Versteeg H., Malalasekra W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method

Подождите немного. Документ загружается.

APPENDIX G 467

The predictions and experiments show good agreement, which illustrates the

capability of CFD in predicting complex ﬂows. The predictions reproduce

the main features of the experiments and, despite the coarse grid, the pre-

dictions agree well with the experimental data.

G.4 Laminar flow in a circular pipe driven by periodic

pressure variations

The problem considered

Many engineering problems involve unsteady behaviour. In some cases,

for instance paint mixing, the ﬂow may be steady but the distribution of a

transported scalar variable changes with time. In the present example we

consider one of the simplest cases of the class of problems with genuinely

unsteady ﬂow ﬁelds: the periodic oscillations of an incompressible laminar

ﬂow in a circular pipe driven by harmonic pressure variations between inlet

and outlet. Blood ﬂows in veins and arteries, pressure waves in oil pipelines

and air ﬂows in intake manifolds of internal combustion engines can be

modelled as periodic duct ﬂows.

The applied pressure difference between the pipe ends is varied

according to

∆P = K cos nt (G.1)

Figure G.7 Comparison of predicted and measured temperature distributions for LLNL test MOD08

ANIN_Z07.qxd 29/12/2006 04:50PM Page 467

The amplitude K is taken to be 50 000 (Pa) and the circular frequency n is

equal to 2

Hz, giving an oscillation period of 1 s. Schlichting (1979) gives

the analytical solution for the axial velocity component u(r, t) as a function of

radius r and time t for periodic laminar ﬂow in a very long pipe as the real

part of the following expression:

(G.2)

In this formula

and L are the ﬂuid density, kinematic viscosity and the

length of the pipe respectively, J

denotes the Bessel function of the ﬁrst

kind of order 0, and i is −1. The general features of the velocity distribu-

tions are dependent on the value of the non-dimensional parameter (n/

)R.

The solution behaviour for small and large values of this parameter will be

discussed below. Here we calculate the ﬂow for two intermediate values of

n by taking the pipe radius R equal to 0.01 m and frequency n as 2

Hz in

conjunction with a constant ﬂuid density of 1000 kg/m

and dynamic vis-

cosity of 0.4 and 0.1 kg/m.s. This yields values of (n/

)R of 1.253 and

2.507 respectively.

CFD simulation

In order to set up a valid comparison between the analytical and ﬁnite

volume solution of this problem we need to consider a pipe of sufﬁcient

length. The boundary layer ﬂow near the inlet of a pipe changes in the down-

stream direction, and in a steady ﬂow the velocity distribution becomes fully

developed after a distance l

given by (Schlichting, 1979)

= 0.25 (G.3)

An estimate of the maximum possible mean velocities X (approximately 4 m/s

here) can be obtained from the Hagen–Poiseuille formula (Schlichting,

1979). This leads to a maximum Reynolds number of 800 and hence a value

of l

of 1 m. Since the ﬂow switches direction in the course of a cycle it is

necessary to employ a computational domain of length greater than two

times l

to ensure that there is always a section of fully developed ﬂow half

way along the duct. In this simulation we use a domain with a length L equal

to 2.5 m and consider the solution in a cross-sectional plane at a distance of

1.25 m from its ends.

The ﬂow is axisymmetric, and we use a grid of 250 axial and 20 radial

nodes distributed uniformly in the z- and r-directions. Figure G.8 shows a

sketch of the solution domain and part of the mesh used. At r = 0 a symme-

try boundary condition ensures that there is no ﬂow across the axis and that

the gradients of all variables in the radial direction are locally zero. At r = R

= 0.01 m the usual wall boundary condition is maintained. The cosinusoidal

driving pressure difference given by Figure G.9 and equation 10.36 is

applied by means of prescribed pressure boundary conditions at z = 0 and

ur t i

int

(, )

=−

⎛

⎝

⎜

⎞

⎠

⎟

−

⎛

⎝

⎜

⎞

⎠

⎟

−

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

468 APPENDIX G

ANIN_Z07.qxd 29/12/2006 04:50PM Page 468

APPENDIX G 469

z = L = 2.5 m. The solution procedure is SIMPLER with fully implicit

time marching; the time step is 1 ms. The parabolic velocity proﬁle of a

steady laminar pipe ﬂow is used as the initial velocity ﬁeld.

