Richard B. Pelz

Reconnection in orthogonally interacting vortex tubes: Direct numerical simulations and quantifications

The three‐dimensional time evolution of two orthogonally offset cylindrical vortices of equal strength is simulated by solving the hyperviscosity‐regularized incompressible Navier–Stokes equations. A Fourier pseudospectral method with a time‐split integration scheme is used for the solution. Four runs with different Reynolds numbers ranging between 690–2100 are performed, each with a resolution of 963 collocation points. The sequence of important physical processes and the evolution of local and global quantities such as vorticity, velocity, and mean‐square strain rate are presented. It is found that the growth rate of the maximum vorticity is at most exponential. The Reynolds number dependence of the time scale of reconnection, the vorticity growth rate, and the time at which the maximum vorticity is attained are examined and differences between the present results and Saffman’s essentially two‐dimensional model predictions are encountered and elucidated. The distributions of the eigenvalues α, β, γ and ...

STRUCTURES AND STRUCTURE FUNCTIONS IN THE INERTIAL RANGE OF TURBULENCE

The deviations from the Kolmogorov 1941 laws of inertial range of turbulence are investigated using the results from the direct numerical simulations of an unforced flow starting from a high-symmetry initial condition by Kida [J. Phys. Soc. Jpn. 54, 2132 (1985)]. The resolution is 3003 points (12003 with symmetries, maximum wavenumber 400 after dealiasing), and the Taylor scale Reynolds number is in the order of 100. The scaling exponents of the pth order longitudinal and lateral structure function (for p between 2 and 16) are computed using different methods with particular focus on a recent method by Benzi and collaborators [Phys. Rev. E 48, R29 (1993); Europhys. Lett. 32, 709 (1995)]. Both longitudinal and lateral scaling exponents deviate considerably from Kolmogorov 1941 (K-41) scaling laws, the lateral deviating much more than the longitudinal. A systematic methodology (strain–enstrophy state) is developed to relate the K-41 deviations to different structures in the field. Enstrophy-dominated struct...

Direct numerical simulation of transition to turbulence from a high‐symmetry initial condition

The three‐dimensional (3‐D) time evolution of a high‐symmetry initial condition [J. Phys. Soc. Jpn. 54, 2132 (1985)] is simulated using a Fourier pseudospectral method for Re=1/ν=500, 1000, 2000, and 5000 with an effective resolution of 10243 collocation points (1713 independent modes, maximum wave number kmax=340). It is found that much before the peak enstrophy is reached, there is a short interval when the local quantities increase sharply. It is also found that during this interval, six vortex dipoles (at the origin) and three dipoles (at the π/2 corner) collapse toward two separate vorticity null points at the opposite corners of the domain in a nearly self‐similar fashion. The coherent vortices break up afterward, followed by a sharp decrease in local quantities. The singularity analysis shows that, within the limits of the resolution, the maximum vorticity scales approximately as (T−Tc)−1, shortly before the breakup. However, the increase in peak vorticity stops at a certain time, possibly due to v...

Symmetry and the hydrodynamic blow-up problem

The problem of whether a spontaneous singularity can occur in finite time in an incompressible inviscid fluid flow is addressed. As suggested by previous numerical simulations, candidate flows are restricted to be invariant under the octahedral group of symmetries and to have a compact vortex tube in the fundamental domain

The parallel Fourier pseudospectral method

Abstract Parallel algorithms of the Fourier pseudospectral method are presented for the solution of the unsteady, incompressible Navier-Stokes equations. The only major operation that requires parallelization is the multidimensional FFT. In tests performed on a 1024-node hypercube computer, an efficiency of about 83 % is obtained for a three-dimensional problem with mesh size 128 3 . The all-FORTRAN code requires 17 s per timestep, rivalling rates obtained from optimized codes on current supercomputers.

