Three-dimensional Navier-Stokes Turbulence Research Articles

The production of uncertainty in three-dimensional Navier–Stokes turbulence

We derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier–Stokes turbulence and show that uncertainty is increased by strain rate compression and decreased by strain rate stretching. We use three different direct numerical simulations (DNS) of non-decaying periodic turbulence and identify a similarity regime where (a) the production and dissipation rates of uncertainty grow together in time, (b) the parts of the uncertainty production rate accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (c) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy, (d) the uncertainty energy spectrum's evolution is self-similar if normalised by the uncertainty's average uncertainty energy and characteristic length and (e) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (a), (b) and (c) imply that the average uncertainty energy grows exponentially in this similarity time range. The Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on random large-scale sweeping of dissipative eddies. In the two DNS cases of statistically stationary turbulence, this exponential growth is followed by an exponential of exponential growth, which is, in turn, followed by a linear growth in the one DNS case where the Navier–Stokes forcing also produces uncertainty.

Small-Scale Isotropy and Ramp-Cliff Structures in Scalar Turbulence.

Passive scalars advected by three-dimensional Navier-Stokes turbulence exhibit a fundamental anomaly in odd-order moments because of the characteristic ramp-cliff structures, violating small-scale isotropy. We use data from direct numerical simulations with grid resolution of up to 8192^{3} at high Péclet numbers to understand this anomaly as the scalar diffusivity, D, diminishes, or as the Schmidt number, Sc=ν/D, increases; here ν is the kinematic viscosity of the fluid. The microscale Reynolds number varies from 140 to 650 and Sc varies from 1 to 512. A simple model for the ramp-cliff structures is developed and shown to characterize the scalar derivative statistics very well. It accurately captures how the small-scale isotropy is restored in the large-Sc limit, and additionally suggests a possible correction to the Batchelor length scale as the relevant smallest scale in the scalar field.

Intermittency, cascades and thin sets in three-dimensional Navier-Stokes turbulence

Visual manifestations of intermittency in computations of three-dimensional Navier-Stokes fluid turbulence appear as the low-dimensional or “thin” filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. In order to study this phenomenon, the first task of this paper is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale with integer dimension and, in the light of the occurrence of thin sets, to discuss the mechanism of how the fluid might find the smoothest, most dissipative class of solutions rather than the most singular.

Combustion characteristics of hydrogen-air mixture in pulse detonation engines

The literature review reveals that the supersonic combustion wave remains a helpful method for improving the efficiency of pulse detonation engines (PDEs). A number of experimental studies have been conducted on detonation waves, which are formed by deflagration-to-detonation (DDT) transition waves. In this work, a straight PDE tube with a length of 1200 mm and a circular cross section measuring 60 mm in diameter is considered for combustion analysis. The combustion mechanism of stoichiometric hydrogen-air mixture is modeled by a three-dimensional Navier-Stokes turbulence model with a one-step reduced chemical kinetic reaction model using ANSYS Fluent software. The detonation tube contains obstacles with various blockage ratios of 0.5, 0.6 and 0.7 with 60 mm spacing (S) between them. Initial boundary conditions of 0.1 MPa pressure and 293 K temperature are applied to the hydrogen-air mixture to initiate combustion. The objective of the present work is to analyze the combustion flame generation and development of a stable detonation wave in the PDE tube. The flame rapidly develops and accelerates due to the burning of unburnt fuel particles in the leading zone and reduces the DDT run-up length.

Steep Cliffs and Saturated Exponents in Three-Dimensional Scalar Turbulence.

The intermittency of a passive scalar advected by three-dimensional Navier-Stokes turbulence at a Taylor-scale Reynolds number of 650 is studied using direct numerical simulations on a 4096^{3} grid; the Schmidt number is unity. By measuring scalar increment moments of high orders, while ensuring statistical convergence, we provide unambiguous evidence that the scaling exponents saturate to 1.2 for moment orders beyond about 12, indicating that scalar intermittency is dominated by the most singular shocklike cliffs in the scalar field. We show that the fractal dimension of the spatial support of steep cliffs is about 1.8, whose sum with the saturation exponent value of 1.2 adds up to the space dimension of 3, thus demonstrating a deep connection between the geometry and statistics in turbulent scalar mixing. The anomaly for the fourth and sixth order moments is comparable to that in the Kraichnan model for the roughness exponent of 4/3.

Optimal subgrid scheme for shell models of turbulence.

