12th  International  Symposium  on  Turbulence  and  Shear  Flow  Phenomena  (TSFP12)  Osaka,  Japan  (Online),  July  19-­22,  2022    ON THE SPECTRAL DECOMPOSITION OF SKEWNESS IN CANONCIAL AND
ACTUATED TURBULENT BOUNDARY LAYERS
S. Midya,
Institute for Flow Physics and Control
University of Notre Dame
Notre Dame, IN
Samaresh.Midya.1@nd.edu

F. O. Thomas
Institute for Flow Physics and Control
University of Notre Dame
Notre Dame, IN
fthomas@nd.edu

S. V. Gordeyev
Institute for Flow Physics and Control
University of Notre Dame
Notre Dame, IN
sgordeye@nd.edu

	
  
	
  

	
  

ABSTRACTThe skewness of the streamwise velocity is an
important parameter in turbulent boundary layer flows.
It is intrinsically related to the mechanism of quadraticenergy transfer between different scales of motion andhas also been shown to be related to the modulation ofnear-wall turbulence by large scales. In this paper, theskewness in both a canonical and actuated zero pressure
gradient turbulent boundary layer is spectrally
decomposed via the real part of the bispectrum. It isshown that the real part of the bispectrum allows theindividual triad interactions contributing to theskewness to be analyzed. These measurements arepresented for a range of wall-normal locations
associated with positive, zero and negative skewness.
The individual contributions are also summed to obtain
partial and cumulative sums of the skewness as afunction of frequency. Complementary conditional
sampling measurements demonstrate that the main
contribution to boundary layer skewness is intimately
associated with ejection-sweep events and the degree ofasymmetry of their characteristic velocity signatures.
Actuation is used to document the influence of outerlayer large-scale structures on the spectral content ofskewness.

decomposed signal given by u+ = uL+ + uS+ where subscripts L
and S denote large- and small-scale contributions, respectively,
they showed that a significant contribution to the skewness isthe cross term 3"#$ "%$ !!where an overbar indicates a time-meanquantity. Given this importance, this paper examines the
spectral decomposition of the skewness in a zero-pressure
gradient turbulent boundary layer. This decomposition isachieved via the real part of the bispectrum and provides ameasure of the modal contribution to the skewness at different
representative wall-normal locations. The relation between theskewness and the bispectrum is described in the following
section.

INTRODUCTION
It is clear that the skewness of the streamwise velocity
fluctuation is an important parameter in turbulent boundary
layers (TBL). It is intrinsically related to the quadratic energy
transfer mechanism since only triadically coupled modes makenon-zero contributions to the skewness (Duvvuri & Mckeon,2015). Skewness is also intimately related to the interactions
between different spatial scales present in the turbulent flow.
Hutchins and Marusic (2007a,b) demonstrated that near wall
turbulence is modulated by the outer large-scale motions. Thisamplitude modulation of near wall turbulence wascharacterized by Mathis et al (2009, 2011). Their study
correlated a low-pass filtered outer region signal with theenvelope function of the near-wall, small-scale velocity
fluctuations obtained via a Hilbert transform. The resulting
profile of the normalized amplitude modulation correlation
coefficient was found to bear a very strong resemblance to thecorresponding velocity skewness profile. Using a scale

Here, ![##] denotes an expected value, !"#* a temporal Fouriertransform of a time series x(t) and the superscript ! denotes acomplex conjugate. In the application of the bispectrum to be
used in the boundary layer here, the frequencies have beennormalized by the large eddy frequency ( !"* ≡ !" %&%/() ).
Owing to its symmetry properties (e.g Rosenblatt and Ness
(1965), Kim and Powers 1979, Elgar (1987)) it is sufficient tocompute the bispectrum for the triangular frequency domainshown in Figure 1. This triangular region is associated withtriad sum interactions. The correspodning two difference
interactions involving the same wave triad would also map to
the same point in this domain. For the triangular region shownin Figure 1, the third moment is related to the real part of thebispectrum by the relation (Rosenblatt and Ness (1965), Elgar
and Guza (1985)),

