numerical investigations of turbulent free surface flows … · surface flows using level set...
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NUMERICAL INVESTIGATIONS OF TURBULENT FREE
SURFACE FLOWS USING LEVEL SET METHOD
AND LARGE EDDY SIMULATION
By
Wusi Yue, Ching-Long Lin, and Virendra C. Patel
IIHR Technical Report No. 435
IIHR—Hydroscience & Engineering College of Engineering The University of Iowa
Iowa City, Iowa 52242-1585
December 2003
ABSTRACT
A numerical model for large eddy simulation (LES) of free surface flows is developed
by coupling level set method with the filtered incompressible Navier-Stokes equations.
The level set method transforms a free-surface flow problem into an air-water two-phase
flow problem on a fixed grid. The free surface is implicitly captured by the zero level set
function. The incompressible Navier-Stokes equations are discretized on a non-staggered
boundary-fitted grid based on the finite volume method and advanced with a four-step
fractional step method in time. The level set equations are solved with high-order
Essentially Non-Oscillatory schemes. The numerical model is first validated and verified
by some model free-surface flow problems, such as a two-dimensional travelling solitary
wave and two- and three-dimensional broken dams.
Turbulent flows in an open-channel, with a train of two-dimensional fixed dunes
on the bottom wall, are numerically simulated with the free surface being treated in
two different ways, as a fixed undisturbed plane surface, and as a freely deformable
air-water interface. The former case is studied by the single-phase LES with stress-
free boundary condition on the free surface, establishing a baseline for the latter case.
In the latter case, the level set method is coupled with LES to simulate the air-water
interface in a two-phase flow model. The numerical predictions by both cases are in
good agreement with experimental data. Complex flow patterns on the free surface are
revealed in both cases, such as surface-upwellings and downdrafts. Both cases show
that the “quadrant-two” events dominate the production of the Reynolds shear stress,
and streaky structures appear in the wall layer. The secondary peaks in the profiles of
the streamwise component of turbulence intensities, measured in the experiment, are
predicted in the latter case while they are largely absent in the first case. In LES, three
subgrid-scale (SGS) models, namely Smagorinsky, dynamic Smagorinsky, and dynamic
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two-parameter models, are applied in the latter case. The latter two models predict
similar turbulence statistics, while Smagorinsky model shows very dissipative results.
The effects of flow depth on the free surface are investigated by simulating two flow
depths. A prominent phenomenon in shallow-water flow is the absence of near-wall
streaky structures.
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TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Numerical Methods for Free-surface Flows . . . . . . . . . . . 31.2.2 Turbulent Open-channel Flows . . . . . . . . . . . . . . . . . 7
1.3 Research Objectives and Overview of Report . . . . . . . . . . . . . 9
2 MATHEMATICAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Smoothing of Two-phase Flow System . . . . . . . . . . . . . . . . . 132.2 Coupling of Level Set Method with Navier-Stokes Equations . . . . . 152.3 Reinitialization of Level Set Function and Mass Conservation . . . . 172.4 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 NUMERICAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Numerical Methods for Incompressible Navier-Stokes Equations . . . 233.2 Numerical Schemes for Level Set Function . . . . . . . . . . . . . . . 26
3.2.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Reinitialization . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Computation of Surface Tension . . . . . . . . . . . . . . . . . . . . 303.4 Restriction of Time Step . . . . . . . . . . . . . . . . . . . . . . . . 31
4 VERIFICATION AND VALIDATION . . . . . . . . . . . . . . . . . . . . 34
4.1 2D Lid-driven Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . 344.2 Validation of Level Set Method . . . . . . . . . . . . . . . . . . . . . 35
4.2.1 Reinitialization of a Circle . . . . . . . . . . . . . . . . . . . 354.2.2 Zalesak’s Problem . . . . . . . . . . . . . . . . . . . . . . . . 354.2.3 Circular Fluid Element Stretching . . . . . . . . . . . . . . . 36
4.3 Free Surface Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.1 2D Laminar Open-channel Flow . . . . . . . . . . . . . . . . 384.3.2 A Travelling Solitary Wave . . . . . . . . . . . . . . . . . . . 394.3.3 2D Dam-Breaking . . . . . . . . . . . . . . . . . . . . . . . . 414.3.4 3D Dam-Breaking . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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5 LES OF TURBULENT OPEN-CHANNEL FLOW OVER A DUNE MOD-ELLED WITH PLANE FREE SURFACE . . . . . . . . . . . . . . . . . . 69
5.1 Description of the Simulated Flow . . . . . . . . . . . . . . . . . . . 695.2 Time-averaged Results . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.1 Mean Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . 715.2.2 Turbulence Intensities and Reynolds Shear Stress . . . . . . . 735.2.3 Friction and Pressure Coefficients . . . . . . . . . . . . . . . 765.2.4 Higher-order Turbulence Statistics . . . . . . . . . . . . . . . 785.2.5 Quadrant Analysis . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Instantaneous Flow Field . . . . . . . . . . . . . . . . . . . . . . . . 805.3.1 Reattachment . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.2 Flow Structures on Free Surface . . . . . . . . . . . . . . . . 815.3.3 Streaky Structures Near the Wall . . . . . . . . . . . . . . . . 825.3.4 3D Vortical Structures . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 LES-LSM SIMULATION OF TURBULENT OPEN-CHANNEL FLOWOVER A DUNE WITH FREELY DEFORMABLE FREE SURFACE . . 120
6.1 Description of the Simulated Flow . . . . . . . . . . . . . . . . . . . 1206.2 Deep-Water Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.1 Mean Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2.2 Instantaneous Flow Field . . . . . . . . . . . . . . . . . . . . 125
6.3 Shallow-Water Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.3.1 Mean Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . 1276.3.2 Instantaneous Flow Field . . . . . . . . . . . . . . . . . . . . 128
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . 158
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 160
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
APPENDIX
MOVIES IN APPENDED CD . . . . . . . . . . . . . . . . . . . . . . . . 169
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LIST OF TABLES
Table
4.1 Area error after one period for a circular fluid in the time-reversed swirlingdeformation flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Computational cases of solitary waves with different wave amplitude . . . 41
6.1 Comparison of normalized free-surface velocity (< u > /U0) betweendeep- and shallow-water flows . . . . . . . . . . . . . . . . . . . . . . . . 127
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LIST OF FIGURES
Figure
1.1 Schematic of open-channel flow over a dune-bed. . . . . . . . . . . . . . 12
3.1 Schematic of curvature definitions and unit normal on a 2D cell. . . . . 33
4.1 Non-uniform grid for 2D lid-driven cavity flow. . . . . . . . . . . . . . . 46
4.2 Streamlines and velocity vectors of 2D lid-driven cavity flow at steady state. 47
4.3 Comparison of velocity components (u, w) along center lines with bench-mark data. Symbols, Ghia et al., 1982; solid lines, grid of 48×48; dashedlines, grid of 24× 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Residual of pressure vs. iteration numbers. Solid lines, grid of 48 × 48;dashed lines, grid of 24× 24. . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Non-uniform grid for circle reinitialization. . . . . . . . . . . . . . . . . 49
4.6 Level set contours at t = 25. Contour interval ∆φ = 2.0; dashed lines,uniform grid; dotted lines, non-uniform grid. . . . . . . . . . . . . . . . . 49
4.7 Contour of φ = 0 at t = 25. Dotted line, initial (t = 0); dashed line,uniform grid; thin solid line, non-uniform grid. . . . . . . . . . . . . . . 50
4.8 Non-uniform grid for Zalesak’s problem (slotted disk rotation), grid of100× 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.9 Zalesak’s problem (rotation of a slotted disk), grid of 100×100. The diskboundaries by different advection schemes after one revolution (t = 628);∆t = 1 is used in both uniform grid and non-uniform grid. . . . . . . . . 51
4.10 Zalesak’s problem (rotation of a notched disk), grid of 200 × 200, ∆x =∆t = 0.45. Dotted lines, initial position of the slotted disk; solid lines,only the third order ENO evolution scheme is applied; dashed lines, boththe evolution and reinitialization schemes are applied. . . . . . . . . . . . 51
4.11 Stretching of a circular fluid element in a swirling deformation flow. (a)The fluid element at the initial state; (b) the stretched fluid element att = 100; (c) the stretched fluid element at t = 200; (d) the stretched fluidelement at t = 300; in (b) (c) and (d), solid lines, uniform grid of 200×200;dashed lines, uniform grid of 100 × 100; dashdot lines, non-uniform gridof 100× 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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4.12 Reversed circular fluid element after one period T . (a) T = 250; (b)T = 500; dotted lines, initial contour of the circular fluid; solid lines,uniform grid of 200×200; dashed lines, uniform grid of 100×100; dashdotlines, non-uniform grid of 100× 100. . . . . . . . . . . . . . . . . . . . . 53
4.13 Schematic of 2D open-channel flow. . . . . . . . . . . . . . . . . . . . . . 54
4.14 Computational velocity vector field. Grid of 40× 40. . . . . . . . . . . . 54
4.15 Comparison of velocity profiles in water and air regions. Solid lines, ana-lytical solutions; dashed lines, grid of 40× 80; dotted lines, grid of 40× 40. 55
4.16 Shear stress profiles in water and air. Solid line, grid of 40 × 80; dashedline, grid of 40× 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.17 Illustration of the formation, travelling, and run-up of a solitary wave inan enclosed canal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.18 Travelling of a solitary wave. (a) Travelling waves; (b) typical velocity field. 57
4.19 Damping rate of solitary waves. . . . . . . . . . . . . . . . . . . . . . . . 58
4.20 Wave run-up versus incident wave amplitude. . . . . . . . . . . . . . . . 58
4.21 Schematic of 2D dam-breaking. . . . . . . . . . . . . . . . . . . . . . . . 59
4.22 Non-uniform grid for 2D dam-breaking. . . . . . . . . . . . . . . . . . . . 59
4.23 Two-dimensional broken dam. (a) Surge front position s versus non-dimensional time; (b) remaining water column height h versus non-dimensionaltime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.24 Free-surface position- (left pictures) and velocity vectors (right pictures)at selected times by the uniform grid; the shadow areas in the left figuresrepresent the water, the lines in the right figures represent the free-surfacepositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.25 Momentum components in the computational domain at selected times.(a) x-momentum at T = 2.6; (b) z-momentum at T = 2.6; (c) x-momentumat T = 8.0; (d) z-momentum at T = 8.0; black lines stand for the free-surface positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.26 Time history of mass errors in 2D and 3D broken-dam. . . . . . . . . . . 63
4.27 Effects of the thickness of interface on the spreading velocity of surgefronts, where ε is the half thickness of interface. . . . . . . . . . . . . . . 64
4.28 Schematic of 3D broken-dam. . . . . . . . . . . . . . . . . . . . . . . . . 64
4.29 Three-dimensional broken-dam. Time history of (a) Surge front positions;(b) remaining water column height h. . . . . . . . . . . . . . . . . . . . . 65
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4.30 Free-surface positions (shallow areas in the left pictures) and velocityvectors at the center plane of the container (right picture, lines representthe free-surface positions) at selected times for the periodic boundary casewithout surface tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.31 Free-surface positions (shallow areas in the left pictures) and velocityvectors at the center plane of the container (right picture, lines representthe free-surface positions) at selected times for the wall boundary casewithout surface tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.32 Close-up views of surge fronts and rear views of entrained air bubbles inFig. 4.30g and 4.31g, respectively. (a) and (c) for periodic boundaries(Fig. 4.30g); (b) and (d) for wall boundaries (Fig. 4.31g). . . . . . . . . . 68
5.1 Open-channel flow over a fixed 2D dune with undisturbed plane surface. 86
5.2 Grid cross-section for open-channel flow over a fixed 2D dune. . . . . . . 86
5.3 Time-averaged field. (a) Velocity vectors and streamlines; (b) pressurecontours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4 Free-surface position implied by the calculated pressure distribution. . . . 88
5.5 Time-averaged profiles at selected streamwise stations. (a) x componentof velocity; (b) y component of velocity; (c) z component of velocity; (d)pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.6 Comparison of u profiles among LES, RANS (Ryu 2003), and LDV exper-iment (Balachandar et al. 2003) at selected streamwise stations: squares,experiment; solid lines, present LES; dashed lines, RANS. . . . . . . . . 89
5.7 Comparison of w profiles among LES and RANS at selected streamwisestations: solid lines, present LES; dashed lines, RANS. . . . . . . . . . . 90
5.8 Logarithmic velocity profiles in wall units at selected streamwise stations. 91
5.9 Comparison of rms of u′ among LES and LDV experiment at selectedstreamwise stations, solid lines, present LES; squares, experiment. . . . 92
5.10 Comparison of rms of w′ among LES and LDV experiment at selectedstreamwise stations, solid lines, present LES; squares, experiment. . . . 93
5.11 Comparison of Reynolds shear stress −u′w′ among LES, RANS and LDVexperiment at selected streamwise stations. Solid lines, present LES;squares, experiment; dashed lines, RANS. . . . . . . . . . . . . . . . . . 94
5.12 Turbulence intensity contours. (a) < u′ >rms; (b) < w′ >rms; (c) <v′ >rms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.13 Time-averaged contours. (a) TKE; (b) < −u′w′ >. . . . . . . . . . . . . 96
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5.14 Profiles at selected streamwise stations. (a) rms of u′; (b) rms of v′; (c)rms of w′; (d) Reynolds shear stress −u′w′; (f) Reynolds shear stress−u′v′; (g) TKE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.15 Schematic of (a) wall shear stress at dune bed; (b) local unit normal ofdune bed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.16 Coefficients of friction and pressure. Solid lines, present LES; dashedlines, RANS; •, LDV experiment. (a) Cf ; (b) Cp . . . . . . . . . . . . . . 99
5.17 RMS of coefficients of friction and pressure. . . . . . . . . . . . . . . . . 100
5.18 Contour of rms of pressure fluctuation. . . . . . . . . . . . . . . . . . . 100
5.19 Skewness of u, v and w at selected streamwise stations, where label “o”represents zero crossing point of Su. . . . . . . . . . . . . . . . . . . . . 101
5.20 Flatness of u, v and w at selected streamwise stations. . . . . . . . . . . 102
5.21 Schematic of quadrant analysis. (a) quadrant-correspondent motions; (b)“Hole” region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.22 Fractional contribution to the Reynolds shear stress −u′w′ at selectedstreamwise stations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.23 Fractional contribution to Reynolds shear stress −u′w′ by sorting out thehole H = 2 at selected streamwise stations. . . . . . . . . . . . . . . . . 105
5.24 Absolute contribution to Reynolds shear stress −u′w′ at selected stream-wise stations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.25 Instantaneous reattachment position within a time window. . . . . . . . 107
5.26 Instantaneous streamlines behind the dune within a time window. Rrepresents reattachment point. . . . . . . . . . . . . . . . . . . . . . . . . 108
5.27 A time sequence of instantaneous velocity fluctuation fields of u′ and w′
in the middle spanwise planes, where u′ and w′ are normalized by U0. . 109
5.28 Instantaneous velocity fluctuation fields of v′ and w′ in selected streamwiseplanes, where v′ and w′ are normalized by U0. . . . . . . . . . . . . . . . 110
5.29 Top view of free surface. Vector of u − U0 and v; light gray, positive w;dark gray, negative w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.30 Top view of free surface. Streamlines of u− U0 and v. . . . . . . . . . . 112
5.31 Contours of u′ on surface z+b = 9 in a time sequence, where z+
b is thevertical distance from the dune bed in wall units. . . . . . . . . . . . . . 113
5.32 Contours of u′ on surface z+b = 40 in a time sequence, where z+
b is thevertical distance from the dune bed in wall units. . . . . . . . . . . . . . 114
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5.33 Contours of w′ on surface z+b = 9 in a time sequence, where z+
b is thevertical distance from the dune bed in wall units. . . . . . . . . . . . . . 115
5.34 Contours of w′ on surface z+b = 40 in a time sequence, where z+
b is thevertical distance from the dune bed in wall units. . . . . . . . . . . . . . 116
5.35 Isosurface of vorticity components, dark gray, negative value; light gray,positive value. (a) Ωx; (b) Ωy; (c) Ωz. . . . . . . . . . . . . . . . . . . . 117
5.36 Isosurface of λ2 = −200 in a time sequence. . . . . . . . . . . . . . . . . 118
5.37 Closeup view of Fig. 5.36 sliced by streamwise planes. Vectors of v and win the sliced planes. (a) t0, slice at x/λ = 0.225; (b) t0 + 0.25L/uτ , sliceat x/λ = 0.51; (c) t0 + 0.5L/uτ , slice at x/λ = 0.425. . . . . . . . . . . . 119
6.1 Streamlines in the middle plane (dashed lines represent free surfaces). (a)DSM; (b) DTM; (c) SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Free-surface elevation in the middle plane. . . . . . . . . . . . . . . . . . 132
6.3 Coefficients of friction and pressure. (a) Cf ; (b) Cp . . . . . . . . . . . . 133
6.4 Comparison of mean u profiles between different SGS models and exper-iment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.5 Comparison of mean w profiles. . . . . . . . . . . . . . . . . . . . . . . . 135
6.6 Comparison of rms of u′ profiles between different SGS models and ex-periment. Dashed lines represent local free-surface positions for DSM. . 136
6.7 Comparison of rms of v′ profiles. Dashed lines represent local free-surfacepositions for DSM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.8 Comparison of rms of w′ profiles between different SGS models and ex-periment. Dashed lines represent local free-surface positions for DSM. . 138
6.9 Comparison of the Reynolds shear stress −u′w′ between different SGSmodels and experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.10 Time-averaged TKE contours. Dashed lines represent free surface. . . . 140
6.11 Absolute quadrant contribution to the Reynolds shear stress −u′w′ at fourselected streamwise stations. Vertical dotted lines represent free-surfacepositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.12 A time sequence of instantaneous velocity fluctuation fields of u′ and w′
in the middle spanwise planes, where u′ and w′ are normalized by U0.Dashed lines represent free surfaces; label A represents stagnation pointor line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.13 Contours of u′ on surface z+b = 9 in a time sequence, where z+
b is thevertical distance from the dune bed in wall units. . . . . . . . . . . . . . 143
x
6.14 Contours of w′ on surface z+b = 9 in a time sequence. . . . . . . . . . . . 144
6.15 Contours of v′ on surface z+b = 9 in a time sequence. . . . . . . . . . . . 145
6.16 Isosurface of λ2 = −200 in a time sequence. Shadow areas represent freesurface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.17 Free-surface evolution in deep-water flow. D represents downdraft; Urepresents upwelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.18 Free-surface elevation in the middle plane. . . . . . . . . . . . . . . . . . 148
6.19 Comparison of mean u profiles between shallow- and deep-water flows.Dotted lines represent free-surface positions; • represents the grid cellimmediately below the surface transition zone. . . . . . . . . . . . . . . . 149
6.20 Comparison of rms of u′ profiles between shallow- and deep-water flows.Dotted lines represent free-surface positions. . . . . . . . . . . . . . . . . 150
6.21 Comparison of rms of w′ profiles between shallow- and deep-water flows.Dotted lines represent free-surface positions. . . . . . . . . . . . . . . . . 151
6.22 Absolute contribution to the Reynolds shear stress −u′w′ by four quad-rants at four selected streamwise stations. . . . . . . . . . . . . . . . . . 152
6.23 A time sequence of instantaneous velocity fluctuation fields of u′ and w′
in the middle spanwise planes, where u′ and w′ are normalized by U0
(dashed lines represent free surfaces). . . . . . . . . . . . . . . . . . . . 153
6.24 Contours of u′ on surface z+b = 11 in a time sequence, where z+
b is thevertical distance from the dune bed in wall units. . . . . . . . . . . . . . 154
6.25 Contours of w′ on surface z+b = 11 in a time sequence. . . . . . . . . . . 155
6.26 Isosurface of λ2 = −100 in a time sequence. . . . . . . . . . . . . . . . . 156
6.27 Free-surfaces evolution in shallow-water flow. D represents downdraft; Urepresents upwelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
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NOMENCLATURE
Alphabetical Symbols
Amax amplitude of solitary wave
C coefficient of SGS model
Cf friction coefficient
Cg wave speed
Cp pressure coefficient
D material derivative operator
Dd diagonal diffusion operator
d normal distance to free surface
det determinant
Fr Froude number
Fu, Fv, Fw Flatness of u, v and w
G(x, x′) kernal filter function
gi gravity acceleration component
H(φ) Heaviside function
HQ threshold value of conditional sampling
h dune height
J Jacobian
L water depth
ly spanwise length of dune
xii
Nx, Nz normal components to free surface
n unit normal vector to free surface
nx, nz unit normal components to free surface
O(·) of the order of
p pressure
pref reference pressure
Qi event fraction in quadrant i (1, 2, 3 and 4)
Re Reynolds number
sign(a) sign of variable a
Sij (resolved) strain rate tensor
Su, Sv, Sw Skewness of u, v and w
Tmn mesh skewness tensor
t time
U0 velocity at free surface (time-averaged in turbulent flow)
Um volume flux components, m = 1, 2, 3
U , V , W volume flux components
ui velocity components, i = 1, 2, 3
uτ friction velocity
u, v, w velocity components
x1, x2, x3 Cartesian coordinates
x, y, z Cartesian coordinates
zb local vertical distance from dune bed
z+
b = zbuτ
ν, zb in wall units
xiii
Greek Symbols
∆ grid filter width
∆t time step
δ differencing operator, variation
δ(φ) delta function
Γ spatial position of free surface
κ curvature
λ dune wave length
λ2 the second largest eigenvalue
σ coefficient of surface tension
µ dynamic molecular viscosity
ν kinematic molecular viscosity
νt eddy viscosity
Ωx, Ωy, Ωz components of vorticity
φ level set function
ψ pressure-like function
ρ density
ε half thickness of interface
ξm curvilinear coordinates, m = 1, 2, 3
ξ, η, ζ curvilinear coordinates
Superscripts
∗ intermediate quantity
−1 inverse
xiv
n, n+ 1 time step
′ fluctuation quantity
Subscripts
a air
g gas
i, j, k indices in the x, y, z directions
l liquid
m, n indices in the computational domain
w water
Other Symbols
∂ partial derivative operator
∇ gradient operaror
(·) grid-scale filtering
˜(·) test-scale filtering
|(·)| absolute value
< · > time-average
< · >rms root-mean-square
Abbreviations
2D Two-dimensional
3D Three-dimensional
DNS direct numerical simulation
DSM dynamic Smagorinsky model
xv
DTM dynamic two-parameter model
ENO Essentially Non-Oscillatory
FSM fractional step method
LDV Laser Doppler Velocimetry
LES large eddy simulation
LSM level set method
MAC marker and cell
PDE partial differential equation
PIV Particle Image Velocimetry
PLIC piecewise linear interface calculation
QUICK Quadratic Upwind Interpolation for Convective Kinematics
RANS Reynolds-Averaged Navier-Stokes equations
RMS, rms root mean square
SM Smagorinsky model
SGS Subgrid-scale
TDMA tridiagonal-matrix algorithm
TKE turbulent kinetic energy
TVD total variation diminishing
VOF Volume-of-Fluid
xvi
1
CHAPTER 1
INTRODUCTION
1.1 Background
Free-surface motions arise in many engineering applications and geophysical flows,
such as cavitation, boiling, crystal growth, bubbly flows, ship hydrodynamics, and the
motions of rivers, lakes, and seas, to name a few. There are numerous theoretical,
experimental and numerical studies of free surface flows, dating back to Young and
Laplace’s work in the early 1800s in which the interface between two fluids is represented
as a surface of zero thickness endowed with surface tension (Anderson et al. 1998). The
Young-Laplace equation relates pressure jump across an interface with surface tension
and curvature. Quantitative description of physical processes on free surfaces remains
far from complete due to the complexities of deforming and breaking surfaces, nonlinear
effects, multiple time and length scales, mass and heat transfer, surface wave interactions
with underlying flows, free surface turbulence, and other such phenomena.
