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

STATISTICALLY CONSISTENT RESOLVENT-BASED RECONSTRUCTION
OF TURBULENT CHANNEL FLOWS FROM LIMITED MEASUREMENTS

Vamsi Krishna Chinta
Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089.
vchinta@usc.edu

Mitul Luhar
Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089.
luhar@usc.edu

ABSTRACT

statistically consistent reconstruction of turbulent flows fromlimited measurements. Specifically, we attempt spatiotemporal reconstruction from a limited number of non-time-resolved
two-dimensional, two-component (2D-2C) velocity measurements (c.f., basic PIV). The velocity field is projected onto
resolvent modes and the amplitudes and phases are optimized
to satisfy statistical constraints. The velocity field can thenbe reconstructed in the homogeneous spatial directions as wellas in time. Because the resolvent modes are derived directlyfrom the governing Navier-Stokes equations, and statisticalconstraints are enforced, the reconstructed flow field is expected to be physically and statistically consistent.
For proof-of-concept tests of this method, we use DNS
data for turbulent channel flow at Reτ = uτ h/ν = 1000 available from the JHTDB (Graham et al., 2016) as the input or
the surrogate for experimental PIV measurements. Here, uτ
is the friction velocity, h is the channel half-height, and ν iskinematic viscosity. For consistency with typical PIV systems,
we use only 2D-2C velocity data that is sampled uniformly intime and space. We consider reconstruction from measurements made in a single streamwise-wall normal (x − y) plane
as well as measurements made simultaneously in the x − y andy − z planes, where z is the spanwise direction. We evaluate
how increasing the number of snapshots affects reconstruction
performance. In addition, we compare reconstructions for two
different resolvent models: one which makes use of the molecular viscosity (ν) and another that includes the eddy viscosity(νT ) in the input-output transfer function Morra et al. (2019).
This work complements prior literature on resolventbased reconstruction. Prior studies have successfully reconstructed the velocity field from time-resolved point measurements in band-limited flows such as inclined square cylinder
wakes (Gómez et al., 2016) and jets (Beneddine et al., 2017)by projecting the velocity field on to resolvent modes. Recent
studies reconstruct the full 3D velocity field for turbulent channel flow from spatiotemporally-resolved velocity field snapshots at a single wall-normal location (Illingworth et al., 2018)
or from wall-based measurements of shear stress and pressure(Amaral et al., 2021). The key distinction in the present workis the use of limited measurements that are not spatially or temporally resolved to reconstruct a broadband turbulent flow.

This study develops a resolvent-based framework that canbe used to reconstruct the velocity field for turbulent channel
flow from limited measurements. For reconstruction, resolvent
modes computed at different frequency and wavenumbers arepooled into a large database. The Forward Regression with Orthogonal Least Squares (FROLS) algorithm is used to sequentially identify modes that best represent the input data. A composite loss function is defined with the error for the secondorder statistics imposed as a soft constraint to the reconstruction error for velocity field. The amplitude and phase of theidentified modes are then optimized to minimize this composite loss function. After calibration, the velocity field can bereconstructed at arbitrary spatiotemporal resolution using theweighted resolvent modes. The resulting model enables estimation of out-of-plane components of velocity and pressure aswell as reconstruction in other spanwise locations. For proofof-concept tests of this method, direct numerical simulation(DNS) data for turbulent channel flow from the Johns Hopkins
Turbulence Database (JHTDB) are used. Results indicate thatusing measurements from multiple planes results in reduced
error for the untrained variables (i.e., non-measured velocity
components, pressure, out-of-plane data). Using an eddy viscosity model to compute resolvent modes results in smootherstatistical profiles, but the velocity field reconstructions do not
reproduce small-scale features.

