Stokes Layer Thickness Research Articles

Mean flow and turbulence in unsteady canopy layers

Non-stationarity is the rule in the atmospheric boundary layer (ABL). Under such conditions, the flow may experience departures from equilibrium with the underlying surface stress, misalignment of shear stresses and strain rates, and three-dimensionality in turbulence statistics. Existing ABL flow theories are primarily established for statistically stationary flow conditions and cannot predict such behaviours. Motivated by this knowledge gap, this study analyses the impact of time-varying pressure gradients on mean flow and turbulence over urban-like surfaces. A series of large-eddy simulations of pulsatile flow over cuboid arrays is performed, programmatically varying the oscillation amplitude $\alpha$ and forcing frequency $\omega$ . The analysis focuses on both longtime-averaged and phase-dependent flow dynamics. Inspection of longtime-averaged velocity profiles reveals that the aerodynamic roughness length $z_0$ increases with $\alpha$ and $\omega$ , whereas the displacement height $d$ appears to be insensitive to these parameters. In terms of oscillatory flow statistics, it is found that $\alpha$ primarily controls the oscillation amplitude of the streamwise velocity and Reynolds stresses, but has a negligible impact on their wall-normal structure. On the other hand, $\omega$ determines the size of the region affected by the unsteady forcing, which identifies the so-called Stokes layer thickness $\delta _s$ . Within the Stokes layer, phase-averaged resolved Reynolds stress profiles feature substantial variations during the pulsatile cycle, and the turbulence is out of equilibrium with the mean flow. Two phenomenological models have been proposed that capture the influence of flow unsteadiness on $z_0$ and $\delta _s$ , respectively.

Parameterization of oscillating boundary layers in lakes and coastal oceans

Oscillating turbulent bottom boundary layers (BBLs) occur in lakes and coastal oceans. At the mesoscale level, their kinematics are usually characterized by assuming either laminar or steady turbulent flow, and applying analytical solutions or semi-empirical correlations; e.g. log-law, Stokes’ second problem, inertial dissipation method (IDM), Batchelor fitting to temperature microstructure method (TMM). To investigate the ability of these models to capture oscillating turbulent BBLs, we have performed large eddy simulations (LES) and direct numerical simulations (DNS) for Reynolds numbers (Reδs; based on the Stokes layer thickness) between 20 and 3600. The velocity profiles showed logarithmic behavior throughout the cycle, in fully turbulent flows (error less than 2% for Reδs>3000), but the log-law was only accurate during turbulent bursts in the later stage of the acceleration phase for Reδs<550. Stokes’ second problem predicted the velocity profile in laminar or disturbed laminar flow (errors less than 10% for Reδs<500). At high Reynolds number (Reδs=3600), LES shows that the dissipation of turbulent kinetic energy from the IDM is more accurate than that from the log-law, particularly during changes in flow direction. The difference between the dissipation estimated from field observations (using IDM and TMM) and from idealized LES, however, suggest that measurement errors may predominate over uncertainties in simulation of boundary conditions and methodological errors in field applications.

Non-modal transient growth of disturbances in pulsatile and oscillatory pipe flows

Abstract

Dissimilar heat transfer enhancement in a fully developed laminar channel flow subjected to a traveling wave-like wall blowing and suction

A series of numerical simulations of heat and momentum transfer in a fully developed incompressible laminar channel flow subjected to a traveling wave-like wall blowing and suction is conducted. By systematically changing the wavelength and the phase speed of a traveling wave, their impacts on the Stanton number, the friction coefficient and their ratio, i.e., the analogy factor, are evaluated at different Reynolds numbers. Significant dissimilar heat transfer enhancement is confirmed when a downstream traveling wave is applied. It is also found that such a control input remains advantageous even when the power consumption for applying the control input is taken into consideration. In order to analyze the dissimilar responses of the velocity and thermal fields to the applied control input, we introduce the influential layer thickness and the magnitude of the Reynolds shear stress and the convective heat flux. It is shown that the influential layer thicknesses for the velocity and thermal fields are kept relatively similar, and can be well correlated with the Stokes layer thickness determined by the temporal period of the wave and the fluid viscosity for fast traveling waves. In contrast, significant difference in their magnitudes is confirmed. Phase and budget analyses of the coherent velocity and thermal fluctuations reveal that the continuity constrain on the velocity field is the primary reason for dissimilar heat transfer enhancement.

