Time-dependent Velocity Profiles Research Articles

Inertially driven transient response in polymeric liquids

We report here a computational study, with inertial contributions included, of the time-dependent velocity and shear stress profiles following flow start-up from rest for a polymeric liquid. Earlier workers solved similar fluid-mechanical start-up problems with viscoelastic character represented by a single Maxwell element. They found significant oscillatory departures from the steady-state velocity profile that persist throughout the entire period of stress growth to steady state. Such results are unsettling since they imply that the usual assumptions about velocity fields for transient tests of polymeric liquids are seriously in error. It turns out, however, that such oscillations are rapidly suppressed when a broad distribution of relaxation times is employed, the physically appropriate situation for polymeric liquids. The initial shear wave oscillations decay quickly compared with the time to reach the stress steady state when realistic descriptions of their viscoelastic character are employed. We note that a combination of inertial contributions with sparse relaxation spectra in fluid-mechanical modeling can lead to spurious velocity and stress field predictions for polymeric liquid flows.

Time-dependent directional solidification of binary Al–Cu alloys in the initial transient

Abstract At the start of directional solidification a complex interaction between the movement of the solidification front, the concentration pile-up ahead of the front and the temperature gradient at the solid – liquid interface takes place. Experiments with binary Al-Cu1.3 wt.% were performed in this regime in a Bridgman – Stockbarger type gradient furnace allowing solidification with a planar solid – liquid interface. The time-dependent velocity and temperature profiles were measured during the experiments, whereas the concentration profiles were analysed from the processed samples. The experimental results were compared to predictions from numerical simulations based on the model of Coriell et al. Good agreement is obtained in the description of time-dependent diffusive, planar growth in the initial transient.

Local velocity measurements in heterogeneous and time-dependent flows of a micellar solution

We present and discuss the results of pointwise velocity measurements performed on a viscoelastic micellar solution made of cetyltrimethylammonium bromide and sodium salicylate in water, respectively, at the concentrations of 50 and 100 mmol. The sample is contained in a Couette device and subjected to flow in the strain controlled mode. This particular solution shows shear banding and, in a narrow range of shear rates at the right end of the stress plateau, apparent shear thickening occurs. Time-dependent recordings of the shear stress in this range reveal that the flow has become unstable and that large sustained oscillations of the shear stress and of the first normal stresses difference emerge and grow in the flow. Local pointwise velocity measurements clearly reveal a velocity profile typical of shear banding when the imposed shear rate belongs to the plateau, but also important wall slip in the entire range of velocity gradients investigated. In the oscillations regime, the velocity is recorded as a function of time at a fixed point close to the rotor of the Couette device. The time-dependent velocity profile reveals random fluctuations but, from time to time, sharp decreases much larger than the standard deviation are observed. An attempt is made to correlate these strong variations with the stress oscillations and a correlation coefficient r is computed. However, the small value found for the coefficient r does not allow us to draw a final conclusion as concerns the correlation between stress oscillations and velocity fast decreases.

Stability of an axially accelerating string subjected to frictional guiding forces

Abstract The dynamic response of an axially translating continuum subjected to the combined effects of a pair of spring supported frictional guides and axial acceleration is investigated; such systems are both non-conservative and gyroscopic. The continuum is modeled as a tensioned string translating between two rigid supports with a time-dependent velocity profile. The equations of motion are derived with the extended Hamilton's principle and discretized in the space domain with the finite element method. The stability of the system is analyzed with the Floquet theory for cases where the transport velocity is a periodic function of time. Direct time integration using an adaptive step Runge–Kutta algorithm is used to verify the results of the Floquet theory. This approach can also be employed in the general case of arbitrary time-varying velocity. Results are given in the form of time history diagrams and instability point grids for different sets of parameters such as the location of the stationary load, the stiffness of the elastic support, and the values of initial tension. This work showed that presence of friction adversely affects stability, but using non-zero spring stiffness on the guiding force has a stabilizing effect. This work also showed that the use of the finite element method and Floquet theory is an effective combination to analyze stability in gyroscopic systems with stationary friction loads.

