How to Find Drag Force from Skin Friction

Question

Just wondering if someone can please can give me some idea, as to how I can solve this question that is not related to current study. I haven't been able to work out where to start with it just yet.

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 1

The problem you cite in the question is a pretty standard homework problem in a graduate fluid mechanics course (it might be also an undergraduate course but less likely). Therefore, I would be reluctant to provide the solution.

What is important to remember when dealing with this problem is that it is crucial to attend the lecture at which the subject was covered. Otherwise, it will be difficult to understand how to solve the problem since the books are not very clear on the subject.

Regarding your question - where to start at - I would recommend Robert H. Nunn "Intermediate fluid mechanics" text book. In the 1989 edition (probably the first edition of the book) I would refer you to the paragraph 13.5 "Flat plate boundary layer flows" on p.255.

Brief introduction to the problem.

It was found that when a fluid flows over a flat plate, almost all flow velocity reduction happens in a narrow layer adjacent to the plate. This layer is called boundary layer. That is why in order to find drag force which fluid exerts on the plate, one should consider only narrow layer of the flow adjacent to the plate, i.e. boundary layer.

People have considered it. And revealed that drag force can very conveniently be calculated using "skin friction coefficient" $c_f$ in general and "friction drag coefficient" $C_f$ (or $C_D$ sometimes) in particular.

There are two formulas for friction drag coefficient: one if the ENTIRE boundary layer (over the entire length of the plate) is laminar; the other one is if the ENTIRE boundary layer is turbulent.

It was revealed that in most cases of flows over a plate, boundary layer can start as laminar and then at some length of the plate become turbulent. And then it is turbulent to the end of the plate.

How to deal with it, if we have only formulas for entire laminar and entire turbulent boundary layers? There is one way (not very difficult) of dealing with it, about which one may read in the Nunn's book, for instance.

It is clear from the statement of your problem, that at some point the boundary layer of your flow converts from laminar to turbulent. It is clear because it is stated that at the trailing edge (i.e. at the end of the plate), $Re = 10^7$ which is higher then critical $Re$ at which conversion occurs: $Re_{crit} = 5·10^5$. Hence, you are dealing with a combined boundary layer.

Having found friction drag coefficient for a combined boundary layer (partially laminar - partially turbulent), one can find drag force as: $$F_D = C_f·(0.5·\rho · U_{free stream}^2·A_{plate})$$ Here:

$\rho$ - density of the fluid

$U_{free stream}$ - flow velocity long before it hits the plate

$A_{plate}$ - plate's area

NOTE:

1) $C_f$ - is, actually, called either drag coefficient or friction drag coefficient (sometimes is denoted as $C_D$) - not skin friction coefficient as in the statement of your problem; friction drag coefficient is the coefficient written for the entire length of the plate, whereas skin friction coefficient ($c_f$) is written for a point on the plate; i.e. $C_f$ is an integral of $c_f$

2) You are given critical $Re$ in order to determine at which point the boundary layer converts from laminar to turbulent. You will need it when you'll be calculating combined friction drag coefficient. In order to do it you will need to use the formula: $$x_{crit} = \frac{Re_{crit} · \mu}{\rho · U_{free stream}}$$ Here:

$\mu$ - dynamic coefficient of viscosity of the fluid (the thing which is called "dynamic viscosity" in your question)

$Re_{crit} = 5·10^5$ - $Re$ at which boundary layer becomes turbulent

$x_{crit}$ - length at which boundary layer becomes turbulent

3) You can find $U_{free stream}$, since you know $Re$ at the end of the plate and the length of the plate.

I wish you good luck in your endeavor to write an expression for $C_D$ for a combined boundary layer. While you'll be doing it, I will be glad to answer you theoretical questions if any occur.

