Flows Infected by Disturbance… P M V Subbarao Professor Mechanical Engineering Department I I T...
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Transcript of Flows Infected by Disturbance… P M V Subbarao Professor Mechanical Engineering Department I I T...
Flows Infected by Disturbance…
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Turbulent Flows
The Infection
This change, however, does not occur suddenly.
The instability triggers a transition process, which is characterized by its
intermittently laminar-turbulentnature.
Instability may drastically change the
flow pattern from laminar to turbulent
6
Definition
• A Fluid motion in which velocity, pressure, and other flow quantities fluctuate irregularly in time and space.
• “Turbulent Fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be observed.” - Hinze
• “Turbulence is due to the formation of point or line vortice on which some component of the velocity becomes infinite.:”
-Jean Leray
Time
What is turbulence?
• Unsteady, aperiodic motion in which all three velocity components fluctuate, mixing matter, momentum, and energy.
First Methods on Analyzing Turbulent Flow
Reynolds (1895) decomposed the velocity field into a time average motion and a turbulent fluctuation
- Likewise
stands for any scalar: u, v, w, T, p, where:
)(x,y,z,τuU(x,y,z)u(x,y,z,τ(
d
1
Time averaged Scalar
Averaging Navier Stokes equations
ρ ρ ρ
uUu
p Pp
v'Vv
w'Ww
Substitute into Steady incompressible Navier Stokes equations
Continuity equation:0
z
w
y
v
x
u
Instantaneous velocity
Averagevelocity
fluctuationaround averagevelocity
time
0
z
w')(W
y
v')(V
x
u')(U
0
z
w'
y
v'
x
u'
z
W
y
U
x
U
Averaging of Continuity Equations
0
z
w'
y
v'
x
u'
z
W
y
V
x
U
d
10
1
d
d
1
0
z
W
y
V
x
U 0
z
W
y
V
x
U
Average x-momentum Equation
uμ z
u
y
u
x
uμ
2
2
2
2
2
2
vu )v(u
=0 continuity
2
2
2
2
2
2
z
u
y
u
x
uμ
x
p
z
uw
y
uv
x
uu
τ
uρ
uμ x
pvu
τ
uρ
Write x-momentum equations in a short format:
Short format of momentum equation in x direction:
)v(u
uμ x
pvu
τ
uρ
vVuUvu
vuVuvUVV
vuVuvUVU
vuVU
kwujvuiuuvu ˆˆˆ
z
wu
y
vu
x
uu
z
wu
y
vu
x
uu
z
wu
y
vu
x
uuVU