Time-periodic Flow Research Articles

Monolithic One-Shot Optimization for Time-Periodic Flows Using Harmonic Balance

Adjoint methods for aerodynamic design optimization have been a considerable area of research in computational fluid dynamics. Several approaches have been developed to fully couple the flow, adjoint, and design equations for steady flows. However, only limited attempts have been made to apply those to unsteady flows using a monolithic approach. This study aims to fully couple the primal, adjoint, and design equations based on a harmonic balance technique allowing the unsteady flow optimization to be performed in a single monolithic computation. The harmonic balance is the appropriate method for such problems as it reformulates unsteady time-periodic problems as “mathematically steady,” where one does not need to store the entire time history of the primal solution. The novel unsteady technique will be applied to a time-periodic lid-driven cavity. Here, the Reynolds number is taken to be the design variable so as to match a target vorticity distribution at the first subtime level of the time-spectral (harmonic balance) solution. Finally, mean drag minimization of a rotationally oscillating cylinder in crossflow is performed. In all cases, it is shown that the proposed method is quite efficient, requiring less than a couple of hundred total iterations to identify the “optimum” conditions.

Efficient computation of time-periodic compressible flows with spectral techniques

A systematic approach to parametrically analyze compressible time-periodic flows is proposed. Instead of time-stepping simulations, a Fourier–Galerkin strategy is adopted, which consists of the projection of the periodic solution onto a truncated Fourier series. Starting from the compressible Navier–Stokes equations, a truncated nonlinear problem coupling the 2N+1 fields, representing the Fourier discretization, is derived. This nonlinear problem is solved by Newton iteration and a set of efficient algorithms for the resolution of the arising inner linear systems is proposed. Compared to alternative methods, the formulation is free of any numerical constraint affecting performance, e.g. a CFL-like condition. It is also free of aliasing effects arising in other spectral techniques. The efficiency of the method is illustrated for two configurations where sound is radiated due to a flow instability, namely the flow around a circular cylinder and the flow through two successive apertures, i.e. the hole-tone configuration.

Time periodic electroosmotic flow in a pH-regulated parallel-plate nanochannel

In this paper, the separation of variables method is applied to investigate the effects of solution pH, background salt concentration and AC electric field frequency on time periodic electroosmotic flow in a pH-regulated parallel-plate nanochannel. The surface charge is generated by the protonation and deprotonation of the functional group SiOH. The background salt is KCl. The pH value of the solution is adjusted by HCl and KOH. Analytical and semi-analytical solutions for electric potential and velocity distributions are obtained. The results show that the electric potential caused by the electric double layer depends greatly on the solution pH and background salt concentration. The amplitudes of the velocity and flow rate of the time periodic electroosmotic flow decrease with the background salt concentration and increase with the deviation of the solution pH from the isoelectric point. In a nanochannel having a height less than 100 nm, the electroosmotic velocity amplitude is not affected by the AC electric field frequency because the oscillating Reynolds number is much less than unity.

Time-periodic flows of magnetic fluids

Time-periodic flows of magnetic fluids

Invariant manifold approach for quantifying the dynamics of highly inertial particles in steady and time-periodic incompressible flows.

The dynamics of finite-sized particles with large inertia are investigated in steady and time-dependent flows through the numerical solution of the invariance equation, describing the spatiotemporal evolution of the slow/inertial manifold representing the effective particle velocity field. This approach allows for an accurate reconstruction of the effective particle divergence field, controlling clustering/dispersion features of particles with large inertia for which a perturbative approach is either inaccurate or not even convergent. The effect of inertia on heavy and light particles is quantified in terms of the rate of contraction/expansion of volume elements along a particle trajectory and of the maximum Lyapunov exponent for systems exhibiting chaotic orbits, such as the time-periodic sine-flow on the 2D torus and the time-dependent 2D cavity flow.

