12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

CORRELATION BETWEEN VELOCITY AND HEAT TRANSFERNEAR REATTACHMENT POINT BEHIND A BACKWARD FACING STEP

Shunsuke Yamada
Department of Mechanical Engineering
National Defense Academy
Hashirimizu, 1-10-20, Yokosuka, Kanagawa,239-8686, Japan
yamadas@nda.ac.jp

Yuki Funami
Department of Mechanical Engineering
National Defense Academy
Hashirimizu, 1-10-20, Yokosuka, Kanagawa,239-8686, Japan
funami@nda.ac.jp

Hajime Nakamura
Department of Mechanical Engineering
National Defense Academy
Hashirimizu, 1-10-20, Yokosuka, Kanagawa, 239-8686, Japannhajime@nda.ac.jp

ABSTRACTIn this study, the mechanism of heat transfer enhancement
near the reattachment point behind a backward facing step is
experimentally investigated at Reynolds number of 5400. The
unsteady fluctuations of the velocity and temperature are
simultaneously measured by using 2 dimensions and 3components (2D-3C) Particle Image Velocimetry (PIV) andthe high-speed infrared (IR) thermography. To discuss the
temporal-spatio correlations of the velocity and the heattransfer fluctuations, 3 components of the velocity and the heattransfer coefficient are calculated from direct cross-correlation
method and unsteady heat conduction equation. Also, thesetime history data are analysed by using continuous wavelet
transform and two-point correlation. As a result, the flow
behaviour induced by the streamwise vortex and heat transfer
enhancement present a high correlation with thecharacteristics frequency.

and Sung, 2001). However, there are little reports about thethermal fluctuation, and it is not enough to clarify the periodic
phenomenon of the flow and thermal fields.
In order to understand the temporal-spatio periodicity of
the flow and thermal behaviour, it is required to measure withthe high temporal-spatio resolutions. We suggested thesimultaneous measurement of the flow and the thermal fieldusing 2D-3C PIV and a high-speed infrared (IR)thermography combined system (Yamada and Nakamura,2016). The correlation of each velocity in the xz cross section
and heat transfer coefficient on the heated wall were
investigated behind a backward facing step.
In this paper, the temporal-spatio fluctuations of thevelocity in yz cross section under the step and temperature on
the heated wall is investigated near the reattachment point at
ReH of 5400. Also, the temporal-spatio correlated relationshipbetween the velocity and heat transfer coefficient is analysedby using continuous wavelet transform and two-point
correlation.

INTRODUCTION
The insight of the heat transfer in the separated andreattachment flow is very important to develop the coolingtechnique for the electric devices, nuclear reactor, turbine
blade and many heat transfer devices. Many researchers haveinvestigated the aspects of fluid and thermal dynamics over a
backward facing step as the simple model of the separated andreattachment flow. Recently, the detail review of this flow has
been summarized by Chen et al., (2018). Many studies focus
on the enhancement of the heat transfer, and the flow andthermal fluctuations downstream of a backward facing step,
because there is the complicated flow that included the free
shear layer by flow separation, vortex motion andreattachment flow. Vogel and Eaton, (1984) reported that thelocation of the maximum peak of the heat transfer hascorresponded to that of root-mean square of the skin friction
coefficient. Also, Avancha and Pletcher, (2002) indicated thatthe heat transfer has been relationship with the thickness of theviscous sublayer. On the other hand, the flow fluctuation near
the reattachment point has the characteristics frequency of
fu0/H of about 0.2 and 0.06, at which the fluctuating motion of
the vortex and the flapping in the shear layer appear near thereattachment point (Driver et al., 1987, Le et al., 1997 and Lee

