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Register a new userAnalysis of one assumption of the Navier-Stokes equations
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V.A. Budarin
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This article analyses the assumptions regarding the influence of pressure forces during the calculation of the motion of a Newtonian fluid. The purpose of the analysis is to determine the reasonableness of the assumptions and their impact on the results of the analytical calculation. The connections between equations, causes of discrepancies in exact solutions of the Navier-Stokes equations at low Reynolds numbers and the emergence of unstable solutions using computer programs are also addressed. The necessity to complement the well-known equations of motion in mechanical stress requires other equations are substantive. It is shown that there are three methods of solving such a problem and the requirements for the unknown equations are described. Keywords: Navier-Stokes, approximate equation, closing equations, holonomic system.
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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.
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy develop nearly singular solutions at the origin. This nearly singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in cite{Hou-euler-2021}. One important feature of the potential Euler singularity is that the solution develops nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Navier-Stokes equations develop nearly singular scaling properties with maximum vorticity increased by a factor of $10^7$. Moreover, the nearly self-similar profiles seem to be very stable to the small perturbation of the initial data. However, the 3D Navier-Stokes equations with our initial data do not develop a finite time singularity due to the development of a mild two-scale structure in the late stage, which eventually leads to viscous dominance over vortex stretching. To maintain the balance between the vortex stretching term and the diffusion term, we solve the 3D Navier-Stokes equations with a time-dependent viscosity roughly of order $O(|log(T-t)|^{-3})$ in the late stage. We present strong numerical evidence that the 3D Navier-Stokes equations with such time-dependent viscosity develop a finite time singularity.
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.
We present a comprehensive study of the statistical features of a three-dimensional time-reversible Navier-Stokes (RNS) system, wherein the standard viscosity $ u$ is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter $mathcal{R}_r$, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale $ell_{rm f}$ and the total energy $E_0$. We find that the system exhibits a transition from a high-enstrophy phase at small $mathcal{R}_r$, where truncation effects tend to produce partially thermalized states, to a hydrodynamical phase with low enstrophy at large $mathcal{R}_r$. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous phase transition, where $mathcal{R}_r$ indeed behaves as a thermodynamic control parameter, e.g., a temperature, the enstrophy plays the role of an order parameter, while the symmetry breaking parameter $h$ is (one over) the truncation scale $k_{rm max}$. We find that the signatures of the phase transition close to the critical point $mathcal{R}_r^star$ can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti, Phys.Lett.A, 223, 1996 could hold true. Our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes (NS) equations with viscosity chosen so as to match the average value of the reversible viscosity.
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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