Two-dimensional Incompressible Navier-Stokes Equations Research Articles

Free-falling autorotating plate - A coupled fluid and flight mechanics problem

A computational method coupling the three-degrees-of-freedom flight mechanics equations and the twodimensional, time-dependent Navier-Stokes equations was developed which could be used to predict the autorotating characteristics of a free-falling, two-dimensional flat plate. The two-dimensional incompressible Navier-Stokes equations were cast in a body-fixed coordinate system. The corresponding velocities were cast in an inertial reference system. The Navier-Stokes equations were represented by implicit finite differences and solved using a successive-overrelaxation iteration technique. The resulting aerodynamic coefficients were entered into the three-degrees-of-freedom flight mechanics equations. The system of ordinary differential equations describing the flight mechanics was solved using an Adams-Bashforth explicit method to predict the movement of the plate. New values of the boundary conditions for the Navier-Stokes equations solver were obtained from this movement of the plate. The process was repeated to advance the solution in time. The flight path of a freefalling autorotating plate was predicted using the computational procedure and the validity of the overall approach demonstrated by comparison with experiment.

High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method

The vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10,000 and meshes consisting of as many as 257 × 257 points. For Re = 1000, the (129 × 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional grid-clustering coordinate transformations.

Viscous motion in an oceanic circulation model

The barotropic motion of a viscous fluid in a laboratory simulation of ocean circulation may be modelled by Beards ley's vorticity equations. It is established here that these equations have unique smooth solutions which depend continuously on initial conditions. To avoid a boundary condition which involves an integral operator, the vorticity equations are replaced by an equivalent system of momentum equations. The system resembles the two-dimensional incompressible Navier-Stokes equations in a rotating reference frame. The existence of unique generalized solutions of the system in a square domain is established by modifying arguments used by Ladyzhenskaya for the Navier-Stokes equations. Smoothness of the solutions is then established by modifying Golovkin's arguments, again originally for the Navier- Stokes equations. A numerical procedure for solving the vorticity equations is discussed, as are the effects of reentrant corners in the domain modelling islands and peninsulae.

Studies of the merging of vortices

Finite difference equations of the two-dimensional unsteady incompressible Navier Stokes equations are constructed to study the merging of vortices. Comparison with asymptotic solutions show that the latter remain as good approximations even when the viscous cores of the vortices overlap each other.

The bid method for the steady-state Navier-Stokes equations

The Biharmonic Driver (BID) method uses direct (non-iterative) linear biharmonic solvers to obtain solutions to the nonlinear two-dimensional steady-state incompressible Navier-Stokes equations. Low Reynolds number recirculating flows are solved in 6–8 non-time-like iterations. Modifications are suggested for flow-through problems, and comparisons are made with other semidirect methods.

Numerical Simulation of Two-Dimensional Turbulence

The two-dimensional incompressible Navier-Stokes equations are numerically integrated in order to test the validity of Kraichnan's predictions on the structure of two-dimensional turbulence. The numerically simulated turbulence is produced by an artificial forcing function, constructed from a randomly varying set of Fourier modes lying within a narrow spectral band near scalar wavenumber ke. For the case of turbulence maintained by a constant rate of energy injection, Kraichnan predicts the development of two power-law energy spectra, a k−5/3 range for frequencies less than ke, and a k−3 range for those greater than ke. The numerical results are consistent with these predictions, although truncation and aliasing errors restrict the useful spectral range of the calculations to less than that desirable for a fully satisfactory test. The predictions and results are apparently relevant to several areas of geophysical fluid dynamics and to the correct interpretation of other numerical simulation experiments.