ENVIRONMENTAL AND INDUSTRIAL CFD SIMULATIONS

Turbulence models in the environmental flow

Zbyněk Jaňour Institute of Thermomechanics AS

CR, Dolejškova 5 Prague 8, 182 00, Czech Republic,

2

Overview• Introduction,• Equations,• Turbulence,• Atmospheric Boundary Layer,• Closure Problem,• Models,• Boundary Conditions,• Applications• Conclusion

3

Introduction

• The most fluid on the world belongs to the atmosphere and the ocean,

• Geophysical fluid dynamics

4

Introduction

5

Equations

Inertial coordinate system

Reference coordinate system

6

EquationsInertial coordinate system:

Continuity equation

The equation of motion

The energy equation

,0)(

vt

.3,2,1,

)(

iforx

fx

vv

t

v

j

iji

k

kii

.

kkk

p Rx

Tk

xtd

pd

td

Tdc

7

EquationsReference coordinate system:

R – perpendicular distance from the rotation axis,

The last term on the r.h.s. can be included into the gravitation force

.2 2 Rvtd

vd

td

vd

rI

I

8

Equations

Reference coordinate system:Continuity equation

The equation of motion

Coriolis force:

or

,0)(

vt

.21

vvGptd

vd

,cos2

,sin2

,cos2sin2

uC

uC

wvC

z

y

x

,0

,

,

z

y

x

C

ufC

vfC

f~10-4

9

Turbulence• Is the atmosphere turbulent?According to Tennekes, Lumley: A First Course in Turbulence the

turbulence flow has following characters:

• Irregular - Y,

• Diffusive - Y,

• Large Re 109 - Y,

• 3D vorticity fluctuations - Y,

• Dissipative needs energy supply - Y/N,

• Continuum - Y,

• Turbulent flows are flows - Y

10

Turbulence

Wake behind a jet – turbulent / nonturbulent ?The answer: It is not flow; it is a picture of the former turbulent

wake

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TurbulenceEnergy sources:

– Atmospheric Boundary Layer (ABL)– Free atmosphere:

• Clouds,

• Clear-Air Turbulence (CAT)

12

Turbulence

Characteristic scale:– Velocity U,– Length in horizontal direction L,– Length in vertical direction H,– -pressure P,

13

Turbulence

14

Turbulence

15

Turbulence

Turbulent flow - L~102

– Atmospheric Boundary Layer (ABL)– Free atmosphere:

• Clouds,

• Clear-Air Turbulence (CAT)

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Turbulence

• The ABL:

Layer of air directly above the Earth surface in which effects of the surface (friction, heating and cooling) are felt on time scales less than a day, and in which significant fluxes of momentum, heat or matter are carried by turbulent motions on scale of the order of the depth of the boundary layer or less

17

Turbulence

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Turbulence• Cloud

Cumulus-type cloud associated with thunderstorm:

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TurbulenceCAT• Shear turbulence without visible manifestations. • It occurs outside of clouds,• In only about 20% of the free atmosphere below 12 km,• is even less common above 12 km and occurs in only about 2% near 17 km,• It generally occurs in stable conditions, • It has not cased severe structure damage of aircraft.

20

Turbulence

Atmospheric turbulence differs from most

laboratory turbulence in:– Heat convection coexists with mechanical

turbulence,– The rotation of the earth becomes important for

many problems

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Atmospheric Boundary Layer (ABL)

The ABL is the region in which the large-scale flow of the free atmosphere adjusts to the boundary condition imposed by the earth´s surface

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ABL

Small-scale maximum - turbulent peakLarge-scale maximum - synoptic peakSpectral gap around 1 cycle/hour

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ABL

Fluctuations with frequency smaller than 0.1 cycle/km belongs to the mean value

Fluctuations with frequency large than 0.1 cycle/km belongs to the turbulent fluctuations

+ Reynolds conditions

´

'

24

Equations

.0)(

Vt

TktvTVt

Tcp

)()(0

.)()( CKcvCVt

CC

25

Closure problem

),()(1

)(1

)(1

)(1

0

03

03

00

0

k

jkj

k

iki

k

ijk

k

jik

i

j

j

ij

ji

iikjjki

ikjjkikjikjik

ji

x

Uuu

x

Uuu

x

u

x

u

x

u

x

up

T

tug

T

tuguu

upupuuuUuuxt

uu

jiij uu0 ,0 ipi utcg .0 ii uch

New dependent variables:

Closure problem, etc.

