Hanjalic MF 2 2008 Lect 2 Equations

7/30/2019 Hanjalic MF 2 2008 Lect 2 Equations

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2. Fundamental Equations of Fluid and Heat/Mass Flow

(Field Description)

Conservation laws for control volume in differential form Source terms and Constitutive relations Classification of equations Boundary conditions Reynolds decomposition and RANS equations

DIPARTIMENTO DI MECCANICA ED AERONAUTICA

MAINSKI FAKULTET

U SARAJEVU

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Conservation Laws for Dynamic andConservation Laws for Dynamic and

Thermodynamic SystemsThermodynamic Systems

All conservation laws for a Closed System (Control Mass) can beexpressed in a general mathematical form

( ) = msmsor, as a time rate

0lim

= = &

tms

d

dt twhere: is a conservable quantity (extensive property) in a mass system,

and is the source/sink (cause of change) of .

Note:m Vdm dV = = where:

is the intensive property in the (closed mass) system , and m andV denote the mass and volume of the considered close system, resp.

Summary of conservation laws:

Q - WeE = m eenergy

FFtvM = m vmomentum

0 (or r)1mmass

&

r&

Q W& &


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Concept of Control VolumeConcept of Control Volume

Closed System(Control Mass)

Open System(Control Volume)

Inflow

Outflow

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( )x xv

( )x x x +v

( , , )P x y z

V x y z =x

xy

z

y

z

Elementary (differential) control volume


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( )x xv

( )x x x +v

( , , )P x y z

V x y z =x

xy

z

y

z


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XY

Z

Intake (100

Tilt)

Exhaust (100

Tilt)

( )x xv

( )x x x +v

( , , )P x y z

V x y z =x

xy

z

y

z



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Conservation Laws (CL) for a Control Volume (CV)Conservation Laws (CL) for a Control Volume (CV)

Since we are interested in the differential form of the CL, recall that

m V

dm dV = =

Note: at a particular time instant we can choose the control volume tocoincide with the mass system so that mass system = control volume but,

= = && &m V

dm dV

= = & & &m V

dm dV = = m V

dm dV

( )dm v.dA=rr

& ( )dV v.dA=rr&

masssystem controlvolume

d d

dt dt

CVi e

i e

d

dt

= + & & &

We now need to account for inflow and outflow of the variable through allopen parts of the Control boundary -identifiable inlets (is) and exists (es)

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We replace the extensive properties in the generic formulationby volume/surface integrals in terms of intensive properties

Transformation of conservation equationsTransformation of conservation equations

for a control volume (CV)for a control volume (CV)

( )CVCV CV

dVd ddV

dt dt t

= =

( ).

i Aii

v dA =

rr&

( ).e Aeev dA =

rr&

( ). .( )i e A CVCVi e

v dA v dV = = rr r& &

The conservation equation is now written as

( ).( )

CV CV CV

dVv dV dV

t

= +

r

&

inv

r

exvr

inA

r exA

r

CV


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( ).( )v

t

= +

r

&( )

( )jj

vt x

= +

&

Conservation equation for a control volume (CV)Conservation equation for a control volume (CV)

in a generic differential formin a generic differential formVector notations Index notations (for Cartesian coordinates)

StrongStrongconservative formconservative form

WeakWeakconservative formconservative form

( . )D

vDt t

= + =

r&

j j

Dv

Dt t x

= + =

&

Note: for mass conservation =m, =/m=1 and =0D/Dtis the material (substantial) derivative.

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Source terms and Constitutive relationsSource terms and Constitutive relations

( . )]m A m AV V A CV dV dV dA dV = = + = + & & & & & &

where m& = an internal source per unit mass (e.g. of heat due to chemicalreaction, electric or magnetic heating in energy eqn, and gravitational

or other body forces in momentum equation)

A&

= a surface source, i.e. diffusion flux through the Control Surfacee.g. heat conduction in energy equation, viscous and pressure

forces in momentum equation

The surface sources is a molecular transfer and can often be expressedin terms of the gradient of the property (Constitutive relations):

A L = &

whereL is the molecular transport coefficient


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Basic constitutive relationsBasic constitutive relations

Conduction flux (Fourier Law),L== thermal conductivity:

A q T = = r

& or ii

Tq

x

=

Species (mass) molecular diffusion (Fick Law),L=D= mass diffusivity:

"A m C = = r

& D"i

i

Cm

x

= D

( )2

.3

T

A p v v v = = + +

T I Ir r r

&

Momentum flux (pressure and viscous stress) (Newton-Poisson Law)L== dynamic viscosity:

2 12

3 3

ji kij ij ij ij ij kk ij

j i k

vv vp p S S

x x x

= + + = +

or

or

where 1

2

jij

j i

vvS

x x

= +

is the mean rate of strain.

