Dimensionless Acceleration Research Articles

Numerical Investigation of the Effects of Particle Density and Dimensionless Acceleration on Segregation in 2D Vertically Vibrated Binary Granular Mixtures

The present work aims to investigate the influences of particle density and dimensionless acceleration on segregation phenomena in 2D binary granular mixtures by using contact dynamics method. The granular samples were prepared by randomly depositing two species of regular pentagonal particles inside a rectangular container. These two particle species are the same in size, but differ in terms of particle density. Different levels of vertical vibration were then applied to the mixtures. The simulation results were systematically analyzed by using segregation coefficient (H), segregation pattern, and velocity field related to convection mechanism. It was found that segregation phenomena are not observed in this study. This could be explained by the fact that a value of friction coefficient between interparticle and that value between wall and particle is set to be the same, that is why no segregation was found in the present study. Due to the friction, only convection rolls were observed for all the density ratios when dimensionless acceleration is greater than 2.

Starting Flow by Pitch Acceleration of Flat-Plate Foil

This paper presents the investigation of the starting flow in association with pitch acceleration using transient laminar-flow computation and an unsteady inviscid-flow theory. A thin flat-plate foil pitched at the leading edge is modeled in a two-dimensional incompressible fluid at chord Reynolds numbers of the order of . Several acceleration amplitudes () and acceleration durations () are maneuvered using a modified canonical kinematic equation at reduced pitch rates from 0.196 to 0.763. Comparisons between computational and theoretical data reveal that the formation of the starting vortex is determined by the acceleration amplitude scaled by the square of the convective time. The corresponding starting force coefficient that scales with the dimensionless acceleration amplitude is proportional to the acceleration angle , which decreases with increasing reduced pitch rate. The starting vortex, once formed at the trailing edge, is shed at the end of the acceleration period. The quasi-steady theory that includes only pitch motion as the strength of the dipolar sheet for apparent-mass flow can also yield an acceptable approximation to capture the apparent-mass force coefficient spike in acceleration.

Convection behavior of ellipsoidal particles in a quasi-two-dimensional bed under vertical vibration

ABSTRACT Ellipsoidal particles are anisotropic in shape. Production practice often involves the distribution and convection of ellipsoidal particles under external excitation. However, few studies have focused on convection in ellipsoidal particles. Herein, we experimentally investigated the convection behavior of ellipsoidal particles in a two-dimensional bed under vertical vibration. Several patterns were observed at the dimensionless vibration acceleration (Г) values of 1–6 within the frequency (ƒ) range of 12–70 Hz. The displacement vectors of single roll, asymmetrical double rolls and symmetrical double rolls were drawn. At fixed Г, the system changed from global convection to local convection with increased ƒ. To describe convection intensity, the expansion of bed and the volume fraction of convection region were proposed. Results showed that higher Г or lower ƒ led to stronger convection intensity. The relationship between the pattern and input energy distribution was further discussed. The arrangement of the ellipsoidal-particle long axis along the streamline or at an angle was observed under weak vibration. Two necessary conditions for this phenomenon were observed, namely, nonspherical particles and low input energy.

Thin‐layer inertial effects in plasticity and dynamics in the Prandtl problem

AbstractEspecially in metal forming, large plastic deformation occurs in thin plates. The problem of compressing dies is analyzed to evaluate the spreading of a thin layer in between. The velocity of dies is a given function in time so that the kinematics of the process is known. This problem can be considered as a generalization of the classical Prandtl problem by taking inertial effects into account and introducing dimensionless parameters as internal variables depending on time. The first parameter is purely geometric corresponding to the thin‐layer approximation; the second and the third parameters are dimensionless velocity and acceleration during the dies getting pressed. We use singular asymptotic expansions of unknown functions and study how these parameters vary preceding the dies of moment. Depending on this relation, the dynamic corrections to the quasistatic solution is a part of various terms of the asymptotic series. The corresponding analytical investigation both for general case and for particular typical regimes of plates motion is carried out.

