0° to 90° due to lower transpiration intensity. The G(t) decreases with the increase in θ from 0° to 90° for wall suction due to the damping of T–S modes. However, an opposite trend has been observed for injection due to the amplification of T–S modes. The uniform profile is found to be more effective than the non-uniform profiles of suction and injection in terms of their impact on the stability of the boundary layer. The 2D spatial structures of the optimal perturbations are damped and flow becomes modally stable as the spanwise wavenumber (β) is increased.

Transition control of the blasius boundary layer using linear robust control theory

The paper considers control system design for linearized three-dimensional perturbations about a nominal laminar boundary layer over a flat plate (the Blasius profile). The objective is prevention of the laminar to turbulent transition using appropriate inputs, outputs, and feedback controllers. They are synthesized with a view to reducing transient energy growth, a known precursor to important transition scenarios. The linearized Navier–Stokes equations are reduced to the Orr–Sommerfeld and Squire equations with wall-normal velocity actuation entering through the boundary conditions on the wall. The sensor output is taken to be the wall-normal derivative of the wall-normal vorticity measured on the plate. Several multivariable output controllers are examined, including simple constant gain output feedback, loop transfer recovery, and H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\\infty }$$\\end{document} loop shaping. Reduced order compensators are developed using balanced truncation and analyzed for robustness using the gap metric between reduced order models and full order models. It is demonstrated that the level of minimum transient energy growth that can be achieved is similar for these diverse controller methodologies but falls short of that which can be achieved using optimal state feedback.

Algebraic growth of 2D optimal perturbation of a plane Poiseuille flow in a Brinkman porous medium

The impact of various factors on the stability of a plane porous Poiseuille flow is investigated in detail. Both modal and non-modal linear stability analyses are used to study the effects of the porosity of the porous media and the ratio of effective viscosity to the fluid viscosity. The present analysis includes solving the linearized Navier-Stokes equation in the form of the Orr-Sommerfeld (O-S) type equation by applying the 2D perturbation to the basic mean flow. The Chebyshev collocation method is used to solve the Orr-Sommerfeld equation numerically. Through modal analysis, the accurate values of the critical triplets (αc,Rc,cc), the eigen-spectrum, the growth rate curves, and the marginal stability curves are studied. Then, by using non-modal analysis, the transient energy growth G(t) of two-dimensional optimal perturbations, the ε-pseudospectrum of the non-normal O-S operator (L), and the regions of stability, instability, and potential instability of the fluid flow system are investigated in detail. The collective results of both modal and non-modal analysis show that the porous parameter and the ratio of effective viscosity to the fluid viscosity have stabilizing effects on the fluid system due to the postponement of the onset of stability.

Optimal Transient Energy Growth of Two-Dimensional Perturbation in a Magnetohydrodynamic Plane Poiseuille Flow of Casson Fluid

Abstract The study investigates the influence of the Casson fluid parameter and the spanwise uniform magnetic field on the onset of instability against infinitesimal disturbances in an electrically conducting fluid flow between two parallel nonconducting rigid plates. The fourth-order linearized disturbance equation governing stability is solved using the spectral collocation method with Chebyshev-based polynomials. The aim is to analyze in detail the effect of the parameters involved in the problem using both modal and nonmodal linear stability analysis. The modal analysis provides accurate values of the critical Reynolds number, critical wave number, and critical wave speed, denoted as critical triplets (Rc, αc, cc). Additionally, it examines the eigen-spectrum, growth rate curves, and neutral stability curves. On the other hand, the nonmodal analysis investigates the transient energy growth G(t) of two-dimensional (2D) optimal perturbations, the pseudospectrum of the non-normal Orr–Sommerfeld (O–S) operator (ℒ), and the regions of stability, instability, and potential instability of the fluid flow system. The extensive examination of both long-term behavior through modal analysis and short-term behavior through nonmodal analysis reveals that the Hartmann number (Ha) acts as a stabilizing agent, delaying the onset of instability. Conversely, the Casson parameter (η) acts as a destabilizing agent, advancing the onset of instability. The results obtained here are verified to be in good agreement with the existing literature in the absence of the Casson fluid flow parameter.

Regeneration mechanisms of oblique and streamwise vortices in oblique transition

Oblique transition can be triggered by introducing componentwise optimal perturbations into a plane Couette flow. It is shown that in the middle stage of the transition, oblique vortices are generated again through transient spanwise velocity energy growth in the presence of developed streamwise streaks. In the following stage, streamwise independent kinetic energies of cross-streamwise velocities are intensified due to the nonlinear contribution caused by amplified oblique vortices together with the pressure transport, and the increased cross-stream velocity fluctuations can strengthen the lift-up effect. As a result, the streamwise vortices are formed again, and the streamwise streaks are even enhanced prior to their final breakdown to turbulence.

Non-normality and transient growth in stall flutter instability.

