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

THREE-DIMENSIONAL STRUCTURAL ASPECTS OF THE ATTACHEDTURBULENT BOUNDARY LAYER FLOW OVER A HILL

Julie E. Duetsch-Patel, Aldo Gargiulo, William J. Devenport, and K. Todd Lowe
Kevin T. Crofton Department of Aerospace and Ocean Engineering
Virginia TechBlacksburg, VA, 24060, USA
jduetsch@vt.edu, galdo@vt.edu, devenport@vt.edu, kelowe@vt.edu

ABSTRACT

grid scale (SGS) model to model the effects of the small scales
on the resolved flow. Accurately resolving or modeling nearwall motions is critical because production, dissipation, turbulent kinetic energy, and Reynolds-stress anisotropy all peak
below x2+ = 20, and as such, all meshes and filtering for LES
must be on the order of one viscous unit. Because of this, wallresolved LES is infeasible for higher Reynolds number applications, requiring wall-modeled LES. Like RANS turbulencemodels, existing wall models are largely built on equilibrium
assumptions, which are not valid in the vast majority of applicable 3D flows. These assumptions can be built into the model
directly, for example through assuming an equilibrium stressor velocity distribution, or indirectly through assumptions regarding eddy viscosity and inner-layer Reynolds stress.

The BeVERLI (Benchmark Validation Experiments for
RANS/LES Investigations) Hill bump model, designed specifically for validation experiments, was tested in the Virginia
Tech Stability Wind Tunnel to collect validation experiment
data on the three-dimensional (3D) boundary layer flow overa 3D hill. Laser Doppler velocimetry measurements on thebump model were used to study the mean flow and turbulence
structure and evaluate the impact of pressure gradient and curvature upon the total shear stress in the boundary layer andevaluate the impact of pressure gradient and curvature upon
the total shear stress behavior in the near-wall region. From
analysis of the BeVERLI Hill flow, including the boundary
layer just upstream of the hill, and comparison with the 3D
flow around a wing-body junction of Ölçmen et al. (2001), it
is shown that there is no region in which the total shear stress
is approximately constant, unlike the typical behavior for 2D
equilibrium flows.

Equilibrium two-dimensional boundary layers typicallyhave a region near the wall where the total shear stress is nearly
constant, approximately 0.95 ≤ τtotal /τw ≤ 1 for sufficiently
high Reynolds numbers. Devenport & Lowe (2022) demonstrate the derivation of 2D zero-pressure gradient boundary
layer equations that produce this behavior, showing that as theviscous shear stress decreases rapidly outside the viscous sublayer, the turbulent shear stress increases to balance out the
total shear stress. In non-zero pressure gradient flows, a linear
gradient of the total shear stress will be present instead.

INTRODUCTION
Among the most well-known features of threedimensional turbulent boundary layers (3D TBLs) are the lag
between the shear-stress direction vector and the velocitygradient vector direction, a reduction in near-wall turbulent
kinetic energy and Reynolds stresses, and a reduction in
the turbulence parameter A1 below its typical 2D valueof 0.15. Because of this, 3D TBLs present a particularchallenge to modeling, as the stress/strain misalignment
invalidates the assumptions behind any turbulence models
that assume an isotropic eddy viscosity. This limits the utilityof Reynolds-Averaged Navier Stokes (RANS) computational
fluid dynamics (CFD) for flows of engineering interest, evenas CFD is increasingly used for high-impact decisions in
vehicle design. Improved modeling of 3D flows depends upona better understanding of 3D boundary layers and turbulence
to improve the assumptions that go into turbulence models.
The flow over a wall-mounted bump is a common test
case to the goal of improved computational fidelity (e.g. Byun
& Simpson (2006); Bell et al. (2012)), as the simple geometry
generates a flow field with pressure gradient, curvature effects,
and three-dimensional boundary layer separation and reattachment. All these features are generally challenging for CFD andare very common in practical applications.
Unlike RANS, large-eddy simulation (LES) resolves thelarger turbulence scales and models the small scales that cannot be resolved on the grid. As a result, LES requires a sub-Equilibrium approximations to the thin-boundary layer
equations are often made to relax strict assumptions on the velocity profile that would otherwise be required for LES (Bose& Park (2018)). These approximations also lead to the assumption of a constant total stress region in the near-wall region of the boundary layer. However, most flows of engineering interest involve strong pressure gradients in both thestreamwise and lateral direction and other non-equilibrium
phenomena, which will invalidate these assumptions.
This work presents results from experiments studying
the three-dimensional turbulent attached and separated flow
over the BeVERLI (Benchmark Validation Experiment for
RANS/LES Investigations) Hill. The flow over this modelexperiences three-dimensional pressure gradients of varying
sign and magnitude and skewing and three-dimensional flow
effects. We study the evolution of the boundary layer fromupstream of the hill to over the hill itself, specifically the impact of curvature, 3D pressure gradients, and skewing upon
the total shear stress distribution in the boundary layer. It will
be shown that streamwise and lateral pressure gradients andskewing significantly impact the flow, reducing the turbulent
shear stress throughout the boundary layer and resulting in noconstant shear stress region for any of the stations studied.

