[Paper Review] Steady streaming between two vibrating planes at high Reynolds numbers

Konstantin Ilin, Andrey Morgulis|arXiv (Cornell University)|Aug 12, 2011

Nanofluid Flow and Heat Transfer22 references3 citations

TL;DR

This paper develops an asymptotic theory for steady streaming between two vibrating planes at high Reynolds numbers, using the Vishik-Lyusternik method to derive averaged Navier-Stokes equations with a Stokes drift correction. The key result is an analytical solution for unidirectional, plane-parallel flows induced by co-propagating traveling waves, extending peristaltic pumping theory to high Reynolds number regimes.

ABSTRACT

We consider incompressible flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier-Stokes equations in the limit when both the amplitude of vibrations and the thickness of the Stokes layer are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions, for the averaged flow. In the leading order, the averaged flow is described by the stationary Navier-Stokes equations with an additional term which contains the leading-order Stokes drift velocity. In a slightly different context (for a flow induced by an oscillating conservative body force), the same equations had been derived earlier by Riley (2001). The general theory is applied to two particular examples of steady streaming induced by transverse vibrations of the walls in the form of standing and travelling plane waves. In particular, in the case of waves travelling in the same direction, the induced flow is plane-parallel and the Lagrangian velocity profile can be computed analytically. This example may be viewed as an extension of the theory of peristaltic pumping to the case of high Reynolds numbers.

Motivation & Objective

• To develop a systematic asymptotic expansion for viscous incompressible flows between two transversely vibrating walls in the limit of small amplitude and small Stokes layer thickness, with Reynolds number scaling such that $ R_s \sim 1 $.
• To derive the averaged (steady streaming) equations and boundary conditions valid up to the flow boundary, accounting for nonlinear interactions in the leading-order flow.
• To extend the classical theory of peristaltic pumping to high Reynolds number flows by analyzing flows induced by traveling and standing waves.
• To determine optimal wall deformation profiles that maximize the total volume flux through the channel.
• To identify the limitations of the current asymptotic framework, particularly for short-wavelength vibrations and high $ R_s $.

Proposed method

• Employ the Vishik-Lyusternik method to construct an asymptotic expansion in the small parameter $ \epsilon $, which measures the ratio of wall displacement amplitude to channel height.
• Use a multiple-scale approach to separate fast time oscillations from slow averaged dynamics, retaining nonlinear terms in the leading-order equations.
• Derive the averaged Navier-Stokes equations with an additional term proportional to the leading-order Stokes drift velocity, capturing the net steady flow.
• Apply the method to two specific cases: standing waves (producing recirculating flows) and co-propagating traveling waves (producing unidirectional flows).
• Solve the resulting asymptotic equations analytically in the case of traveling plane waves, yielding explicit expressions for the Lagrangian velocity profile.
• Use Fourier series expansions of wall deformations to express the averaged velocity field for general periodic vibrations and optimize flux by selecting optimal harmonic components.

Experimental results

Research questions

• RQ1 What is the structure of the steady streaming flow induced by transverse vibrations of two parallel walls when the streaming Reynolds number $ R_s \sim 1 $?
• RQ2 How does the asymptotic solution account for the non-decaying boundary layer behavior near vibrating walls when the Stokes layer thickness is comparable to the wall displacement amplitude?
• RQ3 Can the averaged flow induced by traveling harmonic waves be solved analytically, and what is the resulting velocity profile?
• RQ4 What wall deformation profile maximizes the total volume flux through the channel for a given vibration amplitude?
• RQ5 How does the flow structure change with increasing wavelength for different types of wall vibrations (bending vs. contraction/expansion)?

Key findings

• For co-propagating traveling waves, the steady streaming flow is unidirectional and two-dimensional, with a symmetric Lagrangian velocity profile that is minimum at the center and maximum near the walls.
• In the short-wavelength limit, the velocity profile becomes nearly constant across the channel except in thin boundary layers near the walls, where it sharply increases.
• For long waves, the velocity variation across the channel diminishes, and the maximum velocity grows rapidly for contraction/expansion waves but decays for bending waves as the wavelength increases.
• The optimal wall deformation for maximizing volume flux corresponds to a single harmonic: the first harmonic for contraction/expansion waves, and a higher harmonic (e.g., $ m=4 $ for $ k=1 $) for bending waves.
• For counter-propagating traveling waves, the averaged flow evolves from a shear flow with 'cat’s eyes' vortices at short wavelengths to a complex pattern of alternating vortices at long wavelengths.
• The theory breaks down for horizontal length scales much smaller than the channel height, indicating the need for double boundary layer models in the short-wavelength regime.