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 Cd - drag coefficient

Cs - concentration on the seagrass surface Cw - concentration in the water column D - molecular diffusivity D - depth

DBL - diffusive boundary layer

& - diffusive boundary layer thickness

&D - diffusive boundary layer (=DBL)

5i-inertial sublayer or logarithmic (log) layer

5v-viscous sublayer

Fd - friction or viscous drag

Fp - form or pressure drag g - acceleration due to gravity

H - water depth

H - wave height h - canopy height

J -flux k - von Karman constant l - length scale X - wavelength m - mass

/ - molecular or dynamic viscosity p - hydrostatic or dynamic pressure Q - volume flow rate p - density

REI - relative wave exposure index

Re - Reynolds number

Recrit - critical Reynolds number

St - Stanton number

T - wave period t - shear stress to - boundary shear stress t w - wall shear stress u - current velocity ut - friction velocity

Uk - critical velocity

Uo - free stream velocity v-kinematic viscosity x - horizontal distance x - principal flow direction y - cross-stream direction z - vertical direction or depth zo - roughness height

1979; White, 1999; Kundu and Cohen, 2002). In the absence of motion, seawater is described by: (i) density (p, i.e. mass/volume), which is used preferentially over mass (m) in fluids; (ii) kinematic viscosity (v) which is a measure of how easily the fluid will flow (i.e. v = /x/p, where / 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 = pgz, 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 (dp/dz) as result of gradients in water surface elevation or depth (dz/dx ; where x is the horizontal distance) and/or density (dp/ 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. dp/dz) that lead to expansion, instabilities, and mixing. Other sources include inputs of freshwater and other chemical constituents (i.e. dp /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 = V2 pu2), which, when added together with the hydrostatic pressure, is constant along a flow streamline (i.e. Bernoulli's principle).

Bernoulli's principle, which states that the sum of the hydrostatic pressure and dynamic pressure along a streamline are constant (Vogel, 1994), helps to explain flow-induced pressure changes (i.e. lift) that occur within, around, and under seagrass canopies (e.g. Nepf and Koch, 1999). Drag is another important force that acts downstream of obstacles. It has two additive components: (i) the friction or viscous drag that exists due to the interaction of the obstacle's surface with the water, which can be defined algebraically (i.e. Fd = V2CdpAu2, where Cd is the drag coefficient, a shape and flow dependent constant); and (ii) the dynamic, form or pressure drag (Fp) that exists under high flows when flows separate from boundaries, which cannot be expressed algebraically and must, therefore, be determined empirically. As u increases, the dynamic drag contributes a disproportionate fraction of the total drag. It is important to note that drag is a force that operates opposite to the flow direction in that it "sucks" a moving object upstream or a stationary object downstream.

Water flow can exhibit a number of different properties that depend on the temporal and spatial scales under investigation. Water flow could either be smooth and regular as if the fluid flows in layers (i.e. laminar flow) or rough and irregular as if the flow is "chaotic" (i.e. turbulent flow). This depends on the velocity and the length scale (i.e. temporal and spatial scale, respectively) under investigation as defined by the Reynolds number (Re = lup / x or more simply Re = lu/v; where l is the length scale appropriate for the hypothesis being tested). Re, which is the non-dimensional ratio of inertial to viscous forces in a fluid, defines four regimes that grade into one another: (i) creeping flow (Re ^ 1), which occurs at very low flows and spatial scales such as those experienced by individual bacteria cells; (ii) laminar flow (1 < Re < 103) as defined above; (iii) transitional flow (Re - O(103); i.e. of the order of 103), which involves the production of eddies and disturbances in the flow and is characterized by a critical Re (Recrit) defined for a particular geometry and flow; and (iv) fully turbulent flow (Re ^ 103). Associated with these flow patterns are important differences related to the fluid dynamic forces (e.g. friction vs. pressure drag) and mass transfer processes (diffusion vs. advection) that operate under the different regimes (see below; White, 1999; Kundu and Cohen, 2002). Moreover, because Re is scale dependent, it is possible to experience multiple flow regimes simul taneously in the flow field depending on the spatial scale under investigation. Consequently, flow is almost always turbulent at large spatial scales such as seagrass beds, but it can also be laminar on the scale of seagrass leaves and flowers (e.g. Ackerman and Okubo, 1993; Koch, 1994). This not-so-subtle distinction can influence the application and interpretation of physiological and ecological processes in seagrass canopies (see Section III).