The effect of advance ratio on the aerodynamics of revolving wings

Recent studies have demonstrated that a quasi-steady model closely matches the instantaneous force produced by an insect wing during hovering flight. It is not clear, however, if such methods extend to forward flight. In this study we use a dynamically scaled robotic model of the fruit fly Drosophila melanogaster to investigate the forces produced by a wing revolving at constant angular velocity while simultaneously translating at velocities appropriate for forward flight. Because the forward and angular velocities were constant wing inertia was negligible, and the measured forces can be attributed to fluid dynamic phenomena. The combined forward and revolving motions of the wing produce a time-dependent free-stream velocity profile, which suggests that added mass forces make a contribution to the measured forces. We find that the forces due added mass make a small, but measurable, component of the total force and are in excellent agreement with theoretical values. Lift and drag coefficients are calculated from the force traces after subtracting the contributions due to added mass. The lift and drag coefficients, for fixed angle of attack, are not constant for non-zero advance ratios, but rather vary in magnitude throughout the stroke. This observation implies that modifications of the quasi-steady model are required in order to predict accurately the instantaneous forces produced during forward flight. We show that the dependence of the lift and drag coefficients upon advance ratio and stroke position can be characterized effectively in terms of the tip velocity ratio--the ratio of the chordwise components of flow velocity at the wing tip due to translation and revolution. On this basis we develop a modified quasi-steady model that can account for the varying magnitudes of the lift and drag coefficients. Our model may also resolve discrepancies in past measurements of wing performance based on translational and revolving motion.

Some Hydrodynamic Models for Studying the Displacement of One Liquid by Another. Part II. Flow Within a Wedge

We construct and solve in an analytical way a model to describe the displacement process of one liquid by another. By neglecting surface tension and assuming simple shear flows, we find an explicit analytical solution for the time-dependent velocity profiles and discharges in a wedge-shaped geometry. Our formalism permits us to discern the presence of the 3 different time scales involved in this process that may have an influence over the whole displacement time. Finally, we discuss the qualitative similarity of this process and a diffusion process that has been used previously for fitting experimental results.

Some Hydrodynamic Models for Studying the Displacement of One Liquid by Another. Part I. Cylindrical Flow

We construct and solve in an analytical way a model to describe the displacement process of one liquid by another. By neglecting surface tension and assuming simple shear flows, we find an explicit analytical solution for the time-dependent velocity profiles. Our formalism permits us to discern the presence of the 3 different time scales involved in this process that may have an influence over the whole displacement time.

Magnetically Linked Star‐Disk Systems. II. Effects of Plasma Inertia and Reconnection in the Magnetosphere

Following on the results of the companion paper, we study the processes occurring in a force-free magnetosphere of a magnetically linked star-disk system as it expands owing to the rotation of the disk and, in finite time, approaches an open state. In particular, we use the self-similar solution for the magnetic field evolution to calculate the time-dependent plasma density and velocity profiles. We also perform an asymptotic analysis of the plasma velocity and density at the apex of an expanding field line as it approaches the open state. We use the resulting expressions to investigate how and when the inertial effects in the magnetosphere will intervene to slow down the expansion and to invalidate the force-free assumption. We find that the field opening drastically reduces the density near the apex of the expanding field lines (typically elongated in a direction making a ~60° angle with respect to the rotation axis) while at the same time creating a pronounced density enhancement near the rotation axis. The former effect is conducive to the triggering of microinstabilities in the current layer that forms along the direction of elongation, whereas the latter appears to be related to a mechanism for the formation of an axially outflowing condensation previously identified in axisymmetric numerical simulations. The appearance of a second outflow component along the disk surface in these simulations can also be reproduced by our model. Finally, we discuss the possibility that the expanding field lines reconnect across the current layer before they open up, leading to a quasi-periodic process of field-line expansion and reconnection.

Flow measurement on an oscillating pipe flow near the entrance using the UVP method

The authors have carried out a study to investigate and clarify the characteristics of purely oscillating pipe flows over the developing region. The main objective of this study is to establish the method of time-dependent velocity profiles obtained by the ultrasonic velocity profile (UVP) measurement method. First, the relationship between the test fluids and the microparticles, as reflectors of ultrasonic pulses, was investigated. In addition, the relationship between the sound speeds of the test fluids and the wall materials was studied. Second, the UVP was used to obtain the instantaneous velocity profiles in oscillating pipe flows, and the developing characteristics of the flows were analyzed. Finally, the “entrance length” (by analogy with a unidirectional pipe flow) required for oscillating pipe flows was analyzed by examining the amplitude of the harmonic spectral components of the oscillating frequency. A fast Fourier transform (FFT) is proposed as the applicable method to estimate the “entrance length”. From the Fourier transform of the velocity on the centerline, nonlinear oscillation of fluid occurs in the “entrance length” of the oscillating flows, and the viscous dissipation of the higher-order velocity harmoncis determines the entrance region. The “entrance length” can be obtained from the dissipation length of the third-order harmonic. These results prove that the UVP method is highly applicable to carry out the flow measurement in the “entrance length” of oscillating pipe flow.

