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Web Topic 7.2
Hydrodynamic stimuli

Introduction

Hydrodynamic stimuli are created when an object moves through a fluid or a fluid moves past an object. The geometries of the stimuli and their persistence after being created vary with the size of the object, the relative velocities of the fluid and object, and the viscosity of the fluid. Although this sounds like a simple relationship, increasing one or more of these parameters do not simply scale up the stimuli, but instead can produce discrete changes in their geometries and life times. Fluid hydrodynamics turns out to be a fairly complicated area of physics. It is not an esoteric topic, however, as it is crucial to our understanding of weather patterns, the design of airplanes, helicopters, and sailboats, the mechanisms by which aquatic invertebrates and fish swim and birds, bats, and insects fly, the shapes of pine cones and flowers, the formation of spiral galaxies, and the behavior of black holes in space.

Reynold’s numbers

Although the interaction between fluids and objects can produce quite varied hydrodynamic stimuli, molecular fluids such as air or water behave similarly once the appropriate scale corrections are made. This scaling is achieved by computing a dimensionless Reynold’s number for the situation. The magnitude of the Reynold’s number allows us to provide at least a qualitative prediction about what kinds of hydrodynamic stimuli will be created by the interaction of an object and a fluid when they are moving at different velocities.

As discussed in Web Topic 6.4, the Reynold’s number is a ratio of the inertial forces in a fluid (which tend to maintain movement of the fluid) to viscous forces (which tend to slow down fluid movements). It is computed as the product of the effective size L of an object (in meters) and the relative velocity v of the object and the fluid (m/sec) divided by the kinetic viscosity of the fluid (m2/sec). The latter is a measure of the resistance of a fluid to spreading at the ambient temperature. It is computed by dividing the absolute (dynamic) viscosity of the fluid μ, (the forward momentum of a given area of moving fluid measured in Newton-seconds/m2) by the density of the fluid ρ (kg/m3). Algebraically, the Reynold’s number, Re is thus equal to

Water versus air

The density of water at room temperature is about 800–900 times that of air. The absolute (dynamic) viscosity of water is about 51 times higher than that of air. As a result, the same relative velocity of fluid and object and the same object size result in a Reynolds number in water that is about 15–17 times higher than that in air. Put another way, we expect air and water to show similar hydrodynamic behaviors when either the object or the flow rate (or more accurately, their product) is 15 times smaller in water. In practice, typical flow rates in air (e.g. wind) are 10–15 times faster than in water (e.g. stream currents). Animals of similar size in air and water thus tend to experience similar patterns of hydrodynamic stimuli (Vogel 1994).

Some examples of Reynold’s numbers for moving objects in air (blue) and water (red):

Characterizing hydrodynamic stimuli

Hydrodynamic stimuli are traditionally generated by inserting an object of interest into a steady and linear flow of fluid. Wind tunnels are often used for studies with air, and water sluices for aquatic studies. In more recent studies, live animals are allowed to move through a static fluid (air or water), and the hydrodynamic effects of these movements studied. In either case, the challenge is to make visible the resulting eddies, vortices, and other movements of the fluid. Ideally, one would track the path of an individual fluid particle over time, and then repeat this effort with additional particles that began at different initial locations in the fluid. When no object is present in a steady fluid flow, each such particle should trace out a parallel straight line as it flows through the tunnel or sluice. These trajectories are called pathlines. Placing an object in a fluid flow will distort what otherwise would be straight and parallel pathlines. It will also create a mosaic of locations in front, around, and downstream from the object where fluid pressures and fluid velocities (both magnitude and direction) differ from each other and from the average for the fluid as a whole. Movement of an animal in a static fluid has similar effects. How can these be characterized?

There are a number of ways to visualize hydrodynamic perturbations in fluids (http://en.wikipedia.org/wiki/Flow_visualization). Several currently popular methods include:

Vortices

When water in a flowing stream encounters an obstacle, the result is often a swirling eddy. Eddies are one example of a vortex. In a vortex, pathlines become circular or spiral instead of linear or gently curvilinear. There are several kinds of vortices:

Examples of naturally occurring vortices can be found online at http://gfm.aps.org/.

