Coastal Lagoons - Chapter 6 ppsx

Modeling Concepts

Boris Chubarenko, Vladimir G. Koutitonsky,

Ramiro Neves, and Georg Umgiesser

CONTENTS

6.1 Introduction

6.2 Numerical Discretization Techniques

6.2.1 Computational Grid

6.2.2 Control Volume Approach

6.2.3 Numerical Calculation of Advection

6.2.3.1 Spatial Approach

6.2.3.1.1 Linear Approach

6.2.3.1.2 Upstream Stepwise Approach

6.2.3.1.3 Quadratic Upwind Approach (QUICK)

6.2.3.2 Temporal Approach

6.2.4 Taylor Series Approach

6.2.4.1 Time Discretization

6.2.4.2 Spatial Discretization

6.2.5 Stability and Accuracy

6.2.5.1 Introductory Example

6.2.5.2 Stability

6.2.5.3 The Need for a Fine Resolution Grid

6.3 Pre-Modeling Analysis and Model Selection

6.3.1 Hydrographic Classification

6.3.1.1 Morphometric Parameters

6.3.1.2 Hydrological Parameters

6.3.2 Description of Forcing Factors

6.3.2.1 General Hierarchy of Driving Forces

6.3.2.2 Water Budget Components

6.3.2.2.1 Surface Evaporation Budget

6.3.2.2.2 Ocean–Lagoon Exchange Budget

6.3.2.3 Heat Budget

6.3.3 Pre-Estimation of Spatial and Temporal Scales

6.3.3.1 Flushing Time

6.3.3.1.1 Integral Flushing Time

6.3.3.1.2 Local Flushing Time

6.3.3.2 Surface and Bottom Friction Layers

6.3.3.3 Time Scales of Current Adaptation

6

L1686_C06.fm Page 231 Monday, November 1, 2004 3:39 PM

© 2005 by CRC Press

6.3.3.3.1 Wind Driven Current

6.3.3.3.2 Equilibrium Current Structure

6.3.3.3.3 Gradient Flow Development

6.3.3.4 Wind Surge

6.3.3.5 Seiches or Natural Oscillations of a Lagoon Basin

6.3.3.6 Wind Waves

6.3.3.7 Coriolis Force Action

6.3.4 Objectives of Modeling

6.3.5 Recommendations for Model Selection

6.3.5.1 Selection Possibilities for Hydrodynamic

and Transport Models

6.3.5.2 Possible Simplifications in Spatial Dimensions

6.3.5.3 Possible Simplification in the Physical Approach

6.3.5.4 Possible Simplification According to the Task

To Be Solved

6.3.5.5 Computer, Data, and Human Resources

6.4 Model Implementation

6.4.1 Bathymetry and the Computational Grid

6.4.1.1 Laterally Integrated Models

6.4.1.2 Horizontal Resolution Models

6.4.2 Initial Conditions

6.4.3 Boundary Conditions

6.4.4 Internal Coefficients: Calibration and Validation

6.5 Model Analysis

6.5.1 Model Restrictions

6.5.1.1 Physical Restrictions

6.5.1.2 Numerical Restrictions

6.5.1.3 Subgrid Processes Restrictions

6.5.1.4 Input Data Restrictions

6.5.2 Sensitivity Analysis

6.5.3 Calibration

6.5.4 Validation

Acknowledgments

References

Note: The term modeling is used in this chapter in the sense of “numerical

modeling.” Physical modeling, conceptual modeling, or numerical model￾ing will only be used explicitly in relevant cases.

6.1 INTRODUCTION

In Chapter 3, the concept of transport equation was introduced, starting from the

concepts of control volume and accumulation rate of a property inside this control

volume. Diffusive and advective fluxes were also defined to account for exchanges

between the control volume and its neighborhood, and the concept of evolution equation

was introduced by adding sources and sinks to the transport equation. A “model” is

L1686_C06.fm Page 232 Monday, November 1, 2004 3:39 PM

© 2005 by CRC Press

built on the same concepts. Its implementation requires the definition of at least one

control volume, the calculation of the fluxes across its boundary, and the calculation of

the source and sinks using values of the state variables inside the volume. The number

of dimensions of the model depends on the importance of relevant property gradients.

