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https://doi.org/10.1007/s10409-020-00994-9 R E S E A R C H P A P E R

Wall shear-stress extraction by an optical flow algorithm with a sub-grid formulation

The Hung Tran1·Lin Chen2,3

Received: 16 March 2020 / Revised: 19 May 2020 / Accepted: 7 July 2020

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract

In this study, we developed a novel optical-flow algorithm for determining the wall shear-stress on the surface of objects.

The algorithm solves the thin-oil-film equation using a numerical scheme that recovers local features neglected by smoothing filters. A variational formulation with a smoothness constraint was applied to extract the global shear-stress fields. The algorithm was then applied to scalar images generated using direct numerical simulation (DNS) method, which revealed that the errors were smaller than those of conventional methods. The application of the proposed algorithm to recover the wall shear-stress on a low-aspect-ratio wing and on an axisymmetric boattail model taken as examples in this study showed a strong potential for analysing shear-stress fields. Compared to the methods used in previous studies, proposed method reveals more local features of separation line and singular points on object surface.

Keywords Wall shear-stress·Optical flow·Thin-oil-film equation·Sub-grid model

1 Introduction

1

The wall shear-stress, along with pressure and temperature,

2

are important parameters in fluid mechanics. In fact, the

3

skin-friction drag due to the shear-stress for conventional

4

aircraft during cruising can reach around 50% of the total

5

drag [1]. In addition, the most complicated features of the

6

wall shear-stress occur near the separation and reattachment

7

regions, where flow behaviour suddenly changes. A good

8

understanding of the shear-stress is important for analysing

9

flow behaviour and for developing the most appropriate drag

10

reduction strategy.

11

The lines formed by wall shear-stress coincide with

12

streamlines and determine the average flow fields near the

13

interaction surfaces of the object and the fluid [2]. How-

14

ever, measuring the shear-stress is significantly complicated.

15

Previously, point measurement techniques, such as the Pitot

16

B

The Hung Tran thehungmfti@gmail.com

1 Faculty of Aerospace Engineering, Le Quy Don Technical University, Hanoi 10000, Vietnam

2 Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China

3 School of Aeronautics and Astronautics, University of Chinese Academy of Sciences, Beijing 100049, China

tube, hot-film sensors [3], sub-layer fence [4] and wall hot- 17

wire [5], have been applied for analysing the shear-stress at 18

some positions on the surface. Clearly, the application of a 19

sensor significantly disturbs the flow and reduces the accu- 20

racy of measurement [6]. An oil-flow visualization technique 21

was developed and widely applied for visualizing surface 22

shear-stress [7,8]. However, that technique allows only qual- 23

itative measurement of skin-friction fields. 24

Recently, developments in technology and computational 25

processes have provided an advantageous tool for analysing 26

shear-stress fields. Shear-sensitive liquid crystal [9, 10], 27

surface-stress-sensitive film [11] and global-luminescent oil 28

film [12], which are based on the analyses of oil-film images, 29

enable to extract the global shear-stress fields. In those 30

measurement techniques, global-luminescent-oil-film skin- 31

friction measurement, which solves the thin-oil-film equation 32

on the image plane, shows a simple setup and high potential 33

for extracting the skin-friction fields. The experimental setup 34

for measuring the global-luminescent-oil-film skin friction 35

is analogous to that used in conventional oil-flow visualiza- 36

tion. In data processing, skin friction is calculated through 37

optical-flow velocity and intensity of the luminescent oil. The 38

technique uses Horn and Schunck [13] method as an initial 39

solution for numerical calculation. By performing a calibra- 40

tion process in advance, the quantitative wall shear-stress can 41

be obtained. Application of the technique in a delta wing [14], 42

(2)

low-aspect-ratio rectangular wing [16], wing-junction flow

43

[15] and boattail models [17,18] have shown good potential

44

of the skin-friction fields. However, since a smoothing filter

45

is applied for both the Horn and Schunck [13] and Liu et al.

46

[12] estimations, the skin-friction topology is highly smooth

47

and the small local features are not well preserved. One way

48

to improve the accuracy of the measurement is to increase

49

the resolution of the camera or to focus on the measurement

50

region. However, those methods are time-consuming and are

51

not suitable for large flow-field regions.

52

In this study, we propose a novel optical-flow algorithm

53

for extracting the wall shear-stress fields. The main role of

54

the algorithm is to recover the local features, which is often

55

neglected by smoothing filters. Similar to the technique pre-

56

sented by Liu et al. [12], the wall shear-stress is extracted

57

from sequential images obtained during the oil-flow visu-

58

alization process. Additionally, the thin-oil-film equation is

59

solved using an additional constraint. However, a Gaussian

60

filter is applied to the governing equation. Consequently,

61

the large surface features are solved while local features are

62

recovered using a sub-grid scalar function. This methodology

63

is analogous to the large eddy simulation (LES) algorithm.

64

We present a detailed explanation of this technique in Sect.2

65

and in Sects. 3 and 4 we describe our validation of the

66

technique based on synthetic and experimental data from

67

low-aspect-ratio wings and boattail models, respectively.