INTRODUCTION
Turbulent flows are characterized by broadbanded fluctuations in the velocity field in both space and time, whichmakes acquisition of spatiotemporally resolved measurements
challenging. For instance, particle image velocimetry (PIV), a
standard technique for acquiring spatial velocity fields, is often restricted to a limited field-of-view and not typically timeresolved. Point sensors such as hot-wire anemometers (HWA)
can yield time-resolved velocity measurements, but do not provide any spatial information. Previous studies have tried tobridge this gap by reconstructing the flow field between two
consecutive PIV snapshots using physics-based models (Krishna et al., 2020) or by using Bayesian inference to fuse nontime-resolved PIV snapshots with time-resolved HWA measurements (Wang et al., 2021). Recent studies have also reconstructed turbulent flow fields from limited measurements
using machine learning techniques (e.g., Fukami et al., 2021;
Kim et al., 2021).
In this work, we develop a resolvent-based framework for

METHODS
Resolvent-based reconstruction
In the resolvent framework, the Navier-Stokes equations
are reformulated into a linear forcing-response system that in-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
available and the large number of candidate resolvent modes inthe database, i.e., the likelihood of over-fitting. Thus, we seeka sparse solution in which only a small number of coefficientsare non-zero. For this we use the FROLS algorithm (Billings,2013), which sequentially identifies the resolvent modes thatbest represent the input training data and calibrates their amplitude and phase. Here, we use a slightly modified version
which selects a full resolvent mode instead of the real or imaginary component of it. This allows us to reconstruct flow fields
with arbitrary phase shifts between the modes. This implementation is similar to the one used in Chinta et al. (2022). For
all the cases considered in this paper, reconstruction is carriedout with the first 200 resolvent modes (or 400 basis functions)
identified by the FROLS algorithm.

terprets the nonlinear term as a forcing and the velocity andpressure fields as the response (McKeon, 2017). For turbulent channel flows that are homogeneous in the streamwise (x)and spanwise (z) directions and statistically stationary in time(t), the governing equations can be Fourier transformed andthe forcing-response system can be formulated independently
for each wave-number frequency triplet k = {kx , kz , ω}. The
transfer function between the input forcing and output statevariables for each k is the resolvent operator Hk . A gainbased decomposition of the resolvent operator then identifies
high gain forcing and response modes. The response modes
are travelling wave structures, which serve as the candidate
basis functions onto which we project the velocity field. An
eddy viscosity model is included in resolvent operator to incorporate the effect of Reynolds stresses (Morra et al., 2019).
Resolvent modes are then computed for a channel geometry for the following streamwise and spanwise wavelength(λx , λz ) and mode speed (c+ ) ranges: 0.125 < |λx |/h < 16,
0.125 < λz /h < 16, and 0 < c+p ≤ 22.63. A database of 53040
resolvent modes is compiled. To limit database size only rank
1 modes are used.
We assume that the total velocity field can be represented
as a linear combination of the resolvent modes:
Enforcing statistical constraints
Although the reconstructed flow field (û = Db) is expected to represent the input data with reasonable accuracy,
there is no guarantee that the resulting flow field will be statistically sound. Thus, we solve an additional optimization problem to enforce statistical consistency. Specifically, we minimize the following composite cost function:
C = λ C1 + (1 − λ ) C2
û(x, y, z,t) = ℜ ∑ αk ũk (y) exp (i(kx x + kz z − ωt)) . (1)
|
{z
}
k

where λ is a hyper-parameter to adjust the fit towards the velocity field or the statistics, andpk

Here, ũk (y) is the resolvent mode computed for wavenumberfrequency combination k = (κx , κz , ω). The complex coefficients αk determine the amplitude and phase of the mode.
Note that each term in the summation comprises two basis
functions and two coefficients corresponding to the real andimaginary components of ũk and αk , respectively. Thus, M/2
resolvent modes yield M basis functions and M coefficients.
The reconstruction problem essentially amounts to tuning
the coefficients αk using measurement data. For this, the 3D
flow fields for each resolvent mode, pk , are evaluated with thesame spatial and temporal dimensions at the input data to formcandidate basis functions dm in a large database D. For instance, for reconstruction from 2D-2C data in the x − y plane,
the streamwise and wall normal components of velocity (u, v)
are extracted with the same spatial and temporal resolution as
the PIV-like input data. The PIV-like input velocity data are
now projected on to these basis functions:

 


b1
u1



 u2   b2 
d1 d2 · · · dM  .  ,
 .  ≈
 ..   .. bMuN| {z }|{z} | {z }C1 = ∥u − û∥2F /∥u∥2F ,basis f unctionsc
C2 = ∥u2 − u2 ∥2 /∥u2 ∥2 ,(4)

represent cost functions for the velocity field and second-order
statistics. Note that the velocity components and statisticalquantities used for training can be different depending on thedata available. As an example, for reconstruction from 2D-2C
velocity snapshots in the x − y plane, we have u = (u, v) and
u2 = (u2 , v2 , uv). For this case, second-order statistics for theidentified and calibrated resolvent modes are given byb
 ∗ u2ũ ũ
|αk |2  k∗ k 
c2  b ṽk ṽku = v2  = ∑
ℜ
2
∗ ṽ
k
ũ
b
k k
uv

(5)

where ũk and ṽk respectively represent the streamwise andwall-normal components of ũk , and a superscript ∗ denotes thecomplex conjugate. Note that second-order statistics for turbulent channel flow, u2 , can be computed directly from the measurements or obtained from other sources (including models).
We use the multistart optimization routine in MATLAB to findthe solution, i.e., the set of coefficients α = {αk }) that minimize the composite cost function in Eq. 3: α = arg min C (α ′ ).



input data

(3)

(2)

coe f f icientsα′or equivalently, u ≈ Db. Here, u represents the sequence of N
input velocity field snapshots reshaped into a column vector, Drepresents the database of M basis functions (or M/2 different
resolvent modes) and b is a column vector of coefficients forthe basis functions.

Numerical Evaluation
The full DNS dataset acquired from JHTDB composes
256 snapshots obtained at intervals of δt + = 0.0625, for boththe x − y and y − z planes. Following standard notation, thesuperscript + denotes normalization with respect to uτ and ν.
Thus, the total time interval between the first and last snapshots is T + = 256 × δt + ≈ 16. For the x − y plane, we use
a uniformly sampled grid with Nx × Ny = 32 × 64 extending

Identifying dominant resolvent modes
Standard least squares method cannot be used for the regression problem in Eq. 2 due to the limited measurement data2

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

over a region of size 2h × h. For the y − z plane, we use a
Ny × Nz = 64 × 32 grid extending over a region of size h × 2h.
Only the first and last DNS snapshots are used to reconstruct
the flow field. The intervening snapshots are used to quantify reconstruction accuracy using separate error metrics forthe observed (trained) variables and the unobserved (watched)variables. For reconstruction from 2D-2C data in the x − yplane, the trained components are the velocity fields (u, v) andthe statistical quantities (u2 , v2 , uv). The watched variables are
the spanwise velocity component and pressure (w, p), and thestatistical quantities (w2 , p2 , wp). Integrated error metrics forthe trained and watched quantities are defined as
1 ∥u − û∥F ∥v − v̂∥F
+
,
2
∥u∥F
∥v∥F


1 ∥w − ŵ∥F ∥p − p̂∥Fεu(watch) =
+
,
2
∥w∥F
∥p∥F

This paper evaluates reconstruction for the 5 different data
scenarios listed in Table 1. These cases systematically evaluate reconstruction accuracy for different measurement planes,
viscosity models, training snapshot numbers, as well as thehyper-parameter (λ ) that sets the relative weighting for the velocity field and statistics in the cost function used for the optimization algorithm (Eq. 3).
Case 1 represents the baseline case in which reconstruction is carried out using the molecular viscosity (ν) resolvent
modes from N = 2 snapshots obtained in the x − y plane witha time interval of T + ≈ 16. Case 2 builds on Case 1 to consider reconstruction from simultaneous measurements in boththe x−y and y−z planes. Case 3 makes use of the eddy viscosity (νT ) resolvent model for reconstruction. Case 4 considers
a higher value for the hyper-parameter (λ = 0.9 vs. λ = 0.5
for the remaining cases). This emphasizes reconstruction of
the velocity field. Finally, Case 5 evaluates reconstruction accuracy from N = 8 snapshots of the velocity field.
Reconstruction accuracy is evaluated for both the recostructions from FROLS (which does not take into accountstatistical constraints) and from constrained optimization separately. Since the watched variables and statistics are notused for either the FROLS or optimization, the reconstruction error for these quantities provides a sense for whether
the resolvent-based reconstruction framework being developed
here is able to generate volumetric reconstructions from limited planar data.

