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2011 Sajid
Chapter
Dr Muhammad Sajid
Assistant Professor
NUST, SMME.
Reference Text:Fundamental of
Computational Fluid
Dynamics, J. Anderson.
Email: [email protected]
Tel: 9085 6065
Computational Fluid
Dynamics
4-Nov-130
3 Governing Equations
Review
Conservation of mass
Conservation of LinearMomentum
Navier Stokes Equation
Computational Fluid Dynamics
Introduction
CFD is fundamentally based on the governing
equations of fluid dynamics.
These equations represent mathematical
statements of the laws of physicsregarding the
conservation of mass, momentum and energy.
The purpose of this chapter is to introduce thederivation and discussion of these equations.
All of CFD is based on these equations; we must
therefore begin our understanding at the most
basic description of the fluid flow processes.
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Computational Fluid Dynamics
Introduction
After these equations are obtained, forms suited
for use in formulating CFD solutions will be
highlighted.
At the end of this chapter, some of the mysteries
surrounding CFD based predictions of fluid flow
problems will be replaced with an understanding
of the equations governing the fluid transport.
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2
Computational Fluid Dynamics 3
Review of basic concepts
Fluid properties:
Physical laws are stated in terms of parameters like v, a etc.
Let represent a fluid parameter/property and the amountof that parameter per unit mass, i.e. = m.
is extensive property and is an intensive property.
Fluid element is a volume stationary in space,
Fluid particle is a volume of fluid moving with the flow. A fluid particle in motion experiences two rates of
changes:
Due to changes in the fluid as a function of time.
Due to the fact that it moves to a different location in the fluidwith different conditions.
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Computational Fluid Dynamics 4
Review of basic concepts
The sum of these two rates of changes for aproperty per unit mass (extensive) is called
the totalor substantivederivative D/Dt:
With dx/dt=u, dy/dt=v, dz/dt=w, this results in:
dt
dz
zdt
dy
ydt
dx
xtDt
D
4-Nov-13
.u
tDt
D
w
vu
u zyx
zw
yv
xu
tDt
D
Computational Fluid Dynamics 5
Review of basic concepts
Fluid element and properties The behavior of the fluid is
described in terms of macroscopicproperties: Velocity u.
Pressure p.
Density r.
Temperature T. Energy E.
dy
dx
dz(x,y,z)
1 1
2 2W E
p pp p x p p x
x xd d
Properties at faces are expressed as first
two terms of a Taylor series expansion,
e.g. for p : and
Properties are averages of a sufficiently largenumber of molecules.
A fluid element can be thought of as the smallestvolume for which the continuum assumption isvalid.
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Computational Fluid Dynamics
Review of basic concepts
Field representation The flow field of a fluid can be thought of as being
comprised of a large number of finite sized fluid particleswhich have mass, momentum, internal energy, and otherproperties.
The distribution of fluid parameters (r, v, P& a) overspace and time is called field representation.
Velocity field Representation of fluid velocity as function of spatial
coordinates and time, V = f(x,y,z,t).
4-Nov-13
6
k),,,(j),,,(i),,,( tzyxwtzyxvtzyxuV
222 wvuVV
Computational Fluid Dynamics
Review of basic concepts
System:A fixed quantity of matter, and no mass is allowed to
cross the system boundary.
Control Volume:A space of interest where mass can cross the boundary,
cv.
Control Surface: The boundary of the control volume is called the controlsurface, cs.
Reynolds transport theorem
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7
dAnVVdtDt
D
cscv
sys
rr
()()()
V
tDt
D
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Computational Fluid Dynamics
Review of basic concepts
Lagrangian approach: Fluid particles are tagged/identified
and their properties are determined
as they move in space.
Eulerian approach:
Fluid properties are determined at
fixed points in space as fluid flows
by.
Governing equations can bederived using each method and
converted to the other form.
4-Nov-13
8
Computational Fluid Dynamics 10
The governing equations include the followingconservation laws of physics: Conservation of mass.
Newtons second law: the change of momentumequals the sum of forces on a fluid particle.
First law of thermodynamics (conservation of energy):
rate of change of energy equals the sum of rate of heataddition to and work done on fluid particle.
The fluid is treated as a continuum. For lengthscales of, say, 1m and larger, the molecularstructure and motions may be ignored.
Governing equations4-Nov-
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Computational Fluid Dynamics
Conservation of mass
Time rate of change of system mass = 0
For a fixed and nondeforming control volume:
t - dt, t (coincident), t + dt
From Reynolds transport theorem, we have.
