Axisymmetric travelling waves in annular sliding Couette flow at finite and asymptotically large Reynolds number

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (2019). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous-resistive magneto-hydrodynamic equations in plane Couette flow have been utilized. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of the plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll-streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in Deguchi (2019). When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of 'self-sustained shear driven dynamos' at the zero-external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in Deguchi (2019). Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions. arXiv:1809.03855v2 [physics.flu-dyn]

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-speed shear-driven dynamos. Part 2. Numerical analysis

Deguchi

2019

“…Interestingly, the wave packet seen in the time periodic solution is fully localised in the streamwise direction, namely when the local amplitude of the wave becomes small the local wavelength increases so that the flow heads back towards its undisturbed state. It is noteworthy that the longest possible wavelength of the nonlinear wave is of O(R) as Deguchi & Walton (2013a) showed. This flow regime is a two-dimensional counterpart of Deguchi, Hall & Walton (2013), and is therefore governed by Prandtl's boundary layer equations but with an O(1) vertical scale and driven by an unknown pressure gradient that is determined as part of the solution.…”

Section: Conclusion and Discussionmentioning

confidence: 97%

“…Smith & Bodonyi (1982a) and Bodonyi, Smith & Gajjar (1983), using as a basis the asymptotic structure governing linear disturbances near the upper neutral branch, formalised this work and extended it to the situation where the critical layer becomes strongly nonlinear and moves away from the boundary. Much of this 'nonlinear critical layer theory' was formulated originally in the context of boundary-layer flows, but can easily be extended to parallel flows in channels (Bodonyi et al 1983), pipes (Smith & Bodonyi 1982b, Deguchi & Walton 2013a and annular geometries (Walton 2003, Deguchi & Walton 2013b, with the latter two Deguchi-Walton papers demonstrating the excellent agreement between the asymptotics and 2D Navier-Stokes computations, provided the Reynolds number is sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of nonlinear Tollmien–Schlichting waves in a high-speed channel flow

Deguchi

Walton

2018

Plane Poiseuille flow has long served as the simplest testing ground for Tollmien-Schlichting wave instability. In this paper, we provide a comprehensive comparison of equilibrium Tollmien-Schlichting wave solutions arising from new high resolution Navier-Stokes calculations and the corresponding predictions of various large Reynolds number asymptotic theories developed in the last century, such as double deck theory, viscous nonlinear critical layer theory and strongly nonlinear critical layer theory. The theories excellently predict the relatively small amplitude behaviour of the numerical solutions at Reynolds numbers of order 10 6 and above, whilst for larger amplitudes our computations suggest the need for further asymptotic theories to be developed.

On the breakdown of Rayleigh’s criterion for curved shear flows: a destabilization mechanism for a class of inviscidly stable flows

Hall

2013

The stability of a high-Reynolds-number flow over a curved surface is considered. Attention is focused on spanwise-periodic vortices of wavelength comparable with the boundary layer thickness. The wall curvature and the Görtler number for the flow are assumed large, so that stable or unstable vortices of wavelength comparable with the boundary layer thickness are dominated by inviscid effects. The growth or decay rate determined by using the viscous correction to the inviscid prediction is found to give a good approximation to the exact value for a range of wavenumbers. For the stable configuration, it is shown that the flow can be destabilized by arbitrarily small wall waviness, with the critical configuration involving only geometric properties of the flow. This destabilization mechanism has consequences for a number of flows of aerodynamic interest, particularly those where surface waviness under loading occurs. Control strategies based on how the curvature is distributed are discussed, and it is demonstrated how small regions of varying curvature can be used to remove the exponentially growing modes. The development of the parametric instabilities is continued into the strongly nonlinear regime. The nonlinear evolution is found to be governed by a generalized form of the cubic amplitude equation, generally referred to as the Stuart-Landau equation. The novel feature of the new equation is that the nonlinearity is not in the form of a power of the amplitude but is fixed by the solution of an associated nonlinear partial differential equation boundary value problem. Numerical solutions of the nonlinear partial differential system that fix the disturbance amplitude are given. The mechanism of destabilization described is shown to be directly relevant to a number of flows that are inviscidly stable in the presence of body forces. Thus it is shown how Rayleigh-Bénard convection or Taylor vortices can, at high Rayleigh or Taylor numbers, be destabilized by asymptotically small modulation in time or space.