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

CONVECTION VELOCITY IN DRAG-REDUCED OSCILLATING PIPE FLOW

Daniel Coxe
School for Engineering of Matter, Transport and Energy
Arizona State University
501 E Tyler Mall, Tempe, AZ 85281
dcoxe@asu.edu

Yulia Peet
SEMTE
Arizona State University
ypeet@asu.edu

Ronald Adrian
SEMTE
Arizona State University
rjadrian@asu.edu

ABSTRACT

simulated flow domain. Later we shall use 0 < y = R − r < R to
denote the distance from the pipe wall. Periodicity in x is a consequence of using the periodic inflow/outflow boundary condition,ui (x, r, θ ) = ui (x + L, r, θ ), i = 1, 2, 3, a condition that is thought
to emulate flow that is statistically homogeneous in x provided
L/R is sufficiently large. The sufficiency condition is not well understood, nor is the approach to statistically homogeneous flowin an infinitely long pipe. One idea is that L is sufficiently large
if there is a range L0 << L − L0 in which the spatial correlation
coefficient ρ(∆x, ∆t = 0) = 0, and that range L0 is a substantial
fraction of L. Conceptually, structures in this range would be uncorrelated with structures at the inlet or the outlet, suggesting thatthat the inlet/outlet conditions have no effect on structure in thisrange. However, uncorrelatedness does not imply statistical independence, so the inference must be considered weak.

Direct Numerical Simulations (DNS) of turbulent pipe flow
with periodic inflow/outflow boundary conditions at the ends ofa 12 diameter long pipe domain are performed at Reynolds numbers=170, 360 and 720. We simulate standard pipe flow with motionless wall and pipe flow in which drag is reduced by oscillating
the wall in the azimuthal direction. We present space-time correlations of the streamwise velocity with separation in the streamwise
direction and examine the effect of periodic in-flow/out-flow. It isfound that the turbulent fluctuations maintain significant correlation well beyond the time it takes to flow through the simulation
domain, implying that streamwise periodicity is not a benign condition, at least for the 12 diameter domain length. We computethe convection velocity from the space-time correlation using themethod of Wills (1964) and a new method obtained from the autocorrelations with pure spatial separation and pure time separation,
quantities which are much less computationally demanding than
the full space-time correlation. The new method uses the the values of the separation in space and time at which the respective
correlations first cross zero. Fair agreement is found between thetwo methods, although the method of zeroes is sensitive to noisein the correlation functions. We also find that the temporal coherence of large-scale motions increases markedly in space and timein the frame of the convection velocity when the drag is reduced
by wall oscillation.

The convection velocity of turbulence describes the average
speed at which turbulent structures (eddies) move downstream. Its
principal use occurs in applications of Taylor’s Frozen Field Hypothesis (Taylor (1938)), an approximation that makes it possible
to estimate the two-point correlation function with spatial separation in terms of the two-point correlation function with timedelay. The latter correlation is found readily from experimental
time-series data which are the natural outcome of physical experiments using a velocity probe at a fixed point. If the probe has highfrequency response, the inferred two-point spatial correlation has
high resolution in space. Conversely the Frozen Field Hypothesis
can also be used to convert highly resolved spatial data computedby numerical simulation into temporal data. The Hypothesis assumes that the velocity structure of the turbulence does not change
during the time it convects past the probe (Townsend (1980)). It
is variously assumed to be equal to the local mean velocity or thebulk mean velocity, but neither value is necessarily correct a priori, and this ambiguity is a major source of error in the correlation functions inferred using Taylor’s Hypothesis Del Alamo &Jiménez (2009)),Zaman & Hussain (1981)).

Introduction
Space-time correlation provides interesting insights notfound in space or time correlations, especially in flows having
strong mean velocities. In this paper we consider flow in an oscillating pipe. The space-time auto-correlation coefficient of streamwise velocity fluctuation U with streamwise separation, ∆X, andtime-delay, ∆t, is given byρuu (r, ∆x, ∆t) =⟨u(r, θ , x,t)u(r, θ , x + ∆x,t + ∆t)⟩x,θ ,t
⟨u2 ⟩

(1)

