Entropy Generators in Simplest Flow P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Identification of the Fourth Irreversibility…

Induction & Deduction of Concepts

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

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

The Concrete Part With a vorticity 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.

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. 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

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. • T0 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.

What was Stagnation Temperature At Columbia Breakup Loss Of Signal at: 61.2 km altitude ~18.0 Mach Number T∞ ~ 243 K

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

The Ideal Simple Flow Reversible flow : No vorticity generation between x & y, An isentropic flow…. It was not till Daniel Bernoulli published his celebrated treatise in the eighteenth century that the science of fluid Flow really began to get off the ground – G.I. Taylor

Bernoulli equation The Bernoulli equation can be obtained as a scalar product of the Euler differential equation and a differential displacement vector. For steady flow cases, the differential distance along the particle path is identical with a distance along a streamline. Thus, the multiplication of Euler equation with the differential displacement , gives:

The terms in above equation must be rearranged as follows. The first term starts with a scalar product of two vectors, eliminating the Kronecker delta and utilizing the Einstein summation convention results in: For Steady flow this will be zero. The temporal variation of kinetic entry is zero.

Integrating above Equation results in:

Along a Stream line For incompressible flows which is the Bernoulli equation.

The Greatest Natural Flow

Barotropic Atmosphere The term barotropic is derived from the Greek baro, relating to pressure, and tropic, changing in a specific manner: that is, in such a way that surfaces of constant pressure are coincident with surfaces of constant temperature or density. A barotropic atmosphere is one in which the density depends only on the pressure, ρ = ρ(p). The isobaric surfaces are also surfaces of constant density. For an ideal gas, the isobaric surfaces will also be isothermal if the atmosphere is barotropic. Along a isobar : The geostrophic wind is independent of height in a barotropic atmosphere.

Geostrophic Wind Patterns Geostrophic wind is defined a horizontal velocity field,

Prediction of Simple Wind Patterns The knowledge of the pressure distribution at any time determines the geostrophic wind. This is true for large-scale motions away from the equator. The geostrophic wind becomes the total horizontal velocity to within 10–15% in mid latitudes.