Time-periodic pulse electroosmotic flow of Jeffreys fluids through a microannulus

Abstract In this article, we investigate the time-periodic pulse electroosmotic flow (EOF) of Jeffreys fluids through a microannulus. By using the Laplace transform method, the velocity expression of the pulse EOF is derived. The effect of some variables on the time it takes for the fluid to go from a static state to a flowing state is analyzed. We find that increasing the relaxation time λ ¯ 1 {\bar{\lambda }}_{\text{1}} and decreasing the inner and outer radius ratio α \alpha will result in longer time for the fluid to reach the flowing state, but the retardation time λ ¯ 2 {\bar{\lambda }}_{\text{2}} and the inner and outer zeta potential ratio β \beta have little effect on it. The impact of some related parameters on the pulse EOF velocity for different inner and outer radius ratios ( α \alpha ) is discussed in detail. The results show that for a smaller inner and outer radius ratio α \alpha , the velocity amplitude increases with the relaxation time λ ¯ 1 {\bar{\lambda }}_{\text{1}} and decreases with the retardation time λ ¯ 2 {\bar{\lambda }}_{\text{2}} . As the inner and outer radius ratio α \alpha increases, the effect of relaxation time λ ¯ 1 {\bar{\lambda }}_{\text{1}} on velocity amplitude gradually weakens or even becomes insignificant, and the effect of the retardation time λ ¯ 2 {\bar{\lambda }}_{\text{2}} on the velocity amplitude remains unchanged. Moreover, the velocity amplitude will decrease with the increase in the inner and outer radius ratio α \alpha and its change range will expand from the electric double layer near the annular wall to the entire flow region.

Time periodic pulse electroosmotic flow of Jeffreys fluids through a parallel plate microchannel

AbstractThe purpose of this paper is to investigate the time periodic pulse electroosmotic flow (EOF) of Jeffreys fluids through a parallel plate microchannel. The driving mode of pulse EOF here is considered as a rectangle pulse. Using the method of Laplace transform, the analytical solution of velocity and volumetric flow rate of pulse EOF is derived. The effects of pertinent dimensionless parameters on the velocity and volumetric flow rate are discussed in detail. Results show that the pulse width is a valuable parameter for analyzing the influence of relaxation time and retardation time on magnitude of velocity and volumetric flow rate when the fluid flow reaches the steady state. Increasing the relaxation time reduces he magnitude of velocity and volumetric flow rate at low value pulse width , yet the trend is opposite for larger pulse width . The increase of retardation time will reduce the magnitude of velocity and volumetric flow rate no matter whether the value of pulse width is large or not. Furthermore, the frequency of velocity profile and volumetric flow rate is different because of the different pulse width . With the increase of pulse width , the different change frequency of velocity profile and volumetric flow rate slows down, which means a long cycle time. And as the relaxation time and pulse width grow, the time required for the fluid flow to reach a steady state will become longer. The conclusions have theoretical significance for biofluid based microfluidic transport systems driven by pulsed electric field.

Non-stationary rarefied gas flow in a plane channel with a series of oscillating barriers

Present paper is dedicated to the numerical study of transient free-molecular gas flow through the microchannel with a series of barriers, oscillating perpendicular to the axis of the channel. Phases of barriers motion are arranged in a specific way to obtain a “wave-like” motion of barriers system. Time-dependent flow rate pattern is studied for different values of problem parameters — oscillation frequencies, number of barriers, distance between them, pressure drop in the channel. It is obtained that pattern of gas flow rate significantly depends on the ratio of characteristic “wavespeed” of barriers system motion and the thermal speed of molecules. It is shown that frequency of output pulsatile flow may be significantly higher than the frequency of barriers mechanical oscillations, depending on the number of barriers, which can be used in practice for creation of high-frequency pulsatile flows. Similarly to authors previous paper Kosyanchuk, et al. (2017) it was obtained that motion of barriers system can lead to separation of gas mixtures with different molecular masses. In contrast to “quasi-stationary” (time-periodic) flow regime (at times around 100-1000 oscillation periods), it is shown that in transient regime (at times less than 10-100 oscillation periods) separation factor can be significantly higher. Moreover, separation effect in transient regime is observed even for the values of defining parameters, which do not grant separation in quasi-stationary flow regime.

Conjugate and cut points in ideal fluid motion

Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which ‘pull’ nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows—a class of stationary solutions of the Euler equations—on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow.

