2.5
TURBULENT JET
Consider outflow from a plane jet with the following profiles for the mean flow and its vertical
derivative. Recall that U (x, y) = Us (x)f (⇠) where ⇠ = y/`(x).
Left: Mean self-similar profile f (⇠) at any distance downstream. Middle: df /d⇠. Right: uv
The self-preservation equation for a jet is



Z
Z
` dUs 2
d` 0 ⇠
` dUs 0 ⇠
1 0
f
f
f d⌘
f
f d⌘ =
g
(2.3)
Us dx
dx
Us dx
RT
0
0
a) How does the downstream length scale ` change? To answer this, each of the terms in the
square brackets must be independent of x. Hint: Only one of them is needed to determine
` changes downstream.
(2 marks)
b) The velocity scale is determined by the constant momentum integral
Z 1
Z 1
2
2
M=
U (x, y)dy = [Us `]
f 2 d⇠
1
1
How does the velocity scale Us change with distance downstream?
(5 marks)
c) Sketch how a turbulent jet grows downstream and include mean velocity profiles at two
downstream locations (x ⇡ 30 and x ⇡ 60).
(15 marks)
d) Explain what is meant by the constant stress layers and where they are in a turbulent jet.
Recall that a jet can be viewed as two mixing layers.
(5 marks)
(25 marks total)
11
ES4410 2012
------------------------------------------------------------------------------------------------------5.
a) Write down the Navier-Stokes equations, introduce non-dimensional
variables, and derive the expression for Reynolds Number. Explain the physical
meaning of Re.
(13 points)
b) Consider a circular (3-dimensional, axisymmetric) turbulent jet. Assume that the
mean axial velocity in the jet is described by a self-similar expression
⎛ r ⎞
V = VC ( z ) ⋅ f ⎜⎜
⎟⎟ , where r is the radius and z is the axial coordinate. Using the
⎝ δ (z ) ⎠
appropriate conservation law, find how VC depends on z.
(12 points)
You may find some useful information available in the Appendix.
(25 points total)
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