Method to Use Conservations Laws in Fluid Flows…… P M V Subbarao Professor Mechanical...
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Transcript of Method to Use Conservations Laws in Fluid Flows…… P M V Subbarao Professor Mechanical...
Method to Use Conservations Laws in Fluid Flows……
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Mathematics of Reynolds Transport Theorem
The Field quantity
• The field quantity f(x,t) may be a zeroth, first or second order tensor valued function.
• Namely, as mass, concentration, velocity vector, and stress tensor.
• A fluid element of given initial volume (dV0) mya change its volume and/or change it surface (ds0) with a given time, while moving through the flow field.
• This is due to various experiences by the element namely, dilatation, compression and deformation.
• Let us consider the same fluid particles at any time and therefore, it is called the material volume.
The volume integral of the quantity f(x,t):
This is a function of time only.
The integration must be carried out over the varying volume V(t).
The material change of the quantity F(t) is expressed as:
Since the shape of the volume V(t) changes with time, the differentiation and integration cannot be interchanged.
tV
dVtxftF ,
tV
dVtxfDt
D
Dt
tDF,
This analogy permits the transformation of the time dependent volume V(t) into the fixed volume V0 at time t = 0 by using the Jacobian transformation function:
odVJtdV
With this operation it is possible to interchange the sequence of differentiation and integration:
The chain differentiation of the expression within the parenthesis results in
0
0,V
JdVtxfDt
D
Dt
tDF
0
0
,
V
dVDt
JtxfD
Dt
tDF
0
0,,
V
dVDt
DJtxf
Dt
txDfJ
Dt
tDF
tV
dVtxfDt
D
Dt
tDF,
Introducing the material derivative of the Jacobian function and simplifying based on derivatives of transformation functions
JvJvt
J
Dt
DJ ..
0
0.,,
V
JdVvtxfDt
txDf
Dt
tDF
This equation permits the back transformation of the fixed volume integral into the time dependent volume integral.
tV
dVvtxfDt
txDf
Dt
tDF
.,,
tV
dVvtxftxfvt
txf
Dt
tDF
.,,.,
The chain rule applied to the second and third term yields:
tV
dVvtxft
txf
Dt
tDF
,.,
The second volume integral in above equation can be converted into a surface integral by applying the Gauss' divergence theorem:
tstV
dsnvtxfdVt
txf
Dt
tDFˆ.,
,
dVvtxfdVt
txf
Dt
tDF
tVtV
,.
,
This Equation is valid for any system boundary with time the dependent volume V(t) and surface s(t) at any time.Also valid at the time t = t0 , where the volume V = VC and the surface s = sC assume fixed values. We call VC and sC the control volume and control surface.These control surfaces can be inlets or exits.
tstV
dsnvtxfdVt
txf
Dt
tDFˆ.,
,
tststV inletexit
dsnvtxfdsnvtxfdVt
txf
Dt
tDFˆ.,ˆ.,
,
m
j ts
n
i tstV jinletiexit
dsnvtxfdsnvtxfdVt
txf
Dt
tDF
11,,
ˆ.,ˆ.,,
Thermodynamic form of RTT
n
iiinlet
n
iiexit
CMCV
FFdt
tdF
dt
tdF
1,
1,
V
CV dVtxfF ,
inletinlet A
in
A
inlet AdvfAdvfF ..
exitexit A
exit
A
exit AdvfAdvfF ..
Steady State Steady Flow Thermodynamic Model
n
iiinlet
n
iiexit
CV
FFdt
tdF
1,
1,
V
CV dVxfF
inletinlet A
in
A
inlet AdvfAdvfF ..
exitexit A
exit
A
exit AdvfAdvfF ..
Uniform State Uniform Flow Thermodynamic Model
n
iiinlet
n
iiexit
CMCV
FFdt
tdF
dt
tdF
1,
1,
V
CV dVtfF
inletinlet A
in
A
inlet AdvfAdvfF ..
exitexit A
exit
A
exit AdvfAdvfF ..
Mass Flow Balance in Stationary Frame of Reference
• The conservation law of mass requires that the mass contained in a material volume V=V(t), must be constant:
tV
dVm
Consequently, above equation requires that the substantial changes of the above mass must disappear:
Mass contained in a material volume
0
tV
dVDt
D
Dt
Dm
Using the Reynolds transport theorem , the conservation of mass, results in:
0.
tVtV
dVvt
dVDt
D
This integral is zero for any size and shape of material volume. Implies that the integrand in the bracket must vanish identically. The continuity equation for unsteady and compressible flow is written as:
0.
vt
This Equation is a coordinate invariant equation. Its index notation in the Cartesian coordinate system given is:
0
i
i
x
v
t
0
3
3
2
2
1
1
x
v
x
v
x
v
t
Continuity Equation in Cylindrical Polar Coordinates
• Many flows which involve rotation or radial motion are best described in Cylindrical Polar Coordinates.
• In this system coordinates for a point P are r, and z.
The velocity components in these directions respectively are vr ,v and vz. Transformation between the Cartesian and the polar systems is provided by the relations,
22 yxr
x
y1tan
The gradient operator is given by,
zz
rrr
ˆˆ1ˆ
As a consequence the conservation of mass equation becomes,
0
11
z
vv
rr
vr
rtzr
Continuity Equation in Cylindrical Polar CoordinatesSpherical polar coordinates are a system of curvilinear coordinates
that are natural for describing atmospheric flows.Define to be the azimuthal angle in the x-y -plane from the x-axis with 0 < 2 . to be the zenith angle and colatitude, with 0 < r to be distance (radius) from a point to the origin.
The spherical coordinates (r,,) are related to the Cartesian coordinates (x,y,z) by
222 zyxr
x
y1tan
r
z1cos