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

ON THE STRUCTURE OF TURBULENCE FIELDS IN A SEPARATEDFLOW AROUND A FINITE WING; ANALYSIS USING DIRECT NUMERICALSIMULATION

Juan Carlos Bilbao-Ludena, George Papadakis
Department of Aeronautics
Imperial College London
Exhibition Rd, South Kensington, London SW7 2BX
jcb4717@ic.ac.uk, g.papadakis@imperial.ac.uk

ABSTRACT

these simulations, shear layers and separated boundary layers have been reported as the leading sources of turbulencein the tip-flow (Jiang et al. (2008)). The study of Garmann& Visbal (2017) reports both instantaneous and time-average
flows, and shows that the formation, separation, and entrainment of shear-layer substructures within the tip vortex feeding
sheet are stationary whereas interactions with the wake are unsteady. Smith & Ventikos (2021) conducted a DNS study, andpresented details related to the roll-up process and the behaviorof mean axial velocity and mean pressure profiles at relatively
low angles of attack.
Despite the recent progress in the fundamental understanding of the general flow features, there is relatively little
discussion in the interaction of vortical structures with the developed turbulence field. The main goal of this work is therefore to conduct a thorough analysis of the turbulence production mechanisms in the very near-field of a squared-tip finite
wing, including regions far from self-similarity; these are seldom examined and analytical approaches do not apply. In
comparison to some of the aforementioned studies, in thiswork, we aim to study a flow with a multi-scale character andmultiple developing secondary flows. This necessitates the useof DNS as a research tool. The revealed magnitudes and location of these mechanisms will lay the foundation for better
understanding and modeling/control strategies of these highly
non-equilibrium vortical flows.

We investigate the spatial distribution and production
mechanisms of turbulent kinetic energy around a NACA 0018
wing with square wingtip profile at Rec = 104 and 10◦ angleof attack with the aid of Direct Numerical Simulation (DNS).
The analysis focuses on the highly inhomogeneous region
around the tip and the near wake; this region is highly convoluted, strongly three-dimensional and far from being selfsimilar. The flow separates close to the leading edge creatinga large, open, recirculation zone around the central part of thewing. In the proximity of the tip, the flow remains attached
but another smaller recirculation zone forms closer to the trailing edge; this zone strongly affects the development of main
wing tip vortex. We find that the three-dimensional flow separation at the sharp tip close to the leading edge plays an important role on subsequent vortical flow development on thesuction side. The production of turbulent kinetic energy andReynolds stresses is also investigated and discussed in conjunction with the identified vortex patterns with a special focus on the role of vortex stretching/compression. The detailed
analysis of the mechanisms that sustain turbulent kinetic energy improves our understanding of these highly three dimensional, non-equilibrium flows and can lead to better actuation
methods to manipulate these flows.

INTRODUCTION
The flow physics of the near wingtip field is known to
be extremely complex due to its highly turbulent and threedimensional nature comprising multiple cross-flows. Morespecifically, wingtip flows involve roll-up of vortex sheets,
shear layer instabilities, and boundary layer separation (Green(2012)). It is known that traditional turbulence models that employ the eddy-viscosity concept, cannot capture the effect ofrotation (and the associated misalignment between the stressand stain-rate tensors) that characterize vortical flows, and thishas led to over-predictions of turbulence production in flows
involving round tip profiles (Churchfield & Blaisdell (2009)).
Recent works have investigated the three-dimensional
flow around low-aspect-ratio finite wings at low Reynolds
number (laminar flow) (Taira & Colonius (2009a,b); Zhang
et al. (2020b)), focusing mainly on post-stall conditions andthe effect of different sweep angles (Zhang et al. (2020a)).
High-fidelity simulations at higher Reynolds numbers in thepresence of turbulence have been made (Uzun & Hussaini(2010); Garmann & Visbal (2017); Jiang et al. (2008)). In

