# Reynolds—Transport Theorem (RTT)

```(RTT)
• Most principles of fluid mechanics are adopted from solid
mechanics, where the physical laws dealing with the time
rates of change of extensive properties are expressed for
systems.
• In fluid mechanics, it is usually more convenient to work
with control volumes, and thus there is a need to relate the
changes in a control volume to the changes in a system.
• The relationship between the time rates of change of an
extensive property for a system and for a control volume is
expressed by the Reynolds transport theorem (RTT), which
provides the link between the system and control volume
• RTT is named after the English engineer, Osborne
Reynolds (1842– 1912), who did much to advance its
application in fluid mechanics.
F
net external force
on an object


m
mass of the object
a
acceleration
First law of thermodynamics
Ein  Eout 
net energy intering
a system
Esystem
change in the total
energy of the system
 cp 
m
 T
mass of the object
• The laws in their basic forms are stated in terms of
system a system (or Object): a collection of matter of
fixed identity (always coame atoms or molecules and
always containing same mass no matter moves and
interacts with its surrounding).
• System approach is NOT very suitable for the
applications in fluid mesh is hard to identify and track a
group of fluid since moves quite instead, control
volume (CV) approach is more often used in fluid mesh.
How to choose a control volume?
• CV is arbitrarily chosen by fluid dynamicist, however,
selection of CV can either simplify or complicate analysis;
• Clearly define all boundaries. Analysis is often simplified
if CS is normal to flow direction;
• Only CS conditions needed! Do not require detailed
information inside CV.
• Clearly identify all fluxes crossing the CS;
• Clearly identify forces &amp; torques of interest acting on the
CV and CS.
Reynolds—Transport Theorem (RTT)
Bsys,t  BCV ,t (the system and CV concide at time t)
Bsys,t  t  BCV ,t  t  B,t  t  B,t 
Bsys ,t  t  Bsys ,t
t

dBsys
dt
BCV ,t  t  BCV ,t

t
B ,t 
t
t
dBCV

 Bin  Bout
dt
t

B ,t 
B,t  t  b1m,t  t  b11V,t  t  b11V1 tA1
B,t  t  b2m,t  t  b2 2V,t  t  b2 2V2 tA2
b11V1 tA1
 b11V1 A1
t 0
t

0
t
t
B ,t  t
b  V tA
Bout  B  lim
 lim 2 2 2 2  b2 2V2 A2
t 0
t 0
t
t
Bin  B  lim
dBsys
dt

B ,t 
t
 lim
dBCV
 b11V1 A1  b2 2V2 A2
dt
t
t
Reynolds—Transport Theorem (RTT)
• the time rate of change of
the property B of the system
is equal to the time rate of
change of B of the control
volume plus the net flux of B
out of the control volume by
mass crossing the control
surface.
Bnet  Bout Bin   bV ndA (inflow if negative)
CS
Reynolds—Transport Theorem (RTT)
The total amount of property B within the control volume
must be determined by integration:
BCV    bdV
CV
Therefore, the system-to-control- volume transformation
for a fixed control volume:
dBsys
d
   bdV   bV ndA
CS
dt
dx CV
Material derivative (differential analysis):
Db b

 (V  )
Dt t
General RTT, non-fixed CV (integral analysis):
dBsys
dt

d
 bdV    bV ndA

CS
dx CV
Reynolds—Transport Theorem (RTT)
• Interpretation of the RTT:
– Time rate of change of the property B of the system is
equal to (Term 1) + (Term 2)
– Term 1: the time rate of change of B of the control
volume
– Term 2: the net flux of B out of the control volume by
mass crossing the control surface


(  b)dV '   bV ndA
CV
CS
dt
t
dBsys
Moving control volume
Fixed, moving, and deforming
control volumes
-For moving CV, use relative
velocity in the surface integral;
-For deforming CV, use relative
velocity on all the deforming
control surfaces,
W = V −Vcv
W = V −Vcs
For moving and/or deforming control volumes,
dBsys
d

 bdV    bVr ndA

CS
dt
dx CV
• Where the absolute velocity V in the second term is
replaced by the relative velocity Vr = V –VCS
• Vr is the fluid velocity expressed relative to a coordinate
system moving with the control volume.
```