Efficient simulation of non-hydrostatic free-surface flow

Efficient simulation of non-hydrostatic free-surface flow Wave simulation with interpolation of the vertical pressure profile

M. van Reeuwijk

M.Sc. Thesis, May 2002 Delft University of Technology Faculty of Civil Engineering and Geosciences Section Fluid Mechanics

Graduation Committee prof. dr. ir. G.S. Stelling dr. ir. P. Wilders dr. ir. R.E. Uittenbogaard prof. dr. ir. J.A. Battjes

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Abstract A numerical non-hydrostatic 2DV free surface flow model has been developed. The equations are based on the Navier-Stokes equations without friction, and are solved in the σ-co-ordinate system. In the model, the number of pressure layers can be chosen independently of the number of horizontal velocity layers. This extra degree of freedom allows, at minimal computational effort, for a high vertical resolution of the horizontal velocities, which is needed for the simulation of density currents or transport problems. With splines, a continuous pressure function is constructed, using the discrete pressure in the vertical. This pressure function is used to estimate the pressure in-between the pressure layers, such as dictated by multiple velocity layers. The model is applied for simulation of short wave propagation. The behavior of a standing wave in a closed basin is simulated, as well as wave propagation over a trapezoidal bar. It is shown that the model allows for a significant reduction of computational effort, while maintaining a high resolution.

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Preface & Acknowledgements This report contains a study on efficient simulation of non-hydrostatic free surface flow. It is the result of my M.Sc. project at the Faculty of Civil Engineering and Geosciences at Delft University of Technology, which was carried out at the consulting and research institute WL | delft hydraulics. Working on the subject was challenging, frustrating, euphorical, depressing but most of all fun to do. I appreciated the open and informal culture at WL | delft hydraulics, which gave me access to the large knowledge of flow modeling available inside WL | delft hydraulics. I would like to thank my supervisors, who gave me the freedom to follow my intuition, while still guiding and inspiring me. Finally, I would like to thank my family and friends for their emotional support. They helped me to take distance from the subject, especially in periods when I was stuck for weeks. I couldn’t have done this without them. Delft, May 2002, Maarten van Reeuwijk

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Table of contents ABSTRACT ...........................................................................................................................................................................iii PREFACE & ACKNOWLEDGEMENTS ...........................................................................................................................v 1

INTRODUCTION..........................................................................................................................................................1 1.1 1.2 1.3 1.4

2

OVERVIEW OF FLOW AND WAVE SIMULATIONS ........................................................................................................1 EFFICIENT SIMULATION OF WEAKLY HYDRODYNAMIC FLOW ...................................................................................2 OBJECTIVES .............................................................................................................................................................3 CONTENTS OF THE REPORT .......................................................................................................................................3

FORMULATION OF THE ORIGINAL FIXED LAYER MODEL .........................................................................5 2.1 INTRODUCTION ........................................................................................................................................................5 2.2 MODEL DEFINITIONS ................................................................................................................................................5 2.3 PREPARATION OF THE 2DV NAVIER STOKES EQUATIONS FOR THE MODEL. .............................................................6 2.4 DISCRETISATION OF THE EQUATIONS .......................................................................................................................8 2.5 CONSISTENCE OF THE MODEL’S KINEMATIC BOUNDARY CONDITIONS ......................................................................9 2.6 SOLUTION OF THE SYSTEM .......................................................................................................................................9 2.6.1 A 4D matrix for storage of 2D pressure gradient coefficients. ...........................................................................9 2.6.2 Substitution of the momentum equations in the local continuity equation........................................................10

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FORMULATION OF THE MODEL IN SIGMA COORDINATES ......................................................................13 3.1 INTRODUCTION ......................................................................................................................................................13 3.2 THE CONCEPT OF THE SIGMA TRANSFORMATION ....................................................................................................14 3.3 TRANSFORMATION OF THE 2DV NAVIER STOKES EQUATIONS ...............................................................................14 3.3.1 Definition of the new vertical velocity ..............................................................................................................14 3.3.2 Formulation of first order time and space derivatives......................................................................................15 3.3.3 Transformation of the equations.......................................................................................................................16 3.4 DISCRETISATION OF THE EQUATIONS .....................................................................................................................16 3.5 CONSISTENCE OF THE MODEL’S KINEMATIC BOUNDARY CONDITIONS AND THE LOCAL CONTINUITY EQUATION ....19 3.6 SOLUTION METHOD ................................................................................................................................................19 3.7 VERIFICATION OF THE SIGMA MODEL .....................................................................................................................21 3.7.1 Standing wave in a closed basin. ......................................................................................................................21 3.7.2 Propagation of a solitary wave in a channel ....................................................................................................22 3.7.3 The Beji-Battjes experiment..............................................................................................................................24 3.7.4 Comparison to the original model for the Beji-Battjes experiment ..................................................................27 3.8 COMPARING THE NUMERICAL AND ANALYTICAL PROPAGATION SPEED FOR VARIOUS WAVE LENGTHS. .................28 3.8.1 Introduction ......................................................................................................................................................28 3.8.2 Setup of the experiment.....................................................................................................................................28 3.8.3 Experimental results. ........................................................................................................................................29 3.8.4 Cr: a parameter that couples the wave shortness to the layer thickness...........................................................30 3.9 CONCLUSIONS ........................................................................................................................................................32

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EFFICIENT SIMULATION BY INTERPOLATION OF THE VERTICAL PRESSURE PROFILE. ..............33 4.1 INTRODUCTION ......................................................................................................................................................33 4.2 RELATION BETWEEN PRESSURE AND VELOCITY LAYER VARIABLES .......................................................................34 4.3 SPLINES AS BASIS FUNCTIONS FOR THE INTERPOLATION ........................................................................................35 4.4 INTERPOLATION WITH ONE PRESSURE LAYER .........................................................................................................36 4.4.1 The consistency condition for one pressure layer.............................................................................................36 4.4.2 Interpolation with a first and third order spline ...............................................................................................37 4.5 REQUIREMENTS OF THE INTERPOLANT FOR MORE THAN ONE PRESSURE LAYER. ....................................................38 4.5.1 The consistency condition for more than one pressure layer............................................................................39 4.6 FIRST ORDER SPLINE INTERPOLATION. ...................................................................................................................40 4.7 FOURTH ORDER SPLINE INTERPOLATION ................................................................................................................41 4.8 GENERAL SOLUTION METHOD FOR THE SPLINE COEFFICIENTS ................................................................................44 4.8.1 Definition of vectors for the complete vertical..................................................................................................44 4.8.2 Hermitian method in the complete vertical.......................................................................................................45

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4.8.3 Coupling the analytical function Q to the pressure gradients.......................................................................... 45 4.9 INCORPORATION OF THE INTERPOLATION IN THE MODEL ....................................................................................... 47 4.9.1 Outline of the solution algorithm...................................................................................................................... 47 4.9.2 Operator S ........................................................................................................................................................ 48 4.9.3 Calculation of the pressure term in the horizontal momentum equation of a velocity layer. ........................... 48 4.9.4 Derivation of the continuity equation for a pressure layer............................................................................... 49 4.9.5 The advection scheme....................................................................................................................................... 50 4.10 CONCLUSIONS ....................................................................................................................................................... 50 5