68

2 Wall shear-stress diagnostic

69

2.1 Thin-oil-film equation and optical flow methods

70

An oil-film skin-friction measurement technique is applied

71

to extract the wall shear-stress. In detail, a thin lumines-

72

cent oil film is painted on the surface of objects prior to

73

the experimental process. The variation of oil-film thickness

74

during experiments is recorded by a high-speed camera for

75

post-processing. The relation between the wall shear-stress

76

vectorsτwand the thickness of the oil filmhcan be obtained

77

from the thin-oil-film equation [19,20]. Clearly, if the tempo-

78

ral and spatial variations of oil-film thicknesshare measured,

79

the wall shear-stressτwcan be recovered as:

80

∂h

∂t +∇ · h2τw

2μ(pρg)h3 3μ

0. (1)

81 82

Here,pis the distribution of the wall static pressure on the

83

oil film,ρis the oil density,gis a gravity vector andμis the

84

dynamic viscosity of the oil.

85

The thickness of oil filmhis considered to be proportional

86

to the luminescent intensityI,i.e.,hκI,where the parame-

87

terκis a constant. Note that the above assumption was widely

88

applied for determining the wall shear-stress [12,21]. The

89

uncertainty of the shear-stress results associated with using 90

that relation is less than 12%. The uncertainty can be reduced 91

by painting the object surface in white and by using suffi- 92

ciently thin oil layer [22]. By applying that relation, Eq. (1) 93

becomes: 94

∂I

∂t +∇ ·(Iu)0, (2) 9596

where u refers to the optical-flow velocity vectors, which is 97

written as follows: 98

u κI

2μτw(pρg2I2

3μ . (3) 10099

Equation (2) shows the relation between the wall shear- 101

stress and optical-flow velocity vectors of the luminescent 102

oil. In most cases, the pressure gradient and gravity terms are 103

around two orders of magnitude smaller than the shear-stress 104

term and are neglected [6,23,24]. The uncertainty related 105

to the neglect of the pressure gradient is often less than 2% 106

[25]. However, near the separation position, where the pres- 107

sure gradient is high and the wall shear-stress is close to zero, 108

the effect becomes large.Appendix Ashows our evaluation 109

of the effect of the pressure gradient on the wall shear- 110

stress. Clearly, when the pressure gradient is sufficiently high 111

(dP/dx> 105Pa/m), its effect should be considered. 112

When the optical-flow velocity vectors are obtained, the 113

wall shear-stress can be recovered using Eq. (3). By conduct- 114

ing a calibration process in advance, the quantitative wall 115

shear-stress fields can be obtained. Since this study focuses 116

on the ability of the proposed technique to extract the skin- 117

friction topology, we do not perform a calibration. However, 118

the wall shear-stress in region of small pressure-gradient can 119

be found from the boundary layer profile using a logarithmic 120

formulation. The indirect calibration process can be found in 121

previous study by Tran et al. [18]. 122

2.2 Determination of optical-flow velocity vectors 123

Previously, the optical-flow technique processed each pixel 124

of the image frames. Additionally, a specific filter is required 125

to smooth the luminescent intensity prior to the numerical 126

process [13]. The small-scale components, which are limited 127

by the spatial resolution of images and by filtering process, 128

were often neglected. However, the sub-grid scale motions 129

evidently affect large components and final solutions [26, 130

27]. To recover these local features, LES method is applied 131

in this study. In fact, the LES was widely applied for compu- 132

tational fluid dynamics in previous studies [28–30]. In detail, 133

a specific filter, which is often Gaussian in type, is applied 134

to Eq. (2). The cut-off wavelengthkc in the optical flow is 135

(3)

depended on the filter size¯ of the image by the following

136

equation:

137

kc π

¯. (4)

138 139

The optical-flow velocity vectors and image intensity can

140

be expressed as: u ¯u + u,I ¯I + I,whereu¯,I¯are filter

141

components, u,Iare sub-filtered portion of the velocity and

142

image intensity. The filter components can be solved from

143

the optical-flow equation, while small-scale components are

144

simulated by appropriate function.

145

By applying the filter, Eq. (2) becomes:

146

∂I¯

∂t +∇ ·(uI)0. (5)

147 148

Similar the LES method in computational fluid dynamics,

149

the component inside the divergence operator can be rewrit-

150

ten as: uI ¯uI¯+τs, whereτsis the sub-grid scalar flux [26]:

151

τs=L+C+RandL ¯uI¯− ¯uI¯is Leonard stress,R uI

152

is sub-grid Reynolds stress andC ¯Iu+Iu is cross stress.¯

153

Using the above relation, Eq. (5) can be expressed as:

154

∂I¯

∂t +∇ ·(u¯I¯) +∇ ·τs0. (6)

155 156

The sub-grid scalar fluxτscontains interaction between

157

filter resolved components and unresolved components,

158

which are missed in the oil-film images. Since the sub-grid

159

scale affects the resolved term, it cannot be neglected. The

160

sub-grid term is determined by the balance of energy transfer

161

between the small-scale and the large-scale features. Previ-

162

ously, major works have been conducted for modelling theτs

163

[26,27]. In this study, we proposed to apply sub-grid scalar

164

flux model developed by Cassisa et al. [31] for solving scalar

165

transport equations:τsDt∇ ¯I, whereDt is the turbu-

166

lence diffusion coefficient. Additionally, a simple viscosity

167

model developed by Smagorinsky [32] is used to estimateDt

168

as bellow:

169

Dt cs¯2

√2Prsgs

∂u¯

∂y+ ∂v¯

∂x

. (7)

170 171

Here,u, vare two components of the optical-flow veloc-

172

ity vectorsu,¯ Prsgs is the sub-grid Prandtl number (Prsgs

173

0.1–1) and cs is a constant coefficient (cs 0.1–0.2).