Energy loss and developing length during reciprocating flow in a pipe with a free-end

Direct numerical simulations of reciprocating pipe flow in a straight pipe with a free-end are presented. The range of amplitudes and frequencies studied span the laminar regime and the beginning of transition toward a conditionally turbulent flow. Two primary results are reported: the measurement of the flow development length and the loss of energy, both due to the presence of the free-end. Two regimes of flow are identified with distinct length scales. For low frequencies, the development length scales with the pipe diameter D. However, for higher frequencies, the development length scales with the Stokes layer thickness δ=2ν/ω. The energy loss is studied by calculating the viscous dissipation function, indicating where energy is lost, and allowing the energy lost due to the presence of the free-end to be isolated. While strong vortices are formed and convected from the exit, most of the energy they dissipate is lost within a few pipe diameters of the exit. It is shown that these trends continue even as the amplitude and frequency are increased so that the flow begins to transition from a laminar toward a turbulent state. Two modes of instability are observed in the Stokes layers near the free-end, one short wavelength mode with a wavelength set by the Stokes layer thickness and the other long wavelength mode with a wavelength set by the amplitude of the oscillatory flow. These modes are related to those observed in the fully developed oscillatory flow.

Phase-Lags' Radial Variations Between Velocity, Shear Stress, and Pressure Gradient in Ultrahigh Frequency Pulsating Turbulent Flows

Abstract Two-dimensional numerical simulations of wall-bounded turbulent pulsating flow driven by a sinusoidal velocity through a circular smooth tube are carried out. These computations for a Womersley number α ranged from 0.7 to 2069 and a dimensionless frequency ω+ ranged from 1.2×10−5 to 33.5. The aim of this study is to calculate the phase lag inside the unsteady turbulent boundary layer and across the tube. The phase lag of the velocity and shear stress with respect to the pressure gradient is deduced. Also, the instantaneous logarithmic layer and the turbulent parameters are analyzed. It is found that capturing the phase lag near the wall depends on resolving the Stokes layer thickness δst. At ultrahigh frequencies, the centerline velocity was delayed from the pressure gradient and wall shear stress by 45 deg and 90 deg, respectively. Consequently, the velocity and shear stress lagged behind the pressure gradient by 90 deg and 280 deg at the core of the tube, respectively, and by 45 deg at the wall. Thus, the trend of the radial phase lag increases toward the tubes center for ω+&amp;gt;0.06, which contrasts with that at low frequencies. When a reversed flow is caused by increasing the amplitude of the imposed oscillations, the phase lag is not affected noticeably by this increment. The radial phase lag is kept constant outside the oscillatory boundary layer at high frequencies because the radial gradient of the axial velocity has vanished.

Estimation of shear rate change in vertically oscillating non-Newtonian fluids: Predictions on particle settling

This study investigates the effect of oscillatory motion on velocity and shear rate change within different non-Newtonian, slightly viscoelastic fluids oscillated in a vertical U-shaped circular pipe. An estimation to quantify the influence on particle settling in an oscillatory environment is also presented. Flow visualization using particle image velocimetry (PIV) technique was deployed to compare the two-dimensional velocity field in the vertical plane of the U-tube axis and the resulting shear rate change in three different non-Newtonian fluids. The experiments were performed in a 1.2 m high, 50 mm diameter transparent test section, at room temperature (21 ± 0.5 °C) and atmospheric pressure.A piston was driven at harmonic motion via a gas buffer, to provide the driving force for the test fluids at four different low frequencies ranging from 0.1 – 0.75 Hz and at three different piston oscillation amplitude ratios of A = a/D = 0.3, 0.4 and 0.5, where a is the displacement amplitude of the piston and D is the pipe diameter. Oscillatory Reynolds number (Reδ) based on Stokes layer thickness was used as the criteria for determining the specific flow regime. 36 different experimental cases were tested within the range of 2 < Reδ < 34, and all the experimental cases exhibited the laminar flow regime.The study reveals that the axial velocity amplitude along the pipe centerline increases with the increasing frequency and with increasing oscillation amplitude irrespective of the non-Newtonian fluid type. The thickness of the shear region decreases with the increase of frequency. The change of shear rate is maximum near the wall region of the pipe and that is achieved at the maximum position of the phase cycle, where the axial velocity also possesses its highest magnitude. The most viscous and the least elastic fluid have reported a maximum reduction of viscosity as an effect of the oscillatory motion and the viscosity reduction becomes insignificant when the non-Newtonian fluids become less viscous.

