JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year HIGH SPEED AERODYNAMICS Previous Question Paper (March - 2021 - R18)

The Mach number (M) is a dimensionless quantity defined as the ratio of the local flow velocity (v) to the local speed of sound (a) in the medium:

To derive the integral form of the energy equation for 3-D inviscid compressible flow, consider a fixed control volume (CV) with a control surface (CS). The first law of thermodynamics states that the rate of change of total energy within the CV plus the net rate of energy flux out of the CS equals the rate of heat addition plus the rate of work done on the system.

The total energy per unit mass (e) is defined as the sum of internal energy, kinetic energy, and potential energy:

The integral form is expressed as:

For an inviscid flow, the work done by surface forces is restricted to pressure work. Since the flow is inviscid, there are no viscous shear stresses, and the energy equation accounts for the transport of enthalpy (h = u + p/ρ) across the control surface. The final integral form is:

Where ρ is density, v is the velocity vector, n is the outward unit normal vector, Q̇ is the rate of heat transfer, and Ẇ_shaft is the rate of shaft work done on the fluid.

A normal shock wave is a thin region of extreme gradients where flow properties change abruptly. To derive the pressure ratio, we apply the conservation laws for mass, momentum, and energy across a control volume enclosing the shock, assuming adiabatic, one-dimensional, steady flow of a calorically perfect gas.

The governing equations are:

Using the equation of state p = ρRT and the definition of the Mach number M = u/a, where a = sqrt(γRT), we relate the properties across the shock. By substituting the continuity and energy equations into the momentum equation and utilizing the normal shock relations for Mach number, we arrive at the pressure ratio expression:

This expression demonstrates that the pressure ratio is a function only of the upstream Mach number M₁ and the ratio of specific heats γ. For M₁ = 1, p₂/p₁ = 1, representing a Mach wave. As M₁ increases, the pressure ratio increases significantly, indicating a strong compression process.

The importance of this expression lies in its role in aerodynamic design:

• Performance Prediction: It allows engineers to calculate the sudden pressure rise and subsequent drag increase experienced by supersonic aircraft and engine intakes.
• Shock-Boundary Layer Interaction: The high pressure gradient across the shock can cause boundary layer separation, which is critical for analyzing stall and control surface effectiveness.
• Intake Design: It is essential for designing supersonic inlets (e.g., diffusers) where shocks are used to decelerate flow efficiently while minimizing total pressure loss.
• Structural Loading: The abrupt pressure change imposes significant structural loads on airframes, requiring precise calculation to ensure vehicle integrity.