VERIFICATION OF THE INTERPOLATION MODEL....................................................................................... 51 5.1 INTRODUCTION ...................................................................................................................................................... 51 5.2 NORM FOR THE QUALITY OF THE VERTICAL DISTRIBUTION OF THE HORIZONTAL VELOCITY .................................. 51 5.2.1 Error of a single periodic signal ...................................................................................................................... 51 5.2.2 Mean total error along the vertical .................................................................................................................. 52 5.3 STANDING WAVE IN CLOSED BASIN........................................................................................................................ 53 5.3.1 Convergence of the vertical distribution of the horizontal velocities. .............................................................. 54 5.3.2 Residual currents for longer simulations ......................................................................................................... 57 5.4 THE BEJI-BATTJES EXPERIMENT ............................................................................................................................ 60 5.5 CONCLUSIONS ....................................................................................................................................................... 62

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PERFORMANCE OF THE INTERPOLATION MODEL ..................................................................................... 65 6.1 INTRODUCTION ...................................................................................................................................................... 65 6.2 METHOD OF MEASUREMENT .................................................................................................................................. 65 6.3 PERFORMANCE OF THE MODELS............................................................................................................................. 66 6.3.1 Sigma model ..................................................................................................................................................... 66 6.3.2 Interpolation by first order spline .................................................................................................................... 66 6.3.3 Interpolation by fourth order spline ................................................................................................................. 68 6.4 COMPARISON BETWEEN SIGMA AND INTERPOLATION MODEL ................................................................................ 69 6.5 A CASE STUDY: INDICATION OF REQUIRED TIME FOR SIMULATION OF ONE HOUR OF WAVE RUN-UP AT PETTEN..... 69 6.6 CONCLUSIONS ....................................................................................................................................................... 72

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CONCLUSIONS AND RECOMMENDATIONS..................................................................................................... 73 7.1 7.2

CONCLUSIONS ....................................................................................................................................................... 73 RECOMMENDATIONS ............................................................................................................................................. 74

REFERENCES ..................................................................................................................................................................... 73 LIST OF MAIN SYMBOLS................................................................................................................................................ 75

Appendices A

INCREASING WAVE AMPLITUDES FOR WAVE BOUNDARY CONDITION OF SHORT WAVES WITH HYDROSTATIC PRESSURE ASSUMPTION ............................................................. APP.1

B

BOUNDARY CONDITIONS FOR SHORT WAVES .............................................................................. APP.5

C

A DERIVATION OF THE SIGMA MODEL BASED ON LAYER AVERAGING ............................ APP.15

D

FIRST ORDER UPWIND ADVECTION SCHEME.............................................................................. APP.21

E

DERIVATION OF THE NORM FOR PERIODIC SIGNAL QUALITY EVALUTATION .............. APP.25

F

RESULTS OF THE BEJI-BATTJES EXPERIMENT FOR THE INTERPOLATION MODEL WITHOUT ADVECTION......................................................................................................................... APP.29

G

ANALYTICAL EXPRESSIONS FOR THE ACCURACY OF PRESSURE APPROXIMATION BY STRAIGHT LINES. ............................................................................................................................ APP.33

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1

Introduction

1.1

Overview of flow and wave simulations Most problems in hydraulic and coastal engineering involve seas, estuaries, rivers or lakes. These problems are characterized by nearly horizontal flow, with length scales far exceeding the water depth. Various models have been developed to simulate these flows, such as Delft3D-FLOW (WL | delft hydraulics) or TRIWAQ (Rijkswaterstaat). These models are based on the 3D non-compressible Navier Stokes equations with a mobile free surface. The main assumption made is that the vertical accelerations of the water are neglected: the hydrostatic pressure assumption. This reduces the vertical momentum equation to the relation for hydrostatic pressure. It limits the scope of these models to long time and length scales. Just long waves (with the wave length significantly larger than the water depth) are described correctly, which is the reason they are also referred to as ‘shallow water’ models. Short waves play an important role as well. Although the time and length scales are smaller than for long waves, short waves introduce longshore currents (through radiation stresses), water level setup and increased bottom friction. This means their effect cannot be neglected for large scale simulations. Generally a short wave model is run after the flow field is obtained by a shallow water model. This is a one-way coupling, where the flow field changes the wave field, and not the other way around. Optionally an off-line coupling can be made to incorporate the non-linear interaction (bottom friction, radiation stresses) into the shallow water simulation. Various types of models have been developed for describing short wave propagation. A widely used type is the spectral model, such as SWAN (TU Delft) or HISWA (WL | Delft Hydraulics). The wave propagation is described in space and the wave interaction in the frequency domain. These codes can be easily applied for large domains, since the space and time step can be quite large. The model gives information about the spectral wave energy distributions and the ensemble-averaged wave properties rather than information about individual waves. Another type of short wave model is the class of Boussinesq models. These models are based on an a priori assumption of the vertical distribution of the horizontal velocity, and depth averaging of the Navier-Stokes equations. These models occur in various types and forms, for example (Borsboom, 1998) or (Borsboom et. al., 2000). The waves are simulated in time space, and thus give information about individual waves. However, since depth averaged equations are used, no direct information about the vertical distribution of the horizontal velocity is available. However, the strongest point of Boussinesq models is also their weakest. Wave problems can be efficiently tackled, but other non-hydrostatic phenomena cannot be described with these models, such as flows near bed discontinuities, rapidly varying bed slopes or flow around hydraulic structures. Correct calculation of these effects requires the solution of the full 3D incompressible Navier Stokes equations with a free surface. Various papers have been published on direct numerical simulation of non-hydrostatic flows, such as Casulli and Stelling (1998) and Casulli (1999). Zijlema (2000) reported the extension of TRIWAQ (Rijkswaterstaat) with the effect of the non-hydrostatic pressure, using a pressure correction method. Weilbeer and Jankowski (2000) extended the finite element model TELEMAC3D with the effect of the non-hydrostatic pressure. Stansby and Zhou (1998) implemented a non-hydrostatic model in sigma coordinates based on a pressure correction method, allowing non-hydrostatic effects to be solved where they are significant. Stelling and Van Kester (2000) present a new vertical approximation based on a Hermitian method with horizontal layers.

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Efficient Numerical Simulation of Non-Hydrostatic Waves

These models are very promising, since they provide an integral model for both hydrostatic and non-hydrostatic flows, not limited to any category of phenomena. The biggest challenge is that non-hydrostatic effects take place on smaller time and length scales. This means the space and time step must be reduced, which causes simulations to take more computer time. In addition, the pressure distribution in the vertical is unknown as well, which leads to another increase in computer time. For large-scale simulations, it is essential this increase in computer time stays within reasonable limits. Therefore emphasis on efficient algorithms is required when developing codes for non-hydrostatic flow.