	  THE SPECTRAL DECOMPOSITION OF SKEWNESS
Like the spectral decomposition of the second moment via
the autospectral density, one can decompose the skewness infrequency domain using the third-order spectral estimate
known as the bispectrum. The bispectrum captures the triadic
interactions between the frequencies present in the flow. The
bispectrum is defined as, 	
  

(

)

⎡!C ! ! ⎤⎥⎦⎣⎢B XXX f1 , f2 = E ⎢ X f X f X f ⎥
*

*

*

*

*

1

2

where

f * = f1* + f2*

(

)

( (

	
  

	
  

))

( (

E u3(t ) =12∑ ∑ Re B xxx f1* , f2* +6∑ Re B xxx f1* , f2*

1  f1* f2*<f1*

f1*

))

	
  

12th  International  Symposium  on  Turbulence  and  Shear  Flow  Phenomena  (TSFP12)  Osaka,  Japan  (Online),  July  19-­22,  2022    %This equation can be normalized by !"#$
	  !!in	  order	
  to	
  directly	  relate	  the	
  real	  part	  of	
  the	
  bispectrum	
  to	
  skewness.	
  The	
  real	
  
part	
  of	
  the	
  bispectra	
  to	
  be	
  presented	
  were	  computed	
  form	  %signals	
  normalized	
  by	
  !"#$.	
  !!

modal velocity is shown in Figure 10 of Lozier et al (2020,2022).
For the canonical turbulent boundary layer at !"# = 1770,!!
the Reynolds number was sufficiently low that the premultiplied 1-D wavenumber spectra showed no dominant outerpeak. In this manner, the ALSSA could be used to introduce
periodic coherent structures into the outer region as describedin Lozier et al (2021, 2022). This allowed examination of theirinfluence on the nature and spectral decomposition of
skewness.

Figure 1. Region of computation of the auto-bispectrum. Only
the sum interaction region is shown.

Figure 2. Schematic of the ALSSA actuator mounted on theboundary layer development plate.

EXPERIMENTAL FACILITY
The experiments were conducted in a subsonic, open-return
wind tunnel in the Hessert Laboratory at the University ofNotre Dame. The wind tunnel has an inlet with a contraction
ratio of 20:1. The inlet has a series of 12 turbulence
management screens to condition the flow resulting infreestream turbulence levels of less than 0.1% for frequencies
greater than 10 Hz. The test section is 0.61 m by 0.61 m incross section and 1.83 m in length. A flat boundary layer
development plate 2 meters in length with an elliptic leadingedge covered with distributed sandgrain roughness was
installed in the tunnel. The plate spanned the full width of thetest section. Hot-wire traverses confirmed canonical zeropressure gradient TBL development on the plate. Table 1
summarizes the boundary layer parameters measured 1.42 m
downstream of the plate leading edge.

Figure 3 compares skewness profiles for the canonical andplasma-actuated turbulent boundary layer cases. This figure
shows similar skewness profile shapes for both cases.
However, somewhat higher positive skewness occurs near thewall for the plasma actuated case. Both profiles reach near zeroskewness at y+ = 15 with the plasma actuated case showing
greater negarive skewness in the lower logarithmic region.
0.7

J
O

Skewness

0.6
0.5

J

0.4

O

0.3

O

0.1

-0.1

!

"#

$%

&'

H

()*

()%

35 mm

7.0 m/s

0.298 m/s

0.0037

1.3

1,770

690

-0.2

J
O
J
OO
OO
J
O
JO
O O
J

-0.3
5

Plasma actuated

10

O
J

O
J

O
OOO OJJOJ OJ J J
J

100

200

y+

!

Figure 3. Skewness profiles for the canonical and plasmaactuated turbulent boundary layers.