We discuss a theoretical framework to define an optimal subgrid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of Kolmogorov formulated for similar quantities for the original three-dimensional Navier-Stokes turbulence. We also propose a series of systematic approximations to the optimal model by assuming different degrees of correlations across scales among amplitudes and phases of consecutive multipliers. We show numerically that such low-order closures work well, reproducing all known properties of the large-scale dynamics including anomalous scaling. We found small but systematic discrepancies only for a range of scales close to the subgrid threshold, which do not tend to disappear by increasing the order of the approximation. We speculate that the lack of convergence might be due to a structural instability, at least for the evolution of very fast degrees of freedom at small scales. Connections with similar problems for large eddy simulations of the three-dimensional Navier-Stokes equations are also discussed.

Unsteady turbulence cascades

We have run a total of 311 direct numerical simulations (DNSs) of decaying three-dimensional Navier-Stokes turbulence in a periodic box with values of the Taylor length-based Reynolds number up to about 300 and an energy spectrum with a wide wave-number range of close to -5/3 power-law dependence at the higher Reynolds numbers. On the basis of these runs, we have found a critical time when (i) the rate of change of the square of the integral length scale turns from increasing to decreasing, (ii) the ratio of interscale energy flux to high-pass filtered turbulence dissipation changes from decreasing to very slowly increasing in the inertial range, (iii) the signature of large-scale coherent structures disappears in the energy spectrum, and (iv) the scaling of the turbulence dissipation changes from the one recently discovered in DNSs of forced unsteady turbulence and in wind tunnel experiments of turbulent wakes and grid-generated turbulence to the classical scaling proposed by G. I. Taylor [Proc. R. Soc. London, Ser. A 151, 421 (1935)1364-502110.1098/rspa.1935.0158] and A. N. Kolmogorov [Dokl. Akad. Nauk SSSR 31, 538 (1941)]. Even though the customary theoretical basis for this Taylor-Kolmogorov scaling is a statistically stationary cascade where large-scale energy flux balances dissipation, this is not the case throughout the entire time range of integration in all our DNS runs. The recently discovered dissipation scaling can be reformulated physically as a situation in which the dissipation rates of the small and large scales evolve together. We advance two hypotheses that may form the basis of a theoretical approach to unsteady turbulence cascades in the presence of large-scale coherent structures.

Multilocality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence.

Using the fusion-rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator "multilocality." The resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry. Our main result is that the fusion-rules hypothesis implies both locality and multilocality in both the IR and UV limits for the downscale energy cascade of three-dimensional Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two-dimensional Navier-Stokes turbulence. We stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term-by-term perturbative locality of diagrammatic theories or closure models that involve only two-point correlation and response functions.

Estimating intermittency in three-dimensional Navier–Stokes turbulence

The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addressed. Under the assumption that the three-dimensional Navier–Stokes equations have a global attractor it is nevertheless shown that solutions can potentially behave differently in two distinct regions of space–time$\mathbb{S}$±where$\mathbb{S}$−is comprised of a union of disjoint space–time ‘anomalies’. If$\mathbb{S}$−is non-empty it is dominated by large values of |∇ω|, which is consistent with the formation of vortex sheets or tightly coiled filaments. The local number of degrees of freedom±needed to resolve the regions in$\mathbb{S}$±satisfies$\mathcal{N}^{\pm}(\bx,\,t)\lessgtr 3\sqrt{2}\,\mathcal{R}_{u}^{3},$, whereu=uL/ν is a Reynolds number dependent on the local velocity fieldu(x,t).

Geometric features of the mixing of passive scalars at high Schmidt numbers.

The mixing of passive scalars of decreasing diffusivity, advected in each case by the same three-dimensional Navier-Stokes turbulence, is studied. The mixing becomes more isotropic with decreasing diffusivity. The local flow in the vicinity of steepest negative and positive scalar gradients is in general different, and its behavior is studied for various values of the scalar diffusivity. Mixing approaches monofractal properties with diminishing diffusivity. We consider these results in the context of possible singularities of scalar dissipation.

