Computer Fluid Dynamics - cvut.czusers.fs.cvut.cz/~zitnyrud/CFD6.pdfflow of a fluid over bluff...

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 E181107CFD6

2181106

Some figures and equations are copied from Wikipedia

TurbulenceCFD6

( ) ( ( ) ) M

uu u p e u S

tρ µ∂ + ⋅∇ = −∇ + ∇⋅ ∇ +

∂

( ( )) ( ( ) ) M

uuu p e u S

tρ µ∂ + ∇ = −∇ + ∇ ⋅ ∇ +

∂

i

Source of nonlinearities of NS equations are inertial and viscous terms

for incompressible fluids the equivalent conservative form

( ( ) ) ( ( ) ) M

uuu u u p e u S

tρ µ∂ + ∇ − ∇ = −∇ + ∇ ⋅ ∇ +

∂

i i

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.

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.

TurbulenceCFD6

2

( )Re

uu uD

u

ρ ρµ µ∇= =

∇

i

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

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 Tρ

2'u

2 2 21( ' ' ' )

2k u v w= + +

2( )/

3I k u=

2 2 2, , , , ,' , ' , ' , ' ', ' 't xx t yy t zz t xy t xzu v w u v u wτ ρ τ ρ τ ρ τ ρ τ ρ= = = = =

Turbulent eddies - scalesCFD6

E(κ)

κ=2πf/u1/L 1/η

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 ε)

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

is converted to heat by friction

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 dκ κ∞

= = ∫

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

Kolmogorov scale η decreases with the increasing Re


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

3

4νηε

= ντε

= 4u νε=

Length scale Time scale velocity scale

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

2 2 2 2

3 3[ ] [ ][ ]

1 2 2

0 3

m m mm

s s s

α β

α β α β

α β α β

η ν ε

α βα β

+

+

=

= =

= += +

3 / 4

1/ 4

αβ

== −

Example tutorialCFD6

Derive time scale of the smallest turbulent eddies

ντε

=

Time scale

2 2 2 2

3 3[ ] [ ][ ]

0 2 2

1 3

m m ms

s s s

α β

α β α β

α β α β

τ ν ε

α βα β

+

+

=

= =

= +− = +

1/ 2

1/ 2

αβ

== −


Wikipedia (abbreviated)

Turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristiclength 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 uporiginating smaller eddies, and the kinetic energy of the initial large eddy is divided into thesmaller eddies that stemmed from it. The energy is passed down from the large scales of themotion to smaller scales until reaching a sufficiently small length scale such that the viscosityof 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, thesmall 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 theboundaries (the size characterizing the large scales will be denoted as L).

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

DNS Direct Numerical SimulationCFD6

ηηηη 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 (Kolmgorov) 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 LuN

L τ

ηε

η ν ν ν

>

> = = = =

3u

Lε ≃

ReuL

τ ν=Re

uL

ν=

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

(based upon dimensional ground)

114 k2100 M4650230000

63 k150 M145061600

47 k40 M80030800

32 k6.7 M38012300

No.oftime steps

No.of grid points in DNS

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

9/4

3/4_

Re

Re

DNS

time steps

N

N

τ

τ

∼

∼

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


AbstractThe 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

Ronge-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 STEP

FLOW USING 2ND- AND 4TH-ORDER DISCRETIZATION

ADNAN 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-facing

step with a fully developed channel flow utilized

as a time-dependent inflow condition. Numerical solutions using a

fourth-order compact (Hermitian) scheme, which was formulated directly

for a non-equidistant and staggered grid in [1] are compared with numerical

solutions using the classical second-order central scheme. The results

from LES (using the dynamic subgrid scale model) are evaluated against a

corresponding DNS reference data set (fourth-order solution).

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

Transition Laminar -TurbulentCFD6

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


How to indentify whether the flow is laminar or turbulent ?