174

Consequently, turbulence diffusion coefficientDt could be

175

obtained directly from the optical-flow velocity of the previ-

176

ous interactions. For better estimation, the mean value ofDt

177

is calculated after each interaction and is applied to all pixels

178

in the calculation domain.

179

When the special change of velocity vectoruis small, the 180

divergence of the velocity vectors can be considered as zero. 181

Using that assumption, Eq. (6) becomes: 182

∂I¯

∂t +u¯· ∇ ¯IDtI¯0, (8) 183184

whereis the Laplace operator. Note that the divergence- 185

free of velocity vectors is not always satisfied for experimen- 186

tal data. However, as shown inAppendix B, the divergence 187

of velocity is sufficiently smaller by comparison to velocity 188

vector term and it can be neglected in numerical process. 189

To reduce the error due to assumption of divergence-free of 190

velocity, the thin oil-film layer with low viscosity should be 191

applied for the measurement. 192

To solve Eq. (8) with two unknown optical-flow velocity 193

components (u¯,v), a variational formulation with a smooth-¯ 194

ness constraint is applied. In the conventional technique 195

developed by Horn and Schunck [13], the smoothness con- 196

straint, which is called the regulation term, is considered to 197

be the magnitude of the velocity gradient: 198

JR

Ω

∇ ¯u(x,t)2dx. (9) 199200

Previously, Liu et al. [12] used the same regulation term 201

as that proposed by Horn and Schunck [13]. Note that the 202

magnitude of the velocity gradient is equivalent to the penalty 203

of divergence and vorticity magnitudes at the same rate [33]. 204

JR

Ω

∇ · ¯u(x,t)2+∇ × ¯u(x,t)2 dx. (10) 205206

However, with respect to the detailed flow behaviour, the 207

above two components are not always equal. In this study, 208

we proposed separated the divergence and vorticity penalties 209

by introducing a smoothing term as: 210

JR

Ω

α1∇ · ¯u(x,t)2+α2∇ × ¯u(x,t)2 dx. (11) 211212

whereα1andα2are Lagrange multipliers that are selected 213

prior to the numerical process. 214

The first term in Eq. (11) is characterized for divergence of 215

the optical-flow velocity vector and the second component 216

is characterized for rotation of vortex structure. Note that 217

in optical-flow method, the observation term and regulation 218

term should have similar order of magnitude. When the coef- 219

ficientα1is significantly large (α1α2), the divergence of 220

the optical-flow velocity vectors in Eq. (11) becomes small 221

for balancing the observation and regulation terms. Conse- 222

quently, the estimated results show only vortex structure [34]. 223

Conversely, whenα2is large, the divergence of the velocity 224

(4)

is dominated in the estimated results. Clearly, depending on

225

detailed object, the divergence and vorticity components of

226

skin-friction fields could be different. The selection of two

227

multiplier coefficients provides clear physical meaning and is

228

flexible in selection, particularly when the initial information

229

of flow can be observed from oil-film distribution.

230

Finally, the optical-flow velocity can be found by mini-

231

mizing the observation and regulation terms as follows:

232

J(u)¯

Ω

∂I¯

∂t +u¯· ∇ ¯IDtI¯ 2

dx1dx2

233

+

Ω

α1∇ · ¯u(x,t)2+α2∇ × ¯u(x,t)2 dx1dx2. (12)

234 235

InAppendix C, we investigate the effect of the Lagrange

236

multipliers on the error of measurement. The results show

237

that the solution is not significantly sensitive to the selection

238

of a Lagrange multiplier in a certain range. However, since

239

the solutions of variational method depend on Lagrange mul-

240

tipliers, the numerical results should be always compared to

241

experimental observations to check the reasonability.

242

A numerical process for calculating skin friction is pre-

243

sented inAppendix D. This variational method enables the

244

extraction of velocity vectors from a pair of images, which

245

is called a snapshot solution. By averaging the snapshot

246

solutions obtained at different times, the mean optical-flow

247

velocity vectors can be obtained. Since the wall shear-stress

248

and optical-flow velocity vectors can be obtained using

249

Eq. (3), proposed method can be used to extract the wall

250

shear-stress as well as the velocity vectors from the scalar

251

sequence images.

252

3 Validation of the method for synthetic

253

images

254

3.1 Images data acquisition

255

In this section, proposed method is applied to recover the

256

optical-flow velocity from synthetic scalar images generated

257

by the direct numerical simulation (DNS) method [35]. Here,

258

the dye concentration on the image represents homogeneous

259

and isotropic two-dimensional turbulent flow at Reynolds

260

numberRe3000. Since the DNS database shows exact

261

solution of turbulent flow with a large scale range of energy

262

spectrum, it has been widely used in previous studies for dif-

263

ferent algorithms in recovering velocity vectors such as for

264

scalar transport equation [31,36] stochastic transport equa-

265

tions [37] and wavelet-based optical-flow technique [38]. The

266

summary of synthetic images is shown in Table1.