EXPERIMENTAL SETUP AND METHODWind tunnel and co-ordinate system
The experiments were carried out in a blower-type windtunnel having a rectangular cross section of 400 mm in heightand 150 mm in width. Figure 1(a) shows the test section, andthe present experiments was conducted by turning it upside
down considering buoyancy effect. A flat plate of 900 mm inlength was located so that the channel height upstream of a
backward facing step was 100 mm. The turbulent boundary
layer was created by a tripping wire of 2 mm in diameter at
downstream of 20 mm from the edge of a flat plate. The stepheight, H, was 20 mm. The expansion ratio, ER was 1.2, andthe aspect ratio in the spanwise direction based on this stepheight 7.5. The bottom wall of the step side behind the stepedge was heated for heat transfer measurement.
The origin of the coordinate system is at mid-span of thebottom wall of the step edge in Fig. 1(b). In this coordinate
system, the flow velocity in the streamwise, transverse andspanwise directions (x, y and z) corresponds to u, v, w,
respectively. The main stream velocity, u0, is fixed at 4 m/s
upstream of this step. The streamwise turbulent intensity of1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
the mainstream is approximately 0.5 %. Reynolds number,
ReH, based on the step height and the mainstream velocity is
at about 5400.
Figure 1(b) shows a schematic diagram of the heated plate.
The heated plate was fabricated from acrylic resin and had 7
sections removed. These sections were covered with the thintitanium foil sheets (61 mm in length, 100 mm in width, and2.1 m in thickness) arranged in the streamwise direction. In
the heated plate, air layers of 1 mm in thickness exist between
the heated titanium foil and the aluminium plate (see Fig. 1(b)).
Since the titanium foil and the aluminium plate are heated tothe almost same temperature, the heat conduction loss inside
the test plate is suppressed. The temperature inside the heatedaluminium plate was measured by thermocouples (T C) at x =62.5, 124.5, 186.5, 248.5, 310.5 and 372.5 mm. Both sides ofthese thin foils were tightly adhered to electrodes with a highconductivity bond in order to suppress contact resistance. 7
sheets of these foils were heated by direct current at theconstant heat flux of approximately 410 W/mm2. The titanium
foil sheets are stretched in response to tension, because thethermal expansion coefficient of the heated acrylic resin isgreater than that of the titanium foil. The temperature of theheated foil was measured using a high-speed IR thermograph,so that high-speed thermal fluctuation has been observed dueto the low heat capacity of these thin foils.

thermography, were at 1000 frames per second (fps). The
physical sizes of the velocity and the temperature images arein the range of z/H from -1.5 to 1.5 under the step at x/H = 4.5mm near the reattachment point, and x/H from 3.1 to 6.0 andz/H from -1.5 to 1.5 on the heated wall. The heat transfer
coefficient was calculated from unsteady heat conduction
equation using the time history data of temperature
distributions.
2D-3C PIV system
The 2D-3C PIV system consisted of a Nd: YAG laser withlight-sheet optics and 2 CMOS cameras of resolution 512 ×
256. Air flow was seeded with the DOS (dioctyl sebacate)particle of an average diameter of about 1 m. The seeding
density was about 10 particles per interrogation area in thelaser light sheet of about 1mm in the thickness and the size ofthe interrogation was 16 × 16 pixels with 50 % overlap. The
time intervals between the pair images were set to 50 s at themain stream velocity u0 = 4 m/s, and 4000 pair images were
captured by 2 cameras arranged in stereo. 3 velocity
components were calculated from the direct-cross correlation
method and the camera function by using commercial
software (ProVISION-XS ver. 3.13, IDT)
Calculation procedure of convective heat transfer
The temperature distributions, Tw on the heated wall
photographed by IR Thermograph was obtained from thefollowing equation:

Experimental apparatus of 2D-3C PIV and Highspeed IR thermograph
In Fig. 1(c), the shutters of 2 cameras (Motion Scope M3,
IDT), IR thermograph (SC4000, FLIR), and the emission of
Nd: YAG laser (HFYA-10SD, SEIKA Digital Image Corp.)
are synchronized by the timing controller. As shown in Fig.
1(d), the high-speed cameras and the IR thermograph werearranged. The camera frame rates for PIV and the IR

