Direct Interaction Approximation Research Articles

Modal time correlation functions for homogeneous isotropic turbulence in a projection operator method

Applying a projection operator method to homogeneous isotropic turbulence in the Lagrangian and Eulerian specifications of the flow field, we obtain a closure equation, which is called the similarity approximation (SA) equation, for the time correlation function. We compare its solutions with the time correlation function obtained from direct numerical simulation in the Eulerian specification and with that obtained from direct interaction approximation in the Lagrangian specification. Results of the comparison show that the SA equation can express time correlation for homogeneous isotropic turbulence in the Lagrangian and Eulerian specifications.

Universality of Modal Time Correlation Functions in Medium Scale

We investigate the universality of modal time correlation functions using a closure equation for the normalized dimensionless time correlation function. As a candidate for a new universal function for turbulence, we propose a solution to the closure equation in the case of the critical value of the wavenumber, at which the decay form of the time correlation function changes from exponential to oscillatory exponential. The solution is compared with the normalized dimensionless time correlation functions obtained from numerical results for one-dimensional turbulence, such as the Kuramoto–Sivashinsky equation and that obtained from the direct interaction approximation for three-dimensional Navier–Stokes turbulence. As a result of the comparison, we provide evidence to show that the normalized dimensionless time correlation function is universal in the case of the critical value of the wavenumber.

Improvement of the constant temperature anemometer and measurement of energy spectra in a turbulent jet

This paper presents a rearranged electrical circuit for a constant temperature anemometer (CTA), along with details of calibration results and velocity measurement results for a plane jet and square jet that were obtained by using the rearranged CTA. In this rearranged CTA, the ratio of the electrical resistance of the Wheatstone bridge was set at one, and the feedback circuit used two operational amplifiers whose gain-bandwidth product and slew rate were 110MHz and 20V/μs, respectively. The results of a frequency response test showed that the roll-off frequency of the rearranged CTA was 20kHz for a 5-μm hot-wire and 40kHz for a 3-μm hot-wire, given a free stream flow velocity of 20m/s; those of a basic CTA were 5kHz and 6kHz, respectively. It was also found that the energy spectra measured by the rearranged CTA yielded a power spectrum that agreed with the profile derived numerically from the Lagrangian direct-interaction approximation (LDIA) theory in the range of non-dimensional wave numbers smaller than 0.5. These results indicate that the rearranged CTA could be used for more accurate investigations of fine-scale structures in turbulent flows. Further, the profile obtained by the LDIA theory was valid and useful for discussing the turbulent flows.

On the strength of the nonlinearity in isotropic turbulence

AbstractTurbulence governed by the Navier–Stokes equations shows a tendency to evolve towards a state in which the nonlinearity is diminished. In fully developed turbulence, this tendency can be measured by comparing the variance of the nonlinear term to the variance of the same quantity measured in a Gaussian field with the same energy distribution. In order to study this phenomenon at high Reynolds numbers, a version of the direct interaction approximation is used to obtain a closed expression for the statistical average of the mean-square nonlinearity. The wavenumber spectrum of the mean-square nonlinear term is evaluated and its scaling in the inertial range is investigated as a function of the Reynolds number. Its scaling is dominated by the sweeping by the energetic scales, but this sweeping is weaker than predicted by a random sweeping estimate. At inertial range scales, the depletion of nonlinearity as a function of the wavenumber is observed to be constant. At large scales it is observed that the mean-square nonlinearity is larger than its Gaussian estimate, which is shown to be related to the non-Gaussianity of the Reynolds-stress fluctuations at these scales.

Improved Estimation of Protein-Ligand Binding Free Energy by Using the Ligand-Entropy and Mobility of Water Molecules

We previously developed the direct interaction approximation (DIA) method to estimate the protein-ligand binding free energy (ΔG). The DIA method estimates the ΔG value based on the direct van der Waals and electrostatic interaction energies between the protein and the ligand. In the current study, the effect of the entropy of the ligand was introduced with protein dynamic properties by molecular dynamics simulations, and the interaction between each residue of the protein and the ligand was also weighted considering the hydration of each residue. The molecular dynamics simulation of the apo target protein gave the hydration effect of each residue, under the assumption that the residues, which strongly bind the water molecules, are important in the protein-ligand binding. These two effects improved the reliability of the DIA method. In fact, the parameters used in the DIA became independent of the target protein. The averaged error of ΔG estimation was 1.3 kcal/mol and the correlation coefficient between the experimental ΔG value and the calculated ΔG value was 0.75.

