Tony Burden’s Lecture Notes on Turbulence, Spring 2007
Chapter 5. Mean Kinetic Energy
This chapter is based on the NavierStokes equation in the form,
ρ
∂u
i
∂t
+
ρ u
l
∂u
i
∂x
l
=

∂p
∂x
i
+
∂
∂x
l
2
μs
il
,
where,
s
ij
=
1
2
∂u
i
∂x
j
+
∂u
j
∂x
i
,
is the rateofstrain tensor. The corresponding form of Reynolds’ equation is,
ρ
∂U
i
∂t
+
ρ U
l
∂U
i
∂x
l
=

∂P
∂x
i
+
∂
∂x
l

ρ u
i
u
l
+
∂
∂x
l
2
μS
il
.
These forms are closer to the fundamental mechanics of a Newtonian fluid.
The density of mean kinetic energy
The total mean kinetic energy (
r¨
orelseenergi
) is the sum of the kinetic energy of the
mean velocity and the mean kinetic energy of the turbulence;
1
2
ρ

u

2
=
1
2
ρ u
i
u
i
=
1
2
ρ U
i
U
i
+
1
2
ρ u
i
u
i
=
1
2
ρ
U
2
+
ρK,
where
K
=
1
2
u
i
u
i
=
1
2
u
2
+
v
2
+
w
2
is the mean kinetic energy of the
turbulence which is often denoted by
k
.
The kinetic energy of the mean flow
The scalar product of the mean velocity with Reynolds’ equation,
i.e.
,
U
i
∂U
i
∂t
+
· · ·
=
∂
∂t
(
1
2
U
i
U
i
)
+
· · ·
,
yields,
∂
∂t
(
1
2
U
2
)
+
U
l
∂
∂x
l
(
1
2
U
2
)
=

∂
∂x
l
1
ρ
P U
l
+
u
i
u
l
U
i

2
νS
il
U
i
+
u
i
u
l
∂U
i
∂x
l

2
νS
il
∂U
i
∂x
l
.
50
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The discussion which is given below of the corresponding terms in the equation for the
mean kinetic energy of the turbulence shows that; the first group of terms in the r.h.s.
(
HL
) are transport terms; the second term is
P
, the transfer of energy from the mean
flow to the turbulence; and the third term is viscous dissipation of the kinetic energy of
the mean velocity, which is negligible at high (local) Reynolds numbers.
The dynamical equation for the fluctuating velocity
The derivation of a dynamical equation for the fluctuating part of the velocity,
u
i
, is
an important step in the derivation of a governing equation for the mean kinetic energy
of the turbulence.
Subtraction of Reynolds’ equation for the mean velocity,
U
i
, from
NavierStokes equation for the instantaneous velocity,
u
i
=
U
i
+
u
i
, yields,
∂u
i
∂t
+
U
l
∂u
i
∂x
l
=

u
l
∂U
i
∂x
l

∂
∂x
l
u
i
u
l
+
∂
∂x
l
u
i
u
l

1
ρ
∂p
∂x
i
+
∂
∂x
l
{
2
νs
il
}
.
The mean kinetic energy of the turbulence
The scalar product of the fluctuating part of the velocity,
u
i
, with the dynamical equa
tion for
u
i
above yields,
∂K
∂t
+
U
l
∂K
∂x
l
=

u
i
u
l
∂U
i
∂x
l

∂
∂x
l
1
2
u
i
u
i
u
l
+
1
ρ
p u
l

2
ν s
il
u
i

2
ν
s
il
∂u
i
∂x
l
.
This equation expressing the energy budget of turbulence can be written in the form,
∂K
∂t
+
U
l
∂K
∂x
l
=
P 
∂
∂x
l
J
l

ε,
in which the terms in the r.h.s. (
HL
) are:
P
=

u
i
u
l
∂U
i
∂x
l
production of turbulence by the mean flow;

∂
∂x
l
J
l
=

div
J
transport terms (see next page for the flux
J
);
ε
= 2
ν
s
il
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 Spring '11
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 Fluid Dynamics, Kinetic Energy, Strain

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