Two Important Solutions of Euler Equation

1. Two Important Solutions of Euler Equation P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Reversible and Irreversible Solutions to an Inviscid Flow……

2. Propagation of Infinitesimal Disturbance in a Fluid • Any small disturbance in a fluid travels as a sound wavel • The speed of sound can be obtained is the simplest solution to Euler equation.. Speed of sound in a fluid medium Speed sound in a perfect gas medium Mach’s measure of flow speed =

3. Role of Speed of Sound in High Speed Flows • Speed of sound in a fluid medium establishes an interaction between microscopic and macroscopic kinetic energies of the fluid molecules. • Every decelerating molecule will gain microscopic energy and lose macroscopic kinetic energy. • These interactions are understood through definition of Stagnation Properties. • Stagnation property is defined as property of a high speed fluid, when it is isentropically bought to rest. • Stagnation enthalpy, it is defined as the maximum possible microscopic kinetic energy of a fluid.

4. Stagnation Properties (Perfect Gas)

5. In general for an adiabatic Flow Field the Stagnation Temperature is defined by the relationship Stagnation Temperature is constant throughout an adiabatic flow field. • T0 is also sometimes referred to at Total Temperature • T is sometimes referred to as Static Temperature • Stagnation temperature is a measure of the total Kinetic Energy of the flow Field. • Largely responsible for the high Level of heating that occurs on high speed aircraft or reentering space Vehicles …

6. The Columbia Disaster !!!!

7. The Cause • Approximately 82 seconds after launch from Kennedy Space Center's LC-39-A, a suitcase-size piece of thermal insulation foam broke off the External Tank (ET), striking Columbia's left wing Reinforced Carbon-Carbon (RCC) panels. • At the time of the foam strike, the orbiter was at an altitude of about 66,000 feet (20 km; 13 mi), traveling at Mach 2.46 (1,870 miles per hour or 840 m/s). • As demonstrated by ground experiments conducted by the Columbia Accident Investigation Board, this likely created a 6-to-10-inch (15 to 25 cm) diameter hole, allowing hot gases to enter the wing when Columbia later reentered the atmosphere.

8. Mach’s Analysis of the Problem Loss Of Signal at 61.2 km altitude T∞ ~ 243 K ~18.0 Mach Number

9. Ideal gas Variable Properties Real gas

10. The Jet Plane Vampire • The first to exceed a speed of 500 miles per hour. • A total of 3,268 Vampires were built in 15 versions, including a twin-seat night fighter, trainer and a carrier-based aircraft designated Sea Vampire. • DH108 was a newer version was built and released for test. • Initially DH 108 behaved very nicely. • As the speed was stepped up in was unsuspectingly drawn closer to an invisible wall in the sky. • It was unknown to anyone. • One evening the pilot hit this wall and the plane was disappered.

11. Finite Disturbance in a simplest flow • A control volume for this analysis is shown, and the gas flows from left to right. • The control volume is having a very narrow width. • No chemical reactions. • There is no friction or heat loss across this CV. • NO work transfer is possible as this is so narrow. • The increase of the entropy is fundamental to any thing that happens in this world.

12. Propagation of Finite Disturbance in 1D inviscid Adiabatic Ideal Gas Flow

13. Solution of Simultaneous Equations • If the conditions upstream are known, then there are four unknown conditions downstream. • A system of four unknowns and four equations is solvable. • There exist two solutions because of the quadratic nature of the equations. • Both of these two possible solutions may or may not feasible. • Thermodynamics dictates the feasibility of these solutions. • Changes in pressure, temperature, volume cannot detect the possible solution. • The only tool that helps in identifying the possible solution is change in entropy. • For the adiabatic process, the entropy must increase or remain constant.

14. Conservation of mass: The relation between Upstream & Downstream Mach Numbers Mach number is a true measure of speed !!! Replace velocity by Mach number.

15. Conservation of momentum: Introduce Mach number in place of speed

16. Combined study of Conservation of Mass and Momentum &

17. Conservation of energy Energy equation in terms velocity of sound for a perfect gas Replace velocity by Mach number

18. The Unified Equation for the Vorticity Generator Simplified Energy Equation : Combined Mass & Momentum Equation : Combined Mass, Momentum and Energy Conservation :

19. An Unified Template for 1DS Euler flow Reversible flow : No vorticity generation between x & y, No increase in entropy of adiabatic passive Euler flow of pure substance

20. The Hidden Entropy Generator : The Ghost !!! With a vorticity (entropy) generator between x & y, The Normal Shock !!!! This equation relates the downstream Mach number to the upstream Mach Number. It can be used to derive pressure ratio, the temperature ratio, and density ratio.

21. The Nature of Irreversible Phenomenon g =constant=1.4 My Mx

22. Nature of Normal Shock • The flow across the shock is adiabatic and the stagnation temperature is constant across a shock. • The effect of increase in entropy across a shock will result in a decrease of stagnation pressure.

23. Normal Shock : A zero weight Compressor

24. Mach Number Vs Static Temperature : Normal Shock

25. Mach Number Vs Density : Normal Shock • This feasible only if

26. Change in entropy for an ideal gas Obtain a pressure temperature relation across shock. &

27. Shock is possible only in supersonic flows Infeasible M

28. Travel Cycle for Modern Spacecrafts

29. What was the Surface Temperature At Columbia Breakup?

30. Rentering Space Craft