COMPUTATIONAL DETAILSIn the present work, the flow around a NACA 0018 wingwith square wingtip is simulated at a chord-based Rec = 104(value relevant to small and medium-scale unmanned aircrafts)
and 10◦ angle of attack (AoA). The simulation has been performed with the Finite Volume Method (FVM) in-house code
PANTARHEI which solves the incompressible Navier-Stokes
equations using the fractional step method, with incrementalpressure correction to satisfy the continuity equation. The aspect ratio of the wing is equal to 2 and the domain extends
one chord length in the spanwise direction beyond the wingtip. The mesh comprises around 63 million cells which areclustered in the vicinity of the airfoil, around the wing tip andthe wake. The thickness of the cell closest to the wall has∆y+ < 0.5 and the grid spacings in the streamwise and spanwise directions have ∆x+ < 3 and ∆z+ < 1 respectively (theplus superscript, + , indicates wall units, as usual). The DNS
quality of the mesh was further checked by computing the ra-1

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

Figure 1. a) Contours of the ratio V 1/3 /η in the y − z planeat x/C = 1.1, the location of fully developed turbulence. The
white line represents the Q-value (see next section for definition) and the red dot the projected location of the wingtip
corner. (b) Plot of V 1/3 /η along the dotted blue line shownin (a) at y/C = 0.04. (c) Contour at x − y plane at z/C = 0.5(quarter-span).

Figure 2. Contours of the mean axial velocity ⟨U1 ⟩ at equally
spaced planes x/C = 0.2, ..., 1.3. The purple-coloured contour
line represents ⟨U1 ⟩ = 0.

tio of the grid size (equal to the cubic root of the cell volume,
V 1/3 ) to the Kolmogorov length scale, η. Fig. 1(a) shows acontour plot of this ratio in the z − y plane downstream of thetrailing edge; the maximum value is below 2.5. In panel (b),the values along the dashed line shown in panel (a) are found tobe below 2. Fig. 1(c) shows contours in the x − y plane locatedat quarter span; again the values do not exceed 2.5.

Visualization of isosurfaces of the time-average Qcriterion from different viewing angles enables us to understand better the structures that originate in the pressure area
and the tip region. Fig. 3 (a) shows clearly one vortex thatstarts at the flat tip side and grows. This vortex is designated as
Left Vortex (LV) and arises due to the spanwise separation of
the shear layer emanating from the pressure side of the wing.
Near the sharp edge on the suction side, a concentrated vortex starts developing at around x/C = 0.1, it grows and subsequently weakens as it approaches the trailing edge; we designate this vortex as V0. The evolution of this vortex can be visualized better in Fig. 3(c); we see that V0 has decayed shortly
after the trailing edge. Starting at the same initial plane as V0,
the Right Vortex (RV) develops in parallel but its trace is lost ata short distance downstream of the trailing edge. The vorticalregion R0 is defined as the area where the separated shear layers taking part in the roll-up process amalgamate and readjust
downstream of the tip. The main trailing vortex V1 originates
at R0 and persists throughout the computational domain. V2 isanother vortex that originates also from R0 but merges further
downstream with V1.
The organization of the vortex structure topology at
x/C = 0.75 is shown in Fig. 3(b) and shows good agreementwith visualizations reported in experimental (Katz & Galdo(1989); Bailey et al. (2006); Giuni (2013); Birch et al. (2003))
as well as computational studies (Smith & Ventikos (2021))
despite employing different Rec and geometrical setups.
We explore next the origins of the wake tip vortices andtheir evolution. We consider the streamwise vorticity ⟨Ωx ⟩ andthe vorticity intensification ⟨Ω j ⟩⟨Sx j ⟩, which can be decomposed into vortex stretching (or axial elongation) and tilting
terms,RESULTS
Main Features of the Near Wake velocity fieldand vortical structures
Contours of the time-averaged streamwise velocity at different streamwise planes are shown in Fig. 2. The flow separates at x/C ≈ 0.2 and does not reattach on the suction sideof the wing. Close to the leading edge, the separating region
occupies most of the span, but once the secondary flow in the
wingtip starts to develop, the left boundary is displaced inboard. This creates a recirculating bubble with the outboard
end forming an angle with the streamwise direction; this will
be seen more clearly in subsequent figures. A 3D separated
region with a similar trapezoidal shape has been also observedin Garmann & Visbal (2017) for a wing with a rounded tip
profile. It is very interesting to see that at x/C ≈ 0.8 anothersmall recirculating bubble (indicated by a deep blue color) appears close to the wingtip; this bubble closes in the near wakeat x/C ≈ 1.2. Between the two recirculation regions, a jet withpositive velocity appears. As will be seen later, the shear layers in the wall-normal and spanwise directions play a very important role in the turbulence production and distribution of
Reynolds stresses across the span. The flow has a wake-likebehaviour, i.e ⟨U1 ⟩ < 1 at all planes (the angular brackets indicate time-averaging). The only region where ⟨U1 ⟩ > 1 is observed slightly above the wake where the velocity magnitude
reaches a value of approximately 1.1.
The Q-criterion was used to identify the emerging coherent structures of the finite wing flow. The identification method
λ2 (Jeong & Hussain (1995)) was also implemented and resulted in similar structures. Q is defined as the second invarianti, i.e. Q = 0.5 Ωi j Ωi j − Si j Si j ,of velocity gradient tensor ∂U∂xj
where Ωi j and Si j are the components of the instantaneous rotation and strain-rate tensors respectively. Values Q > 0 denote a rotation-dominated region, which can be used for vortex
identification purposes.

