Jawaharlal Nehru Technological University Hyderabad B.Tech II LOW SPEED AERODYNAMICS Previous Question Paper (July/August - 2021 - R16)

To determine the functional dependence of the aerodynamic force F, we identify the relevant physical variables: force F, characteristic length l, fluid density ρ, velocity V, dynamic viscosity μ, and speed of sound a. There are n = 6 variables and m = 3 fundamental dimensions (Mass [M], Length [L], Time [T]). According to Buckingham's pi theorem, there are n - m = 3 dimensionless pi groups.

Choosing ρ, V, and l as repeating variables, we form the groups:

The functional relationship is expressed as f(π₁, π₂, π₃) = 0, which can be rewritten as π₁ = φ(π₂, π₃). Substituting the groups, we get:

This confirms that the aerodynamic force coefficient is a function of the Reynolds number and the Mach number.

Euler’s equation is derived from the Navier-Stokes equations by assuming the fluid is inviscid (viscosity μ = 0). When viscosity is present, it fundamentally alters the flow physics in the following ways:

• Boundary Layer Formation: Viscosity enforces the no-slip condition at solid surfaces, creating a thin boundary layer where velocity gradients are high. Euler's equation fails to predict this, as it assumes slip at the wall.
• Skin Friction Drag: Viscosity leads to shear stresses τ = μ(du/dy). Euler's equation neglects these stresses and therefore cannot account for skin friction drag.
• Flow Separation: Viscous effects cause the flow to lose kinetic energy near the wall in the presence of an adverse pressure gradient, leading to boundary layer separation. Euler's equation predicts attached flow even when physical flow is separated.
• Energy Dissipation: Viscosity converts mechanical energy into internal energy (heat) through internal friction. Euler's equation assumes isentropic flow, failing to capture the entropy production associated with viscous dissipation.