The Reynolds Transport Theorem
ø Correlation between System (Lagrangian) concept
↔ Control-volume (Eulerian) concept
for comprehensive understanding of fluid motion?
Reynolds Transport Theorem
Let’s set a fundamental equation of physical parameters
B = mb where B: Fluid property which is proportional to
amount of mass (Extensive property)
b: B per unit mass (Independent to the mass)
(Intensive property)
e.g. a) If (Linear momentum): Extensive property
then, (Velocity): Intensive property
b) If (Kinetic energy): Extensive property
then, : Intensive property
i. B of a system Bsys at a given instant,
=
where : Volume of ith fluid particle
And Time rate of change of Bsys,
=
ii. B of fluid in a control volume Bcv
=
and
=
ø Relationship between and : Reynolds Transport Theorem
Derivation of the Reynolds Transport Theorem
Consider 1-D flow through a fixed control volume shown
a) At time t, Control volume (CV) & System (SYS): Coincide
b) At (after ), CV: fixed & SYS: Move slightly
F Fluid particles at section (1): Move a distance
F Fluid particles at section (2): Move a distance
F I : Volume of Inflow (entering CV)
F II : Volume of Outflow (leaving CV)
That is, SYS (at time t) = CV
SYS (at time ) = CV – I + II
Or if B: Extensive fluid property, then
Bsys(t) = Bcv(t) (at time t)
(at time )
Then, Time rate of change in B can be;
=
=
In the limit ,
Left-side: = (according to Lagrangian Concept)
1st term on Right-side: = =
2nd term on Right-side: (4.13)
because
where : Area at section (1)
: Velocity at section (1)
3rd term on Right-side: (4.12)
because
Relationship between the time rate of change of and
\
: Special version of Reynolds transport theorem
- Fixed CV with one inlet and one outlet
- Velocity normal to Sec. (1) and (2)
General expression of Reynolds Transport Theorem
Consider a general flow shown
At time t, CV & SYS: Coincide
At time , CV: Fixed & SYS: Move slightly
F Still valid,
but : Different
ø What are ?
1) : Net flowrate of B leaving CV (Outflow)
across the control surface between II and CV ()
B across the area element on
where (Fluid volume leaving CV across
Then, the time rate of B across
By integrating over the entire ,
2) : Net flowrate of B entering CV (Inflow)
across the control surface between I and CV ()
By the similar manner,
(because )
Finally, Net flowrate of B across the entire CS ()
=
: General expression of Reynolds Transport Theorem
PHYSICAL INTERPRETATION
: Time rate of change of an extensive B of a system
F Lagrangian concept
: Time rate of change of B within a control volume
F Eulerian concept
: Net flowrate of B across the entire control surface
F Correlation term – Motion of a fluid
c.f. Comparison with the definition of Material Derivative
: Time rate of change of a property of fluid particle F Lagrangian concept
: Time rate of change of a property at a local space F Eulerian concept: Unsteady effect
: Change of a property due to the fluid motion
F Correlation term – Convective effect
ø Reynolds Transport Theorem
F Transfer from Lagrangian viewpoint to Eulerian one (Finite size)
ø Special cases
1. Steady Effects.
F Any change in property B of a system
= Net difference in flowrates entering CV and leaving CV
2. Unsteady Effects.
F Any change in property B of a system
= Change in B within CV
+ Net difference in flowrates entering and leaving CV
e.g. For 1-D flow
Constant
Choose (Momentum), and thus
(Inflow of B = Outflow of B)
\ = : No convective effect
Reynolds Transport Theorem for a moving control volume
: Valid for a stationary CV
In case of moving control volume as shown,
Consider a constant velocity of CV =
ø Reynolds transport theorem
: Relation between a system and CV, (Neglect the surrounding)
F Velocity of a system: Defined w.r.t. the motion of CV
F Relative velocity of a system:
where : Absolute velocity of a system
Finally,
: Valid for a stationary or moving CV with constant