εu(train) =

(6)

and


b
b
b 2
1  ∥u2 − u2 ∥2 ∥v2 − v2 ∥2 ∥uv − uv∥,++εu2 (train) =3∥uv∥2∥u2 ∥2∥v2 ∥2ccc 2
1  ∥w2 − w2 ∥2 ∥p2 − p2 ∥2 ∥up − up∥.
++εu2 (watch) =3∥up∥2∥w2 ∥2∥p2 ∥2(7)

RESULTS AND DISCUSSION
Here the variables shown with a (ˆ) are the reconstructed velocity or statistical components, while the unhatted variables are
the DNS ground ‘truth’. Similarly, for reconstructions fromdata in both the x − y and y − z planes, the trained variables
are the streamwise and wall-normal velocity components (u, v)
for the x − y plane, and the wall-normal and spanwise velocity
components (v, w) for the y − z plane. Statistical constraints
are imposed on the quantities (u2 , v2 , w2 , uv). The watchedcomponents are (w, p) in the x − y plane and (u, p) in the y − zplane. The watched statistical quantities are (p2 , up). In thiscase, the error metrics in Eqs. 6-7 are modified to reflect thedifferent trained and watched quantities.

Table 2. Reconstruction errors in the velocity field and statistical quantities for the trained and watched variables. For eachcase, the top row shows reconstruction error after the FROLS
algorithm is used to identify and calibrate 200 resolvent mode
amplitudes based on the training snapshots. The bottom row(in parentheses) shows reconstruction error after the optimization algorithm modifies the amplitudes to enforce statisticalconstraints.
Case
1

Table 1. Cases discussed in this study. In the second column,
ν represents resolvent analysis with the molecular viscosity
while νT is for an eddy viscosity model. N is the number ofsnapshots used for training. The fourth column indicates theplanes and velocity data available for training. λ is the hyperparameter used in constrained optimization, see Eq. 3

2

3

4

Case

Visc. Model

N

Planesλ1

ν2

x-y (u,v)

0.5

2

ν

2

x-y (u,v), y-z (v,w)

0.5

3

νT

2

x-y (u,v), y-z (v,w)

0.5

4

ν

2

x-y (u,v), y-z (v,w)

0.9

5

ν

8

x-y (u,v), y-z (v,w)

0.55εu(train)εu(watch)

εu2 (train)

εu2 (watch)

0.36

760.712.82 × 104

1.23 × 104(0.46)(2.07)(0.15)(1.14)

0.51

1.15

12.50

224.16(0.60)(1.25)(0.30)(1.54)

0.56

2.16

6.78 × 109

2.51 × 1010

(0.86)(0.94)(0.18)(0.81)

0.51

1.15

12.50

224.16(0.54)(1.14)(0.49)(1.78)

0.66

1.00

12.50

224.16(0.75)(1.01)(0.31)(1.79)

Table 2 shows reconstruction errors in the trained andwatched variables for the training snapshots, along with theerrors for the trained and watched statistical quantities. For
each case, the top row shows the error from FROLS. The second row (in parenthesis) shows the error after constrained opti-

3

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

b)
0.9

2.5

Case 1Case 20.8

Case 32

Case 4

0.7

Case 5
0.6

1.5

0.5
1
0.4
0.3

0.5
0

5

10

15

0

5

10

15

Figure 1. Error evolution for trained variables in the x − y plane (a) and y − z plane (b). Solid lines represent reconstructions fromFROLS and dashed lines from constrained optimization.

