Time-periodic Flow Research Articles

Spatial Decay of the Vorticity Field of Time-Periodic Viscous Flow Past a Body

We study the asymptotic spatial behavior of the vorticity field, $\omega(x,t)$, associated to a time-periodic Navier-Stokes flow past a body, $\mathscr B$, in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, $\mathcal R$, $\omega$ decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by $\bar{\omega}$ its time-average over a period and by $\omega_P:=\omega-\bar{\omega}$ its purely periodic component, we prove that inside $\mathcal R$, $\bar{\omega}$ has the same algebraic decay as that known for the associated steady-state problem, whereas $\omega_P$ decays even faster, uniformly in time. This implies, in particular, that "sufficiently far" from $\mathscr B$, $\omega(x,t)$ behaves like the vorticity field of the corresponding steady-state problem.

Secondary instability in thin film flows under an inclined plane: growth of lenses on spatially developing rivulets

The response of a thin film flowing under an inclined plane, modelled using the lubrication equation, is studied. The flow at the inlet is perturbed by the superimposition of a spanwise-periodic steady modulation and a decoupled temporally periodic but spatially homogeneous perturbation. As the consequence of the spanwise inlet forcing, the so-called rivulets grow downstream and eventually reach a streamwise-invariant state, modulated along the direction perpendicular to the flow. The linearized dynamics in the presence of a time-harmonic inlet forcing shows the emergence of a time-periodic flow characterized by drop-like structures (so-called lenses) that travel on the rivulet. The spatial evolution is rationalized by a weakly non-parallel stability analysis. The occurrence of the lenses, their spacing and thickness profile, is controlled by the inclination angle, flow rate, and the frequency and amplitude of the time-harmonic inlet forcing. The faithfulness of the linear analyses is verified by nonlinear simulations. The results of the linear simulations with inlet forcing are combined with the computations of nonlinear travelling lenses solutions in a double-periodic domain to obtain an estimate of the dripping length, for a large range of conditions.

Control of Mass Flow-Rate of Viscoelastic Fluids Through Time-Periodic Electro-Osmotic Flows in a Microchannel

Abstract In the present research, we address the implications of the pulsating electric field on controlling mass flow-rate characteristics for the time-periodic electro-osmotic flow of a viscoelastic fluid through a microchannel. Going beyond the Debye-Hückel linearization for the potential distribution inside the Electric Double Layer, the Phan-Thien-Tanner constitutive model is employed to describe the viscoelastic behaviour of the fluid. The analytical/semi-analytical expressions for the velocity distribution corresponding to a steady basic part, and a transient perturbed part are obtained by considering periodic pulsations in the applied electrical field. Our results based on sinusoidal pulsations reveal that enhanced shear thinning characteristics of the viscoelastic fluids show higher amplitude of pulsations with the oscillations in the velocity gradients primarily contrived within the Electric Double Layer region. The amplitude of mass flow rates increases with increasing the viscoelastic parameter , whereas, the phase lag displays a reverse trend. The analysis for an inverse problem is extended where the required magnitude of electric field pulsations for a target mass flow rate in the form of sinusoidal pulsations. It is found that with increasing shear-thinning characteristics of the viscoelastic fluid, there is a progressive reduction in the required electric field strength to maintain an aimed mass flow rate. Besides, required electric fields for controlled mass flow with triangular and trapezoidal pulsations are also determined.

Time-Periodic Weak Solutions to Incompressible Generalized Newtonian Fluids

In this study we are interested in the Navier–Stokes-like system for generalized viscous fluids whose viscosity has a power-structure with exponent q. We develop an existence theory of time-periodic three-dimensional flows.

Experimental study on influence of oscillatory heat flux on heat transfer of R-134a time-periodic subcooled flow boiling in annular duct

