University:
Widener University
Course:
ESSC 103 | Intriduction To Earth Science
Academic year:

2023
Views:

57

Pages:

1
Author:

yevenarchitrave

We can solve the equations for homogeneous, incompressible, two–dimensional ﬂow d p u + f k̂ × u = −∇ + ν∇2 u dt ρ ∇·u=0 or u = − ∂ ∂ ψ, v= ψ ∂y ∂x using a streamfunction u = −∇ × k̂ψ(x, y, t) From the divergence of the momentum equations, we ﬁnd ∇2 p ∂ ∂ ui uj = ∇ · f ∇ψ − ρ ∂xi ∂xj so, given ψ, we can ﬁnd the velocities and the pressure p/ρ. From the curl of the momentum equations, we ﬁnd the vorticity equation d q = ν∇2 q dt with ˆ·∇×u=f + q =f +k ∂ ∂ v− u = f + ∇2 ψ ∂x ∂y Given ψ, we can ﬁnd q and then evaluate the advection and diﬀusion terms to step q forward in time. Inverting the Laplace operator allows us to calculate ψ at the new time.