NAME: PERSON #:LAB SECTION:

EAS 230 – Fall 2017 – PP1 Page 1 of 8

EAS-230 Fall 2017

Programming Project # 1 (PP1)

Score: /100

Directions:

1. Your three m-files must be named as in the deliverables section of this assignment.

2. All three m-files (two scripts and one function) must be uploaded to UBlearns before 11:59 PM on

Monday, Nov. 13, 2017 (Please do not upload files ending in “.m~”, “.sav” or “.mat”.) A pdf copy of the

report (part 3) must also be uploaded to UBlearns on the same due date.

3. A paper copy (the report) including your codes, the results and analysis must also be written (a

template of the report will be posted on UBlearns). Your report must be turned-in at the start of your

first lecture of the week of Monday, Nov. 13, 2017 (Mon or Tue, depending on your section). Be sure

to write your name, person # and lab section on every page.

4. You must write your own code and follow all instructions to get full credit. You are not allowed to

use codes or scripts found on the internet or any other references.

5. You must use good programming practices, including indentation, commenting your functions

scripts and choosing meaningful variable names to make your programs self-documenting.

6. It is your responsibility to make sure that your functions/scripts work properly and are free of

errors by utilizing the resources at your disposal.

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EAS 230 – Fall 2017 – PP1 Page 2 of 8

Background, [1]:

The friction factor in fully developed turbulent pipe flow depends on the Reynolds number and the

relative roughness 𝜀/𝐷, which is the ratio of the mean height of roughness of the pipe to the pipe

diameter. The functional form of this dependence cannot be obtained from a theoretical analysis, and

all available results are obtained from painstaking experiments using artificially roughened surfaces

(usually by gluing sand grains of a known size on the inner surfaces of the pipes). Most such

experiments were conducted by Prandtl’s student, J. Nikuradse in 1933, followed by the works of

others. The friction factor was calculated from the measurements of the flow rate and the pressure

drop. The experimental results obtained are presented in tabular, graphical, and functional forms

obtained by curve-fitting experimental data. In 1939, Cyril F. Colebrook (1910 –1997) combined the

available data for transition and turbulent flow in smooth as well as rough pipes into the following

implicit relation known as the Colebrook equation

1

√𝑓= −2.0 log (

𝜀/𝐷

3.7+

2.51

Re √𝑓) (1)

This equation is only applicable for turbulent flow, Re ≥ 4000. For laminar and transitional flow,

Re < 4000, the friction coefficient can be determined as

𝑓 =64

Re (2)

We must note that the logarithm in Equation (1) is a base 10 rather than a natural logarithm. In 1942,

the American engineer Hunter Rouse (1906 –1996) verified Colebrook’s equation and produced a

graphical plot of f as a function of ε and the product Re √𝑓. He also presented the laminar flow

relation and a table of commercial pipe roughness. Two years later, Lewis F. Moody (1880 –1953)

redrew Rouse’s diagram into the form commonly used today.

The now famous Moody chart is shown in Figure 1. It presents the Darcy friction factor for pipe flow

as a function of the Reynolds number and 𝜀/𝐷 over a wide range. It is probably one of the most widely

accepted and used charts in engineering.

At very large Reynolds numbers (to the right of the dashed line on the chart) the friction factor curves

corresponding to specified relative roughness curves are nearly horizontal, and thus the friction

factors are independent of the Reynolds number (Fig. 1 ). The flow in that region is called fully rough

turbulent flow or just fully rough flow.

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EAS 230 – Fall 2017 – PP1 Page 3 of 8

NAME: PERSON #:LAB SECTION:

EAS 230 – Fall 2017 – PP1 Page 4 of 8

Deliverables:

You are required to reproduce the friction coefficient, f , as a function of Reynolds number, Re, for

laminar flow and as a function of Re and the relative roughness, 𝜺/𝑫, for turbulent flow. Your data

must be presented in both Table format and chart format (Moody chart shown in Fig.1) by doing the

following:

1. (30 pts) Create a script file named FCLT_UBITName.m to tabulate the value of the friction

coefficient for both laminar and turbulent flows. _UBITName is your UBITName as in your

email UBITName@ buffalo.edu.

a. Create a variable RelRough and assign to it all values of the dimensionless roughness 𝜺/𝑫

shown in Figure 2 starting from 𝜺

𝑫= 𝟎 , for smooth pipes, till

𝜺

𝑫= 𝟎.05. The element values

of the vector RelRough must be exactly the same as those values shown in Figure 2 next

to each curve (see the values inside the dashed rectangle).

