Rotating Navier-Stokes Equations Research Articles

Global solutions to the rotating Navier–Stokes equations with large data in the critical Fourier–Besov spaces

AbstractWe consider the initial value problem for the 3D incompressible Navier–Stokes equations with the Coriolis force. The aim of this paper is to prove the existence of a unique global solution with arbitrarily large initial data in the scaling critical Fourier–Besov spaces (, ), provided that the size of the Coriolis parameter is sufficiently large. Moreover, if the initial data additionally belong to the scaling sub‐critical spaces, we obtain an explicit relationship between the initial data and the Coriolis force, which ensures the existence of a unique global solution.

Near-resonance approximation of rotating Navier–Stokes equations

We formalise the concept of near resonance for the rotating Navier–Stokes equations, based on which we propose a novel way to approximate the original partial differential equation (PDE). The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size in any H s space with . Such approximations retain many more 3-mode interactions, and are thus more accurate, than the conventional exact-resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any use of small divisors (so that all estimates depend smoothly on the torus’s aspect ratios). The key estimate hinges on the counting of integer solutions of Diophantine inequalities rather than Diophantine equations. Using a range of novel ideas, we handle rigorously and optimally challenges arising from the non-trivial irrational functions in these inequalities. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear, oscillatory dynamics in real-world applications.

Large Time Behavior of Solutions to the 3D Rotating Navier–Stokes Equations

We consider the large time behavior of the solutions for the initial value problem of the Navier–Stokes equations with the Coriolis force in the three-dimensional whole space. We show the L^p temporal decay estimates with the dispersion effect of the Coriolis force for the global solutions. Moreover, we prove the large time asymptotic expansion of the solutions behaving like the first-order spatial derivatives of the integral kernel of the corresponding linear solution.

Interaction of an Upwelling Front with External Vortices: Impact on Cross-shore Particle Exchange

Coastal upwellings, due to offshore Ekman transport, are more energetic at the western boundaries of the oceans, where they are intensified by incoming Rossby waves, than at the eastern boundaries. Western boundary upwellings are often accompanied by a local vortex field. The instability of a developed upwelling front and its interaction with an external vortex field is studied here with a three-dimensional numerical model of the hydrostatic rotating Navier-Stokes equations (the primitive equations). The baroclinic instability of the front leads to the growth of meanders with 100-200 km wavelength, in the absence of external vortex. On the $$f$$ -plane, these waves can break into a row of vortices when the instability is intense. The $$\beta$$ -effect is stabilizing and strongly decreases the amplitude of meanders. Simulations are then performed with a front initially accompanied by one or several external vortices. The evolutions in this case are compared with those of the unstable jet alone. On the $$f$$ -plane, when an external vortex is close to the front, this latter sheds a long filament which wraps up around the vortex. This occurs over a period similar to that of the instability of the isolated front. Cyclones are more efficient in tearing such filaments offshore than anticyclones. On the $$\beta$$ -plane, the filaments are short and turbulence is confined to the vicinity of the front. At long times, waves propagate along the front, thus extending turbulence alongshore. The initial presence of a vortex alley leads to a stronger destabilization of the front and to a larger cross-shore flux than for a single vortex, with many filaments and small vortices pushed far offshore. In the ocean, this cross-shore exchange has important consequences on the local biological activity.

Global solvability of the rotating Navier-Stokes equations with fractional Laplacian in a periodic domain

We consider existence of global solutions to equations for three-dimensional rotating fluids in a periodic frame provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. In the spatially decaying case, Koh et al. (J Differ Equ 256:707–744, 2014) showed existence for the large times of solutions of the rotating Euler equations provided by the large Coriolis force. In this case the resonant equation does not appear anymore. In the periodic case, however, the resonant equation appears, and thus the main subject in this case is to show existence of global solutions to the resonant equation. Research in this direction was initiated by Babin et al. (Indiana Univ Math J 48:1133–1176, 1999) who treated the rotating Navier-Stokes equations on general periodic domains. On the other hand, Golse et al. (Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains. Instability in Models Connected with Fluid Flows. I. Springer, New York, pp 301–338. arXiv:0704.0337v1 , 2008) considered bursting dynamics of the resonant equation in the case of a cylinder with no viscosity. Thus we may not expect to show global existence of solutions to the resonant equation without viscosity in the periodic case. In this paper we show existence of global solutions for fractional Laplacian case (with its power strictly less than the usual Laplacian) in the periodic domain with the same period in each direction. The main ingredient is an improved estimate on resonant three-wave interactions, which is based on a combinatorial argument.