As indicated above, the flow conditions become more complicated when water approaches a boundary (e.g. seagrass canopy, leaves, or seafloor, depending on the scale) or any obstacle for that matter. The water cannot normally penetrate boundaries, except for the most porous ones (see reviews in Boudreau and Jorgensen, 2001; Okubo et al., 2002), and more importantly, the water molecules directly next to a boundary stick to the boundary rather than slip by it. This no-slip condition leads to the development of a velocity gradient perpendicular to the boundary (Fig. 1), as the velocity at the boundary will be zero relative to the free stream velocity (U0). As the water flows downstream, the velocity gradient will grow in size and a slower moving layer of fluid will develop next to the boundary, which is referred to as the boundary layer under turbulent conditions, otherwise technically it is a deformation layer (Prandtl and Tietjens, 1934). This boundary layer, which is defined by velocities <0.99 U0, has a thickness of S that is relatively small and can be expressed as a function of Re and x. Initially it appears laminar in nature, but the boundary layer will become turbulent when the local Re (Rex = ux/v) approaches a critical value of 3 to 5 x 105, in the case of a flat plate oriented parallel to the flow. In nature, this transition is accelerated by the presence of roughness or obstacles on the boundary (Schlichting, 1979; Nikora et al., 2002; Fig. 1) including undulations on macroalgal blades (Hurd and Stevens, 1997). In addition to the streamwise structure in a fully developed boundary layer, there is important vertical structure as well. The first layer directly adjacent to the boundary is the viscous sublayer (Sv ^ 10v/u* where u* is the friction velocity, which is a velocity scale that provides an indication of the mass transfer within the boundary layer) in which the forces (or stresses if surface forces are considered) are largely viscous, and consequently the mass transfer in this layer is slow and dominated by diffusion, especially within the thin diffusional sublayer (also called the diffusive boundary layer; DBL) at the bottom of this

Diffusive boundary layer

Fig. 1. Velocity (U) gradient/profile adjacentto smooth (left) andrough (right) boundaries. Weak currents (solidline) generate arelatively thick boundary layer (1) when compared with boundary layers (2) generated by faster currents (dashed curve). Names of boundary layer zones are provided for the fast flowing water velocity profile. When the boundary (such as a seagrass leaf) is rough (e.g. due to the presence of epiphytic organisms), a roughness height (arrow) extends the boundary layer farther into the water column. Consequently, the flux of nutrients and carbon from the water column to the boundary is reduced.

Diffusive boundary layer

Fig. 1. Velocity (U) gradient/profile adjacentto smooth (left) andrough (right) boundaries. Weak currents (solidline) generate arelatively thick boundary layer (1) when compared with boundary layers (2) generated by faster currents (dashed curve). Names of boundary layer zones are provided for the fast flowing water velocity profile. When the boundary (such as a seagrass leaf) is rough (e.g. due to the presence of epiphytic organisms), a roughness height (arrow) extends the boundary layer farther into the water column. Consequently, the flux of nutrients and carbon from the water column to the boundary is reduced.

layer (5D ^ v/u2(v/D)-a, where D is the molecular diffusivity and a is a constant equal to 1/2 or 1/3; (Lorke et al., 2003). It is important to note that ^ Sv, which relates to the fact that the molecular diffusion of momentum (i.e. v ~ 10-6 m2 s-1) is much larger than the molecular diffusion of a scalar quantity like CO2 (i.e. D ~ 10-9 m2 s-1). The next layer is the inertial sublayer or logarithmic (log) layer (5 ^ 0.155), which is a region of exponentially increasing velocity; hence, it is dominated by inertial forces (or stresses) and mass transfer occurs through turbulent advection. The outer layer of the boundary layer is the largest layer, and it represents a transition to the free stream flow (it is referred to as the Ekman layer in situations where the Coriolis force causes rotation of the flow; Fig. 1). Boundary layers exist embedded in one another as they are defined by spatial scale (e.g. Ackerman, 1986; Boudreau and Jorgensen, 2001); consequently, it is possible to define boundary layers around plant epiphytes, flowers, leaves, canopies, and the benthos. In this sense, there is a benthic boundary layer (BBL) above the seagrass canopy, and separate boundary layers around individual shoots, leaves, flowers, and the smaller constituents described above. In addition, it is important to note that there may also be boundary layers generated by other types of water motion (e.g. wave current boundary layers), but this topic is beyond the scope of this review. For biolog ically relevant information on this topic see Denny (1988).

