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CFD Equations
Chapter 1
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Navier-Stokes Equations, Conservation of Mass, and the Energy Equation
• Definition of the Equations
• The Continuity Equation
• The Stress-Strain Relation
• Forms of the Equations
• Important Properties
• Dimensionless Parameters
• Dimensionless Equations
• The Energy Equation
• Dimensionless Energy Equation
• Rotational Frames of Reference
• Swirl
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• It is common to use vector and tensor notation to describe the equations compactly. Sometimes, within a single equation it is convenient to use different notation for different terms.
• Some ways to express the Continuity Equation (conservation of mass):
• Scalar Equation:
where U,V,W are velocities in orthogonal x,y,z directions
• Vector form: where V is the vector of velocity
0
z
W
y
V
x
U
t
0
Vt
Conservation of Mass (and notation…)
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• Indicial notation - repeated subscript means summing the terms:
• ui represents velocities in the three xi directions
• The substantial derivative is particularly useful in describing transport. The operator is:
• and its use yields a fourth expression of continuity:
0
i
i
x
u
t
VtDt
D
0VρDt
ρD
Conservation of Mass… (con’t)
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• Note that for the case of constant density:
• The means divergence of the velocity is zero in constant property flows. This is often checked as a condition of accuracy or convergence in computational fluids.
• It is fairly common for the continuity equation to serve as a link in determining the pressure in computational algorithms. Generally, you can assume that the Navier-Stokes equation provides the velocities in response to the pressure field.
0
i
i
x
uV
Remarks on Continuity
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• The velocity or its gradient is put in terms of the pressure gradient in some fashion. In such a case, the density change must be put in terms of pressure.
• Introduce the Bulk modulus:
So that
• This term is responsible for the rate at which sound waves will propagate. The higher the value of K, the faster the propagation of the wave.
pt
p
t
ρρ
T
PK
Kt
p
t
More Remarking
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S
sound
PV
Remarks (continued)
• In the literature, speed of sound is given by an expression quite similar to the Bulk modulus . . .
• The “bulk modulus parameter” used in FLOTRAN is specified by the user for incompressible transient flows:
• When the compressible formulation is used, the value is calculated by FLOTRAN based on the Ideal Gas Law
P
p
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• The Navier-Stokes Equations come from applying the Conservation of Momentum to the flow of a Newtonian fluid (The characteristics of which will be described shortly!).
• Begin with the Momentum Equation - Newton’s Law of Motion
• Represents three equations for the three orthogonal directions I=1,2,3 (e.g.; x,y,z in Cartesian space)
• Acceleration (due to) Body Forces and Surface Forces
• Vector Expression
i
iji
j
iji
xB
x
uu
t
u
Conservation of Momentum
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Types of Terms in the Navier-Stokes Equations
• Acceleration Terms:
– Non-Linear
– Continuity Equation Imbedded
– Treated as advection transport
• Body Force Terms:
– Gravity
– Rotating Coordinate System
– Effects of Magnetic Fields
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Training ManualTypes of Terms (continued)
• Surface Force Terms:
– Normal Pressure (Mechanical, Thermodynamic)
– Shear Stresses
• Treated as diffusion terms
• The “Navier-Stokes Equations” are the momentum equations as formulated for a Newtonian Fluid
– Next, just what is a Newtonian Fluid?
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• The three postulates of Stokes (1845) lead to the description of the Newtonian Fluid:
– 1. The fluid is continuous and isotropic.
– 2. The stress tensor is at most a linear function of the strain rate.
– 3. With zero strain, the deformation laws must reduce to the hydrostatic pressure condition.
• The resulting relationship is:
absolute viscosity
second coefficient of viscosity (rarely considered)
• This relationship is valid for all gases and most common fluids.
k
kij
i
j
j
iijij x
u
x
u
x
up
The Stress-Strain Relationship for the Newtonian Fluid
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The Second Coefficient of Viscosity Treatment
• Our Goal: Get rid of this irritating term . . .
1. Consider the fluid Incompressible. So, continuity reduces to the divergence of velocity, and the term containing vanishes.
2. Assume the term is small anyway and is neglected (this might not be true near shock waves).
3. Stokes Hypothesis. Based on requiring the thermodynamic and mechanical pressures to be the same . . .
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• Mechanical Pressure is the average compression stress on an element of fluid. This in turn is a tensor invariant, so it can be expressed in the principle stress directions.