A Light-Weight Vibrational Motor Powered Recoil Robot That Hops Rapidly Across Granular Media

Abstract A 1-cm coin vibrational motor fixed to the center of a 4-cm square foam platform moves rapidly across granular media at a speed of up to 30 cm/s or about 5 body lengths/s. Fast speeds are achieved with dimensionless acceleration number, similar to a Froude number, up to 50, allowing the light-weight 1.4 g mechanism to remain above the substrate, levitated and propelled by its kicks off the surface. The mechanism is low cost and moves across granular media without any external moving parts. With 2-s exposure, we photograph the trajectory of the mechanism with an LED fixed to the mechanism. Trajectories can exhibit period doubling phenomena similar to a ball bouncing on a vibrating table top. A two-dimensional robophysics model is developed to predict mechanism trajectories. We find that a vertical drag force is required in the model to match the height above the surface reached by the mechanism. We attribute the vertical drag force to suction from air flow below the mechanism base and through the granular substrate. Our numerical model suggests that horizontal speed is maximized when the mechanism is prevented from jumping high off the surface. In this way, the mechanism resembles a galloping or jumping animal whose body remains nearly at the same height above the ground during its gait. Our mechanism and model illustrate that speed and efficiency of light-weight hoppers on granular media can be affected by aerodynamics and substrate permeability.

THE SPECIFICITIES OF KINEMATICS OF ROTOR-PISTON ENGINE FOR A NEW CONSTRUCTION

The general structure and principle of operation of the rotary-piston engine of the new design 12RPE-4,4/1,75 with adjustable spool air distribution are given. The design of the engine combines the advantages of piston and rotor engines, provides low values of the specific consumption of the working fluid and high values of efficiency. Considered a rotary-piston engine due to its technical and operational advantages has a wide range of applications in various industries and as part of power plants for various purposes, for example, in the schemes of combined power plants of vehicles. A fairly simple and compact design with a low weight of the proposed engine ensures its low price in manufacturing, as well as reliability and simplicity in operation. The rotor-piston engine due to the uniform placement of the cylinders in the rotor and the low weight of parts that move back and forth gradually, has high balance and the ability to start at any position of the central rotor. The design of the engine, as for all rotary engines, implies the absence of dead volume. That is, a virtually dead volume corresponds to the push operation at the intake pressure in the compressor cycle. In contrast to the existing serial air motors with a spool air distribution, the considered design of a rotary-piston engine has the ability to adjust the valve timing and engine operating conditions due to the degree of filling of the cylinder. Regulation of the valve timing over a fairly wide range is ensured by turning the central adjusting camshaft. In addition, the central adjusting cam allows you to change the direction of rotation of the central rotor. The kinematic scheme is considered, which differs significantly from the classical scheme of a crank mechanism. A kinematic analysis of the rotor-piston engine motion pattern is carried out. For further calculation of the acting forces in the engine, the dependencies of the definition of dimensionless displacement, velocity and acceleration of the piston depending on the angle of rotation of the central rotor φ are proposed. The coefficients of the harmonic series in these dependencies are determined.

An Experimental Investigation on Segregation Process of Binary Materials Based on Various Vibration Conditions

The mixing and segregation process of binary steel-glass mixture within a vertically vibrated containerat different frequencies are investigated in this paper. The image processing techniques areemployed to track the convection flow of granular materials.Six different amounts of frequency (20, 25, 30, 35, 40, and50 Hz) for certain amplitudes are used to study the effects of miscellaneous vibration conditions on the velocity fields of moving beads.The results show that increasing frequency with constant amplitude has a positive effect on increasing the velocity movement of the beads in addition, the decreasing of amplitude at almost constant dimensionless vibrational acceleration cause the velocity movement of the beads to be increased.The granular bed was divided into three horizontal parts and a separate study was provided for each one.

The effect of wall-normal gravity on particle-laden near-wall turbulence

We performed two-way coupled direct numerical simulations of turbulent channel flow with Lagrangian tracking of small, heavy spheres at a dimensionless gravitational acceleration of 0.077 in wall units, which is based on the flow condition in the experiment by Gerashchenko et al. (J. Fluid Mech., vol. 617, 2008, pp. 255–281). We removed deposited particles after several collisions with the lower wall and then released new particles near the upper wall to observe direct interactions between particles and coherent structures of near-wall turbulence during gravitational settling through the mean shear. The results indicate that when the Stokes number is approximately 1 on the basis of the Kolmogorov time scale of the flow ($St_{K}\approx 1$), the so-called preferential sweeping occurs in association with coherent streamwise vortices, while the effect of crossing trajectories becomes significant for $St_{K}>1$. Consequently, in either case, the settling particles deposit on the wall without strong accumulation in low-speed streaks in the viscous sublayer. When particles settle through near-wall turbulence from the upper wall, more small-scale vortical structures are generated in the outer layer as low-speed fluid is pulled farther in the direction of gravity, while the opposite is true near the lower wall.