The simplest model is the “zero-dimensional” model. In this model, there is no

spatial variability, and only one control volume needs to be considered. At the other

extreme of complexity is the three-dimensional (3D) model, which is required when

properties vary along the three spatial dimensions. Whatever the number of its

dimensions, a model must include the following elements:

• Equations

• Numerical algorithm

• Computer code

The order of the items in this list can also be considered the order of their chrono￾logical development. Hydrodynamic equations are based on mass, momentum, and

energy conservation principles, which were presented in Chapter 3. These have been

known for more than 100 years. Actually, numerical algorithms used to solve hydro￾dynamic models were attempted even before the existence of computers. The analytical

equations and the numerical algorithms developed before the existence of computers

allowed the rapid development of modeling starting in the 1960s, when computers were

made available to a small scientific community. Since that time, models and the mod￾eling community have evolved exponentially. Modern integrated computer codes have

done more for interdisciplinarity than 100 years of pure field and laboratory work.

The number of implementations of a model to solve various problems increases

the knowledge of the range of validity of the model equations. The accuracy of the

numerical algorithm is better known and confidence in the results increases. At that

time, the major source of errors in the results is the existence of mistakes in the data

files. Once the model equations, algorithms, and results are validated, the next priority

is the development of a user-friendly graphical interface that simplifies the use of the

model by nonspecialists. This reduces the errors of input files and simplifies the checking

of those files. This chapter presents the concepts and methodologies used to build models

and to understand their functioning.

6.2 NUMERICAL DISCRETIZATION TECHNIQUES

Computers can solve only algebraic equations. Analytic equations, integral or dif￾ferential, must be discretized into algebraic forms. The procedure followed depends

on the form of the analytical equation to be solved. The control volume approach

is best for the integral form of evolution equations, while the Taylor series is best

suited for differential equations.

6.2.1 COMPUTATIONAL GRID

The calculation of fluxes across a control volume surface is simpler if the scalar

product of the velocity by the normal to each elementary area (face) composing that

L1686_C06.fm Page 233 Monday, November 1, 2004 3:39 PM

© 2005 by CRC Press

surface remains constant in each of them. The control volume that makes that

calculation simpler must have faces perpendicular to the reference axis. If rectangular

coordinates are used, the control volume generating the simpler discretization is a

parallelepiped. In the case of a large oceanic model, a suitable control volume will

have faces laying on meridians and parallels.

In depth-integrated models, also called two-dimensional or 2D horizontal mod￾els, the upper face of the control volume is the free surface and the lower face is

the bottom. In three-dimensional or 3D models, a control volume occupies only part

of the water column and its shape depends on the vertical coordinate used. In coastal

lagoons, Cartesian and sigma-type coordinates (or a combination of both) are the

most commonly used coordinates.

The ensemble of all control volumes forms the computational grid. In finite￾difference-type grids, control volumes are organized along spatial axes and a struc￾tured grid is obtained. In contrast, typical finite-element grids are nonstructured. The

latter are more difficult to define, but they are more flexible, thus allowing some

variability in the spatial resolution. Figure 6.1 shows an example of a very general

finite-difference-type grid using several discretizations in the vertical direction.

A system can be considered one-dimensional (1D) if properties change only

along one physical dimension. In this case, control volumes can be aligned along

the line of variation and one spatial coordinate is enough to describe their locations.

Properties are considered as being constants across control volume faces perpendic￾ular to that axis. Fluxes across the faces not perpendicular to that axis are null or

have no net resultant.

6.2.2 CONTROL VOLUME APPROACH

Control volumes used in numerical models have the same meaning as the derivation

of the evolution equation in Chapter 3. A discretization is adequate if it generates a

simple calculation algorithm while maintaining the accuracy of the results. The

FIGURE 6.1 Example of a grid for a three-dimensional (3D) computation. Two vertical

domains are used. The upper domain uses a sigma coordinate. The lower one uses a Cartesian.

L1686_C06.fm Page 234 Monday, November 1, 2004 3:39 PM

© 2005 by CRC Press

simpler calculation is obtained if properties can be considered as being constant inside

the control volume and along parts of its surface. To make this possible without com￾promising accuracy, the control volume must be as small as possible; a fine-resolution

grid is needed.

In a 1D model, properties can be stored into 1D arrays (vectors). Adjacent

elements of a generic element i are i – 1 on the left side and i + 1 on the right

side (Figure 6.2). The length of a control volume must be small enough to allow

properties in its interior to be represented by the value at its center. In that case,

equations deduced in Section 3.2 apply and the rate of accumulation in volume i will

be given by

where ∆t is the time step of the model. This equation is simplified if the volume

remains constant in time. This is not the case in most coastal lagoons subjected to

changing winds and it is certainly not the case in tidal lagoons.

Exchanges between i volume and neighboring ones are accounted for by advec￾tive and diffusive fluxes. Their calculation requires some hypotheses. Let us consider

Figure 6.2 and define the distances between the faces (spatial step) and the location

points where other auxiliary variables are defined as shown in Figure 6.3. The net

advective gain of matter to volume i is given by

where while the diffusive flux, using the approach of Chapter 3, is

given by

FIGURE 6.2 Example of one-dimensional (1D) grid.

