Subcritical bifurcations of shear flows

The bifurcation and instability of nonisothermal annular Poiseuille flow (NAPF) of air as well as water is studied. We have emphasized the impact of a gap between cylinders in terms of curvature parameter (C) for axisymmetric as well as nonaxisymmetric disturbances. The results from the linear stability analysis reveal that the first azimuthal mode acts as a least stable mode of the NAPF of air for relatively small values of C. In this situation, even though for some values of C, the NAPF has supercritical bifurcation, but the same flow may experience subcritical bifurcation under zero azimuthal mode. It has also been observed that for relatively larger values of the Reynolds number (Re) and lower values of C, the NAPF under axisymmetric disturbance always exhibits subcritical bifurcation. However, for small values of Re, the NAPF exhibits only supercritical bifurcation. The finite amplitude analysis predicts only supercritical bifurcation of NAPF of water. The influence of nonlinear interaction of different harmonics on the amplitude profile as well as kinetic energy spectrum is investigated. The amplitude profile possesses a jump in the vicinity of a point where the type of bifurcation is changed. In the subcritical regime, the induced shear production due to modification of the gradient production acts as a main destabilizing factor balanced by the gradient production of kinetic energy.

A weakly nonlinear analysis of the bifurcation of the stratified Ekman boundary-layer flow near a critical bulk Richardson number is conducted and compared to a similar analysis of a continuously stratified parallel shear flow subject to Kelvin–Helmholtz instability. Previous work based on asymptotic expansions and predicting supercritical bifurcation at Prandtl number $Pr\lt 1$ and subcritical bifurcation at $Pr\gt 1$ for the parallel base flow is confirmed numerically and through fully nonlinear temporal simulations. When applied to the non-parallel Ekman flow, weakly nonlinear analysis and fully nonlinear calculations confirm that the nature of the bifurcation is dominantly controlled by $Pr$, although a sharp threshold at $Pr= 1$ is not found. In both flows the underlying physical mechanism is that the mean flow adjusts so as to induce a viscous (respectively diffusive) flux of momentum (respectively buoyancy) that balances the vertical flux induced by the developing instability, leading to a weakening of the mean shear and mean stratification. The competition between the former nonlinear feedback, which tends to be stabilizing, and the latter, which is destabilizing and strongly amplified as $Pr$ increases, determines the supercritical or subcritical character of the bifurcation. That essentially the same competition is at play in both the parallel shear flow and the Ekman flow suggests that the underlying mechanism is valid for complex, non-parallel stratified shear flows.

Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garget al.(Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio$\beta \lessapprox 0.785$), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ($\beta \gtrapprox 0.785$), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical$\beta \approx 0.855$. The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of$\beta$, the Landau coefficient scales with the inverse of the Weissenberg number ($Wi$) when$Wi$is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.

A comparative study of 1D and 2D approaches for simulating flows at right angled dividing junctions

An unsteady mathematical model for predicting flow divisions at a right-angled open-channel junction is presented. Existing dividing models depend on a prior knowledge of a constant flow regime. In addition, their strong nonlinearity does not guarantee compatibility with the St. Venant solutions in the context of an internal boundary condition treatment. Assuming zero crest height at the junction region, a side weir model explicitly introduced within the one-dimensional St. Venant equations is used to cope with the two-dimensional pattern of the flow. An upwind implicit numerical solver is employed to compute the new governing equations. The performance of the proposed technique in predicting super-, trans-, and subcritical flow bifurcations is illustrated by comparing with experimental data and/or theoretical predictions. In all the tests, lateral-to-upstream discharge ratios ( Rq ) are successfully reproduced by the present technique with a maximum error magnitude of less than 9%.

We analyze the effect of a small inclination on the well-studied problem of two-dimensional binary fluid convection in a horizontally extended closed rectangular box with a negative separation ratio, heated from below. The horizontal component of gravity generates a shear flow that replaces the motionless conduction state when inclination is not present. This large-scale flow interacts with the convective currents resulting from the vertical component of gravity. For very small inclinations the primary bifurcation of this flow is a Hopf bifurcation that gives rise to chevrons and blinking states similar to those obtained with no inclination. For larger but still small inclinations this bifurcation disappears and is superseded by a fold bifurcation of the large-scale flow. The convecton branches, i.e., branches of spatially localized states consisting of counterrotating rolls, are strongly affected, with the snaking bifurcation diagram present in the noninclined system destroyed already at small inclinations. For slightly larger but still small inclinations we obtain small-amplitude localized states consisting of corotating rolls that evolve continuously when the primary large-scale flow is continued in the Rayleigh number. These localized states lie on a solution branch with very complex behavior strongly dependent on the values of the system parameters. In addition, several disconnected branches connecting solutions in the form of corotating rolls, counterrotating rolls, and mixed corotating and counterrotating states are also obtained.

A self-sustaining process conjectured to be generic for wall-bounded shear flows is investigated. The self-sustaining process consists of streamwise rolls that redistribute the mean shear to create streaks that wiggle to maintain the rolls. The process is analyzed and shown to be remarkably insensitive to whether there is no-slip or free-slip at the walls. A low-order model of the process is derived from the Navier–Stokes equations for a sinusoidal shear flow. The model has two unstable steady solutions above a critical Reynolds number, in addition to the stable laminar flow. For some parameter values, there is a second critical Reynolds number at which a homoclinic bifurcation gives rise to a stable periodic solution. This suggests a direct link between unstable steady solutions and almost periodic solutions that have been computed in plane Couette flow. It is argued that this self-sustaining process is responsible for the bifurcation of shear flows at low Reynolds numbers and perhaps also for controlling the near-wall region of turbulent shear flows at higher Reynolds numbers.

