In the first year of research, an in-house, parallel, hybrid spectral/finite difference solver of the
incompressible Navier-Stokes equations under the Boussinesq approximation was developed with
the goal of performing large-eddy simulations (LES) of turbulent flows in the shallow coastal ocean.
The solver uses a second order time-accurate fractional step method; horizontal directions are
discretized spectrally via fast Fourier transforms and the vertical direction is discretized via high
order (fifth and sixth order) finite difference schemes. Finite differences allow for grid stretching in
the vertical in order to resolve expected strong vertical gradients in density and velocity. The
solver is equipped with a subgrid-scale stress parameterization, suitable for large-eddy simulations
(LES). Boundary conditions can be assigned as prescribed velocity components or as prescribed
shear stresses at the bottom and top bounding surfaces perpendicular to the vertical direction.
Validation of the solver was completed in the second year of research. In order to test the solver
and its subgrid-scale parameterization, both were used in LES of turbulent channel flow at
Reynolds number of 180 (based on friction velocity and channel half-width). LES using the inhouse solver was in good agreement with published, well-known direct numerical simulations
(DNS) results in terms of mean streamwise velocity, mean Reynolds shear stress, root mean square
of velocity fluctuations, and one-dimensional streamwise and spanwise energy spectra.
Furthermore, the solver alone (without a subgrid-scale parameterization) was used in DNS of the
same problem, but with a finer grid than the LES. As expected, DNS results using the in-house
solver were nearly indistinguishable from the published, well-known DNS results.
The in-house solver was then equipped with parameterization terms required for LES of Langmuir
circulation often observed in the mixed layer of the deep ocean and more recently observed in
shallow coastal regions. In order to study the dynamics of Langmuir turbulence, the solver was also
upgraded to include computation of every term present in the transport equations of the Reynolds
stress tensor and turbulent kinetic energy. A suite of external, post-processing routines was written
in order to reduce the multi-processor solver output containing time averages of each term of the
transport equations. A second suite of post-processing routines was written to reconstruct the flow
field (via fast Fourier transforms) on finer grids than LES grids in order to visually inspect the
large-scale structures in Langmuir turbulence (see Figure 2).
Numerous LES were performed simulating Langmuir circulation in deep water and shallow water
and Langmuir circulation generated by small and large amplitude surface waves. Characteristics of
the dynamics and structure of these cases were analyzed leading to clear differences between each
case. Furthermore, differences between Langmuir-dominated turbulence and shear-dominated
turbulence were recognized (see Figures 1 and 2). For comparison with field observations, we
performed LES of Langmuir circulation in shallow water generated by (1) surface gravity waves
with wavelength 6 times the water depth, frequency 0.8/s, and amplitude 0.6m and (2) wind stress
at 0.1N/m2. LES results showed good agreement with observations in terms of the Lumley invariant
map (Figure 1B) and in terms of the large-scale flow structure (Figure 2D-F). More recent LES of
Langmuir circulation with the in-house solver indicate that the results of Figures 1 and 2 become
independent of Reynolds number (Re) at sufficiently high Re, an important finding given that the
shear flow in the field observations is characterized by very high Reynolds numbers impossible to
simulate (while resolving the bottom boundary layer as our simulations do) even with the fastest
computers available today. Current LES focuses on the effect of stable stratification on Langmuir
circulation.
A
B
A
Figure 1: Trajectories of invariants of the Reynolds stress anisotropy tensor in the Lumley
triangle; values are horizontal plane and time averages, moving from the bottom (x/δ = -1) to
the surface (x/δ = 1) of the computational domain. δ=mid-depth. Trajectories of the shear
dominated case (A) and Langmuir-dominated case (B) bear no resemblance. Trajectory of case
(B) is similar to that recorded by observations.
Figure 2: Color-contoured maps of fluctuating down-wind (uτ), vertical (w), and cross-wind (u┴) velocity
components for shear-dominated turbulence (A-C) and Langmuir-dominated turbulence with conditions
matching field observations (D-F): the plane shown is perpendicular to the wind direction, looking down-wind.
Fluctuating velocities, averaged (<>) over time and down-wind direction, are normalized by the down-wind
component of bulk velocity.