⟨Ω j ⟩Sx j = ⟨Ωx ⟩⟨Sxx ⟩ + ⟨Ωy ⟩⟨Sxy ⟩ + ⟨Ωz ⟩⟨Sxz ⟩ .
| {z }|{z}vortex stretching(1)

vortex tilting

From conservation of angular momentum, a vortex tube un∂ ⟨U ⟩dergoing mean stretching, i.e. ⟨Sxx ⟩ = ∂ x 1 > 0, will become1more narrow and have increased streamwise vorticity. On theother hand, under mean compression i.e. ⟨Sxx ⟩ < 0, the vorticity tube will grow thicker, resulting in a reduced rotational
rate (Wu et al. (2015)). Fig. 4 presents (filled) contours of2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
the large bubble originating from the leading edge. The levelsof turbulent production and k are small in the V0 core, most
likely due to the stabilizing effect of solid body rotation (Zeman (1995)).
We now proceed to study the production terms of the
Reynolds stresses of ⟨u1 u1 ⟩ and ⟨u2 u2 ⟩, given by Pope (2000),

R−s
P11
= −2



∂ ⟨U1 ⟩
∂ ⟨U1 ⟩
∂ ⟨U1 ⟩
+ ⟨u1 u2 ⟩+ ⟨u1 u3 ⟩⟨u1 u1 ⟩
∂ x1
∂ x2
∂ x3
(2)



∂ ⟨U2 ⟩
∂ ⟨U2 ⟩
∂ ⟨U2 ⟩
R−s
P22
= −2 ⟨u2 u1 ⟩+ ⟨u2 u2 ⟩+ ⟨u2 u3 ⟩
∂ x1
∂ x2
∂ x3
(3)
The magnitude of ⟨u3 u3 ⟩ (and its production) is smaller incomparison with the other two stresses and is not investigated.
R−s
R−s
Figures 6 and 7 show isosurfaces of P11
and P22
respectively. In each plot, isosurfaces of the most importantcomponents that make up the production terms are also plot∂ ⟨U ⟩ted. Fig. 6 indicates that the term −2⟨u1 u2 ⟩ ∂ x 1 is by far the2largest contributor of P11 , compare panels (b) and (c). This
is of course expected since both the separating shear layer on
the suction side and the attached flow in the pressure side are∂ ⟨U ⟩
associated with strong shear ∂ x 1 (note however that the pro2
duction is stronger in the suction side because flow separation
results in higher shear). It is also interesting to notice the lo∂ ⟨U ⟩calised contribution of the normal term, −2⟨u1 u1 ⟩ ∂ x 1 , which1
wraps around vortex V0, as can be seen in panel (a).
R−sis shown in panel (b) of Fig. 7.
The production term P22

Figure 3. (a) Contours of the time-average ⟨Q⟩ at equally
spaced planes x/C = 0.1, 0.2, ..., 1.4. with designated vorticalstructures. (b) Organization of the vortex structure topology
at x/C = 0.75. (c) Visualization and designation of vorticalstructures using the iso-surface ⟨Q⟩ = 5.

