Remark: foils with „black background“ could be skipped, they are aimed to the more advanced courses

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2013

Computer Fluid Dynamics E181107CFD5

2181106

Transport equations,Navier Stokes equationsFourier Kirchhoff equation

Cauchy’s EquationsCFD5

_ _ _Du

pressure force viscous stress gravity centrifugal forcesDt

mass

acceleration Sum of forces on fluid particle

V S Vp

Dudv n ds gdv

Dt

u P

total stress

flux

=V S V

Ddv n ds dv

Dt

P

MOMENTUM transport

Newton’s law (mass times acceleration=force)

Pressure forces on fluid element surfaceCFD5

x

x

y

z

zSW

T

E

N

B

x

y

( )2

1x

p xy z p

xn

px y z

x

Resulting pressure force acting on sides W and E in the x-direction

( )2

1x

p xy z p

xn

V S V

Dudv npds gdv

Dt

Viscous forces on fluid element surfaceCFD5

( )yxxx zxx y zx y z

Resulting viscous force acting on all sides (W,E,N,S,T,B) in the x-direction

x i ix i ixi

f n

f n n

x

x

y

z

z

x

y

( )2

1

xxxx

x

xy z

xn

( )2

1

yxyx

y

yx z

y

n

( )2

1, 0

xxxx

x y z

xy z

xn n n

( )2

1

yxyx

y

yx z

y

n

( )2

1

zxzx

z

zx y

zn

S

E

T

W

N

( )2

1

zxzx

z

zx y

zn

B

Balance of forces CFD5

_ _Du

x y z pressure force viscous stress gravityDt

yxx xx zxx

y xy yy zyy

yzxzz zzz

Du pS

Dt x x y z

Du pS

Dt y x y z

Du pS

Dt z x y z

M

Dup S

Dt

[N/m3]

Cauchy’s equation of momentum

balances (in fact 3 equations)

Balance of forces CFD5

M

Dup S

Dt

Do you remember? This is a generally valid relationship( )

Du

Dt t

( ) M

uuu p S

t

Therefore the following formulations are equivalent (mathematically but not from the point of view of numerical solution - CFD)

conservative formulation using momentum as the unknown variable (suitable for

compressible flows, shocks…)

formulation with primitive variables,u,v,w,p. Suitable for incompressible flows (very large

speed of sound)

Balance of ENERGY CFD5

2 2 21( )

2

_

E i u v w

DE heat mechanical workDt

TOTAL ENERGY transport

Total energy [J/kg]

Internal energy all form of energies (chemical, intermolecular, thermal) independent of coordinate

system

Kinetic energy [J/kg]

Heat added to FE by diffusion only (convective transport is

included in DE/Dt)

Power of mechanical forces is velocity times force. Internal

forces are pressure and viscous forces.

Balance of ENERGY CFD5

E Fourier's law

q k T

PHeat flux by conduction

Power done by total stresses ( u)

D

Dt

P

Heat conduction CFD5

x

x

y

z

zSW

T

E

N

B

x

y

( )2

1

xx

x

q xy z q

xn

( )2

1

xx

x

q xy z q

xn

Heat transfer by conduction is described by Fourier’s lawx

y

z

Tq k

xT

q ky

Tq k

z

q k T

( ) ( ) ( ) ( )DE T T T

q k T k k kDt x x y y z z

Mechanical work - pressureCFD5

x

x

y

z

zSW

T

E

N

B

x

y

( )2

1x

pu xy z pu

xn

( )2

1x

pu xy z pu

xn

( ) ( )

( ) ( ) ( )

DEk T pu

DtT T T

k pu k pv k pwx x y y z z

Mechanical work - stressesCFD5

( ) ( ) ( )

( )

( )

( )

xx xy xz

yx yy yz

zx zy zz

DEk T pu u

DtT

k pu u v wx x

Tk pv u v w

y y

Tk pw u v w

z z

x

x

y

z

zSW E

N

B

x

y

( )2

1

xxxx

x

u xy z u

xn

( )2

1

xxxx

x

u xy z u

xn

( )2

1

zxzx

z

u zx y u

zn

T

( )2

1

zyzy

z

v zx y v

zn

The situation is more complicated because not only the work of normal but also shear stresses must be included.

