Maximum Transient Energy Growth Research Articles

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.

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%.

Optimal control of growth of instabilities in Taylor–Couette flow

The present work aims to achieve optimal control of instabilities in a standard Taylor–Couette flow. The motivation of the present study is to reduce the disturbance growth and delay the transition process to turbulence. We numerically employ control using a stability modifier, namely, wall transpiration. In the non-modal stability framework, we form a state-space model employing control actuation by means of periodic suction/blowing of fluid from the walls. The study is conducted for two cases of flow rotations: (i) counter-rotating cylinders and (ii) the stationary outer cylinder with inner cylinder rotating. The parametric study was performed with varying radii ratios, Reynolds numbers (Re), axial (α), and azimuthal (n) wavenumbers. The time evolution of governing equation is written in terms of perturbation velocities in radial (r) and azimuthal (θ) directions. The optimal feedback control is obtained using a linear quadratic regulator controller and feed backed to the system to reduce the maximum optimal growth of the instabilities in the flow. The perturbation kinetic energy is taken as the measure of the amplification of disturbances and used as the cost function to be minimized. We use Chebyshev spectral collocation method to discretize the equations and variational method to calculate the optimal growth. We studied four different parametric cases of radii ratios (η=r1r2= 0.1, 0.25, 0.5, 0.75), with angular velocity (Ω2Ω1) ratio fixed as μ=−1 and μ = 0. We choose the subcritical wavenumbers that led to a maximum transient energy growth corresponding to a Reynolds number ≈ 0.65 times the critical Reynolds number for the case of counter-rotation. For the case of the stationary outer cylinder, we showed the effect of the control in the modal analysis framework. The presented control technique resulted in a maximum of 72% reduction in the growth rate and the typical growth of perturbation energy.

Nonlinear stability analysis of transitional flows using quadratic constraints

The dynamics of transitional flows are governed by an interplay between the non-normal linear dynamics and quadratic nonlinearity in the incompressible Navier-Stokes equations. In this work, we propose a framework for nonlinear stability analysis that exploits the fact that nonlinear flow interactions are constrained by the physics encoded in the nonlinearity. In particular, we show that nonlinear stability analysis problems can be posed as convex feasibility and optimization problems based on Lyapunov matrix inequalities, and a set of quadratic constraints that represent the nonlinear flow physics. The proposed framework can be used to conduct global stability, local stability, and transient energy growth analysis. The approach is demonstrated on the low-dimensional Waleffe-Kim-Hamilton model of transition and sustained turbulence. Our analysis correctly determines the critical Reynolds number for global instability. For local stability analysis, we show that the framework can estimate the size of the region of attraction as well as the amplitude of the largest permissible perturbation such that all trajectories converge back to the equilibrium point. Additionally, we show that the framework can predict bounds on the maximum transient energy growth. Finally, we show that careful analysis of the multipliers used to enforce the quadratic constraints can be used to extract dominant nonlinear flow interactions that drive the dynamics and associated instabilities.

Control-Oriented Model Reduction for Minimizing Transient Energy Growth in Shear Flows

A linear nonmodal mechanism for transient amplification of perturbation energy is known to trigger subcritical transition to turbulence in many shear flows. Feedback control strategies for minimizing this transient energy growth can be formulated as convex optimization problems based on linear matrix inequalities. Unfortunately, solving the requisite linear matrix inequality problem can be computationally prohibitive within the context of high-dimensional fluid flows. This paper investigates the utility of control-oriented reduced-order models to facilitate the design of feedback flow control strategies that minimize the maximum transient energy growth. An output projection onto proper orthogonal decomposition modes is used to faithfully capture the system energy. Subsequently, a balanced truncation is performed to reduce the state dimension, while preserving the system’s input-output properties. The model reduction and control approaches are studied within the context of a linearized channel flow with blowing and suction actuation at the walls. Controller synthesis for this linearized channel flow system becomes tractable through the use of the proposed control-oriented reduced-order models. Further, the resulting controllers are found to reduce the maximum transient energy growth compared with more conventional linear quadratic optimal control strategies.