Existence and analyticity of mild solutions for the 3D rotating Navier–Stokes equations

We study the three-dimensional Navier–Stokes equations in the rotational framework. By using the Littlewood–Paley analysis technique and the dispersive estimates for the Coriolis linear group \(\{e^{\pm i\Omega t\frac{D_3}{|D|}}\}_{t\in \mathbb {R}}\), we prove unique existence of global mild solutions to initial value problem and mild solutions to time-periodic problem of the rotating Navier–Stokes equations under some precise conditions, respectively, which permit the initial velocity and the time-periodic external force to be arbitrarily large provided that the speed of the rotation is fast enough. These results improve the related ones obtained in Iwabuchi and Takada (Math Ann 357:727–741, 2013) and Koh et al. (Adv diff Equ 19:857–878, 2014), and the result on the time-periodic problem can also be regarded as an enhancement and complement of that in Iwabuchi and Takada (J Evol Equ 12:985–1000, 2012). Meanwhile, based on the so-called Gevrey estimates, which are motivated by the work of Foias and Temam (J Funct Anal 87:359–369, 1989), we particularly verify that the obtained mild solutions are analytic in the spatial variables.

A finite element dimensional splitting method for 3D rotating Navier-Stokes equations

In this paper, a finite element splitting method for rotating Navier-Stokes equations is established. It is well known that the channel in turbo-machinery is bounded by two adjacent blades, top surface (shroud), bottom surface (hub), inlet and outlet. The channel is divided into several sub-channels by a series of the surfaces which are isomorphic to the blades surface. Based on a special curvilinear coordinate system, the velocity is approximated by the Hermite finite element with 3 degree and the pressure is approximated by the Lagrange finite element with 1 degree on the rotating direction. The Navier-Stokes equations are expressed into the sum of membrane and bending operators. Applying finite element expansion for the velocity and pressure leads to the finite element algebraic equations whose unknown variable are the approximated function value of the velocity and pressure and derivative value of the velocity with respect to transverse variable at the nodes. Each surface represents a two dimensional manifold in the channel. There are 8 unknown functions on each two dimensional manifold and total unknown functions satisfy a system of partial differential equations. In particular, they are called boundary layer equations on positive blade surface and negative blade surface. The system can be solved by a two scale parallel algorithm which is proposed in this paper.

Global well-posedness for the 3D rotating Navier–Stokes equations with highly oscillating initial data

In this paper, we prove the global well-posedness for the 3D rotating Navier-Stokes equations in the critical functional framework. Especially, this result allows to construct global solutions for a class of highly oscillating initial data.

On the local solvability in Morrey spaces of the Navier–Stokes equations in a rotating frame

We prove local-in-time (non-uniform) solvability for the rotating Navier–Stokes equations in Morrey spaces Mp,μσ(R3). These spaces contain singular and nondecaying functions which are of interest in statistical turbulence. We give an algebraic relation between the size of existence time and angular velocity Ω. The evolution of velocity u is analyzed in suitable Kato–Fujita spaces based on Morrey spaces. We show the asymptotic behavior uΩ→w in L∞(0,T;Mp,μσ(R3)) as Ω→0, where w is the solution for the Navier–Stokes equations with the same data u0. Particularly, for μ=3−p, the solution is approximately self-similar for small |Ω|, when u0 is homogeneous of degree −1.

Dimension splitting method for the three dimensional rotating Navier-Stokes equations

In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces \(\Im \) and is decomposed by a series of surfaces \(\Im _i \) into several sub-domains, which are called the layers of the flow. Every interface \(\Im _i \) between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on \(\Im _i \), Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on \(\Im _i \), another one is called the bending operator taking value in the normal space on \(\Im _i \). Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface \(\Im _i \) is obtained, which is called the two-dimensional three-component (2D–3C) Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D–3C Navier-Stokes equations is provided, and some approximate methods for solving 2D–3C Navier-Stokes equations are presented.

Stochastic Three-Dimensional Rotating Navier–Stokes Equations: Averaging, Convergence and Regularity

We consider stochastic three-dimensional rotating Navier–Stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems.

The boundary integral method for the linearized rotating Navier–Stokes equations in exterior domain

In this paper, we apply the boundary integral method to the linearized rotating Navier–Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and an infinite domain, we obtain a coupled problem by the linearized rotating Navier–Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence and uniqueness of solution. Finally, we study the finite element approximation for the coupled problem and obtain the error estimate between the solution of the coupled problem and its approximation solution.

The boundary integral method for the steady rotating Navier–Stokes equations in exterior domain (I): the existence of solution

In this paper, we apply the boundary integral method to the steady rotating Navier–Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and a infinite domain, we obtain a coupled problem by the steady rotating Navier–Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence of solution in a convex set.

