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

DIRECT NUMERICAL SIMULATION OF ADVERSE PRESSURE GRADIENTTURBULENT BOUNDARY LAYER UP TO Reθ ' 8000
Hussein Rkein
Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille,
UMR 9014 -LMFL- F-59000, Lille, France
hussein.rkein@univ-lille.fr

Jean-Philippe LAVAL
Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille
UMR 9014 -LMFL- F-59000, Lille, France
jean-philippe.laval@univ-lille.fr

ABSTRACT

dient to focus on the condition before or at the edge of separation. Moreover, only few DNS are available at Reynolds numbers sufficient to decouple the near wall from the outer regionof the boundary layer (Lee (2017)). A recent wall resolved
LES of APG TBL at Reynolds number up to Reθ ' 8700 withmild adverse pressure gradient has been performed by Pozuelo
et al. (2022). The effect of a strong adverse pressure gradienthas been studied using a DNS of converging diverging channel
flow at two Reynolds numbers by Marquillie et al. (2008) andMarquillie et al. (2011). It has been shown from these studies
that a strong peak of production in the adverse pressure gradient region was due to the strengthening of the near-wall streaks
instability. However, the Reynolds number was too low to uncouple this near wall effect to the consequence of the APG in
the outer region. The aim of the present work is to fill thisgap by analysing a new direct numerical simulation (DNS) of
APG TBL at moderate and not constant pressure gradient. The
present work will focus on velocity scaling as well as streamwise large scale structures and their contribution to Reynolds
stresses.

A direct numerical simulation of a moderate adverse pressure gradient turbulent boundary layer flow on a flat plate was
performed on a fairly long domain with a Reynolds number
from Reθ ∼ 2000 to Reθ ∼ 8000. The mean flow and Reynolds
stress statistics are analysed. The mean streamwise velocity
profile does not exhibit a clear log region even at the highest
Reynolds number. The Reynolds stresses exhibit a well defined peak in the outer region which moves away from the wall
before to stabilise near 0.45 boundary layer thickness. Theseouter peaks are the consequences of an excess of production
over dissipation. The large scales structures analysed by meansof two point correlations of streamwise fluctuating velocity are
shown to be strongly modified by the adverse pressure gradient.

Introduction
The accurate prediction of turbulent boundary layer(TBL) remains a significant challenge in many practical problems including engineering design of vehicles, internal aerodynamics or turbomachinery. In order to better understand thephysics of such flows, it was natural to concentrate first on flat
plate zero pressure gradient (ZPG) TBL. However, ZPG conditions are nearly never encountered in real-life applications as
the majority of flow problems encounter the effect of complex
pressure gradient on flat or curved walls. The applicability of
the knowledge from ZPG TBLs to decelerating boundary layers is still rather limited. Despite many simulations and experiments on adverse pressure gradient (APG) TBLs, there is still
no consensus and clear understanding of the main effects ofa streamwise adverse pressure gradient, its magnitude and its
upstream history on turbulence statistics and large scale coherent structures. Despite many proposals, no universal velocity
and length scale are able to successfully re-scale the turbulence statistics for a wide range of flow parameters. In orderto study the effect of these APG flow parameters, a large variety of experimental and numerical data are needed to cover awide range of pressure gradient and Reynolds numbers. Mostof detailed databases have been obtained for adverse pressure
gradient flows in equilibrium (see Kitsios et al. (2017)) either for weak pressure gradient or very strong pressure gra-

Numerical simulation
A DNS of APG turbulent boundary layer on a flat plate
was performed on a fairly large domain with a Reynolds number up to Reθ ∼ 8000. The DNS was simulated using an inlet condition for velocity from a precursor DNS of ZPG TBL
at Reθ ∼ 2200 (see Solak & Laval (2018)). As the original
DNS was on a smaller domain, the inlet velocity fields havebeen extrapolated in the normal direction and periodized by afactor of 4 in the spanwise direction. However, to avoid theexact repetition in the results because of duplication of inlet
planes, a spatial-temporal noise of small amplitude was added
to the inlet velocity components similar to the numerical forcing used by Schlatter et al. (2012). The APG is imposed byprescribing the wall normal velocity estimated from a potential
flow solution like in Kitsios et al. (2017). Neumann Boundary
conditions are used for the streamwise velocity by imposing
a zero spanwise vorticity. A fringe region has been appliedon the last 3% of the computational domain in order to force
the laminarity of the outgoing flow. The pressure gradient waschosen to be comparable with an APG TBL experiment from1