Unity maximum transient energy growth of heat-driven acoustic oscillations

Transient energy growth of acoustic disturbances may trigger thermoacoustic instability in a non-normal thermoacoustic system. In this work, minimizing transient energy growth of heat-driven acoustic oscillations in an open-ended thermoacoustic system is considered. For this, a state-space thermoacoustic model with an acoustically compact heat source and distributed monopole-like actuators is developed. The heat source gives rise to the mean temperature jump, as experimentally measured. It is modeled with a modified King’s Law. Coupling the unsteady heat release model with a Galerkin series expansion of the acoustic waves present enables the time evolution of flow disturbances and acoustical energy to be calculated, thus providing a platform on which to gain insight on the system’s transient stability behaviors and the non-normal response of the system to the dynamic actuators. It is first shown that implementing a linear-quadratic regulator (LQR) leads to the system being asymptotically stabilized. However, the LQR optimization strategy fails in eliminating the transient growth. This finding is consistent with Pseudospectra analysis of the present system. In order to achieve unity maximum transient growth, a Lyapunov-based optimization strategy is systematically designed. It is found that this optimization strategy achieves both exponential decay of the acoustical energy and unity maximum transient growth. Furthermore, the sound pressure level is reduced by approximately 25dB. In addition, the number of the actuators K is shown to be related to the mode number N as K=N.

Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

Transient energy growth of acoustic disturbances in triggering self-sustained thermoacoustic oscillations

Thermoacoustic instability occurs in many modern combustion systems. It most often arises due to the coupling between unsteady heat release and acoustic waves. Transient energy growth of acoustic disturbances could trigger thermoacoustic instability in a non-normal combustion system. In this work, transient energy growth analysis of a modelled choked combustor with a gutter confined is conducted. The non-normal interaction between acoustic disturbances and the anchored V-shaped flame is studied first. The thermoacoustic system is shown to be non-normal and characterized by non-orthogonal eigenmodes. Transient energy growth analysis is then performed to gain insights on its finite-time stability behaviour, which cannot be predicted by classical linear theory. To characterize the non-normality, two different energy measures are defined and estimated. One involves with acoustic travelling waves. The other is concerned with not only the travelling waves but the monopole-like flame. Comparison is then made between the two measures. It is found that the maximum transient energy growth of combustion-excited oscillations is about 102−104 times greater than that of acoustic disturbances. Furthermore, the ‘critical’ time taken to reach the maximum transient growth rate is about half of period of the fundamental mode, which is about 90% shorter than that when only acoustic disturbances are considered. In addition, the most ‘dangerous’ location at which the flame is more susceptible to thermoacoustic instability is estimated. Finally, experiments are conducted on an open-ended thermoacoustic system. It is found that transient growth of flow disturbances can trigger nonlinear limit cycle oscillations.

Linear Stability Analysis and Transient Growth in Steady and Pulsatile Flows in a Constricted Channel

A three-dimensional direct stability analysis of the flow in a straight channel with a smooth constriction with 50% occlusion is performed. Two-dimensional steady and long-period pulsatile flows are investigated with respect to instabilities and transition to turbulence. The steady base flow is asymmetric for the Reynolds number range considered in this work, with critical value for linear instability of Re = 558.7 to three-dimensional perturbations. Although the steady flow is linearly stable for two-dimensional perturbations at Re = 700, it presents a maximum transient energy growth of the order of 107. The pulsatile flow becomes linearly unstable at Re = 249.7 to three-dimensional perturbations. The optimal initial condition that triggers the maximum growth of a two-dimensional perturbation occurs for a disturbance applied shortly after the peak velocity of the pulse for Re = 200 with a growth of the order of 103, the same order found for Re = 400, but applying the perturbation in the beginning of the pulse.