Total energy transport CFD5

[W/m3]( ) ( ) ( ) E

DEk T pu u S

Dt

This is scalar equation for total energy, comprising internal energy

(temperature) and also kinetic energy.

( ) E

DEk T pu u S

Dt

Fourier Kirchhoff equationCFD5

[W/m3]

1( )

2 ( ) ( ) ( ) E

D i u uk T pu u S

Dt

Kinetic energy can be eliminated from the total energy equation

using Cauchy’s equation multiplied by velocity vector (scalar product, this is the way how to obtain a scalar equation from a vector equation)

12

M

D u uDuu u p u u SDt Dt

(1)

(2)

Subtracting Eq.(2) from Eq.(1) we obtain transport equation for internal energy

( ) : E M

Dik T p u u S u S

Dt

Fourier Kirchhoff equationCFD5

Interpretation using the First law of thermodynamics

di = dq - p dv

( ) : E M

Dik T p u u S u S

Dt

Heat transferred by conduction into FE Expansion cools

down working fluid

This term is zero for incompressible fluid

Dissipation of mechanical energy to heat by viscous friction

Dissipation termCFD5

:

x x xxx xy xz

y y yyx yy yz

z z zzx z

iij

i j

y zz

j

u u u

x y z

u u u

x y z

u u u

x

x

z

uu

y

Heat dissipated in unit volume [W/m3] by viscous forces

Dissipation termCFD5

U=ux(H)y

1 1 1( )

2 2 2x

xy yx x y y x

ue e u u

y

1( ( ) )

2Te u u

1: : ( ( ) ) :

2Tu u u e

This identity follows from the stress tensor symmetry Rate of deformation

tensor

: : xy yx yx xy xyu e e e

Example: Simple shear flow (flow in a gap between two plates, lubrication)

Example tutorialCFD5

Gap width H=0.1mm, U=10 m/s, oil M9ADS-II at 00C

=3.4 Pa.s, =105 1/s, =3.4.105 Pa, = 3.4.1010 W/m3

At contact surface S=0.0079 m2 the dissipated heat is 26.7 kW !!!!

U=10 m/sy

H=0.1 mm

D=5cm

L=5cm

Rotating shaft at 3820 rpm

Fourier Kirchhoff equationCFD5

[W/m3]

Internal energy can be expressed in terms of temperature as di=cpdT or di=cvdT. Especially simple form of this equation holds for liquids when cp=cv and divergence of velocity is zero (incompressibility constraint):

( ) : i

DTc k T u SDt

An alternative form of energy equation substitute internal energy by enthalpy

( ) ( ) i

DH pk T p u u S

Dt t

where total enthalpy is defined as 1

2

pH i u u

Pressure

energy

Thermal energy

Kinetic energy

Example tutorialCFD5

Calculate evolution of temperature in a gap assuming the same parameters as previously (H=0.1 mm, U=10 m/s, oil M9ADS-II). Assume constant value of heat production term 3.4.1010 W/m3, uniform inlet temperature T0=0oC and thermally insulated walls, or constant wall temperature, respectively.

Parameters: density = 800 kg/m3, cp=1.9 kJ/(kg.K), k=0.14 W/(m.K).

Approximate FK equation in 2D by finite differences. Use upwind differences in convection terms

( ) :DTc k T uDt

T0=0

x

y

SummaryCFD5

( ) : i

DTc k T u SDt

M

Dup S

Dt

( ) 0ut

Mass conservation

(continuity equation)

Momentum balance

(3 equations)

Energy balance

State equation F(p,T,)=0, e.g. =0+T

Thermodynamic equation di=cp.dT

formulation with primitive variables u,v,w,p suitable for incompressible flows

Constitutive equationsCFD5

Macke

Constitutive equationsCFD5

Constitutive equations represent description of material properties

Kinematics (rate of deformation) – stress (dynamic response to deformation)

p

Viscous stresses affected by fluid flow. Stress is in fact momentum flux due to molecular diffusion

1( ( ) )

2Te u u

rate of deformation is symmetric part of gradient of velocity)

