New omega vortex identification method

A new vortex identification criterion called W-method is proposed based on the ideas that vorticity overtakes deformation in vortex.The comparison with other vortex identification methods like Q-criterion and λ_2-method is conducted and the advantages of the new method can be summarized as follows:(1) the method is able to capture vortex well and very easy to perform;(2) the physical meaning of W is clear while the interpretations of iso-surface values of Q and λ_2 chosen to visualize vortices are obscure;(3)being different from Q and λ_2 iso-surface visualization which requires wildly various thresholds to capture the vortex structure properly, W is pretty universal and does not need much adjustment in different cases and the iso-surfaces of W=0.52 can always capture the vortices properly in all the cases at different time steps, which we investigated;(4) both strong and weak vortices can be captured well simultaneously while improper Q and λ_2 threshold may lead to strong vortex capture while weak vortices are lost or weak vortices are captured but strong vortices are smeared;(5) W=0.52 is a quantity to approximately define the vortex boundary. Note that, to calculate W, the length and velocity must be used in the non-dimensional form. From our direct numerical simulation, it is found that the vorticity direction is very different from the vortex rotation direction in general 3-D vortical flow,the Helmholtz velocity decomposition is reviewed and vorticity is proposed to be further decomposed to vortical vorticity and non-vortical vorticity.     

Least-squares spectral element solution of incompressible Navier–Stokes equations with adaptive refinement

Least-squares spectral element solution of steady, two-dimensional, incompressible flows are obtained by approximating velocity, pressure and vorticity variable set on Gauss–Lobatto–Legendre nodes. Constrained Approximation Method is used for h- and p-type nonconforming interfaces of quadrilateral elements. Adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient. Effective use of p-refinement to overcome poor mass conservation drawback of least-squares formulation and successful use of h- and p-refinement together to solve problems with geometric singularities are demonstrated. Capabilities and limitations of the developed code are presented using Kovasznay flow, flow past a circular cylinder in a channel and backward facing step flow.     

Penetrative ferroconvection via internal heating in a saturated porous layer with constant heat flux at the lower boundary

A model for penetrative ferroconvection via internal heat generation in a ferrofluid saturated porous layer is explored. The Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid- paramagnetic and insulated to temperature perturbations, while at upper stress-free boundary a general convective-radiative exchange condition on perturbed temperature is imposed. The resulting eigenvalue problem is solved numerically using the Galerkin method. It is found that increasing in the dimensionless heat source strength N_s, magnetic number M₁ Darcy number Da and the non-linearity of magnetization parameter M₃ is to hasten, while increase in the ratio of viscosities Λ, Biot number Bi and magnetic susceptibility χ is to delay the onset of ferroconvection. Further, increase in Bi, Da⁻¹ and N_s and decrease in Λ, M₁ and M₃ is to diminish the dimension of convection cells.     

A nonlinear scenario for development of vortex layer instability in gravity field

A Hamiltonian version of contour dynamics is formulated for models of constant-vorticity plane flows with interfaces. The proposed approach is used as a framework for a nonlinear scenario for instability development. Localized vortex blobs are analyzed as structural elements of a strongly perturbed wall layer of a vorticity-carrying fluid with free boundary in gravity field. Gravity and vorticity effects on the geometry and velocity of vortex structures are examined. It is shown that compactly supported nonlinear solutions (compactons) are candidates for the role of particle-like vortex structures in models of flow breakdown. An analysis of the instability mechanism demonstrates the possibility of a self-similar collapse. It is found that the vortex shape stabilizes at the final stage of the collapse, while the vortex sheet strength on its boundary increases as (t ₀ ? t)^?1, where t ₀ is the collapse time.     

Influence of metallic nanoparticles in water driven along a wavy circular cylinder

In the present study, simultaneous effects of metallic nanoparticles and magnetohydrodynamic due to stagnation point flow of nanofluid along a wave circular cylinder is presented. The effect of induced magnetic field is incorporated to deal the boundary and thermal boundary layer domain. Mathematical modelling for momentum and energy equation is constructed that is based upon three different kinds of nanoparticles namely: copper (Cu), Titanium di oxide (TiO₂), and alumina (Al₂O₃) within the working fluid water. Each mixture is analysed at the individual level and made comparison amongst all the mixture to examine the resistance and thermal conductivity of nanofluid within the boundary layer region. The solutions are exposed via boundary value problem using shooting method along with the Runge-Kutta-Fehlberg method. The characteristics of emerging parameters for the fluid flow and heat transfer are discussed through graphs and tables. The effects of ϕ (nanoparticle volume fraction) on heat transfer and shear stress at the wall are analysed in detail. It is finally concluded that by increasing the ratio of nanoparticles there is a significant increase in the temperature but slight decrease in the velocity profile.     