Minimizing transient energy growth of nonlinear thermoacoustic oscillations

Abstract Thermoacoustic oscillations triggered by transient energy growth of flow disturbances are wanted in thermoacoustic prime movers or cooling systems. However, they are undesirable in many combustion systems, such as aero-engines, gas turbines, boilers and furnaces. In this work, minimizing transient energy growth of flow disturbances in a thermoacoustic system with Dirichlet boundary conditions is considered. For this, a simplified thermoacoustic model of a premixed laminar flame with an actuator is developed. It is formulated in state-space by expanding acoustic disturbances via Galerkin series and linearizing flame model and recasting it into the classical time-lag N - τ for controllers implementation. As a linear-quadratic-regulator (LQR) is implemented, the thermoacoustic oscillations exponentially decay. However, it is associated with transient energy growth of flow disturbances. To minimize the transient energy growth, a strict dissipativity controller designed basing on LQR is then implemented. Comparison is then made between the performances of these controllers. It is found that the strict dissipativity controller achieves both exponential decay of the thermoacoustic oscillations and unity maximum transient energy growth.

Effect of heat source on transient energy growth analysis of a thermoacoustic system

Transient growth of disturbances in a non-normal combustion system could trigger thermoacoustic instability. In this work, the effect of heat source on triggering thermoacoustic instability in a system with Dirichlet or Neumann boundary conditions is investigated. For this a thermoacoustic model of a premixed laminar flame is developed. It is formulated in state-space by expanding flow disturbances via Galerkin series and coupling with a flame model, thus providing a platform on which to gain insight on the system stability behaviors. Transient energy growth analysis is then performed by discretizing the flame model into distributed monopole-like sound sources. It is shown that the thermoacoustic system is non-normal and associated with transient energy growth. Parametric studies are then conducted to study the effects of (1) the flame-acoustics interaction index I, (2) the eigenmode number N, (3) the mean temperature ratio T between pre- and after-combustion regions on the system stability behavior and non-normality. It is found that the maximum transient energy growth Gmax is increased with increased I. Furthermore, the thermoacoustic system with more eigenmodes is found to be associated with a larger Gmax.

Assessing non-normal effects in thermoacoustic systems with mean flow

In this paper, non-normal interactions in a thermoacoustic system are studied, using a low-order expansion of the state variables in terms of eigenmodes. The thermoacoustic eigenmodes are determined as solutions of the Helmholtz equation or the linearized Euler equations, respectively, in the presence of a time-lagged heat source. Subsequently, non-normal effects are evaluated in a post-processing analysis based on the computed eigenmodes. In the case where the eigenmode analysis is based on the linearized Euler equations, effects of a non-zero mean flow velocity can be taken into account. The energy associated with the eigenmodes may then contain contributions of convected entropy and vorticity modes as well as the acoustic field. The notion of transient growth of perturbation energy is thus extended from an expression based on the classical acoustic energy density to a form based on a generalized disturbance energy. The expansion in terms of eigenmodes is computationally efficient, making the approach potentially applicable to complex, 3D configurations including non-trivial boundary conditions and spatio-temporal distributions of heat release fluctuations. In the present paper, the method is applied to a 1D configuration that consists of a duct including a 1D heat source, followed by a choked isentropic nozzle. It is shown that for such a case, it is essential to include the contribution of entropy perturbations in the calculation of the optimal initial perturbation and the maximum transient energy growth. Subsequently, the impact of increasing mean flow Mach number and increasing strength of flame/acoustic interaction on non normal effects is assessed in a parameter study.

Computing the maximum transient energy growth

The calculation of maximum transient energy growth is a problem of interest in several areas of science and engineering. An algorithm that guarantees the calculation of this measure to an arbitrary accuracy in a finite number of steps is proposed for finite-dimensional linear-time-invariant dynamical systems. The algorithm is illustrated with a numerical example.

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.

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.

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.

A note on the stability of slip channel flows

We consider the influence of slip boundary conditions on the modal and nonmodal stability of pressure-driven channel flows. In accordance with previous results by Gersting [“Hydrodynamic stability of plane porous slip flow,” Phys. Fluids 17, 2126 (1974)] but in contradiction with the recent investigation of Chu [“Instability of Navier slip flow of liquids,” C. R. Mec. 332, 895 (2004)], we show that the slip increases significantly the value of the critical Reynolds number for linear instability. The nonmodal stability analysis, however, reveals that slip has a very weak influence on the maximum transient energy growth of perturbations at subcritical Reynolds numbers. Slip boundary conditions are therefore not likely to have a significant effect on the transition to turbulence in channel flows.