Three-dimensional Linear Stability Analysis Research Articles

Multimode theory of electron hole transverse instability

We present Vlasov–Poisson three-dimensional linear stability analysis of an initially planar electron hole structure, solving for the distribution function by integration along unperturbed orbits. The non-sinusoidal potential perturbation shape (parallel to$B$) is expanded in eigenfunctions of the adiabatic Poisson operator, generalizing the prior assumption of a rigid shift of the equilibrium. We show that the shiftmode is then modified by a second discrete mode plus an integral over a continuum of wave-like modes. A rigorous treatment shows that the continuum can be approximated effectively by a single mode that satisfies the external wave dispersion relation, thus making the perturbation a weighted sum of three modes. We find numerically the solution for the complex instability frequency, and the corresponding three mode amplitudes determining the perturbation eigenmode. This multimode analysis refines the accuracy of the prior single-mode results, giving slightly higher growth rates at most parameters, as expected from the extra mode shape freedom. Oscillating modes near stability boundaries have larger mode distortions which help explain particle-in-cell simulations that observe instability up to${\sim }20$ % beyond the prior shiftmode thresholds, and narrowing of the perturbation. At high magnetic field, the multimode analysis predicts a reduction of the already small growth rate.

Global modes of variable-viscosity two-phase swirling flows and their triadic resonance

The linear and nonlinear dynamics of two-phase swirling flows produced by the Grabowski–Berger profile under the influence of a viscosity stratification are investigated. We perform axisymmetric nonlinear simulations and fully three-dimensional linear global stability analysis of the flow for several swirl numbers $S$ and viscosity ratios $\tilde {\mu }$ . They are accompanied by nonlinear three-dimensional simulations and subsequent modal analysis using the bispectral mode decomposition, recently introduced by Schmidt (Nonlinear Dyn., vol. 102, issue 4, 2020, pp. 2479–2501). We find a pronounced destabilising effect of the viscosity stratification on both the onset of axisymmetric vortex breakdown and helical instability that is linked to the required shear stress continuity across the interface. Consequently, destabilisation is shifted to lower $S$ as compared with an equivalent flow with uniform viscosity. Further, the stability analysis reveals the simultaneous destabilisation of two global modes with wavenumbers $m=1$ and $m=2$ that have harmonic frequencies. The analysis of the nonlinear flow reveals a strong triadic resonance between these modes that governs the nonlinear dynamics and leads to a rapid departure from the linear dynamics. At larger swirl, the bifurcation of additional modes initiates an interaction cascade by means of triadic resonance which is elucidated by the bispectral analysis. It leads to the emergence of a variety of additional modes in the nonlinear flow. This study contributes to an improved understanding of the influence of viscosity stratification on the onset of vortex breakdown and the destabilisation of global modes. Further, it provides a clear picture of the dynamics of swirling flows with a codimension-two point and related triadic interaction of two global modes at harmonic frequencies and wavenumbers.

The bimodal instability of thermal convection in a tall vertical annulus

We pursue a three-dimensional linear stability analysis to investigate the convective stability in a cylindrical annulus imposed with a horizontal temperature gradient. The investigation focuses on the stability of both axisymmetric and asymmetric modes in a wide range of Prandtl number Pr and the radii ratio η between the inner to outer cylinders. Results show that, for the axisymmetric mode, the stability strongly depends on the Prandtl number. For 0≤Pr≤1.4, the instability sets in as the shear mode that the instability enhances as η increases while it is virtually independent of Pr. The critical Grashof number can be approximated by Grc(η)≈103×4η−0.68+3. For 1.5≤Pr≤12.5, the buoyant mode appears and competes with the shear mode to predominate the stability under various Pr and η, namely, the bimodal instability occurs. The transition between the two modes occurs at the specific radii ratio η1=0.21Pr0.727−0.31. For Pr≥12.6, the shear mode vanishes and the buoyant mode prevails. The asymmetric modes are less prevailing than the axisymmetric mode for small Prandtl numbers, except that the mode of an azimuthal wavenumber equal to 1 becomes dominant in two small regions of η. When the Prandtl number is large, the axisymmetric mode predominates over the asymmetric mode except when η is very small. Otherwise, the asymmetric mode of a larger azimuthal wavenumber is less prevailing.