Specimen results

Figures G.9 and G.10 compare the numerical and analytical solutions half

way along the pipe at time intervals of 0.125 s. The ﬁnite volume solution is

studied after three pressure cycles, allowing time for the initial transients to

die out. It is clear from the solution that the agreement between numerical

and analytical solutions is generally excellent. There are minor discrepancies

in the simulation with (n/

)R = 2.507 during those parts of the solution

cycle where the ﬂow near the boundary moves in the opposite direction

to that in the core of the pipe. These can be explained by the fact that the

local pressure gradient

∂

x is somewhat different from the overall pres-

sure gradient ∆p/L due to energy losses between the inlet and the solution

cross-section.

Figure G.8 Solution domain and

a part of the mesh for simulation

of periodic laminar pipe ﬂow

Figure G.9 Imposed transient

pressure cycle

ANIN_Z07.qxd 29/12/2006 04:50PM Page 469

Overall ﬂow behaviour can be explained by considering appropriate

expressions (Abramowitz and Stegun, 1964) of the Bessel function J

in the

analytical solution.

For very slow oscillations (n/

)R → 0 we obtain

u(r, t) = (R

− r

) cos nt (G.4)

This exhibits the parabolic velocity distribution of a steady, fully developed,

laminar pipe ﬂow with a periodic time variation. The amplitude depends on

the ﬂuid viscosity, and the oscillations are in phase with the driving pressure

difference. For fast oscillations (n/

)R →∞we have

u(r, t) = sin nt − exp − (R − r) sin nt − (R − r) (G.5)

Expression (G.5) contains two sinusoidal terms, the ﬁrst of which is

independent of viscosity. It describes the ﬂow in the central core of the pipe,

which has a uniform velocity distribution with an amplitude inversely pro-

portional to the oscillation frequency and a phase lag of

/2 radians behind

the excitation force. The amplitude and the phase of the second term are

viscosity dependent. The term decays quickly with distance (R − r) from the

pipe wall due to the exponential factor. It can be shown that this boundary

layer ﬂow lags behind the driving pressure difference by

/4 radians. The

phase difference between the core and the boundary layer gives rise to an

annular ﬂow pattern during fast oscillations. It is clear that the results of

Figures G.10 and G.11 exhibit the main characteristics of the slow and fast

solution respectively.

470 APPENDIX G

Figure G.10 Velocity

distribution for periodic

laminar pipe ﬂow with

(n/

)R = 1.253

ANIN_Z07.qxd 29/12/2006 04:50PM Page 470

APPENDIX G 471

The above ﬂow can be comfortably calculated on a workstation, but this

success should not mislead the prospective user of commercial CFD codes.

Other types of unsteady ﬂow problems with complex geometries and/or

ﬂuid physics such as turbulent intake manifold ﬂows (Chen, 1994), pulsed

combustion (Benelli et al., 1992), transient free convective cooling of warm

crude oil in storage tanks (Cotter and Charles, 1993) or hydrodynamic insta-

bilities such as vortex shedding require very large computing resources.

Often such ﬂow calculations are only practical within reasonable time limits

on dedicated large computers with advanced architecture and specially

adapted algorithm structures.

Figure G.11 Velocity

distribution for periodic

laminar pipe ﬂow with

(n/

)R = 2.507

ANIN_Z07.qxd 29/12/2006 04:50PM Page 471

Chapter 1

Gottlieb, D. and Orszag, S. A. (1977). Numerical Analysis of Spectral Methods: Theory

and Applications, SIAM, Philadelphia.

Hastings, C. (1985). Approximations for Digital Computers, Princeton University

Press, Princeton, NJ.

Patankar, S. V. (1980). Numerical Heat Transfer and Fluid Flow, Hemisphere

Publishing Corporation, Taylor & Francis Group, New York.

