JNTU B.Tech II AERODYNAMICS - I Previous Question Paper (September - 2021 - R18)

To derive the momentum equation for 2-D inviscid flow, consider an infinitesimal fluid element of dimensions dx and dy in the x-y plane. For an inviscid fluid, the only forces acting on the element are pressure forces and body forces (such as gravity). According to Newton's second law, the net force acting on the fluid element in a given direction equals the rate of change of momentum in that direction.

Applying the momentum principle in the x-direction:

Where u is the velocity component in the x-direction, p is the pressure, ρ is the fluid density, and f_x is the body force per unit mass. Expanding the material derivative du/dt for a 2-D flow, we obtain:

Similarly, applying the momentum principle in the y-direction for the velocity component v:

These are the Euler equations of motion for a 2-D inviscid flow. The derivation relies on the assumption that viscosity is zero, meaning there are no shear stresses acting on the faces of the fluid element. The terms on the left represent the convective and local acceleration of the fluid particle, while the terms on the right represent the pressure gradient force and the external body force acting on the control volume.

In low-speed aerodynamics, flows are typically assumed to be incompressible, inviscid, and adiabatic. The concept of entropy serves as a fundamental thermodynamic indicator of flow reversibility. For an adiabatic, inviscid flow (Euler flow), the entropy of a fluid particle remains constant along a streamline, satisfying the condition of isentropic flow. In low-speed regimes, where Mach numbers are significantly below 0.3, density variations are negligible, and the flow behaves as if it were incompressible. Entropy helps define the thermodynamic state of the gas; since low-speed flows involve minimal pressure changes that do not cause significant temperature gradients or heat transfer, the entropy change is effectively zero, justifying the use of Bernoulli’s equation and the assumption of constant density.

The International Standard Atmosphere (ISA) is a mathematical model that defines the vertical distribution of pressure, temperature, and density in the Earth's atmosphere, providing a common reference for aircraft performance and aerodynamic calculations. It assumes the air behaves as a perfect gas and is in hydrostatic equilibrium.

The model is divided into layers based on the temperature lapse rate:

1. Troposphere (0 to 11,000 m): The temperature decreases linearly with altitude at a lapse rate of 6.5 K/km. The temperature at sea level is defined as 288.15 K.
2. Stratosphere (11,000 to 20,000 m): The temperature remains constant at 216.65 K, creating an isothermal layer.

Mathematically, the model relies on the hydrostatic equation:

Combined with the ideal gas law (p = ρRT), this allows for the derivation of pressure and density ratios as functions of altitude. In the troposphere, where temperature T varies as T = T₀ - αh, the pressure ratio is expressed as:

where α is the lapse rate, g is gravitational acceleration, and R is the specific gas constant for air. This model provides the standard values for sea-level pressure (101,325 Pa) and density (1.225 kg/m³), which are essential for normalizing aerodynamic coefficients and comparing flight test data across different environmental conditions.