a) A sports car travels at a speed of 250 miles/h on the ground.

(i) Discuss the numerical approaches that are available, e.g. DNS, LES, RANS, etc., to simulate this particular flow and how the turbulence is dealt with if these methods are used including their advantages and disadvantages.

(ii) Set up a mathematical model for the flow using the RANS approach with all the necessary equations, boundary conditions and properties of the fluid ready to be implemented in a computer code to simulate the flow. You need to fully discuss the assumptions you have made and the justifications of the approach, equations and models employed.

(iii) Discuss the turbulence model that you have chosen, including its background, advantages and limitations and make reference to the literature.

b) Figure 1 below is a sketch of a methane (CH₄) combustion furnace. The furnace is vertically-fired and has a square cross section of 0.3 m x 0.3 m. The height of the combustion chamber is 1.0 m. The fuel (methane) enters the furnace through a circular bluff body burner installed at the bottom. The diameter of the bluff body is 0.05 m, and the diameter of the fuel jet hole is 0.004m. Instead of using air, a mixture of oxygen and carbon dioxide is used for the combustion of the methane. The mass fractions of oxygen and carbon dioxide are 27% O₂ and 73% CO₂, respectively. The fuel jet velocity is 120 in/s, and the velocity of the mixture of oxygen and carbon dioxide is 40 m/s, both at a temperature of 300K. The furnace is operated at the ambient pressure. The furnace walls are refractory-lined and the exhaust is at the top of the furnace.

Provide a mathematical model using the RANS method which describes the reacting flow in the furnace. You may assume that the flame is steady, the chemical reaction is fast, and the radiative heat transfer can be ignored.

Your model should consider at least the following:

(i) In addition to giving ALL the governing fluid flow equations, discuss how turbulence, turbulence-chemical interactions, chemical reactions and species transport are dealt with. State and justify fully any assumptions you make.

(ii) A schematic diagram showing the locations of ALL boundaries. Then state and justify ALL necessary boundary conditions employed.

(iv) Discuss the main assumptions and limitations of the turbulence-chemical interaction model employed, and the RANS modelling method in general, for the particular reacting flow.