Impact of seismic-type shock parameters on the soil-structure interaction effect in the USC mining region

The paper deals with seismic-type surface and building vibrations randomly occurring as a result of rockbursts in mining regions (random events as with earthquakes). The focus is on the problem of ground vibrations transmission to building foundation – it is one of very important phenomenon associated with soil-structure interaction effect. One of the ways of estimation of possible differences between simultaneously developing free-field vibrations next to a building and building foundation vibrations, i.e. using a ratio of response spectra (RRS), is applied to this study. Analysis concerns the ratio of dimensionless and dimensional acceleration response spectra (β and Sa) – denoted as RRS(β) and RRS(Sa), respectively. Horizontal vibrations parallel to the transverse (x) and longitudinal (y) axis of the representative (typical) two-storey, masonry office building are discussed. Calculations are based on the results of in situ surface vibration measurements performed in the seismically active Upper Silesian Coalfield (USC) mining area in Poland (long-term, full-scale monitoring). Evaluation of the dependence of the curves of ratio of response spectra on some parameters corresponding to mine-induced vibrations (i.e. epicentral distance, mining shock energy, peak value of free-field vibrations) is executed. From the obtained results, it can be definitely concluded that the influence of the most important mining tremor parameters (i.e. epicentral distance, mining shock energy, peak value of free-field vibrations) on the ratio of response spectra calculated in the successive ranges of these parameters, is clearly observed.

Thermal fluid characteristics of pulsating heat pipe in radially rotating thin pad

The thermal performances of a radially rotating pulsating heat pipe (RPHP) formulated by the interconnected 1 × 4 mm2 channels in the thin pad were investigated. With the 50% filling ratio of water and the specific geometric configurations, the selected parameters controlling the thermal-fluid phenomena by the present study were boiling number (Bn), dimensionless centrifugal acceleration (Ω) and thermal resistance of condenser (Rth,cond). The vapor-liquid flow images, heat transfer rates and thermal resistances of the pad-type RPHP were detected at the rotor speeds of 0, 100, 300 and 500 rev/min. The matrix of test conditions was formulated by four Bn, four Rth,cond and four Ω for both sets of flow visualization and heat transfer tests. As Ω increased from 0 to 27.8, the intermittent vapor slugs gradually deformed toward the tiny bubbles to incur the respective Nusselt number reductions and elevations over the evaporator and condenser of the RPHP. As the dominance of condenser heat transfer rate in constructing the overall thermal resistance (Rth) took precedence to its evaporator counterpart, the increases of Ω led to the Rth reductions for the present RPHP. The empirical correlations using Bn, Ω and Rth,con as the controlling parameters were devised to evaluate Rth and the regionally averaged Nusselt numbers over the evaporator and condenser of the present RPHP for relevant applications.

Forced and Mixed Convection Heat Transfer at High Pressure and High Temperature in a Graphite Flow Channel

High pressure/high temperature forced and mixed convection experiments have been performed with helium and nitrogen at temperatures and pressures up to 893 K and 64 bar, respectively. The test section had a 16.8 mm ID flow channel in a 108 mm OD graphite column. Flow regimes included turbulent, transitional, and laminar flows with the inlet Reynolds numbers ranging from 1500 to 15,000. Due to strong heating, the local Reynolds number decreased by up to 50% over the 2.7 m test section. In addition, heat transfer degradation and flow laminarization caused by intense heating led to Nusselt numbers 20–50% lower than the values given by the modified Dittus–Boelter and modified Gnielinski correlations. Flow laminarization criteria were considered based on a dimensionless acceleration parameter (Kv) and buoyancy parameter (Bo*). Upward turbulent flows displayed higher wall temperatures than downward flows, due to the impact of flow laminarization which is not expected to affect buoyancy-opposed flows. Laminar Reynolds number flows presented an opposite behavior due to the enhancement of heat transfer for buoyancy-aided flows. At low Reynolds numbers, downward flows displayed higher and lower wall temperatures in the upstream and downstream regions, respectively, than the upward flow cases. In the entrance region of downward flows, convection heat transfer was reduced due to buoyancy leading to higher wall temperatures, while in the downstream region, buoyancy-induced mixing caused higher convection heat transfer and lower wall temperatures.

Vertical separation criterion of binary particles under external excitation

Vertical separation occurs in the binary granular system and varies with most factors of external excitation and particle properties, that is complex to explain and predict. We propose that it can be discussed from two aspects: 1) the influence of the specific external-excitation condition and 2) the separation response of the system itself. Based on the percolation and filling behaviors of particles, a vertical separation criterion is presented to discuss the above two aspects with vibration frequency f, dimensionless vibration acceleration Г, airflow velocity u, particle size ratio φd, density ratio φρ and total mass ratio φM when the container size and the granular system's packing height are fixed. In our criterion, Brazil nut separation (BN, large particles rise to the top), reverse Brazil nut separation (RBN, large particles sink to the bottom), and their crossover line (It can present as a MIX distribution where particles mix uniformly) are discussed. It's found that the system with strong separation response (a low φM/φd3φρ) tends to show a strong BN or RBN separation under appropriate conditions of vibration and/or airflow. Conversely, the system with weak separation response (a high φM/φd3φρ) inclines to separate weakly or mix uniformly.

