Flow Past A Circular Cylinder.docx

Flow past a circular cylinder Name: Animesh Phatowali Roll no: 17AE30005

Group: 6

Aim: To record and observe the pressure distribution along the cylinder and plot 𝑐𝑝 𝑣𝑠 𝜃 of the flow. Theory: We can write the flow past a cylinder as a sum of uniform stream flow and the doublet i.e. Flow past a cylinder = uniform stream + doublet Now we know that the velocity components are given by 𝑣𝜃 = 𝑢∞ (1 +

𝑎2 ) sin 𝜃 𝑅2

𝑣𝑅 = 𝑢∞ (1 −

𝑎2 ) cos 𝜃 𝑅2

for 𝑣𝑅,𝑠 = 0 , 𝑣𝜃𝑠 = 2𝑢∞ sin 𝜃 Now using the Bernoulli’s equation, we get, 1 2 1 1 2 𝑝∞ + 𝜌𝑢∞ = 𝑝𝑠 + 𝜌𝑢𝑠2 = 𝑝𝑠 + 𝜌𝑢∞ 4 sin2 𝜃 2 2 2 1 2 ⇒ 𝑝𝑠 − 𝑝∞ = 𝜌𝑢∞ (1 − 4 sin2 𝜃) 2 Now the pressure coefficient is given by 𝑐𝑝 =

𝑝𝑠 − 𝑝∞ 1 2 2 𝜌𝑢∞

At stagnation point again using Bernoulli’s equation, 1 1 2 𝑝0 + 𝜌𝑢02 = 𝑝∞ + 𝜌𝑢∞ 2 2 1 2 𝜌𝑢 = 𝑝∞ − 𝑝0 [∵ 𝑎𝑡 𝑠𝑡𝑎𝑔𝑛𝑎𝑡𝑖𝑜𝑛 𝑝𝑜𝑖𝑛𝑡 𝑝𝑠 = 𝑝0 𝑎𝑛𝑑 𝑢𝑠 = 𝑢0 = 0 ] 2 ∞ 𝑝𝑠 − 𝑝∞ ∴ 𝑐𝑝 = = 1 − 4 sin2 𝜃 𝑝∞ − 𝑝0 Apparatus required: Blow down wind tunnel, manometer, circular cylinder [with a hole which is connected to manometer with a tube, and 360 degree dial.

Procedure: 

Measure 𝑝0 𝑎𝑛𝑑 𝑝𝑠 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣∞



Rotate the cylinder to measure the pressure variation at an interval of 10°

Table:

GROUP-1/2 Θ

Ps

Pr

P∞

0

334

336

322

10

334

336

322

20

330

336

322

30

326

336

322

40

320

336

322

50

314

336

322

60

310

336

322

70

308

336

322

80

308

336

322

90

308

336

322

100

309

336

322

110

309

336

322

120

309

336

322

130

309

336

322

140

309

336

322

150

308

336

322

160

308

336

322

170

308

336

322

180

308

336

322

190

308

336

322

200

308

336

322

210

308

336

322

220

308

336

322

230

308

336

322

240

308

336

322

250

309

336

322

260

309

336

322

270

308

336

322

280

308

336

322

290

307

336

322

300

308

336

322

310

312

336

322

320

318

336

322

330

324

336

322

340

330

336

322

350

334

336

322

360

334

336

322

Group – 3,4 Degree

Pr(mm)

Pinfinty(mm) Ps(mm)

0

358

328

356

10

358

328

354

20

358

328

346

30

358

328

336

40

358

328

332

50

358

328

302

60

358

328

290

70

358

328

288

80

358

328

290

90

358

328

292

100

358

328

292

110

358

328

292

120

358

328

292

130

358

328

292

140

358

328

290

150

358

328

290

160

358

328

290

170

358

328

290

180

358

328

290

190

358

328

290

200

358

328

290

210

358

328

291

220

358

328

291

230

358

328

292

240

358

328

292

250

358

328

293

260

358

328

293

270

358

328

293

280

358

328

290

290

358

328

286

300

358

328

288

310

358

328

302

320

358

328

320

330

358

328

334

340

358

328

346

350

358

328

354

360

358

328

356

Group – 5,6 Θ

Ps

Pr

P∞

0

381

382

334

10

379

382

334

20

365

382

334

30

340

382

334

40

312

382

334

50

295

382

334

60

274

382

334

70

266

382

334

80

268

382

334

90

270

382

334

100

274

382

334

110

272

382

334

120

272

382

334

130

272

382

334

140

272

382

334

150

270

382

334

160

270

382

334

170

270

382

334

180

270

382

334

190

270

382

334

200

270

382

334

210

270

382

334

220

270

382

334

230

270

382

334

240

272

382

334

250

272

382

334

260

272

382

334

270

270

382

334

280

268

382

334

290

266

382

334

300

270

382

334

310

290

382

334

320

320

382

334

330

344

382

334

340

366

382

334

350

378

382

334

360

380

382

334

Table 𝑐𝑝 𝑣𝑠 𝜃 Θ

Cp(theo.)

Cp(prac)

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360

1 0.879385 0.532089 0 -0.6527 -1.3473 -2 -2.53209 -2.87939 -3 -2.87939 -2.53209 -2 -1.3473 -0.6527 0 0.532089 0.879385 1 0.879385 0.532089 0 -0.6527 -1.3473 -2 -2.53209 -2.87939 -3 -2.87939 -2.53209 -2 -1.3473 -0.6527 -1.8E-15 0.532089 0.879385 1

1 0.957447 0.659574 0.12766 -0.46809 -0.82979 -1.2766 -1.44681 -1.40426 -1.3617 -1.2766 -1.31915 -1.31915 -1.31915 -1.31915 -1.3617 -1.3617 -1.3617 -1.3617 -1.3617 -1.3617 -1.3617 -1.3617 -1.3617 -1.31915 -1.31915 -1.31915 -1.3617 -1.40426 -1.44681 -1.3617 -0.93617 -0.29787 0.212766 0.680851 0.93617 0.978723

Observation and calculations:

Chart Title 1.5 1 0.5 0 -0.5 0

50

100

150

200

250

300

350

400

-1 -1.5 -2 -2.5 -3 -3.5 Cp(theo.)

Cp(prac)

Results: The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field in fluid dynamics. The experiment has shown that the flow over a cylinder can be classified as a potential flow and the same has been verified by plotting a graph of Cp vs θ. Only for laminar flow the values of Cp obtained match with those obtained theoretically. In the experiment, as shown by the graph the pressure is lowest from 90 & 270 degrees and the velocity at these points is maximum according to the equations. The velocity is also completely tangential at these points. The graph being symmetrical about θ=180 proves that there is no lift generated.

Discussions: 1. How Cp will vary with Reynold’s number? Ans. Cp have a little change in the change of Reynold’s number 2. Why Cp from experiment is different from potential flow theory? Ans. In the region of 𝜃 = 90° − 270° the viscous force is dominant. So it suppress the pressure flow and thus the Cp value doesn’t match with theoretical value. 3. What pressure you are measuring in your experiment? Ans. We are measuring static pressure in the experiment