In order to introduce organized large-scale spanwise
vorticity into the boundary layer, a plasma-based Active LargeScale Structure Actuator (ALSSA) was developed and used(see Lozier et al, 2020, 2022). The ALSSA uses AC-DBD
plasma to introduce organized large-scale coherent structures
into the TBL. Figure 2 shows a schematic of the setup. The
actuator consists of thin plate of streamwise length d (theboundary layer thickness), spanwise length 8d and maximum
thickness of 0.05d, with its trailing 0.3d linearly tapered. The
contoured elliptic leading edge of the plate was located at a
streamwise distance of x =140 cm from the boundary layer
development plate leading edge. The plate was placed at a wall
normal location of h = 0.3d (h+ = 200). The spanwise lengthchosen ensured spanwise homogeneity. The plate was made of
Ultem (dielectric material) to facilitate the plasma generation.
The DBD plasma was operated with a 4kV carrier frequency
with 32kV peak-to-peak voltage and was modulated with asquare wave at a frequency of 80Hz (f* = 0.4) with 50% duty
cycle. This introduced a series of spanwise vortices at passage
frequency f* = 0.4 into the TBL at y+ = 200. The resulting	
  Canonical

J
J
O

0.2

0
Table 1. Turbulent Boundary Layer ParametersO

SPECTRAL DECOMPOSITION OF SKEWNESS IN THE
TURBULENT BOUNDARY LAYER
The location y+ = 8 is taken as representative of the positive
skewness that is observed in the near-wall region of turbulent
boundary layers over a wide range of Reynolds numbers andthe spectral decomposition for the canonical boundary layercase is shown in Figure 4a. The real part of the bispectrum ispresented in an !"* , !%* ,&&sub-domain (highlighted by the coloredrectangular region in Figure 1) that is a smaller subset of thefull domain in order to highlight the region of significant modal
content. For all bispectral computations, a sampling frequency*of 30 kHz (!"#$%&'= 150) , an FFT blocksize of 8192 and total
1300 blocks were used for the calculations. A series of
convergence tests were performed in order to verify that all
bispectra were fully converged.
The variable-interval-time-averaging (VITA) conditional
sampling technique of Blackewelder and Kaplan (1976) was
also applied but a distinction was made in the conditional
sampling algorithm between ejection-sweep (ES) events

2  12th  International  Symposium  on  Turbulence  and  Shear  Flow  Phenomena  (TSFP12)  Osaka,  Japan  (Online),  July  19-­22,  2022    leading to positive versus negative skewness. The ESsignatures that gives rise to negative skewness are denoted by-Ve and those giving rise to positive skewness are denoted+Ve. For comparative purposed Figure 4b presents theprobability density functions (PDF) at y+ = 8 for VITA ES
event frequency, f*, that result in either positive or negativeskewness. The PDF for positive skewness events is indicatedin red while that associated with negative skewness is indicatedin green and overlays the +Ve with sufficient transparency sothat its PDF can still be discerned. It is quite clear that at y+=8
positive skewness ES events are much more frequent which is
fully consistent with the positive skewness shown in Figure 3
for the y+=8 location.

respectively. As in the canonical TBL, the dominant triad
interactions involve involve !"* $ and !"* < 0.3 as indicated by thedashed triangular region in Figure 5a. The peak triad
interactions in this region are associated with integer multiples*of !"#$
	
  !!.	  A	  main	  difference	
  of	
  the	
  spectral	
  decomposition	
  for	  the	
   actuated	
   TBL	
   are	   the	   indicated	
   triad	
   interactions	
  associated	
  with	  actuation	
  frequency	
  ! * !!
=	
  0.4	
  and	  a	
  range	
  of	
  frequencies	
   extending	   to	
  ! * !!
=	
   1.5	
   as	
   indicated	
   by	
   the	   arrows	
  in	
   Figure	
   5a.	
   These	   triad	
   interactions	
   are	   associated	
   with	  the	
  production	
  of	
  positive	  skewness.	
  	
  
a.)

b.)

