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Special Section: Computational Fluid Mechanics and Fluid–Structure Interaction

Large-Eddy Simulation of Shallow Water Langmuir Turbulence Using Isogeometric Analysis and the Residual-Based Variational Multiscale Method

[+] Author and Article Information
Andrés E. Tejada-Martínez, Ido Akkerman, Yuri Bazilevs

Civil & Environmental Engineering,  University of South Florida, Tampa, FL 33620, e-mail: aetejada@usf.eduCoastal & Hydraulics Laboratory, US Army Engineer Research and Development Center, Vicksburg, MS 39180-6133; Department of Structural Engineering,  University of California, San Diego, La Jolla, CA 92093, e-mail: iakkerman@ucsd.eduDepartment of Structural Engineering,  University of California, San Diego, La Jolla, CA 92093, e-mail: jbazilevs@ucsd.edu

J. Appl. Mech 79(1), 010909 (Dec 13, 2011) (12 pages) doi:10.1115/1.4005059 History: Received March 12, 2011; Revised April 20, 2011; Published December 13, 2011; Online December 13, 2011

We develop a residual-based variational multiscale (RBVMS) method based on isogeometric analysis for large-eddy simulation (LES) of wind-driven shear flow with Langmuir circulation (LC). Isogeometric analysis refers to our use of NURBS (Non-Uniform Rational B-splines) basis functions which have been proven to be highly accurate in LES of turbulent flows (Bazilevs, Y., 2007, Comput. Methods Appl. Mech. Eng., 197 , pp. 173–201). LC consists of stream-wise vortices in the direction of the wind acting as a secondary flow structure to the primary, mean component of the flow driven by the wind. LC results from surface wave-current interaction and often occurs within the upper ocean mixed layer over deep water and in coastal shelf regions under wind speeds greater than 3 m s−1 . Our LES of wind-driven shallow water flow with LC is representative of a coastal shelf flow where LC extends to the bottom and interacts with the sea bed boundary layer. The governing LES equations are the Craik-Leivobich equations (Tejada-Martínez, A. E., and Grosch, C. E., 2007, J. Fluid Mech., 576 , pp. 63–108; Gargett, A. E., 2004, Science, 306 , pp. 1925–1928), consisting of the time-filtered Navier-Stokes equations. These equations possess the same structure as the Navier-Stokes equations with an extra vortex force term accounting for wave-current interaction giving rise to LC. The RBVMS method with quadratic NURBS is shown to possess good convergence characteristics in wind-driven flow with LC. Furthermore, the method yields LC structures in good agreement with those computed with the spectral method in (Thorpe, S. A., 2004, Annu. Rev. Fluids Mech., 36 , pp. 584 55–79) and measured during field observations in (D’Alessio, S. J., , 1998, J. Phys. Oceanogr., 28 , pp. 1624–1641; Kantha, L., and Clayson, C. A., 2004, Ocean Modelling, 6 , pp. 101–124).

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Figures

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Figure 1

Sketch of Langmuir circulation spanning the upper ocean mixed layer

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Figure 2

Computational domain

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Figure 3

Instantaneous snapshot of iso-contours of wall-normal (vertical) velocity fluctuations in flow with LC on the 64 × 66 × 64 quadratic NURBS mesh described earlier

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Figure 4

Color maps of instantaneous downwind velocity fluctuation u1’ on the downwind-crosswind plane at mid-depth in flows with and without C-L vortex forcing (i.e., with and without LC). Results are from the simulations on the 48 × 50 × 48 quadratic NURBS mesh described earlier.

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Figure 5

Crosswind-vertical variation of velocity fluctuations vi ′ (defined in Eq. 26) in flow with LC. Results are from the simulation on the 48 × 50 × 48 quadratic NURBS mesh.

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Figure 6

Crosswind-vertical variation of partially averaged velocity fluctuations vi ′ (defined in Eq. 26) in flow without LC. Results are from the simulation on the 48 × 50 × 48 quadratic NURBS mesh.

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Figure 7

Crosswind-vertical variation of velocity instantaneous velocity fluctuations ui ′ (in cm/s). Results are from the simulation on the 48 × 50 × 48 quadratic NURBS mesh. Computational velocities have been made dimensional with wind stress friction velocity recorded in the field during episodes of full-depth LC [6]. Field measurements were made in a 15-meters deep water column under a wind stress of 0.1 N/m2 .

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Figure 8

Crosswind-vertical variation of velocity instantaneous velocity fluctuations ui ′ (in cm/s) during episode of full-depth Langmuir cells measured during field experiments of Gargett and Wells [6] using a bottom-mounted, upward-facing acoustic Doppler current profiler (ADCP). Field measurements were made in a 15 ms-deep water column under a wind stress of 0.1 N/m2 . This figure is courtesy of Ann Gargett.

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Figure 9

Convergence of mean downwind velocity in flow with LC. Quadratic NURBS meshes were used for all cases.

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Figure 10

Convergence of turbulent kinetic energy (TKE) in flow with LC. Quadratic NURBS meshes were used for all cases.

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Figure 11

Mean downwind velocity in flows with and without LC. The 48 × 50 × 48 quadratic NURBS mesh was used for both flows.

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Figure 12

Mean downwind velocity versus wall-normal (vertical) direction in wall (plus) units in flows with and without LC. The 48 × 50 × 48 quadratic NURBS mesh was used for both flows. Note that y+  = uτ y/v.

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