Two-dimensional Base Flow Research Articles

Arbitrary-order sensitivities of the incompressible base flow and its eigenproblem

First-order sensitivities and adjoint analysis are used widely to control the linear stability of unstable flows. Second-order sensitivities have recently helped to increase accuracy. In this paper, a method is presented to calculate arbitrary high-order sensitivities based on Taylor expansions of the incompressible base flow and its eigenproblem around a scalar parameter. For the incompressible Navier–Stokes equations, general expressions for the sensitivities are derived, into which parameter-specific information can be inserted. The computational costs are low since, for all orders, a linear equation system has to be solved, of which the left-hand-side matrix stays constant and thus its preconditioning can be exploited. Two flow scenarios are examined. First, the cylinder flow equations are expanded around the inverse of the Reynolds number, enabling the prediction of the two-dimensional cylinder base flow and its leading eigenvalue as a function of the Reynolds number. This approach computes accurately the base flow and eigenvalue even in the unstable regime, providing, when executed subsequently, a mean to calculate unstable base flows. This case gives a clear introduction into the method and allows us to discuss its constraints regarding convergence behaviour. Second, a small control cylinder is introduced into the domain of the cylinder flow for stabilization. Higher-order sensitivity maps are calculated by modelling the small cylinder with a steady forcing. These maps help to identify stabilizing areas of the flow field for Reynolds numbers within the laminar vortex shedding regime, with the required number of orders increasing as the Reynolds number rises. The results obtained through the proposed method align well with numerically calculated eigenvalues that incorporate the cylinder directly into the grid.

Global instability of the interaction between an oblique shock and a laminar boundary layer

This study investigates the oblique shock-wave/boundary-layer interaction with a Mach number of 2.15 and a Reynolds number of 1×105. Both global stability analysis and direct numerical simulation are used to reveal the global instability characteristics and three-dimensional details of the incident shock flow. The results of global stability analysis indicate that stationary global instability occurs when the shock angle exceeds the critical angle 31.8°. At a shock angle equal to 33°, an additional unstable mode appears, which is oscillatory at large wavelength and gradually dominant when the wavelength decreases. As the wavelength is further reduced, the mode and its conjugation evolve into two stationary modes with different growth rates. A global instability criterion for incident shock flow is established based on the triple deck theory, which determines the instability only through free-stream conditions and shock angles. A direct numerical simulation is performed for the 32° shock angle case. It is found that secondary separation occurs during the nonlinear growth, which is absent in the two-dimensional base flow. Moreover, the separated flow undergoes a secondary perturbation growth, during which the dominant spanwise wavelength is doubled and the flow structures change significantly. The flow oscillates around a quasi-steady state in the end, indicating that a stationary unstable mode can develop unsteadiness without external disturbances.

Stability of flow in a channel with repeated flow-facing wedge-shaped protrusions

Motivated by an interest in inciting instabilities and mixing for heat transfer enhancement in ducts, flow in a channel with repeated wedge-shaped protrusions is considered for various blockage ratios (wedge height to duct height), pitch (distance between wedges) and wedge angles. The stability of the two-dimensional base flow and its dependence on the geometric parameters is elucidated through a global linear stability analysis. A linearly unstable two-dimensional mode was found, contrasting similar confined flow set-ups. However, the primary instability is a three-dimensional mode manifesting as counter-rotating streamwise vortices over the wedge tip. Analysis of the kinetic energy budget indicates a lift-up mechanism leading to instability, with the dominant energy gain of the global three-dimensional mode due to shear in the base flow. Structural sensitivity and receptivity of the instability to momentum forcing identifies the core of the instability and locations important for flow control. An endogeneity approach is used to show that the local perturbation pressure gradient component dominates the distribution of the local contribution to the growth rate of the linear global eigenmode in most cases considered, despite its net contribution being identically zero. Weakly nonlinear Stuart–Landau analysis reveals that the primary bifurcation is supercritical across all tested geometric parameter combinations. This is consistent with the finding of low linear transient growth amplifications at subcritical Reynolds numbers, being orders of magnitude lower than in similar channel flow set-ups.

