A study on the formation of bed forms in rivers and coastal waters

A STUDY ON THE FORMATION OF BED FORMS IN RIVERS AND COASTAL WATERS MA PEIFENG B.Eng, M.Eng, SJTU M.Eng, NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 i ACKNOWLEDGEMENTS First and foremost, I would like to express my gratitude to my two supervisors, Professor Chan Eng Soon and Professor Ole Madsen, for their guidance and support Without encouragement and generous support given by Professor Chan, I would not be able to start my PhD study, for which I am grateful I was extremely fortunate to have Professor Madsen supervise my thesis work I greatly appreciate his help in sharing with me a lot of his expertise and research insight I also wish to acknowledge the financial support from the Defense Science and Technology Agency of Singapore for my research at the Tropical Marine Science Institute during the past years I would like to thank my former colleagues at TMSI and all my friends from NUS for making my days at NUS more enriching and enjoyable Finally, I want to dedicate this accomplishment to my wife and also my parents, for their patience, love and steadfast support and encouragement ii CONTENTS SUMMARY v LIST OF TABLES .vii LIST OF FIGURES viii LIST OF SYMBOLS xi CHAPTER ONE INTRODUCTION 1.1 Background 1.2 Literature review 1.2.1 Studies on bed form generation in open channels 1.2.2 Studies on sand wave formation in coastal waters 1.2.3 Weaknesses in previous studies 1.3 Motivations 1.4 Limitations of linear instabilty analysis 14 1.5 Objectives 14 1.6 Thesis outline 15 CHAPTER TWO BED-LOAD SEDIMENT TRANSPORT MODEL 16 2.1 General formulation 16 2.2 Determination of friction angles 19 2.3 Validation of the model 21 2.3.1 Bed-load transport in steady channel flow 22 2.3.2 Bed-load transport induced by unsteady wave motion on horizontal beds 24 i 2.3.3 Bed-load transport induced by unsteady wave motion on sloping beds 29 2.4 Summary of bed-load formulation 31 CHAPTER THREE THE ESSENCE OF BED INSTABILITY 33 3.1 General mechanism 33 3.2 Perturbed bed-load sediment transport rate 36 CHAPTER FOUR MODELS FOR SLOPE FACTORS 43 4.1 Fredsøe’s (1974) formula 44 4.2 Slope factor in our conceptual bed-load model 47 4.3 Validation on slope factor with experimental data of King (1991) 48 CHAPTER FIVE MODELS FOR PERTURBED BED SHEAR STRESS 60 5.1 Governing equations and boundary conditions 60 5.1.1 Governing equations 61 5.1.2 Boundary conditions 62 5.2 Models for eddy viscosity ν t 64 5.2.1 Linear varying eddy viscosity 64 5.2.2 Constant eddy viscosity 65 5.3 Base flow solutions 66 5.3.1 Steady river flow 66 5.3.2 Oscillatory tidal base flow solution 69 5.4 Perturbed flow models 75 5.4.1 Equations for linear perturbed flow 75 5.4.2 Perturbed flow solution with constant eddy viscosity: Slip velocity model 78 5.4.2.1 Governing equation 78 5.4.2.2 Boundary conditions 80 5.4.2.3 Effects of the perturbations of eddy viscosity and slip factors 85 5.4.2.4 Numerical Methodology 89 ii 5.4.2.5 Model tests 89 5.4.3 Perturbed Flow with linearly varying eddy viscosity: GM-model 95 5.4.3.1 Potential base flow 96 5.4.3.2 Potential perturbed flow solution 97 5.4.3.3 Perturbed velocity solution within bottom boundary layer 100 5.4.3.4 Model test 108 5 Comparison of the linear models with experimental data 110 5.5.1 Comparison with Richards’ (1980) model 111 5.5.2 Validation with experimental data 111 5.6 