Influence of Centrifugal Buoyancy in Thermal Convection within a Rotating Spherical Shell

The dynamo action, which is of importance in the study of the geomagnetism mechanism, is considered to be caused by the convection structure formed inside a rotating spherical shell. This convection structure elongated in the rotation axis is generated by the action of both heat and rotation on the fluid inside a spherical shell. In this study, we analyzed thermal convection in such a rotating spherical shell and attempted to understand the phenomenon of this convective structure. It is known that each value of the Prandtl number, the Ekman number and the Rayleigh number and their balance are important for the generation of such convective structure. We fixed these three parameters and considered the effect of centrifugal buoyancy as the Froude number additionally. To investigate how the effects of centrifugal buoyancy affect the convective structure, we carried out both three-dimensional numerical simulations and linear stability analyses. In particular, we focused on the transition from axisymmetric flow to non-axisymmetric flow having wavenumbers in the toroidal direction and investigated both growth rate and phase velocity of the disturbance. It was found that axisymmetric flow tends to be maintained as the effect of centrifugal buoyancy increases.

Local stability analysis of homogeneous and stratified Kelvin–Helmholtz vortices

We perform a three-dimensional short-wavelength linear stability analysis of numerically simulated two-dimensional Kelvin–Helmholtz vortices in homogeneous and stratified environments at a fixed Reynolds number of $Re = 300$ . For the homogeneous case, the elliptic instability at the vortex core dominates at early times, before being taken over by the hyperbolic instability at the vortex edge. For the stratified case of Richardson number $ Ri = 0.08$ , the early-time instabilities comprise a dominant elliptic instability at the core and a hyperbolic instability influenced strongly by stratification at the vortex edge. At intermediate times, the local approach shows a new branch of (convective) instability that emerges at the vortex core and subsequently moves towards the vortex edge. A few more convective instability bands appear at the vortex core and move away, before coalescing to form the most unstable region inside the vortex periphery at large times. In addition, the stagnation point instability is also recovered outside the periphery of the vortex at intermediate times. The dominant instability characteristics from the local approach are shown to be in good qualitative agreement with the results based on global instability studies for both homogeneous and stratified cases. A systematic study of the dependence of the dominant instability characteristics on $ Ri $ is then presented. While $ Ri = 0.1$ is identified as most unstable (with convective instability being dominant), another growth rate maximum at $ Ri = 0.025$ is not far behind (with the hyperbolic instability influenced by stratification being dominant). Finally, the local stability approach is shown to predict the potential orientation of the flow structures that would result from hyperbolic and convective instabilities, which is found to be consistent with three-dimensional numerical simulations reported previously.

Critical Modes of Thermoelectric Convection Instabilities in a Vertical Slot

Investigation of the natural convection in a dielectric fluid confined inside a vertical slot subject to a constant horizontal temperature gradient and a high-frequency electric field was revisited. The thermoelectric coupling produces a dielectrophoretic force which contains a term called electric buoyancy associated with an electric gravity. Similar to the Archimedean buoyancy, this force can induce thermo-electro-convective instability in a stable fluid flow. Including the feedback effect of the electric field on the flow, a three-dimensional linear stability analysis is performed to characterize the instability induced by this force. Different critical modes are found depending on the intensity of the electric field: stationary hydrodynamic modes, oscillatory thermal modes and stationary electric modes. The threshold of the electric modes is independent of the Prandtl number in contrast with the threshold of the hydrodynamic and thermal modes. The obtained results are compared with those of previous studies.

Three-dimensional instability of a flow past a sphere: Mach evolution of the regular and Hopf bifurcations

A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics.

Successive bifurcations in a fully three-dimensional open cavity flow

The transition to unsteadiness of a three-dimensional open cavity flow is investigated using the joint application of direct numerical simulations and fully three-dimensional linear stability analyses, providing a clear understanding of the first two bifurcations occurring in the flow. The first bifurcation is characterized by the emergence of Taylor–Görtler-like vortices resulting from a centrifugal instability of the primary vortex core. Further increasing the Reynolds number eventually triggers self-sustained periodic oscillations of the flow in the vicinity of the spanwise end walls of the cavity. This secondary instability causes the emergence of a new set of Taylor–Görtler vortices experiencing a spanwise drift directed toward the spanwise end walls of the cavity. While a two-dimensional stability analysis would fail to capture this secondary instability due to the neglect of the lateral walls, it is the first time to our knowledge that this drifting of the vortices can be entirely characterized by a three-dimensional linear stability analysis of the flow. Good agreements with experimental observations and measurements strongly support our claim that the initial stages of the transition to turbulence of three-dimensional open cavity flows are solely governed by modal instabilities.

