Non-modal Stability Theory Research Articles

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%.

Non-modal stability analysis of magneto-hydrodynamic flow in a single pipe

A complete understanding of the stability of fluid flows under varying magnetic field profiles is imperative for achieving control of plasma and operating fluids in the blankets of future fusion reactors. In this context, the primary objective of this study is to investigate the influence of varying magnetic profiles on the flow regime of a generic fluid, which is representative of both thermonuclear plasma and conductive fluids within a nuclear fusion reactor. To this aim in this work non-modal stability theory is adopted to perform stability analysis of a magneto-hydrodynamic (MHD) flow in an infinite circular pipe in order to study the effects of the magnetic field on the fluid dynamics of the pipe flow. In particular, the effects on the general stability of two magnetic field profiles are compared with the reference case of a pipe Poiseuille flow without magnetic field. Firstly, the classic modal stability technique is employed to study asymptotical stability. Then, non-modal stability analysis is applied to magneto-hydrodynamic pipe flow to study the system's response for a finite time immediately after a perturbation. Fourier–Chebyshev Petrov–Galerkin spectral method is used to compute the eigenvalues and pseudospectra of the weak formulation associated with the linearised system. Investigations on the dependence of spectra and transient growths on the specific magnetic profiles are conducted for different values of perturbation wave numbers. The obtained results show that in general the magnetic field has an effect of stabilization on the system, which depends on the specific magnetic profile considered. In addition, the non-modal stability analysis reveals that the inclusion of the magnetic field mitigates the effects of perturbations also in the short term, a phenomenon that cannot be seen using only modal stability analysis.

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.

Transient growth and dissipative exceptional points.

In the context of non-Hermitian photonics, we study the physics of transient growth in coupled waveguide systems that exhibit higher-order exceptional points. We demonstrate the counterintuitive effect of transient growth despite the decaying spectrum, which is a direct consequence of the underlying modal nonorthogonality. Eigenvalue analysis fails to capture the power dynamics and thus we have to rely on methods of nonmodal stability theory, namely singular value decomposition and pseudospectra. The relation between the order of the exceptional point and transient growth is also examined. Our work provides a general methodology that can be applied to any non-Hermitian system that contains complex elements with more loss than gain, and exploits the boundaries of transient amplification in dissipative environments.

The response to external excitations in thermocapillary liquid layers

The responses to external excitations in thermocapillary liquid layers are investigated by non-modal stability theory. The maximum amplification of input signals is measured by a response function depending on the perturbation velocity and temperature. There can be rather large amplifications in subcritical flows at both small and large Prandtl numbers (Pr). For small Pr, the response increases significantly with both the Reynolds number (R) and the Biot number (Bi) but decreases with Pr. The optimal response is achieved when the perturbation is almost a spanwise wave. The response properties for the linear flow and the return flow are similar. The amplification is caused by a combination of the lift-up mechanism, Orr mechanism, and external forcing. However, for large Pr, large amplifications could only be found in the return flow, while the variations of response with Pr and Bi are opposite to those at small Pr. The optimal response propagates in the streamwise direction. The amplification is caused by the thermocapillary effect.

Stability of ablation flows in inertial confinement fusion: Nonmodal effects.

Fast transient growth of hydrodynamic perturbations due to nonmodal effects is shown to be possible in an ablation flow relevant to inertial confinement fusion (ICF). Likely to arise in capsule ablators with material inhomogeneities, such growths appear to be too fast to be detected by existing measurement techniques, cannot be predicted by any of the methods previously used for studying hydrodynamic instabilities in ICF, yet could cause early transitions to nonlinear regimes. These findings call for reconsidering the stability of ICF flows within the framework of nonmodal stability theory.

Stability analysis of a zero-dimensional model of PWR core using non-modal stability theory

In this work, non-modal stability theory is used to study the non-linear dynamics of the Pressurised Water Reactor (PWR), using a zero-dimensional model. Usually, stability analysis for nuclear reactors is performed using modal methods. Instead, the non-modal approach used in the present work offers a different perspective. This approach considers the so-called short-term behaviour following a disturbance instead of the asymptotic (long-term) one. In the model, point-reactor kinetics with six groups delayed neutrons is used for neutronics, whereas thermal-hydraulics is treated using a two-region model (fuel and coolant). The non-orthogonality of the dynamic matrix of the obtained system shows the possibility of transient energy growth following a disturbance, and the proposed stability analysis offers proof of this. A sensitivity analysis of some parameters is then carried out to study how their variation influences the short-term response of the system. The results of this analysis allow for a ranking of the parameters, allowing the identification of the most significant ones regarding system stability.