Flame Front Curvature Distributions in a Turbulent Premixed Flame Zone

Distributions of flame front curvature obtained by laser sheet tomography agree with those derived from numerical simulations of passive flame propagation within three-dimensional Navier-Stokes turbulence. The experimental configuration is that of grid turbulence impinging upon a plate which stabilizes a premixed methane/air flame, planar images of the flame allow construction of flame curvature as a function of flame location within the spatial zone that contains products and reactants. In the simulations the flame burning velocity is twice the turbulence intensity and the Reynolds number based on the computed Taylor length scale is approximately 55. The computed flame geometry and flame strain rate are obtained as a function of location based on the mean progress variable (defined by the passive surface displacement or by the scalar fluctuations defined over transverse planes). The shape of the mean progress variable profile compares well with experiment and with two reaction-diffusion models of propagation (KPP and an independent Gaussian model). From the simulations planar slices are created in order to provide curvature information which is directly comparable to the experimental data. Distributions of curvature, based on planar information, exhibit a change with location in the turbulent flame zone: an overall positive curvature (convex to the reactants) at the front to a negative value at the rear; however, this behavior is composed of positive curvature (which by itself has an average value with no spatial variation) and negative curvature (which increases in magnitude with distance from the front). A single length scale allows a good match between experimental and computed curvature throughout the flame zone. The passive flame simulations show the most probable flame shape to be cylindrical, and this feature, allows the planar information to be scaled in order to match the curvature distributions based on three dimensional information. The scaling factor is obtained by observing a cylinder with planar slices at all possible angles.

Scaling of Low-Order Structure Functions in Homogeneous Turbulence.

High-resolution direct numerical simulation data for three-dimensional Navier-Stokes turbulence in a periodic box are used to study the scaling behavior of low-order velocity structure functions with positive and negative powers. Similar to high-order statistics, the low-order relative scaling exponents exhibit unambiguous departures from the Kolmogorov 1941 theory and agree well with existing multiscaling models. No transition from normal scaling to anomalous scaling is observed. {copyright} {ital 1996 The American Physical Society.}

Intermittency and anomalous scaling of passive scalars in any space dimension.

We establish exact inequalities for the structure-function scaling exponents of a passively advected scalar in both the inertial-convective and viscous-convective ranges. These inequalities involve the scaling exponents of the velocity structure functions and, in a refined form, an intermittency exponent of the convective-range scalar flux. They are valid for three-dimensional Navier-Stokes turbulence and satisfied within errors by present experimental data. The inequalities also hold for any ``synthetic'' turbulent velocity statistics with a finite correlation in time. We show that for time-correlation exponents of the velocity smaller than the ``local turnover'' exponent, the scalar spectral exponent is strictly less than that in Kraichnan's [Phys. Rev. Lett. 72, 1016 (1994); Phys. Fluids 11, 945 (1968); J. Fluid Mech. 64, 737 (1974); 62, 305 (1974)] soluble ``rapid-change'' model with velocity \ensuremath{\delta} correlated in time. Our results include as a special case an exponent inequality derived previously by Constantin and Procaccia [Nonlinearity 7, 1045 (1994)], but with a more direct proof. The inequalities in their simplest form follow from a Kolmogorov-type relation for the turbulent passive scalar valid in each space dimension d. Our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars. These are compared with the relations implied by ``fusion rules'' hypothesized for scalar gradients. \textcopyright{} 1996 The American Physical Society.

Numerical and analytical estimates for the structure functions in two-dimensional magnetohydrodynamic flows

In two-dimensional magnetohydrodynamic turbulence, the Kraichnan–Iroshnikov dimensional analysis suggests a linear scaling law for the exponents ζp=p/4 of the structure functions for the Elsässer variables z±=u±B. Numerical simulations are presented and higher order structure functions are calculated using the extended self-similarity hypothesis of Benzi et al. [Phys. Rev. E 48, R29 (1993)]. In addition, an estimate for the first structure function ζ1≥1/4 is derived using a geometric technique introduced by Constantin and Procaccia [Phys. Rev. E 47, 3307 (1993)] in the the context of the transport of a passive scalar in three-dimensional Navier–Stokes turbulence.

Renormalization-group theory for the eddy viscosity in subgrid modeling.

Renormalization-group theory is applied to incompressible three-dimensional Navier-Stokes turbulence so as to eliminate unresolvable small scales. The renormalized Navier-Stokes equation now includes a triple nonlinearity with the eddy viscosity exhibiting a mild cusp behavior, in qualitative agreement with the test-field model results of Kraichnan. For the cusp behavior to arise, not only is the triple nonlinearity necessary but the effects of pressure must be incorporated in the triple term. The renormalized eddy viscosity will not exhibit a cusp behavior if it is assumed that a spectral gap exists between the large and small scales.

Similar solutions in modified cumulant expansion

A method of matching is applied to the energy spectrum equations in the modified cumulant expansion theory for Burgers and two- and three-dimensional Navier–Stokes turbulence, in order to investigate the asymptotic behavior of the energy spectrum in the limit of large Reynolds number and time. It is shown analytically that the spectrum has a similar structure with respect to both Reynolds number and time, and that there is no single similarity law which is valid throughout the wavenumbers, but instead there are several subranges of the wavenumbers which have different similarity laws.