Fluid Mechanics (C.V. analysis) - uni
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Transcript Fluid Mechanics (C.V. analysis) - uni
Fluid Mechanics (C.V. analysis)
Dept. of Experimental Orthopaedics and Biomechanics
Bioengineering
Reza Abedian (M.Sc.)
Analysis of a Fluid Sys. Flowing through a C.V.
• We use this general representation of a flowing F.S. through a C.V.to
develop a relationship between a F.S. and a C.V. known as:
– The control volume approach, or
– The Raynolds transport theorem
•
Note: the C.V. approach is also called the Eulerian Approach in contrast to the
Lagrangian approach which describes the motion of individual particle as a function of
time
Analysis of a Fluid Sys. Flowing through a C.V.
• Let X = total amount of a fluid property (Mass, Energy, Momentum)
– At time t, C.V. and C.S. coincide, (Xs)t= (XC.V.)t
– At time t + dt , F.S has moved and changed shape
inC .V . volume of fluid entered C.V. in dt carrying
X Cin.V .
Cout.V . volume of fluid leaving C.V. in dt carrying
X Cout.V .
• Thus the amount of Xs at time t + dt is
• Subtraction
( X s )t t ( X C .V . )t t X Cout.V . X Cin.V .
( X s )t t ( X s )t ( X C .V . )t t ( X C .V . )t t X Cout.V . X Cin.V .
X s X C .V . X Cout.V . X Cin.V .
• Derivation
X s X C .V . X Cout.V . X Cin.V .
t
t
t
t
Continuation of Derivation of Reynold’s Transport
Theorem:
• Last result
X s X C .V . X Cout.V . X Cin.V .
t
t
t
t
• Take limit as t 0
• Reynold’s Transport Theorem
dX s dX C .V . dX Cout.V . dX Cin.V .
dt
dt
dt
dt
Analyzing the Theorem
dX s dX C .V . dX Cout.V . dX Cin.V .
dt
dt
dt
dt
1
2
3
4
1: rate of change of total amount of extensive property X within the
moving fluid system
2: rate of change of X contained within the fixed control volume
3: rate of outflow of X through the control surface downstream
4: rate of inflow of X through the control surface upstream
3 – 4 net rate of out flow of X passing through the C.S
• EQUATION STATES that the difference between the rate of change of
X within the system and that within the control volume is equal to the
net rate of outflow from the control volume
dX s dX C .V . dX Cout.V . dX Cin.V .
dt
dt
dt
dt
Why is the Reynold’s transport theorem Important?
• Laws of physics (Newtons 2nd law or conservation of momentum,
conservation of energy and mass) apply to fluid systems…
• It is difficult to follow a fluid system as if flows (Lagrangian approach)
• It is easier to observe fluid characteristics (density, velocity,
pressure, momentum, etc at fixed points in space that is using a
C.V. (Eulerian approach)
• Thus we need a way to relate the behavior of a fluid system (S) to
the quantities that can be observed within a control volume
• Solution: the Reynold’s Transport Theorem
dX s dX C .V . dX Cout.V . dX Cin.V .
dt
dt dt
dt
System
Quantity
C .V . related
Quantities
Equation of Continuity
• No flow across the stream tube sides
• Only flow through the ends of streamtube
• Let X=m in Reynold’s transport theorem
• Conservation of mass
dms dmC .V . dmCout.V . dmCin.V .
dt
dt
dt
dt
dms
0
dt
Equation of Continuity
dmC .V . dmCin.V . dmCout.V . d
d
(t , x, y, z )
( C .V . .) C .V . C .V .
dt
dt
dt
dt
dt
t
mCout.V . 2 .Volume2 2 .(v2 t ) A2
mCout.V .
2 v2 A2
t
t 0
dmCout.V .
2 v2 A2
dt
dmCin.V .
Similarly :
1v1 A1
dt
C .V .
2 A2 v2 1 A1v1
t
1 A1v1 2 A2 v2 C .V .
t
0
net mass in flow
in C .V . per unit time
rate of change
of density within C .V .
Differential Form of the Equation of Continuity
• Mass flux: mass flow rate = mass/area.time
• u,v and w are all positive and velocities in each directions are as
follows:
velocity u
– In upstream face in x density
– In downstream face x
u
velocity
u
dx
x
density dx
x
fluxes
mC .V . .dx.dy.dz
(1)
dmCin.V .
udydz vdxdz wdxdy
( 2)
dt
dmCout.V .
u
v
w
dx u dx dydz
dy v dy dxdz
dz w
dz dxdy
dt
x
x
y
y
z
z
u
u
udydz vdxdz wdxdy
dxdydz u
dxdydz
(dx) 2 dydz
x
x
x x
flowrate in
v
v
w
w
dxdydz v
dxdydz
(dy ) 2 dxdz
dxdydz w
dxdydz
(dz ) 2 dxdy
y
y
y y
z
z
z z
dmCin.V .
dxdydz ( u ) ( v) ( w) higher order terms are negligible
dt
y
z
x
•
Reynold´s Transport Theprem:
dms
dmC .V .
dm out C .V . dm in C .V .
dt
dt
dt
dt
in
dm in C .V .
dm C .V .
0
( dxdydz)
d
u
v
w
x
t
dt
y
z
dt
d
•
t
d
x u y v z w
u
v
w 0
t
x
y
z
d
Differential form of the continuity equation for any kind of flow
3
Vector analysis notation
• In Cartesian cordinates (x,y,z) the unit vectors are
e x iˆ
e y ˆj
e z kˆ
• We use scalar functions Q(x,y,z,t) such as density or temperature as
well as vector functions F(x,y,z,t) such as velocity and momentum
• The „del“ or „nabla“ operator is defined as:
iˆ
ˆ ˆ
j
k
x
y
z
• Gradient of a scalar function:
ˆ ˆ
grad iˆ
j
k
x
y
z
• Divergence of a vector function:
F
F
F
y
div.F .F iˆ ˆj kˆ . Fxiˆ Fy ˆj Fz kˆ x
z
y
z
x
y
z
x
Application to fluid dynamics:
• Let q be the velocity vector of a flow in Cartesian coordinates and ρ
the density,consider the vector flux ρq:
q uiˆ vˆj wkˆ
• Then the divengence of ρq is:
div.( q ) .( q ) u v w
x
y
z
• Then the differential form of the continuity equation is:
.( q ) 0
t
• Rule of derivatives of product:
u v w
u
u
v w
v
w
x
y
z
x y z x
y
z
T1
T1 .q
T2
ˆ ˆ ˆ
v
w
uiˆ vˆj wkˆ .
i
j
k q.
x
y
z
x
y
z
u
v w .q q.
x
y
z
T2 u
Special form of the continuity equation:
• For an incompressible flow:
const .
0
t
0 .q 0
.q 0
u v w
0
x y z
Vorticity
• Another operation related to the „del“ or „Nabla“ operator is the Curl
or Rotational of a vector function F=(FxFyFz)
iˆ
curl .F rot .F F
x
cross
product
Fx
F Fy
iˆ z
y
z
ˆj
y
Fy
kˆ
z
Fz
F
ˆj Fz Fx kˆ y Fx
z x
y
x
• The curl operation is used to define the vorticity of a flow
q
• The vorticity is related to the angular velocity if a fluid particles
• An irrotational flow is one that has no angular velocity, for a 2D flow
2
in xy plane:
iˆ
F
x
u
ˆj
y
v
kˆ
v u
kˆ
z
x y
0
irrotation al flow
v u
0
x y