Chapter 4 Flashcards
What is a pathline?
A pathline is a trajectory of a fluid particle during its motion. Usually it is drawn by a solid line/curve without any vector sign on it. Draw diagram.
What are the relations used to compute pathlines?
u=dx/dt , v=dy/dt
Equation for typical velocity profile?
V(arrow)= ui + vj
What are streamlines?
Streamlines are tangents drawn to the velocity vector at every point in the fluid domain. Usually they are drawn by using solid lines/curves with vector signs on them. Drawn diagram.
What is the relation used to compare streamlines?
dy/dx = v/u
What is a stagnation point?
A point where fluid velocity is zero.
What happens at the tip of a pitot tube?
There is a stagnation point.
What is the Bernoulli Equation?
P1 + 0.5.rho.v1^(2) + gz1 = P2 + 0.5.rho.v2^(2) + gz2
Equation to find initial velocity of a particle when we have a stagnation point somewhere in the flow?
v1 = sqrt(2deltaP/rho)
What are the relations for streamfunction?
u=∂ψ/∂y , v=-∂ψ/∂x
How to prove that the stream function relations are valid?
(mass) Flow Conservation Law: ∂u/∂x + ∂v/∂y = 0
LHS: ∂u/∂x + ∂v/∂y = ∂/∂x(∂ψ/∂y) + ∂/∂y(-∂ψ/∂x)
= ∂^(2)ψ/∂x∂y - ∂^(2)ψ/∂y∂x = 0 - RHS
From calculus, if f(x,y) is a complete function, what is df(x,y)?
df(x,y) = (∂f/∂x)dx + (∂f/∂y)dy
Starting from the streamline relations, can you obtain a relation for ψ?
dy/dx = v/u udy=vdx udy=vdx=0 (∂ψ/∂y)dy + (∂ψ/∂x)dx = 0 => dψ(x,y)=0 ∫dψ =∫0 ψ=C (c=c1=ψ1... etc)
How to use streamfunction to get flow rate?
(Example with constant x, draw diagram)
Q=Vol./time=(Distance/time).area = u x A
Qab=∫(from y1 to y2) u dy= ∫(∂ψ/∂y)dy = ∫dψ
[dψ = (∂ψ/∂y)dy + (∂ψ/∂x)dx.. on AB: x=const. => dx=0 => (∂ψ/∂y)dy = dψ]
=> Qab = ∫dψ = [ψ] = ψ2 -ψ1