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022/(∂C p /∂ x1 )Table 1. LDV data locations of interest for this study, focusing on Windward (W) stations. ΠW SS = (∂C p /∂ x3 )
W SS
W SS
is the ratio of pressure gradients in local wall-shear-stress coordinates at ReH = 250, 000, γFA is the mean flow angle closest to the wall
relative to the global x1 axis, δ0 /R is the ratio of the empty tunnel boundary layer thickness at the location of the bump height to themagnitude of the local radius of curvature at each station, and βXi = (δin∗ f low /τw,in f low )(∂ P/∂ Xi ) is the Clauser parameter.

Station

x1 /H

x2 /H

x3 /H

uτ /U∞Min x2+Max x2+γFA (◦ )δ0 /|R|

ΠW SS

βX1

βX3

W1

-4.1

0

0

0.034

3.8

919

0

0

0

1.3

0

W2

-1.55

0.02

-1.55

0.037

5.3

1209

-38.9

0.38

-0.02

-3.1

0.9

W3

-1.42

0.07

-1.42

0.038

2.0

1510

-37.4

0.36

-0.10

-1.7

2.4

W5

-0.80

0.70

-0.80

0.043

4.3

1683

16.0

0.12

-3.3

-4.4

5.4

EXPERIMENTAL SETUP

During post-processing, outlying points from histograms were
removed. Nominal x2 -heights at each point in the profile werecorrected by fitting the nearest-wall data to U1+ = x2+ for datain the linear sublayer. The measurement volume was approximately 45 µm in diameter, and measurements were achievable
at a minimum and maximum height above the wall of 0.1 mm
and 30 mm, respectively. The flow was seeded using a mineral oil smoke machine (MDG MAX 3000 APS) with particles
ranging in size from 0.5 to 1 µm, resulting in Stokes numbers
small enough to capture the full range of relevant scales.
Data was collected at 13 locations on the bump surface,
as well as at the inflow to the test section and on the flat wall
just upstream of the hill. For this study, four locations are ofinterest at ReH = 250, 000, as shown in Table 1, where ΠW SS is
the pressure-gradient ratio parameter in local wall-shear-stresscoordinates proposed by Lozano-Durán et al. (2020) to assessthe degree to which the flow is non-equilibrium. These selected stations all contain data in the linear sublayer, allowing
for a direct calculation of τw from the velocity gradient.
Instrumentation and statistical uncertainty of the resultsare δ (U1 /U∞ ) = 0.012, δ (U2 /U∞ ) = 0.007, δ (U3 /U∞ ) =
2
−3
2
′2
0.010, δ (u′2
1 /U∞ ) = 1.2 × 10 , δ (u2 /U∞ ) = 3.0 ×

Measurements were performed in the Virginia Tech Stability Wind Tunnel (SWT) in a configuration as depicted inFig. 1. The bump geometry is defined by Gargiulo et al.
(2020) and, from the two-dimensional view, consists of a mirrored fifth-degree polynomial with a flat top in between. The
model has a length of 0.93472 m and a height of 0.1869 m
and exhibits 90 degree rotational symmetry, allowing for different orientations of the model with respect to the incoming
flow. For these experiments, the bump was oriented as shownin Fig. 1 with a pair of rounded corners directly aligned withthe approaching freestream. This orientation is referred to as
the 45◦ yaw case (Gargiulo et al. (2020)). Coordinate systems
labeled Xi indicate global coordinates, with the origin at thecenter of the hill, while coordinate systems labeled lowercase
xi indicate local wall-shear-stress coordinates, with the origin
at the wall-location of the profile.
Two CNC-milled models were used: One with 135 pressure taps installed in the surface and one with slots for clear
windows for laser Doppler velocimetry (LDV) measurements
along the fifth-degree polynomial curves. The models were
scanned after manufacturing and found to match the design
geometry to within ±0.40 mm and ±0.30 mm, respectively.

