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€ The Reynolds Transport Theorem
Ú Correlation between System (Lagrangian) concept
↔ Control-volume (Eulerian) concept
for comprehensive understanding of fluid motion?
y 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)
r
r
e.g. a) If B = mV (Linear momentum): Extensive property
r r
then, b = V (Velocity): Intensive property
1
b) If B = mV 2 (Kinetic energy): Extensive property
2
1
then, b = V 2 : Intensive property
2
i. B of a system Bsys at a given instant,
∑ bi ( ρiδVi ) = ∫sys ρbdV
Bsys = lim
δV → 0 i
δmi for ith fluid particle in the system
where δVi : Volume of ith fluid particle
And Time rate of change of Bsys,
dBsys
dt
=
(
d ∫ sys ρbdV
)
dt
ii. B of fluid in a control volume Bcv
Bcv = lim
∑ bi ( ρiδVi ) = ∫cv ρbdV
δV → 0 i
and
d (∫cv ρbdV )
dBcv
=
dt
dt
Ú Relationship between
dBsys
dt
and
Only difference from
B of a system
dBcv
: Reynolds Transport Theorem
dt
€ Derivation of the Reynolds Transport Theorem
Consider 1-D flow through a fixed control volume shown
Fixed control surface at t (coincide with a system boundary)
System boundary at t + δt
a) At time t, Control volume (CV) & System (SYS): Coincide
b) At t + δt (after δt ),
CV: fixed
& SYS: Move slightly
) Fluid particles at section (1): Move a distance dl1 = V1δt
) Fluid particles at section (2): Move a distance dl2 = V2δt
) I : Volume of Inflow (entering CV)
) II : Volume of Outflow (leaving CV)
That is,
SYS (at time t)
= CV
SYS (at time t + δt )
= CV – I + II
Or if B: Extensive fluid property, then
Bsys(t) = Bcv(t)
(at time t)
Bsys (t + δt ) = Bcv (t + δt ) − BI (t + δt ) + BII (t + δt )
(at time t + δt )
Then, Time rate of change in B can be;
δBsys Bsys (t + δt ) − Bsys (t )
= Bcv(t), at time t
=
δt
δt
Bcv (t + δt ) − BI (t + δt ) + BII (t + δt ) − Bsys (t )
=
δt
B (t + δt ) − Bcv (t ) BI (t + δt ) BII (t + δt )
−
+
= cv
δt
δt
δt
In the limit δt → 0 ,
Left-side:
δBsys DBsys
=
δt
Dt
(according to Lagrangian Concept)
⎛
⎞
∂⎜ ∫ ρbdV ⎟
B
(
t
+
δ
t
)
−
B
(
t
)
∂
B
⎝
⎠
cv
= cv = cv
1st term on Right-side: lim cv
δt → 0
∂t
δt
∂t
B (t + δt )
2nd term on Right-side: B&in = lim I
= ρ1 A1V1b1
δt → 0
δt
(4.13)
because BI (t + δt ) = ( ρ1δV1 )b1 = ρ1 A1V1b1δt
where A1 : Area at section (1)
V1 : Velocity at section (1)
B (t + δt )
3rd term on Right-side: B& out = lim II
= ρ 2 A2V2b2
δt → 0
δt
(4.12)
because BII (t + δt ) = ( ρ 2δV2 )b2 = ρ 2 A2V2b2δt
Relationship between the time rate of change of Bsys and Bcv
∴
DBsys
Dt
=
∂Bcv &
∂B
+ Bout − B&in = cv + ρ 2 A2V2b2 − ρ1 A1V1b1
∂t
∂t
: Special version of Reynolds transport theorem
- Fixed CV with one inlet and one outlet
- Velocity normal to Sec. (1) and (2)
y General expression of Reynolds Transport Theorem
Consider a general flow shown
At time t,
CV & SYS: Coincide
At time t + δt , CV: Fixed & SYS: Move slightly
∂Bcv &
+ Bout − B&in
Dt
∂t
) Still valid,
but B& out & B&in : Different
DBsys
=
Ú What are B& out & B&in ?
