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]