Flow by shallow water equation

Update: June 1, 2017
OpenFOAM 4.x

Case directory

$FOAM_TUTORIALS/incompressible/shallowWaterFoam/squareBump

Summary

The shallow water equation is used to calculate the flow in a system where the velocity distribution in the depth direction can be ignored. The equations to be solved are as follows.

{\partial\over\partial t}(h \vec{u})+\nabla\cdot(h \vec{u}^T \vec{u})+f\times h\vec{u} = - \left \vert \vec{g} \right \vert h \nabla(h + h_0) + \tau^\omega - \tau^b

{\partial\over\partial t}(h + h_0)+\nabla\cdot(h \vec{u}) = 0

where h and u are the fluid depth (fluid surface height) and velocity vector, respectively. The f is the Coriolis force, g is the acceleration of gravity, h0 is the deviation from the mean fluid depth, and τω and τb are the stresses from the fluid surface and bottom, respectively.

Fluid flows in from the region "inlet" (red part) at a velocity of (0.1, 0, 0) m/s and out from the region "outlet" (green part). The walls of the channel (blue part) are set to slip wall conditions, and the calculation is performed as a 2-dimensional problem with a single mesh in the Z direction. The fluid depth is set to 0.009 m only in the center of the channel and 0.01 m in the other areas.

Model geometry Model geometry
Definition of fluid depth Definition of fluid depth

The meshes are as follows.

Mesh Meshes

To visualize the fluid depth, Check "h" in the "Properties" tab on ParaView.

Check h Check h

The calculation result is as follows.

Depth distribution of fluid (h)
Depth distribution of fluid at 0 second (h) Depth distribution of fluid at 0 second (h)
Depth distribution of fluid at 1 second (h) Depth distribution of fluid at 1 second (h)
Depth distribution of fluid at 100 seconds (h) Depth distribution of fluid at 100 seconds (h)

We can see how the wave is generated at the central projection and then approaches the steady state.

Commands

cp -r $FOAM_TUTORIALS/incompressible/shallowWaterFoam/squareBump squareBump
cd squareBump

blockMesh
shallowWaterFoam

paraFoam

Calculation time

4.87 seconds *Single, Inter(R) Core(TM) i7-2600 CPU @ 3.40GHz 3.40GHz

Reference