Vortex ring formation at tube and orifice openings

The formation, at tube and orifice openings, of vortex rings generated by a piston moving with velocity proportional to time to some power m, is considered. The expansion of the axisymmetric generating flow about the circular forming edge is used in conjunction with the similarity theory of edge vortex growth to model the ring formation process. For large Reynolds numbers the ring diameter and circulation are not strongly dependent on the piston velocity profile. However, the ring viscous subcore shows peaks in the tangential velocity profile only if m < (π–θ)/(2π–θ), where θ is the edge forming angle

Large-eddy simulation and wall modelling of turbulent channel flow

We report large-eddy simulation (LES) of turbulent channel flow. This LES neither resolves nor partially resolves the near-wall region. Instead, we develop a special near-wall subgrid-scale (SGS) model based on wall-parallel filtering and wall-normal averaging of the streamwise momentum equation, with an assumption of local inner scaling used to reduce the unsteady term. This gives an ordinary differential equation (ODE) for the wall shear stress at every wall location that is coupled with the LES. An extended form of the stretched-vortex SGS model, which incorporates the production of near-wall Reynolds shear stress due to the winding of streamwise momentum by near-wall attached SGS vortices, then provides a log relation for the streamwise velocity at the top boundary of the near-wall averaged domain. This allows calculation of an instantaneous slip velocity that is then used as a ‘virtual-wall’ boundary condition for the LES. A Kármán-like constant is calculated dynamically as part of the LES. With this closure we perform LES of turbulent channel flow for Reynolds numbers Re_τ based on the friction velocity u_τ and the channel half-width δ in the range 2 × 10^3 to 2 × 10^7. Results, including SGS-extended longitudinal spectra, compare favourably with the direct numerical simulation (DNS) data of Hoyas & Jiménez (2006) at Re_τ = 2003 and maintain an O(1) grid dependence on Re_τ

Remark on a result of D. Dritschel

A hypothesis put forward by D. Dritschel [J. Fluid Mech. 94, 511 (1988)], namely that an isolated symmetrical disturbance on a uniform vortex patch will filament in time proportional to the inverse square of the disturbance amplitude, is subject to independent testing using a nonintrusive numerical method. The hypothesis that the trend is maintained to substantially smaller amplitudes than were originally considered by Dritschel is both supported and verified. The results may be interpreted as providing limited evidence that contour smoothness is maintained in filamentation and that corner formation does not occur up to the time of wave overturning

On singularity formation in three-dimensional vortex sheet evolution

It is shown that if a doubly-infinite vortex sheet has cylindrical shape and strength distributions at some initial time, then this property is retained in its subsequent evolution. It is also shown that in planes normal to the generator of the cylindrical sheet geometry, the nonlinear evolution of the sheet is the same as that of an equivalent strictly two-dimensional sheet motion. These properties are used to show that when an initially flat vortex sheet is subject to a finite-amplitude, three-dimensional normal mode perturbation, weak singularities develop along lines which are oblique to the undisturbed velocity jump vector at a time that can be inferred from an extension of Moore's [Proc. R. Soc. A 365, 105 (1979)] result for two-dimensional motion

On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions

For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions

On the dynamics of the collapse of a diffusion-flame hole

The collapse dynamics of a diffusion-flame hole in the presence of a counterflow are studied. We construct unsteady solutions of the one-dimensional edge-flame model of Buckmaster (1996), in which heat and mass transverse losses are algebraic. The flame structure is determined in the classical limit of large activation energy. Solutions for both planar and axisymmetric strain geometry are considered for the particular case of unity Lewis number. It is shown that the final stage of the edge-flame collapse is determined by a dominant balance between the time rate of change of the mass fractions (and temperature) and diffusion, giving a self-similar structure in which the size of the edge-flame hole approaches zero, to leading (zeroth) order, as a 1/2-power of time. This solution suggests an expansion of the full model equations in 1/2-powers of time that allows detailed analysis of the effects of side losses and flow distribution in the edge-flame collapse process. It is found that side loss effects are apparent at the first order, whereas convection by the counterflow is first felt during collapse at the second order in the fractional-time expansion. Numerical integrations of the governing equations are found to verify the analytic results

Pressure spectra for vortex models of fine-scale homogeneous turbulence

Pressure spectra at large wave numbers are calculated for Lundgren–Townsend vortex models of the fine scales of homogeneous turbulence. Specific results are given for the Burgers vortex and also for the Lundgren-strained spiral vortex. For the latter case, it is found that the contribution to the shell-summed spectrum produced by the interaction between the axisymmetric and nonaxisymmetric components of the velocity field is proportional to k^–7/3 (k=||k|| is the modulus of the wave number) in agreement with Kolmogorov-type dimensional arguments. Numerical estimates of the dimensionless prefactors for this component are obtained in Kolmogorov scaling variables and comparisons are made with results from the Batchelor–Kolmogorov theory, and with experimental measurement

On the two-dimensional stability of the axisymmetric Burgers vortex

The stability of the axisymmetric Burgers vortex solution of the Navier–Stokes equations to two-dimensional perturbations is studied numerically up to Reynolds numbers, R=Gamma/2pinu, of order 104. No unstable eigenmodes for azimuthal mode numbers n=1,..., 10 are found in this range of Reynolds numbers. Increasing the Reynolds number has a stabilizing effect on the vortex

Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability

We first study the linear stability of an interface between two fluids following the passage of an imploding or exploding shock wave. Assuming incompressible flow between the refracted waves following shock impact, we derive an expression for the asymptotic growth rate for a three-dimensional combination of azimuthal and axial perturbations as a function of the Atwood ratio, the axial and azimuthal wave numbers, the initial radial position and perturbation amplitude of the interface, and the interface velocity gain due to the shock interaction. From the linearized theory, a unified expression for the impulsive asymptotic growth rate in plane, cylindrical, and spherical geometries is obtained which clearly delineates the effects of perturbation growth due to both geometry and baroclinic vorticity deposition. Several different limit cases are investigated, allowing recovery of Mikaelian's purely azimuthal theory and Richtmyer's plane model. We discuss the existence of three-dimensional perturbations with zero growth, typical of curvilinear geometries, as first observed by Mikaelian. The effect of shock proximity on the interface growth rate is studied in the case of a reflected shock. Analytical predictions of the effect of the incident shock strength and the perturbation wave numbers are then compared with results obtained from highly resolved numerical simulations of cylindrical imploding Richtmyer–Meshkov instability for ideal gases. A parallel is made with the instability growth in spherical and plane geometry. In particular, we propose a representation of the perturbation growth by considering the volume of the perturbed layer. This volume is found to grow faster in the plane case than in the imploding cylindrical geometry, among other results

On initial-value and self-similar solutions of the compressible Euler equations

We examine numerically the issue of convergence for initial-value solutions and similarity solutions of the compressible Euler equations in two dimensions in the presence of vortex sheets (slip lines). We consider the problem of a normal shock wave impacting an inclined density discontinuity in the presence of a solid boundary. Two solution techniques are examined: the first solves the Euler equations by a Godunov method as an initial-value problem and the second as a boundary value problem, after invoking self-similarity. Our results indicate nonconvergence of the initial-value calculation at fixed time, with increasing spatial-temporal resolution. The similarity solution appears to converge to the weak 'zero-temperature' solution of the Euler equations in the presence of the slip line. Some speculations on the geometric character of solutions of the initial-value problem are presented