• From the expression for the principle stresses:
• Which leads to:
• So Stokes Hypothesis assumes away the problem . . .
33
1 zzyyxxkkmechanicalP
Vx
upxx
2
VpPmechanical
3
2
03
2
Second Coefficient (continued)
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• Next, look at the acceleration terms:
• The last two terms are the velocity multiplied by the continuity equation, making them vanish. The resulting acceleration term allows a more compact statement of the equation
j
j
ii
j
ij
i
iii
j
jii
x
u
xx
u
xx
u
xx
pB
x
uu
t
u
j
ji
j
ij
i
j
jii
x
u
tu
x
uu
t
u
x
uu
t
u
j
j
ii
j
ij
i
iii
i
x
u
xx
u
xx
u
xx
pB
Dt
Du
Navier-Stokes Equations
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Training ManualALE Formulation
• The Momentum Equation (Navier-Stokes Equation) as presented assumes that the mesh is stationary.
• The Arbitrary Lagrangian Eulerian formulation (ALE) modifies the equations to account for mesh motion.
• This is required in Transient Fluid Structure Interaction problems when the fluid problem domain is changing with time.
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ALE Formulation - Momentum Equation
• The governing NS equations must be modified to reflect this mesh motion:
balance Mass 0u
balance Momentum fp-uuw-uu 2
t
Mesh velocity
balance Mass 0u
balance Momentum fp-uuuu 2
t
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The Non-Dimensionalization of the Equations
• Put equations into comparative context. This aids in determining which terms are important, given some flow conditions and properties.
– Properties of Fluid
– Reference Conditions
– Boundary Conditions
– Dimensionless Parameters
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Express the conditions as non-dimensional multipliers of the reference conditions.
The Basic Properties of the Fluid
Density
Absolute viscosity
Second coefficient of viscosity
Coefficient of Thermal Expansion
S Surface Tension
Reference quantities:
Vo Magnitude of the reference velocity
k Thermal conductivity
CP Specific Heat at constant pressure
CV Specific Heat at constant volume
l Mean free path
o, o, To Reference values
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• Relate the acceleration term to the viscous term.
• Neglect body forces such as gravity for the moment. This is appropriate for “high speed” gas flow, for example.
• Reynold’s Number:
• The Dh is the hydraulic diameter for internal flow:
hVD
Re
meterWettedPeri
FlowAreaDh
*4
Basic Properties (continued)
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Training ManualBasic Properties (continued)
• Characteristic length for external flows
– Chord length of airfoil
– Distance from the leading edge
• The Reynolds number is the ratio of advection (transport by virtue of the velocity) to the transport by diffusion.
• Density and viscosity are relevant fluid properties Dh and V are conditions of the problem.
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• Kinematic Viscosity is the only property in the Kinematic Expression of the Reynold’s Number
• Some Kinematic Viscosity's (meter2/sec) - 20CGlycerin 5.0E-4 Kerosene2.5E-6
SAE30 Oil 2.5E-4 Water 1.0E-6
SAE10 Oil 1.0E-4 Benzene 7.0E-7
Air 1.8E-5 Mercury 1.5E-7
Crude Oil 1.0E-5
• For a given set of conditions, the Kinematic viscosity varies amongst fluids as the inverse the Reynold’s Number does.
v
VDhRe
v
Kinematic Viscosity
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L
tVt;L
ρ
ρρ;
μ
μμ
V
uu;
L
xx
o**
o
*
o
*
o
i*i
i*i
Reference Conditions for Non-Dimensionalization
• Choose constant reference values for density and viscosity.
• Choose free stream velocity Vo and a Length scale L.
• Relate the distance, velocity, pressure, and properties to reference values.
• The details of the derivations are provided in the Chapter 1 Appendix (the end of this chapter).
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Training ManualNon-Dimensional Momentum Equation
• The Reynolds number Re signals the relative importance of the advection and the diffusion contributions.
