Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

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 2010

Turbulent flows,…

Computer Fluid Dynamics E181107CFD62181106

Some figures and equations are copied from Wikipedia

TurbulenceCFD6

( ) ( ( ) ) Mu u u p e u St

( ( )) ( ( ) ) Mu uu p e u St

Source of nonlinearities of NS equations are inertial and viscous terms

for incompressible fluids the equivalent conservative form

( ( ) ) ( ( ) ) Mu uu u u p e u St

The convection term is a quadratic function of velocities – source of

nonlinearities and turbulent phenomena

Instabilities Rayleigh Benard convectionCFD6

Horizontal liquid layer heated from below is in still (viscous forces attenuate small disturbances) until the buoyancy exceeds a critical level RaL. Then the more or less regular cells with circulating fluid are formed.

>1100L

Tu

Tb

Even if the stability limit was exceeded the flow pattern is steady and remains laminar. At this stage the nonlinear convective term is not so important. Only if RaL>109 (approximately) eddies start to be chaotic, velocity fluctuates and the flow is turbulent.

http://en.wikipedia.org/wiki/File:B%C3%A9nard_cells_convection.ogv

http://upload.wikimedia.org/wikipedia/commons/f/f5/ConvectionCells.svg

Instabilities Karman vortex streetCFD6

Repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid over bluff bodies. A vortex street will only be observed above a limiting Re value of about 90.

L

Even if the stability limit was exceeded the flow pattern is steady and remains laminar.

http://en.wikipedia.org/wiki/File:Vortex-street-animation.gif

TurbulenceCFD6

2

( )Re uu uDu

Relative magnitude of inertial and viscous terms is Reynolds number

Increasing Re increases nonlinearity of NS equations. This nonlinearity leads to sensitivity of NS solution to flow disturbances.

Laminar flow: Re<Recrit

Turbulent flow: Re>Recrit

Distarbunce is attenuated

Distarbunce is amplified

http://en.wikipedia.org/wiki/Reynolds_number#Transition_Reynolds_number

Turbulence - fluctuationsCFD6

Turbulence can be defined as a deterministic chaos. Velocity and pressure fields are NON-STATIONARY (du/dt is nonzero) even if flowrate and boundary conditions are constant. Trajectory of individual particles are extremely sensitive to initial conditions (even the particles that are very close at some moment diverge apart during time evolution). .

Velocities, pressures, temperatures… are still solutions of NS and energy equations, however they are nonstationary and form chaotically oscillating vortices (eddies). Time and spatial profiles of transported properties are characterized by fluctuations

'u u u Actual value at given time and space

Mean value

fluctuation

Turbulence - fluctuationsCFD6

Statistics of turbulent fluctuations

Mean values (remark: mean values of fluctuations are zero)

rms (root mean square)

Kinetic energy of turbulence

Intensity of turbulence

Reynolds stresses

, , ,u p T

2'u

2 2 21 ( ' ' ' )2

k u v w

2( ) /3

I k u

2 2 2, , , , ,' , ' , ' , ' ', ' 't xx t yy t zz t xy t xzu v w u v u w

Turbulent eddies - scalesCFD6

Large energetic eddies (size L) break to smaller eddies. This transformation is not

affected by viscosity

Inertial subrange (inertial effects dominate and spectral energy depends

only upon wavenumber and )

Smallest eddies (size is called Kolmogorov scale) disappear, because kinetic energy is

converted to heat by friction

E()

=2f/u1/L 1/

Spectral energy of turbulent eddies

wavenumber (1/size of eddy)

Kinetic energy of turbulent fluctuations is the sum of energies of turbulent eddies of different sizes

/ /

0

1 ( )2 i ik u u E d

Typical values of frequency f~10 kHz, Kolmogorov scale ~0.01 up to 0.1 mm

Kolmogorov scale decreases with the increasing Re

2/3

5/3( )E C

http://upload.wikimedia.org/wikipedia/commons/f/fb/Jet.jpg

Viscous subrange - scalesCFD6

Kolmogorov scales (the smallest turbulent eddies) follow from dimensional analysis, assuming that averything depends only upon the kinematic viscosity and upon the rate of energy supply in the energetic cascade (only for small isotropic eddies, of course)

34

4u

Length scale Time scale velocity scale

These expressions follow from dimension of viscosity [m2/s] and the rate of energy dissipation [m2/s3]

