External flow -- lift and drag

We have learned that when an object moves through a fluid, it feels resistive forces that point in a direction opposite to the motion — viscosity and drag, with the relative strength of the two forces being determined by the properties of the fluid and the object's speed and size via the Reynolds number.

What is both interesting and useful to organisms moving in a fluid is that if the object is not symmetric around the line of motion (a sphere or a cylinder moving along its axis), the fluid can have an additional affect: a force in a direction perpendicular to the direction of motion called lift. 

Animals can take advantage of this force arising from a flow of fluid around them (an external flow) to help them get around when they fly or swim. The classic example of lift is the net force that results perpendicular to an airplane or birds wing, when air flows over that wing. 

One way to get a sense of the mechanism responsible for lift is to consider the energy of fluid flow. When following a bit of fluid, that analysis (Bernoulli's principle) can show that when the fluid is made to flow faster (for example, by a narrowing of a pipe), the pressure must drop.

The argument is made that curvature of the top surface of the wing causes the flow to speed up and hence reduce the pressure relative to the flow over the bottom surface of the wing. This net pressure difference results in the upward lift that can balance the gravitational force and enables an eagle to fly or a penguin to swim. (This argument is not quite correct and the situation is more complex — often the case in the study of fluid flow! — but it gives a decent qualitative insight.)

A model of the lift force

If we think about a wing, there are a number of forces being applied to the wing as shown in the figure at the right.  In the simplest application of lift, an animal will use it to glide, maintaining a constant velocity, even in the presence of gravity.

If the animal is gliding with a constant velocity in the direction of the dotted red arrow, from Newton's 2nd law we know that the forces must balance (since there is no acceleration).

Our free-body diagram is shown in the figure at the right. The forces on the wing (used to represent the entire object) include the downward gravitational force (the weight of the flyer), the drag force on the wing (opposite to the direction of motion), and the lift, perpendicular to the wing. In the steady state, the forces of lift and drag will counterbalance the gravitational force to maintain a constant velocity.

The drag and lift are dependent on the characteristics of the wing. A phenomenological model that works well for the force of lift shows that it is well-described by an equation that looks very much like our equation for drag:

$$F^{lift}_{air \rightarrow wing} = {1 \over 2} C_l \rho A v^2$$

where $C_l$  is a shape dependent lift factor, $ρ$ is the density of the fluid, $A$ is the wing’s area (as viewed from on top) and $v$ is relative speed of wing and fluid.

Making a wing have a larger area provides greater lift, as does moving faster. The latter fact will make sense when you think about an aircraft taxiing down a runway. It does not start to rise up off the ground until it reaches a certain speed. More speed gives the plane more lift and enables it to become airborne. Another (somewhat counter-intuitive) thing to note from this equation is that a denser fluid provides more lift. Therefore there is greater lift in water than in air (though it's harder to move as fast since the drag is also greater).

Comparing different animals: Wing loading

The wing needs to provide enough lift that it can compensate for the downward force of gravity on the animal’s weight.  Since the upward lift is proportional the area of the wings and the downward force is the weight. The ratio of the weight to the wing area gives us a way to think about comparing flying or gliding animals. The ration of weight to wing area is called the wing loading:

$$WL = \frac{\mathrm{weight}}{\mathrm{area}} = \frac{mg}{A} =   \frac{\rho_{\mathrm{animal}}Vg}{A} \propto \frac{V}{A}$$

where $m$ is the animal’s mass, given by its density times its volume, $g$ is gravitational acceleration and $A$ is the animal’s wing area. Wing loading describes the balance between gravity and lift, with smaller values occurring for those animal’s which can more readily glide (and ultimately fly). This quantity is not dimensionless, but scales as the animal’s volume to its area. This scaling is proportional to length³ / length² which is of the order of magnitude of animal length.  Therefore, smaller animals will have smaller ratios of V/S and have an easier time staying aloft.  Here is a nice video about flying squirrels.

Thrust

 Many animals will want to do more than just glide, they will want to propel themselves forward through a fluid. This is possible if the angle at which the lift is generated is tilted forward such that there is a net forward thrust on the animal. We can see how this works by looking at the free-body diagrams for the balance of forces.

In diagram A is shown the balance of forces for the glide. The forces of lift and drag  add to produce the blue vector. That vector balances the object's weight leading to a net force of zero so the velocity remains constant. If the animal changes its angle of flight without changing its speed, the free-body diagram becomes that show in B. The lift+drag force no longer cancels the weight. The result is shown in C. The lift+drag force adds to the weight to produce the net force shown by the heavy black arrow: the thrust — a net force that accelerates the animal.

Biological implications of moving through external flows

Biological organisms make all kinds of adaptations to living in flows.  Both the kind of flow (laminar vs turbulent) and the velocity gradient can impact how an organism interacts with a fluid or a fluid interacts with an organism.  Some small organisms actually live in the boundary layer, and take advantage of the slower flow velocities there. These include larvae of various insects such as mayflies or caddis flies, as well as some beetle larvae and some limpets.  Slower flows cause less drag on the organism, making it easier to hang on and stay put.  Other organisms work to extend at least part of their body outside the boundary layer where they have access to the flow, and therefore more items to eat.  Examples of this include black fly larvae that have feeding fans for gathering plankton out of the flow and barnacles that extend their cirri or limbs.  Check out this video to see barnacles feeding.

Organisms can also take advantage of flow using the flow for dispersal.  They can send out gametes as a tree does to disperse its pollen or a dandelion does to disperse its seeds.  For the latter, the seeds are carefully designed with a sort of parachute which holds them aloft to travel further from the parent plant.  Animals or plants can also send out olfactory cues to attract others.  A female moth emits pheromones to attract males for mating.  Flowers give off chemicals to attract bees, moths or even hummingbirds to pollinate them.

Some organisms have acquired special adaptations so that they can efficiently move through a fluid.  This occurs for animals that glide or fly in air or that swim through water.  A wide variety of animals can glide, including frogs, lizards, squirrels and fish, by acquiring some extra flaps of tissue to help catch the air.  However, there have been just four kinds of animals that have acquired the ability to fly: extinct pterosaurs, birds, insects and bats.  Of the last three, each has acquired the wings necessary to fly by a separate evolutionary path.  These wings have large surface areas, sufficient to generate enough lift and can be beat up and down frequently enough to generate thrust or forward motion.  Notably, these flying animals are all relatively small.  Large birds, such as the ostrich can no longer generate sufficient lift to get their large bodies off the ground.

For several reasons, swimming is far easier than flying.  First, the lift provided is larger because of the larger density of water versus air.  Second, the animal is somewhat buoyant in water, and certainly much more so than it is in air, due to the fact that its density is so close to that of water.  This buoyancy helps offset a large fraction of the gravitational force in water.  So most of the effort of swimming can be put in to generating thrust.  This is typically done in one of three ways.  First, animals can jet through the water, by expelling fluid behind them to propel themselves forward.  This is common for squid, which are fast jetters, as well as jellyfish, which are very slow jetters.  Second, animals can use drag, using appendages or even cilia to essentially row themselves through the water.  The forward moving stroke extends the cilia out fully to apply a lot of drag to the fluid while the recovery stroke pulls the cilia in to minimize the applied drag.  The net effect is forward motion due to the difference in drag on the fluid.  Third, animals can actually “fly” underwater, generating lift in a direction so as to propel themselves forward.

Julia Gouvea and Kerstin Nordstrom 10/23/13
Joe Redish 2/28/19