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.com1 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
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 −(∇p−ρg)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−(∇p−ρg)κ2I2
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
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:τs −Dt∇ ¯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¯· ∇ ¯I−DtI¯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
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¯· ∇ ¯I−DtI¯ 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
parameters,¯ csandPrsgs, 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
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 ofk−5/3[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
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
340by proposed method
3414.1 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 α1/α2 is also an important feature in this 361
study. If α1/α21, the estimated results show a domina- 362 tion of solenoidal motion, whereas atα1/α21, 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
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
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
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, #Z−4, whereas
448
proposed method shows #N2, #S3, #Z−4. 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
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 (ϕ1,ϕ2)(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−,#Z−2,#Z3−,#Z−4 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
576In 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
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
608on wall shear-stress results
609The 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 ofV∞22 m/s andV∞45 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
V∞22 m/s, the pressure gradient atV∞45 m/s changes 635
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
parameterκofκ2×10−5. 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 (∇p−ρg). (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(∇p−ρg)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:aV∞22 m/s,bV∞45 m/s
However, atV∞45 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
672velocity vectors
673In 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
Fig. 16 Magnitude of mean optical-flow velocity and mean divergence of velocity in the region ofx/c0 to 1 andy/c −0.15 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 fromα110−4
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
