Transonic flow

Abstract

Transonic flow, as a special case of compressible flow past two and three dimensional objects is considered. First, local linearization approach for steady thin airfoil theory is introduced to obtain approximate analytical solutions. Then, unsteady analytical solutions are provided for simple harmonically heaving-plunging and/or pitching airfoils. Numerical solutions to full non-linear potential flow past shockless airfoils are also given. Furthermore, comparisons of the surface pressure distributions for the conventional and the supercritical airfoils are provided and the significance of the supercritical airfoils is discussed. The aerodynamic performance of supercritical airfoils at off-design conditions is also given. A general approach to the unsteady transonic aerodynamics is given in terms of low transonic, high transonic and low supersonic considerations. The differences among the three are discussed with emphasis on the real and imaginary parts of the surface pressure distributions. The ‘shock doublet’ concept and its significance for the low transonic case are also provided. The shock related transonic phenomena such as flutter, buffeting and aileron buzz are briefly mentioned. The viscous effects on the transonic aerodynamics are considered and the concept of ‘transonic dip’ and its effect on the onset of buffeting is provided. Characteristics for the enveloping curves of the separation points, critical Mach numbers, drag divergence and the onset of buffeting with respect to the lift vs. Mach number are presented. Steady transonic flow over thin wings is provided with its historical development including a pioneering computational work involving the Navier–Stokes solutions over Wing-C. Unsteady transonic considerations for the finite wings are provided for the representative wings for which the measurements are given by AGARD as data bases. Finally, wing-body interactions in terms of the wave drag is studied with introduction of the area rule.

Problems and Questions

• 6.1 Using the energy equation, obtain the linearized form of the relation between the perturbation potential and speed of sound, Eq. 6.1.

• 6.2 Based on the order of magnitude analysis, show that the time derivative of the perturbation potential is negligible compared to x derivative.

• 6.3 Obtain Eq. 6.11 from 6.10a,b by inverse Laplace transform using the convolution integral.

• 6.4 Show that at constant angle of attack the perturbation potential is given by Eq. 6.13.

• 6.5 Show that at constant angle of attack the surface pressure coefficient is given by Eq. 6.14.

• 6.6 Show that lower surface pressure is given by Eq. 6.15.

• 6.7 Show that the sectional lift coefficient depends on the angle of attack as expressed in Eq. 6.17.

• 6.8 Obtain the sectional moment coefficient and center of pressure using Eq. 6.16

• 6.9 Compare the surface pressure coefficient obtained with Dowell method using x ₀ = b as expansion point and e ₀ = 0.12 and f ₀ = 2.4/b with the pressure coefficient obtained using Stahara–Spreiter method for the 6% thick Guderly airfoil. Discuss the choice of x ₀ = b.

• 6.10 Solve Eq. 6.21 in Laplace domain, and obtain the expression 6.22 by inverse transform to give the boundary condition.

• 6.11 Obtain Eq. 6.23 from 6.24 by the limiting procedure as M _∞ → 1.

• 6.12 Show that for simple harmonically heaving plunging thin airfoil the surface pressure expression is given by Eq. 6.25 as the free stream Mach number approaches 1.

• 6.13 Using the values of Example 6.1, plot the phase lag of the surface pressure coefficient along the chord for a heaving plunging thin airfoil.

• 6.14 Find the amplitude of the (i) sectional lift coefficient and (ii) the sectional moment coefficient about the leading edge using the data given in Example 6.1.

• 6.15 Using Eq. 2.15 expressed for the velocity potential, obtain Eq. 6.26 for compressible steady flows.

• 6.16 What is a ‘shock doublet’ in unsteady transonic flow?

• 6.17 Discuss the ‘transonic dip’ phenomenon for the swept wing in a transonic unsteady flow.

• 6.18 What is the function of transonic dip in unsteady transonic flow?

• 6.19 Comment on the ‘area rule’ for the wing fuselage in transonic flow.