Figure 2: Values of the relative roughness in Figure 1.

b. Your script must determine the value of the friction coefficient f (FC variable in MATLAB)

as a function of Re for every element of the vector RelRough. Noting that for laminar flow,

the friction coefficient f is a function of Re only while for turbulent flow, it is a strong

function of both Re and 𝜺/𝑫. You must create a variable Re that represents the x-axis in

Figure 1, noting that the Re axis is logarithmic. For 600 < 𝑅𝑒 < 4000 , you can assume laminar flow and the friction coefficient FC can be determined from equation (2) as a

function of Reynolds number only. Start with a value of the laminar flow Reynolds number ReL of 600 and increase it constantly by 100 to 3900. You can assign your friction factor

values for laminar flow to a variable called FCL. You must note that FCL is a vector

(function of Re only).

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EAS 230 – Fall 2017 – PP1 Page 5 of 8

c. For turbulent flow (Re ≥ 4000), Equation (2) must be used to determine the turbulent

friction coefficient f (FCT). You must note that f is implicit and appears in both sides of

Equation (1). Accordingly, to determine the friction coefficient f numerically, an iterative

technique must be developed. You must also note that f for turbulent flow is a function of both Re and 𝜺/𝑫. Create a new variable ReT = [4000 5000 6000 … 10000 20000 30000

… 100000 200000 300000 … 1000000…] and follow this pattern until 108. Use good

programming practices and Do Not write the elements of the variable explicitly. For every value of RelRough, FCT must be determined for all the values of ReT. You must note that

FCL is a vector (function of Re only) while FCT is a matrix (function of Re and 𝜀/𝐷).

d. To develop the iterative technique for determining the turbulent friction coefficient,

Equation (1) can be rewritten as

𝑓 = 1/ [−2.0 log (𝜀/𝐷

3.7+

2.51

Re √𝑓)]

2

(3)

You can start by entering a guess value for f (Use 𝑓 = 1 as a first guess), and accordingly

the right hand side can be determined and the result will be assigned to the variable f in the left hand side. You can use variable names of FCT_guess or FCT_old for the guessed

value of f and FCT_cal or FCT_new for the calculated value of f. The function should then

compare between the guessed value of f and the calculated value of f. If they are close to each other by a specific tolerance (1E-14), the iteration must stop and the calculated value

of f must be saved. Otherwise, the iteration procedure must continue taking the calculated

value from the previous iteration as the new guess value of the current iteration and

continue as before. In order to avoid infinite loop in case of divergence, you must set a

maximum number of iterations to 30. If this number was reached the function should stop

and display a message “The program failed to converge in the allowable number of

iterations”.

e. After calculating the friction coefficient for both laminar and turbulent flow, you must

display your data in Table format, as shown below, for:

i. Laminar flow

ii. Turbulent flow

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Friction Coefficient for Laminar Flow in pipes*1000

(Re < 4000)

------------------------------------------------

Reynolds # | Friction Coefficient

------------------------------------------------

600 | 106.66667

-------------------------------------------------

700 | 91.42857

-------------------------------------------------

………………………………………….

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EAS 230 – Fall 2017 – PP1 Page 6 of 8

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Friction Coefficient for Turbulent Flow in pipes*1000

(Re >= 4000)

------------------------------------------------

Reynolds# | Relative Roughness

------------------------------------------------

| 0 | 1E-6 | 5E-6 | ---

------------------------------------------------

4000 | ---- | ---- | ---- | ---

-------------------------------------------------

5000 | ---- | ---- | ---- | ---

-------------------------------------------------

………………………………………….

