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

MEASUREMENTS OF 3D TRANSPORT BOUNDARIES WITH URBANCANOPIES USING MAGNETIC RESONANCE IMAGING

Andrew J. Banko
Ty Homan
Civil & Mechanical Engineering
Civil & Mechanical Engineering
United States Military Academy
United States Military Academy
752 Thayer Rd, West Point, NY 10996 752 Thayer Rd, West Point, NY 10996
andrew.banko@westpoint.edu
ty.homan@westpoint.edu

Michael J. Benson
Christopher J. Elkins
Civil & Mechanical Engineering
Mechanical Engineering
United States Military Academy
Stanford University
752 Thayer Rd, West Point, NY 10996 488 Escondido Mall, Stanford, CA 94305
michael.benson@westpoint.edu
celkins@stanford.edu

ABSTRACT

non-intuitive plume spread is enhanced by mean flow transport boundaries, defined as topological surfaces of the flow
field that inhibit scalar transport perpendicular to the boundary. However, these structures are poorly quantified due tolimitations of conventional optical- and probe-based measurement techniques.
Experimental advancements using magnetic resonanceimaging provide 3D, three-component velocity and passive
scalar concentration data on volumetric Cartesian grids (Elkins& Alley, 2007; Banko et al., 2020). These data allow for thequantitative evaluation of flow within complex geometries at
millions of points without the need for optical access. The
techniques have recently been applied to urban canopy flows(Shim et al., 2019; Homan et al., 2021), and bridge a gap between conventional lab data in idealized urban settings andfield data at sparse measurement locations.
Transport boundaries have been previously extracted foroceanic, atmospheric, and biological flows, among others.
They are based on the theory of Lagrangian Coherent Structures (LCS), which separate the flow into materially distinct
regions (Haller, 2015). One example of an LCS is the Finite Time Lyapunov Exponent (FTLE) field (Shadden et al.,2005). In steady flows, the mass flux perpendicular to an FTLE
ridge is negligibly small. Ridges of the FTLE field are therefore suitable candidates for mean flow transport boundaries
in urban canopy flows, because they define two-dimensional
manifolds (surfaces) with approximately zero mean scalar flux
across them.
In this work, we use MRI measurements to quantify the
3D topology of transport boundaries in a realistic urban canopy
model. Transport boundaries are extracted as ridges of the
FTLE field and compared to measurements of passive scalardispersion through the canopy.

Measurements of mean flow transport boundaries in arealistic urban canopy model are conducted using magnetic
resonance imaging (MRI) techniques. A transport boundary is defined as a ridge of the Finite Time Lyapunov Exponent (FTLE) field computed using the mean velocity. It isfound that ridges of the backwards FTLE field, which identify hyperbolic manifolds exhibiting exponential convergenceof fluid trajectories, act as barriers to scalar transport. The
MRI measurements enable the three-dimensional characterization of transport boundary topology and explain observed
cross-wind and vertical scalar transport behaviour. Despite thefact that the FTLE field only guarantees negligibly small meanscalar flux perpendicular to the ridge, the structures appear remarkably robust to turbulent mixing. By contrast, the forwards
FTLE field, which describes exponentially separating trajectories, does not inhibit turbulent mixing. This observation is
explained by analyzing a simple cellular flow field with both
stable and unstable manifolds.

INTRODUCTION
Understanding and predicting turbulent scalar dispersion
from street-level sources within the urban canopy is importantfor determining emergency response, assessing air quality, andproviding input to neighborhood and regional scale pollution
models. Flow within the canopy is governed by mechanicaldispersion due to the mean flow around buildings and turbulent
dispersion (Belcher, 2005). The flow topology is fundamentally three-dimensional (3D) and characterized by interacting
separation bubbles, convergence zones, and non-equilibriumboundary layers (Britter & Hanna, 2003). These features represent a significant modeling challenge. Laboratory, field data,and high-fidelity simulations have demonstrated non-intuitive
plume spread, including up-wind or cross-wind transport andsharp variations in concentration across street intersections(Carpentieri et al., 2012). For instance, Homan et al. (2021)
describe the interaction of a plume with a 3D urban canopy
flow and observe plume displacements larger than the geometric footprint of various buildings. We hypothesize that

URBAN CANOPY DATA
Experimental Methods
Experiments are performed on a 1:2206 scale model of
the Oklahoma City downtown business district circa 2003
based on the JU2003 field tests. The test section was 3D