Transient growth in thermocapillary liquid layers

Transient growth in thermocapillary liquid layers is examined by nonmodal stability theory. Rather large transient growth occurs in subcritical flows at small Prandtl numbers. The growth decreases with Prandtl number but increases with Biot number, while its energy comes from the basic flow.

Heat transfer enhancement strategies in a swirl flow minichannel heat sink based on hydrodynamic receptivity

The swirl flow minichannel heat sink has shown to be a promising alternative for thermal management of high heat flux applications, such as electronics and concentrated photovoltaics. Effective heat transfer enhancement strategies for this device are identified by studying the receptivity of temperature disturbances to a momentum forcing input. Steady state laminar flow is calculated numerically and experimental measurements are carried out to validate the results for subcritical Reynolds numbers. Using the framework of nonmodal stability theory, a harmonically driven linear perturbation problem is formulated, and the methodology to apply the local and parallel flow approximations based on order of magnitude arguments is detailed. The input-output response of temperature perturbations to forcing of the radial, azimuthal, and wall-normal momentum components is calculated for a range of wavenumbers, waveangles and temporal frequencies. The largest amplification is presented by streamwise vortices and streaks, followed by axisymmetric inward travelling waves, and then by streamwise propagating waves. Micromachining the channel walls with streamwise spiral grooves is proposed as a heat transfer enhancement technique. Excitation of streamwise independent structures in the wall-normal direction is expected, therefore maximum amplification should be obtained. Due to its simple implementation, we also propose using a pulsating flow rate as a heat transfer enhancement technique. Receptivity results for streamwise propagating waves of radial forcing show a response curve with moderate amplification for a wide range of actuation frequencies. Experimental work is conducted to measure the performance of the swirl flow channel heat sink using flow pulsations at the range of forcing frequencies suggested by the receptivity study. Compared to the unforced case, a lower wall temperature (up to 5°C cooler) was observed with pulsations, at the same imposed heat flux and flow rate. To get the same wall temperature as in the unforced case, a pumping power reduction of up to 26.6% was observed, and using the same pumping power resulted in up to a 10.3% Nusselt enhancement. Hydrodynamic receptivity was successfully used to identify effective heat transfer enhancement strategies, resulting in a significant performance improvement for the swirl flow channel heat sink. This physics based approach can be extended to other techniques, for instance, to select the wavelength of a wavy surface, the periodicity of surface roughness elements, or the frequency of acoustic vibrations.

Stabilities in plane Poiseuille flow of Herschel–Bulkley fluid

Linear stability in plane Poiseuille flow of a yield-stress shear-thinning fluid is considered. The rheological behavior of the fluid is described by the Herschel–Bulkley model. The effect of shear-thinning on the stability is investigated using the energy method and the nonmodal stability theory. The result of the energy method shows that with the increase of shear thinning, the critical energy Reynolds number decreases for both the streamwise and spanwise disturbances. For the nonmodal stability, we focus on the response to initial conditions by examining the energy growth function G(t). For a Herschel–Bulkley fluid, it is found that there can be a rather large transient growth even though the linear operator of the plan Poiseuille flow has no unstable eigenvalue. The results show that the shear thinning plays an important role in determining the energy growth rate and the structure of the disturbance with optimal transient growth.

