Transcript Large-eddy simulation and its extension to finite element

The structure of turbulence in a shallow water wind-driven
shear current with Langmuir circulation
Andrés E. Tejada-Martínez and Chester E. Grosch
Center for Coastal Physical Oceanography
Department of Earth, Ocean and Atmospheric Sciences
Old Dominion University
Norfolk, Virginia
Observed structure of Langmuir cells
x3
x1
x2
Courtesy of J. Smith, UCSD
Negatively buoyant algae aligned in rows by Langmuir circulation off the
coast of the Bahamas (courtesy of D. Zimmerman, ODU)
The filtered Navier-Stokes equations
ui
0
xi
• Continuity:
Craik-Leibovich
vortex forcing
SGS stress
• Momentum:
 ij( r ) d
ui
ui
1   1  ui
1
s
uj





u

2
2 ijk j k
t
x j
 xi Re xi
x j
LaT
2
1 (r )
     kk
3
u1s 
U H
Re  
2
cosh[2k ( x3  H )]
2 sinh2 (kH )
LaT 
u2s  u3s  0
U
US
Subgrid-scale (SGS) stress
 ij( r )  uiu j  uiu j
SGS stress:
Smagorinsky model for the SGS stress:

(r )d
ij

Eddy viscosity:
(CS ) 2
(r )
ij
1
  kk( r ) ij  2vT Sij
3
1  ui u j 

Sij  

2  x j xi 
 T  (CS )2 | S |
| S | 2 Sij Sij
is computed dynamically using the Germano identity (Germano
et al., Phys. Fluids, 1991)
LES of Langmuir cells in wind-driven channel
x3
L1  4h
L2  (8 / 3)h
 surface
x2
x1
x3  h
H  2h
x3  h
no-slip wall
• Surface stress is applied such that Re  u h /  180
• Craik-Leibovich vortex forcing is added to the filtered momentum equations (LES
equations) to account for Langmuir cells (Lc)
• Two simulations were performed: 1) Langmuir forcing, 2) no Langmuir forcing
• Parameters in simulation with Lc are derived from wave and wind conditions during
field observations of Lc:
LaT  0.7,
  6H
LES of Langmuir cells (mean streamwise velocity)
• Enhanced mixing due to Langmuir circulation tends to homogenize mean
streamwise velocity
LES of LC (instantaneous streamwise velocity fluctuation)
x2
x1
ui  ui  ui
No Langmuir forcing
With Langmuir forcing
LES of Langmuir cells (streamwise-time averages)
x3
x2
x1
spanwise vel.
fluctuations
wall-normal vel.
fluctuations
streamwise vel.
fluctuations
No Langmuir forcing
With Langmuir forcing
LES of Langmuir cells (Reynolds stress components)
With Langmuir forcing
No Langmur forcing
LES of Langmuir cells (Lumley’s triangle)
No Langmuir forcing
With Langmuir forcing
• Presence of Langmuir cells greatly affects the state of the turbulence near the
bottom and near the surface
• Trajectory of Lumley map for case with Langmuir forcing agrees well with
observed data especially near the bottom
• See tomorrow’s talk at 9:40am: “Characteristics of Langmuir turbulence observed
in shallow water,” Judith Wells and Ann Gargett
Turbulent kinetic energy budget terms (near bottom)
With Langmuir forcing
No Langmuir forcing
Turbulent kinetic energy budget terms (near surface)
With Langmuir forcing
No Langmuir forcing
Final Remarks
• First observations and simulations of Langmuir circulation covering
entire water column
• Secondary flow structures of simulations characteristic of Langmuir
circulation agree well with observations
• Reynolds stresses of simulations agree well with observations in lower
region of the water column
• More recent work explores effects of: 1) Reynolds number
2) domain size
3) subgrid-scale parameterization
4) grid size
5) rotation
6) stratification
Domain length dependence
x3
x2
x1
streamwise vel.
fluctuations
spanwise vel.
fluctuations
wall-normal vel.
fluctuations
Extended domain
Original domain
Reynolds number dependence
Re = 180
Re = 395
SGS model (parameterization) dependence
Dynamic
Smagorinsky
Dynamic mixed
Grid dependence
32x64x97
48x96x145