chapter 2 - cfd

National Key Laboratory of Science and Technology on Aerodynamic Design and Research

# Lecture 2 ： The Governing Equations

Department of Fluid Mechanics, School of Aeronautics, Northwestern Polytechnical University, Xi’an, China

Presented by Zhonghua HanE-mail: [email protected]

29.03.2012

CFD Course for International Mixed Class

National Key Laboratory of Science and Technology on Aerodynamic Design and Research School of

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Chapter 2 Governing Equations

2.1 Introduction All of CFD is based on the fundamental governing equations of fluid dynamics – the continuity, momentum and energy equations. These are corresponding to the fundamental physical principles as follows:

- Mass is conserved;

- Newton’s second law, F = ma;

- Energy is conserved.

The purpose of this chapter is to derive these equation. ☆

Why?

- You have no chance to be a CFD expert if you don’t know these equations

- Different forms of governing equations.23/4/28 2


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Partial differential equations of conservation form

Non-conservation form

Fundamental physical principles

Mass is conserved

Newton’s second law

Fixed finite control volume

Moving finite control volume

Fixed infinitesimally small volume

Moving infinitesimally small volume

Models of the flow

Forms of these equations particularly suited for CFD Boundary conditions: (a) Inviscid ; (b) Viscous

Energy is conserved

Non-conservation form Continuity equation

Momentum equations

Energy equation

Governing equations

Road Map for Chapter 2

Integral equations of conservation form

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2.2 Models of the Flow

To derive the basic equations of fluid motion, the following philosophy is always followed:

1. Choose the appropriate fundamental physical principle from the law of physics, such as - Mass is conserved; - Newton’s second law, F = ma; - Energy is conserved; ☆

2. Apply the fundamental physical principle to a suitable model of the flow;

3. Derive the mathematical equations, which embody these physical principles

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2.2.1 Finite Control Volume

② ① Fixed finite control volume

Moving finite control volume

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2.2.2 Infinitesimally Fluid Element

③ ④ Fixed infinitesimally small element

Moving infinitesimally small element

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☆ Different models of flow lead to different forms of governing equations


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2.2.3 Some Comments

- In general aerodynamic theory, whether we deal with

conservation or non-conservation forms of equations is irrelevant;

- One form can be obtained from the other, only through simple

mathematical manipulations;

- For CFD, it is important to choose the form of the equations. The

nomenclature distinguishing these two forms of equations is firstly

introduced by CFD researchers.

- …


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2.3 The Substantial Derivative (Time Rate of Change Following a Moving Fluid Element)

( ) ( ) ( )

( ) (higer-order terms)

x x y y z zx y z

t tt

2 1 2 1 2 1 2 11 11

2 11

),,,(),,,(),,,(

tzyxwwtzyxvvtzyxuu

kwjviuV

( , , , )( , , , )( , , , )

x y z tx y z tx y z t

1 1 1 1 1

2 2 2 2 2

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112

12

1

12

12

112

12

112

12

)()(

)()(

)()(

tttzz

z

ttyy

yttxx

xtt

DtD

tttt

12

12lim12

tz

wy

vx

uDtD

...

DT T T T Tu v wDt x y z t

☆


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zw

yv

xu

tDtD

Definition of Substantial Derivative

zk

yj

xi

)(

VtDt

D Local derivative

( V ) Convective derivativet

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☆


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Substantial Derivative & Total Derivative

( ( ), ( ), ( ), )x t y t z t t

d x y zdt x t y t z t t

u v wx y z t

Vt

u v w

DDt

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☆


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DT

DV VV

1

Physical meaning: the time rate of change of volume of a moving fluid element, per unit volume

2.4 The Divergence of the Velocity: Its Physical Meaning

u v wVx y z

V ui vj wk

i j kx y z

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Net mass flow out of control volume through surface S

2.5 Continuity Equation

Physical principle: Mass is conserved

2.5.1 Model of the Finite Control volume Fixed in Space

time rate of decrease of mass inside the control volume

=

B = C

dS

V

dV

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☆


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S

B V dS

V

dVt

C

V S

dV V dSt

0

An integral form of the continuity equation A conservation form, derived from the flow model fixed in pace

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☆


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2.5.2 Model of the Finite Control Volume Moving with the Fluid

0V

dVDtD

dSnqdVt

dVDtD

bSVV

A frequently used formula :

bq

denotes the velocity the boundary ：

An integral form of the continuity equationA nonconservation form

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2.5.3 Model of an infinitesimally Small Element Fixed in Space (Assignment)

0 V

t

2.5.4 Model of an infinitesimally Small Element Moving with the Fluid (Assignment)

0 VDtD

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2.5.5 All Equations Are One (Assignment) Four equations : 2 integral and 2 partial differential equations

or 2 conservation and 2 nonconservation forms

Four forms of the same equation.

