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Chapter 2

Conservation Laws of Fluid Motion Julaporn Benjapiyaporn

Mechanical Engineering, KKU.

2.1 Governing equations of fluid flow and heat transfer

Three fundamental physical principles

– Mass is conserved.

– Newton’s second law.

– Energy is conserved.

All of CFD is based on the fundamental governing equations of fluid dynamics

—the continuity, momentum and energy equations.

“If you do not physically understand the meaning and significance of each term

in these equations—then how can you even hope to properly interpret the CFD

results obtained by numerically solving these equations?”

analysis of fluid flows at macroscopic length scale (1μm and larger)

macroscopic properties; velocity ( u i v j w k = + +u

), pressure ( p),

density ( ρ ), temperature (T ), their space and time derivatives ( , , , x y z t ∂ ∂ ∂ ∂∂ ∂ ∂ ∂

)

all fluid properties are functions of space and time:

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

x y z t p x y z t pT x y z t T

x y z t

ρ ρ ≡≡≡≡u u

The element under consideration is so small that fluid properties at the faces

can be expressed accurately enough by means of the first two terms of Taylor

series expansion:

2 32 31 1

2 21 12 2 2 3

truncation error

( ) ( )( ) ( ) ( ) ...2! 3!

x x x x x x x x x

δ δ φ φ φ φ δ φ δ

∂ ∂ ∂± = ± + ± +∂ ∂ ∂

Where φ is a property of such an element.

Models of the Flow

1. Choose the appropriate fundamental physical principles from the law of

physics,

2. Apply these to a suitable model of the flow.

3. Extract the mathematical equations which embody such physical principles.


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•

Finite Control Volume

•

Infinitesimal Fluid Element

2.1.1 The Substantial Derivative

Derive by considering an infinitesimally fluid element moving with the flow.

zw

y y

xu

t Dt

D

∂∂

+∂∂

+∂∂

+∂∂

= ρ ρ ρ ρ ρ

)( ∇⋅+∂∂= u

t Dt D Substantial Derivative: the time rate of change following

a moving fluid element

t ∂∂

local derivative: the time rate of change at a fixed point

)( ∇⋅u

convective derivative: the time rate of change due to the movement

of the fluid element

Substantial derivative is the same as the total from calculus.

⇒= ),,,( t z y x ρ ρ zw y y xut dt d

∂

∂

+∂

∂

+∂

∂

+∂

∂

=

ρ ρ ρ ρ ρ

CV1. Finite CV fixed in space with the

fluid moving through it (conservation

form): fixed volume and vary mass.

CV2. Finite CV moving with the

fluid (non-conservation form):fixed mass and vary volume.

Inf1. Inf. FE fixed in space with the

fluid moving through it (conservation

form): fixed volume and vary mass.

Inf2. Inf. FE moving with the

fluid (non-conservation form):fixed mass and vary volume.


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For an arbitrary conserved property:

( ) ( )( )uuu

ρ φ φ ρ ρ

φ φ

ρ ρφ ρφ

⋅∇+⋅∇+⎟ ⎠

⎞⎜⎝

⎛ ∂∂

+∂∂

=⋅∇+∂

∂

t t t

)(

( ) Dt D

t t

φ ρ ρ

ρ φ φ

φ ρ =⎥⎦

⎤⎢⎣

⎡⋅∇+∂

∂+⎥⎦

⎤⎢⎣

⎡⋅∇+∂

∂= uu

Thus, the rate change of x-, y- and z-momentum and energy per unit volume can be

defined by substitution φ by u, v, w and E , respectively.

( ) Dt

Duu

t

u ρ ρ

ρ =⋅∇+

∂∂

u)(

( ) Dt

Dvv

t

v ρ ρ

ρ =⋅∇+

∂∂

u)(

( ) Dt

Dww

t

w ρ ρ

ρ =⋅∇+

∂∂

u)(

and ( ) Dt

DE E

t

E ρ ρ

ρ =⋅∇+

∂∂

u)(

2.1.2 The Divergence of the Velocity

Derive by considering a finite control volume moving with the flow.