A nonlinear approach to transition in subcritical plasmas with sheared flow

In many plasma systems, introducing a small background shear flow is enough to stabilize the system linearly. The nonlinear dynamics are much less sensitive to sheared flows than the average linear growth rates, and very small amplitude perturbations can lead to sustained turbulence. We explore the general problem of characterizing how and when the transition from near-laminar states to sustained turbulence occurs, with a model of the interchange instability being used as a concrete example. These questions are fundamentally nonlinear, and the answers must go beyond the linear transient amplification of small perturbations. Two methods that account for nonlinear interactions are therefore explored here. The first method explored is edge tracking, which identifies the boundary between the basins of attraction of the laminar and turbulent states. Here, the edge is found to be structured around an exact, localized, traveling wave solution that is qualitatively similar to avalanche-like bursts seen in the turbulent regime. The second method is an application of nonlinear, non-modal stability theory which allows us to identify the smallest disturbances which can trigger turbulence (the minimal seed for the problem) and hence to quantify how stable the laminar regime is. The results obtained from these fully nonlinear methods provide confidence in the derivation of a semi-analytic approximation for the minimal seed.

Linear stability analysis of magnetohydrodynamic duct flows with perfectly conducting walls.

The stability of magnetohydrodynamic flow in a duct with perfectly conducting walls is investigated in the presence of a homogeneous and constant static magnetic field. The temporal growth and spatial distribution of perturbations are obtained by solving iteratively the direct and adjoint governing equations with respect of perturbations, based on nonmodal stability theory. The effect of the applied magnetic field, as well as the aspect ratio of the duct on the stability of the magnetohydrodynamic duct flow is taken into account. The computational results show that, weak jets appear near the sidewalls at a moderate magnetic field and the velocity of the jet increases with the increase of the intensity of the magnetic field. The duct flow is stable at either weak or strong magnetic field, but becomes unstable at moderate intensity magnetic field, and the stability is invariance with the aspect ratio of the duct. The instability of magnetohydrodynamic duct flow is related with the exponential growth of perturbations evolving on the fully developed jets. Transient growth of perturbations is also observed in the computation and the optimal perturbation is found to be in the form of streamwise vortices and localized within the sidewall layers. By contrast, the Hartmann layer perpendicular to the magnetic field is irrelevant to the stability issue of the magnetohydrodynamic duct flow.

Stability of surfactant-laden droplet spreading over an inclined heterogeneous substrate

For the spreading of a droplet covered with insoluble surfactant over an inclined random heterogeneous substrate, the base state and disturbance evolution equations for the film thickness and surfactant concentration are derived using the lubrication theory. Stability of the droplet spreading on the heterogeneous substrates, and effects of dimensionless parameters as well as the internal mechanism are investigated based on the non-modal stability theory. Results show that the disturbances of film thickness and surfactant concentration exihibit a double-hump shape and the crests lie in thinning regions at the bottom of the droplet. With increasing disturbance wave number, the stability of the droplet spreading is enhanced, but the maximal disturbance transfers from the right-side to the left-side of the droplet bottom. Increasing inclined angle leads to severe instability distinctly. Improving topographical height or increasing wave number of the substrate leads the spreading stability to firstly enhance and then to weaken. The droplet evolution displays an unstable spreading under smaller capillary number, with the maximum disturbance to be increasing before decreasing.

Non-modal stability analysis and transient growth in a magnetized Vlasov plasma

Collisionless plasmas, such as those encountered in tokamaks, exhibit a rich variety of instabilities. The physical origin, triggering mechanisms and fundamental understanding of many plasma instabilities, however, are still open problems. We investigate the stability properties of a 3-dimensional collisionless Vlasov plasma in a stationary homogeneous magnetic field. We narrow the scope of our investigation to the case of Maxwellian plasma and examine its evolution with an electrostatic approximation. For the first time using a fully kinetic approach we show the emergence of the local instability, a transient growth, followed by classical Landau damping in a stable magnetized plasma. We show that the linearized Vlasov operator is non-normal leading to the algebraic growth of the perturbations using non-modal stability theory. The typical time scales of the obtained instabilities are of the order of several plasma periods. The first-order distribution function and the corresponding electric field are calculated and the dependence on the magnetic field and perturbation parameters is studied. Our results offer a new scenario of the emergence and development of plasma instabilities on the kinetic scale.

