Katsunori Yoshimatsu

Coherent vortices in high resolution direct numerical simulation of homogeneous isotropic turbulence: A wavelet viewpoint

Coherent vortices are extracted from data obtained by direct numerical simulation (DNS) of three-dimensional homogeneous isotropic turbulence performed for different Taylor microscale Reynolds numbers, ranging from Reλ=167 to 732, in order to study their role with respect to the flow intermittency. The wavelet-based extraction method assumes that coherent vortices are what remains after denoising, without requiring any template of their shape. Hypotheses are only made on the noise that, as the simplest guess, is considered to be additive, Gaussian, and white. The vorticity vector field is projected onto an orthogonal wavelet basis, and the coefficients whose moduli are larger than a given threshold are reconstructed in physical space, the threshold value depending on the enstrophy and the resolution of the field, which are both known a priori. The DNS dataset, computed with a dealiased pseudospectral method at resolutions N=2563,5123,10243, and 20483, is analyzed. It shows that, as the Reynolds number inc...

Wavelet-based coherent vorticity sheet and current sheet extraction from three-dimensional homogeneous magnetohydrodynamic turbulence

and the current sheets present in the total fields while retaining only a few percent of the degrees of freedom. The incoherent vorticity and current density are shown to be structureless and of mainly dissipative nature. The spectral distributions of kinetic and magnetic energies of the coherent fields only differ in the dissipative range, while the corresponding incoherent fields exhibit near-equipartition of energy. The probability distribution functions of total and coherent fields for both vorticity and current density coincide almost perfectly, while the incoherent fields have strongly reduced variances. Studying the energy flux confirms that the nonlinear dynamics is fully captured by the coherent fields only.

COHERENT VORTICITY SIMULATION OF THREE-DIMENSIONAL FORCED HOMOGENEOUS ISOTROPIC TURBULENCE *

‡ Abstract. Coherent vorticity simulation (CVS) is a multiscale method to compute incompressible tur- bulent flows based on the wavelet filtered Navier-Stokes equations. At each time step the vorticity field is decomposed into two orthogonal components using an orthogonal wavelet basis: the coherent vorticity, cor- responding to the coefficients whose modulus is larger than a threshold, and the remaining incoherent vorticity. The threshold value only depends on the total enstrophy, which evolves in time, and on the maximal resolution, which remains constant. The induced coherent velocity is computed from the coherent vorticity using the Biot- Savart kernel. To compute the flow evolution one only retains the coherent wavelet coefficients and some of their neighbors in space, scale, and direction, which define the safety zone. Two different strategies are studied to minimize at each time step the number of degrees of freedom to be computed, either by increasing the threshold value or by reducing the width of the safety zone. Their efficiency is compared for a three-dimensional forced homogeneous isotropic turbulent flow at initial Taylor microscale Reynolds number Rλ ¼ 153. The qual- ity of the results is assessed in comparison to a direct numerical simulation of the same flow. It is found that, as long as a safety zone is present, CVS well preserves the statistical predictability of the turbulent flow (even the vorticity and velocity probability distribution functions) with a reduced number of degrees of freedom.

Intermittency and geometrical statistics of three-dimensional homogeneous magnetohydrodynamic turbulence: A wavelet viewpoint

Scale-dependent and geometrical statistics of three-dimensional incompressible homogeneous magnetohydrodynamicturbulence without mean magnetic field are examined by means of the orthogonal wavelet decomposition. The flow is computed by direct numerical simulation with a Fourier spectral method at resolution 5123 and a unit magnetic Prandtl number. Scale-dependent second and higher order statistics of the velocity and magnetic fields allow to quantify their intermittency in terms of spatial fluctuations of the energy spectra, the flatness, and the probability distribution functions at different scales. Different scale-dependent relative helicities, e.g., kinetic, cross, and magnetic relative helicities, yield geometrical information on alignment between the different scale-dependent fields. At each scale, the alignment between the velocity and magnetic field is found to be more pronounced than the other alignments considered here, i.e., the scale-dependent alignment between the velocity and vorticity, the scale-dependent alignment between the magnetic field and its vector potential, and the scale-dependent alignment between the magnetic field and the current density. Finally, statistical scale-dependent analyses of both Eulerian and Lagrangian accelerations and the corresponding time-derivatives of the magnetic field are performed. It is found that the Lagrangian acceleration does not exhibit substantially stronger intermittency compared to the Eulerian acceleration, in contrast to hydrodynamic turbulence where the Lagrangian acceleration shows much stronger intermittency than the Eulerian acceleration. The Eulerian time-derivative of the magnetic field is more intermittent than the Lagrangian time-derivative of the magnetic field.