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

Figure 1. Snapshot of the spanwise vorticity in a streamwise wall-normal plane on the full simulation domain. δmax is the maximum
boundary layer thickness at the end of the simulation domain (before the fringe region).

Statistical results

Cuvier et al. (2017) on a long ramp in the LMFL wind tunnel
at Reynolds number up to Reθ ∼ 23000. It was not possible to reproduce the exact same pressure gradient but the aim
is rather to study the Reynolds number effect on a case withcomparable APG magnitude.

The mean streamwise velocity profiles of APG TBL areplotted in Fig. 2 at several streamwise positions and comparedwith the profiles of ZPG TBL at comparable Reynolds numbers. As already observed the APG leads to a reduction of thelog region and an extension of the wake region as comparedto the ZPG case at the same Reynolds (either Reθ or Reτ ). A
detailed analysis of the log-law scaling using a diagnostic plot
indicates that the profiles exhibit no clear plateau but rather amaximum that moves toward the wall for the lowest Reynolds
numbers (Reθ < 3000) and then moves away from the wall likethe square root of Reθ when moving downstream. This peakdoes not seem to flatten for any Reynolds numbers indicatingthat no log region can be identified even at (Reθ > 7000).
As already noticed for APG turbulent boundary layers
a second outer peak of the streamwise Reynolds stresses increases with the Reynolds and with pressure gradient to exceed
the first inner peak. The inner peak is only weakly affected by
the APG as can be seen in figure 3 which includes a comparison of Reynolds stresses of zero pressure gradient TBL atequivalent Reynolds numbers. The position and magnitude of
the outer peak of the three turbulence intensity profiles are almost identical for y+ > 30 which indicates a common physics
for all velocity components. Lozano-Durán & Bae (2019) have
hypothesised that the wall is not the element that organises themomentum-carrying eddies in wall turbulence, whose intensities and sizes are controlled instead by the mean production
rate of turbulent kinetic energy with no explicit reference to the
distance to the wall. As a consequence, they proposedpa characteristic velocity scale based on the shear stress u∗ = −huviand tested this scaling on channel flows with different mean
velocity profiles obtained by different boundary conditions. In
the present results of APG TBL the Reynolds stress profiles
normalised by the shear stress velocity u∗ are constant for wall
distances 0.15 < y/δ < 0.8 and for all streamwise positions.
These constant values 2, 1.3 and 1.8 for the streamwise, wallnormal and spanwise Reynolds stress respectively are the same
as for zero pressure gradient TBL of Sillero & Moser (2013)at similar Reynolds numbers (not shown).
In order to understand the physics associated with thegrowth of the outer peak, the Reynolds Stress tensor budget
has been computed along the full simulation domain. The
two main components for the turbulent kinetic energy (TKE)budget (production and dissipation) and their sum (the TKE
source) are plotted in Fig. 5 for 3000 < Reθ < 7000. As the
production is mainly controlled by huvi∂U/∂ y the outer peakof the production starts to form at Reθ = 4000 at the sametime as the peak of huvi and other Reynolds stresses compo-

The DNS has been simulated using the open-sourcecode Incompact3D developed by Laizet & Lamballais (2009).
Sixth-order compact finite difference schemes are used forthe spatial discretization and third-order Adams-Bashforth
scheme for the time advancement using a time step ∆t + =0.0125 based on friction velocity at the outlet. The main parameters of the simulation are given in Table 1. The simulation is well resolved on the full boundary layer thickness (thenormal grid resolution at the top of the TBL ∆y+is of theδorder of 4.7). This allows us to perform detailed and accurate statistics of both large scales and small dissipative scales.
A large database of 100 TB was recorded with the results on
70 characteristic times T = δmax /U∞ after the transient. It
is composed of 211 three-dimensional velocity and pressure
fields as well as a time evolution (every 1.37 wall unit time) of
a 3D restricted domain in the downstream region of the flow
(6818 < Reθ < 7582) and three normal-spanwise planes. A
snapshot of the spanwise vorticity in the full simulation domain is shown in Figure 1.

