Time-periodic Flow Research Articles

A mapping method for distributive mixing with diffusion: Interplay between chaos and diffusion in time-periodic sine flow

We present an accurate and efficient computational method for solving the advection-diffusion equation in time-periodic chaotic flows. The method uses operator splitting, which allows the advection and diffusion steps to be treated independently. Taking advantage of flow periodicity, the advection step is solved using a mapping method, and diffusion is “added” discretely after each iteration of the advection map. This approach results in the construction of a composite mapping matrix over an entire period of the chaotic advection-diffusion process and provides a natural framework for the analysis of mixing. To test the approach, we consider two-dimensional time-periodic sine flow. By comparing the numerical solutions obtained by our method to reference solutions, we find qualitative agreement for large time steps (structure of concentration profile) and quantitative agreement for small time steps (low error). Further, we study the interplay between mixing through chaotic advection and mixing through diffusion leading to an analytical model for the evolution of the intensity of segregation with time. Additionally, we demonstrate that our operator splitting mapping approach can be readily extended to three dimensions.

Oscillatory subcooled flow boiling heat transfer of R-134a and associated bubble characteristics in a narrow annular duct due to flow rate oscillation

An experiment is conducted here to investigate how an imposed time periodic flow rate oscillation in the form of a triangular wave affects the long time subcooled flow boiling heat transfer and associated bubble characteristics of refrigerant R-134a in a horizontal narrow annular duct. In the experiment the mean R-134a mass flux G¯ varies from 200 to 500kg/m2s, imposed heat flux ranges from 0 to 45kW/m2, and the amplitude of the mass flux oscillation changes from 0 to 30% of G¯ with the period of the mass flux oscillation varied from 20 to 120s for the inlet liquid subcooling ranging from 0 to 6°C. The duct gap is fixed at 2.0mm. The results indicate that the inlet liquid subcooling significantly affects the oscillatory flow boiling heat transfer characteristics. Besides, when the imposed heat flux is close to that for the onset of stable flow boiling, intermittent flow boiling appears. The intermittent boiling prevails in a very different range of the Boiling number for a change in the inlet subcooling. Moreover, in the subcooled boiling the heated wall temperature, bubble departure diameter and frequency, and active nucleation site density also oscillate periodically in time. Furthermore, in the persistent boiling at high imposed heat flux the resulting Tw oscillation is stronger for a higher inlet liquid subcooling and for a longer period and a larger amplitude of the mass flux oscillation. And for a larger amplitude of the mass flux oscillation, stronger temporal oscillations in dp, f and nac are noted. Finally, a flow regime map is provided to delineate the boundaries separating different boiling regimes for the oscillatory R-134a subcooled flow boiling in the annular duct.

An efficient harmonic balance method for unsteady flows in cascades

In this paper, nonlinear unsteady flows in cascades arising from self-excited blade vibrations are investigated using a mixed time/frequency domain harmonic balance technique. The time-periodic flow is computed at several sub-time levels that are equally spaced over a single period. The time derivative term in the unsteady Navier–Stokes equations is approximated by a pseudo-spectral operator, which couples solutions at different sub-time levels. Compared to the classical time-accurate approach, the present technique enjoys an improved computational efficiency. In addition, using complex periodic boundary conditions, the computational domain can be reduced to a grid spanning only a single blade in the cascade, which further improves the computational efficiency. With the use of the pseudo-spectral operator and pseudo-time marching, the problem can be treated as steady state and convergence acceleration techniques such as local time stepping, implicit residual smoothing and multigrid can be used. To demonstrate the efficiency and accuracy of the technique, we present results for the Tenth Standard and the Eleventh Standard Configurations.

Stability of a time-periodic flow incorporating a simple model for noise

A horizontal layer of Boussinesq fluid is known to be susceptible to vortex instability when the lower surface of the fluid is subjected to a time-periodic heating of sufficient strength. In this paper a quasi-steady technique is used to trace the stability of the layer right from a standard linear theory to the problem when the vortices are so strong so as to change the thermal profile at leading order. Furthermore, this system is used to assess the possible influence of the noise that inevitably arises in many physical and laboratory situations. Even though in many problems a fluid flow is assumed to be time-periodic with one dominant driving frequency, in practice many cases are actually contaminated by small components at higher frequencies, which are often multiples of the primary. Here we make a crude model of noise by taking the temperature of the lower surface of the fluid layer to be the sum of two time-dependent periodic functions of differing frequencies. It is shown that the presence of the second frequency can have a dramatic effect on the evolution of a vortex and only a small element of noise appears to be sufficient to generate vortices that penetrate as much as three times further into the layer compared with the situation when the noise is absent.

