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

RESPONSE OF HIGH REYNOLDS NUMBER TURBULENT PIPE FLOW TO
THE PRESENCE OF AN AXISYMMETRIC BODY

Ian E. Gunady
Liuyang Ding
Marcus Hultmark
Department of Mechanical
Department of Mechanical
Department of Mechanicaland Aerospace Engineering and Aerospace Engineering and Aerospace Engineering
Princeton University
Princeton University
Princeton University
Princeton, NJ 08544
Princeton, NJ 08544
Princeton, NJ 08544
igunady@princeton.edu
liuyangd@princeton.edu
hultmark@princeton.edu

Alexander J. Smits
Department of Mechanicaland Aerospace Engineering
Princeton University
Princeton, NJ 08544
asmits@princeton.edu

ABSTRACT

is placed at the centerline of the Superpipe facility at Princeton University. The flow far upstream and downstream of theBOR is high Reynolds number, fully developed turbulent pipe
flow, and the evolution of turbulence through the different conditions encountered around the body of revolution is explored.
This work furthers our understanding of wall-bounded turbulence in complex flows, and provides a challenging test casefor turbulence models (Visonneau et al., 2022).

Effects from pressure gradients, streamline convergence/divergence, and streamline curvature are often encountered in engineering wall-bounded flows. For example, as theflow passes over a fuselage or hull, it encounters a favorablepressure gradient and streamline curvature and divergence as
it is accelerated past the nose. The flow then relaxes over the
midbody of the vehicle before encountering an adverse pressure gradient and streamline curvature and convergence overthe stern. Past the vehicle, the flow forms a wake that subsequently decays.

EXPERIMENTAL SETUPScaling laws and modeling approaches developed forcanonical wall-bounded turbulence may require modifications
when such non-equilibrium conditions are present. Previousstudies that have explored the effects of pressure gradientsand surface curvature have typically studied these effects separately. For example, Harun et al. (2013) showed that an adverse pressure gradient energizes turbulence while a favorablepressure gradient suppresses turbulence. Similarly, Smits et al.
(1979); Muck et al. (1985) and Hoffmann et al. (1985) notedthat convex surface curvature has a stabilizing effect on turbulent boundary layers while concave surface curvature has adestabilizing effect. Also, Nagib & Chauhan (2008) found thatthe slope of the log law region may be affected by pressure
gradient and flow geometry. After strong perturbations suchas a change in roughness, a short region of surface curvature,or a short region of separated flow, a non-monotonic second
order recovery is often observed where the initial recovery of
the u2s profile overshoots the equilibrium profile (Smits et al.,
1979; Ding et al., 2021). Experiments on the effects of simultaneous perturbations, such as streamline curvature and divergence with pressure gradient, are very rare, and most of theexisting experiments on strong perturbations have been conducted at only moderate Reynolds numbers, so the effects ofhigh Reynolds numbers relevant for many engineering applications are also largely unknown.

Experiments are conducted in the Superpipe facility modified to accommodate the axisymmetric body. The pipe has an
inner diameter D = 2R = 129 mm and a development lengthof 200D upstream of the first measurement location. Further details of the facility are given by Zagarola & Smits(1998). The Reynolds numbers ReD based on pipe diameterand bulk velocity of the incoming fully developed turbulent
pipe flow are varied between 0.2 × 106 and 5 × 106 , corre+
sponding
p to Re = Ruτ /ν between 4, 500 and 80, 000, whereuτ = τw /ρ, τw is the wall shear stress, ρ is the density, andν is the kinematic viscosity.
The geometry of the axisymmetric body follows closely
the geometry of a similar experiment described by Ding et al.
(2019) and Saxton-Fox et al. (2019) in a water pipe at a lower
Reynolds number. The body is held at the centerline of theSuperpipe by a 2D airfoil-shaped sting. The area blockage ratio of the body in this work is γ = d 2 /D2 = 2/9 and future
experiments will also include γ = 1/9 and 1/3. The body iscomposed of three parts: the bow, midbody, and stern. Figure 2 shows an outline of the BOR profile with vertical lines
separating the bow, midbody, and stern. The bow’s profile is a
prolate spheroid with length lb = 4D/3 satisfyingx2y2+=12lb (d/2)2

In this work, an axisymmetric body of revolution (BOR)