mization. Starting with the baseline case, where a single x − yplane is used, the errors for the training snapshots from FROLS
are reasonably low (εu(train) < 0.4) and the flow fields, notshown here for brevity, are qualitatively similar to the DNS
truth. However, the errors for the statistical quantities are several orders of magnitude higher, ∼ O(104 ). After optimization
the error in the statistics decreases to O(10−1 ) with a slightincrease from 0.36 to 0.46 in the error for the trained components of the velocity field. It is interesting to note that the error
for the watched variables (w, p) decreases substantially, from
760 to 2.
Building on this case, we next use simultaneous measurements from both x − y and y − z planes. The error evolution for
the reconstructions in the x − y and y − z plane are shown in
Fig. 1(a) and (b), respectively. Reconstruction error for Case 2
in x − y plane varies little over the time horizon, with a meanvalue (εu(train) ≈ 0.5) which is very close to the values reported for the training snapshots in Table 2. After optimization, the errors for both the trained and watched variables increases slightly, while the errors for the statistical quantities
again decrease substantially. For instance, εu2 (train) decreases
from 12.5 to 0.30. It is important to note that the use of two
orthogonal planes results in a significant decrease in the error
for watched quantities even before optimization. The error for
the watched components of velocity field changes from 760
for Case 1 to 1.15 for Case 2. Similarly, reconstruction error
for the statistical quantities decreases by over two orders ofmagnitude for Case 2 relative to Case 1.
When the νT resolvent model is used (Case 3), reconstruction errors for the velocity field from FROLS are comparable to those for the ν resolvent model (Case 2). However, errors for the statistical quantities (∼ O(1010 )) are several ordersof magnitude larger than that for Case 2 (∼ O(104 )). After optimization, the errors for the statistical quantities are reduced to∼ O(10−1 ). However, the error for the velocity field rises significantly when compared to Case 2. Figure 2 compares reconstructed statistics from FROLS for Case 2 and the constrained
optimization results for Cases 2 and 3 with the DNS ‘truth’.
As discussed before, the statistics from FROLS are several orders of magnitude higher, while the reconstructed statistics after optimization qualitatively match the DNS profiles. Notethat the statistics corresponding to Case 3 are smoother andmore closely approximate the DNS profiles compared to Case
2. This is consistent with the observations of Morra et al.
(2019), who showed that the νT resolvent model yields bet-

ter reduced representations for turbulence statistics. This isbecause the resolvent modes computed with an eddy viscosity
model have greater support in the wall-normal direction. This
results in smoother statistics. However, this also smooths out
fine-scale features in the reconstructions which result in larger
errors for the velocity fields.
Figure 3 compares the reconstructed flow fields after optimization for Case 2 with the corresponding snapshots from
DNS. Subplots (a)-(b) show the trained velocity components
in the x − y plane (u, v) for one of the 2 training snapshots.
Subplots (c)-(d) show the watched variables in the x − y plane,i.e., spanwise velocity and pressure. Subplots (e)-(f) show thevelocity fields reconstructed for a x − y plane that is offset by∆z/h = 0.125 in the spanwise direction from the training snapshots. In other words, panels (c)-(f) show reconstructions forquantities that are not available for training the model. The reconstructions are labelled with a (ˆ). As expected, reconstructions for the training components (u, v) show good qualitative
agreement with the DNS snapshots, though some fine-scalefeatures are missing (see Fig. 3(a,b)). We do not see significant
similarity in structure for the untrained components, i.e., (w, p)
in Fig. 3(c,d). However, the magnitudes are captured. Reconstructions in the spanwise offset plane (Fig. 3(e,f)) again do
not match the DNS data but the magnitudes are within range.
A comparison of Cases 1 and 2 shows that reconstruction
error for the trained velocity components increases when using data from two orthogonal planes. However, errors for thewatched variables and statistical quantities are reduced significantly in comparison to those obtained using data from a single
plane. This suggests that using simultaneous measurements inthe x − y and y − z planes may help in estimating unobserved(untrained) variables and potentially reconstructing volumetricflow fields from limited planar data.
Next, we consider the influence of the hyper-parameter(λ ) used in Eq. 3 on errors for the velocity field and statisticalquantities. Table 2 shows that, when compared to Case 2 (λ =0.5), reconstruction errors for the velocity field for Case 4 (λ =0.9) are slightly better after optimization. However, errors forthe statistical quantities are slightly worse. A similar trend
can be seen for both the trained and watched variables. Thisis expected from Eq. 3. A higher value for λ places greater
emphasis on the loss from the flow field, which improves thevelocity reconstructions but compromises the statistics.
Finally, Case 5 considers the effect of using more training
snapshots. We still use 2 snapshots for training the FROLS re-4

