[Rob Wilk]

I did a Civil Engineering course some years ago and this is a question from my textbook that I haven't been able to solve.

A flat plate 1m wide and 4m long moves with a velocity of 4m/s parallel to its longer sides through still air of kinematic viscosity 1.5 * 10^-5 m^2/s and density 1.21 kg/m^3. On one side of the plate an initially laminar boundary layer is formed. On the other side the leading edge is roughened and the boundary layer may be considered to be entirely turbulent. Assuming a critical Reynolds number of 5 * 10^5 determine the ratio of drag forces on the two sides.

For laminar flow CD = 1.46/ (Rex)^1/2 and for turbulent flow CD = 0.074/ (Rex)^1/5

How do you work out Rex ?

and where does critical Reynolds number come into it ?

I really would like to get an understanding on this,
If someone knows about this, it will be much appreciated that you could please reply to.

Also On one side of the plate it says that an initially laminar boundary layer is formed".
Does this mean that this side of the plate has laminar and turbulent flow ?


This Question has now been solved.


Here is the Answer


First 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 )


Note: Leading edge is the start of the plate where we measure dimensions from


Re critical = (Rho * v * 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 = kinematic viscosity * density = 1.5 * 10^-5 * 1.21 = 0.00001815 kg / ms


Re critical * mu = (Rho * v * x critical)


x critical = (Re critical * mu) / (Rho * v)


x critical = (5 * 10^5 * 0.00001815) / (1.21 * 4)


x critical = 1.875 m


Because the plate is 4 m long, this means that 1.875 m along the plate the flow changes from laminar to turbulent.

The last 2.125 m of the plate is turbulent.


Second Part



We will calculate the Drag Force on the initially laminar side.


Because there is a a turbulent portion on this side of the plate, 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

CD = Drag Coefficient


Firstly we need to calculate Drag Coefficient from the formula given.


CD = 0.074 / (Rex)^1/5


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


Rex = (Rho * v * L) / mu


where L = 4 m, as it is the whole length of the plate

and v = velocity of plate


so Rex = (1.21 * 4 * 4) / 0.00001815

Rex = 1,066,666.67


Now we can calculate CD


CD = 0.074 / (1,066,666.67)^1/5

CD = 0.074 / 16.05483

so CD for AC = 0.0046092 (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

Area of plate in AC = 4 * 1 = 4 m^2


(FD) for AC = CD * 0.5 * Rho * v^2 * A for AC

(FD) for AC = 0.0046092 * 0.5 * 1.21 * 4^2 * 4

(FD) for AC = 0.178468 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 Drag coefficient CD.

CD 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 CD

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

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

CD = 0.074 / 13.79729661

so CD 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 = 1.875 * 1 = 1.875 m^2


(FD) for AB = CD * 0.5 * Rho * v^2 * A for AB

(FD) for AB = 0.00536337 * 0.5 * 1.21 * 4^2 * 1.875

(FD) for AB = 0.09734516 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 = 0.178468 - 0.09734516 = 0.081123241 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 = 1.46 / (Re critical)^1/2


CD = 1.46 / (500,000)^1/2

CD = 1.46 / 707.1067812

so CD for AB = 0.002064752 (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 = 1.875 * 1 = 1.875 m^2


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

(FD) for Laminar = 0.002064752 * 0.5 * 1.21 * 4^2 * 1.875

(FD) for Laminar = 0.03747525 Newtons


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


Total Drag force on plate (for Initially Laminar side) = 0.03747525 + 0.081123241

Total Drag force on plate (for Initially Laminar side) = 0.118598486 Newtons


Third Part



Now we will calculate the drag force on the other side, where the leading edge is roughened and the boundary layer may be considered to be entirely turbulent.


Firstly we need to calculate Drag Coefficient for turbulent flow from the formula given.


CD = 0.074 / (Rex)^1/5


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


and from before Rex = 1,066,666.67

CD = 0.074 / (1,066,666.67)^1/5

CD = 0.0046092 ( For Fully Turbulent Side)


Now we can calculate the drag force for this fully turbulent side

(FD) for Turbulent of (FD) for A to C Turbulent


From before Area of plate = A for AC = 4 * 1 = 4 m^2


so (FD) for Turbulent = CD * 0.5 * Rho * v^2 * A for AC

(FD) for Turbulent = 0.0046092 * 0.5 * 1.21 * 4^2 * 4

(FD) for Turbulent = 0.178468 Newtons (Drag force for fully turbulent side of plate)


Final Part


Determining the ratio of drag forces on the two sides, is done by the ratio of the highest force to the lowest force for this question.


Fully Turbulent Side has the highest drag force of 0.178468 Newtons


Initially Laminar Side has the lowest drag force of 0.118598486 Newtons


The ratio of the Drag Forces = 0.178468 / 0.118598486


Ratio of Drag Forces on the two sides = 1.5 to 1