Takeshi Watanabe

Statistics of a passive scalar in homogeneous turbulence

Statistics of a passive scalar with Sc=1 transported by steady homogeneous turbulence at Rλ=427 and Pλ=427 is studied by using high-resolution direct numerical simulation. The Obukhov–Corrsin constant of the three-dimensional scalar spectrum in the inertial-convective range is found to be 0.68±0.04. It is proved that the -law for the scalar-velocity triple correlation holds in both inertial-convective and viscous-convective ranges when Sc>1, and found that the -law is approached with increase in Peclet number. Structure functions of the passive scalar increment and their local scaling exponents are computed as functions of the separation distance, and it is found that there exist two scaling ranges: the inertial-convective range and a narrow precursory range to the viscous-convective range. The scaling exponents in the inertial-convective range are found to be smaller than those of the velocity field and do not saturate, whereas they saturate at about 1.5 in the short precursory range to the viscous-convective range. It is also found that, contrary to the scalar case, the mixed scalar velocity structure function has a well-developed single scaling range. The scalar and scalar dissipation fields are visualized and compared with the kinetic energy dissipation field. The scalar field has a particular shape with a large-scale plateau, sharp cliff and deep valley, a mesa-canyon structure.

Inertial-range intermittency and accuracy of direct numerical simulation for turbulence and passive scalar turbulence

We examine the effects of the variation in dissipation-range resolution on the accuracy of inertial-range statistics and intermittency in terms of the direct numerical simulations of homogeneous turbulence and passive-scalar turbulence by changing the spatial resolution up to 2048 3 grid points while maintaining a constant Reynolds number at R λ ≃ 180 or ≃ 420 and Schmidt number at Sc = 1. Although large fluctuations of the derivative fields depended strongly on K max η and were underestimated when K max η ≃1, where K max is the maximum wavenumber in the computations and η is the mean Kolmogorov length, the behaviour of the spectra and the scaling exponents of the structure functions up to the eighth order in the range of scales greater than 10 η was insensitive to variations in K max η , even when K max η ≃1. The relationship between the spatial resolution and asymptotic tail of the probability density functions of the energy dissipation fields was studied using the multifractal model for dissipation, and the results were confirmed by comparison to the simulation data. Degradation of the statistics arises from modifications to the flow dynamics due to the finite wavenumber cutoff and the use of a coarser filter width for the data, which is obtained using a reasonable accuracy criterion for the flow dynamics. The effect of the former was less than that of the latter for the low-to-moderate-order statistics when K max η ≥1. We also discuss the universality of the inertial-range statistics with respect to variations in the dissipation-range characteristics.

Universality and anisotropy in passive scalar fluctuations in turbulence with uniform mean gradient

Passive scalar and velocity fluctuations in homogeneous isotropic turbulence with or without mean scalar gradient are studied by using high-resolution direct numerical simulation (DNS). The local scaling exponents of the velocity structure functions are newly computed at larger Reynolds number, and the values at their plateau in the inertial range are found to be consistent with the previous DNS and experimental data. The 4/3 law for the velocity–scalar mixed correlation and the high-order structure functions of the passive scalar increment are analyzed in terms of the Legendre polynomial expansion. It is shown theoretically that the contributions from the second order in the expansion can be removed by taking the average over three directions parallel and perpendicular to the mean gradient, and it is found numerically that the contributions higher than the fourth order are negligible. The scaling exponents of the isotropic part of the scalar structure functions under the mean gradient are found to have the well-developed scaling range and are smaller than those in the isotropic case. The different behavior in the local scaling exponents with or without the mean gradient is examined and the universality of the scaling exponents is discussed.

Statistics of transfer fluxes of the kinetic energy and scalar variance

Statistics of transfer fluxes of kinetic energy and scalar variance are studied theoretically and numerically. The degree of localness in wavenumber space for the mean transfer fluxes is computed as a function of scale disparity parameter using a Lagrangian spectral theory and is compared to direct numerical simulation data at high Reynolds number. It is found that although most of the transfer flux is local, due to the interactions among wavenumber components of nearly equal size, the degree of localness in the scalar transfer flux is weaker than that of the energy, especially for interactions with large scale disparity. The difference is due to the absence of a pressure term in the scalar equation. High order statistics of the transfer fluxes are also studied in terms of structure functions and probability density functions in physical space. The scaling exponents of the scalar transfer flux are smaller than those of the kinetic energy flux. The probability density functions near the peak are examined and it is found that scale similarity holds approximately. Implications for subgrid scale modeling are discussed.

