Time-periodic Flow Research Articles

An Analytical Model of the Kelvin–Helmholtz Instability of Transverse Coronal Loop Oscillations

Abstract Recent numerical simulations have demonstrated that transverse coronal loop oscillations are susceptible to the Kelvin–Helmholtz (KH) instability due to the counterstreaming motions at the loop boundary. We present the first analytical model of this phenomenon. The region at the loop boundary where the shearing motions are greatest is treated as a straight interface separating time-periodic counterstreaming flows. In order to consider a twisted tube, the magnetic field at one side of the interface is inclined. We show that the evolution of the displacement at the interface is governed by Mathieu’s equation, and we use this equation to study the stability of the interface. We prove that the interface is always unstable and that, under certain conditions, the magnetic shear may reduce the instability growth rate. The result, that the magnetic shear cannot stabilize the interface, explains the numerically found fact that the magnetic twist does not prevent the onset of the KH instability at the boundary of an oscillating magnetic tube. We also introduce the notion of the loop σ-stability. We say that a transversally oscillating loop is σ-stable if the KH instability growth time is larger than the damping time of the kink oscillation. We show that even relatively weakly twisted loops are σ-stable.

Interfacial Fourier transform shear rheometry of complex fluid interfaces

Nonlinear rheometry of interfaces is very challenging because of the limits of current day instrumentation and the intricate coupling of flows at interfaces and in the bulk. The use of time periodic flows may constitute a first step in addressing this issue. Fourier transform rheology (FTR) measurements with quasi-monolayers at the air-water interface are used in order to assess the suitability of the different devices to investigate nonlinear interfacial shear viscoelasticity. The probe material was a poly (methyl methacrylate) forming a soft glassy interface, whereas complementary measurements were performed with a polystyrene latex suspension forming a predominantly viscous interface at intermediate surface coverages. The obtained data with the magnetic rod rheometer (ISR) were compared against those obtained with the bicone and the double wall ring geometries attached to standard rotational rheometers. In particular, an unexpected appearance of even in addition to odd harmonics is discussed in terms of flow field asymmetry.

Stability of time-periodic flow with laminar boundary-layer separation

The influence of the low-frequency modulation of flow behind a rectangular backward-facing step on the amplitude characteristics of disturbances in the separated laminar boundary layer has been studied. The experimental data were obtained by the method of hot-wire anemometry in a wind tunnel at a low subsonic velocity. Response of the separated flow to the long-wave oscillations generated by a local source of disturbances on the surface of the experimental model was clarified. The low-frequency nonstationarity of the separation region leads to a growth of velocity fluctuations in the separated boundary layer, which dominate the laminar-turbulent transition and the state of the flow in the near-wall region.

Role of Klebanoff modes in active flow control of separation: direct numerical simulations

Our previous research has shown that an active flow control strategy using two-dimensional (2-D) harmonic blowing and suction with properly chosen frequency and amplitude can significantly reduce the separation region, delay transition to turbulence and can even relaminarize the flow. How such effective flow control for transition delay and relaminarization is affected by free-stream turbulence (FST) remains an open question. In order to answer this question, highly resolved direct numerical simulations (DNS) are carried out where very low-amplitude isotropic FST fluctuations are introduced at the inflow boundary of the computational domain. With FST the effectiveness of the flow control is not diminished, and the extent of the separated flow region is reduced by the same amount as for the zero FST case. However, a striking difference observed in the DNS is the fact that in the presence of even very low levels of FST, the flow transitions shortly downstream of the reattachment location of the bubble, contrary to the case without FST. It appears that this different behaviour for even very small levels of FST is caused by an interaction between the high-amplitude 2-D disturbances introduced by the flow control forcing and 3-D Klebanoff modes (K-modes) that are generated by the FST. The streamwise elongated streaks due to the K-modes cause a spanwise-periodic modulation of the basic flow and subsequently of the primary 2-D wave. The disturbances associated with this modulation exhibit strong growth and initiate the breakdown process to turbulence. Linear secondary instability investigations with respect to low-frequency 3-D disturbances are carried out based on the linearized Navier–Stokes equations. The response of the forced flow to the low-frequency 3-D disturbances reveals that the time-periodic base flow is unstable with respect to a wide range of 3-D modes. In particular, the wavelength associated with the spanwise spacing of the K-mode falls into the range of, and is in fact very close to, the most unstable 3-D disturbances. Results from the secondary instability analysis and the comparison with DNS results, support the conjecture that the forcing amplitude has a major impact on the onset and amplification rate of the K-modes: lowering the forcing amplitude postpones the onset of the growth of the K-modes and reduces the growth rate of the K-modes for a given FST intensity. The net effect of these two events is a delay of the transition onset. Nevertheless, the instability mechanism that governs the transition process is the same as previously identified, i.e. interaction of the K-mode and 2-D primary wave. Furthermore, for low levels of FST, the amplitude of the low-frequency K-modes scales linearly with the FST intensity in the approach boundary layer up to the secondary instability regime.

