Periodic Velocity Field Research Articles

Stokes’ second problem and oscillatory Couette flow for a two-layer fluid: Analytical solutions

The unsteady motion of a two-layer fluid induced by oscillatory motion of a flat plate along its length is mathematically analyzed. Two cases are considered: (i) the two-layer fluid is bounded only by the oscillating plate (Stokes’ second problem), and (ii) the two-layer fluid is confined between two parallel plates, one of which oscillates while the other is held stationary (oscillatory Couette flow). In each of the Stokes and Couette cases, both cosine and sine oscillations of the plate are considered. It is assumed that the fluids are immiscible, and that the flat interface between the fluids remains flat for all times. Solutions to the initial-boundary value problems are obtained using the Laplace transform method.The results obtained for the velocity fields for the flows are new and complete. Steady periodic and transient velocity fields are explicitly presented. Transient and steady-state shear stresses at the boundaries of the flows are calculated; the existing literature lacks any such results. The results derived in this paper retrieve previously known results for corresponding single-layer flows. Furthermore, the results obtained are illustrated taking particular example of each of the Stokes’ problem and the Couette flow. With the help of the illustrations, some physical insights into the flows of the particular problems are gained. Again, the results obtained could also be applicable to a problem of heat conduction in a composite solid with sinusoidal temperature variation on the surface.

Experimental Investigation of the Flow and Heat Transfer in a Helically Corrugated Cooling Channel

The detailed flow field and heat transfer were experimentally investigated in a channel with a circular cross section and equipped with a helical rib of low blockage ratio. Stereoscopic particle image velocimetry (S-PIV) was applied in order to measure the three components of the mean and turbulent velocities in the symmetry plane of the channel. Additionally, steady-state liquid crystal thermography (LCT) and infrared thermography were employed in order to study the convective heat transfer coefficient on the wall. Measurements were carried out more than six pitches downstream of the rib origin, presenting periodic velocity and heat transfer fields from this location on. The resulting velocity and heat transfer fields show similarities with those present in channels of plane walls, such as low momentum and heat transfer areas upstream and downstream of the obstacle, and high kinetic energy and heat transfer a few rib heights downstream of the obstacle. On the other hand, the shape of the rib induces a swirling motion with the same sense as the rib. The azimuthal mean velocity is negligible in the core of the pipe, but it increases considerably close to the wall.

Non-Newtonian fluid flow induced by pressure gradient and time-periodic electroosmosis in a microtube

Non-Newtonian fluid flow, based on the Carreau–Yasuda model, is studied in a circular microchannel which is driven by combined pressure gradient and alternating current electroosmosis. Smaller values of the flow behavior index correspond to the larger degrees of shear-thinning, and so the velocity field contracts. The flow field is strongly influenced by the dimensionless frequency. When viscous diffusion is faster than the period of oscillation, the bulk fluid has sufficient time to respond to alternating current electric field, and quasi-steady plug-like velocity profiles may be observed for sufficient large values of the flow behavior index; while a major part of the flow field may be immobile at sufficiently high dimensionless frequencies. When a pressure gradient is applied along the microchannel, the periodic velocity field deviates from the centerline and hence, a non-zero pulsating flow rate is attained. The non-linearity of the fluid behavior is enhanced by the Weissenberg number through the model time constant. Large or small values of shear stress during one period can lead to sharp variation of viscosity. Using non-Newtonian fluids, the voltage and current required to attain desirable characteristics, and therefore, the detrimental effects of Joule heating may be reduced.

Cooldown of Subsea Deadleg With a Cold Spot: Experimental and Numerical Heat-Transfer Analysis

Summary Experimental and numerical heat-transfer analysis was conducted on a T-shaped acrylic-glass pipe, representing a production header in a subsea production system with a vertical deadleg. The header was insulated, while the deadleg was not insulated and carried a cold spot on the top. The experimental conditions were set to mimic those of steady-state production, followed by a 3-hour shutdown (cooldown). The internal fluid temperature and the wall temperature were measured by use of resistance temperature detectors (RTDs) and thermocouples, respectively, while particle image velocimetry (PIV) was used to measure the velocities in the deadleg. It was shown that the mean velocity field during both steady state and cooldown was periodic, with a clockwise and counterclockwise rotation along the deadleg wall. By use of a k–Ω shear-stress transport (SST) Reynolds-averaged Navier Stokes (RANS) model in ANSYS CFX (2013a, b), the thermal field was correctly predicted for 3 hours of cooldown by modeling the cold spot as an isothermal wall. The RANS model was unable to recreate the periodic velocity field observed in the experiment.

Experimental investigation of the vortex breakdown in a lean premixing prevaporizing burner

The spiral vortex breakdown has been experimentally studied in the flow field generated by a lean premixing prevaporizing (LPP) combustor model. The vortex breakdown structures are characterized by a solid-body rotation. Hence, phase-locked laser Doppler velocimeter measurements, performed in a meridional plane, have allowed reconstruction of the periodic velocity and Reynolds stress fields within a time period and in a relative cylindrical frame of reference. Then, the turbulent kinetic energy (TKE) production term has been computed. The velocity distribution analysis indicates that a spiral vortex core wraps around an inner reverse flow zone. Both are characterized by a precessing motion. Then, the main contributions to the TKE production have been identified together with their spatial locations. Their distribution indicates that the maxima are placed in two regions: one located between the inner reverse flow and the spiral vortex and the second between the spiral vortex core and the external nearly quiescent fluid. In the latter, the TKE production term distributions have the features of a Kelvin–Helmholtz-like instability. In the former, the TKE production seems to be related to a Kelvin–Helmholtz-like instability only at the interface between the bubble and the spiral vortex core.

The Effect of Inertia on the Flow and Mixing Characteristics of a Chaotic Serpentine Mixer

As an extension of our previous study, the flow and mixing characteristics of a serpentine mixer in non-creeping flow conditions are investigated numerically. A periodic velocity field is obtained for each spatially periodic channel with the Reynolds number (Re) ranging from 0.1 to 70 and the channel aspect ratio from 0.25 to one. The flow kinematics is visualized by plotting the manifold of the deforming interface between two fluids. The progress of mixing affected by the Reynolds number and the channel geometry is characterized by a measure of mixing, the intensity of segregation, calculated using the concentration distribution. A mixer with a lower aspect ratio, which is a poor mixer in the creeping flow regime, turns out to be an efficient one above a threshold value of the Reynolds number, Re = 50. This is due to the combined effect of the enhanced rotational motion of fluid particles and back flows near the bends of the channel driven by inertia. As for a mixer with a higher aspect ratio, the intensity of segregation has its maximum around Re = 30, implying that inertia does not always have a positive influence on mixing in this mixer.

Steady flows above a quartz crystal resonator driven at elevated amplitude.

A steady flow of liquid was observed above the surface of a quartz crystal microbalance under conditions where the oscillation amplitude exceeded 10 nm. The streaming flow occurs parallel to the displacement vector and is directed towards the center of the plate. It is expected to have applications in acoustic sensing, in microfluidics, and in micromechanics in a wider sense. The flow is caused by the nonlinear term in the Navier-Stokes equation, which can produce a nonzero time-averaged force from a periodic velocity field. Central to the explanation are the flexural admixtures to the resonator's mode of vibration. Unlike pressure-driven flows, the acoustically driven steady flow attains its maximum velocity at a distance of a few hundred nanometers from the surface. It is therefore efficient in breaking bonds between adsorbed particles and the resonator surface. As a side aspect, the flow pattern amounts to a diagnostic tool, which gives access to the pattern of vibration. In particular, it leads to an estimate of the magnitude of the flexural admixtures to the thickness-shear vibration.

Modelling non-linear evolution using Lagrangian perturbation theory re-expansions

We present a new method to calculate formation of cosmological structure in the Newtonian limit. The method is based on Lagrangian perturbation theory plus two key theoretical extensions. One advance involves identifying and fixing a previously ignored gauge-like degree of freedom relating quantities calculated in LPT to those measured by a preferred Friedmann-Robertson-Walker (FRW) observer. Handling this connection between calculational and observer frames is physically essential and ensures a momentum conserving description. The second extension is to systematically re-expand the equations of motion to increase LPT's radius of convergence to the maximum future time prior to orbit crossing. The paper implements a complete algorithm and performs extensive "proof of principle" tests of the new method, including direct comparison to known solutions, evaluation of conserved quantities and formal convergence studies. All are satisfactory. We show convergence is exponential in grid size and Lagrangian order and polynomial in step size. There are three {\it powerful advantages} afforded by the new technique: (1) it employs a smooth representation of all fields and the results are not limited by particle induced shot-noise errors, (2) it permits the numerical error to be controlled by changing Lagrangian order and/or number of steps allowing, in principle, arbitrarily small errors to be achieved prior to orbit crossing and (3) it handles generic cold initial data (any periodic density and velocity fields, including those with initial rotational components). Together, these properties make the new technique well-suited to handle quasi-linear scales where analytic methods and/or numerical simulations fail to provide suitably accurate answers.

Mixing-dynamics of a passive scalar in a three-dimensional microchannel

The mixing of a diffusive passive-scalar driven by electro-osmotic fluid motion in a micro-channel is studied numerically. Secondary time-dependent periodic or random electric fields, orthogonal to the main stream, are applied to generate cross-sectional mixing. This investigation focuses on the mixing dynamics and its dependence on the frequency (period) of the driving mechanism. For periodic flows, the probability density function (PDF) of the scaled concentration settles into a self-similar curve showing spatially repeating patterns. In contrast, for random flows there is a lack of self-similarity in the PDF for the time interval considered. An exponential decay of the variance of the concentration, and associated moments, is found to exist for both periodic and random velocity fields. The numerical results also indicate that measures of chaoticity (in a deterministic chaotic system) decay exponentially in the frequency – at large frequencies – in agreement with the theory.