Transient Flow Boiling and Maldistribution Characteristics in Heated Parallel Channels Induced by Flow Regime Oscillations

Flow boiling provides an effective means of heat removal but can suffer from thermal and hydrodynamic transients that compromise heat transfer performance and trigger device failure. In this study, the transient flow boiling characteristics in two thermally isolated, hydrodynamically coupled parallel microchannels are investigated experimentally. High-speed flow visualization is synchronized to high-frequency heat flux, wall temperature, pressure drop, and mass flux measurements to provide time-resolved characterization. Two constant and two transient heating conditions are presented. For a constant heat flux of 63 kW/m² into each channel, boiling occurs continuously in both channels and the parallel channel instability is observed to occur at 15 Hz. Time-periodic oscillations in the pressure drop and average mass flux are observed, but corresponding oscillations in the wall temperatures are virtually nonexistent at this condition. At a slightly lower constant heat flux of 60 kW/m², boiling remains continuous in one of the channels, but the other channel experiences time-periodic flow regime oscillations between single-phase and two-phase flow. At this condition, extreme time-periodic wall temperature oscillations are observed in both channels with a long period (~7 s) due to oscillations in the severity of the flow maldistribution. For the transient heating conditions, square-wave heating profiles oscillating between different heat flux levels are applied to the channels. Because of their relatively high frequency, the heating transients are attenuated by the microchannel walls, resulting in effectively constant heating conditions and flow boiling characteristics like that of the aforementioned constant heating conditions. This study illustrates the susceptibility of parallel two-phase heat sinks to flow maldistribution, particularly when undergoing transient flow regime oscillations.

Phase-based control of periodic flows

Unsteady bluff-body flows exhibit dominant oscillatory behaviour owing to periodic vortex shedding. The ability to manipulate this vortex shedding is critical to improving the aerodynamic performance of bodies in a flow. This goal requires a precise understanding of how the perturbations affect the asymptotic behaviour of the oscillatory flow and of the ability to control transient dynamics. In this work, we develop an energy-efficient flow-control strategy to alter the oscillation phase of time-periodic fluid flows rapidly. First, we perform a phase-sensitivity analysis to construct a reduced-order model for the response of the flow oscillation to impulsive control inputs at various phases. Next, we introduce a real-time optimal phase-control strategy based on the phase-sensitivity function obtained by solving the associated Euler–Lagrange equations as a two-point boundary-value problem. Our approach is demonstrated for the incompressible laminar flow past a circular cylinder and an airfoil. We show the effectiveness of phase control with different actuation inputs, including blowing and rotary control. Moreover, our control approach is a sensor-based approach without the need for access to high-dimensional measurements of the entire flow field.

Bacteria hinder large-scale transport and enhance small-scale mixing in time-periodic flows

Understanding mixing and transport of passive scalars in active fluids is important to many natural (e.g., algal blooms) and industrial (e.g., biofuel, vaccine production) processes. Here, we study the mixing of a passive scalar (dye) in dilute suspensions of swimming Escherichia coli in experiments using a two-dimensional (2D) time-periodic flow and in a simple simulation. Results show that the presence of bacteria hinders large-scale transport and reduces overall mixing rate. Stretching fields, calculated from experimentally measured velocity fields, show that bacterial activity attenuates fluid stretching and lowers flow chaoticity. Simulations suggest that this attenuation may be attributed to a transient accumulation of bacteria along regions of high stretching. Spatial power spectra and correlation functions of dye-concentration fields show that the transport of scalar variance across scales is also hindered by bacterial activity, resulting in an increase in average size and lifetime of structures. On the other hand, at small scales, activity seems to enhance local mixing. One piece of evidence is that the probability distribution of the spatial concentration gradients is nearly symmetric with a vanishing skewness. Overall, our results show that the coupling between activity and flow can lead to nontrivial effects on mixing and transport.

Transient linear stability of pulsating Poiseuille flow using optimally time-dependent modes

Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr–Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien–Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.

