Non-frictional Source of Entropy Generation… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Rotational Inviscid Flows

Inviscid-compressible Flows Above equation shows that despite the inviscid flow assumption, it 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.

Governing Equations for Inviscid-compressible Flows

Gibbs form of First Law of Thermodynamics For an infinitesimal flow process For a differential displacement,

First Law of Thermodynamics for Inviscid compressible flow 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 For a differential displacement,

The expression in the parentheses on the left-hand side of above equation is the total enthalpy. In the absence of mechanical or thermal energy addition or rejection for an adiabatic flow; h total remains constant. Meaning that its gradient vanishes. Furthermore, for steady flow cases,

Above equation is an important result that establishes a direct relation between the vorticity and the entropy production in inviscid flows. A flow field generates discontinuities as a result of the presence of shock waves. These are responsible for large jumps in velocities. These jumps cause vorticity production and therefore, changes in entropy.

Discontinuities in High Speed Flows

High Speed Jet Engines

Discontinuities Ahead of inlet

Weakening of Discontinuity : Infinitesimal Strength & Reversible Discontinuity When High speed object is either a “point” or “thin rod”, the discontinuity in an invsicd flow, the discontinuity will weaken to an isentropic flow process. This discontinuity is called as Mach Wave.

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 ( )

The Thought Experiment

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.

Speed of sound in a Fluid Flow The speed of sound can be obtained easily

Moving Disturbance In A Fluid As an infinitesimal object moves through a fluid medium it creates pressure waves. Pressure waves travel out at the speed of sound which in term depends on nature of fluid. If the object is traveling significantly slower than sonic velocity, then pressure waves travel out uniformly similar to waves on the surface of a pond.

Moving Disturbance In A Fluid As the object approaches the speed of sound, it begins to catch up with the pressure waves and creates an infinitesimally weak flow discontinuity just ahead of the aircraft

Moving Disturbance In A Fluid As the vehicle breaks the speed of sound, the infinitesimally weak Shock waves begin to add up along a “Mach Line”.

Moving Disturbance In A Fluid As Mach number increases, the strength of the shock wave increases and the Angle of the shockwave becomes increasingly severe Credit: Selkirk College Professional Aviation Program

Mach Waves, Revisited A ‘’point-mass’’ object moving with Supersonic velocity Generates an infinitesimally weak “mach wave”. The direction of flow remains unchanged across Mach wave.

Irreversible Discontinuity in High Speed Inviscid Flows When generating object is larger than a “point”, shockwave is stronger than mach wave …. Oblique shock wave  -- shock angle  -- turning or “wedge angle”

High Angle Objects  max  max

Descent of A Spacecraft

Landing of A Space Craft