267

For data processing, a Gaussian filter with a variance of

268

σ 1 was applied to the images before data processing to

269

Table 1 Parameters of synthetic image

Type 8 bit

Dimension 2π×2π

Grid size (pixels) 256×256

Time between two images (s) 0.1

Number of image for data processing 100

reduce the noise level. This processing also filtered the local 270

features on the images. The filter width¯ can be computed 271

as¯ 2√

3σ 3.5 [27]. Since flow is dominated by vortex 272

structure, the vorticity regulation term is fixed atα20. The 273

Lagrange multiplier for divergence term was selectedα1 274 1, as it had been used in a previous study [31]. 275

In this study, the numerical scheme was used to minimize 276

the difference between the solutions in the current and previ- 277

ous interactions. Clearly, with the initial setup of the constant 278

parameterscsandPrsgs, the turbulence diffusion coeffi- 279

cientDtconverges to fixed values. As shown inAppendix C, 280

the constantscs0.19 andPrsgs0.1, which yield the con- 281

vergent coefficientDt 0.22, provide the smallest average 282

angle error and were selected for numerical process. 283

3.2 Extraction of optical-flow velocity vectors 284

To examine the results of the algorithm, we selected two 285

scalar images at times of 0.39 s and 0.40 s. Figure1shows 286

these two images and Fig. 2 shows the transient vortic- 287

ity maps obtained from different optical-flow algorithms. 288

Clearly, in the Horn and Schunck method [13] and Liu et al. 289

[12] method, the filter was applied for image before data pro- 290

cessing and the vorticity fields were not recovered totally. 291

In the current methods, vorticity fields are close to those 292

obtained by the DNS results (Fig.2a). Furthermore, the vor- 293

ticity vectors are highly smooth, which are much improved 294

from those obtained in previous studies. It can be explained 295

that the turbulence diffusion termDtplays an important role 296

in recovering local features, which are often smoothed by 297

filters during data processing. 298

To determine the measurement uncertainty, the average 299

angle error (AAE) and root mean square error (RMSE) are 300

calculated [39–41]. The formulas for AAE and RMSE are 301

shown as below: 302

A AE 1 N

N n1

arccos

uc(x,t)·ue(x,t) uc(x,t)ue(x,t)

, (13) 303304

R M S E 1 N

N

n1

uc(x,t)−ue(x,t)2. (14) 305306

Here,uc(x,t)(uc, vc, 1.0) is the exact solution in three- 307

dimensional space, ue(x,t) (ue, ve, 1.0) is the estimated 308

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Fig. 1 Vorticity maps generated using different calculation methods:

at0.39 s,bt0.40 s

solution in 3D space andNis total number of pixels. The use

309

of 3D space avoids the problem of zero flow, which could

310

affect theAAEevaluation. Additionally, since theAAEcon-

311

tains an arbitrary constant of 1.0, the trends ofAAEandRMSE

312

could differ.

313

Figure 3 shows AAE andRMSE by different methods. 314

Here, the x-axis indicates the number of images. At time 315

below 1.0 s (image number 10), the errors of the three 316

methods are similar. At time higher 1.0 s, Liu et al. [12] 317

estimation improves its solution accuracy. However, the 318

AAE and RMSE values are obviously smaller for proposed 319

method. It can be explained that the sub-grid scalar function 320

enables to recover local features and the errors are reduced. 321

Figure4shows the average kinetic energy spectra of the 322

horizontal velocity in log–log coordinates as a function of 323

wavelengthk. To obtain those values, the kinetic energy spec- 324

tra were calculated for each pixel and then averaged over the 325

whole image frame. The average kinetic energy spectra will 326

include all wavelengths of the flow [42]. The large scale is 327

featured by low wavelength while the small scale is featured 328

by high wavelength. In the region of small scale, the energy 329

spectra should be a function ofk5/3[43]. Proposed method 330

yields an energy spectrum closer to that obtained by the DNS. 331

The accuracy of the kinetic energy spectrum can be improved 332

Fig. 2 Vorticity maps foraexact DNS solution,bHorn and Schunck method [13],cLiu et al. method [12],dproposed method

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Fig. 3 Evaluation of errors by different calculation methods:aAAE, bRMSE

Fig. 4 Power spectral energies of optical-flow velocity

by the selection of suitable size of filters for the estimation

333

scheme. However, the variance of filter yields a small change

334

ofAAEandRMSE[31]. Clearly, the use of frequency analysis

335

alone is not sufficient for determining the correct flow struc-

336

ture. By showing the pattern of the kinetic energy spectrum,

337

we have simply confirmed that proposed method improves

338

the estimated results for numerical calculation.

339

4 Analysis of experimental results

340

by proposed method

341

4.1 Skin-friction field on low-aspect-ratio wing 342

4.1.1 Experimental setup 343

To examine the ability of proposed method to extract 344

skin-friction fields, examples on experimental images were 345

analyzed. The experiments were conducted in the small wind 346

tunnel at Western Michigan University, USA and the image 347

data set can be downloaded from the Internet.1In that study, 348

a low-aspect-ratio wing (AR1.2) with a cross-section of 349

NACA 0012 was supported at the bottom. The luminescent 350

oil, which is a mixture of silicon oil and UV dye (Petroleum 351

Tracer Concentrate DFSB-K175), was painted onto the upper 352

wing surface and then illuminated by two UV lamps at the 353

top [44]. The oil movement during the experimental process 354

was recorded by a high-speed camera at a frame rate of 25 355

frames per second. Experiments were conducted at a veloc- 356

ity of 20 m/s and the Reynolds number based on the chord 357

length was aroundRe1.7×105. The angle of attack was 358

set to 18º. Details of the experimental process were presented 359

in Liu et al. [44]. 360

The ratio of α12 is also an important feature in this 361

study. If α121, the estimated results show a domina- 362 tion of solenoidal motion, whereas atα121, irrotational 363

motion is dominant [34]. However, a small change in the 364

Lagrange multipliers has little effect on the numerical results 365

[45]. In this case, Lagrange multipliers are selected atα1 366 20 andα218 for the divergence and vorticity components, 367