Gradient of velocity is tensor with components

ji j

i

uu

x

2 ( )u II e

Dynamic viscosity

[Pa.s]

Second viscosity

[Pa.s]

Constitutive equationsCFD5

Rheological behaviour is quite generally expressed by viscosity function

and by the coefficient of second viscosity, that represents resistance of fluid to volumetric expansion or compression. According to Lamb’s hypothesis the second (volumetric) viscosity can be expressed in terms of dynamic viscosity

2 ( )u II e

3 3

1 1

where( ), : ij jii j

II II e e e e

2

3

3 2 0xx yy zztrace u u

This follows from the requirement that the mean normal stresses are zero (this mean value is absorbed in the pressure term)

This is second invariant of rate of deformation tensor – scalar value (magnitude of

the shear rate squared)

Constitutive equationsCFD5

2 ( )( )3

uII e

The simplest form of rheological model is NEWTONIAN fluid, characterized by viscosity independent of rate of deformation. Example is water, oils and air. Can be solved by FLUENT.

More complicated constitutive equations (POLYFLOW) exist for fluids exhibiting

yield stress (fluid flows only if stress exceeds a threshold, e.g. ketchup, tooth paste, many food products),

generalized newtonian fluids (viscosity depends upon the actual state of deformation rate, example are power law fluids )

thixotropic fluids (viscosity depends upon the whole deformation history, examples thixotropic paints, plasters, yoghurt)

viscoelastic fluids (exhibiting recovery of strains and relaxation of stresses). Examples are polymers.

1)2( nIIK

Constitutive equationsCFD5

Ansys POLYFLOW suggests many constitutive models of generalised newtonian fluids and viscoelastic fluids.

Differential models of viscoelastic fluids:

Upper convected Maxwell model UCM, Oldroyd-B, White Metzner, Phan-Thien-Tanner model and other models. The UCM is typical and the simplest

euuDt

D T

2))((

tensorstress of derivative timeconvectiveupper called so is this

The UCM model reduces to Newtonian liquid for zero relaxation time =0.

Integral models of viscoelastic fluids written in a general form:

1

1 2

0 fading Cauchy-Greenmemory strain

( )[ ( ) ( ) ]M s f C t s f C t s ds

Unknowns / EquationsCFD5

There are 13 unknowns:

u,v,w, (3 velocities), p, T, , i, xx, xy,…(6 components of symmetric

stress tensor)

And the same number of equations

Continuity equation

3 Cauchy’s equations

Energy equation

State equation p/=RT

Thermodynamic equation di=cp.dT

6 Constitutive equations

( ) 0ut

( ) : E M

Dik T p u u S u S

Dt

M

Dup S

Dt

2 ( )( )3

uII e

Navier Stokes equationsCFD5

22 ( ( )( )) ( ( )( ( ) )) ( ( )( ) )

3 3Tu

II e II u u II u

2 2

2( ) ( ) ( ( ) )

3

2( ) ( ) ( ( ))

3

2( ) ( )

3

( )3

ij j i kij

i i i i j i k

j i i

i i i j j i

j i i i i

i i i j j i j i j i

j i

i i i j

u u u

x x x x x x x

u u u

x x x x x x

u u u u u

x x x x x x x x x x

u u

x x x x

2 2

3i i

j i j i

u u

x x x x

( ) 2( ( ) ) ( ) ( ) ( ) ( ) ( )

3 3T II

II u II u u u II

These terms are ZERO for incompressible fluids

These terms are small and will be replaced by a parameter sm

Using constitutive equation the divergence of viscous stresses can be expressed

This is the same, but written in the index notation (you cannot make mistakes when calculating derivatives)

Navier Stokes equationsCFD5

( ( ) ) m M

Dup II u s S

Dt

General form of Navier Stokes equations valid for compressible/incompressible Non-Newtonian (with the exception of viscoelastic or thixotropic) fluids

Special case – Newtonian liquids with constant viscosity

2M

Dup u S

Dt

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

( ) ( )

( ) ( )

( ) ( )

x

y

z

u u u u p u u uu v w g

t x y z x x y z

v v v v p v v vu v w g

t x y z y x y z

w w w w p w w wu v w g

t x y z z x y z

Written in the cartesian coordinate

system