[Rob Wilk]

My name is Rob, I am from New Zealand.
I have an interest in Fluid Mechanics and my spare time I like working out questions related to this.
This has been due to me studying Civil Engineering some years ago.
Luckily I have found a way to get the answer to this question below.

There is another way of doing this, which I have mentioned at the bottom of my answer. Unfortunately I am not sure how to do it this way and if any one knows, I'd appreciate if you can explain how it is done.


Air flows parallel to the surface of a smooth flat plate 10m long.
The boundary layer has zero thickness at the leading edge.
The Reynolds number at the trailing edge of the plate is 10^7.
Calculate the total drag force due to skin friction on one side of the plate per unit width?
Assume that for a laminar boundary layer, up to Rex = 5_10^5, the skin friction coefficient is Cf = 1.328 (Rex)^-1/2 and for turbulent boundary layer Cf = 0.074 (Rex)^-1/5.
Take the density of air as 1.2 kg/ m^3 and its dynamic viscosity as 1.8_10^-5 kg/ ms.


Answer


First Part

We need to find the free stream velocity using the equation for the Reynolds Number at the trailing edge of the plate :

ReL = (Rho * U freestream * L) / mu = 10^7


where ReL = Reynolds Number at the trailing edge of the plate = 10^7

Rho = Density of air = 1.2 kg /m^3

L = length of plate from leading to trailing edge.

Note: Leading edge is the start of the plate where we measure length from
Trailing edge is the end of the plate where length is measured to

where mu = dynamic Viscosity = 1.8 * 10^-5

ReL at the end of the plate where x = L is given as 10^7

From here free stream velocity can be found as:

U freestream = (ReL * mu) / (Rho * L)

U freestream = (10^7 * 1.8 * 10^-5 kg/ms) / (1.2 kg/m^3 * 10 m)

U freestream = 15 m/s


Second Part


First of all workout if the initially laminar side changes (transitions) to turbulent at some point along the length of the plate.


To do this we will use the critical Reynolds Number given of 5 * 10^5.


we can work out x critical ( length of plate from leading edge where laminar changes to a turbulent zone )


Re critical = (Rho * U freestream * x critical) / mu


where Re critical = Critical Reynolds Number = 5 * 10^5

Rho = Density of air = 1.21 kg /m^3

x critical = length of plate from leading edge where laminar changes to a turbulent zone

where mu = dynamic Viscosity = 1.8 * 10^-5 or 0.000018 kg/ ms


Re critical * mu = (Rho * U freestream * x critical)


x critical = (Re critical * mu) / (Rho * U freestream)


x critical = (5 * 10^5 * 0.000018) / (1.2 * 15)


x critical = 0.5 m


Because the plate is 10 m long, this means that 0.5 m along the plate the flow changes from laminar to turbulent.

The next 9.5 m of the plate is turbulent.


Final Part


We will calculate the total drag force due to skin friction on one side of the plate per unit width.

As a small part of it is laminar and most of it is turbulent, this is the approach we will take for drag force.


We will treat it as if the plate is fully turbulent, then subtract the turbulent portion for x critical and then add laminar portion for x critical.


To start with we will first calculate Drag Force for the whole plate based on assuming turbulent boundary throughout the whole length of the plate.


AB = Length of plate to the transition point

AC = Length of the whole plate

(FD) for AC = Drag Force for AC

Cf = Skin Friction Coefficient


Firstly we need to calculate Skin Friction Coefficient from the formula given.

Cf = 0.074 / (Rex)^1/5


For this Rex is the Reynolds Number based on the whole length of the plate.

From before Rex = 10^7 for the whole plate.

Now we can calculate Cf


Cf = 0.074 / (10,0000,000)^1/5

Cf = 0.074 / 25.118864

so Cf for AC = 0.00294599 (assuming whole length of plate is turbulent)

Lets say (FD) for AC = Drag Force for whole length of plate AC

and A for AC = Area for whole plate covering length AC per metre span

Area of plate in AC = 10 * 1 = 10 m^2 per metre span


(FD) for AC = Cf * 0.5 * Rho * (U freestream)^2 * A for AC

(FD) for AC = 0.00294599 * 0.5 * 1.2 * 15^2 * 10

(FD) for AC = 3.9770906 Newtons ( This is Drag Force, assuming whole turbulent length of plate )


Now we will subtract the Drag Force for the turbulent part before the transition zone.

This covers the length AB of the plate.

and we will calculate:

(FD) for AB = Drag Force for AB

Firstly we need to calculate Skin Friction Coefficient Cf.

Cf is different because we are using critical Reynolds Number, which is based on critical length before transition.

From before Re critical = Critical Reynolds = 5 * 10^5

Now we can calculate Cf

Cf = 0.074 / (Re critical)^1/5

Cf = 0.074 / (500,000)^1/5

CD = 0.074 / 13.79729661

so Cf for AB = 0.00536337 (assuming turbulent flow in critical length portion before transition)


Lets say (FD) for AB = Drag Force for assumed turbulent length before transition.

and A for AB = Area for whole plate covering length AB

Area of plate in AB = 0.5 * 1 = 0.5 m^2 per metre span


(FD) for AB = Cf * 0.5 * Rho * (U freestream)^2 * A for AB

(FD) for AB = 0.00536337 * 0.5 * 1.2 * 15^2 * 0.5

(FD) for AB = 0.362027 Newtons

( This is Drag Force, assuming turbulance in critical length portion before transition )



The actual Drag Force in Turbulent Zone = [ (FD) for AC - (FD) for AB ]


The Drag force in Turbulent zone = 3.9770906 - 0.362027 = 3.6150636 Newtons for B to C


Now we can calculate the actual force in the laminar zone.

(FD) for Laminar or (FD) for AB for Laminar Zone


For this Laminar flow CD = 1.328 / (Re critical)^1/2


CD = 1.328 / (500,000)^1/2

CD = 1.328 / 707.1067812

so CD for AB = 0.00187807561 (for actual laminar flow before transition)


Lets say (FD) for AB = Drag Force for actual laminar zone before transition.

and A for AB = Area for whole plate covering length AB

Area of plate in AB = 0.5 * 1 = 0.5 m^2 per metre span


(FD) for Laminar = CD * 0.5 * Rho * (U freestream)^2 * A for AC

(FD) for Laminar = 0.00187807561 * 0.5 * 1.2 * 15^2 * 0.5

(FD) for Laminar = 0.12677010373 Newtons


Total Drag force on plate = [ (FD) for AB of Laminar + (FD) for BC of Turbulent ]


Total Drag force on plate = 0.12677010373 + 3.6150636

so Total Drag Force due to Skin Friction on one side of the plate per unit width = 3.7418 Newtons per metre span

Book answer is 3.75 Newtons because it rounded up to the nearest 0.05 Newtons


There is another way of doing this, it involves calculating the combined Skin Friction Coefficient Cf and then calculating Drag force from that
formula FD = 0.5 * Rho * (U freestream)^2 * A
for the whole plate

However I am not sure how to calculate the combined Skin friction Cf doing it this way.