# Computational methods

Computational methods for compressible flows, particularly transonic flow over wings, have been the subject of a very considerable research effort over the past three decades. Substantial progress has been made, although much still remains to be done. A discussion of these methods is beyond the scope of the present book, save to note that for the linearized compressible potential flow Eqn (6.118) panel methods (see Sections 3.5, 4.10 and 5.8) have been developed for both subsonic and supersonic flow. These can be used to obtain approximate numerical solutions in cases with exceedingly complex geometries. A review of the computational methods developed for the full inviscid and viscous equations of motion is given by Jameson.

Exercises

1 A convergent-divergent duct has a maximum diameter of 150 mm and a pitot – static tube is placed in the throat of the duct. Neglecting the effect of the pitot-static tube on the flow, estimate the throat diameter under the following conditions:

(i) air at the maximum section is of standard pressure and density, pressure differ­ence across the pitot-static tube =127 mm water;

(ii) pressure and temperature in the maximum section are 101 300 N m-2 and 100 °С respectively, pressure difference across pitot-static tube = 127 mm mercury.

(Answer: (i) 123 mm; (ii) 66.5 mm)

2 In the wing-flow method of transonic research an aeroplane dives at a Mach

number of 0.87 at a height where the pressure and temperature are 46 500 N m-2 and -24.6 °С respectively. At the position of the model the pressure coefficient is —0.5. Calculate the speed, Mach number, 0.7p M2, and the kinematic viscosity of the flow past the model. _

(Answer: 344m s"1; M = 1.133; 0.1 pM2 = 30 800N nr2; v = 2.64 x 10-3m2s )

3 What would be the indicated air speed and the true air speed of the aeroplane in Exercise 2, assuming the air-speed indicator to be calibrated on the assumption of incompressible flow in standard conditions, and to have no instrument errors?

(Answer: TAS = 274m s"1; IAS = 219m s"1)

4 On the basis of Bernoulli’s equation, discuss the assumption that the compressi­bility of air may be neglected for low subsonic speeds.

A symmetric aerofoil at zero lift has a maximum velocity which is 10% greater than die free-stream velocity. This maximum increases at the rate of 7% of the free – stream velocity for each degree of incidence. What is the free-stream velocity at which compressibility effects begins to become important (i. e. the error in pressure coefficient exceeds 2%) on the aerofoil surface when the incidence is 5°?

(Answer: Approximately 70 m s_1) (U of L)

5 A closed-return type of wind-tunnel of large contraction ratio has air at standard conditions of temperature and pressure in the settling chamber upstream of the contraction to the working section. Assuming isentropic compressible flow in the tunnel estimate the speed in the working section where the Mach number is 0.75. Take the ratio of specific heats for air as 7 = 1.4. (Answer: 242 m s_1) (U of L)