The dynamics and scaling law for particles suspended in shear flow with inertia

2000 ◽  
Vol 423 ◽  
pp. 317-344 ◽  
Author(s):  
E-JIANG DING ◽  
CYRUS K. AIDUN
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Shear Flow ◽  
Scaling Law ◽  
Density Ratio ◽  
Reynolds Numbers ◽  
Creeping Flow ◽  
Elliptical Cylinder ◽  
Body Of Revolution ◽  
Time Periodic

The effect of inertia on the dynamics of a solid particle (a circular cylinder, an elliptical cylinder, and an ellipsoid) suspended in shear flow is studied by solving the discrete Boltzmann equation. At small Reynolds number, when inertia is negligible, the behaviour of the particle is in good agreement with the creeping flow solution showing periodic orbits. For an elliptical cylinder or an ellipsoid, the results show that by increasing the Reynolds number, the period of rotation increases, and eventually becomes infinitely large at a critical Reynolds number, Rec. At Reynolds numbers above Rec, the particle becomes stationary in a steady-state flow. It is found that the transition from a time-periodic to a steady state is through a saddle-node bifurcation, and, consequently, the period of oscillation near this transition is proportional to [mid ]p−pc[mid ]−1/2, where p is any parameter in the flow, such as the Reynolds number or the density ratio, which leads to this transition at p = pc. This universal scaling law is presented along with the physics of the transition and the effect of the inertia and the solid-to-fluid density ratio on the dynamics. It is conjectured that this transition and the scaling law are independent of the particle shape (excluding body of revolution) or the shear profile.

Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers

2008 ◽  
Vol 599 ◽  
pp. 341-362 ◽  
Author(s):  
HANG DING ◽  
PETER D. M. SPELT
Keyword(s):  
Contact Angle ◽  
Reynolds Number ◽  
Steady State ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Critical Value ◽  
Creeping Flow ◽  
Force Balance ◽  
Diffuse Interface ◽  
Critical Conditions

We investigate the critical conditions for the onset of motion of a three-dimensional droplet on a wall in shear flows at moderate Reynolds number. A diffuse-interface method is used for this purpose, which also circumvents the stress singularity at the moving contact line, and the method allows for a density and viscosity contrast between the fluids. Contact-angle hysteresis is represented by the prescription of a receding contact angle θRand an advancing contact angle value θA. Critical conditions are determined by tracking the motion and deformation of a droplet (initially a spherical cap with a uniform contact angle θ0). At sufficiently low values of a Weber number,We(based on the applied shear rate and the drop volume), the drop deforms and translates for some time, but subsequently reaches a stationary position and attains a steady-state shape. At sufficiently large values ofWeno such steady state is found. We present results for the critical value ofWeas a function of Reynolds numberRefor cases with the initial value of the contact angle θ0=θRas well as for θ0=θA. A scaling argument based on a force balance on the drop is shown to represent the results very accurately. Results are also presented for the static shape, transient motion and flow structure at criticality. It is shown that at lowReour results agree (with some qualifications) with those of Dimitrakopoulos & Higdon (1998,J. Fluid Mech. vol. 377, p. 189). Overall, the results indicate that the critical value ofWeis affected significantly by inertial effects at moderate Reynolds numbers, whereas the steady shape of droplets still shows some resemblance to that obtained previously for creeping flow conditions. The paper concludes with an investigation into the complex structure of a steady wake behind the droplet and the occurrence of a stagnation point at the upstream side of the droplet.

