194
E. W. Koch, J. D. Ackerman, J. Verduin and M. van Keulen
vegetation including freshwater plants, seagrasses,
and kelp.
II. Fluid Dynamics: Fundamentals
The general aspects of water flow in aquatic systems
can be understood through a number of fluid dynamic concepts that have been developed largely for
steady state conditions, i.e. when there are no temporal fluctuations in the water flow (Fischer et al.,
Abbreviations
A – cross sectional area
C – celerity or phase velocity of waves
C d – drag coefficient
C s – concentration on the seagrass surface
C w – concentration in the water column
D – molecular diffusivity
D – depth
DBL – diffusive boundary layer
δ – diffusive boundary layer thickness
δ D – diffusive boundary layer (=DBL)
δ I –inertial sublayer or logarithmic (log) layer
δ v –viscous sublayer
F d – friction or viscous drag
F p – form or pressure drag
g – acceleration due to gravity
H – water depth
H – wave height
h – canopy height
J – flux
κ – von Karman constant
l – length scale
λ – wavelength
m – mass
µ – molecular or dynamic viscosity
p – hydrostatic or dynamic pressure
Q – volume flow rate
ρ – density
REI – relative wave exposure index
Re – Reynolds number
Re crit – critical Reynolds number
St – Stanton number
T – wave period
τ – shear stress
τ o – boundary shear stress
τ W – wall shear stress
u – current velocity
u ∗ – friction velocity
U k – critical velocity
U o – free stream velocity
ν–kinematic viscosity
x – horizontal distance
x – principal flow direction
y – cross-stream direction
z – vertical direction or depth
z o – roughness height
1979; White, 1999; Kundu and Cohen, 2002). In the
absence of motion, seawater is described by: (i) density (ρ, i.e. mass/volume), which is used preferentially over mass (m) in fluids; (ii) kinematic viscosity
(ν) which is a measure of how easily the fluid will
flow (i.e. ν = µ/ρ, where µ is the molecular or dynamic viscosity); and (iii) hydrostatic pressure ( p),
which is a function of the depth from the water surface (i.e. p = ρgz, where g is the acceleration due
to gravity and z is the depth—note that the depth
can be the distance from the water surface to the
seafloor or the height from the seafloor to the water
surface; see below). The introduction of energy into
a fluid causes fluid motion, and the motion in natural
systems is generated by pressure gradients (d p/dz)
as result of gradients in water surface elevation or
depth (dz/dx; where x is the horizontal distance)
and/or density (dρ/dz). The major source of this energy input is the sun, which causes winds that lead
to changes in surface elevation (i.e. dz/dx; waves,
currents, and seiches in embayments), and thermal
gradients (i.e. dρ/dz) that lead to expansion, instabilities, and mixing. Other sources include inputs
of freshwater and other chemical constituents (i.e.
dρ/dz), tides and currents due to gravitation and acceleration of the earth-moon and earth-sun systems
(i.e. dz/dx), and the Coriolis force due to the earth’s
rotation (i.e. dz/dx) (Kundu and Cohen, 2002).
The flow in seawater is described with respect
to a fixed Cartesian reference frame (Eulerian perspective) with x defining the principal flow direction, y defining the cross-stream direction, and z
defining the vertical direction. Whereas it is common in geophysics to define z as the depth (i.e. with
respect to the water surface), it is equally appropriate and perhaps more informative to use height
(i.e. defined with respect to the seafloor) as the vertical direction (e.g. Ackerman and Okubo, 1993).
The volume flow rate (Q), as defined by the velocity (u) of the fluid that passes through a given cross
sectional area A (which is usually defined with respect to the x and y; i.e. dxdy), is conserved because seawater is an incompressible fluid. This continuity principle is one of the essential elements of
fluid dynamics, which, among other things, is used
to determine mass balances of water-borne materials (e.g. Hemond and Fechner, 1994). The flow of
water leads to a second type of pressure, the dynamic pressure ( p = 1 / 2 ρu
2 ), which, when added
together with the hydrostatic pressure, is constant
along a flow streamline (i.e. Bernoulli’s principle).