Another important consequence of the no-slip condition at a boundary is the tractile or shearing force that the boundary imparts on the fluid, which is a tangential force causing rotation of the fluid next to the boundary. A boundary or wall shear stress (t 0 or t W) is defined as the quotient of the shearing force and the area of the boundary, and t w = fdu/dz within the viscous (or laminar) sublayer and tw = pu2 in general. In practice, it is difficult to measure the shear force or the aerial extent of the boundary or to apply the algebraic relationships, and thus a number of methods have been developed to measure t directly using force balances in flow chambers or indirectly using velocity gradients based on the law of the wall (u = u2/k ln(z/z0), where k = 0.4 is the von Karman constant, and z0 is the roughness height; see Fig. 1; for a review of techniques and references for the measurement of bed shear stress see Ackerman and Hoover (2001). The velocity gradient method involves applying the law of the wall to the velocities measured in the log layer of the boundary layer. In this case, u2 is equal to k multiplied by the slope of the linear regression of velocity on the natural logarithm transformed distances from the boundary, and z0 is equal to e raised to the value of the x intercept of the same regression. This method has been applied successfully above and within seagrass canopies (e.g. Fonseca and Fisher, 1986; Gambi et al., 1990; Ackerman and Okubo, 1993). It is important to note that other engineering models have been applied to rough canopies such as corals and seagrasses with the direct measurement of canopy friction and the use of the Stanton number, St (uptake rate by the surface/advection over the surface), to determine the efficiency of canopy uptake (e.g. Thomas et al., 2000; Thomas and Cor-nelisen, 2003). Reconfiguration of seagrass canopies under higher flow conditions (e.g. Fonseca et al., 1982; Ackerman, 1986), and/or unsteadiness due to monamis (waving of the canopy; see Section VC) caused by an instability of the mean velocity profile (Ackerman and Okubo, 1993; Ghisalberti andNepf, 2002) and waves (Koch, 1996; Koch and Gust, 1999) represents a challenge to researchers. Even so, t is the preferred form (over u) of expressing hydrody-namic conditions near boundaries (leaves, flowers, sediment etc; see Nowell and Jumars, 1984).

Hydrodynamic conditions in the environment are rarely stationary, especially in wave-dominated sea-grass habitats where a more appropriate characterization of the fluid environment is that it varies in a periodic fashion with each passing wave. Waves represent the movement of energy through a fluid and exhibit a periodic motion, especially when viewed at an interface (e.g. the water surface). In this case, the passing wave (crest followed by trough) causes a submersed object on the surface to move in a circular or orbital fashion, the diameter of which is equal to the wave height (H). The orbital motions also extend downward through the fluid in a series of orbitals that diminish in diameter with depth until a depth (z) of 1/2X (where X is the wavelength) is reached.

The classification of waves can be based on the disturbing force that creates them, the restoring force that destroys them, and their wavelength (Garrison, 2000). The disturbing force is the source of energy that causes the wave, which can be (i) wind stress acting on water surface causing capillary and gravity waves, (ii) the arrival of surge or sea wave causing swell, (iii) wind setup in an embayment creating seiches, (iv) a change in atmospheric pressure causing short-lived storm surge, and (v) large disturbances (landslides, volcanic eruptions, earthquake) that cause seismic waves (or tsunami; the so-called tidal waves that are actually due to gravitational inertial forces). The restoring forces that reduce the disturbance to the water surface include

(a) surface tension due to the molecular cohesion of water molecules, which works for small waves (i.e. X <1.73 cm; capillary waves) and, (b) gravity that operates onlarger waves (i.e. X ^ 1.73 cm). Whereas the wavelength can be used to distinguish differences among the smallest of waves, it really provides a measure of wave size and relationship to energy; the smaller the wavelength, the higher the energy. Some typical relationships include (1) wind waves (X <60150 m), (2) seiches (X is large and a function of the basin size), (3) seismic waves (X <200 km), and (4) tides (X = 1/2 circumference of earth; note that tides are caused by gravity and inertia).

