cfd equations chapter 1. training manual may 15, 2001 inventory #001478 1-2 navier-stokes equations,...

CFD Equations

Chapter 1

May 15, 2001Inventory

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Training Manual

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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Training Manual

• 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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Training Manual

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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Training Manual

• 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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Training Manual

• 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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*

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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• 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**

*



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• 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



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• 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



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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



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• 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



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• 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



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22o

2

o

Fr

TΔβ

V

gLTΔβ

Re

Gr

gL

VFr


• 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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• 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


• Non-dimensionally term by term:

• Pressure term:


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• 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



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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



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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



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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:

cfd equations chapter 1. training manual may 15, 2001 inventory #001478 1-2 navier-stokes equations,...

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