^{1}, Sarah L. Waters

^{2}and Oliver E. Jensen

^{1,a)}

### Abstract

We examine the linear stability of two-dimensional Poiseuille flow in a long channel confined by a rigid wall and a massless damped-tensioned membrane. We seek solutions that are periodic in the streamwise spatial direction and time, solving the homogeneous eigenvalue problem using a Chebyshev spectral method and asymptotic analysis. Several modes of instability are identified, including Tollmien–Schlichting (TS) waves and traveling-wave flutter (TWF). The eigenmode for neutrally stable downstream-propagating TWF in the absence of wall damping is shown to have a novel asymptotic structure at high Reynolds numbers, not reported in symmetric flexible-walled channels, involving a weak but destabilizing critical layer at the channel centerline where the wave speed is marginally greater than the maximum Poiseuille flow speed. We also show that TS instabilities along the lower branch of the neutral curve are modified remarkably little by wall compliance, but can be either stabilized or destabilized by wall damping. We discuss the energy budget underlying TWF and briefly describe the structure of other flow-induced surface instabilities.

P.S.S. acknowledges support from Biotechnology and Biological Sciences Research Council. S.L.W. is grateful for support from the Engineering and Physical Sciences Research Council in the form of an Advanced Research Fellowship (Contract No. EP/D070635/1). O.E.J. acknowledges support from the Leverhulme Trust. We are very grateful to J. Billingham, R. J. Whittaker, and M. Heil for useful discussions.

I. INTRODUCTION

II. THE MODEL

III. NUMERICAL RESULTS

A. Temporal stability

B. Spatial stability

C. Neutral stability curves

IV. LARGE-REYNOLDS-NUMBER ASYMPTOTICS

A. Lower branch Tollmien–Schlichting instabilities

B. Traveling-wave flutter instabilities

1. Primary oscillations in the core

2. The leading-order adjoint problem

3. Primary oscillations in the Stokes layers

4. First-order corrections in the core

5. Second-order corrections in the core

6. Energetics of neutrally stable TWF

C. The stable surface-based modes

V. THE ROLE OF WALL DAMPING

VI. DISCUSSION

### Key Topics

- Flow instabilities
- 39.0
- Reynolds stress modeling
- 27.0
- Viscosity
- 26.0
- Poiseuille flow
- 23.0
- Boundary value problems
- 12.0

## Figures

Poiseuille flow in an asymmetric compliant channel, subject to an external pressure gradient .

Poiseuille flow in an asymmetric compliant channel, subject to an external pressure gradient .

Eigenvalue spectra for an asymmetric compliant channel where all the modes are linearly stable: (a) temporal modes in the -plane for , , , and ; (b) spatial modes in the -plane for , , , and . The labels SD, TS, TWF, , , , and are described in the text. The crosses in panel (b) represent the leading-order asymptotic approximation to the surface-based modes satisfying Eqs. (33b) and (64). The TWF modes are labeled (u) or (d) to show which are propagating upstream or downstream.

Eigenvalue spectra for an asymmetric compliant channel where all the modes are linearly stable: (a) temporal modes in the -plane for , , , and ; (b) spatial modes in the -plane for , , , and . The labels SD, TS, TWF, , , , and are described in the text. The crosses in panel (b) represent the leading-order asymptotic approximation to the surface-based modes satisfying Eqs. (33b) and (64). The TWF modes are labeled (u) or (d) to show which are propagating upstream or downstream.

Eigenfunctions of the four surface-based modes for , at , calculated numerically (solid lines); (a) downstream-propagating TWF; (b) upstream-propagating TWF; (c) downstream-propagating SD; and (d) upstream-propagating SD. The filled black circles on each panel represent the leading-order asymptotic approximation [, Eq. (32a)] normalized using the numerically determined pressure on the compliant wall . The crosses in panel (a) represent the first order correction [, see Eq. (53)] for neutrally stable downstream-propagating TWF.

Eigenfunctions of the four surface-based modes for , at , calculated numerically (solid lines); (a) downstream-propagating TWF; (b) upstream-propagating TWF; (c) downstream-propagating SD; and (d) upstream-propagating SD. The filled black circles on each panel represent the leading-order asymptotic approximation [, Eq. (32a)] normalized using the numerically determined pressure on the compliant wall . The crosses in panel (a) represent the first order correction [, see Eq. (53)] for neutrally stable downstream-propagating TWF.

Neutrally stability curves for , calculated numerically (solid lines): (a) wavenumber-Reynolds number plane; (b) frequency-Reynolds number plane; (c) the corresponding wave speed of the neutrally stable modes against Reynolds number. The dashed line is the asymptotic approximation to the lower branch of TS instabilities (26). The filled black circles represent the asymptotic approximation to neutrally stable TWF instabilities (33b) and (57).