respectively. The ratioα1/α2is around 1, which can be com- 368

pared with the results obtained by Liu et al. [12]. Effect of 369

pressure gradient on skin-friction fields is neglected in the 370

numerical process. 371

To reduce the random noise of the camera, images are 372

filtered by Gaussian filter with the variance ofσ 4, which 373

leads to a filter width of¯ 2√

3σ 0.9. Compared to the 374 DNS calculation, the variance of the Gaussian filter is much 375

greater for reducing the noise. Since the Lagrange multipliers 376

are large for real images, the turbulence diffusion coefficient 377

Dt must be high to balance the observation and regulation 378

terms. Therefore, the same constants cs and Prsgs can be 379

chosen for the DNS calculation (Appendix C). 380

4.1.2 Results and discussion 381

Figure 5a shows the last oil-film image taken during the 382

experimental process. As can be seen, a large-reversed flow 383

occurs on the rear of the wing. The high accumulation of oil 384

film at aroundx/c0.4 indicates the separation position. Two 385 1 http://www.aero.mech.tohoku.ac.jp/study/advanced.htmls

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Fig. 5 Skin-friction topology by the proposed optical-flow algorithm:atypical luminescent oil-film image,brelative skin-friction magnitude, cskin-friction vectors,dskin-friction line on the image surface

concentrations on the side edge at (x/c, y/c)(0.6,±0.4) and

386

two small concentrations near the trailing edge at (x/c, y/c)

387

(1,±0.6) are also evident from the oil-film pattern. On the

388

side edges of the wing, regions of high luminescent intensity

389

are linked to the occurrence of separations on the model sur-

390

face. A reattachment region with low luminescent intensity

391

is located between two separation lines. Interestingly, due

392

to the movement of the luminescent oil, small focuses are

393

observed near the trailing edges of the wing, which could

394

be observed from the development of luminescent oil on the

395

surface.

396

Figure 5b–d shows the magnitudes of the skin-friction

397

coefficient (Cf=τw/q), skin-friction vectors and skin-friction

398

lines, which were averaged from 190 image pairs. Here, thex-

399

andy-axes are normalized by the chord of the wing. To obtain

400

a clear view, we plotted only a limited number of skin-friction

401

vectors and skin-friction lines. The flow structure is clearly

402

indicated by the average streamline on the wing (Fig.5d).

403

A large separation flow occurred atx/c0.35, as indicated 404

by the accumulation of oil in the image (Fig.5a). Due to the 405

effect of the finished wingspan, separation and reattachment 406

lines can be observed on the two side edges of the model. 407

Additionally, the optical-flow results show a focus near the 408

trailing edge of the model, which is highly consistent with 409

the pattern of luminescent oil (Fig.5a). 410

Figure6shows the skin-friction lines obtained by Liu et al. 411

[12] and proposed methods. For a clear view, we note only the 412

different features of the two methods. It is clear that much 413

more detailed information is represented by the proposed 414

formulation. Near the side edge of the wing, a reattachment 415

line between two separation lines is evident via proposed 416

method. However, in the previous formulation, only a reat- 417

tachment line and a separation line are observed. In addition, 418

we also observe a focus near the trailing edge of the model, 419

which are resulted from the moving-out and moving-in of 420

the luminescent oil on the surface. However, in the Liu et al. 421

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Fig. 6 Skin-friction lines byaLiu et al. method [12],bproposed method

method, the flow becomes smooth around the trailing edge.

422

The current numerical scheme is highly consistent with the

423

temporal and spatial development of luminescent oil on the

424

surface.

425

For the detailed skin-friction structure, a zoomed-in view

426

of the region near the lower focus is represented (Fig. 7).

427

Clearly, skin-friction lines obtained by Liu et al. [12] formu-

428

lation smoothed the flow somewhat such that only a saddle

429

is observed. In proposed method, we observe three saddles,

430

a node and a focus. To evaluate the consistency of the skin-

431

friction topology, the Poincare–Bendixson (P–B) index is

432

used for the zoomed-in skin friction region. In fact, P-B index

433

was widely applied in previous studies to examine the skin-

434

friction topology [44]. The P–B formula for singular points

435

inside a Jordan curve is as follows:

436

#N−#S1+(#Z+−#Z)/2, (14)

437 438

Fig. 7 Zoomed-in skin-friction lines near the focus position:aLiu et al.

method [12],bproposed method,czoomed-in skin-friction lines in region outlined by black dashed–line box

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Table 2 Numbers of singular points on the surface

Case studies Numerical scheme #N #S #Z+ #Z #N- #S-1 - (#Z+- #Z)/2

Zoomed-in view of low-aspect ratio wing

Liu et al. method [12] 0 1 0 4 0

Proposed method 2 3 0 4 0

Boattail model Liu et al. method [12] 4 5 0 4 0

Proposed 11 12 0 4 0

Fig. 8 Relative skin-friction values at centerliney/c0

where#N is the total number of nodes and focuses,#Sis the

439

total number of saddles and#Z+ and#Z are the number

440

of positive and negative switch points, respectively. If there

441

is a segment of skin-friction lines move inward to outward

442

across Jordan curve, it is determined a negative switch point

443

#Z. Conversely, if the segment of skin-friction lines move

444

outward to inward, it is determined a positive switch point

445

#Z+.