• The Grashoff number Gr shows the relative importance of buoyancy effects.
ijgTGr
pDt
VD Re
1
Re2
2
32
o
woooo TLgGr
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Training ManualNon-Dimensional Energy Equation
• The Peclet number Pe is the product of the Reynolds number and the Prandtl Pr number and indicates the relative importance of the transport and conduction of energy
i
j
j
i
j
ip x
u
x
u
x
uEcTk
PeDt
DpEc
Dt
DTc
Re
1
o
poo
k
CPr
PrRePe
wopo
o
Tc
VEc
2
Eckert Number
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• Prandtl Number:
• Prandtl Numbers for various fluids:
Mercury 0.024 Water 7.0
Helium 0.70 Benzene 7.4
Air 0.72 Ethyl Alcohol 16
Liquid Ammonia 2.0 SAE30 Oil3500
Freon-12 3.7 Glycerin12,000
Methyl Alcohol 6.8
o
poo
k
CPr
The Energy Equation (continued)
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Training ManualRotating Coordinates
• The governing equations of motion in a rotating reference frame with constant angular velocity.
• This is useful in the analysis of rotating machinery.– Let v be the velocity of an arbitrary point in a fluid with respect
to the rotating coordinate frame which has a constant angular velocity .
– Denote the position of the point, measured with respect to the origin of the rotating coordinate system, as r.
• Computational solutions for rotating coordinates entail solving for velocities relative to the rotating frame.
• Numerical difficulties can arise because terms involving the Coriolis and Centrifugal accelerations can be large.
• This leads to a modification of the pressure variable to include some of these terms.
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• The vector form of the momentum equation in the rotating reference frame with constant viscosity is:
• The general equation in indicial notation:
uPgrvDt
vD 22
i
j
j
i
jii
qprspqirsqpipqi
ii
x
u
x
u
xx
Pg
rux
u
t
u
2
Equations in the Rotating Frame
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yzzxuvz
PgF
Dt
Dw
xyyzwuy
PgF
Dt
Dv
zxxyvwx
PgF
Dt
Du
zyyxzxyxzshearz
yxxzyzxzysheary
xzzyxyzyxshearx
2
2
2
Formulation
• The rotating acceleration will take the form of additional source terms.
• Momentum equations in X,Y,Z Space
• Concentrating on the acceleration terms, denote the shear stress as shown for XYZ Directions:
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• The magnitude of the source terms due to the rotation can present numerical difficulties.
• The centrifugal portion of the additional terms can be put into the previous definition of the pressure.
zzωzωyωωxωω
yyωyωzωωxωω
xxωxωzωωyωωρ5.xgρpp~
2y
2xzyzx
2z
2xyzyx
2z
2yzxyxoiio
Further Modification to the Pressure
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• In the absence of rotation, this modified pressure is the same as defined earlier.
• With a stationary reference frame:
• With a rotating reference frame:
iiostaticabs xgppp 0
rotateoiioabs Ppxgpp ~
Further Modification to the Pressure (continued)
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vu
zyxgw
pF
Dt
Dw
uw
yzxgy
pF
Dt
Dv
wv
xzygx
pF
Dt
Du
xy
yxzxzxozoshearz
zx
zxzyyxoyosheary
yz
zyzxyxoxoshearx
2
~2
~2
~
22
22
22
Further Modification to the Pressure (continued)
• The governing equations in terms of the velocity with respect to the rotating coordinate system and a modified pressure:
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Training ManualA Rotating Test Case
• Flow in the annulus between two cylinders.
– Inner cylinder rotates, the outer is stationary.
• Goal: calculate the static pressure at the inner wall.
• Stationary Frame Boundary Conditions
– Angular velocity of 1 - counter-clockwise direction
– Apply velocity magnitude of 1 on the inner circle.
– Outer wall is stationary; apply pressure as zero
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Training ManualA Rotating Test Case (continued)
• Rotating frame:
– Velocity at the inner cylinder is stationary.
– Outer wall moves with a velocity magnitude of 2 clockwise.
– Modified pressure must be applied to the outer boundary.
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• Equations of motion:
• The following simplifications are possible:
• The continuity equation yields:
v
rr
vv
r
pv
rvv r
rr 2222 2
22
2 21
r
vv
rv
p
rr
vvvv rr
zz vz
pvv 2
0;0;0
zvz
01
rrvrr
Exact Solution Flow Between Rotating Cylinders
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• The radial velocity is zero at the inner and outer radii.
• Deduce that the gradient is also zero there and everywhere.