1.2

Efficient simulation of weakly hydrodynamic flow This report is a preparatory study for the efficient simulation of 3D non-hydrostatic free surface flows. The basis of this study is the model developed by Stelling & Van Kester (2000). Their approach gives accurate results with respect to wave propagation, even with a very limited number of layers. For practical applications such as stratified flow in estuaries generally a large number of layers is required (5-20). Shallow water models such as Delft3D-Flow use the reduced vertical momentum equation (the hydrostatic pressure relation) to sum the horizontal momentum equations over all the layers and thus solve the water level points only. Back substitution of the water levels in the horizontal momentum equation leads to the horizontal velocities for the individual layers. This approach is very efficient. Only water levels are used in the final system of equations instead of all the individual horizontal momentum equations. Imagine that for the situation sketched above, short waves are taken into account as well. A non-hydrostatic model such as Stelling and Van Kester (2000) can be used. If the model is used straightforwardly with the same number of layers, a big increase in computer time will occur. However, as mentioned by Stelling and Van Kester, the accurate prediction of the propagation of the short waves requires just a very limited amount of layers (1-3). This knowledge can be used to improve the efficiency of the model. Generally, wave problems are weakly hydrodynamic, with only a limited deviation from the hydrostatic pressure. This means that a limited number of layers is required for obtaining accurate results for wave propagation. However, for practical problems, as the stratification example sketched above, the required resolution of horizontal velocities in the vertical is much higher. If the number of pressure layers and velocity layers can be chosen independently, the efficiency of the model can be increased. The advantage is that according to the nature of the problem, the ratio between velocity and pressure layers can be chosen. For weakly hydrodynamic problems, a high ratio can be used. For problems with a very hydrodynamic nature the ratio can be chosen equal to unity, the number of pressure layers equal to the number of velocity layers. This hybrid approach looks upon the non-hydrostatic models as a mere extension of hydrostatic models, and it uses the limited hydrodynamic behaviour for increasing the efficiency. Instead of summing over all layers and obtaining equations with only water levels as unknowns, the summation is performed for a single pressure layer. This means that for the example above, three discharges are used for the final system of equations. When the system has been solved, back substitution yields the velocities in the individual layers.

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1.3

Objectives As mentioned before, the non-hydrostatic model developed by Stelling and Van Kester (2000) is used as the basis of this study. Originally the plan was to consider the following topics: 1. Make an implementation of the model in sigma coordinates; 2. Improve the performance of the model by uncoupling the number of velocity and pressure layers; 3. Add the dynamics of breaking waves. Breaking of waves is a non-linear process, generating much turbulence and momentum exchange. Correct description of the breaking process is important for a correct description of the water flow. Although sigma coordinates are very elegant, they are not suited very well for strongly hydrodynamic or diffusion problems (artificial creeping). Bottom and surface discontinuities invoke singularities. Despite these disadvantages the sigma model is used for two reasons. First, the model’s results can be compared to the original fixed z-coordinate model (are the results better / worse and why?). Second, the fixed number of layers allows for an easier implementation of the uncoupling of pressure and velocity layers. As I was getting used to the model, I observed that the boundary condition for incoming waves produced strange results. The amplitude of the waves increased as they propagated into the model. This problem turned out to be more complex than I expected at first. The outcome of this study led to appendices A and B about boundary conditions for short waves. However, this took a considerate amount of time, leaving no time for the dynamics of breaking waves. For this reason this objective was dropped. This means the adjusted objectives are: 1. Make an implementation of the model in sigma coordinates, mainly for research purposes; 2. Improve the performance of the model by uncoupling the number of velocity and pressure layers.

1.4

Contents of the report Chapter 2 provides a description of the original fixed z-coordinate model. Chapter 3 discusses the formulation of the model in sigma coordinates. A derivation of the equations is given, and the model is verified. At the end of Chapter 3, the numerical correspondence to the dispersion relation is evaluated for various wave lengths as well as for a various number of layers. This leads to a quantitative relation between the wave length and the required number of layers. Chapter 4 goes into the details of uncoupling the pressure and velocities. It describes the details of first and fourth order interpolation. In Chapter 5, the interpolation model is verified. Chapter 6 discusses the computational performance of the interpolation model. Chapter 7 contains the conclusions and recommendations.

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Efficient Numerical Simulation of Non-Hydrostatic Waves

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2

Formulation of the fixed layer model

2.1

Introduction The model described in this chapter has been developed by Stelling and Van Kester in 2000. It is a two dimensional vertical (2DV) model for simulating non-hydrostatic flows. The model is based on a new approach for calculating the hydrodynamic pressure throughout the fluid. The original model is reported and verified in Stelling & Van Kester (2000), the model described in this chapter is a slightly modified version of it. The model is based on a (vertical) staggered grid. The new approach of this model is not to use a classic formulation of a staggered grid, with the flow variables (horizontal and vertical velocities u and w) at the cell interfaces and the scalar variables (hydrodynamic pressure q) in the cell centre. Instead, it uses a modified approach, which positions the pressure, and an additional unknown, the vertical pressure gradient, at the cell interface, as is shown in Figure 2-1. w, q,

w

u

q

u

u

u

w

w, q,

(a) Figure 2-1:

2.2

∂q ∂z

(b)

∂q ∂z

Definition sketch of a classic staggered grid (a) and the staggered grid for the new approach (b), which positions the scalar variable q and its vertical gradient at the cell interface. Note the grid is orientated vertically (2DV).

Model definitions The model uses a horizontally oriented x-axis, and vertically orientated z-axis. The horizontal index for the grid points is denoted by m, the vertical index for the grid is denoted by k. The indices m and k increase as x and z increases. This can be seen in Figure 2-2. The new approach positions the pressure at the top of the cell interface and in addition, introduces the vertical pressure gradient as an extra unknown. The pressure gradient will be used as primary unknown, and is coupled to the pressure by:

qk − qk −1 1 ∂q 1 ∂q = + 2 ∂z k 2 ∂z k −1 ∆zk

wm,k , qm,k ,

zm,k

∂q ∂z m,k

m, k

∆zm,

um,k

(2.1)

The introduction of derivates as additional unknowns is referred to as a Hermitian method. A short description of Hermitian methods can be found in Peyret & Taylor (1983). Hermitian methods allow for WL | delft hydraulics & Delft University of Technology

z, k

zm,k-1 x, m Figure 2-2: Cell definition sketch of the staggered grid. The z-coordinates and layer thickness are defined at pressure points. 5

Efficient Numerical Simulation of Non-Hydrostatic Waves

compact schemes with a high order of accuracy. Another way of increasing the order of accuracy is by creating larger stencils, as described in Lele (1992). For the vertical discretisation, a set of strictly horizontal layers is introduced:

{ zk | k = 0,1,..., kmax}

∧

z0 ≤ − d ( x) ≤ ζ ( x, t ) ≤ zkmax

∀x

(2.2)

Note that the strictly horizontal layers cause the number of layers to vary as a function of the depth as well as the surface elevation. Therefore the layer thickness at the surface and at the bottom will be different than the distance between the horizontal layers. In general terms, the layer thickness can be expressed by:

∆zm ,k = min (ζ m , zm ,k ) − max ( d m , zm ,k −1 )

(2.3)

Control volumes are occupied by fluid (wet) as long as ∆zm,k > 0 holds. The index of the top and bottom layer will vary because of the varying geometry and free surface. The top layer of the mth vertical (at xm) will therefore be denoted as ktop(m), the bottom layer as kbot(m). Note that at a u-velocity point the thickness of the layer is not uniquely defined. In the present implementation an upwind approach is used, meaning that the layer thickness is determined by the direction of the flow. For this purpose the operator mu is used, which is defined by:

m m + 1  mu(m) =  m m + 1

if if if if

Um Um Um Um

>0 ζ m +1 = 0 ∧ ζ m < ζ m +1

(2.4)

U indicates the depth-averaged horizontal velocity. With the use of this operator, the layer thickness at a u-point with index m and k is defined as ∆zmu(m),k. This upwind approach guarantees a positive total water depth at water level points.