2
−3
2
−5
′ ′
10−4 , δ (u′2
3 /U∞ ) = 1.7 × 10 , δ (u1 u2 /U∞ ) = 8.1 × 10 ,
δ (u′1 u′3 /U∞ 2 ) = 7.1 × 10−4 , and δ (u′2 u′3 /U∞ 2 ) = 3.2 × 10−4 .
Rigorous uncertainty quantification encompassing additional
uncertainty sources (e.g. article geometry uncertainty effects
and repeatability of measurements) is critical for the BeVERLI
Hill experiments to ensure the results are suitable for CFD validation. Further depth of uncertainty estimations will be provided in future publications.

The model was installed 6.88 m downstream of the 3.18
mm boundary layer trip in the wind tunnel contraction. Near
the inflow to the test section, 1.83 m upstream of the center of the hill (X1 /H = −9.8, X3 /H = 0), the boundary layer
has δ99 = 5.7 cm, δ ∗ = 0.87 cm, and θ = 0.65 cm such thatReθ = 8615 and Reτ = 2763. Measurements were made withthe freestream flow adjusted to ensure height-based Reynolds
numbers ReH of 250, 000 ± 5000 and 325, 000 ± 5000. Without the bump present, the boundary layer thickness at the location of the center of the bump is approximately 7.3 cm at
ReH ≈ 325, 000. Measurements collected at ReH = 250, 000
are the focus of this study.

RESULTS
Pressure Distribution
Fig. 2(a) shows the mean freestream pressure coefficient
contours over the entire hill at a 45◦ orientation at ReH =250, 000. As shown, the pressure gradient changes sign multiple times over the surface in both the streamwise and spanwise
directions as the flow away from the centerline is accelerated
around the sides of the model. These strong streamwise andspanwise pressure gradients and 3D curvature induce strong
near-wall crossflow. LDV profile locations are indicated by
the black circles. The four stations of interest for this study
are labeled. Fig. 2(b) shows a focus view of the stations ofinterest on the hill, including wall-shear stress direction (fromthe mean flow angle) indicated by black vectors, and the mean
flow direction x2+ ≈ 500 indicated by blue vectors. The maximum x2+ captured at each profile is given in Table 1.
The flow begins largely two-dimensional in an adverse

A specialized embedded three-component LDV system
was used to take measurements of the turbulent boundary layerover the bump surface and upstream of the model, as shown in
Fig. 2. The probe was mounted inside the bump, with the laser
beams passing through the 1.5mm anti-reflective curvaturefitting acrylic windows. The Doppler frequency signals are
obtained using AURStudio from AUR, Inc. One block of samples (ranging from 5,000 to 15,000 samples) was taken at eachmeasurement point above the wall. Near-wall points were collected over approximately one minute of sampling time to prevent seed particles from collecting on the acrylic windows,
and thus typically had fewer samples than points taken abovex2 ≈ 0.5 mm. Data further above the surface was collected
over approximately one to three minutes of sampling time.

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
small compared to the impact of those two parameters whenapplied individually. The behavior between the stations pri22′2marily differs in u′2
1 /U∞ . The inner-region peak in u1 /U∞ is
not captured at station W2, and the behavior after the peak islower in magnitude than at nearby station W3.
Station W5 experiences very different turbulence behavior than the other stations due to the different flow histories,
pressure gradients, and local curvature in this region. Here,the flow most notably experiences a decrease in the peak of
2
2
′2
u′2
1 /U∞ , a sharp increase in the magnitude of u3 /U∞ , and2′′
an inflection point in u1 u2 /U∞ in the log layer. This behavior is likely due to the skewed near-wall flow interacting withthe flow at the edge of the boundary layer, which is probably
nearly aligned with the wind tunnel freestream direction.