2 2 2 2

3 3[ ] [ ][ ]

1 2 20 3

m m mms s s

3 / 41/ 4

Viscous scale - tutorialCFD6

Derive time scale of the smallest turbulent eddies

Time scale

2 2 2 2

3 3[ ] [ ][ ]

0 2 21 3

m m mss s s

1/ 21/ 2

Inertial subrange - scalesCFD6

Spectral energy E decreases in the turbulent cascade when turbulent eddies are decomposed to smaller and smaller eddies. Energy transformation is not affected by viscosity in the inertial subrange

2 2

/ /

[1/ ]0 [ / ] spectralwave numberkinetic energy energy is

2 /of fluctuations independent of size of eddieviscosity in the

inertial range

1 ( ) 2 i i

mm s

k u u E d

Dimension analysis in the inertial subrangen

m

CE )(

2 2

2 3[ ] [ ] [ . ]m m n

n m

k Ed

m m ms s m

35

343

32

n

m

2/3

5/3( )E C

And this is the famous Kolmogorov’s law of 5/3


Wikipedia (abbreviated)

Turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. The energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.In his original theory of 1941, Kolmogorov postulated that for very high Reynolds number, the small scale turbulent motions are statistically isotropic. In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as L).

Kolmogorov introduced a hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the kinematic viscosity () and the rate of energy dissipation (). With only these two parameters, the unique length that can be formed by dimensional analysis is

http://en.wikipedia.org/wiki/Kolmogorov

http://en.wikipedia.org/wiki/Reynolds_number


Why and when it is important to know the turbulent scales:

Rate of chemical reactions depends upon the rate of micromixing determined by the frequency and the size of turbulent eddies

Breakup of droplets depends upon the energy and size of turbulent eddies (too large eddies only move but do not break a droplet)

Coalescence of bubbles depends upon collision frequency, collision speed, and these parameters are correlated with the energy dissipation rate and the length microscale l corresponding to radius of bubbles

3/22 )( lCv

Ronnie Andersson and Bengt Andersson. Modeling the Breakup of Fluid Particles in Turbulent Flows. AIChE Journal June 2006 Vol. 52, No. 6

H. Venneker, Jos J. Derksen and Harrie E. A. Van den Akker. Population balance modeling of aerated stirred vessels based on CFD . AIChE J. Volume 48, Issue 4, April 2002, 673–685

DNS Direct Numerical SimulationCFD6

L

In order to resolve all details of turbulent structures it is necessary to use mesh with grid size less than the size of the smallest (Kolmogorov) eddies. N-grid points in one direction should be

3 3 3/4 3 9/43 9/4 9/4

3 9/4 9/4 3/4 ( ) Re

LN

L L L u LuNL

3uL

Re uL Re uL

Velocity scale u in previous expression is related to magnitude of turbulent fluctuations (rms of u’, or k). The Re related to the velocity fluctuation is called turbulent Reynolds number.

(based upon dimensional ground)

No.of grid points in DNS

No.of time steps

12300 380 6.7 M 32 k

30800 800 40 M 47 k

61600 1450 150 M 63 k

230000 4650 2100 M 114 k

Table concerns DNS modelling of channel flow experiments (rewritten from Wilcox: Turbulence modelling, chapter 8).

9/4

3/4_

Re

ReDNS

time steps

N

N

Remark: Re~106 or 107 at flow around a car or flow around wings

http://upload.wikimedia.org/wikipedia/commons/f/fb/Jet.jpg


Abstract The three-dimensional compressible Navier-Stokes equations are approximated by a fifth

order upwind compact and a sixth order symmetrical compact difference relations combined with threestage

Runge-Kutta method. The computed results are presented for convective Mach number Mc =

0.8 and Re = 200 with initial data which have equal and opposite oblique waves. From the computed

results we can see the variation of coherent structures with time integration and full process of instability,

formation of A -vortices, double horseshoe vortices and mushroom structures. The large structures

break into small and smaller vortex structures. Finally, the movement of small structure becomes dominant,

and flow field turns into turbulence. It is noted that production of small vortex structures is combined

with turning of symmetrical structures to unsymmetrical ones. It is shown in the present computation

that the flow field turns into turbulence directly from initial instability and there is not vortex pairing in

process of transition. It means that for large convective Mach number the transition mechanism for

compressible mixing layer differs from that in incompressible mixing layer.