2.3

Preparation of the 2DV Navier Stokes equations for the model The basic equations of the model are the 2DV Navier Stokes equations. For simplicity, the baroclinic pressure, Coriolis acceleration, viscosity terms and surface stress are omitted. The horizontal momentum equation reads:

Du ∂p + =0 Dt ∂x

(2.5)

and the vertical momentum equation reads:

Dw ∂p + = −g Dt ∂z

(2.6)

where p denotes the total normalized pressure, this is the pressure divided by reference density ρ0. This pressure is split up in a hydrostatic part ph and a hydrodynamic part q such that:

p = ph + q ∧

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ph ( x, ζ ( x, t ), t ) = q( x, ζ ( x, t ), t ) = 0

(2.7)

May 2002

The assumptions made in (2.7) indicate that atmospheric (large scale) spatial pressure differences are neglected for this study. Small scale phenomena, such as wave-wind interaction, are neglected as well. For the horizontal momentum, Substitution of (2.7) in (2.5) yields:

Du ∂q ∂ζ + +g =0 Dt ∂x ∂x

(2.8)

and for the vertical momentum equation:

Dw ∂q + =0 Dt ∂z

(2.9)

The position of the free surface is assumed to be a singe-valued function ζ = ζ(x, t), thereby excluding overtopping waves. The kinematic boundary conditions at the bottom and the free surface read:

∂ζ ∂ζ +u ∂t ∂x

(2.10)

∂d =0 ∂x

(2.11)

Free surface (z = ζ ):

w=

Bottom (z = -d ):

w+u

Note that there is no explicit boundary condition for the pressure. The pressures result directly from (2.8), (2.9), (2.10) and (2.11). The local continuity equation will be used in a layer averaged fashion such that the total amount of water in a cell is conserved: zk

zk

∂h u ∂z '  ∂u ∂w  ∫z  ∂x + ∂z  dz ' = ∂kx k + wk − wk −1 − u ∂x z k −1 k −1

(2.12)

with u indicating the layer averaged horizontal velocity and hk = zk – zk-1. Based on fixed layers, the last term is zero in the interior layers, except for the top and bottom layer. Therefore the local continuity equation yields:

 ∂hk uk  ∂x + wk − wk −1 − utop   ∂hk uk + wk − wk −1 = 0   ∂x  ∂hk uk  ∂x + wk − wk −1 − ubot 

∂ζ = 0 if ∂x if ∂d = 0 if ∂x

k = top k ≠ top ∧ k ≠ bot

(2.13)

k = bot

The layer averaged form of the local continuity, particularly the addition of the last term in (2.12) is the main difference with the model described in Stelling & Van Kester (2000). They discretise the regular local continuity equation and make the model conservative numerically, our new approach makes the model conservative analytically and then discretises.

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Efficient Numerical Simulation of Non-Hydrostatic Waves

Integrating the local continuity equation over the water depth gives by Leibniz’ rule: ζ

ζ

∂ ∂ζ ∂d  ∂u ∂w  ∫− d  ∂x + ∂z  dz = ∂x −∫d udz + wtop − wbot − utop ∂x − ubot ∂x Substitution the kinematic boundary conditions (2.10) and (2.11) for w gives the depthaveraged continuity equation: ζ

∂ζ ∂ + udz = 0 ∂t ∂x −∫d

2.4

(2.14)

Discretisation of the equations Given the definition of the staggered grid, the horizontal momentum equation (2.8) is discretised as:

umn +,k1 − umn ,k ∆t

+ ADV (u ) nm ,k + g

ζ mn +1 − ζ mn ∆x

+

n +1 n +1 n +1 n +1 1 ( qm +1,k + qm +1,k −1 ) − ( qm, k + qm ,k −1 ) =0 2 ∆x

(2.15)

The vertical momentum equation is discretised as:

wmn +,k1 − wmn , k ∆t

n +1

+ ADV ( w)

n m ,k

∂q + =0 ∂z m ,k

(2.16)

The advective terms are implemented by an explicit first order upwind scheme, outlined in appendix D. In the future, a momentum conserving scheme can be implemented, which is especially important near local flow discontinuities, such as breaking waves, bores or a discontinuous bottom profile. The explicitness of the scheme imposes a time step limit, the so-called CFL-condition for advection:

 ∆x ∆z  ∆t ≤ min  ,  U w The local layer-averaged continuity equation (2.13) is discretised as: n n +1 n n +1 ∆zmu ( m ), k um , k − ∆z mu ( m −1), k um −1, k

∆x

+ wmn +, k1 − wmn +,k1−1 = 0

(2.17)

Note the operator mu is used here. Recall the layer thickness at u-points is defined by this operator, so this is a volume-conserving implementation. For the top and bottom layer, (2.17) is extended with an extra term. For example, the last term in (2.13) for the bottom layer is discretised as: n n +1 n n +1 ∆zmu ( m ), k um , k − ∆zmu ( m −1), k um −1, k

∆x

n +1 + wmn +, k1 − wmn +,k1−1 − umu ( m ) −1, kbot ( m )

d mu ( m ) − d mu ( m ) −1 ∆x

=0

And for the top layer: n n +1 n n +1 ∆zmu ( m ), k um , k − ∆zmu ( m −1), k um −1, k

∆x 8

n +1 + wmn +, k1 − wmn +,k1−1 − umu ( m ) −1, ktop ( m )

n n ζ mu ( m ) − ζ mu ( m ) −1

∆x

=0

May 2002

The discretised global continuity equation is given by:

ζ mn +1 − ζ mn ∆t

+

ktop ( m )

∑ ( ∆z

k = kbot ( m )

n mu ( m ), k

n n +1 umn +,k1 − ∆zmu ( m −1), k um −1, k ) = 0

(2.18)

The kinematic boundary condition at the free surface is given by: 1 wmn +, ktop (m) −

ζ mn +1 − ζ mn ∆t

n +1 − umu ( m ) −1, ktop ( m )

n n ζ mu ( m ) − ζ mu ( m ) −1

∆x

=0

(2.19)

and at the bottom: n +1 1 wmn +,kbot ( m ) −1 + umu ( m ) −1, kbot ( m )

n n d mu ( m ) − d mu ( m ) −1

∆x

=0

(2.20)

The kinematic boundary condition is defined at the pressure points, so the index of u has to be decreased by unity for obtaining an upwind approach around this pressure point.

2.5

Consistency of the model’s kinematic boundary conditions In this paragraph we show the consistency of the model with respect to the continuity equations and the kinematic boundary conditions. To do this, summing the local continuity equation (2.17) over all layers should result in exactly zero: n n +1 n n +1  ∆zmu  ( m ), k um , k − ∆z mu ( m −1), k um −1, k n +1 n +1   + wm,ktop ( m ) − wm ,kbot ( m ) −1 ∑ ∆x kbot ( m )   n n d mu ( m ) − d mu ( m ) −1 ζ mu ( m ) − ζ mu ( m ) −1 n +1 n +1 u −umu − =0 ( m ) −1, kbot ( m ) mu ( m ) −1, ktop ( m ) ∆x ∆x ktop ( m )

Substitution of the global continuity equation (2.18) gives: n n  n +1  ζ mu ζ mn +1 − ζ mn ( m ) − ζ mu ( m ) −1 n +1 − umu ( m )−1,ktop ( m )  wm, ktop ( m ) −  ∆t ∆x  

 1 d mu ( m ) − d mu ( m ) −1  n +1 −  wmn +,kbot + umu  = 0 m m kbot m ( ) − 1 ( ) − 1, ( )  x ∆   Substitution of the discrete kinematic boundary conditions (2.19) and (2.20) into the expressions for w results in exactly zero. This proves the kinematic boundary conditions are consistent with the local and global continuity equation.