Optimal energy growth in pulsatile channel and pipe flows

Pulsatile channel and pipe flows constitute a fundamental flow configuration with significant bearing on many applications in the engineering and medical sciences. Rotating machinery, hydraulic pumps or cardiovascular systems are dominated by time-periodic flows, and their stability characteristics play an important role in their efficient and proper operation. While previous work has mainly concentrated on the modal, harmonic response to an oscillatory or pulsatile base flow, this study employs a direct–adjoint optimisation technique to assess short-term instabilities, identify transient energy-amplification mechanisms and determine their prevalence within a wide parameter space. At low pulsation amplitudes, the transient dynamics is found to be similar to that resulting from the equivalent steady parabolic flow profile, and the oscillating flow component appears to have only a weak effect. After a critical pulsation amplitude is surpassed, linear transient growth is shown to increase exponentially with the pulsation amplitude and to occur mainly during the slow part of the pulsation cycle. In this latter regime, a detailed analysis of the energy transfer mechanisms demonstrates that the huge linear transient growth factors are the result of an optimal combination of Orr mechanism and intracyclic normal-mode growth during half a pulsation cycle. Two-dimensional sinuous perturbations are favoured in channel flow, while pipe flow is dominated by helical perturbations. An extensive parameter study is presented that tracks these flow features across variations in the pulsation amplitude, Reynolds and Womersley numbers, perturbation wavenumbers and imposed time horizon.

Control of stationary convective instabilities in the rotating disk boundary layer via time-periodic modulation

The control of stationary convective instabilities in the rotating disk boundary layer via a time-periodic modulation of the disk rotation rate is investigated. The configuration provides an archetypal example of a three-dimensional temporally periodic boundary layer, encompassing both the von Kármán and Stokes boundary layers. A velocity–vorticity formulation of the governing perturbation equations is deployed, together with a numerical procedure that utilises the Chebyshev-tau method. Floquet theory is used to determine the linear stability properties of these time-periodic flows. The addition of a time-periodic modulation to the otherwise steady disk rotation rate establishes a stabilising effect. In particular, for a broad range of modulation frequencies, the growth of the stationary convective instabilities is suppressed and the critical Reynolds number for the onset of both the cross-flow and Coriolis instabilities is raised to larger values than that found for the steady disk without modulation. An energy analysis is undertaken, where it is demonstrated that time-periodic modulation induces a reduction in the Reynolds stress energy production and an increase in the viscous dissipation across the boundary layer. Comparisons are made with other control techniques, including distributed surface roughness and compliant walls.

A scalable spectral Stokes solver for simulation of time-periodic flows in complex geometries

Simulation of unsteady creeping flow in complex geometries has traditionally required the use of a time-stepping procedure, which is typically costly and unscalable. To reduce the cost and allow for computations at much larger scales, we propose an alternative approach that is formulated based on the unsteady Stokes equation expressed in the time-spectral or frequency domain. This transformation results in a boundary value problem with an imaginary source term proportional to the computed mode that is discretized and solved in a complex-valued finite element solver using Bubnov-Galerkin formulation. This transformed spatio-spectral formulation presents several advantages over the traditional spatio-temporal techniques. Firstly, for cases with boundary conditions varying smoothly in time, it provides a significant saving in computational cost as it can resolve time-variation of the solution using a few modes rather than thousands of time steps. Secondly, in contrast to the traditional time integration scheme with a finite order of accuracy, this method exhibits a super convergence behavior versus the number of computed modes. Thirdly, in contrast to the stabilized finite element methods for fluid, no stabilization term is employed in our formulation, producing a solution that is consistent and more accurate. Fourthly, the proposed approach is embarrassingly parallelizable owing to the independence of the solution modes, thus enabling scalable calculations at a much larger number of processors. The comparison of the proposed technique against a standard stabilized finite element solver is performed using two- and three-dimensional canonical and complex geometries. The results show that the proposed method can produce more accurate results at 1% to 11% of the cost of the standard technique for the studied cases.

Prediction on steady-oscillatory transition via Hopf bifurcation in a three-dimensional (3D) lid-driven cube

