u x u )c xxxx )bu x u u)a ′ ∂ ′ ∂ ω′ ∂ ∂ ω′ ∂ ∂ ∂ ω ...

ME 639 EXAM 1 April 2004
1) (20 Points) Consider the turbulent flow of an incompressible fluid with Pr=1.
Estimate the order of magnitude of the following quantities in terms of
u, λ, θ and Λ:
∂u′
∂
ω′
∂ω′
∂u′
2
k
i i
j
a ) u′
k
u′
i
b)
c) ω′
i
u′
j
∂x
i
∂x
j∂x
j
∂x
k∂x
k
∂x
i

k
i
i
2
j
k
ij
k
j
k
i
k
j
k
i
j
i
x
x
x
f )
D
x
u
x
u
e)
x
u
x
u
d)
∂
∂
ω′
∂
∂
∂ω′
∂
′
∂
∂
′
∂
∂
′
∂
∂
′
∂
ω′ω′

2) (30 Points) The Burger equations for momentum and hear transfer are
given as
∂u
∂t
+
u
∂u
∂x
=
ν
∂
∂
2
u
2
x
∂T
∂t
u
∂T
∂x
Derive the transport equation for the velocity-temperature correlation
for the burger model. Identify the terms corresponding to production,
dissipation, diffusion and convection. Find the order of magnitude of
different terms.
+
=
ν
2
∂ T
2
∂x
u ′ T′

3) (25 Points) For a laminar
axisymmetric wake flow behind
a sphere, obtain the variations
of Us and l with x. The
equation governing of motion is
given as
U o
r
x
R
U o
∂U
∂x
=
ν
r
∂
∂r
(r
∂U
)
∂r
U s

4) (25 Points) Determine the contribution of eddies of size r to the correlations
∂u′
i
∂ω′
k
∂ω′
k
a) ω′ . For r being , , and evaluate the these
iω′
j
u′
j
b)
Λ λ η
∂x
k
∂x
j
∂x
j
contribution and compare.