1(1)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
as the local bulk velocity Ubx = Ub R2 /(Yt (2R − Yt )) (figure3b). Reynolds normal stress profiles normalized by upstream
friction velocity uτ are shown in figure 4.
The mean flow and Reynolds normal stress profiles atx/R = −1.46 and x/R = −0.70 are almost identical, suggesting that both these locations are far enough upstream to beunaffected by the presence of the BOR and accurately represent the upstream inflow conditions. Further downstream, themean flow and turbulence profiles change as the presence of
the BOR causes a bulk flow acceleration, while a boundary
layer forms over the surface of the BOR. The mean flow profiles clearly show the acceleration of the flow over the bow
section as the cross-sectional area decreases. The Reynolds
normal stress profiles show that as the flow is accelerated overthe nose, the turbulence in the bulk flow is initially suppressed,
while the region near the surface of the pipe thins. The turbulence then grows as the flow enters the straight section. Away
from the wall of the pipe, the Reynolds normal stress profiles shown in figure 4 agree well with PIV data collected by
Ding et al. (2019) in a companion experiment in a water pipe
at Re+ = 3, 550, despite the differences in Reynolds number.
Close to the wall, however, the PIV data were significantly filtered by spatial resolution effects, whereas the current NSTAP
data are not limited in this respect and so are more accurate. Incontrast, the PIV data was able to resolve the flow close to thesurface of the BOR, which was not possible using the current
probe design.
The velocity spectra can help to examine how the energy
content between various scales changes in various regions ofthe flow. Figure 5 shows pre-multiplied streamwise velocity
spectra close to the pipe wall at y/Yt = 0.015 and away fromthe pipe wall at y/Yt = 0.28 and 0.51. We use the local bulk
velocity Ubx , based on the upstream bulk velocity and the geometry of the BOR, as a measure of the local convective velocity. Also, a constant value y/Yt is a rough approximation tofollowing a streamline close to the wall. As already noted infigure 4, at y/Yt = 0.015 we see an initial reduction in overall
energy content followed by an increase at x/R = 2.43 as theflow enters the mid-body section. Further from the wall, at they/Yt = 0.28 or 0.51 locations, there is less turbulent energy anda general shift in the energy distribution towards larger scales
compared to the near-wall location. In this scaling, the spectraat the larger scales (lower frequencies) collapse suggesting thatthe local bulk velocity and the spacing between the BOR andthe pipe wall are the correct scales for the outer layer motions.
The differences in the spectra shown in figure 5 are also
telling. Close to the wall, local bulk velocity is not the rightscale for the energy content as seen by the lack of collapse in
the middle region of the spectra. However, away from the wall
there is some collapse for the x/R = −1.45 and 0.19 locations,suggesting that the energy content away from the wall may
initially scale with local bulk velocity. This is opposed to thenear-wall locations where there is no collapse of overall energy
content. It is also important to note that some of the difference
observed near the wall at various streamwise positions can be
attributed y+ changing due to the changing pressure gradienteven as y/Yt stays constant.

Figure 1: A low-profile sting holding the NSTAP andpitot probe.

where x ∈ [−lb , 0] and d is the diameter of the center body. The
midbody is a straight cylindrical section with length 5D/3, andthe stern section has a length ls = 4D/3 and follows a powerfunction satisfyingdx4
y = ± (1 − 4 )2ls(2)

where x ∈ [0, ls ].
Data are taken using a nanoscale thermal anemometryprobe (NSTAP) (Bailey et al., 2010; Vallikivi et al., 2011; Vallikivi & Smits, 2014) mounted on a probe holder that traverses
in the radial direction. A low profile probe holder shown infigure 1 was used to minimize blockage. The geometry ofthe probe holder allowed measurements close to the surfaceof the pipe, but limited access to locations close to the BOR
surface. The probe was fixed at a given streamwise location,and to gain access to various streamwise locations the stingholding the BOR was fixed to a carriage and rail mounted on
the pipe wall. A linear drive system placed 16D downstreamof the measurement probe allowed the BOR to move 14D inthe streamwise direction relative to the fixed probe. The railand carriage system occupied 0.6% of the pipe cross section
so that its blockage effects were negligible, and the measurement probe was located diametrically opposite the rail system.