Direct numerical simulation of transition and turbulence in compressible mixing layer

FU Dexun , MA Yanwen ZHANG Linbo

Vol 43 No.4, SCIENCE IN CHINA (Series A), April 2000


DNS AND LES OF TURBULENT BACKWARD-FACING STEPFLOW USING 2ND- AND 4TH-ORDER DISCRETIZATIONADNAN MERI AND HANS WENGLE

Abstract. Results are presented from a Direct Numerical Simulation (DNS)and Large-Eddy Simulations (LES) of turbulent flow over a backward-facingstep with a fully developed channel flow utilizedas a time-dependent inflow condition. Numerical solutions using afourth-order compact (Hermitian) scheme, which was formulated directlyfor a non-equidistant and staggered grid in [1] are compared with numericalsolutions using the classical second-order central scheme. The resultsfrom LES (using the dynamic subgrid scale model) are evaluated against acorresponding DNS reference data set (fourth-order solution).

Transition Laminar-TurbulentCFD6

Bacon

velocity

y

Hydrodynamic instability due to prevailing inertial forces (convection term in NS equations) is the cause of turbulence.

Inviscid instabilities

characterised by existence of inflection point of velocity profile

- jets

- wakes

- boundary layers wit adverse pressure gradient p>0

Viscous instabilities

Linear eigenvalues analysis (Orr-Sommerfeld equations)

- channels, simple shear flows (pipes)

- boundary layers with p>0


Poiseuille flow ~ 5700Couette flow – stable?

velocity

y Inflection – source of instability

(max.gradient)

There is no inflection of velocity profile in a pipe, however turbulent regime exists if

Re>2100

http://en.wikipedia.org/wiki/Orr%E2%80%93Sommerfeld_equation


How to indentify whether the flow is laminar or turbulent ?

Experimentally

Visualization, hot wire anemometers, LDA (Laser Doppler Anemometry).

Numerical experimentsStart numerical simulation selected to unsteady laminar flow. As soon as the solution converges to steady solution for sufficiently fine grid the flow regime is probably laminar

Recrit

According to value of Reynolds number using literature data of critical Reynolds number


Stability analysis

Velocity disturbance

Mean (undisturbed)

flow

Production (extracting energy from the mean

flow to fluctuations)

Linear stability analysisMomentum equation for disturbance

linear stability theory can predict when many flows become unstable, it can say very little about transition to turbulence since this progress is highly non-linear


Geometry Recrit

Jets 5-10

Baffled channels

100

Couette flow

300

Cross flow 400

Planar channel

1000

Geometry Recrit

Circ.pipe 2000

Coiled pipe 5000

Suspension in pipe

7000

Cavity 8000

Plate 500000

D

D

D/2

D

D

D

D

D

http://en.wikipedia.org/wiki/Taylor%E2%80%93Couette_flow

Turbulent structures evolutionCFD6

Journal of Fluids and Structures 18 (2003) 305–324Force coefficients and Strouhal numbers of four cylinders in cross flowK. Lama, J.Y. Lib, R.M.C. Soa

Re=200

Re=800

Re<4

Re<40

Re<200 2D von Karman vortex street

Fully developed turbulent flowsCFD6

Free flows (self preserving flows)

Jets Mixing layers Wakes

h yu

umax

umax

umin

uyb

max

( )u yfu h

min

max min

( )u u ygu u b

max

max

constux

constux

Circular jet

Planar jet

x

x

Jet thickness ~x, mixing length ~x see Goertler, Abramovic Teorieja turbulentnych struj, Moskva 1984:

Example - tutorialCFD6

Entrainment in jets (increase of volumetric flowrate)

h yu

umax

max

( )u yfu h

1

max max max0 0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )h

planaryV x u x f dy h x u x f d h x u x F c xh

12 2

max max max0 0

( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( )h

circyV x u x f ydy h x u x f d h x u x F cxh

Planar jet

Circular jet

~x ~1/x

~1/x


Flow at walls (boundary layers)y

Laminar sublayer

Buffer layer

Log law * wu

*

* u yuu yu

Friction velocity

laminar y+<5 u+=y+

buffer 5<y+<30 u+=-3+5lny+

Log law 30<y+<500 u+=lnEy+/

INNER layer (independent of bulk velocities)

OUTER layer (law of wake) max

*

1 ln( )u u y Au

u


Flow at walls (turbulent stresses)t

*yuy

2 2( )

[1 exp( )]26

t m

m

uly

yl y

2 2' ' 24 [1 exp( )]26t wyu v

Prandtl’s model

vanDriest model of mixing length lm

5 30

Example tutorialCFD6

Calculate thickness of laminar sublayer at flow of water in pipe (D=2 cm) at flowrate 1 l/s.

