文档介绍：Problem Set 4B Name: ____________________
Assigned: Tuesday, September 25
Due: Tuesday, October 2
1. A hydraulic jump occurs in an open channel as shown. Consider the flow to be 2D (no
flow in the z-direction). We can see that there are two relevant length scales for the flow,
y1 and y2 . The other physical variables are the two-dimensional flow rate q (has
dimensions L2 /T ), the density ρ, the dynamic viscosity µ , and the acceleration due to
gravity g. Using dimensional analysis, find the non-dimensional functional relationship
between these variables in the form
= ΠΠ
y2 / y1 f ( a , b )
ΠΠ
where a and b are two dimensionless parameters you must find.
y2
y1 flow
2. Consider a body of water that has a free surface (like a lake). Define the surface of the
still water to be z = 0 , with z positive upward. Recall that in the Navier-Stokes equations
= −ρ
we bine the hydrostatic pressure term ps gz with the total pressure p by
= −
defining the dynamic pressure pd p ps :
∂ v
V v v v 1 v 2 v
(1) +V ⋅∇V = −∇p +ν∇ V
∂t ρ d
(a) Suppose you are studying a submarine moving at a steady speed on the surface of the
lake, and that the submarine has length L and speed U. Choose the characteristic pressure
r
to be ρU 2 . Write each dimensionless variable x *, y * , z *,V * , and p * in terms of