Effect of gravity on the stability of viscoelastic thermocapillary liquid layers

Thermocapillary convection appears in many polymer processing operations. In order to understand the effect of gravity for its stability, three-dimensional linear stability analysis is performed for a viscoelastic fluid. The critical Marangoni number is derived as a function of Bond number, Prandtl number and elasticity. When the Prandtl number is large, the increasing of gravity effect often makes the flow more stable, and the coupling of gravity to elasticity and thermocapillary force excites many different kinds of preferred modes. For small Prandtl number, the flow is always destabilized by gravity effect. The work done by gravity becomes a new energy source for perturbation. However, it can be either positive or negative, which is not directly related to the variation of critical Marangoni number with gravity. The effect of gravity on the instability mechanism and the properties of preferred modes are demonstrated for different elasticity.

Three-dimensional stability analysis for a salt-finger convecting layer

A three-dimensional linear stability analysis is carried out for a convecting layer in which both the temperature and solute distributions are linear in the horizontal direction. The three-dimensional results show that, for $Le=3$ and 100, the most unstable mode occurs invariably as the longitudinal mode, a vortex roll with its axis perpendicular to the longitudinal plane, suggesting that the two-dimensional results are sufficient to illustrate the stability characteristics of the convecting layer. Two-dimensional results show that the stability boundaries of the transverse mode (a vortex roll with its axis perpendicular to the transverse plane) and the longitudinal modes are virtually overlapped in the regime dominated by thermal diffusion and the regime dominated by solute diffusion, while these two modes hold a significant difference in the regime the salt-finger instability prevails. More precisely, the instability area in terms of thermal Grashof number $Gr$ and solute Grashof number $Gs$ is larger for the longitudinal mode than the transverse mode, implying that, under any circumstance, the longitudinal mode is always more unstable than the transverse mode.

Electrohydrodynamic instability of miscible core–annular flows with electrical conductivity stratification

AbstractThis paper investigates the electrohydrodynamical instability of two miscible flows in a micro-pipe subject to an axial electric field. There is an electrical conductivity stratification between the two layers. A weak shear flow arises from a constant axial pressure gradient. The three-dimensional linear stability analysis is studied under the assumption of a quasi-steady state. The influences of the conductivity ratio ${\it\eta}$, the interface location $a$, the interface thickness ${\it\delta}$, the Reynolds number $\mathit{Re}$ and the Schmidt number $\mathit{Sc}$ on the linear stability of the flows are investigated. The flow becomes more unstable for a larger conductivity contrast. When the conductivity in the inner layer is larger, the critical unstable mode can be dominated by either the corkscrew mode (the azimuthal wavenumber $m=1$) or the axisymmetric mode ($m=0$), which is dependent on the interface location $a$. It is observed that, when the interface is proximal to pipe’s wall, the critical unstable mode shifts from the corkscrew mode to the axisymmetric mode. When the conductivity is larger in the outer layer, the instability is dominated by the axisymmetric mode. A detailed parametric study shows that the flow is least stable when the interface between the two liquids is located at approximately $a=0.3$ and $a=0.2$ for conductivity ratios of ${\it\eta}=0.5$ and ${\it\eta}=2$ respectively. The flow becomes more stable as the interface becomes thicker, and the shear flow and ionic diffusion are found to have a stabilizing effect due to the enhancement of dissipation mechanisms.

Unsteady asymmetric engulfment regime in a T-mixer

Unsteady flow regimes in a micro T-mixer are investigated. Direct numerical simulations (DNS) show that, in agreement with experimental and numerical results in the literature, when the Reynolds number is increased above a critical value larger than those typical of the steady engulfment regime, the flow remains asymmetric in the mean but becomes periodic in time. The dynamics of this regime is characterized by analyzing the evolution of the three-dimensional vortical structures forming at the confluence of the two inlet flow streams. As the Reynolds number is further increased, the flow remains time-periodic but it continuously switches between a symmetric configuration, similar to that of the vortical regime, and an asymmetric one, close to the engulfment configuration. Three-dimensional linear stability analysis is successively used to characterize the instability leading to the unsteady asymmetric regime, which is also interesting for applications due to its high mixing efficiency. The critical Reynolds number and the instability frequency are in very good agreement with those found in DNS. The sensitivity of this instability to a generic perturbation of the base flow is also investigated. It is shown that the largest sensitivity is to base-flow modifications introduced close to the three-dimensional vortical structures forming at the confluence between the inlet channels. Finally, the sensitivity analysis is specialized to investigate the effect of a generic perturbation of the inlet velocity profile.