Intermittency of velocity time increments in turbulence

We analyze the statistics of turbulent velocity fluctuations in the time domain. Three cases are computed numerically and compared: (i) the time traces of Lagrangian fluid particles in a (3D) turbulent flow (referred to as the dynamic case); (ii) the time evolution of tracers advected by a frozen turbulent field (the static case); (iii) the evolution in time of the velocity recorded at a fixed location in an evolving Eulerian velocity field, as it would be measured by a local probe (referred to as the virtual probe case). We observe that the static case and the virtual probe cases share many properties with Eulerian velocity statistics. The dynamic (Lagrangian) case is clearly different; it bears the signature of the global dynamics of the flow.     

A lower bound for level spacings

We consider a single particle in a negative potential V: H = ?Δ + V(x). A lower bound is found for the quantity ?_Λ ? ?_∞, where ?_∞ is the ground-state energy of H in all space and where ?_Λ is the ground-state energy of H in a bounded domain Λ with Dirichlet (ψ = 0) boundary conditions. Our estimate for ?_Λ ? ?_∞ involves only ?_Λ and the volume, | Λ |, but does not depend upon V or upon the shape of Λ.     

An arbitrary Lagrangian–Eulerian formulation for the numerical simulation of flow patterns generated by the hydromedusa Aequorea victoria

A new geometrically conservative arbitrary Lagrangian–Eulerian (ALE) formulation is presented for the moving boundary problems in the swirl-free cylindrical coordinates. The governing equations are multiplied with the radial distance and integrated over arbitrary moving Lagrangian–Eulerian quadrilateral elements. Therefore, the continuity and the geometric conservation equations take very simple form similar to those of the Cartesian coordinates. The continuity equation is satisfied exactly within each element and a special attention is given to satisfy the geometric conservation law (GCL) at the discrete level. The equation of motion of a deforming body is solved in addition to the Navier–Stokes equations in a fully-coupled form. The mesh deformation is achieved by solving the linear elasticity equation at each time level while avoiding remeshing in order to enhance numerical robustness. The resulting algebraic linear systems are solved using an ILU(k) preconditioned GMRES method provided by the PETSc library. The present ALE method is validated for the steady and oscillatory flow around a sphere in a cylindrical tube and applied to the investigation of the flow patterns around a free-swimming hydromedusa Aequorea victoria (crystal jellyfish). The calculations for the hydromedusa indicate the shed of the opposite signed vortex rings very close to each other and the formation of large induced velocities along the line of interaction while the ring vortices moving away from the hydromedusa. In addition, the propulsion efficiency of the free-swimming hydromedusa is computed and its value is compared with values from the literature for several other species.     

An experimental study of the effects of pulsating and steady internal fluid flow on an elastic tube subjected to external vibration

The results of an experimental study on both pulsating and steady Newtonian fluid flow in an initially stretched rubber tube subjected to external vibration are reported. A circulating loop system was designed to maintain constant hydrostatic pressure throughout the tests so that the influence of external excitation on the fluid flow could be properly distinguished. The effects of fluid flow velocity and initial stretch rates on the dynamic response and damping of the tube conveying fluid were examined, and it was observed that damping ratios increase with increasing flow velocities, and generally decrease with increasing initial stretch rates for the tube conveying fluid. It was also noted that dynamic responses increase with increasing initial stretch rates, and decrease with increasing flow velocities. The effect of external vibration on fluid flow rates is small in a tube with a thickness-to-radius ratio (D_out−D_in)/D_in=0.617. Fluid pressures vary, in terms of frequency and amplitude, with external vibration as well as Womersley number.     

Relativity and quantum mechanics

Conditions under which quantum mechanics can be made compatible with the curved space-time of gravitation theories is investigated. A postulate is imposed in the formv=v _g wherev is the kinematical Hamilton-Jacobi (geometric optic limit) velocity andv _g is the group velocity of the waves. This imposes a severe condition on the possible coordinates in which the Schrödinger form (the coordinate realization) of quantum mechanics can be set up for purposes of calculating observable effects. Some such effects are calculated for a class of theories and are compared with experiments.     

