Constant Vorticity Research Articles

Hydraulic control of flow in a multi-passage system connecting two basins

When a fluid stream in a conduit splits in order to pass around an obstruction, it is possible that one branch will be critically controlled while the other remains not so. This is apparently the situation in Pacific Ocean abyssal circulation, where most of the northward flow of Antarctic bottom water passes through the Samoan Passage, where it is hydraulically controlled, while the remainder is diverted around the Manihiki Plateau and is not controlled. These observations raise a number of questions concerning the dynamics necessary to support such a regime in the steady state, the nature of upstream influence and the usefulness of rotating hydraulic theory to predict the partitioning of volume transport between the two paths, which assumes the controlled branch is inviscid. Through the use of a theory for constant potential vorticity flow and accompanying numerical model, we show that a steady-state regime similar to what is observed is dynamically possible provided that sufficient bottom friction is present in the uncontrolled branch. In this case, the upstream influence that typically exists for rotating channel flow is transformed into influence into how the flow is partitioned. As a result, the partitioning of volume flux can still be reasonably well predicted with an inviscid theory that exploits the lack of upstream influence.

Destabilizing resonances of precessing inertia-gravity waves.

Instabilities in stratified precessing fluid are investigated. We extend the study by Mahalov [Phys. Fluids A 5, 891 (1993)0899-821310.1063/1.858635] in the stably stratified Boussinesq framework, with an external Coriolis force (with rate Ω_{p}) altering the base flow through the distortion of the circular streamlines of the unperturbed axially stratified rotating columns (with constant vorticity 2Ω.) It is shown that the inviscid part of the modified velocity flow (0,Ωr,-2ɛΩrsinφ) and buoyancy with gradient N^{2}(-2ɛcosφ,2ɛsinφ,1) are an exact solution of Boussinesq-Euler equations. Here (r,φ,z) is a cylindrical coordinate system, with ɛ=Ω_{p}/Ω being the Poincaré number and N the Brunt-Väisälä frequency. The base flow is transformed into a Cartesian coordinate system, and the stability of a superimposed perturbation is studied in terms of Fourier (or Kelvin) modes. The resulting Floquet system for the Fourier modes has three parameters: ɛ, N=N/Ω, and μ, which is the angle between the wave vector k and the solid-body rotation axis in the limit ɛ=0. In this limit, there are inertia-gravity waves propagating with frequency ±ω and the resonant cases are those for which 2ω=nΩ, n being an integer. We perform an asymptotic analysis to leading order in ɛ and characterize the destabilizing resonant case of order n=1 (i.e., the subharmonic instability) which exists and for 0≤N<Ω/2. In this range, the subharmonic instability remains the strongest with a maximal growth rate σ_{m}=[ɛ(5sqrt[15]/8)sqrt[1-4N^{2}]/(4-N^{2})]. Stable stratification acts in such a way as to make the subharmonic instability less efficient, so as it disappears for N≥0.5Ω. The destabilizing resonant cases of order n=2,3,4,5 are investigated in detail by numerical computations. The effect of viscosity on these instabilities is briefly addressed assuming the diffusive coefficients (kinematic and thermal) are equal. Likewise, we briefly investigate the case where N^{2}<0 and show that the instability associated to the mode with k_{3}=0 is the strongest.

Stability of waves on fluid of infinite depth with constant vorticity

The stability of periodic travelling waves on fluid of infinite depth is examined in the presence of a constant background shear field. The effects of gravity and surface tension are ignored. The base waves are described by an exact solution that was discovered recently by Hur and Wheeler (J. Fluid Mech., vol. 896, 2020). Linear growth rates are calculated using both an asymptotic approach valid for small-amplitude waves and a numerical approach based on a collocation method. Both superharmonic and subharmonic perturbations are considered. Instability is shown to occur for any non-zero amplitude wave.

The cubic vortical Whitham equation

The cubic vortical Whitham equation is a model for wave motion on a vertically sheared current of constant vorticity in a shallow inviscid fluid. It generalizes the classical Whitham equation by allowing constant vorticity and by adding a cubic nonlinear term. The inclusion of this extra nonlinear term allows the equation to admit periodic, traveling-wave solutions with larger amplitude than the Whitham equation. Increasing vorticity leads to solutions with larger amplitude as well. The stability of these solutions is examined numerically. All moderate- and large-amplitude solutions, regardless of wavelength, are found to be unstable. A formula for a stability cutoff as a function of vorticity and wavelength for small-amplitude solutions is presented. In the case with zero vorticity, small-amplitude solutions are unstable with respect to the modulational instability if kh>1.252, where k is the wavenumber and h is the mean fluid depth.

