Leading Order Down-Stream Asymptotics of Stationary Navier–Stokes Flows in Three Dimensions

2011

SIAM J. Math. Anal.

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Abstract. We construct solutions for the Navier-Stokes equations in three dimensions with a time periodic force which is of compact support in a frame that moves at constant speed. These solutions are related to solutions of the problem of a body which moves within an incompressible fluid at constant speed and rotates around an axis which is aligned with the motion. In contrast to other authors who analyze stationary solutions in a frame of reference attached to the body, the analysis for the present problem is done in a frame which is moving at constant speed but is not rotating. This avoids the unpleasant unbounded linear terms which are present in a description with respect to a rotating frame. 1. Introduction. The classic paper of Weinberger [9] concerning the steady fall of a body in a Navier-Stokes liquid starts with the following definition: "We say a body undergoes a steady falling motion in an infinite viscous fluid if the motion of the fluid as seen by an observer attached to the body is independent of time."One of the interesting possible cases is a body that is falling steadily, and is rotating around an axis that is parallel to the direction in which the body is falling.A first proof of the existence of such solutions for this case has been given only recently in the three papers by Galdi and Silvestre [4,3,2]. Their method for solving the problem is to consider the equations, as proposed by Weinberger, in a frame attached to the body, where the flow is stationary. In this frame the Navier-Stokes equations have an additional linear term with unbounded coefficients, which is due to the transformation into the rotating frame. This complicates the problem considerably when compared to the situation without rotation.It is important to note that even without the rotation the problem is difficult because of the slow decay of the vorticity in the downstream region. This leads to a very strong asymmetry in the behavior at infinity, and the main difficulty is to encode this behavior when choosing function spaces.Once the existence of a solution is established one is interested in giving detailed information concerning its behavior at infinity. As in related problems, this behavior is expected to be independent of the details of the body. It turns out to be possible to use this fact in order to simplify the analysis of the asymptotic behavior, by considering first the problem in the whole space, and to mimic the body by a smooth force of compact support (see the end of this section for details). The case with a body, i.e., the case of an exterior domain, can then be treated in a second step, once the behavior at infinity is understood. For a related problem in two dimensions, this strategy has

Section: )mentioning

confidence: 99%

“…Then we consider, for a given f that is 2π λ -periodic in time, the map N defined by (10) N (ω) = S (C(Kω, ω) + f ) . Downloaded 05/06/14 to 129.194.8.73.…”

Section: )mentioning

confidence: 99%

Section: The Map S It Is Practical To Decompose the Map S From (23) mentioning

confidence: 99%

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Time Periodic Solutions of the Navier–Stokes Equations with Nonzero Constant Boundary Conditions at Infinity

Baalen

2011

SIAM J. Math. Anal.

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“…Theorem 1 For all F 2 C 1 c ( + ) with F su¢ ciently small in a sense to be de…ned below, there exist a vector …eld u = (u 1 ; u 2 ; u 3 ) 2 H 1 ( + ) and a function p satisfying the Navier-Stokes equations (1), (2) in in [21] will also be explored in our forthcoming work [8]. The proof that the functions F, obtained from the original exterior problem by the truncation procedure, are su¢ ciently small to apply the present theorem will also be given in a subsequent paper.…”

Section: Introductionmentioning

confidence: 99%

Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

Guo

Zhou

2013

Z. Angew. Math. Phys.

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We consider the problem of a body moving within an incompressible ‡uid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible NavierStokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at in…nity. Here we prove existence of stationary solutions for this problem for the simpli…ed situation where the body is replaced by a source term of compact support.

“…In [7] the method has been further refined through the use of additional analytical input [22], and the efficiency of the scheme is now widely recognized: in [4] the same ideas have been successfully tested on the semi-infinite flat plate problem and in [31] the scheme has been implemented in the context of lattice Boltzmann simulations. The present generalization of the setup to three dimensions is again based on analytic work [50] and when compared with results obtained using traditional boundary conditions computational times are again typically reduced by several orders of magnitude.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Boundary Conditions for Exterior Stationary Flows in Three Dimensions

Heuveline

2009

J. Math. Fluid Mech.

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Recently there has been an increasing interest for a better understanding of ultra low Reynolds number flows. In this context we present a new setup which allows to efficiently solve the stationary incompressible Navier-Stokes equations in an exterior domain in three dimensions numerically. The main point is that the necessity to truncate for numerical purposes the exterior domain to a finite sub-domain leads to the problem of finding so called “artificial boundary conditions” to replace the conditions at infinity. To solve this problem we provide a vector filed that describes the leading asymptotic behavior of the solution at large distances. This vector field depends explicitly on drag and lift which are determined in a self-consistent way as part of the solution process. When compared with other numerical schemes the size of the computational domain that is needed to obtain the hydrodynamic forces with a given precision is drastically reduced, which in turn leads to an overall gain in computational efficiency of typically several orders of magnitude