The change in the volume in the CV: ( )[ ] ( ) Sd

⋅Δ=⋅Δ=∇ t dS t V unu

The time rate of change of the CV: ( ) ∫∫∫∫ ⋅=⋅ΔΔ=S S

t t Dt

DV Sd Sd

1 uu

Applying the divergence theorem: ( )∫∫ ⋅∇=V

dV Dt DV u

Let the moving CV is shrunk to a very small V δ :( )

( )∫∫ ⋅∇=V

dV Dt

V D

δ

δ u

Assuming that V δ is small enough such that u

⋅∇ is the same value throughout V δ .Then the integral, in the limit as V δ shrinks to zero, is ( ) V δ u

⋅∇ :

( )( ) V

Dt

V Dδ

δ u

⋅∇=

( ) Dt

V D

V

δ

δ

1=⋅∇ u

∇ ⋅u

: The divergence of the velocity is physically the time rate of change of the

volume of a moving fluid element, per unit volume

Mathematical meaning: z

w

y

v

x

u

∂∂

+∂∂

+∂∂

=⋅∇ u

2.1.3 Continuity Equation

Physical principle: Mass is conserved.


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CV1: Model of the Finite CV Fixed in Space (fixed volume and vary mass)

Net mass flow out of CV through surface S = Time rate of decrease of mass inside CV

∫∫∫∫∫ ∂∂

−=⋅V S

dV t

ρ ρ Sd

u

0Sd =⋅+∂∂

∫∫∫∫∫

S V

dV t

u ρ ρ : The conservation form of an integral form of the

continuity equation: fixed CV.

CV2: Model of the Finite CV Moving with the Fluid (fixed mass and vary volume)

0=∫∫∫V

dV Dt

D ρ : The nonconservation form of an integral form of the continuity

equation: moving CV.

Inf1: Model of an Inf. Small Element Fixed in Space (fixed volume and vary mass)

Net mass flow out of CV through surface S = Time rate of decrease of mass inside CV

Fig. 2.1 Mass flows in and out of fluid element

* Rate of increase of mass in fluid element : ( ) x y z x y zt t

ρ ρδ δ δ δ δ δ

∂ ∂=

∂ ∂

*Rate of increase of mass in

fluid element

*Net rate of flow of mass

into fluid element =

Mass

balance:


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* Net rate of flow of mass into fluid element :

1 1 12 2 2

1 1 1

2 2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( )

u u vu x y z u x y z v y x z

x x y

v w wv y x z w z x y w z x y

y z z

u v w x y z x y

t t t

ρ ρ ρ ρ δ δ δ ρ δ δ δ ρ δ δ δ

ρ ρ ρ ρ δ δ δ ρ δ δ δ ρ δ δ δ

ρ ρ ρ δ δ δ ρ δ δ δ

⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞− − + + −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞− + + − − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

∂ ∂ ∂⎛ ⎞= − + + = −∇⎜ ⎟∂ ∂ ∂⎝ ⎠u

i z

( )dxdydzt

dxdydz z

w

y

v

x

u

∂∂

−=⎥⎦

⎤⎢⎣

⎡∂

∂+

∂∂

+∂

∂ ρ ρ ρ ρ )()()(

( ) 0=⋅∇+∂∂

u

ρ ρ

t : The conservation form of a partial differential equation of the

continuity equation: fixed inf. small element.

For an incompressible fluid (i.e. a liquid), ρ = const. and u

⋅∇ = 0.

Inf2: Model of an Inf. Small Element Moving with the Fluid (fixed mass and vary volume)

V m δ ρ δ =

( ) ( )0=+=

Dt

dV D

Dt

DdV

Dt

dV D ρ

ρ ρ

( ) 01 =⎥⎦⎤⎢⎣

⎡+ Dt

V DV Dt

D δ δ

ρ ρ

0=⋅∇+ u

ρ ρ

Dt

D : The nonconservation form of a partial differential equation of

the continuity equation: moving inf. small element.