Transition to turbulence in hypersonic flow over a compression ramp due to intrinsic instability

In this work, a transition process in a hypersonic flow over a cold-wall compression ramp is studied using direct numerical simulation (DNS) and global stability analysis (GSA). The free-stream Mach number and the Reynolds number based on the flat-plate length are 7.7 and $8.6 \times 10^5$ , respectively. The shock-induced pressure rise causes the boundary layer to separate on the flat plate, forming a separation bubble around the corner. Without introducing any external disturbances, the DNS captures the transition to turbulence downstream of flow reattachment. The DNS results agree well with the experimental data as well as theoretical predictions. To uncover the intrinsic instability in the flow system, GSA is employed to investigate the three-dimensionality of the two-dimensional base flow. Several stationary and oscillatory unstable modes are revealed, which result in spanwise periodicity inside and downstream of the separation bubble. The GSA and DNS results indicate that the intrinsic instability of the flow system triggers the formation of streamwise counter-rotating vortices and boundary-layer streaks near reattachment. The downstream transition to turbulence starts from the breakdown of the streamwise vortices and streaks. Moreover, the second harmonic of the most unstable global mode and a broadband low-frequency unsteadiness occur in the saturated flow, which has a significant influence on the transition process. In summary, the present study demonstrates a transition process in a hypersonic compression-ramp flow as a result of the intrinsic instability of the flow system.

Transition control of cylinder wake via Lorentz force

The flow around a cylinder is a typical flow acting as the oscillator, and the perturbations can grow with time at a fixed position. This phenomenon can induce the laminar-turbulent transition in the wake, thus increasing the vibrations of the cylinder as well as the noise in the flow system. There exist three control strategies, i.e. the streamwise Lorentz force from the electromagnetic actuator set on the windward surface named windward control, the leeward surface named leeward control, and the whole surface of cylinder named global control, which are adopted to modify the two-dimensional base flow, thereby reducing the growth rates of perturbations in the wake and changing the transition mode. According to the Floquet stability analysis, it is found that the growth rates of the transition modes A and B present small changes in the windward control, while the growth rates of the two modes decrease with the increase of the control number in the other two control cases. Comparing the inviscid growth rates induced by the elliptic instability and the hyperbolic instability with each other, it is observed that the high inviscid growth rate in the windward control can be similar to those without control, while the inviscid growth rates can decrease with the increase of control number in the other two control cases. Three-dimensional direct numerical simulations are performed to validate the control effects. The results shows that the three-dimensional shape of the wake is changed from mode B to mode A when the actuator is set on the leeward surface or the whole surface of the cylinder. This is consistent with the result from the Floquet stability analysis. In addition, the drag of the cylinder reduces 15.2% for the leeward control and 14.4% for the the global control.

A Discrete Linear Stability Analysis of Two-dimensional Laminar Flow Past a Square Cylinder

The present study focuses on the development of a numerical framework for predicting the onset of vortex sheading due to flow past a square cylinder. For this a discrete linear stability analysis framework for two-dimensional laminar flows have used. Initially the frame work is validating by using the analysis of thermal stability of flows in the discrete numerical sense. The two-dimensional base flow for various values of the controlling parameter (Reynolds number for flow past a square cylinder and Rayleigh number for double-glazing problem) is computed numerically by using the lattice Boltzmann method. The governing equations, discretized using the finite-difference method in two-dimensions and are subsequently written in the form of perturbed equations with two-dimensional disturbances. These equations are linearized around the base flow and form a set of partial differential equations that govern the evolution of the perturbations. The eigenvalues, stability of the base flow and the points of bifurcations are determined using normal mode analysis. The eigenvalue spectrum predicts that the critical Reynolds number is 52 for the flow past a square cylinder. The results are consistent with the previous numerical and experimental observations.

A Discrete Linear Stability Analysis Framework for Two-Dimensional Laminar Flows

A discrete linear stability analysis framework for two-dimensional laminar flows is presented. Using two case studies involving analysis of thermal and laminar flows, the stability of flows in the discrete numerical sense is addressed. The two-dimensional base flow for various values of the controlling parameter (Reynolds number for flow past a square cylinder and Rayleigh number for double-glazing problem) is computed numerically by using the lattice Boltzmann method. The governing equations, discretized using the finitedifference method in two-dimensions and are subsequently written in the form of perturbed equations with twodimensional disturbances. These equations are linearized around the base flow and form a set of partial differential equations that govern the evolution of the perturbations. The eigenvalues, stability of the base flow and the points of bifurcations are determined using normal mode analysis. The eigenvalue spectrum predicts that the critical Reynolds number is 52 and the critical Rayleigh number is 6 1.88×10 for the square cylinder and double-glazing problem, respectively, The results are consistent with the previous numerical and experimental observations.