Boundary layers

When a volume of fluid and a solid object are moving relative to each other, those molecules of fluid closest to the object surface cannot move differentially from the object. They are forced to move with the object and thus act as if “stuck” to the surface. Fluid molecules far enough away from the object will move at the same relative velocity as other nearby molecules of the fluid volume. In between, there is a gradient of relative fluid velocity which goes from zero at the object surface to that shared by most of the molecules in the fluid volume. This intermediate region showing a gradient of velocity around the object is called the boundary layer.

As fluid slows down and collects in a thin boundary layer on the upstream side of an object, molecules that are not too close to the surface flow along pathlines that track the surface shape of the object. At some point along each side of the object’s surface, this fluid stops following the object shape and simply heads off downstream. These are known as the separation points. For very low Reynold’s numbers, the separation points are located well on the rear (downstream) side of the object. As Reynold’s numbers are increased, the separation points move forwards toward the object’s upstream side. This allows an increasing amount of fluid to pool on the downstream side of the object where it can even backflow towards the object, move along lines parallel to its surface, and finally join the downstream flow at the separation points. This circular movement thus generates eddies or vortices downstream from the object. At high enough Reynold’s numbers, the separation points detach from the object and move downstream. This drastically changes the composition and properties of the downstream boundary layer.

Patterns of hydrodynamic stimuli

Consider a static object in a continuous flow of fluid. As noted above, low Reynold’s numbers are obtained when the kinematic viscosity is much greater than the product of relative velocity and object size. Put another way, the resistance of the fluid to spreading in this case exceeds the inertial forces imposed on the fluid by its encounter with the object. When the object is large and/or the relative velocities are high, then inertial forces easily exceed the viscous resistance of the fluid. This is the case for large Reynold’s numbers. Intermediate values result in a more even match between viscous and inertial factors. Depending on the relative influences, there are also two intermediate cases that are easily distinguished. Each of these four situations generates a qualitatively different type of hydrodynamic stimulus (Cf. Feynman 1964 and Vogel 1994):

Animal wakes

Animals that move through water or air are more likely to leave behind a persistent wake if they are large and/or they move quickly. A small copepod that is slowly foraging may leave no wake at all; however, if it is startled and leaps away, it is likely to leave behind some shed vortices that could be detected by an appropriate predator (Yen & Strickler 1996; Videler et al. 2002). The higher the relevant Reynold’s number, the more likely a wake will be left behind and the longer it will persist (Vogel 1994). As noted above, the spacing and persistence of shed vortices could provide information about the size and velocity of the animal creating them. The situation is complicated by the fact that most animals achieve locomotion by repetitively moving some body part: birds and insects flap their wings, copepods oscillate paired appendages, and fish undulate. The frequency composition of persistent wakes can then also be used to extract additional information about the wake’s creator (Bleckmann et al. 1991).

Further Reading:

There are a number of texts that cover hydrodynamics and fluid mechanics. In order of increasing mathematical difficulty, we recommend: Vogel (1994) (an outstandingly readable treatment of fluid dynamics, vortices, and hydrodynamics from a biologist’s point of view. Strongly recommended!); Videler (2005) (This book focuses on animal flight, but provides a useful general introduction to Reynold’s numbers and vortices); Feynman (1964) (chapters 40–41 outline the basic logic of fluid mechanics. The book uses vector math, but Feynman is so good at explaining things that a reader need not be fully fluent with curls and gradients); and Lighthill (1986) (despite the title, this is a more advanced text using the usual vector math associated with fluid mechanics. It is widely cited in the prior books.).

Some published articles on biological aspects of hydrodynamics include Bleckmann et al. 1991; Blickhan et al. 1992; Catton et al. 2007; Dehnhardt et al. 2001; Hanke et al. 2000; Hanke & Bleckmann 2004; Jiang & Osborn 2004; Kim & Choi 2007; Liu et al. 1998; Magnuson 1978; Müller et al. 2000; Müller et al. 2002; Pohlmann et al. 2001; Schulte-Pelkum et al. 2007; Sun & Tang 2002; Webb 1978; and Yen & Strickler 1996.

Literature Cited

Bleckmann, H., T. Breithaupt, R. Blickhan & J. Tautz. 1991. The time course and frequency content of hydrodynamic events caused by moving fish, frogs, and crustaceans. Journal of Comparative Physiology A 168: 749–757.