2.6

Solution of the system

2.6.1

A 4D matrix for storage of 2D pressure gradient coefficients The model uses a four-dimensional matrix to store the coefficients of the pressure gradients. Normally, when models have two or more dimensions a numbering algorithm is required, since all of the unknowns need to be mapped onto a vector. When a 4 dimensional matrix is defined, renumbering is no longer necessary for two dimensional models. This keeps the code legible and less prone to bugs. The matrix A has four indices, and reads:

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9

Efficient Numerical Simulation of Non-Hydrostatic Waves

n +1

1 kmax

∑∑A

l =−1 o = 0

m , k ,l , o

∂q ∂z m +l ,o

In the previous expression the first two indices represent the location m, k the equation is set up for. Theoretically, this matrix can have relations with any other pressure gradient in the domain. For this purpose the two additional indices l, o are used. In order to keep the matrix compact, just relations with the pressure gradients directly west and east of the current cell are allowed, ranging from m – 1 to m + 1. Therefore, the third index is relative, ranging from -1 to 1. -1 indicates the cell to the west, 1 the cell to the east. The last index is used to identify the (absolute) vertical position of the neighbour. A graphical representation of the coefficient matrix is shown in Figure 2-3.

A m ,k ,−1,kmax

A m ,k ,0,kmax

n +1

A m ,k ,1,kmax

∂q ∂z

A m ,k ,1,k

∂q ∂z m ,k

n +1

m , ktop ( m −1)

n +1

A m ,k ,−1,k

A m ,k ,0,k

m,k

A m,k ,−1,k −1

A m ,k ,−1,0

A m,k ,0,k −1

A m ,k ,0,0

A m,k ,1,k −1

n +1

∂q ∂z m,k −1

n +1

∂q ∂z m+1,k −1

A m ,k ,1,0 Figure 2-4: The horizontal momentum equation is coupled to all vertical pressure gradients towards the water surface.

Substitution of the momentum equations in the local continuity equation The pressure gradients are used as primary unknown for the final system of equations. To obtain the system, the equations for horizontal and vertical momentum equations are substituted in the local continuity equation. For the horizontal momentum equation (2.15), the horizontal pressure gradient can be rewritten by substituting (2.1) as:

( 2q

n +1 m +1, k

) (

− ( qmn ++11, k − qmn ++11, k −1 ) − 2qmn +,k1 − ( qmn +,k1 − qmn +, k1−1 )

2 ∆x  n +1 ∆z  qm +1, k −  4 

n m +1, k

10

n +1

∂q ∂z m +1, k

um,k

Figure 2-3: The 4D matrix indicates the relation the equation m, k has with its neighbours. The first two indices are used to indicate the equation, the last two to indicate the neighbour. The coefficient of the upper neighbour west of equation m, k is addressed by Am,k,-1,k.

2.6.2

∂q ∂z m+1,ktop ( m)

)=

n +1  ∂q    n +1 ∆zmn ,k ∂q +    −  qm,k −   4  ∂z m +1, k ∂z m +1, k −1    ∆x n +1

 ∂q n +1 ∂q n +1   +      ∂z m, k ∂z m, k −1  

(2.21)

May 2002

Note that by definition, q = 0 at the free surface. Therefore, qmn +, k1 can be expressed as a sum of vertical pressure gradients by: n +1 m,k

q

n +1 n +1  ∂q 1 ktop ( m ) n  ∂q = − ∑ ∆zm,l  +  ∂z m ,l ∂z m ,l −1  2 l = k +1  

(2.22)

In effect, (2.22) implies the horizontal momentum equation is coupled to all pressure gradients towards the water surface. This is shown in Figure 2-4. With the 4D matrix, the horizontal momentum equation can be formulated as:

u

n +1 m,k

1 ktop ( m )

+∑ l =0

∑ o =0

n +1

U m , k ,l , o

∂q = [ ru ]m ,k ∂z m +l ,o

(2.23)

U contains all the coefficients for the pressure gradient, [ ru ]m , k represents the right hand side of the equation, containing the explicit terms. The vertical momentum equation, (2.16) is expressed in terms of the vertical pressure gradient so needs no further treatment. It is dependent only of the vertical pressure gradient m, k, so the 4D matrix notation is not necessary. This equation can be formulated as: n +1

∂q = [ rw ]m, k ∂z m, k

wmn +, k1 + Wm, k

(2.24)

Substitution of horizontal and vertical momentum equations

n +1 m ,k

w

umn+−11,k

n +1

∂q ∂z m+1,ktop(m)

n +1

∂q ∂z m,k

n +1

∂q ∂z m +1,k

n +1

∂q ∂z m,k −1

n +1

∂q ∂z m +1,k −1

∂q ∂z m −1,k

n +1

umn+,k1

wmn +,k1−1

Figure 2-5:

n+1

n +1

∂q ∂q ∂z m −1,ktop ( m−1) ∂z m,ktop ( m )

∂q ∂z m −1,k −1

n +1

Substitution of the horizontal and vertical momentum equations into the local continuity equation yields a system of equations in terms of the vertical pressure gradients directly west, at the current position and east of the current position.

The final system of linear equations is obtained by substituting the horizontal and vertical momentum equations (2.23) and (2.24) into the local continuity equation (2.17). This results in a system of equations related to Figure 2-5 and given by:

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11

Efficient Numerical Simulation of Non-Hydrostatic Waves

n +1 n 1 ktop ( m )   ∆zmu ∂q ( m ), k  [ ru ]m ,k − ∑ ∑ U m ,k ,l ,o  ∆x  ∂z m +l ,o  l =0 o=0 n +1 1 ktop ( m −1)   ∆z n ∂q − mu ( m −1),k  [ ru ]m −1, k − ∑ ∑ U m −1, k ,l ,o   ∆x ∂z m +l −1,o  l =0 o=0  n +1 n +1   ∂q   ∂q +  [ rw ]m ,k − Wm, k −  [ rw ]m, k −1 − Wm,k −1  =0  ∂z m, k   ∂z m,k −1  

(2.25)

For the top and bottom layers (2.25) needs to be extended with the extra terms, resulting from (2.13). Equation (2.25) is an equation of the form: 1 kmax

∑ ∑ A m , k ,l , o

l =−1 0

n +1

∂q = [ r ]m ,k ∂z m +l ,o

(2.26)

In a vertical, the number of unknowns equals kmax + 1. Equation (2.26) represents kmax linear equations, so another relation is needed close the system. The kinematic boundary condition at the bottom (2.20) is used for this purpose. Substitution of the discretised momentum equations into (2.20), with m* = mu(m) - 1 leads to: n +1

∂q [ rw ]m,kbot ( m)−1 − Wm,kbot ( m)−1 ∂z m,kbot ( m )−1 n +1 1 ktop ( m * )   d m* +1 − d m* ∂q +  [ ru ]m* , kbot ( m ) − ∑ ∑ U m* ,k ,l ,o =0    ∂ z ∆ x * + l ,o l o = 0 = 1 m  

(2.27)

Note that this equation cannot be written in form (2.26) straightforwardly since its indices depend on the flow direction. Now that a closed system of linear equations is obtained the system can be solved. This is done by Gauss-elimination. The algorithm is implemented for the 4D coefficient matrix. The elimination is optimized for the special structure of the matrix, by only sweeping the cells in the neighbouring verticals. As stated in Stelling & Van Kester (2000), the number of operations is O(mmax x kmax3). The algorithm is not efficient for large kmax, but for the purpose of research it is sufficiently quick.