446

As shown in Fig.7, inside the Jordan curveABCDA, the

447

Liu et al. [12] method indicates that #S1, #Z4, whereas

448

proposed method shows #N2, #S3, #Z4. The P–B

449

formula is satisfied for the two methods (Table2). Conse-

450

quently, the topology of the skin-friction fields is consistent in

451

both calculation methods. Since the luminescent oil is highly

452

accumulated near the separation position, the flow pattern

453

near that region could not be clearly observed from the lumi-

454

nescent image taken after experimental process. However,

455

the topology should be clear from temporal development of

456

the oil on the surface. To improve the results, the thin oil

457

layer with low viscosity should be applied.

458

Figure8shows skin-friction coefficients at the centerline

459

y/c0. The skin friction exhibits a large region of separation

460

with negative values [46]. A similar quality is obtained by

461

the proposed and Liu et al. [12] methods. However, near the

462

region ofx/c0.8, the results from two methods are highly

463

different. It can be explained that proposed method shows

464

more sensitive in capture the movement of oil film, which

465

leads to large magnitude of skin-friction near the end of the

466

wing. Note that the effect of the filter and sub-scale motion

467

on the skin-friction pattern is represented for the first time in

468

this study.

469

Fig. 9 Experimental setup for skin-friction measurement on boattail model (from Tran et al. [18])

4.2 Skin-friction vector on an axisymmetric boattail 470

model 471

4.2.1 Experimental setup 472

In this section, proposed method was applied to recover skin- 473

friction fields in an axisymmetric boattail model in low-speed 474

conditions. The experiments were conducted on a square 475

0.3×0.3 m2test section in a low-speed wind tunnel at the 476

Department of Aerospace Engineering, Tohoku University, 477

Japan. Figure9shows the experimental setup for the skin- 478

friction measurement. The axisymmetric model, which has a 479

diameterDof 30 mm and a total length of 8.4D, is supported 480

at the bottom by a strut with a cross-section of airfoil NACA 481

0018. The boattail model has a conical shape, with an angle of 482

18º and length of 0.7D. UV-LEDs and a CMOS camera were 483

fixed to the top for illuminating and recording the movement 484

of the luminescent oil, respectively. Differing from previous 485

experiments in the low-aspect-ratio wing, the luminescent 486

oil used in the boattail model is a mixture of oleic acid and 487

chloro-9,10- bis(phenylethynyl) anthracene (C30H17Cl) dye 488

in the proportion of 1000 ml:1 mg. The luminescent oil is 489

painted onto the top surface of the boattail before turning on 490

the wind tunnel. Experiments were conducted at a velocity 491

of 45 m/s giving based-diameter Reynolds number of around 492

ReD8.89×104. The turbulent intensity of the wind tunnel 493

is less than 0.5% at that flow condition. The camera recorded 494

images at a frame rate of 5 frames per second. 495

(10)

A total of 120 images was used for data processing. In pre-

496

vious studies of two-dimensional and axisymmetric flows,

497

separation and reattachment lines have often formed saddles

498

and nodes. However, those features were not observed in the

499

axisymmetric boattail flows by Tran et al. [18]. To obtain that

500

solenoidal motion, the Lagrange coefficient of the vorticity

501

component should be smaller than the divergence component

502

[34]. Since the Lagrange coefficient serves as a regulation

503

term, a small change inα2/α1has a negligible effect on the

504

numerical results. However, when the ratio ofα2/α1is close

505

to zero, the estimated results show large vortex structure,

506

which is unsuitable for most aerospace applications. In this

507

study, Lagrange multipliers were set atα120 and α2

508

10 for the divergence and vorticity components, respectively.

509

Note that the Lagrange multiplier ofα120 has been widely

510

applied for numerical schemes recovering flow fields in pre-

511

vious studies [17,40]. For comparison, Liu et al. method

512

was also conducted to recover the skin-friction fields with

513

the same Lagrange multiplierα120 of the initial approx-

514

imation. As shown inAppendix C, the different Lagrange

515

multiplier coefficients in a small range have little effect on

516

the processing results. Consequently, it is reasonable to com-

517

pare results of both methods. Pressure gradient terms are

518

neglected in both numerical methods. The other parameters

519

were selected similar to those used in the low-aspect-ratio

520

wing.

521

4.2.2 Results and discussion

522

Figure10shows the last luminescent oil-film image taken

523

after wind tunnel test. Here, the x-axis is normalized by

524

the model diameter and they-axis indicates the polar angle,

525

which was cropped at (ϕ12)(45º, 135º) (see Ref. [17]).

526

An accumulation of luminescent oil is evident at the separa-

527

tion position on the surface. Clearly, flow separation occurs

528

near the boattail conjunction and reattaches further along the

529

surface to form a recirculation region. The flow separates

530

again at the base edge to form a large wake behind the base.

531

Since the skin friction on the rear surface is sufficiently low,

532

we can observe no details of the reattachment position in

533

the images itself. However, the location of the separation is

534

evident and can be used to examine the numerical results.