streamwise strain rate ⟨Sxx ⟩ superposed with isolines of positive vorticity ⟨Ωx ⟩. The arch-like structure that was formed
from the secondary flow around the recirculation zone at the
tip persists up to about x/C = 1.1. This structure carries tworegions of concentrated vorticity, located at approximately thesame wall normal distance, y/C = 0.1, and at spanwise locations z/C = −1 and −0.9. The former patch of vorticity
quickly disappears, because it is located within a region ofcompression (dark contours), but the latter patch grows due tostretching (note the yellow-coloured spot of high acceleration
at z/C = 0.9 in planes x/C = 0.95−1.1). Eventually this patch
will evolve to form the tip vortex V1 that will persist in thewake, see plane x/C = 1.3. The core of this vortex is locatedslightly inboard with respect to the tip, at z/C ≈ −0.9. Thisagrees with the theoretical analysis of Milne-Thomson (1973).
We conjecture that the high strain rate that leads to the intensification of V1 is due to the recovery of the streamwise velocity
downstream of the recirculation bubble that forms around thetip of the trailing edge. It is also interesting to observe thatthe near wall vorticity patch (that forms between V0 and RV)
is immersed in a region of compression shortly downstreamof the trailing edge and quickly disappears (it is detected until x/C ≈ 1.1). Vortex V2 starts to be visible in the lower tip
corner in planes x/C = 0.95 − 1.05 and undergoes stretchingin all planes. It eventually merges with V1, as shown in figure3(c).

∂ ⟨U ⟩The dominant component is −2⟨u2 u2 ⟩ ∂ x 2 , which is shown2
in panel (a). The two other components are not included because their contribution is negligible; this can be attested by
the resemblance of the isosurfaces in the two panels. The production term is maximised downstream of the trailing edge,∂ ⟨U ⟩
where higher local values of negative normal strain ∂ x 2 are2∂ ⟨U3 ⟩
is small, from the continu∂ x3∂ ⟨U2 ⟩
∂ ⟨U1 ⟩
ity equation we get ∂ x = − ∂ x , so the regions of strong2
1∂ ⟨U ⟩−2⟨u2 u2 ⟩ ∂ x 2 are collocated with the areas of rapid stream2∂ ⟨U ⟩
wise velocity recovery, i.e. large ∂ x 1 acceleration (or normal
1

encountered. Assuming thatstrain rate). This is the area shortly downstream of the closureof the recirculation region close to the tip.
In order to probe further the interaction between the elongation rate, ⟨Sxx ⟩, vortex stretching, circulation and productionof turbulent kinetic energy in the near wake, we integrate theaforementioned quantities in a number of cross-steam planes
from x/C = 1.1 − 2.4. The boundary of the integration area
Sz is shown with a closed dashed line in panels (a) and (b) of∂ ⟨U ⟩∂ ⟨U ⟩
Fig. 8, where contours of −2⟨u1 u2 ⟩ ∂ x 1 and −2⟨u2 u2 ⟩ ∂ x 2
2
2
are plotted respectively at plane x/C = 1.2. The locations of
the planes themselves are marked in Fig. 8(c). The three ar∂ ⟨U ⟩eas of large production −2⟨u1 u2 ⟩ ∂ x 1 in panel (a) correspond2
to flow originating from the pressure side (bottom horizontal
region), the suction side (top right area) and between the tworecirculation zones (middle region), see also contours of thetotal production P = −⟨ui u j ⟩⟨Si j ⟩ in Fig. 5. The patches of

Turbulence structure of the separated wakeIn this section, we examine the spatial distribution of turbulent kinetic energy (TKE), k, and its production. The objective is to identify the most intense regions and uncover themain production mechanisms. We focus on the near-field, i.e.
distance 5% − 30%C from the trailing edge. In this region, theflow is far from being self-similar, thus well known viscous
vortex models, such as the Burgers or Sullivan vortex (Green(2012); Wu et al. (2015)), do not apply.
The increased turbulent activity between the two recirculation bubbles can be detected in the Reynolds stresses isosurface plots shown in Fig. 5. Analysis shows that ⟨u1 u1 ⟩and ⟨u2 u2 ⟩ dominate over ⟨u3 u3 ⟩. It can be seen that k and⟨u1 u1 ⟩ isosurfaces wrap around the outer boundary of V0 and∂ ⟨U ⟩

large −2⟨u2 u2 ⟩ ∂ x 2 in panel (b) correspond to the two areas
2
shown in Fig. 7(a).
In panel (d) Fig. 8 we plot the variation along x/C of thesurface integrals of vortex stretching and axial elongation overSz . We also superimpose the circulation ⟨Γx ⟩ (surface integral