Spreading of droplet with insoluble surfactant on corrugated topography

The flow of microscale fluid on a topography surface is a key to further development of MEMS, nanoscience and technology. In the present paper, a theoretical model of the droplet spreading with insoluble surfactant over corrugated topography is established with the lubrication theory, and the evolution equations of film thickness and surfactant concentration in base state and disturbance state are formulated. The droplet dynamics, the nonlinear stability based on nonmodal stability theory, and the effects of topography structure and Marangoni stress are numerically simulated with PDECOL scheme. Results show that the impact of topographical surface is strengthened apparently while the Marangoni stress driven by surfactant concentration is weakened in the mid-late stages of the spreading. The droplet radius on the topography advances faster and the lowest height of liquid/gas interface near the droplet edge reduces remarkably in the intermediate stage compared with those on the flat wall. The quantity of the wavelet similar to the topography increases gradually, with the characteristics of wavelet crest height with time exhibiting a single-hump feature. The spreading stability is enhanced under the disturbance wavenumber of 4, however, is to deteriorate and even to transform into instability when wavenumber increases further. In addition, the reductive Marangoni number, enhancive capillary number, modest Peclet number, the low height of the topography as well as small wavenumber of topography can make contributions to the evident stability of droplet spreading.

Double diffusive natural convection of CO2 in a brine saturated geothermal reservoir: Study of non-modal growth of perturbations and heterogeneity effects

In a recent study (Islam et al., 2013) double diffusive natural convection of CO2 in brine saturated homogeneous permeable geothermal reservoir was investigated. Here we have shown the study of heterogeneous reservoirs. Firstly, we present the non-modal stability theory to analyze time-dependent double diffusive convection. Linear stability analysis is performed and the maximum amplification of perturbations optimized over the entire space of initial perturbations is examined. These provide us onset time of convection (or the critical time) and the structure of the most-amplified perturbations. Secondly, we investigate numerical simulation of an anisotropic and layered reservoir with permeability variations, which is impervious from the sides, and is open to CO2 at the top. We show the propagation of CO2 plumes over long period of time by analyzing different combination of problem parameters: solutal Rayleigh number (100≤Ras≤10,000), the buoyancy ratio (2≤N≤100), Dykstra-Parsons coefficient (0≤Vdp≤0.85) and a fixed Lewis number. How permeability variation in vertical (kz) to horizontal (kx) direction affects the CO2 dissolution is discussed for different ratios of kz/kx (0.1≤kz/kx≤0.8). Results of different cavity aspect ratios (0.5≤A≤2) are also presented. It is found that heterogeneous porous reservoir renders better mass transfer of CO2 than the homogeneous one. CO2 dissolution is enhanced with increasing both Ras and Vdp. N, which represents geothermal effect, has minor impact on overall dissolution process. The results have implications in enhanced oil recovery and CO2 based geothermal systems.

Non-modal stability in Hagen-Poiseuille flow of a Bingham fluid

Linear stability in Hagen-Poiseuille flow of a Bingham fluid is considered. Bingham fluid exhibits a yield stress in addition to a plastic viscosity. A Bingham number B, which describes the ratio of yield and viscous stresses, is used to characterize the behavior of Bingham-Hagen-Poiseuille flow. The effects of B on the stability are investigated using the energy method and the non-modal stability theory. The energy analysis shows that the non-axisymmetric disturbance has the lowest critical energy Reynolds number for all B. The global critical energy Reynolds number Reg increases with B. At sufficient large B, Reg has the order of B1/2. For the non-modal stability, we focus on response to external excitations and initial conditions. The former is studied by examining the ε-pseudospectrum, and the latter is by examining the energy growth function G(t). For the problem of response to external excitations, the maximum response is achieved by non-axisymmetric and streamwise uniform disturbances at the frequency of ω = 0, with a possible choice of the azimuthal wavenumbers of n = 1, 2, or 3. For the problem of response to initial conditions, it is found that there can be a rather large transient growth even though the linear operator of the Bingham-Hagen-Poiseuille flow has no unstable eigenvalue. For small B, the optimal disturbance is in the form of streamwise uniform vortices and streaks. For large B, the optimal disturbance is in the form of oblique waves. The optimal energy growth decreases and the optimal azimuthal wavenumber increases with the increase of B.