Table 1. Parameters of the APG TBL normalised by thequantities (δ , uτ ) at the outlet. Nx , Ny and Nz are the number of grid points in the streamwise, normal and spanwise direction, Lx , Ly and Lz are the dimensions of the domain and∆x+ , ∆y+ , ∆z+ the grid resolution in wall unit (∆y+being theδ
normal grid size at the top of the boundary layer δ ).

Reθ2200 − 8000Reτ

750 − 1400

Nx , Ny , Nz

6401, 1025, 1280

Lx , Ly , Lz

23.3δ , 2.9δ , 2.33δ

++∆x+ , ∆y+min , ∆yδ , ∆z5.1, 1.0, 4.7, 2.4

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.5

40Reθ = 2250
Reθ = 3000
Reθ = 4500
Reθ = 6500
Reθ = 7000
Reθ = 7660 (Reτ = 1307)
Reθ = 4000 (Reτ = 1307)
Reθ = 6500

20
10

0.4

+

−1
(y +dUdy + )

U+

30

U + = y+

0.3
0.2
0.1

0 0
10

101

102

y

a)

103

+

0.0

Reθ = 2250
Reθ = 3000
Reθ = 4500
Reθ = 6500

Reθ = 7000
Reθ = 7660 (Reτ = 1307)
Reθ = 6500
Reθ = 4000 (Reτ = 1307)

101

y+

b)

102

Figure 2. Evolution of the mean velocity profiles at several streamwise positions (a) and the equivalent diagnostic plots to visualise
the log region (b). The statistics from ZPG TBL of Sillero & Moser (2013) at similar Reynolds numbers (Reτ = 1307, Reθ = 4000)
and (Reτ = 2000, Reθ = 6500) are plotted for comparison.
distances from 0.2δ to 0.85δ . The source terms (production+ dissipation) which supply the outer peak of kinetic energy
corresponds to 20% of the production in this range. Due to aslight vertical shift of the production peak (y ' 0.5δ ) with respect to the dissipation peak (y ' 0.4δ ), the maximum source
peak stabilises near 0.55δ .

u+

3
2
1

0.003

10

1

10

2

y+

10

Production+Dissipation
Production
Dissipation

0.002

3

[. . . ] × δ/Ue3

0 0
10

Figure 3. Mean streamwise velocity fluctuation profiles(same legend as figure 2).

0.001
0.000
0.001

Reθ = 3000, (x= 5 δloc )Reθ = 4000, (x= 11 δloc )Reθ = 5000, (x= 18 δloc )Reθ = 6000, (x= 24 δloc )Reθ = 7000, (x= 30 δloc )

0.002

4

0.003
0.0

Reθ = 3000
Reθ = 4000
Reθ = 5000
Reθ = 6000
Reθ = 7000

3

0.2

0.4

0.6

y/δ

0.8

1.0

Figure 5. Production and turbulent dissipation rate of turbulent kinetic energy in outer scaling at several streamwise positions.

2
1

u/u ∗
0
0.0

0.2

v/u ∗
0.4

y/δ

0.6

w/u ∗
0.8

As can be seen from the Reynolds Stress profiles, theouter peak appears near y+ = 100 at Reθ = 3000 and moves
away from the wall to stabilise before the end of the simulation. For mild APG TBL in equilibrium, the outer peak of turbulence intensity has been located at 1.3 displacement thickness (δ1 ) or 0.4 to 0.5 boundary layer thickness (δ ) from thewall (see i.e. Kitsios et al. (2017)). In the present case of APG
out of equilibrium, the peaks of all Reynolds stresses profiles
are shown to evolve similarly to reach this wall distance at a
streamwise position of the orderR of 20 local boundary layer
thicknesses defined by x/δloc = 0x δ (x0 )−1 dx0 (see figure 6b)after the start of the adverse pressure gradient (x = 0 in ourcase). This evolution is in accordance with the experimental
results of Cuvier et al. (2017) for a similar APG TBL overa long ramp at higher Reynolds numbers (Reθ ' 23000) al-