Reassessment of the classical closures for scalar turbulence

In deducing the consequences of the Direct Interaction Approximation, Kraichnan was sometimes led to consider the properties of special classes of nonlinear interactions in degenerate triads in which one wavevector is very small. Such interactions can be described by simplified models closely related to elementary closures for homogeneous isotropic turbulence such as the Heisenberg and Leith models. These connections can be exploited to derive considerably improved versions of the Heisenberg and Leith models that are only slightly more complicated analytically. This paper applies this approach to derive some new simplified closure models for passive scalar advection and investigates the consistency of these models with fundamental properties of scalar turbulence. Whereas some properties, such as the existence of the Kolmogorov–Obukhov range and the existence of thermal equilibrium ensembles, follow the velocity case closely, phenomena special to the scalar case arise when the diffusive and viscous effects become important at different scales of motion. These include the Batchelor and Batchelor–Howells–Townsend ranges pertaining, respectively, to high and low molecular Schmidt number. We also consider the spectrum in the diffusive range that follows the Batchelor range. We conclude that improved elementary models can be made consistent with many nontrivial properties of scalar turbulence, but that such models have unavoidable limitations.

Lagrangian Markovianized field approximation for turbulence

In a previous communication (W.J.T. Bos and J.-P. Bertoglio 2006, Phys. Fluids, 18, 031706), a self-consistent Markovian triadic closure was presented. The detailed derivation of this closure is given here, relating it to the Direct Interaction Approximation and Quasi-Normal types of closure. The time-scale needed to obtain a self-consistent closure for both the energy spectrum and the scalar variance spectrum is determined by evaluating the correlation between the velocity and an advected displacement vector-field. The relation between this latter correlation and the velocity–scalar correlation is stressed, suggesting a simplified model of the latter. The resulting closed equations are numerically integrated and results for the energy spectrum, scalar fluctuation spectrum and velocity–displacement correlation spectrum are presented for low, unity and high values of the Schmidt number.

Reduction of mean-square advection in turbulent passive scalar mixing

Direct numerical simulation data show that the variance of the coupling term in passive scalar advection by a random velocity field is smaller than it would be if the velocity and scalar fields were statistically independent. This effect is analogous to the “depression of nonlinearity” in hydrodynamic turbulence. We show that the trends observed in the numerical data are qualitatively consistent with the predictions of closure theories related to Kraichnan's direct interaction approximation. The phenomenon is demonstrated over a range of Prandtl numbers. In the inertial-convective range the depletion is approximately constant with respect to wavenumber. The effect is weaker in the Batchelor range.

Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence

Statistical dynamical closures for inhomogeneous turbulence described by multi‑field equations are derived based on renormalized perturbation theory. Generalizations of the computationally tractable quasi-diagonal direct interaction approximation for inhomogeneous barotropic turbulent flows over topography are developed. Statistical closures are also formulated for large eddy simulations including subgrid models that ensure the same large scale statistical behavior as higher resolution closures. The focus is on baroclinic quasigeostrophic and three-dimensional inhomogeneous turbulence although the framework is generally applicable to classical field theories with quadratic nonlinearity.

Homogeneous turbulence: an introductory review

A brief review of developments in the theory of homogeneous turbulence over the last 50 years is given, many of these developments stemming from lectures and discussions at the 1961 Marseille Colloquium Mécanique de la Turbulence. The following topics are discussed: Kolmogorov’s 1961 lecture, intermittency and the finite-time singularity problem, the skewness factor paradox, Kraichnan’s direct interaction approximation, helicity and the dynamo problem, two-dimensional turbulence, rapid distortion theory and the passive and active scalar problems.