12

May 2002

3

Formulation of the model in sigma coordinates

3.1

Introduction The so-called sigma transformation consists of a conversion of the vertical axis in such a way that it fits both the free surface and the bottom, as shown in Figure 3-1. The sigma transformation creates a vertically boundary fitted system by scaling the water depth. This makes the handling of free surface and bottom layers easy, since the coordinate system automatically fits them. However, this makes the coordinate system time-dependent, generating extra terms in the balance equations. Although the sigma transformation is very elegant, bottom discontinuities and bores lead to a singular system. Despite this disadvantage, the sigma model is relatively easy to implement, since the number of layers is fixed. Thus it provides an easy platform for interpolation of the vertical pressure profile. In this chapter, a derivation is made, and the model is verified.

σ =0

σ = −1 Figure 3-1:

Example of sigma layers.

Literature provides sufficient information about the sigma transformation, although mainly for shallow water models. The sigma transformation was first introduced by Philips (1957). A clear and thorough introduction of the sigma transformation is given by Van Kester & Uittenbogaard (1993), although this paper does not formulate the full Navier Stokes equations (hydrodynamic pressure is omitted). These are presented by Gaarthuis (1994). A derivation based on tensor analysis is given by Dunsbergen (1994). More general expressions for the Navier Stokes equations under coordinate transformations are given by Wesseling (2001) and Emanuel (2001). Stansby & Zhou (1998) derive a non-hydrostatic model based on a pressure correction method that uses a strictly horizontal approach for the pressure gradient. A special form of the sigma transformation is presented by Zijlema (1998, 2000) for TRIWAQ (Rijkswaterstaat). He introduces the sigma-layers, but does not perform the actual transformation to the σ- domain. All layer averaging is performed in the original domain (x, z, t), with time-dependent borders of the integrals. The reason for this is that TRIWAQ allows hybrid coordinate systems (both fixed and sigma layers in the vertical), which requires a formulation in terms of the vertical coordinate z. The derivation presented in this chapter uses a formulation in terms of z as well. This can be justified by historical reasons only. The original derivation of the model was an analogue of Zijlema (2000), and gradually changed by the influence of the WL philosophy.

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13

Efficient Numerical Simulation of Non-Hydrostatic Waves

3.2

The concept of the sigma transformation The sigma transformation yields a transformation of the vertical z axis. The transformation is performed in such a way that the coordinate system fits both the bottom and free surface, as can be seen in Figure 3-1. The sigma transformation is given by:

σ=

x* = x

z −ζ z −ζ = ζ +d H

t * = t (3.1)

In this equation d represents the depth of the bottom relative to z = 0. H represents the total water depth. This means σ = -1 at the bottom and σ = 0 at the free surface. The advantage of sigma layers is that the number of layers remains fixed, independent of the water depth. For a fixed-layer solution this is not the case. However, the sigma transformation requires a smooth bottom profile, since dσ / dx tends to infinity on steps in the bottom. The fixed layer solution has no problems with steps in the bottom. The sigma transformation is time dependent, because ζ and H occur in (3.1).

3.3

Transformation of the 2DV Navier Stokes equations

3.3.1

Definition of the new vertical velocity In a sigma model an extra vertical velocity ω is introduced, although this does not replace w. The velocity w is the physical velocity component measured in a Cartesian space. The transformed velocity ω is defined relative to the zp(t+∆t) moving sigma coordinate system. zp(t)

The new vertical velocity ω is defined as the rate of change in vertical distance between a water particle and an observer that moves along with the projection of the water particle on a sigma isoline, see Figure 3-2. In mathematical terms:

ω=

Particle path

Lz

zσ(t) zσ(t+∆t)

DLz Dt

σ-isoline

(3.2) Figure 3-2: Definition of vertical velocity ω

Lz is defined as the distance between the σ-line and the particle. This is the difference between the z-coordinate of the particle (zp), and the z-coordinate of the σ-isoline at the location at the particle (zσ). The observer travelling along the σ-line moves with the horizontal speed of the particle. Thus the vertical velocity ω is defined as:

ω=

Dz p Dt

−

Dzσ ∂z ∂z = w− σ −u σ Dt ∂t ∂x

(3.3)

The σ-transformation introduces a coupling between the horizontal and vertical particle speed. The kinematic boundary conditions, in terms of ω, reduce to: Free surface (z = ζ ):

14

w=

∂ζ ∂ζ +u ⇔ ωζ = 0 ∂t ∂x

(3.4)

May 2002

w+u

Bottom (z = -d ):

3.3.2

∂d = 0 ⇔ ω− d = 0 ∂x

(3.5)

Formulation of first order time and space derivatives The first order derivatives with respect to the Cartesian coordinates are expressed in terms of sigma-coordinates as:

∂ ∂t

= x, z

∂ ∂t *

x* ,σ

∂σ ∂t

t * ,σ

∂σ + ∂x

+

∂ ∂ = * ∂x t , z ∂x

x, z

∂ ∂σ

x* ,σ

t,z

∂ ∂σ

t * ,σ

(3.6)

The subscripts show in what space the space and time coordinates are held constant; asterixes indicate the new time and space variables. The derivatives with respect to σ and the sigma-isolines zσ are related by:

∂z ∂ ∂ ∂ = σ =H ∂σ ∂σ ∂zσ ∂zσ

(3.7)

An important relation is the partial derivative of σ with respect to x and t. These can be written as:

∂σ 1 =− H ∂x

∂H  ∂ζ +σ  ∂x  ∂x

1 ∂zσ  =− H ∂x 

∂σ ∂H 1  ∂ζ =−  +σ ∂t ∂t H  ∂t

1 ∂zσ  =− H ∂t 

(3.8) x, z

(3.9) x, z

Substituting (3.7), (3.8) and (3.9) into (3.6) gives:

∂ ∂t

= x, z

∂ ∂t *

∂ ∂ = * ∂x t , z ∂x

x* ,σ

∂zσ ∂t

t * ,σ

∂z − σ ∂x

−

x, z

∂ ∂zσ

x* ,σ

t,z

∂ ∂zσ

t * ,σ

(3.10)

The material derivative must remain invariant under coordinate transformations, provided the time progresses at equal rates (t* = t):

D D = * Dt Dt

(3.11)

Using the definition of the material derivative, and substituting (3.3) and (3.10) gives:

D ∂ = Dt ∂t ∂ = * ∂t

+u x, z

x* ,σ

∂ ∂ +w ∂x t , z ∂z

∂ +u * ∂x

t * ,σ

x ,t

∂ +ω ∂zσ

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(3.12) x ,t *

15

Efficient Numerical Simulation of Non-Hydrostatic Waves

The transformation of the horizontal gradients in sigma coordinates may introduce large truncation errors near a steep bed, resulting from the summation of large terms of opposite sign, see Stelling and Van Kester (1994). The truncation error can cause spurious flows. In the present implementation this approach is not used.