Thus:
• The solution for velocity is:
r
v
dr
dvr
dr
dp
vrdr
dp
2
11
22
21
22
11
1r
r
rr
r
rr
rrv
Solution (continued)
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• Use this to get the solution for pressure.
• The pressure equation is thus:
• Integration yields:
• Evaluate C3 from pressure boundary condition.
1
;
1
,
21
22
2
21
22
22
121
rr
C
rr
rCrC
r
Cv
rC
r
CC
r
C
dr
dp 22
213
21 2
3
22
212
21
2ln2
2C
rCrCC
r
Cp
Solution (continued)
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Training ManualSwirl
• Axisymmetric flow with a component normal to the axisymmetric plane is known as Swirl.
• Note that the flow between rotating cylinders can also be solved with the Swirl option!
VZ Normal to this plane
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Training ManualThe Equations for Swirl
• Swirling flow exists when an axisymmetric flow pattern has an azimuthal flow component
• Conveniently described by cylindrical coordinates with zero gradients (velocity, pressure) in the azimuthal direction.
• In other words, add a swirl component to the axisymmetric equations.
• Useful in coal plant applications where flows have an added rotational component.
• Rotating machinery in axisymmetric geometry (e.g., spinning shaft).
• Swirl velocity boundary conditions:– Inlet component of swirl
– Moving wall (rotating cylinder)
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Training ManualEquations of Motion
• Cylindrical coordinates without dependence.
• Swirl flow does effect the X-R solution.
• Swirl component loosely coupled to other components.
• Coordinate system directions r,,z
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Training ManualMomentum Equations
2
22 1
z
vrv
rrrr
p
r
v
z
vv
r
vv
t
v rr
rz
rr
r
2
21
z
vrv
rrrr
vv
z
vv
r
vv
t
v rzr
2
21
z
vrv
rrrg
z
P
z
vv
r
vv
t
v zzz
zz
zr
z
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Training ManualContinuity Equation
01
z
vrv
rrz
r
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• The Inner cylinder rotates rotates, outer stationary
– z velocity and directional dependence vanishes
– No time dependence, neglect gravity, constant density
• The analytical solution was previously attained...
Swirl Example - Rotating Cylinder
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Training ManualChapter 1 - Appendix A
• What follows are some of the details of the nondimensionalization of the momentum and energy equations
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Training Manual
*
o
**
o
o
*o
*i*
oi VV
LVV
V
Lt
VuρρVVρ
t
uρ
***
*
*2*
VVt
V
L
VVV
t
u ooi
i
oj
j
oio
i
j
j
i
xL
Vu
xL
Vu
Lx
u
x
u
i
j
j
i2
oo
i
j
j
i
x
u
x
uμ
L
Vμ
x
u
x
uμ
Term-by-Term Non-Dimensionalization
• Advection Terms:
• or
• Stress Terms:
• or
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• The Pressure gradient term:
• Combine all of these terms, and move the constants to one place:
• or
i
o2oo
2oo
i xL
pVρVρP
x
p
i
j
j
i
oo
o
x
u
x
u
LVP
Dt
VD
****
*
i
j
j
i
x
u
x
uP
Dt
VD ****
*
Re
1
Term-by-Term Non-Dimensionalization (continued)
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Training Manual
• Natural convection problems, with an absence of an identifiable free stream velocity, call for a reference velocity based on a Reynold’s Number of unity:
• Note that the second coefficient of viscosity has been neglected along with the body forces.
• Now that the form of the shear stress terms is known, for convenience use:
• So that:
ho
oo Dρ
μV
i
j
j
iij x
u
x
u
ijPDt
VD Re
1**
*
Term-by-Term Non-Dimensionalization (continued)
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Training Manual
• Before turning to the Energy Equation, examine the Navier-Stokes equations for low speed flow cases where gravity is important (e.g., natural convection).
• It is observed that effect of density changes may be neglected in all terms except the body force terms.
• This is known as the Boussinesq approximation, and it is commonly used with the assumption of the following form for the density changes:
• where is the thermal expansion coefficient
p
o
T
T
1
1
Term-by-Term Non-Dimensionalization (continued)
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Training Manual
• The Navier-Stokes equation becomes:
• At this point, a change of variables for pressure is invoked. The constant density head is included in the pressure term, which is now expressed in terms of a static pressure with reference pressure such as atmospheric pressure.
ijoo gTpDt
VD
1
ioii
abs
iioatmabs
gx
p
x
p
xgppp
mod
mod
Term-by-Term Non-Dimensionalization (continued)
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• From now on, drop the “mod” in the designation of the static pressure.