4-Nov-13
11
0sysmDt
D sys
sys Vdm r
dAnVVdt
mDt
D
cscv
sys
rr
dAnVVdtDt
D
cscv
sys
rr
Computational Fluid Dynamics
Conservation of mass & Continuity
Or,
The continuity equation is an expression for the
conservation of mass in a control volume.
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13
12
rfacecontrol su
orughof mass th
f flownet rate o
lumecontrol vo
side theof mass in
of changetime rate
systemcoincident
s of theof the mas
of changetime rate
cv
Vdt
lumecontrol vo
side theof mass in
of changetime rate
r
cs
dAnV
rfacecontrol su
roughof mass th
f flownet rate o
r
0 dAnVVdt
cscv
rr
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Computational Fluid Dynamics
Conservation of mass
Consider a differential fluid element.
Let density and velocities components at the
center of the element be , u, v and w.
The volume integral can be expressed as.
Next, we need to find the mass flow rate
through the surfaces
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13
0
dAnVVdtcscv
rr
zyxt
Vdt
cv
dddr
r
Computational Fluid Dynamics
Conservation of mass
If ru is the horizontal mass flux at the center of
the element than at the faces the mass flux in
horizontal direction is.
Net rate of mass flowing through the surfaces is.
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14
zyx
x
uzy
x
x
uuzy
x
x
uu ddd
rdd
drrdd
drr
22
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Computational Fluid Dynamics 15
Conservation of mass
The inflows (positive) and outflows (negative)in all directions are:
x
yz
( ) 1.2
ww z x y
z
rr d d d
( ) 1
.2
vv y x z
y
rr d d d
zyxx
uu ddd
rr
2
1.
)(
zxyy
vv ddd
rr
2
1.
)(
yxzz
ww ddd
rr
2
1.
)(
( ) 1.2
uu x y z
x
rr d d d
4-Nov-13
Computational Fluid Dynamics
Conservation of mass
Mass flow rate in the y and z directions is.
Combining these equations we get.
Thus,
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zyx
y
vddd
r
directiony''inrateflowmassnet
zyxz
w
y
v
x
udAnV
cs
dddrrr
r
rateflowmassnet
0
zyx
z
w
y
v
x
uzyx
tddd
rrrddd
r
zyx
y
wddd
r
directionz''inrateflowmassnet
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Computational Fluid Dynamics 17
Continuity equation
Summing all terms in the previous slide anddividing by the volume dxdydzresults in:
In vector notation:
For incompressible fluids r/ t=0, and the
equation becomes: div u= 0. Alternative ways to write this:
0)()()(
zw
yv
xu
trrrr
0)(
urr
divt
Change in densityNet flow of mass across boundaries
Convective term
0
zw
yv
xu 0
i
i
x
u
4-Nov-13
Computational Fluid Dynamics
CONSERVATION OF MOMENTUM
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Computational Fluid Dynamics 22
Momentum equation in three dimensions
Newtons second law:the sum of forces equals the rate of change ofmomentum.
Rate of increase of momentum along x, y, and zaxis's.
Forces on fluid particles are: Surface forces such as pressure and viscous forces.
Body forces, which act on a volume, such as gravity,centrifugal and electromagnetic forces.
Dt
Dw
Dt
Dv
Dt
Durrr
4-Nov-13
()()()
V
tDt
D
Computational Fluid Dynamics
Conservation of Momentum
The surface forces are due to the stresses,
exerted on the sides of the fluid element.
Stresses are forces per area. = N/m2or Pa.
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23
Two types of stresses:
normal stress (ij), often shown as
(ij) and
shear stress (ij).
i refers to the axis normal to the surface,
j represents the direction of the stress.
Forces in direction of an axis are
positive, else negative.
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Computational Fluid Dynamics
All surface forces acting in the x-direction on thefluid element are:
Conservation of momentum4-Nov-13
24
x
z
y
zyxx
pp ddd )
2
1.(
zyxx
pp ddd )
2
1.(
zyzz
zxzx
ddd
)2
1.(
yxzz
zxzx
ddd
)2
1.(
zxyy
yx
yx ddd
)
2
1.(
zxyy
yx
yx ddd
)
2
1.(
zyxx
xxxx
ddd
)2
1.(
zyxx
xxxx ddd
)
2
1.(
Computational Fluid Dynamics 25
Conservation of Momentum
Set the rate of change of x-momentum for a
fluid particle Du/Dt equal to:
the sum of the forces due to surface stresses
shown in the previous slide, plus
the body forces. These are usually lumped
together into a source term SM:
p is a compressive stress and xxis a tensile
stress.
