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# Fluid Dynamics

Factors that influence flow

Reynolds number

Laminar and turbulent flow

Bernoullis principle

Boundary layer theory

Shape and drag

Resources

Fluid dynamics is the study of the flow of liquids and gases, usually in and around solid surfaces. The flow patterns depend on the characteristics of the fluid, the speed of flow, and the shape of the solid surface. Scientists try to understand the principles and mechanisms of fluid dynamics by studying flow patterns experimentally in laboratories and also mathematically, with the aid of computer models. The two fluids studied most often are air and water. Aerodynamics is used mostly to look at air flow around planes and automobiles with the aim of reducing drag and increasing the efficiency of motion. Hydrodynamics deals with the flow of water in various situations such as in pipes, around ships, and underground. Apart from the more familiar cases, the principles of fluid dynamics can be used to understand an almost unimaginable variety of phenomena such as the flow of blood in blood vessels, the flight of geese in V-formation, and the behavior of underwater plants and animals.

## Factors that influence flow

The viscosity, density, and compressibility of a fluid are the properties that determine how the liquid or gas will flow. Viscosity measures the internal friction or resistance to flow. Water, for instance, is less viscous than honey and so flows more easily. All gases are compressible whereas liquids are practically incompressible and cannot be squeezed into smaller volumes. Flow patterns in compressible fluids are more complicated and difficult to study than those in incompressible ones. Fortunately for automobile designers, air, at speeds less than 220 mph (354 km/h) or one-third the speed of sound, can be treated as incompressible for all practical purposes. Also, for incompressible fluids, the effects of temperature changes can be neglected.

## Reynolds number

The speed of flow is another factor that determines the nature of flow. The speed of flow is either that of a liquid or gas moving across a solid surface or, alternatively, the speed of a solid object moving through a fluid. The flow patterns in either case are exactly the same. That is why airplane designs can be tested in wind tunnels where air is made to flow over stationary test models to simulate the flight of actual planes moving through the air.

The speed of flow is related to the viscosity by virtue of the fact that a faster moving fluid behaves in a less viscous manner than a slower one. Therefore, it is useful to take viscosity and speed of flow into account at the same time. This is done through the Reynolds number named after the English scientist Observe Reynolds (18421912). This number characterizes the flow. It is greater for faster flows and more dense fluids and smaller for more viscous fluids. The Reynolds number also depends on the size of the solid object. The water flowing around a large fish has a higher Reynolds number than water flowing around a smaller fish of the same shape.

As long as the shape of the solid surface remains the same, different fluids with the same Reynolds number flow in exactly the same way. This very useful fact is known as the principle of similarity or similitude. Similitude allows smaller scale models of planes and cars to be tested in wind tunnels where the Reynolds number is kept the same by increasing the speed of air flow or by changing some other property of the fluid. The Ford Motor Company has taken advantage of the principle of similarity and conducted flow tests on cars under water. Water flowing at 2 mph (3.2 km/hr) was used to simulate air flowing at 30 mph (48 km/hr).

## Laminar and turbulent flow

Flow patterns can be characterized as laminar or turbulent. In laminar or streamlined flow, the fluid glides along in layers, which do not mix so that the flow takes place in smooth continuous lines called streamlines. This is what we see when we open a water faucet just a little so that the flow is clear and regular. If we continue turning the faucet, the flow gradually becomes cloudy and irregular. This is known as turbulent flow. This change to turbulence takes place at high Reynolds numbers. The critical Reynolds number at which this change occurs differs from case to case.

## Bernoulli’s principle

An important idea in fluid flow is that of the conservation of mass. This implies that the amount of fluid that goes in through one end of a pipe is the same as the amount of fluid that comes out through the other end. Thus the fluid has to flow faster in narrower sections or constrictions in the pipe. Another important idea, expressed by Bernoullis principle, is that of the conservation of energy.

Daniel Bernoulli (17001782) was the first person to study fluid flow mathematically. He imagined a completely non-viscous and incompressible or ideal fluid in order to simplify the mathematics. Bernoullis principle for an ideal fluid essentially says that the total amount of energy in a laminar flow is always the same. This energy has three components potential energy due to gravity, potential energy due to pressure in the fluid, and kinetic energy due to speed of flow. Since the total energy is constant, increasing one component will decrease another. For instance, in a horizontal pipe in which gravitational energy stays the same, the fluid will move faster through a constriction and will, therefore, exert less pressure on the walls. In recent years, powerful computers have made it possible for scientists to attack the full mathematical complexity of the equations that describe the flow of real, viscous, and compressible fluids. Bernoullis principle, however, remains surprisingly relevant in a variety of situations and is probably the single most important principle in fluid dynamics.

## Boundary layer theory

Even though Bernoullis principle works extremely well in many cases, neglecting viscosity altogether often gives incorrect results. This is because even in fluids with very low viscosity, the fluid right next to the solid boundary sticks to the surface. This is known as the

### KEY TERMS

Boundary layer The layer of fluid that sticks to the solid surface and in which the speed of the fluid decreases.

Compressibility The property that allows a fluid to be compressed into a smaller volume.

Laminar A mode of flow in which the fluid moves in layers along continuous well-defined lines known as streamlines.

Reynolds number A number that characterizes a flow situation and allows flows of different fluids in different situations to be compared.

Turbulent An irregular, disorderly mode of flow.

Viscosity The internal friction within a fluid that makes it resist flow.

Wake The area of low pressure turbulent flow behind a moving body that causes the body to experience resistance to its forward motion.

no-slip condition. Thus, however fast or easily the fluid away from the boundary may be moving, the fluid near the boundary has to slow down gradually and come to a complete stop exactly at the boundary. This is what causes drag on automobiles and airplanes in spite of the low viscosity of air.

The treatment of such flows was considerably simplified by the boundary layer concept introduced by Ludwig Prandtl (18751953) in 1904. According to Prandtl, the fluid slows down only in a thin layer next to the surface. This boundary layer starts forming at the beginning of the flow and slowly increases in thickness. It is laminar in the beginning but becomes turbulent after a point determined by the Reynolds number. Since the effect of viscosity is confined to the boundary layer, the fluid away from the boundary may be treated as ideal.

## Shape and drag

Moving automobiles and airplanes experience a resistance or drag due to the force of air sticking to the surface. Another source of resistance is pressure drag, which is due to a phenomenon known as flow separation. This happens when there is an abrupt change in the shape of the moving object, and the fluid is unable to make a sudden change in flow direction and stays with the boundary. In this case, the boundary layer gets detached from the body, and a region of low pressure turbulence or wake is formed below it. This creates a drag on the vehicle due to the higher pressure in the front. That is why aerodynamically designed cars are shaped so that the boundary layer remains attached to the body longer, creating a smaller wake and, therefore, less drag. There are many examples in nature of shape modification for drag control. The sea anemone, for instance, continuously adjusts its form to the ocean currents in order to avoid being swept away while gathering food.

## Resources

### BOOKS

Fox, Robert W., Alan T. McDonald, Philip J. Pritchard. Introduction to Fluid Mechanics. Danvers, MA: John Wiley & Sons, 2003.

Kundu, Pijush K., and Ira M. Cohen. Fluid Mechanics. San Diego: Academic Press, 2001.

Munson, Bruce R., Donald F. Young, Theodore H. Okiishi. Fundamentals of Fluid Mechanics. Danvers, MA: John Wiley & Sons, 2005.

Sreela Datta