Limitation in theoretical and experimental investigations has made computational
fluid dynamics (CFD) the major means of modelling free-surface motions and exploring
free surface physical processes. The nature of grid used in free-surface simulation divides
the numerical methods into moving-grid or fixed-grid methods. In the moving-grid
method, the free surface is identified as the boundary of a surface-fitted grid. As its
name suggests, the fixed-grid method employs a predefined fixed-grid for computation,
and the free surface is identified by a surface-fitted grid or the value of a scalar function.
The fixed-grid methods are relatively simple and flexible for implementation, without
complicated bookkeeping techniques involved in the moving-grid methods. A detailed
literature review of the numerical methods is presented in the next section. This study
aims to develop a fixed-grid numerical model to simulate free-surface flows.
2
Turbulence interaction with a free surface is present in many environmental and
engineering applications. The transport and dispersion of a passive scalar, such as
salinity and heat, in a natural water body are mainly governed by turbulence. Ship
resistance is partly determined by the interaction of ship boundary layer turbulence and
surface waves. Of special interest in global warming is the mass transfer rate of carbon
dioxide between atmosphere and ocean (Komori et al. 1993), mainly controlled by
free-surface turbulence. Statistical, structural, and dynamic properties of free-surface
turbulence in an open-channel with a flat bed have been the subject of a number of
investigations by experiment (Nakagawa and Nezu 1981; Nezu and Rodi 1986; Rashidi
1997), direct numerical simulation (DNS) (Fulgosi et al. 2003; Borue et al. 1995;
Handler et al. 1993), and large eddy simulation (LES) (Shi et al. 2001; Shen and Yue
2001). These results have revealed important features of free-surface turbulence. For
instance, turbulence energy is redistributed from the vertical direction to the horizontal
plane at the free surface. Large-scale eddies are generated by bursting phenomena in
the wall region and ascend to the free surface and appear as surface-renewal eddies.
Our understanding of turbulence structures in free-surface flows, however, is still very
limited. This is especially so for flows over complex objects and geometries. Most
numerical simulations consider the free surface to be plane slip-wall or a rigid predefined-
shape boundary. This restricts our understanding of true free-surface turbulence, since
the deformation of the free surface determines the nature of free-surface turbulence.
Free-surface motions, such as surges and waves, boiling, and ship wakes, are unsteady
and highly deformed, preventing predefinition of the free surface. To reveal turbulence
statistics and flow structures in highly deformed free-surface flows, accurate capture and
simulation of the instantaneous free surface are necessary. This is the main topic of the
present report.
3
1.2 Literature Review
1.2.1 Numerical Methods for Free-surface Flows
Unsteady flows with free surfaces receive special attention in CFD due to the
challenge of dealing with moving boundaries. A variety of numerical methods have
been developed over the past four decades. They can be classified into moving-grid and
fixed-grid methods. The moving-grid method is basically a Lagrangian method that
treats the free surface as the boundary of a moving surface-fitted grid, structured or
unstructured. It includes strictly Lagrangian methods, free Lagrangian methods, and
mixed Lagrangian-Eulerian methods, etc. (Floryan and Rasmussen 1989).
The free surface remains sharp and is computed precisely in the moving-grid
method. Strictly Lagrangian methods are restricted to well-defined simple surface topol-
ogy and small surface steepness, such as small amplitude water waves and slightly de-
formed air bubbles. To mitigate the difficulty in grid distortion, free Lagrangian meth-
ods (Crowley 1971) allow grids, usually unstructured triangular elements, to switch their
neighbors and reconnect with them under certain conditions. Grid points can be added
and deleted as necessary. This method has been applied in droplet oscillation and droplet
breakup in a shear layer. When grids are highly distorted due to a strongly deformed
free surface, rezoning (or remeshing) is inevitable. Methods using rezoning are referred
to as arbitrary Lagrangian-Eulerian (ALE) methods (Hirt et al. 1987). Flow informa-
tion for the new grid is transferred from the old grid. Excessive numerical diffusion may
be induced by frequent rezoning. The ALE method has been improved and modified by
a number of researchers, making it still an attractive method today.
The fixed-grid method can be further categorized into surface-tracking and surface-
capturing methods. Both use a fixed stationary grid covering liquid and gas regions. In
the surface-tracking method, the free surface is explicitly identified and tracked by pre-
defined markers or interface-fitted grid cells. In the surface-capturing method, the free
surface is implicitly captured by contours of a certain scalar function. In the surface-
tracking method, the governing equations are usually solved only for the liquid and the
4
free surface grid cells. In the surface-capturing method, the equations are solved on both
the liquid and gas regions, i.e., in the frame work of two-phase flows.
The surface-tracking method has variants including front-tracking methods and
marker methods. Front-tracking methods (Glimm et al. 1987; Unverdi and Tryggvason
1992) represent the interface by a connected set of points. An additional unstructured
grid is constructed in the vicinity of the interface to explicitly evolve the interface. A
special numerical algorithm handles the interaction between the unstructured interface
grid and the original fixed Eulerian grid. Restructuring the interface grid must be
performed dynamically during computation. Points are added or subtracted depending
on whether the grid cells are stretched or compressed. This complicates the algorithm
for design and implementation, especially for three-dimensional (3D) cases. Unverdi
and Tryggvason addressed the complication of front tracking, incorporating features of
surface-capturing into it. In their version of the front tracking method, the interface
is still explicitly tracked by the unstructured interface grid but with a finite thickness
in the order of the grid cell size. Fluid properties such as density and viscosity are
smoothed within the transition zone with an indicator function solved by a Poisson
equation. An advantage over the original front tracking methods is that the interaction
between irregular interface and stationary grids is automatically treated. This method
was successfully applied to dendrite solidification, breakup of 3D bubbles and jets, 3D
film boiling, etc. (Tryggvason et al. 2001).
Marker methods include the Marker-And-Cell (MAC) method (Harlow and Welch
1965) and the Volume-of-Fluid (VOF)-family of methods tracking the free surface with
volume markers. MAC-class markers are Lagrangian massless particles moving with the
local fluid velocity to update the free surface front. VOF-family methods employ an aux-
iliary function, volume fraction or color function, as the volume marker. This auxiliary
function is advected with the local velocity field to simulate free surface propagation.
One important procedure in the VOF algorithms is that the surface must be recon-
structed in terms of the volume fraction. The choice of different reconstructed interface
5
geometries distinguishes VOF-family members. The SLIC (simple line interface calcula-
tion) method (Noh and Woodward 1976), also known as the piecewise constant scheme,
reconstructs the interface cutting through a cell as a straight line parallel to a grid
coordinate direction. The interface propagates in the direction normal to the line. In
Hirt & Nichols’ VOF (1981), also known as SOLA-VOF, the interface within the cell
is again forced to align with a grid coordinate direction but could align with different
coordinate directions in different cells depending on the relative magnitude of interface
normal components. Youngs’ VOF (1982), known as the piecewise linear interface cal-
culation (PLIC) method, uses a more accurate interface reconstruction algorithm. The
reconstructed straight line truncating the cell is given a slope determined by the inter-
face normal, approaching the actual interface. Many new reconstruction schemes are
developed based upon PLIC. Rider & Kothe (1998) presented a PLIC reconstruction
by means of a “geometric toolbox”. The reconstruction algorithm is based on volume
conservation. Total volume (or mass) is conserved in VOF-family methods. The re-
construction procedure is, however, mainly a geometric operation in which non-physical
interface breakup may occur and fluid parcels may be merged non-physically into the
interface. This is called the effect of numerical surface tension (Rider and Kothe 1998).
In spite of the inherent drawbacks, VOF-family methods are widely used due to their
relative convenience for implementation and robustness in application to a wide range
of interface topologies.
Many numerical methods are based on the surface-capturing approach, including
phase field, artificial compressibility, level set methods, and others. Phase field meth-
ods describe the interface from a thermodynamics viewpoint based on Van der Waals
hypothesis that the interface equilibrium state is determined by minimization of fluid
free energy. An order parameter, measure of phases, is defined to account for free en-
ergy density, satisfying the Cahn-Hilliard equation (Jacqmin 1999), the extension of Van
der Waals hypothesis to time dependent situation. A great numerical advantage of the
phase-field methods, compared to other surface-capturing methods, is that the order pa-
rameter can be solved by employing a general numerical scheme because Cahn-Hilliard
6
equation is an ordinary advection-diffusion equation. A volume force, the potential gra-
dient representing surface tension, is introduced into the momentum equations to enforce
the incompressibility constraint. The interface is diffused smeared over several grid cells,
that may be subject to thickening or thinning leading to non-physical interface behavior.
Phase field methods naturally handle the topological change of interfaces and have been
applied in multi-phase flows such as the dentritic growth in solidification, and contact
line dynamics. Application on air-water free surface flows has yet to be demonstrated.
Other than the order parameter, density is chosen as the surface indicator scalar in
the surface-capturing method by Kelecy and Pletcher (1997). They incorporate artificial
compressibility in the Navier-Stokes equations with variable density and viscosity. The
free surface is captured as a density-field discontinuity without adopting other tracking
procedures. Inviscid flux is approximated by a slope-limited high-order MUSCL (Mono-
tone Upstream-Centered Schemes for Conservation Laws) scheme to achieve a high order
non-oscillatory solution. They applied this method to the two-dimensional (2D) and 3D
broken dam problems. In this method, surface tension effects can not be estimated
because it is difficult to compute surface curvature.
In the level set method (LSM) (Osher and Sethian 1988), the level set function
is employed as the surface-capturing function. The original notion of LSM is to define
a smooth (at least Lipschitz continuous) function φ(x, t) (level set) that represents an
interface at φ(x, t) = 0 (zero level set) (Osher and Fedkiw 2001). The zero level set
propagates at interface velocity. The interface can then be captured instantaneously by
locating the zero level set. This alleviates the burden of increasing grid resolution at the
interface that plagues many other numerical approaches.
It is important to avoid steep and flat φ gradients, because it is difficult to com-
pute interface curvature around steep gradients of φ, and hard to maintain uniform
interface thickness if the gradient of φ is too flat. It is desirable to define and maintain
φ as a signed distance function to the interface which possesses uniform unit gradient
and uniform thickness around the interface. This greatly facilitates handling topological
7
merging, breaking, and even self-intersecting of interfaces naturally by taking advan-
tage of smoothness of level set function. Interface information, such as orientation and
curvature, is conveniently obtained by examining the zero level set, and so, surface ten-
sion can be accurately estimated. Surface tension is either diffused over the interface
as a δ-function-like volume force in the momentum equations (Sussman et al. 1994) or
treated exactly as a jump condition and incorporated in the pressure Poisson equation
(Liu et al. 2000). Another advantage of LSM is the straightforward extension from two
to three dimensions. LSM has been applied widely in incompressible fluid mechanics
(Chang et al. 1996; Foster and Fedkiw 2001; Iafrati et al. 2001; Sussman et al. 1994;
Sussman and Smereka 1997), 2D and 3D free-surface flows (Yue et al. 2003), detonation
shock dynamics (Aslam et al. 1996), combustion (Smiljanovski et al. 1997), solidifica-
tion (Kim et al. 2000), crystal growth (Smereka 2000), boiling (Son and Dhir 1998), and
etching and deposition (Sethian and Adalsteinsson 1997), to name just a few.
1.2.2 Turbulent Open-channel Flows
Open-channel flows have been extensively investigated by experiment, theory, and
numerical simulation. Unlike closed-channel flows (without a deformable boundary)
driven by pressure, open-channel flows are driven by gravity. Scaling laws for open-
channel flows involve both Reynolds number and Froude number. Similar to the closed-
channel flows, there also exist two flow regions in open-channel flows: the near-wall
region characterized by the friction velocity, and an outer region controlled by free-
surface velocity and flow depth. Turbulence production dominates in the inner region. A
notable influence of the free surface on turbulence is the damping of the vertical velocity
fluctuation near the surface and enhancement of streamwise and spanwise fluctuations
due to the continuity constraint, analogous to flow impingement close to a solid wall.
Eddy viscosity decreases rapidly near the free surface (Nezu and Rodi 1986).
In turbulent flows, large-scale eddies containing most of the turbulent energy are
normally visualized as coherent structures. In natural river and estuary flows these
structures consist mainly of inner region turbulent bursts and outer region large-scale
8
vortical motions (Nezu and Nakagawa 1993). The former are associated with near-wall
ejection and sweep events caused by periodic lift-up and oscillation of the near-wall low-
speed streaks, which have been experimentally visualized by many investigators (Smith
and Metzler 1983; Komori et al. 1989; Rashidi 1997). Outer region large-scale vortical
structures consist of cyclic pulsation, downward vortices, and strong intermittent upward
tilting quasi-streamwise vortices called “kolk” (Jackson 1977; Nezu and Nakagawa 1993;
Kadota and Nezu 1999).
In rivers and estuaries, the bed usually has periodic irregularities, commonly called
dunes, ripples or sand waves (Yalin 1977). Dunes are nearly triangular in shape, with
a straight steep lee face and a slightly curved stoss face, as sketched in Fig. 1.1. The
flow depth is usually measured from the half dune height to the free surface. But for
convenience, it is measured from the dune crest to the free surface and denoted by L.
Dunes originate from the interaction of the bed with the free surface and turbulence.
They are important in river flows since they influence sediment transport and flow
resistance. Large vortical structures behind dunes have been experimentally observed
by investigators (Muller and Gyr 1986; Nezu and Nakagawa 1989; Bennett and Best
1995; Kadota and Nezu 1999).
In an open channel with a bed of dunes, The flow detaches at the dune crest and
reattaches on the stoss face, forming a recirculation zone behind the crest. A bound-
ary layer develops from the reattachment point and grows toward the crest of the next
dune. A mixing layer forms from the dune crest between the main stream and the re-
circulation zone. Instability of the mixing layer eventually evolves into large vortical
structures (Muller and Gyr 1986), carrying significant momentum and determining sed-
iment transport and bedform movement. Some strong vortical structures reach the free
surface forming a “boil” (Kadota and Nezu 1999). A typical boil grows in time as a
protruding circular or oval patch on the water surface, and gradually widens and falls
until it merges with the surroundings (Nezu and Nakagawa 1993). Sediment may be
brought to the water surface by boils, and dispersed to develop and reform dunes. The
present research does not take into account sediment transport, but focuses on turbulent
9
motions and flow structures that impact mass and momentum transport.
Compared to the number of experimental investigations of turbulent flows over
dunes, there are few numerical simulations. Most of them are 2D RANS (Reynolds-
Averaged Navier-Stokes) simulations (Mendoza and Shen 1990; Johns et al. 1993; Yoon
and Patel 1996), with the free surface treated as a symmetry boundary (or shear-free
slip-wall) allowing use of a fixed grid. These simulations provide mean flow quantities,
such as velocity profiles, the Reynolds shear stress, and bed resistance, but fail to provide
detailed information about the turbulent flow structures. As Lyn (2002) commented,
“The numerical predictions have been, if not disappointing, at least not as impressive as
one might desire. It would be facile to blame the discrepancies on deficient turbulence
models, but can they illuminate our understanding and point to directions to pursue?”.
To better understand flow coherent structures and their interactions with the free surface,
either DNS or LES must be employed. A recent DNS on dune-type wavy bed was carried
out to examine the coherent structures and “boil of the first kind” (Hayashi et al. 2003)
in the flow, but the free surface was still treated as a plane slip-wall. The present work
aims to apply LES to turbulent open-channel flows over a train of 2D dunes and simulate
the free surface as a freely deformable air-water interface of a two-phase flow.
1.3 Research Objectives and Overview of Report
To better understand the physical processes of the free surface, and its interaction
with turbulence, the first step is to model the free surface in a realistic way, allowing
it to freely deform under the proper kinematic and dynamic constraints. In this study,
LSM is used to model free surface motions, capturing the sharp free surface on a fixed-
grid. Properties of the free surface, such as curvature, can be computed by taking
advantage of the smoothness of the level set function. Effects of surface tension can
be easily implemented in the model. LSM treats the free surface naturally without
surface reconstruction, making coupling of LSM with the incompressible Navier-Stokes
equations straightforward.
To study turbulence statistics and coherent structures in turbulent free-surface
10
flows, either DNS or LES must be applied. Turbulence is composed of a wide range
of spatial and temporal scales where large-scale motions are determined by geometric
boundaries, and small-scale motions tend to be universal and isotropic, depending more
on molecular viscosity. RANS models the effects of all scales, making it impossible to
resolve the large scale structures that vary greatly among different flows. DNS attempts
to resolve the energy dissipating scales (Kolmogorov scale) of turbulent motions. Its
computational cost is proportional to Reynolds number cubed (Tennekes and Lumley
1972), making DNS an extremely expensive method. From a practical standpoint, DNS
is not a predictive method. Currently, DNS can only be applied for low Reynolds
number (O(104)) turbulent flows with relatively simple geometry. As an intermediate
technique between RANS and DNS, LES attempts to resolve motions accounting for
bulk of the turbulent kinetic energy everywhere in the flow field. LES models the effect
of small scales with a “subgrid-scale (SGS) model” by applying a filtering procedure to
the governing equations. This provides a 3D, time-dependent solution of the filtered
Navier-Stokes equations, and can be applied at relatively high Reynolds numbers in
complex flows. Its cost is significantly less than that of DNS.
This research has two major objectives. The first is to develop a numerical model
coupling LSM with the filtered incompressible Navier-Stokes equations in a curvilin-
ear coordinates. High-order essentially non-oscillatory (ENO) schemes in curvilinear
coordinates are developed to discretize equations of the level set function, namely the
evolution and reinitialization equations. The coupling of LSM with the incompressible
Navier-Stokes equations is first verified and validated in selected laminar free-surface
flows. The computational results are compared with experimental data and theoretical
predictions.
The second objective is to study turbulent open-channel flows over a train of fixed
2D dune by LES. The free surface is treated in two different ways. First, it is modelled as
a fixed undisturbed plane surface with single-phase LES to establish a baseline. Next, it
is simulated as a freely deformable air-water interface by coupling the level set method
and LES. The numerical predictions of turbulence statistics and flow structures are
11
examined and compared with each other, and with the experimental data (Balachandar
et al. 2002; Balachandar et al. 2003). The effect of flow depth on the free surface and
flow structures are also investigated. Different SGS models are applied and compared.
The outline of this report is as follows. Chapter 2 describes mathematical mod-
elling of a two-phase flow system by coupling LSM and the incompressible Navier-Stokes
equations. A brief introduction to LES is presented. Chapter 3 presents numerical
schemes for solving the incompressible Navier-Stokes equations and level set equations.
A four-step fractional step method (FSM) is used to discretize the Navier-Stokes equa-
tions on a non-staggered grid based on the finite volume method (FVM). High-order
ENO schemes are developed for the evolution and reinitialization equations of the level
set function in a curvilinear coordinate system. Chapter 4 presents validation and veri-
fication of the mathematical models and numerical schemes described in Chapters 2 and
3, respectively. Chapter 5 presents numerical simulation of a pressure-driven turbulent
open-channel flow over a train of fixed 2D dunes with a single-phase LES, with the free
surface treated as a symmetry plane (stress-free plane surface). Chapter 6 describes sim-
ulation of the same open-channel flows with a freely deforming free surface using coupled
LSM and LES. Two flow depths are simulated to study the effects of flow depths on the
free surface and flow structures. Also, three SGS models, namely Smagorinsky, dynamic
Smagorinsky, and dynamic two-parameter models, are applied for comparison. Chapter
7 summarizes the major results and recommendations for future work.
13
CHAPTER 2
MATHEMATICAL MODEL
This chapter describes the mathematical model for free surface flows. The model
couples level set method (LSM) with the incompressible Navier-Stokes equations. For
turbulent flows, the coupling is performed on the filtered Navier-Stokes equations. A
brief introduction to LES is also presented.
2.1 Smoothing of Two-phase Flow System
In the level set method, free-surface flows are modelled as immiscible gas-liquid
two-phase flows. The sharp jumps in density and viscosity at gas-liquid interfaces,
however, can cause numerical instabilities if not treated properly. To minimize this
problem, fluid properties, such as density and viscosity, are smeared over a narrow
transition zone on either side of the free surface. The free surface is identified as the
zero level set, i.e. φ(x, t) = 0, where x = (x, z) in two dimensions or (x, y, z) in three
dimensions.
