Calculating the Drag Coefficient C D for a
Cylinder in Cross Flow
Theoretical Basis— Using Surface Pressure Measurements
Consider the area element dA = LRd on the surface of the cylinder as shown below in Fig. 1.
Fig. 1 Measuring the Surface Static Pressure Profile [1]
The component of the pressure–area force on this element projected into the streamwise
direction is dFD = Ps()dA cos = Ps()LR cosdwhere the dependence on  is included
explicitly to emphasize that Ps is not constant. Integrating over the entire surface of the cylinder
to find the pressure drag, and recognizing the axisymmetric nature of the pressure distribution,
we have


F D  2 Ps  LR cos d  LD  Ps cos d
0
(1)
0
Subtracting the constant approach static pressure P1 from the integrand does not change the
magnitude of the drag force since a constant pressure force normal to the surface gives the zero
vector when integrated over the cylinder surface. This is easily seen from

 P cosd  0
1
for P1 = constant.
(2)
0


F D  LD Ps   P1 cosd 
0
where D=2R.
1|Page
(3)
The drag coefficient then is formed by dividing both sides of the equation by the free-stream
dynamic pressure 1 2 U21 and the projected area Aproj = LD
FD
CD 

2
U 1 LD 2


0

Ps ( )  P1
cos  d  C P cos  d
U 12 2

(4)
0
Later, we will use this equation in conjunction with surface static pressure measurements to
determine the cylinder drag coefficient. (Note: Since surface shear forces have been neglected
here, this method ignores the skin friction contribution to the total drag.)
Using Excel Spreadsheets
The first thing that you must do is create a template with all measured parameters (i.e. D, T, Patm, Vel, ,
, etc.). Using this information, calculate the Reynolds number using the cylinder diameter as the
characteristic length.
Next, choose a characteristic interval over which to do the surface pressure
measurements (and subsequent numerical integration).
In this case, every 10 was chosen for
convenience as seen in Fig. 2.
Fig. 2 Excel Spreadsheet Template
2|Page
Object
Cylinder
D (mm)
88.9
Theta (deg) Theta (rad)
0
0
10
0.1745329
20
0.3490659
30
0.5235988
40
0.6981317
50
0.8726646
60
1.0471976
70
1.2217305
80
1.3962634
90
1.5707963
100
1.7453293
110
1.9198622
120
2.0943951
130
2.268928
140
2.443461
150
2.6179939
160
2.7925268
170
2.970597
180
3.1415927
23.33
Vel
(m/s)
8.55
Ps-P1 (H20)
0.063191732
0.053649902
-0.01595052
-0.1104126
-0.22920736
-0.36159261
-0.46394857
-0.48276774
-0.4508667
-0.42651367
-0.43103027
-0.42108154
-0.45676676
-0.44812012
-0.45607503
-0.42600505
-0.43094889
-0.51200358
-0.434611
Ps-P1 (Pa)
15.724631
13.350242
-3.969128
-27.47507
-57.03596
-89.97871
-115.449
-120.1319
-112.1937
-106.1337
-107.2576
-104.7819
-113.6618
-111.5102
-113.4897
-106.0071
-107.2373
-127.407
-108.1486
T1
Patm
(mmHg)
746
Cp
T1
(Celsius)
23.33
Integrand
T1
mu air
rho air
(K)
(Pa-s) (kg/m3)
Re
296.48 1.84E-05 1.16791 48245.79
Weight
Int*Wt
Weight
Cd
Fig. 3 Excel Spreadsheet with Surface Pressure Measurements (shown in red)
Fig. 4 Excel Spreadsheet with Calculated Cp Values
3|Page
Int*Wt
In Figure 3, the surface pressure measurements are added to the spreadsheet noting the units of the
pressure transducer (i.e. in H2O) and making the conversion to SI units (if necessary). Next, the
coefficient of pressure Cp is calculated which is equal to the measured pressure difference divided by the
dynamic pressure. In Figure 4, the column ‘Integrand’ is equal to Cp multiplied by cos as shown in
Equation (4).
Using the trapezoidal rule to perform the numerical integration requires a weighting factor of 2 for all
terms in the summation except the first and last term. If Simpson’s Rule is used instead, then the
weighting factors follow the pattern 1, 4, 2, 4, 2,…, 2, 4, 2, 4, 1 as shown below. More specifically,
applying Simpson’s rule results in the following formula for the integral over the interval:
Simpson’s Rule
Finally then, multiplying the integrand in column F by the appropriate weighting factor and summing up
these terms results in the drag coefficient (i.e. CD) for the cylinder as shown in Figure 5.
A
Object
Cylinder
B
D (mm)
88.9
Theta (deg) Theta (rad)
0
0
10
0.1745329
20
0.3490659
30
0.5235988
40
0.6981317
50
0.8726646
60
1.0471976
70
1.2217305
80
1.3962634
90
1.5707963
100
1.7453293
110
1.9198622
120
2.0943951
130
2.268928
140
2.443461
150
2.6179939
160
2.7925268
170
2.970597
180
3.1415927
C
D
E
23.33
Vel
(m/s)
8.55
Patm
(mmHg)
746
Ps-P1 (H20)
0.063191732
0.053649902
-0.01595052
-0.1104126
-0.22920736
-0.36159261
-0.46394857
-0.48276774
-0.4508667
-0.42651367
-0.43103027
-0.42108154
-0.45676676
-0.44812012
-0.45607503
-0.42600505
-0.43094889
-0.51200358
-0.434611
Ps-P1 (Pa)
15.724631
13.350242
-3.969128
-27.47507
-57.03596
-89.97871
-115.449
-120.1319
-112.1937
-106.1337
-107.2576
-104.7819
-113.6618
-111.5102
-113.4897
-106.0071
-107.2373
-127.407
-108.1486
Cp
0.368356925
0.312735741
-0.092978696
-0.643616559
-1.336094372
-2.107793829
-2.70444666
-2.814147289
-2.628189898
-2.486231353
-2.512559504
-2.454566414
-2.662582527
-2.612179535
-2.658550288
-2.483266471
-2.512085123
-2.9845687
-2.533432273
T1
F
G
H
T1
(Celsius)
23.33
Fig. 5 Using the Trapezoidal Rule and Simpson’s Rule to Calculate CD
References: [1] University of Illinois (UIUC) Fluid Mechanics Laboratory Manual
4|Page
I
J
T1
mu air
rho air
(K)
(Pa-s) (kg/m3)
Re
296.48 1.84E-05 1.16791 48245.79
Trapezoidal
Simpson
Integrand
Weight Int*Wt Weight Int*Wt
0.368356925
1
0.368357
1
0.368357
0.307984584
2
0.615969
4
1.231938
-0.087371393
2
-0.17474
2
-0.17474
-0.557388283
2
-1.11478
4
-2.22955
-1.02350767
2
-2.04702
2
-2.04702
-1.354863799
2
-2.70973
4
-5.41946
-1.352223216
2
-2.70445
2
-2.70445
-0.962494997
2
-1.92499
4
-3.84998
-0.45638039
2
-0.91276
2
-0.91276
-6.66183E-08
2
-1.3E-07
4
-2.7E-07
0.436301498
2
0.872603
2
0.872603
0.839511209
2
1.679022
4
3.358045
1.331291258
2
2.662583
2
2.662583
1.679076584
2
3.358153
4
6.716306
2.036567756
2
4.073136
2
4.073136
2.150571876
2
4.301144
4
8.602288
2.36058785
2
4.721176
2
4.721176
2.941041247
2
5.882082
4
11.76416
2.533432273
1
2.533432
1
2.533432
Cd
1.699881
1.720084