A vortex–wave interaction theory describing the effect of boundary forcing on shear flows

2021 ◽  
Vol 932 ◽  
Author(s):  
Philip Hall
Keyword(s):  
Reynolds Number ◽  
Shear Flows ◽  
Flow Direction ◽  
Wave Interaction ◽  
Propagation Speed ◽  
Instability Criterion ◽  
Stationary Waves ◽  
Critical Wavelength ◽  
High Reynolds ◽  
Forcing Mechanisms

A strongly nonlinear theory describing the effect of small amplitude boundary forcing in the form of waves on high Reynolds number shear flows is given. The interaction leads to an $O(1)$ change in the unperturbed flow and is relevant to a number of forcing mechanisms. The cases of the shear flow being bounded or unbounded are both considered and the results for the unbounded case apply to quite arbitrary flows. The instability criterion for unbounded flows is expressed in terms of the wall forcing and the friction Reynolds number. As particular examples we investigate wall transpiration or surface undulations as sources of the forcing and both propagating and stationary waves are considered. Results are given for propagating waves with crests perpendicular to the flow direction and for stationary waves with crests no longer perpendicular to the flow direction. In the first of those situations we find the instability induced by transpiration waves is independent of the propagation speed. For wavy walls downstream propagation completely stabilises the flow at a critical speed whereas upstream propagation greatly destabilises the flow. For stationary oblique waves we find that the instability is enhanced and a much wider range of unstable wavenumbers exists. For the bounded case with a wall of fixed wavelength we identify a critical wavelength where the most dangerous mode switches from the aligned to the oblique configuration. For the transpiration problem in the oblique configuration a strong resonance occurs when the vortex wavelength coincides with the spanwise wavelength of the forcing.

scholarly journals Movement of a finite body in channel flow

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160164 ◽  
Author(s):  
Frank T. Smith ◽  
Edward R. Johnson
Keyword(s):  
Reynolds Number ◽  
Channel Flow ◽  
Flow Instability ◽  
Flow Direction ◽  
Finite Size ◽  
The Body ◽  
Thin Body ◽  
Shear Stresses ◽  
Unsteady Interaction ◽  
High Reynolds

A body of finite size is moving freely inside, and interacting with, a channel flow. The description of this unsteady interaction for a comparatively dense thin body moving slowly relative to flow at medium-to-high Reynolds number shows that an inviscid core problem with vorticity determines much, but not all, of the dominant response. It is found that the lift induced on a body of length comparable to the channel width leads to differences in flow direction upstream and downstream on the body scale which are smoothed out axially over a longer viscous length scale; the latter directly affects the change in flow directions. The change is such that in any symmetric incident flow the ratio of slopes is found to be cos ⁡ ( π / 7 ) , i.e. approximately 0.900969, independently of Reynolds number, wall shear stresses and velocity profile. The two axial scales determine the evolution of the body and the flow, always yielding instability. This unusual evolution and linear or nonlinear instability mechanism arise outside the conventional range of flow instability and are influenced substantially by the lateral positioning, length and axial velocity of the body.

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

2013 ◽  
Vol 721 ◽  
pp. 58-85 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall ◽  
Andrew Walton
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
High Reynolds Number ◽  
Stokes Equations ◽  
Wave Interaction ◽  
Critical Layer ◽  
Navier Stokes ◽  
Turbulent Spots ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractThe 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 (VWI) theory is used in the long streamwise wavelength limit to continue the development found at order-one wavelengths by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). 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 & Sherwin as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling, which, when approached, causes the flow to localize 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.

Stationary waves on an inclined sheet of viscous fluid at high Reynolds and moderate Weber numbers

1996 ◽  
Vol 307 ◽  
pp. 191-229 ◽  
Author(s):  
Jeng-Jong Lee ◽  
Chiang C. Mei
Keyword(s):  
Dynamical System ◽  
Phase Speed ◽  
Propagation Speed ◽  
Reynolds Numbers ◽  
Homoclinic Orbits ◽  
Finite Amplitude ◽  
Stationary Waves ◽  
Asymptotic Equations ◽  
Wave Propagation Speed ◽  
High Reynolds