On the local topology evolution of a high‐symmetry flow

The local topology evolution of a high‐symmetry, high resolution (effective maximum resolution of 10243 grid points, maximum wave number of 341) incompressible flow simulation having a Reynolds number (=1/ν) of 1000 is investigated. The Q–R invariants of the velocity gradient tensor Aij, the enstrophy, ΩijΩij and the mean‐square strain rate SijSij are computed at an interval when the local maximum vorticity increases drastically. All the analysis of the computations are done on the z=0 plane, where the maximum vorticity and strain are located during the evolution. In the Q–R plane, most of the collocation points evolve towards the lower right corner, a region where strain dominates over vorticity. The pressure Hessian tensor components are computed in the 0 planes. Points with very large strain and no vorticity, which are located along the boundaries separating oppositely signed vortices, are found to have a diagonal pressure Hessian tensor. It is discussed how such a Hessian tensor form can result in a s...

Coupling between anomalous velocity and passive scalar increments in turbulence

Given (and confirmed numerically) that the exponents α and β in the passive scalar and velocity structure functions 〈(ΔT3)a〉∼rα, and 〈(Δu3)a〉∼rβ are anomalous, the scaling of γ in 〈(u(x+r)−u(x))a(T(x+r)−T(x))2a〉∼rγ is investigated. Analytical estimates show that γ cannot be as anomalous as α or β. Numerical computations (Pr=1, Reλ=141) show that γ is closer to β than to α. In addition, the statistical dependence of the velocity and passive scalar differences leads to an enhanced anomaly in γ.

Extended series analysis of full octahedral flow: numerical evidence for hydrodynamic blowup

The power series in time of a full-octahedral flow, a candidate for finite-time blowup of the incompressible Euler equations, is analyzed. Sixty-five Taylor coefficients are found with a precision of 512 bits per Fourier coefficient (154 digits). This work is an extension of Pelz and Gulak (Phys. Rev. Lett. 79 (1997) 1998), where 32 terms were found with half the precision. The solution is found in the form of a power series in time, Fourier series in space with the equation of motion written as a quadratic operation on the Fourier–Taylor coefficients of a single component of vorticity. Particular attention to precision and roundoff is given. In agreement with previous work, Pade approximants of enstrophy series consistently show an isolated, simple pole is located at a real time of about two. Similar findings exist for the series representation of vorticity derivatives at the origin, the purported blowup point. Despite the lack of convergence proofs for Pade resummation, consistent evidence typically yields a valid continuation. Higher-order Sobolev norms of vorticity, however, do not show a similar behavior. The results are also difficult to explain in terms of similarity scalings.

Locally isotropic pressure Hessian in a high‐symmetry flow

Regions in a high‐symmetry flow [J. Phys. Soc. Jpn. 54, 2132 (1985); Phys. Fluids 6, 2757 (1994)] where the pressure Hessian term remains diagonal for all times are identified. It is found that vorticity also vanishes in such regions. The components of the pressure Hessian tensor are equal only at the origin, toward which the vortices approach. For the case of isotropic pressure Hessian with no vorticity, and in the absence of viscosity, the resulting equations for the evolution of the eigenvalues and the Q invariant of the strain rate tensor are integrated numerically and analytically respectively. It is found that all the eigenvalues and the Q invariant diverge to infinity in finite time. In a recent simulation of a high‐symmetry flow [Phys. Fluids 6, 2757 (1994)], it is found that the two (positive) eigenvalues of the strain rate tensor become equal as all the eigenvalues and the Q invariant show trends of divergence and as the vortices approach to the origin.

Numerical investigation of helicity in turbulent flow

Abstract Results of direct numerical simulations of decaying, nearly isotropic turbulence are presented. The angular orientation between vorticity and velocity evolves from a state that is initially random to one in which there is a higher probability of vector alignment. Total helicity evolves in a quantitatively different manner than the energy or ensotrophy, reflecting changes of flow topology. From a random, initially zero-helicity field it is seen that viscosity can be a source of spontaneous helicity generation and reflexional symmetry breaking.