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General form of Differential Conservation LawsGeneral form of Differential Conservation Laws

( ). .m AD

vDt t

= + = +

r

& &

{ {

{

{

Aj m

j jmassmaterial time source surfaceconvectionderivative rateof

sourcechange

D vDt t x x

= + = +

&&

14243


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Momentum (Momentum (

NavierNavier

--StokesStokes

) equations) equations

iv v= r

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Energy equationEnergy equation

ph c T= =in terms of static enthalpy

in terms of temperature T=

( ). .gp

Dh hv h q h

Dt t c

= + = +

r&

j gj j p j

Dh h h hv q

Dt t x x c x

= + = +

&

( ) ( ). .g

p

qDT Tv T T

Dt t c

= + = +

&r

gj

j p j j

qDT T T Tv

Dt t x c x x

= + = +

&

where =/(cp) is the temperature diffusivity.


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Species concentration equationSpecies concentration equation

C=

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Classification of equationsClassification of equations

Conservation equations can often be simplified for specific problems(some terms can be neglected), making it possible to solve equationsanalytically, or by using a simple numerical integration.

In some cases the truncation leads to the well established formsof equations known under separate names

We consider equation classification using two criteria

Physical criteria (based on physical meaning of the terms in equations)

Mathematical criteria (types of equations, method of solution)


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Classification of equationsClassification of equations Physical criteria (based on physical meaning of each term)

{ {

( ){

( ). .mD

v LDt t

= + = +

r&

14243 14243SC DLM

M= material (substantial) derivative (the total change along a streamline)

L = local time rate of change (felt by an observer at a fixed position in

an inertial coordinate frame)

S = source of

D = diffusion (flux ofthrough the surface of the elementary control volume)

C = convection (rate of change felt by the observer moving with the fluidparticle with the local velocity )v

r

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Special forms of Conservation equationsSpecial forms of Conservation equations

2 0 =

2Lt

=

( ) 2.v L = r

Stationary transport ( ) without flowing ( , solids, stationary liquids)with constant material properties (L=const) and without internal source;

M=L+C=0, S=0, Laplace(potential) equation

/ 0t = 0v =r

Non-stationary (transient) transport without flowing and without source(L=const) and without internal source;

C=0, S=0, Diffusion(conduction) equation

Stationary transport with flowing, but without internal source (L=const) andwithout internal source;

L=0, S=0, Convection-diffusionequation:


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Classification of equations: Mathematical criteriaClassification of equations: Mathematical criteria

Partial differential equations can be written as (for 2-dimensional problem)

2 0xx xy yy x yAu Bu Cu Du Eu Fu+ + + + + =

where subscripts denote differentiation, i.e. / ,xu u x= 2 / ,xyu u x y=

Elliptic:2 4 0B AC e.g. Wave propagation eqn (momentum eqn for supersonic flows)

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Boundary and Initial conditionsBoundary and Initial conditions

Transport processes are described by (nonlinear) partial differential equationsof the second order in space and first order in time; Hence

two boundary conditions required for each dependent variable for time-dependent problems, the initial conditions are required

Two types of boundary conditions can be used (depending on the problem):

Dirichlet b.c.: values of is defined on the boundary, i.e. b= b(x,y,z) Neuman b.c.: a flux of related to normal to the boundary

For energy eqn, heat flux at a solid wall can be defined in two ways:

Convective flux:

Radiative flux:

( , , , )b bb

L f x y zn = =

/ nx

For momentum eqn the no-slip b. c. at a solid wall: | 0 ( )b b wv or v v= =r r

0( , , , ) ( , , )( )b bb

Tq f x y z T x y z T T

n

= = =

r

4 40( , , , ) ( , , ) ( )b b

b

Tq f x y z T G x y z T T

n

= = =

r


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Turbulence realisations: needs forTurbulence realisations: needs for statisticalstatisticaldescriptiondescription

Navier-Stokes (N-S) equations are believed to describe exactly the motionof a fluid element in the differential form irrespective of whether fluid motionlaminar or turbulent; but

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Time recording of the axial velocity U1(t)

at a point on the axis of a turbulent jet

(Tong and Wahrhaft 1995)

Time recording of temperature in the

bottom corner of a side-heated cube,

y/H=0.1, (x/H)Ra1/4=0/7 (Opstelten 1994)

Turbulence realisations: featuresTurbulence realisations: features

A time history of velocity (temperature, pressure) at a point in a steadyturbulent flow shows irregular, non-repeatable fluctuations, despite thefact that the time-averaged flow rate is constant;

Navier-Stokes (N-S) equations are believed to describe exactly the motionof a fluid element in the differential form irrespective of whether fluid motionlaminar or turbulent;


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A sequence of instantaneous distributions of velocity across a pipe cross-sectionshows a bunch of non-repeatable wriggled curves (realizations);

