Tutorial EG501V Computational Fluid Dynamics (AY 2023/24)
Tutorial 3. Discretization: start-up flow in a thin fluid layer
In Lecture Notes 2 & Tutorial 2 we derived a parabolic PDE for the flow between two parallel plates. Initially both plates and the fluid between them are at rest. At time equal zero the upper plate starts moving with a constant velocity u0 , see the figure. The PDE reads (Eq. 2.2 in Lecture Notes 2) with ux the flow velocity in x-direction that depends on they-location and time t; ν = μρ is the kinematic viscosity of the fluid and is a constant.
In Lecture Notes 3 it was shown how to discretize a parabolic PDE (there in temperature T) and to eventually come up with an update rule (Eq. 3.5) that describes how to determine the situation at a new time instant (j+1) if the situation at the old time instant (j) is known.
Your assignment
(1) Follow the same procedure as in Lecture Notes 3 to derive an update rule for the velocity ux at time instant j+1 ( ux ,i ,j+1 where the index i stands for they-location in the fluid film).
(2) Consider the specific situation where L=5 mm, u0 =1 m/s, and ν =10-6 m2/s. Discretize y such that △y =1 mm (i.e. divide the distance L in five equal portions) and take time steps of △t =0.1 s. Determine the velocity in the fluid layer after one and two time steps (i.e. at t=0.1 s and t=0.2 s).