two-equation turbulence models for boundary layer ﬂows · program of presentation nse...

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Two-equation turbulence modelsfor boundary layer flows

Hans [email protected]

Baltic Sea Research Institute Warnemunde, Germany

NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 1/33

Program of presentation

NSE Two-equation models

Two-equation models

Boundary conditions

General Ocean Turbulence Model (GOTM)

Examples (observations versus simulations)

Links to sediment modelling

Modelling of Estuarine Turbidity Maxima

Conclusions



� NSE � Two-equation models

Two-equation models

Boundary conditions





Conclusions




� Two-equation models

Boundary conditions





Conclusions





� Boundary conditions





Conclusions






� General Ocean Turbulence Model (GOTM)




Conclusions







� Examples (observations versus simulations)



Conclusions








� Links to sediment modelling


Conclusions









� Modelling of Estuarine Turbidity Maxima

Conclusions









� Modelling of Estuarine Turbidity Maxima

� Conclusions


Navier-Stokes-EquationsContinuity Equation:

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Momentum Equation:

Heat Equation:



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Momentum Equation:

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Heat Equation:



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Momentum Equation:

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Heat Equation:

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Averaging Rules1. Linearity:

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2. Exchange of Derivative and Averaging:

3. Double Averaging:

4. Products of Averages:



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Reynolds-averaged EquationsContinuity Equation:

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Reynolds Equation:

Heat Equation:



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Reynolds Equation:

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Heat Equation:



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Reynolds Equation:

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Heat Equation:

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Reynolds Stress Equation

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Heat Flux Equation

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Algebraic SMCsThe following steps lead to different types of second-moment

closures:

Empirical closures of pressure-strain correlators.

Neglect or simplification of advective and diffusive fluxes

of second-moments.

Neglect of rotational terms in the second-moment

equations.

Neglect of tracer-tracer correlations.

Assumption of local equilibrium for tracer variances.

Boundary layer assumption (neglect of horizontal gradients

and non-hydrostatic effects).



closures:

� Empirical closures of pressure-strain correlators.

Neglect or simplification of advective and diffusive fluxes

of second-moments.


equations.







closures:


� Neglect or simplification of advective and diffusive fluxes

of second-moments.


equations.







closures:



of second-moments.

� Neglect of rotational terms in the second-moment

equations.







closures:



of second-moments.


equations.

� Neglect of tracer-tracer correlations.






closures:



of second-moments.


equations.


� Assumption of local equilibrium for tracer variances.





closures:



of second-moments.


equations.


� Assumption of local equilibrium for tracer variances.

� Boundary layer assumption (neglect of horizontal gradients



Algebraic SMCsTurbulent Fluxes:

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Eddy Viscosity / Eddy Diffusivity:

Shear Number, Buoyancy Number:



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Stability FunctionsKantha & Clayson [1994]:

0

5

10

15

20

0 5 10 15 20

cµ

0

5

10

15

20

0 5 10 15 20

αN

0

5

10

15

20

0 5 10 15 20

α M

0

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20

0.02

0.04

0.04

0.06 0.06

0.08

0.08

0.1

0.1

0.120.14

0.160.18

0

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20

c’µ

0

5

10

15

20

0 5 10 15 20

αN

0

5

10

15

20

0 5 10 15 20α M

0

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20

0.02

0.02

0.04

0.04

0.06

0.06

0.08

0.080.1

0.1

0.12

0.12

0.14

0.14

0.16

0.16

0.180.20.22

0.240.26

0

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20


Stability FunctionsCanuto et al. [2001]:

0

5

10

15

20

0 5 10 15 20

cµ

0

5

10

15

20

0 5 10 15 20

αN

0

5

10

15

20

0 5 10 15 20

α M

0

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20

0.02

0.04

0.06

0.06

0.08 0.08

0.1

0.1

0.12

0

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20

c’µ

0

5

10

15

20

0 5 10 15 20

αN

0

5

10

15

20

0 5 10 15 20α M

0

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20

0.02

0.02

0.04

0.040.06

0.060.08

0.080.1 0.1

0.12

0.120.14

0.140.16

0.160.18

0.18 0.20.2

0.22

0.220.24

0.260

5

10

15

20

0 5 10 15 200

5

10

15

20

0 5 10 15 20


Stability functions (qe)

0

0.1

0.2

0.3

0.4

0.5

-1 -0.5 0 0.5 1

stab

ility

fun

ctio

n

gradient Richardson number

Kantha & Clayson [1994]

a) qe, momentumqe, heat

0

0.1

0.2

0.3

0.4

0.5

-1 -0.5 0 0.5 1

stab

ility

fun

ctio

n

gradient Richardson number

Canuto et al. [2000], model A

c) qe, momentumqe, heat


Exact TKE-Equation

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This TKE equation will be modelled as it is givenabove, the only parameterisations needed are for theturbulent flux terms , for which usually thedown-gradient approxmation is used.


Length scale equations

- �model (Launder and Spalding [1972]):

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- model (Mellor and Yamada [1982]):


Length scale equations

- �model (Launder and Spalding [1972]):

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model (Mellor and Yamada [1982]):

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Total equilibrium ( - �)

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: Steady-state Richardson number.

0

0.1

0.2

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0.4

0.5

0.6

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1

-2.5 -2 -1.5 -1 -0.5 0 0.5 1

KCRHCACB

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Total equilibrium ( - �)

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KCRHCACB

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Total equilibrium ( - )

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Total equilibrium ( - )

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0.3

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Length scale equations (cont’d)Generic length scale equation (Umlauf and Burchard [2002]):

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General relation between , and :


Length scale equations (cont’d)Generic length scale equation (Umlauf and Burchard [2002]):

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General relation between

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Boundary conditionsLaw of the wall:

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GOTM, http://www.gotm.net

Challenge

Aim

The Idea

Key features

Software

Fortran code

Test cases

Forcing

How to run?

Information

What's

Publications

E-mail list

FAQ

User Group

Hot Links

Who's Who?

Guestbook

PPM:

10/26/00 20:48:54

New

GOTM is a one-dimensional numerical modeldeveloped and supported by a core team ofocean modellers. GOTM aims at simulatingaccurately vertical exchange processes in themarine environment where mixing is known toplay a key role. GOTM is freely available underthe GPL (Gnu Public License).

The interested user can download the sourcecode, a set of test cases (Papa, November, Flex,...) and a comprehensive report.

You are warmly invited to join the GOTM mailinglist and send any comments/questions to theGOTM team or become a GOTM contributor. TheGOTM developers are grateful to their sponsors.

Page "www.gotm.net" maintained by webmaster. Last update: 10/28/00 18:10:02


Wave-enhanced layerSimulation with the generic two-equation model byUmlauf and Burchard [2002]:

1

10

100 0.0001 0.001 0.01 0.1 1 10

PSfrag replacements

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Terray et al. [1996]Drennan et al. [1996]

Anis and Moum [1995]Numerical, ��

Numerical, �� ! �

Numerical, �� "

Log-LawNo shear production


Free Convection

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-10 -8 -6 -4 -2 0

PSfrag replacements

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LES

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Temperature

Temperature FluxVariance ofVariance ofVariance of

Dissipation Rate-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-0.2 0 0.2 0.4 0.6 0.8 1

PSfrag replacements

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LES

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TemperatureTemperature Flux

Variance ofVariance ofVariance of


-1

-0.8

-0.6

-0.4

-0.2

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

PSfrag replacements

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LES

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Variance of �

Variance ofVariance of

Dissipation Rate

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.1 0.2 0.3 0.4 0.5 0.6

PSfrag replacements

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Variance of

Variance of �

Variance ofDissipation Rate

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.0001 0.01 1 100

PSfrag replacements

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LES

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Variance ofVariance of

Variance of

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-0.8

-0.6

-0.4

-0.2

0

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PSfrag replacements

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LES

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Variance ofVariance ofVariance of

Dissipation Rate


Lago Maggiore, ItalyObservations and simulations of

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Northern North SeaBathymetry and station map

356˚

356˚

358˚

358˚

0˚

0˚

2˚

2˚

4˚

4˚

6˚

6˚

8˚

8˚

56˚ 56˚

57˚ 57˚

58˚ 58˚

59˚ 59˚

60˚ 60˚

61˚ 61˚

62˚ 62˚

0

100

100

100

200300

356˚

356˚

358˚

358˚

0˚

0˚

2˚

2˚

4˚

4˚

6˚

6˚

8˚

8˚

56˚ 56˚

57˚ 57˚

58˚ 58˚

59˚ 59˚

60˚ 60˚

61˚ 61˚

62˚ 62˚

Norw

ay

Scotland

0˚ 30'

0˚ 30'

0˚ 45'

0˚ 45'

1˚ 00'

1˚ 00'

1˚ 15'

1˚ 15'

1˚ 30'

1˚ 30'

1˚ 45'

1˚ 45'

2˚ 00'

2˚ 00'

59˚ 00' 59˚ 00'

59˚ 15' 59˚ 15'

59˚ 30' 59˚ 30'

59˚ 45' 59˚ 45'

100

100

110

120

120

120

130

150

150

160

0˚ 30'

0˚ 30'

0˚ 45'

0˚ 45'

1˚ 00'

1˚ 00'

1˚ 15'

1˚ 15'

1˚ 30'

1˚ 30'

1˚ 45'

1˚ 45'

2˚ 00'

2˚ 00'

59˚ 00' 59˚ 00'

59˚ 15' 59˚ 15'

59˚ 30' 59˚ 30'

59˚ 45' 59˚ 45'


Northern North SeaWind and Tides

0.00.10.20.30.40.5

Surface stress at station NNSPSfrag replacements

Q / (W m )

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/(N

m

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0.00.10.20.30.40.5

21/1012:00

22/1000:00

22/1012:00

23/1000:00

23/1012:00

Date in 1998

Bed stress at station NNS

ADCPModelPSfrag replacements

Q / (W m ) �

/(N

m

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Northern North Sea


Liverpool BaySection of Temperature and Salinity

1 1

1 1 .5

1 2

1 2 .5

1 3

1 3 .5

1 4

1 4 .5

1 5

1 5 .5

1 6

3 2 .2

3 2 .4

3 2 .6

3 2 .8

3 3

3 3 .2

3 3 .4

3 3 .6

3 3 .8

3 4

3 4 .2

3 4 .4

2 3 .6

2 3 .8

2 4

2 4 .2

2 4 .4

2 4 .6

2 4 .8

2 5

2 5 .2

2 5 .4

2 5 .6

2 5 .8

2 6

2 6 .2

2 6 .4

2 6 .6

0

-20

-40

-60

0

-20

-40

-60

0

-20

-40

-60

0

-20

-40

-60

100 90 80 70 60 50 40 30 20 10 0

100 90 80 70 60 50 40 30 20 10 0

100 90 80 70 60 50 40 30 20 10 0

Fig 2.a Temperature (Degrees C)

Fig 2.b Salinity (PSU)

Fig 2.c Sigma T (kg/m3)

Distance Along Transect (km)

De

pth

(m

)D

ep

th (

m)

De

pth

(m

)

LB2

Rippeth, Fisher, Simpson [2001]


Liverpool BayObserved and simulated temperature and salinity

Simpson, Burchard, Fisher, Rippeth [2002]


Liverpool BayObserved and simulated current velocity



Liverpool BayObserved and simulated dissipation rates



Links to Sediment Modelling

� Lagrangian or Eulerian modelling ?

When is it necessary to apply multi-phase flowtheory ?

How to model flocculation/breaking of flocks ?

How to model the bottom fluxes of SPM fromsediment into the water column ?

How to model the wave-BBL ?




� When is it necessary to apply multi-phase flowtheory ?

How to model flocculation/breaking of flocks ?







� How to model flocculation/breaking of flocks ?








� How to model the bottom fluxes of SPM fromsediment into the water column ?







� How to model the bottom fluxes of SPM fromsediment into the water column ?

� How to model the wave-BBL ?


Conceptual model for ETMs

Jay & Musiak, 1994


ETM computer simulationsBurchard & Baumert, 1998

Ruiz Villareal and Burchard, under progress


http://www.io-warnemuende.de/homepages/burchard/pdf/animations/spm_salt.gif

Conclusions

� With statistical turbulence modelling, we do notlearn about the structure of turbulence but aboutthe impact of turbulence on the flow.

The choices for the length scale equation and thealgebraic second-moment closure areindependent.

Two-equation turbulence models provide a usefultool for investigating and reproducing variousprocesses in boundary layer flows.

Estuarine Turbidity Maxima have beeninvestigated in some detail with these models.


Conclusions


� The choices for the length scale equation and thealgebraic second-moment closure areindependent.

Two-equation turbulence models provide a usefultool for investigating and reproducing variousprocesses in boundary layer flows.



Conclusions



� Two-equation turbulence models provide a usefultool for investigating and reproducing variousprocesses in boundary layer flows.



Conclusions



� Two-equation turbulence models provide a usefultool for investigating and reproducing variousprocesses in boundary layer flows.

� Estuarine Turbidity Maxima have beeninvestigated in some detail with these models.


two-equation turbulence models for boundary layer ﬂows · program of presentation nse...

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