Smith, G. D. (1985). Numerical Solution of Partial Differential Equations: Finite

Difference Methods, 3rd edn, Clarendon Press, Oxford.

Zienkiewicz, O. C. and Taylor, R. L. (1991). The Finite Element Method – Vol. 2:

Solid and Fluid Mechanics, McGraw-Hill, New York.

Chapter 2

Bland, D. R. (1988). Wave Theory and Applications, Clarendon Press, Oxford.

Fletcher, C. A. J. (1991). Computational Techniques for Fluid Dynamics, Vols I and II,

Springer-Verlag, Berlin.

Gresho, P. M. (1991). Incompressible Fluid Dynamics: Some Fundamental

Formulation Issues, Ann. Rev. Fluid Mech., Vol. 23, pp. 413–453.

Issa, R. I. and Lockwood, F. C. (1977). On the Prediction of Two-dimensional

Supersonic Viscous Interactions near Walls, AIAA J., Vol. 15, No. 2, pp. 182–188.

McGuirk, J. J. and Page, G. J. (1990). Shock Capturing Using a Pressure-Correction

Method, AIAA J., Vol. 28, No. 10, pp. 1751–1757.

Schlichting, H. (1979). Boundary-layer Theory, 7th edn, McGraw-Hill, New York.

Shapiro, A. H. (1953). Compressible Fluid Flow, Vol. 1, John Wiley & Sons, New

York.

The Open University (1984). Mathematical Methods and Fluid Mechanics, Course

MST322, The Open University Press, Milton Keynes.

Chapter 3

Abbott, M. B. and Basco, D. R. (1989). Computational Fluid Dynamics – An Introduc-

tion for Engineers, Longman Scientiﬁc & Technical, Harlow.

Anderson, D. A., Tannehill, J. C. and Pletcher, R. H. (1984). Computational Fluid

Mechanics and Heat Transfer, Hemisphere, New York.

Bardina, J., Ferziger, J. H. and Reynolds, W. C. (1980). Improved Subgrid-scale

Models for Large-eddy Simulation, AIAA Paper 80-1357.

Bibliography

ANIN_Z08.qxd 29/12/2006 04:52PM Page 472

BIBLIOGRAPHY 473

Bradshaw, P., Cebeci, T. and Whitelaw, J. H. (1981). Engineering Calculation Methods

for Turbulent Flow, Academic Press, London.

Buchhave, P., George, W. K. Jr and Lumley, J. L. (1979). The Measurement

of Turbulence with the Laser-Doppler Anemometer, Ann. Rev. Fluid Mech.,

Vol. 11, pp. 443–503.

Cebeci, T. (1989). Essential Ingredients of a Method for Low Reynolds-number

Airfoils, AIAA J., Vol. 27, No. 12, pp. 1680–1688.

Champagne, F. H., Pao, Y. H. and Wygnanski, I. J. (1976). On the Two-dimensional

Mixing Region, J. Fluid Mech., Vol. 74, Pt 2, pp. 209–250.

Chen, H.-C. and Patel, V. C. (1988). Near-wall Turbulence Models for Complex

Flows Including Separation, AIAA J., Vol. 26, No. 6, pp. 641–648.

Clark, R. A., Ferziger, J. H. and Reynolds, W. C. (1979). Evaluation of Subgrid-scale

Models Using an Accurately Simulated Turbulent Flow, J. Fluid Mech., Vol. 91,

pp. 1–16.

Comte-Bellot, G. (1976). Hot-wire Anemometry, Ann. Rev. Fluid Mech., Vol. 8,

pp. 209–231.

Craft, T. J., Launder, B. E. and Suga, K. (1996). Development and Application of

a Cubic Eddy Viscosity Model of Turbulence, Int. J. Heat Fluid Flow, Vol. 17,

pp. 108–115.

Deardorff, J. W. (1970). A Numerical Study of Three-dimensional Turbulent

Channel Flow at Large Reynolds Numbers, J. Fluid Mech., Vol. 41, pp. 453–

480.