The classical definition of convection velocity given byWills(1964) (Wills (1964)) is based on the full two-dimensional
space-time correlation . This has resulted in convective velocities
being approximated using comparatively short time-correlations.
Under these conditions the short time-scales of turbulence may appear to behave as though they obey Taylor’s frozen flow hypothesis
and exhibit a distinct non-decaying periodicity with time delay due
to an insufficient time record or streamwise lengths (Wang et al.
(2020)). The modified method of Del Alamo & Jiménez (2009)

where averaging is done by integrating over the stream-wise ( oraxial) direction, x, the azimuth angle, θ , and time,t. The zeromean turbulent velocity fluctuation is a periodic and statisticallyhomogeneous function of 0 < x < L and 0 < θ < 2 ∗ pi and a statistically stationary function of t. r is the radial distance from thecenter-line, R is the radius of the pipe, and L is the length of the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022
non-zero correlation longer than the pipe length. The effect becomes greater with increasing Reynolds number producing weakcorrelations as long as 48R at Re=720 in the non-oscillating pipe.
The coherence length and time are more than doubled when thewall is oscillated to achieve drag reduction.(Compare Figure 1(a)and 1(b).) Clearly, L=24R is insufficient to render the periodic
inflow/outflow condition benign. hen running simulations of turbulent flows with drag reductions, researchers should take careto understand that what is considered a sufficiently long domain
for standard turbulence may be too short for drag reduction. The
space-time auto-correlations for radial velocity are significantly
less converged than the corresponding auto-correlations for the
streamwise velocity. Wall oscillations impart a resonance in thetemporal separation curves most easly seen in Figure 1. The sameis apparent for the radial fluctuation correlation. For comparison,
the streamwise-temporal auto-correlations of radial velocity arepresented in Figure 3. The correlations of radial velocity are significantly more noisey than those of the streamwise velocity.

is also numerically expensive as it requires the co-spectra of nonlinear terms with velocity. In this paper we consider several issues
bearing upon the definition and interpretation of the convection
velocity. Firstly, what is the effect of scale. Do large-scales of
motion travel faster or slower than small scales? Secondly, how is
the convection velocity affected by Reynolds number and pipewall
oscillation? Lastly, how are the length-scale of the spatial correlation and the time-scale of the temporal correlation related to theconvection velocity? This last question leads to a new method of
approximating the convective velocity by the ratio of the spatial
separation at which the spatial correlation function first crossed
zero, ∆x0 , to the temporal delay at which the time correlation firstcrosses zero, ∆t0 .

MethodThe results presented are gathered from Direct Numerical
Simulations of a turbulent pipe flow at three Reynolds numbers:Reτ = 170, 360 and 720. The simulation is carried out using ahighly scalable code Nek5000 (Deville & Mund (2002); Fischer(1997)). The domain of the flow is a locally structured, globally unstructured Cartesian grid. The wall shear stress τw is kept
fixed using a constant forcing term in the streamwise direction
fx = 2τw /ρR (R is the radius of the pipe). Spanwise wall oscillations are introduced to achieve drag reduction. Wall oscillations
are modeled as a velocity boundary condition, which is sinusoidal
in time and invariable in space.
The following terms scale the primitive variables. Ubulk
is the bulk velocity, which, coupled with the pipe diameter D
and viscosity ν describes the bulk Reynolds number of the flow,
D
Rebulk = Ubulk
ν . The pipe length is fixed at L = 24R to encourage structural decorrelation of very large scales in the streamwise
direction (Kim & Adrian (1999)). The selected oscillation param+
eters are kept fixed in the inner units as w+
wall = 10 and Twall = 100.
These values were selected from the available literature (Quadrio& Ricco (2004)). The result of drag reduction under the constantforcing term allows the mean flow rate to increase.
Simulations were run for 10000 ν/τw , and data is collected
every 3.125ν/τw to allow for well resolved temporally averaged
statistics and correlations. Results are interpolated using spectrally accurate interpolation routines (Merrill et al. (2016)) to agrid uniformly distributed in streamwise and azimuthal directions
and that uses Gauss-Legendre-Lobatto nodes in the radial direction. High performance MPI libraries for Python (Otten & Min(2016)) are used to post-process and plot the data. A Hanningwindow was applied to the time record of velocity fluctuations.
Record lengths were varied to achieve a total record length ofTrec ≈ U5L . A sliding window average of width ∆TURbulk = 4 wasbulk
applied to zero spatial separation time delay correlation function
and width ∆X
R = 6 was applied to the zero time delay correlation
function to filter out high frequency oscillation which resulted inspurious early time/space zero crossing.

