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1.
Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T
0 > 0, ν
0 > 0 and a unique continuous family of strong solutions u
ν
(0 ≤ ν < ν
0) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T
0). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary
conditions imposed on curl
u
ν
and curl
2
u
ν
.
相似文献
2.
Yuning Liu Takéo Takahashi Marius Tucsnak 《Journal of Mathematical Fluid Mechanics》2012,14(1):177-195
In this paper we study a mathematical model for the dynamics of vesicle membranes in a 3D incompressible viscous fluid. The
system is in the Eulerian formulation, involving the coupling of the incompressible Navier–Stokes system with a phase field
equation. This equation models the vesicle deformations under external flow fields. We prove the local in time existence and
uniqueness of strong solutions. Moreover, we show that, given T > 0, for initial data which are small (in terms of T), these solutions are defined on [0, T] (almost global existence). 相似文献
3.
Herv�� V. J. Le Meur 《Journal of Mathematical Fluid Mechanics》2011,13(4):481-514
In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the
equations of a Viscoelastic Fluid of Johnson–Segalman type in a domain with a free surface. Managing more general constitutive
laws is also briefly depicted. The 2D geometry is defined by a solid fixed bottom and an upper free boundary submitted to
surface tension. The proof relies on a Lagrangian formulation. First we solve two intermediate problems through a fixed point
using mainly (Allain in Appl Math Optim 16:37–50, 1987) for the Navier–Stokes part. Then we solve the whole Lagrangian problem
on [0, T
0] for T
0 small enough through a contraction mapping. Since the Lagrangian solution is regular enough and the change of coordinates
invertible, we can come back to an Eulerian one. 相似文献
4.
Paulo J. S. A. Ferreira de Sousa José C. F. Pereira 《Theoretical and Computational Fluid Dynamics》2009,23(1):1-14
The dynamics of laminar co-rotating vortex pairs without axial flow have been recently thoroughly studied through theoretical,
experimental and numerical studies, which revealed different instabilities contributing to the decay of the vortices. In this
paper, the objective is to extend the analysis to the case of co-rotating vortices with axial flow at low Reynolds numbers.
A high-order incompressible Navier–Stokes flow solver is used. The momentum equations are spatially discretized on a staggered
mesh by finite differences and all derivatives are evaluated with 10th order compact finite difference schemes with RK-4 temporal
discretization. The initial condition is a linear superposition of two co-rotating circular Batchelor vortices with q = 1. It is found that there is an initial evolution that resembles the evolution that single q = 1 vortices go through. Azimuthal disturbances grow and result in the appearance of large-scale helical sheets of vorticity.
With the development of these instability waves, the axial velocity deficit is weakened. The redistribution of both angular
and axial momentum between the core and the surroundings drives the vortex core to a more stable configuration, with a higher
q value. After these processes, the evolution is somewhat similar to a pair of co-rotating Lamb–Oseen vortices. A three-dimensional
instability develops, with a large band of unstable modes, with the most amplified mode corresponding scaling with the vortex
initial separation distance.
P. J. S. A. Ferreira de Sousa wishes to acknowledge the support of FCT—SFRH/BD/1129/2000 and SFRH/BPD/21778/2005. 相似文献
5.
We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly,
when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges
(as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains. 相似文献
6.
James P. Kelliher 《Journal of Dynamics and Differential Equations》2009,21(4):631-661
We develop the concept of an infinite-energy statistical solution to the Navier–Stokes and Euler equations in the whole plane.
We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes
the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions
in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier–Stokes equations. We then construct an infinite-energy statistical
solution to the Euler equations by making a vanishing viscosity argument. 相似文献
7.
The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method. 相似文献
8.
Sébastien Barré Christophe Bogey Christophe Bailly 《Theoretical and Computational Fluid Dynamics》2008,22(1):65-82
The aerodynamic evolution and the acoustic radiation of elliptic vortices with various aspect ratios and moderate Mach numbers
are investigated by solving numerically the full compressible Navier–Stokes equations. Three behaviours are observed according
to the aspect ratio σ = a/b where a and b are the major and minor semi-axes of the vortices. At the small aspect ratio σ = 1.2, the vortex rotates at a constant angular velocity and radiates like a rotating quadrupole. At the moderate aspect
ratio σ = 5, the vortex is initially unstable. However the growth of instability waves is inhibited by the return to axisymmetry
which decreases its aspect ratio. The noise level becomes lower with time and the radiation frequency increases. For vortices
with larger aspect ratios σ ≥ 6, the return to axisymmetry does not occur quickly enough to stop the growth of instabilities, which splits the vortices.
Various mergers are then found to occur. For instance in the case σ = 6, several successive switches between an elliptic state and a configuration of two co-rotating vortices are observed.