Optimal perturbations of gravitationally unstable, transient boundary layers in porous media

AbstractWe study the linear stability of gravitationally unstable, transient, diffusive boundary layers in porous media using non-modal stability theory. We first perform a classical optimization procedure, using an adjoint-based method, to obtain the perturbations at the initial time $t= {t}_{p} $ that have a maximum amplification at a final time $t= {t}_{f} $. We then investigate the sensitivity of the optimal perturbations to the initial time, ${t}_{p} $, and the final time, ${t}_{f} $, as well as different measures of perturbation amplification. Due to the transient nature of the base state, we demonstrate that there is an optimal initial perturbation time, ${ t}_{p}^{o} $. By rescaling the problem, we develop analytical relationships for the optimal initial time and wavenumber in terms of aquifer properties. We also demonstrate that the classical optimization procedure essentially recovers the dominant perturbation structures predicted by a quasi-steady modal analysis. Although the classical optimal perturbations are mathematically valid, we observe that due to physical constraints, they are unlikely to reflect analogous laboratory experiments. Therefore, we propose a modified optimization procedure (MOP) that constrains the optimization to physically admissible initial perturbation fields. We compare the results of the classical and modified optimization procedures with quasi-steady modal analyses and initial value problems commonly used in the literature. Finally, we validate the predictions of the modified optimization scheme by performing direct numerical simulations (DNS) that emulate the onset of convection in physical systems.

Nonmodal stability in Hagen-Poiseuille flow of a shear thinning fluid

Linear stability in Hagen-Poiseuille flow of a shear-thinning fluid is considered. The non-Newtonian viscosity is described by the Carreau rheological law. The effects of shear thinning on the stability are investigated using the energy method and the nonmodal stability theory. The energy analysis shows that the nonaxisymmetric disturbance with the azimuthal wave number m=1 has the lowest critical energy Reynolds number for both the Newtonian and shear-thinning cases. With the increase of shear thinning, the critical energy Reynolds number decreases for both the axisymmetric and nonaxisymmetric cases. For the nonmodal stability, we focus on two problems: response to external excitations and response to initial conditions. The former is studied by examining the ε pseudospectrum, and the latter by examining the energy growth function G(t). For both Newtonian and shear-thinning fluids, it is found that there can be a rather large transient growth even though the linear operator of the Hagen-Poiseuille flow has no unstable eigenvalue. For the problem of response to external excitations, the optimal response is achieved by disturbance with m=1 for both the Newtonian and non-Newtonian cases. For the problem of response to initial conditions, the optimal disturbance is in the form of streamwise uniform streaks. Being different from the Newtonian case, the azimuthal wave number of the optimal disturbance may be greater than 1 for strongly shear-thinning cases.

Disturbance energy norms: A critical analysis

The question of which norm should be used to characterize the fluctuating disturbance energy in studies on thermoacoustic instabilities remains a topic of continued debate. In this paper we formulate a strategy for developing mathematically consistent norms that do not support spurious growth of disturbance energy in the absence of ‘physical’ sources of energy. A critical analysis of various disturbance energy norms existing in literature is conducted and important conclusions regarding positive definitiveness and susceptibility to exhibiting unphysical growth are drawn. It is shown that the energy norm proposed by Cantrell and Hart [Interaction between sound and flow in acoustic cavities: mass, momentum and energy considerations, Journal of Acoustical Society of America 36 (1964) 697–706] and Myers' first order disturbance energy norm [Transport of energy by disturbances in arbitrary steady flow, Journal of Fluid Mechanics 226 (1991) 383–400] are positive definite if M0<1 and M0<1/γ, respectively, where M0 is the magnitude of the local mean flow Mach number, γ is the ratio of specific heats at constant pressure and volume and the stipulated conditions should be met at every point in the flow domain. Our analysis shows that the disturbance energy norm proposed by Chu [On the energy transfer to small disturbances in fluid flow (part I), Acta Mechanica 1 (1965) 215–234] does not exhibit unphysical growth. It is also shown that this property is not unique to Chu's disturbance energy norm and there exists a family of norms which satisfy this requirement. The analysis also leads to an interesting interpretation of the acoustic energy conservation principle of Cantrell and Hart. The potential held by various disturbance energy norms for exhibiting fictitious growth is quantified using tools from nonmodal stability theory. It is concluded that if the mean flow Mach number is small, Myers' norm is a suitable measure of the disturbance energy.