pressure gradient (APG) at station W1. As the flow approaches
the hill, the APG increases in strength up to the stagnation
region on the front of the model, after which it is accelerated
over the windward face of the hill. Along the sides, stations
W2 and W3 experience similar pressure gradients and pressure
gradient histories. At these stations, the flow direction next to
the wall is inclined nearly 40◦ relative to the upstream flow
direction, indicating strong skewing under the lateral pressure
gradients present upstream of these stations.
In contrast, station W5 experiences an entirely differentpressure gradient and pressure gradient history. The local
streamwise pressure gradient remains favorable, as the nearwall flow direction remains nearly perpendicular to the local
pressure coefficient contours. However, the lateral pressure
gradients have induced milder skewing, pointing in the opposite direction relative to the skewing at W2 and W3. Here, theflow direction next to the wall is inclined 16◦ relative to the
upstream flow direction.

Total Shear Stress
In the wall-shear stress coordinate system, the total shear
stress is composed of the viscous component, µ∂U1 /∂ x2 ,which is maximum in the viscous sublayer, and the turbulent

Mean Velocity and Turbulence22component, ρ(u′1 u′2 + u′2 u′3 )1/2 . The wall-parallel momentum equations can be simplified to focus upon the behaviorof the flow in the limit of being in wall-shear-stress coordinates in the viscous sublayer by ignoring the inertial terms and

LDV measurements were collected in global tunnel coordinates, as shown in Fig. 1, and transformed into local wallshear-stress coordinates for analysis. In this coordinate system, the x1 -axis points in the estimated direction of the wallshear stress vector at the wall via the mean flow angle closest
to the surface, indicated in Fig. 2(b) by black arrows. The
x2 -axis is perpendicular to the local surface curvature, and thex3 -axis completes a right-handed coordinate system.
As shown in Fig. 3, the flow begins largely 2D with nearzero components in U2 and U3 at W1. However, the flow experiences vastly different physics in different regions over the
BeVERLI Hill, with varying levels of skewing and streamwise
acceleration depending on the location on the model, as discussed. Though the U2 and U3 components are near-zero near
the wall due to the wall-shear-stress coordinate system, the U3components for all stations on the bump increase significantly
further away from the surface due to the flow skewing. The U2components are near-zero for all profiles, although all profiles
on the hill experience a small, increasingly negative component of U2 as a function of height above the surface, especiallyat Station W5. This could be due to the local pressure gradientacting to pull the freestream flow down towards the surface atthis station as it negotiates the turn over the hill.
The behavior in the Reynolds stresses in Figs. 4 and 5
is even more varied between stations. Select Reynolds stresscomponents are shown in Fig. 5 normalized on the friction
velocity for comparison with equilibrium 2D TBL behavior.
The u′1 u′3 /U∞2 and u′2 u′3 /U∞2 are approximately zero within uncertainties. Unexpectedly, the flow at W1 does not behave
like a typical 2D APG, as seen when compared to the zeropressure gradient (ZPG) direct numerical simulation (DNS)
data of Schlatter & Örlü (2010) in Fig. 5, as the inner peakdecreases in magnitude instead of rising above the 2D ZPG
level and u′1 u′2 /U∞2 decreases in magnitude relative to the ZPG
DNS. Further work is needed to explain this discrepancy.
In contrast, all three stations on the bump have significant
components of both normal and shear stresses. Stations W2
and W3, located near each other and in a region of similar pressure gradient and pressure gradient histories, experience very
similar turbulence behavior, as well as significant skewing of
the near-wall flow relative to the tunnel X1 -axis and local concave curvature (see Figure 6). While concave curvature generally has a destabilizing effect on a 2D TBL, Baskaran et al.
(1990) found that the combined effects of mild convex andconcave curvature and mean flow three-dimensionality were+ x+
2