New dependent variables

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Model taxonomy• Ensemble-averaged equations

– Integral models,

– First-order closure models,

– Second-order closure models,

– Reynolds-stress models,

• Volume-averaged equations– Large Eddy Simulation (LES)

• Full simulation– Direct Numerical Simulation (DNS)

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Integral models

Reynolds equations are integrated over at least one coordinate direction and the number of independent variables decreases

28

Integral models

Mixed Layer

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Integral models

)´´´´(1

)´´´´(1

)´´´´(1

0

0

0

íi

gmm

íi

gmm

íi

m

vwvwz

UUfdt

dV

uwuwz

VVfdt

dU

twtwzdt

dT

iz

im Tdz

zT

0

1

Where is:

30

Integral models

i

gy

i

gx

ii

iy

m

ii

ix

ii

i

dz

dVS

dz

dUS

Ufvwvwzdt

dzS

dt

Vd

Vfuwuwzdt

dzS

dt

Ud

twtwzdt

dz

dt

Td

´´´´1

´´´´1

´´´´1

0

0

0

Equations for velocity and temperature jumps

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Integral models

dt

dzVvw

dt

dzUuw

dt

dzTtw

ii

ii

ii

´´

´´

´´

Equations for heat and momentum fluxes at the inversion base

9 equations for 10 dependent variables

iiiimmm vwuwtwzVUTVUT ´´,´´,´´,,,,,,,

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Integral models

dt

dzw i

e

Models for zi:

we entrainment velocity

-

-

*..,

2

ugescalevelocityV

V

TzR

s

s

ib

Rb – Richardson number

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First-order closure models

K-models based on the hypothesis of Boussinesq(1877), who suggested that turbulent shearing stress in analogy to viscous stress can be related to the mean strain

Where t is eddy viscosity – new dependent variable

),(i

j

j

itij x

U

x

U

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Eddy viscosity

t = constant - Ekman spiral(1905)

 

 

(Bz)(-Bz)U=V

(Bz)],(-Bz)-[1U=U

0

0

sinexp

cosexp

)(=B 2

1

t

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Eddy viscosity

Notice: ABL thickness 1km t = 10

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Eddy viscosity

Prandt´s model - Blackadar (1962) generalized by Estoqe, Bhumralk (1969) and Yu (1977).

l-mixing length

z0 – roughness length

,G])z

V(+)

z

U[(l=K 1/2222

0

0

zzl

zz=l

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Eddy viscosity

.f

10|V|27=

-50

,0Rifor Ri)+(1=G -2

.0<Rifor )Ri-(1=G 2

.

)z

V(

zg

Ri2

Richardson number

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Two equations models

.2

kcK

-z

kK(

z+

y

kK(

y+

x

kK(

x+G+P=

z

kW+

y

kV+

x

kU+

t

k

kkk

)))

.)))12

231 kc-

z

K(

z+

y

K(

y+

x

K(

x+GP

kRcc=

zW+

yV+

xU+

t f

j

i

i

j

j

i

x

U)

x

U+

x

UK(=P

z

K g -=G

.

)1

(0

P

G-=R

0.7,=C 1.92,=C 1.44,=C 1.3,= 1.0,= 0.09,=C ,)(1

-=

f

321k

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Large Eddy Simulation

• The first large-eddy simulations were performed by Deardorff (1972; 1973;1974), and were later investigated by e.g.,:

• Schemm and Lipps (1976),

• Sommeria (1976),

• Moeng (1984),

• Wyngaard and Brost (1984), Schmidt and Schumann (1989), Mason (1989). Much of the previous work

• LES has been focused on simulations of the convective boundary layers (Nieuwstadt et al., 1992).

• The cloudy boundary layers were simulated by e.g., Sommeria 1976; Deardorff 1980; Moeng 1986; Moeng et al. 1996; Lewellen and Lewellen 1996, Cuijpers and Duynkerke (1993).