Footprints of Lagrangian flow structures in Eulerian concentration distributions in periodic mixing flows

Transport of passive tracers may be described through the spatio-temporal evolution of Eulerian concentration distributions or via the geometrical composition of the Lagrangian flow structure. The present study seeks to deepen insight into the connection between the Eulerian and Lagrangian perspectives by investigating the role of Lagrangian coherent structures (LCSs) in the Eulerian concentration distributions in time-periodic and spatially-periodic mixing flows. Representation of the Eulerian transport by the mapping method, describing realistic transport problems by distribution matrices, admits a generic analysis based on matrix and graph theory. This reveals that LCSs–and the transport barriers that separate them–leave a distinct “footprint” in the eigenmode spectrum of the distribution matrix and, by proxy, of the underlying Eulerian transport operator. Transport barriers impart a block-diagonal structure upon the mapping matrix, where each block matrix A corresponds with a given LCS. Its kind is reflected in the spectrum of A; higher-order periodicity yields a distinct permutation within A. The composition of the distribution matrix versus the Lagrangian flow structure thus predicted is demonstrated by way of examples. These findings increase fundamental understanding of transport phenomena and have great practical potential for e.g. flow and mixing control.

Harmonic balance methods applied to computational fluid dynamics problems

In this paper, we briefly review the classical harmonic balance method, and describe the adaptation of the method required for its application to computational fluid dynamics models of unsteady time periodic flows. We describe several variations of the method including a classical balancing method with pseudo time relaxation, the nonlinear frequency domain form and the time spectral form. We show that the maximum stable Courant–Friedrichs–Lewy (CFL) number for explicit schemes is dependent on the grid reduced frequency, a non-dimensional parameter that depends on the cell size, characteristic wave speed, and the highest frequency retained in the harmonic balance analysis. We apply the harmonic balance methods to several nonlinear unsteady flow problems and show that even strongly nonlinear flows can be modelled accurately with a small number of harmonics retained in the model.

Turbulent Flame Speeds of G-equation Models in Unsteady Cellular Flows

We perform a computationl study of front speeds of G-equation models in time dependent cellular flows. The G-equations arise in premixed turbulent combustion, and are Hamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strain G-equations are also non-convex with degenerate diffusion. The computation is based on monotone finite difference discretization and weighted essentially nonoscillatory (WENO) methods. We found that the large time front speeds lock into the frequency of time periodic cellular flows in curvature-strain G-equations similar to what occurs in the basic inviscid G-equation. However, such frequency locking phenomenon disappears in viscous G-equation, and in the inviscid G-equation if time periodic oscillation of the cellular flow is replaced by time stochastic oscillation.

Asymptotic Behavior of Solutions to the Compressible Navier--Stokes Equation Around a Time-Periodic Parallel Flow

The global in time existence of strong solutions to the compressible Navier--Stokes equation around time-periodic parallel flows in $\mathbb{R}^n$, $n\geq 2$, is established under smallness conditions on Reynolds number, Mach number, and initial perturbations. Furthermore, it is proved for $n=2$ that the asymptotic leading part of solutions is given by a solution of the one-dimensional viscous Burgers equation multiplied by the time-periodic function. In the case $n\geq 3$ the asymptotic leading part of solutions is given by a solution of the $n-1$-dimensional heat equation with the convective term multiplied by the time-periodic function.

Exact solution to the inverse Womersley problem for pulsatile flows in cylindrical vessels, with application to magnetic particle targeting

An exact solution to the inverse Womersley problem was derived for the fully-developed, laminar pulsatile flow of a viscous Newtonian fluid, within a circular cylindrical vessel with rigid walls. In particular, given an arbitrary, time-periodic flow rate, the axisymmetric velocity profile was obtained by means of two neat and computable maps relating the corresponding Fourier coefficients. The study of such an inverse problem is motivated by the fact that flow rate is the main physical quantity which can be actually measured in many practical situations. The hypothesis of a fully-developed flow was deliberately introduced, in order to obtain an analytical solution (otherwise hardly achievable). Despite the intrinsic simplifications associated with the adopted position (which restrict the applicability of our results to 3D finite-length complex domains, and non-Newtonian fluids), the obtained solution provides a benchmark – and at the same time an approximation – for the inverse problem of pulsatile flows, it may serve as a debugging tool for more ambitious numerical approaches based on realistic data, and can also be used as an improved source of boundary data. As expected, the main advantage of our analytical solutions (compared to fully numerical approaches) resides in computational efficiency; this was quantitatively assessed through numerical tests. Moreover, the proposed solution was applied in the context of magnetic particle targeting, to highlight some peculiar effects on particle trajectories and capture efficiency due to pulsatility. Such a transport problem is increasingly drawing the attention of an interdisciplinary community, ranging from physicians to biomedical engineers, physicists and roboticists, thanks to its potential for targeted therapy, up to remote guidance of intravascular devices. More in general, the obtained benchmark solution holds potential for effectively exploitation in an interdisciplinary context.

Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane

We consider the two-dimensional motion of a Navier-Stokes liquid in the whole plane, under the action of a time-periodic body force $F$ of period $T$, and tending to a prescribed nonzero constant velocity at infinity. We show that if the magnitude of $F$, in suitable norm, is sufficiently small, there exists one and only one corresponding time-periodic flow of period $T$ in an appropriate function class.

Numerical Study on Conjugate Conduction–Convection in a Cubic Enclosure Submitted to Time-Periodic Sidewall Temperature

The laminar conjugate conduction-natural convection heat transfer in a cubic enclosure of finite thickness conductive walls and central cavity filled with fluid is comprehensively studied by using recently developed high accuracy temporal-spatial multidomain pseudospectral method. The enclosure is assumed to have one sidewall submitted to time-periodic pulsating temperature and the opposing sidewall constant temperature, and the top, bottom and two lateral sidewalls are adiabatic. The present study is devoted to explore the fluid mechanics and heat transfer mechanisms of the time-periodic conjugate conduction-natural convection in the enclosure, with particular highlights on the heat transfer resonance and back heat transfer phenomena, the perturbation propagation patterns and the three-dimensional characteristics. The computations are performed for wide ranges of controlling parameters of engineering significance, i.e., the dimensionless wall thickness 0 ≤ s ≤ 0.10, the solid–fluid thermal conductivity ratio 10 ≤ k ≤ 50 and diffusivity ratio 0.001 ≤ a ≤ 0.1, and the sidewall temperature pulsating period 1 ≤ P ≤ 103. Numerical results reveal that the time-periodic fluid flow and conjugate heat transfer performances of the enclosure system are greatly affected by the conductive walls and complexly dependent on the controlling parameters. The thickness and thermophysical properties of the conductive walls, together with the pulsating period of the sidewall temperature, govern the sidewall temperature disturbance propagation patterns (amplitude, phase position and speed) within the enclosure. The heat transfer resonance only appears in cases of large diffusivity ratio, but the variation of period-averaged heat transfer rate with respect to the pulsating period is quite different from that of the zero wall thickness enclosure. The back heat transfer exists in region close to the corners formed by either the top or bottom walls and the enclosure hot sidewall, and the former is more remarkable in both scale and duration and is gradually disappearing as the pulsating period increases.

Alternating Current Electro-Osmotic Flow of the Maxwell Fluids Through a Circular Micro-Pipe

AbstractAnalytical solutions are presented for time periodic EOF flow of linear viscoelastic fluids through a cylindrical micro-pipe. The linear viscoelastic fluids used here are described by the general Maxwell model. The solution involves analytically solving the linearized Poisson-Boltzmann equation, together with the Cauchy momentum equation and the general Maxwell constitutive equation considering the depletion effect produced by the interaction between macro-molecules of the Maxwell fluid and the channel surface. The overall flow is divided into depletion layer and bulk flow outside of depletion layer. The velocity expressions of these two layers were obtained, respectively. By numerical computations, the influences of the periodic EOF electric oscillating Reynolds number Re, Deborah number De, depletion layer thickness δ and the viscosity ratio γ of Maxwell to Newtonian fluids on velocity profile are presented. For a prescribed De, the increasing Re will cause large changes of the EOF velocity with decreasing velocity magnitude. For a given Re, large De gives large EOF velocity magnitude. Increasing γ will lead to larger velocity amplitude for a given lower Re. However, at higher Re, the velocity amplitude decreases with the viscosity ratio γ, especially within the depletion layer. In addition, large depletion layer thickness gives small EOF velocity magnitude for fixed Re and De. Finally, the influence of De on energy dissipation is studied. These results provide a detail insight of the flow characteristic of this flow configuration.

Four Decades of Studying Global Linear Instability: Progress and Challenges

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two (and periodic in one) or all three spatial directions. After a brief exposition of the theory, some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton-Krylov time-stepping method into a standard finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers.

Alternating current (AC) electroosmotic flow of generalized Maxwell fluids through a circular microtube

An analytical solution for the time periodic electroosmotic flow (EOF) of the generalized Maxwell model through a circular microtube is presented by solving the linearized Poisson-Boltzmann equation, together with Cauchy momentum equation and General Maxwell constitutive equation. By numerical calculations, we find that for lower Ω and smaller De, classical plug-like profile of Newtonian fluids is reduced. At lower frequency, the flow field can extend to the whole microtube. At higher frequency, however, the velocity amplitude variations away from electric double layer (EDL) almost decrease to zero. In addition, larger De leads to rapid variations of EOF velocity profiles with increased amplitude. The velocity amplitude of microtube is larger than that of plate microchannel by comparison. Key words: Circular microtube, alternating current (AC) electroosmotic flow, generalized Maxwell fluid, normalized oscillation frequency, relaxation time.