Investigation of the steady engulfment regime in a three-dimensional T-mixer

The steady engulfment regime in a fully three-dimensional micro T-mixer is investigated. This regime is of significant interest for applications since it implies high mixing between the flow streams entering the device. Direct numerical simulations are first used to characterize this regime. In particular, the main vortical structures typical of the engulfment regime and their effects on mixing are investigated. Three-dimensional linear stability analysis is successively applied to the characterization of the instability leading to the engulfment regime. The critical Reynolds number and the global unstable mode are first computed for a configuration characterized by fully-developed inlet velocity conditions. The sensitivity of this instability to a generic modification of the base flow is then investigated, thanks to the computation of the mode adjoint to the direct unstable one. Finally, this kind of analysis is specialized to investigate the effect of a perturbation of the velocity distribution at the inlet of the T-mixer. Sensitivity analysis shows that non-fully developed inlet velocity conditions lead to an increase of the critical Reynolds number. More generally, the sensitivity maps can be used for the design of control strategies aimed at promoting or inhibiting the engulfment. An example is provided for a control based on blowing/suction through the mixer walls.

Linear stability of flow in rectangular ducts in the vicinity of the critical aspect ratio

We have performed three-dimensional BiGlobal linear stability analyses of flow in rectangular ducts by using a spectral element method. A linear stability theory is applied to a steady laminar solution and an eigenvalue problem is generated in the matrix form. The critical Reynolds numbers are evaluated for some aspect ratios to describe the neutral curves. It is found that the neutral curves consist of two bifurcation branches. One is the first bifurcation branch where the flow loses its stability with respect to three-dimensional disturbances, while the other is the second bifurcation branch where it recovers its stability. The first and second branches connect at the critical point of As=3.19 and Re=2.47×105, where the saddle–node bifurcation occurs. We have revealed that the discontinuous divergence of the critical Reynolds number to infinity in the vicinity of the critical aspect ratio As is caused by the saddle–node bifurcation.

Revisiting the stability of circular Couette flow of shear-thinning fluids

Three-dimensional linear stability analysis of Couette flow between two coaxial cylinders for shear-thinning fluids with and without yield stress is performed. The outer cylinder is fixed and the inner one is rotated. Three rheological models are used: Bingham, Carreau and power-law models. Wide range of rheological, geometrical and dynamical parameters is explored. New data for the critical conditions are provided for Carreau fluid. In the axisymmetric case, it is shown that when the Reynolds number is defined using the inner-wall shear-viscosity, the shear-thinning delays the appearance of Taylor vortices, for all the fluids considered. It is shown that this delay is due to reduction in the energy exchange between the base and the perturbation and not to the modification of the viscous dissipation. In the non axisymmetric case, contrary to Caton [1], we have not found any instability.

Pipe flow of shear-thinning fluids

Pipe flow of purely viscous shear-thinning fluids is studied using numerical simulations. The rheological behavior is described by the Carreau model. The flow field is decomposed as a base flow and a disturbance. The perturbation equations are then solved using a pseudo-spectral Petrov–Galerkin method. The time marching uses a fourth-order Adams–Bashforth scheme. In the case of an infinitesimal perturbation, a three-dimensional linear stability analysis is performed based on modal and non-modal approaches. It is shown that pipe flow of shear-thinning fluids is linearly stable and that for the range of rheological parameters considered, streamwise-independent vortices are optimally amplified. Nonlinear computations are done for finite amplitude two-dimensional disturbances, which consist of one pair of longitudinal rolls. The numerical results highlight a strong modification of the viscosity profile associated with the flow reorganization. For a given wall Reynolds number, shear-thinning reduces the energy gain of the perturbation. This is due to a reduction of the exchange energy between the base flow and the perturbation. Besides this, viscous dissipation decreases with increasing shear-thinning effects.