Convergence in time-periodic quasilinear parabolic equations in one space dimension

We consider time-periodic quasilinear parabolic equations in the domain {(t,x)∈R2|0<x<r(t),t>0}, where the right boundary r(t) of the spatial interval is a positive function which might be periodic, or asymptotically periodic, or a function tending to infinity, or infinity. We show that, in the first case (that is, r(t) is a periodic function), any bounded solution of the equations converges as t→∞ to a periodic one; in the other three cases, any positive bounded solution converges as t→∞ to a nonnegative periodic one. Using such a result, we study the long time dynamics of the initial-boundary value problem on the half line, as well as the Stefan free boundary problem, of a general heterogeneous reaction–diffusion equation. Also, we use the convergence result to study the long time dynamics of the initial-boundary value problem for a time-periodic (mean) curvature flow equation.

The influence of two-dimensional temperature modulation on nonlinear Marangoni waves in two-layer films

The nonlinear dynamics of waves generated by the deformational oscillatory Marangoni instability in a two-layer film under the action of a two-dimensional temperature modulation on the solid substrate is considered. A system of long-wave equations governing the deformations of the upper surface and the interface between the liquids is presented. The long-wave approach is applied. The nonlinear simulations reveal the existence of different dynamic regimes, including stationary, time-periodic and quasi-periodic flows. The general diagrams of the flow regimes are constructed.

High-frequency thermal-fluidic characterization of dynamic microchannel flow boiling instabilities: Part 1 – Rapid-bubble-growth instability at the onset of boiling

Dynamic flow boiling instabilities are studied experimentally in a single, 500 µm-diameter glass microchannel subjected to a uniform heat flux. Fluid flow is driven through the microchannel in an open-loop test facility by maintaining a constant pressure difference between a pressurized upstream reservoir and a reservoir at the exit that is open to the ambient; the working fluid is HFE-7100. This hydrodynamic boundary condition resembles that of an individual channel in a parallel-channel heat sink where the channel mass flux can vary in time. Simultaneous high-frequency measurement of reservoir, inlet, and outlet pressures, pressure drop, mass flux, inlet and outlet fluid temperatures, and wall temperature is synchronized to high-speed flow visualizations enabling transient characterization of the thermal-fluidic behavior. Part 1 of this study investigates the rapid-bubble-growth instability at the onset of boiling; the effect of flow inertia and inlet liquid subcooling is assessed. The mechanisms underlying the rapid-bubble-growth instability, namely, a large liquid superheat and a large pressure spike, are quantified; this instability is shown to cause flow reversal and can result in large temperature spikes. Low flow inertia exacerbates the rapid-bubble-growth instability by starving the heated channel of liquid replenishment for longer durations and results in severe temperature increases. In the case of high flow inertia or high inlet liquid subcooling, flow reversal is still observed at the onset of boiling, but results in a minimal wall temperature rise because liquid quickly replenishes the heated channel. A companion paper (Part 2) investigates the effect of flow inertia, inlet liquid subcooling, as well as heat flux on the thermal-fluidic oscillations during time-periodic flow boiling that follows the initial incipience at the onset of boiling considered here.