𝑬 = 𝜺𝒇(𝑻𝒘 ) + (𝟏 − 𝜺)𝒇(𝑻𝒂𝒎𝒃 )(1)

where E is the spectral emissive power detected by IR
thermograph, f(T) is the calibrated function of the IR
thermograph as the black body, and  is the spectral emissivityof this foil for IR thermograph. The terms on the right-hand(a) Wind tunnel(b) Heated plate
IR thermograph

PIV measurement arealaser light sheet
flow

433 mmystep heightH = 20 mmzx

IRT measurement area

high-speed camera(c) Simultaneous measurement system(d) Camera arrangement of PIV and IR thermography
Figure 1 Schematic diagram of experimental apparatus.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
side of Eq. (1) denote the emissive powers at the temperatureof the heated wall, Tw, and the ambience, Tamb, respectively. In
this measurement, the distance between IR thermograph andthe heated wall was approximately 400 mm, so that thetransmittance of the air was approximately 1.0.
Figure 2 shows a schematic diagram of the heat transfer
model from the heated plate. The local and instantaneous heattransfer coefficients are obtained from the following equations
that translated unsteady heat conduction equation:𝒉=𝒒̇ 𝒊𝒏 − 𝒒̇ 𝒄𝒅 − 𝒒̇ 𝒓𝒅 − 𝒒̇ 𝒓𝒅𝒊 + 𝑨 − 𝑩
𝑻𝒘 − 𝑻 𝟎

The finite difference method was used to calculate the heattransfer coefficient, h, using Eq. (2). The discretization of thespatial derivative terms of Eq. (3) and the temporal derivative
term of Eq. (4) applied to the second-order central differencewith the mean pixel spacing (x ≈ 0.43, z ≈ 0.56 mm) andthe central difference with the time step of a frame interval(the frame rate of IRT is 1000 fps; t = 1 ms). A low-pass filter
was set to the temporal and spatial directions (cut-off
frequency fcut and wavenumber kcut under 123 Hz and 0.29mm-1), respectively (Nakamura and Yamada, 2013).

(2)

Comparison of experimental and numerical data
Figures 3 show the time-averaged velocity profiles in thestreamwise (a) and transverse (b) directions in yz cross section
at x/H and z/H of about 4.5(near the reattachment point) and
0. For comparison, these figures include the DNS data of Le
et al (1997) at ReH = 5100 and x/H of 6.0. The present
reattachment length is shorter than the typically 2 dimensional
cases in the same Reynolds number. The present and DNS
mean velocities indicate similar tendency, however thedifference between the present and DNS data increase awayfrom the wall in Fig. 3(a) and (b). In the case of the low aspectratio, the vorticity induced by the shear layer spreads toward
the wall and increases the spanwise entrainment anddownwash flow near the center of the step (Shin and Ho, 1994).
Therefore, the reattachment length is shorter than that, and thevelocity profile does not agree with both.
Figure 4 shows the time-averaged local Nusselt number at
ReH = 5400, and LES data of Avacha and Pletcher (2002) at
ReH = 5540. Both profiles present the similar tendency,
however the difference of the maximum value is about 5.4 %.
This is caused by the difference of the shear layer associatedwith the expansion and aspect ratios.

Where
𝑨 = 𝝀𝒕 𝜹𝒕 (

𝝏 𝟐 𝑻𝒘 𝝏 𝟐 𝑻𝒘
+
)
𝝏𝒙𝟐
𝝏𝒛𝟐

(3)

and
𝝏𝑻𝒘
𝑩 = 𝒄𝒕 𝝆 𝒕 𝜹 𝒕 (
)
𝝏𝒕

(4)