Extension to unsteady tidal flow 117 5.6.1 Perturbed tidal flow with the SV-model 118 5.6.2 Perturbed tidal flow with the GM-model 119 5.6.3 Model tests 120 CHAPTER SIX DUNES FORMED IN OPEN CHANNEL FLOW 123 6.1 Sensitivity analysis 123 6.1.1 Froude number 126 6.1.2 Bottom roughness effects 127 6.1.3 Sediment diameter 129 6.1.4 Bottom boundary condition in the SV-model 130 6.1.5 Surface boundary condition 130 6.1.6 SV-model versus GM-model 132 6.2 Application to the prediction of dunes in flumes 133 6.2.1 Experimental data 133 6.2.2 Model predictions 137 6.2.3 Comparison with other slope factor model 144 iii CHAPTER SEVEN SAND WAVES FORMED IN TIDAL FLOWS 147 7.1 A wave-current interaction model 147 7.1.1 Model description 150 7.1.2 Solution procedure 153 7.2 Bed-load transport rates with wind wave effects 157 7.2.1 General formulation 158 7.2.2 Simplified formulation for special cases 163 7.3 Sand waves in the Grådyb tidal inlet channel in the Danish Wadden Sea 166 7.4 Stability analysis in coastal waters with wind waves effects 168 7.4.1 Idealized case study 168 7.4.2 Real case study for combined wave current conditions 175 7.5 Comparisons of case studies with other models 187 7.5.1 The case in Gerkema (2000) 188 7.5.2 The case in Komarova and Hulscher (2000) 189 7.5.3 The case in Besio et al (2003) 190 CHAPTER EIGHT CONCLUSIONS AND FUTURE WORK 192 8.1 Conclusions 192 8.1.1 Flow models 192 8.1.2 Bed-load sediment transport model 194 8.1.3 Stability analysis for bed-load dominated conditions 195 8.2 Future work 201 REFERENCES 203 APPENDIX NUMERICAL SCHEME FOR SV-MODEL 210 iv SUMMARY In the present study, the mechanisms of bed form generation are investigated by using a linear instability analysis approach The linear analysis suggests that under bed-load sediment dominant conditions, two parameters play key roles in bed instability: the slope factor and the perturbed bed shear stress A conceptual bed-load transport model with a well-formulated slope term is introduced in the present study The slope factor formulated in this bed-load model is different from those in all previous bed form studies, in that it is composed of two terms: one dependent on the ratio between critical and the skin-friction shear stresses, the other a constant In contrast to previous studies, the conceptual bed-load transport model and its slope factor used here are validated and strongly supported by some relevant laboratory data A slip velocity model (SV-model) based on constant eddy viscosity assumption has been adopted by most previous sand wave studies to predict the perturbed bed shear stress However, the slip velocity model in most of these studies neglects the correlation between the constant eddy viscosity and the associated slip factor This enables those models to predict very good agreements via tuning the two parameters In the present study, a slip velocity model is also proposed but the proper correlation between the two parameters is retained In addition, another flow model, the GM-model, is also proposed in the present study based on a much more realistic near-bed linearly varying eddy viscosity The validation of the flow models with some experimental data reveals that both flow models tend to under-estimate the magnitude of perturbed shear stress with the GM-model performing slightly better v The models are applied to predict dunes in channel flows and the