Subsurface Flow Generated by a Steady Wind Stress Applied at the Water Surface by Lionel Gurfinkiel A Thesis Submitted to the Faculty of The College of Engineering In Partial Fulfillment of the Requirements for the Degree of Master of Science Florida Atlantic University Boca Raton, Florida August 2003 SUBSURFACE FLOW GENERATED BY A STEADY WIND STRESS APPLIED AT THE WATER SURFACE by Lionel Gurfinkiel This thesis was prepared under the direction of the candidate's thesis advisor, Dr. Manhar Dhanak, Department of Ocean Engineering, and has been approved by the members of his supervisory committee. It was submitted to the faculty of the College of Engineering and was accepted in partial fulfillment of the requirements for the degree of Master of Science. SUPER VISOR Y COMMITTEE: Dr. Manhar Dhanak Dr. P. Ananthakrishnan 0.'1:-~ Dr. Oleg Zikanov Chairman, Department of Ocean Engineering Date II ACKNOWLEDGEMENTS The author would like to extend his utmost thanks and gratitude to Dr. Manhar Dhanak, thesis advisor, for his precious advice in performing this research and for his patience and willingness to take the time to guide and assist in the work involved in completing this thesis. Gratitude is extended to the supervisory committee, Dr. P. Ananthakrishnan for his valuable input throughout this project, and Dr. Oleg Zikanov for his indispensable help in the computer code work and his expertise in this project. A special thanks is given to Dr. Pierre-Emmanuel Guillerm for his patience and his expert advice in the area of computational fluid dynamics. Special thanks are also given to the faculty and staff of the Department of Ocean Engineering at Florida Atlantic University for the opportunity and support in accomplishing this research. This work was supported by the Office of Naval Research under grant N00014-00-l- 0218 (Program Managers: Dr. Thomas Curtin and Dr. Thomas Swean). 1Il ABSTRACT Author: Lionel Gurfinkiel Title: Subsurface Flow Generated by a Steady Wind Stress Applied at the Water Surface Institution: Florida Atlantic University Dissertation Advisor: Dr. Manhar Dhanak Degree: Master of Science Year: 2003 A turbulent water current induced by winds, through a friction force at the sea surface and subjected to the Coriolis force in shallow water was studied. A Large Eddy Simulation model developed by Zikanov et al. is used to solve the Navier-Stokes equations. To define the bottom boundary condition, a drag coefficient parameter, based on the ideas of Csanady, is used to evaluate the shear stress at the bottom. To find a suitable bottom boundary condition for this LES simulation, several cases were considered with change in drag coefficient property. The effect of variation in the depth of the water column was also considered. Variation in surface deflection of the current, variation of the mass flux and distribution of eddy viscosity with depth of the water column are determined. The cases are compared with the case of a deep water column. Numerical results are also compared with field observations. IV TABLE OF CONTENTS TABLE OF FIGURES ...... ... ............................. .. .... ..... ....... .. ............. ...... .......... .. .. ... vii 1 INTRODUCTION ................................................................................................ l 1.1 Objective ........................................................... ............................................ 1 1.2 Background ................................................................................................... 2 1.2.1 Ekman solution ............... ...................................................................... 3 1.2.2 Ekman transport ....................... .... ... .......... .. .... ................................... .. 7 1.2.3 Limitations of the Ekman solution ........................................................ 9 1.3 Relationship with the thesis research ............................................ .............. 10 2 LARGE EDDY SIMULATION METHOD ..................................................... ll 2.1 Comparisons with the Direct Numerical Simulation ................. ... ...... ....... 11 2.2 Presentation of the LES model ................................................................... 12 3 APPROACH ....................................................................................................... 16 3.1 Assumptions .............. .... .................................................... .......................... 16 3.2 Coriolis force ................. .. ......................................................... .... .. ............ 17 3.3 Governing equations ................. ......... .. ............. .......................................... 18 3.4 Boundary conditions applied in deep water ................................................ 19 3.5 Boundary conditions applied in shallow water. ... .... .. .. ... ...... ... ................... 20 4 COMPUTATIONAL METHOD ...................................................................... 25 v 4.1 LES model used ..... ................................................... ........ .................. .. ...... 25 4.2 Numerical method ................................................................................ ....... 27 4.3 Domain of the simulations .......................................................................... 28 5 RESULTS ........................................................................................................... 31 6 DISCUSSION ..................................................................................................... 36 6.1 Original simulated flow in shallow water compared with deep water simulation ... ................................. .......................................................................... .. 36 6.2 Variation of drag coefficient simulations ................................................... 46 6.3 Variation of the depth ........................................... ................... ................. .. 50 6.4 Ekman transport .............. ....................................... ... ... ... .... .. ...................... 55 6.4.1 Mass transport in deep water .............................................................. 55 6.4.2 Mass transport in shallow water ... ..... ................... .............................. 57 6.5 Comparison with the field observations .. .................................... .. ...... ...... 58 7 CONCLUSIONS AND FUTURE WORK ....................................................... 64 APPENDIX ................................................................................................................. 67 REFERENCE ............................................................................................................. 69 VI TABLES OF FIGURES Figure 1-1 Hodograph showing velocity at various depths ........ ............. .............. .... .. 6 Figure 1-2 Perspective view showing velocity decreasing and rotating with increase of depth of an Ekman flow ............................................ ..... .... ......... ... ... ...... .... ...... 8 Figure 2-1. Schematic representation of turbulent motion (left) and the time dependence of a velocity component at a point (right) [7] . ... .... .. ... .. ..... .... .. .... ... 12 Figure 2-2: Filter G(r) : box filter, dashed line; Gaussian filter, solid line; Sharp spectral filter, dot-dashed line ...... ..................................................... ........ .......... 13 Figure 5-1 Horizontally and time-averaged profiles of the mean horizontal current u and v .... .. ............. ................................................................... ... ... .. ... ... ..... ... ...... .. 32 Figure 5-2 Velocity hodograph .. ............. ............... .... ...... ........... .. .. ........................... 33 Figure 5-3: Subgrid grade and Reynolds stresses ...................................................... 33 Figure 5-4: effective viscosity coefficient ............................. ......... ... ... ... ................... 35 Figure 6-1: volume-averaged kinetic energy as a function of time for shallow water flow simulated at latitude 26 oand wind direction being 170° from south-north direction .......................................................... ... .......... ............ ..... .. .... ............. ... 37 Vll Figure 6-2 volume-averaged kinetic energy as a function of time for deep water flow simulated at latitude 26 oand wind direction being 170° from south-north direction ................................... ...................................................... ..................... 38 Figure 6-3: Horizontally and time-averaged profiles of the components of the mean horizontal current u and v of shallow water at latitude 26 oand wind direction being 170° from south-north direction ................................................................ 39 Figure 6-4 Horizontally and time-averaged profiles of the components of the mean horizontal current u and v of shallow water at latitude 26 oand wind direction being 170° from south-north direction ................................................................ 40 Figure 6-5 Speed of current of the simulated flow for shallow water at latitude 26° and wind direction being 170° from south-north direction ................................. 41 Figure 6-6 Speed of current of the simulated flow for deep water at latitude 26° and wind direction being 170° from south-north direction ........................................ 41 Figure 6-7 Effective viscosity coefficient