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

Figure 2. Contours of maximum concentration over the injection cycle with in-plane velocity vectors on a plane nearground level (y/H = 0.15). Data shown are the full experimental resolution. Regions 1, 2, and the intersection are indicated by the white boxes.

is τ = H/U∞ = 0.067 sec. This protocol allows the plume tonearly wash out of the measurement domain between injection
pulses.
Measurements of ensemble-averaged velocity and concentration are conducted using magnetic resonance velocimetry (MRV) and magnetic resonance concentration (MRC).
Data are acquired on a Cartesian grid of voxels for 12 distinct temporal phases across the injection cycle as describedin Homan et al. (2021). The reader is also referred to Elkins& Alley (2007) and Banko et al. (2020) for detailed descriptions of the techniques. The measurement resolution is 1.5mm, or approximately 10 voxels per street width. Both MRV
and MRC use dilute aqueous solutions of CuSO4 as the working fluid. The uncertainty in velocity is approximately 5% of
the freestream velocity. The concentration uncertainty scaleswith the local concentration level, with a noise floor of about
2% of the injected concentration.

Figure 1. Overview of test section geometry and velocity andconcentration data. (a) The urban canopy model. (b) An isosurface of 5% concentration near the end of injection, coloredby height. (c) Velocity magnitude on a plane near ground level(y/H = 0.15). Streamlines displaying characteristic, 3D flow
patterns. Regions of interest 1 and 2 in the building wakes adjacent to the main street and intersection are indicated by thewhite boxes.

Flow Field Overview
Figures 1(b) and 1(c) provide an overview of the concentration and velocity fields in the urban canopy. Additional details can be found in Homan et al. (2021). The concentration
isosurface in Figure 1(b) is plotted after the plume has dispersed several city blocks in the streamwise direction. The
plume disperses along two distinct paths to the west and eastof the Colcord hotel, and several distinct ventilation characteristics are observed between buildings and along street canyons
as evidenced by the streamline patterns in Figure 1(c). These
include: a recirculation region with vertically oriented vorticalmotion (region 1), typical of a building wake; the entrainmentof freestream fluid downwards into the urban canopy where it
spreads at ground level (region 2); and a longitudinal vortex
along the extended street canyon. The freestream entrainment
in region 2 is due to local variations in building height, similar
to a step-up canyon geometry, and shows that the ventilation
patterns are geometrically sensitive.
Figure 2 plots contours of the maximum concentration
over the course of the injection cycle on a plane near groundlevel with in-plane velocity vectors. A large concentration
gradient bisects the intersection adjacent to region 2. This is

printed using stereolithography and integrated into a water
channel with flow conditioning components (diffuser, contraction, and roughness section) to develop a turbulent boundary
layer upstream of the urban canopy as described in Homanet al. (2021). The main flow direction was from south-to-northas shown in Figure 1(a). The freestream velocity was U∞ = 0.3
m/s and the characteristic building height was H = 20 mm, giving a building Reynolds number of ReH = 6,000. While well
below application scale Reynolds number, the flow within the
canopy is fully-turbulent. Moreover, flow separation initiates
at the corners of buildings and therefore the large-scale features of the flow are approximately independent of Reynolds
number.
A passive scalar is injected at ground level upstream of
the Colcord hotel as shown in Figure 1(a). The injection istransient and follows a square waveform with 40% duty cycle
and a period of 1 second. The characteristic advection timescale based on the freestream velocity and building height, H,

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
due to the convergence of clean freestream fluid spreading at
ground level with contaminated fluid transported north fromthe injector along the large street. Similar phenomena are observed along the west side of region 2 and across the shear
layers of separation bubbles next to buildings. The in-plane
vector fields in each region suggest a dividing streamline acts
as a barrier between the clean and contaminated flow, indicating their importance for the mechanical dispersion of theplume. However, these observations are based on a 2D projection of a 3D flow field. In the following sections, we show thatthe topology of the 3D flow field forms 2D manifolds that act
as transport boundaries.