The present study pertains to the experimental study on the R-134a subcooled flow boiling stemming from oscillatory heat flux in narrow annuli oriented horizontally with focus on heat transfer characteristics. The test section used in the experiment is composed of a Pyrex glass tube, which is completely designed to measure the flow boiling heat transfer coefficient (FBHTC). In this experiment, the annular gap of the horizontal duct is in the range of 1–5 mm with governing parameters used for practical applications in air conditioning and refrigerating devices. Particularly, the focus is on the effects of heat flux oscillation mean value (q‾), amplitude (Δq) and period (tp) on the heat transfer features of subcooled flow boiling. The experimental results show that the obvious temperature undershoot takes place during onset of nucleate boiling (ONB) for R-134a oscillating subcooled flow boiling. Increasing either Δq or tp has relatively little effect on the boiling curve and FBHTC at a given heat flux, while the annular gap plays an important role. Moreover, increasing either the amplitude or period of incident heat flux enhances the wall temperature and FBHTC; while an increase in mean heat flux tends to enhance the wall temperature, but has negligible effect on FBHTC for the flow boiling. Then, a flow pattern map is provided to describe the boundary that distinguishes the different boiling modes of the oscillating heat flux to the subcooled R-134a flow. Finally, a parametric analysis is carried out to further understand the effects of Δq and tp on the heating wall temperature.

Particle Accumulation Structures in a 5 cSt Silicone Oil Liquid Bridge: New Data for the Preparation of the JEREMI Experiment

Systems of solid particles in suspension driven by a time-periodic flow tend to create structures in the carrier fluid that are reminiscent of highly regular geometrical items. Within such a line of inquiry, the present study provides numerical results in support of the space experiments JEREMI (Japanese and European Research Experiment on Marangoni flow Instabilities) planned for execution onboard the International Space Station. The problem is tackled by solving the unsteady non-linear governing equations for the same conditions that will be established in space (microgravity, 5 cSt silicone oil and different aspect ratios of the liquid bridge). The results reveal that for a fixed supporting disk radius, the dynamics are deeply influenced by the height of the liquid column. In addition to its expected link with the critical threshold for the onset of instability (which makes Marangoni flow time-periodic), this geometrical parameter can have a significant impact on the emerging waveform and therefore the topology of particle structures. While for shallow liquid bridges, pulsating flows are the preferred mode of convection, for tall floating columns the dominant outcome is represented by rotating fluid-dynamic disturbance. In the former situation, particles self-organize in circular sectors bounded internally by regions of particle depletion, whereas in the latter case, particles are forced to accumulate in a spiral-like structure. The properties of some of these particle attractors have rarely been observed in earlier studies concerned with fluids characterized by smaller values of the Prandtl number.

Coating flows down a vertical fibre: towards the full Navier–Stokes problem

Abstract

Effects of velocity skewness on the linear stability of the oscillatory Stokes layer

The effects of velocity skewness on the oscillating Stokes layer are investigated. Linear stability characteristics for the family of velocity-skewed time-periodic flows are determined using Floquet theory. Neutral stability curves and critical parameter settings for instability and the structure of the eigenfunctions are presented. Velocity skewness establishes a stabilizing effect and increases the critical Reynolds number for the onset of linear instability to larger values than that found in the non-skewed Stokes layer. Solutions indicate that disturbances develop in the direction that the wall velocity achieves a maximum absolute value.

Talk about Several Time Periodic Pulse Electroosmotic Flow of Maxwell Fluid in a Circular Microchannel

Using the method of Laplace transform, analytical expressions are derived for the time periodic pulse electroosmotic flow (EOF) velocity of the triangle and sawtooth of Maxwell fluid in circular microchannel. The solution involves analytically solving the linearized Poisson-Boltzmann (P-B) equation, together with the Cauchy momentum equation and the general Maxwell constitutive equation. By numerical computations of inverse Laplace transform, the effects of electrokinetic width K, relaxation time and pulse width a on the above several pulse EOF velocities are investigated. In addition, we focused on the comparison and analysis of the formulas and graphs between the triangle and sawtooth pulse EOF with the rectangle pulse EOF. The study found that there are obvious differences in formulas and graphs between triangle and sawtooth pulse EOF with rectangle pulse EOF, and the difference mainly depends on the different definitions of the three kinds of time periodic pulse waves. Finally, we also studied the stability of the above three kinds of pulse EOF and the influence of relaxation time on pulse EOF velocity under different pulse widths is discussed. We find that the rectangle pulse EOF is more stable than the triangle and sawtooth pulse EOF. For any pulse, as the pulse width a increases, the influence of the relaxation time on the pulse EOF velocity will be weakened.