Dynamics of a Rotating Sphere on Free Surface of Vibrated Granular Materials**Supported by the CHED-FDP II Program of the Commission on Higher Education of the Philippines.

We investigate the rotational dynamics of a low-density sphere on the free surface of a vertically vibrated granular material (VGM). The dynamical behavior of the sphere is influenced by the external energy input from an electromagnetic shaker which is proportional to ε, where ε is equal to the ratio between the square of the dimensionless acceleration Γ and the square of the vibration frequency f of the container. Empirical results reveal that as the VGM transits from local-to-global convection, an increase in ε generally corresponds to an increase in the magnitudes of the rotational ωRS and translational vCM velocities of the sphere, an increase in the observed tilting angle θbed of the VGM bed, and a decrease in the time twall it takes the sphere to roll down the tilted VGM bed and hit the container wall. During unstable convection, an increase in ε results in a sharp decrease in the sphere’s peak and mean ωRS, and a slight increase in twall. For the range of ε values covered in this study, the sphere may execute persistent rotation, wobbling or jamming, depending on the vibration parameters and the resulting convective flow in the system.

Modified inelastic bouncing ball model of the Brazil nut effect and its reverse

We developed the modified inelastic bouncing ball model (mIBBM) to describe the emergence of the Brazil nut effect (BNE) and its reverse (RBNE) in a vertically-vibrated binary granular mixture. The mIBBM incorporates the container-to-grains force-transmission efficiency (transmissibility) Tr to quantify the dimensionless mean void lifetime $$ \omega \langle \delta t\rangle $$ that acts as the segregation phase indicator where ω is the vibration angular frequency. The mixture is represented as two non-interacting inelastic balls, Ball A and Ball B. Each ball bounces with a time-of-flight τA (or τB) that depends on transmissibility TrA (or TrB) and the dimensionless container acceleration Γ, i.e., τA = τA(Γ, TrA) and τB = τB(Γ, TrB). The ball dynamics are described by the bifurcation diagrams of the dimensionless times-of-flight, ωτA(Γ, TrA) and ωτB(Γ, TrB). The probability-weighted difference $$ \omega \langle \delta t\rangle $$ between branches of the two diagrams is computed and interpreted as follows: $$ \omega \langle \delta t\rangle > 0 $$ (occurrence of BNE), $$ \omega \langle \delta t\rangle < 0 $$ (RBNE) and $$ \omega \langle \delta t\rangle $$ = 0 (no segregation). Segregation phases are revealed as varying shifts and widths of $$ \omega \langle \delta t\rangle $$ across the Γ axis. The phase boundaries in the $$ \omega \langle \delta t\rangle $$ -versus-Γ diagram are sensitive to changes in TrA, TrB and ΔTr = (TrA − TrB). The mIBBM explains why the BNE is a more likely than the RBNE and predicts a segregation phase sequence that is generally consistent with related experimental results taken over a limited ω-range. Additional experiments are needed to enable a more comprehensive and precise evaluation of the mIBBM.

Effects of low-Re pulsatile flow on friction characteristics in bare square array rod bundles

Interest in unsteady flow in floating systems has increased in recent times. This paper presents the results of the experimental investigation on the resistance characteristics of pulsating flow through square array bare rod bundles with P/D = 1.326 and W/D = 1.268 conducted in the range of time-averaged Reynold’s number Reta = 1–7 × 103 and dimensionless acceleration (α) of 0.001–0.013. Pulsatile flow affects the time-averaged friction factor because it changes the velocity field close to the wall. The dimensionless frequency parameter (ω′) relates the frequency to the viscous effects while the dimensionless acceleration (α) is the product of the velocity amplitude (Au), diameter (D) of the channel and the frequency (ω) normalized by the mean velocity (Um). Ratios (Cλ) of the time-averaged friction factor (λp) in pulsatile flow to the friction factor (λs) in a steady flow and (Cp) of their average pressure drops were used to compare the two flows. Cλ and Cp increase with an increase in Au, ω′ and α; but decreases with increase in Reta. The dynamics of the extra turbulence generated as a result of pulsation was also analyzed in the subchannels.

