Robert M. Kerr

Evidence for a Singularity of the Three Dimensional, Incompressible Euler Equations

Three‐dimensional, incompressible Euler calculations of the interaction of perturbed antiparallel vortex tubes using smooth initial profiles in a bounded domain with bounded initial vorticity are discussed. A numerical method that uses symmetries and additional resolution in the direction and location of maximum compression is used to simulate periodic boundary conditions in all directions. For an initial condition that yields singular behavior, the growth of the peak vorticity, the peak axial strain, and the enstrophy production obey (tc−t)−1, and the enstrophy grows logarithmically. The enstrophy growth is associated with the energy spectrum approaching k−3. Self‐similar development and equal rates of collapse in all three directions are shown.

Rayleigh number scaling in numerical convection

Using direct simulations of the incompressible Navier-Stokes equations with rigid upper and lower boundaries at fixed temperature and periodic sidewalls, scaling with respect to Rayleigh number is determined. At large aspect ratio (6:6:1) on meshes up to 288 × 288 × 96, a single scaling regime consistent with the properties of ‘hard’ convective turbulence is found for Pr = 0.7 between Ra = 5 × 10 4 and Ra = 2 × 10 7 . The properties of this regime include Nu ∼ Ra β T with β T = 0.28 ≈ 2/7, exponential temperature distributions in the centre of the cell, and velocity and temperature scales consistent with experimental measurements. Two velocity boundary-layer thicknesses are identified, one outside the thermal boundary layer that scales as Ra −1/7 and the other within it that scales as Ra −3/7 . Large-scale shears are not observed; instead, strong local boundary-layer shears are observed in regions between incoming plumes and an outgoing network of buoyant sheets. At the highest Rayleigh number, there is a decade where the energy spectra are close to k −5/3 and temperature variance spectra are noticeably less steep. It is argued that taken together this is good evidence for ‘hard’ turbulence, even if individually each of these properties might have alternative explanations.

Velocity, scalar and transfer spectra in numerical turbulence

Velocity and passive-scalar spectra for turbulent fields generated by a forced three-dimensional simulation with 1283 grid points and Taylor-microscale Reynolds numbers up to 83 are shown to have convective and diffusive spectral regimes. One-and three-dimensional spectra are compared with experiment and theory. If normalized by the Kolmogorov dissipation scales and scalar dissipation, velocity spectra and scalar spectra for given Prandtl numbers collapse to single curves in the dissipation regime with exponential tails. If multiplied by k⅗ the velocity spectra show an anomalously high Kolmogorov constant that is consistent with low Reynolds number experiments. When normalized by the Batchelor scales, the scalar spectra show a universal dissipation regime that is independent of Prandtl numbers from 0.1 to 1.0. The time development of velocity spectra is illustrated by energy-transfer spectra in which distinct pulses propagate to high wavenumbers.

Simulation of vortex reconnection

Abstract The reconnection of two antiparallel viscous vortices is simulated in a periodic domain. Reynolds numbers based on circulation divided by viscosity range from 1600 to 3200. Symmetries are used along with increased resolution in the direction normal to the dividing plane to reduce the computation requirements. As the Reynolds number is increased a significant increase in enstrophy is not seen, but there is a significant increase in the peak vorticity that is consistent with a singularity of the three-dimensional, incompressible Euler equations in a finite time. The reconnection region is characterized by vortex sheets and large relative helicity in the outer regions of the reconnection. A power law regime is found in energy spectra.

Deterministic forcing of homogeneous, isotropic turbulence

A deterministic low‐wave‐number forcing scheme designed to obtain statistically stationary homogeneous, isotropic turbulence in incompressible flows, and address criticisms of earlier schemes is proposed. Three‐dimensional turbulent kinetic energy spectra collapse well and are more consistent with the experimentally determined Kolmogorov coefficient. Spectra for unforced scalar fields at different Prandtl numbers advected by the forced velocity fields collapse under Batchelor scaling, and do not show as strong a low‐wave‐number anomaly as earlier simulations that use forcing on both the velocity and scalar fields.

Small‐scale properties of nonlinear interactions and subgrid‐scale energy transfer in isotropic turbulence

Using results of direct numerical simulations of isotropic turbulence the subgrid‐scale energy transfer in the physical space is calculated exactly employing a spectral decomposition of the velocity field into large (resolved) and small (unresolved) scales. Comparisons with large‐scale quantities reveal large qualitative correlations between regions of subgrid transfer and the boundaries of regions of large vorticity production. This suggests a novel analysis of the nonlinear term, where it is decomposed into four components determined by four combinations of the resolved and unresolved velocity and vorticity fields. It is found that there is a 90% vector‐correlation between the subgrid transfer and the component of the full transfer associated with the resolved velocity and unresolved vorticity, but that 90% of the total subgrid energy production is determined by the component associated with the unresolved velocity and resolved vorticity. These results suggest subgrid‐scale models that have higher corre...