comparisons between predictions and measurements reveal that the wave numbers predicted by both models are smaller than the measurements The GM-model affords slightly better agreements, but is by no means perfect Due to their importance in coastal waters, the effects of wind waves are taken into account for the first time ever in the present sand wave study The analysis suggests that strong waves cause sand waves to decay, whereas weak and moderate waves may make sand waves grow This prediction is supported by the observation of ephemeral sand waves in a surf zone area along the Florida panhandle Another case study on sand waves along the Danish west coast reveals that the decrease of sand wave height in strong storm conditions during a few days is comparable to the increase of sand wave height by normal wave conditions during a few years This indicates that observed sand wave equilibrium may be a result of balance between short-duration storm wave and long term mean wave conditions Improvements of the present model in future studies, e.g improving the perturbed flow model and the inclusion of suspended load sediment transport, are suggested vi LIST OF TABLES Table 2.1 Allen’s (1970) experiments for natural sands 19 Table 2.2 Allen’s (1970) experiments for glass beads 20 Table 4.1 Summary of Sloping Bed Experiments n & ns = number of runs and number of slopes in the experiment, d = grain size (mm), T = wave period (second), U bm = maximum orbital velocity above wave boundary layer (cm/s), φ m is the corresponding repose angle in degrees obtained from a best fit of (4.11) to data 95% is the range of φ m values within a 95% confidence interval 49 Table 4.2 Hydrodynamic Characteristics of µ cr = τ cr / τ wm , u*wm / w f , γ = the slope factor corresponding to slope effects on critical shear stress, γ = γ + γ =the total slope factor, φ m _ C = tan −1 (1 / γ ) = the computed equivalent “friction angle” 50 Table 5.1 Parameters and results in the experiments and model predictions 113 Table 6.1 Dunes in 8-foot flume for d 50 = 0.28mm ( w f = 3.79cm / s ) 132 Table 6.2 Dunes in 8-foot flume for d 50 = 0.47mm ( w f = 6.69cm / s ) 133 Table 6.3 Dunes in 8-foot flume for d 50 = 0.93mm ( w f = 11.7cm / s ) 136 Table 7.1 Parameters computed by wave-current interaction model for various wave heights and φcw = 170 Table 7.2 Parameters computed by wave current interaction model for various wave directions and 2m height 171 Table 7.3 Scenarios for various wave and current conditions 177 Table 7.4 Parameters computed by wave current interaction model for Scenario with different grain size 178 Table 7.5 Parameters computed by wave current interaction model for Scenario with different sediments 178 Table 7.6 Parameters computed by wave current interaction model different currents 187 vii LIST OF FIGURES Figure 1.1 Sketch of eddy viscosity models and corresponding velocity profiles 10 Figure 1.2 Illustration of transform of spatial varying perturbed flow into temporal domain (a) Perturbed flow over wavy bed; (b) Corresponding wave motion 12 Figure 2.1 Comparison with the MPM model and measurements 23 Figure 2.2 Variation of the ratio between the transport rate computed by the present bed-load transport model and that computed by the Meyer-Peter and Muller’s model versus the ratio between the critical shear stress and the skin friction 24 Figure 2.3 Measured and predicted