Fortunately, we are primarily interested in the averaged (mean) field andeffects (forces on structures, friction, heat and mass transfer);

Instantaneous velocity field: velocity

profiles in a pipe for a number of shots

Time-averaged (mean) velocity profiles

in a pipe for laminar and turbulent flows

Turbulence realisations: statistical averagingTurbulence realisations: statistical averaging

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Statistical description of turbulence:Statistical description of turbulence:

Reynolds decomposition andReynolds decomposition and timetimeaveragingaveraging

' = +

Instantaneous property decomposed into a mean and fluctuation '

If the flow is stationary (or slowly varying with time) the mean property isthe time-averaged (time mean) property, defined as

0

1 ( ) lim ( , )i ix x t dt

= = so that

0

1' lim '( , ) 0ix t dt

= =

Hence, for stationary turbulent flows, it is more appropriate to write the(time) decomposition as

( , ) ( ) '( , )i i ix t x x t = +

Note that an overbar over a variable denotes that the variable has beensubjected to an operator(normalised time integration)


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Reynolds decomposition: ensemble and phase averagingReynolds decomposition: ensemble and phase averaging

If the flow varies in time with a time scale comparable to the turbulence timescales, the time averaging makes little sense: here we can apply ensembleaveraging and define ensemble mean:

1

1 ( , ) lim ( , )ne

ji i

ne e j

x t x tn

=

=

In the case of a periodic flow with a period of it is convenient to use

the phase averaging over np periodsp

1

1 ( , ) lim ( , )

np

i i pnp p j

x t x t jn

=

= +

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Reynolds decomposition: some rulesReynolds decomposition: some rules

Irrespective of the type of averaging, we can now decompose all propertiesof interest:

$ 'i i iv U U u = +

r 'P P p= + ' = +

'T T = + 'H H h= + etc.

fluid velocity: pressure: density:

temperature: enthalpy:

'K K k= +

kinetic energy:

Note:

Multiplication with a constant makes no effect on averaging, i.e.

C C C = =

Differentiation and integration commute with averaging, i.e.

s s s

= =

ds ds ds = =


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Consider two instantaneous variables in a turbulent flow:

' = + ' = +

Reynolds decomposition: rules involvingReynolds decomposition: rules involving twotwovariablesvariables

Addition and subtraction: no effect of averaging

Multiplication: because of nonlinearcharacter of the N-S equations, theproduct of two fluctuating variablesis usually non-zero,thus

= =

( )( ){ {

0 0

' '

' ' ' '

' '

= + + =+ + + =

= +

' ' 0

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ReynoldsReynolds--averaged conservation equations (averaged conservation equations (RANS equationsRANS equations))

Apply Reynolds averaging to the instantaneous conservation equations:

The new term in the box is a new unknown variable (a correlation),which conceals the information lost due to averaging!


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ReynoldsReynolds--averaged conservation equations, cont.averaged conservation equations, cont.

For incompressible flows and the continuity eqn reduces to:'/ 0t ='

' div ' 0j

j

uv v

x

= = =

r r

Hence, the term in the box can be written as

The term has a character of diffusion and represents transport of by uj andcan be lumped with the molecular diffusion so that RANS eqn becomes

Replacing by appropriate variables leads to the RANS equations., ' and &

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ReynoldsReynolds--averaged (averaged (RANSRANS) equations) equations


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RANS equations: physical interpretation ofRANS equations: physical interpretation ofnewnewflux termsflux terms

' ' ti j iju u = turbulent stress tensor (turbulent flux - transport of momentum

per unit volume by velocity fluctuation )'iu

'ju

' ' tp j jc u q = turbulent heat flux vector (turbulent transport of enthalpy per

unit volume by velocity fluctuation )' 'ph c ='ju

' '

"t

j jc u m = turbulent mass flux vector (turbulent transport of mss of aspecies per unit volume by velocity fluctuation )'ju'c

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RANS ApproachRANS ApproachAdvantages and LimitationsAdvantages and Limitations

Advantages:

Averaged RANS conservation equations reduce to a form manageable bythe available computational codes for laminar flows (Navier-Stokes Code)

Details of fluctuating motions (not recoverable by RANS) are usually notneeded for computing mean-flow properties for industrial applications

Disadvantages: Statistical averaging brings about a loss of information (the appearance

of superfluous variables that need to be provided); this is called theClosure Problem. A set of additional semi-empirical algebraic and/ordifferential equations that provide the unknown (superfluous) variablesis referred as a Turbulence Modelurbulence Model

Conventional RANS models cannot capture any spectral features ofturbulence such as interactions among eddies of different sizes or effectsassociated with well-organized coherent eddy motion.

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Hanjalic MF 2 2008 Lect 2 Equations