At a free surface, there are kinematic and dynamic boundary conditions. The
kinematic boundary condition can be interpreted in a Lagrangian way as particles on
the surface always stay on the surface. This can be expressed as the advection of the
level set function,
∂φ
∂t+ u · ∇φ = 0 (2.1)
where u = (u,w) in two dimensions or (u, v, w) in three dimensions is the fluid velocity.
Free surface motion is represented by propagation of the zero level set embedded in the
equation.
The dynamic boundary condition represents the jump in the normal stress at the
free surface balanced by surface tension, known as the Laplace-Young equation,
(Sl − Sg) · n = σκn (2.2)
14
where S = −pI+2µD is the stress tensor, I is the identity matrix, µ is fluid viscosity, D
is the deformation rate tensor, n is the unit normal to the free surface, σ is the surface
tension coefficient, κ is the total curvature of the free surface, and subscripts l and g
represent liquid and gas, respectively.
Similar to approaches employed by Brackbill et al. (1992) and Unverdi & Tryg-
gvason (1992), Eq. (2.2) is implemented in the momentum equations as a volume force.
Surface tension is smoothly distributed over the transition zone that eliminates the jump
in the normal stress at the free surface. The two-phase flow system can then be treated
as a single-fluid system applying the single set of the Navier-Stokes equations to the
whole computational domain.
The level set function φ is initially assigned with a signed distance function,
φ =
−d for x ∈ Ωgas
0 for x ∈ Γ (free surface)
d for x ∈ Ωliquid
(2.3)
where d is the absolute normal distance to the free surface. For immiscible incompressible
fluids, density and viscosity remain constant along the fluid particle trajectories,
Dρ
Dt= 0,
Dµ
Dt= 0 (2.4)
where DDt≡ ∂
∂t+ u · ∇ is the material derivative. Numerical instability may be induced
by direct discretization of Eq. (2.4) due to viscosity and particularly density jumps.
This problem can be minimized by smoothing out density and viscosity in the transition
zone defined as |φ| ≤ ε, where ε, the half thickness of interface, is typically one or two
grid spacing. By defining an infinitely differentiable smoothed Heaviside function H(φ)
(Sussman et al. 1994),
H(φ) =
0 if φ < −ε
12
[1 + φ
ε+ 1
πsin(πφ
ε)]
if |φ| ≤ ε
1 if φ > ε
(2.5)
15
The density and viscosity are smoothed so that they are ρl+ρg
2and µl+µg
2, respectively,
at the free surface (φ = 0),
ρ(φ) = ρg + (ρl − ρg)H(φ)
µ(φ) = µg + (µl − µg)H(φ)(2.6)
Surface tension in Eq. (2.2) is spread over the transition zone as a δ-function-like volume
force in the momentum equations (Chang et al. 1996), which reads
σκ(φ)δ(φ)n (2.7)
where n and κ(φ) can be computed in terms of φ,
n = ∇φ|∇φ|
∣∣∣φ=0
κ(φ) = ∇ · n = ∇ · ∇φ|∇φ|
∣∣∣φ=0
(2.8)
and the delta function δ(φ) is obtained by taking the gradient of the smoothed Heaviside
function
δ(φ) = ∇φH(φ) =
0 if |φ| > ε
12ε
[1 + cos(πφ
ε)]
if |φ| ≤ ε(2.9)
Kinematic and dynamic boundary conditions at the free surface are thus automatically
embedded in the LSM formulation.
2.2 Coupling of Level Set Method with Navier-Stokes Equations
To model immiscible free-surface flows in complex geometries, we consider the in-
compressible Navier-Stokes equations in boundary-fitted curvilinear coordinates. Phys-
ical domain boundaries are accurately represented, and boundary conditions are simply
applied in a transformed computational domain. Numerical fluxes can be conveniently
estimated for non-orthogonal grids, a further advantage of the curvilinear system. To
16
couple with LSM, the incompressible Navier-Stokes equations are modified with variable
density and viscosity. They include a volume force to represent surface tension:
∂Um
∂ξm
= 0 (2.10)
∂(J−1ui)
∂t+
∂(Umui)
∂ξm
= − 1
ρ(φ)
∂
∂ξm
(J−1∂ξm
∂xi
p
)− J−1gi − J−1σκ(φ)δ(φ)∇φ
ρ(φ)
+1
ρ(φ)
∂
∂ξm
(µ(φ)Tmn ∂ui
∂ξn
)
+1
ρ(φ)
∂
∂ξm
(J−1∂ξm
∂xj
∂ξn
∂xi
µ(φ)∂uj
∂ξn
)(2.11)
where ui is the velocity component in Cartesian space, J−1 is the inverse of the Jacobian
defined as
det
(∂xi
∂ξm
)(2.12)
in which xi and ξm are Cartesian and curvilinear coordinates, respectively, det represents
determinant, and gi is the gravitational acceleration component in i-direction. Um is the
volume flux normal to the surface of constant ξm defined as
J−1∂ξm
∂xj
uj (2.13)
Tmn is called the mesh skewness tensor,
J−1∂ξm
∂xj
∂ξn
∂xj
(2.14)
The penultimate on the right of Eq. (2.11) is the primary viscous force. The last term
is the subsidiary viscous force due to viscosity variation and exists only in the transition
zone. Cartesian velocity ui is kept as a dependent variable so that Eq. (2.11) is in
a conservative form (Zang, Street, and Koseff 1994). This facilitates discretization of
equations and eliminates extra source terms from introducing contravariant velocity.
The curvilinear form of the evolution equation of level set function, Eq. (2.1), reads
∂(J−1φ)
∂t+
∂(Umφ)
∂ξm
= 0 (2.15)
Motion of the free surface is embedded in this equation. As level set function information
is needed only within the transition zone, it is not necessary to solve Eq. (2.15) for the
entire domain. It is solved only within the transition zone on either side of the free
17
surface, and is called the “narrow band” approach (Sethian 1999).
In summary, LSM is coupled with the variable fluid property Navier-Stokes equa-
tions by solving Eqs. (2.6), (2.8), (2.11), and (2.15) together.
2.3 Reinitialization of Level Set Function andMass Conservation
A free surface can be mathematically parameterized by a signed distance function
(|∇φ| = 1). Properties of a free surface, such as the unit normal, curvature, etc., can
then be derived from the signed distance function. Though φ is initialized as a signed
distance function from the free surface, Eq. (2.15) does not ensure φ as a signed distance
function as time proceeds. Equation (2.1) can virtually be written as a Hamilton-Jacobi
equation (Crandall and Lions 1983),
∂φ
∂t+ un |∇φ| = 0 (2.16)
where un = u · n.
Equation (2.16) does not preserve the distance function (Gomes and Faugeras
2000). In a complex non-uniform flow field, it is possible for φ to develop steep gradients
from Eq. (2.16), especially when the free surface itself has a steep slope. Consequently,
it is difficult to maintain a finite thickness of the transition zone. Computation of
unit normal and curvature [Eq. (2.8)] is no longer accurate. The surface tension term
becomes a source of numerical instability. For this reason, the Heaviside and delta
functions are not well-behaved. This severely distorts density and viscosity distribution
in the transition zone. Consequently, a significant loss or gain of mass occurs, and
conservation of mass breaks down.
One cure for this problem is to use a reinitialization (or redistancing) procedure for
φ to recover |∇φ| = 1. There are basically two algorithms, the PDE (partial differential
equation)-based approach (Sussman, Smereka, and Osher 1994) and the geometry-based
Fast Marching Method (Sethian 1999). The former solves a nonlinear PDE to a steady
state by an iterative method. The latter solves the Eikonal equation making use of
the efficient Huygens’ principle. Both algorithms successfully achieve redistancing for
18
specific problems. The PDE-based algorithm is employed here.
The signed distance function is obtained by solving for the steady solution of the
PDE∂d∂τ
+ s(d0)(|∇d| − 1) = 0
d0(x, 0) = φ(x, t)(2.17)
where d(x, τ) shares the same zero level set with φ(x, t), τ is an artificial time, and s(d0)
is the smoothed sign function defined as (Peng et al. 1999)
s(d0) =d0√
d02 + (|∇d0| ε)2
(2.18)
where ε is usually one grid spacing.
Equation (2.17) is a nonlinear hyperbolic equation and can be recast as
∂d
∂τ+ F · ∇d = s(d0) (2.19)
where F = s(d0)∇d|∇d| is the characteristic velocity pointing outward from the free surface
so that redistancing starts consistently from the free surface. We apply a narrow band
LSM here and need only to obtain the signed distance function within the transition
zone. Only ε∆τ
iteration steps are needed, where ∆τ is the artificial time step. If ∆τ takes
a quantity equal to one grid spacing, only one or two iterations are required depending
on the transition zone width.
In Eq. (2.17), the free surface captured by the zero level set does not move during
the reinitialization procedure, in theory, because s(0) = 0. However, this is not guaran-
teed in numerical implementation. Mass error may be induced during redistancing. As
a remedy, the volume of fluid in each cell is preserved by adding a volume constraint
in Eq. (2.17). The modified equation becomes (Sussman, Fatemi, Smereka, and Osher
1998)
∂d
∂τ+ s(d0)(|∇d| − 1) = Cδ(d) |∇d| (2.20)
where C =−
∫Ωc
s(d0)(1−|∇d|)δ(d)dΩ∫Ωc
δ2(d)|∇d|dΩand Ωc is the cell volume. After redistancing by solving
Eq. (2.20), φ is re-assigned with d and the next time step starts.
19
2.4 Large Eddy Simulation
LES is used here to simulate 3D turbulent flows. Turbulence has a wide range of
spatial and temporal scales. Large scales of motion are determined mainly by domain
boundaries. Small scales of motion are determined mainly by molecular viscosity and
tend to be more universal and isotropic. LES is based on the fact that the large scales
of motion carry most of the turbulent energy and determine major flow features. LES
seeks to resolve only large scales, but models small scales.
To separate large scales from small scales, LES applies a filtering operation. Flow
variable q is decomposed into a resolved component (large scale) q and a subgrid-scale
component (small scale) q′,
q = q + q′ (2.21)
where q =∫V q(x′)G(x,x′)dx′, G is the kernel filter. An integral of this kind is called
a convolution. Commonly-used filters include the sharp Fourier cutoff, Gaussian, and
tophat filters (Piomelli 1999). A box filter (Zang, Street, and Koseff 1994), i.e. G = 1,
is used in this LES.
The filtered governing equations (2.10) and (2.11) become
∂Um
∂ξm
= 0 (2.22)
∂(J−1ui)
∂t+
∂(Umui)
∂ξm
= − 1
ρ(φ)
∂
∂ξm
(J−1∂ξm
∂xi
p
)− J−1gi − J−1σκ(φ)δ(φ)∇φ
ρ(φ)
+1
ρ(φ)
∂
∂ξm
(µ(φ)Tmn ∂ui
∂ξn
)+
∂
∂ξm
(νt(φ)Tmn ∂ui
∂ξn
)
+1
ρ(φ)
∂
∂ξm
(J−1∂ξm
∂xi
∂ξn
∂xj
µ(φ)∂uj
∂ξn
)(2.23)
In the evolution equation of the level set function, Eq. (2.15), the volume flux Um should
be replaced by the resolved one Um. No modification is necessary for the reinitialization
equation, Eq. (2.20). For open-channel flows, gravitational acceleration components gi
in Eq. (2.23) read
g1 = g sin θ, g2 = 0, g3 = −g cos θ,
20
where θ is the channel inclination angle, and subscripts 1, 2, and 3 correspond to x, y,
and z directions, respectively.
The filtering process produces the terms so-called subgrid-scale (SGS) stress tensor,
τij = uiuj − uiuj, which is usually modelled with an eddy-viscosity model,
τij − δij
3τkk = −2νtSij (2.24)
where νt is the eddy viscosity and Sij is the resolved strain rate tensor.
The role of a SGS model is to remove energy from the resolved scales by generating
an effective eddy viscosity that is proportional to some measure of the turbulent energy
of filtered-out small scales (Piomelli 1999), i.e. νt is computed using information from the
resolved scales. A number of SGS models have been developed, such as Smagorinsky’s
eddy viscosity model, Kraichnan’s spectral eddy viscosity model, the structure-function
model, the scale similarity model, Germano’s dynamic Smagorinsky model (DSM), the
mixed model, etc. Detailed information on these variations can be found in reviews
(Lesieur and Metais 1996; Piomelli 1999). Many SGS models are based on the original
Smagorinsky’s model (SM) (Smagorinsky 1963),
νt = C∆2|S| (2.25)
where |S| =√
2SijSij, ∆ = 3√
∆x1∆x2∆x3, and C is a constant. Values between 0.032
and 0.053 are suggested for isotropic turbulence and change between different flows. A
striking improvement to the Smagorinsky model is DSM with a dynamic procedure to
calculate C (Germano et al. 1991). DSM defines a test filter (denoted by˜), whose width
∆ is larger than the grid width ∆, and assumes the Smagorinsky model to hold on both
grid-filter and test-filter levels with unchanged C. In this study, a DSM based on Lilly’s
modification (Lilly 1992) is employed, computing C as follows (Zang et al. 1993).
C = − LijMij
2∆2MijMij
(2.26)
where
Mij = α2| ˜S| ˜Sij − ˜|S|Sij (2.27)
Lij = −2C∆2Mij (2.28)
21
α =˜∆
∆(2.29)
Typically, ˜∆ is twice ∆, then α = 2. The coefficient C, computed by these procedures,
vanishes as the flow becomes laminar and approaches zero at wall boundaries as a func-
tion of the distance from the wall cubed, providing the correct asymptotic behavior.
DSM is robust to errors in physics due to a built-in mechanism adjusting the dissipa-
tion without substantially modifying the larger scales (Jimenez and Moser 1998). The
dynamic procedures in DSM exert great influence on SGS modelling, altering the SGS
models to match the actual state of the flow, e.g., the Lagrangian dynamic SGS (Meve-
veau et al. 1996), the dynamic localization SGS model (Ghosal et al. 1995), the mixed
model (Zang et al. 1993), and among others.
Another SGS model employed in this research is the dynamic two-parameter mixed
model (DTM) (Salvetti and Banerjee 1994; Salvetti et al. 1997), in which the SGS stress
tensor τij is further decomposed into three terms, namely modified Leonard tensor Lmij ,
modified cross term Cmij , and modified SGS Reynolds tensor Rm
ij .
τij = Lmij + Cm
ij + Rmij (2.30)
where
Lmij = uiuj − ¯ui ¯uj (2.31)
Cmij = uiu′j + u′iuj − ¯uiu′j + u′i ¯uj (2.32)
Rmij = u′iu
′j − u′iu
′j (2.33)
Lmij , representing the resolved part of SGS stress tensor, can be computed directly from
the resolved velocity. Rmij , representing the purely unresolved part of SGS stress tensor,
is modelled with the eddy-viscosity Smagorinsky model according to the same dynamic
procedures in DSM. Cmij is assumed to be proportional to Lm
ij , called scale similarity,
motivated by the observation that both Lmij and Cm
ij stand for the overlaping scales
between resolved and unresolved fields. Thus, the SGS stress tensor is modelled as
τij − δij
3τkk = −2Ct∆
2|S|+ K(Lmij −
δij
3Lm
kk) (2.34)
There are two parameter, Ct and K, to be determined in this SGS model. They are
calculated based on dynamic procedures similar to those in DSM (Salvetti, Zang, Street,
22
and Banerjee 1997).
K =−(Lij − δij
3Lkk)(Hij − δij
3Hkk) + AB
DA2
D− (Hij − δij
3Hkk)2
(2.35)
Ct =AK −B
2∆2D(2.36)
A = (Hij − δij
3Hkk)Mij (2.37)
B = (Lij − δij
3Lkk)Mij (2.38)
D = MijMij (2.39)
Lij = ˜uiuj − ˜ui ˜ui (2.40)
Hij = ˜ui ¯uj − ˜ui˜ui (2.41)
The definition of Mij is the same as in Eq. (2.27). The modified Leonard tensor and scale
similarity terms require addition of the following extra terms on the right of Eq. (2.23):
K∂
∂ξm
[¯Um ¯ui − Umui
](2.42)
It is seen that DTM reduces to DSM when K = 0, and to the dynamic mixed model
(Zang et al. 1993) when K = 1. One improvement of DTM over DSM is that it does
not require alignment of the principal axes of the SGS stress and with those of the
resolved strain rate tensors, an assumption not supported by DNS (Zang et al. 1993).
DTM is also expected to provide more energy backscatter (energy transfer from SGS
to the resolved scale) than DSM due to the addition of the Leonard term. Salvetti et
al. found that DTM improved the agreement with DNS in free surface decay (Salvetti
et al. 1997). SM, DSM, and DTM are applied to the turbulent open-channel flows over
a dune in Chapter 6.
23
CHAPTER 3
NUMERICAL METHODS
In this chapter, the mathematical models presented in Chapter 2 are discretized
for numerical solution. The Navier-Stokes equations are solved by a four step fractional
method (FSM). The evolution and reinitialization equations of the level set function are
solved by high-order ENO schemes.
3.1 Numerical Methods for IncompressibleNavier-Stokes Equations
Numerical solution of the unsteady incompressible Navier-Stokes equations re-
quires special attention because there is no time evolution equation for the pressure.
Constraint of mass conservation is enforced by implicit coupling of the continuity equa-
tion with the pressure. A number of methods address this problem, such as the pseudo-
compressibility method, penalty method, vorticity-streamfunction formulation, the pressure-
correction method, and fractional step method (FSM) or projection method (Shyy and
Mittal 1998). FSM (Chorin 1968) is a popular method and has many variants. The basic
idea of FSM is to decouple pressure from the momentum equations. This is achieved by
introducing an intermediate (non-solenoidal in general) velocity field. The divergence-
free velocity field is obtained by adding pressure gradients to the non-solenoidal field.
FSM is widely used in time-accurate simulation of unsteady flows. There are two issues
with the decoupling process in FSM. One is the boundary conditions for the intermediate
velocity and the other is splitting error (Dukowicz and Dvinsky 1992).
In this study, a finite volume method is employed to solve the incompressible
Navier-Stokes equations (2.11) on a non-staggered grid, in which ui, p, ρ, µ, νt, κ and n
are all defined at cell centers. Um is defined at the centers of cell faces to ensure strong
pressure-velocity coupling and enforce mass conservation in each cell through Eq. (2.10).
A semi-implicit scheme is used for time marching of Eq. (2.11). The Crank-Nicolson
24
scheme advances the diagonal part of the primary viscous terms. All other terms, in-
cluding convective, surface tension, off-diagonal primary viscous, and subsidiary viscous
terms, are marched using the second-order Adams-Bashforth scheme. The resulting
formula is
un+1i − un
i =∆t
J−1
[(3
2En
i −1
2En−1
i ) + Gi(pn+1) +
1
2Dd(u
n+1i + un
i )]
(3.1)
where Eni = Cn
i +Do(uni )+Bn
i +Ds(uni ), Ci represents the i-component of the convective
terms, Dd and Do are, respectively, the diagonal and off-diagonal diffusion operators of
primary viscous terms divided by density, Bi represents the i-component of gravity
acceleration and surface tension forces, Ds is the diffusion operator of the subsidiary
viscous force divided by density, and Gi is the i-component of the negative gradient
operator divided by density.
The convective term Ci is discretized with a modified QUICK scheme (Perng and
Street 1991). The upwind direction is determined by the volume flux Um. Spatial central
difference is applied to all diffusion terms. Equation (3.1) is solved with a four-step FSM
as follows:
Predictor:
(I − ∆t
2J−1Dd)(u
∗i − un
i ) =∆t
J−1
[(3
2En
i −1
2En−1
i ) + Gi(ψn) + Dd(u
ni )
](3.2)
where u∗i is the first intermediate velocity and ψ is a pressure-like variable. The oper-
ator (I − ∆t2J−1 Dd) is approximated by the approximate factorization technique (Zang
et al. 1994) allowing the above linear equations to be solved with a tridiagonal-matrix
algorithm (TDMA) or a periodic TDMA depending on boundary conditions.
First corrector:
ui − u∗i = − ∆t
J−1Gi(ψ
n) (3.3)
25
where ui is the second intermediate velocity.
Solving for the pressure-like variable:
δ
δξl
(T lm
ρ(φ)
δψn+1
δξm
)=
1
∆t
δUl
δξl
(3.4)
where δ is a differencing operator. This is a variable-coefficient Poisson equation, solved
by a multigrid technique (Perng and Street 1991). The inner gradient operator is esti-
mated at cell faces. The outer divergence operator is estimated at the cell center.
Second corrector:
un+1i − ui =
∆t
J−1Gi(ψ
n+1) (3.5)
This procedure is also called the four-step time advancement scheme (Choi and Moin
1994) in which the spatial derivatives are all approximated with second-order central
differencing on a staggered grid. As compared with the Kim & Moin’s FSM (1985), the
pressure gradient is added to the predictor step Eq. (3.2) and another correction step
Eq. (3.3) is added. The resulting relationship between the pressure-like variable ψ and
the true pressure p becomes
Gi(pn+1) = Gi(ψ
n+1)− ∆t
2J−1DdGi(ψ
n+1 − ψn)
= Gi(ψn+1) + O(∆t2). (3.6)
In general, implicit time advancement of diffusion terms results in a splitting error
in FSM, determining the time accuracy of the method. Equation (3.6) implies that this
four-step FSM is of the second order temporal accuracy, as seen by summing Eqs. (3.2),
(3.3) and (3.5). Other fractional step methods, like the popular one by Kim & Moin
(1985), are of only first-order temporal accuracy (Perot 1993). The relationship between
the first intermediate velocity and the true velocity obtained by summing Eqs. (3.3) and
(3.5) reads
un+1i = u∗i +
∆t
J−1Gi(ψ
n+1 − ψn)
= u∗i + O(∆t2) (3.7)
26
Equation (3.7) implies that the physical boundary velocity can be used as the boundary
condition of u∗i in solving Eq. (3.2) without degrading the overall time accuracy. In the
Kim & Moin FSM (1985), the boundary value of u∗i must satisfy a modified equation to
maintain second-order accuracy. Three conclusions on the four-step FSM can be drawn.