Deardorff, J. W. (1973). The Use of Subgrid Transport Equations in a Three-

dimensional Model of Atmospheric Turbulence, Trans. ASME, J. Fluids Eng.,

Vol. 95, Ser. I, pp. 429–438.

Demuren, A. O. and Rodi, W. (1984). Calculation of Turbulence-driven Secondary

Motion in Non-circular Ducts, J. Fluid Mech., Vol. 140, pp. 189–222.

Elghobashi, S. and Truesdell, G. C. (1993). Two-way Interaction between

Homogeneous Turbulence and Dispersed Solid Particles. I. Turbulence

Modiﬁcation, Phys. Fluids A, Vol. 5, No. 7, pp. 1790–1801.

Evangelinos, C., Lucor, D. and Karniadakis, G. E. (2000). DNS-derived Force

Distribution on Flexible Cylinders Subject to Vortex-induced Vibration, J. Fluids

Struct., Vol. 14, No. 3, pp. 429–440.

Ferrante, A. and Elgobashi, S. E. (2004). A Robust Method for Generating Inﬂow

Conditions for Direct Simulations of Spatially-developing Turbulent Boundary

Layers, J. Comput. Phys., Vol. 198, pp. 372–387.

Ferziger, J. H. (1977). Large Eddy Numerical Simulations of Turbulent Flows,

AIAA J., Vol. 15, No. 9, pp. 1261–1267.

Fureby, C., Tabor, G., Weller, H. G. and Gosman, A. D. (1997). A Comparative

Study of Subgrid Scale Models in Homogeneous Isotropic Turbulence, Phys.

Fluids, Vol. 9, No. 5, pp. 1416–1429.

Germano, M. (1986). A Proposal for a Redeﬁnition of the Turbulent Stresses in the

Filtered Navier–Stokes Equations, Phys. Fluids, Vol. 29, pp. 2323–2324.

Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991). A Dynamic Subgrid-

scale Eddy Viscosity Model, Phys. Fluids A, Vol. 3, pp. 1760–1765.

Geurts, B. J. and Leonard, A. (2005). Is LES Ready for Complex Flows? Preprint

http://www.newton.cam.ac.uk/preprints/NI99009.pdf.

Ghosal, S. and Moin, P. (1995). The Basic Equations for the Large Eddy Simula-

tion of Turbulent Flows in Complex Geometry, J. Comput. Phys., Vol. 118,

pp. 24–37.

Gutmark, E. and Wygnanski, I. (1976). The Planar Turbulent Jet, J. Fluid Mech.,

Vol. 73, Pt 3, pp. 465–495.

ANIN_Z08.qxd 29/12/2006 04:52PM Page 473

Hanjaliz, K. (2004). Closure Models for Incompressible Turbulent Flows, Von Karman

Institute Lecture Series Turbulence 2004, Von Karman Institute, Rhode-Saint

Genese, Belgium, http://www.vki.ac.be/educat/lect-ser/2004/turbulence2004/

hanjalic.pdf.

Hoarau, Y., Faghani, D., Braza, M., Perrin, R., Anne-Archard, D. and Ruiz, D.

(2003). Direct Numerical Simulation of the Three-dimensional Transition to

Turbulence in the Incompressible Flow around a Wing, Flow, Turbul. Combust.,

Vol. 71, No. 1–4, pp. 119–132.

Horiuti, K. (1990). Higher-order Terms in the Anisotropic Representation of

Reynolds Stresses, Phys. Fluids A, Vol. 2, No. 10, pp. 1708–1710.

Jiang, X. and Luo, K. H. (2000). Spatial Direct Numerical Simulation of the Large

Vortical Structures in Forced Plumes, Flow, Turbul. Combust., Vol. 64, No. 1,

pp. 43–69.

Jones, W. P. and Whitelaw, J. H. (1982). Calculation of Turbulent Reacting Flows:

A Review, Combust. Flame, Vol. 48, pp. 1–26.