*

5 yu

3

3

4 4 10 1000Re 636620.02 10

VD

2 22

2 4 2 44

1 0.3162 2 25.2[ ]8 Rew

V Vu PaD D

Wall shear stress from Blasius

Turbulent region, well within validity of Blasius correlation

Dimensionless thickness of laminar

sublayer

*28

0.632 0.159[ / ]Re

w Vu m sD

2 2 88 8 85 Re 0.005 0.02 63662 2 10 63662 0.3110000.632 0.001 0.632 0.632

Dy mV

Friction velocity

TIME AVERAGING of turbulent fluctuationsCFD6

Benton


RANS (Reynolds Averaging of Navier Stokes eqs.)

1 ( ) '

( ) ''

' '

t t

t

t t

tt t

t

t dtt

t dt

dt

Time average

Favre’s average

Proof: ( ')( ') ' '

Remark: Favre’s average differs only for compressible substances and at high Mach number flows. Ma>1


Trivial facts

Averaged value of fluctuation is zero

Average value of gradient of fluctuations is zero

Average value of product is not the product of averaged values

' 0 ' 0

x

' 'uv u v u v

( ) ( ) ( ' ')uv u v u v

TIME AVERAGING of NS equationsCFD6

0 0 0u u u

( ) ( ) ( )

( ) ( )) ) ( ' '(

u uu p ut

u ut

u uu p u

Continuity equation

Navier Stokes equations

Reynolds stresses

' 't u u

http://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations

http://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations

TIME AVERAGING of transport equationsCFD6

( ) ( ) ( )

( ) ( ') ( ')) ( u

ut

ut

Turbulent fluxes

' 'tq u

Boussinesq hypothesisCFD6

Turbulent fluxes and turbulent stresses are defined by the same constitutive equations as in laminar flows, just only replacing diffusion coefficients and viscosity by turbulent transport coefficients.

Boussinesq hypothesisCFD6

Rate of deformation based upon gradient

of averaged velocities

' ' tu

' 'i ti

ux

' ' ( ( ) )Ttu u u u

1 ( ( ) )2

TE u u 2t tE

' ' ( )j ii j t

i j

u uu u

x x

Analogy to Fourier law

Analogy to Newton’s law

q T

( ( ) )Tu u

TRANSPORT coefficients-analogyCFD6

It is assumed that the rate of turbulent transport based upon migration of turbulent eddies is the same for momentum, mass and energy, therefore all transport coefficients should be almost the same

tt

t

Prandtl number for turbulent heat transfer

Schmidt number for turbulent mass transfer

t=0.9 at walls

t=0.5 for jets

according to Rodi

Turbulent viscosity modelsCFD6

Botero

Turbulent viscosity modelsCFD6

Turbulent viscosity is not a material parameter. It depends upon the actual velocity field and fluctuations at current point x,y,z. There exist different RANS models for turbulent viscosity prediction

Algebraic models (not reflecting transport of eddies)

1 equation models (transport equation for turbulent viscosity)

2 equations models (viscosity derived from transport equations of other characteristics of turbulent eddies)

Nonlinear eddy viscosity models (v2-f)

RSM Reynolds Stress Modelling (transport equations for components of reynolds stresses)

Algebraic modelsCFD6

Prandtl’s model of mixing length. Turbulent viscosity is derived from analogy with gases, based upon transport of momentum by molecules (kinetic theory of gases). Turbulent eddy (driven by main flow) represents a molecule, and mean path between collisions of molecules is substituted by mixing length.

Excellent model for jets, wakes, boundary layer flows. Disadvantage: fails in recirculating flows (or in flows where transport of eddies is very important).

