Study of Euler Equation P M V Subbarao

Study of Euler Equation P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Reversible and Irreversible Solutions to Inviscid Flow……

General Compressible Euler Equation of Motion Thus, the Euler equation of motion assumes the following form: Above equation shows that despite the inviscid flow assumption, fluid flow contains vorticities that are inherent in viscous flows but special in inviscid flows. The vortices cause additional entropy production in inviscid flows. This can be better explained using the first law of thermodynamics.

Gibbs form of First Law of Thermodynamics for A Fluid Flow For an infinitesimal process A fluid particle A undergone a displacement, during time dt

First Law Analysis of Fluid flow For a differential displacement, with s as the specific entropy, h as the specific static enthalpy and p the static pressure. Inserting the above property changes into the first law of thermodynamics

Combination of First Law & Euler equation Vectorial Entropy equation for Euler flow:

Bridge the gap between Thermodynamics & Fluid Mechanics Define total energy per unit mass of fluid flow as

Energy Conservation Adiabatic Passive Flows • In the absence of mechanical addition or rejection for an adiabatic flow; • htotal remains constant. • Meaning that its gradient vanishes. • Furthermore, for steady flow cases,

Entropy Generation in Inviscid Adiabatic Flows • Above equation is an important result that establishes a direct relation between the vorticity and the entropy production in inviscid flows. • What is the source of vorticy in an inviscid flows? ? • Can a flow field generate vorticity as a result of the any unknown events? • Can these events lead to discontinuities in flow field? • These are responsible for large jumps in velocities. • These jumps cause vorticity production and therefore, changes in entropy.

Normal Shock Past F-18

Reentry Interface Gas Dynamics

Mach’s Measure of Speed • Prof. Mach initiated the art of understanding the basic characteristics of high speed flow. • He proposed that one of the most important variables affecting aerodynamic behavior is the speed of the air flow over a body (V) relative to the speed of sound (c). • Mach was the first physicist to recognize that dependency. • He was also the first to note the sudden and discontinuous changes in the behavior of an airflow when the ratio V/c goes from being less than 1 to greater than 1. Ernst Mach (1838 -1916)

Mach’s Flow Visualization Experiments Ernst Mach's photo of a bullet in supersonic flight Mach was actually the first person in history to develop a method for visualizing the flow passing over an object at supersonic speeds. He was also the first to understand the fundamental principles that govern supersonic flow and their impact on aerodynamics.

The Simplest Flow Geometry y Ma in F low z x

The First & Foremost Solution

Conservation Laws Applied to 1 D Steady disturbance Conservation of Mass: Conservation of Mass for 1 DSF: p, r. . . c-u P+dp, r+dr. . . Change is final -initial C

For infinitesimally small disturbance Assume ideal gas conditions for Conservation of Momentum : For steady flow momentum equation for CV: For steady 1 -D flow :

For infinitesimally small disturbance

Nature of Substance • The expressions for speed of sound can be used to prove that speed of sound is a property of a substance. • Using the momentum analysis : • If it is possible to obtain a relation between p and r, then c can be expressed as a state variable. • This is called as equation of state, which depends on nature of substance.

Steady disturbance in A Perfect Gas Medium p, r. . . c-u P+dp, r+dr. . . C

Speed of sound in ideal and perfect gases • The speed of sound can be obtained easily for the equation of state for an ideal gas because of a simple mathematical expression. • The pressure for an ideal gas can be expressed as a simple function of density and a function molecular structure or ratio of specific heats, namely Mach’s measure of flow speed =

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.

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

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.

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

Conservation of momentum:

Introduce Mach number in place of speed

Combined study of Conservation of Mass and Momentum &

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

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

Discovery of Entropy Generator Reversible flow : No vorticity generation between x & y,

The Culprit !!! 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.

The Nature of Irreversible Phenomenon My g = constant=1. 4 Mx

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.

Mach Number Vs Static Pressure : Normal Shock

Mach Number Vs Static Temperature : Normal Shock

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

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

Shock is possible only in supersonic flows Infeasible M

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. • DH 108 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.

General Adiabatic Flow In general for an adiabatic Flow Field the Stagnation Temperature is defined by the relationship Stagnation Temperature is constant throughout an adiabatic flow field. • T 0 is also sometimes referred to at Total Temperature • T is sometimes referred to as Static Temperature

The Art of Reentry • The difference between Stagnation and Static temperatures is a measure of the Kinetic Energy of the flow Field. • Largely responsible for the high Level of heating that occurs on high speed aircraft or reentering space Vehicles … “stagnation” (total) pressure : Constant throughout Isentropic flow field.

How Nature Makes the Impossible Events into Possible ? !? !? !

Human Inventions : Protection of Space Vehicle Loss Of Signal at: 61. 2 km altitude ~18. 0 Mach Number T∞ ~ 243 K

Mach View of NS Equations • Reference velocity : Velocity of sound in a selected fluid. • Velocity sound, c: For an ideal gas: For a real gas: For an incompressible liquids: Average bulk modulus for water is 2 X 109 N/m 2. In Mach’s view it is not possible to differentiate compressible and incompressible flows with Ma<0. 2. Reduces the resolution of flow solutions….