Estimating of higher order velocity moments and their derivatives in boundary layer by Smoke Image Velocimetry

Abstract

The results of an experimental evaluation of the third-order moments profiles of velocity fluctuations and their partial derivatives in a zero pressure-gradient turbulent boundary layer are presented. Profiles of characteristics are estimated on the basis of the dynamics of two-component instantaneous velocity vector fields measured by the optical method Smoke Image Velocimetry (SIV). Comparison SIV-measurements with the results of measurements by a thermoanemometer and DNS data with similar Reθ and Reτ showed good agreement between the profiles of <u'2v'>⁺, <u'v'2>⁺, ∂<u'2v'>⁺/∂y⁺ и ∂<u'v'2>⁺/∂y⁺ obtained by SIV and DNS.

Full text

Journal of Physics: Conference Series
PAPER
Estimating of higher order velocity moments and
their derivatives in boundary layer by Smoke
Image Velocimetry
To cite this article: N.I. Mikheev et al 2017 J. Phys.: Conf. Ser. 891 012092
View the article online for updates and enhancements.
Related content
Estimating of turbulent velocity fluctuations
in boundary layer with pressure gradient
by Smoke Image Velocimetry
N I Mikheev, A E Goltsman and I I Saushin
-
Research of the boundary layer with an
adverse pressure gradient by the Smoke
Image Velocimetry method
N I Mikheev, I I Saushin and A E Goltsman
-
Technique for velocity vector field
dynamics measurement on the basis of
smoke visualization of flow
N I Mikheev, N S Dushin and I I Saushin
-
This content was downloaded from IP address 80.82.77.83 on 12/11/2017 at 07:50

1
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
1234567890
PTPPE-2017 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 891 (2017) 012092 doi :10.1088/1742-6596/891/1/012092
Estimating of higher order velocity moments and their
derivatives in boundary layer by Smoke Image Velocimetry
N.I. Mikheev1,2, А.Е. Goltsman1, I.G. Salekhova3, I.I. Saushin1
1 Kazan Scientific Center of the Russian Academy of Sciences, Lobachevsky str. 2/31,
Kazan, 420111, Russian Federation
2 Kazan National Research Technical University named after AN Tupolev - KAI, 10,
K.Marx St., Kazan, 420111, Russian Federation
3Kazan Federal University, 18 Kremlyuvskaya street, Kazan, 420008, Russian
Federation
E-mail: an116ya@mail.ru
Abstract. The results of an experimental evaluation of the third-order moments profiles of
velocity fluctuations and their partial derivatives in a zero pressure-gradient turbulent boundary
layer are presented. Profiles of characteristics are estimated on the basis of the dynamics of two-
component instantaneous velocity vector fields measured by the optical method Smoke Image
Velocimetry (SIV). Comparison SIV-measurements with the results of measurements by a
thermoanemometer and DNS data with similar Reθ and Reτ showed good agreement between the
profiles of <uꞌ2vꞌ>+, <uꞌvꞌ2>+, ∂<uꞌ2vꞌ>+/∂y+ и ∂<uꞌvꞌ2>+/∂y+ obtained by SIV and DNS.
1. Introduction
The physical interpretation of third-order velocity moments are the convective transport of second-order
moments by the turbulent velocity fluctuation [1, 2]. Triple correlations play a defining role in the
diffusion term of the turbulent energy balance equation which is the basis for analytical and numerical
methods of investigating the boundary layer. The importance of estimating the triple correlations <uꞌ3>,
<uꞌ2vꞌ>, <uꞌvꞌ2>, <vꞌ3> was noted in the work of Townsend [3] long enough. In describing of complex
unsteady flows by the Reynolds-averaged Navier-Stokes equations, the order of the moments in the
closing equations has a significant effect on the uncertainty of the solution. As an example would be
flows with dominant convective forces [4]. Nevertheless, the inclusion of higher order moments in the
empirical turbulence models is hampered by the insufficient volume of the accumulated experimental
database. The results of estimating of triple correlations in zero pressure-gradient boundary layer
obtained from the results of measurements by a thermoanemometer are presented in [5-13]. The point
method of laser Doppler anemometry for this purpose was used much less often [14-15]. In the case of
estimating higher-order moments from the results of measurement with a thermoanemometer the
uncertainty of the estimate depends essentially on the scale of the measurement, which is determined by
the distance between the wires and their length [16]. Obviously, one of the possible solutions to this
problem is the use of thermoanemometric probes with reduced wire sizes, in particular, for the
estimation of third-order moments in work [13] a thermoanemometer with platinum wire of 60 μm in
length and 2 μm in diameter was used. The second approach to reducing the uncertainty of estimates of
small-scale characteristics is the use of empirical-based algorithms for filtering the signal from