TRANSPORT BOUNDARY DEFINITION
We define transport boundaries using the theory of Lagrangian Coherent Structures (LCSs). An LCS is a 2D material surface that exhibits normally attracting or repelling fluid
trajectories. The mean scalar flux perpendicular to an LCS iszero (or near zero, see numerical computation below). Therefore, LCSs are candidates for transport boundaries within theurban canopy because their properties agree with our qualitative overview of the velocity field and its effect on contaminant dispersion. We focus primarily on LCSs defined from the
FTLE field and also make comparisons to the Stationary LCSs
(SLCSs) described in Teramoto et al. (2013).
The FTLE field is computed from the deformation gradient tensor of the flow map φ T (xo ), obtained by integrating
the system of equations dφ t /dt = U over a finite time interval, T . Here, xo is the initial position of fluid particles andU is the mean velocity vector field. In other words, φ T (xo )is the final positions of fluid particles originating at xo after
a time T . Denote the deformation gradient tensor by Dxo φ T ,and let λiT and ζiT be the ith eigenvalue and eigenvector of the
Cauchy-Green strain tensor, CT = (Dxo φ T )∗ (Dxo φ T ), where(·)∗ indicatespthe transpose. Then, the FTLE field is given
by σ T = ln( max(λi ))/T . Integration forwards in time determines the forwards FTLE field corresponding to exponentially separating trajectories. Integration backwards in time
determines the backward FTLE field corresponding to exponentially converging trajectories.
Figures 3(a) and 3(b) plot the forwards and backwards
FTLE fields computed from the experimental mean velocity
field on a plane near ground level. A 4th order Runge-Kutta
temporal integration scheme and linear spatial interpolation
for the velocity are used to compute the particle trajectories.
Initial positions for the fluid particles are seeded on a uniform Cartesian grid with an isotropic spacing of ∆x = 0.3
mm. The velocity outside the fluid domain is set to zero.
The total integration time is T = 2H/U∞ and the time stepis ∆t = ∆x/(4 · max(|U|)). Second order central differencing
is used to compute the deformation gradient tensor. Ridges, or
local extrema within the wall-normal plane, are nearly parallel to the in-plane velocity vectors. Therefore, the mean flux
across a ridge is small, at least in the 2D projection.
The flux across a ridge is made quantitative in 3D by comparing the FTLE fields to the Stationary LCSs described in
Teramoto et al. (2013). An SLCS is the set of points satisfying: (1) LT ≡ ζ1T · U = 0, (2) λ2T < ε 2 λ1T and λ1T > 0, and (3)|cos−1 (|ζ1T ·n|)| ≤ α. Subscripts 1 and 2 denote the largest andsecond largest eigenvalues, respectively, and the corresponding eigenvectors. Condition (1) guarantees that the direction of
maximal particle separation (or attraction backwards in time)is orthogonal to the local flow direction. Condition 2 ensuresthat rate of separation (attraction) normal to the SLCS dom-

Figure 3. Contours of the FTLE field on a plane near groundlevel (y/H = 0.15) for (a) forwards-in-time and (b) backwardsin-time integration. (c) A comparison between the contours of
the backwards FTLE field and the SLCS shown in red.

inates the rates of separation (attraction) in other directions.
Condition 3 enforces that the flux perpendicular to the SLCS
can be made small by controlling the parameter α. Here, nis the unit normal to the SLCS computed from the gradient of
LT .
Figure 3(c) overlays the backwards SLCS (red) on contours of the backwards FTLE field. The SLCS parameters are
ε = 0.2 and α = 25o . Note the close correspondence between
ridges of the FTLE field and the SLCS. Figure 4 plots the probability distribution (PDF) of the angle between the unit normal
vector to the SLCS and the mean velocity vector. The majority of points are concentrated around 90o , indicating near zero
flux across the SLCS. Therefore, the flux across FTLE ridges

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

0.015
0.1
0.01
0.05
0.005

0

0
0

15

30

45

60

75

90

0

Figure 4. Probability distribution of the angle between thenormal vector to the SLCS and the local velocity vector.

30

60

90

120

150

180

Figure 6. Probability disbtribution of the angle between the
maximum convergence direction and the average concentration gradient. PDFs are computed for the entire flow field andconditioned on proximity to an FTLE ridge.

FTLE ridges. For example, the separation bubbles adjacent tobuildings are bounded by an FTLE ridge that extends the entire building length in the streamwise direction. Likewise, thefreestream flow which spreads at ground level in region 2 isbounded by FTLE ridges which extend a full building width
to the east and west. The eastern ridge bisects the intersection,
as was qualitatively discussed above. These topological structures have a significant impact on the concentration field andare robust to turbulent mixing as shown below.
Conversely, the forwards FTLE ridges are typically located close to building corners and do not penetrate as deeply
into the flow field. Forwards FTLE ridges intuitively terminate at building corners since they demarcate diverging fluid
trajectories. It is observed that forward FTLE ridges do notsuppress turbulent dispersion of the plume, and are therefore
less efficient transport boundaries. For brevity, we focus on the
backwards FTLE field in the remainder of this section. The
final section of the paper describes a model system for understanding the relative importance of turbulent mixing across
backwards and forwards FTLE ridges.