3.3.3

Transformation of the equations Transformation of the horizontal momentum equation (2.8)by substitution of (3.6) results in:

Du * ∂ζ ∂q* ∂q* ∂zσ g + + − Dt * ∂x* ∂x* ∂zσ ∂x*

(3.13)

Transformation of the vertical momentum equation gives:

Dw* ∂q* + =0 Dt * ∂zσ

(3.14)

with q* as the hydrodynamic part of the pressure. It is normalized by dividing by a constant reference density ρ0. The transformed local continuity equation reads:

∂u * ∂w* ∂zσ ∂u * + − =0 ∂x* ∂zσ ∂x* ∂zσ

(3.15)

The transformed global continuity equation reads:

∂ζ ∂ + * * ∂t ∂x

ζ

∫

u *dzσ = 0

(3.16)

z =− d

This concludes the formulation of the 2DV Navier Stokes equations in sigma coordinates. The asterixes for the new space and time variables will be dropped from now on.

3.4

Discretisation of the equations The equations will be discretised on a staggered grid, the same way as in Chapter 2. In the vertical, a set of sigma layers is introduced. This set of layers is independent of the horizontal coordinate x: ∂q z, k wm,k , ωm,k , qm,k , ∂z m,k

{σ

k

k = 0,1,..., kmax} ∧

−1 = σ 0 < σ k < σ k +1 < σ kmax = 0

zm,k

m, k

(3.17)

k is the index for the layers. The sigma coordinate system is boundary fitted. Therefore, the number of layers is fixed. The z-coordinate of the layer interface is given by:

zk = ζ + σ k H

(3.18)

z0 represents the bottom and zkmax the free surface. Note the surface elevation and total water depth are a function of x, therefore zk is a function of x as well.

16

hm,k

um,k

zm,k-1 x, m Figure 3-3: Cell definition sketch of the staggered grid for sigma coordinates. The z-coordinates and layer thickness are defined at pressure points.

May 2002

The layers are numbered from 1 to kmax, with 1 representing the bottom layer and kmax representing the top layer. Figure 3-3 shows a definition sketch for the variables defined on the staggered grid. The layer thickness hm,k and z-coordinates are defined at the pressure points. An upwind approach is applied for the layer thickness and water level at the velocity points, which guarantees a positive total water depth at the pressure points. The depth-averaged velocity U is used for the upwinding. Again, the operator for the upwind approach will be denoted by mu and is defined by:

m  m + 1 mu (m) =  m m + 1

∧ Um > 0 ∧ Um < 0

(2.4)

∧ U m = 0 ∧ ζ m > ζ m +1 ∧ U m = 0 ∧ ζ m < ζ m +1

In a velocity point, the layer thickness equals hmu(m), k. The Hermitian relation (2.1)between the pressure gradient and pressure remains the same and it reads:

qm ,k − qm ,k −1 hm ,k

=

1 ∂q 1 ∂q + 2 ∂z m, k 2 ∂z m, k −1

(3.19)

The horizontal momentum equation (3.13) is discretised as:

umn +,k1 − umn ,k

ζ mn +1 − ζ mn

(q +

n +1 m +1, k

+ qmn ++11,k −1 ) − ( qmn +,k1 + qmn +,k1−1 )

+ ADV (u ) + g ∆t ∆x 2 ∆x n +1 n +1 n +1 n +1 n n  zm +1,k − zm, k + zmn +1,k −1 − zmn ,k −1 ∂q ∂q ∂q 1  ∂q −  + + + =0  4  ∂z m ,k ∂z m ,k −1 ∂z m +1,k ∂z m +1,k −1  2∆x n m,k

(3.20)

The only difference with the discretised horizontal momentum equation is the discretised cross term. The vertical momentum equation is discretised as:

wmn +,k1 − wmn ,k ∆t

n +1

+ ADV ( w)

n m,k

∂q + =0 ∂z m,k

(3.21)

For the advective terms a first order upwind scheme is applied, outlined in Appendix D. Before the local continuity equation (3.15) is discretised, it is formulated in a conservative way: zk

 ∂u * ∂w* ∂zσ ∂u *  ∫  ∂x* + ∂zσ − ∂x* ∂zσ  dz = 0 zk −1  Application of Leibniz’ rule leads to the conservative local continuity equation:

∂hk u *   ∂z + wk − wk −1 −   σ* * ∂x   ∂x

  ∂z u  ( zk ) −  σ*   ∂x

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  u  ( zk −1 )  = 0  

(3.22)

17

Efficient Numerical Simulation of Non-Hydrostatic Waves

(3.22) can be discretised as: n n +1 n n +1 hmu ( m ), k um , k − hmu ( m −1), k um −1, k

∆x For

 δ zn δ zn  + wmn +, k1 − wmn +,k1−1 −  umn +,k1 m, k − umn +, k1−1 m ,k −1  = 0 δx δx  

(3.23)

δz an upwind strategy is applied, for stability reasons. Central differences lead to δx

unstable simulations in regions with large variations in the geometry. By definition, the horizontal velocity component umn +, k1 is defined at pressure points, so it needs to be interpolated from the surrounding horizontal velocities as:

umn +, k1 =

n n +1 n n +1 n n +1 n n +1 hmu ( m −1), k um −1, k + hmu ( m −1), k +1um −1, k +1 + hmu ( m ), k um , k + hmu ( m ), k +1um , k +1

2hmn ,k + 2hmn ,k +1

(3.24)

The global continuity equation is discretised as:

ζ mn +1 − ζ mn ∆t

1 kmax n ∑ ( hmu ( m),k umn+,k1 − hmun (m−1),k umn+−11,k ) = 0 ∆x k =1

+

(3.25)

Finally, the kinematic boundary conditions are discretised. The kinematic boundary condition at the free surface (3.4) leads with (3.30) to: n +1 m , kmax

w

ζ mn +1 − ζ mn

−

∆t

n n +1 n n +1 n hmu ( m −1), kmax um −1, kmax + hmu ( m ), kmax um , kmax δζ m − =0 δx 2hmn ,kmax

(3.26)

The kinematic boundary condition at the bottom (3.5) reads: n +1 m ,0

w

+

n n +1 n n +1 hmu ( m −1),1um −1,1 + hmu ( m ),1um ,1 δ d m , k

δx

2hmn ,1

=0

(3.27)

Although the relative vertical velocity ω does not show up in the momentum and continuity equations, it is hidden in the calculation of the advection terms. The vertical velocity component ω can always be expressed in terms of u and w, since the geometry is handled explicitly. It is calculated from a different version of the layer-averaged continuity equation (3.22). After substitution of the equation of ω (3.3) the following result is obtained:

∂ hk ∂ hk uk + + ωk − ωk −1 = 0 ∂t ∂x

(3.28)

Discretisation of (3.28) gives:

hmn +,k1 − hmn , k ∆t

+

n n +1 n n +1 hmu ( m ), k um , k − hmu ( m −1), k um −1, k

∆x

+ ωmn +,k1 − ωmn +,k1−1 = 0

(3.29)

A comparison of (3.29) to (3.23) shows that the equation for the discretised relative vertical velocity ω reads:

ω

18

n +1 m ,k

n +1 m ,k

=w

−

zmn +,k1 − zmn ,k ∆t

−u

n +1 m ,k

δ zmn ,k δx

(3.30)

May 2002

Here, ω is calculated after the system of equations is solved, when both un+1 and hn+1 are known. Since ω0 = 0 at the bottom, (3.29) can be used to calculate ω1, which can be used to calculate ω2 and so on. In this way an extra check is built in to verify the correctness and accuracy of the model, since calculating ω over all the layers should result in zero at the free surface because of the kinematic boundary condition (3.4).