• Note that since the density in other than the gravity term is taken as the reference density we see that for all the terms:
• Also shown is the expression for the gravitational acceleration.
ijoo gTpDt
VD
oggg
;1
Term-by-Term Non-Dimensionalization (continued)
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Training Manual
• Non-dimensionalize each term in the gravity component, noting that everything on the right hand side was multiplied by the inverse of the constants from the advection term . . .
• The Temperature . . .
• It is customary to non-dimensionalize the temperature in terms of a temperature differential from a reference temperature and a reference temperature delta:
gTV
Loo
oo
2
woowo TTTTTTTT
Term-by-Term Non-Dimensionalization (continued)
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Training Manual
• Finally, put the gravity term into non-dimensional terms:
• And of course, the Grashoff number has been introduced:
• Navier-Stokes Equation for Boussenesq fluid:
gT
GrggT
LT
LV
Lo
o
owooo
o
o
oo
22
3
3
2
2 Re
2
32
o
woooo TLgGr
ijgTGr
pDt
VD Re
1
Re2
Term-by-Term Non-Dimensionalization (continued)
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Training Manual
22o
2
o
Fr
TΔβ
V
gLTΔβ
Re
Gr
gL
VFr
Term-by-Term Non-Dimensionalization (continued)
• The equations simplify slightly for natural convection, because you can assume a Reynold’s Number of unity.
• For forced flow cases, it is natural to ask if the buoyancy terms are important. Usually, the Froude number is consulted:
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Training Manual
• Without further ado . . .
– where Cp is the specific heat
– and the last term is the viscous dissipation function
• It is interesting that the second coefficient of viscosity must be no lower than the value implied by Stokes hypothesis to ensure that the viscous dissipation term is positive.
TkDt
Dp
Dt
DTcp
2
k
k
i
j
j
i
j
i
x
u
x
u
x
u
x
u
The Energy Equation
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Dt
DTc
L
TcV
VLt
D
TDTcc
Dt
DTc
pwopooo
o
woop po
Dt
Dp
L
V
VLt
D
DV
Dt
Dp
oo
o
oo
3
2
The Energy Equation (continued)
• Non-dimensionally term by term:
• Pressure term:
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Training Manual
• Diffusion or conduction term:
• Dissipation term:
• Next, collect the reference terms and find the right dimensionless parameters . . .
TkL
Tk
TTL
kkL
Tk
oo
woo
2
i
j
j
i
j
ioo
i
oj
j
oi
j
oio
x
u
x
u
x
u
L
V
Lx
Vu
Lx
Vu
Lx
Vu
2
2
The Energy Equation (continued)
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• Following the strategy used in the Navier-Stokes Equation, multiply the entire equation by the inverse of the group of reference quantities on the left . . .
• Obviously, it is time to introduce more dimensionless relationships!
i
j
j
i
j
i
wopoo
oo
opoo
o
wopo
pp
x
u
x
u
x
u
TLcV
V
TkLVc
k
Dt
Dp
Tc
V
Dt
DTc
2
2
The Energy Equation (continued)
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• The Eckert Number looks at the relative strengths of kinetic energy and energy storage.
• Peclet Number
• The Peclet Number is the ratio of thermal transport by advection to thermal transport by diffusion.
wopo
o
Tc
VEc
2
PrReo
poo
k
LCVPe
The Energy Equation (continued)
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i
j
j
i
j
ip x
u
x
u
x
uEcTk
PeDt
DpEc
Dt
DTc
Re
1
Non-Dimensional Energy Equation
• Introduce these into the non-dimensionalized equation to get:
• All the terms are important for high speed gas flows.
• For flows below Mach=0.3, the Eckert number is small enough to validate neglecting the pressure term and the viscous dissipation term.
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Pe
Br
Re
Ec
PrEcBr
Non-Dimensional Energy Equation (continued)
• To compare the relative strength of the conduction and dissipation terms, you introduce the Brinkman number. If it is significantly greater than unity, viscous dissipation should be considered: