two-equation turbulence models for boundary layer flows · program of presentation nse...
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![Page 1: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/1.jpg)
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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
![Page 2: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/2.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 3: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/3.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 4: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/4.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 5: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/5.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 6: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/6.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 7: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/7.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 8: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/8.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 9: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/9.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 10: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/10.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 2/33
![Page 11: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/11.jpg)
Navier-Stokes-EquationsContinuity Equation:
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Momentum Equation:
Heat Equation:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 3/33
![Page 12: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/12.jpg)
Navier-Stokes-EquationsContinuity Equation:
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Momentum Equation:
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Heat Equation:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 3/33
![Page 13: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/13.jpg)
Navier-Stokes-EquationsContinuity Equation:
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Momentum Equation:
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Heat Equation:
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NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 3/33
![Page 14: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/14.jpg)
Averaging Rules1. Linearity:
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2. Exchange of Derivative and Averaging:
3. Double Averaging:
4. Products of Averages:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 4/33
![Page 15: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/15.jpg)
Averaging Rules1. Linearity:
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3. Double Averaging:
4. Products of Averages:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 4/33
![Page 16: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/16.jpg)
Averaging Rules1. Linearity:
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3. Double Averaging:
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4. Products of Averages:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 4/33
![Page 17: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/17.jpg)
Averaging Rules1. Linearity:
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3. Double Averaging:
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4. Products of Averages:� � � � � ��
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NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 4/33
![Page 18: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/18.jpg)
Reynolds-averaged EquationsContinuity Equation:
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Reynolds Equation:
Heat Equation:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 5/33
![Page 19: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/19.jpg)
Reynolds-averaged EquationsContinuity Equation:
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Reynolds Equation:
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Heat Equation:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 5/33
![Page 20: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/20.jpg)
Reynolds-averaged EquationsContinuity Equation:
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Reynolds Equation:
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Heat Equation:
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NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 5/33
![Page 21: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/21.jpg)
Reynolds Stress Equation
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NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 6/33
![Page 22: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/22.jpg)
Heat Flux Equation
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NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 7/33
![Page 23: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/23.jpg)
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).
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 8/33
![Page 24: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/24.jpg)
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).
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 8/33
![Page 25: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/25.jpg)
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).
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 8/33
![Page 26: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/26.jpg)
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).
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 8/33
![Page 27: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/27.jpg)
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).
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 8/33
![Page 28: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/28.jpg)
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).
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 8/33
![Page 29: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/29.jpg)
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).
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 8/33
![Page 30: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/30.jpg)
Algebraic SMCsTurbulent Fluxes:
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Eddy Viscosity / Eddy Diffusivity:
Shear Number, Buoyancy Number:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 9/33
![Page 31: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/31.jpg)
Algebraic SMCsTurbulent Fluxes:
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Eddy Viscosity / Eddy Diffusivity:
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Shear Number, Buoyancy Number:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 9/33
![Page 32: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/32.jpg)
Algebraic SMCsTurbulent Fluxes:
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Eddy Viscosity / Eddy Diffusivity:
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Shear Number, Buoyancy Number:
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NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 9/33
![Page 33: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/33.jpg)
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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 10/33
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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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 11/33
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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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 12/33
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Exact TKE-Equation
�
� ��� � � � � ��
� � � ��
�
� � � ����
� � � � �� �
� �� � � � � � � � � ��� �
��
�
��
���� � ��� �� �
��
�
� � � � � � � � � �
��
�
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.
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 13/33
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Exact TKE-Equation
�
� ��� � � � � ��
� � � ��
�
� � � ����
� � � � �� �
� �� � � � � � � � � ��� �
��
�
��
���� � ��� �� �
��
�
� � � � � � � � � �
��
�
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.
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 13/33
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Length scale equations
- �model (Launder and Spalding [1972]):
��� � � ���
� �� �
�� � �
��
� � � � � � � � � � � � � � � � �
- model (Mellor and Yamada [1982]):
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 14/33
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Length scale equations
- �model (Launder and Spalding [1972]):
��� � � ���
� �� �
�� � �
��
� � � � � � � � � � � � � � � � �
-
�
model (Mellor and Yamada [1982]):
��� � � �� � ���
�� ��
� � �� �
�
�� � � � � � � � � � � �
���
�� �
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 14/33
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Total equilibrium ( - �)
� �
� �� ����
� ���
� � � � � � �
� � � � �� �
� �� � � ��
: Steady-state Richardson number.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1
KCRHCACB
PSfrag replacements
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 15/33
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Total equilibrium ( - �)
� �
� �� ����
� ���
� � � � � � �
� � � � �� �
� �� � � ��
: Steady-state Richardson number.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1
KCRHCACB
PSfrag replacements
�� �
�����
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 15/33
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Total equilibrium ( - )
� �
� �� �
��
��
� �
�
� �
� �� � � ��
: Steady-state Richardson number.
0.1
0.15
0.2
0.25
0.3
4.5 5 5.5 6 6.5 7 7.5 8PSfrag replacements
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 16/33
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Total equilibrium ( - )
� �
� �� �
��
��
� �
�
� �
� �� � � ��
: Steady-state Richardson number.