ExperimentallyVisualization, 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 sayvery little about transition to turbulence since this progress is highly non-linear


1000Planar channel

400Cross flow

300Couetteflow

100Baffled channels

5-10Jets

RecritGeometry

500000Plate

8000Cavity

7000Suspension in pipe

5000Coiled pipe

2000Circ.pipe

RecritGeometry

D

D

D/2

D

D

D

D

D

Turbulent structures evolutionCFD6

Journal of Fluids and Structures 18 (2003) 305–324Force coefficients and Strouhal numbers offour 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 y

fu h

= min

max min

( )u u y

gu u b

− =−

max

max

constu

xconst

ux

=

=

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 y

fu h

=

1

max max max

0 0

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

planar

yV x u x f dy h x u x f d h x u x F c x

hη η= = = =∫ ∫ɺ

12 2

max max max

0 0

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

circ

yV x u x f ydy h x u x f d h x u x F cx

hπ η η η= = = =∫ ∫ɺ

Planar jet

Circular jet

~x ~1/√x

~1/x


Flow at walls (boundary layers)y

Laminar sublayer

Buffer layer

Log law * wuτρ

=

*

*

u yuu y

u

ρµ

+ += =

Friction velocity

u+=lnEy+/κ30<y+<500Log law

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

u+=y+y+<5laminarINNER layer

(independent of bulk velocities)

OUTER layer (law of wake) max

*

1ln( )

u u yA

u κ δ− = +

u


Flow at walls (turbulent stresses)τt

*yuy

ρµ

+ =

2 2( )

[1 exp( )]26

t m

m

ul

y

yl y

τ ρ

κ+

∂=∂

= − −

2 2' ' 24 [1 exp( )]26t w

yu vτ ρ κ τ

+

= = − −

Prandtl’smodel

vanDriest model of mixing length lm5 30

Example tutorialCFD6

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

*

5yu ρ

µ=

3

3

4 4 10 1000Re 63662

0.02 10

V

D

ρπ µ π

−

−

⋅ ⋅= = =⋅

ɺ

2 22

2 4 2 44

1 0.3162 2 25.2[ ]

8 Rew

V Vu Pa

D Dτ λρ λρ ρ

π π= = = =

ɺ ɺWall shear stress from Blasius

Turbulent region, well within validity of Blasius correlation

Dimensionless thickness of laminar

sublayer

*

28

0.6320.159[ / ]

Rew V

u m sD

τρ π

= = =ɺ

2 2 88 8 85 Re 0.005 0.02 63662 2 10 636620.31

10000.632 0.001 0.632 0.632

Dy m

V

µ π π π µρ

−⋅ ⋅= = = =ɺ

Friction velocity

TIME AVERAGING of turbulent fluctuationsCFD6

RANS (Reynolds Averaging of Navier Stokes eqs.)

1( ) '

( )

''

' '

t t

t

t t

tt t

t

t dtt

t dt

dt

ρ

ρ

ρρ

+∆

+∆

+∆

Φ = Φ Φ = Φ + Φ∆

ΦΦ = Φ = Φ + Φ

ΦΦ = Φ +

∫

∫

∫

ɶ ɶ

ɶ

Time average

Favre’s average

Proof: ( ')( ') ' 'ρ ρ ρ ρ+ Φ + Φ = Φ + Φ

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

TIME AVERAGING of turbulent fluctuationsCFD6

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 aver aged values

' 0Φ ='

0x

∂Φ =∂

' 'uv u v u v= +

( ) ( ) ( ' ')uv u v u v∇ = ∇ + ∇ i i i

TIME AVERAGING of NS equationsCFD6

0 0 0u u u∇ = ∇ = → ∇ = i i i

( ) ( ) ( )

( ) ( )) ) ( ' '(

u uu p ut

u ut

u uu p u

ρ ρ µ

ρ ρ µ ρ

∂ + ∇ = −∇ + ∇ ∇∂∂ + ∇ = −∇ + − ∇∇ ∇∂

i

i i

i i

Continuity equation

Navier Stokes equations

Reynolds stresses

' 't u uτ ρ= −

TIME AVERAGING of transport equationsCFD6

( ) ( ) ( )

( ) ( ') ( ')) ( u

ut

ut

ρ

ρ ρ

ρ

ρ

∂ Φ + ∇ Φ = ∇ Γ∇Φ∂∂ Φ + ∇ Φ = ∇ Γ ΦΦ∂

− ∇∇

i i

ii i

Turbulent fluxes

' 'tq uρ= − Φ

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 deformationbased upon gradient

of averaged velocities

' ' tuρΦ = −Γ ∇Φ

' 'i ti

ux

ρ ∂ΦΦ = −Γ∂

' ' ( ( ) )Ttu u u uρ µ= − ∇ + ∇

1( ( ) )