Answer 2

Solution

1) Upstream velocity is not given, but it goes into the drag force expression. Hence, free stream velocity is necessary to find: $$Re_L=\frac{\rho · U_{free stream} · L}{\mu}=10^7$$ (Re at the end of the plate where x=L is given: $10^7$). From here free stream velocity can be found as: $$U_{free stream} = \frac{Re_L · \mu}{\rho · L}=\frac{10^7 · 1.8·10^{-5}\frac{kg}{m·s}}{1.2\frac{kg}{m^3} · 10m}=15\frac{m}{s}$$ 2) Just to get appreciation of the boundary layer structure (and to formally prove that we do in fact deal with a mixed laminar + turbulent flow), let's find $x_{crit}$ - position along the plate where initially laminar boundary layer converts to a turbulent boundary layer. $Re_{crit}$ is given, upstream velocity is found in the item 1, hence $x_{crit}$ can be found from the expression for $Re_{crit}$: $$x_{crit} = \frac{Re_{crit} · \mu}{\rho · U_{free stream}}=\frac{5·10^5 · 1.8·10^{-5}\frac{kg}{m·s}}{1.2\frac{kg}{m^3} · 15\frac{m}{s}}=0.5m$$ One can see on the schematic that conversion occurs pretty close to the leading edge. Hence, the effect of initial laminar boundary layer can be neglected in general. The entire boundary layer can be assumed turbulent and expression for turbulent boundary layer (which is given in the statement of the problem) can be safely used. Still, we shall proceed without this assumption.
3) Drag coefficient for the mixed laminar-turbulent boundary layer needs to be found now. If $Re_{crit}$ is assumed to be equal $5·10^5$ (which is given in the statement of the problem) and neglecting effects of the surface roughness, drag coefficient can be found as follows: $$C_{f_{total}}=C_{f_{turb_{0->L}}}-\frac{1750}{Re_L}$$ Here: $C_{f_{total}}$ - the drag coefficient of interest, $C_{f_{turb_{0->L}}}$ - drag coefficient as if the entire boundary layer were turbulent, $\frac{1750}{Re_L}$ - amendment taking into account half a meter of the laminar boundary layer at the trailing edge. $C_{f_{turb_{0->L}}}$ can be found according to the expression given in the statement of the problem: $$C_{f_{turb_{0->L}}}=\frac{0.074}{Re_L^{0.2}}$$ Also, $Re_L$ - Re number at the end of the plate - is given in the statement of the problem. Overall, drag coefficient is equal to: $$C_{f_{total}}=\frac{0.074}{Re_L^{0.2}}-\frac{1750}{Re_L}=\frac{0.074}{(10^7)^{0.2}}-\frac{1750}{10^7}=0.00278$$ 4) Now, drag force acting on one side of the plate can be found as follows: $$F_D = C_{f_{total}}·(0.5·\rho · U_{free stream}^2·A_{plate})$$ Here, $A_{plate} = L·b$ is the plate's area, where $L$ is the plate's length and $b$ is the plate's width. Therefore, drag force per unit width can be found as follows: $$\frac{F_D}{b} = C_{f_{total}}·(0.5·\rho · U_{free stream}^2·L)=0.00278·0.5·1.2\frac{kg}{m^3}·(15\frac{m}{s})^2·10m=3.753\frac{N}{m}$$

Answer: $3.75\frac{N}{m}$

Answer 3

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.

Find the combined Cf Formula for the whole plate

For the laminar part of the plate, Skin Friction Coefficient (Cf) = 1.328 / Rex^1/2

For the turbulent part of the plate, Skin Friction Coefficient (Cf) = 0.074 / Rex^1/5

Now Rex = Critical Reynold Number, where the transition from laminar to turbulent happens and for this Question, Rex = 5 * 10^5

Let L = Total length of plate l = part of L, in which boundary layer is laminar B = Width of plate Rho = Density of air U = Freestream velocity of air, also shown as U freestream

Lets find an expression for Drag Force on length l, when boundary layer is laminar

Drag Force on Laminar part = 1.328 / Rex^1/2 * B * l * ((Rho * U^2) / 2 )

Where Rex is critical Reynolds Number of 5* 10^5 , because it is before the transition to turbulent

Now we will work out expression for the turbulent part

For the overall length L :

Lets find an expression for Drag Force on length L, when boundary layer is turbulent

Drag Force for L, if whole plate is turbulent = 0.074 / ReL^1/5 * B * L * ((Rho * U^2) / 2 )

where ReL = 10^7 for the whole plate

Now lets find an expression for imaginary turbulent l length of the plate.

Drag Force = 0.074 / ReL^1/5 * B * l * ((Rho * U^2) / 2 )