bed-load transport rates averaged over a half wave cycle on a flat bed: (a) data with 0.135mm sediment cases; (b) data without 0.135mm sediment cases The black solid line represents the ratio 1:1 between predictions and measurements and the red lines are the best linear fitted lines 28 Figure 2.4 Measured and predicted bed-load transport rates averaged over a half wave cycle on a sloping bed: (a) data with 0.135mm sediment cases; (b) data without 0.135mm sediment cases The black solid line represents the ratio 1:1 between predictions and measurements and the red lines are the best linear fitted lines 31 Figure 3.1 Perturbed total sediment transport rates and bed waves 35 Figure 3.2 Ratios between bed-load transport rates predicted by the original formula (2.8a) and the formula (3.10) with linearized slope terms 38 Figure 3.3 Illustration of bed state depending on parameters τ s' and µ cr , (a) with slope factor given by (3.10b); (b) with constant slope factor γ = 41 Figure 4.1 Values of γ against µ cr = τ cr / τ b 48 Figure 4.2 Comparisons of bed-load transport ratio q Bβ / q B between bed-load formula with different slope factors and King’s measurements (a) EXP1; (b) EXP2; (c) EXP3; (d) EXP4; (e) EXP5; (f) EXP6; (g) EXP7; (h) EXP8 59 Figure 5.1 Sketch of study domain 61 viii applicable to very long bed forms To improve it, a better eddy viscosity needs to be considered, e.g applying the parabolic eddy viscosity profile over the entire water column or incorporating a turbulent closure model to solve the eddy viscosity With a more advanced eddy viscosity model, the perturbed flow can be solved numerically in a curvilinear coordinate system (such as Richards, 1980; Thorsness, 1975) Another weakness in the present study is related to suspension effects on bed instability predictions As discussed in Chapter 2, the bed-load transport model in the present study can be applied to predict sediment transport in wave motions with u*wm / w f < 2.7 with u*wm the maximum wave shear velocity and w f the fall velocity of sediment particle For sediment transport in steady flow, this criterion should be stricter as the boundary layer is much thicker in steady flow than that in pure wave motion Furthermore, strong storm waves usually violate this criterion, e.g the case of sand waves in surf zone investigated in Chapter has u*wm / w f greater than 3.0 In this case, the suspension effect must be taken into account so that more accurate prediction can be obtained So far, the suspension effect is neglected by most sand wave studies except Besio et al (2006) Although the suspension effect is qualitatively considered by Besio et al (2006), the influence of suspension effects on bed instability prediction is not actually quantified in the study In addition, the suspension effects have been taken into account by Engelund (1970) and Fredsøe (1974) to predict dune and anti-dune formation in alluvial channels However, due to assuming constant eddy viscosity in their studies, the unrealistic near-bed velocity and concentration structures make those predictions physically unreliable In accordance with the above proposed perturbed flow solution, a better way to obtain suspended load sediment transport over a wavy bed is to solve it in a curvilinear coordinate system so that the large gradient of near-bed concentration can be avoided 202 REFERENCES Allen, J.R.L., 1970, “The