2TE u u= ∇ + ∇

2t tEτ µ=

' ' ( )j ii j t

i j

u uu u

x xρ µ

∂ ∂= − +∂ ∂

Analogy to Fourierlaw

Analogy to Newton’slaw

q Tλ= − ∇

( ( ) )Tu uτ µ= ∇ + ∇

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

µ σ=Γ

Prandtl number for turbulent heat transfer

Schmidt number for turbulent mass transfer

σt=0.9 at walls

σt=0.5 for jets

according to Rodi

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 forturbulent 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)

Nonlineasr eddy viscosity models (v2-f)

RSM Reynolds Stress Modelling (transport equations forcomponents of reynolds stresses)

Algebraic modelsCFD6

Prandtl’s model of mixing length. Turbulent viscosity is derived fromanalogy with gases, based upon transport of momentum by molecules (kinetictheory of gases). Turbulent eddy (driven by main flow) represents a molecule, andmean 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

ul

yµ ρ ∂=

∂Mixing length

h

h

y

u(y) 0.075ml h=0.09ml h=

0.07ml h=

[1 exp( )]26m

yl yκ

+

= − −

Circular jet

Planar jet

Mixing layers

Currently used algebraic models are Baldwin Lomax, and Cebecci Smith

2 equations modelsCFD6

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

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

ε (dissipation of kinetic energy) [m2/s3]

ω (specific dissipation energy) [1/s]

2

t

t

k

kCµ

µ ρω

µ ρε

=

=

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

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

3 /u lε ∼

Cµ = 0.09 (Fluent-default)

Tutorial 2 equations modelsCFD6

Derive relationship for turbulent viscosity

2

t

kCµµ ρ

ε= Launder Spalding k-epsilon model

1

1

2

αβγ

== −

=

kα β γµ ρ ε=2 2 2 2 3

2 3 3 2 2 3[ . ] [ ][ ][ ]

Ns kg kg m m kg mPa s

m ms m s s s

α β γ α γ β α

α β γ γ β

+ −

+

= = = =

1

1 2 2 3

1 2 3

αγ β

γ β

=− = + −

= +

k – transport equationCFD6

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

' ' ( ' ' ' )2 2i ik u u u v w= = + +

' ' ( ) ' ' ( )u

u u uu u p u ut

ρ ρ µ∂ + ∇ = − ∇ + ∇ ∇∂

i i i i i i

1' ( ' ')

2

u ku u u

t t t

ρ ρρ∂ ∂ ∂= =∂ ∂ ∂

i i

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

kku p u u e u u u e e u u E

t

ρ ρρ µ µ ρ∂ + ∇ = ∇ − + − − −∂

i i i i

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’) (2µe’) (ρu’u’)

k – transport equationCFD6

Rate of dissipation of kinetic energy of velocity fluctuation

' ' tu k kµ ∇∼

( ) ( ) 2 : 2 ' : 'tt

k

kku k E E e e

t

µρ ρ µ µσ

∂ + ∇ = ∇ ∇ + −∂

i i

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

' ' ' ' '2 ku u u k u kρ ρ= = Γ ∇ i

Example: approximation of Reynolds stress transport term

-ρε

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

µ µερ ρ

= =

' '1' ( )

2j i

iji j

u ue

x x

∂ ∂= +∂ ∂

Laminar and not turbulent viscosity!

σk = 1 Fluent

εεεε – transport equationCFD6

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

kt ε εε

µρε ρε ε µ ρεσ

ε∂ + ∇ = ∇ ∇ + −∂

i i

Transport equation for dissipation ε looks like the k-transport. Just substitute ε for k. Production and dissipation terms are modified by universal constants C1ε and C2ε

This term follows from dimensional analysis

Production term (generator of turbulence) Dissipation

term

! 1[ ] [ ]k s

ε ≡

C1ε = 1.44 C2ε = 1.92 σε= 1.3 Fluent (default)

k-εεεε modifications (RNG, realizable)CFD6

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

t kf fε ε

ε

µρε ερε ε µ εσ

µ ρ∂ + ∇ = ∇ + ∇ + −∂

i i

Corrections of dispersion, production and dissipation terms withthe aim to extent applicability of k-ε for low Reynolds numberflows