where ct, t t and t are specific heat, density, thermal
conductivity and the thickness of the titanium foil,
respectively. And 𝑞̇ 𝑖𝑛 is the input heat fluxes to the foils dueto the Joule heating, 𝑞̇ 𝑐𝑑 is the heat fluxes from the foil due tothe conduction, and 𝑞̇ 𝑟𝑑 and 𝑞̇ 𝑟𝑑𝑖 are the radiation heat fluxes
out of and into the heated plate, respectively. The translated
heat conduction equation, given by Eq. (2), includes bothlateral heat conduction through the foil given by Eq. (3) andthe temporal delay due to the heat capacity of the foil given by
Eq. (4). The heat conduction to the air layer inside the heatedplate is calculated based on the temperature distribution inside
the air layer (thickness a = 1 mm in Fig. 2), which can be
determined by solving the following heat conduction equationof Eq. (5).
𝝏𝑻𝒂𝒒̇ 𝒄𝒅 = −𝝀𝒂 ()|
𝝏𝒚 𝒚=𝟎(5)and𝒄𝒂 𝝆 𝒂𝝏𝑻𝒂
𝝏 𝟐 𝑻𝒂 𝝏 𝟐 𝑻𝒂 𝝏 𝟐 𝑻𝒂
= 𝝀𝒂 ( 𝟐 +
+
)
𝝏𝒕
𝝏𝒙
𝝏𝒚𝟐
𝝏𝒛𝟐

(6)

where ca, a and a are specific heat, density and thermal
conductivity of the air. The temperature of the aluminium
plate measured by the thermocouples (TC) is the boundary
condition for calculating the temperature of the air layer from
Eq. (6). The temperature of the air layer inside the heated plate,Ta, is calculated using the alternative direction implicit (ADI)
method with respect to the x and z directions.

(a) Streamwise direciton

Main stream temperature T0
Thickness of titanium foil t=2.1m
Flow・
Heat convection qyrdi Radiationx Temperature T・q(Outside)
w

q・

in

Input
heat-flux

cv

・
qcd
Heat conduction

・
qrd
Radiation(Inside)Air layer(a 1mm)

Aluminum plate
Thermocouple(TC: T type)(b) Transverse direction
Figure 3. The time-averaged velocity profiles near thereattachment point.

Figure 2. Model of the heat transfer in the heated plate.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
EXPERIMENTAL RESULTS AND DISCUSSIONS
Instantaneous distribution and wavelet analysis
Figure 5 shows the instantaneous velocity vectors, v, w
and contour map u (top), and heat transfer coefficient profile,h (bottom) at x/H of 4.5. In Fig. 5, the instantons flow fieldnear the reattachment point occurs complicated state and theheat transfer has the spatial-periodic fluctuation. Also, thelocal downwash flow is formed by the streamwise vortex.
Wavelet analysis applied to the backward facing step flow
(Lee and Sung, 2001). Continuous wavelet transform was used
for the time history data of the velocities and heat transfer
coefficient, and it is possible to reveal the existence of shortlived and quasi-periodic events. Figures 6 show the time
history of velocities and wavelet coefficient in the streamwise(a) and transverse (b) direction, and heat transfer coefficient(c), respectively. In these figures of the contour map ofwavelet coefficient, these vertical axes indicate the logarithm.
Morlet wavelet function was used in the present analysis. The
peak of wavelet coefficients of velocity and heat transfer
appear in non-dimensional frequency range from about 0.06
to 0.20. In the free shear layer, the frequency appears at f∙H/u0
of about 0.2 due to the large vortex structure. In addition tothis frequency, there is the peak at the low frequency, f∙H/u0
of about 0.06. The low frequency is possibly due to theappearance of the flapping motion (Driver et al, 1987). In thisstudy, it seems that the low frequency near about 0.06
corresponds to the downwash flow induced by the flapping
motion.

Momentum and thermal transport induced by
vorticities
The heat transfer fluctuation is relationship to themomentum transport due to the downwash flow induced by
the streamwise vortex. To detect the vortex in yz crosssectional flow, the vorticity, yz and the shear stress Syz were
calculated from Eq. (7) and (8). The strengths, Qyz of therotation and the shear stress were compered from Eq. (9), and(a) Velocity in the streamwise direciton(b) Velocity in the transverse direciton
Figure 4. Time-averaged local Nuselt number(c) Heat trasfer coefficinet
Figure 6. Time history data (top) and wavelet coefficient(bottom) of each velocity and heat transfer coefficient at
ReH of 5400, and z/H and y/H of 0.0 and 0.41.