Reynolds shear stresses, resulting in U1+ = ∂∂ Px+
+ x+

2

1

2

2

+ x2+ andU3+ = ∂∂ Px+ 22 . If the pressure gradient is assumed to be ap3
proximately constant as a function of height in the viscous sublayer, then an FPG will cause the viscous stress to reduce more
rapidly from the wall in wall units than a ZPG or APG due tothe influence of the pressure term.
The turbulent shear stress is also very sensitive to flow
conditions. The presence of mean streamwise vorticity is
known to impact the quasi-coherent turbulent structures thatare responsible for the production of u′1 u′2 and u′2 u′3 (Johnston & Flack (1996)) and lead to the reduction of these shear
stress components. However, Flack & Johnston (1998) havealso found that τturbulent increases outside of the buffer layer
in regions of increasing three-dimensionality. This can also
be seen at some stations in the data of Ölçmen et al. (2001).
For both experiments, although u′1 u′2 decreased in magnitude
as the three-dimensionality of the flow increased, an increasein the magnitude of u′2 u′3 compensated for this behavior andled to an increase in τturbulent outside of the buffer layer.
The viscous and turbulent shear stress components of thetotal shear stress in the BeVERLI Hill stations of interest arenormalized by τw , computed via uτ , in Fig. 7 and comparedwith the 2D ZPG equilibrium boundary layer DNS of Schlatter & Örlü (2010). As shown in Fig. 7, stations W2 and W3
both experience a region below x2+ ≈ 200 where the turbulent
shear stress is significantly reduced and not able to balanceout the viscous stress it drops outside of the linear sublayer.
This results in a total stress distribution that is far from theequilibrium 0.95 ≤ τ/τw ≤ 1 that is typically present in 2D
ZPG boundary layers and is assumed in some turbulence models. Above x2+ ≈ 200, there is a sharp increase in the turbulent stress, as was also observed by Flack & Johnston (1998)(though in their case, this was observed above x2+ ≈ 50). Thereis, therefore, a region below x2+ ≈ 200 that is dominated by
viscous and wall-effects, and a region above this point of theboundary layer where the effects of the crossflow stress u′2 u′3
begin to contribute more significantly as the flow turns awayfrom the local wall-shear-stress direction. At Stations W2
and W3, this significant drop in turbulent shear stress is expected, as a reduction in turbulent shear stress is often seen3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ilar behavior seen in the BeVERLI Hill flow was also observedin the flat wall flow around a wing-body juncture of Ölçmen
et al. (2001) in regions with similar skewing and pressure gradient behavior. Turbulence models that rely on this equilibrium shear stress assumption will not be able to accurately
predict the behavior of 3D flow.

in 3D flows, where the significant near-wall skewing results
suppresses the pressure-strain term in the turbulent transport.
In contrast, at station W5, the flow behavior is extremely
different due to the different pressure gradient and flow history
in this region. Notably, there is a small region near the wall
where the turbulent shear stress approximately matches the 2D
equilibrium DNS profile. This continues until the edge of thebuffer layer, after which the shear stresses decrease for a portion of the log layer before sharply increasing again, similar tostations W2 and W3. This behavior near the wall may be dueto the favorable streamwise pressure gradient in this region,
the largest magnitude out of all of the stations. FPGs typicallysuppress turbulent fluctuations relative to the wall-shear-stress,
including u′1 u′2 (e.g. Volino (2020)). However, in this case,the strong streamwise FPGs may accelerate the near-wall flowsuch that spanwise motions are inhibited, reducing the mean
streamwise vorticity and near-wall three-dimensionality suchthat u′1 u′2 recovers in the near-wall region.
The profiles are compared in Fig. 8 with the 3D TBL on
a flat wall around a wing-body junction from Ölçmen et al.
(2001) in regions that experience similar pressure gradient andskewing behavior. At Stations 3 and 4 of Ölçmen et al., theflow is extremely skewed (> 30◦ from the upstream flow direction), and streamwise pressure gradient is favorable as it
passes around the side of the wing-body junction. This global
flowfield behavior is similar to that of stations W2 and W3 ofthe BeVERLI flow, and, as shown, results in similar turbulent
shear stress behavior, as shown in Fig. 8(a). However, the dataof Ölçmen et al. is missing in the region in which Stations W2
and W3 begin experiencing a sharp increasing in the turbulent
shear stress, so it is not possible to identify if this behvior isalso present in the juncture flow.
A similar trend follows for Ölçmen et al.’s stations 6 and
7 and the hill’s station W5. At both stations, the flow is muchless skewed than at previous stations as it recovers past thejunction and is also experiencing a stronger streamwise FPG
than further upstream in the flow, similar to W5. Notably,
all three stations experience a region through the buffer layer
where the turbulent shear stress nearly matches the 2D DNS,
before all show a sudden drop in turbulent shear stress above
the buffer layer, and a subsequent increase in above x2+ ≈ 200.
Here, the effects of less severe near-wall skewing relative tothe upstream flow direction, as well as very strong streamwise
FPGs, may inhibit the mean streamwise vorticity such that the
turbulent shear stresses recover locally in this region.