Global stability of multiple solutions in plane sudden-expansion flow

AbstractThe two-dimensional, incompressible flow in a plane sudden expansion is investigated numerically for a systematic variation of the geometry, covering expansion ratios (steps-to-outlet heights) from $0. 25$ to $0. 95$. By means of a three-dimensional linear stability analysis global temporal modes are scrutinized. In a symmetric expansion the primary bifurcation is stationary and two-dimensional, breaking the mirror symmetry with respect to the mid-plane. The secondary asymmetric flow experiences a secondary instability to different three-dimensional modes, depending on the expansion ratio. For a moderately asymmetric expansion only one of the two secondary flows (the connected branch) is realized at low Reynolds numbers. Since the perturbed secondary flow does not deviate much from the symmetric secondary flow, both secondary stability boundaries are very close to each other. For very small and very large expansion ratios an asymptotic behaviour is found for suitably scaled critical Reynolds numbers and wavenumbers. Representative instabilities are analysed in detail using an a posteriori energy transfer analysis to reveal the physical nature of the instabilities. Depending on the geometry, pure centrifugal and elliptical amplification processes are identified. We also find that the basic flow can become unstable due to the effects of flow deceleration, streamline convergence and high shear stresses, respectively.

Three-dimensional instability of the flow over a forward-facing step

AbstractThe global, temporal stability of the two-dimensional, incompressible flow over a forward-facing step in a plane channel is investigated numerically. The geometry is varied systematically covering constriction ratios (step-to-inlet height) from 0.23 to 0.965. A three-dimensional linear stability analysis shows that the stability boundary is a smooth continuous function of the constriction ratio. If the critical Reynolds and wavenumbers are scaled appropriately, they approach a linear asymptotic behaviour for large step heights. The critical mode is found to be stationary and confined to the region of separated flow downstream of the step for all constriction ratios. An energy-transfer analysis reveals that the basic flow becomes unstable due to a combined effect involving lift-up and flow deceleration, leading to a critical mode exhibiting steady streaks. Moreover, the receptivity of the flow to initial as well as to structural perturbations is studied by means of an adjoint analysis.

Three-dimensional stability of a thin film between two approaching drops

In most simulations of the coalescence process, the rupture of the thin film between two drops is assumed to be axisymmetric. In this paper, we examine the possibility of a nonaxisymmetric rupture by carrying out a three-dimensional linear stability analysis of an axisymmetric thin film region. First, the effect of tangential interfacial velocity on the stability of a flat film is analyzed and a scaling analysis is provided to predict the dependence of the critical film thickness on the dimensionless parameters of the problem: the capillary number Ca, the dimensionless Hamaker constant A∗, and the viscosity ratio λ. Multigrid integration and implicit eigensystem solution techniques are used to simultaneously solve the boundary integral and film evolution equations for the disturbance shape and growth rate. The calculations show that a fixed-end disturbance, which decays to zero at the edge of the film, exhibits maximum instability in the axisymmetric mode. On the other hand, the first nonaxisymmetric mode is the most unstable for the class of free-end disturbances that approach the edge of the film with zero slope. Next, the stability calculation is interfaced with the full axisymmetric simulation of two drops approaching in a biaxial extensional flow in order to obtain the correct base-state film shape at each time interval. In this case, the thin film first becomes unstable to a nonaxisymmetric disturbance for both kinds of boundary conditions. The critical thickness from the stability calculation is compared with the critical film thickness obtained earlier from numerical calculations of the collision and interaction of a pair of fully axisymmetric drops and the effect of base-state curvature on film stability is investigated.

Stability analysis of cusped bubbles in viscoelastic flows

Experiments have established that an axisymmetric sharp trailing edge of a gas bubble in a viscoelastic fluid can develop into a three-dimensional ‘knifelike’ shape under certain conditions (high capillary number, large bubble size). A numerical study is conducted to discover the physics of this phenomenon. The axisymmetric deformation of a bubble rising buoyantly in a viscoelastic fluid is simulated by solving the axisymmetric flow equations coupled with the constitutive equations of the finitely extensible nonlinear elastic Chilcott–Rallison (FENE-CR) model. The three-dimensional temporal linear stability analysis of this axisymmetric base state is carried out. The dominant eigenvalue which is indicative of the growth rate of the perturbations is computed. The only unstable eigenmode has azimuthal wavenumber m equal to 2. The corresponding eigenfunction shows that indeed a sharp axisymmetric tail develops a knife-edge form. A further investigation of the energy budget of the disturbances for m = 2 is performed to determine the production and dissipation terms affecting the growth of this instability. It is shown that the normal gradient of the base-state pressure along the free surface plays an important role in the evolution of the instability.