Figure 5. a) Spectral decomposition of skewness at y+=8 forthe actuated TBL. b) PDF of the frequency of positive (+Ve)
and negative (-Ve) skewness producing E-S events.
A summation over the real part of the bispectrum recovers
Figure 8. a) Spectral
decompositionof skewness
at y+ = 8; b)ofPDFs
of the frequency
positive (+}r ) and
Figure4. a) Spectral
decomposition
skewnessat y+=8offor
the skewness. Figure 6 shows the method of summation overnegative (−}r ) skewness producing ES and SE events.
the canonical TBL. b) PDF of the frequency of positive (+Ve)
and negative (-Ve) skewness producing E-S events.

the real part of the bispectrum. The frequency resolution, Δ"#* !!
and Δ"#* !!of the bispectral plot are set equal. As shown in Figure
6a the area is divided using lines connecting ( !"Δ$%* , 0!!
) and(0, #Δ%&* )!!(where ! = 1,2, . . , '!!
). The sum is taken over the 	  Figure 4a shows that the most significant triad interactions
contributing to positive near-wall skewness at y+ = 8 involve!"* $ and !"* < 0.3. That is, the region of most significant triad
interactions is bounded by !"*$ + !&*$ = 0.6 as indicated by thetriangular region delineated by the dashed lines. Figure 4b
shows this to correspond to the frequency range contributing
most to the probability density function for the frequency of+Ve burst events. Based on the VITA conditional sampling, themean positive skewness event frequency at y+ = 8 is*
= 0.05. It may be noted from Figure 4aapproximately !"#$
Figure 9. Partialthat
and cumulative
summation
skewnessfrom the realpart bispectrumof the bispectrum
the highest
peaks ofofthethe
real portionof thelieat ~ , = .
inside the dashed region with triad interactions that involve**+ (//2)%&'(),0
multiples of this frequency:( !/2)%&'(),+,
8! = 2,3,4,5((() = (2,3,4,5 . Figure 4a also shows that weaker
triad interactions involving !"*$ = !&* < 0.7 , which conicides
with the upper limit of frequency content of the PDF for VITA
ES event frequency, also contribute to the positive skewness.
Figure 5a presents the spectal decomposition of skewness at
y+ = 8 for the ALLSA actuated TBL. Figure 5b presents thecorresponmding PDFs at y+ = 8 for VITA ES event frequency,that result in either positive or negative skewness. With
Figure 6. Summation over the real part of the bispectrum toactuation, the most probable frequency for positive andrecover skewness.
**!
=
negative producing ES events is "#$ 0.04 and !"#$ ≈!!0.003,

	
  

3  

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

  
*
*
!"#$
= 0.031 and !"#$!!= 0.029. This is consistent with theobservation from Figure 3 that the skewness is nearly zero at
this wall-normal location. Figure 8a shows that as before, thedominant interactions are clearly associated with the frequency
range shown in the PDFs for +Ve and -Ve events(!"* , !%* < 0.3)!!
. Negative peaks are now present in this region!! !"* $!!< 0.15. Examination of the ordinate scale
for !"* $ < $0.3$andshows that the negative peaks clearly dominate the positive
peaks in this region. 	
  Figure 9a presents the spectral decomposition of skewness
at y+ =15 for the actuated TBL. The corresponding PDFs forthe positive and negative skewness producing ES events areshown in Figure 9b. As in the canonical flow the most probablefrequencies for positive and negative skewness producing**= 0.034 and !"#$!! 0.038. As was
events are comparable: !"#$=
the case for the canonical TBL, negative peaks dominate the!! !"* $!!< 0.15. The influence of thebispectrum for !"* $ < $0.3$and
ALLSA-imposed outer large-scale structures is manifest in
multipe triad interactions between ! * !!= 0.4 and a range of
frequencies extending to ! * !!
= 1.7 although the strongest occur
for ! * !!
< 0.6.