Karniadakis, G. E. (1989). Spectral Element Simulations of Laminar and Turbulent

Flows in Complex Geometries, Appl. Num. Math., Vol. 6, No. 1–2, pp. 85–105.

Karniadakis, G. E. (1990). Spectral Element-Fourier Methods for Incompressible

Turbulent Flows, Comput. Methods Appl. Mech. Eng., Vol. 80, No. 1–3, pp. 367–

380.

Kasagi, N. (1998). Progress in Direct Numerical Simulation of Turbulent Transport

and its Control, Int. J. Heat Fluid Flow, Vol. 19, pp. 125–134.

Klebanoff, P. S. (1955). Characteristics of Turbulence in a Boundary Layer with Zero

Pressure Gradient, NACA Report 1247, National Bureau of Standards,

Washington, DC.

Klein, M., Sadiki, A. and Janicka, J. (2003). A Digital Filter Based Generation

of Inﬂow Data for Spatially Developing Direct Numerical or Large Eddy

Simulations, J. Comput. Phys., Vol. 186, pp. 652–665.

Kleiser, L. and Zang, T. A. (1991). Numerical Simulation of Transition in

Wall-bounded Shear Flows, Ann. Rev. Fluid Mech., Vol. 23, pp. 495–537.

Lam, C. K. G. and Bremhorst, K. A. (1981). Modiﬁed Form of the k–

Model for

Predicting Wall Turbulence, J. Fluids Eng., Vol. 103, pp. 456–460.

Laufer, J. (1952). The Structure of Turbulence in Fully Developed Pipe Flow, NACA

Report 1174, National Bureau of Standards, Washington, DC.

Launder, B. E. (1989). Second-moment Closures: Present and Future?, Int. J. Heat

Fluid Flow, Vol. 10, pp. 282–300.

Launder, B. E. and Sharma, B. I. (1974). Application of the Energy-Dissipation

Model of Turbulence to the Calculation of Flow near a Spinning Disc, Lett. Heat

Mass Transfer, Vol. 1, pp. 131–137.

Launder, B. E. and Spalding, D. B. (1974). The Numerical Computation of

Turbulent Flows, Comput. Methods Appl. Mech. Eng., Vol. 3, pp. 269–289.

Launder, B. E., Reece, G. J. and Rodi, W. (1975). Progress in the Development of

a Reynolds-stress Turbulence Closure, J. Fluid Mech., Vol. 68, Pt 3, pp. 537–

566.

Lele, S. K. (1997). Computational Aeroacoustics: A Review, AIAA Paper 97-0018.

Leonard, A. (1974). Energy Cascade in Large-eddy Simulations of Turbulent Fluid

Flows, Adv. Geophys., Vol. 18A, pp. 237–248.

Lesieur, M. and Métais, O. (1996). New Trends in Large-eddy Simulations of

Turbulence, Ann. Rev. Fluid Mech., Vol. 28, pp. 45–82.

Leslie, D. C. and Quarini, G. L. (1979). The Application of Turbulence Theory

to the Formulation of Subgrid Modelling Procedures, J. Fluid Mech., Vol. 91,

pp. 65–91.

474 BIBLIOGRAPHY

ANIN_Z08.qxd 29/12/2006 04:52PM Page 474

BIBLIOGRAPHY 475

Lilly, D. K. (1966). On the Application of the Eddy Viscosity Concept in the Inertial

Sub-range of Turbulence, NCAR Report No. 123.

Lilly, D. K. (1967). The Representation of Small-scale Turbulence in Numerical

Simulation Experiments, Proceedings of the IBM Scientiﬁc Computing Symposium

on Environmental Science, p. 195.

Lilly, D. K. (1992). A Proposed Modiﬁcation of the Germano Subgrid-scale Closure

Method, Phys. Fluids A, Vol. 4, pp. 633–635.

Lumley, J. L. (1970). Toward a Turbulent Constitutive Equation, J. Fluid Mech.,

Vol. 41, pp. 413–434.

Lumley, J. L. (1978). Computational Modelling of Turbulent Flows, Adv. Appl.