535

Figure11shows the skin-friction topologies obtained by

536

the two numerical methods. Highly consistent structures are

537

obtained by Liu et al. [12] and proposed methods. In detail, a

538

separation line presents near the conjunction and a reattach-

539

ment line occurs at aroundx/D0.2. These separation and

540

reattachment lines are formed by nodes and saddles. How-

541

ever, in Liu et al. method, only the main characteristics near

542

the separation and reattachment lines can be observed and

543

there are a limited number of singular points. As shown in

544

Fig.11b, proposed method shows more detailed informa-

545

tion around the separation and reattachment regions, which

546

Fig. 10 Luminescent oil-film images taken after wind tunnel test (V 45 m/s,ReD8.89×104, boattail angle of 18º)

highly agrees with previous observations by Roshko and 547

Thomke [47] for separation flow on axisymmetric bodies 548

and by Chen et al. [48] for back-step flow. Since pressure 549

gradient and gravity terms are neglected in both numerical 550

formulations, the different features of flow fields near sepa- 551

ration and reattachment regions should come from numerical 552

schemes themselves. 553

Clearly, the selection of the Lagrange multipliers and other 554

parameters is sufficient for recovering the flow fields. The 555

numerical scheme also indicates that at a fixedα1 20, a 556

small change inα2has little effect on the average flow fields. 557

It can be explained that the regulation term serves to smooth 558

the solution, which is not as crucial as the observation term 559

and does not strongly affect the solutions [36]. However, for 560

better estimation, a detailed scheme for optimizing Lagrange 561

multipliers should be considered in future work. 562

Table 2 lists the singular points in Jordan curve 563

ABCDEFA. In both methods, four negative switch points 564

#Z1,#Z2,#Z3,#Z4 can be observed. It is evident that 565

the numbers of nodes and saddles are significantly differ- 566

ent in the two studies. The P–B index is satisfied for both 567

calculation schemes. 568

Figure12shows the average skin-friction distribution on 569

the centerline by the two methods. Here, the skin-friction val- 570

ues are normalized at the positionx/D −0.06. The negative 571

value indicates a region of reversed flow. A highly consis- 572

tent skin-friction coefficient is observed for both methods. 573

Proposed method, therefore, shows a high ability to extract 574

skin-friction fields. 575

5 Conclusion

576

In this study, a simple optical-flow algorithm was devel- 577

oped for wall shear-stress measurement on the object surface. 578

This method solves the thin-oil-film equation by a varia- 579

(11)

Fig. 11 Skin-friction lines:aLiu et al. [12],bproposed methods

Fig. 12 Average skin-friction distribution on the centerlineϕ90º

tional formulation with a smoothness constraint. Differing

580

from previous studies of wall shear-stress measurement, the

581

proposed formulation accounts for the effect of a smooth-

582

ing filter and sub-grid scale on the skin-friction topology.

583

Additionally, the regulation term is separated into divergence

584

and vorticity components, which provides more flexibility

585

regarding details of the flow condition. The validation of DNS 586

images revealed good potential for the recovery of vorticity 587

fields. By comparison to previous methods, proposed method 588

reduces remarkably average angle error and root mean square 589

error. The application of experimental data to a low-aspect- 590

ratio wing and an axisymmetric boattail model demonstrated 591

that proposed method provided a much greater level of detail 592

regarding the structure of near-wall flow than the previous 593

method reported by Liu et al. [12], such as separation lines, 594

saddles, nodes and focuses. Those observable local features 595

from history of luminescent oil, which were often smoothed 596

in previous studies, were recovered well by proposed method. 597

The method can also be applied to capture the velocity fields 598

from scalar images taken during the experimental process. 599

Acknowledgements The authors would like to thank Professor Keisuke 600

Asai and Professor Taku Nonomura at the Department of Aerospace 601

Engineering, Tohoku University, Sendai, Japan for their support dur- 602

ing the experimental processes. We also would like to thank Professor 603

Tianshu Liu at Department of Mechanical and Aerospace Engineering, 604

Western Michigan University, Michigan, USA for his support of images 605

dataset in numerical process. This work was supported by Le Quy Don 606

Technical University and Duy Tan University, Hanoi, Vietnam. 607

Appendix A. Effect of pressure gradient

608

on wall shear-stress results

609

The current study neglects the effects of pressure gradient 610

and gravity on the wall shear-stress. This assumption holds 611

in principle for most results except those near the separation 612

position, where pressure is highly variable and the wall shear- 613

stress is close to zero. Clearly, the pressure gradient can affect 614

the shear-stress value as well as streamline around the sur- 615

face. However, for most cases of two-dimensional flow, the 616

pressure gradient in the direction normal to the flow fields is 617

often small and its effect on the normal flow can be neglected. 618

The oil flow in the normal direction can be considered to 619

follow the direction of the boundary-layer motion. In this 620

section, the effect of pressure on the shear-stress component, 621

which is parallel with the flow direction, is evaluated. 622

Experiments were conducted on a boattail model of 20º at 623

two velocities ofV22 m/s andV45 m/s. The exper- 624

imental setup for the shear-stress measurement was the same 625

as that for the boattail model of 18º in Sect.4.2. Additionally, 626

the pressure on the vertical surface was measured by silicon 627

tubes. Details of the experimental setup are available in the 628

paper by Tran et al. [49]. 629

Figure 13 shows the pressure distribution on the boat- 630

tail surface for the different velocities. Clearly, the minimum 631

pressure occurs near the shoulder, where the flow separated. 632

The change in pressure leads to a high pressure gradient in 633

this region, as shown in Fig.14. By comparison to the case of 634

V22 m/s, the pressure gradient atV45 m/s changes 635

(12)

Fig. 13 Pressure coefficient on boattail surface

Fig. 14 Pressure gradient on boattail surface

greatly. It should be noted that the pressure gradient atV

636

45 m/s is much higher for a conventional flow in subsonic

637

conditions, ranging within the region of 102–104Pa/m [50].