1) B& out : Net flowrate of B leaving CV (Outflow)
across the control surface between II and CV ( CS out )
B across the area element δA on CSout
δB = bρδV = bρ (V cos θδt )δA
where δV (Fluid volume leaving CV across δA 
= δlnδA = δl cos θδA = (Vδt cos θ )δA
Then, the time rate of B across δA
ρbδV
( ρbV cos θδt )δA
= lim
= ρbV cos θδA
δt → 0 δt
δt → 0
δt
δB& out = lim
By integrating over the entire CS out ,
B& out = ∫CS
out
dB& out = ∫CS
out
ρbV cos θ dA = ∫CS
out
r
ρbV ⋅ nˆ dA
2) B& out : Net flowrate of B entering CV (Inflow)
across the control surface between I and CV ( CSin )
By the similar manner,
r
B&in = − ∫CS ρbV cos θ dA = − ∫CS ρbV ⋅ nˆ dA
in
(because
π
2
<θ <
3π
)
2
in
Finally, Net flowrate of B across the entire CS ( = CSin + CS out )
B& out − B&in = ∫CS
r
out
r
ρbV ⋅ nˆdA − (− ∫CS ρbV ⋅ nˆ dA)
in
r
= ∫CS ρbV ⋅ nˆ dA
DBsys
Dt
=
r
r
∂Bcv
∂
+ ∫CS ρbV ⋅ nˆ dA = ∫cv ρbdV + ∫CS ρbV ⋅ nˆ dA
∂t
∂t
: General expression of Reynolds Transport Theorem
€ PHYSICAL INTERPRETATION
y
DBsys
Dt
: Time rate of change of an extensive B of a system
) Lagrangian concept
∂
∫ ρbdV : Time rate of change of B within a control volume
∂t cv
) Eulerian concept
r
y ∫CS ρbV ⋅ nˆ dA : Net flowrate of B across the entire control surface
y
) Correlation term – Motion of a fluid
c.f. Comparison with the definition of Material Derivative
D( ) ∂( )
∂( )
∂( )
∂( ) ∂( ) r
=
+u
+v
+w
=
+ (V ⋅ ∇)(
Dt
∂t
∂x
∂y
∂z
∂t
y
y
)
D( )
Dt
: Time rate of change of a property of fluid particle
∂( )
∂t
: Time rate of change of a property at a local space
r
y (V ⋅ ∇)
) Lagrangian concept
) Eulerian concept: Unsteady effect
: Change of a property due to the fluid motion
) Correlation term – Convective effect
Ú Reynolds Transport Theorem
) Transfer from Lagrangian viewpoint to Eulerian one (Finite size)
Ú Special cases
DBsys
1. Steady Effects.
Dt
=
r
∂
ρ
bd
V
ρ
b
V
+
⋅ nˆ dA
∫
∫
cv
CS
∂t
) Any change in property B of a system
= Net difference in flowrates B& entering CV and leaving CV
2. Unsteady Effects.
∂
∫ ρbdV ≠ 0
∂t CV
) Any change in property B of a system
= Change in B within CV
+ Net difference in flowrates B& entering and leaving CV
e.g. For 1-D flow
r
V = V0 (t )iˆ
ρ = Constant
r r
r
r
r
Choose B = mV (Momentum), and thus b = B / m = V = V0 (t )iˆ
r
r
∫CS ρbV ⋅ nˆdA = ∫CS ρ (V0iˆ)V ⋅ nˆdA
= ∫(1) ρ (V0iˆ )(−V0 )dA + ∫( 2) ρ (V0iˆ )(V0 )dA + ∫side ρ (V0iˆ )(V0 cos 90o )dA
= − ρV0 2 Aiˆ + ρV0 2 Aiˆ = 0 (Inflow of B = Outflow of B)
∴
DBsys
Dt
=
∂
∫ ρbdV
∂t CV
: No convective effect
€ Reynolds Transport Theorem for a moving control volume
DBsys
Dt
=
r
∂
ρ
bd
V
ρ
b
V
+
⋅ nˆ dA
∫
∫
cv
CS
∂t
: Valid for a stationary CV
In case of moving control volume as shown,
r
Consider a constant velocity of CV = Vcv
Ú Reynolds transport theorem
: Relation between a system and CV, (Neglect the surrounding)
) Velocity of a system: Defined w.r.t. the motion of CV
r r r
) Relative velocity of a system: W = V − VCV
r
where V : Absolute velocity of a system
Finally,
r
∂
= ∫cv ρbdV + ∫CS ρbW ⋅ nˆ dA
Dt
∂t
DBsys
r
: Valid for a stationary or moving CV with constant Vcv