Mech., Vol. 18, pp. 123–176.

Lumley, J. L. (1989). Whither Turbulence? Turbulence at the Crossroads, Lecture Notes

in Physics No. 357, Springer-Verlag, Berlin.

Lund, T. S., Wu, X. and Squires, K. D. (1998). Generation of Turbulent Inﬂow

Data for Spatially-developing Boundary Layer Simulations, J. Comput. Phys.,

Vol. 140, pp. 233–258.

Luo, K. H., Bellan, J., Delichatsios, M. and Nathan, G. S. (2005). Axis Switching

in Turbulent Buoyant Diffusion Flames, Proc. Combust. Inst., Vol. 30, No. 1,

pp. 603–610.

Marsden, A. L., Vasilyev, O. V. and Moin, P. (2002). Construction of Commutative

Filters for LES on Unstructured Meshes, J. Comput. Phys., Vol. 175, pp. 584–

603.

McMillan, O. J. and Ferziger, J. H. (1979). Direct Testing of Subgrid-scale Models,

AIAA J., Vol. 17, pp. 1340–1346.

Meneveau, C. and Katz, J. (2000). Scale-invariance and Turbulence Models for

Large-eddy Simulation, Ann. Rev. Fluid Mech., Vol. 32, pp. 1–32.

Menter, F. R. (1992a). Performance of Popular Turbulence Models for Attached and

Separated Adverse Pressure Gradient Flow, AIAA J., Vol. 30, pp. 2066–2072.

Menter, F. R. (1992b). Improved Two-equation k–

Turbulence Models for

Aerodynamic Flows, NASA Technical Memorandum TM-103975, NASA

Ames, CA.

Menter, F. (1994). Two-equation Eddy-viscosity Turbulence Model for Engineering

Applications, AIAA J., Vol. 32, pp. 1598–1605.

Menter, F. (1997). Eddy-viscosity Transport Equations and their Relation to the k–

Model, Trans. ASME, J. Fluids Eng., Vol. 119, pp. 876–884.

Menter, F. R., Kuntz, M. and Langtry, R. (2003). Ten Years of Industrial

Experience with the SST Turbulence Model, Proceedings of the Fourth

International Symposium on Turbulence, Heat and Mass Transfer, Begell House,

Redding, CT.

Moin, P. (1991). Towards Large Eddy and Direct Simulation of Complex Turbulent

Flows, Comput. Methods Appl. Mech. Eng., Vol. 87, No. 2–3, pp. 329–334.

Moin, P. (2002). Advances in Large Eddy Simulation Methodology for Complex

Flows, Int. J. Heat Fluid Flow, Vol. 23, pp. 710–720.

Moin, P. and Kim, J. (1997). Tackling Turbulence with Supercomputers, Sci. Am.,

Vol. 276, No. 1, pp. 62–68.

Moin, P. and Mahesh, K. (1998). Direct Numerical Simulation: A Tool in

Turbulence Research, Ann. Rev. Fluid Mech., Vol. 30, pp. 539–578.

Monin, A. S. and Yaglom, A. M. (1971). Statistical Fluid Mechanics: Mechanics of

Turbulence, Vol. 1, MIT Press, Cambridge, MA.

Nakayama, Y. (ed.) (1988). Visualised Flow, Pergamon Press, Oxford.

Naot, D. and Rodi, W. (1982). Numerical Simulation of Secondary Currents in

Channel Flow, J. Hydraul. Div. ASCE, Vol. 108 (HY8), pp. 948–968.

ANIN_Z08.qxd 29/12/2006 04:52PM Page 475

Orlandi, P. and Fatica, M. (1997). Direct Simulations of Turbulent Flow in a Pipe

Rotating about its Axis, J. Fluid Mech., Vol. 343, pp. 43–72.

Orszag, S. A. and Patera, A. T. (1984). A Spectral Element Method for Fluid

Dynamics: Laminar Flow in a Channel Expansion, J. Comput. Phys., Vol. 54,

p. 468.

