GD Midterm 1 Review - Section 4 Flashcards
If u > 0, then cp__ 0
< – Pressure drops with acceleration
If u < 0, then cp ___ 0
> – Pressure rises with deceleration
What assumptions are made about the velocity perturbations (u, v, w) relative to the freestream velocity (U∞) that allow the gas dynamic equations to be linearized?
It is assumed that the perturbation velocities are small compared to U∞.
This permits neglecting products of perturbation velocities (and products with their derivatives), i.e. terms like (u/U∞)² and (u/U∞)uₓ are considered negligible.
How are the total velocity components expressed in terms of the freestream velocity and the perturbation velocities?
ũ = U∞ + u,
ṽ = v,
w̃ = w
How do the streamlines and pressure distribution behave for subsonic flow over a wavy wall?
What is the significance of the phase relationship between the wall contour and the pressure coefficient?
In subsonic flow, the streamlines are harmonic and remain in phase with the wall contour, while their amplitude decays exponentially with distance from the wall.
The pressure distribution, however, is 180° out-of-phase with the wall—high speed (at the peaks) corresponds to low pressure and vice versa—which leads to no net drag (d’Alembert’s paradox).
What is d’Alembert’s paradox in the context of subsonic flow over a wavy wall, and why does it arise in the linearized analysis?
d’Alembert’s paradox refers to the prediction that the net drag on a body in an inviscid, incompressible (or linearly approximated subsonic) flow is zero because the pressure forces on the front and back of the body exactly cancel. This arises due to the symmetric, harmonic pressure distribution obtained in the linearized analysis.
Why does the governing potential equation become hyperbolic for supersonic flows, and what are the implications for the solution (e.g., propagation of disturbances)?
For supersonic flow, (1 − M∞²) is negative, so the governing equation changes character to a hyperbolic type (similar to the wave equation). This means that disturbances are propagated along characteristic lines (Mach lines) and affect only a limited region downstream of the wall.
What are Mach waves in supersonic flow, how is the Mach angle defined, and how do they relate to the generation of wave drag over a wavy wall?
Mach waves are the characteristic lines along which disturbances propagate in supersonic flow. The Mach angle, μ∞, is defined by sin μ∞ = 1/M∞. In the linearized supersonic solution, properties remain constant along these lines. Because the pressure distribution is no longer symmetric (with a 90° phase lead relative to the wall), a net drag force—known as wave drag—develops.
What are Mach lines in the context of supersonic flow over a wavy wall, and how are they mathematically defined for the f-wave solution?
Mach lines are the characteristic lines along which disturbances propagate in a supersonic flow. For the f-wave solution, they are defined by the equation:
x - d_{\infty} y = constant
d_{\infty} = 1 / \sqrt{M_\infty^2 - 1}
What is the difference between the f-wave and g-wave solutions for Mach lines in supersonic flow, and which one is physically acceptable for flow from left to right?
The f-wave solution has the form
𝑥−𝑑∞𝑦=constant
indicating disturbances propagate downstream (left-running Mach lines). The g-wave solution is given by
𝑥 + 𝑑∞𝑦 = constant
which corresponds to upstream propagation.
For flow from left to right in supersonic conditions, only the f-wave solution is physically valid, because disturbances cannot propagate upstream.
How do Mach lines influence the pressure distribution on a wavy wall in supersonic flow?
In supersonic flow, the pressure, velocity, and potential remain constant along the Mach lines. These characteristic lines transmit disturbances without damping, resulting in a pressure distribution that is shifted (typically leading the wall by 90 degrees ) relative to the wall contour. This asymmetry in pressure leads to a net wave drag on the body.
What is the phase difference between the cp between supersonic and subsonic flow?
In subsonic flow, the pressure coefficient cp is 180° out-of-phase with the wall contour, meaning that when the wall is at a peak, is at a trough (and vice versa). In contrast, for supersonic flow the cp distribution leads the wall contour by 90°, so the maximum (or minimum) pressure occurs 90° ahead of the wall’s peak (or trough) .This phase difference is crucial because it results in no net drag (d’Alembert’s paradox) for subsonic flow, while the 90° phase lead in supersonic flow produces an asymmetry that generates wave drag.