Колебания неосесимметричного цилиндра в заполненной жидкостью полости, совершающей вращательные осцилляции

The behavior of a cylindrical solid of nonaxisymmetric cross section in a horizontal cylindrical cavity filled with a viscous incompressible fluid under rotational vibrations is experimentally investigated. The effect of the vibrational suspension of a heavy nonaxisymmetric cylinder near the bottom of the vibrating cavity was found and studied at a distance comparable with the Stokes layer thickness. The character of oscillations of the cylinder and its interaction to the wall of the cavity is studied by high-speed video registration of process.

Comparison of oscillatory flow conditions in Newtonian and non-Newtonian fluids using PIV and high-speed image analysis

Oscillatory flow in pipelines is common in many industrial applications, including oil well drilling. An experimental investigation of oscillatory flow inside a vertical U-shaped circular pipe is presented in this paper to mimic the practical scenario takes place within a vertical oil well. Flow visualization was deployed to compare the velocity distributions in Newtonian (deionized water) and non-Newtonian fluid (a mixture of three water-based polymeric liquids). The experiments were performed in a 1.2 m high, 50 mm diameter transparent test section, at room temperature (21 °C) and atmospheric pressure.Particle image velocimetry (PIV) technique was used to obtain non-invasive instantaneous flow velocity profiles. Based on local velocities, the streamwise (axial) velocity component within the pipe across its diameter was determined and the cross-sectional average velocity together with the normalized axial velocity were also calculated. In addition, high-speed motion pictures were used to determine the displacement of the air-liquid interface at the top of the U-tube limb. This enabled comparison of the flow field with the overall volumetric oscillating flow. A piston was driven at harmonic motion via a gas buffer, to provide the driving force for the test fluids at four different low frequencies ranging from 0.1 to 0.75 Hz. Oscillatory Reynolds number (Reδ) based on Stokes layer thickness was used as the criteria for determining the specific flow regime. According to the literature, the critical value for the oscillating Reynolds number was considered to be 500, and with this as reference all the experimental cases were within the laminar regime. Eight different experimental cases were tested within the ranges of (4 <Reδ < 116) and Womersley number (3 <Wo < 55).At higher frequencies, the viscous effects for the Newtonian fluid are confined to the Stokes layer and the central core of the velocity profile is plug-type. At the same time, higher frequencies resulted with increased velocity amplitudes. The viscous resistance of the non-Newtonian fluid and the presence of shear layer contribute to an uneven velocity profile across the pipe cross-section. Cross-sectional average velocity provides a more complete picture of the kinematic structure of oscillating flow and its dynamic distribution across the cross-section. Non-Newtonian fluid tends to achieve higher normalized axial velocities compared to that for Newtonian fluid, which is more or less equals to unity.The study was partly motivated by challenges associated with operational procedures during drilling and maintenance of petroleum wells. Furthermore, this study is also part of a comprehensive study aimed at investigating the influence of low frequency oscillations on particle settling in non-Newtonian drilling fluids.

Dissipative-convergent intermittency in dynamics of tidal waves in estuaries

The effect of dissipative-convergent intermittency in dynamics of tidal waves in estuaries is described. The effect manifests itself in spatial modulation of the tide amplitude. The effects of convergent channel and turbulent friction are competitive in dynamics of tidal waves in bays, gulfs, and estuaries. This manifests itself in the alternating excess of the action of one effect over another as the tidal wave penetrates deep the estuary. The effect of dissipative-convergent intermittency is manifested most completely in “strange” bays in which the mean water depth is commensurable with the Stokes layer thickness. It is shown that the spatial distribution of the tidal wave amplitude has a minimum in the convergent channel. If the minimum point lies beyond the estuary, then the effect of convergent channel predominates and the wave height increases to the vertex. If the minimum point lies in the neighborhood of the estuary vertex then the effect of turbulent friction predominates along the entire bay length and the wave height decreases to the bay vertex. Finally, if the minimum point finds itself in the middle part of the estuary, then the case of the “strange” bays arises, namely, in such bays the wave height decreases to crossing the minimum point and begins increases again after its passage. The local increase in the tidal wave amplitude can be manifested further in the mouth of the inflowing river.