E. W. Koch, J. D. Ackerman, J. Verduin and M. van Keulen
vegetation including freshwater plants, seagrasses,
and kelp.
II. Fluid Dynamics: Fundamentals
The general aspects of water flow in aquatic systems
can be understood through a number of fluid dynamic concepts that have been developed largely for
steady state conditions, i.e. when there are no temporal fluctuations in the water flow (Fischer et al.,
Abbreviations
A – cross sectional area
C – celerity or phase velocity of waves
C d – drag coefficient
C s – concentration on the seagrass surface
C w – concentration in the water column
D – molecular diffusivity
D – depth
DBL – diffusive boundary layer
δ – diffusive boundary layer thickness
δ D – diffusive boundary layer (=DBL)
δ I –inertial sublayer or logarithmic (log) layer
δ v –viscous sublayer
F d – friction or viscous drag
F p – form or pressure drag
g – acceleration due to gravity
H – water depth
H – wave height
h – canopy height
J – flux
κ – von Karman constant
l – length scale
λ – wavelength
m – mass
µ – molecular or dynamic viscosity
p – hydrostatic or dynamic pressure
Q – volume flow rate
ρ – density
REI – relative wave exposure index
Re – Reynolds number
Re crit – critical Reynolds number
St – Stanton number
T – wave period
τ – shear stress
τ o – boundary shear stress
τ W – wall shear stress
u – current velocity
u ∗ – friction velocity
U k – critical velocity
U o – free stream velocity
ν–kinematic viscosity
x – horizontal distance
x – principal flow direction
y – cross-stream direction
z – vertical direction or depth
z o – roughness height
1979; White, 1999; Kundu and Cohen, 2002). In the
absence of motion, seawater is described by: (i) density (ρ, i.e. mass/volume), which is used preferentially over mass (m) in fluids; (ii) kinematic viscosity
(ν) which is a measure of how easily the fluid will
flow (i.e. ν = µ/ρ, where µ is the molecular or dynamic viscosity); and (iii) hydrostatic pressure ( p),
which is a function of the depth from the water surface (i.e. p = ρgz, where g is the acceleration due
to gravity and z is the depth—note that the depth
can be the distance from the water surface to the
seafloor or the height from the seafloor to the water
surface; see below). The introduction of energy into
a fluid causes fluid motion, and the motion in natural
systems is generated by pressure gradients (d p/dz)
as result of gradients in water surface elevation or
depth (dz/dx; where x is the horizontal distance)
and/or density (dρ/dz). The major source of this energy input is the sun, which causes winds that lead
to changes in surface elevation (i.e. dz/dx; waves,
currents, and seiches in embayments), and thermal
gradients (i.e. dρ/dz) that lead to expansion, instabilities, and mixing. Other sources include inputs
of freshwater and other chemical constituents (i.e.
dρ/dz), tides and currents due to gravitation and acceleration of the earth-moon and earth-sun systems
(i.e. dz/dx), and the Coriolis force due to the earth’s
rotation (i.e. dz/dx) (Kundu and Cohen, 2002).
The flow in seawater is described with respect
to a fixed Cartesian reference frame (Eulerian perspective) with x defining the principal flow direction, y defining the cross-stream direction, and z
defining the vertical direction. Whereas it is common in geophysics to define z as the depth (i.e. with
respect to the water surface), it is equally appropriate and perhaps more informative to use height
(i.e. defined with respect to the seafloor) as the vertical direction (e.g. Ackerman and Okubo, 1993).
The volume flow rate (Q), as defined by the velocity (u) of the fluid that passes through a given cross
sectional area A (which is usually defined with respect to the x and y; i.e. dxdy), is conserved because seawater is an incompressible fluid. This continuity principle is one of the essential elements of
fluid dynamics, which, among other things, is used
to determine mass balances of water-borne materials (e.g. Hemond and Fechner, 1994). The flow of
water leads to a second type of pressure, the dynamic pressure ( p = 1 / 2 ρu
2 ), which, when added
together with the hydrostatic pressure, is constant
along a flow streamline (i.e. Bernoulli’s principle).