Time-Dependent Laminar Backward-Facing Step Flow in a Two-Dimensional Duct

1988 ◽  
Vol 110 (3) ◽  
pp. 289-296 ◽  
Author(s):  
F. Durst ◽  
J. C. F. Pereira
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Separated Flow ◽  
Flow Region ◽  
Reynolds Numbers ◽  
Steady State Flow ◽  
Backward Facing Step ◽  
Step Flow ◽  
Recirculating Flow ◽  
State Flow

This paper presents results of numerical studies of the impulsively starting backward-facing step flow with the step being mounted in a plane, two-dimensional duct. Results are presented for Reynolds numbers of Re = 10; 368 and 648 and for the last two Reynolds numbers comparisons are given between experimental and numerical results obtained for the final steady state flow conditions. In the computational scheme, the convective terms in the momentum equations are approximated by a 13-point quadratic upstream weighted finite-difference scheme and a fully implicit first order forward differencing scheme is used to discretize the temporal derivatives. The computations show that for the higher Reynolds numbers, the flow starts to separate on the lower and upper corners of the step yielding two disconnected recirculating flow regions for some time after the flow has been impulsively started. As time progresses, these two separated flow regions connect up and a single recirculating flow region emerges. This separated flow region stays attached to the step, grows in size and approaches, for the time t → ∞, the dimensions measured and predicted for the separation region for steady laminar backward-facing flow. For the Reynolds number Re = 10 the separation starts at the bottom of the backward-facing step and the separation region enlarges with time until the steady state flow pattern is reached. At the channel wall opposite to the step and for Reynolds number Re = 368, a separated flow region is observed and it is shown to occur for some finite time period of the developing, impulsively started backward-facing step flow. Its dimensions change with time and reduce to zero before the steady state flow pattern is reached. For the higher Reynolds number Re = 648, the secondary separated flow region opposite to the wall is also present and it is shown to remain present for t → ∞. Two kinds of the inlet conditions were considered; the inlet mean flow was assumed to be constant in a first study and was assumed to increase with time in a second one. The predicted flow field for t → ∞ turned out to be identical for both cases. They were also identical to the flow field predicted for steady, backward-facing step flow using the same numerical grid as for the time-dependent predictions.

scholarly journals Entrance Effects on Diffused Film-Cooling Holes

1998 ◽  
Author(s):  
A. Kohli ◽  
K. A. Thole
Keyword(s):  
Reynolds Number ◽  
Film Cooling ◽  
Gas Turbines ◽  
High Performance ◽  
Computational Study ◽  
Density Ratio ◽  
Reynolds Numbers ◽  
Supply Channel ◽  
Cooling Hole ◽  
Film Cooling Holes

Film-cooling is a widely used method of prolonging blade life in high performance gas turbines and is implemented by injecting cold air through discrete holes on the blade surface. Most experimental research on film-cooling has been performed using round holes supplied by a stagnant plenum. This can be quite different from the actual turbine blade conditions in that a crossflow may be present whereby the internal channel Reynolds number could be as high as 90,000. This computational study uses a film-cooling hole that is inclined at 35° with respect to the mainstream and is diffused at the hole exit by 15°. An engine representative jet-to-mainstream density ratio of two was simulated. The test matrix consisted of fourteen different cases that were simulated for the two different blowing ratios in which the following effects were investigated: a) the effect of the orientation of the coolant supply channel relative to the cooling hole, b) the effect of the channel Reynolds number, and c) the effect of the metering length of the cooling hole. Results showed that the orientation of the coolant supply had a large effect whereby the worst orientation, in terms of a reduced adiabatic effectiveness, was predicted when the channel supplying the cooling hole was perpendicular to the mainstream. For this particular orientation, higher laterally averaged effectiveness occurred at lower channel Reynolds numbers and with the hole having a short metering length.

Simulations of Droplet Collisions in Shear Flow

2012 ◽  
Author(s):  
Orest Shardt ◽  
J. J. Derksen ◽  
Sushanta K. Mitra
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Capillary Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Simple Shear Flow ◽  
Number Range ◽  
Droplet Collisions ◽  
Critical Capillary Number ◽  
Boltzmann Method

When droplets collide in a shear flow, they may coalesce or remain separate after the collision. At low Reynolds numbers, droplets coalesce when the capillary number does not exceed a critical value. We present three-dimensional simulations of droplet coalescence in a simple shear flow. We use a free-energy lattice Boltzmann method (LBM) and study the collision outcome as a function of the Reynolds and capillary numbers. We study the Reynolds number range from 0.2 to 1.4 and capillary numbers between 0.1 and 0.5. We determine the critical capillary number for the simulations (0.19) and find that it is does not depend on the Reynolds number. The simulations are compared with experiments on collisions between confined droplets in shear flow. The critical capillary number in the simulations is about a factor of 25 higher than the experimental value.