Seagrasses experience each of these types of waves, but the most common are wind waves, swell, and tides (tides can be viewed as long waves). Wind waves develop from capillary waves to gravity waves as a function of the wind strength and direction and the fetch (length of the unrestricted zone over which the wind stress operates). Wind waves are affected by local wind conditions, and are generally of a short period, T (T is time it takes for a wave to pass a fixed point). Wave action has a direct impact on the ecosystem, with obvious effects on sediment transport, boundary layer processes and physical stresses (Denny, 1988; Koch and Gust, 1999). There has also been some suggestions that fetch (relative wave exposure index) is an important factor affecting seagrass on a landscape level (e.g. Fonseca and Bell, 1998; Hovel etal., 2002; Krause-Jensenetal.,2003).

Just as the size of the wave is determined by the wavelength, the shape of the orbit is determined by the water depth. In deep water (i.e. z > 1/2X) the orbits are circular, whereas in shallow water (i.e. z < 1 /20X) the orbits become elliptical or flatter due to the influence of the bottom. Intermediate waves (i.e. 1/20X < z < 1/2X) are more complicated as they combine characteristics of deep and shallow water waves. Deep water waves travel at a celerity or phase velocity C = VgX/2n or X/T (~1.56 T), but shallow water waves are slower due to the influence of the bottom and travel at C = „fgz (or 3.1 -Jz), which is why waves build up in shallow areas (Denny, 1988). Waves travel in a wave train, which is a progression of groups of waves of similar X from the same origin. Energy is lost by the leading wave, which eventually dissipates, but a new trailing edge wave is created from this energy. In deep water, the waves progress with C a X but the wave train has a group velocity of C a 1/2X, whereas in shallow water the celerity of the individual waves slow until the wave and group celerities are equal. It is important to note that waves and wave trains are not isolated from one another; circumstances can lead to destructive interference with calm periods between wave trains and constructive interference with the generation of large waves including rogue wavees due to the convergence of many waves.

As indicated above, a number of changes occur as waves enter the nearshore and ultimately reach the shore. The waves become shallow water waves as the wave train encounters the friction (shear stress) of the bottom or seagrass canopy, the wave orbits become more elliptical in shape near the bottom, and the wave crests become more pronounced. This can lead to wave-induced transport in a process referred to as Stokes drift, which may be of considerable importance in many coastal environments (Monismith and Fong, 2004). The steepness of the wave becomes unstable if it is greater than 1:7 (H:X) and the water at the crest begins to travel faster than the water near the bottom and it will break into a plunging wave, spilling wave, or surging wave depending on the steepness and topography of the bottom. Since waves approach the shore at different angles, they are unlikely to break simultaneously and may refract from the original direction leading to the complexity of waves experienced in coastal seagrass beds (Koch and Gust, 1999). Realistically, the fluid dynamic conditions within these nearshore regions are affected by a number of factors including tides and wind waves, all of which lead to significant changes in the surface elevation and water flow within seagrass beds. The general predictions are that seagrasses, like other benthic vegetation, increase the bottom shear stress and hence have a wave dampening affect (see Section V.B., below). This process has been relatively well characterized for coastal kelp forests (see review in Okubo et al., 2002), but has yet to be examined in a thorough manner for seagrasses. Clearly, additional efforts are needed in this area.

Whereas, the ultimate goal of studying fluid dynamic concepts is to better understand ecological processes in seagrasses, it is important to note that vegetative flows remain the most complex and difficult flows to describe and understand (Raupach et al., 1991; Finnigan, 2000). Therefore, applications in vegetated flows have typically involved steady state condition (i.e. non turbulent), although some progress in unsteady flows has been made with respect to seaweeds (Gaylord and Denny, 1997). Fortunately, this realization provides a challenge to those interested in the biological, chemical, geological, and physical processes that occur in seagrass systems.

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