All the Equations are One

0Sd =⋅+∂∂

∫∫∫∫∫

S V

dV t

u ρ ρ CV1, conservation form, fixed CV

Since the CV is fixed, the limits of integration are constant:

0Sd =⋅+∂∂

∫∫∫∫∫

S V

dV t

u ρ ρ

Applying the divergence theorem: ( ) 0=⋅∇+∂∂

∫∫∫∫∫∫V V

dV dV t

u

ρ ρ

or ( ) 0=⎥⎦⎤

⎢⎣

⎡ ⋅∇+∂∂

∫∫∫ dV t V

u

ρ ρ

( ) 0=⋅∇+∂∂ u ρ ρ

t Inf1, conservation form, fixed inf. small element


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( )

0=⋅∇+∇⋅+∂∂

⋅∇

u

uu

ρ

ρ ρ ρ

t

0=⋅∇+ u

ρ ρ

Dt

D Inf2, nonconservation form, moving inf. small element.

Integral versus Differential Form of the Equations

The integral form of the equations allows for the presence of

discontinuities inside the fixed CV

The differential form of the governing equations assumes the flow

properties are differentiable, hence continuous.

The integral form of the equations is considered more fundamental

than the differential form.

This consideration becomes of particular importance when calculatinga flow with real discontinuities, such as shock waves.

2.1.4 Momentum equation

Physical principle: Newton’s second law ( aF m= )

* Rate of increase of x-, y- and z-momentum per unit volume: , , Du Dv Dw

Dt Dt Dt ρ ρ ρ

*Forces:

•

surface forces: act directly on the surface of the fluid element

- pressure forces, p (an inviscid fluid: the only surface force is due to the

pressure)

z

p

y

p

x

p

∂∂

−∂∂

−∂∂

− ,,

- viscous forces, τ ij: the stress component acts in the j-direction on a

surface normal to the i-direction

* normal stress: time rate of change of volume of the fluid element

z y x

zz yy xx

∂∂

∂

∂

∂∂ τ τ τ

,,

* shear stress: time rate of change of the shearing deformation of the fluid

element

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂

∂

∂∂

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂

∂

∂

∂⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂∂

∂

∂

y x z x z y

yz xz

zy xy zx

yx τ τ τ τ τ τ

,,,,,

*Rate of increase of momentum of

fluid element

*Sum of forces on

fluid element= Newton’s

second law:


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In most viscous flow, normal stresses are much smaller than shear stresses and

many times are neglected.

Normal stresses become important when the normal velocity gradients are very

large such as inside a shock wave.

•

body forces: act directly on the volumetric mass of the fluid element oract at a distance

- gravity force

- centrifugal force

- Cariolis force

- electromagnetic force

Fig. 2.2 Stress components on three faces of fluid element

Fig. 2.3 Stress components in the x-direction

Example of the net surface force in the x-direction per unit volume:

( ) yx xx zx p

x y z

τ τ τ ∂∂ − + ∂+ +

∂ ∂ ∂

Momentum equations for a viscous flow (Navier-Stokes equations) in

nonconservation form.

( ) (2.2a)

yx xx zx Mx

p DuS

Dt x y z

τ τ τ ρ

∂∂ − + ∂= + + +

∂ ∂ ∂

include as source terms, ( ) M S φ

⎫⎪⎬⎪⎭


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( )

(2.2b) xy yy zy

My

p DvS

Dt x y z

τ τ τ ρ

∂ ∂ − + ∂= + + +

∂ ∂ ∂

( ) (2.2c)

yz xz zz Mz

p DwS

Dt x y z

τ τ τ ρ

∂∂ ∂ − += + + +

∂ ∂ ∂

Example of body force: 0, 0, Mx My Mz

S S S g ρ = = = −

Navier-Stokes equations in conservation form: substitution these terms in the left sideof the equations

( )( )u

u

t

u

Dt

Du ρ

ρ ρ ⋅∇+

∂∂

= ,( )

( )u

vt

v

Dt

Dv ρ

ρ ρ ⋅∇+

∂∂

= ,( )

( )u

wt

w

Dt

Dw ρ

ρ ρ ⋅∇+

∂∂

=

Newtonian fluids: the viscous stresses are proportional to the rates of

deformation (velocity gradients) – fluid in all aerodynamic problems.