1.0

Figure 4. Mean velocity fluctuation
profiles normalised byp
the shear stress velocity u∗ = −huvi at 5 steamwise positions and Reynolds numbers.

nents. The emergence of an outer peak of dissipation is delayed to Reθ > 5000 which corresponds to 7 local boundary
layer thickness downstream. Further downstream, both production and dissipation exhibit an outer peak but the production always exceed the dissipation on a wide range of wall3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
though the evolution and magnitude of the pressure gradientare not exactly the same. As the two APG flows do not encounter the same history of pressure gradient which extends
over a long domain, on which the boundary layer thickness
evolves significantly, the choice of local boundary layer thicknesses to normalise the streamwise position seems more pertinent than the averaged one. The extent of the simulation being
larger than the experimental one (based on δloc ), the peaks of
Reynolds stresses profiles continue to move away for the wall
to stabilise near 1.5 displacement thickness or 0.45 boundary
layer thickness despite the fact that the pressure gradient starts
to decrease at x/δloc = 20. These results shows that, for amild pressure gradient, a minimum of 30 local boundary layer
thicknesses is needed for the outer peak position to stabilise
even when the pressure gradient does not evolve significantly
as in the present DNS case. It has been shown that the positionof this outer peak is very close to the most external inflection
point of the mean velocity profile observed in APG flows (seeShah et al. (2010), George et al. (2012)). This observationis consistent with the present results. However the mean velocity gradient being almost constant in a wide range of wall
distance this inflection point is not well defined and it is notpossible to conclude that it coincides with the normal positionof the Reynolds stress outer peaks.

2.00

v + /δ
ypeak
v + /δ
ypeak
1

u + /δ
ypeak
u + /δ
ypeak
1

1.75

5

uv + /δ
ypeak
uv + /δ
ypeak
1

4

1.50
1.25

β

3

1.00
0.75

2

0.50

1

0.25
0.00

0

2.00

5

10

u + /δ
ypeak
99
u + /δ
ypeak
1

1.75

15

20

x/δloc25v + /δ
ypeak
99
v + /δ
ypeak
1

30

35

40

0

5

uv + /δ
ypeak
99
uv + /δypeak
1

U∞ = 9ms −1

4

1.50
1.25

β

3

1.00
0.75

2

Large scale structures

0.50

The increase of the outer peak of turbulence intensity
at large Reynolds number ZPG TBL is associated with thepresence of the very large scale structures of the streamwise
fluctuating velocity. These structures are usually decomposedon wall attached structures (self-similar or not) and wall detached ones (see Hwang et al. (2020)). The outer peak observed in APG TBL flows is more pronounced and located approximately at the same wall distance. A decomposition of
the Reynolds stresses between low and high speed structures
shows that for the same volume fraction, the kinetic energy ofthe high-speed streamwise structures is increased by the effectof APG while the low speed one remains almost unchanged.
The two-point correlations of streamwise velocity fluctuations
for the current APG TBL have been compared with the DNS
results of a ZPG TBL at a lower Reynolds number (see figure 7). The effect of APG is shown to increase the inclinationand size of the large scale structures. Moreover, the shape ofthe correlation function in the buffer layer upstream of the fix
point indicates a different organisation in this region with respect to the ZPG TBL. When the fix point is just above the inner peak of u0 (y+o > 50) associated with near-wall streaks theiso-contours extend much more upstream of the position xo of
fix the point and the near-wall part of the iso-contours linked
to the near wall streaks clearly detached from the upper partof the structures (associated with the outer peak of u0 ). The
iso-contours also extend Further downstream and away fromthe wall to reach 0.8δ . In Figure 8, the two-point correlation
function of the streamwise velocity fluctuation conditioned by
the sign of u0 (xo , yo ) and normalised by uτ at the fix point
are plotted for different altitudes yo of the fix point. These
results reveal the different behaviours with high-speed largescale structures (when conditioned by u0 (xo , yo ) > 0) which extends more downstream for yo < 100+ and more upstream foryo > 100+ . As both conditioned correlations are normalisedby uτ the highest iso-contours show that the variance of positive u0 is larger than the variance in the negative one for all
wall distances yo . More detailed statistics of each individual
structure would be needed to better assess the effect of APG