Electroviscous effect and electrokinetic energy conversion in time periodic pressure-driven flow through a parallel-plate nanochannel with surface charge-dependent slip

In this paper we analytically investigate the electroviscous effect and electrokinetic energy conversion in the time periodic pressure-driven flow of an incompressible viscous Newtonian liquid through a parallel-plate nanochannel with surface charge-dependent slip. Analytical and semi-analytical solutions for electric potential, velocity and streaming electric field are obtained and are utilized to compute electrokinetic energy conversion efficiency. The results show that velocity amplitude and energy conversion efficiency are reduced when the effect of surface charge on slip length is considered. The surface charge effect increases with zeta potential and ionic concentration. In addition, the energy conversion efficiency is large when the ratio of channel half-height to the electric double layer thickness is small. The boundary slip results in a large increase in energy conversion. Higher values of the frequency of pressure pulsation lead to higher values of the energy conversion efficiency. We also obtain the energy conversion efficiency in constant pressure-driven flow and find that the energy conversion efficiency in periodical pressure-driven flow becomes larger than that in constant pressure-driven flow when the frequency is large enough.

Experimental comparison and visualization of in-tube continuous and pulsating flow boiling

This experimental study investigated the application of fluid flow pulsations for in-tube flow boiling heat transfer enhancement in an 8 mm smooth round tube made of copper. The fluid flow pulsations were introduced by a flow modulating expansion device and were compared with continuous flow generated by a stepper-motor expansion valve in terms of the time-averaged heat transfer coefficient. The cycle time ranged from 1 s to 7 s for the pulsations, the time-averaged refrigerant mass flux ranged from 50 kg m−2 s−1 to 194 kg m−2 s−1 and the time-averaged heat flux ranged from 1.1 kW m−2 to 30.6 kW m−2. The time-averaged heat transfer coefficients were reduced from transient measurements immediately downstream of the expansion valves with 2 K and 20 K subcooling upstream, resulting in inlet vapor qualities at 0.05 and 0.18, respectively, and covered the saturated flow boiling range up to the dry-out inception. Averaged results of the considered range of vapor qualities, refrigerant mass flux and heat flux showed that the pulsations at low cycle time (1 s) improved the time-averaged heat transfer coefficients by 5.6% and 2.2% for the low and high subcooling, respectively. However, the pulsations at high cycle time (7 s) reduced the time-averaged heat transfer coefficients by 1.8% and 2.3% for the low and high subcooling, respectively, due to significant dry-out when the flow-modulating expansion valve was closed. Furthermore, the flow pulsations were visualized by high-speed camera to assist in understanding the time-periodic flow regimes and the effect they had on the heat transfer performance.

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.

Effects of planar shear on the three-dimensional instability in flow past a circular cylinder

A Floquet stability analysis has been carried out in order to investigate how a planar shear in wake flow affects the three-dimensional (3D) instability in the near-wake region. We consider a circular cylinder immersed in a freestream with planar shear. The cylinder was implemented in a Cartesian grid system by means of an immersed boundary method. Planar shear tends to promote the primary instability, known as Hopf bifurcation where steady flow bifurcates into time-periodic flow, in the sense that its critical Reynolds number decreases with increasing planar shear. The effects of planar shear on the 3D instability are different depending on the type of 3D instability. The flow asymmetry caused by the planar shear suppresses a QP-type mode but generates a C-type mode. The conventional A and B modes are stabilized by the planar shear, whereas mode C is intensified with increasing shear. The criticality of each 3D mode is discussed, and the neutral stability curves for each 3D mode are presented. The curren...