Non-newtonian fluids – blood flow

For newtonian fluid: τ ij is expressed as functions of the linear and volumetric

deformation rates.

linear deformation rate: , , xx yy zzu v w

e e e x y z

∂ ∂ ∂= = =

∂ ∂ ∂

1 1 1, ,

2 2 2 xy yx xz zx yz zy

u v u w v we e e e e e

y x z x z y

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= = + = = + = = +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

volumetric deformation rate: z

w

y

v

x

u

∂

∂+

∂

∂+

∂

∂=⋅∇ u

Two constants of proportionality:

the (first) dynamic viscosity, μ -- to relate stresses to linear deformations, and

the second viscosity, λ -- to relate stresses to the volumetric deformation.

u

⋅∇+∂∂

= λ μ τ x

u xx 2 , u

⋅∇+

∂∂

= λ μ τ x

v yy 2 , u

⋅∇+

∂∂

= λ μ τ x

w zz 2

, , xy yx xz zx yz zyu v u w v w

y x z x z yτ τ μ τ τ μ τ τ μ

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= = + = = + = = +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

For gases, 23

λ μ = − For incompressible liquid: 0=⋅∇ u

Substitution τ ij into the momentum equations yields the Navier-Stokes equations:

( ) ( ) (2.4a) Mx

Du p u v wu S

Dt x x x y x w x x ρ μ μ μ μ λ

⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + ∇ ⋅ ∇ + + + + ∇ ⋅ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦u

( ) ( ) (2.4b) My

Dv p u v wv S

Dt y x y y y w y y ρ μ μ μ μ λ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= − + ∇ ⋅ ∇ + + + + ∇ ⋅ +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

u


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( ) ( ) (2.4c) Mz Dw p u v w

w S Dt z x z y z z z z

ρ μ μ μ μ λ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + ∇ ⋅ ∇ + + + + ∇ ⋅ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

u

The complete Navier-Stokes equations in conservation form:

2( ) ( ) ( ) ( )2

Mx

u u uv uw p u

t x y z x x x

v u u wS

y x y z z x

ρ ρ ρ ρ λ μ

μ μ

∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ + + = − + ∇ ⋅ +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

⎡ ⎤⎛ ⎞ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦

u

(2.4aa)

2( ) ( ) ( ) ( )

2 My

v uv v vw p v u

t x y z y x x y

v w vS

y y z y z

ρ ρ ρ ρ μ

λ μ μ

⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂+ ∇⋅ + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟

∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

u

(2.4bb)

2( ) ( ) ( ) ( )

2 Mz

w uw vw w p u w

t x y z z x z x

w v wS

y y z z z

ρ ρ ρ ρ μ

μ λ μ

⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ + + = − + +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞+ + + ∇ ⋅ + +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎣ ⎦u

(2.4cc)

2.1.5 Energy equation

Physical principle: Energy is conserved .

* Rate of increase of energy of fluid particle: DE

Dt ρ

Net rate of work done by surface forces acting on x-, y- and z-direction are:

( )[ ( )] ( )

-direction:( ) [ ( )] ( )

-direction:

( )( ) [ ( )]-direction:

yx xx zx

xy yy zy

yz xz zz

uu p u

x x y zv v p v

y x y z

ww w p z

x y z

τ τ τ

τ τ τ

τ τ τ

∂∂ − + ∂

+ +∂ ∂ ∂∂ ∂ − + ∂

+ +∂ ∂ ∂

∂∂ ∂ − ++ +

∂ ∂ ∂

*Total rate of work done on the fluid particle by surface forces:

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

( )( ) ( )

yx xy yy zy xx zx

yz xz zz

u v v vu u p

x y z x y zww w

x y z

τ τ τ τ τ τ

τ τ τ

∂ ∂ ∂ ∂∂ ∂−∇ ⋅ + + + + + +

∂ ∂ ∂ ∂ ∂ ∂∂∂ ∂

+ + +∂ ∂ ∂

u

first law of

thermod namics:

*Rate of increase

of energy of

fluid

*Net rate of heat

added to

fluid element+=

*Net rate of work

done on

fluid element


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Energy flux due to heat conduction:q q q

x y z

∂ ∂ ∂− − − = −∇ ⋅

∂ ∂ ∂q

& by Fourier’s law: k T = − ∇q

* Net rate of heat added to fluid particle due to heat conduction: ( )k T −∇ ⋅ = ∇ ⋅ ∇q

Energy equation in nonconservation form:

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

( )( ) ( ) ( ) (2.3a)

yx xy yy zy xx zx

yz xz zz E

u v v vu u DE p

Dt x y z x y zww w

k T S x y z

τ τ τ τ τ τ ρ

τ τ τ

∂ ∂ ∂ ∂∂ ∂= −∇ ⋅ + + + + + +

∂ ∂ ∂ ∂ ∂ ∂∂∂ ∂

+ + + + ∇ ⋅ ∇ +∂ ∂ ∂

u

Where 2 2 212( ) E i u v w= + + + and the gravitational potential energy is included in S E .

