On the Environmental Realizability of Algebraically Growing Disturbances and Their Relation to Klebanoff Modes

Goldstein, M. E.; Wundrow, David W.

doi:10.1007/s001620050057

Cited by 49 publications

(56 citation statements)

References 32 publications

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“…Goldstein et al (1992) pointed out that the forced motion remains linear only within a distance much smaller than min R Λ , −1 to the leading edge. The continued development downstream is nonlinear, governed by the nonlinear boundary-layer equations, if R Λ 1, or by the so-called nonlinear boundaryregion equations which include cross-flow ellipticity if R Λ = O(1), as was shown by Goldstein & Wundrow (1998). Numerical solutions reveal that nonlinearity leads to a strong vorticity concentration along the spanwise direction causing the viscous boundary layer to separate for R Λ 1, but the streamwise velocity profile of the distorted boundary layer does not become inflectional.…”

Section: Introductionmentioning

confidence: 99%

Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances

Riccó

Luo

2011

J. Fluid Mech.

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We investigate the influence of free-stream vortical disturbances on the evolution and instability of an incompressible laminar boundary layer, focusing on components of sufficiently long wavelength, which are known to penetrate into the boundary layer to generate streamwise elongated streaks. The free-stream disturbance is assumed to be sufficiently strong (but still of small amplitude) that the induced streaks acquire an O(1) streamwise velocity in the region where the boundary-layer thickness becomes comparable with the spanwise wavelength of the perturbation. The formation and evolution of the streaks are governed by the nonlinear unsteady boundary-region equations supplemented by appropriate upstream and far-field boundary conditions. This initial-boundary-value problem is solved for the special case where the free-stream disturbance is modelled by a pair of oblique vortical modes with the same frequency but opposite spanwise wavenumbers. Nonlinearity is found to inhibit the response. The nonlinear interaction alters the meanflow profile appreciably, the shape of which is in quantitative agreement with experimental measurements. Wall-normal inflection points are detected in the instantaneous streamwise velocity profiles. The secondary stability analysis indicates that in the presence of free-stream disturbance with an intensity of 2.8%, the resulting streaky boundary layer becomes inviscidly unstable. The characteristic frequency, phase and group velocities, and growth rate of unstable sinuous modes are found to be in broad agreement with recent experiments. The present theoretical framework allows in principle a quantitative relation to be established between the characteristics of free-stream turbulence and the secondary instability, and this relation may be exploited to develop an efficient and physics-based approach for predicting bypass transition.

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Section: Introductionmentioning

confidence: 99%

Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances

Riccó

Luo

2011

J. Fluid Mech.

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“…In the context of boundary-layer transition, the linearized version of the equations are relevant to the transient growth of streaks; see Luchini (2000), Higuera and Vega (2009), whilst the full nonlinear equations were solved by Zuccher, Tumin and Reshotko (2006) to compute optimal nonlinear streaks in boundary layers. The linearized version of the equations are also relevant to the receptivity of boundary layers to free stream turbulence; see for example Goldstein and Wundrow (1998), Goldstein and Sescu (2008). The key property of these equations is that they are parabolic in the flow direction and elliptic in the spanwise direction.…”

Section: Introductionmentioning

confidence: 99%

The emergence of localized vortex–wave interaction states in plane Couette flow

2013

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The recently understood relationship between high Reynolds number vortex-wave interaction theory and computationally-generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex-wave interaction theory, which we now refer to as VWI, is used in the long streamwise wavelength limit to continue the development found at order one wavelengths by Hall and Sherwin (2010). The asymptotic description given reduces the Navier-Stokes equations to the so-called boundary-region equations for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from O(h) to O(Rh) where R is the Reynolds number and 2h is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave-vortex states. The solutions we calculate have an asymptotic error proportional to R −2 when compared to the full Navier-Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel so that the roll part of the flow is driven throughout the flow rather than as in Hall and Sherwin (2010) as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling which when approached causes the flow to localise and take on similarities with computationally-generated or experimentally-observed turbulent spots. In effect the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier-Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

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“…The relatively mild singularity in this solution is smoothed out by viscous effects in the linearized boundary layer solution of region II. The mean boundary layer is of the Blasius type at large downstream distances and an analysis similar to the one carried out by Crow [6] shows that the streamwise velocity in this region behaves like [10] …”

Section: (C) High-turbulence Reynolds Number (The Leading Edge Problem)mentioning

confidence: 63%

“…The paper begins by considering the general situation shown in figure 2, but assumes, for simplicity, that the flow is incompressible [9][10][11]. The fundamental length-scale Λ is set by the (idealized) upstream grid, which is presumed to generate weak free-stream turbulence with characteristic amplitude O(εU ∞ ).…”

Section: Overall Flow Structurementioning

confidence: 99%

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Effect of free-stream turbulence on boundary layer transition

Goldstein

2014

Phil. Trans. R. Soc. A.

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This paper is concerned with the transition to turbulence in flat plate boundary layers due to moderately high levels of free-stream turbulence. The turbulence is assumed to be generated by an (idealized) grid and matched asymptotic expansions are used to analyse the resulting flow over a finite thickness flat plate located in the downstream region. The characteristic Reynolds number R λ based on the mesh size λ and free-stream velocity is assumed to be large, and the turbulence intensity ε is assumed to be small. The asymptotic flow structure is discussed for the generic case where the turbulence Reynolds number εR λ and the plate thickness and are held fixed (at O (1) and O (λ), respectively) in the limit as and ε →0. But various limiting cases are considered in order to explain the relevant transition mechanisms. It is argued that there are two types of streak-like structures that can play a role in the transition process: (i) those that appear in the downstream region and are generated by streamwise vorticity in upstream flow and (ii) those that are concentrated near the leading edge and are generated by plate normal vorticity in upstream flow. The former are relatively unaffected by leading edge geometry and are usually referred to as Klebanoff modes while the latter are strongly affected by leading edge geometry and are more streamwise vortex-like in appearance.

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On the Environmental Realizability of Algebraically Growing Disturbances and Their Relation to Klebanoff Modes

Cited by 49 publications

References 32 publications

Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances

Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances

The emergence of localized vortex–wave interaction states in plane Couette flow

Effect of free-stream turbulence on boundary layer transition

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