638 639

In flow measurement technique, the maximum thickness

640

of oil film layer is often in from 30μm to 100μm [25]. To

641

estimate the effect of the pressure gradient, the oil thickness

642

of 50μm was used in this study, which yields a constant

643

parameterκofκ2×105. The oil viscosity isμ0.0276

644

(kg/ms) and the oil density isρ895 kg/m3. From Eq. (3),

645

the wall shear-stress can be estimated as follows:

646

τw,s 2μ κIu +2κI

3 (pρg). (A1)

647 648

In other words, we write the wall shear-stress as the sum

649

ofτw,s τw,0+τw,pg. Here,τw,0 2κμIu is the shear-stress

650

term neglecting the effect of the pressure gradient and gravity

651

terms, andτw,pg 2κ3I(pρg)is the shear-stress term

652

formed by pressure gradients and gravity.

653

Figure15shows the results obtained using the two skin-

654

friction components for different velocities. The main effect

655

occurs near the separation position, where the pressure gra-

656

dient is high and the wall shear-stress is close to zero. The

657

effects of pressure and gravity are very small and have a

658

negligible effect on the wall shear-stress atV 22 m/s.

659

Fig. 15 Wall shear-stress with consideration of the pressure gradient effect:aV22 m/s,bV45 m/s

However, atV45 m/s, the effect of the pressure is very 660

high near the separation position, which causes the separation 661

position to move forward. These results are highly consistent 662

with previous observations by Zhang [50] and Squire [51], 663

whereby the separation positions changed less than 5% in 664

a region with a high pressure gradient. Clearly, the effect of 665

pressure gradient leads to early flow separation. However, the 666

change in the shear-stress tendency is negligible. The effect 667

of the pressure gradient can be reduced by the use of a thinner 668

oil layer. To obtain a more highly accurate skin-friction cal- 669

culation, further experimental setups and numerical schemes 670

should be considered. 671

Appendix B. Divergence-free of optical-flow

672

velocity vectors

673

In this study, the optical-flow velocity vectors were consid- 674

ered as divergence-free parameters. However, this assump- 675

tion is not always satisfied. The divergence-free optical-flow 676

velocity can be tested by the relation between the wall shear- 677

stress and the luminescent intensity. Specifically, the relation 678

between the optical-flow velocity, luminescent intensity and 679

wall shear-stress is as follows: 680

∇ ·u ∇ · κI

2μτw

, (B1) 681682

(13)

Fig. 16 Magnitude of mean optical-flow velocity and mean divergence of velocity in the region ofx/c0 to 1 andy/c0.15 to 0.15

or

683

∇ ·u κI 2μ∇ ·τw+

κI

2μ

·τw. (B2)

684 685

This relation can be used to determine whether the optical-

686

flow velocity is in fact divergence-free.

687

To obtain the above relation, we used the data of low-

688

aspect-ratio wing presented in Sect. 4.1. Here, the skin-

689

friction fields and luminescent intensity is averaged in the

690

region ofx/c 0 to 1 andy/c −0.15 to 0.15. Figure16

691

shows the magnitude of the mean velocity, which was calcu-

692

lated by u 2κμIτw,and the mean divergence of the velocity

693

obtained using Eq. (B2). The results clearly show that the

694

divergence of the velocity is sufficiently small and can be

695

neglected in the numerical process.

696

Appendix C. Lagrange multipliers

697

and sub-grid coefficient selections

698

Although Lagrange multipliers are very important param-

699

eters for numerical schemes, there is no criteria for their

700

selection. These parameters are often selected based on the

701

numerical results, which should be consisted with experi-

702

mental observations and previous knowledge. Woodiga [6]

703

showed that the skin-friction fields on a wing surface are

704

not sensitive to the Lagrange multipliers. The problem could

705

occur when the Lagrange multipliers are too small or too

706

large, which could lead to some artificial features or smooth-

707

ing of the flow, respectively.

708

To evaluate the effect of the Lagrange multipliers on the

709

numerical results, we conducted tests on two systematic

710

images att10.39 s andt20.40 s, as shown in Sect.3.

711

In that case, the vorticity regulation coefficient is set toα2

712

0, while divergence regulation is changed fromα1104

713

to 104. Figure16 shows the effect of the Lagrange multi-

714

pliers on the AAE. Generally, Lagrange multipliers play as

715

smoothing function, which has less effect on the solution

716

Fig. 17 Effect of Lagrange multiplier on the AAE

Fig. 18 Effect of the final turbulent diffusion coefficientDton the AAE

than the observation term [36]. Consequently, a change in 717

the Lagrange multipliers in the range from 1 to 100 has only 718

a small effect on the AAE. However, the problem will occur 719

when the Lagrange multipliers are sufficiently large, which 720

affects the solution and leads to a high AAE (Fig.17). 721

In the next step, the effect ofcsandPrsgson the solution 722

is considered. By changingcs andPrsgsin the ranges ofcs 723

0.1–0.2 and Prsgs 0.1–1, we obtained the turbulence 724

diffusion coefficientDt changed from 0.01 to around 0.25. 725

Here, the smallest and highest values ofDt correspond to 726

(cs, Prsgs)=(0.1, 1) and (cs, Prsgs)=(0.2, 0.1), respectively. 727

Figure 18 shows the AAE as a function ofDt. The AAE 728

atDt0 (cs 0), which corresponds to the Horn–Struck 729

solution, is also plotted. 730

From these results, it is clear that AAE decreases with 731

increases in Dt up toDt0.22, which corresponds to (cs, 732

Prsgs)=(0.19, 0.1), and then it increases again. These results 733

strongly agree with those observed by Cassisa (2011), where 734

the minimum AAE corresponded toDt0.2. Note that the 735

coefficientcs0.19 has also been used in previous studies 736

by Chen et al. [36] on particle-image velocimetry data to 737

recover local features. 738

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