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Coastal Lagoons - Chapter 3 pdf
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Mô tả chi tiết
Physical Processes
Georg Umgiesser and Ramiro Neves
CONTENTS
3.1 Introduction to Transport Phenomena
3.2 Fluxes and Transport Equation
3.2.1 Velocity and Diffusivity in Laminar and Turbulent
Flows and in a Numerical Model
3.2.2 Advective Flux
3.2.3 Diffusive Flux
3.2.4 Elementary Area and Elementary Volume
3.2.5 Net Flux across a Closed Surface
3.3 Transport and Evolution
3.3.1 Rate of Accumulation
3.3.2 Lagrangian Form of the Evolution Equation
3.3.3 Eulerian Form of the Evolution Equation
3.3.4 Differential Form of the Transport Equation
3.3.5 Boundary and Initial Conditions
3.4 Hydrodynamics
3.4.1 Conservation Laws in Hydrodynamics
3.4.1.1 Conservation of Mass
3.4.1.2 Conservation of Momentum
3.4.1.2.1 The Euler Equations
3.4.1.2.2 The Euler Equations in a Rotating
Frame or Reference
3.4.1.2.3 The Navier-Stokes Equations
3.4.1.3 Conservation of Energy
3.4.1.4 Conservation of Salt
3.4.1.5 Equation of State
3.4.2 Simplification and Scale Analysis
3.4.2.1 Incompressibility
3.4.2.2 The Hydrostatic Approximation
3.4.2.3 The Coriolis Force
3.4.2.4 The Reynolds Equations
3.4.2.5 The Primitive Equations
3.4.3 Special Flows and Simplifications in Dimensionality
3.4.3.1 Barotrophic 2D Equations
3.4.3.2 1D Equations (Channel Flow)
3
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3.4.4 Initial and Boundary Conditions
3.4.4.1 Initial Conditions
3.4.4.2 Conditions on Material Boundaries
3.4.4.3 Conditions on Open Boundaries
3.4.4.4 Conditions on the Sea Surface and the Sea Bottom
3.5 Boundary Processes
3.5.1 Bottom Processes
3.5.1.1 Bottom Shear Stress
3.5.1.2 Other Bottom Processes
3.5.2 Solid Boundary Processes
3.5.3 Free Surface Processes
3.5.3.1 Mass Exchange
3.5.3.2 Momentum Exchange
3.5.3.3 Energy Exchange
3.5.4 Cohesive and Noncohesive Sediment Processes
Bibliography
3.1 INTRODUCTION TO TRANSPORT PHENOMENA
Chapter 2 concluded that the calculation of spatio-temporal distribution of major
components of a lagoon’s hydrogeomorphological unit and biocoenose is important
for the description of the structure and function dynamics (productivity and carrying
capacity) of the lagoon system and, consequently, for sustainable management. This
concept is required to understand the transport phenomena that describe the evolution
of properties due to fluid motion (advection) and/or molecular and turbulent dynamics (diffusion). In the case of turbulent flows the small-scale motion of the fluid
particles is actually random, and this nonresolved advection is also treated as diffusion (eddy diffusion).
A mathematical description of the transport phenomena (transport equations) is
based on the concept of conservation principle, which is valid in any application.
Conservation principle can be stated as
{The rate of accumulation of a property inside a control volume}
= {what flows in minus what flows out}+{production minus consumption}
Using this conservation principle, transport equations for any property inside a
control volume can be derived if production and consumption mechanisms are known
and if the control volume and transport processes are quantified. The control volume
is presented as the largest volume for which one can consider the interior properties
as uniformly distributed as well as fluxes across the surface.
In previous coastal lagoon studies the control volume was often implicitly
defined as the whole lagoon. Concepts of residence and flushing time were derived
from this global approach (see Chapter 5 for details). In that case, only fluxes at
the boundaries were required. This type of integral approach cannot describe
gradients and is consequently not sufficient to support process-oriented research
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or management. For these purposes the system has to be divided into homogeneous
parts (control volumes) and fluxes between them have to be calculated. This
approach requires a numerical model. The number of space dimensions required
to describe the control volumes equals the number of dimensions of the model.
The resolution of the transport equations in practical situations has been made
possible using numerical methods and computers (see Chapter 6 for details).
Before the advent of computers transport processes had to be studied using empirical formulations (derived from experiments) or analytical solutions in simple
geometries or boundary conditions.
This chapter presents a general transport equation (also called an evolution equation) based on the concepts of (1) control volume, (2) advective flux, and (3) diffusive
flux. Based on this generic equation, equations for hydrodynamics, temperature, salinity, and suspended sediments are also introduced. Special flows and simplification of
dimensionality and boundary processes and conditions, particularly for coastal
lagoons, are described in detail for use in lagoon modeling studies.
3.2 FLUXES AND TRANSPORT EQUATION
3.2.1 VELOCITY AND DIFFUSIVITY IN LAMINAR AND TURBULENT
FLOWS AND IN A NUMERICAL MODEL
For transport purposes, fluids are considered a continuum system. Velocity is defined
in a macroscopic way based on the concept of continuum system. Because fluids
are not a real continuum, system velocity cannot describe transport processes at the
molecular scale. The nonrepresented processes are represented by diffusion.
Although the concept of velocity is well known, it is reconsidered for modeling
purposes.
Diffusion in laminar flows occurs from movements at a molecular scale not
represented by the velocity. In turbulent flows, velocity, as defined for laminar flows,
becomes time dependent, changing at a frequency that is too high to be represented
analytically. As a consequence time average values must be considered, following
the Reynolds approach (see Section 3.4.2.4 for details). Transport processes, not
described by this average velocity, are represented by turbulent diffusion (using an
eddy diffusivity, which is several orders of magnitude higher than molecular diffusivity). More information on this topic is given in Section 3.4.
Most numerical models use grids with spatial and time steps larger than those
associated with turbulent eddies. Again, processes not resolved by velocity computed
by models have to be accounted for by diffusion (subgrid diffusion).
The box represented in Figure 3.1 is commonly used to illustrate the concept of
diffusion in laminar flows. The same box could be used to illustrate the concept of
eddy diffusion in turbulent flows or subgrid diffusion in numerical models. In
molecular diffusion white and black dots represent molecules, while in other cases
they represent eddies. In the initial conditions (stage (a) in Figure 3.1) two different
fluids are kept apart by a diaphragm. Molecules inside each half-box move randomly,
with velocities not described by our model (Brownian or eddy). When the diaphragm
is removed, particles from each side keep moving, resulting in the possibility of
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