Kabir Bakhshaei - Navier-Stokes Equations

CFD Programming

Navier-Stokes Equations

To describe how the pressure, velocity, density, and temperature of a moving fluid are related. On this page, I have programmed the following five problems with a method entitled 'Vorticity-Stream Function'.

Viscous Incompressible Flow Over a Backward-Facing Step
Viscous Incompressible Flow in the Lid-Driven Cavity
Flow in an Open Channel Confluence
The Effect of Inlet and Outlet Location in a Storage Tank
Flow through a Sudden Expansion

Vorticity-Stream Function formulation for 2D Incompressible Flows

The stream function-vorticity formulation was among the first unsteady, incompressible Navier–Stokes algorithms. For incompressible two-dimensional flows with constant fluid properties, the Navier–Stokes equations can be simplified by introducing the stream function ψ and vorticity ω as dependent variables.

1_Flow Over a Backward-Facing Step:

Backward-Facing Step (BFS) flow is one representative separation flow model, which is of significance in both theoretical and engineering development. Various applications for BFS flow can be found in our daily life, such as the airfoils at large attack angles, separation flow behind a vehicle, and also the flow around a boat or a building.

1-Stream Function for Flow Over a Backward-Facing Step

1-Pressure Distribution for Flow Over a Backward-Facing Step

1_Distribution of Velocity Vector for Flow Over a Backward-Facing Step

1-Stream Line for Flow Over a Backward-Facing Step

2_Flow in the Lid-Driven Cavity Problem

The lid-driven cavity consists of a square cavity filled with fluid. At the top boundary, a tangential velocity is applied to drive the fluid flow in the cavity. The remaining three walls are defined as no-slip conditions; that is, the velocity is 0 and it is a popular problem within the field of computational fluid dynamics (CFD) for validating computational methods. While the boundary conditions are relatively simple, the flow features created are quite interesting and complex.

2_Stream Function in the Lid-Driven Cavity

2_Pressure Distribution in the Lid-Driven Cavity

2_Distribution of Velocity Vector in the Lid-Driven Cavity

3_Flow in an Open Channel Confluence

Flow connection in channels is a phenomenon which frequently happens in rivers, water, and drainage channels. The phenomenon appears to be more complex in rivers than in channels, especially at the y-junction bed joint because of the interaction between the tributary and the main river.

3_Stream Function in an Open Channel Confluence

3_Pressure Distribution in an Open Channel Confluence

3_Distribution of Velocity Vector in an Open Channel Confluence

4_The Effect of Inlet and Outlet Location in a Tank

4_Stream Function in a Storage Tank

4_Pressure Distribution in a Storage Tank

4_Distribution of Velocity Vector in a Storage Tank

5_Flow through a Sudden Expansion

One of the cases where the fluid flow goes through an abrupt change of geometry is the flow through a sudden expansion. Due to this sudden geometrical abnormality, the head loss and flow instability contributes to the minor flow losses associated with internal flows.