JNTU B.Tech III HIGH SPEED AERODYNAMICS Previous Question Paper (October - 2020 - R16)

For a one-dimensional, steady, adiabatic flow of a calorically perfect gas, the energy equation is derived from the First Law of Thermodynamics applied to a control volume. Neglecting body forces and heat transfer, the steady-flow energy equation states that the stagnation enthalpy remains constant along a streamline:

Substituting the enthalpy for a perfect gas, h = c_pT, we obtain:

Dividing by c_pT, we express this in terms of the Mach number (M = u/a, where a = sqrt(γRT)):

To deduce the isentropic relations between stagnation (subscript 0) and sonic (subscript *) conditions, we set M = 1 at the sonic state. At this condition, T = T*, p = p*, and ρ = ρ*.

For temperature, substituting M = 1 into the energy equation gives:

Using the isentropic relation p/p₀ = (T/T₀)^γ/(γ-1), we find the pressure ratio:

Using the isentropic relation ρ/ρ₀ = (T/T₀)^1/(γ-1), we find the density ratio:

For a compressible, inviscid fluid flow, the momentum equation is derived by applying Newton's Second Law to a differential control volume of fluid. Considering a control volume of size dx, dy, dz, the net force acting on the fluid element is equal to the rate of change of momentum.

The forces acting on the element include pressure forces and body forces (such as gravity). Summing the forces in the x-direction:

Expanding the substantial derivative, the general vector form of the momentum equation (Euler's equation) is:

Where V is the velocity vector, p is pressure, ρ is density, and f represents body forces. In Cartesian coordinates, the x-component is:

The physical meaning of the terms is as follows:

• Local Acceleration (∂u/∂t): Represents the change in velocity at a fixed point in space over time.
• Convective Acceleration (u(∂u/∂x) + v(∂u/∂y) + w(∂u/∂z)): Represents the change in velocity due to the fluid particle moving from one position to another in a non-uniform velocity field.
• Pressure Gradient Term (-∂p/∂x): Represents the net force exerted by the surrounding fluid due to pressure variations, acting from high pressure to low pressure.
• Body Force Term (ρfₓ): Represents external forces acting on the bulk of the fluid, such as gravity or electromagnetic forces.

The left-hand side represents the inertial force per unit volume, while the right-hand side represents the sum of surface and body forces acting on the fluid element.