Flow resistance of low-frequency pulsatile turbulent flow in mini-channels

Plate-type nuclear reactor fuel is currently getting increasing attentions as it features excellent heat transfer ability and compact structure. To gain an insight into the influences of channel size and pulsatile parameters on flow resistance characteristics, steady and pulsatile turbulent experimental investigation were performed. Three channels with varying height were used. The covered ranges of time-averaged Re Reta=(0.5–2) ×104, dimensionless frequency sqrtω’= 0.3–3.2, and pulsatile velocity amplitude Av= 0.04–0.93. A normalized parameter, friction factor ratio C =λta/λta, was proposed to denote the effects of flow fluctuation on time-averaged friction factor, where λta and λta are time-averaged and steady friction factor. The results show that in channel I (40 × 2 mm2), the time-averaged friction factor is remarkably larger than the steady values in the ranges of 1.5<sqrtω’<3.2.. The friction factor ratio C increases with the increasing sqrtω’ and Av, decreases with the increasing Reta. Further analysis indicated the friction factor ratio C is a function of dimensionless acceleration α*=4Avω′2/Reta. As the channel height increasing, the time-averaged friction factor decreases gradually to the same values as that of steady flows in the present ranges of sqrtω’. Finally, correlations to predict time-averaged friction factor were proposed.The effect of channel size is inferred to associate with the fractions of turbulent Stokes layer thickness to half-height of channels. In mini-channels, the turbulence Stokes layer is almost filled the entire flow area. As channel height increasing, the turbulence, which generated in near wall layer, cannot timely propagates into pipe core region before decaying-which cause the impact of superimposed unsteadiness fades rapidly. The effect of dimensionless frequency is interpreted as the response time of fluid to the rapid changes of pulsatile pressure gradient.

Steady streaming around a cylinder pair

The steady streaming motion that appears around a pair of circular cylinders placed in a small-amplitude oscillatory flow is considered. Attention is focused on the case where the Stokes layer thickness at the surface of the cylinders is much smaller than the cylinder radius, and the streaming Reynolds number is of order unity or larger. In that case, the steady streaming velocity that persists at the edge of the Stokes layer can be imposed as a boundary condition to numerically solve the outer streaming motion that it drives in the bulk of the fluid. It is investigated how the gap width between the cylinders and the streaming Reynolds number affect the flow topology. The results are compared against experimental observations.

Hydrodynamic effects at the entry of tidal waves into estuaries

The phenomenon of an increase in tidal wave height in cone-shaped estuaries is studied. The effect of estuary narrowing in the direction of tidal wave propagation (the hydrodynamic effect of confusor) is among the factors amplifying the tide. An opposite effect of turbulent friction, whose manifestation increases with decreasing bay’s depth, conversely, reduces tide amplitude because of the dissipation of tidal wave energy. Stokes diffusion layer also plays a significant role in the formation of wave transformation regime. In an estuary with a median depth, which is much greater than the Stokes layer thickness, the confusor effect is stronger and tide amplitude increases at estuary head. At depths lesser than Stokes layer thickness, the turbulent friction dominates over the confusor effect and the amplitude of tidal wave decreases at the head of the estuary. The depths of the order of Stokes layer thickness cause an interesting intermediate phenomenon—at the entrance into the estuary, first the effect of friction manifests itself, resulting in a decrease in the amplitude of tidal wave, but later, the effect of confusor starts dominating, and the amplitude of tidal wave again increases toward estuary head. When the period of tidal wave coincides with seiche period, a resonance enhancement of seiche oscillations takes place in the estuary.

