Navier-Stokes Equations and Turbulence

Since the 1970s the use of statistical solutions of the Navier–Stokes equations has led to a number of rigorous results for turbulent flows. This paper reviews the concept of a statistical solution, its role in the mathematical foundation of the theory of turbulence, some of its successes, and the theoretical and applied challenges that still remain. The theory is illustrated in detail for the particular case of a two-dimensional flow driven by a uniform pressure gradient. Introduction It is believed that turbulent fluid motions are well modelled by the Navier–Stokes equations. However, due to the complicated nature of these equations, most of our understanding of turbulence relies to a great extent on laboratory experiments and on heuristic and phenomenological arguments. Nevertheless, a number of rigorous mathematical results have been obtained directly from the Navier–Stokes equations, particularly in the last two decades. Of great interest in turbulence theory are mean quantities, which are in general well behaved, in contrast to the corresponding instantaneous values, which tend to vary quite dramatically in time. The treatment of mean values, however, is a delicate problem, as remarked by Monin & Yaglom (1975). In practice time and space averages are the most generally used, while in theory averages with respect to a large ensemble of flows avoid some analytical difficulties and have a more universal character.

The Lagrangian Averaged Navier-Stokes (LANS) equations are a recently derived approximation to the Navier-Stokes equations. Existence of global solutions for the LANS equation has been proven for initial data in the Sobolev space $H^{3/4,2}(\mathbb{R}^3)$ and in the Besov space $B^{n/2}_{2,q}(\mathbb{R}^n)$. In this paper, we use an interpolation-based method to prove the existence of global solutions to the LANS equation with initial data in $B^{3/p}_{p,q}(\mathbb{R}^3)$ for any $p>n$.

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier–Stokes equations. Noticeably, we also find the volume-averaged dissipation $\unicode[STIX]{x1D700}_{r}$ used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) in the resulting system of equations, because it is related to dissipation correlations.

PERIODIC AND NONPERIODIC MOTIONS OF HOMOGENEOUS TURBULENCE

We consider columnar solutions of the three dimensional incompressible Navier–Stokes equations in a slab type domain. We call the solution u a columnar flow when for some scalar function γ and φ. In this paper, we obtain the global existence of columnar flows in Sobolev spaces if is nonnegative. We also show that a solution blows up in finite time if the initial data is radial and the negative part of is dominant. Furthermore, we present nontrivial exact blow-up solutions. These exact solutions blow up at every point of the domain and the blow-up rate of these solutions in the W1,p norm for is (T − t)−1 near the blow-up time t = T > 0.

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0 for the Navier–Stokes equation and a non-negative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (with respect to the measure [Formula: see text]) over any time interval [0, T], T>0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text]-norm, at a rate that is independent of (ṵ0, ψ0) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the Kolmogorov–Riesz theorem to Banach-space-valued Sobolev spaces.

This paper addresses the problem of designing low-order and linear robust feedback controllers that providea prioriguarantees with respect to stability and performance when applied to a fluid flow. This is challenging, since whilst many flows are governed by a set of nonlinear, partial differential–algebraic equations (the Navier–Stokes equations), the majority of established control system design assumes models of much greater simplicity, in that they are: firstly, linear; secondly, described by ordinary differential equations (ODEs); and thirdly, finite-dimensional. With this in mind, we present a set of techniques that enables the disparity between such models and the underlying flow system to be quantified in a fashion that informs the subsequent design of feedback flow controllers, specifically those based on the$\mathscr{H}_{\infty }$loop-shaping approach. Highlights include the application of a model refinement technique as a means of obtaining low-order models with an associated bound that quantifies the closed-loop degradation incurred by using such finite-dimensional approximations of the underlying flow. In addition, we demonstrate how the influence of the nonlinearity of the flow can be attenuated by a linear feedback controller that employs high loop gain over a select frequency range, and offer an explanation for this in terms of Landahl’s theory of sheared turbulence. To illustrate the application of these techniques, an$\mathscr{H}_{\infty }$loop-shaping controller is designed and applied to the problem of reducing perturbation wall shear stress in plane channel flow. Direct numerical simulation (DNS) results demonstrate robust attenuation of the perturbation shear stresses across a wide range of Reynolds numbers with a single linear controller.

Over the past few decades, a host of theoretical evidence has surfaced that suggest a connection between theories of gravity and the Navier-Stokes (NS) equation of fluid dynamics. It emerges out that a theory of gravity can be treated as some kind of fluid on a particular surface. Motivated by the work carried out by Bredberg et al. [6], our paper focuses on including certain modes to the vacuum solution which are consistent with the so called hydrodynamic scaling and discuss the consequences, one of which appear in the form of Damour Navier Stokes (DNS) equation with the incompressibility condition. We also present an alternative route to the results by considering the metric as a perturbative expansion in the hydrodynamic scaling parameter ϵ and with a specific gauge choice, thus modifying the metric. It is observed that the inclusion of certain modes in the metric corresponds to the solution of Einstein’s equations in presence of a particular type of matter in the spacetime. This analysis reveals that gravity has both the NS and DNS description not only on a null surface, but also on a timelike surface. So far we are aware of, this analysis is the first attempt to illuminate the possibility of presenting the gravity dual of DNS equation on a timelike surface. In addition, an equivalence between the hydrodynamic expansion and the near-horizon expansion has also been studied in the present context.