On a Circular Cylinder in a Uniform Planar Shear Flow at Subcritical Reynolds Number

2002 ◽  
Author(s):  
D. Sumner ◽  
O. O. Akosile
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Shear Flow ◽  
Lift Force ◽  
Experimental Studies ◽  
Reynolds Numbers ◽  
Low Velocity ◽  
Static Pressure Distribution ◽  
Planar Shear ◽  
The Mean

An experimental investigation was conducted of a circular cylinder immersed in a uniform planar shear flow, where the approach velocity varies across the diameter of the cylinder. The study was motivated by some apparent discrepancies between numerical and experimental studies of the flow, and the general lack of experimental data, particularly in the subcritical Reynolds number regime. Of interest was the direction and origin of the steady mean lift force experienced by the cylinder, which has been the subject of contradictory results in the literature, and for which measurements have rarely been reported. The circular cylinder was tested at Reynolds numbers from Re = 4.0×104 − 9.0×104, and the dimensionless shear parameter ranged from K = 0.02 − 0.07, which corresponded to a flow with low to moderate shear. The results showed that low to moderate shear has no appreciable influence on the Strouhal number, but has the effect of lowering the mean drag coefficient. The circular cylinder develops a small steady mean lift force directed towards the low-velocity side, which is attributed to an asymmetric mean static pressure distribution on its surface. The reduction in the mean drag force, however, cannot be attributed solely to this asymmetry.

The effect of weak Brownian rotations on particles in shear flow

1971 ◽  
Vol 46 (4) ◽  
pp. 685-703 ◽  
Author(s):  
L. G. Leal ◽  
E. J. Hinch
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Probability Distribution ◽  
Shear Flow ◽  
Equilibrium Distribution ◽  
Steady State Probability ◽  
Bulk Properties ◽  
Closed Orbits ◽  
Basic Equations

Axisymmetric particles in zero Reynolds number shear flow execute closed orbits. In this paper we consider the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit. After presenting the basic equations, we derive an expression for the equilibrium distribution. This result is then used to calculate some bulk properties for a suspension of such particles, and these predicted properties are compared with available experimental observation.

Transitory Behavior of Synthetic Jets

2005 ◽  
Author(s):  
Michael Amitay ◽  
Florine Cannelle
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Input Signal ◽  
Mass Flux ◽  
Reynolds Numbers ◽  
Synthetic Jet ◽  
Synthetic Jets ◽  
Hot Wire ◽  
Stroke Length ◽  
Zero Net Mass Flux

The transitory behavior of an isolated synthetic (zero net mass flux) jet was investigated experimentally using PIV and hot-wire anemometry. In the present work, the synthetic jet was produced over a broad range of length- and time-scales, where three formation frequencies, f = 300, 917, and 3100Hz, several stroke lengths (between 5 and 50 times the slit width) and Reynolds numbers (between 85 and 408) were tested. The transitory behavior, following the onset of the input signal, in planes along and across the slit was measured. It was found that the time it takes the synthetic jet to become fully developed depends on the stroke length, formation frequency and Reynolds number. In general, the transients consist of four stages associated with the merging of vortices in both cross-stream and spanwise planes that grow in size, which lead to the pinch off of the leading vortex before the jet reaches its steady-state.