DNS study of particle-bed–turbulence interactions in an oscillatory wall-bounded flow

Particle-resolved direct numerical simulations (DNS) are performed to investigate the behaviour of an oscillatory flow field over a rough bed, corresponding to the experimental set-up of Keiller &amp; Sleath (J. Fluid Mech., vol. 73 (04), 1976, pp. 673–691) for transitional and turbulent flows over a range of Reynolds numbers (95–400) based on the Stokes-layer thickness. It is shown that the roughness modulates the near-bed turbulence, produces streamwise horseshoe structures which then undergo distortion and breaking, and therefore reduces the large-scale anisotropy. A fully developed equilibrium turbulence is observed in the central part of the oscillation cycle, with two-component turbulence in the near-bed region and cigar-shaped turbulence in the outer region. A double averaging of the flow field reveals spatial inhomogeneities at the roughness scale and alternate paths of energy transport in the turbulent kinetic energy (TKE) budget. Contrary to the unidirectional, steady flow over rough beds, bed-induced production terms are important and comparable to the shear production term. It is shown that the near-bed velocity and pressure fluctuations are non-Gaussian, a result of critical importance for the modelling of incipient motion of sediment grains.

The influence of edge geometry on end-correction coefficients in micro perforated plates.

Global expressions are proposed for end-correction coefficients in micro perforated plates (MPPs) using non-dimensional parameters. MPPs are sound absorbers with small perforation diameters such that the Stokes boundary layers fill up almost the entire perforation. Sound absorption does not only occur within the perforation, but also takes place just outside of it. The latter contribution plus the outside inertia effect on the transfer impedance of the MPP are referred to as end-corrections. In order to determine them, an analytical solution employing the very thin Stokes layer assumption has been derived. However, this assumption requires empirical coefficients in the end-corrections for accurate results. To explore the effects of various parameters a numerical model is used. This model is verified with open-end reflection coefficient measurements. The most prominent result from this study is that compared to plate thickness, the ratio of perforation diameter to Stokes layer thickness (Shear number) and edge geometry affect the end-correction coefficients more significantly. The effect of plate thickness can be neglected for practical purposes, therefore, expressions for the end-corrections in terms of Shear number and edge geometry are provided. The relative error of these expressions is <3% compared to the numerical results.

Characterization of Oscillatory Boundary Layer over a Closely Packed Bed of Sediment Particles

Lack of accurate criteria for onset of incipient motion and sediment pickup function remain two of the biggest hurdles in developing better predictive models for sediment transport. To study pickup and transport of sediment, it is necessary to have a detailed knowledge of the small amplitude oscillatory flow over the sediment layer near the sea bed. Fully resolved direct numerical simulations are performed using fictitious domain approach (Apte &amp; Finn, 2012) to investigate the effect of a sinusoidally oscillating flow field over a rough wall made of regular hexagonal pack of spherical particles. The flow arrangement is similar to the experimental data of Keiller &amp; Sleath (1976). Transitional and turbulent flows at Reδ = 95, 150, 200 (based on the Stokes layer thickness, [Formula: see text] are studied. Turbulent flow is characterized in terms of coherent vortex structures, Reynolds stress variation and PDF distributions. The nature of unsteady hydrodynamic lift forces on sediment grains and their correlation to sweep and burst events is also reported. The dynamics of the oscillatory flow over the sediment bed is used to understand the mechanism of sediment pick-up.

Comparison between spatial and temporal wall oscillations in turbulent boundary layer flows

AbstractDirect numerical simulations have been performed to study the drag reduction resulting from spatial oscillations of a segment of the wall under a turbulent boundary layer. The oscillating motion is imposed by utilizing a streamwise modulated spanwise wall forcing. The results are compared with earlier simulations using temporal oscillations with an identical segment and forcing amplitudes, and with a frequency related to the wavelength through a convective velocity. Two different oscillation amplitudes with equal oscillation wavelength have been used, which allows for a direct comparison between a relatively weak and strong forcing of the flow. The weaker forcing results in 25 % drag reduction while the stronger forcing, with twice the amplitude, yields 41 % drag reduction. Comparison with the temporal cases reveals drastically improved energy savings for the spatial oscillation technique, in accordance with earlier channel flow investigations. The streamwise variation of spanwise shear is shown to follow the analytical solution to the laminar Navier–Stokes equations derived under the assumption of constant friction velocity. Furthermore, the spanwise velocity profiles at various phases are compared with the analytical solution, and show very good agreement. The downstream development of the spatial Stokes layer thickness is theoretically estimated to be ${\sim }{x}^{1/ 15} $, in general agreement with the simulation data. The spatial variation of the spanwise Reynolds stress is investigated and compared with the variation in time for the temporal wall forcing cases. The controversy regarding a zero or non-zero production of spanwise Reynolds stress in the temporal case is elucidated. In addition, comparison with the spatial case reveals that a second production term originating from the downstream variation of the spanwise wall velocity has a negative contribution to the production, and hence relates to the larger drag reduction in the case of spatial forcing.