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 4. Emergence of the wake tip vortices V1 and V2. Contours of streamwise strain rate ⟨Sxx ⟩ superposed to coloured isolines of
positive ⟨Ωx ⟩. The solid white line indicates ⟨Q⟩ = 5.

incides with the location where ⟨U1 ⟩ = 0. Note also the faster∂ ⟨U ⟩∂ ⟨U ⟩
decay rate of −2⟨u1 u2 ⟩ ∂ x 1 compared to −2⟨u2 u2 ⟩ ∂ x 2 .
22While circulation is (almost) constant, both production terms
decay because of the solid body rotation velocity profile found∂ ⟨U ⟩
in columnar vortices. The fast decay of −2⟨u1 u2 ⟩ ∂ x 1 be2
tween x/C ≈ 1.2 − 1.4 is probably due to the flow recovery
away from the recirculation region that smooths the streamwise velocity in the y direction.

CONCLUSIONSDNS of a half-span NACA 0018 wing with square wingtip
profile was performed at a Rec = 104 and 10◦ angle of attack.
A thorough analysis of the main vorticity and turbulent production mechanisms above the wing and in the near wake wascarried out. The flow was found to separate close to the leading edge and did not reattach on the suction side of the wing.
Close to the leading edge, the separating region occupied mostof the span, but once the secondary flow in the wingtip started
to develop, the left boundary of the recirculating bubble wasdisplaced inboard. In the proximity of the tip, the flow remained attached, but another recirculation zone appeared close
to the trailing edge.
The emergence of the main wingtip vortex V1 from theamalgamation and interaction of the different vortices was also
elucidated. The presence of the recirculation zone close to thetrailing edge resulted in the distortion of the shear layers originating from the pressure side. The distribution of turbulent
kinetic energy around the main wing tip vortex V1 was foundto be highly asymmetric and high values were concentrated
on the right of its axis. Apart from the main separating shear
layer, patches of strong production were also found in the area
between the two recirculation bubbles in the near wake.
Analysis of the production terms of individual Reynolds
stresses has enabled us to uncover the main production mecha∂ ⟨U ⟩nisms. It was found that −2⟨u1 u2 ⟩ ∂ x 1 is one of the dominant2
terms responsible for maintaining high levels of ⟨u1 u1 ⟩. Thisis the standard production term for wall-bounded flows. Ad∂ ⟨U ⟩ditionally, the wall-normal stress source term −2⟨u2 u2 ⟩ ∂ x 2
2
was responsible for maintaining ⟨u2 u2 ⟩ primarily in the spiralwakes in the near field.

Figure 5. Isosurfaces of (a) k, (b) ⟨u1 u1 ⟩, (c) ⟨u2 u2 ⟩ and (d)⟨u3 u3 ⟩ at a value of 0.05, 0.05, 0.04 and 0.02 respectively.
The contour values correspond to (a) ≈ 45%, (b) ≈ 55%, (c)≈ 49%,(d) ≈ 29% of their respective positive maximum value
at x/C=1.2 in the z-y plane. (e) k - blue contour (0.5) andturbulence production - red contour (0.2). (f) Turbulence production P colored by k. Hereby, the contour values correspond to ≈ 55% and ≈ 31% of the maximum value of k and
P at x/C=1.2 in the z-y plane. All values are superimposedon ⟨Q⟩ = 5.

Rof ⟨Ωx ⟩). RThis panel shows that the peaks of Sz ⟨Ωx ⟩⟨Sxx ⟩ dSz ,⟨Γx ⟩ and Sz ⟨Sxx ⟩ dSz are not in phase. The axial elongation
term peaks at the streamwise location where ⟨U1 ⟩ = 0 (marked
with the vertical dashed line). This is the result of flow recovery, as the flow comes out of the recirculation region aroundthe tip. The vortex stretching term peaks slightly further downstream. The latter term causes the growth of circulation, ⟨Γx ⟩which peaks at about x/C ≈ 1.5. Note the very weak decay of⟨Γx ⟩ with respect to the other two terms. The only mechanism
responsible for the decay for x/C > 2.0 is viscous diffusion,
which is a slow term and explains the persistence of V1 until
the exit of the computational domain. Panel (e) shows that the∂ ⟨U ⟩
production component due to shear −⟨u1 u2 ⟩ ∂ x 1 peaks ear2

∂ ⟨U ⟩

lier compared to the normal component −2⟨u2 u2 ⟩ ∂ x 2 and it
2reaches a higher value. The peak of the former term almost co-4

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

Figure 6.