Investigation of Periodically Unsteady Flow in a Radial Pump by CFD Simulations and LDV Measurements

The periodically unsteady flow fields in a low specific speed radial diffuser pump have been investigated both numerically and experimentally for the design condition (Qdes) and also one part-load condition (0.5Qdes). Three-dimensional, unsteady Reynolds-averaged Navier–Stokes equations are solved on high-quality structured grids with the shear stress transport turbulence model by using the CFD (computational fluid dynamics) code CFX-10. Furthermore, two-dimensional laser Doppler velocimetry (LDV) measurements are successfully conducted in the interaction region between the impeller and the vaned diffuser, in order to capture the complex flow with abundant measurement data and to validate the CFD results. The analysis of the obtained results has been focused on the behavior of the periodic velocity field and the turbulence field, as well as the associated unsteady phenomena due to the unsteady interaction. In addition, the comparison between CFD and LDV results has also been addressed. The blade orientation effects caused by the impeller rotation are quantitatively examined and detailedly compared with the turbulence effect. This work offers a good data set to develop the comprehension of the impeller-diffuser interaction and how the flow varies with relative impeller position to the diffuser in radial diffuser pumps.

Calculating effective diffusivities in the limit of vanishing molecular diffusion

In this paper we study the problem of the numerical calculation (by Monte Carlo methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity field, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume-preserving property of the equations of motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non-trivial time-correlation structure and when inertial effects cannot be neglected. The method of modified equations is used to explain failings of Euler-based methods. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.

Unsteady Aperiodic Behavior of a Periodically Disturbed Tip-Leakage Vortex

unsteadyperiodicvorticalinflow representativeof anarrayofstator necklace vortices.Thedownstream tip-leakage vortex is disturbed periodically by the vortical inflow shed by those vortex generators. Phase-averaged two-point correlations are measured near the trailing edge of the cascade. The measurement results clearly show that organized, intense, large-scale aperiodic motion exists at each phase time within the tip-leakage-vortex region. The aperiodic fluctuations obtained by linear stochastic estimation have a magnitude of less than � 0:15U1 in all three velocity directions and appear to be organized large-scale structures. The streamwise fluctuations appear as coexisting “rush” and “retard” regions with different shapes, and the crossflow fluctuations appear to be clockwise/ anticlockwise or doubletlike rotating vortical structures with different strengths and center positions. These aperiodic fluctuations make the estimated instantaneous velocity field significantly different from the phaseaveraged periodic velocity field.

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

The probability density function (PDF) for a decaying passive scalar advected by a deterministic, periodic, incompressible fluid flow is numerically studied using a variety of random and coherent initial scalar fields. We establish the dynamic emergence at large Péclet numbers of a broad-tailed PDF for the scalar initialized with a Gaussian random measure, and further explore a rich parameter space involving scales of the initial scalar field and the geometry of the flow. We document that the dynamic transition of the PDF to a broad-tailed distribution is similar for shear flows and time-varying nonsheared flows with positive Lyapunov exponent, thereby showing that chaos in the particle trajectories is not essential to observe intermittent scalar signals. The role of the initial scalar field is carefully explored. The long-time PDF is sensitive to the scale of the initial data. For shear flows we show that heavy-tailed PDFs appear only when the initial field has sufficiently small-scale variation. We also connect geometric features of the scalar field with the shape of the PDFs. We document that the PDF is constructed by a subtle balance between spatial regions of strong and weak shear in conjunction with the presence of small-scale scalar variation within the weak shear regions. For cellular flows we document a lack of self-similarity in the PDFs when periodic time dependence is present, in contrast to the self-similar decay for time independent flow. Finally, we analyze the behavior of the PDFs for coherent initial fields and the parametric dependence of the variance decay rate on the Péclet number and the initial wavenumber of the scalar field.

Drift and diffusion in a periodic two-dimensional velocity field without attractors

We consider the transport of overdamped particles in a two-dimensional periodic velocity field. This field possesses extended lines of fixed points where the deterministic motion stops. Additive noise makes the lines penetrable and results in an oscillatory motion along tori. We characterize the stochastic motion by the probability distribution density, the stationary mean velocity, and the mean times of escape from bounded domains. For intermediate noise intensities, the fluctuations enhance the transport of the particles compared to the deterministic case. A fast dichotomic modulation of asymmetry enhances fluxes.

