AE 2303AERODYNAMICS-II Dr.S.Elangovan
Introduction • Review of prerequisite elements • Perfect gas • Thermodynamics laws • Isentropic flow • Conservation laws • Speed of sound • Analogous concept • Derivation of speed of sound • Mach number
Review of prerequisite elements Entropy • Perfect gas: Equation of state For calorically perfect gas Entropy changes?
Review of prerequisite elements Cont. Forms of the 1st law The second law
Review of prerequisite elements Cont. If ds=o For an isentropic flow
flow 1 2 If r is constant (incompressible): Review of prerequisite elements Cont. Conservation of mass (steady flow): Rate of mass enters control volume Rate of mass leaves control volume =
Net force on gas in control volume Rate momentum leaves control volume Rate momentum enters control volume = - flow 1 2 Euler equation (frictionless flow): Review of prerequisite elements Cont. Conservation of momentum (steady flow):
heat transfer work transfer energy transfer due to mass flow Review of prerequisite elements Cont. Conservation of energy for a CV (energy balance): • Basic principle: • Change of energy in a CV is related to • energy transfer by heat, work, and energy in • the mass flow.
Most important form of energy balance. Review of prerequisite elements Cont. • Analyzing more about Rate of Work Transfer: • work can be separated into 2 types: • work associated with fluid pressure as mass entering or leaving the CV. • other works such as expansion/compression, electrical, shaft, etc. • Work due to fluid pressure: • fluid pressure acting on the CV boundary creates force.
For calorically perfect gas (dcp=dcv=0): flow 1 2 For adiabatic flow (no heat transfer) and no work: Review of prerequisite elements Cont.
Review of prerequisite elements Cont. Conservation laws Conservation of mass (compressible flow): Conservation of momentum (frictionless flow): Conservation of energy (adiabatic):
Group Exercises 1 • Given that standard atmospheric conditions for air at 150C are a pressure of 1.013 bar and a density of 1.225kg, calculate the gas constant for air. Ans: R=287.13J/kgK • The value of Cv for air is 717J/kgK. The value of R=287 J/kgK. Calculate the specific enthalpy of air at 200C. Derive a relation connecting Cp, Cv, R. Use this relation to calculate Cp for air using the information above. Ans: h=294.2kJ/kgK,Cp=1.004kJ/kgK • Air is stored in a cylinder at a pressure of 10 bar, and at a room temperature of 250C. How much volume will 1kg of air occupy inside the cylinder? The cylinder is rated for a maximum pressure of 15 bar. At what temperature would this pressure be reached? Ans: V=0.086m2, T=1740C.
Sound wave Sound wave Speed of sound Sounds are the small pressure disturbances in the gas around us, analogous to the surface ripples produced when still water is disturbed Sound wave moving through stationary gas Gas moving through stationary sound wave
Speed of sound cont. Derivation of speed of sound Combination of mass and momentum Conservation of mass For isentropic flow Conservation of momentum Finally
Mach Number M=V/a M<1 Subsonic M=1 Sonic M>1 Supersonic M>5 Hypersonic Distance traveled = speed x time = 4at Distance traveled = at Zone of silence Source of disturbance Region of influence If M=0
Mach Number cont. Original location of source of disturbance Source of disturbance If M=0.5
Mach Number cont. ut Original location of source of disturbance ut ut ut Direction of motion Source of disturbance Mach wave: If M=2
Normal and Oblique Shock • A shock wave (also called shock front or simply "shock") is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium (solid, liquid, gas or plasma) or in some cases in the absence of a material medium, through a field such as the electromagnetic field.
Shock waves are characterized by an abrupt, nearly discontinuous change in the characteristics of the medium. Across a shock there is always an extremely rapid rise in pressure, temperature and density of the flow. In supersonic flows, expansion is achieved through an expansion fan. A shock wave travels through most media at a higher speed than an ordinary wave.
Unlike solutions (another kind of nonlinear wave), the energy of a shock wave dissipates relatively quickly with distance. Also, the accompanying expansion wave approaches and eventually merges with the shock wave, partially canceling it out. Thus the sonic boom associated with the passage of a supersonic aircraft is the sound wave resulting from the degradation and merging of the shock wave and the expansion wave produced by the aircraft.
Thus the sonic boom associated with the passage of a supersonic aircraft is the sound wave resulting from the degradation and merging of the shock wave and the expansion wave produced by the aircraft.
When a shock wave passes through matter, the total energy is preserved but the energy which can be extracted as work decreases and entropy increases. This, for example, creates additional drag force on aircraft with shocks.
Oblique Shock • An oblique shock wave, unlike a normal shock, is inclined with respect to the incident upstream flow direction. • It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and compresses.
The upstream streamlines are uniformly deflected after the shock wave. The most common way to produce an oblique shock wave is to place a wedge into supersonic, compressible flow. Similar to a normal shock wave, the oblique shock wave consists of a very thin region across which nearly discontinuous changes in the thermodynamic properties of a gas occur. While the upstream and downstream flow directions are unchanged across a normal shock, they are different for flow across an oblique shock wave.
It is always possible to convert an oblique shock into a normal shock by a Galilean transformation.
EXPANSIONWAVES,RAYLEIGH AND FANNO FLOW • A Prandtl-Meyer expansion fan is a centered expansion process, which turns a supersonic flow around a convex corner. • The fan consists of an infinite number of Mach waves, diverging from a sharp corner. In case of a smooth corner, these waves can be extended backwards to meet at a point.
Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible to turn the flow away from itself through a single "shock" wave because it will violate the second law of thermodynamics. Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, temperature and density decrease. Since the process is isentropic, the stagnation properties remain constant across the fan.
Prandtl-Meyer Function • θ2 − θ1 = ν(M2) − ν(M1)
Rayleigh flow • Rayleigh flow refers to diabetic flow through a constant area duct where the effect of heat addition or rejection is considered. Compressibility effects often come into consideration, although the Rayleigh flow model certainly also applies to incompressible flow. For this model, the duct area remains constant and no mass is added within the duct. Therefore, unlike Fanno flow, the stagnation temperature is a variable.
The heat addition causes a decrease in stagnation pressure, which is known as the Rayleigh effect and is critical in the design of combustion systems. Heat addition will cause both supersonic and subsonic Mach numbers to approach Mach 1, resulting in choked flow. Conversely, heat rejection decreases a subsonic Mach number and increases a supersonic Mach number along the duct. It can be shown that for calorically perfect flows the maximum entropy occurs at M = 1. Rayleigh flow is named after John Strutt, 3rd Baron Rayleigh.
Solving the differential equation leads to the relation shown below, where T0* is the stagnation temperature at the throat location of the duct which is required for thermally choking the flow. • These values are significant in the design of combustion systems. For example, if a turbojet combustion chamber has a maximum temperature of T0* = 2000 K, T0 and M at the entrance to the combustion chamber must be selected so thermal choking does not occur, which will limit the mass flow rate of air into the engine and decrease thrust. • For the Rayleigh flow model, the dimensionless change in entropy relation is shown below.
Fanno flow • Fanno flow refers to adiabatic flow through a constant area duct where the effect of friction is considered.Compressibility effects often come into consideration, although the Fanno flow model certainly also applies to incompressible flow. For this model, the duct area remains constant, the flow is assumed to be steady and one-dimensional, and no mass is added within the duct. The Fanno flow model is considered an irreversible process due to viscous effects. The viscous friction causes the flow properties to change along the duct. The frictional effect is modeled as a shear stress at the wall acting on the fluid with uniform properties over any cross section of the duct.
For a flow with an upstream Mach number greater than 1.0 in a sufficiently long enough duct, deceleration occurs and the flow can become choked. On the other hand, for a flow with an upstream Mach number less than 1.0, acceleration occurs and the flow can become choked in a sufficiently long duct. It can be shown that for flow of calorically perfect gas the maximum entropy occurs at M = 1.0. Fanno flow is named after Gino Girolamo Fanno.
TRANSONIC FLOW OVER WING • In aerodynamics, the critical Mach number (Mcr) of an aircraft is the lowest Mach number at which the airflow over a small region of the wing reaches the speed of sound.
For all aircraft in flight, the airflow around the aircraft is not exactly the same as the airspeed of the aircraft due to the airflow speeding up and slowing down to travel around the aircraft structure. At the Critical Mach number, local airflow in some areas near the airframe reaches the speed of sound, even though the aircraft itself has an airspeed lower than Mach 1.0. This creates a weak shock wave. At speeds faster than the Critical Mach number:
drag coefficient increases suddenly, causing dramatically increased drag • in aircraft not designed for transonic or supersonic speeds, changes to the airflow over the flight control surfaces lead to deterioration in control of the aircraft.
In aircraft not designed to fly at the Critical Mach number, shock waves in the flow over the wing and tail plane were sufficient to stall the wing, make control surfaces ineffective or lead to loss of control such as Mach tuck. The phenomena associated with problems at the Critical Mach number became known as compressibility. Compressibility led to a number of accidents involving high-speed military and experimental aircraft in the 1930s and 1940s.
Drag Divergence Mach Number • The drag divergence Mach number is the Mach number at which the aerodynamic drag on an airfoil or airframe begins to increase rapidly as the Mach number continues to increase. This increase can cause the drag coefficient to rise to more than ten times its low speed value.
The value of the drag divergence Mach number is typically greater than 0.6; therefore it is a transonic effect. The drag divergence Mach number is usually close to, and always greater than, the critical Mach number. Generally, the drag coefficient peaks at Mach 1.0 and begins to decrease again after the transition into the supersonic regime above approximately Mach 1.2.
The large increase in drag is caused by the formation of a shock wave on the upper surface of the airfoil, which can induce flow separation and adverse pressure gradients on the aft portion of the wing. This effect requires that aircraft intended to fly at supersonic speeds have a large amount of thrust.
In early development of transonic and supersonic aircraft, a steep dive was often used to provide extra acceleration through the high drag region around Mach 1.0. In the early days of aviation, this steep increase in drag gave rise to the popular false notion of an unbreakable sound barrier, because it seemed that no aircraft technology in the foreseeable future would have enough propulsive force or control authority to overcome it. Indeed, one of the popular analytical methods for calculating drag at high speeds, the Prandtl-Glauert rule, predicts an infinite amount of drag at Mach 1.0.
Two of the important technological advancements that arose out of attempts to conquer the sound barrier were the Whitcomb area rule and the supercritical airfoil. A supercritical airfoil is shaped specifically to make the drag divergence Mach number as high as possible, allowing aircraft to fly with relatively lower drag at high subsonic and low transonic speeds. These, along with other advancements including computational fluid dynamics, have been able to reduce the factor of increase in drag to two or three for modern aircraft designs