Fluid Mechanics

I

INTRODUCTION

Fluid Mechanics, physical science dealing with the action of fluids at rest or in motion, and with applications and devices in engineering using fluids. Fluid mechanics is basic to such diverse fields as aeronautics chemical, civil, and mechanical engineering, meteorology, naval architecture, and oceanography.

Fluid mechanics can be subdivided into two major areas, fluid statics, which deals with fluids at rest, and fluid dynamics, concerned with fluids in motion. The term hydrodynamics is applied to the flow of liquids or to low-velocity gas flows where the gas can be considered as being essentially incompressible. Aerodynamics is concerned with the theory of flight, and compressible fluid flow or gas dynamics with the behavior of gases under flow conditions, where velocity and pressure changes are sufficiently large to require inclusion of the compressibility effects.

Applications of fluid mechanics involve all kinds of flow machinery, including jet propulsion, hydraulics, turbine, compressors, and pumps. Hydraulics mainly concerns machines and structures such as hydraulic turbines, dams, and hydraulic pressures, using water or other liquids.

II

FLUID STATICS OR HYDROSTATICS

A fundamental characteristic of any fluid at rest is that the force exerted on any particle within the fluid is the same in all directions. If the forces were unequal, the particle would move in the direction of the resultant force. It follows that the force per unit area, or the pressure exerted by the fluid against the walls of an arbitrarily shaped containing vessel, is perpendicular to the interior walls at every point. If the pressure were not perpendicular an unbalanced tangential force component would exist and the fluid would move along the wall.

This concept was first formulated in a slightly extended form by the French mathematician and philosopher Blaise Pascal in 1647. Known as Pascal’s law, it states that the pressure applied to an enclosed fluid is transmitted equally in all directions and to all parts of the enclosing vessel, if pressure changes due to the weight of the fluid can be neglected. This law has extremely important applications in hydraulics.

The top surface of a liquid at rest in an open vessel will always be perpendicular to the resultant forces acting on it. If gravity is the only force, the surface will be horizontal. If other forces in addition to gravity act, then the “free” surface will adjust itself. For instance, if a glass of water is spun rapidly about its vertical axis, both gravity and centrifugal forces will act on the water and the surface will form a parabola that is perpendicular to the resultant force. If gravity is the only force acting on a liquid contained in an open vessel, the pressure at any point within the liquid is directly proportional to the weight of a vertical column of that liquid. This, in turn, is proportional to the depth below the surface and is independent of the size or shape of the container. Thus the pressure at the bottom of a pipe about 2.5 cm (about 1 in) in diameter and about 15 m (about 50 ft) high that is filled with water is the same as the pressure at the bottom of a lake about 15 m (about 50 ft) deep. Similarly, a pipe about 30 m (about 100 ft) long that is filled with water, and slanted so that the top is only about 15 m (about 50 ft) above the bottom vertically, will have the same pressure exerted at the bottom of the pipe even though the distance along the pipe is much longer. The weight of a column of fresh water about 30 cm (about 12 in) high and with a cross section of about 6.5 sq cm (about 1 sq in) is about 0.196 kg (about 0.433 lb) and this will be the pressure exerted at the bottom. A column about 30 cm (about 12 in) high and about 0.093 sq m (about 1 sq ft) in cross section will weigh 144 times as much, but the pressure, which is force per unit area, will remain identical. The pressure at the bottom of a mercury column about 30 cm (about 12 in) high will be 0.196 × 13.6 = 2.07 kg per 6.5 sq cm (1 sq in) as mercury is 13.6 times as heavy as water.

The second important principle of fluid statics was discovered by the Greek mathematician and philosopher Archimedes. The so-called Archimedes’ principle states that a submerged body is subject to a buoyancy force that is equal to the weight of the fluid displaced by that body. This explains why a heavily laden ship floats; its total weight equals exactly the weight of the water that it displaces, and this weight exerts the buoyant force supporting the ship.

A point at which all forces producing the buoyant effect may be considered to act is the center of buoyancy and is the center of gravity of the fluid displaced. The center of buoyancy of a floating body is directly above its center of gravity. The greater the distance between these two, the more stable the body.

Archimedes’ principle also makes possible the determination of the density of an object that is so irregular in shape that its volume cannot be measured directly. If the object is weighed first in air and then in water, the difference in weights will equal the weight of the volume of the water displaced, which is the same as the volume of the object. Thus the weight density of the object (weight divided by volume) can readily be determined. In very high precision weighing, both in air and in water, the displaced weight of both the air and water has to be accounted for in arriving at the correct volume and density.

III

FLUID DYNAMICS OR HYDRODYNAMICS

This branch of fluid mechanics deals with the laws of fluids in motion; these laws are considerably more complex and, in spite of the greater practical importance of fluid dynamics, only a few basic ideas can be discussed here.

Interest in fluid dynamics dates from the earliest engineering application of fluid machines. Archimedes made an early contribution by his invention of the screw pump, the pushing action of which is similar to that of the corkscrewlike device in a meat grinder. Other hydraulic machines and devices were developed by the Romans, who not only used Archimedes’ screw for irrigation and mine pumping but also built extensive aqueduct systems, some of which are still in use. The Roman architect and engineer Vitruvius first described the verticle waterwheel, a technology that revolutionized corn milling, during the 1st century bc.

Despite the early practical applications of fluid dynamics, little or no understanding of the basic theory existed, and development lagged accordingly. After Archimedes, more than 1800 years elapsed before the next significant scientific advance was made by the Italian mathematician and physicist Evangelista Torricelli, who invented the barometer in 1643, and formulated Torricelli’s law, which related the efflux velocity of a liquid through an orifice in a vessel to the liquid height above it. The major spurt in the development of fluid mechanics had to await the formulation of Newton’s laws of motion by the English mathematician and physicist Isaac Newton. These laws were applied to fluids first by the Swiss mathematician Leonhard Euler, who derived the basic equations for a frictionless, or inviscid, fluid.

Euler first recognized that dynamical laws for fluids can only be expressed in a relatively simple form if the fluid is assumed incompressible and ideal, that is, if the effects of friction or viscosity can be neglected. Because, however, this is never the case for real fluids in motion, the results of such an analysis can only serve as an estimate for those flows where viscous effects are small.

A

Incompressible and Inviscid, or Frictionless, Flows

These flows follow Bernoulli’s principle, named after the Swiss mathematician and scientist Daniel Bernoulli. The principle states that the total mechanical energy of an incompressible and inviscid flow is constant along a streamline. Streamlines are imaginary flow lines that are always parallel to the local direction of the flow, and that for steady flow are also the lines followed by individual fluid particles. Bernoulli’s principle leads to an interrelationship between pressure effects, velocity effects, and gravity effects, and indicates that the velocity increases as the pressure decreases. This principle is important in nozzle design and in flow measurements.

B

Viscous Flows, Laminar and Turbulent Motion

The first carefully documented friction experiments in low-speed pipe flow were carried out independently in 1839 by the French physiologist Jean Leonard Marie Poiseuille, who was interested in the characteristics of blood flow, and in 1840 by the German hydraulic engineer Gotthilf Heinrich Ludwig Hagen. An attempt to include the effects of viscosity into the mathematical equations was made first in 1827 by the French engineer Claude Louis Marie Navier, and independently by the British mathematician Sir George Gabriel Stokes, who in 1845 perfected the basic equations for viscous incompressible fluids. These are now known as the Navier-Stokes equations, and they are so complex that they can be applied only to simple flows. One such flow is that of a real fluid through a straight pipe. Here Bernoulli’s principle is not applicable because part of the total mechanical energy is dissipated as a result of viscous friction, resulting in a pressure drop along the pipe. The equations suggest that this pressure drop for a given pipe and a given fluid should be linear with the flow velocity. Experiments first conducted near the middle of the 19th century showed that this was only true for low velocities; at higher velocities, the pressure drop was more nearly proportional to the square of the velocity. This problem was not resolved until 1883 when the British engineer Osborne Reynolds showed the existence of two types of viscous flows in pipes. At low velocities the fluid particles follow the streamlines (laminar flow) and results match the analytical prediction. At higher velocities the flow breaks up into a fluctuating velocity pattern or eddies (turbulent flow) in a form that cannot be fully predicted even today. Reynolds also established that the transition from laminar to turbulent flow was a function of a single parameter that has since become known as the Reynolds number. If the Reynolds number, which is the product of velocity, fluid density, and pipe diameter, divided by the fluid viscosity, is less than 2100, the pipe flow will always be laminar; at higher values it will normally be turbulent. The concept of a Reynolds number is basic to much of modern fluid mechanics.

Turbulent flows cannot be evaluated solely from computed predictions and depend on a mixture of experimental data and mathematical models for their analysis, with much of modern fluid-mechanics research still being devoted to better formulations of turbulence. The transitional nature from laminar to turbulent flows and the complexity of the turbulent flow can be observed as cigarette smoke rises into very still air. At first it rises in a laminar streamline motion but after some distance it becomes unstable and breaks up into an intertwining eddy pattern.

C

Boundary Layer Flows

Before about 1860 the engineering interest in fluid mechanics was limited almost entirely to water flows. The development of the chemical industry during the latter part of the 19th century directed attention to other liquids and to gases. Interest in aerodynamics began with the studies of the German aeronautical engineer Otto Lilienthal in the last decade of the 19th century and saw major advances following the first successful powered flight by the American inventors Orville and Wilbur Wright in 1903.

The complexity of viscous flows, especially turbulent flows, severely restricted progress in fluid dynamics until the German engineer Ludwig Prandtl recognized in 1904 that many flows could be divided into two principal regions. The region close to the surface consists of a thin boundary layer where the viscous effects are concentrated and where the mathematical model can be greatly simplified. Outside the boundary layer viscous effects can be disregarded and the simpler mathematical equations for inviscid flows can be used. The boundary-layer theory has made possible much of the development of modern aircraft wings and the design of gas turbines and compressors. The boundary-layer model not only permitted a much simplified formulation of the Navier-Stokes equations in the region close to the body surface but also led to further developments of the flow of inviscid fluids that can be applied outside the boundary layer. Much of the modern development of fluid mechanics was made possible by the boundary-layer concept and it has been carried out by such key contributors as the Hungarian-born American aeronautical engineer Theodore von Kármán, and the German mathematician Richard von Mises, by the British physicist and meteorologist Sir Geoffrey Ingram Taylor.

D

Compressible Flows

Interest in compressible flows started with the development of steam turbines by the British inventor Charles Algernon Parsons, and the Swedish engineer Carl Gustaf Patrik de Laval during the 1880s. Here high-speed flow of steam within flow passages was first encountered and the need for efficient turbine design led to improved compressible flow analyses. Modern advances, however, had to wait for the stimulus of successful gas turbine and jet engine development in the 1930s. The early interest in high-speed flows over surfaces arose in the study of ballistics, for which an understanding of the motion of projectiles was needed. Major developments started near the end of the 19th century, involving Prandtl and his students, among others, and increased after the introduction of high-speed aircraft and rockets in World War II.

One of the basic principles of compressible flows is that the density of a gas changes when the gas is subjected to large velocity and pressure changes. At the same time its temperature also changes, leading to more complex means of analysis. The flow behavior of a compressible gas depends on whether the flow velocity is smaller or greater than the velocity of sound. The velocity of sound is the name given to the propagation velocity of a very small disturbance, or pressure wave, within the fluid. For a gas it is proportional to the square root of the absolute temperature. For instance, air at 20° C, or 293° on the Kelvin, or absolute, scale (68° F), has a sound velocity of 344.65 m per sec (1130 ft per sec). If the flow velocity is less than the sound velocity (subsonic flow), pressure waves can be transmitted throughout the whole fluid to adjust the flow that rushes toward an object. Thus the subsonic flow approaching an airplane wing will adjust itself some distance upstream to flow smoothly over the surface. In supersonic flow, pressure waves cannot travel upstream to readjust the flow. As a result, the air rushing toward a wing in supersonic flight will not be prepared for the impending disturbance the wing will cause. Instead, it has to redirect very suddenly in the proximity of the wing, where a sharp compression or shock is coupled with the redirection. The noise associated with this sudden shock causes the sonic boom of aircraft flying at supersonic speeds. Compressible flows are often identified by the Mach number, which is the ratio of the flow velocity divided by the sound velocity. Supersonic flows therefore have a Mach number greater than 1.