Figure 5. Instantaneous velocity (topside) and heat transfer
coefficient (bottom) at ReH of 5400.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
the zone above a certain threshold, Qth was defined as thevortex region, Qvortex in Eq. (10)𝛀𝒚𝒛 (𝒚, 𝒛, 𝒕) =𝟏 𝝏𝒘 𝝏𝒗(− )𝟐 𝝏𝒚 𝝏𝒛(7)

𝐒𝒚𝒛 (𝒚, 𝒛, 𝒕) =𝟏 𝝏𝒘 𝝏𝒗(+ )𝟐 𝝏𝒚 𝝏𝒛(8)

𝟏
𝟐
𝟐(−|𝑺𝒚𝒛 | + |𝛀𝒚𝒛 | )𝟐(9)𝐐𝒚𝒛 (𝒚, 𝒛, 𝒕) =𝐐𝒗𝒐𝒓𝒕𝒆𝒙 (𝒚, 𝒛, 𝒕) = {𝟏 𝒊𝒇 𝑸𝒚𝒛 > 𝑸𝒕𝒉
𝟎 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆(10)

From reult of Qvortex, it is possible to creat the isosurface ofvortcity in the yz cross section. In this study, a certain
threshold, Qth was more than 8000 [1/s2]. Also, the region of
the high mometum trasport was detected from Reynolds stress,
uv and uw. Similar to determination of this vortex region, theisosurface of the mometum trasport was created from thesethresholds of Reynolds stress less than uv of 0 m2/s2 and morethan (uw)2 of 0.02 m4/s4.
Figures 7 show the contour map of the time history of theheat trasfer coefficient at x/H of 4.5 (a), the isosurface of thevortex rigion (b) and the negative and high Reynolds stressregion, uv and uw (c), respectively. In Fig. 7(b) and (c), the redand blue isosurfaces indicate the counter-crokweise and thecrockwise of the rotation, and the green and magenta
isosurfaces the negative and the high regions of the Reynolds
stress, uv and uw. As shown in Fig. 7(a), the thermal spot andthermal streaks apper on the heated wall. In Fig. 7(b), as wellas heat trasfer coefficient, the vorticies have the streakly
structure toward the time, and are arranged periodically in the
spanwise direction. Figure 7(c) shows that the negative andhigh Reynolds stress regions, uv, uw apper near the thremal
spots and streaks.
Two-point correlation of flow and thermal
fluctuations
To investigate flow and thermal structure quantitatively, a
two-point correlation coefficient, Rnn was calculated from Eq.
(11).

(a) Heat trensferr on the heated wall

𝑹𝒏𝒏 (𝒚, 𝚫𝒛, 𝚫𝒕) =𝒏′ (𝒚, 𝒛, 𝒕) − 𝒏′ (𝒚, 𝒛 + 𝚫𝒛, 𝒕 + 𝚫𝒕)√̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝒏′𝟐 (𝒚, 𝒛, 𝒕)√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝒏′𝟐 (𝒚, 𝒛 + 𝚫𝒛, 𝒕 + 𝚫𝒕)(11)

