Great Innovations are possible through General Understanding …. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Thermodynamic View of Navier-Stokes Equations

The Navier-Stokes Equations : : Thermodynamically Irreversible Flows The momentum equation for a general linear (newtonian) viscous fluid is now obtained by substituting the stress relations, into Newton's law. In vector form, we obtain In thermodynamics view, these are inherently irreversible flows. Friction & viscosity are basic culprits of irreversibility. What if the viscosity is negligible?

Euler Equation of Motion For the special case of inviscid flow (no viscosity), Replacing the convective by the following vector identity Identify that the convective acceleration is expressed in terms of the gradient of the kinetic energy

Simplification of Body Forces First rearrange the gravitational acceleration vector by introducing a scalar surface potential z. The gradient of z has the same direction as the unit vector in – x 3 direction. Furthermore, it has only one component that points in the negative x 3 -direction. As a result, we may write.

Thus, the Euler equation of motion assumes the following form: 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. General Compressible Euler Equation of Motion

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

Vector Calculus of First Law 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 vector equation:

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,

Entropy Generation in Inviscid 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 can 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 ( )

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 x y z Main Flow

The Thought Experiment : Well behaving flow

Conservation Laws Applied to 1 D Steady disturbance Conservation of Mass: c-u p,  C P+dp,  d  Conservation of Mass for 1DSF: Change is final -initial

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 , 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 Medium c-u p,  C P+dp,  d 

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, g namely