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 Stability of plane Couette flow for high Reynolds number

1983 ◽  
Vol 24 (3) ◽  
pp. 350-357 ◽  
Author(s):  
G. B. Davis ◽  
A. G. Morris
Keyword(s):  
Reynolds Number ◽  
Experimental Evidence ◽  
Linear Stability ◽  
Couette Flow ◽  
Stability Theory ◽  
High Reynolds Number ◽  
Numerical Technique ◽  
Linear Stability Theory ◽  
Plane Couette Flow ◽  
High Reynolds

AbstractExperimental evidence shows that plane Couette flow becomes unstable when the Reynolds number R reaches certain critical values. Linear stability theory does not predict these observations and has been unable to locate these instabilities. A Chebyshev/QR numerical technique is used to investigate much higher values of R than those previously tested. In particular, values of R up to 108 are confidently tested, whereas previously values of R up to only 2 × 104 have been considered.

scholarly journals Catastrophic Phase Inversion in High-Reynolds-Number Turbulent Taylor-Couette Flow

2021 ◽  
Vol 126 (6) ◽  
Author(s):  
Dennis Bakhuis ◽  
Rodrigo Ezeta ◽  
Pim A. Bullee ◽  
Alvaro Marin ◽  
Detlef Lohse ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Phase Inversion ◽  
High Reynolds Number ◽  
Taylor Couette ◽  
High Reynolds ◽  
Taylor Couette Flow

Fine Scale Structure of High Reynolds Number Taylor-Couette Flow

2007 ◽  
Author(s):  
W. He ◽  
M. Tanahashi ◽  
T. Miyauchi
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
High Reynolds Number ◽  
Flow Fields ◽  
Fine Scale ◽  
Azimuthal Direction ◽  
Taylor Vortices ◽  
Taylor Couette ◽  
High Reynolds ◽  
Taylor Couette Flow

Direct numerical simulation (DNS) has been conducted to investigate turbulence transition process and fine scale structures in Taylor-Couette flow. Fourier-Chebyshev spectral methods have been used for spatial discretization and DNS are conducted up to Re = 12000. With the increase of Reynolds number, fine scale eddies are formed in a stepwise fashion. In relatively weak turbulent Taylor-Couette flow, fine scale eddies elongated in the azimuthal direction appear near the outflow and inflow boundaries between Taylor vortices. These fine scale eddies in the outflow and inflow boundaries are inclined at about −45/135 degree with respect to the azimuthal direction. With the increase of Reynolds number, the number of fine scale eddies increases and fine scale eddies appear in whole flow fields. The Taylor vortices in high Reynolds number organize lots of fine scale eddies. In high Reynolds number Taylor-Couette flow, fine scale eddies parallel to the axial direction are formed in sweep regions between large scale Taylor vortices. The most expected diameter and maximum azimuthal velocity of coherent fine scale eddies are 8 times of Kolmogorov scale and 1.7 times of Kolmogorov velocity respectively for high Reynolds Taylor-Couette flow. This scaling law coincides with that in other turbulent flow fields.

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.

Fine Structure of Vortex Sheet Rollup by Viscous and Inviscid Simulation

1991 ◽  
Vol 113 (1) ◽  
pp. 31-36 ◽  
Author(s):  
G. Tryggvason ◽  
W. J. A. Dahm ◽  
K. Sbeih
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Large Scale ◽  
Stokes Equations ◽  
Flow Region ◽  
Vortex Sheet ◽  
Navier Stokes ◽  
Vortex Sheets ◽  
Limiting Behavior ◽  
High Reynolds

Numerical simulations of the large amplitude stage of the Kelvin-Helmholtz instability of a relatively thin vorticity layer are discussed. At high Reynolds number, the effect of viscosity is commonly neglected and the thin layer is modeled as a vortex sheet separating one potential flow region from another. Since such vortex sheets are susceptible to a short wavelength instability, as well as singularity formation, it is necessary to provide an artificial “regularization” for long time calculations. We examine the effect of this regularization by comparing vortex sheet calculations with fully viscous finite difference calculations of the Navier-Stokes equations. In particular, we compare the limiting behavior of the viscous simulations for high Reynolds numbers and small initial layer thickness with the limiting solution for the roll-up of an inviscid vortex sheet. Results show that the inviscid regularization effectively reproduces many of the features associated with the thickness of viscous vorticity layers with increasing Reynolds number, though the simplified dynamics of the inviscid model allows it to accurately simulate only the large scale features of the vorticity field. Our results also show that the limiting solution of zero regularization for the inviscid model and high Reynolds number and zero initial thickness for the viscous simulations appear to be the same.

Sign in / Sign up

Export Citation Format

Share Document