Backwards FTLE Results
Figure 5 illustrates the spatial correspondence between
the backwards FTLE ridges and the concentration field. FTLE
contours on wall parallel planes are overlaid with a 3% isocontour of maximum concentration. The plane near ground levelshows that little contaminant mixes across the ridges bounding
the separation bubble on the west side of the Colcord Hotel.
The FTLE ridge which bisects the intersection also separates
clean and contaminated fluid, and redirects the plume eastward
as previously described. The ridge on the west side of region 2
nominally separates clean and contaminated fluid, but the concentration isocontour crosses a portion of the FTLE ridge. In
fact, this is partly due to 3D mean flow effects as described in
the following section.
A plane halfway up the Colcord hotel shows similar correlations between the backwards FTLE ridges and the boundaries between clean and contaminated fluid. Notably, closed
FTLE contours on the east side of the measurement domain
enclose contaminated fluid that is transported vertically out of
the street canyon in the tall building wakes.
Figure 6 summarizes these observations by plotting the
PDF of the angle between the maximum convergence direction (eigenvector ζ1T ) and the average concentration gradient.

Figure 5. Contours of the backwards FTLE field overlaid
with a 3% maximum concentration isocontour (red). The interior of the contour is shaded for clarity. (a) A plane nearground level (y/H = 0.15). (b) A plane near y/H = 0.5.

is near zero as well. Future work will investigate the effectof experimental uncertainty on the flux. For the remainder ofthe paper, we define LCSs by thresholding the FTLE fields tolarge values.

TRANSPORT BOUNDARY EFFECTS ON 3D DISPERSION PATTERNSThe FTLE ridges in Figure 3 coincide with the convergence zones, separation bubbles, and stagnation points described previously. These structures extend outwards into theflow field by a distance similar to the characteristic building
length scales. This is particularly evident for the backwards

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ries of the forwards FTLE field would amplify the effects of
turbulent dispersion across the FTLE surfaces.
A simple cellular flow field is constructed to illustrate
this hypothesis. Consider the mean velocity field defined by
U = −sin(2πx)cos(2πy) and V = cos(2πx)sin(2πy) on theunit square x ∈ [0, 1] and y ∈ [0, 1]. This flow field forms four
recirculating cells separated by the line x = 0.5 (a stable manifold), and the line y = 0.5 (an unstable manifold). Additional
stable and unstable manifolds can similarly be identified along
the domain boundaries. A passive scalar is initialized at 100%concentration in one quandrant of the domain and allowed toadvect and diffuse over a time T = 1. The scalar diffusivity is√D = 0.0025. This implies a diffusion length scale oflD ≡ D · T = 0.05, or 5% of the domain. A no-flux condition
is applied at the domain boundaries.
Figure 9 plots the final concentration field overlaid withvelocity vectors. The scalar diffused across both the stableand unstable manifolds, but the net flux perpendicular to thestable manifold is reduced due to the converging flow. Therefore, the concentration gradient remains large near the central
stagnation point. Concentration which diffuses across the unstable manifold is eventually advected away from either sideof the downstream stagnation points and disperses rapidly into
the adjacent quadrants. In fact, a comparison of Figures 3 and5 show evidence of this behavior in the wake of the Colcord
Hotel. Contaminant is transported across the forwards FTLE
ridge extending from the downstream building corner and fills
region 1.

Figure 7. Isosurfaces of the backwards FTLE field coloredby height. The isosurface level is σ T = 20. Two isometricviews are shown.

When conditioned on proximity to an FTLE ridge, the PDF
becomes peaked towards angles of 0o and 180o . This indicates that the concentration gradient is preferentially alignedwith the direction normal to the FTLE ridge. In other words,
the concentration increases or decreases most rapidly moving
across the ridge. Therefore, the FTLE ridges act as transport
boundaries which inhibit scalar flux normal to their surface.

