scholarly journals An operator splitting scheme for the stream-function formulation of unsteady Navier–Stokes equations

2006 ◽  
Vol 53 (3) ◽  
pp. 417-442 ◽  
Author(s):  
C. I. Christov ◽  
X.-H. Tang
Keyword(s):  
Stream Function ◽  
Stokes Equations ◽  
Operator Splitting ◽  
Navier Stokes ◽  
Splitting Scheme ◽  
Navier Stokes Equations ◽  
Function Formulation ◽  
Stream Function Formulation

scholarly journals Numerical Implementations for 2D Lid-Driven Cavity Flow in Stream Function Formulation

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
K. Poochinapan
Keyword(s):  
Reynolds Number ◽  
Stream Function ◽  
Stokes Equations ◽  
Operator Splitting ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Function Formulation ◽  
High Reynolds ◽  
Stream Function Formulation

The aim of this paper is to study the properties of approximations to nonlinear terms of the 2D incompressible Navier-Stokes equations in the stream function formulation (time-dependent biharmonic equation). The nonlinear convective terms are numerically solved by using the method with internal iterations, compared to the ones which are solved by using explicit and implicit schemes (operator splitting scheme Christov and Marinova; (2001)). Using schemes and algorithms, the steady 2D incompressible flow in a lid-driven cavity is solved up to Reynolds number Re =5000 with second-order spatial accuracy. The schemes are thoroughly validated on grids with different resolutions. The result of numerical experiments shows that the finite difference scheme with internal iterations on nonlinearity is more efficient for the high Reynolds number.

scholarly journals Stochastic 2D Incompressible Navier-Stokes Solver Using the Vorticity-Stream Function Formulation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Mohamed A. El-Beltagy ◽  
Mohamed I. Wafa
Keyword(s):  
Stream Function ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Function Formulation ◽  
Mean And Variance ◽  
The Mean ◽  
Stream Function Formulation

A two-dimensional stochastic solver for the incompressible Navier-Stokes equations is developed. The vorticity-stream function formulation is considered. The polynomial chaos expansion was integrated with an unstructured node-centered finite-volume solver. A second-order upwind scheme is used in the convection term for numerical stability and higher-order discretization. The resulting sparse linear system is solved efficiently by a direct parallel solver. The mean and variance simulations of the cavity flow are done for random variation of the viscosity and the lid velocity. The solver was tested and compared with the Monte-Carlo simulations and with previous research works. The developed solver is proved to be efficient in simulating the stochastic two-dimensional incompressible flows.

A Transformation for Imposing the Rigid Bodies on the Solution of the Vorticity-Stream Function Formulation of Incompressible Navier–Stokes Equations

2019 ◽  
Vol 17 (09) ◽  
pp. 1950063
Author(s):  
Mohammadali Badri ◽  
Fereidoun Sabetghadam
Keyword(s):  
Numerical Method ◽  
Stream Function ◽  
Stokes Equations ◽  
Rigid Bodies ◽  
Time Dilation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Modified Equations ◽  
Function Formulation ◽  
Stream Function Formulation

A new penalization method is proposed for implementing the rigid bodies on the solution of the vorticity-stream function formulation of the incompressible Navier–Stokes equations. The method is based upon an active transformation of dependent variables. The transformation may be interpreted as time dilation. In this interpretation, the rigid body is considered as a region where the time is dilated infinitely, that is, time is stopped. The transformation is introduced in the vorticity and stream function equations to achieve a set of modified equations. The, in the modified equations, the time dilation of the solid region is approached to infinity. The mathematical and physical properties of the modified equations are investigated and implementation of the no-slip and no-penetration conditions are justified. Moreover, a suitable numerical method is presented for the solution of the modified equations. In the proposed numerical method, time integration is performed via the Crank–Nicolson method, and the semi-discrete equations are spatially discretized via second-order finite differencing on a uniform Cartesian grid. The method is applied to the fluid flow around a square obstacle placed in a channel, a sudden flow perpendicular to a thin flat plate, and the flow around a circular cylinder. The accuracy of the numerical solutions is evaluated.

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