Intermittency in passive scalar turbulence under the uniform mean scalar gradient

Small scale statistics of a passive scalar convected by turbulence under the uniform mean scalar gradient is studied by high-resolution direct numerical simulation. It is found that the scaling exponents of the structure functions of scalar increments in parallel and perpendicular directions to the mean gradient are the same and saturate approximately 1.3 at large order, and that they are dependent on scalar injection scheme at large scales within the Reynolds numbers studied. Tails of the probability density functions for the scalar increment in the inertial convective range are well fitted by a scaling form inferred from the saturation and the tail of the one point scalar probability density function.

An 'ideal' form of decaying two-dimensional turbulence

In decaying two-dimensional Navier–Stokes turbulence, Batchelors similarity hypothesis fails due to the existence of coherent vortices. However, it is shown that decaying two-dimensional turbulence governed by the Charney–Hasegawa–Mima (CHM) equation (∂/∂ t )(∇ 2 φ−λ 2 φ)n + J (φ, ∇ 2 φ) = D , where D is a damping, is described well by Batchelors similarity hypothesis for wave numbers k [Lt ] λ (the so-called AM regime). It is argued that CHM turbulence in the AM regime is a more ‘ideal’ form of two-dimensional turbulence than is Navier–Stokes turbulence itself.

Scalar flux spectrum in isotropic steady turbulence with a uniform mean gradient

The scaling law of a scalar flux spectrum (velocity-scalar cospectrum) in the inertial convective range of passive scalar turbulence under a uniform mean scalar gradient is examined using direct numerical simulation with a resolution of up to 20483 grid points. When the Reynolds number Reλ is increased up to Reλ=585, the scalar flux spectrum tends to obey the power law k−7∕3, as predicted by Lumley [J. Atmos. Sci. 21, 99 (1964); Phys. Fluids 10, 855 (1967)], with a nondimensional constant of Cuθ=1.50±0.08 at Reλ=585. The Reλ effect on the scaling of the scalar flux spectrum is well compensated using the mean molecular destruction of the scalar flux ϵ¯uθ. The Reλ dependence of Cuθ is also compared with the results of previous studies, and its asymptotic state at an infinite Reynolds number is discussed.

Power and nonpower laws of passive scalar moments convected by isotropic turbulence.

The scaling behavior of the moments of two passive scalars that are excited by two different methods and simultaneously convected by the same isotropic steady turbulence at R_{λ}=805 and Sc=0.72 is studied by using direct numerical simulation with N=4096^{3} grid points. The passive scalar θ is excited by a random source that is Gaussian and white in time, and the passive scalar q is excited by the mean uniform scalar gradient. In the inertial convective range, the nth-order moments of the scalar increment δθ(r) do not obey a simple power law, but have the local scaling exponents ξ_{n}^{θ}+β_{n}log(r/r_{*}) with β_{n}>0. In contrast, the local scaling exponents of q have well-developed plateaus and saturate with increasing order. The power law of passive scalar moments is not trivial. The universality of passive scalars is found not in the moments, but in the normalized moments.

Defects and Photonic Wells in One-Dimensional Photonic Lattices

The concept of local photonic lattice (PL) is introduced in 1D photonic lattices (PLs) to interpret the defects as donor-like or acceptor-like within any band gap of the host PL. Based on the concept of the local PL, an analysis is presented on the tunability of a movable defect which would be used as a tunable frequency filter. By further extending the local PL, we introduce a 1D photonic well composed of the well-like 1D PL sandwiched by the barrier-like 1D PLs. Similarity is investigated between the 1D photonic wells and the semiconductor quantum wells (QWs). It is shown that the bound states are well described by the effective “mass” approximation in analogy to the QWs. These bound states appear as sharp peaks in the transmission spectrum of the finite size 1D photonic well.

Power-law spectra formed by stretching polymers in decaying isotropic turbulence

The spectral dynamics of isotropic decaying turbulence with polymer additives is numerically investigated using a hybrid Eulerian-Lagrangian approach with making use of large-scale parallel computation. We found that the kinetic energy and pressure variance spectra obeyed the power law E(k) ∼ k −α and E p (k) ∼ k −β in the scale range below the Kolmogorov length l K when the turbulence sufficiently decayed while the Weissenberg number W i remained greater than unity. The exponents α and β were found to be α = 4.1 − 4.6 and β = 2.8 − 3.2, respectively, and were found to decrease with increasing W i . We discuss the similarities and differences between the present results and the results of previous experimental and numerical studies for elastic turbulence, which is characterized by W i ≫ 1 and a Reynolds number below unity.