2t m

uly

Mixing length

h

h

y

u(y) 0.075ml h0.09ml h

0.07ml h

[1 exp( )]26myl y

Circular jet

Planar jet

Mixing layers

Currently used algebraic models are Baldwin Lomax, and Cebecci Smith

ndeformatio

of rate

2 1][s

mtt

2 equations modelsCFD6

Two equation models calculate turbulent viscosity from the pairs of turbulent characteristics k-, or k-, or k-l (Rotta 1986)

k (kinetic energy of turbulent fluctuations) [m2/s2]

(dissipation of kinetic energy) [m2/s3]

(specific dissipation energy) [1/s]

2

t

t

k

kC

Wilcox (1998) k-omega model, Kolmogorov (1942)

Jones,Launder (1972), Launder Spalding (1974) k-epsilon model

3 /u l

C = 0.09 (Fluent-default)

Tutorial 2 equations modelsCFD6

Derive relationship for turbulent viscosity

2

tkC

Launder Spalding k-epsilon model

11

2

k 2 2 2 2 3

2 3 3 2 2 3[ . ] [ ][ ][ ]Ns kg kg m m kg mPa sm ms m s s s

1

1 2 2 31 2 3

k – transport equationCFD6

All two equation models (and also Prandtl’s one equation model) are based upon transport equation for the kinetic energy of turbulence 2 2 21 1' ' ( ' ' ' )

2 2i ik u u u v w

' ' ( ) ' ' ( )uu u uu u p u ut

1' ( ' ')2

u ku u ut t t

( ) ( ' ' 2 ' ' ' ' ') 2 ' : ' ' ' :2

k ku p u u e u u u e e u u Et

Transport equation for k is derived in a similar way like the transport equation of mechanical energy by multiplying NS equation by vector of velocities (unlike mechanical energy only by the vector of velocity fluctuations)

Transport of pressure viscous stress reynolds stress rate of dissipation turbul.production

(p’) (2e’) (u’u’)

k – transport equationCFD6

Rate of dissipation of kinetic energy of velocity fluctuation

' ' ~ tu k k

( ) ( ) 2 : 2 ' : 'tt

k

k ku k E E e et

Dispersion terms (viscous and reynolds stresses) and production term can be expressed using turbulent viscosity (Boussinesq)

' ' ' ' '2 ku u u k u k

Example: approximation of Reynolds stress transport term

-

2 2' : ' ' 'ij ije e e e

' '1' ( )2

j iij

i j

u uex x

Laminar and not turbulent viscosity!

k = 1 Fluent

– transport equationCFD6

1 2( ) ( ) ( 2 : )ttCu E E C

kt

Transport equation for dissipation looks like the k-transport. Just substitute for k. Production and dissipation terms are modified by universal constants C1 and C2

This term follows from dimensional analysis

Production term (generator of turbulence) Dissipation

term

! 1[ ] [ ]k s

C1 = 1.44 C2 = 1.92 = 1.3 Fluent (default)

k- modifications (RNG, realizable)CFD6

1 221( ) (( ) ) ( 2 : )ttu C E E C

t kf f

Corrections of dispersion, production and dissipation terms with the aim to extent applicability of k- for low Reynolds number flows

( ) (( ) ) 2 :tt

k

k ku k E Et

Functions of turbulent Reynolds 2

Re tt

kC

Effective viscosity (RNG)CFD6

2

0 (laminar)

(high Re)

ef

ef t

k

C kk

Blending formula for effective viscosity

2[1 ]ef

C k

- estimateCFD6

Dissipated energy in a mixing tank

3 5 ~ 1PPon d

Power number

d

n

2 2 4 2~ ~F d u d n 5 2~tM Fd d n

5 3~tP M d n

Check units of dissipation [W/kg =m2/s3]

k- boundary conditionsCFD6

*3

wuy

k and must be specified at inlets. Estimate of kinetic energy k is based either upon measurement (anemometers) or experience (from estimated intensity of turbulence I). Dissipation is estimated from correlations for power consumption estimates.

Values of k and at wall (must be also defined as boundary conditions) can be approximated by wall functions

*2

wukC

Friction velocity

* wu

Distance of the nearest boundary

node from wall

This is implemented in majority of CFD programs

Reynolds stresses (k-)CFD6

2' ' 2 ( ) 2 23

i i ii i t ii t

i i i

u u uu u k k k

x x x

Constitutive equation for Reynolds stresses must be modified with respect to isotropic pressure

2' ' ( )3

jitij i j t ij

j i

uuu u kx x

This term is zero for incompressible liquidsRemark: Turbulent stresses determine

kinetic energy of turbulent fluctuations 0u

Assesment of k- modelsCFD6

Problems and erroneous results can be expected

Unconfined flows (wakes, jets). k- model overestimates dissipation, therefore jets are overdumped.