Display only 5 values of the relative roughness per page. You must notice that the values of the

friction coefficient to be displayed in the table for both laminar and turbulent flow must be the

values calculated from your script after being multiplied by 1000 for better display.

2. (20 pts) Use the techniques you developed before for determining the friction coefficient for

laminar and turbulent flow in a User Defined Function (UDF) called fcoeff_UBITName.m.

The function must take Re and 𝜺/𝑫 as input arguments and returns the friction coefficient f for

either laminar or turbulent flow as the output argument. Note that fcoeff_UBITName.m will

use Equation (2) for the laminar friction coefficient and the iterative technique developed in

part 1.d for determining the turbulent friction coefficient from Equation (1).

3. (50 pts)

a. (30 pts) You are now required to reproduce Figure 1 (the Moody chart) with all of its details using the function fcoeff_UBITName(). You must write a new script file called

MoodyCh_UBITName.m, which may be a modification of FCLT_UBITName.m script you

developed for part 1. Use only 8 points for the laminar flow Reynolds number ReL = [600

700 800 900 1000 2000 3000 3900] for the plot of the laminar flow line in the Figure. Use

good programming practice and Do Not write the elements explicitly. For turbulent flow, use

the same values for vector RelRough you have developed for FCLT_UBITName.m script.

For turbulent Reynolds number, ReT, you must assigned new values that develop a smooth

curve. You can use the function logspace() (see the help) with 1000 elements or more to

cover all the range of turbulent Reynold number 4000 ≤ Re𝑇 ≤ 1 × 108. You must note that logspace() starts with 103 in this range. For calculating the friction factor either for

laminar flow or for turbulent flow, your script MoodyCh_UBITName.m must use(call) the

function fcoeff_UBITName() to determine the friction coefficient for every value of Re

and 𝜺/𝑫. You can assign your friction factor values for laminar flow to a variable called FCL

and the friction factor values for turbulent flow to a variable called FCT, as before. After

creating all the variables, create your plot using the appropriate plotting function. You must note that, if you are plotting x vs. Y where x is a vector and Y is a matrix (m×n), the length of

x must be equal to the number of rows of Y, (m), and MATLAB will be plotting x vs. Y column

by column, i.e., (x, C1Y), (x, C2Y), …, (x, CnY). You can develop your figure in

color using line width of 1. You must annotate your figure to be as close as to the Moody chart

in Figure 1 with all the details such as the axes labels, the axes limits, the tic labels, etc. You

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EAS 230 – Fall 2017 – PP1 Page 7 of 8

must add texts for the 𝜀/𝐷 of every curve. The figure caption, as a text, must also be shown,

as well as, the table of roughness for various materials.

b. (20 pts) The dashed line that determines the start of the fully rough turbulent flow region

(see the red dashed line in Fig. 3) can be plotted once its intersection points (the red crosses

in Fig. 3), with all relative roughness 𝜺/𝑫 curves, starting from 0.00005 to 0.05, are determined. In order to numerically do this in your script MoodyCh_UBITName.m, you are

first required to produce smooth curves for every value of 𝜺/𝑫 by using enough points (≥

1000 on log Re scale) and then scan every curve to determine the point at which the curve

starts to be almost horizontal (not a function of Re). You can do that by determining the

relative change in f as

|𝑓2−𝑓1|

𝑓1 (4)

Once this change is less than a certain tolerance (10-4), you can assume that the curve is

horizontal and you must determine the coordinate of this point (Re, f) for every curve of 𝜺/𝑫.

The line connecting these points will be the dashed line in Figure 1 and you are required to

add this graph to your plot using line width of 2.

References:

[1] Y. Cengel and J. Cimbala, “Fluid Mechanics Fundamentals and Applications,” 2nd edition, McGraw-Hill, 2010.

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EAS 230 – Fall 2017 – PP1 Page 8 of 8

Figure 3: The start of the completely rough turbulent region.