On three-dimensional free surface water flows with constant vorticity

&lt;p style='text-indent:20px;'&gt;We present a survey of recent results on gravity water flows satisfying the three-dimensional water wave problem with constant (non-vanishing) vorticity vector. The main focus is to show that a gravity water flow with constant non-vanishing vorticity has a two-dimensional character in spite of satisfying the three-dimensional water wave equations. More precisely, the flow does not change in one of the two horizontal directions. Passing to a rotating frame, and introducing thus geophysical effects (in the form of Coriolis acceleration) into the governing equations, the two-dimensional character of the flow remains in place. However, the two-dimensionality of the flow manifests now in a horizontal plane. Adding also centripetal terms into the equations further simplifies the flow (under the assumption of constant vorticity vector): the velocity field vanishes, but, however, the pressure function is a quadratic polynomial in the horizontal and vertical variables, and, surprisingly, the surface is non-flat.&lt;/p&gt;

Numerical investigation of depth-varying currents on ship hydrodynamics in confined water

Vessels can operate in unpredictable environments depending on the geographical area and weather conditions. One example of conditions a vessel might not be assessed against is the presence of depth-varying currents, which are particularly relevant in confined waters where currents can be created due to tidal influences, or short fetches in inland waterways. The possible presence of depth-varying currents motivates a numerical assessment of the effects of sheared currents on the hydrodynamic performance of the KRISO Container Ship (KCS) in confined waters. The results highlight that exploiting currents, such as those generated by tides could be used to improve the energy efficiency of vessels considerably. These currents present significant possibilities for voyage optimisation based on geographical and meteorological conditions. The results specific for the KRISO container ship point to resistance reductions when the current assists ship motions, accompanied by considerable decreases in sinkage and trim. Conversely, when currents oppose ship motion, resistance, sinkage and trim can increase by a factor of 3 depending on the strength and shape of the depth-varying current. The results also show that a current with constant vorticity, a case frequently used in the literature to investigate the impact of sheared currents, creates the biggest increase and decrease for inhibiting and assisting currents, respectively.

A New Three-Dimensional Residual Flow Theory and Its Application to Brewer–Dobson Circulation in the Middle and Upper Stratosphere

Abstract This study formulates three-dimensional (3D) residual flow, treating both stationary and transient waves. The zonal and meridional momentum equations contain four terms: the geostrophic wind tendency, Coriolis force for the residual horizontal flow, product of the geostrophic wind and potential vorticity other than the constant planetary vorticity, and friction. The thermodynamic equation contains three terms: the potential temperature tendency, advection of the basic potential temperature by the residual vertical flow, and diabatic heating. The zonal mean of the 3D residual flow equals the time mean of the residual flow of the transformed Eulerian-mean equations. The new residual flow is the sum of that derived by Plumb for transient waves and the quadratic terms of the time-mean fields, which correspond approximately to the Stokes correction due to stationary waves. The 3D residual flow and momentum equations are symmetric in the zonal and meridional directions, in contrast with those formulated by Kinoshita et al., which treat the time-mean zonal-mean zonal wind as the basic wind. The newly derived formulas are applied to the climatology of the 3D structure of the deep branch of the Brewer–Dobson circulation. In the Northern Hemisphere in December–February, the residual flows are directed inward toward the polar vortex strongly over east Siberia, where the downward flow is maximized, and weakly over the Atlantic; meanwhile, they are directed outward from the vortex over North America and Europe. A longitudinal dependence of the poleward flow is also observed in the Southern Hemisphere in June–August.

Ondas rotacionales de gravedad-capilaridad generadas por el paso de una perturbación

Nonlinear gravity-capillary waves generated by the passage of a pressure distribution over a sheared channel with constant vorticity are investigated. The problem is modeled using the full Euler equations. The harmonic part of the velocity field is formulated in a canonical domain through the use of the conformal mapping, which flattens the fluid domain onto a strip. The Froude number is considered to be nearly-critical and the Bond number critical. The shear effect changes drastically the pattern of the generated waves for large times. Moreover, depending on the intensity of the vorticity, the wave solutions can become smoother with small amplitudes.