Fluid fields due to many-body hydrodynamic interactions

We consider the characteristics of the unsteady flows, generated in the incompressible, viscous fluid by the hydrodynamic interactions between a finite number of solid spheres. The non-stationary properties of the velocity fields, due to the local inertia of the fluid, are discussed. To take into account the local acceleration effects, we apply the time-dependent Green's tensor. The four different types of the time-dependent velocity profiles, generated at different régimes of the low Reynolds number hydrodynamic interactions, are regarded.

Spatial decay estimates for heat-conducting micropolar fluids

Abstract Two sorts of spatial decay estimates are established for describing the spatial behaviour of the solutions for the flow of a heat-conducting micropolar fluid in a semi-infinite cylindrical pipe. The lateral surface is thermal insulated and the adherence of the fluid to the lateral boundary is assumed. At the finite end of the pipe a time-dependent velocity and angular velocity profile are prescribed together the temperature field.

曲がり円管内振動層流助走区間流れの可視化解析

Pictures have been presented for oscillatory laminar flow in the inlet region of circular curved pipes with long, straight tangents. Visualization of the flow was realized by the hydrogen bubble method and the time-dependent velocity profiles were grasped with reasonable accuracy. Their features substantiate the behavior of the developing oscillatory flow in dependence on the Womersley number α. Influences of the curvature radius ratio Rc and the Dean number D on the development are also investigated. From LDV measurements executed with reference to the visualization data, moreover, the inlet lengths of both the curved pipe and the upstream tangent are obtained in a wide range of the parameters and the formulas for them are presented as a function of D/α 2·Rc1/2, which is effective when α>4. The inlet length of the upstream tangent is 1.5 times as long as one of the curved pipe.

開口端を有するダクトにおける脈動流に関する研究

The purpose of the present paper is to study the characteristics of flow and gas exchange in the pulsating flow in the respiratory net work both by flow visualization and by numerical simulation.The pulsating flow in a two-dimensional duct was visualized by the smoke-wire method, and its time-dependent velocity profile in the duct was measured with reasonable accuracy. The behavior of the pulsating flow around the duct intake was observed by the tobacco-smoke method and it was shown that the breathing air causes the streaming that brings about gas exchange.The numerical results confirmed the experimental results and predicted the behavior of gas diffusion at the intake.

Quasi-steady dissipative nonlinear critical layer in a stratified shear flow

When a wave with small but finite amplitude ε propagates towards the CL, where the effects of nonlinearity and dissipation are essential, the jump of mean vorticity over the CL appears. For the dynamically stable stratified shear flow with the gradient Richardson number Ri≳1/4 the jump of vorticity has the same order as the undisturbed one [J. Fluid Mech. 233, 25 (1991)]. The process of formation of the flow with this substantial jump of vorticity (or ‘‘break’’ of the velocity profile) in the CL is studied at large time after beginning of the process. The transition region between the CL and the undisturbed flow, the dissipation boundary layer (DBL), is shown to be formed. Its thickness grows in time proportional to √t (t being time), and the CL moves towards the incident wave. When the jump of the wave momentum flux over the CL is constant in time, the flow characteristics can be found in the most simple way. The velocity profile in the DBL appears to be self-similar, the displacement of the CL is proportional to √t and the values of vorticity at the both sides of the CL do not depend on time and they are determined only by the constant wave momentum flux. It is shown that, to provide the constant jump of the wave momentum flux the amplitude of the wave radiated by the source in the undisturbed flow region should vary in a certain complicated manner, because it reflects from the time-dependent (broadening) velocity profile in the DBL. On the other hand, the wave momentum flux from the steady source (for example, the corrugated wall) depends on time. When the coefficients of reflection from the CL (R) and from the DBL (r) are small, this dependence is weak and the wave and flow parameters depending on time are found as series in R and r. The wave–flow interaction for this case is studied.

