Boundary layer flow of air over water on a flat plate

Nelson, John J.; Alving, Amy E.; Joseph, Daniel D.

doi:10.1017/s0022112095000309

Cited by 33 publications

(45 citation statements)

References 4 publications

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“…The iterative procedure ensures convergence of the interface height and velocity profiles at every downstream location, ξ . The solution at large ξ agrees with the asymptotic behaviour described in Nelson et al (1995) and provides the mean profile for the numerical evaluation of the continuous-spectrum modes.…”

Section: The Two-fluid Theoretical Formulationsupporting

confidence: 75%

On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers

Zaki

Saha

2009

J. Fluid Mech.

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Studies of vortical interactions in boundary layers have often invoked the continuous spectrum of the Orr-Sommerfeld (O-S) equation. These vortical eigenmodes provide a link between free-stream disturbances and the boundary-layer shear -a link which is absent in the inviscid limit due to shear sheltering. In the presence of viscosity, however, a shift in the dominant balance in the operator determines the structure of these eigenfunctions inside the mean shear. In order to explain the mechanics of shear sheltering and the structure of the continuous modes, both numerical and asymptotic solutions of the linear perturbation equation are presented in single-and two-fluid boundary layers. The asymptotic analysis identifies three limits: a convective shear-sheltering regime, a convective-diffusive regime and a diffusive regime. In the shear-dominated limit, the vorticity eigenfunction possesses a three-layer structure, the topmost being a region of exponential decay. The role of viscosity is most pronounced in the diffusive regime, where the boundary layer becomes 'transparent' to the oscillatory eigenfunctions. Finally, the convective-diffusive regime demonstrates the interplay between the the accumulative effect of the shear and the role of viscosity. The analyses are complemented by a physical interpretation of shearsheltering mechanism. The influence of a wallfilm, in particular viscosity and density stratification, and surface tension are also evaluated. It is shown that a modified wavenumber emerges across the interface and influences the penetration of vortical disturbances into the two-fluid shear flow.

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Section: The Two-fluid Theoretical Formulationsupporting

confidence: 75%

On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers

Zaki

Saha

2009

J. Fluid Mech.

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“…The continuity of the streamwise velocity then yieldŝ 12) with the shape of the interface f (x) determined byλ + f 2 = const, as follows from the mass conservation within the film; cf. Nelson et al (1995). 14) hold, similar to the usual triple-deck solution but with the extra contribution U 01 in the horizontal velocity and, more significantly for buoyant fluids, with the vertical pressure variation of order ε 0 .…”

Section: Derivation Of the Viscous/inviscid Interaction Equationsmentioning

confidence: 57%

“…Then the steady boundary layer unaffected by instabilities can be treated as predominantly independent of the film flow; see e.g. Nelson et al (1995). In the bulk of the boundary layer, where the vertical coordinate y 1 = yε −4 0 is of O (1), the velocity components and pressure can be written in the form…”

Section: Derivation Of the Viscous/inviscid Interaction Equationsmentioning

confidence: 99%

“…The lower-branch TS waves are slow, their phase speed is an O Re −1/8 fraction of the free-stream velocity. In order to examine the effect of a wall coating on the TS instability we shall assume the film and the TS-wave viscous layer to be of comparable thickness (the assumption seems appropriate since sufficiently far away from the region of film generation the film becomes asymptotically thin compared to the boundary layer; see Nelson, Alving & Joseph 1995). The equations derived on this basis are found to describe both the TS waves of the main boundary layer modified by the film and interfacial waves.…”

Section: Introductionmentioning

confidence: 99%

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Instabilities in a high-Reynolds-number boundary layer on a film-coated surface

Timoshin

1997

J. Fluid Mech.

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A high-Reynolds-number asymptotic theory is developed for linear instability waves in a two-dimensional incompressible boundary layer on a flat surface coated with a thin film of a different fluid. The focus in this study is on the influence of the film flow on the lower-branch Tollmien-Schlighting waves, and also on the effect of boundary-layer/potential flow interaction on interfacial instabilities. Accordingly, the film thickness is assumed to be comparable to the thickness of a viscous sublayer in a three-tier asymptotic structure of lower-branch Tollmien-Schlichting disturbances. A fully nonlinear viscous/inviscid interaction formulation is derived, and computational and analytical solutions for small disturbances are obtained for both TollmienSchlichting and interfacial instabilities for a range of density and viscosity ratios of the fluids, and for various values of the surface tension coefficient and the Froude number. It is shown that the interfacial instability contains the fastest growing modes and an upper-branch neutral point within the chosen flow regime if the film viscosity is greater than the viscosity of the ambient fluid. For a less viscous film the theory predicts a lower neutral branch of shorter-scale interfacial waves. The film flow is found to have a strong effect on the Tollmien-Schlichting instability, the most dramatic outcome being a powerful destabilization of the flow due to a linear resonance between growing Tollmien-Schlichting and decaying capillary modes. Increased film viscosity also destabilizes Tollmien-Schlichting disturbances, with the maximum growth rate shifted towards shorter waves. Qualitative and quantitative comparisons are made with experimental observations by Ludwieg & Hornung (1989).

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“…Hence, sidewall friction and water setup caused by wind shear may reduce the effective surface flow. Nelson et al 39 formulated a nonsimilar analytic solution for laminar airflow shearing over a thin water layer on a flat plate. The solution showed that the height of the surface film was proportional to x 1/4 and the surface current calculated using their method is only about 0.14% of the free-stream velocity.…”

Section: -11mentioning

confidence: 99%

On the spatial linear growth of gravity-capillary water waves sheared by a laminar air flow

2005

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The initial growth of mechanically generated small amplitude water waves below a laminar air stream was examined numerically and experimentally in order to explore the primary growth mechanism, that is, the interfacial instability of coupled laminar air and water flows. Measurements of the laminar velocity profile in the air over the water surface were found to be consistent with Lock's ͓Q. J. Mech. Appl. Math. 4, 42 ͑1951͔͒ theory. This profile was then used to calculate the spatial growth rates by solving the Orr-Sommerfeld equations. The simulation shows that the growth of the boundary layer affects the exponential growth of water waves along the fetch. The sensitivity of the growth rate is observed to vary by a factor of 2 for changes in the laminar velocity profile as small as 2% at the water surface. This indicates that the interfacial instability is strongly influenced by the wind-induced surface current. A laminar airflow was produced in the wind tunnel over mechanically generated monochromatic gravity-capillary water waves with the ka value in the order of 10 −3 . The novel experiment was designed to measure the minute changes in the wave slope and phase velocity simultaneously using a highly sensitive reflected twin laser beam technique. Agreement between linear theory and experiments for the spatial development of wave height and phase velocity suggests that the linear instability mechanism determines the initial stages of development of small-scale water waves.

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Boundary layer flow of air over water on a flat plate

Cited by 33 publications

References 4 publications

On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers

On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers

Instabilities in a high-Reynolds-number boundary layer on a film-coated surface

On the spatial linear growth of gravity-capillary water waves sheared by a laminar air flow

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