Effect of Heat Dissipation on Thermocapillary Convection of Low Prandtl Number Fluid in the Annular Pool Heated from Inner Cylinder

This paper presents a series of numerical simulations on the effect of heat dissipation on the thermocapillary convection of a low Prandtl number (Pr = 0.011) fluid in an annular pool heated from its inner cylinder. The radius ratio and the aspect ratio of the annular pool set as η = 0.5 and e = 0.1, respectively. The results show that with the increase of Biot number, the isotherms of the two-dimensional basic flow are seriously compressed near the hot inner cylinder and the thermocapillary flow pattern will be impaired. With the increase of Marangoni number, the thermocapillary convection first evolves from the basic flow to a three-dimensional stationary flow with longitudinal stationary stripes on the free surface, and then to a three-dimensional oscillatory flow featured with the combination of stationary stripes and azimuthal waves, which is caused by the mutual effect of the radial convection, azimuthal local flow and heat dissipation. After the flow destabilization, the amplitude of temperature fluctuation increases and the wave number decreases with the increase of Marangoni number when the free surface is adiabatic. Furthermore, the temperature fluctuation pattern is changed due to the surface heat dissipation. In addition, with the increase of Biot number, the wave number almost keeps constant when Marangoni number is small, but it increases significantly at a large Marangoni number.

Stability of a jet moving in a rectangular microchannel.

We study numerically the basic flow and linear stability of a capillary jet confined in a rectangular microchannel. We consider both the case where the interface does not touch the solid surfaces and that in which the jet adheres to them with a contact angle slightly smaller than 180^{∘}. Given an arbitrary set of values of the governing parameters, the fully developed (parallel) two-dimensional basic flow is calculated and then the growth rate of the dominant perturbation mode is determined as a function of the wave number. The flow is linearly stable if that growth rate is negative for all the wave numbers considered. We show that when the coflowing stream viscosity is sufficiently small in terms of that of the jet, there is an interval of the flow rate ratio Q for which the jet adheres to the walls or not depending on whether the flow is established by decreasing or increasing the value of Q. When the distance between the interface and the channel wall is of the order of the jet radius, the jet is unconditionally unstable. However, for sufficiently small interface-to-wall distances, the viscous stress can dominate the capillary pressure and fully stabilize the flow. Our results suggest that the capillary modes are suppressed and the flow becomes stable when the jet adheres to the channel walls. The combination of the above results indicates that, under certain parametric conditions, stable or unstable jets can be formed depending on whether the experimenter sets the flow rate ratio by decreasing or increasing progressively the jet flow rate while keeping constant that of the outer stream. Our theoretical predictions for the stablity of a coflow in a rectangular channel are consistent with previous experimental results [Humphry et al., Phys. Rev. E 79, 056310 (2009)PLEEE81539-375510.1103/PhysRevE.79.056310].

Non-modal stability analysis of low-Re separated flow around a NACA 4415 airfoil in ground effect

In this numerical–theoretical study, we perform a linear non-modal stability analysis of the separated flow around a NACA 4415 airfoil over a no-slip ground at low Reynolds numbers (300⩽Re⩽500) and high angles of attack (12∘⩽α⩽20∘). We find that: (i) the strength of the recirculation zone behind the airfoil is a key parameter controlling the absolute/convective nature of the instability in the boundary layer downstream; (ii) when Re, α or the ground clearance increases, the energy gain also increases, with the optimal perturbations switching from being three dimensional to two dimensional; and (iii) classical hairpin vortices, or Klebanoff modes, can be produced by three-dimensional optimal perturbations on a two-dimensional steady base flow containing a laminar separation bubble. Knowledge of the spatiotemporal features of the optimal mode could aid the design of advanced strategies for flow control. This study offers new insight into the transient growth behavior of airfoil–ground flow systems at low Re and high α, contributing to a better understanding of the ground-effect aerodynamics of small insects and micro aerial vehicles.