Convective velocities
The black dotted line passing through the high correlation
region shows a line with the slope equal to the convective velocity given by the method of Wills (1964), defined below in the
Equation (2). Since the method of Wills involves the behavior of
the correlation function at small separations, it is dominated by
the contributions of the small scales. On the other hand, the low
correlation region corresponds to large-scale structures, which are
known to influence the log-layer turbulence (Hutchins & Marusic(2007); Guala et al. (2006)). One sees that the slope of the dashedline is greater than the slope of the zero contour line, indicatingsmall scales moving faster than large scales. In the current paper,
two methods for calculating convective velocity are presented andcompared.
Method of Wills. The first method is the classical method of
calculating convective velocity (Wills (1964)), which defines it as
the slope of the line that passes through the location of the maximum correlation of the streamwise-temporal correlation function.
This is found by finding ∆t through a process of a linear regressionof the points (∆x, ∆t, r) for each value of streamwise separation ∆x
and wall-normal location r as
∂ ρui ui (∆x, ∆t, r)|∆xi = 0.
∂ ∆t

(2)

From the equation 2 a set of (∆xi , ∆ti ) where the correlation is
maximized is gathered. Assuming a linear fit a linear least squares
algorithm can be applied with the slope cui ui = a1 .
[∆xi ] = [a0 , a1 ]⊺ [1, ∆ti ]Results
Streamwise-temporal correlations(3)

Method of Zeroes. In the second method, which is a new
method that we propose (referred to as the method of “zero matching”), we approximate the convection velocity through the ratio of
the length scale (Ix ) in the streamwise direction to the temporalscale (It ), where the convective velocity ⟨cui ui ⟩θ is defined asContours of the space-time auto-correlation coefficients ofthe streamwise velocity, ui are presented in Figure 1 for for a single, representative wall-normal location y+ = 110. This locationis chosen to represent the region of the outer-layer that is above
the logarithmic layer. The solid black lines indicate zero correlation, and the gray regions correspond to correlation coefficients
between 0-7.5% (nominally ’low’ correlation). Dark red denotes
the region of nominally ’high’ correlation (90-100%). Streamwise
periodicity of the velocity field causes the correlation coefficientsto be periodic in ∆x, and the use of Fourier transforms to represent
the velocity in time causes the correlation coefficients to also be
periodic in ∆t. We see that the temporal coherence of the velocity in a frame moving with the convective velocity is greater than
the flow through-time, 24R/Ubulk , leading to regions of weak, but⟨cui ui ⟩θ (r) =∆x ρui ui (∆x, ∆t = 0) = 0
Ix (r)=It (r)∆t ρui ui (∆x = 0, ∆t) = 0(4)

In Equation (4), the streamwise length scale Ix (r) is definedas the distance to the first zero crossing of the streamwise correlation function at zero time delay, and the temporal scale It (r) is
defined as the distance to the first zero crossing of the temporalauto-correlation function at zero spatial separation. A cubic spline2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022
interpolation with bisection is used to determine the numerical approximation of the first zero crossing. Figure 2 presents convective velocities obtained from Equation (2) and Equation (4) for
streamwise fluctuations, and Figure 4 presents the corresponding
data for radial fluctuations. Observations regarding drag reduction
show that for the lower Reynolds numbers, convective velocities
are significantly increased with respect to the bulk mean velocity.

Taylor, G.I. 1938 The spectrum of turbulence. Proc. of the Royal
Society of London 164 (919), 476–490.
Townsend, A. 1980 The structure of turbulent shear flow. Cambridge university press.
Wang, H., Huang, W. & Xu, C. 2020 Space–time characteristics ofturbulence in minimal flow units. Phys. Fluids 32 (12), 125103.
Wills, J.A.B. 1964 On convection velocities in turbulent shear
flows. J. of Fluid Mechanics 20 (3), 417–432.
Zaman, K. & Hussain, A. 1981 Taylor hypothesis and large-scalecoherent structures. J. of Fluid Mech. 112, 379–396.