Blickhan, R., C. Krick, D. Zehren, W. Nachtigall & T. Breithaupt. 1992. Generation of a vortex chain in the wake of a subundulatory swimmer. Naturwissenschaften 79: 220–221.

Catton, K.B., D.R. Webster, J. Brown & J. Yen. 2007. Quantitative analysis of tethered and free-swimming copepodid flow fields. Journal of Experimental Biology 210: 299–310.

Dehnhardt, G., B. Mauck, W. Hanke & H. Bleckmann. 2001. Hydrodynamic trail-following in harbor seals (Phoca vitulina). Science 293: 102–104.

Feynman, R.P. 1964. The Feynman lectures on physics (II): mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

Hanke, W. & H. Bleckmann. 2004. The hydrodynamic trails of Lepomis gibbosus (Centrarchidae), Colomesus psittacus (Tetraodontidae) and Thysochromis ansorgii (Cichlidae) investigated with scanning particle image velocimetry. Journal of Experimental Biology 207: 1585–1596.

Hanke, W., C. Brucker & H. Bleckmann. 2000. The ageing of the low-frequency water disturbances caused by swimming goldfish and its possible relevance to prey detection. Journal of Experimental Biology 203: 1193–1200.

Hwang, J.S. & R. Strickler. 2001. Can copepods differentiate prey from predator hydromechanically? Zoological Studies 40: 1–6.

Jiang, H.S. & T.R. Osborn. 2004. Hydrodynamics of copepods: A review. Surveys in Geophysics 25: 339–370.

Kim, D. & H. Choi. 2007. Two-dimensional mechanism of hovering flight by single flapping wing. Journal of Mechanical Science and Technology 21: 207–221.

Lighthill, J. 1986. An Informal Introduction to Theoretical Fluid Mechanics. Oxford, U.K.: Clarendon Press.

Liu, H., C.P. Ellington, K. Kawachi, C. Van den Berg & A.P. Willmott. 1998. A computational fluid dynamic study of hawkmoth hovering. Journal of Experimental Biology 201: 461–477.

Magnuson, J.J. 1978. Locomotion by scombrid fishes: hydromechanics, morphology, and behavior. In Fish Physiology. Vol. VII. Locomotion (Hoar, W.S. & D.J. Randalll, eds.). New York: Academic Press. pp. 239–313.

Müller, U.K., E.J. Stamhuis & J.J. Videler. 2000. Hydrodynamics of unsteady fish swimming and the effects of body size: comparing the flow fields of fish larvae and adults. Journal of Experimental Biology 203: 193–206.

Müller, U.K., E.J. Stamhuis & J.J. Videler. 2002. Riding the waves: the role of the body wave in undulatory fish swimming. Integrative and Comparative Biology 42: 981–987.

Pohlmann, K., F.W. Grasso & T. Breithaupt. 2001. Tracking wakes: The nocturnal predatory strategy of piscivorous catfish. Proceedings of the National Academy of Sciences of the United States of America 98: 7371–7374.

Schulte-Pelkum, N., S. Wieskotten, W. Hanke, G. Dehnhardt & B. Mauck. 2007. Tracking of biogenic hydrodynamic trails in harbour seals (Phoca vitulina). Journal of Experimental Biology 210: 781–787.

Sun, M. & H. Tang. 2002. Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. Journal of Experimental Biology 205: 55–70.

Videler, J.J. 2005. Avian Flight. Oxford, U.K.: Oxford University Press.

Videler, J.J., U.K. Muller & E.J. Stamhuis. 1999. Aquatic vertebrate locomotion: wakes from body waves. Journal of Experimental Biology 202: 3423–3430.

Videler, J.J., E.J. Stamhuis, U.K. Muller & L.A. van Duren. 2002. The scaling and structure of aquatic animal wakes. Integrative and Comparative Biology 42: 988–996.

Vogel, S. 1994. Life in Moving Fluids: The Physical Biology of Flow. Princeton, N.J.: Princeton University Press.

Webb, P.W. 1978. Hydrodynamics: nonscombroid fish. In Fish Physiology. Vol. VII. Locomotion (Hoar, W.S. & D.J. Randalll, eds.). New York: Academic Press. pp. 189–237.

Yen, J. & J.R. Strickler. 1996. Advertisement and concealment in the plankton: What makes a copepod hydrodynamically conspicuous? Invertebrate Biology 115: 191–205.

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