Comparing Finite-Time Lyapunov Exponents and Lagrangian Descriptors for identifying phase space structures in a simple two-dimensional, time-periodic double-gyre model

This paper compares the advantages, limitations, and computational considerations of using Finite-Time Lyapunov Exponents (FTLEs) and Descriptors (LDs) as tools for identifying barriers and mechanisms of fluid transport in two-dimensional time-periodic flows. These barriers and mechanisms of transport are often referred to as Lagrangian Coherent Structures, though this term often changes meaning depending on the author or context. This paper will specifically focus on using FTLEs and LDs to identify stable and unstable manifolds of hyperbolic stagnation points, and the Kolmogorov-Arnold-Moser (KAM) tori associated with elliptic stagnation points. The background and theory behind both methods and their associated phase space structures will be presented, and then examples of FTLEs and LDs will be shown based on a simple, periodic, time-dependent double-gyre toy model with varying parameters.

Attainability of time-periodic flow of a viscous liquid past an oscillating body

A body $$\mathscr {B}$$ is started from rest by a translational motion in an otherwise quiescent Navier–Stokes liquid filling the whole space. We show, for small data, that if after some time $$\mathscr {B}$$ reaches a spinless oscillatory motion of period $$\mathcal T$$ , the liquid will eventually execute also a time periodic motion with the same period $$\mathcal T$$ . This result is a suitable generalization of the famous Finn’s starting problem for steady states, to the case of time-periodic motions.

On the performance of harmonic balance method for unsteady flow with oscillating shocks

The harmonic balance method (HBM) is a promising alternative for simulating time-periodic unsteady flows; however, its performance in strong nonlinear problems has not been fully investigated. In this paper, we focus on the unsteady flow with oscillating shocks through which several unique aspects regarding the accuracy and efficiency of the HBM are addressed. First, different from most existing studies, influences of the number of harmonics and the number of time samples on the solution are separately analyzed. It is found that the mode factor plays a role in accurately describing the unsteady physics, while the sampling factor is associated with the numerical oscillations in shock oscillation regions. Second, based on the recently developed Jacobian-free Newton–Krylov method with a simplified preconditioner and comparison with other methods, the efficiency of the present HBM and its relationship with the two factors are studied. The computational cost is found to increase almost linearly with the increase of the number of harmonics, but nonlinearly with the number of time samples. Finally, noticing the fact that fewer harmonics (time samples) can ensure accurate solutions in smooth regions, while many more are required in shock oscillation regions, a non-uniform HBM is developed by adopting various numbers of harmonics (time samples) throughout the whole flow domain. The superiority of this method compared with the traditional uniform HBM is demonstrated, showing a good application prospect in large-scale unsteady flows with strong nonlinearities.

An averaging scheme for the efficient approximation of time-periodic flow problems

We study periodic solutions to the Navier-Stokes equations. The transition phase of a dynamic Navier-Stokes solution to the periodic-in-time state can be excessively long and it depends on parameters like the domain size and the viscosity. Several methods for an accelerated identification of the correct initial data that will yield the periodic state exist. They are mostly based on space-time frameworks for directly computing the periodic state or on optimization schemes or shooting methods for quickly finding the correct initial data that yields the periodic solution. They all have a large computational overhead in common. Here we describe and analyze a simple averaging scheme that comes at negligible additional cost. We numerically demonstrate the efficiency and robustness of the scheme for several test-cases and we will theoretically show convergence for the linear Stokes problem.

Heat transfer in electrokinetic micro-pumps under the influence of various oscillatory excitations

Thermal characteristics of time-periodic electroosmotic flow are analyzed in a micro-annulus under the influence of various alternating electric fields. Representative hydrodynamic and thermal quantities, i.e. volumetric flow rate and Nusselt number, demonstrate oscillatory behaviors approaching a quasi-steady state if few periods of time elapse. An important parameter named dimensionless frequency, which normally affects the diffusion mechanism, is responsible for the penetration depth of momentum and energy into the fluid in the radial direction. The higher the dimensionless frequency, the smaller the transport phenomena diffuse radially into the bulk fluid; while the advection mechanism is intensified and consequently Nusselt number is increased. The cooling-mode mean Nusselt number is slightly smaller than the heating- mode one. A variety of waveforms is examined in the present research; their performances are then compared together using proper measures. A strength index is introduced to evaluate the relative ability of each individual excitation in a half period of time. A key parameter named thermal/frictional index is also utilized to assess the total effectiveness of the system in different circumstances especially when the basic excitation functions are applied. Unlike the vigorous square waveform, the sawtooth one is the most efficient.