Shear banding in large amplitude oscillatory shear (LAOStrain and LAOStress) of soft glassy materials

We study theoretically shear banding in soft glassy materials subject to large amplitude time-periodic shear flows, considering separately the protocols of large amplitude oscillatory shear strain, large amplitude square or triangular or sawtooth strain rate, and large amplitude oscillatory shear stress. In each case, we find shear banding to be an important part of the material's flow response for a broad range of values of the frequency $\omega$ and amplitude of the imposed oscillation. Crucially, and highly counterintuitively, in the glass phase this persists even to the lowest frequencies accessible numerically (that seems consistent with it persisting even when $\omega\to 0$), although the soft glassy rheology model we use has an underlying monotonic constitutive curve of shear stress as a function of shear rate, and is therefore unable to support shear banding as its steady state response at constant shear rate. We attribute this to the repeated competition, within each flow cycle, of glassy aging and flow rejuvenation. Besides reporting significant banding in the glass phase, where the flow curve has a yield stress, we also observe it at noise temperatures just above the glass point, where the flow curve is of power law fluid form. Thus, our results suggest a predisposition to shear banding in flows of even extremely slow time-variation, for both aging yield stress fluids, and for power law fluids with sluggish relaxation timescales. We show that shear banding can have a pronounced effect on the shape of the Lissajous-Bowditch curves that are commonly used to fingerprint complex fluids rheologically. We therefore counsel caution in seeking to compute such curves calculations that impose upfront a homogeneous shear flow, discarding the possibility of banding. We also analyze the stress response to the imposed strain waveforms in terms of a `sequence of physical processes'.

Residual diffusivity in elephant random walk models with stops

We study the enhanced diffusivity in the so called elephant random walk model with stops (ERWS) by including symmetric random walk steps at small probability $\epsilon$. At any $\epsilon > 0$, the large time behavior transitions from sub-diffusive at $\epsilon = 0$ to diffusive in a wedge shaped parameter regime where the diffusivity is strictly above that in the un-perturbed ERWS model in the $\epsilon \downarrow 0$ limit. The perturbed ERWS model is shown to be solvable with the first two moments and their asymptotics calculated exactly in both one and two space dimensions. The model provides a discrete analytical setting of the residual diffusion phenomenon known for the passive scalar transport in chaotic flows (e.g. generated by time periodic cellular flows and statistically sub-diffusive) as molecular diffusivity tends to zero.

Time-delayed feedback technique for suppressing instabilities in time-periodic flow

Time-delayed feedback technique for suppressing instabilities in time-periodic flow

Nonlinear dynamics of an elliptic vortex embedded in an oscillatory shear flow.

The nonlinear dynamics of an elliptic vortex subjected to a time-periodic linear external shear flow is studied numerically. Making use of the ideas from the theory of nonlinear resonance overlaps, the study focuses on the appearance of chaotic regimes in the ellipse dynamics. When the superimposed flow is stationary, two general types of the steady-state phase portrait are considered: one that features a homoclinic separatrix delineating bounded and unbounded phase trajectories and one without a separatrix (all the phase trajectories are bounded in a periodic domain). When the external flow is time-periodic, the ensuing nonlinear dynamics differs significantly in both cases. For the case with a separatrix and two distinct types of phase trajectories: bounded and unbounded, the effect of the most influential nonlinear resonance with the winding number of 1:1 is analyzed in detail. Namely, the process of occupying the central stability region associated with the steady-state elliptic critical point by the stability region associated with the nonlinear resonance of 1:1 as the perturbation frequency gradually varies is investigated. A stark increase in the persistence of the central regular dynamics region against perturbation when the resonance of 1:1 associated stability region occupies the region associated with the steady-state elliptic critical point is observed. An analogous persistence of the regular motion occurs for higher perturbation frequencies when the corresponding stability islands reach the central stability region associated with the steady-state elliptic point. An analysis for the case with the resonance of 1:2 is presented. For the second case with only bounded phase trajectories and, therefore, no separatrix, the appearance of much bigger stability islands associated with nonlinear resonances compared with the case with a separatrix is reported.

Quantifying the tangling of trajectories using the topological entropy.