Prandtl number dependence of Nusselt number in direct numerical simulations

The dependence of the Nusselt number Nu on the Rayleigh Ra and Prandtl Pr number is determined for 10 4 Ra 7 and 0.07 Pr < 7 using DNS with no-slip upper and lower boundaries and free-slip sidewalls in a 8 × 8 × 2 box. Nusselt numbers, velocity scales and boundary layer thicknesses are calculated. For Nu there are good comparisons with experimental data and scaling laws for all the cases, including Ra 2/7 laws at Pr = 0.7 and Pr = 7 and at low Pr , a Ra 1/4 regime. Calculations at Pr = 0.3 predict a new Nu ∼ Ra 2/7 regime at slightly higher Ra than the Pr = 0.07 calculations reported here and the mercury Pr = 0.025 experiments.

Non‐Gaussian statistics in isotropic turbulence

Several measures of non‐Gaussian behavior in simulations of decaying isotropic turbulence are compared with predictions of the direct‐interaction approximation (DIA) at an initial Rλ≈35. The quantities studied include the variances and wavenumber power spectra of (a) the total nonlinear term in the Navier–Stokes equation, (b) the time derivative of the velocity at a point, (c) pressure fluctuations, and (d) vorticity and dissipation fluctuations. The direct‐interaction approximation gives a good quantitative prediction of the variance of the time derivative and the variance of total nonlinear term, and a fair qualitative prediction of the power spectrum associated with the latter. But DIA totally fails to capture the non‐Gaussian statistics associated with pressure fluctuation and vorticity spottiness. Some discussion is given of demands that vorticity and dissipation statistics place upon theories of tubulence at moderate and high Reynolds numbers.

Development of enstrophy and spectra in numerical turbulence

Decaying isotropic turbulence with initial Taylor microscale Reynolds number (Rλ≤258) is studied via direct numerical simulations (DNS), with spectral resolution ≤2563. DNS results are compared with two‐point closure, in the form of the direct interaction approximation (DIA) and the test field model (TFM). The goals of this study are to understand the time‐dependence of enstrophy and spectra as they evolve from random initial conditions, and to assess and interpret differences between DNS and closure. Two time scales are identified in the DNS. The first is that for the development of normalized enstrophy production (velocity derivative skewness) and is independent of Rλ. The second is that for the saturation of the enstrophy which follows after a longer period of near exponential growth and is strongly Rλ dependent. For ν≠0, the DIA represents the time development of both integral quantities, such as enstrophy and spectra, with surprising accuracy in spite of its lack of invariance to random large‐scale s...

3D Euler about a 2D symmetry plane

Initial results from new calculations of interacting anti-parallel Euler vortices are presented. The objective is to understand the origins of singular scaling presented by Kerr (1993) with different core profiles, develop more robust analysis for identifying singular behaviour and to develop criteria for when calculations should be terminated. If a localised three-dimensional perturbation is added to an analytic vortex core, then smoothed with a symmetric hyperviscous filter (the first two steps of the initialisation of Kerr (1993)), then an anomalous region of negative vorticity forms in the lee of the primary vortex as in Fig. 1. This is the primary difference between between the initial condition of Kerr (1993) and Hou and Li (2006). At late times the anomalous vorticity becomes sandwiched between the two primary vortices, requiring extra resolution. The additional Chebyshev mapping of Kerr (1993) removes this anomaly, which we have reproduced with a purely Fourier expansion by adding positive vorticity as a function of z as in Fig 2. Results from a similar initial condition with additional initial stretching of the shape of the vortex cores in x are given. The tools for identifying singular versus regular behaviour have been reexamined as well as the criteria for terminating the calculations. Kerr (1993) assumed the simplest power law (or similar) growth of peak vorticity, enstrophy and enstrophy production. A new approach is to: a) Infer a singular time tc for a quantity and its directly determined time derivative, b) Use this to estimate the power-law behaviour as t → tc and c) Find the pre-factor. If the estimated singular time tc increases as the calculation proceeds, then regular behaviour is favoured. Fig. 3 shows results using the global enstrophy Ω and enstrophy production Ωp = Ω from the data used for Kerr (1993). A new conclusion si that Ω ∼ (tc − t)−γΩ as t→ tc with γΩ ≈ 0.5 rather than the logarithmic growth suggested by Kerr (1993). New results are consistent, but do not go as far in time. It is found that numerical depletion of circulation in the symmetry plane is the best indication of when the calculations become underresolved. Current results imply that such simple behaviour either does not occur or occurs at such late times that it would be unreachable using Fourier methods. Figure 1: ωy in the symmetry plane from a test initial condition with only step 1) and 2) (a high-wavenumber filter). Note the large negative vorticity in the lee (right) of the primary vortex as in HouLi (2006) (Fig. 2) and how this is entrained underneath the primary vortex.