avalanching of granular solids on dune and similar slopes”, J Geology, V.78, pp.326-351 Allen, J.R.L 1982, “Sedimentary structures: their character and physical basis, Volume 1.” Elsevier Scientific, 1982, pp.593 Aliotta S and Perillo G.M.E., 1987, “A sand wave field in the entrance to Bahia Blanca estuary, Argentina”, Marine Geology, V76, 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52 McLean, S R., 1990, “The stability of ripples and dunes”, Earth-Science Reviews, V29, pp.131-144 53 Meyer-Peter, E., and Mueller, R 1948, “Formulas for bed-load transport.” Proceedings, 2nd Meeting of International Association for Hydraulic Research, Stockholm, Sweden, 26pp 54 Nèmeth A.A., Hulscher S.J.M.H., and de Vriend H.J., 2002, “Modeling sand wave migration in shallow shelf seas”, Cont Shelf Res., V22, pp.2795-2806 55 Nèmeth A.A., 2003, “Modeling offshore sand waves”, PhD thesis, The Netherlands, Print Partners Ipskamp BV, Enschede, 140pp 56 Off T., 1963, “Rhythmic linear sand bodies caused by tidal currents”, Bulletin of the American association of petroleum geologists, V47, N2, pp.324-341 57 Richards, K.J., 1980, “The formation of ripples and dunes on an erodible bed.”, J Fluid Mech., V99, pp.597-618 58 Schielen R., Doelman A, and de Swart H.E., 1993, “On the nonlinear dynamics of free bars in straight channels.” J Fluid Mech., V252, pp 325-356 59 Terwindt J.H.J., 1971, “Sand waves in the Southern Bight of the North Sea”, Marine Geology, V10, pp.51-67 60 Thorsness, C B., 1975, “Transport phenomena associated with flow over a solid wavy surface.”, PhD thesis, University of Illinois at Urbana-Champaign, pp282 61 Vittori G., and Blondeaux P., 1990, “Sand ripples under sea waves, Part2: Finite-amplitude development”, J Fluid Mech., V218, pp 19-39 62 Wilson, K C., 1966, “Bed-load transport at high shear stress.” J Hydraul Div., Am Soc, Civ, Eng., 92(6), 49-59 208 63 Yalin, M.S., 1977, “Mechanics of Sediment Transport.”, Pergamon, Oxford 64 Zilker D P., Cook G W and Hanratty T J., 1977, “Influence of the amplitude of a solid wavy wall on turbulent flow Part Non-separated flows”, J Fluid Mech., V82, pp.29-51 209 APPENDIX NUMERICAL SCHEME FOR SV-MODEL Computational domain and grid system To apply finite difference method, we first discretize the domain ≤ z ≤ h , as sketched in figure blow, into mm1 segments with the grid point m = denoting the bottom boundary and m = mm1 = mm + representing the surface boundary, and the internal grid points are from ≤ m ≤ mm that are unknowns to be computed Since there is a fourth order term in the equation, we introduce two so-called ghost points, m = −1 outside the bottom boundary and m = mm2 = mm + outside the surface boundary The values of function F at the two ghost points and the two boundary points, can be expressed by the values of F at adjacent internal grid points via the four boundary conditions Bottom -1 mm-1 mm1 mm Surface mm2 Sketch of computational domain (black: internal points; blue: boundary points; red: ghost points) Numerical discretization of governing equation Eemploying a second order central finite difference scheme, the different order derivatives of function F at the grid point m , i.e z = m ⋅ ∆z with ∆z = h / mm1 can be written by Fz = dF Fm +1 − Fm −1 = dz 2∆z (1a) 210 Fzz = Fzzz = dFz Fm+1 + Fm−1 − Fm = dz ∆z dFzz Fm + − Fm +1 + Fm −1 − Fm − = dz ∆z F − 4Fm+1 + 6Fm − 4Fm−1 + Fm−2  dF  Fzzzz =  zz  = m+2 ∆z  dz  zz (1b) (1c) (1d) Then the governing equation (5.65) can be discretized, at grid point m F + Fm −1 − Fm   ik u ⋅ m +1 − k u Fm − u zz Fm  ∆z   F + Fm−1 − Fm  F − Fm+1 + Fm + Fm−1 + Fm−2  = ν c ⋅  m+2 − 2k ⋅ m+1 + k Fm  ∆z ∆z   or Fm−2 ⋅ a1m + Fm−1 ⋅ a m + Fm ⋅ a3m + Fm+1 ⋅ a m + Fm+ ⋅ a1m = (2a) (2b) where a1m = νc ∆z a2m = − a 3m 4ν c 2k (ν c − 2ν n ) iku m + − ∆z ∆z ∆z 6ν c 4k (ν c − 2ν n ) 2iku m = − +ν c k + + ik u m + iku zzm 2 ∆z ∆z ∆z (3a) (3b) (3c) with u m = u @ z = m∆z , and u zzm = u zz @ z = m∆z It can be observed from differential equation (2a-b) that the coefficient matrix is a non-symmetrical banded complex matrix with five bands To obtain the expressions of F−1 , F0 , Fmm1 , Fmm2 , we need to use the boundary conditions Bottom boundary conditions Kinematic boundary condition (5.66c) is discretized to be F0 = −u 0b (4) 211 Dynamic boundary condition with linear friction (5.68a) gives F1 + F−1 − F0 ~ F1 − F−1 −s ⋅ + k F0 = (~u zb − u zzb ) s 2 ∆z ∆z (5a) or [ ( ( ) ) ] ( F−1 = (~u zb − u zzb )Ab − k − / ∆z F0 − / ∆z − ~ / 2∆z F1 s s / / ∆z + ~ / 2∆z s ) (5b) By using (4), (5b) can be written as F−1 = c1 F1 + c (6) where ( [ )( ( ) c1 = ~ / 2∆z − / ∆z / / ∆z + ~ / 2∆z s s c2 = ~u zb − u zzb + u 0b k − / ∆z / / ∆z + ~ / 2∆z s s )] ( ) (6a) Similarly, from nonlinear condition (5.68b), we can obtain F−1 = c1n F1 + c n (7) where c1n = (~ u 0b / ∆z − / ∆z ) / (1 / ∆z + ~ u 0b / ∆z ) s s c2 n = 2~u zb u b − u zzb + u 0b (k − / ∆z ) / (1 / ∆z + ~ u b / ∆z ) s s [ ] (7a) With the boundary condition (4) and (6) or (7), we can re-write differential equation (2b) at the point m = and m = , as At m = , we have, from (2b), F−1 ⋅ a11 + F0 ⋅ a 21 + F1 ⋅ a31 + F2 ⋅ a 21 + F3 ⋅ a11 = (8a) Introduce (4) and (6) to (8a) we obtain, for linear friction condition, F1 ⋅ (a31 + c1a11 ) + F2 ⋅ a 21 + F3 ⋅ a11 = u 0b Ab ⋅ a 21 − c a11 (8b) For nonlinear friction condition, the equation will be readily obtained by just replacing c1 , c with c1n , c n At m = , we have, from (2b), 212 F0 ⋅ a12 + F1 ⋅ a 22 + F2 ⋅ a32 + F3 ⋅ a 22 + F4 ⋅ a12 = (9a) Introduce (4) to (9a) we obtain, for both linear and nonlinear friction conditions, F1 ⋅ a 22 + F2 ⋅ a32 + F3 ⋅ a 22 + F4 ⋅ a12 = u 0b Ab ⋅ a12 (9b) Rigid lid surface boundary condition From (5.87a), we have Fmm1 = (10) From (5.87b), we have Fmm + Fmm − Fmm1 + k Fmm1 = ∆z (11a) By virtue of (10), (11a) would be Fmm2 = − Fmm (11b) Thus, we can obtain the differential equation near the surface boundary, At m = mm , we have, from (2b), Fmm − ⋅ a1,mm + Fmm −1 ⋅ a ,mm + Fmm ⋅ a 3,mm + Fmm1 ⋅ a 2, mm + Fmm ⋅ a1,mm = (12a) Introduce (10) and (11b) to (12a) we have Fmm − ⋅ a1,mm + Fmm −1 ⋅ a 2,mm + Fmm ⋅ (a3,mm − a1,mm ) = (12b) At m = mm − , we have, from (2b), Fmm −3 ⋅ a1, mm −1 + Fmm − ⋅ a 2,mm −1 + Fmm −1 ⋅ a 3, mm −1 + Fmm ⋅ a 2,mm −1 + Fmm1 ⋅ a1,mm −1 = (13a) with (10), we have Fmm −3 ⋅ a1, mm −1 + Fmm − ⋅ a ,mm −1 + Fmm −1 ⋅ a3,mm −1 + Fmm ⋅ a 2,mm −1 = (13b) Free surface boundary condition From (5.85a), we have 213 Fmm + Fmm − Fmm1  u zzs + k −  u0s ∆z    Fmm1 =   (14a) To discretize (5.85b), we re-write the third order derivative (1) at the boundary point m = mm1 to be Fzzz = (Fzz )mm1 − (Fzz )mm ∆z Fmm − 3Fmm1 + 3Fmm − Fmm −1 = ∆z Introduce this to (5.85b), we have Fmm − Fmm + Fmm − Fmm −1 ⋅ν c ∆z F − Fmm + mm ⋅ ( ν c − 4ν n )k − iku s 2∆z   g  + Fmm ⋅ ik  + u zs   =      u0s [ ] (14b) Equations (14a) and (14b) can be written as e11 Fmm2 + e12 Fmm1 = e13 Fmm (15) e21 Fmm2 + e22 Fmm1 = e23 Fmm + e24 Fmm−1 Where e11 = / ∆ z e12 = k − u zzs − / ∆z u0s (15a) e13 = e11 [ ] e 21 = ν c / ∆ z + ( ν c − 4ν n )k − iku s / ∆ z  g  e 22 = − 3ν c / ∆ z + ik   u + u zs    0s  e 23 = 3ν c / ∆ z − ( ν c − 4ν n )k − iku s / ∆ z [ ] (15b) e 24 = −ν c / ∆ z Thus, from (15), we can obtain 214 Fmm1 = Fmm e13 e22 − e23 e12 e24 e12 ⋅ Fmm − ⋅ Fmm −1 = e01 ⋅ Fmm + e02 ⋅ Fmm −1 e11e22 − e12 e21 e11e22 − e12 e21 e e −e e e11e24 = 11 23 13 21 ⋅ Fmm + ⋅ Fmm −1 = e03 ⋅ Fmm + e04 ⋅ Fmm −1 e11e22 − e12 e21 e11e22 − e12 e21 (16) Hence, with free surface boundary condition (16), we can re-write the differential equation for the points near surface boundary as following At m = mm , we have, from (2b), Fmm − ⋅ a1,mm + Fmm −1 ⋅ a ,mm + Fmm ⋅ a 3,mm + Fmm1 ⋅ a 2, mm + Fmm ⋅ a1,mm = (17a) Introduce (16) to (12a) we have Fmm− ⋅ a1,mm + Fmm−1 ⋅ (a 2,mm + e02 ⋅ a 2,mm + e04 ⋅ a1,mm ) + Fmm ⋅ (a3,mm + e01 ⋅ a 2, mm + e03 ⋅ a1,mm ) = (17b) At m = mm − , we have, from (12b), Fmm −3 ⋅ a1, mm −1 + Fmm − ⋅ a 2,mm −1 + Fmm −1 ⋅ a 3, mm −1 + Fmm ⋅ a 2,mm −1 + Fmm1 ⋅ a1,mm −1 = (18a) with (16), we have Fmm−3 ⋅ a1,mm−1 + Fmm− ⋅ a 2, mm−1 + Fmm−1 ⋅ (a3,mm−1 + e02 ⋅ a1,mm−1 ) + Fmm ⋅ (a 2,mm−1 + e01 ⋅ a1,mm−1 ) = (18b) Numerical linear system and solver So far, with help of boundary conditions, we replaced the values of boundary points and the ghost points in the differential equation (2b) for all internal points, i.e ≤ m ≤ mm To solve the linear system, we write the system in matrix form as Alm ⋅F m = Bl (19) where, in general, 215 a1, m a  2,m  Alm = a 3,m a  2,m +1 a1, m+  if l = m − if l = m − if l = m (20) if l = m + if l = m + and Bl = In particular, the values of Alm and Bl near boundaries will be different from those in (20) due to the incorporation of boundary conditions For different boundary conditions, the relevant values that need to be modified are, A11 = a3,1 + c1 a11 (Linear friction condition), or (21a) A11 = a3,1 + c1n a11 (Nonlinear friction condition) (21b) Amm ,mm = a3,mm − a1, mm (Rigid lid surface condition) or (21c) Amm ,mm = a3,mm + e01 ⋅ a ,mm + e03 ⋅ a1,mm Amm −1,mm = a ,mm + e02 ⋅ a 2,mm + e04 ⋅ a1,mm Amm ,mm−1 = a ,mm −1 + e01 ⋅ a1,mm−1 (Free surface condition) (21d) Amm −1,mm−1 = a3,mm−1 + e02 ⋅ a1,mm−1 B1 = u b Ab ⋅ a ,1 − c a1,1 (Linear friction condition), or (21e) B1 = u 0b Ab ⋅ a 2,1 − c n a1,1 (Nonlinear friction condition) (21f) B2 = u b Ab ⋅ a1, (21g) Therefore, with the coefficient matrix Alm and right hand side array Bl given in the above, the complex linear system (19) can be readily solved In this study, we use a solver from “LAPACK” (http://www.netlib.org/lapack/) library, which is specialized in solving linear complex system with general banded coefficient matrix 216 ... origin, classification and modeling in continental shelf seas Among all types of natural sandy bed forms, dunes in alluvial rivers and sand waves in coastal waters have relatively large size and. .. by the observation of ephemeral sand waves in a surf zone area along the Florida panhandle Another case study on sand waves along the Danish west coast reveals that the decrease of sand wave... formation in coastal waters In contrast to dune studies, almost all sand wave studies are carried out using rotational flow models A mathematical analysis on sand wave formation has been conducted