Numerical simulation of the evolution of 

Tollmien–Schlichting waves over finite compliant panels

Davies, Christopher; Carpenter, Peter W.

doi:10.1017/s0022112096004636

Cited by 112 publications

(137 citation statements)

References 43 publications

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“…In the spectral collocation approach this is inadequate and leads to a degenerate system of equations, see Ehrenstein & Peyret (1989). A different technique has to be adopted and here we use an approach pioneered by Davies & Carpenter (1997) in which some integral constraints are used instead. These integral relations can be derived from Navier-Stokes equations by integration.…”

Section: Alternative Boundary Conditionsmentioning

confidence: 99%

Numerical solution of the Navier–Stokes equations for the flow in a cylinder cascade

Gajjar

Azzam

2004

J. Fluid Mech.

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A numerical study of the steady, two-dimensional incompressible flow past a cascade of circular cylinders is presented. The Navier-Stokes equations are written in terms of the streamfunction and vorticity and solved using a novel numerical technique based on using the Chebychev collocation method in one direction and high-order finite differences in the other direction. A direct solver combined with NewtonRaphson linearization is used to solve the discrete equations. Steady flow solutions have been obtained for large Reynolds numbers, far higher than those obtained previously, and for varying gap widths between the cylinders. Three distinct types of solutions, dependent on the gap width, have been found. Comparisons with theoretical predictions for various flow quantities show good agreement, especially for the narrow gap width case. However, existing theories are unable to explain the solution properties which exist for intermediate gap widths.

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Section: Alternative Boundary Conditionsmentioning

confidence: 99%

Numerical solution of the Navier–Stokes equations for the flow in a cylinder cascade

Gajjar

Azzam

2004

J. Fluid Mech.

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“…Since the existence of TS waves was confirmed by Schubauer & Skramstad (1948), numerous studies aiming to stabilise or destabilise the TS modes have been carried out in order to explore and explain different paths to transition. If the growth of the TS waves is reduced or completely suppressed, and providing no other instability mechanism comes into play, it has been suggested transition could be postponed or even eliminated (Davies & Carpenter 1996). Despite roughness elements being traditionally seen as an impediment to the stability of the flat plate boundary layer, recent research has shown this might not always be the case.…”

Section: Motivation Behind the Study Of Steps In Boundary Layersmentioning

confidence: 99%

Influence of localised smooth steps on the instability of a boundary layer

Lombard

Sherwin

2017

J. Fluid Mech.

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We consider a smooth, spanwise-uniform forward-facing step defined by a Gauss error function of height 4 %-30 % and four times the width of the local boundary layer thickness δ 99 . The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the twodimensional Tollmien-Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward-facing smooth steps is conducted by linear analysis. The results indicate that for a TS wave with a frequency F ∈ [140, 160] (F = ων/U 2 ∞ × 10 6 , where ω and U ∞ denote the perturbation angle frequency and free-stream velocity magnitude, respectively, and ν denotes kinematic viscosity), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward-facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20 % of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified, and thereby a destabilisation effect is introduced. Therefore, the stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward-facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favourably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter F = 150 and in the absence of other external perturbations, two forward-facing smooth steps of height 5 % and 12 % of the boundary layer thickness delayed the H-type transition scenario and completely suppressed for the K-type transition. By considering Gaussian white noise with both fixed and random phase shifts, it is demonstrated by DNS that transition is postponed in time and space by two forward-facing smooth steps.

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“…The compliant section of the channel is finite in length, and so there is a second junction at x = 2 between the compliant and rigid sections. This problem was investigated by Davies & Carpenter (1997) using direct numerical simulations. Davies & Carpenter (1997).…”

Section: Introductionmentioning

confidence: 99%

“…This problem was investigated by Davies & Carpenter (1997) using direct numerical simulations. Davies & Carpenter (1997).…”

Section: Introductionmentioning

confidence: 99%

A wave driver theory for vortical waves propagating across junctions with application to those between rigid and compliant walls

et al. 2009

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A theory is described for propagation of vortical waves across alternate rigid and compliant panels. The structure in the fluid side at the junction of panels is a highly vortical narrow viscous structure which is idealized as a wave driver. The wave driver is modelled as a ‘half source cum half sink’. The incoming wave terminates into this structure and the outgoing wave emanates from it. The model is described by half Fourier–Laplace transforms respectively for the upstream and downstream sides of the junction. The cases below cutoff and above cutoff frequencies are studied. The theory completely reproduces the direct numerical simulation results of Davies & Carpenter (J. Fluid Mech., vol. 335, 1997, p. 361). Particularly, the jumps across the junction in the kinetic energy integral, the vorticity integral and other related quantities as obtained in the work of Davies & Carpenter are completely reproduced. Also, some important new concepts emerge, notable amongst which is the concept of the pseudo group velocity.

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Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels

Cited by 112 publications

References 43 publications

Numerical solution of the Navier–Stokes equations for the flow in a cylinder cascade

Numerical solution of the Navier–Stokes equations for the flow in a cylinder cascade

Influence of localised smooth steps on the instability of a boundary layer

A wave driver theory for vortical waves propagating across junctions with application to those between rigid and compliant walls

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