▪ Abstract We review the direct numerical simulation (DNS) of turbulent flows. We stress that DNS is a research tool, and not a brute-force solution to the Navier-Stokes equations for engineering problems. The wide range of scales in turbulent flows requires that care be taken in their numerical solution. We discuss related numerical issues such as boundary conditions and spatial and temporal discretization. Significant insight into turbulence physics has been gained from DNS of certain idealized flows that cannot be easily attained in the laboratory. We discuss some examples. Further, we illustrate the complementary nature of experiments and computations in turbulence research. Examples are provided where DNS data has been used to evaluate measurement accuracy. Finally, we consider how DNS has impacted turbulence modeling and provided further insight into the structure of turbulent boundary layers.

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DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research

Astrophysical magnetic fields and nonlinear dynamo theory

The immune system, adaptation, and machine learning

Publisher Summary This chapter presents an overview of the theory of homogeneous turbulence. Homogeneous isotropic turbulence is an idealized concept of turbulence, assumed to be governed by a statistical law that is invariant under arbitrary translation (homogeneity), rotation, or reflection (isotropy) of the coordinate system. This is an idealization of real turbulent motions, which are observed in nature or produced in a laboratory, and generally have much more complicated structures. This idealization was first introduced by Taylor to the theory of turbulence and used to reduce the formidable complexity of the statistical expression of turbulence, and thus make the subject feasible for theoretical treatment. Turbulence combines the difficulty of a strongly nonlinear dynamical system with that of a nonconservative system. The presence of energy dissipation due to viscosity deprives turbulence of the opportunity of being in a thermal equilibrium state. In this chapter, concepts of quasi-normality of large-scale motions are explained. A brief discussion on zero cumulant approximation and modified zero cumulant approximation is presented. Turbulence of other dimensions is also explained.

Theory of Homogeneous Turbulence

Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

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The phenomenology of small-scale turbulence

Two time-periodic solutions with genuine three-dimensional structure are numerically discovered for the incompressible Navier–Stokes equation of a constrained plane Couette flow. One solution with strong variation in spatial and temporal structure exhibits a full regeneration cycle, which consists of the formation and breakdown of streamwise vortices and low-velocity streaks; the other one, of gentle variation, represents a spanwise standing-wave motion of low-velocity streaks. These two solutions are unstable and the corresponding periodic orbits in the phase space are connected with each other. A turbulent state wanders around the strong one for most of the time except for occasional escapes from it. As a result, the mean velocity profile and the root-mean-squares of velocity fluctuations of the plane Couette turbulence agree very well with the temporal averages of those of this periodic motion. After an occasional escape from the strong solution, the turbulent state reaches the gentle periodic solution and returns. On the way back, it experiences an overshoot accompanied by strong turbulence activity like an intermittent bursting phenomenon.

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Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst

An analysis of the data of a direct numerical simulation of a forced incompressible isotropic turbulence at a high Reynolds number (Rλ≊180) is made to investigate the interaction among three Fourier modes of wave numbers that form a triangle. The triad interaction is classified into six types according to the direction of the energy transfer to each Fourier mode: (+,+,−), (+,−,+), (+,−,−), (−,+,+), (−,+,−), (−,−,+), where the + (or −) denotes the energy gain (or loss) of the modes of the largest, the intermediate, and the smallest wave numbers in this order. The last three types of the interaction are very few. In the first type of the interaction, a comparable amount of energy is exchanged typically among three modes of comparable magnitude of wave numbers. In the second and third types, the magnitudes of the larger two wave numbers are comparable and much larger than the smallest one, and a great amount of energy is exchanged between the former two. This behavior of the triad interaction agrees very wel...

Triad interactions in a forced turbulence

A new method is proposed to extract the axes of tubular vortices in turbulence. Loci of sectional local minimum of the pressure associated with the advection acceleration are traced numerically. It is applied to a homogeneous isotropic turbulence to demonstrate that swirling motions actually exist around these axes. The present method is shown to be superior to several typical ones commonly used in identifying tubular vortices.

Identification of Tubular Vortices in Turbulence

The time development of an infinitesimal disturbance and the relation to the spatiotemporal intermittency of a developed turbulence is studied by solving numerically the Navier–Stokes equation and its linearized form. A high‐symmetric flow is integrated by the use of the spectral method with 3403 effective modes to realize a developed turbulent field with Reynolds number of about 186 which is in statistically stationary state. Energy transfer in the wave‐number space is not uniform in time but quiescent and active periods of energy transfer and of the energy dissipation rate occur repeatedly in a large‐scale eddy‐turnover time. Vorticity tends to be concentrated in long thin tubelike regions and the energy dissipation field forms double peaks around a tube, which resembles a flow around a nonaxisymmetric Burgers vortex. Both the energy and the enstrophy of the disturbance field increase exponentially in time with growth rate comparable to the reciprocal of the time scale of the peak wave number of the energy dissipation spectrum. The energy spectrum of the disturbance field shows an equipartition of energy at small wave numbers including the inertial range. The disturbance vorticity field has a dipole structure at the position of a vorticity concentrated tube. The direction of this dipole structure is orthogonal to that of the energy dissipation field.

Spatiotemporal intermittency and instability of a forced turbulence

The velocity field of the Burgers one-dimensional model of turbulence at extremely large Reynolds numbers is expressed as a train of random triangular shock waves. For describing this field statistically the distributions of the intensity and the interval of the shock fronts are defined. The equations governing the distributions are derived taking into account the laws of motion of the shock fronts, and the self-preserving solutions are obtained. The number of shock fronts is found to decrease with time t as t−α, where α (0 [les ] α < 1) is the rate of collision, and consequently the mean interval increases as tα. The distribution of the intensity is shown to be the exponential distribution. The distribution of the interval varies with α, but it is proved that the maximum entropy is attained by the exponential distribution which corresponds to α = ½. For α = ½, the turbulent energy is shown to decay with time as t−1, in good agreement with the numerical result of Crow & Canavan (1970).

Shigeo Kida

Papers

Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst

Triad interactions in a forced turbulence

Identification of Tubular Vortices in Turbulence

Spatiotemporal intermittency and instability of a forced turbulence

Statistical mechanics of the Burgers model of turbulence