An Approach to Glassy Systems with Direct-Interaction Approximation

1Department of Mechanical and Aerospace Engineering, Tottori University, Tottori 680-8552, Japan 2Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan 3Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan 4Laboratory of Physics, College of Science and Technology, Nihon University, Funabashi 274-8501, Japan 5Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara 229-8558, Japan

A statistical theory for inhomogeneous turbulent flow formulated in the mean-Lagrangian coordinates

An analytical way for investigating the inhomogeneous turbulent flow has been proposed by introducing Lagrangian-like view. The present method is developed based on TSDIA (Two-Scale Direct Interaction Approximation) which is one of the few theoretical works for the inhomogeneous turbulent flow formulated in the Eulerian frame work and has fundamental inconsistency with the general covariance under the coordinate transformation. The present work reached certain success in the following two aspects. First, the algebraic representation for the Reynolds stress derived by the present method has been proved to be generally covariant. Second the algebraic Reynolds stress is in much better agreement with the result of a DNS of the channel flow.

Wave turbulent statistics in non-weak wave turbulence

In wave turbulence, which is made by nonlinear interactions among waves, it has been believed that statistical properties are well described by the weak turbulence theory, where separation of linear and nonlinear time scales derived from weak nonlinearity is assumed. However, the separation of the time scales is often violated. To get rid of this inconsistency, closed equations are derived in wave turbulence without assuming the weak nonlinearity according to Direct-Interaction Approximation (DIA), which has been successful in Navier–Stokes turbulence. The DIA equations is a natural extension of the conventional kinetic equation to not-necessarily-weak wave turbulence.

Application of statistical dynamical turbulence closures to data assimilation

We describe the development of an accurate yet computationally tractable statistical dynamical closure theory for general inhomogeneous turbulent flows, coined the quasi-diagonal direct interaction approximation closure (QDIA), and its application to problems in data assimilation. The QDIA provides prognostic equations for evolving mean fields, covariances and higher-order non-Gaussian terms, all of which are also required in the formulation of data assimilation schemes for nonlinear geophysical flows. The QDIA is a generalization of the class of direct interaction approximation theories, initially developed by Kraichnan (1959 J. Fluid Mech. 5 497) for isotropic turbulence, to fully inhomogeneous flows and has been further generalized to allow for both inhomogeneous and non-Gaussian initial conditions and long integrations. A regularization procedure or empirical vertex renormalization that ensures correct inertial range spectra is also described. The aim of this paper is to provide a coherent mathematical description of the QDIA turbulence closure and closure-based data assimilation scheme we have labeled the statistical dynamical Kalman filter. The mathematical formalism presented has been synthesized from recent works of the authors with some additional material and is presented in sufficient detail that the paper is of a pedagogical nature.

Theory of particle diffusion in electrostatic turbulent plasma using extended direct-interaction approximation

AbstractThe theory of particle diffusion in an electrostatic turbulent plasma is formulated by applying the direct-interaction approximation (DIA) to subensemble-averaged functions instead of conventional ensemble-averaged ones. This theory approximately incorporates the Lagrangian description into the DIA through decorrelation trajectories. The running diffusion coefficient is shown to be calculated by solving a nonlinear ordinary differential equation together with an equation for decorrelation trajectories and by averaging initial conditions at the starting point of trajectories.

Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter

We investigate the scale-locality of subgrid-scale (SGS) energy flux and interband energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments, and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial range. Interband energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of “local transfer by nonlocal triads,” with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all individual wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, nonlocal triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the −4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan’s abridged Lagrangian-history direct-interaction approximation and test-field model closures. We support our results with numerical data from a 5123 pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We also discuss a rigorous counterexample of Eyink [Physica D 78, 222 (1994)], which showed that nonlocal wavenumber triads may dominate in the sharp spectral flux (but not in the SGS energy flux for graded filters). We show that this mathematical counterexample fails to satisfy reasonable physical requirements for a turbulent velocity field, which are employed in our proof of scale locality. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation modeling of turbulent flows.