2.5.6 Integral versus Differential Forms of Equations There is a subtle difference between the integral and differential forms of the governing equations:

- The integral form of the governing equations allows for the presence of discontinuities inside the fixed control volume.

- The differential form of the governing equations assumes the flow properties are differentiable.

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☆


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Apply Newton’s second law to the infinitesimally small element moving with the fluid (this model is particularly convenient ), in x direction:

xx maF

2.6 Momentum Equations

Physical principle: Newton’s second law, F= ma


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gravitational forceBody force

magnetic force

pressure F, , )Surface force

viscous, , , )xx yy zz

xy xz yz

normal stress（shear stress（

x x( body force ) f ( dxdydz ) （ 2.45)

Forces on a Moving Fluid Element

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Shear and Normal Stresses in Viscous Flows

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x(Net surface force) = [ ( ) ]

[( ]

[( ) ]

[(

xxxx xx

yxyx yx

zzx

pp p dx dydzx

dx ) dydzx

dy dxdzy

) ]xzxdz dxdy

z

（ 2.46)

[ ]yxxx zxx x

pF dxdydz f dxdydzx x y z

（ 2.47)

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xzxyxxx fzyxx

pDtDu

xDuma dxdydzDt

（ 2.50a)

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zzzyzxz

yzyyyxy

xzxyxxx

fzyxz

pDtDw

fzyxy

pDtDv

fzyxx

pDtDu

For the infinitesimally small moving element, apply Newton’s second in x, y, z directions, respectively, one can obtain nonconservation differential equations ：

（ 2.50c)

（ 2.50b)

（ 2.50a)

Called Navier-Stokes equations in honor of two men: the French man M. Navier and the English man G. Stokes.

☆


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Through mathematical manipulation, one can obtain the conservation differential momentum equations as follows

zzzyzxz

yzyyyxy

xzxyxxx

fzyxz

pVwtw

fzyxy

pVvtv

fzyxx

pVutu

)()(

)()(

)()(

（ 2.56)

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( ) (2.57a)

( ) (2.57b)

( ) (2.57c)

(2.57e)

xx

yy

zz

xy yx

x

uVxvVywVz

v ux y

2

2

2

(2.57f)

(2.57g)

z zx

yz zy

u wz xv ux y

In the late of 17th century, Newton stated that shear stress in a fluid is proportional to the time rate of strain, i.e. velocity gradients. Such fluids are called Newtonian fluid. For such fluid, Stokes in 1845 obtained

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32

is the molecular viscosity coefficient and the second viscosity coefficient. Stokes made the hypothesis as

Substituting Eqs. （ 2.57) into (2.56), we obtain the complete Navier-Stokes equations in conservation form (to be continued):

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xfxw

zu

z

yu

xv

yxuV

x

xp

zuw

yuv

xu

tu

)(

)()2(

)()()()( 2

(2.58a)

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yfzv

yw

z

yvV

yyu

xv

x

yp

xvw

yv

xuv

tv

)(

)2()(

)()()()( 2

(2.58b)

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2

2 z

( w ) ( uw ) ( vw ) ( w ) pt x y z z

u w w v( ) ( )x z x y y z

w( V ) fz z

(2.58c)

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Physical principle: Energy is conserved

When applied to flow model of a fluid element moving with the flow, the first law states that