1. The splitting error is second-order accurate in time.
2. The pressure-like variable is of second-order temporal accuracy of the true pressure.
3. There is no special treatment for boundary condition of u∗i to maintain a consistent
second-order temporal accuracy.
A similar four step FSM proposed by Ferziger and Peric (1997) advances the pres-
sure gradient with the Crank-Nicolson method instead of the implicit Euler method in
Eq. (3.1). Our experience shows that this version of FSM may become unstable (as in
free surface motion induced by a submerged hydrofoil (Yue 2001) simulated by the body
force method (Verzicco et al. 2000)).
3.2 Numerical Schemes for Level Set Function
3.2.1 Evolution
As mentioned in Chapter 2, Eq. (2.15) is of the Hamilton-Jacobi type. Disconti-
nuities in derivatives are easily produced even when initial conditions are smooth. In
general such solutions are not unique. Numerical schemes must be specifically designed
to converge to a unique viscosity solution satisfying the entropy condition and singling
out a physically generalized solution (Crandall and Lions 1983). The key idea to the
entropy condition is monotonicity preservation. Such schemes include total variation di-
minishing (TVD), ENO and weighted ENO (WENO) schemes. TVD schemes generally
degenerate to first-order accuracy at smooth extrema while ENO and WENO main-
tain global high-order accuracy (Jiang and Wu 1999). The ENO scheme was originally
developed for hyperbolic conservation laws (Harten et al. 1987) and later extended
to Hamilton-Jocobi equations (Shu and Osher 1989) motivated by the observation of
27
the close relationship between conservation laws and Hamilton-Jocobi equations. ENO
chooses the locally “smoothest” stencil among several candidates to approximate nu-
merical fluxes at cell faces. Numerical viscosity is adjusted adaptively based on the local
smoothness of the solution to eliminate Gibbs phenomenon, or spurious oscillation, near
the discontinuity. ENO schemes maintain a uniform high-order accuracy even at a dis-
continuity. The higher-order ENO scheme is obtained inductively on the lower-order
ENO using a hierarchy of divided differences. This makes implementation of high-order
ENO schemes relatively straightforward. The multi-dimensional ENO scheme can be
conveniently extended from a one-dimensional ENO scheme in a dimension-by-dimension
way.
Equation (2.15) is advanced with a third order TVD Runge-Kutta scheme (Shu
and Osher 1989) which is total variation (TV) stable.
φ(1) = φn − ∆t
J−1R(φn)
φ(2) =3
4φn +
1
4φ(1) − ∆t
4J−1R(φ(1)) (3.8)
φn+1 =1
3φn +
2
3φ(2) − 2∆t
3J−1R(φ(2))
where R(φ) = δ(Uiφ)δξi
.
Let (U, V,W ) and (ξ, η, ζ) denote components of Ui and ξi, where i = 1, 2, and
3, respectively. The grid distance is set on the transformed computational grid to unity
in all dimensions, i.e., ∆ξ = ∆η = ∆ζ = 1. Spatial operator R is discretized for the
control volume (i, j, k) in a conservative manner,
δ(Umφ)
δξm
= (Uφ)i+ 12,j,k−(Uφ)i− 1
2,j,k +(V φ)i,j+ 1
2,k−(V φ)i,j− 1
2,k +(Wφ)i,j,k+ 1
2−(Wφ)i,j,k− 1
2
(3.9)
Volume fluxes U , V and W are defined at the cell faces. φ is defined at the cell center.
Cell face values of φ are thus constructed by the third order ENO interpolation scheme
(Shu and Osher 1989).
Denote
28
δφ−i = φi,j,k − φi−1,j,k, δφ0i = φi+1,j,k − φi,j,k, δφ+
i = φi+2,j,k − φi+1,j,k
δ2φ−i = φi−2,j,k − 2φi−1,j,k + φi,j,k, δ2φ0i = φi−1,j,k − 2φi,j,k + φi+1,j,k
δ2φ+i = φi,j,k − 2φi+1,j,k + φi+2,j,k, δ2φ++
i = φi+1,j,k − 2φi+2,j,k + φi+3,j,k
The second order ENO is formulated as
φ(2)
i+ 12,j,k
= φup
i+ 12,j,k
+1
2max[sign(Ui+ 1
2,j,k), 0]m(δφ−i , δφ0
i )
+1
2min[sign(Ui+ 1
2,j,k), 0]m(δφ0
i , δφ+i ) (3.10)
where
φup
i+ 12,j,k
=
φi,j,k if Ui+ 12,j,k ≥ 0
φi+1,j,k otherwiseis the first-order upwind value.
m(a, b) =
a if |a| ≤ |b|b otherwise
sign(a) =
1 if a > 0
0 if a = 0
−1 if a < 0
The third order ENO is formulated as
φ(3)
i+ 12,j,k
= φ(2)
i+ 12,j,k
+1
3max[sign(Ui+ 1
2,j,k), 0]max[c−i , 0]m(δ2φ−i , δ2φ0
i )
+1
2min[c−i , 0]m(δ2φ0
i , δ2φ+
i )
+1
3min[sign(Ui+ 1
2,j,k), 0]1
2max[c+
i , 0]m(δ2φ0i , δ
2φ+i )
+ min[c+i , 0]m(δ2φ+
i , δ2φ++i ) (3.11)
where
c−i = c(δφ−i , δφ0i )
c+i = c(δφ0
i , δφ+i )
29
c(a, b) =
1 if |a| ≤ |b|−1 otherwise
Other cell-face values of φ, such as φi,j+ 12,k, φi,j,k+ 1
2, etc., are approximated in the same
way.
3.2.2 Reinitialization
Clearly, Eq. (2.20) is also a Hamilton-Jacobi equation. A second-order ENO scheme
is applied here.
Denote
dx ≡ δdδx
, ∆x−i ≡ xi,j,k − xi−1,j,k, ∆x+i ≡ xi+1,j,k − xi,j,k
(1) Derivative approximation
First-order approximation:
d(1)−x =
∂ξ
∂x(di,j,k − di−1,j,k) +
∂η
∂x(di,j,k − di,j−1,k) +
∂ζ
∂x(di,j,k − di,j,k−1) (3.12)
d(1)+x =
∂ξ
∂x(di+1,j,k − di,j,k) +
∂η
∂x(di,j+1,k − di,j,k) +
∂ζ
∂x(di,j,k+1 − di,j,k) (3.13)
Second-order approximation:
d(2)−x = d(1)−
x +∆x−
2
δ2d−
δx2(3.14)
d(2)+x = d(1)+
x − ∆x+
2
δ2d+
δx2(3.15)
where
δ2d−
δx2= minmod(d1, d2) (3.16)
δ2d+
δx2= minmod(d2, d3) (3.17)
minmod(a, b) =
sign(a) min(|a| , |b|) if a · b > 0
0 otherwise(3.18)
30
d1, d2 and d3 are central difference approximations of δ2dδx2 ≡ ∂ξm
∂xδ
δξm
(∂ξn
∂xδdδξn
)on sten-
cils 1(xi−2,j,k, xi−1,j,k, xi,j,k), 2(xi−1,j,k, xi,j,k, xi+1,j,k), and 3(xi,j,k, xi+1,j,k, xi+2,j,k), respec-
tively. d(2)−y , d(2)+
y , d(2)−z , and d(2)+
z are computed in the same way.
(2) Compute |∇d|Let
a = d(2)−x , b = d(2)+
x
c = d(2)−y , d = d(2)+
y
e = d(2)−z , f = d(2)+
z
Define
a+ = max(a, 0), a− = min(a, 0)
and the same subscripts for b, c, d, e, and f .
The computation of |∇d| is performed based on Godunov’s method (Sussman et al.
1994),
|∇d| =
D+ if s(d0) > 0
D− if s(d0) < 0
0 otherwise
(3.19)
where
D+ =√
max(a2+, b2−) + max(c2
+, d2−) + max(e2+, f 2−)
D− =√
max(a2−, b2+) + max(c2−, d2
+) + max(e2−, f 2+)
Apply the upwind scheme for Eq. (2.20) to obtain
s(d0) |∇d| = max[s(d0), 0]D+ + min[s(d0), 0]D− (3.20)
Equation (2.20) is also advanced in time with the third order TVD Runge-Kutta scheme (3.8).
3.3 Computation of Surface Tension
Surface tension effects are confined to the neighborhood of the free surface. Care
must be taken in calculating surface tension. The key is the numerical evaluation of
curvature in terms of Eq. (2.8). Similar to the MAC-like method by Brackbill et al.
(1992), components of unit normal are defined at cell faces on a staggered-grid while
31
curvature is defined at cell center (Fig. 3.1). For brevity, only the 2D formulation is
presented here. A 3D extension is straightforward. Denote
Nx ≡ ∂φ∂x
, Nz ≡ ∂φ∂z
Components of unit normal n are then calculated
nx =Nx
|∇φ|(i− 12,k)
, nz =Nz
|∇φ|(i,k− 12)
(3.21)
where Nx and nx are defined at the left cell face (i− 12, k), and Nz and nz are defined at the
bottom cell face (i, k− 12). Curvature κ is defined at the cell center (i, k). Denominators
in Eqs. (3.21) are defined as
|∇φ|(i− 12,k) =
√N2
x,i− 12,k
+ N2z,i− 1
2,k
(3.22)
|∇φ|(i,k− 12) =
√N2
x,i,k− 12
+ N2z,i,k− 1
2
(3.23)
where Nz,i− 12,k and Nx,i,k− 1
2are obtained by linear interpolations.
Nx,i,k− 12
=1
4[Nx,i,k−1 + Nx,i+1,k−1 + Nx,i,k + Nx,i+1,k] (3.24)
Nz,i− 12,k =
1
4[Nz,i−1,k + Nz,i−1,k+1 + Nz,i,k + Nz,i,k+1] (3.25)
Once unit normals nx and nz are calculated, the curvature is determined from
κ =∂ξ
∂x(nx,i,k−nx,i−1,k)+
∂ζ
∂x(nx,i,k−nx,i,k−1)+
∂ξ
∂z(nz,i,k−nz,i−1,k)+
∂ζ
∂z(nz,i,k−nz,i,k−1)
(3.26)
Extrapolation is used on boundaries where necessary.
3.4 Restriction of Time Step
Since the convective, surface tension, gravity, off-diagonal primary viscous, and
subsidiary viscous terms in Eq. (3.2) are advanced with an explicit scheme in time, the
32
time step must be restricted to enforce numerical stability. According to Brackbill et al.
(1992), the time step for surface tension reads
∆tσ = minΩδ
√(ρl + ρg)∆h3
4πσ(3.27)
where ∆h = min(∆x, ∆y, ∆z) and Ωδ represents the transition zone where δ(φ) > 0.
The time step for the convective terms must satisfy the Courant-Friedrichs-Lewy (CFL)
condition,
∆tu = minΩ
J−1C|U |+ |V |+ |W | (3.28)
where C = 0.5 and Ω represents the whole computational domain. Time step restrictions
due to gravity and subsidiary viscous terms are
∆tg = minΩ
√∆z
g(3.29)
∆tµ = minΩδ
ρ∆h2
2µ(3.30)
The eventual time step restriction is then
∆tn+1 = min(∆tu, ∆tσ, ∆tg, ∆tµ) (3.31)
33
.
κ(i, k)
Nz(i, k-1/2)
nx(i-1/2, k) ..
nz(i, k-1/2)
Nx(i-1/2, k)
Figure 3.1: Schematic of curvature definitions and unit normal on a 2D cell.
34
CHAPTER 4
VERIFICATION AND VALIDATION
This chapter describes verification and validation of the mathematical models pre-
sented in Chapter 2 and numerical schemes developed in Chapter 3. The benchmark
case, 2D laminar lid-driven cavity flow, is used to validate the four step FSM. Three
other cases, namely reinitialization of a static circle, Zalesak’s problem (rotation of a
slotted disk), and stretching of a circular fluid element, are used to verify and vali-
date LSM. The verify and validation of the coupling of LSM with the incompressible
Navier-Stokes equations are performed on a 2D laminar open-channel flow, travelling of
a solitary wave, and 2D and 3D dam breaking flows. Most computations reported here
are performed on a Dell Dimension 8200 PC with Pentium 1.8G CPU, and some are
carried out on ITS PC cluster, all at The University of Iowa.
4.1 2D Lid-driven Cavity Flow
The four-step FSM presented in section 3.1 is validated by calculation of a 2D
laminar lid-driven cavity single-phase flow on a unit square with two non-uniform grids,
48 × 48 and 24 × 24 (Fig. 4.1). The Reynolds number is Re = U0L/ν = 1000 for both
grids, where U0 is the lid velocity, L is the cavity length, and ν is the fluid kinematic
viscosity. Computation ends after 10,000 time steps for both grids. Streamline patterns
for primary, secondary, and corner vortices are shown in Fig. 4.2. They are in agreement
with those obtained by Ghia et al. (1982). Velocities along vertical and horizontal
centerlines (u and w, respectively) are compared with the benchmark data (Ghia et al.
1982) in Fig. 4.3. Results for the 48 × 48 grid are in excellent agreement with the
benchmark data. Those for the 24 × 24 grid show a small discrepancy in the vicinity
of the turning points of the curves, but are in good agreement around the core of the
cavity. The numerical convergence rate of the four step FSM is shown in Fig. 4.4.
35
Pressure residual is measured from the residual of Poisson equation (3.4). The coarse
grid shows faster numerical convergence rate than the fine grid, as expected. This case
demonstrates the high-order accuracy and fast convergence of the four-step FSM.
4.2 Validation of Level Set Method
4.2.1 Reinitialization of a Circle
Reinitialization is a key procedure in LSM. To validate the numerical scheme de-
scribed in section 3.2.2, the reinitialization procedure is applied to a stationary circle
with an initially discontinuous level set function. The domain size is 100 × 100. A
uniform and a non-uniform grid (Fig. 4.5) are employed. Both have 100 × 100 points.
The center of the circle is at the domain center. The circle radius is 30. The level set
function is initially assigned a value of +100 outside the circle and −100 inside the circle.
Figure 4.6 shows the contours of φ at t = 25 time units. φ is redistanced as a signed
function (|∆φ| = 1) in the whole domain except at the circle center where a singularity
exists. Reinitialization on both grids produces the same results. Figure 4.7 shows that
contour φ = 0 is not altered from the initial position (the thick dashed line) through
reinitialization. This is desirable for area preservation. This case demonstrates that the
accuracy of the reinitialization scheme on a non-uniform grid is very close to that on a
uniform grid.
4.2.2 Zalesak’s Problem
The rotating slotted disk problem (Zalesak 1979) has become a benchmark for
testing advection schemes. A slotted solid disk rotates around the center with constant
angular velocity. This problem is used to measure the diffusive error of the third-order
ENO scheme for level set function evolution. The slotted disk has a radius of 15 and
a slot width of 6. It is initially located at (50, 75) in the 100 × 100 domain. Angular
velocity Ω is set to 0.01 so that the disk returns to its original position every 628 (≈ 200π)
time units. Diffusive errors can be evaluated by checking the degree of disk boundary
36
distortion. Three grids are employed for comparison, a uniform 100 × 100 grid, a non-
uniform 100 × 100 grid (Fig. 4.8), and a refined 200 × 200 uniform grid. Time step
∆t = 0.5 is used for the refined uniform grid and ∆t = 1.0 for the other two. Since this
is a pure advection problem with a constant angular velocity, it is expected that a good
evolution scheme without reinitialization should adequately preserve disk geometry. To
illustrate the effects of different schemes on preserving complex geometries, different
numerical schemes are applied for approximation of φ at cell-faces in the evolution
equation (Eq. (3.9)), namely the second-order ENO scheme Eq. (3.10), the third-order
ENO scheme of Eq. (3.11), and the third-order QUICK scheme.
The second-order ENO is very diffusive with the slot totally smeared out in Fig. 4.9.
Thus, a higher order scheme with at least third-order accuracy is required for complex
boundary evolution. The third-order ENO evolves the circular boundary and the slot
boundaries quite accurately without significant distortion except near the sharp corners.
QUICK performs as well as a third-order upwind scheme, the slot boundary deviates
from its original position to some degree and a small distortion near the slot top is
observed. QUICK tends to generate over- and under-shoots near a discontinuity. With
the non-uniform grid, disk boundary advection is improved at the corners and the slot
top due to finer grid resolution. Figure 4.10 shows the rotation obtained by the third-
order ENO scheme on the fine grid 200 × 200 at t = 0, 157, 314, 471, and 628. Disk
boundaries even near the corners are advected accurately. The reinitialization scheme
with area preservation is applied for comparison. A slight improvement is observed.
This case demonstrates that high-order-accuracy ENO, at least third-order, is required
for evolution of complex interfaces.
4.2.3 Circular Fluid Element Stretching
A circular fluid element is placed in a swirling shear flow field within a unit square
described by
ϕ =1
πsin2(πx) sin2(πz) (4.1)
37
where ϕ is the stream function. The fluid is stretched into a thin filament by the
shearing velocity field. This case provides a challenging test for surface-tracking and
surface-capturing methods. Rider and Kothe (1998) and Rudman (1997) used it to
evaluate their VOF methods. To use the three grids employed in the Zalesak’s problem,
the same domain size is adopted here. The circle is initially centered at (50, 75) with
radius 15. The solenoidal velocity field becomes
u = − sin2(πx
100) sin(
πz
50), w = sin2(
πz
100) sin(
πx
50) (4.2)
A time step of 0.5 is used for the two 100× 100 grids, and 0.25 for the 200× 200 grid.
Figure 4.11 shows the stretching process of the fluid element at t = 0, 100, 200, and
300 by the three grids. The circular fluid is drawn out into a filament by the shearing
flow, and becomes thinner over time. There is no appreciable difference between the two
100× 100 grids at t = 100 and 200, but the non-uniform grid preserves the filament tail
with better resolution. The uniform 200 × 200 grid preserves the filament area much
better than the coarse grids. There is no significant filament breakup, especially for the
200×200 grid, as was found in the VOF method results (Rider and Kothe 1998; Rudman
1997) due to the numerical surface tension inherent in the reconstruction procedure.
To evaluate the area-preservation errors and the interface advection and deforma-
tion accuracy, the velocity field of Eq. (4.2) is multiplied by cos πtT
, so that the stretching
process is time-reversed (Leveque 1996), where T is the prescribed reversal period. The
flow slows down and the fluid is stretched out during 0 < t < T2. The flow reverses
direction and the fluid shrinks back during T2
< t < T . The fluid element is expected to
resume its initial circular shape at t = T . Two periods, T = 250 and 500, are chosen to
estimate errors. The fluid element in all three grids return to their original circular shape
with slight deformation after one period of T = 250, shown in Fig. 4.12. For T = 500,
the fluid element in the 200× 200 grid nearly returns to the circular shape while those
in the other two grids are deformed. This occurs because the fluid is stretched severely
for T = 500 and the accuracy of fluid interface advection degrades if the grid is not fine
38
Table 4.1: Area error after one period for a circular fluid in the time-reversed swirlingdeformation flow
area error (%)
Grid T=250 T=500
uniform (100× 100) grid 0.687 0.09
non-uniform (100× 100) grid 0.42 -1.635
uniform (200× 200) grid 0.038 1.36
enough. Table 4.1 shows area errors calculated by
εA =A(t)− A(0)
A(0)(4.3)
where A(t) =∫Ω H(φ)dΩ is the total fluid element area at time t. The total fluid
element area is well preserved in all cases. The area for T = 500 on the uniform
100×100 grid appears to be better preserved than others, though its interface is severely
deformed. Thus, the present LSM resolves stretched interfaces on the grid-cell scale
without introducing significant artificial surface tension effects.
4.3 Free Surface Flows
4.3.1 2D Laminar Open-channel Flow
The simplest open-channel flow is a laminar flow on an inclined flat bed driven by
gravity (Fig. 4.13). There is an analytical solution for this flow,
u(z)
U0
=z
L(2− z
L) (4.4)
where U0 = gL2 sin θ/2νw is free surface velocity, L is water depth, and the subscript
w denotes water. The inclination angle θ = 10−4 rad is chosen here, corresponding to
Reynolds number Re = U0L/νw = 31, viscosity ratio νa/νw = 15, and density ratio
ρa/ρw = 1.2 × 10−3. These ratios are also employed in all subsequent free-surface flow
cases. Subscript a denotes air. Air density is so small compared to water, that it is
expected that air motion is mainly governed by free-surface velocity. When the top
boundary of the computational domain is set with a no-slip wall, the flow of air is like
39
a Couette flow (Guyon et al. 2001).
u(z)
U0
=z
L(4.5)
Two uniform grids, 40 × 80 and 40 × 40, are used for simulation. Computation
starts with a stationary velocity field. The final steady velocity field is achieved when
the profiles at all streamwise cross-sections become identical (Fig. 4.14). Velocity profiles
in water regions for the present computation and the analytical solution, Eq. (4.4), are
compared in Fig. 4.15 (a). Computation results are grid independent and in excellent
agreement with the analytical solution. The computational velocity profiles in the air
region are compared with the Couette-flow solution, Eq. (4.5), in Fig. 4.15 (b). The
linear profiles of air velocity imply that the air is fully driven by the free-surface velocity.
Normalized shear stress (2τ/ρwU20 ) in water and air regions is shown in Fig. 4.16. Shear
stress vanishes at the free surface and is much smaller in the air than in the water,
implying that air motion has negligible effect on the water.