Energy equation in conservation form by substitution these terms in the left side of the

equations : ( )( ) DE E

E Dt t

ρ

ρ ρ

∂= + ∇ ⋅∂ u

By multiplying the x-, y- and z-momentum equation (2.2a-2.2c) by u, v and w,

Respectively, adding the results together, and subtracting from (2.3), energy equation

in terms of internal energy only, the kinetic energy term has dropped out:

( )

i

xx yx zx xy yy zy

xz yz zz E M

S

Di u u u v v v p k T

Dt x y z x y zw w w

S x y z

ρ τ τ τ τ τ τ

τ τ τ

∂ ∂ ∂ ∂ ∂ ∂= − ∇ ⋅ + ∇ ⋅ ∇ + + + + + +

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂

+ + + + − ⋅∂ ∂ ∂

u

u S

For an incompressible fluid, i = cT and 0∇ ⋅ =u

.

For Newtonian model for viscous stress, the energy equation completely in terms of

the flow field variables by repressing the viscous stresses in terms of the velocity

gradients:

( ) (2.5)i

Di p k T S

Dt ρ = − ∇ ⋅ + ∇ ⋅ ∇ +Φ+u

2 2 22 2 2

22 ( )u v w u v u w v w

x y z y x z x z yμ λ

⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + + + + + + + ∇ ⋅⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

Φ

⎣ ⎦

u

Where Φ is the dissipation function, arises from irreversible viscous work, representsa source of internal energy due to deformation work on the fluid particle.

2.2 Conservative form of the governing equation of fluid flow

Incompressible -- density constant

Compressible -- density varies

Equations for Viscous Flow (the Navier-Stokes Equations) include the dissipative,

transport phenomena of friction, thermal conduction, and/or mass diffusion.


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For an unsteady, three-dimensional, compressible viscous flow:

mass: ( ) 0t

ρ ρ

∂+ ∇ =

∂u

i (2.1)

x-momentum:( )

( ) ( ) x Mx

u pu u S S

t x τ

ρ ρ μ

∂ ∂+ ∇ = − + ∇ ∇ + +

∂ ∂

u

i i (2.4a)

y-momentum:( )

( ) ( ) y My

v pv v S S

t y τ

ρ ρ μ

∂ ∂+ ∇ = − + ∇ ∇ + +

∂ ∂u

i i (2.4b)

z-momentum:( )

( ) ( ) x Mz

w pw w S S

t z τ

ρ ρ μ

∂ ∂+ ∇ = − + ∇ ∇ + +

∂ ∂u

i i (2.4c)

e.g., ( ) x

u v wS

x x y x w x xτ

μ μ μ λ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + ∇⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

u

i

internal energy: ( )

( )

( ) ii

i p k T S t

ρ

ρ

∂

+ ∇ ⋅ = − ∇ + ∇ ∇ + +Φ∂ u u

i i (2.5)

Equation of state ( , ) p p T ρ = and ( , )i i T ρ = , e.g., perfect gas p RT ρ = and vi C T =

Equations for Inviscid Flow (the Euler Equations) Exclude the dissipative, transport

phenomena of friction, thermal conduction, and/or mass diffusion.

For an unsteady, three-dimensional, compressible inviscid flow:

mass: ( ) 0t

ρ ρ

∂+ ∇ =

∂u

i (2.1)

x-momentum:( )

( ) Mx

u pu S

t x

ρ ρ

∂ ∂+ ∇ = − +

∂ ∂u

i (2.4a’)

y-momentum:( )

( ) My

v pv S

t y

ρ ρ

∂ ∂+ ∇ = − +

∂ ∂u

i (2.4b’)

z-momentum:( )

( ) Mz

w pw S

t z

ρ ρ

∂ ∂+ ∇ = − +

∂ ∂u

i (2.4c’)

internal energy: ( )( )

i

ii p S

t

ρ ρ

∂+ ∇ ⋅ = − ∇ +

∂u u

i (2.5’)

Comments on the Governing Equations

1.