On flow simplification occurring in viscous three-dimensional water flows with constant non-vanishing vorticity

We show that a viscous steady water flow of constant non-vanishing vorticity situated below a free surface (assumed to have the shape of a two-dimensional time dependent graph) and above a flat bottom has to be two-dimensional; that is, while satisfying the viscous three-dimensional water wave equations, it turns out that the free surface, the velocity field and the pressure present no variation in one of the horizontal directions. Moreover, the vorticity must have only one non-zero component which points in the horizontal direction orthogonal to the direction of the surface wave propagation. Compared with previous works, our study here has the advantage that it considers the Navier–Stokes equations and also takes into account the normal and tangential stress boundary conditions.

Miles’ mechanism for generating surface water waves by wind, in finite water depth and subject to constant vorticity flow

The Miles’ theory of wave amplification by wind is extended to the case of finite depth h and a shear flow with (constant) vorticity Ω. Vorticity is characterised through the non-dimensional parameter ν=ΩU1/g, where g the gravitational acceleration, U1 a characteristic wind velocity. The notion of ’wave age’ is generalised to account for the effect of vorticity. Several widely used growth rates are derived analytically from the dispersion relation of the wind/water interface, and their dependence on both water depth and vorticity is derived and discussed. Vorticity is seen to shift the maximum wave age, similar to what was previously known to be the effect of water depth. At the same time, a novel effect arises and the growth coefficients, at identical wave age and depth, are shown to experience a net increase or decrease according to the shear gradient in the water flow.

Large-Amplitude Steady Downstream Water Waves

We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling gravity waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded.

Vortex Solution in Elliptic Coordinates

Abstract Vortices (flows with closed elliptic streamlines) are exact nonlinear solutions to the compressible Euler equation. In this contribution, we use differential geometry to derive the transformations between Cartesian and elliptic coordinates, and show that in elliptic coordinates a constant vorticity flow reduces to μ ˙ = 0 and ν ˙ = const along the streamline μ 0 that matches the vortex eccentricity.

Wake behind circular cylinder excited by spanwise non-uniform disturbances

We studied a modification of wake behind a circular cylinder using a plasma actuator. The plasma actuators were arranged in the spanwise direction of the cylinder to give temporal periodic disturbances having Strouhal number St = 0.18-2.3 with a burst ratio BR = 20 and 40%. The Reynolds number was set in a rage of Re = 4200 to 8400. Two types of plasma actuator were prepared; one is a single strip of the actuator placed at each side of the cylinder to give a spanwise uniform disturbance, and another is an array of small piece of actuators placed at the same location to create a spanwise non-uniform disturbance with temporal phase difference, φ = 0 or π, between adjacent electrodes. A conventional two-component PIV and stereo PIV was used to measure the flow field. Figure 1 shows the instantaneous spanwise component of vorticity at Re = 4200 evaluated by two-component PIV. Under no disturbance condition, the laminar shear layer extends straight to around x / d = 1.5 and then forms a wake vortex, as shown in Fig.1(a). In the case of spanwise non-uniform forcing with St = 1.09 and φ =π, rapid roll up of the initial shear layer leads to arrangement of wake vortices closer to the cylinder., as shown in Fig.1(b). With higher Strouhal number case with St = 1.09 and φ = 0, shown in Fig.1(c), a series of fine scale vortices are generated behind both side of the cylinder without forming regular Karman vortices. The spanwise non-uniform forcing was effective to suppress the formation of large scale vortices just behind the cylinder. Figure 2 shows surface of constant vorticity magnitude and vortex lines under St =1.09 and φ = π case. These were computed from a phase-averaged threecomponents velocity field evaluated by stereo PIV. The value of the surface was selected to display the boundary layer formed on the cylinder, and the vortex lines are selected to visualize the vortex structure formed in the following shear layer. A bundle of vortex lines are shaped in a wavy pattern along spanwise direction with 180 degrees out of phase to the adjacent bundle upstream of downstream. This structure, so called ‘chain-line fence structure’ was already found in planar free shear layer [Nygaard, K.J. and Glezer, A., 1990, Phys. Fluids A, 2, 461] and planar jet [Sakakibara, J., Anzai, T., 2001, Phys. Fluids, 13, 1541], but it became evident to create it in the wake of circular cylinder in this study.

