**8. References**


Turbulent Boundary Layer Models: Theory and Applications 237

Nezu, I.E. & Nakagawa, H., 1993. Turbulence in open-channel flows. IARH/AIRH

Reynolds, O., 1895. On the dynamical theory of incompressible viscous fluids and the

Rodi, W., 1984. Turbulent models and their applications in *Hydraulics – A state of the art* 

Rotta, J.C., 1951. Statistische theorie nichthamagener turbulenz, *Zeitschrift f. Physik*, Bd. 192,

Sato, S., Mimura, N. & Watanabe, A., 1984. Oscilatory boundary layer flow over rippled

Sheng, Y.P., 1984. A turbulent transport model of coastal processes. Proc. 19th *Conf. Coastal* 

Schiestel, R., 1993. *Modélisation et simulation des écoulements turbulents*, Hermes, Paris, ISBN:

Silva, P.M.C.A., 2001. *Contribution for the study of sedimentary dynamics in coastal regions*, Ph.D

Sleath, J.F.A., 1987. Turbulent oscillatory flow over rough beds, *Journal of Fluid Mechanics*,

Sleath, J.F.A., 1991. Velocities and shear stresses in wave-current flows, *Journal of Geophysical* 

Soulsby, R.L., Hamm, L., Klopman, G., Myrhaug, D., Simons, R.R. & Thomas, G.P., 1994.

Sumer, B.M., Jensen, B.L. & L. Fredsøe, L., 1987. Turbulence in oscillatory boundary layers.

Swart, D.H., 1974. *Offshore sediment transport and equilibrium beach profiles*. Delft Hydraulics

Tanaka, H. & Thu, A., 1994. Full-range equation of friction coefficient and phase difference

Tran-Thu, T., 1995. *Modélisation numérique de l'interaction houle-courant-sédiment*, Ph.D thesis,

Wave-current interaction within and outside the bottom boundary layer, *Coastal* 

Roache, P.J., 1976. *Computational Fluid Dynamics*. Eds. Hermosa Publishers, New Mexico. Rodi, W., 1980. *Turbulence Models and their Application in Hydraulics - A State-of-the-Art* 

determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels *Phil. Trans. Roy. Soc. London Ser. A*, vol. 174, 935-

determination of the criterion, *Phil. Trans. Roy. Soc. London Ser. A*, vol. 186, 123-

Prandtl, L., 1925. Über die ausgebildete Turbulenz. *Z. Angew, Math. Mech*., Vol. 5, 136-139. Reynolds, O., 1883. On the experimental investigation of the circumstances which

Monograph Series, 3ª Ed. Balkema.

*Review*, I.A.H.R – Publication.

pp. 547-572, and Bd. 131, 51-77.

*Eng*., 2380-2396.

2-86601-371-9.

182, 369-409.

*Eng*., 21, 41-69.

Lab., Publ. 131.

*review*, 2nd Edition, Bookfield Publishing, 1-36.

beds. Proc 19th *Conf. Coastal Engineering*, 2293-2309.

thesis, University of Aveiro, Portugal (in Portuguese).

In *Advances in Turbulence*, Springer, Heidelberg, 556-567.

in a wave-current boundary layer, *Coastal Eng*., 22, 237-254. Tennekes, H. & Lumley, J.L., 1972. *A first course in turbulence*, MIT Press.

Université Joseph Fourier – Grenoble, France (in French).

Sleath, J.F.A., 1984. *Sea bed Mechanics*. Eds. Wiley-Interscience.

*Research*, Vol. 96, No. C8, 15, 237-15, 244.

Rodi, W., 1993. *Turbulence models and their application in hHydraulics*, Balkema.

982.

164.


236 Advanced Fluid Dynamics

Fourniotis, N.Th., Dimas, A.A & Demetracopoulos, A.C., 2006. Spatial development of

Huynh-Thanh, S. & A. Temperville, A.,1991. A numerical model of the rough turbulent

Jensen, B.L., Sumer, B.M. & Fredsøe, J., 1989. Turbulent oscillatory boundary layers at high

Jonsson, I.J. & Carlsen, N.A., 1976. Experimental and theoretical investigations in an oscillaory turbulent boundary layer, *Journal of Hydraulics Research*, 14(1), 45-60. Kamphuis, J.W., 1975. Friction factor under oscillatory waves, *J. Waterw. Port Coastal Ocean* 

Kaneda, Y. & Ishihara, T., 2006. High-resolution direct numerical simulation of turbulence,

Kolmogorov, A.N., 1941. *The local structure of turbulence in incompressible viscous fluid for very* 

Launder, B.E. & Spalding, D.B., 1972. *Lectures in mathematical models of turbulence*, Academic

Lesieur, M., 1997. Turbulence in fluids: Third revised and enlarged edition, *Fluid Mechanics and its Applications*, Vol. 40, Kluwert Academic Publishers, ISBN 0-7923-4415-4. Lewellen, W.S., 1977. Use of the invariant modelling, in *Handbook of turbulence*, Plenum

Lumley, J.L., 1996. Fundamental aspects of incompressible and compressible turbulent

Monin, A.S. & Yaglom, A.M., 1971. *Statistical Fluid Mechanics: Mechanics of Turbulence*, Vol. 1,

Moser, R.D., 2006. On the validity of the continuum approximation in high Reynolds

Moulden, T.H., W. Frost & Garner, A.H., 1978. The complexity of turbulent fluid motion, *Handbook of turbulence*, W. Frost and T.H. Moulden, Eds, Plenum Press

McComb, W.D.: http://www.ph.ed.ac.uk/acoustics/turbulence/ (last acces May 2011). Mohammadi, B. & Pironneau, 1994. *Analysis of the k-epsilon turbulence model*, John Wiley e

The MIT Press, Edited by John L. Lumley, ISBN: 0-262-13062-9.

flows, *Simulation and Modeling of Turbulent Flows*, Cap. 1. ICASE/LaRC Series in Computacional Science and Engineering. Edited by Gatski, Hussaini e Lumley.