Time-averaged transport in oscillatory squeeze flow of a viscoelastic fluid

Periodically-driven flows are known to generate non-zero, time-averaged fluxes of heat or solute species, due to the interactions of out-of-phase velocity and temperature/concentration fields, respectively. Herein, we investigate such transport (a form of the well-known Taylor--Aris dispersion) in the gap between two parallel plates, one of which oscillates vertically, generating a time-periodic squeeze flow of either a newtonian or Maxwellian fluid. Using the method of multiple time-scale homogenization, the mass/heat balance equation describing transport in this flow is reduced to a one-dimensional advection--diffusion--reaction equation. This result indicates three effective mechanisms in the mass/heat transfer in the system: an effective diffusion that spreads mass/heat along the concentration/temperature gradient, an effective advective flux, and an effective reaction that releases or absorbs mass/heat - in the time-averaged frame. Our results demonstrate that there exist resonant modes under which the velocity peaks when the dimensionless plate oscillation frequency (embodied by the Womersley number, the ratio of the transient inertia to viscous forces) approaches specific values. As a result, transport in this flow is significantly influenced by the dimensionless frequency. On the one hand, the effective, time-averaged dispersion coefficient is always larger than the molecular diffusivity, and is sharply enhanced near resonance. The interaction between fluid elasticity and the oscillatory forcing enhances the efficiency of transport in the system. On the other hand, the identified effective advection and reaction mechanisms may transport mass/heat from regions of high concentration/temperature to those of low concentration/temperature, or vice versa, depending on the value of dimensionless frequency.

Linearized lattice Boltzmann Method for time periodic electro-osmotic flows in micro- and nanochannels

Time periodic electro-osmosis (TPEO) is a popular means to pump liquids or manipulate species of interest in today’s micro- and nanofluidic devices. In this article, we propose a double distribution-function lattice Boltzmann (LB) model to describe its oscillatory flows coupled with electrokinetics in micro- and nanochannels. To remove advective effects, we derive the LB model from a linearized Boltzmann Bhatnagar–Gross–Krook-like equation and formulate its equations depending on the alternating current (AC) frequency, instead of time. This treatment facilitates a direct comparison of the LB results to experimental measurements in practical applications. We assessed accuracy of the proposed frequency-based Linearized LB model by simulating time periodic electro-osmotic flows (TPEOFs) with a thin and a thick electric double layer (EDL) at different Stokes parameters. The results are in excellent agreement with analytical solutions. The model was used to simulate TPEOFs with various EDL thicknesses and those driven by an AC electric field combined with an oscillatory pressure gradient. The simulations show distinct distributions of the electric potential and solution velocity subject to different length ratios and frequency ratios in the flows and interesting flow responses to compounding influences of the applied electric and mechanical driving fields. Importantly, diverse vortex patterns and vorticity variations were also revealed for TPEOFs in heterogeneously charged channels. These results demonstrate that the LB model developed in this article can well capture rich TPEO flow characteristics in micro- and nanochannels. It is effective for design and optimization of TPEO-based micro- and nanofluidic devices.

Dynamics of nonisothermal two-thin-fluid-layer systems subjected to harmonic tangential forcing under Rayleigh–Taylor instability conditions

The stability of a nonisothermal system consisting of two superimposed fluid layers: a thin liquid film layer and a gas layer sandwiched between differentially heated horizontal solid plates in the gravity field, is investigated. The system is assumed to be subjected to the Rayleigh–Taylor instability (RTI) with the Marangoni effect that either enhances the RTI or opposes it and to the tangential harmonic vibration of the upper substrate. A set of reduced evolution equations is derived based on the weighted-residual integral boundary layer approach, and the investigation is carried out in the framework of this set. The base state of the system represents a time-periodic flow, and its linear stability analysis is carried out using the Floquet theory in the large-time limit. The nonlinear dynamics of the system is investigated numerically in the case of either a static or vibrating substrate. Among the possible outcomes of the nonlinear dynamics, there is the emergence of ruptured states of the liquid film with rupture taking place at either the upper or lower substrate and also the emergence of saturated continuous flows of the liquid film. We also find that the nonlinear dynamics of the system is consistent with the results of the linear stability analysis in terms of enhancement or attenuation of interfacial distortion.