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity Omega , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

Constant vorticity equatorial flows beneath surface solitary waves with centripetal forces

In this paper, we consider equatorial flows with Coriolis and centripetal forces, with constant vorticity beneath a two-dimensional solitary wave and above a flat bed in the -plane approximation equations. We prove that the flow must be steady, two-dimensional, decayed and the vorticity may have only one nonzero component, which points in the horizontal direction orthogonal to the direction of wave propagation. Moreover, based on the vorticity equation and analytical techniques, we prove that the free surface of the flow exhibiting constant horizontal vorticity can only be two-dimensional.

Aspects of holography of Taub-NUT- AdS4 spacetimes

In this paper we consider aspects of the holographic interpretation of Taub-NUT-${\mathrm{AdS}}_{4}$. We review our earlier results which show that ${\mathrm{TNAdS}}_{4}$ gives rise to a holographic three-dimensional conformal fluid having constant vorticity. We then study the holographic relevance of the Misner string by considering bulk scalar fluctuations. The scalar fluctuations organize naturally into representations of the $SU(2)\ifmmode\times\else\texttimes\fi{}\mathbb{R}$ isometry algebra. If we require the string's invisibility we obtain a Dirac-like quantization relating the frequency of the scalar field modes to the NUT charge. As the latter quantity determines the total vorticity flux of the boundary fluid, we argue that such an assumption allows for a holographic interpretation of ${\mathrm{TNAdS}}_{4}$ as a nondissipative superfluid whose excitations are quantized vortices. Alternatively, if we regard the Misner string as a physical object, as has recently been advocated for thermodynamically, the aforementioned quantization conditions are removed, and we find that ${\mathrm{TNAdS}}_{4}$ corresponds to a holographic fluid whose dissipative properties are probed as usual by the complex quasinormal modes of the bulk fluctuations. We show that such quasinormal modes are, perhaps surprisingly, organized into infinite-dimensional nonunitary representations of the isometry algebra.

Method and experimental study of cell manipulation based on swirl

The development of a multifunctional noncontact manipulation method is desired in cell manipulation. In this paper, we theoretically study and experimentally demonstrate the method of cell manipulation based on swirl. We analyze the swirl flow field features and the cell manipulation mechanism. The swirl is constructed by jetting fluid through three microtubules, and the cells are manipulated using the low pressure of stagnation point and the constant vorticity in the flow field. The experimental results indicate that the cell can be trapped robustly using the stagnation point of swirl. Moreover, the cell rotates with the same angular velocity at any position of the swirl flow field, and the cell is transported along with the movement of the stagnation point. Finally, we extend the application of swirl to manipulate multiple cells. Overall, this method can achieve trapping, rotating, transporting, pairing and enriching for cells, which realizes multifunctional cell manipulation.

Stratospheric vacillations in a minimal contour dynamics model

AbstractA low-dimensional dynamical system that describes dynamical variability of the stratospheric polar vortex is presented. The derivation is based on a linearized, contour-dynamics representation of quasigeostrophic shallow water flow on a polar f-plane. The model consists of a single linear wave mode propagating on a near-circular patch of constant potential vorticity (PV). The PV jump at the vortex edge serves as an additional degree of freedom. The wave is forced by surface topography, and interacts with the vortex through a simplified parameterization of diabatic wave/mean flow interaction. The approach can be generalized to other geometries.The resulting three-component system depends on four non-dimensional parameters, and the structure of the steady state solutions can be determined analytically in some detail. Despite its extreme simplification, the model exhibits variability that is closely analogous to the Holton-Mass model, a well-known and more complex dynamical model of stratospheric variability. The present model exhibits two stable steady solutions, one consisting of a strong vortex with a small amplitude wave and the second consisting of a weak vortex with a large amplitude wave. Periodic and aperiodic limit cycles are also identified, analogous to similar solutions in the Holton-Mass model. Model trajectories also exhibit a number of behaviors that have been identified in observations. A key insight is that the time-mean state of the vortex is predominantly controlled by the properties of the linear mode, while the strength of the topographic forcing plays a far weaker role away from bifurcations.

Constant vorticity water flows in the modified equatorial $$\beta $$-plane approximation

Constant vorticity water flows in the modified equatorial $$\beta $$-plane approximation

The decay of Hill's vortex in a rotating flow

Abstract