∂ ⟨U ⟩

Production terms of ⟨u1 u1 ⟩ (a) The magenta iso-surface represents −2[⟨u1 u1 ⟩ ∂ x 1 ] = 0.4 with pink lines representing the1∂ ⟨U1 ⟩
∂ ⟨U ⟩= −2. (b) The purple iso-surface represents −2[⟨u1 u2 ⟩ ∂ x 1 ] =∂ x1
2∂ ⟨U ⟩
0.5, blue lines represents the contours of shear strain coming from the top wing surface with ∂ x 1 = −7 and orange lines ⟨u1 u2 ⟩ = 0.01,
2∂ ⟨U ⟩
the red line represents shear from the bottom wing surface with ∂ x 1 = 7 and the corresponding orange line ⟨u1 u2 ⟩ = −0.01 and (d)2R−stotal production of ⟨u1 u1 ⟩: iso-surface P11= 0.3 colored by ⟨u1 u1 ⟩. Hereby, the contour values correspond to (a) ≈ 45%, (b)contours of ⟨u1 u1 ⟩ = 0.03 and blue line representing compression≈ 36% and (c) ≈ 27% of their respective positive maximum value at x/C = 1.2 (at x/C = 0.9 for (a)) in the z-y plane. All values aresuperimposed on ⟨Q⟩ = 5.

Figure 7.

∂ ⟨U ⟩

Production terms of ⟨u2 u2 ⟩ (a) isosurface of −2⟨u2 u2 ⟩ ∂ x 2 = 0.3, blue lines represent contours2∂ ⟨U2 ⟩
= −3, −4 and pink
∂ x2

R−s
= 0.3 colored by ⟨u1 u1 ⟩. The contour values correspond to (a) ≈ 61% and (b)line represents ⟨u2 u2 ⟩ = 0.04 (b) isosurface of P22≈ 64% of their respective positive maximum value at x/C=1.2 in the z-y plane. All values are superimposed on ⟨Q⟩ = 5.

The vortex stretching rate, axial elongation, streamwise
vorticity and production terms were integrated over square yz surfaces surrounding the V1 vortex, and plotted against thestreamwise direction x/C. When the stretching rate became
negligible, the circulation (integral of streamwise vorticity)reached a peak and then started to decay very slowly, explaining the persistence of V1 until the end of the computational
domain. The production terms become negligible at x/C ≈ 2
as expected, because the velocity profile within the core of V1(solid body rotation) cannot sustain turbulence. The detailed
analysis presented aims to contribute to a better understandingof separated vortical flows and can hopefully lead to actuation
strategies to manipulate them. Future work will also include
analysis of the statistics of small-scale turbulent fluctuations
undergoing large-scale straining due to the strong shear causedby the main wake.

Acknowledgements
The authors are grateful to the ARCHER2 UK National Supercomputing Service (https://www.archer2.ac.uk),
with access via the UK Turbulence Consortium (EPSRC
grant EP/R029326/1). Additional computational resourceswere provided by the UK Materials and Molecular Modelling
Hub, which is partially funded by EPSRC (EP/P020194/1
and EP/T022213/1) and Imperial College London. The firstauthor would like to acknowledge the financial support by
the National Program of Scholarships and Educational Credit(PRONABEC) of Peru as part of the Beca Generación Bicentenario PhD scholarship.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.25
0.2
0.15
0.1
0.05
0
1

1.2

1.4

1.6

1.8

2

2.2

2.4

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.01
0.008
0.006
0.004
0.002
0

∂ ⟨U ⟩∂ ⟨U ⟩

Figure 8. (a) Contours of −2[⟨u1 u2 ⟩ ∂ x 1 ] (b) −2[⟨u2 u2 ⟩ ∂ x 2 ] superimposed with the integration surface Sz at x/C = 1.2. (c)22Streamwise locations of the integration surfaces superimposed on ⟨Q⟩ = 5. Streamwise evolution of quantities related to (d) the
vorticity field and (e) the production of Reynolds stresses. The quantities shown are integrated over the surface Sz marked in panels (a)and (b). The vertical dashed line indicates the streamwise location where ⟨U1 ⟩ = 0.

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