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Register a new userA correspondence between the multifractal model of turbulence and the Navier-Stokes equations
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We study a correspondence between the multifractal model of turbulence and the Navier-Stokes equations in $d$ spatial dimensions by comparing their respective dissipation length scales. In Kolmogorovs 1941 theory the key parameter $h$, which is an exponent in the Navier-Stokes invariance scaling, is fixed at $h=1/3$ but is allowed a spectrum of values in multifractal theory. Taking into account all derivatives of the Navier-Stokes equations, it is found that for this correspondence to hold the multifractal spectrum $C(h)$ must be bounded from below such that $C(h) geq 1-3h$, which is consistent with the four-fifths law. Moreover, $h$ must also be bounded from below such that $h geq (1-d)/3$. When $d=3$ the allowed range of $h$ is given by $h geq -2/3$ thereby bounding $h$ away from $h=-1$. The implications of this are discussed.
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We accomplish two major tasks. First, we show that the turbulent motion at large scales obeys Gaussian statistics in the interval 0 < Rlambda < 8.8, where Rlambda is the microscale Reynolds number, and that the Gaussian flow breaks down to yield place to anomalous scaling at the universal Reynolds number bounding the inequality above. In the inertial range of turbulence that emerges following the breakdown, the effective Reynolds number based on the turbulent viscosity, Rlambda* assumes this same constant value of about 9. This scenario works also for the emergence of turbulence from an initially non-turbulent state. Second, we derive expressions for the anomalous scaling exponents of structure functions and moments of spatial derivatives, by analyzing the Navier-Stokes equations in the form developed by Hopf. We present a novel procedure to close the Hopf equation, resulting in expressions for zetan in the entire range of allowable moment-order, n, and demonstrate that accounting for the temporal dynamics changes the scaling from normal to anomalous. For large n, the theory predicts the saturation of zetan with n, leading to two inferences: (a) the smallest length scale etan = LRe-1 << LRe-3/4, where Re is the large-scale Reynolds number, and (b) velocity excursions across even the smallest length scales can sometimes be as large as the large scale velocity itself. Theoretical predictions for each of these aspects are shown to be in quantitative agreement with available experimental and numerical data.
A dynamic procedure for the Lagrangian Averaged Navier-Stokes-$alpha$ (LANS-$alpha$) equations is developed where the variation in the parameter $alpha$ in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where $alpha$ is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-$alpha$ procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic energy decay are captured more accurately as compared with the LANS-$alpha$ simulations using a fixed $alpha$. In order to evaluate the applicability of the dynamic LANS-$alpha$ model in anisotropic turbulence, a priori test of a turbulent channel flow is performed. It is found that the parameter $alpha$ changes in the wall normal direction. Near a solid wall, the length scale $alpha$ is seen to depend on the distance from the wall with a vanishing value at the wall. On the other hand, away from the wall, where the turbulence is more isotropic, $alpha$ approaches an almost constant value. Furthermore, the behavior of the subgrid scale stresses in the near wall region is captured accurately by the dynamic LANS-$alpha$ model. The dynamic LANS-$alpha$ model has the potential to extend the applicability of the LANS-$alpha$ equations to more complicated anisotropic flows.
This paper has been withdrawn by the authors for adding some results.
We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier-Stokes. The consequences on the applicability of the regularizations as SGS models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-alpha model are compared to two previously employed regularizations, LANS-alpha and Leray-alpha (at Re ~ 3300, Taylor Re ~ 790) and to a DNS. We derive the Karman-Howarth equation for both the Clark-alpha and Leray-alpha models. We confirm one of two possible scalings resulting from this equation for Clark as well as its associated k^(-1) energy spectrum. At sub-filter scales, Clark-alpha possesses similar total dissipation and characteristic time to reach a statistical turbulent steady-state as Navier-Stokes, but exhibits greater intermittency. As a SGS model, Clark reproduces the energy spectrum and intermittency properties of the DNS. For the Leray model, increasing the filter width decreases the nonlinearity and the effective Re is substantially decreased. Even for the smallest value of alpha studied, Leray-alpha was inadequate as a SGS model. The LANS energy spectrum k^1, consistent with its so-called rigid bodies, precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in resolution. However, that this same feature reduces its intermittency compared to Clark-alpha (which shares a similar Karman-Howarth equation). Clark is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than alpha, whereas high-order intermittency properties for larger values of alpha are best reproduced by LANS-alpha.
The concept of continuous topological evolution, based upon Cartans methods of exterior differential systems, is used to develop a topological theory of non-equilibrium thermodynamics, within which there exist processes that exhibit continuous topological change and thermodynamic irreversibility. The technique furnishes a universal, topological foundation for the partial differential equations of hydrodynamics and electrodynamics; the technique does not depend upon a metric, connection or a variational principle. Certain topological classes of solutions to the Navier-Stokes equations are shown to be equivalent to thermodynamically irreversible processes.
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