0.25
0.00

1

0

5

10

x/δloc

15

20

25

0

Figure 6. Evolution of outer peak wall normal position of the
Reynolds stresses with respect to the boundary layer thickness(δ ) and the displacement thickness (δ1 ) for (top) the present
DNS of APG TBL (bottom) the experimental results of a TBL
flow over a ramp at −5◦ from Cuvier et al. (2017) with comparable magnitude although different evolution of adverse presesure gradient β = δ1uU2 e dUdx (dash lines, right axis) and higherτ
Reynolds number (Reθ = 23000). x/δloc is the position in
number of local boundary layer thickness from the start of the
APG (x = 0).

on these large scale structures. However such statistics would
require much more uncorrelated velocity fields than available
for the current DNS.

Conclusions
A new DNS of APG TBL over a flat plate has been presented. This simulation is long enough to study the evolutionof the Reynolds stress profiles and to follow the growth andstabilisation of the outer peak. This outer peak is located at thesame position for all Reynolds stresses and takes more than 25
local boundary layer thicknesses to stabilise near 1.3 boundary
layer displacement thicknesses or 0.45δ . The Reynolds stressprofiles scale almost perfectly with the shear stress velocity
profiles u? on a wide range of wall distances from 0.15δ up
to 0.8δ , independently of the Reynolds numbers. The APG isshown to have a strong effect on the upstream part of streamwise fluctuating velocity inclined structures as compared to the
ZPG case.

4

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

y/δ

0.8

0.1
0.15
0.2
0.3
0.4
0.6
0.9

0.4
0.0
−3

−2

−1

0
(x − x0)/δ

1

2

3

Figure 7. Comparison of the iso-contours of the two-point correlation of the streamwise velocity fluctuation at yo = 0.08δ between
the present APG case at Reθ = 7240 (continuous lines) and the ZPG case at Reθ = 2068 (dash-dotted lines) from Solak & Laval (2018).

y/δ

0.8

0.4

0.0

2.0

y/δ

0.8

2.0

y/δ

0.8

y/δ

0.8

y/δ

0.8

2.0

1.0

0.5

1.0

0.5

yo /δ = 0.128, yo+ = 150

1.5

1.0

0.5

yo /δ = 0.214, yo+ = 250

1.0
1.5
2.0
3.0
4.0

0.4

0.0

1.5

2.0

0.5

yo /δ = 0.085, yo+ = 100

1.0
1.5
2.0
3.0
4.0

0.4

0.0

1.5

2.0

1.0

yo /δ = 0.042, yo+ = 50

1.0
1.5
2.0
3.0
4.0

0.4

0.0

1.5
1.0
1.5
2.0
3.0
4.0

0.4

0.0

yo /δ = 0.016, yo+ = 20

1.0
1.5
2.0
3.0
4.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

(x − xo )/δ

(x − xo )/δ

(x − xo )/δ

(x − xo )/δ

(x − xo )/δ

Figure 8. Iso-contours of the two-point correlations of the streamwise velocity fluctuation u0 conditioned by u0 (xo , yo ) > 0 (continuouslines) and u0 (xo , yo ) < 0 (dash lines) for APG case at xo such that Reθ (xo ) = 7240. The correlation are normalised by u2τ at the fix point(xo , yo )512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Acknowledgement5989–6015.
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This work was granted access to the HPC resources of
IDRIS under the allocation 021741 made by GENCI (Grand
Equipment National de Calcul Intensif).

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