region between a pair of consecutive lines and is denoted by!" #(where!!
! = 1,2, . . , '!!
). These values can then be plotted as a*!!
partial sum as shown in Figure 6b. The abscissa is denoted !"#**and its resolution Δ"#$ !!equals Δ"# !!
. Thus, the area under the
curve of !" s!!recovers the skewness.
0.5

Bispectral Sum

0.4
0.3

Canonical 50X Partial Sum
Canonical Cummulative Sum

0.2

Actuated 50X Partial Sum
0.1

Actuated Cummulative Sum

0
0

0.5

1

1.5

2

2.5

∗
!"#

3

3.5

4

4.5

5

Figure 7. Partial and cumulative bispectral sums for thecanonical and actuated TBL at y+ =8.
Figure 7 compares the partial and cumulative bispectral
sums for the canonical and actuated TBLs at y+=8. Both*exhibit similar partial bispectral sum variations with !"#.	
  !! The	  dominance	
   of	
   triad	
   interactions	   producing	
   positive	  skewness	
   is	
   readily	
   apparent.	
   The	   canonical	
   case	
   peaks	
  *!! 0.6	
   and	   the	   actuated	
   case	   near	  !"#* =0.5.	
  !! In	
   both	
  
near	
  !"#
=	
  
cases	
   this	
   is	
   associated with the dominant triad interactions
within the dashed regions shown in Figures 4a and 5a. 	
  Figure 8a presents the spectral decomposition of skewness
at y+ = 15 for the canonical TBL. The corresponding PDFs forthe frequency of positive (+Ve) and negative (-Ve) skewness
producing ES events is shown in Figure 8b. It is clear from thisfigure that at this wall normal location the relative frequency ofboth type of events is now comparable. More specifically, 	
  

a.)

a.)
b.)

Figure 9. a) Spectral decomposition of skewness at y+=15 forthe actuated TBL. b) PDF of the frequency of positive (+Ve)
and negative (-Ve) skewness producing E-S events.
Figure 10 compares the partial and cumulative bispectral
sums for the canonical and actuated TBLs at y+=15. The
dominance of the negative skewness producing triad
interactions shown in the dashed triangular regions in Figures
8a and 9a is apparent and both the actuated and canonical*!!
partial sums show negative peak values centered near !"#=0.2.
The primary difference between the two cases is associatedwith multiple triad interactions between ! * !!
= 0.4 and a range of
frequencies extending to ! * !!
= 1.7 for the actuated case. This
gives rise to positive peaks in the actuated case partial sum
leading to nearly zero skewness. In contrat, the canonical casecummulative sum remains negative.

b.)

Figure 8. a) Spectral decomposition of skewness at y+=15 forthe canonical TBL. b) PDF of the frequency of positive (+Ve)
and negative (-Ve) skewness producing E-S events.

	
  

4  

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

  
0.04
0.02
0
-0.02
-0.04
-0.06

*
producing evetns ( !"#$%= 0.053). Consistent with this, triad
interactions associated with negative skewness dominate thespectral decomosition over the frequency band associated with
the PDFs of ES events. The effect of the actuated large-scale
structures imposed by ALLSA is also now readlily apparent
and gives rise to triad interactions producing positive skewness
for !"* $!!< 0.4 and !"* !!> 0.2. 	
  Canonical 50X Partial Sum
Canonical Cummulative Sum

-0.08
-0.1
-0.12
-0.14

Actuated 50X Partial Sum
Actuated Cummulative Sum
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

a.)

5

∗
!"#

Figure 10. Partial and cumulative bispectral sums for thecanonical and actuated TBL at y+ =15.

a.)

b.)

Figure 12. a) Spectral decomposition of skewness at y+=60 for
the actuated TBL. b) PDF of the frequency of positive (+Ve)
and negative (-Ve) skewness producing E-S events.

b.)