Orszag, S. A. and Patterson, G. S. (1972). Numerical Simulation of Three-dimensional

Homogeneous Isotropic Turbulence, Phys. Rev. Lett., Vol. 28, pp. 76–79.

Patel, V. C., Rodi, W. and Scheuerer, G. (1985). Turbulence Models for Near-

wall and Low Reynolds Number Flows: A Review, AIAA J., Vol. 23, No. 9,

pp. 1308–1319.

Patera, A. T. (1986). Advances and Future Directions of Research on Spectral

Methods, Proceedings of Computational Mechanics – Advances and Trends.

Presented at the Winter Annual Meeting of the American Society of Mechanical

Engineers, AMD Vol. 75, pp. 411–427.

Peyret, R. and Krause, E. (eds) (2000). Advanced Turbulent Flow Computations, CISM

Courses and Lectures No. 395, International Centre for Mechanical Sciences,

Springer-Verlag, Vienna.

Poinsot, T. J., Haworth, D. C. and Bruneaux, G. (1993). Direct Simulation and

Modeling of Flame-Wall Interaction for Premixed Turbulent Combustion,

Combust. Flame, Vol. 95, pp. 118–132.

Pope, S. B. (1975). A More General Effective Viscosity Hypothesis, J. Fluid Mech.,

Vol. 72, pp. 331–340.

Rivlin, R. S. (1957). The Relation between the Flow of Non-Newtonian Fluids and

Turbulent Newtonian Fluids, Q. Appl. Math., Vol. 15, pp. 212–215.

Rodi, W. (1980). Turbulence Models and their Application in Hydraulics – A State of the

Art Review, IAHR, Delft, The Netherlands.

Rodi, W. (1991). Experience with Two-layer Models Combining the k–

Model with

a One-equation Model near the Wall, AIAA Paper 91-0216.

Rogallo, R. S. and Moin, P. (1984). Numerical Simulation of Turbulent Flows,

Ann. Rev. Fluid Mech., Vol. 16, pp. 99–137.

Rogers, M. M. (2002). The Evolution of Strained Turbulent Plane Wakes, J. Fluid

Mech., Vol. 463, pp. 53–120.

Schlichting, H. (1979). Boundary-layer Theory, 7th edn, McGraw-Hill, New York.

Schumann, U. (1975). Subgrid Scale Model for Finite Difference Simulations

of Turbulent Flows in Plane Channels and Annuli, J. Comput. Phys., Vol. 18,

pp. 376–404.

Scotti, A., Meneveau, C. and Lilly, D. K. (1993). Generalized Smagorinsky Model

for Anisotropic Grids, Phys. Fluids A, Vol. 5, pp. 1229–1248.

Serre, E., Bountoux, P. and Launder, B. E. (2002). Direct Numerical Simulation

of Transitional Turbulent Flow in a Closed Rotor-stator Cavity, Flow, Turbul.

Combust., Vol. 69, No. 1S, pp. 35–50.

Serre, E., Tuliszka-Sznitko, E. and Bountoux, P. (2004). Coupled Numerical and

Theoretical Study of the Flow Transition between a Rotating and a Stationary

Disk, Phys. Fluids, Vol. 16, No. 3, pp. 688–706.

Shih, T.-H., Liou, W. W., Shabbir, A., Yang, Z. and Zhu, J. (1995). A New k–

Eddy-viscosity Model for High Reynolds Number Turbulent Flows – Model

Development and Validation, Comput. Fluids, Vol. 24, No. 3, pp. 227–238.

Smagorinsky, J. (1963). General Circulation Experiments with the Primitive

Equations. I. The Basic Experiment, Mon. Weather Rev., Vol. 91, No. 3,

pp. 99–164.

So, R. M. C., Lai, Y. G., Zhang, H. S. and Hwang, B. C. (1991). Second-order Near-

wall Turbulence Closures: A Review, AIAA J., Vol. 29, No. 11, pp. 1819–1835.

476 BIBLIOGRAPHY

ANIN_Z08.qxd 29/12/2006 04:52PM Page 476