Construction of Euclidean (QED)2 via lattice gauge theory

Let ν=det_ren(1+K _g ) be the renormalized Matthews-Salam determinant of (QED)₂, where \(K_g = ieA_{g,} S = \left( {\sum {\gamma _\mu \partial } _\mu + m} \right)^{ - 1} \) is euclidean fermion propagator of one of the following boundary conditions: (1) free, (2) periodic at ?Λ, Λ=[?L/2;L/2]², (3) anti-periodic at ?Λ, and \(A_g (x) = (\sum \gamma _\mu A_\mu (x))g(x)\) . Hereg(x)=1 ifxεΛ₀=[?r/2,r/2]² с Λ and 0 otherwise. Then we show

1. νεL ^p (dμ(A)), p>0. Further we prove a new determinant inequality which holds for the QED, QCD-type models containing fermions. This enables us to prove:
2. Z(Λ₀)=∫νdμ(A)≦exp[c|Λ₀|]. Similar volume dependence is shown for the Schwinger functions.

     

The influence of magnetic field on short-wavelength instability of Riemann ellipsoids

We address the question of stability of the so-called S-type Riemann ellipsoids, i.e. a family of Euler flows in gravitational equilibrium with the vorticity and background rotation aligned along the principal axis perpendicular to the flow. The Riemann ellipsoids are the simplest models of self-gravitating, tidally deformed stars in binary systems, with the ellipticity of the flow modelling the tidal deformation. By the use of the WKB theory we show that mathematically the problem of stability of Riemann ellipsoids with respect to short-wavelength perturbations can be reduced to the problem of magneto-elliptic instability in rotating systems, studied previously by Mizerski and Bajer [K.A. Mizerski, K. Bajer, The magneto-elliptic instability of rotating systems, J. Fluid Mech. 632 (2009) 401-430]. In other words the equations describing the evolution of short-wavelength perturbations of the Riemann ellipsoids considered in Lagrangian variables are the same as those for the evolution of the magneto-elliptic-rotational (MER) waves in unbounded domain. This allowed us to use the most unstable MER eigenmodes found in Mizerski et al. [K.A. Mizerski, K. Bajer, H.K. Moffatt, The α-effect associated with elliptical instability, J. Fluid Mech., 2010 (in preparation)] to provide an estimate of the characteristic tidal synchronization time in binary star systems. We use the idea of Tassoul [J.-L. Tassoul, On synchronization in early-type binaries, Astrophys. J. 322 (1987) 856-861] and that the interactions between perturbations significantly increase the effective viscosity and hence the energy dissipation in an Ekman-type boundary layer at the surface of the star. The results obtained suggest that if the magnetic field generated by (say) the secondary component of a binary system is strong enough to affect the flow dynamics in the primary, non-magnetized component, the characteristic tidal synchronization time can be significantly reduced.     

On the damping of unsteady flow by compliant boundaries

This paper concerns unsteady motion of an incompressible inviscid fluid near a flexible surface which, in responding to the surface pressure field, absorbs energy. The modification of the flow consequent on energy removal at the boundaries is examined. Energy absorption always occurs when the mechanical surface properties include an element of dissipation. But surface dissipation is not essential; surface waves have a similar property. Unsteady fluid induced forces excite surface waves which carry with them energy that must have originated in the flow. The question of how flow characteristics change as energy is gradually given up to the boundary is examined through a particular model problem from which it becomes evident that surface motion draws vorticity towards the surface. The model chosen is that of a rectilinear vortex adjacent to a weakly responding boundary. Surface motion induces a velocity perturbation which is shown to move the vortex towards the surface whenever the fluid gives energy to that surface.     

Effects of superficial gas velocity on process dynamics in bioreactors

Present work analyzes the flow hydrodynamics and mass transfer mechanisms in double Rushton and CD-6 impeller on wide range (0.0075–0.25 m/s) of superficial gas velocity (v _g) in a gas-liquid phase bioreactor by employing computational fluid dynamics (CFD) technique. The volume averaged velocity magnitude and dissipation rate are found higher with increasing superficial gas velocity. Higher relative power draw (P _g/P ₀) is predicted in CD-6 than the Rushton impeller but no significant difference in volume averaged mass transfer coefficient (k _L a) observed between these two types of impeller. The ratio of power draw with mass transfer coefficient has been found higher in CD-6 impeller (25–50 %) than the Rushton impeller.     