Neutrally stability curves for , calculated numerically (solid lines): (a) wavenumber-Reynolds number plane; (b) frequency-Reynolds number plane; (c) the corresponding wave speed of the neutrally stable modes against Reynolds number. The dashed line is the asymptotic approximation to the lower branch of TS instabilities (26). The filled black circles represent the asymptotic approximation to neutrally stable TWF instabilities (33b) and (57).

Variation in the critical wavenumber for lower branch TS instability against dimensionless membrane tension, shown for , 1, 10, and 100. The corresponding result for a rigid channel [obtained by Bogdanova and Ryzhov (Ref. 52)] is illustrated using black circles.

Variation in the critical wavenumber for lower branch TS instability against dimensionless membrane tension, shown for , 1, 10, and 100. The corresponding result for a rigid channel [obtained by Bogdanova and Ryzhov (Ref. 52)] is illustrated using black circles.

Eigenfunction of the downstream-propagating TWF mode at , computed numerically (solid lines) where (temporally unstable); (a) the cross-stream variation in ; (b) the variation in across the Stokes’ layer adjacent to the wall at ; (c) the cross-stream variation in ; and (d) the flow structure including the weak critical layer in the center of the channel. The filled black circles in [(a)–(c)] correspond to the inviscid core solution Eq. (32a) and the crosses in (b) to the inner solution (43) in the Stokes’ layer close to . The leading-order wave speed from the asymptotic approximation (33b) and (57) is .

Eigenfunction of the downstream-propagating TWF mode at , computed numerically (solid lines) where (temporally unstable); (a) the cross-stream variation in ; (b) the variation in across the Stokes’ layer adjacent to the wall at ; (c) the cross-stream variation in ; and (d) the flow structure including the weak critical layer in the center of the channel. The filled black circles in [(a)–(c)] correspond to the inviscid core solution Eq. (32a) and the crosses in (b) to the inner solution (43) in the Stokes’ layer close to . The leading-order wave speed from the asymptotic approximation (33b) and (57) is .

The numerically computed wave speed (solid lines) of neutrally stable TWF modes for , (corresponding to Fig. 4) and plotted against wavenumber . The filled black circles are the leading-order asymptotic approximation [Eq. (33b)]; the crosses also include the first order correction (50).

The numerically computed wave speed (solid lines) of neutrally stable TWF modes for , (corresponding to Fig. 4) and plotted against wavenumber . The filled black circles are the leading-order asymptotic approximation [Eq. (33b)]; the crosses also include the first order correction (50).

Neutrally stable TWF mode for and at , normalized so that the amplitude is ; (a) snapshot of the wall position; (b) contour plot of the instantaneous kinetic energy flux due to nonlinear Reynolds stresses ; (c) contour plot of the rate of energy loss ; insets show (c)(i) contour plot of the rate of energy loss across the Stokes layer close to and (c)(ii) contour plot of across the inviscid core. In (b) and (c) the thick solid line corresponds to the zero contour and in the shaded (unshaded) regions the plotted quantity is positive (negative). [(d) and (e)] Eigenfunctions of the averaged integrands and for a neutrally stable TWF mode scaled to illustrate (d) the variations in the Stokes layer close to ; (e) the variation across the center of the channel.

Neutrally stable TWF mode for and at , normalized so that the amplitude is ; (a) snapshot of the wall position; (b) contour plot of the instantaneous kinetic energy flux due to nonlinear Reynolds stresses ; (c) contour plot of the rate of energy loss ; insets show (c)(i) contour plot of the rate of energy loss across the Stokes layer close to and (c)(ii) contour plot of across the inviscid core. In (b) and (c) the thick solid line corresponds to the zero contour and in the shaded (unshaded) regions the plotted quantity is positive (negative). [(d) and (e)] Eigenfunctions of the averaged integrands and for a neutrally stable TWF mode scaled to illustrate (d) the variations in the Stokes layer close to ; (e) the variation across the center of the channel.

Neutral stability curves of the downstream propagating traveling wave flutter mode for for , , , 1, 10, and .

Neutral stability curves of the downstream propagating traveling wave flutter mode for for , , , 1, 10, and .

Neutral stability curves calculated numerically for and of the local modes (a) in the wavenumber-Reynolds number plane; (b) frequency-Reynolds number plane. The filled black circles are the predictions of a corresponding one-dimensional model (Ref. 4). The dashed lines are the large asymptotes to the one-dimensional model [Eq. (68)] and the dot-dashed line is the large asymptote to the lower branch TS mode [Eq. (26)].

Neutral stability curves calculated numerically for and of the local modes (a) in the wavenumber-Reynolds number plane; (b) frequency-Reynolds number plane. The filled black circles are the predictions of a corresponding one-dimensional model (Ref. 4). The dashed lines are the large asymptotes to the one-dimensional model [Eq. (68)] and the dot-dashed line is the large asymptote to the lower branch TS mode [Eq. (26)].

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