The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow

The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration $\unicode[STIX]{x1D70E}$) of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the exterior horizontal, as it advances ($\unicode[STIX]{x1D70E}&gt;0$) or retreats ($\unicode[STIX]{x1D70E}&lt;0$) from an initially stationary and horizontal strip of inviscid incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied by Needham et al. (Q. J. Mech. Appl. Maths, vol. 61 (4), 2008, pp. 581–614). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D707}$ (where $\unicode[STIX]{x1D707}=1+\unicode[STIX]{x1D70E}\tan \unicode[STIX]{x1D6FC}$), with a bifurcation in structure as $\unicode[STIX]{x1D707}$ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (J. Fluid Mech. vol. 841, 2018, pp. 146–166). In particular, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$, the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times [1,\infty )$ or a collapsing jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times (0,1)$. However, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{-}$, the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{n}^{\ast }\in (0,\unicode[STIX]{x03C0}/2)$ ($n=1,2,\ldots$).

Experimental investigations of density-induced segregation in binary granular mixtures under vertical vibrations

The objective of this work is aimed to investigate experimentally the influences of density ratio on segregation of cylindrical granular mixtures under vertical vibrations. Polytetrafluoroethylene, Polyamide, Polyethylene terephthalate, Balsa woods and Hinoki woods were used as the cylindrical particles, with 10 mm in diameter and 60 mm in length. The granular sample is composed of 1000 particles with two different materials with each type of 500 particles. All particles are periodically filled into the rectangular container with size 300 x 300 x 60 mm, as the initial condition. Density ratios are 2.73, 3.36, 5.27, and 13.63. The experimental operated by vertical shaker for 10 minute to achieve a steady state condition with same amplitude at 5 mm. The dimensionless vibration accelerations changed by frequency for 5 experimental, experiment the dimensionless vibration accelerations = 1, 2, 3, 4, and 5. The beginning state and the end state collected data by shooting the image. The segregation is considered by average height or the segregation coefficient. The result showed that for the density ratios of 2.73 and 3.36 for all dimensionless acceleration values, the segregation does not occur. On the other hand, for density ratio more than 5.27, the segregation occur by the way that the higher density particles move downward to the bottom of the sample, while the lower density particles move up to the top of the container. The segregation occurs when the dimensionless acceleration is more than 2 and the segregation is clearer when the dimensionless acceleration increases.

Planar granular shear flow under external vibration.

We present results from a planar shear experiment in which a two-dimensional horizontal granular assembly of pentagonal particles sheared between two parallel walls is subjected to external vibration. Particle tracking and photoelastic measurements are used to quantify both grain scale motion and interparticle stresses with and without imposed vibrations. We characterize the particle motion in planar shear and find that flow of these strongly interlocking particles consists of transient vortex motion with a mean flow given by the sum of exponential profiles imposed by the shearing walls. Vibration is applied either through the shearing surface or as bulk vertical vibration of the entire shearing region with dimensionless accelerations Γ=A(2πf)^{2}/g≈0-2. In both cases, increasing amplitude of vibration A at fixed frequency f leads to failure of the force network, reduction in mean stress, and a corresponding reduction in imposed strain. Vibration of the shearing surface is shown to induce the preferential slipping of large-angle force chains. These effects are insensitive to changes in frequency in the range studied (f=30-120 Hz), as sufficiently large displacements are required to relieve the geometrical frustration of the jammed states.

Effects of the moment of inertia on the energy dissipation and convection motion of particles in a horizontally vibrated monolayer

Convection and energy dissipation in a horizontally vibrated granular systems is an indispensable phenomenon of nature due to its importance in fields of engineering development. A monolayer of granular particles of size d= 8.00 ± 0.02mm with density ρ= 1.6g/cm3 is filled in a rectangular container of size Lx= 25cm, Ly= 25cm, Lz=18cm and vibrated horizontally with 5Hz, 10Hz and 15Hz frequencies; the dimensionless acceleration Γ varies from 0.8 to 1.4. Four different regions namely the Solid Region, Cluster Region, Gassy Region and Empty Region, as well as a 3D state when particles jump from one corner of the container, were observed in the experiment. The moment of inertia played a significant role in the energy dissipation and convection motion. We relate the moment of inertia with the angular momentum L and the linear velocity vt of the particles as I∝(Lvt ). Convection motion can be observed when the product of the angular momentum and linear velocity (Lvt) ˃ 2.24X104gcm3/s2 and the dimensionless acceleration Г ≥ 1.2. There is no effect of frequency on the 3D state; this state depends only on the dimensionless acceleration Г.

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.