Conclusions
The convective velocity profile calculated with the method ofzero matching (Equation (4)) shows a reasonable agreement withthe classical method. Near the centerline of the pipe the convective velocity tends towards the centerline velocity. Drag reduction shows a tendency to increase the convective velocity with respect to the bulk mean velocity for streamwise fluctuations. Thisis shown both using Method 1 and Method 2. This shows that themethod presented is consistent with the classical method. Somespurious spikes in Method 2, especially for Reτ = 360, are attributed to a difficulty to reliably converge on the first zero crossing for a correlation function, which is oscillatory and crosses anabscissa multiple times.
The Method of zeroes gives a reasonably good agreement
throughout the domain for the convective velocity related to
streamwise fluctuations. Radial fluctuations, however, exhibit a
distinct retardation. This is hypothesized to be due to what is observed in isotropic turbulence where the velocity correlation function of a fluctuation transverse to the axis of correlation quickly
goes negative and positive again (De Karman & Howarth (1938))
due to continuity requirements. Thus the streamwise length scale
based on a first zero crossing is shorter.

Acknowledgements
This work has been partially supported by NSF CBET grant
1944568.

REFERENCES
De Karman, T. & Howarth, L. 1938 On the statistical theory of
isotropic turbulence. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences 164 (917),
192–215.
Del Alamo, J.C. & Jiménez, J. 2009 Estimation of turbulent convection velocities and corrections to taylor’s approximation. J.
of Fluid Mech. 640, 5–26.
Deville, M.O., Fischer P. & Mund, E.H. 2002 High-Order Methods for Incompressible Fluid FLow. Cambridge University
Press.
Fischer, P. 1997 An overlapping schwarz method for spectral element solution of the incompressible Navier-Stokes equations.
J. Comp. Phys. 133, 84–101.
Guala, M., Hommema, S.E. & Adrian, R.J. 2006 Large-scaleand very-large-scale motions in turbulent pipe flow. J. of Fluid
Mech. 554, 521–542.
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary
layers. J. of Fluid Mech. 579, 1–28.
Kim, K.C. & Adrian, R.J. 1999 Very large-scale motion in theouter layer. Physics of Fluids 11 (2), 417–422.
Merrill, B., Peet, Y.T., Fischer, P. & Lottes, J. 2016 A spectrally
accurate method for overlapping grid solution of incompressible navier–stokes equations. J. of Comp. Phys. 307, 60–93.
Otten, M., Gong J. Mametjanov A. Vose A. Levesque J. Fischer P.
& Min, M. 2016 An mpi/openacc implementation of a highorder electromagnetics solver with gpudirect communication.
The Int. J. of High Perf. Comp. App. 30 (3), 320–334.
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent
drag reduction through spanwise wall oscillations. J. of Fluid
Mech. 521, 251–271.

3

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

ρuxux(∆x?, ∆t?, y + = 110)

ρuxux(∆x?, ∆t?, y + = 110)
0.9

10

0.9

10

0.6

0.3

0

0.0
−0.3

−5

∆x? = z/R

∆x? = z/R

0.6
5

−0.6

−10
−40

−20

0

∆t? = ∆t ubulk
R

20

0.3

0

0.0
−0.3

−5

−0.6

−10

−0.9

40

5

−50

−25

0

∆t? = ∆t ubulk
R

(a)
?

?

25

−0.9

50

(b)
?

ρuxux(∆x , ∆t?, y + = 110)+

ρuxux(∆x , ∆t , y = 110)
0.9

10

0.9

10

0.6

0.3

0

0.0
−0.3

−5

∆x? = z/R

∆x? = z/R

0.6
5

−0.6

−10
−40

−20

0

20

∆t? = ∆t ubulk
R

0.3

0

0.0
−0.3

−5

−0.6

−10

−0.9

40

5

−50

−25

0

∆t? = ∆t ubulk
R

(c)

25

−0.9

50

(d)

ρuxux(∆x?, ∆t?, y + = 110)

ρuxux(∆x?, ∆t?, y + = 110)
0.9

10

0.9

10

0.6

0.3

0

0.0
−0.3

−5

∆x? = z/R

∆x? = z/R

0.6
5

−0.6

−10
−50

−25

0

∆t? = ∆t ubulk
R

25

0.3

0

0.0
−0.3

−5

−0.6

−10

−0.9

50

5

−50

−25

0

25

∆t? = ∆t ubulk
R

(e)