4.3.2 A Travelling Solitary Wave
Propagation of a solitary wave is a simple and practical free-surface problem well
studied experimentally and numerically. A travelling solitary wave in a canal (Fig. 4.17)
is simulated to examine if LSM can predict correct viscous damping and run-up on the
vertical wall. Here, L is the still water depth, and subscripts a and w denote air and
water, respectively. The channel size is 20L×2L. The theoretical wave speed Cg =√
gL
is set as 1.0m/s. The Reynolds number Re = CgL/νw = 5 × 104. A 200 × 120 grid is
used. Grid-distance is uniform in the x-direction, and within the range (0, A0) in the
z-direction. It then expands to the top and bottom boundaries. The half thickness of the
interface ε is fixed at twice the grid-distance ∆z. To generate a solitary wave, Laitone’s
analytical approximation (Ramaswamy 1994) may be used. Instead, here a still water
surface with a Boussinesq profile (The et al. 1994), initially in hydrostatic balance, is
suddenly released from the left vertical wall,
40
A(x, 0) = A0/ cosh2
(√3A0
2x
)(4.6)
Water starts moving due to the pressure difference in the horizontal direction between
the contiguous grids of air and water, and a wave is formed. After t = 6.0 s, the wave
has escaped the influence of the left wall boundary and may be regarded as a solitary
wave. This is set as the initial time of the solitary wave propagation. Grid points cluster
between 0 ≤ z ≤ A0, with an interval of ∆z = 0.01L to resolve the wave. Due to the
large density ratio of air and water, the top boundary condition has negligible effect on
the solitary wave motion. A no-slip boundary condition is applied for simplicity.
Figure 4.18(a) shows the travelling solitary wave and its climb at the right vertical
wall for the case A0/L = 0.4. A slight damping of the wave amplitude due to viscous
effects is evident. The wave speed, measured from Fig. 4.18(a), is 1.05, close to the
theoretical value of 1.0. Figure 4.18(b) shows a typical velocity field at t = 4.0 s. A
vortex is observed centered at the wave top. A movie showing the wave progression is
contained in the CD appended to the report.
To quantify the wave viscous damping characteristics, three waves with different
initial amplitudes are computed, and the computational results compared with those
predicted by Mei’s perturbation theory (Mei 1989):
A− 1
4max = A
− 14
0max + 0.08356
νw
C12g L
32
12
Cgt
L(4.7)
where Amax is the solitary wave amplitude, and A0max is the amplitude at the initial
state. The smaller A0max/L is, the better the numerical prediction agrees with the
perturbation theory, as shown in Fig. 4.19. This is because Eq. (4.7) is valid only for
A0max/L ≤ 0.1.
Another quantity for comparison is the wave run-up (the highest point) on the
right vertical wall. Nine cases (Table 4.2) with different initial wave amplitudes are
computed and the run-up at the right wall boundary is measured. Computational results
41
Table 4.2: Computational cases of solitary waves with different wave amplitude
Case A0 Ac
1 0.1 0.0485
2 0.2 0.094
3 0.3 0.138
4 0.4 0.18
5 0.5 0.22
6 0.6 0.26
7 0.7 0.30
8 0.8 0.336
9 0.9 0.373
are compared with the experimental data (Chan and Street 1992) in Fig. 4.20. Ac in
the x-axis is the solitary wave amplitude in the horizontal center of the computational
domain (Fig. 4.17). Agreement between computation and experiment is very good for
Ac/L < 0.3. After Ac/L > 0.3, the experimental data exhibits scatter. These results
demonstrate that the present LSM accurately predicts viscous damping characteristics
without introducing significant numerical damping effects.
To further quantify numerical errors, the mass error is defined as
εM =M(t)−M(0)
M(0)(4.8)
where M(t) =∫Ω ρH(φ)dΩ is the total water mass at time t. The numerical mass error is
smaller than 0.01% at t = 20 s for case A0 = 0.4L and is 0.0085% for the case A0 = 0.2L
at the same time, indicating that the present numerical scheme accurately conserves
mass.
4.3.3 2D Dam-Breaking
The collapse of a water column on a rigid horizontal plane is sometimes called a
broken-dam or dam-breaking problem. It simulates the abrupt failure of a dam, in which
42
an initially blocked still water column spreads out suddenly after the barrier is removed.
This problem has been studied experimentally (Martin and Moyce 1952) to investigate
the spreading velocity of the surge front and the falling rate of the water column. The
water motion was recorded by cine-photography at about 300 frames per second. In one
of the experimental cases, the square water column has length a = 214
inch. This case
is employed here to validate the present LSM. It was also studied by Kelecy & Pletcher
(1997) in their numerical simulation by an artificial compressibility method.
The computational domain is 5a × 1.25a, same as that employed by Kelecy &
Pletcher (Fig. 4.21); s and h denote the surge front position and the remaining wa-
ter column height, respectively. These parameters measure the spreading velocity and
falling rate of the water column. The numerical experiments are performed in a closed
container with wall boundaries. There are no confinements on the top and the right in
the experiment. A uniform 200×50 grid and a non-uniform 160×40 grid are used. Grid
points are clustered near the left, the right, and the bottom walls and at the top and
right boundaries of the initial water column (Fig. 4.22). The half thickness of interface
ε is fixed with twice the grid spacing ∆x for the uniform grid, and 1.12 times that for
the non-uniform grid. The still water column is initially in hydrostatic balance. The
surface tension effect is examined by keeping or removing the surface tension term in
Eq. (2.11). Time is non-dimensionalized by tg =√
a/g in all plots for the 2D and 3D
broken-dam flows.
Figure 4.23(a) compares surge fronts predicted by the present LSM with Martin
& Joyce’s experiment. The surface tension effect on the surge fronts is indistinguish-
able in the figure. The non-uniform grid results deviate from those of the uniform grid
only slightly in the final stage. This difference is well within experimental uncertain-
ties. The water spreading velocity is quite accurately predicted by the present LSM.
Figure 4.23(b) compares the remaining water column height predicted by the present
LSM with the experiment. Agreement is excellent. Surface tension and grid differences
do not significantly affect the results.
Snapshots of water motion and velocity fields in the whole computational domain,
43
obtained with the uniform grid are displayed in Fig. 4.24. The water column collapses
and accelerates toward the air due to the pressure difference between adjacent water and
air along the right boundary of the initial water column. The largest pressure difference
is at the right lower corner of the water column where water is greatly accelerated
and moves rapidly along the bottom wall. Air is entrapped, forming bubbles, when the
surge front is reflected from the right wall and falls into the bottom pool (Fig. 4.24f). An
elongated thin surge is created by surge front splashing (Fig. 4.24g). Velocity vector fields
reveal a large vortex formed in the air region near the water surface that accompanies
the surface motion at all times. The strongest motion, the largest velocity, always occurs
in the air region near the surge front. Though the induced air velocity is comparable
to and even higher than that of the water, the air momentum is substantially smaller
(Fig. 4.25). This implies that the effects of air motion on water can be neglected.
The present computational results are thus in excellent agreement with the experiment,
though top and right boundary conditions are different.
Figure 4.26 shows the time history of mass errors for the uniform and non-uniform
grids. In general, mass is conserved quite well given the fast-transient surface motion and
large free-surface topology changes. Mass error can be further reduced by a finer grid. To
study the effect of interface thickness on computational results, five different thicknesses
are applied to the uniform grids. The surface tension term is included. Figure 4.27
shows the effect of the half thickness of interface ε on surge front spreading speed. The
results are not very sensitive to interface thickness, but the larger the interface thickness,
the greater the deviation from experimental data. Though not shown here, there is very
little variation in the remaining water column heights among these five thicknesses.
4.3.4 3D Dam-Breaking
To demonstrate the capability of simulating 3D free-surface flows with the present
LSM, the cubic-water-column-breaking (Fig. 4.28) is computed. The computational
domain size is 5a × a × 1.25a in the x, y, and z directions, respectively. A uniform
44
200 × 24 × 50 grid is used. The top, bottom, left, and right boundaries of the com-
putational domain are prescribed as walls. Two types of boundary conditions for the
lateral boundaries (y-direction), no-slip and periodic, are considered. The former means
that the computational domain is an enclosed container. The latter implies that the
computational domain is a short segment of a wide container, i.e., it is essentially 2D
flow. These simulations assess side boundary effects on surge front structures and air
entrapment.
As in the 2D case, the spreading velocity and the falling rate of the water column
are examined. Figures 4.29 (a) and (b) show these quantities in good agreement with
experimental data. The surge front and the remaining water column height in these
figures are measured at the center plane in the spanwise direction. There is no obvious
difference in the results obtained for different side boundary conditions. As in the 2D
cases, surface tension does not have a significant effect on the results.
Figure 4.30 shows snapshots of water surface positions, and velocity fields at the
center plane in the spanwise direction, at selected times for periodic side boundaries. The
free surface shapes are essentially 2D. Figure 4.31 shows results with wall boundaries.
The flow does not show obvious three-dimensionality until T = 8.0 (Fig. 4.31g). Air
entrapment and the elongated thin surge formation due to surge-front splashing, as in
the 2D cases, are observed in Figs. 4.30 and 4.31. Figure 4.32 shows close-up views of
the surge fronts and rear views of the air bubbles observed in Figs. 4.30g and 4.31g.
The surge front in Fig. 4.32(a) is basically 2D due to the periodic condition of the side
boundaries. It has a paw-like shape in Fig. 4.32(b) due to wall boundary layer effects
of the side boundaries. The entrapped air bubble in Fig. 4.32(c) for the periodic case
shows a quasi-cylindrical shape. In the wall boundary case, it displays a symmetric
horse-shoe shape in Fig. 4.32(c). Much of the air is concentrated in the center of the
bubble. Despite the three-dimensionality of the free surface in Fig. 4.31g, the sliced
free-surface shape at the center plane in the spanwise direction (the right vector picture)
is very similar to that in Fig. 4.30g. Mass errors in both 3D cases are shown in Fig. 4.26.
Mass preservation is somewhat better than in the 2D cases. An animation showing the
45
water surface motion is contained in the appended CD.
4.4 Summary
Mathematical models presented in Chapter 2 and numerical schemes developed in
Chapter 3 are widely verified and validated by a series of benchmark test cases. The
four-step FSM is validated by a 2D laminar lid-driven cavity flow. Computational results
are in good agreement with benchmark data even for the very coarse grid, showing a
high order accuracy and fast convergence rate of the four-step FSM.
The numerical schemes for LSM developed for general curvilinear coordinates are
verified and validated by several benchmark cases, namely reinitialization of a static
circle, Zalesak’s problem, and stretching of a circular fluid element. By comparing the
computational results, it is found that, first, the accuracy of reinitialization scheme on a
non-uniform grid is similar to that on a uniform grid. Second, at least third-order accu-
racy ENO is required for the evolution equation to capture complex interfaces. Third,
the circular interface, severely deformed by a swirling shear flow, is accurately evolved
and resolved on the grid-cell scale without introducing significant artificial surface ten-
sion effects.
Computations of benchmark cases, such as 2D laminar open-channel flow, soli-
tary wave travelling, 2D and 3D dam-breaking, show that free-surface characteristics
and motions are accurately predicted by the present model coupling LSM and the in-
compressible Navier-Stokes equations. Air motion is mainly governed by free-surface
velocity and plays negligible influence on water motion. Mass is well conserved in all
cases, demonstrating success of 3D volume preservation in the reinitialization process.
69
CHAPTER 5
LES OF TURBULENT OPEN-CHANNEL FLOW OVER A DUNEMODELLED WITH PLANE FREE SURFACE
In this chapter the single-phase LES with DSM, without LSM, is applied to devel-
oped turbulent open-channel flow over a train of identical 2D dunes on the bottom wall
(in a periodic array). The free surface is treated as a symmetry plane (fixed flat surface).
This case is used to validate the present LES by comparing computational results with
the LDV and PIV experiments recently conducted at IIHR (Balachandar et al. 2002;
Balachandar et al. 2003). Turbulence statistics and large-scale flow structures are exam-
ined. This case also establishes a baseline for comparison with the numerical simulation
with a freely deformable free surface in the next chapter.
5.1 Description of the Simulated Flow
It is observed in laboratory experiments and in rivers and lakes that the surface of
a mobile bed is made up of statistically periodic irregularities, called sand waves. They
exert considerable influence on granular material transport and flow resistance. All
sand waves originate from geometric discontinuities on the bed. Their shape depends
on several factors, including flow rate, sediment size, flow depth, and turbulence eddy
structures. Based on the Froude number (Fr), sand waves are roughly classified into
three categories (Simons and Richardson 1963; Yalin 1977):
1. Ripples at low flow rate (Fr << 1).
2. Dunes at subcritical flow (Fr < 1).
3. Antidunes at supercritical flow (Fr > 1).
Where Fr = U0/√
gL, U0 is free-surface velocity, L is water depth, and g is gravity
acceleration. Ripples are short steep features usually present in beds composed of fine-
grained sediments. Dunes are longer features of less steepness and usually appear in
70
beds of coarser sediments (Maddux 2002). Turbulent open-channel flow over dunes is
the subject of this study.
An idealized flow in an open-channel, with an infinite number of identical 2D fixed
dunes on the bottom, is investigated. The flow is considered periodic in the streamwise
and spanwise directions. In the recent experiments of Balachandar et al. (2002, 2003),
LDV and PIV measurements were made on the 17th dune of a train of 22 dunes mounted
in a laboratory flume. As noted in Section 1.2.2, most previous numerical simulations of
flow over dunes are based on 2D RANS formulations with eddy viscosity calculated by
either the k − ε or k − ω model, with the free surface treated as an undisturbed plane
boundary. Difficulties associated with such simulations include the proper modelling of
the effects of alternating adverse and favorable pressure gradients, recirculating flow, and
free-surface boundary conditions. Also, RANS equations cannot predict turbulent-burst
phenomena and large-scale turbulent eddy structures that are primarily responsible for
momentum and sediment transport. LES does not suffer greatly from these difficulties
and provides detailed turbulence statistics and eddy structures.
In this chapter, the single-phase LES is used to study turbulent open-channel
flow over a fixed 2D dune with undisturbed plane free surface. There are two ways to
numerically simulate this kind of space-periodic flow. One is to impose a head loss or
pressure drop and calculate the flow rate. The other is to specify the flow rate and
determine the head loss. The first approach is employed here.
Fig. 5.1 shows the flow geometry over a fixed 2D dune. The origin of the coordinate
system is at the dune crest. The Reynolds number, based on flow depth L and free-
surface velocity U0 at the inlet to the solution domain, is 5.7×104. The Froude number,
based on U0 and L, is 0.44. Dune height h is 20 mm. Non-dimensional dune wavelength
is λ/h = 20, and water depth is L/h = 6. The channel width (in y direction) is ly/h = 7.
A non-uniform 80×32×64 grid in the respective x, y and z directions is used. Figure 5.2
shows the grid in the x − z (vertical) plane. The grid is uniformly distributed in the
spanwise (y) direction. The first grid points are less than four wall units (ν/uτ , where
uτ is the mean friction velocity measured at x/h = 18) from the dune bed.
71
Periodic boundary conditions are imposed at the streamwise and spanwise bound-
aries, and no-slip boundary condition is applied on the bottom. A symmetry boundary
condition (allowing slip velocity) is applied at the “free surface” which is assumed to be
flat. It is recognized that such a condition may be inappropriate in shallow water where
the free surface is subject to significant deformation (see next chapter). The present
simulation also corresponds to the flow in a closed channel formed by reflection of the
solution domain in the symmetry plane. Because of this analogy, the flow is driven by
a mean pressure gradient. In the simulation, this mean pressure gradient is adjusted to
match the experimental Reynolds number. The flow reaches a statistically steady state
when the sum of the pressure and viscous resistance is in balance with the imposed pres-
sure difference between the inlet and outlet. The LES solution is continued for about
15 large-eddy turnover times (L/uτ , where uτ is the time- and space-averaged friction
velocity), and 1000 data sets are stored to calculate the statistics. This proved adequate
to obtain statistically stationary mean velocity profiles and turbulence quantities, such
as the Reynolds shear stress.
5.2 Time-averaged Results
This section presents the time-averaged results in the y = ly/2 mid-plane for
comparison with the LDV experiments. The symbol < · > denotes time-averaging over
1000 data sets. The free-surface velocity U0 at the top boundary at the inlet to the
solution domain is used as the reference velocity. Recent 2D RANS simulation results
of Ryu (2003) with the k − ω turbulence model are plotted for comparison. A 80 × 64
grid was used in the RANS simulation.
5.2.1 Mean Flow Field
Figure 5.3 (a) shows time-averaged streamlines and velocity vectors. For clarity,
only every other point in the streamwise direction is shown. Flow separates at the
dune crest and forms a recirculation zone on the lee side of the dune. The predicted
reattachment point (point of zero friction coefficient) is at x/h = 4.4, close to the
72
experimental estimate of 4.5. A new wall boundary layer develops from the reattachment
point and grows towards the next dune crest. Mean pressure contours are shown in
Fig. 5.3 (b). The mean uniform pressure gradient driving the flow is subtracted from
the pressure in this and subsequent figures in this chapter. An adverse pressure gradient
appears on the lee side of the dune decelerating the flow. A favorable pressure gradient
occurs after the reattachment point on the dune stoss side, accelerating the flow. The
highest pressure is found some distance downstream of the reattachment point, around
x/h = 6, ascending straight to the free surface and implying an elevation of the free
surface.
Figure 5.4 illustrates the free-surface elevation implied by the pressure variation
along the plane of symmetry (the top boundary) assuming hydrostatic balance. The
highest elevation of the free surface coincides with the profile of the pressure maximum
noted above. The largest free-surface deformation provided by the present LES is about
4% of the dune height. This is a little lower than that provided by the RANS calculation.
The free-surface shapes calculated by the two simulations are similar.
Figure 5.5 shows mean profiles of u, v, w, and p at x/h = 2, 4, 5, 6, 12, and 18.
Note that zb is the local vertical distance measured from the dune bed (Fig. 5.1). The
u profiles show that the free-surface velocity is not uniform, as would be expected from
the pressure distribution (and the implied free-surface elevation of Fig. 5.4). Figure 5.5
(b) shows that mean spanwise velocity component (v) is negligible compared to u and w,
indicating that the mean flow is basically 2D. Fig. 5.5 (c) shows that ∂w/∂z is negative
after x/h = 6, implying a positive ∂u/∂x (accelerating flow) due to the continuity
constraint. At x/h = 12, there is zero pressure variation in the vertical direction. This
station demarcates negative and positive vertical pressure gradient ∂p/∂z, also seen in
Fig. 5.4.
Figure 5.6 compares mean u profiles predicted by the present LES with the RANS
results of Ryu (2003) and LDV experiments of Balachandar et al. (2003) at six stream-
wise stations. Overall agreement of LES and RANS with the experiment is rather good,
demonstrating the capability of these methods for predicting the mean velocity. The
73
velocity profile at the station x/h = 12 is under-predicted by both LES and RANS,
although the LES profile is closer in magnitude as well as shape to the experimental
data. This may be due to the fact that station x/h = 12 is a demarcation of positive
and negative ∂p/∂z, as seen in Fig. 5.5 (d). Grid refinement around x/h = 12 may be
necessary to better resolve this transition.
Figure 5.7 shows mean vertical velocity profiles at the six stations. w is positive
at x/h = 12 and 18 due to the flow acceleration resulting from the contraction of the
channel. There is some disagreement between the LES and RANS predictions in the
recirculating flow region at stations x/h = 2 and 4. RANS predicts lower peaks than
LES, but similar profiles in the outer layer. Given the small magnitude of this component
of velocity, however, the agreement between two simulations is considered acceptable.
To determine if the law-of-the-wall (log-law) applies anywhere in such a complex
flow, the mean velocity profile is normalized by wall variables, < u+ >= <u><uτ >
, z+ =
zb<uτ >ν
, and plotted in Fig. 5.8.The local friction velocity is obtained from the slope of
the velocity profile of the bed, in the usual manner. The near-wall velocity profiles at all
the selected stations from the present LES fall on the viscous sublayer curve, u+ = z+,
but none lies on the line showing the usual logarithmic law. The profile lies below the
log-law at x/h = 8, and moves upward and lies above the log-law after x/h = 10.5. The
averaged favorable pressure gradient dp/dx between 7 < x/h < 15 on the dune stoss side
is −1.1 × 10−2, as seen in Fig. 5.3 (b), far beyond the mean pressure gradient driving
the flow (−6.6 × 10−4), where p = p/ρU20 and x = x/h. We can thus conclude that
the standard log-law does not apply in this complex flow due to large favorable pressure
gradients that are present.
5.2.2 Turbulence Intensities and Reynolds Shear Stress
Figure 5.9 shows comparison of the streamwise component of turbulence intensities
(root mean square of u′, e.g., < u′ >rms=<√
(u− < u >)2 >) by the present LES and
the LDV experiment at the six selected stations. At x/h = 2 and all subsequent stations
there is a prominent peak (labeled A) in the turbulence level. This peak broadens and
74
its magnitude decreases with downstream distance (note that the chosen stations are
not equally spaced), until it resembles the secondary peak (labeled B) at x/h = 2. This
secondary peak (B) at x/h = 2 also broadens with distance until it practically disappears
by x/h = 12. The secondary peak (B) at x/h = 2 is, in fact, the remanent of the primary
peak A of the previous dune and represents the upstream history of the flow over the
current dune. Under different flow conditions, therefore, it is possible for the solution to
show peaks beyond the secondary peak, representing effects of more than one upstream
dune. At x/h = 12, there is a noticeable peak (labeled C) very close to the wall. This
peak is more pronounced at x/h = 18, and grows into the peak A of the downstream
dune. The near wall peaks at these stations, far from reattachment, resemble those in
simple boundary layer and pipe flows and suggest that the local flow structure may be
similar to these flows. In general, it is found that the space-periodic LES solution reveals
a very complex flow pattern that evolves over a number of dunes.
The prominent peak in turbulence intensity is associated with the thin shear layer
that lies at the top of the recirculation zone (see Fig. 5.3 (a)) and originates at the dune
crest. However, as will be seen from later results, major contributors to the peak in the
turbulence intensity are large-scale organized structures that arise from flow separation
at the crest.
It is seen from Fig. 5.9 that the LES simulations reproduce most of the features
found in the measurements although there are some quantitative differences. Foremost
among these is the magnitude of the secondary peak B associated with the primary peak
A of the upstream dune. The experiments show a more pronounced peak compared to
that predicted by LES. This leads to the conclusion that the present LES does not
properly preserve the upstream history. In other words, the simulation dissipates the
energy from the previous dune much too quickly.
Figure 5.10 compares the vertical component of turbulence intensity (root mean
square of w′) by the present LES and the LDV experiment. Again the peaks are labeled.
Agreement is good except the primary peak A at x/h = 2. Peaks A in < w′ >rms at
stations x/h = 2, 4, 5, and 6 occur at almost the same location as those in < u′ >rms.