ACKNOWLEDGEMENTS
The authors thank BeVERLI collaborators Vignesh Sundarraj, Tom Hallock, Thomas Ozoroski, Dr. Chris Roy, and Dr.
Aurelien Borgoltz for their assistance with wind tunnel testing
and thoughtful discussion on the data presented. We also thankNASA, particularly Dr. Michael Kegerise, Dr. Mujeeb Malik,
and Dr. Christopher Rumsey, for their support under grants
80NSSC18M0146 and 80NSSC22M0061 and the Aerospaceand Ocean Engineering Machine Shop staff for leading the
fabrication and instrumentation hardware efforts.

REFERENCES
Baskaran, V, Pontikis, Y G & Bradshaw, P 1990 Experimental
investigation of three-dimensional turbulent boundary layers on ’infinite’ swept wings. J. Fluid Mech 211, 95–122.
Bell, James H, Heineck, James T, Zilliac, Gregory, Mehta, Rabindra D & Long, Kurtis R 2012 Surface and flow fieldmeasurements on the FAITH hill model. In 50th AIAA
Aerospace Sciences Meeting, pp. 2012–0704. Nashville,
TN: American Institute of Aeronautics and Astronautics.
Bose, Sanjeeb T. & Park, George Ilhwan 2018 Wall-Modeled
Large-Eddy Simulation for Complex Turbulent Flows. Annual Review of Fluid Mechanics 50, 535–561.
Byun, Gwibo & Simpson, Roger L. 2006 Structure of ThreeDimensional Separated Flow on an Axisymmetric Bump.
AIAA Journal 44 (5), 999–1008.
Devenport, William J. & Lowe, K. Todd 2022 Equilibriumand non-equilibrium turbulent boundary layers. Progress in
Aerospace Sciences 131 (100807), 1–50.
Flack, K. A. & Johnston, J. P. 1998 Near-wall flow in a threedimensional boundary layer on the endwall of a 30◦ bend.
Experiments in Fluids 24, 175–184.
Gargiulo, Aldo, Vishwanathan, Vidya, Fritsch, Daniel J,
Duetsch-Patel, Julie E, Szoke, Mate, Borgoltz, Aurelien,
Devenport, William J, Roy, Christopher J & Lowe, Kevin T2020 Examination of Flow Sensitivities in Turbulence
Model Validation Experiments. In AIAA SciTech 2020 Forum, pp. 2020–1583. Orlando.
Johnston, J. P. & Flack, K. A. 1996 Review - Advances inThree-Dimensional Turbulent Boundary Layers With Emphasis on the Wall-Layer Regions. Journal of Fluids Engineering 118, 219–232.
Lozano-Durán, Adrián, Giometto, Marco G., Park, George Ilhwan & Moin, Parviz 2020 Non-equilibrium threedimensional boundary layers at moderate Reynolds numbers. Journal of Fluid Mechanics 883 (A20), 1–42.
Ölçmen, S M, Simpson, R L & George, J 2001 Some
Reynolds number effects on two-and three-dimensional turbulent boundary layers. Experiments in Fluids 31, 219–228.
Schlatter, Philipp & Örlü, Ramis 2010 Assessment of direct
numerical simulation data of turbulent boundary layers.
Journal of Fluid Mechanics 659, 116–126.
Volino, Ralph J 2020 Turbulence structure in non-equilibriumboundary layers with favorable and adverse pressure gradients. International Journal of Heat and Fluid Flow 86.

CONCLUSION
LDV measurements of the flow upstream of and on thethree-dimensional BeVERLI Hill model were used to analyze
the fluid physics at those stations. As shown, the selected stations experience significantly different mean flow and turbulence behavior due to differing pressure gradients, radii of curvature, and history effects at each station. Of these effects,
local pressure gradient and skewing appear to be the strongest
influences on the turbulence at each station.
Analysis of the BeVERLI Hill data and comparison withthe 3D TBL on a flat wall around a wing-body junction fromÖlçmen et al. (2001) show there is no region in the flow overthe BeVERLI Hill where equilibrium assumptions regarding
the shear stress distribution are valid. This is especially severein strongly skewed flow, such as at stations W2 and W3. In
accelerated regions of the flow with milder flow skewing, suchas at station W5, the turbulent stresses below the buffer layer
may approximately follow an equilibrium distribution, before
diverging significantly from this behavior above x2+ ≈ 30. Sim-4

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

𝑊2
𝑊3

𝑊1

𝑊5

(𝑏)

(𝑎)
Figure 1. BeVERLI Hill geometry and global
coordinate system. The side wall is shown as
the floor to accurately portray the right-handed
coordinate system.