( ) (( ) ) 2 :tt

k

kku k E E

t

µρ ρ µµ ρεσ

∂ +∇ =∇ + ∇ + −∂

i i

Functions of turbulent Reynolds 2

Re tt

k

Cµ

νεν ν

= =

Effective viscosity (RNG)CFD6

2

0 (laminar)

(high Re)

ef

ef t

k

C kk µ

µ µ

ρµ µ µ

µε

→ →

→ ∞ → =

Blending formula for effective viscosity

2[1 ]ef

C kµρµ µ

µ ε= +

εεεε - estimateCFD6

3 23

PPo n d

dε ρ= =

Dissipated energy in a mixing tank

3 51

PPo

n dρ= ≈

Power number

d

n

2 2 4 2F d u d nρ ρ∼ ∼

5 2tM Fd d nρ= ∼

5 3tP M d nω ρ= ∼

k-εεεε boundary conditionsCFD6

*3

w

u

yε

κ=

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

*2

w

uk

Cµ

= Friction velocity

* wuτρ

=

Distance of the nearest boundary

node from wall

This is implemented in majority of CFD programs

Reynolds stresses (k -εεεε)CFD6

2' ' 2 ( ) 2 2

3i i i

i i t ii ti i i

u u uu u k k k

x x xρ ρ µ ρ δ µ ρ∂ ∂ ∂− = − = + − = −

∂ ∂ ∂

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

2' ' ( )

3ji

tij i j t ijj i

uuu u k

x xτ ρ µ ρ δ

∂∂= − = + −∂ ∂

This term is zero for incompressible liquidsRemark: Turbulent stresses determine

kinetic energy of turbulent fluctuations 0u∇ =i

Assesment of k -εεεε modelsCFD6

Problems and erroneous results can be expected

Unconfined flows (wakes, jets). k-ε model overestimatesdissipation, 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 kε εε

µρε ερε ε µ ρεσ

∂ + ∇ = ∇ ∇ + −∂

i i

( ) ( ) 2 :tt

k

kku k E E

t

µρ ρ µ ρεσ

∂ +∇ =∇ ∇ + −∂

i i

2( )t u

y

µερ

∂=∂

2 2

1

( )t Cu

y Cε

ε

µερ

∂=∂

These terms are identically zero in homogeneous turbulence

Further reading C1ε = 1.44 C2ε = 1.92 σk = 1 σε= 1.3

Reynolds Stress Models (RSM)CFD6

' 'ij i jR u u=

( )ij tij ij ij ij ij

DRP R

Dt

ν εσ

= + ∇ ∇ − + Π + Ωi

production

diffusion

Dissipation

Pressure transport

Centrifugal forces

2

3 i jε δ

6 transport equations

1 dissipation

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

t kε εε

µρε ερε ε µ ρεσ

∂ + ∇ = ∇ ∇ + −∂

i i

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 dξ ξ ξ ξ ξ ξΩ Ω

= − ∆ = ∆ −∫∫∫ ∫∫∫Convolution, Green’s

function G

LES G-function exampleCFD6

3( , ) 1/ for | | / 2

( , ) 0 for | | / 2

G x x

G x x

ξ ξξ ξ

− ∆ = ∆ − < ∆− ∆ = − ≥ ∆

∆

/2

/2

( , ) /x

x

u t dξ ξ+∆

−∆

∆∫

x

u

LES filtering - propertiesCFD6

u u≠Basic difference in comparison with time averaging

( ) ( , )u u

G u x t dx x x

ξ ξ ξ∂ ∂ ∂= = −∂ ∂ ∂ ∫∫∫

LES NS equationsCFD6

0i

i

u

x

∂ =∂

2

( ( ))i ii j

j i k k

u upu u

t x x x xρ µ∂ ∂∂ ∂+ = − +

∂ ∂ ∂ ∂ ∂

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 u

t x x x xρ µ τ∂ ∂∂ ∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂

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

2

1( )

2

ij T ij

jiij

j i

E

uuE

x x

τ µ=

∂∂= +∂ ∂

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( ) :T sC E Eµ ρ= ∆

Computer Fluid Dynamics - cvut.czusers.fs.cvut.cz/~zitnyrud/CFD6.pdfflow of a fluid over bluff...

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