A Dynamical Monte Carlo Simulation of a Collisionless Shock

We report results from a new time-dependent, self-consistent simulation of a plane-parallel collisionless shock, including the acceleration of energetic particles. In this simulation we model the basic physics of scattering off magnetic irregularities with a prescribed scattering algorithm. This algorithm simply requires isotropy and conservation of energy in a particle's local flow frame during a scattering. A single constant parameter, the scattering time, characterizes the scattering. Since the scattering time is taken to be constant and the same for all particles, this implies that a particle's mean free path is proportional to its velocity. The collective behavior of a large number of such particles models a collisionless fluid with a background magnetic field parallel to the shock normal. <P />We begin the simulation with a gas of particles having both a thermal distribution and a net motion toward a massive reflecting wall. The wall, similar to a piston moving supersonically through a stationary gas, causes a shock wave to develop. The shock evolves and propagates across the box (away from the wall) in a configuration similar to that used in hybrid simulations. A power-law tail of accelerated particles develops as the shock evolves. Furthermore, the flow pattern is modified by the energetic particles, as seen in earlier hybrid simulations. <P />We present time-dependent shock simulation results, including density and velocity profiles, phase-space particle plots, and spectra. We also present a few tests of the validity of this new method, as well as an analysis of the time-dependent evolution of the subshock.

Theory of transient creep in the L1 2 intermetallic compounds displaying the yield strength anomaly

Abstract A theory of transient creep in the L1 2 intermetallic alloys displaying the yield strength anomaly is presented. The theory is based on the hypothesis that the mobile a 〈101〉 superdislocations undergo a pinning-depinning transition, displaying critical behavior at a critical stress. The areal-velocity scaling function, which describes the average, time-dependent velocity profile of a dislocation as it propagates through the crystal, is introduced. The theory is compared with preliminary experiments. It is argued that the measurement of transient creep allows one to determine information concerning the deformation history of the sample.

The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M2) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM2 to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM2= 0 in order to derive a solution with very small divergence can with good result be carried through withM2 as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatΔt · (i/(M Δx)−v/Δx2) is in the stability region of the time stepping method, and asM2=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.

Spatial characteristics of dynamic properties of modulated wavy vortex flow in a rotating Couette system

Time-dependent velocity profiles Vz(z, t) have been measured in a cylindrical Couette system for a range of Reynolds numbers corresponding to wavy vortex flow (WVF) and modulated wavy vortex flow (MWV). The spatial distribution of the power spectrum was obtained and the behaviour of the dynamic properties of WVF and MWV was investigated with respect to Reynolds number (Re) at four corners of cells (inside-outside and Inflow-Outflow). All modes of f1, f2 and B were found to be equally strong and broad at different corners for lower Re number. At large Re, however, f2 remains broad only at the outside-Inflow corner, while B is strong at Outflow. It was also found that B disappears at the outside-Inflow corner for R* > 18. These results suggest that the dynamic properties of WVF and MWV would have a spatial dependence within a cell structure scale.

START-UP FLOW IN A PIPE FOLLOWING THE SUDDEN IMPOSITION OF A CONSTANT FLOW RATE

The transient flow in a long pipe following the sudden imposition of a constant flow rate is considered. An exact analytical solution of the axisymmetric Navier-Stokes equations in terms of an infinite series of Bessel functions is derived. Time-dependent velocity profiles, as well as time histories of the axial pressure-gradient and the wall-friction, are presented. It is observed that the start-up time required to reach steady state is significantly shorter than if the start-up flow results from a suddenly imposed constant pressure gradient. The “Annular Jet Effect” observed experimentally by Kataoka et al. (1975) is not exhibited by the present solution.

Numerical simulation of time-dependent heat transfer in oscillating pipe flow

The problem of Enhanced Axial Heat Transfer (EAHT) in oscillatory pipe flow has been numerically studied. Time-dependent velocity and temperature profiles and Lagrangian and tidal displacements at various Wormersley numbers are obtained. The enhanced axial heat flux for different tidal displacements is calculated. The wall thickness effect in EAHT is also studied, and the optimum wall thickness is shown to be about 20% of the pipe radius in the test case studied. In addition, the tuning effect in axial heat flow is examined, the results indicating good agreement with analytic predictions. The numerical studies show that EAHT can be a very effective tool for those problems in which a large amount of heat must be transported without an accompanying convective mass exchange.