3D Flow Topology
Figure 7 shows isosurfaces of the FTLE field colored byheight within the urban canopy to show the 3D structure of thesurfaces. Sheet-like structures demarcate the separation bubbles on the sides of buildings. The boundaries generated by theentrained flow within region 2 are curved and impact the vertical plume transport in addition to its lateral dispersion. Figure
8 illustrates this behaviour by plotting contours of peak concentration on a streamwise vertical plane that cuts through the
FTLE extending into the intersection. The plume is deflectedvertically over the intersection structure creating a bubble ofclean flow near the ground that is eventually filled in by wallnormal mixing downstream of the intersection.
Two additional streamwise vertical planes cutting throughthe western FTLE surface are also depicted in Figure 8. The
maximum concentration contours again show that the plumeis dispersed vertically by the transport boundary. Due tothe downstream building and the FTLE surface generated bythis building’s separation bubble, the contaminated flow is advected back towards ground level on the downstream side of
the FTLE surface and recirculates upstream. Therefore, thedownstream side of the transport boundary experiences diluteconcentrations due to turbulent mixing across the ridge, butalso due to mean advection by the complex 3D flow. This
explains the why the concentration contours cross the FTLE
ridge in Figure 5.

CONCLUSIONSThe effect of 3D flow topology on passive scalar dispersion within a realistic urban canopy model was studied experimentally. Magnetic resonance imaging was used to obtain
volumetric fields of ensemble-averaged velocity and concentration for a transient release in a scale model of the Oklahoma City downtown business district circa 2003. Mechanicaldispersion of the plume was enhanced due to the presence ofmean flow transport boundaries. Transport boundaries correspond to 2D manifolds with zero mean scalar flux, formed by
the global topology of the flow field.
Two different definitions were considered based on thetheory of Lagrangian Coherent Structures and computed from
the MRI data: the Finite Time Lyapunov Exponent Field(FTLE) and the Stationary Lagrangian Coherent Structure(SLCS). Ridges of the FTLE field agree with zero level setsof the SLCSs, and were shown to have small mean flux normal to the surfaces. Deviations from zero mean flux are likely
due to experimental uncertainty and resolution, and will be a
topic of future investigation.
The backwards FTLE field was correlated with sharp gradients in the concentration field and forms transport boundaries in the urban canopy. The MRI measurements provided
unique insight into the 3D structure of the FTLE surfaces andtheir effect on dispersion. Prominent features included theshear layers bounding separation bubbles around buildings andregions of strong flow convergence due to complex ventilation
patterns induced by the heterogeneous urban geometry. Bothlateral and vertical dispersion were enhanced due to the deflection of the main plume by these structures.
While the backwards FTLE surfaces were remarkably robust to turbulent mixing, the forwards FTLE surfaces did not
clearly demarcate clean and contaminated fluid. This observation was explained by analyzing the dispersion of a diffusive
passive scalar contaminant in a cellular flow field.

A Model System for Understanding Turbulent
Dispersion
The turbulent scalar flux through the FTLE surfaces is
not necessarily zero because the FTLE is defined using themean velocity field. Yet, observations suggest that the turbulent flux is small, especially for the backwards FTLE field.
This could be because turbulent fluctuations which cross thetransport boundary are advected by the strong convergence
flow back towards the boundary, keeping contaminated flownear the hyperbolic manifolds. Figure 9(a) shows a diagramof this process. Conversely, exponentially separating trajecto-

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 8. Contours of maximum concentration on streamwise vertical planes with in-plane velocity vectors. Isocontours of the
backwards FTLE field at σ T = 20 are shown in black. Locations of each plane are shown in the left-hand-side subfigures.

REFERENCES
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Homan, Ty A, Benson, Michael J, Banko, Andrew J, Elkins,
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transient release. Building and Environment 205, 108163.
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2005 Definition and properties of lagrangian coherent
structures from finite-time lyapunov exponents in twodimensional aperiodic flows. Physica D: Nonlinear Phenomena 212 (3-4), 271–304.
Shim, Gawoon, Prasad, Dipak, Elkins, Christopher J, Eaton,
John K & Benson, Michael J 2019 3d mri measurementsof the effects of wind direction on flow characteristics andcontaminant dispersion in a model urban canopy. Environmental Fluid Mechanics 19 (4), 851–878.
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Figure 9. Model for turbulent diffusion across hyperbolic Lagrangian coherent structures. (a) Illustration of the pathline fora fluid element near a stable manifold. (b) Cellular flow fieldwith passive scalar advection and diffusion.

Future work will assess the prevalence of transport boundaries across a wider variety of urban canopy flow datasets developed by the authors using MRI. Studies using large eddy
simulations (LES) or conventional optical experimental techniques could focus on the turbulence dynamics near the mean
flow transport boundaries. MRI data provide rigorous validation cases for LES and also identify regions of interest withincomplex, 3D flow fields for additional experiments. Finally,
identifying transport boundaries may be important for future
model development, because models must accurately predict
their locations in order to capture mechanical dispersion.

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