Flow dynamics in longitudinally grooved duct

Flow in a finite-width rectangular duct with a corrugated top-bottom wall has been studied. The primary goal is to establish geometries that allow early flow destabilization at a possibly low drag increase. The flow is assumed periodic in the streamwise direction and bounded by the duct sidewalls in the spanwise direction; the top and bottom wall corrugations have a form of sinusoidal waves oriented transversely to the flow and form longitudinal grooves; i.e., the lines of constant elevation (or phase) are parallel to the direction of the flow. The analysis is performed up to the Reynolds numbers resulting in the formation of secondary states. The first part of the analysis is focused on the properties of the two-dimensional base flow. Mainly, the dependence of hydraulic losses and drag reducing properties on duct’s geometry is characterized. The second part of the analysis discusses the onset of the three-dimensional travelling wave instability over a wide spectrum of geometric configurations. Linear stability is investigated by means of the direct numerical simulation of the Navier-Stokes equations. Critical conditions for the onset of instabilities at a range of geometric parameters are determined. Finally, the nonlinear saturation of unstable modes and the resulting secondary flows are examined. We have shown that in the state resulting from the nonlinear saturation of the disturbance, the flow becomes more complex while the drag reducing properties of the base flow can be maintained.

Stability of the wakes of cylinders with triangular cross-sections

The stability of the wakes of cylinders with triangular cross-sections at incidence is investigated using Floquet stability analysis to elucidate the effects of cylinder inclination on the dominant flow instability. The upper limit of the Reynolds numbers (scaled by the height projected by the cylinder in this study) at which the wake of the two-dimensional base flow is time periodic is$Re\approx 140$for most cylinder inclinations, exceeding which the flow becomes aperiodic, restricting the range of Reynolds numbers permitted for the stability analysis. Two different instability modes are predicted to manifest as the first-occurring mode at various cylinder inclinations – a regular mode possessing perturbation structures consistent with mode A dominates the wakes of cylinders at inclinations$\unicode[STIX]{x1D6FC}\lesssim 34.6^{\circ }$and$\unicode[STIX]{x1D6FC}\gtrsim 55.4^{\circ }$, with a subharmonic mode consistent with mode C emerging as the primary mode in the wakes of the cylinder at the intermediate range of inclinations. For all inclinations, the mode B branch is not detected within the range of Reynolds numbers examined. The peak instability growth rates corresponding to mode A for all cylinder inclinations describe a linear variation with$(Re-Re_{A})/Re_{A}$, where$Re_{A}$is the mode A transition Reynolds number, while those corresponding to mode C vary only approximately linearly. The generalized trend most pertinently shows mode C to develop more rapidly than mode A at inclinations which permit it. Examination of the near wake of the two-dimensional time-periodic base flow demonstrates the dependence of the development and intensity of mode C on imbalances in the flow solution over each shedding period, directly implying that the two-dimensional base flow solutions deviate from the half-period-flip map as the cylinder inclination is increased. The degree of asymmetry of the two-dimensional base flow relative to the ideal half-period-flip map is quantified using several measures. The results show distinctly different trends in these asymmetry measures between inclinations where modes A or C are dominant, agreeing with results from the stability analysis. The nature of the predicted instability modes at transition are also investigated by applying the Stuart–Landau equation, showing the transitions to be supercritical for all cylinder inclinations, with mode C being consistently more strongly supercritical than mode A.

Flow transition and hydrothermal wave instability of thermocapillary-driven flow in a free rectangular liquid film

Thermocapillary-driven flow in a free liquid film, which has two gas-liquid interfaces, is experimentally investigated. Silicone oil of 5 cSt is employed as the test fluid. Two-dimensional basic flow known as ‘double-layered flow’ after Ueno and Torii (2010) is realized under small-enough Marangoni number Ma, the non-dimensional number to describe the intensity of thermocapillary effect, under the geometry considered in the present study. The flow exhibits a transition from the two-dimensional steady flow state to the three-dimensional oscillatory state when the intensity of the imposed thermocapillary effect along the free surfaces exceeds the threshold. In this oscillatory regime, two types of hydrothermal wave instabilities are observed: the traveling-wave flow and the standing-wave flow. We especially focus on the traveling-wave instability and compare it with a hydrothermal wave in a liquid layer with a single free surface investigated by other researchers.

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the Orr-Sommerfeld and Squire (OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a re-orthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes (N-S) equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is pro-posed. The algorithm is then applied to a one-dimensional base flow (the plane Poiseuille flow) and a two-dimensional base flow (the plane Poiseuille flow with a low-speed streak) in a channel. For one-dimensional cases, the effects of the spanwise width of the chan-nel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is character-ized by high-and low-speed streaks and the corresponding streamwise vortical structures. The lift-up mechanism that induces the transient growth of perturbations is discussed. The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one-and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the cross-check method.

Stability of a radially stretching disk beneath a uniformly rotating fluid

The steady radial stretching of a disk beneath a rigidly rotating flow with constant angular velocity is considered. The steady base flow is determined numerically for both a stretching and a shrinking disk. The convective instability properties of the flow are examined using temporal stability analysis of the governing Rayleigh equation, and typically for small to moderate radial wave numbers, the range of azimuthal wave numbers β , over which the flow is unstable increases for both a stretched and a shrinking disk, compared to the unstretched case. The inviscid absolute instability properties of the resulting base flows are also examined using spatiotemporal stability analysis. For suitably large stretching rates, the flow is absolutely unstable in only a small range of positive β . For small stretching rates there exists a second region of absolute instability for a range of negative β values. In this region the “effective” two-dimensional base flow, comprised of a linear combination of the radial and azimuthal velocity profiles that enter the Rayleigh equation calculation, has a critical point (unlike for β > 0 ) that can dominate the absolute instability growth rate contribution compared to the shear layer component. A similar behavior is found to occur for a radially shrinking disk, except these profiles have a strong shear layer structure and hence are more unstable than the stretching disk profiles. We thus find for a suitably large shrinking rate the absolute instability contribution from the critical point becomes subdominant to the shear layer contribution.

Linear stability of confined flow around a 180-degree sharp bend

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Optimal perturbations in time-dependent variable-density Kelvin–Helmholtz billows

We analyse the influence of the specific features of time-dependent variable-density Kelvin–Helmholtz (VDKH) roll-ups on the development of three-dimensional secondary instabilities. Due to inertial (high Froude number) baroclinic sources of spanwise vorticity at high Atwood number (up to 0.5 here), temporally evolving mixing layers exhibit a layered structure associated with a strain field radically different from their homogeneous counterpart. We use a direct-adjoint non-modal linear approach to determine the fastest growing perturbations over a single period of the time-evolving two-dimensional base flow during a given time interval $[t_{0},T]$. When perturbations are seeded at the initial time of the primary KH mode growth, i.e. $t_{0}=0$, it is found that additional mechanisms of energy growth are onset around a bifurcation time $t_{b}$, a little before the saturation of the primary two-dimensional instability. The evolution of optimal perturbations is thus observed to develop in two distinct stages. Whatever the Atwood number, the first period $[t_{0},t_{b}]$ is characterised by a unique route for optimal energy growth resulting from a combination of the Orr and lift-up transient mechanisms. In the second period $[t_{b},T]$, the growing influence of mass inhomogeneities raises the energy gain over the whole range of spanwise wavenumbers. As the Atwood number increases, the short spanwise wavelength perturbations tend to benefit more from the onset of variable-density effects than large wavelength ones. The extra energy gain due to increasing Atwood numbers relies on contributions from spanwise baroclinic sources. The resulting vorticity field is structured into two elongated dipoles localised along the braid on either side of the saddle point. In return they yield two longitudinal velocity streaks of opposite sign which account for most of the energy growth. This transition towards three-dimensional motions is in marked contrast with the classic streamwise rib vortices, so far accepted as the paradigm for the transition of free shear flows, either homogeneous or not. It is argued that the emergence of these longitudinal velocity streaks is generic of the transition in variable-density shear flows. Among them, the light round jet is known to display striking side ejections as a result of the loss of axisymmetry. The present analysis helps to renew the question of the underlying flow structure behind side jets, otherwise based on radial induction between pairs of counter-rotating longitudinal vortices (Monkewitz & Pfizenmaier, Phys. Fluids A, vol. 3 (5), 1991, pp. 1356–1361). Instead, it is more likely that side ejections would result from the convergence of the longitudinal velocity streaks near the braid saddle point. When the injection time is delayed so as to suppress the initial stage of energy growth, a new class of perturbations arises at low wavenumber with energy gains far larger than those observed so far. They correspond to the two-dimensional Kelvin–Helmholtz secondary instability of the baroclinically enhanced vorticity braid discovered by Reinaud et al. (Phys. Fluids, vol. 12 (10), pp. 2489–2505), leading potentially to another route to turbulence through a two-dimensional fractal cascade.