Figure 13 compares the partial and cumulative bispectral sums
for the canonical and actuated TBLs at y+=60. This location iswithin the lower logarithmic region of the mean velocity
profile (the center of the logarithmic region is given by! " = 15&'() = 102 ). In this case the influence of theintroduced large-scale structures is quite significant. The
canonical case is dominated by triad interactions producing
negative skewness over a wide range in frequency. In contrast,
the influence of the imposed large-scale structures gives rise toa series of triad interactions (Figure 12a) that produces a*= 0.4.)!!
positive peak in the partial skewness summation at !"#It
is also apparent that the imposed structures exacerbate thenegative triad interactions such that the cumulative sum for theactuated case is more negative than in the canonical flow.

Figure 11. a) Spectral decomposition of skewness at y+=60 for
the canonical TBL. b) PDF of the frequency of positive (+Ve)
and negative (-Ve) skewness producing E-S events.
The spectral decomposition of skewness for the canonical
TBL at y+ = 60 is shown in Figure 11a and demonstrates a clear
dominance of triad interactions producing negative skewness.
The corresponding PDFs obtained at y+ = 60 for the frequency
of both +Ve, positive and -Ve, negative skewness producing
ES events are presented in Figure 11b. This figure shows thesignificantly increased frequency of negative skewness
producing events relative to those producing positive skewness.
*
The mean frequency for +Ve events is !"#$
= 0.049 and for -Ve*
events, !"#$% = 0.076 at this location. The dominant interactions
shown in the spectral decomposition involve multiples of themean frequencies for both +Ve and -Ve events withinteractions producing negative skewness dominating.
Figure 12a presents the spectral decomposition of skewness
for the actuated TBL at y+ = 60. The corresponding PDFs forthe frequency of ES events producing positive and negativeskewness are shown in Figure 12b. As was the cse in the
canonical flow, the mean frequency of negative skewness*
events ( !"#$%= 0.085) exceeds that for positive skewness	  !∗ = 0.4

Canonical 50X Partial Sum

Bispectral Sum

0.15

Canonical Cummulative Sum

0.1

Actuated 50X Partial Sum

0.05

Actuated Cummulative Sum

0
-0.05
-0.1
-0.15
-0.2
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 13. Partial and cumulative bispectral sums for thecanonical and actuated TBL at y+=60.

5  512th  International  Symposium  on  Turbulence  and  Shear  Flow  Phenomena  (TSFP12)  Osaka,  Japan  (Online),  July  19-­22,  2022    Figure 14 presents the phase-averaged skewness (meanremoved) in the actuated boundary layer. Here ! denotes theperiodic modal component due to the imposed periodic
actuation and u the residual turbulence. This figure clearlyshows that the artifically introduced outer large-scale structuresat y+ =200 modulate the skewness. Although this effect extends
to near the wall, it is much more pronounced in the logarithmic
region. Hence, as shown in the spectral decomposition
measurements, even in the actuated boundary layer the
skewness is largely dominated by discrete ES events associatedwith buffer layer streamwise vorticity.