2
1234567890
PTPPE-2017 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 891 (2017) 012092 doi :10.1088/1742-6596/891/1/012092
measurement errors [16-21]. Nevertheless, the question of the uncertainty of estimating higher-order
moments from the results of measurement with a thermoanemometer is still open [22].
However, if we return to the main reason for the relevance of the experimental estimate of triple
correlations, namely, to the rate of advection of turbulent energy by turbulent motion
'''
1,
2i i j
j
uuu
x



(1)
there are new obvious problems and uncertainties associated with the determination of spatial gradients
from the measurement results by point methods, so this estimate is potentially more correct to be
performed on the basis of the measurement results by field methods.
The profiles of triple correlations estimated from measurements of the field optical method Particle
Image Velocimetry (PIV) were performed in [23]. Unfortunately, according to the authors of this paper
the profiles of spatial gradients of the third-order moments were not reliable and were not performed.
This can be explained by the limitation of the PIV method on the balance between spatial resolution and
uncertainty in the measurement of small-scale turbulence structures [24-36]. In part, this problem was
solved in two new spatial measurement techniques "Shake-The-Box" (STB) [37] and VIC+ [38].
However, these two approaches have not yet been tested to estimate higher-order moments of the
velocity fluctuations.
The use of the high-speed optical method for measuring the instantaneous velocity fields Smoke
Image Velocimetry (SIV) [39, 40] due to the multiply higher tracer concentration compared to PIV and
PTV (which helps reduce measurement noise with increasing spatial resolution) allows estimating the
fields of small-scale characteristics with a sufficient degree of accuracy, for example, [40] presents an
estimate of the dissipation term for the conservation equation of turbulent energy with a spatial
resolution to 1.6-2.0 of the Kolmogorov scale and temporal resolution of 7 kHz. The results of the
evaluation of the second and third orders moments and their derivatives in a turbulent boundary layer
obtained on the basis of SIV-measurements are presented in this paper.
2. Experimental setup
The structure of air flow in a smooth plane channel has been studied using the optical method of Smoke
Image Velocimetry (SIV). Experimental setup is shown in figure 1. The setup arrangement and flow
parameters were very close to the ones in the experiments by Willert [41], in which turbulence
characteristics in the developed turbulent boundary layer were studied using PIV technique. The test
section 1 was a rectangular 75×150 mm2 channel with the length of 1 m and a smooth inlet 10 with 6:1
contraction. A turbulence generating grid 9 with 5 mm cell size, 1.2 mm steel wire diameter and 36%
solidity was mounted downstream of the smooth inlet 1. A 50-mm long strip of abrasive P24 (ISO 6344-
2) 12 was glued onto the channel perimeter. This provided fully developed turbulent boundary layer in
the measurement area during the experiments. Channel walls were made of transparent materials (glass
and polycarbonate). Stable air flow rate downstream of the test section was provided by a regulating
gate 11 and a 1.3 m3 receiver tank 2 mounted upstream of the latter. Flow rate was measured by an
ultrasonic flowmeter 3 IRVIS RS4-Ultra mounted downstream of the receiver tank. The relative error
in flow rate did not exceed 1%.
To visualize the flow pattern, the air-aerosol mixture (MT-Gravity fluid with medium fog density
and average particle size of 0.1…5 μm; Safex aerosol generator 5) was supplied from the preparation
chamber 4 to the channel inlet. The measurement area 6 was illuminated by a continuous diode-pumped
solid-state laser KLM-532/5000-h 7. The flow pattern in the channel symmetry plane at the distance of
L = 0.7 m from the turbulence generating grid 9 (figure 1) was recorded by a monochrome high-speed
camera Fastec HiSpec 8 with the frame resolution of 665×110 pixel (scaling factor of 0.0625 mm/pixel),
frame rate f = 7083 1/s, and recording time of 3.5 s. The camera was equipped with a Navitar 1’’F/0,95
lens (focal length 25 mm, manual focus).
Flow velocity fields were measured by optical SIV technique based on digital processing of flow
pattern video recordings. Here, velocity vector fields were estimated by the analysis of turbulent
structure displacements visualized by smoke. Profiles of velocity and turbulent fluctuations were

3
1234567890
PTPPE-2017 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 891 (2017) 012092 doi :10.1088/1742-6596/891/1/012092
estimated from 16×16 pixel windows as well. Maximum displacement of turbulent structures between
two consecutive frames was 10 pixels (at the boundary layer edge). Image resolution in y+ coordinates
was 1 pixel = 0.8 y+.
Figure 1. Experimental setup
3. Results and discussion
Table 1 presents the main characteristics of the investigated turbulent boundary layer. Profiles of
turbulent fluctuation intensity and triple correlation were written in wall coordinates:
' ' '
'''
3
,,
i i j
i i j
u
yu
y u u u u
uu





(2)
where dynamic velocity, uτ, was estimated by using the results of velocity measurement at a point within
the viscous sublayer.
Figure 2 shows the profiles of triple correlations <uꞌ2vꞌ>+ and <uꞌvꞌ2>+ at close values of Reθ and
Reτ estimated by the results of SIV measurements, by the thermoanemometer [7] and obtained by the
results of DNS [42]. The profile <uꞌ2vꞌ>+ estimated from the results of SIV-measurements qualitatively
and quantitatively agrees well with the results of DNS. In the viscous sublayer negative values of
<uꞌ2vꞌ>+ monotonically decreasing at a distance from the wall are observed. In part of transition the
viscous sublayer to the buffer region near the coordinate y+≈8, the local extremum is clearly observed
on the <uꞌ2vꞌ>+ profiles. In the buffer layer, the profile <uꞌ2vꞌ>+ monotonically increases to the coordinate
y+≈30 where the second local extremum is observed. Further from this coordinate to the edge of
boundary layer, <uꞌ2vꞌ>+ decreases monotonically to an asymptotic zero value. A similar behavior is
observed for the profile <uꞌvꞌ2>+ but both the SIV and DNS results indicate a shift of the second
extremum into the depth of the logarithmic sublayer. The described features of the evolution of the
profiles along the thickness of boundary layer are not found on the profiles estimated from the results
of measurements by the thermoanemometer [7], however, triple correlations have a similar order. The
graphs also clearly demonstrate the advantage of the field optical method over the point method with
respect to the spatial frequency of measurements.
The profiles of the triple correlation derivatives <uꞌ2vꞌ>+ and <uꞌvꞌ2>+ (figure 3) taken along the
normal to the wall describe the transfer of turbulent energy by turbulent velocity fluctuations. The
quantities ∂<uꞌ2vꞌ>+/∂y+ and ∂<uꞌvꞌ2>+/∂y+ describe respectively the transport (diffusion) of the double
correlations uꞌ2 and uꞌvꞌ over the thickness of boundary layer [23]. The profiles of the partial differentials
<uꞌ2vꞌ>+ and <uꞌvꞌ2>+ in figure 3 are obtained by approximating the DNS, SIV, and thermoanemometric
measurements using a three-point central difference scheme. The trends of profiles ∂<uꞌ2vꞌ>+/∂y+ and
∂<uꞌvꞌ2>+/∂y+ estimated by the results of SIV-measurements, in spite of the relatively large spread of
discrete values, are generally agree satisfactorily with DNS results. The application of the SIV-
measurement technique made it possible to estimate the location of the extreme point and the sign of
∂<uꞌ2vꞌ>+/∂y+ and ∂<uꞌvꞌ2>+/∂y+ with good agreement with the DNS results. At the same time, the profiles
∂<uꞌ2vꞌ>+/∂y+ and ∂<uꞌvꞌ2>+/∂y+ estimated from the results of measurements by the thermoanemometer
[7] showed a satisfactory agreement with DNS results only in order of magnitude.
4. Conclusions
For the considered turbulent boundary layer, the optical method of SIV made it possible to estimate the
profiles of <uꞌ2vꞌ>+ and <uꞌvꞌ2>+ with a reasonable degree of accuracy in comparison with the results of
DNS [42] with similar Reτ. Profiles ∂<uꞌ2vꞌ>+/∂y+ и ∂<uꞌvꞌ2>+/∂y+ estimated by the results of SIV-
measurements, in spite of the available scatter of values, on the whole describe DNS results well enough.

4
1234567890
PTPPE-2017 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 891 (2017) 012092 doi :10.1088/1742-6596/891/1/012092
Table 1. Characteristics of turbulent boundary layer measured
Shear velocity at the wall (s-1)
2767
Momentum thickness, θ (mm)
1.58
Dynamic velocity, uτ (m/s)
0.204
Friction factor, Cf
0.005
Velocity at the channel axis, U∞ (m/s)
4.06
Reθ
425
Turbulent boundary layer thickness, δ99% (mm)
15.88
Reτ
214
Displacement thickness, δ* (mm)
2.44
Figure 2. Profiles of triple correlations; - DNS [42] (Reτ=180); - hot wire [7] (Reθ=706); - SIV
(Reτ=214, Reθ=425)
Figure 3. Profiles of triple correlation derivatives; - approximation of DNS results [42] (Reτ=180);
- approximation of hot wire results [7] (Reθ=706); - SIV (Reτ=214, Reθ=425)
Acknowledgements
The work was supported by the Russian Science Foundation (project no.16-19-10336).
References
[1] Dean RB, Bradshaw P 1976 Journal of fluid mechanics 78(04), 641-676
[2] Shah MK, Agelinchaab M, Tachie MF 2008 Experimental Thermal and Fluid Science 32(3) 725-
747