RESULTS
Data were taken on an axial-radial plane intersecting theaxis of the pipe between x/R = −1.5 and 12.5 (about 4R downstream of the stern), where x/R = 0 is located at the nose of
the BOR and R is the pipe radius. The wall-normal direction
is denoted by y/R where y/R = 0 is the pipe wall. For the data
presented here, the Superpipe facility was operated at a bulk
velocity Ub = 10.4 m/s, Reynolds number based on pipe diameter ReD = 490, 000, friction Reynolds number Re+ = 10, 000,and upstream friction velocity uτ = 0.43 m/s. Figure 2 shows
points where data are taken as well as the BOR profile andpipe wall. These data are collected using a NSTAP with non
dimensional length l + = l/lν = 9.4 where lν is the viscous
length scale based on the upstream friction velocity.

Upstream and Bow
Straight Development Section

The mean flow at various streamwise locations upstreamof the BOR and in the region of the bow are shown in figure
3. The wall-normal coordinate y is normalized by Yt whichis equal to the pipe radius R upstream and downstream of theBOR, and elsewhere equal to the distance between the pipe
wall and the surface of the BOR. The mean velocity is shownnormalized by the upstream bulk velocity (figure 3a) as well

After the bow region of the BOR, the flow enters thestraight cylindrical midbody region where bulk flow acceleration and streamline curvature no longer have an affect. Figure 6a shows growth of the low-momentum near-wall shear
layer and acceleration of the flow in the middle of the annulus. Given enough development length, the mean flow profile2

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

Figure 2: An axial-radial cross section of the BOR and pipe wall are shown. Points where data are taken are denoted bydots. x/R describes the streamwise location relative to the nose of the BOR and y/R denotes position relative to the pipewall. Vertical lines on the BOR separate the bow, straight cylindrical, and stern sections of the BOR.

(a) Mean flow normalized by upstream bulk velocity Ub .

Figure 4: Reynolds normal stress upstream of the BOR

and in the region of the bow.

cover in the BOR’s wake and the shear layer close to the pipe
wall continues to thicken as the flow recovers towards fully
developed pipe flow.

CONCLUSIONSNSTAP data are taken for a canonical axisymmetric geometry placed in the Princeton Superpipe facility. The geometry features favorable and adverse pressure gradients, streamline convergence and divergence, and streamline curvature.
The body used in this work is composed of three sections: thebow, straight cylindrical section, and stern. As the flow accelerates past the bow section, turbulence is suppressed andthe near-wall shear layer shrinks. Analysis of spectra suggestthat in the bow region, the large turbulent scales with the local
bulk velocity and spacing between the BOR and pipe wall. In
the straight cylindrical section, the flow relaxes and the nearwall shear layer grows before encountering the stern where theshear layer grows again as the flow decelerates and expands.
A pronounced velocity deficit grows in the stern region before recovering in the wake of the BOR. Results in this workalign well with previous results obtained using PIV in a water pipe of a similar geometry (Ding et al., 2019; Saxton-Fox
et al., 2019). This work provides data close to the pipe wall
where PIV experiences near-wall filtering. Future analysis ofthese data includes a deeper look into Reynolds normal stressand spectra in the straight cylindrical section, the stern, andthe wake of the BOR along with a more robust scaling for themean velocity profiles and spectra. Future works will also traverse a range of Reynolds numbers and BOR blockage ratios.

(b) Mean flow normalized by local bulk velocity Ub,x .
Figure 3: Mean flow upstream of the BOR and in theregion of the bow.

would relax to fully developed annular flow profile. However,the flow encounters the stern section of the BOR well before
this can happen.

Stern and Wake
Figure 6b shows mean flow profiles in the stern region ofthe BOR and in the wake. In the stern, there is a bulk deceleration of the flow as well as a pronounced velocity deficit aroundthe centerline that grows in the stern region as seen in the profiles from x/R = 6.59 to x/R = 8.45. As the velocity deficit isgrowing, the shear layer near the pipe wall thickens due to theadverse pressure gradient and streamline divergence. Between
x/R = 9.28 and x/R = 12.49 the velocity deficit begins to re-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) Midbody(a) y/Yt = 0.015(b) Stern and wake(b) y/Yt = 0.28

Figure 6: Mean flow normalized by upstream bulk veloc-

ity for the midbody, stern, and wake of the BOR.

REFERENCES
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ACKNOWLEDGEMENTS
I.E.G. was supported by the National Defense Scienceand Engineering (NDSEG) Fellowship Program. Additional
support was received under ONR Grants N00014-17-1-2309
and N00014-19-1-2301 (Program Manager Peter Chang).

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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