Very difficulty to solve analytically a coupled system of nonlinear PDEs.2. Difference between conservation&nonconservation forms is just the LHS.

3. Conservation form called divergence form, due to ( ) ρ ∇ ⋅ u

or ( )u ρ ∇ ⋅ u

.

4.

Normal and shear stresses are functions of velocity gradients.

5. Five equations. of six unknowns ( ρ , p, u, v, w, e); use a perfect gas

assumption in aerodynamics: p RT ρ = seventh unknown (T ), need the caloric equation of state (

ve c T = )

6. In modern CFD, Navier-Stokes equations. means a solution of a viscous flow

problem using the full governing equations.

7. Euler equations. means a solution of a inviscid flow problem using the full

governing equations.


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2.3 Boundary and initial conditions

* Although the governing equations are the same, the flow fields are quite different

due to the boundary conditions (include initial conditions).

* The real driver for any particular solution is the boundary condition.

* Any numerical solution of the governing flow equations must be made to see astrong and compelling numerical representation of the proper boundary conditions.

Auxiliary conditions are specified in three ways:

1) Dirichlet condition

( , ) ( , ) or ( , ) constant x y f x y x yφ φ = =

2) Neumann condition

, ,

( , ) or constant x y x y

d d f x y

dn dn

φ φ = =

3) Mixed condition

, ,

( , ) ( , ) or ( , ) constant x y x y

d d c x y f x y c x y

dn dn

φ φ φ φ + = + =

Proper boundary conditions for a viscous flow:

• boundary layer flow – viscosity important close to surface

• separated flow – viscosity important everywhere

No-slip condition at the stationary surface with the moving flow*Velocity: 0u v w= = = (at the surface)

*Temperature:

1. Fixed wall temperature;wT T = (at the wall)

2. Instantaneous heat flux to the wall;w

w

T q k

n

∂⎛ ⎞= −⎜ ⎟∂⎝ ⎠ (at the wall)

3. Adiabatic wall; 0w

T

n

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠ (at the wall)

The no-slip conditions for a continuum viscous flow are associated with velocity and

temperature at the wall.

Pressure and density at the wall, fall out as part of the solution.

Proper boundary conditions for an inviscid flow:

There is no friction at the surface to promote its “sticking”.

Hence, the flow velocity at the wall is nonzero value.

For a nonporous wall (no mass flow into or out of the wall), the flow at the surface is

tangent to the wall: n 0⋅ =u (at the surface)


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The magnitude of the velocity, temperature, pressure and density at the wall, fall out

as part of the solution.

Various types of BCs:

- Duct flow inflow & outflow boundaries (Fig. 2.4)

- Aerodynamic body freestream conditions (upstream, downstream, above, below)

(Fig. 2.5)

Initial conditions for unsteady flow

• Everywhere in the solution region ρ , u and T must be given at time t = 0

Boundary conditions for unsteady and steady flows

• On solid wall, u = uw (no-slip condition)

T = T w

(fixed temperature) or (fixed heat flux)w

k T n q∂ ∂ = −

• On fluid boundaries, inlet: ρ , u and T must be known as a function of

position

outlet: and (stress continuity)n n t t p u n F u n F μ μ + ∂ ∂ = ∂ ∂ =

The subject of “proper numerical implementation of the physical boundary

conditions” in CFD is very important and is the subject of much current CFD

research.

Fig. 2.4 Boundary conditions for an internal flow problem (Duct flow)


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Fig. 2.5 Boundary conditions for an external flow problem (Aerodynamic body)

Special geometrical features of the solution region (Fig. 2.6):

•

Symmetry boundary condition: ∂φ /∂n = 0

• Cyclic boundary condition: φ 1 = φ 2

Fig. 2.6 Examples of flow boundaries with symmetry and cyclic conditions


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2.4 Differential and integral forms of the general transport equations

Let φ is a general variable, the conservative form of all fluid flow equations are

( )( ) ( ) (2.6)S

t

φ

ρφ ρφ φ

∂+ ∇ = ∇ Γ∇ +

∂

u

i i (Γ is a diffusion coefficient)

rate of change + convection = diffusion + source

Finite volume method: the integration of the governing equations over a control

volume CV :