A conservative Fourier-finite-element method for solving partial differential equations on the whole sphere

Solving transport equations on the whole sphere using an explicit time stepping and an Eulerian formulation on a latitude–longitude grid is relatively straightforward but suffers from the pole problem: due to the increased zonal resolution near the pole, numerical stability requires unacceptably small time steps. Commonly used workarounds such as near-pole zonal filters affect the qualitative properties of the numerical method. Rigorous solutions based on spherical harmonics have a high computational cost. The numerical method we propose to avoid this problem is based on a Galerkin formulation in a subspace of a Fourier-finite-element spatial discretization. The functional space we construct provides quasi-uniform resolution and high-order accuracy, while the Galerkin formalism guarantees the conservation of linear and quadratic invariants. For N2 degrees of freedom, the computational cost is 𝒪(N2logN), dominated by the zonal Fourier transforms. This is more than with a finite-difference or finite-volume method, which costs 𝒪(N2), and less than with a spherical harmonics method, which costs 𝒪(N3). Differential operators with latitude-dependent coefficients are inverted at a cost of 𝒪(N2). We present experimental results and standard benchmarks demonstrating the accuracy, stability and efficiency of the method applied to the advection of a scalar field by a prescribed velocity field and to the incompressible rotating Navier–Stokes equations. The steps required to extend the method towards compressible flows and the Saint-Venant equations are described. Copyright © 2009 Royal Meteorological Society

Solvability of the 3D rotating Navier–Stokes equations coupled with a 2D biharmonic problem with obstacles and gradient restriction

A coupled system by the 3D rotating Navier–Stokes equations with a mixed boundary condition and a 2D biharmonic problem with two obstacles and the gradient restriction is investigated in this paper. Using the Schauder’s fixed point theorem, we show the existence of a strong solution for a sufficiently large viscosity ν and sufficiently small data.

On the rotating Navier-Stokes equations with mixed boundary conditions

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.

Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data

We establish a global existence result for the rotating Navier-Stokes equations with nondecaying initial data in a critical space which includes a large class of almost periodic functions. The scaling invariant function space we introduce is given as the divergence of the space of 3× 3 fields of Fourier transformed finite Radon measures. The smallness condition on initial data for global existence is explicitly given in terms of the Reynolds number. The condition is independent of the size of the angular velocity of rotation. Mathematical Subject Classification (2000). Primary: 76D05, 76U05, Secondary: 35Q30, 28B05, 28C05.

Variational inequality for the rotating Navier–Stokes equations with subdifferential boundary conditions

A steady viscous incompressible fluid through a rotating channel satisfies the rotating Navier–Stokes equations. Subdifferential boundary conditions are imposed. A variational inequality is derived for this system. Furthermore, under some assumptions, the weak solution of the variational inequality exists and is unique.

Navier-Stokes equations in a rotating frame in ${\mathbb R}^3$ with initial data nondecreasing at infinity

Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter $\Omega$ and initial data nondecreasing at infinity. In contrast to the non-rotating case ($\Omega=0$), it is shown for the problem with rotation ($\Omega \neq 0$) that Green's function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to $L^1({\mathbb R}^3)$. Moreover, the corresponding integral operator is unbounded in the space $L^{\infty}_{\sigma}({\mathbb R}^3)$ of solenoidal vector fields in ${\mathbb R}^3$ and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in $L^{\infty}_{\sigma}({\mathbb R}^3)$. Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which consists of $L^{\infty}$ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space ${\dot B}_{\infty,1}^0$ which is smaller than $L^\infty$ but still contains various periodic and almost periodic functions. This restriction of initial data to $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which is a subspace of $L^{\infty}_{\sigma}({\mathbb R}^3)$ is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. Using the rotation transformation, we also obtain local in time solvability of the classical 3D Navier-Stokes equations in ${\mathbb R}^3$ with initial velocity and vorticity of the form $\mbox{\bf{V}}(0)=\tilde{\mbox{\bf{V}}}_0(y) + \frac{\Omega}{2} e_3 \times y$, $\mbox{curl} \mbox{\bf{V}}(0)=\mbox{curl} \tilde{\mbox{\bf{V}}}_0(y) + \Omega e_3$ where $\tilde{\mbox{\bf{V}}}_0(y) \in L^{\infty}_{\sigma,a}({\mathbb R}^3)$.

Optimal shape design for blade's surface of an impeller via the Navier–Stokes equations

AbstractIn this paper, a new principle of geometric design for blade's surface of an impeller is established. This is a thorough mathematical investigation of a shape control problem for the optimal design of an impeller's blade while 3D rotating Navier–Stokes equations are as the state equations in the control problem. The objective is to design the blade such that the viscous drag is minimized. Establishing a new curvilinear coordinate system, we depart from a standard domain transformation commonly used in the engineering literature and utilizing it we rephrase the design problem in one on a fixed domain. Certain advantages of our approach as no need for mesh regeneration and node redistribution and 3D mesh quality improvement are evident. Furthermore, in new coordinate system, it is easy to obtain gradient of optimal functional with respect to the surface of blade and to derive Euler–Lagrange equations for optimal solution which is an elliptic boundary value problem of fourth order. Moreover, we give the conjugate gradient algorithm for the control problem. Copyright © 2005 John Wiley & Sons, Ltd.