Pressure transport term (u’p’) is neglected (errors in flows characterised by high pressure gradients)

Curved boundaries or swirling flows

Fully developed flows in noncircular ducts should be characterized by secondary flows due to anisotropy of normal Reynolds stresses (these features cannot be predicted by linear viscosity models)

Problems in buoyancy driven flows

Discrepancies?CFD6

Special case: Steady unidirectional flow, constant density, homogeneous turbulence

1 2( ) ( ) ( 2 : )ttu C E E C

t k

( ) ( ) 2 :tt

k

k ku k E Et

2( )t uy

2 2

1

( )t Cuy C

These terms are identically zero in homogeneous turbulence

Further reading C1 = 1.44 C2 = 1.92 k = 1 = 1.3

Reynolds Stress Models (RSM)CFD6

' 'ij i jR u u

( )ij tij ij ij ij ij

DRP R

Dt

production

diffusion

DissipationPressure transport

Centrifugal forces

23 ij

6 transport equations

1 dissipation

1 2( ) ( ) ( 2 : )ttu C E E C

t k

Large Eddy Simulation (LES)CFD6

Demuth

Large Eddy Simulation (LES)CFD6

Instead of time averaging of NS equations, spatial fluctuation filtering of NS equations (only small eddies are removed, motion of large eddies is calculated).

Filter is realized by convolution integral

( , ) ( , ) ( , ) ( , ) ( , )u x t G x u t d G u x t d

Convolution, Green’s

function G

LES G-function exampleCFD6

3( , ) 1/ for | | / 2( , ) 0 for | | / 2

G x xG x x

Δ

/2

/2

( , ) /x

x

u t d

x

u

LES filtering - propertiesCFD6

u uBasic difference in comparison with time averaging

( ) ( , )u u G u x t dx x x

LES NS equationsCFD6

0i

i

ux

2

( ( ))i ii j

j i k k

u upu ut x x x x

Continuity equation without changes

( ')( ') ' ' ' '

' ' ' 'i j i i j j i j i j i j i j

i j i j i j i j i j i j

u u u u u u u u u u u u u u

u u u u u u u u u u u u

Lij Leonard stresses

Cij cross stresses

Rij SGS (sub grid stresses)

LES NS equationsCFD6

( ( )) ( )i ii j ij

j i j j

u upu ut x x x x

Leonard stresses are usually neglected because they are of the second order – almost the same as discretisation error. Cross and subgrid stresses are usually modeled together using turbulent viscosity approach.

It looks like Reynolds stresses

LES Smagorinski SGSCFD6

123

1 ( )2

ij T ij kk ij

jiij

j i

E

uuE

x x

Smagorinski model of subgrid stresses is almost the same as the Prandtl’s model of mixing length, only instead of the mixing length is substituted a filter size.

2( ) 2 :T sC E E

Smagorinski’s constant Cs=0.1 in Fluent

Remark: in the vicinity of wall the Cs mixing length is limited by y (karman constant times the distance from wall)

LES RNG SGSCFD6

21/3(1 max(0, ))s eff

eff C

2( ) 2 :s rngC E E

Crng=0.157 in Fluent

For small s the effective viscosity equals the laminar viscosity , while eff=s holds for a high level of turbulence.

Crng=100 in Fluent

LES boundary conditionsCFD6

Inlet: Simulated noise of velocities (normal distribution superposed to specified intensity of turbulence I)

Wall: For a fine mesh resolving laminar sublayer a linear (laminar) velocity profile is assumed, otherwise velocity is calculated from the buffer layer model

1 ln u yu Eu

Von Karman constant 0.41Friction velocity

Fluent turbulent modelsCFD6

Spalart Almaras (viscosity transport 1 PDE)

k- dissipation of k (standard+RNG+realizable)

k- specific dissipation (standard+SST shear stress transport)

k-kl- transition model laminarturbulent

v2-f boundary layer detachment (low Re)

RSM

DES (Detached Eddy Simulation, Spalart Almaras+realizable+SST), like LES at wall

LES (Large Eddy Simulation)

Fluent example flow in a pipeCFD6

Material: water-liquid (fluid)

Property Units Method Value(s) --------------------------------------------------------------- Density kg/m3 constant 998.20001 Cp (Specific Heat) j/kg-k constant 4182 Thermal Conductivit w/m-k constant 0.6 Viscosity kg/m-s constant 0.001003 Molecular Weight kg/kgmol constant 18.0152

Tube geometry: L=0.5m, D=0.04m.