The present results show that the initial value of the aspect ratio yields the relative weight between the return to axisymmetry
which stabilizes the vortex and the growth of instabilities which tends to split it. Moreover the noise generated by the vortices
is also calculated using the analytical solution derived by Howe (J. Fluid Mech. 71:625–673, 1975) and is compared with the
reference solution provided by the direct computation. This solution is found to be valid for σ = 1.2. An extended solution is proposed for higher aspect ratios. Finally, the pressure field appears weakly affected by
the switches between the two unstable configurations in the case σ = 6, which underlines the difficulty to detect the split or the merger of vortices from the radiated pressure. This study
also shows that elliptic vortices can be used as a basic configuration of aerodynamic noise generation.
相似文献
9.
We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields
are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This
toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided
the initial data belong to a “large” set in the Sobolev space H
1. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the
class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions
of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show
how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear)
inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits
a short and simple proof of the existence of strong solutions for all time. 相似文献
10.
11.
12.
Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier–Stokes Dynamics
J. Andrzej Domaradzki Kuo Chieh Loh Patrick P. Yee 《Theoretical and Computational Fluid Dynamics》2002,15(6):421-450
We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated
Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress
tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each
Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations
are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is
evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding
results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications
of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in
terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution.
Received 27 April 2001 and accepted 16 December 2001 相似文献
13.
Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have
been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary
solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata
(J Math Fluid Mech 7:339–367, 2005), in L
p
spaces for p ≥ 3. In this article, we first extend their result to the case
\frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using
our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with
a nonzero velocity at infinity. 相似文献
14.
In this article we present a Ladyženskaja–Prodi–Serrin Criteria for regularity of solutions for the Navier–Stokes equation
in three dimensions which incorporates weak L
p
norms in the space variables and log improvement in the time variable. 相似文献
15.
Imran Akhtar Ali H. Nayfeh Calvin J. Ribbens 《Theoretical and Computational Fluid Dynamics》2009,23(3):213-237
Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on
a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using
a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity
and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant
of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes
equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear
dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding.
We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to
home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation
and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared
with the CFD results. 相似文献
16.
Yasunori Maekawa 《Journal of Mathematical Fluid Mechanics》2008,10(1):89-105
In this paper we establish spatial decay estimates for derivatives of vorticities solving the two-dimensional vorticity equations
equivalent to the Navier–Stokes equations. As an application we derive asymptotic behaviors of derivatives of vorticities
at time infinity. It is well known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the
initial vorticity is integrable. We show that each derivative of the vorticity also behaves asymptotically as that of the
Oseen vortex.
相似文献
17.
We consider non-linear viscous shallow water models with varying topography, extra friction terms and capillary effects, in
a two-dimensional framework. Water-depth dependent laminar and turbulent friction coefficients issued from an asymptotic analysis
of the three-dimensional free-surface Navier–Stokes equations are considered here. A new proof of stability for global weak
solutions is given in periodic domain Ω = T2, adapting the method introduced by J. Simon in [15] for the non-homogeneous Navier–Stokes equations. Existence results for
such solutions can be obtained from this stability analysis. 相似文献
18.
Sergio Pirozzoli 《Theoretical and Computational Fluid Dynamics》2009,23(1):55-62
A class of exact solutions of the Navier–Stokes equations is introduced to model the fine-scale, tubular structures of isotropic
turbulence. The model vortices exhibit slow algebraic fall-off of the induced velocity, and accurately reproduce the velocity
signatures observed in DNS and experiments. The proposed model has interesting implications for the theoretical analysis of
turbulence, supporting the view that the inertial range energy scaling may have a link with the near-singular velocity field
induced by vortex tubes produced by the roll-up of vortex sheets.
相似文献
19.
Yoshiyuki Kagei 《Journal of Mathematical Fluid Mechanics》2011,13(1):1-31
Asymptotic behavior of solutions to the compressible Navier–Stokes equation around the plane Couette flow is investigated.
It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L
2 Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time
in L
2 norm as solutions of an n − 1 dimensional linear heat equation with a convective term. 相似文献
20.
Magnus Fontes 《Journal of Mathematical Fluid Mechanics》2010,12(3):412-434
In this work we study the fully nonhomogeneous initial boundary value problem for the two-dimensional time-dependent Navier–Stokes
equations in a general open space domain in R2 with low regularity assumptions on the initial and the boundary value data. We show that the perturbed Navier–Stokes operator
is a diffeomorphism from a suitable function space onto its own dual and as a corollary we get that the Navier–Stokes equations
are uniquely solvable in these spaces and that the solution depends smoothly on all involved data. Our source data space and
solution space are in complete natural duality and in this sense, without any smallness assumptions on the data, we solve
the equations for data with optimally low regularity in both space and time. 相似文献