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

Optimization of heat‐transfer rate into time‐periodic two‐dimensional Stokes flows

AbstractThe periodic boundary displacement protocol leading to the optimum wall‐to‐fluid heat‐transfer rate, or to the most efficient mixing rate, in 2‐D annular Stokes flows is determined by calculating the steady periodic velocity and temperature fields. To obtain the steady periodic state one usually solves the dynamical system obtained after the spatial coordinates have been discretized. Here, we calculate the steady periodic state using an implicit method based on the discretization of the time coordinate over a period and the asymptotic regime is enforced by the periodicity condition in the computed temperature field. The obtained system of equations is solved using a Newton‐type iterative algorithm with invariant Jacobian. At each iteration step, the sparse linearized system is solved using a multi‐grid algebraic technique of rapid convergence. From a computational point of view and for the problem considered here, this method is an order of magnitude faster than the one based on a spatial discretization. Copyright © 2006 John Wiley & Sons, Ltd.

Conditional averaging of PIV plane wake data using a cross-correlation approach

A new approach is investigated for conditional averaging of a series of random snapshots of periodic velocity fields obtained from PIV measurements. This approach departs from conventional conditional averaging approaches that require a reference signal for extracting phase information and uses the cross-correlation between the velocity fields as a phase identifier. The methodology is applied on the natural and forced turbulent wake of a circular cylinder, the latter serving also as a validation/verification case. The results show that the vortex formation and shedding processes in the wake are drastically affected by inflow oscillations and reveal the underlying vortex dynamics. Additionally, detailed velocity measurements are reported for the two cases investigated.

Periodic homogenization for inertial particles

We study the problem of homogenization for inertial particles moving in a periodic velocity field, and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large scale, long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. To achieve this we use a formal multiple scale expansion in the scale parameter. This expansion relies on the hypo-ellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties studied. In particular, an expansion in terms of the non-dimensional particle relaxation time τ (the Stokes number) is shown to co-incide with the known result for passive (non-inertial) tracers in the singular limit τ → 0 . This requires the solution of a singular perturbation problem, achieved by means of a formal multiple scales expansion in τ Incompressible and potential fields are studied, as well as fields which are neither, and theoretical findings are supported by numerical simulations.

Colored particle tracking method for mixing analysis of chaotic micromixers

Micromixers have a variety of applications in chemical and biological processes, becoming an important component in microfluidic systems. The present work aims at understanding detailed mixing behaviour of micromixers by developing a numerical analysis scheme, which ultimately facilitates efficient micromixer design. A systematic numerical method has been developed, enabling visualization of detailed mixing patterns and quantification of the mixing performance in chaotic micromixers. The overall numerical scheme is named ‘colored particle tracking method’ (CPTM), consisting of three steps: (i) a flow analysis to obtain a periodic velocity field of a periodic mixing protocol by the Galerkin/least-squares (GLS) method; (ii) a particle tracking step, particles being labeled by a specific color at the inlet according to fluid species, to obtain a distribution of colored particles at the end of the final period; (iii) a quantification of the degree of mixing from the obtained particle distribution. For the last step we propose a new mixing measure based on the information entropy. The CPTM has successfully been applied to three examples of micromixers with patterned grooves to evaluate their mixing performance both qualitatively and quantitatively. The CPTM seems promising as a practically attractive numerical scheme for mixing analysis of chaotic micromixers.

TEST-CASE NO 11B: STRETCHING OF A CIRCLE IN A VORTEX VELOCITY FIELD (N)

Two advection numerical tests dedicated to strong interface stretching are proposed. The first test case considers a single vortex cen- tered in a square box, whereas in the second one a periodic multi-vortex velocity field is generated in a square cavity. An initially circular concen- tration shape is distorted during n time iterations. Then, the flow is reversed during n time iterations to recover the initial shape. After 2n iterations, the theoretical solution of the scalar advection problem should be identical to the initial interface shape. 1. PRACTICAL SIGNIFICANCE AND INTEREST OF THE TEST-CASE The accuracy and overall quality of front tracking and front capturing methods is of ma- jor importance for fundamental and industrial research simulations devoted to multiphase flows. Two advection numerical tests dedicated to strong interface stretching are proposed here. Our objective is to estimate the sensitivity of interface tracking methods to free sur- face deformations and to quantify mass conservation in strongly varying interface shape problems. The first test-case considers a single vortex centered in a square box whereas in the second problem, a periodic multi-vortex velocity field is generated in a square cavity. In an analytical velocity field, an initially circular concentration shape of radius R0 is dis- torted during n time iterations until interface structures of characteristic length less than R0=20 are generated. Then, the flow field is reversed and a same calculation is performed during an equal duration to recover the initial cylindrical shape. After 2 n iterations, the theoretical solution of the scalar advection problem is the initial interface condition. The works of Rider & Kothe (1995) or Rudman (1997) scrutinize numerical solutions provided by Volume Of Fluid, Level-Set or Front Tracking approaches on these two tests.