Time correlation functions in a similarity approximation for one-dimensional turbulence

The projection operator formalism yields a time evolution equation for the time correlation function Un(t) of the chaotic modes of interest in terms of the memory function Gamman(t). On the assumption of similarity between Un(t) and Gamman(t), this equation leads to a closed equation for Un(t), which yields the asymptotic behavior of the time correlation function Un(t) and the corresponding power spectrum In(omega) analytically. Thus it turns out that the time correlation function takes the algebraic form 1/(1+t2) for t-->0 as predicted previously, and can be classified into three decay forms for t-->infinity according to the wave number kn: the exponential decay e(-t), the oscillatory exponential decay e(-t) cos t, and the oscillatory power-law decay t(-3/2) cos t. All the corresponding power spectra form a dual structure which is Lorentzian as omega-->0 and decays exponentially as omega-->infinity. In the entire domain 0<or=t<infinity, solutions to the closed equation are quite consistent with the numerical results for small kn, while they are consistent with those for large kn, except for the phase. In the case that the integral time scale of Un(t) is equal to that of Gamman(t), the closed equation is identical to the direct interaction approximation equation for fluid turbulence in the limit kn-->infinity.

Reassessment of the classical turbulence closures: the Leith diffusion model

The diffusion model for turbulent energy transfer proposed by Leith is reconsidered from the viewpoint of Markovianized analytical closures based on the direct interaction approximation. We show that the Leith diffusion model represents a subset of the nonlinear interactions; making this connection to analytical closure suggests significant improvements of the Leith model without significantly increasing its analytical complexity. Similar ideas also lead to improved versions of the classical Kovasznay and Heisenberg models. The new models are applied to dissipation range dynamics, the “bottleneck” phenomenon, and the existence of thermalized tails in the truncated Euler equations. As an example of transient spectral dynamics, the models are applied to the development of a Kolmogorov spectrum under steady forcing.

Mechanisms of countergradient diffusion in turbulent combustion

The mechanism of countergradient diffusion of chemical species and heat in turbulent combustion is sought with the aid of the results by the two-scale direct-interaction approximation. The deviation of the Reynolds stress and the turbulent fluxes of chemical species, heat, and mass from their gradient-diffusion representations is related to the Lagrange derivatives of mean velocity and scalars. Their relative magnitude to the gradient-diffusion parts paves the way for explaining the countergradient diffusion. Towards engineering applications, these theoretical findings are converted to a Reynolds-averaged model. It consists of a closed system of equations for the mean density, the mean velocity, the mean internal energy, and the mean scalar, with the turbulence equations for the kinetic energy and its dissipation rate supplemented. This system is tested in a turbulent premixed flame and is shown to reproduce some of the characteristics pointed out by the direct numerical simulation.

Statistical dynamical subgrid-scale parameterizations for geophysical flows

Simulations of both atmospheric and oceanic circulations at given finite resolutions are strongly dependent on the form and strengths of the dynamical subgrid-scale parameterizations (SSPs) and in particular are sensitive to subgrid-scale transient eddies interacting with the retained scale topography and the mean flow. In this paper, we present numerical results for SSPs of the eddy–topographic force, stochastic backscatter, eddy viscosity and eddy–mean field interaction using an inhomogeneous statistical turbulence model based on a quasi-diagonal direct interaction approximation (QDIA). Although the theoretical description on which our model is based is for general barotropic flows, we specifically focus on global atmospheric flows where large-scale Rossby waves are present. We compare and contrast the closure-based results with an important earlier heuristic SSP of the eddy–topographic force, based on maximum entropy or statistical canonical equilibrium arguments, developed specifically for general ocean circulation models (Holloway 1992 J. Phys. Oceanogr. 22 1033–46). Our results demonstrate that where strong zonal flows and Rossby waves are present, such as in the atmosphere, maximum entropy arguments are insufficient to accurately parameterize the subgrid contributions due to eddy–eddy, eddy–topographic and eddy–mean field interactions. We contrast our atmospheric results with findings for the oceans. Our study identifies subgrid-scale interactions that are currently not parameterized in numerical atmospheric climate models, which may lead to systematic defects in the simulated circulations.