2.7 Energy Equations

Rate of change of energy inside fluid element

Net flux of heat into element

Rate of work done on element due to body and surface forces

= +

A = B + C

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( up ) ( up )up ( up dx ) dydz dxdydzx x

yx yxyx yx

( u ) ( u )( u dy ) u ) dxdz dxdydz

y y

zx zxzx zx

( u ) ( u )( u dz ) u ) dxdy dxdydzz z

xx xxxx xx

( u ) ( u )( u dx ) u ) dydz dxdydz

x z

- The net rate of work done by pressure in x direction （ adhe and bcgf) ：

- The net rate of work done by normal stresses in x direction (adhe and bcgf) ：

- The net rate of work done by shear stresses in x direction （ hefg and dabc) ：

-The net rate of work done by shear stresses in x direction （ dcgh and abfe) ：

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☆


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dxdydzVf

dxdydzz

wy

wx

w

zv

yv

xv

zu

yu

xu

zwp

yvp

xupC

zzyzxz

zyyyxy

zxyxxx

)()()(

)()()(

)()()()()()(

In total, the net rate of work done on the moving fluid element is the sum of the surface force contributions in x, y, z directions, as well as the body force contribution

( )pV

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☆


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xq dydz

Define as volumetric heat addition per unit mass. Note that the mass of the moving element is , we obtain：dxdydz

q

Volumentic heating of element = qdxdydz

The heat transferred by thermal conduction into the moving fluid element across face adhe is

is the heat transferred in the x direction per unit area by thermal conduction xq

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x xx x

q qq q dx dydz dxdydzx x

－＝－

The heat transferred by thermal conduction into the moving fluid element across face bcgf is

xx

qq dx dydzx

Thus the net heat transferred in the x direction into the fluid element by thermal conduction is

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B yx zqq qq dxdydz

x y z

－＋＋

Taking into account heat transfer in the y and z directions across the other faces,

yx zqq q dxdydz

x y z

= －＋＋Heating of fluid element by thermal conduction

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dxdydzVeDtDA

2

2

The time rate change of total energy (internal and kinetic energy)

e The internal energy due to the random molecular motion

V 2

2 The kinetic energy due to the transitional motion of fluid element

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( ) ( ) ( )

( )( ) ( )( ) ( ) ( )

( ) ( ) ( )

yxxx zx

xy yy zy

D V T T Te q k k kDt x x y y z z

uu uup vp wpx y z x y z

v v vx y z

2

2

( )( ) ( ) yzxz zzww w f V

x y z

The final form of energy equation (nonconservation)

（ 2.66）23/4/28 39


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( / ) (2.67)

( / ) (2.68)

yxxx zxx

xy yy zyy

D u pu u u u ufDt x x y z

D v pv v v v vfDt y x y z

2

2

2

2

( / ) (2.69)

yzxz zzz

D w pw w w w wfDt z x y z

2 2

To convert into one involving De/Dt , Multiply Eqs. （ 2.56a) 、 (2.56b) 、 (2.56c) by u 、 v, w, respectively.


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Through some manipulations, the energy equation can be simplified to

( ) ( ) ( )

( )

xx yx zx

xy yy zy

xz yz zz

De T T Tq k k kDt x x y y z z

u v w u u upx y z x y zv v vx y zw w wx y z

（ 2.71）The kinetic energy and the body force terms have dropped out☆


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)()()(

)(

)()()(

yw

zv

xw

zu

xv

yu

zw

yv

xu

zw

yv

xup

zTk

zyTk

yxTk

xq

DtDe

zyzxyx

zzyyxx

（ 2.72）

Since , , ,one can obtain

yx xy zx xz yz zy

Other forms in terms of enthalpy h and total enthalpy h0 can be obtained by similar manipulations

☆


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( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

De T T Tq k k kDt x x y y z z

u v w u v wpx y z x y z

u v wy y z

u v u w v wy x z x z y

2

2 2 2

2 2 2

2 2 2（ 2.73）

Apply Eqs. (2.57a) to (2.72) in order to express the viscous stresses in terms of velocity gradient


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( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

e T T TeV q k k kt x x y y z z

u v w u v wpx y z x y z

u v wy y z

2

2 2 22 2 2

( ) ( ) ( )

u v u w v wy x z x z y

2 2 2

The energy equation can be written in conservation form：

（ 2.79）


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The conservation form of energy equation in term of total energy：

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

V V T T Te e V q k k kt x x y y z z

up vp wpx y z

2 2

2 2

( )( ) ( )

( ) ( ) ( )

( )

yxxx zx

xy yy zy

xz

uu ux y z

v v vx y z

w

( ) ( )yz zzw w f V

x y z

（ 2.81）


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2.8 Summary of the Governing Equations for Fluid Dynamics: with Comments

2.8.1 Navier-Stokes euqations

2.8.2 Euler Equations

2.8.3 Comments on the Governing Equations（ 1 ） The y are coupled system of nonlinear partial differential equations,

and hence are very difficult to solve analytically. To date, there is no close-form solution to these equations.

（ 2 ） For the momentum and energy equations, the differences between the nonconservation and conservation forms of the equations is just the left-hand side.

（ 3 ） Note that the conservation forms of equations contain terms on the left-hand side which include the divergence of some quantity. For this reason, the conservation form of governing equations is sometimes called divergence form.


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（ 4 ） The normal and shear stress terms in these equations are functions of the velocity gradients.

（ 5 ） Examine the equations, we have 5 equations in terms if 7 unknown flow-field variables, . In aerodynamic, it is generally reasonable to assume that the gas is a perfect gas (which assume that intermolecular forces are negligible). For a perfect gas, the equations of state.

where R is the specific gas constant. A seventh equation to close the

entire system must be a thermodynamic relation between state variables. For example

For a calorically perfect gas (constant specific heats)

Tewvup ,,,,,,

RTp

Tce

pTee

v

),(


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（ 6 ） Navier-Stokes equations are referred to as the whole system of equations in modern CFD, not only to the momentum equations for a viscous flow.

（ 7 ） Similarly, Euler equations are referred to as the whole system of equations in modern CFD, not only to the momentum equations for a inviscid flow.


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2.9 Physical Boundary Conditions

Wall boundary ：For viscous flow: non-slip condition, or zero relative velocity between the wall surface and the gas immediately at the wall

u=v=w=0

Boundary condition on the gas temperature ： (1)

(2)

(3) Adiabatic wall

Inviscid flow: flow velocity adjacent to the wall must be tangent to the way

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( ) ww w

w

qT Tq k orn n k

0)(

wnT

0nV

T T w


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Other boundary conditions

- Outer flow: far-field boundary condition

- Internal flow: inflow and outflow

Numerical Boundary conditions

If the problem involves an aerodynamic body immersed in a known free stream, then the boundary conditions applied to a distance infinitely far upstream, above, blow and downstream of the body is simply the given free condition.

Numerically, due to the limitation of computational resource, the outer boundary of the computational domain has to be limited to a finite distance away from the wall boundary. In this case, the one-dimensional Riemann boundary condition is often applied.

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Other numerical boundary conditions:

- Symmetric

- periodic conditions

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( ) 0 wpn

Question: is this a boundary condition for solving of Navier-Stokes equations?

☆


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2.10 Forms of the Governing Equations Particullay Suited for CFD: Comments on Conservation Form, Shock Fitting and Shock Capturing

The governing equations in conservation form can be written as

(2.93) JzH

yG

xF

tU

)2

(2Ve

wvu

U


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xzxyxx

xz

xy

xx

wvuxTkpuuVe

wuvu

pu

u

F

)2

(2

2

yzyyyx

yz

yy

xy

wvuyTkpvuVe

wvpv

uvv

G

)2

(2

2


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zzzyzx

zz

zy

zx

wvuzTkpwwVe

pwvwuww

H

)2

(2

2

qwfvfuffff

J

zyx

z

y

x

)(

0


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zzzyzx

zz

zy

zx

wvuzTkpwwVe

pwvwuww

H

)2

(2

2

qwfvfuffff

J

zyx

z

y

x

)(

0


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(2.99) zH

yG

xFJ

tU

Time-marching method

Flux variables & primitive variables

2)2/( 2222 wvuVee

ww

vv

uu


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Spatial marching methods, such as

(2.110) zH

yGJ

xF


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Benefit of using conservation form

- Numerically and computationally convenient;

- Generally more suited for CFD

Conservation Form of Governing Equations:

- Strong conservation form

- weak conservation form

Question : Conservation and non-conservation forms, which one is more suitable for flow with shock wave?

☆


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Shock capturing and shock fitting methods：


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Hand on Practice

Use any existing code or software to simulate the

flow past an RAE 2822 airfoil at following flow

condition:

Ma = 0.729, al = 2.79, Re = 6.5E6

Assuming that this is an academic research project,

try to show me your capability as a scientist.

Make a presentation with sides no more than 10

and the results should be prepared according to the

standard required by top-level journal publication.

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xzxyxxx fzyxx

pDtDu

Added by 0)(

uV

t

)()(

)()(

)(

Vutu

uVut

uVtu

uVtDt

Du

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chapter 2 - cfd

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