JNTU B.Tech III HIGH SPEED AERODYNAMICS Previous Question Paper (February - 2022 - R18)

To derive the momentum equation for a compressible flow, we consider a control volume fixed in space through which a fluid flows. Applying Newton's second law to a differential fluid element of volume dV = dx dy dz, the rate of change of momentum is equal to the sum of forces acting on the element.

The momentum equation in the x-direction is derived by considering the surface forces (pressure and viscous stresses) and body forces (such as gravity). The net force in the x-direction is given by the sum of pressure gradients and viscous stress gradients:

Applying the Reynolds Transport Theorem or considering the control volume approach, the rate of change of momentum for the fluid element is expressed as:

Equating the inertial terms to the sum of forces per unit volume, we obtain the Navier-Stokes equation for compressible flow in the x-direction:

Where:

• ρ is the fluid density.
• u, v, w are velocity components.
• p is the static pressure.
• μ is the dynamic viscosity and λ is the second coefficient of viscosity.
• ∇·V represents the velocity divergence, accounting for compressibility.

Similar equations are derived for the y and z directions by cyclic permutation of the velocity components and coordinates. These equations, combined with the continuity and energy equations, form the complete set of governing equations for compressible viscous flow.

Compressible flow regimes are classified based on the Mach number (M), which represents the ratio of flow velocity to the local speed of sound. The regimes are defined as follows:

• Incompressible flow (M < 0.3): Density variations are negligible, and the fluid is treated as having constant density.
• Subsonic flow (0.3 < M < 0.8): Compressibility effects become significant, but no shock waves are present.
• Transonic flow (0.8 < M < 1.2): Mixed flow regions exist where both subsonic and supersonic flow occur, typically characterized by the presence of shock waves.
• Supersonic flow (1.2 < M < 5.0): The entire flow field is supersonic, and flow changes occur through oblique shock waves and expansion fans.
• Hypersonic flow (M > 5.0): High-temperature effects, such as chemical dissociation and ionization, become dominant.

To derive the speed of sound (a) in a perfect gas, consider a small pressure disturbance propagating through a stationary fluid in a tube. By applying the continuity and momentum equations across a control volume moving with the disturbance, we relate the pressure change (dp) to the density change (dρ). For an isentropic process, the speed of sound is defined as the square root of the derivative of pressure with respect to density at constant entropy:

For a perfect gas undergoing an isentropic process, the relationship between pressure and density is given by p = Cρ^γ, where γ is the ratio of specific heats. Differentiating this with respect to ρ gives ∂p/∂ρ = γ(p/ρ). Substituting the ideal gas law (p = ρRT) into this expression yields:

Here, γ is the ratio of specific heats (1.4 for air), R is the specific gas constant, and T is the absolute temperature. This equation demonstrates that the speed of sound in a perfect gas depends solely on its temperature.