Flow instability in baffled channel flow

Flow instability in baffled channel flow, where thin baffles are mounted on both channel walls periodically in the direction of the main flow, has been numerically investigated. The geometry considered here can be regarded as a simple model for finned heat exchangers. The aim of this investigation is to understand how baffle interval (L) and Reynolds number (Re) influence the flow instability. With a fixed baffle length of one quarter of channel height (H), ratios of baffle interval to channel height (RB=L/H) between 1 and 4 are considered. The critical Reynolds number of the primary instability, a Hopf bifurcation from steady flow to time-periodic flow, turned out to be minimum when RB=3.08. The friction factor (f) is strongly correlated with the critical Reynolds number for RB⩽2.5. For the particular cases of RB=1.456 and RB=1.0, we performed Floquet stability analysis in order to study the secondary instability through which time-periodic two-dimensional flow bifurcates into three-dimensional flow. The results obtained in this investigation are in good agreement with those computed from full simulations, and shed light on understanding and controlling flow characteristics in a finned heat exchanger, quite beneficial to its design.

Molecular dynamics simulations of oscillatory Couette flows with slip boundary conditions

The effect of interfacial slip on steady-state and time-periodic flows of monatomic liquids is investigated using non-equilibrium molecular dynamics simulations. The fluid phase is confined between atomically smooth rigid walls, and the fluid flows are induced by moving one of the walls. In steady shear flows, the slip length increases almost linearly with shear rate. We found that the velocity profiles in oscillatory flows are well described by the Stokes flow solution with the slip length that depends on the local shear rate. Interestingly, the rate dependence of the slip length obtained in steady shear flows is recovered when the slip length in oscillatory flows is plotted as a function of the local shear rate magnitude. For both types of flows, the friction coefficient at the liquid-solid interface correlates well with the structure of the first fluid layer near the solid wall.

Observability of periodic lines in three-dimensional lid-driven cylindrical cavity flows

This study employs three-dimensional particle-tracking velocimetry (3D-PTV) for experimental investigation of the existence and properties of periodic lines in 3D lid-driven time-periodic flows inside a cylindrical cavity. These periodic lines, consisting of material points that periodically return to their initial position, play a central role in the transport properties of laminar flows, yet their existence has so far been demonstrated only in numerical simulations. The formation and characteristics of periodic lines are inextricably linked with spatiotemporal symmetries of the flow. 3D-PTV measurements determined that relevant symmetries, identified with previous symmetry analyses, are satisfied within experimental error bounds. These measurements subsequently isolated periodic lines in the designated symmetry planes, thus offering first experimental evidence of their physical existence and their fundamental reliance on symmetries. Experimental periodic lines are topologically equivalent to those in simulated flows with identical symmetries and exhibit the same response to changes in forcing conditions. The laboratory experiments by these observations bridge the gap from theoretical and numerical predictions on periodic lines to real 3D flows.

Comment on “Ordering of Small Particles in One-Dimensional Coherent Structures by Time-Periodic Flows”

A Comment on the Letter by D. O. Pushkin, D. E. Melnikov, and V. M. Shevtsova Phys. Rev. Lett. 106, 234501 (2011). The authors of the Letter offer a Reply.Received 1 July 2011DOI:https://doi.org/10.1103/PhysRevLett.108.249401© 2012 American Physical Society

DECAY PROPERTIES OF SOLUTIONS TO THE LINEARIZED COMPRESSIBLE NAVIER–STOKES EQUATION AROUND TIME-PERIODIC PARALLEL FLOW

Decay estimates on solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow are established. It is shown that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in L2 norm as an (n - 1)-dimensional heat kernel. Furthermore, it is proved that the asymptotic leading part of solutions is given by solutions of an (n - 1)-dimensional linear heat equation with a convective term multiplied by time-periodic function.

Passive scalar decay in chaotic flows with boundaries

This paper considers the long-time decay rate of a passive scalar in two-dimensional flow. The focus is on the effects of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain . The boundary conditions on the passive scalar are imposed on the boundary of by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar is exponential in time with a decay rate γ(κ), where κ is the molecular diffusivity. Scaling laws of the form γ(κ) ≃ Cκα for small κ are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, α ≃ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that α ≃ 1/2 for a plane layer, but find α ≃ 2/3 in a square subdomain where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain , the exponent measuring the decay of the scalar field is α ≃ 1/2 for a slip flow condition and α ≃ 3/4 for a no-slip condition. The scaling law exponents α for chaotic time-periodic flows are compared with those for similarly constructed random flows.