combined effect of modulation by outer large-scale structures
whose effect on skewness grows with distance from the wall.
The increased effect of large-scale modulation with distancefrom the wall is apparent in the phase averaged skewness
shown in Figure 14.
REFERENCES
Blackwelder, R.F., Kaplan, R.E. On the wall structure of
the turbulent boundary layer, 1976, J. Fluid Mech., Vol. 76, pp.
89-112.
Duvvuri, S., and McKeon, B., “Phase relations in a forced
turbulent boundary layer: implications for modelling of highReynolds number wall turbulence”. Philos Trans A Math PhysEng Sci., 375(2089), 20160080, 2017.
Elgar, S., 1987, Relationships Involving Third Moment andBispectra of a Harmonic Process, IEEE Transactions on
Acoustics and Signal Processing, Vol. ASSP-35, No. 12, pp
1725-1726.
Hutchins and I Marusic. Evidence of very long meanderingfeatures in the logarithmic region of turbulent boundary layers.
J. Fluid Mech., 579:1–28, 2007a.
N. Hutchins and I. Marusic. Large-scale influences in nearwall turbulence. Philos. Trans. R. Soc. A, 365(1852):647–664,
2007b.
Kim, Y. C. and Powers, E. J., "Digital Bispectral Analysis
and Its Applications to Nonlinear Wave Interactions," in IEEE
Transactions on Plasma Science, vol. 7, no. 2, pp. 120-131,
June 1979, doi: 10.1109/TPS.1979.4317207.
Lozier, M., Thomas, F.O., and Gordeyev, S. 2020.
Streamwise Evolution of Turbulent Boundary Layer Response
to Active Control Actuator. AIAA SciTech Forum, AIAAPaper 2020-0097.
Lozier, M.,
Thomas, F. O. and
Gordeyev,
S.V.,“Experimental Studies of Boundary Layer Dynamics via
Active Manipulation of Large-Scale Structures,“ Procedings ofthe 12th Turbulent Shear Flow Phenomena (TSFP12), Osaka,
Japan, 2022.
Mathis, R., Hutchins, N., and Marusic, I., 2009, Largescale amplitude modulation of the small-scale structures inturbulent boundary layers, J. Fluid Mech., Vol. 681, p. 311.
Mathis, R., Marusic, I., Hutchins, N. and Sreenivasan, K.
R., 2011, The relationship between the velocity skewness andthe amplitude modulation of the small scale by the large scalein turbulent boundary layers, Physics of Fluids, 23, 121702.
Midya, S.,“On the spectral decomposition of skewness of
streamwise velocity in a zero pressure gradient turbulent
boundary layer,“ PhD Thesis, University of Notre Dame, 2021.
Rosenblatt, M.; Ness, J. W. Van. Estimation of the
Bispectrum. Ann. Math. Statist. 36 (1965), no. 4, 1120--1136.
doi:10.1214/aoms/1177699987.

!#" − ! "!%"!

Figure 14. Phase-averaged skewness (mean removed) in theactuated turbulent boundary layer.
CONCLUSION

The real part of the bispectrum is used to spectrally
decompose the skewness of streamwise fluctuating velocity inboth a canonical zero pressure gradient TBL and for the case inwhich the ALLSA actuator (Lozier et al, 2020, 2022)
introduces organized vortical motions into the TBL at y+ =200.
In both cases !"# !! = 1,770. The Reynolds number issufficiently low that a naturally occurring dominant outer
turbulence intensity peak does not exist in the 1D premultiplied streamwise wavenumber spectrum (Midya, 2021).
Comparison of the spectral decomposition for the natural andactuated turbulent boundary layer is performed at three wallnormal locations: y+ = 8, which represents a region of positive
skewness in the near-wall region; y+ = 15 which corresponds tothe wall normal location of peak turbulence intensity and forwhich the skewness is near zero; y+ = 60 which represents thelowerlogarithmic
region.
Supporting
conditional
measurements show that the triad interactions responsible fornear-wall skewness are intimately tied to the frequencies
associated with positive and negative skewness producing
ejection-sweep events. This suggests the important role playedby buffer layer streamwise vorticity in near-wall TBL
skewness. Furthermore, the imposition of large-scale vorticalstructures into the logarithmic region via ALLSA has little
effect on the spectral distribution of skewness at y+ = 8 andonly a modest effect at y+ = 15 with the observed triad
interactions remaining dominated by those associated withpositive or negative skewness producing ejection-sweep events.
In contrast, the effect of the actuation becomes quite significant
at y+ = 60 where triad interactions associated with the presenceof coherent structures with passage frequency f* = 0.4 become
readily apparent and significantly effect the skewness.
These results suggest a near-wall autonomous mechanism
associated with the production of buffer layer vorticity whoserole in near-wall skewness dominates. Along with this is the

	
  

	
  

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