Simple Two- and Three-Dimensional Flow Problems of the Navier-Stokes Equations

2016 ◽  
pp. 423-484
Author(s):  
Kolumban Hutter ◽  
Yongqi Wang
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Three Dimensional Flow ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Dimensional Flow

Solution of three-dimensional incompressible flow problems

1977 ◽  
Vol 82 (2) ◽  
pp. 309-319 ◽  
Author(s):  
S. M. Richardson ◽  
A. R. H. Cornish
Keyword(s):  
Stream Function ◽  
Incompressible Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Vector Potentials

A method for solving quite general three-dimensional incompressible flow problems, in particular those described by the Navier–Stokes equations, is presented. The essence of the method is the expression of the velocity in terms of scalar and vector potentials, which are the three-dimensional generalizations of the two-dimensional stream function, and which ensure that the equation of continuity is satisfied automatically. Although the method is not new, a correct but simple and unambiguous procedure for using it has not been presented before.

Three-Dimensional Flow and Pressure Patterns in a Hydrostatic Journal Bearing Pocket

1997 ◽  
Vol 119 (4) ◽  
pp. 711-719 ◽  
Author(s):  
M. J. Braun ◽  
M. B. Dzodzo
Keyword(s):  
Journal Bearing ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Flow Regimes ◽  
Axial Direction ◽  
Outward Bound ◽  
Navier Stokes ◽  
Flow Structures ◽  
Three Dimensional Flow ◽  
Navier Stokes Equations

The laminar flow in a hydrostatic pocket is described by a mathematical model that uses the three-dimensional Navier-Stokes equations written in terms of the primary variables, u, v, w, and p. Using a conservative formulation, a finite volume multiblock method is applied through a collocated, body fitted grid. The flow is simulated in a shallow pocket with a depth/length ratio of 0.02. The flow structures obtained and described by the authors in their previous two dimensional models are made visible in their three dimensional aspect for both the Couette, and the jet dominated flows. It has been found that both flow regimes formed central and secondary vortical cells with three dimensional corkscrew-like structures that lead the fluid on an outward bound path in the axial direction of the pocket. In the Couette dominated flow the position of the central vortical cell center is at the exit region of the capillary restrictor feedline, while in the jet dominated flow a flattened central vortical cell is formed in the downstream part of the pocket. It has also been determined that a fluid turn around zone occupies all the upstream space between the floor of the pocket and the runner, thus preventing any flow exit through the upstream exit of the pocket. The corresponding pressure distribution under the shaft for both flow regimes is presented as well. It was clearly established that both for the Couette, and the jet dominated cases the pressure varies significantly in the pocket in the circumferential direction, while its variation is less pronounced axially.

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