Prandtl-Meyer Function

How is the Prandtl-Meyer function introduced in relation to an oblique shock, as described in Chapter 15?

What is the definition of the Mach line angle, µ, in a field of supersonic flow?

In the alternative approach to deriving the Prandtl-Meyer function using cylindrical coordinates, what remarkable conclusion is reached about the tangential velocity, Uθ?

How is the change in deflection angle (from ν₁ to ν₂) calculated using the Prandtl-Meyer function for a change in Mach number from M₁ to M₂?

What occurs physically when the deflection angle of a supersonic flow exceeds the maximum possible turning angle predicted by the Prandtl-Meyer function?

In Example 15.1, air with a temperature of 20 degrees Celsius and speed of 450 m/sec flows over a bend. What is the initial Mach number of the flow, assuming k=1.4 and R=287 J/KgK?

Based on the table in Example 15.1 for an initial Mach number of 1.31, what is the value of the Prandtl-Meyer function, ν?

In Example 15.1, the flow with an initial Prandtl-Meyer angle of 6.4449 degrees undergoes an expansion through a 20.0-degree inclination. What is the new angle, ν₂, after the bend?

From the results table in Example 15.1, what is the final Mach number corresponding to the new Prandtl-Meyer angle of 26.4449 degrees?

In the context of the supersonic d'Alembert's Paradox discussed in Section 15.5, what is the formula for the drag, D, on the two-dimensional diamond-shape body?

How is the maximum turning angle, ν_max, for a Prandtl-Meyer expansion theoretically obtained?

According to Equation (15.36), the maximum of the deflection point and the maximum turning point for a Prandtl-Meyer expansion are a function of what single property?

In Example 15.1, what is the calculated fan angle for the 20-degree expansion?

For the flat body with an angle of attack shown in Figure 15.8, what conditions are mentioned for the slip condition?

In the reverse problem of Example 15.2, gas with k=1.67 and an initial Mach number of 1.4 expands until the pressure is P₂ = 1.0 bar from an initial pressure of P₁ = 1.2 bar. What is the calculated bend angle, Δν?

What is the key assumption made in the 'rigorous model' presented in Section 15.2.1 (Alternative Approach) that simplifies the governing equations?

In Example 15.3, a flat plate is at a 4-degree angle of attack in a Mach 3.3 flow with k=1.3. What is the value of the Prandtl-Meyer function, ν, on the expansion side before the 4-degree turn?

In Example 15.3, with an initial state of M=3.3 and k=1.3, what is the pressure ratio P₃/P₁ on the expansion side of the flat plate after the 4-degree turn?

According to the final result of the alternative derivation, what is the formula for the Mach number M in the turning area as a function of the angle θ and specific heat ratio k?

What happens to the total Mach number during a Prandtl-Meyer expansion?

According to the discussion in Section 15.3, why is the maximum turning angle for a Prandtl-Meyer expansion much larger than the maximum deflection point for an oblique shock?

In Example 15.1, the width of a two-dimensional tunnel is initially 0.1 m. What is the calculated width of the tunnel, x₂, after the 20-degree expansion bend?

What physical quantity does the variable ν represent in the Prandtl-Meyer equations?

In the derivation of the drag on a diamond airfoil (Section 15.5), the force is calculated based on the pressures on the front and back surfaces. Why is there a pressure difference between P₂ (after the front shock) and P₄ (after the rear expansion)?

When does the Prandtl-Meyer deflection angle v have to match the definition of the angle that is chosen where θ = 0 when M = 1?

In the geometrical explanation in Section 15.2, what are the 'typical simplifications for geometrical functions' that are used?

What is the relationship between the radial velocity (Ur) and the stagnation enthalpy (h₀) as derived in Section 15.2.1?

In Example 15.3, what is the approximate value of the drag coefficient, d_d, for the flat plate at a 4-degree angle of attack?

According to the energy equation for a perfect gas with constant specific heat, k, used in Section 15.2.1, what is the expression for enthalpy, h(θ)?

What does the text suggest is always true about drag when a body is in a supersonic flow, as concluded in Section 15.6?

In Example 15.3, the final conclusion about the lift and drag coefficients is that:

What is the physical interpretation of the constant of integration in the Prandtl-Meyer function v(M) as described in Section 15.2.1?

In the derivation from Section 15.2.1, after establishing that Uθ = c and relating velocities, the analysis arrives at Equation (15.27). What does this equation represent?

What is the reverse function for the turning angle θ as a function of Mach number, as given by Equation (15.32)?

In the geometrical explanation (Section 15.2), the change in velocity in the flow direction, dx, for an infinitesimal turn is given as:

The analysis of a flat thin plate at an angle of attack (Section 15.6) is an example of combining which two supersonic flow phenomena?

What is the Mach number for a gas with k = 1.3 after it has undergone a Prandtl-Meyer expansion of 66.3100 degrees from a sonic state?

According to Figure 15.6, which shows the Prandtl-Meyer angle as a function of Mach number for k=1.4, what happens to the rate of change of the angle as the Mach number increases significantly?

Why is it stated in Section 15.5 that there is a paradox (d'Alembert's Paradox) for ideal inviscid incompressible flow but not for supersonic flow?

What is the pressure ratio P/P₀ for a Mach number of 1.4 and k=1.67, as given in the table for Example 15.2?

In the context of the alternative derivation, what is the expression for the ratio dU/U, where U is the total velocity?

What does the text say about a 'detachment' point in the context of the Prandtl-Meyer model?

In the expression dv = sqrt(M²-1)dM² / (2M²(1 + ((k-1)/2)M²)), what does the term 'dv' represent?

According to the final equation for the lift coefficient d_L in Example 15.3, what is its approximate value?

What recommendation does the author make regarding the use of the Prandtl-Meyer function near a sharp turning point due to boundary layer effects?

In the combination of an oblique shock and isentropic expansion, such as a flat plate at an angle of attack, the flow properties are determined on the upper (expansion) surface using the Prandtl-Meyer function and on the lower (compression) surface using what?

What is the key difference between the simplified geometrical model and the rigorous model for deriving the Prandtl-Meyer function, as mentioned in Section 15.2.2?

In the geometrical derivation of the Prandtl-Meyer function, the relationship tan µ = dU / (Udv) is used. What does this physically represent?

The total energy equation on a streamline is given as h(θ) + (Ur² + Uθ²)/2 = h₀. What does h₀ represent?

For the flat plate in Example 15.3, why is a weak shock assumed for the oblique shock calculation on the lower surface?