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

We perform BiGlobal linear stability analysis on flow past a NACA0012 airfoil at 16° angle of attack and Reynolds number ranging from 400 to 1000. The steady-state two-dimensional base flows are computed using a well-tested finite difference code in combination with the selective frequency damping method. The base flow is characterized by two asymmetric recirculation bubbles downstream of the airfoil whose streamwise extent and the maximum reverse flow velocity increase with the Reynolds number. The stability analysis of the flow past the airfoil is carried out under very small spanwise wavenumber β = 10−4 to approximate the two-dimensional perturbation, and medium and large spanwise wavenumbers (β = 1–8) to account for the three-dimensional perturbation. Numerical results reveal that under small spanwise wavenumber, there are at most two oscillatory unstable modes corresponding to the near wake and far wake instabilities; the growth rate and frequency of the perturbation agree well with the two-dimensional direct numerical simulation results under all Reynolds numbers. For a larger spanwise wavenumber β = 1, there is only one oscillatory unstable mode associated with the wake instability at Re = 400 and 600, while at Re = 800 and 1000 there are two oscillatory unstable modes for the near wake and far wake instabilities, and one stationary unstable mode for the monotonically growing perturbation within the recirculation bubble via the centrifugal instability mechanism. All the unstable modes are weakened or even suppressed as the spanwise wavenumber further increases, among which the stationary mode persists until β = 4.

Streak instability and generation of hairpin-vortices by a slotted jet in channel crossflow: Experiments and linear stability analysis

Streaks and hairpin-vortices are experimentally generated in a laminar plane Poiseuille crossflow by injecting a continuous jet through a streamwise slot normal to the crossflow, with air as the working media. Small disturbances form stable streaks, however, higher disturbances cause the formation of streaks which undergo instability leading to the generation of hairpin vortices. Particular emphasis is placed on the flow conditions close to the generation of hairpin-vortices. Measurements are carried out in the cases of natural and phase-locked disturbance employing smoke visualisation, particle image velocimetry, and hot-wire anemometry, which include, the dominant frequency, wavelength, and the disturbance shape (or eigenfunctions) associated with the coherent part of the velocity field. A linear stability analysis for both one- and two-dimensional base-flows is carried out to understand the mechanism of instability and good agreement of wavelength and eigenfunctions are obtained when compared to the experimental data, and a slight under-prediction of the growth-rates by the linear stability analysis consistent with the final nonlinear stages in transitional flows. Furthermore, an energy analysis for both the temporal and spatial stability analysis revels the dominance of the symmetric varicose mode, again, in agreement with the experiments, which is found to be governed by the balance of the wallnormal shear and dissipative effects rather than the spanwise shear. In all cases the anti-symmetric sinuous modes governed by the spanwise shear are found to be damped both in analysis and in our experiments.

Global Linear Instability of Flow Through a Converging–Diverging Channel

The global linear stability, where we assume no homogeneity in either of the spatial directions, of a two-dimensional laminar base flow through a spatially periodic converging–diverging channel is studied at low Reynolds numbers. A large wall-waviness amplitude is used to achieve instability at critical Reynolds numbers below ten. This is in contrast to earlier studies, which were at lower wall-waviness amplitude and had critical Reynolds numbers an order of magnitude higher. Moreover, our leading mode is a symmetry-breaking standing mode, unlike the traveling modes which are standard at higher Reynolds numbers. Eigenvalues in the spectrum lie on distinct branches, showing varied structure spanning the geometry. Our global stability study suggests that such modes can be tailored to give enhanced mixing in microchannels at low Reynolds numbers.