Mxzxyxxx S
zyx
p
Dt
Du
r
)(
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Computational Fluid Dynamics 26
Conservation of Momentum
Similarly for y- and z-momentum:
Mxzxyxxx S
zyx
p
Dt
Du
r
)(
My
zyyyxyS
zy
p
xDt
Dv
r
)(
Mzzzyzxz
Sz
p
yxDt
Dw
)( r
4-Nov-13
Computational Fluid Dynamics
Cauchys equation of motion
It is an expression for the acceleration, the
body forces, the pressure gradient forces, and
the viscous forces coupled with Newtons 2nd
Law:
Note that this is an incredibly succinct
equation, and is much more complicated than
it at first appears.
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ij
PkgDt
vDrr
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Computational Fluid Dynamics
Cauchys equation of motion
The full expanded version can be written foreach component,x, y, and zas:
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28
zyxzPkg
zvv
yvv
xvv
tv
zyxy
P
z
vv
y
vv
x
vv
t
v
zyxx
P
z
vv
y
vv
x
vv
t
v
zyyyxyzz
zy
zx
z
zyyyxyy
z
y
y
y
x
y
zxyxxxx
z
x
y
x
x
x
rr
r
r
Computational Fluid Dynamics
Cauchys equation of motion
Things to notice about these equations: only the zcomponent equation has a body force,
because gravity only works in the zdirection;
thex- and y-component equations are identical exceptfor subscripts;
the equations cannot be solved in their present form
because the stresses have not yet been recast interms of velocities.
In order to solve this equation for a specificsubstance, like a fluid, we need to substituteexpressions involving velocity for the viscousstress gradients.
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Computational Fluid Dynamics
Constitutive Relationship for Viscous Fluids
The mechanical behavior of every substance canbe described in terms of stress and strain, and the
relationship between these variables is called a
constitutive relationship.
Generally, these must be determined through
experiments and differ for every type of substance
(i.e., rock, plastic, fluid, gas, etc.).
Mathematically, the relationship is between the
stress tensor and the strain tensor (for rigidsolids) or strain-rate tensor (for fluids).
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30
Computational Fluid Dynamics
Constitutive Relationship for Viscous Fluids
Strain is a measure of distortion and the strain-rate tensor has nine components just like thestress tensor:
The diagonal entries , , and representnormal strain rates (elongation, contraction) andthe off-diagonal strains represent shear strainsrates. Remember
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zzzyzx
yzyyyx
xzxyxx
ij
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Computational Fluid Dynamics
Strain and strain rate
The strain rates in the strain-rate tensor can be describedin terms of velocity gradients.
Consider a small elongate element of fluid moving in the
x-direction with a non-constant velocity.
The element is stretching as it is moving, resulting in a
normal strain-rate in thex-direction.
The element has a length ofxand undergoes strain
which stretches it tox+ xin the time t. xis the
small elongation ofxand is always smaller thanx.
The strain xxis:
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x
xxx
d
d
Computational Fluid Dynamics
Strain and strain rate
There are different ways to determine the
relationship between strain rate and velocity gradients.
First Method:
The only way the box can deform (i.e., stretch; i.e.,
strain) is if the right-hand side moves faster than the
left-hand slide, which happens when > 0.
The rateof strain will equal because this is the
amount by which the right-hand side is moving faster
than the left-hand side.
So we can deduce the answer as: =
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Computational Fluid Dynamics
Strain and strain rate
Second Method: Take the expression for strain = and
divide by the time increment, t;
The term in parentheses is the rate at which the
increment xgrows with time or differential velocity,
the difference in velocity between the left and right
sides of the original box.
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x
v
x
vx
xt
x
xt
xxxxxx
11
d
d
d
d
x
vxv
x
vxvvv
t
x xL
x
LLR
d
d
x
xxx
d
d
t
x
xtx
x
t
xx
d
d
d
d
d
d 1
Computational Fluid Dynamics
Strain and strain rate
Similar arguments show that the other diagonal
elements in the strain-rate tensor, are:
This is an example how just one of the 9
components of the strain-rate tensor is related to
the gradients of velocity in thex, y, and zdirections.
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z
v
y
vz
zz
y
yy
;
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Computational Fluid Dynamics
Strain and strain rate
The full strain-rate tensor can be expressed in terms ofvelocity gradients as follows:
This is a symmetric tensor across the diagonal elements. This strain-rate tensor is valid for all materials, including fluids.
It expresses the strain rates as a function of the velocity
gradients, and is constructed entirely from geometry.
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z
v
y
v
z
v
x
v
z
v
y
v
z
v
y
v
x
v
y
v
x
v
z
v
x
v
y
v
x
v
zzyzx
zyyyx
zxyxx
zzzyzx
yzyyyx
xzxyxx
ij
2
1
2
1
2
1
2
1
2
1
2
1