Transitions and instabilities of flow in a symmetric channel with a suddenly expanded and contracted part

2001 ◽  
Vol 434 ◽  
pp. 355-369 ◽  
Author(s):  
J. MIZUSHIMA ◽  
Y. SHIOTANI
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Nonlinear Stability ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Critical Conditions ◽  
Stable Steady State ◽  
Weakly Nonlinear ◽  
Symmetric Channel ◽  
Steady State Solutions

Transitions and instabilities of two-dimensional flow in a symmetric channel with a suddenly expanded and contracted part are investigated numerically by three different methods, i.e. the time marching method for dynamical equations, the SOR iterative method and the finite-element method for steady-state equations. Linear and weakly nonlinear stability theories are applied to the flow. The transitions are confirmed experimentally by flow visualizations. It is known that the flow is steady and symmetric at low Reynolds numbers, becomes asymmetric at a critical Reynolds number, regains the symmetry at another critical Reynolds number and becomes oscillatory at very large Reynolds numbers. Multiple stable steady-state solutions are found in some cases, which lead to a hysteresis. The critical conditions for the existence of the multiple stable steady-state solutions are determined numerically and compared with the results of the linear and weakly nonlinear stability analyses. An exchange of modes for oscillatory instabilities is found to occur in the flow as the aspect ratio, the ratio of the length of the expanded part to its width, is varied, and its relation with the impinging free-shear-layer instability (IFLSI) is discussed.

Unsteady drag on a sphere at finite Reynolds number with small fluctuations in the free-stream velocity

1991 ◽  
Vol 233 ◽  
pp. 613-631 ◽  
Author(s):  
Renwei Mei ◽  
Christopher J. Lawrence ◽  
Ronald J. Adrian
Keyword(s):  
Reynolds Number ◽  
Free Stream ◽  
Low Frequency ◽  
Reynolds Numbers ◽  
Free Stream Velocity ◽  
Creeping Flow ◽  
Stream Velocity ◽  
Velocity Difference ◽  
Long Time ◽  
The Time Domain

Unsteady flow over a stationary sphere with small fluctuations in the free-stream velocity is considered at finite Reynolds number using a finite-difference method. The dependence of the unsteady drag on the frequency of the fluctuations is examined at various Reynolds numbers. It is found that the classical Stokes solution of the unsteady Stokes equation does not correctly describe the behaviour of the unsteady drag at low frequency. Numerical results indicate that the force increases linearly with frequency when the frequency is very small instead of increasing linearly with the square root of the frequency as the classical Stokes solution predicts. This implies that the force has a much shorter memory in the time domain. The incorrect behaviour of the Basset force at large times may explain the unphysical results found by Reeks & Mckee (1984) wherein for a particle introduced to a turbulent flow the initial velocity difference between the particle and fluid has a finite contribution to the long-time particle diffusivity. The added mass component of the force at finite Reynolds number is found to be the same as predicted by creeping flow and potential theories. Effects of Reynolds number on the unsteady drag due to the fluctuating free-stream velocity are presented. The implications for particle motion in turbulence are discussed.

Pulsatile Blood Flow in a Channel of Small Exponential Divergence—III. Unsteady Flow Separation

1977 ◽  
Vol 99 (2) ◽  
pp. 333-338 ◽  
Author(s):  
Daniel J. Schneck
Keyword(s):  
Reynolds Number ◽  
Blood Flow ◽  
Steady State ◽  
Flow Separation ◽  
Reynolds Numbers ◽  
Critical Value ◽  
Flow Oscillation ◽  
Pulsatile Blood Flow ◽  
Steady Streaming ◽  
Flow Through

Analysis of pulsatile flow through exponentially diverging channels reveals the existence of critical mean Reynolds numbers for which the flow separates at a downstream axial station. These Reynolds numbers vary directly with the frequency of flow oscillation and inversely with the rate of channel divergence. Increasing the Reynolds number above its critical value results in a rapid upstream displacement of the point of separation. For a tube of fixed geometry, periodic unsteadiness causes flow separation to occur at lower Reynolds numbers and upstream of a corresponding steady-state situation. The point of separation moves progressively downstream, however, towards its steady-state location, as the frequency of oscillation increases. These results are discussed as consequences of the nonlinear steady streaming phenomenon described in an earlier paper.

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