Subcritical route to turbulence via the Orr mechanism in a quasi-two-dimensional boundary layer
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The link to the online abstract of this manuscript, accepted in Phys. Rev. Fluids, is this https URL.
A subcritical route to turbulence via purely quasi-two-dimensional mechanisms, for a quasi-two-dimensional system composed of an isolated exponential boundary layer, is numerically investigated. Exponential boundary layers are highly stable, and are expected to form on the walls of liquid metal coolant ducts within magnetic confinement fusion reactors. Subcritical transitions were detected only at weakly subcritical Reynolds numbers (at most $\approx 70$% below critical). Furthermore, the likelihood of transition was very sensitive to both the perturbation structure and initial energy. Only the quasi-two-dimensional Tollmien-Schlichting wave disturbance, attained by either linear or nonlinear optimisation, was able to initiate the transition process, by means of the Orr mechanism. The lower initial energy bound sufficient to trigger transition was found to be independent of the domain length. However, longer domains were able to increase the upper energy bound, via the merging of repetitions of the Tollmien-Schlichting wave. This broadens the range of initial energies able to exhibit transitional behaviour. Although the eventual relaminarization of all turbulent states was observed, this was also greatly delayed in longer domains. The maximum nonlinear gains achieved were orders of magnitude larger than the maximum linear gains (with the same initial perturbations), regardless if the initial energy was above or below the lower energy bound. Nonlinearity provided a second stage of energy growth by an arching of the conventional Tollmien-Schlichting wave structure. A streamwise independent structure, able to efficiently store perturbation energy, also formed.Cite
A subcritical route to turbulence is found in a quasi-two-dimensional exponential boundary layer, driven by the action of the Orr mechanism on a Tollmien-Schlichting (TS) wave. A finite band of initial disturbance energies triggers a nonlinear contortion of the TS wave, producing sufficient nonlinear growth for a transition to turbulence. Remarkably though, the resulting turbulent state is never indefinitely sustained. Larger initial energies disrupted the transition process, leading to rapid relaminarization, while the growth generated by disturbances with smaller initial energies is insufficient to reach the turbulent attractor.
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We consider a smooth, spanwise-uniform forward-facing step defined by a Gauss error function of height 4 %–30 % and four times the width of the local boundary layer thickness . The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the two-dimensional Tollmien–Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward-facing smooth steps is conducted by linear analysis. The results indicate that for a TS wave with a frequency ( , where and denote the perturbation angle frequency and free-stream velocity magnitude, respectively, and denotes kinematic viscosity), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward-facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20 % of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified, and thereby a destabilisation effect is introduced. Therefore, the stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward-facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favourably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter and in the absence of other external perturbations, two forward-facing smooth steps of height 5 % and 12 % of the boundary layer thickness delayed the H-type transition scenario and completely suppressed for the K-type transition. By considering Gaussian white noise with both fixed and random phase shifts, it is demonstrated by DNS that transition is postponed in time and space by two forward-facing smooth steps.
Destabilisation
Wavenumber
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In single-fluid boundary layers, streaks can amplify at sub-critical Reynolds numbers and initiate early transition to turbulence. Introducing a wall film of different viscosities can appreciably alter the stability of the base flow and, in particular, the transient growth of the perturbation streaks. The formalism of seminorms is used to identify optimal disturbances which maximize the kinetic energy in the two-fluid flow. An examination of optimal growth over a range of viscosity ratios of the film relative to the outer flow reveals three distinct regimes of amplification, each associated with a particular combination of the eigenfunctions. In order to elucidate the underlying amplification mechanisms, a model problem is formulated: An initial value problem is solved using an eigenfunction expansion and is used to compute the evolution of pairs of eigenfunctions. By appropriately selecting the pair, the initial value problem qualitatively reproduces the temporal evolution of the optimal disturbance, and provides an unambiguous explanation of the dynamics. Two regimes of transient growth are attributed to the evolution of the interface mode along with free-stream vortical modes; the third regime is due to the evolution of the interface and a discrete mode. The results demonstrate that a lower-viscosity film can effectively reduce the efficacy of the lift-up mechanism and, as a result, transient growth of disturbances. However, another mechanism of amplification of wall-normal vorticity arises due to the deformation of the two-fluid interface and becomes dominant below a critical viscosity ratio.
Eigenfunction
Inviscid flow
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Hagen–Poiseuille equation
Laminar-turbulent transition
Linear stability
Hydrodynamic stability
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Oscillatory Stokes flows, with zero mean, are subjected to subcritical transition to turbulence. The maximal energy growth of perturbations is computed in the subcritical regime through an optimisation method. The results show strong amplifications during half a period. The energy transfer from the base flow involves an Orr mechanism with two-dimensional vorticity waves, and the maximum energy scales exponentially with the Reynolds number. Nonlinear simulations show that low-energy perturbations are sufficient to trigger turbulent flow.
Transient (computer programming)
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We perform a three-dimensional stability analysis of the Kelvin-Helmholtz billow, formed upon a shear-layer between two fluids with a density ratio of 3. We begin with two- dimensional simulations of the temporally evolving mixing- layer yielding the unsteady base flow fields. The Reynolds number is 3000 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis validated against previous results in the homogeneous case. Among the least stable modes, we retain those growing faster than the primary wave thus ensuring the validity of the quasi-steady approach. The spectrum of rapidly growing modes is analyzed and shown to exhibit a typical two-dimensional mode, in addition to core-centered and braid-centered ones. These modes are developing on the braid lying on the light side. There the flow evolves toward a sharp vorticity ridge due to a baroclinic vorticity source concentrated on a steep density-gradient. For the present density contrast, the wave-length of the two-dimensional instability is ten times shorter than the one of the primary wave. Its amplification rate competes well against the one of the braid-centered least-stable three-dimensional mode. The numerical continuation of the non-linear development of this particular mode is carried out from two starting points along the roll-up of the primary wave. We describe secondary small-scale roll-ups due to a Kelvin-Helmholtz mechanism favored by the strain field. This mode is demonstrated consistent with finite Reynolds number mixing-layers. We are also able to discuss its precedence against transverse modes thus contributing to the complex picture of the transition of the variable-density shear-layer.
Kelvin wave
Linear stability
Wavenumber
Hydrodynamic stability
Helmholtz equation
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Supercritical flow
Parametrization (atmospheric modeling)
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The asymptotic suction boundary layer (ASBL) is a parallel shear flow that becomes turbulent in a bypass transition in parameter regions where the laminar profile is stable. We here add a temperature gradient perpendicular to the plate and explore the interaction between convection and shear in determining the transition. We find that the laminar state becomes unstable in a subcritical bifurcation and that the critical Rayleigh number and wave number depend strongly on the Prandtl number. We also track several secondary bifurcations and identify states that are localized in two directions, showing different symmetries. In the subcritical regime, transient turbulent states which are connected to exact coherent states and follow the same transition scenario as found in linearly stable shear flows are identified and analyzed. The study extends the bypass transition scenario from shear flows to thermal boundary layers and shows the intricate interactions between thermal and shear forces in determining critical points.
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