75
Unlike < u′ >rms, secondary peaks B of < w′ >rms are very weak except at x/h = 2,
indicating that the vertical component of turbulence intensity is not significantly affected
by the upstream dunes. The LES results show that < w′ >rms approaches zero at the
free surface. No experimental data is available, however, due to the difficulty in making
measurement close to the free surface.
The Reynolds shear stress < −u′w′ > profiles are shown in Fig. 5.11. Here, the
results of the RANS calculation by Ryu (2003) are also plotted. The peaks, where they
appear, are also labeled as before. LES results are in agreement with the experiment, but
RANS predicts much lower peaks at stations x/h = 2, 4, 5, and 6. The secondary peaks
(B) are somewhat better predicted by the RANS calculation. Under-prediction of the
primary peaks by RANS is due to its inability to take explicit account of large organized
structures which occur in the shear layer. The fact that RANS gives somewhat better
prediction of the the secondary peaks (B) indicates the general suitability of the k − ω
model for complex flows. The resolved Reynolds shear stress is zero at the free surface
at all the stations, satisfying the shear free condition there (note that zero resolved shear
stress is enforced by the symmetry plane boundary condition).
Figures 5.12 and 5.13 show field contours of the three turbulence intensities, tur-
bulence kinetic energy (TKE), i.e., < (u′2 +v′2 +w′2)/2 > and the Reynolds shear stress
< −u′w′ >. Strong turbulence generated by the eddies shed from the dune crest is
convected downstream by the mean flow and dissipated at the dune bed. The strongest
turbulence intensities occur around the core of the recirculation zone. Above the line
connecting the dune crests, turbulence intensities tend to be homogeneous in the stream-
wise direction. This implies that the effect of dune geometry on turbulence intensity is
not felt above some distance from the dune.
Figure 5.14 shows superimposed profiles of turbulence intensities, < u′ >rms, <
v′ >rms, and < w′ >rms, Reynolds shear stresses < −u′w′ > and < −u′v′ >, and TKE
at the six streamwise stations. All profiles nearly collapse to similar curves beyond a
distance of approximately 2.5h, indicating that the mean flow at this distance from the
dune is practically unaffected by the local shape of the dune. Thus, for some purposes the
76
flow may be divided into two regions, the near-wall flow that changes in the x-direction
and an outer layer that is relatively free from wall disturbance.
The largest near-wall turbulence intensities, < −u′w′ >, and TKE occur at station
x/h = 4, immediately before the reattachment point. This indicates the largest turbu-
lence production there, and possible turbulence bursting: violent, intermittent eruption
of fluid away from the bed caused by passage of one or more tilted, quasi-streamwise
vortices (Robinson 1991). The Reynolds shear stress component < −u′v′ > is negligi-
ble compared to the primary component < −u′w′ >, confirming the two-dimensionality
of the mean flow. This also signifies sufficient statistical sample size for this simula-
tion. < u′ >rms is larger than < v′ >rms, and the latter is larger than < w′ >rms,
as observed in plane open-channel flow (Nezu and Nakagawa 1993). This result invali-
dates a frequent assumption in RANS simulations and experimental measurements that
< v′ >rms≈< w′ >rms in estimating TKE. The spanwise turbulence intensity < v′ >rms
increases at the free surface for all the stations (Fig. 5.14 (b)) where the vertical vanishes
(Fig. 5.14 (c)). This suggests a redistribution of TKE among its three components.
5.2.3 Friction and Pressure Coefficients
The presence of dunes on the bed of a channel dramatically increases the stream re-
sistance. Computation of the local wall shear stress τw ( Fig. 5.15) requires consideration
of the dune geometry, i.e.,
τw = (τx, τz) · n (5.1)
where n is the unit normal and is computed as follows (Calhoun 1998; Cui 2000)
n =1√
1 +(
dzdx
)2
(− dzdx
1
)(5.2)
and the components of the shear stress are related to the rates of strain at the wall by
τx =2µ√
1 +(
dzdx
)2(−S11
dz
dx+ S13) (5.3)
77
τz =2µ√
1 +(
dzdx
)2(−S13
dz
dx+ S33) (5.4)
Sij =1
2
(∂ui
∂xj
+∂uj
∂xi
), i = 1, 2, 3 (5.5)
Figure 5.16 shows the time-averaged coefficients of wall friction and pressure along the
dune, Cf = 2τw/ρU20 and Cp = 2(p − pref )/ρU2
0 , calculated by LES and RANS, where
pref is the wall pressure and U0 is the free-surface velocity, both at the flow inlet. The
only available experimental value of Cf (determined from a Clauser chart, i.e., a fit with
the log-law) is at x/h = 18. It lies between the values predicted by LES and RANS.
The region of negative Cf represents the mean recirculating or reverse flow region.
Positions of Cf = 0 represent separation or reattachment points. Clearly, RANS predicts
a reattachment point (5.2h) farther than LES (4.4h). The reattachment position is
difficult to accurately measure by experiment. For the present case, Balachandar et al.
(2003) estimated it to be around 4.5h, close to the LES prediction. Both LES and RANS
predict a small Cf drop around x/λ = 0.78 where the bed slope changes abruptly from
5.0o to 1.8o (see Fig. 5.1). The pressure coefficient Cp is much higher than Cf , indicating
that pressure drag dominates the dune resistance. The highest Cp is around x/h = 6,
downstream of the reattachment point, demarcating flow deceleration and acceleration.
Figure 5.17 shows the root-mean-square of fluctuations in friction and pressure co-
efficients calculated from the LES simulation. Such information is not readily obtained
by experiment, but may be helpful for developing models that connect local flow prop-
erties with sediment transport at the bed. < C ′p >rms magnitude is much larger than
that of < C ′f >rms. The similarity in their shapes after reattachment, in spite of the
differences in the distributions of Cf and Cp, may be an indication of a high level of
flow organization in the wall boundary layer. The highest < C ′p >rms takes place around
x/h = 4, not the position of the highest Cp, ahead of reattachment. < C ′f >rms peaks
at 2.2h and 3.2h. Both gradually decrease toward the next crest.
Figure 5.18 shows the contours of root-mean-square of pressure fluctuations. The
peak value occurs in the recirculation zone immediately below the crest line, suggesting
78
a great change in instantaneous pressure. High values of < p′ >rms contours emanate
from the crest and extend downstream, ending at the dune bed.
5.2.4 Higher-order Turbulence Statistics
Third order moments, or skewness factors, of the velocity fluctuations, e.g., Su =<
u′3 > / < u′ >3rms, are plotted in Fig. 5.19. Sv should be zero if sufficient samples
are collected to compute the average because of the flow homogeneity in the spanwise
direction. The non-zero Sv in the figure indicates somewhat marginal sampling for these
high-order statistics. The skewness Su is mostly opposite in sign to Sw, and positive
away from the wall. This suggests that the major contribution to the Reynolds shear
stress < −u′w′ > comes from the second quadrant events (u′ < 0 and w′ > 0, see
section 5.2.5 below). Much closer to the wall, the fourth quadrant events (u′ > 0 and
w′ < 0) appear to dominate. The zero crossing points of Su (labeled o) at the stations
x/h = 2, 4, 5, and 6 occur near the peak position of < u′ >rms. This was also observed
in plane turbulent Couette flow (Komminaho et al. 1996), but the reason is unclear yet.
Fourth order moments, or flatness factors, of velocity fluctuations, e.g., Fu =<
u′4 > / < u′ >4rms , are plotted in Fig. 5.20. Peaks of Fu at the stations x/h = 2, 4,
5, and 6 take place at the same locations as those for Su in Fig. 5.19, and are far from
the Gaussian value of three, indicating high turbulence intermittency. At the stations
x/h = 12 and 18, there is high Fw near the dune bed, similar to that in turbulent
plane channel flows (Kim et al. 1987; Komminaho et al. 1996), indicating the highly
intermittent character of the vertical velocity in the wall layer.
5.2.5 Quadrant Analysis
Quadrant analysis based on conditional sampling (Lu and Willmarth 1973) is used
to detect turbulent bursts (composed of ejection and sweep events) in wall-bounded
turbulence. As shown in Fig. 5.21a, a second-quadrant event indicates a low-speed
(u′ < 0, v′ > 0) ejection-like motion and a fourth-quadrant event indicates a high-speed
(u′ > 0, v′ < 0) sweep-like motion. A first-quadrant event represents reflected high-speed
79
fluid motion from the sweep and is called outward interaction. A third-quadrant event
represents low-speed fluid motion pushed back from an ejection and is called wallward
interaction (Wallace et al. 1972). In quadrant analysis, the product −u′w′ is sorted into
four quadrants in the u′, w′ plane by
< −u′w′ >i=< −Siu′w′ >, (5.6)
where Si is the quadrant sorting function defined as
Si =
1 if (u′, w′) falls into the ith quadrant
in the u′, w′ plane, i = 1, 2, 3, and 4.
0 otherwise
(5.7)
Figure 5.22 shows the fractional contribution to the Reynolds shear stress < −u′w′ > at
the six selected stations, where the quadrant event Qi =< −u′w′ >i / < −u′w′ > is a
normalized value. It is found that Q1 and Q3 events are almost equal and generally make
small negative contributions. The Q2 and Q4 events make large positive contributions,
with Q2 contributing a little more than Q4. There are peaks in all four events above
the crest line (zb/h between 1 and 2) at stations x/h = 2 and 4. Figure 5.22 gives
information about the relative contribution but no information on the magnitude because
the quadrant fraction sorted by Eq. (5.7) contains all magnitudes. Insufficient sampling
size may result in a large fraction of a small Reynolds shear stress, such as that at the
free surface and near the bed.
To better understand fractional event contributions to the Reynolds shear stress
from different quadrants, the sorting function Si should exclude relatively weak signals.
The crossed area in Fig. 5.21, or a “hole” (Lu and Willmarth 1973), represents relatively
weak signals not correlated with turbulence bursts, which should be sorted out. The
sorting function Si is then changed to
Si =
1 if |u′w′| > H < u′ >rms< w′ >rms and (u′, w′) falls into
the ith quadrant in the u′, w′ plane, i = 1, 2, 3, and 4.
0 otherwise
(5.8)
80
where H is the hole size. The fractional contribution to the Reynolds shear stress
by sorting out the “hole” region from the four quadrant events is shown in Fig. 5.23.
Clearly, Q2 events dominate at almost all the locations, except the near wall regions
at the stations x/h = 2, 4, and 5 where Q4 events account for a greater contribution.
Contributions from Q1 and Q3 events are very weak at most locations. Q1 and Q3
events characterize basically quiescent motions. This is even clearer by showing the
absolute contribution of each quadrant in Fig. 5.24 where all quadrants are normalized
by the reference velocity. Q4 events are only comparable to Q2 events at x/h = 2 but
much weaker than Q2 events at other stations.
5.3 Instantaneous Flow Field
One of the main advantages of LES is that it offers a full view of instantaneous
turbulent flow structures and a deeper insight into turbulence transport mechanisms.
Visualization of instantaneous flow can aid identification of turbulent eddy structures.
In the following an attempt is made to identify salient features of the principal flow
structures.
5.3.1 Reattachment
Figure 5.25 shows time variation of the reattachment point determined by the
zero instantaneous wall shear stress in the mid-channel plane. The reattachment point
fluctuates over a streamwise distance of 3 to 8 times dune height from the crest. Re-
call that the mean position of reattachment is at 4.5h. Similar fluctuations have been
reported by Nezu et al. (1993) (3 to 9 times dune height) on a dune with a different
geometry and flow conditions. The most intriguing feature in Fig. 5.25 is the remarkable
regularity of the slow downstream movement of the reattachment point followed by a
very rapid movement upstream. There appears to be an almost periodic motion with a
non-dimensional time (tuτ/h) interval of about 1.2.
Figure 5.26 displays instantaneous streamlines in a small region close to the lee
81
side of the dune in a consecutive time sequence. The oscillatory pattern of recirculation
zone is clearly visualized. The reattachment point (labeled R) fluctuates with time.
Unlike the mean streamlines in Fig. 5.3 (a), the recirculation zone consists of several
co-rotating vortices in different sizes and shapes. It seems that they arise from a roll-up
of the shear layer coming from the separation at the crest. This is confirmed by a movie
of instantaneous velocity vectors (in the appended CD) revealing organization of well-
defined large-vortical (coherent) structures produced in the shear layer downstream of
flow separation from the dune crest. These structures are almost periodic, rotate in the
clockwise sense and weaken as they travel along the dune. The PIV data (Balachandar
et al. 2002) confirm the existence of such vortices.
Figure 5.27 shows a time sequence of velocity fluctuation fields (u′ and w′) in the
x-z plane of middle-channel. Strong Q2 and Q4 events dominate the near wall motion,
illustrating the occurrence of turbulence ejections (Q2-relative) and sweeps (Q4-relative).
Q2 events dominate the near-bed flow in Fig. 5.27 (b), while Q4 events cover almost
half the dune in Fig. 5.27 (c). The strong Q2 events around the reattachment point
in Fig. 5.27 (b) imply large-scale motions emanating from there. Near-wall Q4 events
increase bed shear stress, and may be strongly linked to sediment transport (Drake
et al. 1988). Q2 events are said to play an important role in sediment suspension of
geophysical flows (e.g., Jackson 1977, Zedler and Street 2001).
Figure 5.28 shows the velocity fluctuation fields (v′ and w′) in the y-z plane at
six selected streamwise stations. The presence of turbulence eddies indicates the three-
dimensionality of the instantaneous flow field. Strong fluctuations at stations x/h = 4
and 6 again suggest occurrence of turbulence ejections and sweeps around the reattach-
ment point.
5.3.2 Flow Structures on Free Surface
To examine flow patterns at the free surface, it is useful to plot the velocity vectors
relative to the free surface with components (u − U0, w) and take a view from above.
This kind of velocity transformation has the advantage of preserving the relative shear
82
between adjacent flow structures (Adrian et al. 2000). Figure 5.29 shows successive
snapshots of the velocity field (u− U0, v) superimposed on the contours of the vertical
velocity w two grid points below the free surface (note that w is zero at the free surface
imposed by the boundary condition). Dark zones represent negative w, and light zones
show positive w. One prominent flow pattern is surface upwelling (labeled U) where
high positive w is accompanied by divergent flow as indicated by the velocity vectors
radiating outwards. Another flow pattern is surface depressions of downdrafts (labeled
D) where negative w is accompanied by converging flow. Single vortex (labeled V) and
vortex-pair (labeled VP) are also observed on the free surface. The vortex-pairs appear
to occur in regions of surface downdrafts. Similar surface flow patterns are also observed
in the experiments in turbulent open-channel flow over a flat bed (Banerjee 1994).
Figure 5.30 shows surface streamlines. Among flow characteristics revealed by
the streamlines are lines of convergence (labeled C) and lines of divergence (labeled D).
Convergence lines are associated with the downdraft motion while divergence lines occur
near the zones of upwelling. All surface vortical structures appear to have spiral instead
of closed shapes. This is because the reference velocity U0 is not equal to the velocities
of the cores of the vortices. A movie showing the evolution of free surface flow structures
is contained in the appended CD.
5.3.3 Streaky Structures Near the Wall
A characteristic of near-wall in flat-wall turbulent boundary layers are the sublayer
streaks, which are believed to trigger turbulence bursts in a process of lifting, oscillating,
and breaking (Robinson 1991). Streaky structures near a flat wall have been visualized
by many investigators through experiments and numerical simulations. To examine
whether streaky structures exist near the dune bed in this complex flow, Fig. 5.31 plots
the contours of u′ at zb/h = 0.0225 or z+b = 9 (where z+
b = zbuτ/ν) in the same
time sequence as in Fig. 5.29. Low-speed streaks (u′ < 0) are observed, lining up and
alternating with high-speed fluid (u′ > 0) on the stoss side of the bed downstream of
reattachment. These spatially organized streaks, always present in this region, are the
83
well-known coherent structures. Organized motions, however, are absent in the near-wall
recirculation zone. High- and low-speed motions are present in the dune trough where
turbulent eddies break down without preference of a specific direction. Short, low-speed
stripes at the crest are immediately disrupted by the strong mixing layer, producing 3D
turbulence structures.
Figure 5.32 shows contours of u′ at a higher location, zb/h = 0.1 or z+b = 40, in the
same time sequence. Low-speed streaks still remain. The spacing between the streaks
is nearly constant at times t0 and t0 + 0.5d/uτ , suggesting that these streaks are rather
thick structures.
Contours of w′ also show elongated structures at z+b = 9 (Figure 5.33). Unlike u′ in
Fig. 5.31, the streamwise elongated stripes of w′ emerge on the dune trough, indicating
that the vertical motion is more spatially organized than the streamwise motion in the
recirculation zone. w′ contours on the dune lee side in Fig. 5.33 correspond to the
opposite sign of the u′ contours in Fig. 5.31, suggesting the occurrence of turbulence-
bursts on the lee side of the dune. Streaky structures of w′ contours are even more
evident at z+b = 40 (Fig. 5.34). Corresponding to low-speed streak positions in Fig. 5.32,
the w′ streaks in Fig. 5.34 are composed of positive contours. They are ejection-related
motions.
5.3.4 3D Vortical Structures
Large-scale vortical structures play an important role in heat and momentum trans-
fer, and sediment transport and dispersion in complex turbulent flows. Identification
of coherent vortical motion is not only useful for understanding turbulent motion, but
crucial in the development of viable turbulence models for complex flows (Jeong and
Hussain 1995).
Vorticity is a traditional measure of vortical structures. Figure 5.35 illustrates iso-
surfaces of the three components of vorticity at a specific time. The vertical component
of vorticity is much weaker than the other two, indicating that large-scale flow struc-
tures are almost horizontal. The vorticity does not distinguish shear layers and vortical
84
motions, however. The mixing layer that originates from the dune crest and the wall
layers near the dune bed produce very large spanwise vorticity (Fig. 5.35 (b)). Vorticity
is thus inappropriate for identifying vortical structures in a strong shear flow.
A more efficient way to identify vortical structures is the λ2 method of Jeong and
Hussain (1995) in which the negative λ2, the second largest eigenvalue of the tensor
SikSkj +ΩikΩkj, is employed to capture vortex core. Here Sij and Ωij are the symmetric
and antisymmetric parts of the deformation rate tensor ∂ui
∂xj, i.e., Sij = 1
2
(∂ui
∂xj+ ∂uj
∂xi
)and
Ωij = 12
(∂ui
∂xj− ∂uj
∂xi
).
Figure 5.36 shows snapshots of the large-scale vortical structures identified by
isosurfaces of λ2 = −200 at the same time sequence as in Figure 5.29. The effects of
shear and mixing layers are absent in these structures, indicating the effectiveness of the
λ2 method in capturing vortical structures. In comparison to Fig. 5.35 (a), the tube-like
vortical structures in Fig. 5.36 (a) are orientated in the streamwise direction. These
quasi-streamwise large-scale eddies are continuously produced in the recirculation zone.
They tilt upward, and travel downstream. Most of them are eventually dissipated before
arriving at the next dune crest. A movie showing the evolution and dissipation of these
large-vortical structures is contained in the appended CD.
Figure 5.37 shows closeup views of Fig. 5.36 superimposed on the instantaneous
cross stream velocity vectors (v, w) on streamwise slices. The isosurfaces of λ2 cross
the cores of strong vortices. There are interactions between these vortical structures,
such as coalition (Figure 5.37 (c)). Cross-stream velocity is upward between two vortical
structures, seen in Figures 5.37 (b) and (c), resulting from counter-rotating neighboring
vortical structures. This may be responsible for vertical transport of sediment (Zedler
and Street 2001).
5.4 Summary
In this chapter, LES was applied on turbulent open-channel flow over a fixed
2D dune with the free surface treated as a plane surface. Turbulence statistics and
instantaneous flow structures were examined. Computational results agree well with the
85
LDV experimental data, demonstrating the accuracy of the present LES. In contrast to
a turbulent open-channel flow over a plane bed, e.g., (Komori et al. 1993; Borue et al.
1995), the law-of-the-wall does not hold in this complex flow. Quadrant analysis shows
that second quadrant events dominate the production of the Reynolds shear stress.
Visualization of instantaneous velocity fluctuation fields shows occurrence of turbulence-
bursts. There are complex flow patterns on the free surface, such as upwellings, down-
drafts, vortices, vortex-pairs, convergence lines, etc. Coherent structures are present
within a layer near the dune bed following flow reattachment. Visualization of λ2 iso-
surfaces clearly shows these coherent large-scale vortical structures and their evolution
and dissipation processes.
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CHAPTER 6
LES-LSM SIMULATION OF TURBULENT OPEN-CHANNEL FLOWOVER A DUNE WITH FREELY DEFORMABLE FREE SURFACE
In this chapter, turbulent flow in an open-channel, with a train of identical 2D
dunes (the same geometry as in Chapter 5) on the bottom, is simulated with the nu-
merical model coupling LES and LSM. LSM allows the free surface to be treated as
a freely deformable interface of air-water two-phase flow. Comparison of the results
with the plane-surface solutions of Chapter 5 enables evaluation of the effect of the free
surface boundary condition on the turbulence statistics and flow structures. Effects of
different SGS models are assessed by comparing simulations with three SGS models,
namely Smagorinsky model (SM), dynamic Smagorinsky model (DSM), and dynamic
two-parameter model (DTM). The effect of flow depth is investigated by simulating for
two flow depths: a deep-water flow (nominal flow depth of 6h), and a shallow-water flow
(nominal flow depth of 3h).
6.1 Description of the Simulated Flow
There are quite a few numerical simulations of turbulent flows with a free-surface,
but most represent the free surface as a plane shear-free boundary, as in Chapter 5.
Very few attempts have been made to represent the physical free surface as it moves
and deforms, particularly in turbulent flow on complex geometry with LES or DNS. The
present model coupling LSM and LES is developed for this purpose. The motion of the
free surface is represented by the evolution of the zero level set function. Free-surface
kinematic and dynamic boundary conditions, including surface tension, are naturally
incorporated into the numerical model as described in Chapter 3.
