Pure Potential Flow Research Articles

Stability of flow and the transition to turbulence around a quartz tuning fork in superfluidHe4at very low temperatures

We have studied the transition between pure potential flow and turbulent flow around a quartz tuning fork resonator in superfluid $^{4}\mathrm{He}$ at millikelvin temperatures. Turbulent flow is identified by an additional drag force on the fork prongs due to the creation of quantized vortices. When driven at a constant driving force amplitude, the transition to turbulence causes an abrupt decrease in the velocity amplitude of the prongs. For a range of driving forces, continuous switching is observed between the two flow states. We have made a statistical study of the switching characteristics and of the lifetimes of the unstable states. We find a characteristic velocity ${v}^{\ensuremath{\star}}$ which separates quasistable turbulent flow at higher velocities and quasistable potential flow at lower velocities. We show that the potential-to-turbulent flow transition is driven by random processes involving remanent vortices pinned to the prongs.

Hysteresis, Switching and Anomalous Behaviour of a Quartz Tuning Fork in Superfluid 4He

We have been studying the behaviour of commercial quartz tuning forks immersed in superfluid 4He and driven at resonance. For one of the forks we have observed hysteresis and switching between linear and non-linear damping regimes at temperatures below 10 mK. We associate linear damping with pure potential flow around the prongs of the fork, and non-linear damping with the production of vortex lines in a turbulent regime. At appropriate prong velocities, we have observed metastability of both the linear and the turbulent flow states, and a region of intermittency where the flow switched back and forth between each state. For the same fork, we have also observed anomalous behaviour in the linear regime, with large excursions in both damping, resonant frequency, and the tip velocity as a function of driving force.

Horizontal circulation and jumps in Hamiltonian wave models

Abstract. We are interested in the modelling of wave-current interactions around surf zones at beaches. Any model that aims to predict the onset of wave breaking at the breaker line needs to capture both the nonlinearity of the wave and its dispersion. We have therefore formulated the Hamiltonian dynamics of a new water wave model, incorporating both the shallow water and pure potential flow water wave models as limiting systems. It is based on a Hamiltonian reformulation of the variational principle derived by Cotter and Bokhove (2010) by using more convenient variables. Our new model has a three-dimensional velocity field consisting of the full three-dimensional potential velocity field plus extra horizontal velocity components. This implies that only the vertical vorticity component is nonzero. Variational Boussinesq models and Green–Naghdi equations, and extensions thereof, follow directly from the new Hamiltonian formulation after using simplifications of the vertical flow profile. Since the full water wave dispersion is retained in the new model, waves can break. We therefore explore a variational approach to derive jump conditions for the new model and its Boussinesq simplifications.

A study of two-dimensional flow past regular polygons via conformal mapping

In this paper we study two-dimensional flow around regular polygons with an arbitrary but even number of edges N and one apex pointing to the free stream, with comparison to circular-cylinder flow. Both inviscid flow and low-Reynolds-number viscous flow are addressed. For inviscid flow, we obtained the exact solution for pure potential flow through Schwarz–Christoffel transformation, with the emphasis on the role of edge number, N, on the flow details. We also studied the behaviour, stationary lines and stability of vortex pair and found new stationary lines compared to circular cylinder. For viscous flow we derived the equation of stream function in the mapped (circle) domain, based on which approximate expressions for the critical Reynolds numbers and Strouhal number, as functions of the edge number, are obtained. The Reynolds number is based on the diameter of the circumscribed circle. For the steady flow, the first critical Reynolds number is a monotonically decreasing function of N, while N → ∞ corresponds to that for circular cylinder. The bifurcation point is ahead of the bifurcation point for circular cylinder. For unsteady flow, the critical Reynolds number for vortex shedding and the Strouhal number are both monotonically decreasing functions of N.