(a) 𝑈1 /𝑈∞

Figure 2. Freestream pressure coefficient contours over the hill surface at 𝑅𝑒𝐻 =250,000 in global coordinates. Black circles show LDV profile locations. Black
vectors indicate mean flow direction at the wall, while blue vectors indicate mean flow
direction at 𝑥2+ ≈ 500. (a) Full pressure coefficient contour on the surface. (b) Focus
view of the three stations of interest on the bump. Here the pressure coefficient color
range is limited to better show the pressure contours in this region.

(c) 𝑈3 /𝑈∞(b) 𝑈2 /𝑈∞

Figure 3. Components of the mean velocity in local wall-shear-stress coordinates, normalized by the freestream velocity. (a) 𝑈1 /𝑈∞ . (b)𝑈2 /𝑈∞ . (c) 𝑈3 /𝑈∞ .

2(c) തതതത
𝑢3′2 /𝑈∞2(b) തതതത
𝑢2′2 /𝑈∞2(a) തതതത
𝑢1′2 /𝑈∞′ ′2തതതതതത(d) 𝑢
1 𝑢2 /𝑈∞2(f) തതതതതത
𝑢2′ 𝑢3′ /𝑈∞′ ′2തതതതതത(e) 𝑢
1 𝑢3 /𝑈∞

2 . (b)Figure 4. Components of the Reynolds stress in local wall-shear-stress coordinates, normalized by the freestream velocity. (a) തതതത𝑢1′2 /𝑈∞
′2
′2
′ ′
′ ′
′ ′
2
2
2
2
2
തതതത
തതതത
തതതതതത
തതതതതത
തതതതതത
𝑢2 /𝑈∞ . (c) 𝑢3 /𝑈∞ . (d) 𝑢1 𝑢2 /𝑈∞ . (e) 𝑢1 𝑢3 /𝑈∞. (f) 𝑢2 𝑢3 /𝑈∞ .

5

+

+

2
(a) തതതതത
𝑢1′(b) തതതതതത
𝑢1′ 𝑢2′+2(c) തതതതത
𝑢3′

Figure 5. Selected Reynolds stresses in viscous units and compared to the DNS data of Schlatter and Schlatter and Örlü (2010). (a)
+
+
+
2
2
തതതതത
𝑢1′ . (b) തതതതതത
𝑢1′ 𝑢2′ . (c) തതതതത
𝑢3′ .

Figure 6. The ratio of the empty tunnel boundary layer thickness
Figure 7. Comparison of viscous and turbulent shear stressat the location of the center of the hill, 𝛿0 , to the local magnitude
components at the four stations of interest, compared to the 2D DNSof the 3D radius of curvature, |𝑅|. 𝛿0 /|𝑅| > 0 indicates concaveof Schlatter and Örlü (2010) at 𝑅𝑒𝜃 = 4060. Open symbols indicate
curvature, while 𝛿0 /|𝑅| < 0 indicates convex curvature. Symbols
viscous stress components, 𝜕𝑈1+ /𝜕𝑥2+ , and filled symbols indicate
indicate the location of the hill stations on the local curvature.
+2+2
turbulent shear stress components, √തതതതതത
𝑢1′ 𝑢2′ + തതതതതത
𝑢2′ 𝑢3′ .

(b)

(a)

Figure 8. Comparison of turbulent and viscous shear stress behavior of selected stations on the BeVERLI Hill with the 2D
equilibrium DNS of Schlatter and Örlü (2010) and the 3D flat-wall TBL of Ölçmen et al. (2001). (a) Stations W1, W2, and W3of the BeVERLI Hill compared with stations 2 and 3 of Ölçmen et al. (2001). (b) Station W5 of the BeVERLI Hill comparedwith stations 6 and 7 of Ölçmen et al. (2001).

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