The SGS model that accounts for unresolved small-scale motions is a very im-
portant factor for LES accuracy. A realistic SGS model should possess the following
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features. First, the eddy viscosity should be determined dynamically and locally. Sec-
ond, the model must have the correct asymptotic behavior near a wall and in laminar
flow. Third, the model should allow energy backscatter in a physically realistic way.
Fourth, the principal axes of the SGS stress tensor are not necessarily aligned with
those of the resolved strain tensor, i.e., it is not necessary to require a local balance be-
tween shear production and dissipation in the SGS TKE equation (Sullivan et al. 1994).
In the present study, SM (Smagorinsky model), DSM (dynamic Smagorinsky model),
and DTM (dynamic two-parameter model) SGS models are employed for comparison.
Not all of these possess the aforementioned features
Unlike the idealized flow considered in Chapter 5, real open-channel flow is driven
by gravity. Hence, pressure in Eq. (2.23) is the true pressure instead of the reduced
pressure in Chapter 5. Developed and space-periodic turbulent flow over a 2D dune
(same as in Chapter 5) is considered here. Two different flow depths are simulated
to investigate the interaction between the bed and the free surface. In the deep-water
case, the computational domain extends to 8h in the vertical (z) direction to cover the
water and air regions. An initially flat free surface starts to deform as the gravity force
is applied. DSM is employed as a baseline case. The water depth at the flow inlet
is around 6.6h in a time-averaged sense. The Reynolds number, based on this water
depth and free surface velocity, is 5.8 × 104, close to the flow in Chapter 5. The same
coordinate system as Fig. 5.1 is used. Spanwise width is still 7h. A 80× 48× 84 grid in
the respective x, y, and z directions is used. The vertical grid spacing between 5h and
7h (around the free surface) is uniform, ∆z = h/10.
The top boundary for air is set as a no-slip wall given that air momentum is neg-
ligible compared to water. Periodic boundary conditions are imposed at the streamwise
and spanwise boundaries. The gravity component, g sin θ, is applied for water only. Flow
reaches a statistically steady state when pressure and viscous resistance components on
the dune bed are in balance with the gravity force of water. The solution is continued for
about 15 large-eddy turnover times to record 1000 subsequent data sets for calculation
of mean flow quantities and turbulence statistics.
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The channel is inclined at an angle of 7.8×10−4 rad (θ in Fig. 1.1) in both deep- and
shallow-water cases. To save computing time, solutions with DTM and SM are initiated
from the developed flow solution with DSM. When DSM and DTM are applied, the
gravity force component, g sin θ, is imposed in the water region excluding the two grid-
cells in the transition zone of LSM, i.e., φ > ε, to avoid a velocity overshoot at free
surface. This is because the filtering process described in Chapter 2 assumes a constant
density, while density varies in the transition zone. When SM is applied, the gravity
force is imposed on φ > 0 (whole water region), without inducing velocity overshoot at
free surface. This is because there is no explicit filtering process in SM.
The deep-water flow is described in the next section. The results of the shallow-
water flow are presented in Section 6.3.
6.2 Deep-Water Flow
6.2.1 Mean Flow Field
This section presents the comparison of time-averaged results by SM, DSM, and
DTM in the mid-plane (y = ly/2 ) of the channel. The reference free-surface velocity U0
is measured at the grid point immediately below the transition zone of LSM at the flow
inlet location.
Figure 6.1 shows the mean streamlines predicted by the three different SGS models.
DSM and DTM predict similar recirculation zones, while SM, with a prescribed eddy
viscosity, predicts a larger one. Since the solutions by DTM and SM are initialized from
the developed flow field by DSM, the mean free-surface positions are not located at the
same place. There are some deviations from the 6.6h of the DSM solution. In Fig. 6.2,
the free-surface elevations are all shifted to 6h at the flow inlet to compare surface shape
and magnitudes with the solution of Chapter 5. In the figure, the line named “Plane-
surface” refers to the solution of Chapter 5 while the lines named “Free-surface” refer
to the solutions of this chapter. The same names are also used in the subsequent figures
of this chapter. The result by DSM is close to the plane-surface solution in magnitude,
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0.04h. DTM predicts a magnitude of 0.035h, and SM 0.03h. All solutions in this chapter
show farther locations of surface elevation maxima than the plane-surface solution.
Figure 6.3 compares the time-averaged coefficients of wall friction and pressure, Cf
and Cp, respectively, along the dune bed. All solutions of this chapter show higher Cf and
lower Cp than the plane-surface solution. The Cf prediction by SM is much higher than
the others, indicating the high near-wall dissipation of SM. After the Cf drop location,
however, predictions by DSM and DTM approach the plane-surface solution and coincide
with the only experimental value. DSM and DTM predict the same reattachment point
(Cf = 0), at 5h, larger than the plane-surface solution (4.4h). SM predicts a farther
reattachment point, 6h. Unlike the plane-surface solution, Cp predicted in this chapter
is negative on the lee side of the dune, accelerating the flow. This causes a slight surface
dip near the flow inlet in Fig. 6.2.
Figure 6.4 compares the mean u profiles by the three SGS models with the LDV
experiments of Balachandar et al (2003) at six streamwise stations. Overall agreement is
good. The velocity profiles at x/h = 12 are under-predicted from the experimental data
by all SGS models, similar to the plane-surface solution. DSM gives the best near-wall
agreement with the experimental data, implying that DTM provides excessive energy
backscatter near the bed. The velocity profiles in the air region are linear, indicating
that the air motion is basically driven by the free-surface velocity, like a Couette flow.
In the following, most of the results with the SM model are no longer considered as they
are the most inaccurate.
Figure 6.5 compares the mean w profiles with the plane-surface solution. They are
very close at x/h = 4, 12, and 18. The predictions of this chapter are lower than the
plane-surface solution at x/h = 2, but higher at x/h = 5 and 6.
Figure 6.6 compares the streamwise turbulence intensity (u′) at the six streamwise
stations with the experimental data. There are prominent peaks (labeled A) at all
the stations, in good agreement with the experimental data. Unlike the plane-surface
solution in Fig. 5.9, the secondary peaks (labeled B) are predicted at x/h = 2, 4, 5, and 6,
suggesting that they are partly induced by the free-surface deformation. The magnitudes
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of the predicted secondary peaks are still smaller than those of the experiment, implying
that the main contribution to the secondary peaks may be the remanent of the upstream
primary peaks A, as discussed in Sec. 5.2.2. As in the plane-surface solution, the near-
wall peaks (labeled C) also appear at x/h = 12 and 18. They grow into the peaks A
of the downstream dune. A noticeable feature in the figure is the enhancement of u′
below the free surface at all the stations. Since no gravity force is applied in the surface
transition zone of LSM, u′ decreases in that zone. u′ increases and forms a peak (labeled
D) in the air region at x/h = 2 and the subsequent stations. The distribution of u′ in
the air region is similar to that expected in a plane shear flow generated by a moving
surface and a stationary outer boundary.
Figure 6.7 compares the spanwise turbulence intensity (v′) with the plane-surface
solution. Though the mean v is almost zero, v′ is comparable to u′ in magnitude, but
the v′ profiles are not much affected by the free-surface conditions. The enhancement of
v′ below the surface transition zone is also observed.
Figure 6.8 compares the vertical turbulence intensity (w′) at the six streamwise
stations. The predicted profiles show values larger than the experimental data but
similar shapes. Near-wall peaks (labeled A) are greater than the plane-surface solution
(see Fig. 5.10). w′ decreases to zero at the free surface, indicating the damping of the
vertical motion by the free surface. The fact that w′ is damped and u′ and v′ are enhanced
near the free surface shows that turbulent kinetic energy is redistributed among its three
components, i.e., from the vertical component to the horizontal ones. There are weak
peaks of w′ (labeled D) in the air region, possibly induced by the wall and free-surface
turbulence.
Figure 6.9 shows the profiles of the Reynolds shear stress −u′w′. Both primary
(labeled A) and secondary peaks (labeled B) are predicted. They are greater than
the plane-surface solution in Figure 5.11. A near-wall peak (labeled C) is observed at
x/h = 2, not found in the plane-surface solution. In the air region, the Reynolds shear
stress is negligible.
125
Figure 6.10 shows the contours of mean TKE with the DSM, DTM and SM solu-
tions. They all show that the largest TKE occurs in the recirculation zone. Above some
distance to the dune crest, the flows tend to be homogenous in the streamwise direction,
as was observed in the plane-surface solution.
Figure 6.11 compares the absolute values of the two quadrant events, Q2 and Q4
at the six streamwise stations. Q1 and Q3 events are both around zero and not shown
in the figure. The Q2 peaks occur around zb/h = 1 at x/h = 4, 5, and 6. The Q4 peaks
occur closer to the wall, around zb/h = 0.4 at x/h = 4, 5, and 6. Magnitudes of the Q2
peaks are more than two times of the Q4 peaks. The large peaks between x/h = 4 and
6 imply the occurrence of large coherent structures within that region, as shown later in
this chapter.
To sum up, DSM and DTM predict very similar mean flow quantities and tur-
bulence statistics, while SM is very dissipative near the wall. The following results for
instantaneous flow field are based only on the DTM model.
6.2.2 Instantaneous Flow Field
Figure 6.12 displays velocity fluctuation fields (u′ and w′) in a time sequence. Q2
events dominate the flow field. Unlike the plane-surface solution in Fig. 5.27, strong
Q2 events occur near the free surface. Clearly, the vertical motion is damped by the
free surface. The streamwise motion at the free surface is enhanced, consistent with the
observation in Fig. 6.6. Between Q2 and Q4 events there is a stagnation point or line
(labeled A). The large velocity fluctuations in the air region in Figure 6.12 (c) (labeled
B) contribute to the peaks D in Figs. 6.6 and 6.8.
Figure 6.13 shows the contours of u′ at zb/h = 0.04 or z+b = 13 in the same time
sequence as in Fig. 6.12. Low-speed streaks (u′ < 0) are lined up and alternate with
high-speed fluid (u′ > 0) on the stoss side of the dune. These spatially organized streaks
are present from z+b = 8 to 80, suggesting that they are rather thick. These coherent
structures resemble those in the plane-surface solution of Chapter 5. Figure 6.14 shows
the contours of w′ at zb/h = 0.04 or z+b = 13. The streamwise elongated stripes of
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positive w′ appear almost along the whole dune bed, from z+b = 8 to 170. The contours
of spanwise velocity fluctuation v′ are shown in Fig. 6.15 at z+b = 13. The streaky
structures composed of negative v′ are also observed, and are present from z+b = 3 to 32.
Figure 6.16 illustrates the evolution of large vortical structures identified by isosur-
faces of λ2 (= −200) in the same time sequence as above. Similar to the plane-surface
solution in Fig. 5.36, the quasi-streamwise elongated vortical structures dominate the
near-wall flow. Most are generated in the recirculation zone and dissipated quickly, but
some are convected to the crest of the next dune. These large structures rarely reach
the free surface, thus the free surface is not significantly disturbed by them. A movie
showing the evolution of these large vortical structures is contained in the appended CD.
Figure 6.17 shows a magnified view of the free surface. Surface ripples and waves
are clearly visualized. Some surface flow patterns identified in the plane-surface solution
of Chapter 5 are clearly revealed here, such as upwellings (labeled U) and downdrafts
(labeled D). An animation of the free surface evolution is contained in the appended
CD.
6.3 Shallow-Water Flow
To investigate the effect of flow depth on the free surface and flow structures, a
shallow-water flow (of initial depth 3h) is simulated. Only DSM is employed in this LES.
The vertical extent of the computational domain is 5h, covering both air and water. The
dune geometry and spanwise width are the same as those in the deep-water flow. The
time-averaged free surface at the flow inlet is around 3.24h. The Reynolds number,
based on this water depth and the free-surface velocity, is about 1.07× 104, lower than
the deep-water flow. A 80× 48× 72 grid in the respective x, y, and z directions is used.
Turbulence statistics and mean flow quantities are calculated over 1000 data samples
recorded after the flow reaches a statistically steady-state.
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Table 6.1: Comparison of normalized free-surface velocity (< u > /U0) between deep-and shallow-water flows
free-surface velocity
Water depth x/h = 2 x/h = 4 x/h = 5 x/h = 6 x/h = 12 x/h = 18
Deep 1.0 0.99 0.98 0.975 0.98 1.0
Shallow 1.1 1.05 1.0 0.975 0.9 0.96
6.3.1 Mean Flow Field
Figure 6.18 shows the time-averaged free-surface elevation. There is a dip in the
surface, 0.014h, downstream of the crest, greater than the dip in the deep-water flow.
The highest surface elevation is about 0.03h. Note that the Reynolds number is five
times smaller than the deep water flow.
Figure 6.19 shows a comparison of the mean streamwise velocity between the deep-
and shallow-flows at the six streamwise stations. The near-wall profiles in the two flows
are very close to each other except at x/h = 12, indicating the near-wall similarity of the
mean velocity independent of the water depth and Reynolds number. The maximum
velocity occurs immediately beneath the surface transition zone of LSM, and is labeled
•. Velocity decreases in the surface transition zone. There are peaks in the air region in
the shallow-water flow, indicating strongly disturbed air motion by the free surface and
the top wall. If the maximum velocity is considered as the “real” free-surface velocity,
it is more non-uniform in the shallow-water flow than in the deep-water flow, as shown
in Table 6.1.
Figure 6.20 compares the rms of u′ between the shallow- and deep-water flows. The
profile shapes are very similar, but the streamwise turbulence intensity in the shallow-
water flow is considerably higher than that in the deep-water flow even though the
Reynolds number of the shallow-water is five times smaller. There is a sharp increase of
u′ at the free surface at the stations x/h = 2, 4, 5, and 6, but a decrease at the stations
x/h = 12 and 18, in the shallow-water flow. If the surface transition zone is neglected,
128
u′ is enhanced at the “water” surface at all stations in both flows. It is noted that the
secondary peaks are absent in the shallow-water flow. This may be because the near-bed
turbulence is so strong and the flow depth is so low that there is not sufficient space
for the free surface to adjust the flow. The peaks appearing in the air region are much
greater than in the deep-water flow, indicating significant influence of the free surface
on the air motion.
Figure 6.21 compares the rms of w′ between the shallow- and deep-water flows.
The vertical turbulence intensity in the shallow-water flow is also higher than that in
the deep-water flow. In both flows, w′, similar to the plane-surface solution of Chapter
5, approaches zero at the free surface, suggesting redistribution of turbulence kinetic
energy among its three components, i.e., transfer from the vertical to the horizontal
directions. The peaks in the air region again indicate the stronger air motion induced
by the free-surface in the shallow-water flow.
Figure 6.22 compares the absolute values of the four quadrant events at four se-
lected stations. The plots are placed in a natural quadrant position. Same as in the
deep-water flow, Q1 and Q3 events are very weak, around zero. The Q2 peaks occur
between x/h = 1 and 2, while the Q4 peaks occur closer to the dune bed. Magnitudes
of the Q2 peaks are more than two times of those of the Q4 peaks, suggesting the dom-
inance of Q2 events in the flow. In the outer layer of the flow and the air region, the
Reynolds shear stress is negligible.
6.3.2 Instantaneous Flow Field
Figure 6.23 displays a time sequence of velocity fluctuation fields (u′ and w′). Q2
and Q4 events spread over most of the flow field. Q2 and Q4 events appear near the
free surface. It is clearly seen that the vertical motion is damped at the free surface.
Consequently, large velocity disturbance is induced in the air near the free surface,
causing the peaks in the mean flow field (Fig. 6.19) and turbulence intensities (Fig. 6.20
and Fig. 6.21).
Figure 6.24 shows the contours of u′ at zb/h = 0.04 or z+b = 11 in the same time
129
sequence as in Fig. 6.23. Unlike the deep-water flow, there are no streaky structures
appearing in the wall layer. Figure 6.25 shows the contours of w′ at z+b = 11. The
streamwise elongated stripes appear in the recirculation zone and extend some distance
downstream, but are absent farther away. This, together with Fig. 6.24, suggests that
coherency is strongly disturbed in the wall layer.
Figure 6.26 shows the large vortical structures identified by the isosurfaces of λ2
(= −100) in the same time sequence as above. Similar to the deep-water flow in Fig. 6.16,
quasi-streamwise elongated tube-like vortical structures dominate the near-wall flow.
Due to the low free-surface position, some of them reach and deform the free surface.
A movie showing the evolution of these vortical structures is contained in the appended
CD.
Figure 6.27 shows a magnified view of the free surface. The surface dip from
the flow inlet is clearly seen. Similar to the deep-water flow, downdrafts (labeled D)
and upwellings (labeled U) are present on the free surface, suggesting that these flow
patterns are characteristics of free surface in turbulent open-channel flows. A movie
showing free-surface variation is contained in the appended CD.
6.4 Summary
The model coupling LSM and LES was applied to simulate developed turbulent
open-channel flows over a fixed 2D dune with freely deformable free surface. The results
were compared with the plane-surface solution of the previous chapter and experimental
data. Three SGS models, namely Smagorinsky model, dynamic Smagorinsky model,
and dynamic two-parameter model, were applied for comparison. Two flow depths were
simulated to investigate the effects of flow depth on the free surface and flow structures.
In the deep-water flow, the predictions of mean flow quantities and turbulence
statistics are in agreement with experimental data. Especially, there is a significant im-
provement of the streamwise component of turbulence intensities over the plane-surface
solution of the previous chapter, suggesting that the secondary peaks are partly con-
tributed by the free-surface deformation. In contrast to the plane-surface solution, there
130
are strong Q2 events near the free surface. Similar to the plane-surface solution, near-
wall streaky structures appear in the deep-water flow, indicating the flow coherency in
the wall layer. Some free-surface flow patterns identified in the plane-surface solution
are revealed in the deep-water flow, such as upwellings and downdrafts. The computa-
tional results show that DSM and DTM predict very similar mean flow quantities and
turbulence statistics, while SM is very dissipative near the wall.
Turbulence intensities in the shallow-water flow are greater than those in the deep-
water flow, though the Reynolds number is five time smaller. The air is strongly per-
turbed by the free surface. The secondary peaks are absent in the shallow-water flow,
suggesting that the near-bed turbulence is so strong and the flow depth is so low that
there is not sufficient space for the free surface to adjust the flow. Another prominent
phenomenon in the shallow-water flow is the absence of near-wall streaky structures,
implying that coherency is strongly disturbed in the wall boundary layer. Unlike the
deep-water flow, some of large vortical structures reach and deform the free surface.
Free-surface flow patterns, such as upwellings and downdrafts, are also observed in the
shallow-water flow.
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CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
7.1 Conclusions
The level set method (LSM) is coupled with the incompressible Navier-Stokes
equations in general curvilinear coordinates to simulate free-surface flows on a fixed
grid. The free surface is treated as air-water interface in a two-phase flow. The kinematic
condition of free surface is represented by the evolution of the zero level set function.
Surface tension is diffused into a volume force, embedding the dynamic condition of free
surface in the Navier-Stokes equations. Large eddy simulation (LES) is incorporated in
the model to simulate turbulent free-surface flows, with the option of three SGS models,
namely Smagorinsky model (SM), dynamic Smagorinsky model (DSM), and dynamic
two-parameter model (DTM).
A four-step fractional step method (FSM) is implemented to advance solution of
the incompressible Navier-Stokes equations in time. In comparison to traditional three-
step FSMs, the splitting error in this FSM is of the second-order in time. The physical
boundary velocity can be used as boundary conditions of the intermediate velocity,
maintaining a consistent second-order temporal accuracy. High-order essentially non-
oscillatory (ENO) schemes are developed in general curvilinear coordinates, verified
and validated by some benchmark cases, such as Zalesak’s problem and stretching of
a circular fluid element. Computations on some benchmark free-surface flows, such as
2D laminar open-channel flow, a travelling solitary wave, and 2D and 3D dam-breaking,
show the accurate predictions of free-surface motions by the present model coupling
LSM and the incompressible Navier-Stokes equations.
Turbulent open-channel flow over a fixed 2D dune is simulated with the free surface
treated in two different ways. First, it is modelled as a fixed undisturbed plane surface.
159
Next, it is simulated as a freely deformable air-water interface. In the former case, the
single-phase large eddy simulation is used, establishing a baseline for the latter case.
In the latter case, the model coupling level set method and large eddy simulation is
employed to simulate the air-water two-phase flow. The numerical predictions in both
cases are in good agreement with the experimental data. Complex flow patterns on the
free surface, such as surface-upwellings and downdrafts, are revealed. It is found that the
second quadrant burst events dominate the production of the Reynolds shear stress in
both cases. Streaky structures appear in the wall layer, indicating that flow coherency
does not seem to be much affected by the free-surface condition. The reason for the
above similarities in these two cases may be because the flow is deep enough for the free
surface not to significantly affect the flow structures especially near the wall.
In spite of the above similarities, there are also some difference in the predictions by
the two simulations. For example, the secondary peaks in the profiles of the streamwise
component of turbulence intensities are absent in the simulation with a plane surface,
but predicted by the simulation with a freely deformable surface, suggesting that the
free-surface deformation partly contributes to the secondary peaks. However, the major
contribution to the secondary peaks may be from the remanent of the primary peaks
of the upstream dune. This hypothesis can be confirmed by simulating flow over two
dunes in the solution domain. The simulation with a freely deformable surface predicts
longer mean reattachment point than the simulation with a plane surface, and also larger
Reynolds shear stress near the dune bed, demonstrating the importance of free-surface
condition.
To investigate the effects of flow depth on the free surface and flow structures, a
shallow-water flow with the flow-depth one half of the deep-water flow is simulated. The
channels have the same slope, resulting in the Reynolds number of the deep-water flow
five-times larger than the shallow-water flow. Even though the Reynolds number and
the flow depth are different in the two flows, some flow similarities are observed between
them. For instance, the near-wall profiles of mean streamwise velocity are very similar.
The second quadrant burst events dominate the production of the Reynolds shear stress.
160
Upwellings and downdrafts on the free surface are stronger in the shallow-water flow.
