Historical Note: Bernoulli & Euler

1. Historical Note: Bernoulli & Euler Daniel Bernoulli (1700 – 1782) was born in Groningen, Netherlands. He was the member of remarkable mathematician family, father and uncle both were noted mathematician and physicists. The entire family was swiss and made Basel, Switzerland as home. He gave insight on kinetic theory of gases in his book Hydrodynamica (1738), also about jet propulsion, manometers, flow in pipes etc. Bernoulli’s theorem better understood by his father, both did not understand that pressure is a point property. Leonhard Euler (1707 – 1783) was also a swiss mathematician. He became one of mathematical giant of history & his contribution to fluid dynamics are of interest. He was a close friend of Bernoulli and a student of his father, Euler was influenced by the work of Bernoulli's in hydrodynamics & he originated the concept of pressure acting at a point in a gas. He came up with differential equation for a fluid accelerated by pressure, the same equation as Eq. (4.8) driven in the chapter. Bernoulli’s equation has been obtained as Eq. (4.9)

2. The Pitot Tube Henri Pitot (1695 – 1771) borne in Aramon, France, the inventor of Pitot Tube. He began his career as astronomer and mathematician. In 1724, he became interested in hydraulics and in particular, in the flow of water in rivers and canals. He was not satisfied with the way velocity of floating objects was measured. He devised an instrument consisting of two tubes. One was simply a straight tube open at one end, which was inserted vertically into water (to measure static pressure) and the other was a tube with one end bent at right angles, with the open end facing directly into the flow (to measure total pressure). He used this instrument in 1732, to measure water flow in a river. It was thought at that time that the water flow velocity increases with depth. Pitot reported stunning and correct results, measured with this instrument, that in reality the flow velocity decreases as the depth increased. Hence Pitot tube was introduced with style. French Captain A. Eteve in Jan 1911, later by British engineer deployed Pitot Tube on an airplane for the first time. It then evolved into the primary instrument for flight speed measurement.

3. The First Wind Tunnels Aerodynamics is an empirically based discipline. Discovery and development by experimental means have been its lifeblood. Wind Tunnel has been the work horse for such experiments. Today, most aerospace industrial, government and university labs have a complete spectrum of wind tunnels ranging from low subsonic to hypersonic speeds. Evolution of wind tunnels goes back more than 400 years when Leonardo Vinci near the beginning of 16th century said that “since the action of the medium upon the body is the same whether the body moves in medium ore the particles of the medium impinge with the same velocity upon body”. The lift and drag of an aerodynamics body are the same whether it moves through the stagnant air at 100 miles/hour or whether the air moves over the stationary body at 100 miles/hour. First wind tunnel in history was designed and built by Francis Wenham in Greenwich, England in 1871. thereafter, many wind tunnels were made all over the world, and all of them were low speed to start with (essentially in incompressible flow) but as the airplane speed increased, new wind tunnels with higher velocity capability were made. First supersonic wind tunnel was developed by Dr A Busemann at Germany in mid 1930s. First hypersonic wind tunnel was operated by NACA at Langley in 1947.

4. Saying of Prof. Albert F Zahm 1912 • Theoretical fluid dynamics, being a difficult subject, is for convenience, commonly divided into branches, one treating of frictionless or perfect fluids, the other treating of viscous or imperfect fluids. The frictionless fluid has no existence in nature, but is hypothesized by mathematicians in order to facilitate the investigation of important laws and principles that may be approximately true of viscous or natural fluids.

7. Mass Flow Rate

8. Mass Flow Rate • Within some problem domain, the amount of mass remains constant --mass is neither created nor destroyed. • The mass of any object is simply the volume that the object occupies times the density of the object. For a fluid (a liquid or a gas) the density, volume, and shape of the object can all change within the domain with time. And mass can move through the domain. • If the fluid initially passes through an area A at velocity V, we can define a volume of mass to be swept out in some amount of time t. The volume v is: • v = A * V * t • A units check gives area x length/time x time = area x length = volume. The mass m contained in this volume is simply density r times the volume. • m = r * A * V * t • To determine the mass flow rate mdot, we divide the mass by the time. The resulting definition of mass flow rate is shown on the slide in red. • mdot = r * A * V • From Newton's Second Law of Motion, the aerodynamic forces on an aircraft (lift and drag) are directly related to the change in momentum of a gas with time. The momentum is defined to be the mass times the velocity, so we would expect the aerodynamic forces to depend on the mass flow rate past an object.

9. Mass Flow Rate • The thrust produced by a propulsion system also depends on the change of momentum of a working gas. The thrust depends directly on the mass flow rate through the propulsion system. • Considering the mass flow rate equation, it would appear that for a given area, we could make the mass flow rate as large as we want by setting the velocity very high. However, in real fluids, compressibility effects limit the speed at which a flow can be forced through a given area.

11. Bernoulli's equation • In the 1700s, Daniel Bernoulli investigated the forces present in a moving fluid. This slide shows one of many forms of Bernoulli's equation. The equation appears in many physics, fluid mechanics, and airplane textbooks. • The equation states that the static pressure ps in the flow plus the dynamic pressure, one half of the density r times the velocity V squared, is equal to a constant throughout the flow. We call this constant the total pressure pt of the flow. • Thermodynamics is the branch of science which describes the macro scale properties of a fluid. One of the principle results of the study of thermodynamics is the conservation of energy; within a system, energy is neither created nor destroyed but may be converted from one form to another. • Assuming a steady, inviscid flow we have a simplified conservation of energy equation in terms of the enthalpy of the fluid: • ht2 - ht1 = q - wsh • where ht is the total enthalpy of the fluid, q is the heat transfer into the fluid, and wsh is the useful work done by the fluid.

12. Applications of Bernoulli's Equation • The fluids problem shown on this slide is low speed flow through a tube with changing cross-sectional area. For a streamline along the center of the tube, the velocity decreases from station one to two. Bernoulli's equation describes the relation between velocity, density, and pressure for this flow problem. Since density is a constant for a low speed problem, the equation at the bottom of the slide relates the pressure and velocity at station two to the conditions at station one. • Along a low speed airfoil, the flow is incompressible and the density remains a constant. Bernoulli's equation then reduces to a simple relation between velocity and static pressure. The surface of the airfoil is a streamline. Since the velocity varies along the streamline, Bernoulli's equation can be used to compute the change in pressure. The static pressure integrated along the entire surface of the airfoil gives the total aerodynamic force on the foil. This force can be broken down into the lift and drag of the airfoil. • Bernoulli's equation is also used on aircraft to provide a speedometer called a pitot-static tube. A pressure is quite easy to measure with a mechanical device. In a pitot-static tube, we measure the static and total pressure and can then use Bernoulli's equation to compute the velocity.

14. Isentropic Flow • As a gas is forced through a tube, the gas molecules are deflected by the walls of the tube. • If the speed of the gas is much less than the speed of sound of the gas, the density of the gas remains constant and the velocity of the flow increases. • As the speed of the flow approaches the speed of sound we must consider compressibility effects on the gas. The density of the gas varies from one location to the next. • If the flow is very gradually compressed (area decreases) and then gradually expanded (area increases), the flow conditions return to their original values. We say that such a process is reversible. • From a consideration of the second law of thermodynamics, a reversible flow maintains a constant value of entropy. • Engineers call this type of flow an isentropic flow; a combination of the Greek word "iso" (same) and entropy. • Isentropic flows occur when the change in flow variables is small and gradual, such as the ideal flow through the nozzle shown above. • If a supersonic flow is turned abruptly and the flow area decreases, shock waves are generated and the flow is irreversible. The isentropic relations are no longer valid and the flow is governed by the oblique or normal shock relations.

16. Mach Number • The ratio of the speed of the aircraft to the speed of sound in the gas determines the magnitude of many of the compressibility effects. • Aerodynamicists have designated it with a special parameter called the Mach number in honor of Ernst Mach, a late 19th century physicist who studied gas dynamics. The Mach number M allows us to define flight regimes in which compressibility effects vary. • Subsonic conditions occur for Mach numbers less than one, M < 1 . For the lowest subsonic conditions, compressibility can be ignored. • As the speed of the object approaches the speed of sound, the flight Mach number is nearly equal to one, M = 1, and the flow is said to be transonic. • Supersonic conditions occur for Mach numbers greater than one, 1 < M < 3. Compressibility effects are important for supersonic aircraft, and shock waves are generated by the surface of the object. For high supersonic speeds, 3 < M < 5, aerodynamic heating also becomes very important for aircraft design. • For speeds greater than five times the speed of sound, M > 5, the flow is said to be hypersonic. At these speeds, some of the energy of the object now goes into exciting the chemical bonds which hold together the nitrogen and oxygen molecules of the air. At hypersonic speeds, the chemistry of the air must be considered when determining forces on the object. The Space Shuttle re-enters the atmosphere at high hypersonic speeds, M ~ 25. Under these conditions, the heated air becomes an ionized plasma of gas and the spacecraft must be insulated from the high temperatures.

17. As an object moves through the atmosphere, the gas molecules of the atmosphere near the object are disturbed and move around the object. Aerodynamic forces are generated between the gas and the object. The magnitude of these forces depend on the shape of the object, the speed of the object, the mass of the gas going by the object and on two other important properties of the gas; the viscosity, or stickiness, of the gas and the compressibility, or springiness, of the gas. • Aerodynamic forces depend in a complex way on the viscosity of the gas. As an object moves through a gas, the gas molecules stick to the surface. This creates a layer of air near the surface, called a boundary layer, which, in effect, changes the shape of the object. • The flow of gas reacts to the edge of the boundary layer as if it was the physical surface of the object. To make things more confusing, the boundary layer may separate from the body and create an effective shape much different from the physical shape. • To make it even more confusing, the flow conditions in and near the boundary layer are often unsteady (changing in time). The boundary layer is very important in determining the drag of an object. To determine and predict these conditions, aerodynamicists rely on wind tunnel testing and very sophisticated computer analysis.

18. Aerodynamic forces also depend in a complex way on the compressibility of the gas. As an object moves through the gas, the gas molecules move around the object. If the object passes at a low speed (typically less than 200 mph) the density of the fluid remains constant. • For high speeds, some of the energy of the object goes into compressing the fluid and changing the density, which alters the amount of resulting force on the object. This effect becomes more important as speed increases. Near and beyond the speed of sound (about 330 m/s or 700 mph on earth), shock waves are produced that affect the lift and drag of the object. Again, aerodynamicists rely on wind tunnel testing and sophisticated computer analysis to predict these conditions. • The effects of compressibility and viscosity on lift are contained in the lift coefficient and the effects on drag are contained in the drag coefficient. For propulsion systems, compressibility affects the amount of mass that can pass through an engine and the amount of thrust generated by a rocket or turbine engine nozzle.