Surface Waves in a Square Container Due to Resonant Horizontal Oscillations

Resonantly forced water waves in a square container due to its horizontal oscillations are examined. The excited waves are assumed to be gravity waves for infinite depth. Using the reductive perturbation method and including the effect of a linear damping, we derive an evolution equation for the complex amplitudes of two degenerate resonant modes. When the angle θ between the direction of the oscillations and that along one of the sidewalls of the container is 0° or 45°, we obtain planar stationary solutions without the rotation of wave pattern as well as a pair of non-planar ones associated with the clockwise or anti-clockwise rotation. If 0° < θ< 45° , however, no planar stationary solution exists, and the symmetry between these non-planar solutions for θ=0° or 45° is broken. We find the pitchfork bifurcations of the stationary solution for θ=0° and 45°, and also the Hopf and saddle-node bifurcations of this solution for 0° ≤θ≤45°. Furthermore, periodic or chaotic solutions exist within the parameter re...

Symmetry Breaking of Radially Outgoing Flow between Two Parallel Disks

The instability and transition of radially outgoing flow between two parallel disks with two symmetrically attached inlet pipes through which fluid enters are investigated theoretically and experimentally assuming an incompressible and axisymmetric flow field. The flow is steady and symmetric with respect to the center plane between the two disks at small Reynolds numbers, but becomes asymmetric above a critical Reynolds number. Numerical simulations and the linear stability analysis reveal that steady asymmetric flows result from the instability of the steady symmetric flow. The axisymmetry and the transition were confirmed in our experiments with flow visualization.

Coherent vorticity and current density simulation of three-dimensional magnetohydrodynamic turbulence using orthogonal wavelets

A simulation method to track the time evolution of coherent vorticity and current density, called coherent vorticity and current density simulation (CVCS), is developed for three-dimensional (3D) incompressible magnetohydrodynamic (MHD) turbulence. The vorticity and current density fields are, respectively, decomposed at each time step into two orthogonal components, the coherent and incoherent fields, using an orthogonal wavelet representation. Each of the coherent fields is reconstructed from the wavelet coefficients whose modulus is larger than a threshold, while their incoherent counterparts are obtained from the remaining coefficients. The two threshold values depend on the instantaneous kinetic and magnetic enstrophies. The induced coherent velocity and magnetic fields are computed from the coherent vorticity and current density using the Biot–Savart kernel. In order to compute the flow evolution, one should retain not only the coherent wavelet coefficients but also their neighbors in wavelet space, and the set of those additional coefficients is called the safety zone. CVCS is performed for 3D forced incompressible homogeneous MHD turbulence without mean magnetic field for a magnetic Prandtl number equal to unity and with 2563 grid points. The quality of CVCS is assessed by comparing the results with a direct numerical simulation. It is found that CVCS with the safety zone well preserves the statistical predictability of the turbulent flow with a reduced number of degrees of freedom. CVCS is also compared with a Fourier truncated simulation using a spectral cutoff filter where the number of retained Fourier modes is similar to the number of the wavelet coefficients retained by CVCS. It is shown that the wavelet representation is more suitable than the Fourier representation, especially concerning the probability density functions of vorticity and current density.

Primary Patterns in Faraday Surface Waves at High Aspect Ratio

The pattern selection in Faraday surface water waves caused by the instability of the state of no waves is examined under the assumption of high aspect ratio. First, nonlinear evolution equations for the amplitudes of resonant capillary-gravity waves with different directions of wavenumber vector are derived. These equations include cubic nonlinearity and the effects of viscous damping and parametric forcing obtained from the energy equation. Then, using the method of a center manifold, quintic amplitude equations for unstable modes are derived in which cubic damping and forcing are included. From these equations, it is concluded that the squares are stable for sufficiently short waves, the hexagons and the 8-fold quasipatterns are stable when the wavelength is within two intermediate regions, and the stripes are always unstable.

Influence of vortex dynamics and structure on turbulence statistics at large scales

The question whether the vortex dynamics and structure at small scales have significant influence on the statistics at large scales is addressed on the basis of quantitative comparison of two turbulent fields. One is a reference field generated by direct numerical simulation of turbulence of an incompressible fluid obeying the Navier-Stokes (NS) equation in a periodic box. The other is an artificial field in which the coherent vortical structures at small scales (∼η) that could be formed by the NS dynamics in the reference field are destroyed by an artificial computational operation, where η is the Kolmogorov micro-length scale. The comparison of the two fields suggests that the statistics at larger scale (≫η) are not sensitive to the exact vortex dynamics and structure, at least in the case studied here.

Coherent Vortex Simulation: application to 3D homogeneous isotropic turbulence

Self-organization and formation of active regions in fully developed turbulent flow are observed in many physical quantities, such as vorticity or energy dissipation. The regions are not distributed homogeneously and exhibit spatial intermittency. Wavelet analysis is a prominent tool to allow a sparse representation of intermittent fields [1]. Wavelets, well-localized functions both in physical and spectral space, decompose a given flow field into scale-space contributions.