Different from the deep-water flow, the streaky structures are absent in the wall layer
of the shallow-water flow, indicating that near-wall flow coherency is destroyed. The
secondary peaks in the profiles of the streamwise component of turbulence intensities are
absent in the shallow-water flow, suggesting that the near-bed turbulence is so strong
and the flow depth is so low that there is not sufficient space for the free surface to
adjust the flow. The air is strongly perturbed by the free surface in the shallow-water
flow, indicating stronger free-surface motions than deep-water flow. In the shallow-water
flow, the large tube-like vortical structures are closer to the free surface, causing stronger
interaction between the bed and the free surface.
7.2 Recommendations for Future Work
1. In level set method, mass conservation may break down when the free surface is
subject to rapid topological change. The volume constraint added in the reinitial-
ization equation in Chapter 2 does not exactly “freeze” the interface in numerical
discretization, causing mass loss or gain. A more refined method to numerically
freeze the interface during reinitialization will greatly improve mass conservation
problem in level set method. This is a topic to be considered in future development
of the level set method.
2. Other aspects of interest in the future development of the level set method include
the following. First, LSM based on unstructured grids or finite element methods
will be very useful for free-surface flows subject to irregular obstacles or boundaries.
Second, the PDE-based reinitialization scheme is computationally expensive. A
more efficient scheme is desirable. The geometry-based Fast Marching Method
(Sethian 1999) may be one choice. Third, WENO may be used to replace ENO
for higher accuracy and robustness (Jiang and Peng 2000).
3. To ascertain why the present numerical simulations underestimate the secondary
peaks in the profiles of the streamwise component of turbulence intensities in the
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turbulent flow over a dune, it is necessary to simulate the flow over two identical
dunes in the solution domain so that the residual turbulence in the first dune can
be transported to the second one.
4. To achieve better agreement with the experimental data at the station x/h = 12
in Chapters 5 and 6, a grid refinement around this station may be necessary.
5. In the present simulations of turbulent flow over a 2D dune, the side boundaries
are periodic, while the experiments are bounded by side walls. The effects of the
side walls on the flow structures remains to be investigated. Large eddy simulation
of such a wall-bounded open-channel flow is computationally expensive, however.
6. Though the single-phase large eddy simulation has already been parallelized with
MPI (message-passing-interface), parallelization of the model coupling level set
method and large eddy simulation will enable applications with fine grids and
more complex geometries.
7. In the shallow-water case considered here, the variation of free-surface elevation is
still small. It may be of interest to simulate even shallower flows (flow-depth of 2h
or less) to study interaction between the bed and the free surface.
8. Natural dunes are often three-dimensional and the flow features may be quite
different from 2D dunes. An experimental investigation of flow over fixed, artificial,
sinuous-crested 3D dunes was recently carried out by Maddux (2002). It would be
of great interest to perform a numerical simulation of such a flow.
BIBLIOGRAPHY
Adrian, R. J., C. D. Meinhart, and C. D. Tomkins (2000). Vortex organization in theouter region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.
Anderson, D. M., G. B. McFadden, and A. A. Wheeler (1998). Diffuse-interface meth-ods in fluid mechanics. Ann. Rev. Fluid Mech. 30, 139–165.
Aslam, T. D., J. B. Bdzil, and D. S. Stewart (1996). Level set methods applied tomodelling detonation shock dynamics. J. Comput. Phys. 126, 390–409.
Balachandar, R., B.-S. Hyun, C. Polatel, and V. C. Patel (2003). Structure of flowover a fixed dune and effect of depth. Submitted to J. Hydr. Eng .
Balachandar, R., C. Polatel, B.-S. Hyun, K. Yu, C.-L. Lin, W. Yue, and V. C. Patel(2002). LDV, PIV and LES investigation of flow over a fixed dune. In Proceedingsof the Symposium held in Monte Verita: Sedimentation and Sediment Transport,pp. 171–178. Kluwer Academic Publisher.
Banerjee, S. (1994). Upwellings, downdrafts, and whirlpools: Dominant structures infree surface turbulence. Appl. Mech. Rev. 47, S166–172.
Bennett, S. J. and J. L. Best (1995). Mean flow and turbulence structure over fixedtwo-dimensional dunes: implications for sediment transport and bedform stability.Sedimentology 42, 491–513.
Borue, V., S. A. Orszag, and I. Staroselsky (1995). Interaction of surface waves withturbulence: direct numerical simulations of turbulent open-channel flow. J. FluidMech. 286, 1–23.
Brackbill, J. U., D. B. Kothe, and C. Zemach (1992). A continuum method for mod-elling surface tension. J. Comput. Phys. 6, 68–94.
Calhoun, R. J. (1998). Numerical investigations of turbulent flow over complex terrain.Ph. D. Thesis, Stanford University.
Chan, R. K. and R. L. Street (1992). A computer study of finite amplitude waterwaves. J. Comput. Phys. 100, 335–354.
Chang, Y. C., T. Y. Hou, B. Merriman, and S. Osher (1996). A level set formulationof Eulerian interface capturing methods for incompressible fluid flows. J. Comput.Phys. 124, 449–464.
Choi, H. and P. Moin (1994). Effects of the computational time step on numericalsolutions of turbulent flow. J. Comput. Phys. 113, 1–4.
Chorin, A. J. (1968). A numerical solution of the Navier-Stokes equations. Math.
163
Comput. 22, 745–762.
Crandall, M. and P. Lions (1983). Viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc. 277, 1–42.
Crowley, W. P. (1971). Flag: A free-Lagrangian method for numerically simulatinghydrodynamic flows in two dimensions. In Lecture Notes in Physics 8, pp. 37–43.Springer-Verlag.
Cui, J. (2000). Large-eddy simulation of turbulent flow over rough surfaces. Ph. D.Thesis, The University of Iowa.
Drake, T, G., R. L. Shreve, W. E. Dietrich, P. J. Whiting, and L. B. Leopold (1988).Bedload transport of fine gravel observed by motion-picture photography. J. FluidMech. 192, 193–217.
Dukowicz, J. K. and A. S. Dvinsky (1992). Approximate factorization as a high ordersplitting for the implicit incompressible flow equations. J. Comput. Phys. 102,336–347.
Ferziger, J. H. and M. Peric (1997). Computational methods for fluid dynamics.Springer.
Floryan, J. M. and H. Rasmussen (1989). Numerical methods for viscous flows withmoving boundaries. Appl. Mech. Rev. 89, 323–341.
Foster, N. and R. Fedkiw (2001). Practical animation of liquids. SIGGRAPH 21,15–22.
Fulgosi, M., D. Lakehal, S. Banerjee, and V. De Angelis (2003). Direct numericalsimulation of turbulence in a sheared air-water flow with a deformable interface.J. Fluid Mech. 482, 319–345.
Germano, M., U. Piomelli, P. Moin, and W. H. Cabot (1991). A dynamic subgrid-scaleeddy viscosity model. Phys. Fluids A 3, 1760–1765.
Ghia, U., K. N. Ghia, and C. T. Shin (1982). High-resolution for incompressible flowusing the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48,387–411.
Ghosal, S., T. S. Lund, P. Moin, and W. H. Cabot (1995). A dynamic localizationmodel for large-eddy simulation of turbulent flows. J. Fluid Mech. 286, 229–255.
Glimm, J., O. M. Bryan, R. Menikoff, and D. H. Sharp (1987). Front tracking appliedto Rayleigh-Taylor instability. SIAM J. Sci. Stat. Computing 7, 230–251.
Gomes, J. and O. D. Faugeras (2000). Reconciling distance functions and level sets.J. Visual Communication and Image Representation 11, 209–223.
Guyon, E., J.-P. Hulin, L. Petit, and C. D. Mitescu (2001). Physical hydrodynamics.Oxford University Press.
Handler, R. A., T. F. Swean Jr, R. I. Leighton, and J. D. Swearingen (1993). Length
164
scale and the energy balance for turbulence near a free surface. AIAA J. 31, 1998–2007.
Harlow, F. H. and J. E. Welch (1965). Numerical study of large-amplitude free surfacemotions. Phys. Fluids 8, 2182–2189.
Harten, A., B. Engquist, S. Osher, and S. Chakravarthy (1987). Uniformly high-orderaccurate essentially non-oscillatory schemes. J. Comput. Phys. 71, 231–303.
Hayashi, S., T. Ohmoto, and K. Takikawa (2003). Direct numerical simulation ofcoherent vortex structures in an open-channel flow over dune type wavy bed. J.Hydrosci. Hydrau. Eng. 21, 1–10.
Hirt, C. W., A. A. Amsden, and J. L. Cook (1987). Uniformly high-order accurateessentially non-oscillatory schemes. J. Comput. Phys. 71, 231–303.
Hirt, C. W. and B. D. Nichols (1981). Volume of fluid (VOF) methods for the dynamicsof free boundaries. J. Comput. Phys. 39, 201–225.
Iafrati, A., A. D. Mascio, and E. F. Campana (2001). A level set technique applied tounsteady free surface flows. Int. J. Numer. Meth. Fluids 35, 281–297.
Jackson, R. (1977). Sedimentological and fluid-dynamic implications of the turbulentbursting phenomenon in geophysical flows. J. Fluid Mech. 77, 531–560.
Jacqmin, D. (1999). Calculation of two-phase Navier-Stokes flows using phase-fieldmodeling. J. Comput. Phys. 155, 96–127.
Jeong, J. and F. Hussain (1995). On the identification of a vortex. J. Fluid Mech. 285,69–94.
Jiang, G. and D. Peng (2000). Weighted ENO schemes for Hamilton-Jacobi equations.Siam J. Sci. Comput. 21, 2126–2143.
Jiang, G. and C. Wu (1999). A high-order WENO finite difference scheme for theequations of ideal magnetohydrodynamics. J. Comput. Phys. 150, 561–594.
Jimenez, J. and R. D. Moser (1998). Large-edduy simulations: where are we and whatcan we expect? AIAA J. 38, 605–612.
Johns, B., R. L. Soulsby, and J. Xing (1993). A comparison of numerical model ex-periments of free surface flow over topography with flume and field observations.J. Hydr. Res. 31, 215–228.
Kadota, A. and I. Nezu (1999). Three-dimensional structure of space-time correlationon coherent vortices generated behind dune crest. J. Hydr. Res. 37, 59–79.
Kelecy, F. J. and R. H. Pletcher (1997). The development of a free surface capturingapproach for multidimensional free surface flows in closed containers. J. Comput.Phys. 138, 939–980.
Kim, J. and P. Moin (1985). Application of a fractional-step method to incompressibleNavier-Stokes equations. J. Comput. Phys. 59, 308–323.
165
Kim, J., P. Moin, and R. Moser (1987). Turbulence statistics in fully developed channelflow at low Reynolds number. J. Fluid Mech. 177, 133–166.
Kim, Y. T., N. Goldenfeld, and J. Dantzig (2000). Computation of dendritic mi-crostructures using a level set method. Phys. Rev. E 62, 2471–2474.
Komminaho, J., A. Lundbladh, and A. V. Johansson (1996). Very large structures inplane turbulent Couette flow. J. Fluid Mech. 320, 259–285.
Komori, S., Y. Murakami, and H. Ueda (1989). The relationship between surface-renewal and bursting motions in an open-channel flow. J. Fluid Mech. 203, 103–123.
Komori, S., R. Nagaosa, Y. Murakami, S. Chiba, K. Ishii, and K. Kuwahara (1993). Di-rect numerical simulation of three-dimensional open-channel flow with zero-sheargas-liquid interface. Phys. Fluids A 5, 115–125.
Lesieur, M. and O. Metais (1996). New trends in large-eddy simulation of turbulence.Annu. Rev. Fluid Mech. 28, 45–82.
Leveque, R. J. (1996). High-resolution conservative algorithms for advection in in-compressible flow. SIAM J. Numer. Analysis 33, 627–665.
Lilly, D. K. (1992). A proposed modification of the Germano subgrid scale closuremethod. Phys. Fluids A 4, 633–635.
Liu, X. D., R. P. Fedkiw, and M. Kang (2000). A boundary condition capturingmethod for Poisson’s equation on irregular domains. J. Comput. Phys. 160, 151–178.
Lu, S. S. and W. W. Willmarth (1973). Measurements of the structure of the Reynoldsstress in a turbulent boundary layer. J. Fluid Mech. 60, 481–511.
Lyn, D. A. (2002). Flow and transport over dunes. J. Hydr. Eng. 128, 726–728.
Maddux, T. B. (2002). Turbulent open channel flow over fixed three-dimensional duneshapes. Ph. D. Thesis, University of California, Santa Barbara.
Martin, J. C. and W. J. Moyce (1952). An experimental study of the collapse of liquidcolumns on a rigid horizontal plate. Philos. Trans. Roy. Soc. London Ser., A 244,312–324.
Mei, C. C. (1989). The applied dynamics of ocean surface waves. World Scientific.
Mendoza, C. and H. W. Shen (1990). Investigation of turbulent flow over dunes. J.Hydr. Eng. 116, 459–477.
Meveveau, C., T. S. Lund, and W. H. Cabot (1996). A Lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech. 319, 353–381.
Muller, A. and A. Gyr (1986). On the vortex formation in the mixing layer behinddunes. J. Hydr. Res. 24, 359–375.
166
Nakagawa, H. and I. Nezu (1981). Structure of space-time correlations of burstingphenomena in an open-channel flow. J. Fluid Mech. 104, 1–43.
Nezu, I. and H. Nakagawa (1989). Turbulent structure of backward-facing step flowand coherent vortex shedding from reattachment in open channel flows. In Turbu-lent Shear Flows 6, pp. 313–337.
Nezu, I. and H. Nakagawa (1993). Turbulence in open-channel flows. A.A.Balkema.
Nezu, I. and W. Rodi (1986). Open-channel flow measurements with a laser doppleranemometer. J. Hydr. Eng. 112, 335–355.
Noh, W. F. and P. Woodward (1976). SLIC (simple line interface calculation). Invan der Vooren and P. Zandbergen (Eds.), Lecture Notes in Physics 59, pp. 330–340. Springer-Verlag.
Osher, S. and R. P. Fedkiw (2001). Level set method: an overview and some recentresults. J. Comput. Phys. 169, 463–502.
Osher, S. and J. Sethian (1988). Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49.
Peng, D., B. Merriman, S. Osher, H. Zhao, and M. Kang (1999). A PDE-based fastlocal level set method. J. Comput. Phys. 155, 410–438.
Perng, C. Y. and R. L. Street (1991). A coupled multigrid-domain-splitting techniquefor simulating incompressible flows in geometrically complex domains. Int. J. Nu-mer. Meth. Fluids 13, 269–286.
Perot, J. B. (1993). An analysis of the fractional step method. J. Comput. Phys. 108,51–58.
Piomelli, U. (1999). Large-eddy simulation: achievements and challenges. ProgressAero. Sci. 35, 335–362.
Ramaswamy, B. (1994). Numerical simulation of unsteady viscous free surface flow.J. Comput. Phys. 90, 396–430.
Rashidi, M. (1997). Burst-interface interactions in free surface turbulent flows. Phys.Fluids 9, 3485–3501.
Rider, W. J. and D. B. Kothe (1998). Reconstructing volume tracking. J. Comput.Phys. 141, 112–152.
Robinson, S. K. (1991). Coherent motions in the turbulent boundary layer. Ann. Rev.Fluid Mech. 23, 601–639.
Rudman, M. (1997). Volume-tracking methods for interfacial flow calculations. Int.J. Numer. Meth. Fluids 24, 671–691.
Ryu, D.-N. (2003). Analysis of 2D turbulent heat/fluid flows by k− ω model. privatecommunication.
167
Salvetti, M. V. and S. Banerjee (1994). A priori tests of a new dynamic subgrid-scalemodel for finite-difference large-eddy simulation. Phys. Fluids 7, 2831–2847.
Salvetti, M. V., Y. Zang, R. L. Street, and S. Banerjee (1997). Large-eddy simula-tion of free-surface decaying turbulence with dynamic subgrid-scale models. Phys.Fluids 9, 2405–2419.
Sethian, J. A. (1999). Level set method and fast marching methods. Cambridge Uni-versity Press.
Sethian, J. A. and D. Adalsteinsson (1997). An overview of level set methods foretching, deposition, and lithography development. IEEE Transaction on Semicon-ductor Manufacturing 10, 167–184.
Shen, L. and D. K. P. Yue (2001). Large-eddy simulation of free-surface turbulence.J. Fluid Mech. 440, 75–116.
Shi, J., T. G. Thomas, and J. J. R. Williams (2001). Free-surface effects in openchannel flow at moderate Froude and Reynolds numbers. J. Hydr. Res. 38, 465–474.
Shu, C. W. and S. Osher (1989). Efficient implementation of essentially non-oscillatoryshock-capturing schemes. J. Comput. Phys. 83, 32–78.
Shyy, W. and R. Mittal (1998). Solution methods for the incompressible Navier-Stokesequations. In R. W. Johnson (Ed.), The Handbook of Fluid Dynamics, pp. 31. CRCPress.
Simons, D. B. and E. V. Richardson (1963). Forms of bed roughness in alluvial chan-nels. Trans. Amer. Soc. Civ. Eng. 128, 284–302.
Smagorinsky, J. (1963). Gerneral circulation experiments with the primitive equations.I. The basic experiment. Monthly Weather Review 91, 99–165.
Smereka, P. (2000). Spiral crystal growth. Physica D 138, 282–301.
Smiljanovski, V., V. Moser, and R. Klein (1997). A capturing-tracking hybrid schemefor deflagration discontinuities. Combustion Theory and Modelling 1, 183–215.
Smith, C. R. and S. P. Metzler (1983). The characteristics of low-speed streaks in thenear-wall region of a turbulent boundary layer. J. Fluid Mech. 129, 27–54.
Son, G. and V. K. Dhir (1998). Numerical simulation of film boiling near criticalpressures with a level set method. J. Heat Transfer 120, 183–192.
Sullivan, P. P., J. C. McWilliams, and C. H. Moeng (1994). A subgrid-scale model forlarge-eddy simulation of planetary boundary-layer flows. Boundary-Layer Meteo-rology 71, 247–276.
Sussman, M., E. Fatemi, P. Smereka, and S. Osher (1998). An improved level setmethod for incompressible two-phase flows. Computers & Fluids 27, 663–680.
Sussman, M. and P. Smereka (1997). Axisymmetric free boundary problems. J. Fluid
168
Mech. 341, 269–294.
Sussman, M., P. Smereka, and S. Osher (1994). A level set approach for computingsolutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159.
Tennekes, H. and J. L. Lumley (1972). A first course in turbulence. The MIT Press.
The, J. L., G. D. Raithby, and G. D. Stubley (1994). Surface-adaptive finite-volumemethod for solving free surface flows. Numerical Heat Transfer, Part B 26, 367–380.
Tryggvason, G., B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han,S. Nas, and Y.-J. Jan (2001). A front-tracking method for the computations ofmultiphase flow. J. Comput. Phys. 169, 708–759.
Unverdi, S. and G. Tryggvason (1992). A front-tracking method for viscous, incom-pressible, multi-fluid flows. J. Comput. Phys. 100, 25–37.
Verzicco, R., J. Mohd-Yusof, P. Orlandi, and D. Haworth (2000). Lareg eddy simula-tion in complex geometric configurations using boundary body forces. AIAA J. 38,427–433.
Wallace, J. M., H. Eckelmann, and R. S. Brodkey (1972). The wall region in turbulentshear flow. J. Fluid Mech. 54, 39–48.
Yalin, M. S. (1977). Mechanics of sediment transport (2nd Edition). Pergamon Press.
Yoon, J. Y. and V. C. Patel (1996). Numerical model of turbulent flow over sanddune. J. Hydr. Eng. 122, 10–18.
Youngs, D. L. (1982). Time-dependent multi-material flow with large fluid distortion.In K. W. Morton and M. J. Baines (Eds.), Numerical Methods for Fluid Dynamics,pp. 273–285. Academic.
Yue, W. (2001). Numerical simulation of free surface waves induced by submergedhydrofoil. Unpublished .
Yue, W., C.-L. Lin, and V. C. Patel (2003). Numerical simulation of unsteady multi-dimensional free surface motions by level set method. Int. J. Numer. Meth. Fluids,in press .
Zalesak, S. T. (1979). Fully multidimensional flux-corrected transport algorithms forfluids. J. Comput. Phys. 31, 335–362.
Zang, Y., R. L. Street, and J. R. Koseff (1993). A dynamic mixed subgrid-scale modeland its application to turbulent recirculating flows. Phys. Fluids A 5, 3186–3196.
Zang, Y., R. L. Street, and J. R. Koseff (1994). A non-staggered grid, fractional stepmethod for time-dependent incompressible Navier-Stokes equations in curvilinearcoordinates. J. Comput. Phys. 114, 18–33.
Zedler, A. and R. L. Street (2001). Large-eddy simulation of sediment transport:currents over ripples. J. Hydr. Eng. 127, 444–452.
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APPENDIX
MOVIES IN APPENDED CD
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Some movies generated from the numerical simulations in this thesis are included
in the appended CD. These movies are in AVI video format, which can be played by
most media players on Windows and Linux platforms. The description of the movie files
are as follows:
solitary-wave.avi Travelling of a solitary wave.
dam3d.avi Free-surface evolution of 3D broken-dam. Side boundaries are walls.
chap5-lambda2.avi Isosurfaces of λ2 (= −200) in the open-channel flow with undis-
turbed plane surface.
chap5-freesurface.avi Motions on the free surface in the open-channel flow with undis-
turbed plane surface. The mean free-surface velocity U0 is subtracted from the
streamwise velocity, like a view moving with the free surface. The color contours
of vertical velocity w refer to the plane two grid-cells beneath the free surface.
chap5-vector.avi Velocity vectors of u and w around the dune trough in the middle
plane of the open-channel flow with undisturbed plane surface.
chap6-deep-lambda2.avi Isosurfaces of λ2 (= −200) in the deep-water open-channel
flow with freely deformable free surface.
chap6-deep-freesurface.avi Magnified free-surface position in the deep-water open-
channel flow with freely deformable free surface.
chap6-shallow-lambda2.avi Isosurfaces of λ2 (= −100) in the shallow-water open-
channel flow with freely deformable free surface.
chap6-shallow-freesurface.avi Magnified free-surface position in the shallow-water
open-channel flow with freely deformable free surface.