Vi

Vi−1

Vi+1

Accumulation Rate = − + () () VC VC

t

i i

t t

i i

∆ t

∆

QC QC i i i i

t t

−− ++

=

− ( ) 1

2 1

2 1

2 1

2

*

Q uA i ii − −− 1 =2 1

2 1

2

−( ) −

+











 + ( ) −

+









 − −  −

−

=

+ +

+

+

=

ν ν i i

i i

i i

t t

i i

i i

i i

t t

A C C

x x

A C C

x x 1

2 1

2 1

2 1

2

1

1

2 1

1

1

2 1 () ()

* *

∆∆ ∆∆

L1686_C06.fm Page 235 Monday, November 1, 2004 3:39 PM

© 2005 by CRC Press

In these equations, t

* is a time interval between t and t + ∆t, to be defined

according to criteria outlined in the next paragraph. is the concentration on

the interface between elements i and i – 1 and will be specified later. Combining

the three equations, we obtain:

(6.1)

In order to introduce the Taylor series discretization methods and to analyze

stability and accuracy concepts, let us consider a simplified version of Equation (6.1).

Consider the particular case of a channel with uniform and permanent geometry and

regular discretization. The cross section (A), volume (V), and discharge are constant.

Assume that diffusivity can be considered constant. Under these conditions,

Equation (6.1) becomes

(6.2)

where U is the constant cross-section average velocity and ∆x is the ratio between

the volume and the average cross section. This is the most popular form of the

transport equation but, as shown above, it is applicable only to particular conditions.

Additional approaches are required to calculate the advective flux, because the

concentration is defined at the center of the control volumes and not at the faces. These

approaches and their numerical consequences are described in the next sections.

FIGURE 6.3 Generic control volume in a 1D discretization.

Ci−1 C i Ci+1

Vi+1 Vi

Vi−1

Qi−1/

2 Qi+1/

2

νi− 1/

2 νi−1/

2

Ai−

∆xi−1

Ai+1/

2

∆xi+1 ∆xi

1/

2

Ci−1

2

() ()

() (

*

*

VC VC

t

QC QC

A C C

x x

A C C

x

i i

t t

i i

t

ii ii

t t

i i

i i

i i

t t

i i

i i

i

+

−− ++

=

− −

−

−

=

+ +

+

− = − ( )

− ( ) −

+











 + ( ) −

∆

∆

∆∆ ∆

1

2 1

2 1

2 1

2

1

2 1

2 1

2 1

2

1

1

2 1

1

1

2

ν ν

+











 +

=

∆xi

t t

1)

*

C C

t

U

C C

x

C CC

x

i

t t

i

t i i

t t

i ii

t t + − +

=

− +

=

− =  −









 +  − +



 





∆

∆∆ ∆

1

2 1

2 1 1

2

2

* *

ν

L1686_C06.fm Page 236 Monday, November 1, 2004 3:39 PM

© 2005 by CRC Press

6.2.3 NUMERICAL CALCULATION OF ADVECTION

6.2.3.1 Spatial Approach

Three common approaches are used to estimate concentration values at control

volume faces:

• Linear approach

• Upstream stepwise approach

• Quadratic upwind approach (QUICK)

6.2.3.1.1 Linear Approach

In the linear approach it is assumed that:

Assuming a discretization where the grid size is uniform, it is easily seen that this

approach generates central differences as obtained using the Taylor series (see

Section 6.2.4).

6.2.3.1.2 Upstream Stepwise Approach

In this case, it is assumed that the concentration at the left face is

This discretization respects the transportivity property of advection. This property

states that advection can transport properties only downstream or that information

comes only from upstream. The linear approach does not respect this property

because volume i will get information of downstream concentration through the

average process. The violation of this property can generate instabilities and will

create conditions to obtain negative values of the concentration. The upstream

discretization avoids this limitation but, as shown in the following paragraphs, it can

introduce unrealistic numerical diffusion.

6.2.3.1.3 Quadratic Upwind Approach (QUICK)

The quadratic upwind approach, or QUICK scheme, is an attempt at a compromise

between respecting the transportivity property and keeping numerical diffusion at

low values. In this case, it is assumed that the concentration distribution around a

point follows a quadratic distribution centered on the upstream side of the face

C Cx C x

x x i

ii i i

i i

−

− −

−

= +

+ 1

2

1 1

1

∆ ∆

∆ ∆

Q CC

Q CC

i ii

i ii

>⇒ = ( )

<⇒ = ( )

− −

−

0

0

1

2

1

2

1

L1686_C06.fm Page 237 Monday, November 1, 2004 3:39 PM

© 2005 by CRC Press