Jawaharlal Nehru Technological University Hyderabad B.Tech II LOW SPEED AERODYNAMICS Previous Question Paper (March - 2022 - R16)

The Navier-Stokes equations represent the conservation of momentum for a viscous, Newtonian fluid. Starting from Newton's Second Law, F = ma, applied to a fluid control volume, the momentum equation in Cartesian coordinates is derived by balancing inertial forces, pressure forces, body forces, and viscous forces.

For an incompressible flow, the momentum equation in the x-direction is:

Similar equations exist for the y and z directions. The terms represent:

• Inertial terms (Left hand side): Represent the local acceleration and convective acceleration of the fluid particle.
• Pressure gradient (-∂p/∂x): Represents the net force due to pressure differences acting on the fluid element.
• Viscous terms (μ∇²u): Represent the internal friction forces due to the fluid's viscosity, where μ is the dynamic viscosity.
• Body force (ρgₓ): Represents external forces like gravity acting on the fluid mass.

Buckingham's Pi Theorem states that if a physical process is governed by n variables involving m fundamental dimensions, it can be expressed as a relationship between n-m dimensionless groups (Pi terms).

In aerodynamics, the aerodynamic force F (lift or drag) typically depends on the following variables: fluid density (ρ), flow velocity (V), characteristic length (l), dynamic viscosity (μ), and speed of sound (a). Thus, F = f(ρ, V, l, μ, a). With n=6 variables and m=3 fundamental dimensions (Mass, Length, Time), we expect 6-3 = 3 dimensionless Pi terms.

Using the repeating variables ρ, V, and l, the resulting dimensionless groups are:

• Reynolds Number (Re = ρVl/μ): Represents the ratio of inertial forces to viscous forces, crucial for determining flow regime (laminar vs turbulent).
• Mach Number (M = V/a): Represents the ratio of inertial forces to elastic forces, determining compressibility effects.
• Force Coefficient (C = F / (0.5ρV²l²)): Represents the nondimensionalized aerodynamic force, such as the lift coefficient (C_L) or drag coefficient (C_D).

This theorem allows aerodynamicists to perform wind tunnel testing on scale models by ensuring these dimensionless parameters match those of the full-scale prototype.