Jawaharlal Nehru Technological University Hyderabad B.Tech II AERODYNAMICS - I Previous Question Paper (July/August - 2021 - R15)

For geometrically similar aerodynamic systems, the aerodynamic force F acting on a body depends on the fluid density ρ, flow velocity V, characteristic length L, fluid viscosity μ, and the speed of sound a. Mathematically, this is expressed as: F = f(ρ, V, L, μ, a).

Applying the Buckingham Pi Theorem, we identify the number of variables (n=6) and the number of fundamental dimensions (m=3: mass, length, time). This results in n-m = 3 non-dimensional groups. Choosing ρ, V, and L as repeating variables, we derive the following dimensionless parameters:

The force coefficient C_F (representing either C_L or C_D) is therefore a function of the Reynolds number and Mach number: C_F = f(Re, M). Since the systems are geometrically similar, the shape and orientation (angle of attack) are identical, meaning the functional form f remains constant. Thus, for any two geometrically similar bodies, the lift and drag coefficients are uniquely determined by the similarity parameters Re and M, ensuring dynamic similarity when these values are matched.

For a viscous incompressible flow, the momentum equation is derived from Newton's Second Law applied to a fluid element, resulting in the Navier-Stokes equations. Considering a control volume of fluid with density ρ, velocity components u, v, w, and pressure p, the forces acting on the element include pressure forces, viscous forces, and body forces (such as gravity).

The momentum balance in the x-direction is given by:

Where μ is the dynamic viscosity and ∇² is the Laplacian operator. For an incompressible flow, the continuity equation ∇ · V = 0 must be satisfied. The derivation follows these steps:

• Apply Newton's Second Law (F = ma) to a differential fluid element of volume dx dy dz.
• Express the acceleration as the material derivative of the velocity vector: DV/Dt = ∂V/∂t + (V · ∇)V.
• Account for surface forces using the stress tensor σ_ij, which for a Newtonian fluid is defined as σ_ij = -pδ_ij + τ_ij, where τ_ij = μ(∂u_i/∂x_j + ∂u_j/∂x_i).
• Substitute the stress tensor into the momentum conservation equation: ρ DV/Dt = ∇ · σ + ρg.
• Simplify using the incompressibility condition (∇ · V = 0), which reduces the viscous term ∇ · τ to μ∇²V.

The final vector form of the Navier-Stokes equation for incompressible flow is: