scholarly journals Decay Estimates for Steady Solutions of the Navier--Stokes Equations in Two Dimensions in the Presence of a Wall

2012 ◽  
Vol 44 (5) ◽  
pp. 3346-3368 ◽  
Author(s):  
Christoph Boeckle ◽  
Peter Wittwer
Keyword(s):  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Decay Estimates ◽  
Navier Stokes Equations ◽  
Steady Solutions

scholarly journals Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling

Fluids ◽  
2021 ◽  
Vol 6 (12) ◽  
pp. 425
Author(s):  
Jeremiah S. Lane ◽  
Benjamin F. Akers
Keyword(s):  
Stream Function ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fixed Point Algorithm ◽  
Slip Boundary ◽  
Thermal Blooming ◽  
Steady Solutions

This research investigates laser-induced convection through a stream function-vorticity formulation. Specifically, this paper considers a solution to the steady Boussinesq Navier–Stokes equations in two dimensions with a slip boundary condition on a finite box. A fixed-point algorithm is introduced in stream function-vorticity variables, followed by a proof of the existence of steady solutions for small laser amplitudes. From this analysis, an asymptotic relationship is demonstrated between the nondimensional fluid parameters and least upper bounds for laser amplitudes that guarantee existence, which accords with numerical results implementing the algorithm in a finite difference scheme. The findings indicate that the upper bound for laser amplitude scales by O(Re−2Pe−1Ri−1) when Re≫Pe, and by O(Re−1Pe−2Ri−1) when Pe≫Re. These results suggest that the existence of steady solutions is heavily dependent on the size of the Reynolds (Re) and Peclet (Pe) numbers, as noted in previous studies. The simulations of steady solutions indicate the presence of symmetric vortex rings, which agrees with experimental results described in the literature. From these results, relevant implications to thermal blooming in laser propagation simulations are discussed.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

Perturbation Procedures for Nonlinear Viscous Flows

1983 ◽  
Vol 50 (2) ◽  
pp. 265-269
Author(s):  
D. Nixon
Keyword(s):  
Perturbation Theory ◽  
Shock Waves ◽  
Transonic Flow ◽  
Stokes Equations ◽  
Viscous Flows ◽  
Two Dimensions ◽  
Experimental Results ◽  
Navier Stokes ◽  
Navier Stokes Equations

The perturbation theory for transonic flow is further developed for solutions of the Navier-Stokes equations in two dimensions or for experimental results. The strained coordinate technique is used to treat changes in location of any shock waves or large gradients.

A Numerical Study of the Hydrodynamics of Reversing Flows Around a Cylinder

1994 ◽  
Vol 116 (4) ◽  
pp. 202-208 ◽  
Author(s):  
K. Nakajima ◽  
Y. Kallinderis ◽  
I. Sibetheros ◽  
R. W. Miksad ◽  
K. Lambrakos
Keyword(s):  
Experimental Data ◽  
Finite Element Method ◽  
Stokes Equations ◽  
Numerical Study ◽  
Offshore Structures ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Random Behavior ◽  
Marching Scheme

A numerical study of the nonlinear and random behavior of flow-induced forces on offshore structures and experimental verification of the results are presented. The numerical study is based on a finite-element method for the unsteady incompressible Navier-Stokes equations in two dimensions. The momentum equations combined with a pressure correction equation are solved employing fourth-order artificial dissipation with a nonstaggered grid, instead of the more commonly used staggered meshes. The solution is advanced in time with a combined explicit and implicit marching scheme. Emphasis is placed on study of reversing flows around a cylinder. Comparisons with experimental data evaluate accuracy and robustness of the method.

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