The non-normal nature and transient growth in amplitude and energy of a pitch-plunge aeroelastic system undergoing dynamic stall are explored in this paper through numerical and supporting experimental studies. Wind tunnel experiments, carried out for a canonical pitch-plunge aeroelastic system in a subsonic wind tunnel, show that the system undergoes stall flutter instability via a sub-critical Hopf bifurcation. The aeroelastic responses indicate a transient growth in amplitude and energy-possibly triggering the sub-criticality, which is critical from the purview of structural safety. The system also shows transient energy growth followed by decaying oscillation for certain initial conditions, whereas sustained limit cycle oscillations are encountered for other initial conditions at flow speeds lower than the critical speed. The triggering behavior observed in the wind tunnel experiments is understood better by resorting to study the numerical model of the nonlinear aeroelastic system. To that end, a modified semi-empirical Leishman-Beddoes dynamic stall model is adopted to represent the nonlinear aerodynamic loads of the pitch-plunge aeroelastic system. The underlying linear operator and its pseudospectral analysis indicate that the aeroelastic system is non-normal, causing amplification in amplitude and energy for a short period.

Effect of waveform on turbulence transition in pulsatile pipe flow

Pulsatile flow in a straight pipe is a model system for unsteady internal flows in industrial engineering and physiology. In some parameter regimes, the laminar flow is susceptible to helical perturbations, whose transient energy growth scales exponentially with the Reynolds number (Re). In this paper, we link the transient growth of these perturbations to the instantaneous linear instability of the laminar flow. We exploit this link to study the effect of the waveform on turbulence transition by performing linear stability and transient growth analyses of flows driven with different waveforms. We find a higher-energy growth in flows driven with longer low-velocity phases as well as with steeper deceleration and acceleration phases. Finally, we perform direct numerical simulations and show that cases with larger transient growth transition faster to turbulence and exhibit larger turbulence intensities. However, these same cases are also more prone to relaminarisation once turbulence has been established. This highlights that, in pulsatile flows, the linear mechanisms responsible for turbulence transition are distinctly different from the nonlinear mechanisms sustaining turbulence.

Nonmodal and modal analyses of a flow through inhomogeneous and anisotropic porous channel

A study of nonmodal and modal stability analyses of a three-dimensional Poiseuille flow through a channel is performed where the bounding walls are partially covered by inhomogeneous and anisotropic porous layers. The evolution equations corresponding to normal velocity and normal vorticity are derived for fluid and porous layers to decipher the transient disturbance energy growth. The modal stability analysis reveals that the anisotropy parameter has a stabilizing impact, while the coefficient of inhomogeneity shows a peculiar behaviour on the most unstable shear mode. The nonmodal stability analysis predicts that the transient disturbance energy growth exists and attenuates with the increase in the value of the anisotropy parameter and larger values of the coefficient of inhomogeneity; however, it intensifies for smaller values of the coefficient of inhomogeneity.

Subharmonic eigenvalue orbits in the spectrum of pulsating Poiseuille flow

Spectral degeneracies where eigenvalues and eigenvectors simultaneously coalesce, also known as exceptional points, are a natural consequence of the strong non-normality of the Orr–Sommerfeld operator describing the evolution of infinitesimal disturbances in parallel shear flows. While the resonances associated with these points give rise to algebraic growth, the development of non-modal stability theory exploiting specific perturbation structures with much larger potential for transient energy growth has led to waning interest in spectral degeneracies. The appearance of subharmonic eigenvalue orbits, recently discovered in the periodic spectrum of pulsating Poiseuille flow, can be traced back to the coalescence of eigenvalues at exceptional points. We present a thorough analysis of the spectral properties of the linear operator to identify exceptional points and accurately map the prevalence of subharmonic eigenvalue orbits for a large range of pulsation amplitudes and frequencies. This information is then combined with solutions of the linear initial value problem to analyse the impact of the appearance of these orbits on the temporal evolution of linear disturbances in pulsating Poiseuille flow. The periodic amplification phases are shown to be heralded by repeated non-normal growth bursts that are intensified by the formation of subharmonic orbits involving the leading eigenvalues. These bursts are associated with the change of alignment of the perturbation from the decaying towards the amplified branch of the subharmonic eigenvalue orbits in a so-called branch transition process.

Feedback control of transitional shear flows: sensor selection for performance recovery

The choice and placement of sensors and actuators is an essential factor determining the performance that can be realized using feedback control. This determination is especially important, but difficult, in the context of controlling transitional flows. The highly non-normal nature of the linearized Navier-Stokes equations makes the flow sensitive to small perturbations, with potentially drastic performance consequences on closed-loop flow control performance. Full-information controllers, such as the linear quadratic regulator (LQR), have demonstrated some success in reducing transient energy growth and suppressing transition; however, sensor-based output feedback controllers with comparable performance have been difficult to realize. In this study, we propose two methods for sensor selection that enable sensor-based output feedback controllers to recover full-information control performance: one based on a sparse controller synthesis approach, and one based on a balanced truncation procedure for model reduction. Both approaches are investigated within linear and nonlinear simulations of a sub-critical channel flow with blowing and suction actuation at the walls. We find that sensor configurations identified by both approaches allow sensor-based static output feedback LQR controllers to recover full-information LQR control performance, both in reducing transient energy growth and suppressing transition. Further, our results indicate that both the sensor selection methods and the resulting controllers exhibit robustness to Reynolds number variations.

Global Stability of Fluid Flows Despite Transient Growth of Energy.

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct nonquadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the the energy stability threshold found by Orr in 1907 [The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid, Proc. R. Ir. Acad. Sect. A 27, 69 (1907)]. This is the first global stability result for any flow that surpasses the energy method.