A theory is described for the nonlinear waves on the surface of a thin film flowing down an inclined plane. Attention is focused on stationary waves of finite amplitude and long wavelength at high Reynolds numbers and moderate Weber numbers. Based on asymptotic equations accurate to the second order in the depth-to-wavelength ratio, a third-order dynamical system is obtained after changing to the frame of reference moving at the wave propagation speed. By examining the fixed-point stability of the dynamical system, parametric regimes of heteroclinc orbits and Hopf bifurcations are delineated. Extensive numerical experiments guided by the linear analyses reveal a variety of bifurcation scenarios as the phase speed deviates from the Hopf-bifurcation thresholds. These include homoclinic bifurcations which lead to homoclinic orbits corresponding to well separated solitary waves with one or several humps, some of which occur after passing through chaotic zones generated by period-doublings. There are also cases where chaos is the ultimate state following cascades of period-doublings, as well as cases where only limit cycles prevail. The dependence of bifurcation scenarios on the inclination angle, and Weber and Reynolds numbers is summarized.

scholarly journals High Reynolds Number Turbulence Model of Rotating Shear Flows

1983 ◽  
Vol 26 (219) ◽  
pp. 1534-1541 ◽  
Author(s):  
Shigeaki MASUDA ◽  
Hide S. KOYAMA ◽  
Ichiro ARIGA
Keyword(s):  
Reynolds Number ◽  
Turbulence Model ◽  
High Reynolds Number ◽  
Shear Flows ◽  
High Reynolds ◽  
Rotating Shear Flows

The evolution of a localized vortex disturbance in external shear flows. Part 1. Theoretical considerations and preliminary experimental results

1995 ◽  
Vol 289 ◽  
pp. 159-177 ◽  
Author(s):  
Vladimir Levinski ◽  
Jacob Cohen
Keyword(s):  
Boundary Layers ◽  
Shear Flows ◽  
Flow Direction ◽  
Linear Equations ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Experimental Results ◽  
Instability Criterion ◽  
Hairpin Vortices ◽  
Plane Normal

The evolution of a finite-amplitude three-dimensional localized disturbance embedded in external shear flows is addressed. Using the fluid impulse integral as a characteristic of such a disturbance, the Euler vorticity equation is integrated analytically, and a system of linear equations describing the temporal evolution of the three components of the fluid impulse is obtained. Analysis of this system of equations shows that inviscid plane parallel flows as well as high Reynolds number two-dimensional boundary layers are always unstable to small localized disturbances, a typical dimension of which is much smaller than a dimensional length scale corresponding to an O(1) change of the external velocity. Since the integral character of the fluid impulse is insensitive to the details of the flow, universal properties are obtained. The analysis predicts that the growing vortex disturbance will be inclined at 45° to the external flow direction, in a plane normal to the transverse axis. This prediction agrees with previous experimental observations concerning the growth of hairpin vortices in laminar and turbulent boundary layers. In order to demonstrate the potential of this approach, it is applied to Taylor-Couette flow, which has additional dynamical effects owing to rotation. Accordingly, a new instability criterion associated with three-dimensional localized disturbances is found. The validity of this criterion is supported by our experimental results.

scholarly journals Frequency–wavenumber mapping in turbulent shear flows

2015 ◽  
Vol 783 ◽  
pp. 166-190 ◽  
Author(s):  
Roeland de Kat ◽  
Bharathram Ganapathisubramani
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Shear Flows ◽  
Spatial Scales ◽  
Turbulent Shear ◽  
Data Set ◽  
Frequency Spectra ◽  
Taylor’S Hypothesis ◽  
Turbulent Shear Flows ◽  
High Reynolds

Spatial turbulence spectra for high-Reynolds-number shear flows are usually obtained by mapping experimental frequency spectra into wavenumber space using Taylor’s hypothesis, but this is known to be less than ideal. In this paper, we propose a cross-spectral approach that allows us to determine the entire wavenumber–frequency spectrum using two-point measurements. The method uses cross-spectral phase differences between two points – equivalent to wave velocities – to reconstruct the wavenumber–frequency plane, which can then be integrated to obtain the spatial spectrum. We verify the technique on a particle image velocimetry (PIV) data set of a turbulent boundary layer. To show the potential influence of the different mappings, the transfer functions that we obtained from our PIV data are applied to hot-wire data at approximately the same Reynolds number. Comparison of the newly proposed technique with the classic approach based on Taylor’s hypothesis shows that – as expected – Taylor’s hypothesis holds for larger wavenumbers (small spatial scales), but there are significant differences for smaller wavenumbers (large spatial scales). In the range of Reynolds number examined in this study, double-peaked spectra in the outer region of a turbulent wall flow are thought to be the result of using Taylor’s hypothesis. This is consistent with previous studies that focused on examining the limitations of Taylor’s hypothesis (del Álamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5–26). The newly proposed mapping method provides a data-driven approach to map frequency spectra into wavenumber spectra from two-point measurements and will allow the experimental exploration of spatial spectra in high-Reynolds-number turbulent shear flows.

The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow

2014 ◽  
Vol 750 ◽  
pp. 99-112 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
High Reynolds Number ◽  
Shear Flows ◽  
Stokes Equations ◽  
Entangled States ◽  
Plane Couette Flow ◽  
Long Wavelength ◽  
Nonlinear Equilibrium ◽  
High Reynolds

AbstractThe relationship between nonlinear equilibrium solutions of the full Navier–Stokes equations and the high-Reynolds-number asymptotic vortex–wave interaction (VWI) theory developed for general shear flows by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) is investigated. Using plane Couette flow as a prototype shear flow, we show that all solutions having $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(1)$ wavenumbers converge to VWI states with increasing Reynolds number. The converged results here uncover an upper branch of VWI solutions missing from the calculations of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). For small values of the streamwise wavenumber, the converged lower-branch solutions take on the long-wavelength state of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58–85) while the upper-branch solutions are found to be quite distinct, with new states associated with instabilities of jet-like structures playing the dominant role. Between these long-wavelength states, a complex ‘snaking’ behaviour of solution branches is observed. The snaking behaviour leads to complex ‘entangled’ states involving the long-wavelength states and the VWI states. The entangled states exhibit different-scale fluid motions typical of those found in shear flows.

scholarly journals On the Flow Along Swept Leading Edges

1967 ◽  
Vol 18 (2) ◽  
pp. 165-184 ◽  
Author(s):  
M. Gaster
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Wave Length ◽  
Leading Edge ◽  
Propagation Speed ◽  
Reynolds Numbers ◽  
Amplification Ratio ◽  
Tollmien Schlichting ◽  
High Reynolds ◽  
Attachment Line

SummaryFlight tests on the Handley Page suction wing showed that turbulence at the wing root can propagate along the leading edge and cause the whole flow to be turbulent. The flow on the attachment line of a swept wing was studied in a low speed wind tunnel with particular reference to this problem of turbulent contamination.The critical Reynolds number, RθL, of the attachment-line boundary layer for the spanwise spread of turbulence was found to be about 100 for sweep angles in the range 40°–60°. A device was developed to act as a barrier to the turbulent root flow so that a clean laminar flow could exist outboard. This device was shown to be effective up to an Rθ of at least 170, so that experiments were possible on a laminar boundary layer at Reynolds numbers above the lower critical value. A spark was used to introduce spots of turbulence into the attachment-line boundary layer and the propagation speeds of the leading and trailing edges were measured. The spots expanded, the leading edge moving faster than the trailing edge, at high Reynolds numbers, and contracted at low values.The behaviour of Tollmien-Schlichting waves was also investigated by exciting the flow with sound emanating from a small hole on the attachment line. Measurements of the perturbation phase and amplitude were made downstream of the source and, although accurate values of wave length and propagation speed could be found, difficulties were experienced in evaluating the amplification ratio. Nevertheless, all small disturbances decayed at a sufficient distance from the source hole up to the highest available Reynolds number of 170.

On the instability of vortex–wave interaction states

2016 ◽  
Vol 802 ◽  
pp. 634-666 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Reynolds Number ◽  
Coherent Structures ◽  
Time Scale ◽  
Stokes Equations ◽  
Wave Interaction ◽  
Navier Stokes ◽  
Edge States ◽  
Asymptotic Approach ◽  
Navier Stokes Equations ◽  
High Reynolds

In recent years it has been established that vortex–wave interaction theory forms an asymptotic framework to describe high Reynolds number coherent structures in shear flows. Comparisons between the asymptotic approach and finite Reynolds number computations of equilibrium states from the full Navier–Stokes equations have suggested that the asymptotic approach is extremely accurate even at quite low Reynolds numbers. However, unlike the situation with an approach based on solving the full Navier–Stokes equations numerically, the vortex–wave interaction approach has not yet been developed to study the instability of the structures it describes. In this work, a comprehensive study of the different instabilities of vortex–wave interaction states is given and it is shown that there are three different time scales on which instabilities can develop. The most dangerous type is a rapidly growing Rayleigh instability of the streak part of the flow. The least dangerous type is a slow mode operating on the diffusion time scale of the roll–streak part of the flow. The third mode of instability, which we will refer to as the edge mode of instability, occurs on a time scale midway between those of the other two modes. The existence of the latter mode explains why some exact coherent structures can act as edge states between the laminar and turbulent attractors. These stability results are compared to results from Navier–Stokes calculations.

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