Steady streaming of fluid in the entrance region of a tube during oscillatory flow

Abstract

To gain understanding and insight of entrance-flow phenomena, we solve a fluid dynamics problem involving an oscillating flat velocity profile (with zero mean flow) at the end (entrance) of a semi-infinite, rigid tube. A successive-approximation method is used. Analytic solutions are given for the first-order (linear) approximation. Time averages of products of these first-order solutions for velocity components and their spatial derivatives are used to approximate the nonlinear terms in the second-order equations, which are solved numerically. The time-averaged second-order results demonstrate steady bidirectional streaming in the entrance region. Solutions are given to an illustrative problem with a Reynolds number of 20 and a Womersley unsteadiness parameter of (200)1/2. At this low Reynolds number, the streaming effects are small, with the magnitudes of the calculated streaming velocities less than three percent of the amplitude of the velocity specified at the entrance. The calculated magnitudes of streaming velocities would be expected to increase at higher Reynolds numbers; however, the convergence of the second-order solution would be less certain at higher Reynolds numbers. The steady streaming is related to first-order velocity variations that exist near the entrance of the tube.

Streaming Caused by Oscillatory Flow in Peripheral Airways of Human Lung

“…Gaver and Grotberg surveyed the oscillatory flow in a tapered channel under conditions of fixed stroke volume and reported a bidirectional streaming in the channel by means of both calculation and experiment [9]. The similar bidirectional steady streaming was also predicted by Goldberg et al in their analysis of oscillatory flow at the entrance region of a semi-infinite tube of circular cross section [10].…”

Analysis of the oscillatory liquid metal flow in an alternate MHD generator

“…The steady streaming is induced by the non-linear Reynolds stresses in the boundary layer that appear due to the axial dependence of the streamwise velocity, produced in this case by the existence of the non-uniform field. In hydrodynamic flows, steady streaming appears, for instance, at the entrance of a rigid tube when a zero-mean oscillatory flow is imposed [23] or in the classic problem of oscillating bluff bodies [22,24]. The persistence of the steady streaming beyond the boundary layer is one of the distinctive aspects of this class of oscillatory flows [25].…”

Behavior of oscillatory tube flow at liquid-gas interfaces

“…Nicolas et al 27 ) or near end-walls. 9 In order to prove that the flow far away from the boundaries is not influenced thereof, the velocity profile was measured in the middle of the tube by PIV and compared with the analytical solution. Fig.…”

“…Far away from the top and bottom interfaces the flow is independent from the axial position in the tube and an analytical solution for the flow exists. 11,26 Near the free surface, the tangential stress at the interface induces steady streaming 9,27,28 which is investigated here in detail. The steady streaming patterns are analyzed by means of flow visualization as well as Particle Image Velocimetry (PIV) measurements under varying boundary conditions.…”

“…7 The occurrence of streaming in oscillatory flow is forced by Reynolds stress gradients which result in an additional body force term. 8,9 Since the body force terms are proportional to the Reynolds number it can be assumed that the streaming increases with increasing Reynolds number. Riley 10 introduced a modified Reynolds number R s = 2 · α 2 which denotes the magnitudes of mean convective effects to mean viscous effects.…”