The topological bifurcation of the flow in non-equilibrium magnetized plasmas is demonstrated by a turbulence simulation. A system with two generic sources of turbulence, the gradients of density and parallel flow, is considered. Topological index of the flow is introduced, in order to indicate the chirality of flow pattern. We here report that the turbulence-driven flow forms the structure of co-axial helixes with opposite chirality. By changing the source of plasma particles, which modifies the density gradient, the transition between three turbulent states is obtained. In addition to the two turbulent states, which are dominated by the drift wave and the D'Angelo mode, respectively, the new state is found. In this third state, fluctuations are driven by both of the free energy sources simultaneously, and compete with the others. The result illustrates the generic feature of turbulence flow generation in non-equilibrium magnetized plasmas.

Nonlinear analysis on the natural convection between vertical plates in the presence of a horizontal magnetic field

Instability, bifurcation and nonlinear dynamics of Poiseuille flow in fluid overlying an anisotropic and inhomogeneous porous domain

The relative influence of momentum diffusivity and thermal diffusivity, in terms of the Prandtl number (Pr), on the finite-amplitude instability of a non-isothermal annular Poiseuille flow (NAPF) is analyzed. The limiting value of the growth of instabilities under nonlinear effects is studied by deriving a cubic Landau equation. Emphasis is given especially on studying the impact of the low Prandtl number and the curvature parameter (C) on the bifurcation and the pattern variation of the secondary flow for both axisymmetric and non-axisymmetric disturbances. The finite-amplitude analysis predicts that in contrast to NAPF of water or fluid with Pr ≥ O(1) where the flow is supercritically unstable, the NAPF of low Pr fluids, particularly liquid metals, has shown both supercritical and subcritical bifurcation in the vicinity as well as away from the critical point. The nonlinear interaction of different harmonics for the liquid metal predicts a lower heat transfer rate than those by the laminar flow model, whereas for a fluid with Pr > 2, it is the other way. The maximum heat transfer takes place for the considered minimum value of C. For fluids with low Pr, a probable lower critical Rayleigh number is obtained. The corresponding variation in neutral stability curves as a function of wavenumber reveals that the instability that is supercritical for some wavenumber may be subcritical or vice versa at other nearby wavenumbers. The structural feature of the pattern of the secondary flow under the linear theory differs significantly from those of the secondary flow under nonlinear theory away from the bifurcation point. This is a consequence of the intrinsic interaction of different harmonics that are responsible for the stabilizing or the destabilizing nature of different components in the disturbance kinetic energy balance.

It is well-established that shear flows in a periodic strip are linearly unstable for the incompressible Navier Stokes equations provided the viscosity is small enough. In this article, under a natural spectral assumption which is satisfied for convex or concave analytic flows, we prove that shear flows undergo a Hopf bifurcation near their upper marginal stability curve. In particular, near this curve, there exist solutions which are periodic in t t and x x .

Localised turbulent structures such as puffs or oblique stripes are building blocks of the intermittency regimes in subcritical wall-bounded shear flows. These turbulent structures are investigated in incompressible pressure-driven annular pipe flow using direct numerical simulations in long domains. For low enough radius ratio $\eta$, these coherent structures have a dynamics comparable to that of puffs in cylindrical pipe flow. For $\eta$ larger than 0.5, they take the shape of helical stripes inclined with respect to the axial direction. The transition from puffs to stripes is analysed statistically by focusing on the axisymmetry properties of the associated large-scale flows. It is shown that the transition is gradual : as the azimuthal confinement relaxes, allowing for an azimuthal large-scale component, oblique stripes emerge as predicted in the planar limit. The generality of this transition mechanism is discussed in the context of subcritical shear flows.

Onset of double-diffusive Rayleigh-Bénard convection of a moderate Prandtl number binary mixture in cylindrical enclosures

We study numerically the flow around a spherical droplet set fixed in a linear shear flow with moderate shear rates ( $Sr\leq 0.5$ , $Sr$ being the ratio between the velocity difference across the drop and the relative velocity) over a wide range of external Reynolds numbers ( $0.1<{{Re}}\leq 250$ , ${{Re}}$ based on the slip velocity and the viscosity of the external fluid) and drop-to-fluid viscosity ratios ( $0.01\leq \mu ^\ast \leq 100$ ). The flow structure, the vorticity field and their intrinsic connection with the lift force are analysed. Specifically, the results on lift force are compared with the low- ${{Re}}$ solution derived for droplets of arbitrary $\mu ^\ast$ , as well as prior data at finite ${{Re}}$ available in both the clean-bubble limit ( $\mu ^\ast \to 0$ ) and the solid-sphere limit ( $\mu ^\ast \to \infty$ ). Notably, at ${{Re}}=O(100)$ , the lift force exhibits a non-monotonic transition from $\mu ^\ast \to 0$ to $\mu ^\ast \to \infty$ , peaking at $\mu ^\ast \approx 1$ . This behaviour is related to an internal three-dimensional flow bifurcation also occurring under uniform-flow conditions, which makes the flow to evolve from axisymmetric to biplanar symmetric. This flow bifurcation occurs at low-but-finite $\mu ^\ast$ when the internal Reynolds number ( ${{Re}}^i$ , based on the viscosity of the internal fluid) exceeds approximately 300. In the presence of shear, the corresponding imperfect bifurcation enhances the extensional rate of the flow in the wake. Consequently, the streamwise vortices generated behind the droplet can be more intense compared with those behind a clean bubble. Given the close relation between the lift and these vortices, a droplet with ${{Re}}=O(100)$ and $\mu ^\ast \approx 1$ typically experiences a greater lift force than that in the inviscid limit.