where the fluctuations of each velocity and heat transfer
coefficient are substituted for n´. Also, these analysis data
were used in the non-dimensional time range t∙u0/H fromabout 160 to 180 when the peak of the wavelet coefficient
appeared in Fig. 6. Contour map of two-point correlation
coefficient of heat transfer coefficient is shown in Fig. 8.
Figures 9 shows the contour map of two-point correlation
coefficient of each velocity at y/H of 0.4(a) and 0.8(b). In Fig.
8, the high peak value of the correlation coefficient isconfirmed at the origin, and there are the weak peaks at z/Hof about -1.0 and 1.0, and t u0/H of 0.0. Figures 9(a) present
that, as same as the heat transfer correlation, the fluctuatingcorrelations of the streamwise, transverse and spanwise
velocities appear at z/H of about -1.0, 0 and1.0, and thecorrelation peak of the streamwise and transverse velocity
fluctuations are stretched toward the non-dimensional time.
Also, these peaks of the transverse velocity fluctuations havemore clearly and higher than that of the streamwise andspanwise velocity fluctuations. However, in Fig. 9(b), these
special peaks of the velocity fluctuations become weak lessthan that of the velocity fluctuations in Fig. 9(a). As a result,the special correlation of the heat transfer corresponds to thevelocity fluctuation at y/H of 0.4, and the non-dimensionalcorrelation lengths z/H is about 1.0. Keating et al., (2004)
reported that the thermal spot corresponds to the intermittent
downwash flow near the reattachment region.

(b) Volticity in the yz cross section and heat treansfer(c) Reynolds stress in the yz cross section and heat treansfer
Figure 7. The instataneous of the vortex structure and thermla
flucutation.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
SUMMARY
In this study, the velocities in yz cross section near thereattachment point and the temperature on the heated wall
were simultaneously measured by 2D-3C PIV and IR
thermography at ReH of 5400. The time history of thestreamwise vortex and the heat transfer coefficient were
calculated from the velocity and temperature profiles. Using
continuous wavelet transform and two-point correlation, thecharacteristic frequency and the temporal-spatio correlationsof the velocity and heat transfer were investigated.
Consequently, the frequency peaks of the velocity and the heattransfer coefficient appear in the non-dimensional periodic
range from 0.06 to 0.2, and the spatial correlation length in the
spanwise direction are at z/H of about 1.0, when the highwavelet coefficients appear at y/H of 0.4.

with heat transfer and property variance", Int. J. Heat andFluid Flow, Vol. 23, pp. 601-614.
Chen, L., et al., 2018, "A review of backward-facing step(BFS) flow mechanisms, heat transfer and control", Therm.
Sci. Eng. Prog., Vol. 6, pp. 194-216.
Driver, M. D., Lee Seegmiller H., and Marvin, G. J., 1987,"Time-dependent behaviour of a reattaching shear layer",
AIAA J., Vol. 25, No. 7, pp. 914-6919.
Keating, A., Piomelli, U., Bremhorst, K., and Nešić, S.,
2004, "Large-eddy simulation of heat transfer downstream ofa backward-facing step", J. Turbulence, Vol. 5, N20.
Le, H., Moin, P., and Kim, J., 1997, "Direct numericalsimulation of turbulent flow over a backward-facing step", J.
Fluid Mech., Vol. 330, pp. 349-374.
Lee, I., and Sung, J., 2001, "Characteristics of wallpressure fluctuations in separated flows over a backwardfacing step: Part ⅠⅠ: Unsteady wavelet analysis", Exp. Fluids,
Vol. 30, pp. 273-282.
Nakamura, H., and Yamada, S., 2013, "Quantitativeevaluation of spatio-temporal heat transfer to a turbulent air
flow using a heated thin-foil", Int. J. Heat Mass Transf., Vol.
64, pp. 892-902.
Shin, C., and Ho, C-H., 1994, "Three-dimensional
recirculation flow in a backward facing step", J. Fluids Eng.,
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Yamada, S., and Nakamura, H., 2016, " Construction of
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REFERENCES
Avancha, V. R. R., and Pletcher, H, R., 2002, "Large eddy
simulation of the turbulent flow past a backward-facing step

Figure 8. Contour map of the two-point correlation of h.

(a) Corrrelation coeffieint at y/H = 0.4(left: u´, cener: v´, rigtht: w´)(b) Corrrelation coeffieint at y/H = 0.8 (left: u´, cener: v´, rigtht: w´)
Figure 9. Contour map of two-point correaltion of velocity fluctuations.

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