0.1
0.15
0.2
0.25
0.3
4.5 5 5.5 6 6.5 7 7.5 8PSfrag replacements
��
�� ���� � �
� � �
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 16/33
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Length scale equations (cont’d)Generic length scale equation (Umlauf and Burchard [2002]):
�� � � � � �� � ��
� ��
� �� � � �
�
� � �� � � � �
� � � � � �� � � � � � � � � � �
General relation between , and :
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 17/33
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Length scale equations (cont’d)Generic length scale equation (Umlauf and Burchard [2002]):
�� � � � � �� � ��
� ��
� �� � � �
�
� � �� � � � �
� � � � � �� � � � � � � � � � �
General relation between
�
, � and
�:
� � �� � �
�
� � � ��
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 17/33
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Boundary conditionsLaw of the wall:
� ����
�
��
�
�� � �
�
� � ����
�
��
�
��
� � � � ��
�
� ����
� ��
�� � � ��
Breaking surface waves:
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 18/33
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Boundary conditionsLaw of the wall:
� ����
�
��
�
�� � �
�
� � ����
�
��
�
��
� � � � ��
�
� ����
� ��
�� � � ��
Breaking surface waves:
� ����
� � ��� � � ���
� � ����
� � ��� � � �� �
�
� ����
� � ��� � � �
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 18/33
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Boundary conditionsLaw of the wall:
� ����
�
��
�
�� � �
�
� � ����
�
��
�
��
� � � � ��
�
� ����
� ��
�� � � ��
Breaking surface waves:
� ����
� � ��� � � ���
� � ����
� � ��� � � �� �
�
� ����
� � ��� � � �
� � � � � � � � � � � � � � � � � � � � � � �
� � � � �
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 18/33
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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
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 19/33
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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
�� �� �
��� �
� �� � �� ��� �
Terray et al. [1996]Drennan et al. [1996]
Anis and Moum [1995]Numerical, ��� ���� � ���
Numerical, ��� � �� � ! �
Numerical, ��� ���� � � "
Log-LawNo shear production
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 20/33
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Free Convection
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
-10 -8 -6 -4 -2 0
PSfrag replacements
� �� �
KCRHCACB
LES
� ���� ��� � ���
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
� �� �
KCRHCACB
LES
���� � � � � �� � ���
TemperatureTemperature Flux
Variance ofVariance ofVariance of
Dissipation Rate-1.2
-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
� �� �
KCRHCACB
LES
���� � � � �� ��
TemperatureTemperature Flux
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
� �� �
KCRHCACB
LES
���� � � � ��� ��
TemperatureTemperature Flux
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
� �� �
KCRHCACB
LES
�� � � � � � � ��
TemperatureTemperature Flux
Variance ofVariance of
Variance of
�
Dissipation Rate-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
PSfrag replacements
� �� �
KCRHCACB
LES
�� ���
TemperatureTemperature Flux
Variance ofVariance ofVariance of
Dissipation Rate
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 21/33
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Lago Maggiore, ItalyObservations and simulations of
�
and � (Stips et al. [2002])
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 22/33
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Lago Maggiore, ItalyObservations and simulations of
�
and � (Stips et al. [2002])
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 22/33
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Lago Maggiore, ItalyObservations and simulations of
�
and � (Stips et al. [2002])
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 22/33
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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'
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 23/33
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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'
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 23/33
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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'
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 23/33
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Northern North SeaWind and Tides
0.00.10.20.30.40.5
Surface stress at station NNSPSfrag replacements
Q / (W m )
�
/(N
m
�� )
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
�� )
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 24/33
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Northern North Sea
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 25/33
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Northern North Sea
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 25/33
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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]
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 26/33
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Liverpool BayObserved and simulated temperature and salinity
Simpson, Burchard, Fisher, Rippeth [2002]
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 27/33
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Liverpool BayObserved and simulated current velocity
Simpson, Burchard, Fisher, Rippeth [2002]
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 28/33
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Liverpool BayObserved and simulated dissipation rates
Simpson, Burchard, Fisher, Rippeth [2002]
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 29/33
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Liverpool BayObserved and simulated dissipation rates
Simpson, Burchard, Fisher, Rippeth [2002]
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 29/33
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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 ?
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 30/33
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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 ?
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 30/33
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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 ?
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 30/33
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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 ?
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 30/33
![Page 70: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/70.jpg)
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 ?
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 30/33
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Conceptual model for ETMs
Jay & Musiak, 1994
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 31/33
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ETM computer simulationsBurchard & Baumert, 1998
Ruiz Villareal and Burchard, under progress
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 32/33
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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.
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 33/33
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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.
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 33/33
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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.
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 33/33
![Page 76: Two-equation turbulence models for boundary layer flows · Program of presentation NSE Two-equation models Two-equation models Boundary conditions General Ocean Turbulence Model](https://reader034.vdocument.in/reader034/viewer/2022042910/5f40bf3ccb88957d117f00f3/html5/thumbnails/76.jpg)
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.
NOPP Sediment Modelling Workshop, Williamsburg, Virginia, Sept. 30 - Oct. 2, 2002 – p. 33/33