Immersed boundary method for the simulation of 2D viscous flow based on vorticity–velocity formulations

A new immersed boundary method based on vorticity–velocity formulations for the simulation of 2D incompressible viscous flow is proposed in present paper. The velocity and vorticity are respectively divided into two parts: one is the velocity and vorticity without the influence of the immersed boundary, and the other is the corrected velocity and the corrected vorticity derived from the influence of the immersed boundary. The corrected velocity is obtained from the multi-direct forcing to ensure the well satisfaction of the no-slip boundary condition at the immersed boundary. The corrected vorticity is derived from the vorticity transport equation. The third-order Runge–Kutta for time stepping, the fourth-order finite difference scheme for spatial derivatives and the fourth-order discretized Poisson for solving velocity are applied in present flow solver. Three cases including decaying vortices, flow past a stationary circular cylinder and an in-line oscillating cylinder in a fluid at rest are conducted to validate the method proposed in this paper. And the results of the simulations show good agreements with previous numerical and experimental results. This indicates the validity and the accuracy of present immersed boundary method based on vorticity–velocity formulations.     

On some hydrodynamical aspects of quantum mechanics

In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the complex polynomial (i.e. Borel-Weil) realization of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dimensional geometric quantum mechanics, the Schrödinger velocity field on projective Hilbert space is divergence-free (being Killing with respect to the Fubini-Study metric) and fulfils the stationary Euler equation, with pressure proportional to the Hamiltonian uncertainty (squared). We explicitly determine the critical points of the pressure of this “Schrödinger fluid”, together with its vorticity, which turns out to depend on the spacings of the energy levels. These results follow from hydrodynamical properties of Killing vector fields valid in any (finite dimensional) Riemannian manifold, of possible independent interest.     

An Eulerian approach for computing the finite time Lyapunov exponent

We propose efficient Eulerian methods for approximating the finite-time Lyapunov exponent (FTLE). The idea is to compute the related flow map using the Level Set Method and the Liouville equation. There are several advantages of the proposed approach. Unlike the usual Lagrangian-type computations, the resulting method requires the velocity field defined only at discrete locations. No interpolation of the velocity field is needed. Also, the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain. The computational complexity of the algorithm is O(Δx^?(d+1)) with d the dimension of the physical space. Since there are the same number of mesh points in the x–t space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only O(1) time. We also extend the algorithm to compute the FTLE on a co-dimension one manifold. The resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold.     

A buoyant flow structure in a magnetic field: Quasi-steady states and linear-nonlinear transitions

The confined evolution of a buoyant blob of fluid subject to a vertical magnetic field is investigated in the limit of low magnetic Reynolds number. When the applied magnetic field is strong, the rise velocity of the blob is small. As the vorticity diffuses along the magnetic field lines, a quasi-steady state characterised by a balance between the work done by buoyancy and Ohmic dissipation is eventually reached at time tqs∼(L²/δ²)τ, where L is the axial dimension of the fluid domain, δ is the radius of the buoyant blob and τ is the magnetic damping time. However, when the applied magnetic field is weak or the axial length is sufficiently large compared to the blob size, the growth of axial velocity eventually makes the advection of vorticity significant. The typical time for the attainment of this nonlinear phase is , where N₀ is the magnetic interaction parameter at time t=τ. The order-of-magnitude estimates for the timescales tqs and tnl are verified by computational experiments that capture both the linear and nonlinear phases.     

Effective field theory for massive gravitons and gravity in theory space

We introduce a technique for restoring general coordinate invariance into theories where it is explicitly broken. This is the analog for gravity of the Callan-Coleman-Wess-Zumino formalism for gauge theories. We use this to elucidate the properties of interacting massless and massive gravitons. For a single graviton with a Planck scale M_Pl and a mass m_g, we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Λ parametrically above m_g. Our methods allow for a transparent understanding of the many peculiarities associated with massive gravitons, among them the need for the Fierz-Pauli form of the Lagrangian, the presence or absence of the van Dam-Veltman-Zakharov discontinuity in general backgrounds, and the onset of non-linear effects and the breakdown of the effective theory at large distances from heavy sources. The natural sizes of all non-linear corrections beyond the Fierz-Pauli term are easily determined. The cutoff scales as Λ∼(m_g⁴M_Pl)^1/5 for the Fierz-Pauli theory, but can be raised to Λ∼(m_g²M_Pl)^1/3 in certain non-linear extensions. Having established that these models make sense as effective theories, there are a number of new avenues for exploration, including model building with gravity in theory space and constructing gravitational dimensions.