−0.9

50

(f)

Figure 1: Space-time auto-correlation of streamwise velocity at y+ = 110 above the log-layer for Reτ = 170 (a),(b);Reτ = 360 (c),(d); Reτ = 720 (e),(f). The left-hand plots give the results for flow with no pipe oscillation (zero dragreduction), and the right-hand plots present the results with oscillation (non-zero drag reduction). Black dashed lines
indicate the line of best fit associated with the convection velocity of Wills (2), and the black dotted line is from themethod of zeroes. (4). The solid black line corresponds to the zero-contour level.
y

y

1

0

0.4 R 0.6

0.2

y
0.8

cux/Ubulk

1.0

cux/Ubulk

0.8

1

50

y

100
+

(a) Reτ = 170

150

0

1.0

0.4 R 0.6

0.2

0.8

1.0

cux/Ubulk

0.4 R 0.6

0.2

1

100

y

200
+

(b) Reτ = 360

300

0

200

400

y+

600

(c) Reτ = 720

Figure 2: These figures show the approximation of mean convective velocities of the streamwise fluctuations. Black solidlines are standard pipe flow and blue dashed are drag reduced. We compare the mean velocity profile (lines with nomarkers), Method 1 (Method of Wills, triangles) and Method 2 (zero matching, squares).
4

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

ρurur (∆x?, ∆t?, y + = 110)

ρurur (∆x?, ∆t?, y + = 110)
0.9

10

0.9

10

0.6

0.3

0

0.0
−0.3

−5

∆x? = z/R

∆x? = z/R

0.6
5

−0.6

−10
−40

−20

0

∆t? = ∆t ubulk
R

20

40

5

0.3

0

0.0
−0.3

−5

−0.6

−10

−0.9

−50

−25

0

∆t? = ∆t ubulk
R

(a)

25

−0.9

50

(b)

ρurur (∆x?, ∆t?, y + = 110)

ρurur (∆x?, ∆t?, y + = 110)
0.9

10

0.9

10

0.6

0.3

0

0.0
−0.3

−5

∆x? = z/R

∆x? = z/R

0.6
5

−0.6

−10
−40

−20

0

20

∆t? = ∆t ubulk
R

0.3

0

0.0
−0.3

−5

−0.6

−10

−0.9

40

5

−50

−25

0

∆t? = ∆t ubulk
R

(c)

25

−0.9

50

(d)

ρurur (∆x?, ∆t?, y + = 110)

ρurur (∆x?, ∆t?, y + = 110)
0.9

10

0.9

10

0.6

0.3

0

0.0
−0.3

−5

∆x? = z/R

∆x? = z/R

0.6
5

−0.6

−10
−50

−25

0

∆t? = ∆t ubulk
R

25

50

5

0.3

0

0.0
−0.3

−5

−0.6

−10

−0.9

−50

−25

0

25

∆t? = ∆t ubulk
R

(e)

−0.9

50

(f)

Figure 3: Space-time auto-correlation coefficient of radial velocity at y+ = 110 for Reτ = 170 (a),(b), Reτ = 360 (c),(d),Reτ = 720 (e),(f). The left-hand plots give the results for flow with no pipe oscillation (zero drag-reduction), and theright-hand plots present the results with oscillation (non-zero drag reduction) Black dashed lines indicate the line of best
fit associated with the convection velocity of Wills (2), and the black dotted line is from the method of zeroes (4). The
solid black line corresponds to the zero-contour level. The noise in the correlations indicates the difficulty in resolving atrue convective velocity.
y

y

1

0

0.4 R 0.6

0.2

y
0.8

cur /Ubulk

1.0

cur /Ubulk

0.8

1

50

y

100
+

(a) Reτ = 170

150

0

1.0

0.4 R 0.6

0.2

0.8

1.0

cur /Ubulk

0.4 R 0.6

0.2

1

100

y

200
+

(b) Reτ = 360

300

0

200

400

y+

600

(c) Reτ = 720

Figure 4: These figures show the approximation of mean convective velocities of the radial fluctuations. Black solid lines

are standard pipe flow and blue dashed are drag reduced. We compare the mean velocity profile (lines with no markers),
Method 1 (Method of Wills, triangles) and Method 2 (zero matching, squares).
5