Experimental study on bubble characteristics of time periodic subcooled flow boiling in annular ducts due to wall heat flux oscillation

The significant effects of imposed wall heat flux oscillation on bubble characteristics of subcooled flow boiling with R-134a in horizontal annular ducts are experimentally performed in details. Specific emphasis has been applied on the bubble dynamics of the time periodic subcooled flow boiling characteristics influenced by the mean levels, amplitudes, and periods of the wall heat flux oscillations. The obtained experimental results have disclosed that the bubble parameters like the departure diameter, frequency and active nucleation site density, do oscillate periodically in time and are at the same frequency as the imposed wall heat flux oscillation. Nevertheless, these parameters also increase at long periods, high amplitudes of imposed heat flux oscillation or high values of mean imposed heat flux. Furthermore, the bubble size reduced after the time lag when the imposed heat flux decreases with time.

Physical topology of three-dimensional unsteady flows with spheroidal invariant surfaces.

Scope is the response of Lagrangian flow topologies of three-dimensional time-periodic flows consisting of spheroidal invariant surfaces to perturbation. Such invariant surfaces generically accommodate nonintegrable Hamiltonian dynamics and, in consequence, intrasurface topologies composed of islands and chaotic seas. Computational studies predict a response to arbitrary perturbation that is dramatically different from the classical case of toroidal invariant surfaces: said islands and chaotic seas evolve into tubes and shells, respectively, that merge into "tube-and-shell" structures consisting of two shells connected via (a) tube(s) by a mechanism termed "resonance-induced merger" (RIM). This paper provides conclusive experimental proof of RIM and advances the corresponding structures as the physical topology of realistic flows with spheroidal invariant surfaces; the underlying unperturbed state is a singular limit that exists only for ideal conditions and cannot be achieved in a physical experiment. This paper furthermore expands existing theory on certain instances of RIM to a comprehensive theory (supported by experiments) that explains all observed instances of this phenomenon. This theory reveals that RIM ensues from perturbed periodic lines via three possible scenarios: truncation of tubes by (i) manifolds of isolated periodic points emerging near elliptic lines or by either (ii) local or (iii) global segmentation of periodic lines into elliptic and hyperbolic parts. The RIM scenario for local segmentation includes a perturbation-induced change from elliptic to hyperbolic dynamics near degenerate points on entirely elliptic lines (denoted "virtual local segmentation"). This theory furthermore demonstrates that RIM indeed accomplishes tube-shell merger by exposing the existence of invariant surfaces that smoothly extend from the tubes into the chaotic shells. These phenomena set the response to perturbation-and physical topology-of flows with spheroidal invariant surfaces fundamentally apart from flows with toroidal invariant surfaces. Its entirely kinematic nature and reliance solely on continuity and solenoidality of the velocity field render the comprehensive theory and its findings universal and generically applicable for (arbitrary perturbation of) basically any incompressible flow-in fact any smooth solenoidal vector field-accommodating spheroidal invariant surfaces.

Time-resolved characterization of microchannel flow boiling during transient heating: Part 1 – Dynamic response to a single heat flux pulse

Microchannel flow boiling is an attractive approach for the thermal management of high-heat-flux electronic devices that are often operated in transient modes. In Part 1 of this two-part study, the dynamic response of a heated 500 μm channel undergoing flow boiling of HFE-7100 is experimentally investigated for a single heat flux pulse. Three heat flux levels exhibiting highly contrasting flow behavior under constant heating conditions are used: a low heat flux corresponding to single-phase flow (15 kW/m2), an intermediate heat flux corresponding to continuous flow boiling (75 kW/m2), and a very high heat flux which exceeds critical heat flux and would cause dryout if applied continuously (150 kW/m2). Transient testing is conducted by pulsing between these three heat flux levels and varying the pulse duration. High-frequency measurements of heat flux, wall temperature, pressure drop, and mass flux are synchronized to high-speed flow visualizations to characterize the boiling dynamics during the pulses. At the onset of boiling, the dynamic response resembles that of an underdamped mass-spring-damper system subjected to a unit step input. During transitions between single-phase flow and time-periodic flow boiling, the wall temperature temporarily over/under-shoots the eventual steady operating temperature (e.g., by up to 20 °C) thus demonstrating that transient performance can extend beyond the bounds of steady performance. It is shown that longer duration high-heat-flux pulses (up to ~50% longer in some cases) can be withstood when the fluid in the microchannel is initial boiling, relative to if it is initially in the single-phase flow regime, despite being at an initially higher heat flux and wall temperature prior to the pulse.