Velocity at inlet (uniform) u=1m/s

Mesh 50 x 20, compression 2 (last/first)

6

1 0.04 1000Re 400000.001

1 0.04 1.1Re 237818.5 10

water

air

WALL

AXISVELOCITY

INLET

PRESSURE OUTLET

4

4

0.316 0.5 1000 1400.042 40000

0.316 0.5 1.1 0.3110.042 2378

water

air

p Pa

p Pa

Pap arla 185.0min


Voda

DisplayVectorsDef.ModelVisc. Def.Solv.Axisym. Def.Boundaryu=1


AirLaminar Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 0.61464286

Spalart Almaras Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 0.62425083

k-epsilon Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 1.2050774

RNG Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 1.2050774

Realizable Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 1.1221648

k-omega Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 0.78146631

WaterSpalart Almaraz Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 141.57076




4

4

0.316 0.5 1000 1400.042 40000

0.316 0.5 1.1 0.3110.042 2378

water

air

p Pa

p Pa

Pap arla 185.0min

ax

max

140 500 6400.311 0.55 0.866

waterm

air

p Pap Pa

ResultsSurface average (pressure at

inlet)


053.0

8.2204.0140

4

*

w

w

u

PaLDp

Buffer layer 5<y+<30

Wall functions must be used as boundary conditions at wall

26001.0

100040/02.0053.0

y

this value was calculated for finer mesh with 40 cells in radial direction

Fluent example flow in a pipeCFD6/ Journal File for GAMBIT 2.4.6, Database 2.4.4, ntx86 SP2007051421/ Identifier "pipe2dn"/ File opened for write Mon Nov 28 08:28:27 2011.identifier name "pipe2dn" new nosavepreviousface create width 1 height 0.02 xyplane rectanglewindow matrix 1 entries 1 0 0 0 0 1 0 0 0 0 1 0 -0.2624999880791 \ 0.2624999880791 -0.1590357869864 0.1590357869864 -1 1face move "face.1" offset 0.5 0.01 0undo begingroupedge delete "edge.1" keepsettings onlymeshedge mesh "edge.1" firstlast ratio1 1.981 intervals 100undo endgroupundo begingroupedge delete "edge.3" keepsettings onlymeshedge modify "edge.3" backwardedge mesh "edge.3" firstlast ratio1 2 intervals 100undo endgroupundo begingroupedge delete "edge.4" keepsettings onlymeshedge modify "edge.4" backwardedge picklink "edge.4"edge mesh "edge.4" lastfirst ratio1 2 intervals 40undo endgroupundo begingroupedge delete "edge.2" keepsettings onlymeshedge mesh "edge.2" firstlast ratio1 0.5 intervals 40undo endgroupface mesh "face.1" mapphysics create btype "WALL" edge "edge.3"physics create btype "AXIS" edge "edge.1"physics create btype "VELOCITY_INLET" edge "edge.4"physics create btype "PRESSURE_OUTLET" edge "edge.2"solver select "FLUENT/UNS"export fluent5 "pipe2dn.msh" nozvalsave/ File closed at Sat Nov 26 13:56:19 2011, 1.39 cpu second(s), 3331408 maximum memory.save name "pipe2dn.dbs"export fluent5 "pipe2dn.msh" nozval/ File closed at Mon Nov 28 08:31:30 2011, 0.50 cpu second(s), 4639176 maximum memory.

the simplest way how to modify geometry is the journal file modification (L=1m, mesh 100x40)

2

1 2 1 2( )2up p L LD

By comparing results for tubes of different lengths (L1,L2) it is possible to eliminate the entrance effect


Papp mm 1405.01 Water L=0.5mSpalart Almaraz Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 141.57076




Water L=1mSpalart Almaraz Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------

velocity_inlet.3 282.RNG Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------

velocity_inlet.3 410.

k-epsilon Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------

velocity_inlet.3 445

k-omega Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------

velocity_inlet.3 481

141Pa error 1%

165Pa error 18%

171Pa error 22%

223Pa error 59%

RSM failed, negative pressure drop predicted -230.0078 Pa !

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

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