2. Consider a large industrial fan where the input power, W, depends on the fan impeller diameter, D, fluid viscosity, u, fluid density, p, volumetric flow rate, Q, and blade rotational rate, w. a) What are the appropriate dimensionless groups (Pi terms) if we consider D, p, and Q as repeating (reference) variables. b) If a model fan has a diameter, Dm, that is half of the prototype fan diameter, D, , what should be the ratio of the model blade rotation rate, Wm , to prototype, wp, to have dynamic similitude. Assume the same fluid (air) properties; that is viscosity and density are the same in both cases.

Fluid Mechanics Problem   Note: Fluids are assumed to be at 20oC.

5. Consider a Pitot probe in high-speed flow.(c) You are flying through air at static p∞ = 0.8 atm [abs]. Your velocimeter broke,but you are still receiving the following direct stagnation pressure readings (atm[abs]) from the Pitot probe mounted on your aircraft’s noise: i) 3.152 ii) 1.357 iii) 38.428 iv) 0.8 Find the freestream Mach number corresponding to each of the pressure readings. (d) If we are to believe reading (iv), can we step out of the aircraft safely? Why?

% ParametersD = 0.1; % Diameter of the tube (m)L = 1.0; % Length of the tube bundle (m)N = 8; % Number of tubes in the bundleU = 1.0; % Inlet velocity (m/s)rho = 1.2; % Density of the fluid (kg/m^3)mu = 0.01; % Dynamic viscosity of the fluid (Pa.s) % Define the grid size and time stepdx = D/10; % Spatial step size (m)dy = L/10; % Spatial step size (m)dt = 0.01; % Time step size (s) % Calculate the number of grid points in each directionnx = ceil(D/dx) + 1;ny = ceil(L/dy) + 1; % Create the velocity matrixU_matrix = U * ones(nx, ny); % Perform the iterationsfor iter = 1:100    % Calculate the velocity gradients    dUdx = (U_matrix(:, 2:end) - U_matrix(:, 1:end-1)) / dx;    dUdy = (U_matrix(2:end, :) - U_matrix(1:end-1, :)) / dy;        % Calculate the pressure gradients    dpdx = -mu * dUdx;    dpdy = -mu * dUdy;        % Calculate the change in velocity    dU = dt * (dpdx / rho);        % Update the velocity matrix    U_matrix(:, 2:end-1) = U_matrix(:, 2:end-1) + dU;        % Apply…