( )( ) ( )

CV CV CV CV

dV dV dV S dV t

φ

ρφ ρφ φ

∂+ ∇ = ∇ Γ∇ +

∂∫ ∫ ∫ ∫u

i i

Using Gauss’ divergence theorem:

CV A

dV dA∇ =∫ ∫a n a

i i

Unsteady problem:

( ) ( ) ( )t CV t A t A t CV

dV dt dAdt dAdt S dV dt t

φ ρφ ρφ φ

Δ Δ Δ Δ

⎛ ⎞∂+ = Γ∇ +⎜ ⎟

∂ ⎝ ⎠∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫n u n

i i

Steady state problem: ( ) ( ) A A CV

dA dA S dV φ

ρφ φ = Γ∇ +∫ ∫ ∫n u n

i i

2.5 Classification for simple partial differential equations

Consider a general second-order PDE in 2-D co-ordinates x and y.

2 2 2

2 20 (2.7) A B C D E F G

x x y y x y

φ φ φ φ φ φ

∂ ∂ ∂ ∂ ∂+ + + + + + =

∂ ∂ ∂ ∂ ∂ ∂

Where A to G are constant coefficients.

Three categorised of PDE:

2

2

2

elliptic PDE: 4 0 parabolic PDE: 4 0hyperbolic PDE: 4 0

B AC B AC B AC

⎧ − ⎪⎩

The classification depends only on the highest-order derivatives in each independent

variable.

Example:

2 2 2

2 2 2elliptic equation: steady conduction problem: (1-D) 0, (2-D) 0

T T T

x x y

∂ ∂ ∂= + =

∂ ∂ ∂

•

boundary-value problem

2 2 2

2 2 2 parabolic equation: unsteady conduction (1-D) , (2-D)

T T T T T

t x t x yα α

⎛ ⎞∂ ∂ ∂ ∂ ∂= = +⎜ ⎟∂ ∂ ∂ ∂ ∂

⎝ ⎠

• initial-boundary-value problem


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2 22

2 2hyperbolic equation: vibration problem

T T c

t x

∂ ∂=

∂ ∂

• initial-boundary-value problem

Steady inviscid compressible flow past a slender body:

( )2 2

2

2 21 0

T T M

x y

∂ ∂− + =

∂ ∂

elliptic: 0 parabolic: 0hyperbolic: 0

M M M

⎪⎩

Applications: the flow about an aerofoil or turbine

The governing equation can change its type in different parts of the computational

domain is one of the major complicating factors in computing transonic flow, with

the occurrence of regions of subsonic (M < 1) and supersonic (M > 1) flow.

2.6 Dynamic similarity

Two flow are dynamically similar if the non-dimensional numbers that govern the

flows have the same value.

Consider the wave motion generates by a ship of length L traveling at a speed U ∞,start with z-momentum equation:

2 2 2

2 2 2

1w w w w p w w wu v w g

t x y z z x y z

μ

ρ ρ

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

Non-dimensional variables are introduced as: * / , * / , * ,/ x x L y y L z z L= = =

2* / , * / , * / , * / and * ( ) ./t U t L u u U v v U w w U p p p U ρ ∞ ∞ ∞ ∞ ∞ ∞= = = = = −

2 2 2

2 2 2 2

* * * * ** * *

* * * * *

w w w w p w w w gLu v w

t x y z z U L x y z U

μ

ρ ∞ ∞

⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

There are two non-dimensional groups:( )

1/ 2Re and Fr

U L U

gL

ρ

μ

∞ ∞= =

• Reynolds number indicates the relative magnitude of the inertia and viscous

forces.

•

Froude number provides a measure of the relative importance of inertia and

gravity forces.

• Two incompressible viscous flows involving free surfaces are dynamically

similar if they have the same values of Re and Fr even though the values of

U ∞ or L or μ / ρ are different for the two flows.

References

Versteeg, H. K. and Malalasekera, W. (1995), An introduction to Computational Fluid

Dynamics, Longman, Malaysia.

Anderson, J. D., Jr. (1995), Computational Fluid Dynamics, The basics withapplications, International Edition, McGraw-Hill, Singapore.

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