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Fluid Mechanics
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Southern Methodist University
Bobby B. Lyle School of Engineering
CEE 2342/ME 2342 Fluid Mechanics
Roger O. Dickey, Ph.D., P.E.
III. BASIC EQS. OF HYDRODYNAMICS
B. Control Volume Theory – Reynolds Transport Theorem
Reading Assignment:
Chapter 4 Fluid Kinematics, Sections 4.3 to 4.5
B. Control Volume Theory – Reynolds
Transport Theorem
Definitions
(i) System a fixed set of fluid particles that
may move, flow, and interact with its
environment:
Gas
Compressed
System of gas molecules
(ii) Control Volume (CV) a three
dimensional region of space chosen for
studying a fluid flow field.
(iii) Control Surface (CS) a mathematical
surface in space enclosing and defining the
physical boundaries of the control volume.
(iv) Flux of a vector field ≡ the “amount” of
some vector quantity passing through a
hypothetical mathematical surface per unit
area. The surface may be either open or
closed (e.g., a control surface enclosing a
control volume immersed in a fluid
velocity field).
Notice that a system is a Lagrangian concept,
while the control volume employs a Eulerian
reference frame.
Judicious selection of control volume
boundaries, i.e., the control surface, relative to
a system of interest often makes analysis of fluid
flow phenomena simpler, and the results more
useful. Analysts have complete freedom in
selection of these boundaries.
Vector Flux Analogy –
Imagine a crowd of people standing shoulder-to-
shoulder in straight columns marching at speed v
toward an arena gateway having width w, but at an
arbitrary angle, , to the opening:
v
w
Also let,
= number of people per unit area [#/L2]
Q = rate at which people pass through the
doorway [#/T]
Case 1 –
People marching straight toward the opening,
i.e., = 0. Then, notice that:
wvQ 1
LL
people #
T
L
T
people #2
Case 2 (a) Effective Width –
People march toward the doorway at angle >
0, then the opening appears to the people as
having an “effective width,” e = w cos :
we
cos
grearrangin , cos
2
2
wvQ
wvQ
LL
people #
T
L
T
people #2
Case 2 (b) Normal Component of V –
Only the component of the marching velocity,
V, perpendicular or normal to the doorway
yields people “flux” through the opening, vn = v
cos :
wvn
cos
grearrangin , cos
2
2
wvQ
wvQ
LL
people #
T
L
T
people #2
vV
Notice that when =0, cos =1:
required as , 12 wvQQ
To put Case 2 (b) into vector notation, define
as the unit normal vector to the hypothetical
plane surface across the doorway opening. Then,
by definition of the dot product:
Thus, the previous result for Q2 can be written:
1
cos ˆ ˆ nVnV
n̂
vcosˆ vnV
wQ
wvQ
nV ˆ
cos
2
2
n̂V
Mass Flux in a Fluid Velocity Field –
Extend the vector flux analogy to a fixed, closed
control surface enclosing an arbitrary 3-
dimensional control volume in space immersed
in fluid velocity field . Define an outward
pointing unit normal vector (i.e., positive
algebraic sign when directed outward), then for
any differential area element, dA, on the control
surface the outflowing mass flow rate is given
by:
n̂
V
md dAdAVmddQmd n nV ˆ
Integrating over the entire CS, encompassing
various regions where the velocity vector is
directed inward, outward, and tangent to the CS
yields the net mass flow rate, :
is called net mass flow rate because:
CS
dAm nV ˆ
m
Mass flux, i.e., mass flow rate per unit control surface area
m
outward directed is when0ˆ (i) VnV
y
x
cos > 0cos < 0
cos < 0 cos > 0
0cos 9090
0cosˆ ˆ nVnVOutflow across CS is positive outflow
y
x
cos > 0cos < 0
cos < 0 cos > 0
0cos 27090 0cosˆ ˆ nVnV
Inflow across CS is negative outflow
inward directed is when0ˆ (ii) VnV
CS theo tangent tis when0ˆ (iii) VnV
n̂
VCS dA
0cos 90or 90
00 ˆ ˆ nVnV
y
x
cos > 0cos < 0
cos < 0 cos > 0
No mass transport across the CS
Reynolds Transport Theorem –
Consider the system for analysis as an arbitrarily
selected fluid element of mass, m, immersed in a
fluid velocity field, V. Choose the volume
occupied by the fluid element at some arbitrary
instant in time, t, as the control volume. Then, of
course, the surface of the fluid element at time t
becomes the control surface.
Let B represent the total amount of any extensive
physical property of the fluid element—velocity,
acceleration, mass, kinetic energy, momentum,
etc.—and let b represent the amount of the
physical property per unit mass of fluid. Then, of
course:
bmB
Now consider the material derivative of B at
time t, providing narrative definitions for
various combinations of terms using system-CV-
CS terminology:
z
Bw
y
Bv
x
Bu
t
B
Dt
DB
Time rate of change in B
within the CV
Instantaneous net transport rate of B across the CS
Total rate of change in B at time t within the system of mass m
Transforming the material derivative into an
equivalent narrative equation:
CS theacross B
of ratetransport
net ousInstantane
CV thewithin
in change
of rate Time
at time mass of
element fluid thewithin
in change of rate Total
B
tm
B
MaterialDerivative
Surface Integralover the CS
Volume Integralover the CV
Reconverting to mathematical symbolism
involving volume and surface integrals yields
Reynolds Transport Theorem (RTT):
CSCV
dAbVdbtDt
DBnV ˆ
Flux of B, i.e., transport rate of B per unit area
Amount of B per unit volume
[Equation 4.19, p. 183]