Massive computations of three-dimensional (3D) flow in subcritical flow state are carried out to investigate the transition from 3D steady flow to time-periodic oscillatory flow via Hopf bifurcation in a cube. Previous prediction of the steady-oscillatory transition in published works has been summarized. The transition point in lid-driven cavity has yet to come to consensus as the critical parameters are predicted in a wide range of values. The 3D unsteady governing equations are solved numerically using the spectral collocation method (SCM) in combination with the artificial compressibility method (ACM), SCM-ACM, a combined method with high accuracy that we developed. The temporal velocity evolution, amplitude evolution and dimensionless angular frequency are investigated. The accurate critical Reynolds number Recr is predicted by Richardson extrapolation. Results show that the initial conditions have significant effect to obtain the velocity which gradually approaches to steady state in the subcritical flow state, and appropriate initial conditions are given to efficiently obtain stable results by gradually increasing Re. Exponential decay of velocity amplitude becomes weaker with the increase of Re, and the decay rate is strictly linear against Re, while the dimensionless angular frequency remains constant. The Recr is affected by the number of nodes, which has also been observed by Feldman and Gelfgat [1] and Kuhlmann and Albensoeder [2], while the critical frequency is unaffected by the number of nodes in the present results. It also indicates that the Recr and the constant dimensionless angular frequency can be innovatively predicted with more accuracy and much fewer nodes. The improved point of steady-oscillatory transition in 3D lid-driven cube is Recr=1916.6 via Hopf bifurcation by Richardson extrapolation with dimensionless angular frequency ω=0.5752.

Asymmetric boundary conditions and their relevance in minimal passive mass species transport in time-periodic electro-osmotic flows

A set of dimensionless groups along with constraints on their values are formulated to serve as an indication of the parametric combinations, leading to minimal enhancement of streamwise passive species transport in time-periodic electro-osmotic microchannel flows with asymmetric hydrodynamic slip and zeta potential boundaries. While the slip lengths and zeta potentials are mostly treated as uncoupled in the electrokinetic transport literature, we incorporate into the analysis wall surface charge/potential coupled apparent slip lengths and find that more than one minimal enhancement instances may be achieved during a zeta potential sweep, provided that the uncharged surface slip length ratio is large enough.

Mixing driven by transient buoyancy flows.II. Flow dynamics

The chaotic advection of the interface between two miscible liquids inside a closed cavity, generated by a damped oscillatory buoyancy-driven (BD) regular flow field, is investigated experimentally for BD mixing. The Lagrangian history of interface motion, determined using the planar laser-induced fluorescence and the photographic full-field view method, is contrasted against the Eulerian flow field measured from particle image velocimetry. Chaotic advection stretches and folds the interface at an early stage to produce an asymmetric pairwise Rayleigh–Taylor (RT) morphology (RTM) structure from long wavelength RT instability and short-time Richtmyer–Meshkov instability and its fractal interface structure at a high impulsive-Reynolds number. The mechanism of folding, from global bifurcation of the flow field, caused by a hyperbolic point, served as an organizing center for multiple vortex interactions. The intermediate-stage kinematics of the RTM structure exhibits RT mixing and shows unfolding of the lamellar structure from the net effect of stretching, folding, and molecular diffusion prior to its breakdown; and it has a probabilistic outcome of exhibiting topological transitions through a breakup of the RTM structure in phase space from necking singularity and pinch-off, indicating sensitivity to the initial conditions. The effectiveness of mixing determined from mixing efficiency is contrasted against mechanical and lamellar models of mixing. The determination of topological entropy, from an approximate Gaussian distribution of the interface length stretch, yields time scale for information decay comparable to time scale for which a low-order horseshoe map emerges from flow, indicating local chaos of the interface. The late-stage breakdown of the RTM structure from internal and wall collision drives the interaction between advection and diffusion, which indicates that critical mixing time scales as the logarithmic of Peclet number, comparable to time-periodic sine flow and blinking vortex flow chaotic mapping models.

Modeling of Fluid Flow in a Flexible Vessel with Elastic Walls

We exploit a two-dimensional model (Ghosh et al. in Q J Mech Appl Math 71(3):349–367, 2018; Kozlov and Nazarov in Dokl Phys 56(11):560–566, 2011, J Math Sci 207(2):249–269, 2015) describing the elastic behavior of the wall of a flexible blood vessel which takes interaction with surrounding muscle tissue and the 3D fluid flow into account. We study time periodic flows in an infinite cylinder with such intricate boundary conditions. The main result is that solutions of this problem do not depend on the period and they are nothing else but the time independent Poiseuille flow. Similar solutions of the Stokes equations for the rigid wall (the no-slip boundary condition) depend on the period and their profile depends on time.