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Boundary Conditions

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Boundary ConditionsThe equations of motion has to be supplemented with initial and boundary conditions – in many papers the conditions are not introduced

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Boundary ConditionsIn limited-area atmospheric models the

surface - S is the only physical boundary of the solution domain. All other boundaries are purely computational

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Boundary Conditions on the surface

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Boundary Conditions on the surface

Two methods:•Boundary conditions on the surface +

modification of the equations of motion for small turbulence Reynold number+ increasing number of grid points near the wall,

•Wall function

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Boundary Conditions-wall function

)zz(

1=u)V+U( 1*

2

122

0

ln

for 30 < z1u*/v < 100 c

u=k* 2

z

u=1

* 3

0=zC i

wTT

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Roughness length- experience

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Roughness length-models

Petersen: z0 = D f H, where D 0.5, f = Af / AT

48

Boundary Conditions on the top of the ABL

ti0 for 0Ck=W=V ,U=U ,

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Outlet Boundary Conditions

t for 0Cx

=x

=Wx

=Vx

=Ux

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Inlet Boundary Conditions

Dirichlet condition determined from:

• In-situ measurement – a very few data sets,

• Universal profiles:– Ekman spiral,– Power law,– ….

-mostly for horizontally homogeneous surface

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Boussinesq approximation

limited-area

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Boussinesq approximation

,´~

´~´~ += p,p+p=p ,+= 000

Large scale flow Small scale fluctuation

Turbulent fluctuation

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Boussinesq approximation

Hydrostatic approximation: , 00 g- =

dz

pd

Geostrophic approximation:,Vk -f= p

100H

0

Large scale flow:

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Boussinesq approximation

Shallow water approximation:(incompressible)

Continuity equation

,0

k

k

x

u

Anelastic approximation 00

kk

ux

Small scale fluctuation:

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Boussinesq approximationReynolds equations:

,´

0

fVz

UK

zy

UK

yx

UK

x+

p

x- =

z

UW

y

UV

x

UU+

t

U

,´

0

fUz

VK

zy

VK

yx

VK

x+

p

y- =

z

VW

y

VV

x

VU

t

V

,´

00

gz

WK

zy

WK

yx

WK

x+

p

z- =

z

WW

y

WV

x

WU

t

W

,´´´´´´´

00 pcz

K

zy

K

yx

K

x =

zW

yV

xU+

t

Notices: •Small scale fluctuation of the pressure and potential temperature,•Buoyant force instead gravitational force,•Incompressible case

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Boussinesq approximation

F´=0 for i

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Application

58

Application

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ApplicationDispersion from linen source inside the street canyon- FLUENT

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Application

experiment k- model RNG k- model

Dispersion from linen source inside the street canyon- FLUENT

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Application

• laser sheet- DANTEC,

• The recordings from the video camera for values of the Reynolds number of Re U0H/(2.3 x 104; 2.3 x 105),

Smoke generator

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Application

• External velocity Ug=1.5m/s,

• liquid is drawn from the cavern into the external stream,

63

Application

• External velocity Ug=4.0m/s

64

Application

65

Application

•UABL is similar to the flow over a rough surface, with a large roughness length z0 and a defined surface heat

flux QG ;•The horizontally homogeneous atmospheric boundary layer horizontal length scale - L

A simple model of the UABL

66

Application

67

Application

00,1

0,20,3

0,40,5

0,60,7

0,80,9

1

-20 -15 -10 -5 0(U-Ug)/U*

Z/Hb

numerical simulation

power

82412

82800

82812

82312

82712

mean profile

•radiosounding launched in Barcelona,

•indifferent stratification

•influence of topography is more important across Internal-Sub-Layer

•artificial mean profile determined from the data sets seems to be more suitable for comparison;

68

Application

00,1

0,20,3

0,40,50,6

0,70,8

0,91

-20 -15 -10 -5 0 5(U-Ug)/U*

Z/Hb

EVORA-60500

EVORA-60600

EVORA-60618

EVORA-60709

EVORA-60718

EVORA-60715

EVORA-60712

vypocet

calculation-u*=0.4

•radiosounding launched in Évora,

•indifferent stratification

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Application

00,1

0,20,30,40,5

0,60,70,8

0,91

-20 -15 -10 -5 0(U-Ug)/U*

Z/Hb

PowervypocetPRAGUE-9:40PRAGUE-10:00PRAGUE-10:20PRAGUE-10:40PRAGUE-11:00PRAGUE-11:20PRAGUE-11:40PRAGUE-12:00

•sodar measurement in Prague,

•without stratification assessment

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Application

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ApplicationAlgebraic turbulence models

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Application

Plume from point sources in south east Giant Mountains

Algebraic turbulence models

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4. Conclusions

• Eddy viscosity models appears:– Quite satisfactory in neutral or stable ABL;– Fail in convective situations;

• Reynolds stress models are more suitable,

• Boundary Conditions are complicated and important task