a) Explain, with the aid of sketches as appropriate, the term “Reynolds decomposition” in the context of turbulent flows.

b) Show how energy transfer by turbulent fluctuations normal to the flow can lead to the following expression for convective heat transfer in the fluid.

𝑞 = −𝑐P𝜌(𝑎 + 𝜀𝐻) dT/dy

Where q is the heat flux, cp is the specific heat capacity,  is the density,  is the ̅̅̅̅̅̅̅̅

molecular diffusivity, H is the eddy diffusivity of heat (= (𝑇′𝑣′)), T is the temperature and y is the coordinate normal to the flow.

c) A Laval (converging-diverging) nozzle, of circular cross section, fed from a large high pressure reservoir, is used to accelerate a flow such that the Mach number in a test section downstream of the nozzle is 3.2.

i) If the diameter of the throat of the nozzle is 25 mm, calculate the cross-sectional area of the test section.

ii) If the air pressure and density in the test section are 100 kPa and 1.12 kg/m3, respectively, calculate the actual velocity in the test section.

Assume the air behaves as an ideal gas and take, Specific isobaric heat capacity of air = 1.005 kJ/kg K Specific isochoric heat capacity of air = 0.718 kJ/kg K Specific ideal gas constant of air = 0.287 kJ/kg K