An Experimental Study for simulation of Pneumatic Conveying. 2nd Report. Lift and Drag Applied on a sphere in High Reynolds Number Linear Shear Flows.

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.

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

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.234 ◽

2014 ◽

Vol 750 ◽

pp. 99-112 ◽

Cited By ~ 27

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.

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