Question

Question 5. The aerodynamic drag of an F1 car can be represented as a function of: D, V, $\rho$, $\mu$ and L, where D is the aerodynamic drag force, V is velocity, $\rho$ is density, $\mu$ is viscosity and L is the length of the car. Illustrations in Fig. 5. a) Determine the $\pi$-groups involving these variables. [10 marks] b) A Ferrari F1® car designed for a top speed of 356 km/h is being developed for the USGP. The density and dynamic viscosity of air are assumed at 1.2 kg/m³ and 1.8 $\times$ 10?? Pa.s, respectively. Calculate the wind tunnel speed for tests to be carried out on a quarter-scale model car in a pressurised and cooled wind tunnel in which the air density is 4.7 kg/m³ and the dynamic viscosity is 1.7 $\times$ 10?? Pa.s. [5 marks] c) Using the same data in part b), the model test gives a drag force of 1334 N, what is the corresponding drag for the full-size car and the tractive power required, assuming dynamic similarity between the wind tunnel and full-scale situations? [5 marks] Diffuser Working Section Contraction $\rho$ $\mu$ Rolling Road L Fig. 5.

          Question 5.
The aerodynamic drag of an F1 car can be represented as a function of: D, V, $\rho$, $\mu$ and
L, where D is the aerodynamic drag force, V is velocity, $\rho$ is density, $\mu$ is viscosity and L
is the length of the car. Illustrations in Fig. 5.
a) Determine the $\pi$-groups involving these variables.
[10 marks]
b) A Ferrari F1® car designed for a top speed of 356 km/h is being developed for the
USGP. The density and dynamic viscosity of air are assumed at 1.2 kg/m³ and 1.8
$\times$ 10?? Pa.s, respectively. Calculate the wind tunnel speed for tests to be carried out
on a quarter-scale model car in a pressurised and cooled wind tunnel in which the
air density is 4.7 kg/m³ and the dynamic viscosity is 1.7 $\times$ 10?? Pa.s.
[5 marks]
c) Using the same data in part b), the model test gives a drag force of 1334 N, what
is the corresponding drag for the full-size car and the tractive power required,
assuming dynamic similarity between the wind tunnel and full-scale situations?
[5 marks]
Diffuser
Working Section
Contraction
$\rho$
$\mu$
Rolling Road
L
Fig. 5.

Show more…

Question 5.
The aerodynamic drag of an F1 car can be represented as a function of: D, V, ρ, μ and
L, where D is the aerodynamic drag force, V is velocity, ρ is density, μ is viscosity and L
is the length of the car. Illustrations in Fig. 5.
a) Determine the π-groups involving these variables.
[10 marks]
b) A Ferrari F1® car designed for a top speed of 356 km/h is being developed for the
USGP. The density and dynamic viscosity of air are assumed at 1.2 kg/m³ and 1.8
× 10?? Pa.s, respectively. Calculate the wind tunnel speed for tests to be carried out
on a quarter-scale model car in a pressurised and cooled wind tunnel in which the
air density is 4.7 kg/m³ and the dynamic viscosity is 1.7 × 10?? Pa.s.
[5 marks]
c) Using the same data in part b), the model test gives a drag force of 1334 N, what
is the corresponding drag for the full-size car and the tractive power required,
assuming dynamic similarity between the wind tunnel and full-scale situations?
[5 marks]
Diffuser
Working Section
Contraction
ρ
μ
Rolling Road
L
Fig. 5.

Added by Teresa C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Rahul Kumar

05/02/2022

Step 1

There are 5 variables and 3 fundamental dimensions (M, L, T). Therefore, according to the Buckingham Pi theorem, there will be 5 - 3 = 2 dimensionless $\pi$ groups. Show more…

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Question 5. The aerodynamic drag of an F1 car can be represented as a function of: D, V, p, µ and L, where D is the aerodynamic drag force, V is velocity, p is density, µ is viscosity and L is the length of the car. Illustrations in Fig. 5. a) Determine the $\pi$-groups involving these variables. [10 marks] b) A Ferrari F1® car designed for a top speed of 356 km/h is being developed for the USGP. The density and dynamic viscosity of air are assumed at 1.2 kg/m³ and 1.8 x $10^{-5}$Pa.s, respectively. Calculate the wind tunnel speed for tests to be carried out on a quarter-scale model car in a pressurised and cooled wind tunnel in which the air density is 4.7 kg/m³ and the dynamic viscosity is 1.7 x $10^{-5}$ Pa.s. [5 marks] c) Using the same data in part b), the model test gives a drag force of 1334 N, what is the corresponding drag for the full-size car and the tractive power required, assuming dynamic similarity between the wind tunnel and full-scale situations? [5 marks]

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Transcript

00:01 In this question, there is given that there is a model of an automobile which is tested in a fresh water.

00:15 And there is a prototype of this model which is tested in air.

00:24 Now there are two things.

00:27 The first one is about dynamic similarity.

00:31 That is dynamic similarity.

00:34 Now, for any body or any flow over a body to have a dynamic similarity resembling with each other, then reynolds number must be equal.

01:02 That is, reynolds number is basically rho vd by mu, or one can say rho vl by mu, where row is the density, v is the velocity and lc is the characteristics length, which is equals to the diameter for the pipe and for other cross -section it depends on other empirical or derivative formula.

01:38 Now this mu is viscosity, that is dynamic viscosity.

01:48 So the condition required to ensure dynamic similarity between the model and prototype, that is the flow in which it is tested, that means if the model is tested in the fresh water and if we let's say the density comes 1000 and another parameters like velocity, the length, characteristics, length and new, the whole set that is taking the ratio of this and finding the reynolds number.

02:19 So, renault number of model must be equal to renault number of prototype in order so that the both model and prototype must be dynamic similar to each other.

02:40 Now let's say that the reynolds number for the model will be equals to rho 1, v1, l1 by mu1.

02:53 And reynolds number for the prototype will be equal to r2 v2l2 divided by mu2.

03:04 Assuming that the both are dynamic similar.

03:08 Now for water we take density equals to 1 ,000.

03:14 And for air, that is the prototype is tested in air.

03:19 So density of air is about 1 kg per meter cube.

03:25 That is water is 1 ,000 times more density.

03:28 Than air.

03:29 Now coming to this, let's say this mu1 comes here, so we have to take mu1 upon mu 2.

03:38 Now the density, sorry, the viscosity of water with respect to air is around 50 times.

03:47 That is mu water, divided by mu air, is around 50.

03:53 That means water is 50 times more viscous than the air.

03:57 So we can write here into 50.

04:04 Also in the question, it is given that the model is 1 5th of the prototype.

04:15 So we can write l1 is equals to 1 by 5 of l2.

04:23 So l1 by l2 can be written as 1 by 5...

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