We present a simple method to efficiently compute a lower limit of the topological entropy and its spatial distribution for two-dimensional mappings. These mappings could represent either two-dimensional time-periodic fluid flows or three-dimensional magnetic fields, which are periodic in one direction. This method is based on measuring the length of a material line in the flow. Depending on the nature of the flow, the fluid can be mixed very efficiently which causes the line to stretch. Here, we study a method that adaptively increases the resolution at locations along the line where folds lead to a high curvature. This reduces the computational cost greatly which allows us to study unprecedented parameter regimes. We demonstrate how this efficient implementation allows the computation of the variation of the finite-time topological entropy in the mapping. This measure quantifies spatial variations of the braiding efficiency, important in many practical applications.

Linear and nonlinear perturbation analysis of the symmetry breaking in time-periodic propulsive wakes.

The two-dimensional and time-periodic wake flows produced by a pitching foil are investigated numerically for a fixed flapping amplitude. As the flapping frequency is increased, three regimes are identified in the time-marching nonlinear simulations. The first regime is characterized by nondeviated wake flows with zerotime-averagedlift. In the second regime, the wake flow is slightly deviated from the streamwise direction and the time-averaged lift is slightly positive or negative. The third regime is characterized by larger deviations of the wake, associated with larger values of both the time-averaged lift and the thrust. The transition from the first to the second regime is examined by performing a Floquet stability analysis of the nondeviated wake. A specific method is introduced to compute the time-periodic, nondeviated wake when it is unstable. It is found that one synchronous antisymmetric mode becomes unstable at the critical frequency where deviation occurs. Investigation of its instantaneous and time-averaged characteristics show that it acts as a displacement mode translating the nondeviated wake away from the streamwise direction. Finally, it is demonstrated that the transition from the second to the third regime is linked to nonlinear effects that amplify both antisymmetric and symmetric perturbations around the foil.

A Foundation for Modeling Time-Periodic Groundwater Flow

The authors of a new book on groundwater hydraulics and hydrology describe a theoretical foundation for modeling time-periodic groundwater flow.

Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method

AbstractWe study the residual diffusion phenomenon in chaotic advection computationally via adaptive orthogonal basis. The chaotic advection is generated by a class of time periodic cellular flows arising in modeling transition to turbulence in Rayleigh-Bénard experiments. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, the solutions of the advection-diffusion equation develop sharp gradients, and demand a large number of Fourier modes to resolve, rendering computation expensive. We construct adaptive orthogonal basis (training) with built-in sharp gradient structures from fully resolved spectral solutions at few sampled molecular diffusivities. This is done by taking snapshots of solutions in time, and performing singular value decomposition of the matrix consisting of these snapshots as column vectors. The singular values decay rapidly and allow us to extract a small percentage of left singular vectors corresponding to the top singular values as adaptive basis vectors. The trained orthogonal adaptive basis makes possible low cost computation of the effective diffusivities at smaller molecular diffusivities (testing). The testing errors decrease as the training occurs at smaller molecular diffusivities. We make use of the Poincaré map of the advection-diffusion equation to bypass long time simulation and gain accuracy in computing effective diffusivity and learning adaptive basis. We observe a non-monotone relationship between residual diffusivity and the amount of chaos in the advection, though the overall trend is that sufficient chaos leads to higher residual diffusivity.

Interactions of a Low-Aspect-Ratio Cantilevered Dynamic Pin with a Boundary Layer

The interaction of a low-aspect-ratio finite-span pin with a laminar flow over a flat plate was investigated experimentally. The pin was either static or dynamically oscillated in the wall-normal direction at a driving frequency that was either equivalent to the natural shedding frequency or its subharmonic. The mean height of the pin was either equal to 1 or 1.5 times the local boundary-layer thickness. The study focused on the formation and development of two main vortical structures: the arch-type vortex and the horseshoe vortex. It was shown that the arch vortex combined with the counter-rotating horseshoe vortex. The shedding of this coupled phenomenon was investigated under static and dynamic conditions to provide comparisons between their corresponding flowfield behaviors. Under dynamic conditions, the arch-type vortex periodically shed from the pin’s free end and advected downstream, interacting with the horseshoe vortex, which resulted in complex, three-dimensional, and time-periodic flow patterns. The dynamics of these flow structures were determined to be dependent upon the driving frequency and actuation amplitude.