5
1234567890
PTPPE-2017 IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 891 (2017) 012092 doi :10.1088/1742-6596/891/1/012092
[3] Townsend AA 1951 Proc. Camb. Phil. Soc. 47, 375-395
[4] Jeyapaul E, Coleman GN, Rumsey CL 2015 International Journal of Heat and Fluid Flow 54 14-
27
[5] Murlis J 1975 The structure of a turbulent boundary layer at low Reynolds number (Doctoral
dissertation, University of London)
[6] Murlis J, Tsai HM, Bradshaw P 1982 Journal of Fluid Mechanics 122, 13-56
[7] Erm LP 1988 Low-Reynolds-number turbulent boundary layers (Ph.D. thesis, University of
Melbourne)
[8] Erm LP, Joubert PN 1991 Journal of Fluid Mechanics 230 1-44
[9] Bruns J, Dengel P, Fernholz HH 1992 Institutsbericht 2 92
[10] Fernholz HH, Finleyt PJ 1996 Progress in Aerospace Sciences 32(4) 245-311
[11] Krogstadt PÅ, Antonia RA 1999 Experiments in fluids 27(5) 450-460
[12] Morrison JF, McKeon BJ, Jiang W, Smits AJ 2004 Journal of Fluid Mechanics 508 99-131.
[13] Bailey SC, Kunkel GJ, Hultmark M, Vallikivi M, Hill JP, Meyer KA, Smits AJ 2010 Journal of
Fluid Mechanics 663 160
[14] Tachie MF, Bergstrom DJ, Balachandar R 2003 Experiments in fluids 35(4) 338-346
[15] Flack KA, Schultz MP, Shapiro TA 2005 Physics of Fluids 17(3) 035102.
[16] Talamelli A, Segalini A, Örlü R, Schlatter P, Alfredsson PH 2013 Experiments in fluids 54(4)
1496.
[17] Hutchins N, Nickels TB, Marusic I, Chong MS 2009 Journal of Fluid Mechanics 635 103-136
[18] Chin C, Hutchins N, Ooi ASH, Marusic I 2009 Meas Sci Tech 20 115401
[19] Chin C, Hutchins N, Ooi ASH, Marusic I 2010 Exp Fluids 50 1443–1453
[20] Monkewitz PA, Duncan RD, Nagib HM 2010 Physics of Fluids 22(9) 091701
[21] Smits AJ, Monty JP, Hultmark M, Bailey SCC, Hutchins N, Marusic I 2011 Journal of Fluid
Mechanics 676 41–53
[22] Alfredsson PH, Örlü R, Schlatter P 2011 Experiments in fluids 51(1) 271-280
[23] Shah M K, Agelinchaab M, Tachie MF 2008 Experimental Thermal and Fluid Science 32(3) 725-
747
[24] Etebari A, Vlachos P 2005 Experiments in Fluids 39 1040-1050
[25] Sheng J, Meng H, Fox RO 2000 Chem Eng Sci 55(20) 4423–4434
[26] Saarenrinne P, Piirto M 2000 Experiments in Fluids 29 300–307
[27] Sharp KV, Adrian RJ 2001 AIChE J 47(4) 766–778
[28] Baldi S, Yianneskis M 2003 Ind Eng Chem Res 42(26) 7006–7016
[29] Racina A, Kind M 2006 Experiments in Fluids 41(3) 513–522
[30] Tanaka T, Eaton JK 2007 Experiments in Fluids 42(6) 893–902
[31] Lavoie P, Avallone G, De Gregorio F, Romano GP, Antonia RA 2007 Experiments in Fluids
43(1) 39–51
[32] Atkinson C, Coudert S, Foucaut JM, Stanislas M, Soria J 2011 Experiments in fluids 50(4) 1031-
1056
[33] Kähler CJ, Scharnowski S, Cierpka C 2012 Experiments in fluids 52(6) 1641-1656
[34] Tokgoz S, Elsinga G, Delfos R, Westerweel J 2012 Experiments in fluids 53(3) 561–583
[35] Charonko J, Prestridge K 2016 18th International Symposium on Applications of Laser
Techniques to Fluid Mechanics, Lisbon, Portugal
[36] Foucaut J-M, Cuvier C, Stanislas M, George WK 2016 Progress in Wall Turbulence 2, Series 23
[37] Schanz D, Gesemann S, Schröder A 2016 Experiments in fluids 57(70)
[38] Schneiders JFG, Scarano F 2016 Experiments in fluids 57(139)
[39] Mikheev NI, Dushin NS 2016 Instruments and Experimental Techniques 59(6) 880–887
[40] Mikheev NI, Goltsman AE, Saushin II, Dushina OA 2017 Experiments in fluids
doi: 10.1007/s00348-017-2379-x
[41] Willert CE 2015 Experiments in fluids 56(17)
[42] Mansour NN, Kim J, Moin P 1988 Journal of Fluid Mechanics 194 15-44