MT4509 — Fluid Dynamics
Example sheet 2
Flow kinematics II
代做流体动力学 1. Consider the flow given by (u, v) = (x − Vt, y). (i) Find and sketch the streamlines at t = 0. What happens to the streamlines for t > 0?
1.
Consider the flow given by (u, v) = (x − Vt, y).
(i) Find and sketch the streamlines at t = 0. What happens to the streamlines for t > 0?
(ii) Show the flow cannot be incompressible.
(iii) Find the particle path originating at (1, 1) at t = 0. Sketch the path when V = 1.
(iv) If we placed a stationary rigid horizontal boundary at y = 0 what boundary condition would we need to use there? Does the flow field satisfy this condition?
2. 代做流体动力学
For the unsteady two-dimensional flow (u, v) = (a cosωt, a sinωt) where a and ω are positive constants.
(i) Find the equation for the streamlines at any time t. Sketch the streamlines for t = 0 and indicate what happens to them as t increases.
(ii) Find the equation of the particle path that starts at x0 at t = 0, sketch the shape of the path.
3. 代做流体动力学
A flow in three dimensions has velocity field u = (cx, cy, −2cz) where c is a positive constant.
(i) Find and sketch the streamlines for the plane z = 0.
(ii) Show the flow is incompressible.
(iii) Find and sketch the streamline through (1, 1, 1).
4. 代做流体动力学
The streamfunction for two-dimensional irrotational flow in a corner of angle π/n is
ψ = ARn sin n∅,
in polar coordinates.
(i) Find the velocity components uR and u∅ in polar coordinates.
(ii) Find the velocity components u and v in Cartesian coordinates.
(iii*) Sketch the streamlines (think about behaviour as R → ∞, and for constant R).
(iv) What happens when n < 1, and why is this unphysical? What might happen in a real fluid?