Hydrodynamic Tweezers: Impact of Design Geometry on Flow and Microparticle Trapping

Here we explore the role of microfabricated device geometry on frequency-dependent low Reynolds number steady streaming flow and particle trapping behavior. In our system, flow and particle trapping is induced near an obstruction or cavity located in an otherwise rectilinear oscillating flow of frequency ω and amplitude s in a fluid of kinematic viscosity ν. This work expands prior studies to characterize nine distinct obstruction/cavity geometries. The imaged microeddy flows show that the device geometry affects the eddy number, shape, structure, and strength. Comparison of measured particle trap locations with the computed eddy flow structure shows that particles trap closer to the wall than the eddy core. Trapping strength and location are controlled by the geometry and the oscillation frequency. In most cases, the trapping behavior is linearly proportional to the Stokes layer thickness, δ(AC) ~ O((ν/ω)(1/2)). We show that steady streaming in microfluidic eddies can be a flexible and versatile method for noncontact microparticle trapping, and hence we call this class of devices "hydrodynamic tweezers".

The laminar generalized Stokes layer and turbulent drag reduction

This paper considers plane channel flow modified by waves of spanwise velocity applied at the wall and travelling along the streamwise direction. Both laminar and turbulent regimes for the streamwise flow are studied. When the streamwise flow is laminar, it is unaffected by the spanwise flow induced by the waves. This flow is a thin, unsteady and streamwise-modulated boundary layer that can be expressed in terms of the Airy function of the first kind. We name it the generalized Stokes layer because it reduces to the classical oscillating Stokes layer in the limit of infinite wave speed. When the streamwise flow is turbulent, the laminar generalized Stokes layer solution describes well the space-averaged turbulent spanwise flow, provided that the phase speed of the waves is sufficiently different from the turbulent convection velocity, and that the time scale of the forcing is smaller than the life time of the near-wall turbulent structures. Under these conditions, the drag reduction is found to scale with the Stokes layer thickness, which renders the laminar solution instrumental for the analysis of the turbulent flow. A classification of the turbulent flow regimes induced by the waves is presented by comparing parameters related to the forcing conditions with the space and time scales of the turbulent flow.

On the linear instability of a finite Stokes layer: Instantaneous versus Floquet modes

The linear instability of a finite Stokes layer is investigated numerically. The aim is to clarify the relationship between the instantaneous instability and the recently discovered Floquet global instability [P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 464, 394 (2002); P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 556, 1 (2006)], and their role in causing transition to turbulence. For the first time, a numerical approach based on initial-value calculations is able to confirm the global instability, which occurs as the Reynolds number R (based on the Stokes-layer thickness) exceeds the critical value Rc=708. In both supercritical (R&amp;gt;Rc) and subcritical (R&amp;lt;Rc) regimes, strong instantaneous growth occurs during the deceleration phase. By projecting the disturbance at an arbitrary instant to instantaneous eigenfunctions, it is shown that the energy of the disturbance is primarily carried by the most unstable instantaneous mode. During the main growing phase, both the spatial structure and the growth rate agree well with those of the instantaneous mode; during the decaying phase part of the energy is continuously transferred to the precursor (i.e., the backward continuation in time) of the most unstable mode existing in the next cycle. The most unstable mode including its continuation therefore constitutes the crucial part of a Floquet mode. Unstable Floquet modes exist despite that none of instantaneous modes is periodic, implying that the commonly held premise that a Floquet mode is equivalent to a periodic instantaneous mode is erroneous. It is also found that in the subcritical regime, the flow responds very sensitively to small surface roughness, modeled by a wavy wall. The sensitivity can be measured by a response function, and our calculations show that significant disturbances can be generated by surface imperfections less than 0.1 μm in typical experimental setups. The response becomes stronger as R increases, and tends to infinity as the wavenumber of the wavy wall approaches that of the critical Floquet mode at Rc, indicating that roughness with a suitable wavelength can excite a global mode through resonance, and the large response may be attributed to a detuned resonance with a global mode.