*large Reynolds numbers*. Proceedings of the USSR Academy of Sciences 30: 299–303. (Russian), translated into English by Kolmogorov, Andrey Nikolaevich (July 8, 1991). *The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers*. Proceedings of the Royal Society of London, Series A:

Reynolds numbers, *Journal of Fluid Mechanics*, 206, 265-297.

Mathematical and Physical Sciences 434 (1991): 9–13.

Press London and New York, ISBN:0-12-438050-6.

Hinze, J., 1975. *Turbulence* (2nd Edition), McGraw-Hill Classic Textbook Reissue Series. Huynh-Thanh, S., 1990. *Modélisation de la couche limite turbulente oscillatoire générée par* 

1032.

French).

93-100.

*Eng*., 101 (WW2), 135-144.

*Journal of Turbulence*, Vol. 7, No. 20.

Publishing Corp., Vol. 1, 237-280.

number turbulence, *Phys. Fluids* 18, 078105.

Sons, ISSN:0298-3168.

turbulent open-channel flow over bottom with multiple consecutive dunes. *Proceedings of* the International Conference River Flow 2006, Lisbon, Portugal, 1023-

*l'interaction houle-courant en zone côtière*, Ph.D thesis, INP – Grenoble, France (in

boundary layer in combined wave and current interaction. In: R.L. Soulsby and R. Betess (Editors), *Sand Transport in Rivers, Estuaries and the Sea*, Balkema, Rotterdam,


**1. Introduction**

correspondence could be made.

with the nonlinear term |*ϕ*|

(2003).

Turbulence is of vital interest and importance to the study of fluid dynamics Pope (1990). In classical physics, turbulence was first studied carefully for *incompressible* flows whose evolution was given by the Navier-Stokes equations. One of the most celebrated results of incompressible classical turbulence (CT) is the existence of an inertial range with the cascade of kinetic energy from large to small spatial scales until one reaches scale lengths on the order of the dissipation wave length and the eddies/vortices are destroyed. The Kolmogorov kinetic

**Unitary Qubit Lattice Gas Representation** 

George Vahala1, Bo Zhang1, Jeffrey Yepez2, Linda Vahala3 and Min Soe4

**of 2D and 3D Quantum Turbulence** 

*1College of William & Mary 2Air Force Research Lab 3Old Dominion University 4Rogers State University* 

*USA* 

**11**

Independently, quantum turbulence (QT) was being studied in the low-temperature physics community on superfluid <sup>4</sup>*He* Pethick & Smith (2009); Tsubota (2008). However, as this QT dealt with a two-component fluid (an inviscid superfluid and a viscous normal fluid interacting with each other) it considered phenomena not present in CT and so no direct

With the onset of experiments in the Bose Einstein condensation (BEC) of dilute gases, we come to a many body wave function that at zero temperature reduces to a product of one-body distributions. The evolution of this one-body distribution function *ϕ*(**x**, **t**) is given by the

BEC gas at temperature *T* = 0. Eq. (2) is ubiquitous in nonlinear physics: in plasma physics and astrophysics it appears as the envelope equation of the modulational instability while in nonlinear optics it is known as the Nonlinear Schrodinger (NLS) equation Kivshar & Agrawal

The quantum vortex is a topological singularity with the wave function |*ϕ*| → 0 at the vortex core Pethick & Smith (2009). A 2*π* circumnavigation about the vortex core leads to an integer multiple of the fundamental circulation about the core: i.e., the circulation is quantized. Thus

*<sup>i</sup>∂t<sup>ϕ</sup>* <sup>=</sup> −∇2*<sup>ϕ</sup>* <sup>+</sup> *<sup>a</sup>*(*g*|*ϕ*<sup>|</sup>

Gross-Petaevskii (GP) equation Gross (1963); Pitaevskii (1961):

*<sup>E</sup>*(*k*) <sup>∼</sup> *<sup>k</sup>*−5/3 (1)

<sup>2</sup>*ϕ* arising from the weak boson-boson interactions of the dilute

<sup>2</sup> <sup>−</sup> <sup>1</sup>)*ϕ*, (2)

energy spectrum in this inertial range follows the power law in wave number space.

Tran-Thu, T. & Temperville, A., 1994. Numerical model of sediment transport in the wavecurrent interaction, in *Modelling of Coastal and Estuarine Processes*, Eds. F. Seabra-Santos & A. Temperville, 271-281.

Wilcox, D.C., 1993. *Turbulent modelling of CFD*. DCW Industries, La Canada, Calif.