3.5

Consistency of the model’s kinematic boundary conditions and the local continuity equation In this paragraph the consistency of the model will be shown with respect to the continuity equations and the kinematic boundary conditions. To do this, summing the local continuity equation (3.23) over all layers should result in exactly zero: n n +1 n n +1 n  hmu  n +1 δ zmn ,k  ( m ), k um , k − hmu ( m −1), k um −1, k n +1 n +1 n +1 δ z m , k −1 w w u u + − − −    ∑ , , 1 m ,k m , k −1 m k m k −    = 0  ∆x x x δ δ k =1   

kmax

and its simplification gives: n n +1 n n +1  hmu ( m ), k um , k − hmu ( m −1), k um −1, k  ∑ ∆x k =1 

kmax

 δζ n n +1 n +1 n +1 n +1 δ d  + wm ,kmax − wm ,0 − um ,kmax m − um ,0 m = 0 δx δx 

Substitution of the global continuity equation (3.25) gives:

 n +1 ζ mn +,k1 − ζ mn ,k δζ mn   n +1 n +1 δ d mn  n +1 − um,kmax  wm,kmax − −w +u =0 ∆t δ x   m ,0 m ,0 δ x   Note that the substitution of the discrete kinematic boundary conditions (3.26) and (3.27) into the expressions for w results in exactly zero. This proves that the kinematic boundary conditions are consistent with the local and global continuity equation.

3.6

Solution method The pressure gradients are used as the primary unknown to be solved. The horizontal momentum (3.20) can be written as a function of the pressure gradient vector in the vertical as:

u

n +1 m,k

1 kmax

+ ∑ ∑ U m , k ,l , o l =0 o =0

n +1

∂q = [ ru ]m, k ∂z m +l ,o

(3.31)

Note that U is the 4D matrix as defined in paragraph 2.6.1. The essence of (3.31) is that the horizontal momentum equation for index m, k is a function of all pressure gradients at the pressure points of index m and m + 1. The vertical momentum equation (3.21) can be written as: n +1

n +1 m,k

w

+ Wm, k

∂q = [ rw ]m, k ∂z m, k

(3.32)

Substitution of the momentum equations (3.31) and (3.32) into the local continuity equation (3.23) leads to a large stencil with 12 unknown pressure points. The stencil is shown in Figure 3-4. WL | delft hydraulics & Delft University of Technology

19

Efficient Numerical Simulation of Non-Hydrostatic Waves

n +1

∂q ∂z m+1,ktop(m)

n +1

∂q ∂z m +1,k +1

n +1

∂q ∂z m +1,k

n +1

∂q ∂z m +1,k −1

n +1

∂q ∂z m+1,k − 2

n +1

∂q ∂z m,k +1

n +1

∂q ∂z m,k

∂q ∂z m −1,k −1

n +1

∂q ∂z m,k −1

n +1

∂q ∂z m,k − 2

∂q ∂z m −1,k +1 n+1 m−1,k +1

n +1 m, k +1

u

u n +1 m,k

w

umn+−11,k

n +1 m−1, k −1

u

Substitution of horizontal and vertical momentum equations

∂q ∂z m −1,k

n +1

n +1

umn+,k1

wmn +,k1−1

n +1 m, k −1

u

∂q ∂z m −1,k − 2 Figure 3-4:

n+1

n +1

∂q ∂q ∂z m −1,ktop ( m −1) ∂z m,ktop ( m )

n +1

n +1

Substitution of the horizontal and vertical momentum equations into the local continuity equation yields a system of equations in terms of the vertical pressure gradients directly west, at the current position and east of the current position.

Substitution leads to a system of equations with the pressure gradients as the only unknown variables: n +1 n +1 1 kmax 1 kmax   n   ∂q ∂q h  [ ru ]m, k − ∑ ∑ U m, k ,l ,o  − hmu ( m −1), k  [ ru ]m −1, k − ∑ ∑ U m, k ,l ,o  ∂z m +l ,o  ∂z m −1+l ,o  l =0 o =0 l =0 o =0   ∆x n +1 n +1   ∂q   ∂q (3.33) +  [ rw ]m ,k − Wm, k −  [ rw ]m, k −1 − Wm,k    ∂z m, k   ∂z m,k −1   n mu ( m ), k

n +1 1 kmax    δ zmn , k ∂q   [ ru ]m ,k − ∑ ∑ U m, k ,l ,o   ∂z m +l ,o  δ x l =−1 o = 0    − =0 + n 1 1 kmax  δ zmn ,k −1  ∂q     −  [ ru ]m,k −1 − ∑ ∑ U m ,k −1,l ,o ∂z  l =− o = 1 0 m − + l o 1 ,  δx   

U and [ ru ]m ,k represent the coefficients for umn +, k1 . These coefficients can be obtained by substituting (3.31) into (3.24) for all 4 velocities. Note that U uses all three relative positions for the third index l, instead of two. (3.33) is an equation of the form:

20

May 2002

n +1

1 kmax

∑∑A

l =−1 0

m , k ,l , o

∂q = [ r ]m ,k ∂z m +l ,o

(3.34)

For closing the system, the kinematic boundary condition at the bottom (3.27) is used again. Substitution of the discretised momentum equations into (3.27) leads to:

[ rw ]m,0 − Wm,0

3.7

n +1 1 ktop ( m −1)  n   ∂q  hmu ( m −1),1  [ ru ]m −1,1 − ∑ ∑ U m ,1,l ,o  n +1  ∂z m −1+l ,o   δ d l =0 o=0 1  ∂q  m + =0   n +1 (3.35) ktop ( m ) 1 δ x ∂z m,0 2hmn ,1    q ∂  n h r U + − [ ]    mu ( m ),1  u m ,1 ∑ ∑ m ,1,l ,o ∂z   l =0 o=0 m +l ,o    

Verification of the sigma model The sigma model is verified in the same way as the original z-coordinate model. The tests performed are: • Simulation of a standing wave in a closed basin • Simulation of the propagation of a solitary wave in a channel • Simulation of the Beji-Battjes experiment Furthermore, the original and sigma model will be compared for the Beji-Battjes experiment.

3.7.1

Standing wave in a closed basin This test demonstrates the correct prediction of the period of a standing non-hydrostatic wave in a closed basin. The basin is 10 meters deep and has a length of 20 meters.

d = 10 m

ζ0 = 0.1 m

B = 20 m Figure 3-5:

Geometry of the basin

For this situation the vertical accelerations of the standing wave are of the same order as the horizontal, therefore the hydrostatic pressure assumption does not hold. The propagation velocity c is no longer independent of frequency, as for long waves, but it is coupled by the dispersion relation for infinitesimal amplitude without current:

c=

ωw k

=

g tanh kd k

(3.36)

Here, ωw represents the angular velocity, k the wave number and d the still water depth. Note that for shallow water waves (kd ∞:

lim ξ = 1⋅ k →∞

App.36

0−

1 (1 + ( n − 1) ⋅ 0 ) 1 2n = 2n ( 0 − 1)

(G.21)

Appendix

1

kd → ∞ ⇒ ξ =

0.1

1 2n kd = 0.01

0.01 ξ [-]

kd = 0.1 0.001

kd = 0.5 kd = 1

0.0001

kd = 2 kd = 10

0.00001

kd = 50 0.000001

kd = ∞

0.0000001 1

2

3

4

5

6

7

8

9

10

number of layers n [-]

Figure G-2:

Definitive measure for the estimation by straight lines of the horizontal momentum exchange. Note the measure goes to zero when kd tends to zero.

WL | delft hydraulics & Delft University of Technology

App.37