Global regularity for a model of three-dimensional Navier–Stokes equation

The Incompressible Navier-Stokes Equations (NSEs) are on the list of Millennium Problems, to prove their existence and uniqueness of solutions. The NSEs can be formulated in a periodic 3D domain, where they are termed the Periodic Navier Stokes (PNS) Equations, and can be treated on a subspace spanning a 3-dimensional torus, or T3. Treating the PNS Equations in T3-space, this article demonstrates that a decaying of turbulence occurs in the 3D case for the z component of velocity when non-smooth initial conditions are considered for x, y components of velocity and that ‘vorticity’ sheets in the small scales of 3D turbulence dominate the flow to the extent that non-smooth temporal solutions exist for the z velocity for smooth initial data for the x, y components of velocity. Unlike the Navier-Stokes equations, which have no anti-symmetric vorticity tensor, there are new governing equations which have vorticity tensor and can be decomposed into a rotational part(Liutex), antisymmetric shear and compression and stretching. It is shown that under these recent findings, that there is a strong correlation between vorticity and vorticies for (PNS).

In this paper, we study the asymptotic behavior of solutions to the threedimensional incompressible Navier-Stokes equations (NSE) with periodic boundary conditions and potential body forces. In particular, we prove that the Foias-Saut asymptotic expansion for the regular solutions of the NSE in fact holds in all Gevrey classes . This strengthens the previous result obtained in Sobolev spaces by Foias-Saut. By using the Gevrey-norm technique of Foias-Temam, the proof of our improved result simplifies the original argument of Foias-Saut, thereby, increasing its adaptability to other dissipative systems. Moreover, the expansion is extended to Leray-Hopf weak solutions and the exponential decay rate for each of its remainders is calculated explicitly.

The purpose of this note is to present new results on linear transport equations with initial values in $W^{1,r}(\Omega)$ spaces, and coefficients in $W^{s+1,p}(\Omega)^N$, where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ ($N \geq 2$) and $sp=N$. As an application, we also give refined regularity and uniqueness results for weak solutions of the two-dimensional density-dependent incompressible Navier-Stokes equations.

One-dimensional wave turbulence

Historically, the mass conservation and the classical Navier–Stokes equations were derived in the co-moving reference frame. It is shown that the mass conservation and Navier–Stokes equations are Galilean invariant—they are valid in any arbitrary inertial reference frame. From the mass conservation and Navier–Stokes equations, we can derive a wave equation, which contains the speed of pressure wave as its parameter. This parameter is independent of the speed of the source—the fluid element velocity. The speed of pressure wave is determined from the thermodynamic equation of state of the fluid, which is reference frame independent. It is well known that Lorentz transformation ensures wave speed invariant in all inertial frames, and the Lorentz invariance holds for different inertial observers. Based on these arguments, general Navier–Stokes equations (conservation law for the energy–momentum) can be written in any arbitrary inertial reference frame, they are transformed from one reference frame into another with the help of the Lorentz transformation. The key issue is that the Lorentz factor is parametrized by the local Mach number. In the instantaneous co-moving reference frame, these equations will degrade to the classical Navier–Stokes equations—the limit of the non-relativistic ones. These extended equations contain a square of the Lorentz factor. When the local Mach number is equal to one (the Lorentz factor approaches infinity), the extended Navier–Stokes equations will embody an intrinsic singularity, meaning that the transitions from the subsonic flow to the supersonic flow will happen. For the subsonic flow, the square of the Lorentz factor is positive, while for the supersonic flow, the square of the Lorentz factor becomes a negative number, which represents that the speed of sound cannot travel upstream faster than the flow velocity.

Oseen equations, which are linear equations, show up as an auxiliary problem in many numerical approaches for solving the Navier–Stokes equations. Applying an implicit method for the temporal discretization of the Navier–Stokes equations requires the solution of a nonlinear problem in each discrete time. Likewise, the steady-state Navier–Stokes equations are nonlinear. Applying in either situation a so-called Picard method (a fixed point iteration) for solving the nonlinear problem, leads to an Oseen problem in each iteration, compare Sect. 6.3 The application of semi-implicit time discretizations to the Navier–Stokes equations leads directly to an Oseen problem in each discrete time, see Remark 7.61. Altogether, Oseen problems have to be solved in many methods for simulating the Navier–Stokes equations. In addition, some parts of the theory of the Oseen equations are used in the analysis of the Navier–Stokes equations, e.g., for the uniqueness of a weak solution of the steady-state Navier–Stokes equations in Theorem 6.20. For these reasons, the analysis and numerical analysis of Oseen problems is of fundamental interest.