Vortex equilibrium in flows past bluff bodies

The equilibrium conditions of a point vortex in the separated flow past a locally deformed wall is studied in the framework of the two-dimensional potential flow. Equilibrium locations are represented as fixed points of the vortex Hamiltonian contour line map. Their pattern is ascribable to the Poincare-Birkhoff fixed-point theorem. An ‘equilibrium manifold', representing the generalization of the Foppl curve for circular cylinders, is defined for arbitrary bodies. The property ∂ω/∂ψ ˜ =0 holds on it, with ψ ˜ being the stream function and ω the streamline slope of the pure potential flow. A ‘Kutta manifold' is defined as the locus of vortices in flows that separate at a prescribed point (Kutta condition). The existence of standing vortices that satisfy the Kutta condition is discussed for symmetric bodies. On the basis of an asymptotic expansion of the equilibrium manifold, Kutta manifold and body geometry, it is shown that different classes of symmetric bodies exist which are ranked by the number of allowable standing vortices that satisfy the Kutta condition

Stability of Laminar and Turbulent Flow of Superfluid4He at mK Temperatures around an Oscillating Microsphere

The flow of superfluid helium around a vibrating microsphere is investigated at temperatures between 1 K and 25 mK. At small oscillation amplitudes pure potential flow is observed, the linear drag force on the sphere being determined only by ballistic quasiparticle scattering below 0.7 K with phonons contributing exclusively below 0.5 K. At larger oscillation amplitudes a strongly nonlinear drag force gives evidence of stable turbulent flow if at least 0.6 pW are transferred from the sphere to the turbulent superfluid. In an intermediate range of amplitudes (or driving forces) both flow patterns are unstable and intermittent switching between both is observed below 0.5 K. We have recorded time series of this switching phenomenon at constant drives and temperatures lasting up to 36 hours. We have made a statistical analysis of the times series by means of reliability theory. The lifetime of the turbulent phases grows with increasing drive and diverges at a critical value (or at least becomes unmeasurably long). Stability of the laminar phases in the intermediate regime depends on the excess velocity of the sphere above the critical velocity. Metastable laminar phases are observed above the critical velocity having a mean lifetime limited to 25 minutes by natural background radioactivity which occasionally produces local vorticity due to ionization of the liquid. Finally, it is suggested that the breakdown of potential flow belongs to the class of “system failure” experiments which is well known in reliability testing and whose statistical properties are described by extreme-value theory.

Observation of Torque Exerted by Pure Superflow

Pure superflow is observed to exert a torque in flowing about an object of arbitrary shape. The magnitude of such torque agrees in every respect with expectations based on pure potential flow theory. The present results indicating pure potential flow for superfluid are considered to constitute the complementary experiment to the earlier Craig-Pellam investigation of zero lift for superflow. We consider that these two experiments taken together establish completely the pure potential flow property for superfluid by verifying that (a) zero force, but (b) a prescribed torque are exerted upon an object of arbitrary shape and orientation. As a by-product to the present investigation, an independent evaluation of $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$ for liquid helium is obtained in conformity with earlier measurements.

Motion of a Viscous Liquid Past an Ellipsoid

A symmetrical flow pattern due to the motion of a viscous liquid past an ellipsoidal body is determined. A unitary treatment of the flow problem is possible in which the pure potential flow, the slow viscous flow and the type of flow postulated in the Prandtl theory are seen to be limiting cases of an exact solution. The flow pattern for the case of the sphere and of the elliptic cylinder have been derived as special cases. The method consists in extending the equations of hydrodynamics by the introduction of certain fictitious body forces which are eventually made to vanish in the limit.

Mechanism of Superfluid Flow

THEORIES of superfluidity usually assume the possibility of pure potential flow, provided that the flow velocity is less than some critical value (at which the creation of excitations first becomes possible)1,2. If we consider the flow of helium (at T = 0° K.), for example through a tube or along a Rollin film, it is implied that the boundary layer is slipping over the solid wall. However, the first few layers of helium atoms would not be expected to take part in this motion, as they are firmly bound by the van der Waals' attractions of the wall. The picture therefore has to be revised slightly to one in which there is a velocity discontinuity (that is, a vortex sheet) close to, but not coincident with, the solid wall. But a continuous vortex sheet implies a rather drastic change in the character of the wave function, over a distance of the order of one interatomic distance a, and hence implies a large energy per unit area, of the order of h 2/ma4 (where m is the mass of an atom).

Observation of Perfect Potential Flow in Superfluid

The lift on an airfoil in a velocity field of perfect superfluid flow within liquid helium ii has been found to vanish for sufficiently low velocity. Thus the flow is pure potential flow with no circulation, and the viscosity boundary condition of zero velocity at the trailing edge (Kutta condition) which is normally introduced does not apply. We believe this to be the first illustration of absolutely perfect hydrodynamic flow, i.e., one in which the viscosity vanishes identically and therefore asserts no influence on the flow field.