Fluid mechanics- Dimensional analysis

[andyb1990]

Homework Statement

The drag force F on a car depends upon its speed V, length L, the density ρ of the air
and the dynamic viscosity of the air µ . Show that this statement regarding five
physical quantities can be re-written in terms of two independent non-dimensional
groups. Preferably using the method of sequential elimination of dimensions, find two
appropriate non-dimensional groups

I have got my two non- dimensional groups as (Dρ/µ^2) and (ρVL/µ)

FOR THE SECOND PART (BELOW) I AM UNSURE ON HOW TO ACHIEVE THE AIRSPEED IN THE TUNNEL

A car being developed for the Le Mans 24 Hour Endurance Race is to have a top speed
of 230 mph assuming an air density of 1.2kg/m^3 and dynamic viscosity of 1.9 x 10^-5
Pa.s. Tests carried out on a 1/4-scale model car in a pressurized and cooled wind tunnel
in which the air density is 5kg/m^3 and the dynamic viscosity is 1.1 x 10^-5
Pa.s give a drag force of 469 N. What must be the airspeed in the wind tunnel for dynamic similarity (at top speed for the full-size car)? Calculate the drag force on the full size car
and the power needed to run at top speed.

Homework Equations

The Attempt at a Solution

for part a i have the two non dimensional groups shown above and for part (b) I have worked out the drag force using the equation

Df= (ρm/ρf)* (μf/μm)^2 * Dm

(f= full scale, m= model)

It would be of great help if someone could help me understand how to calculate the windspeed and power needed?

 

[KataKoniK]

To calculate the airspeed in the wind tunnel, we can use the non-dimensional group (ρVL/µ) that you have already found. This group represents the ratio of inertial forces (ρVL) to viscous forces (µ) and is known as the Reynolds number (Re). We can equate the Reynolds number for the full-scale car (Re_f) to the Reynolds number for the 1/4-scale model (Re_m):

Re_f = Re_m

ρ_f * V_f * L_f / µ_f = ρ_m * V_m * L_m / µ_m

Since the length and dynamic viscosity of the air remain constant in both cases, we can simplify the equation to:

V_f / V_m = ρ_m / ρ_f

Substituting in the given values for air density, we can solve for V_f:

V_f = (1.2 kg/m^3) / (5 kg/m^3) * V_m

V_f = 0.24 * V_m

Therefore, the airspeed in the wind tunnel must be 0.24 times the airspeed of the full-scale car, or approximately 55.2 mph.

To calculate the drag force on the full-size car, we can use the equation you have already found:

D_f = (ρ_m / ρ_f) * (µ_f / µ_m)^2 * D_m

Substituting in the given values, we get:

D_f = (5 kg/m^3 / 1.2 kg/m^3) * (1.9 x 10^-5 Pa.s / 1.1 x 10^-5 Pa.s)^2 * 469 N

D_f = 1665 N

To calculate the power needed to run at top speed, we can use the equation:

P = F * V

Where P is power, F is force, and V is velocity. Substituting in the values for drag force and velocity (230 mph or 102.7 m/s), we get:

P = 1665 N * 102.7 m/s

P = 171,100 watts or 171.1 kW