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

30

10
FROLS
Opt. (Case 2)
Opt. (Case 3)
DNS

20

30

2

8
6

1.5
20

0

0.5

-2
0

-4
1

4

10

-6

10
0.5

2
0
0

500

0
1000

0

2.5

80

0

500

-0.5

-8

0
1000

-10

15

10

6

5

4

0

2

-5

0

0

500

-1
1000

100
2
80

60
10

1.5

60

40
1

40

0.5

20
0
0

500

0
1000

5
20
0
0

500

0
1000

-10
0

500

-2
1000

Figure 2. Reconstructed statistics from FROLS for Case 2 and constrained optimization for Cases 2 and 3. The blue scale on the left
corresponds to the FROLS reconstructions with the orange scale on the right corresponds to the optimization results and DNS profiles.
Note the substantial difference in magnitude from the FROLS profiles to the optimization and DNS profiles. The statistics from eddy
viscosity model are smoother and more closely approximate the DNS profiles.

Figure 3. Reconstructed velocity and pressure fields after optimization for Case 2 are compared with DNS data. Subplots (a)-(b)show reconstructions for the observed (trained) components of velocity in the x − y plane. Subplots (c)-(d) show reconstruction for theunobserved (watched) velocity and pressure fields. Subplots (e)-(f) show reconstructions at a different x − y plane (∆z/h = 0.125) for
the trained components (u, v)

5

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

constructions. However, after the set of 200 resolvent modes
is identified, the data sequence is augmented with 6 additional snapshots obtained at the same spatio-temporal resolution. The coefficients are then optimized to minimize the loss
from these 8 snapshots and the statistics. Figure 1 shows thatthe error after optimization is significantly increased for thex − y plane when reconstruction is carried out from 8 snapshots. For the orthogonal (y − z) plane, reconstruction error
only increases slightly (see the change in dotted vs. dashedlines). Note that the error shown in Table 2 for Case 5 considers all 8 snapshots in the data sequence. However, the timeseries in Fig. 1 only shows the reconstruction between the first2 snapshots.
In general, errors for reconstructions from the x − y plane
are lower in comparison to errors for the y − z plane. Thisis due to the fact that turbulent channel flow is advectiondominated. Since the flow advects in the x−direction, having
training snapshots in the x − y plane helps identify appropriate
traveling wave resolvent modes. When interpolated in time using Eq. 1, these resolvent modes propagate downstream withthe phase speed c = ω/kx . While the snapshots in the y − zplane provide some insight into spanwise structure, they may
not contain the information necessary to characterize oblique
wave motion. This indicates that measurements made in thehorizontal x − z plane may yield improved out-of-plane reconstructions. Note that reconstruction errors generally increase
with increasing input data (see Cases 1, 2, and 5). This may be
attributed to the fact that all reconstructions are carried out using just 200 resolvent modes. A larger set of resolvent modes
may be necessary to accurately represent the input data frommultiple planes and snapshots.

The authors thank Dr. Ketan Savla for fruitful discussions. VKC is grateful for financial support from the graduate school at University of Southern California through the
Provost fellowship.

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CONCLUSIONS
The results presented in this paper show that projection
onto resolvent modes can yield useful reconstructions for wallbounded turbulent flows when very limited measurements areavailable. The FROLS algorithm efficiently identifies resolvent modes that best represent the data from a large database.
However, additional constraints are needed to ensure that thereconstructions are statistically sound. Errors from FROLS
reconstructions are quite high for unobserved (watched) variables and statistical quantities when 2D-2C data from a single
x − y plane are used as input data. After optimization to minimize a composite loss function that accounts for the input data
as well as second-order statistics, these errors decrease significantly. The use of simultaneous measurements in x − y andy − z planes also reduces this error. The use of an eddy viscosity resolvent model leads to an increase in reconstruction error
for the input velocity snapshots but greatly improves the statistics. In general, reconstruction error for unobserved quantities (e.g., out of plane velocity, pressure) reduces as statistical constraints are introduced or simultaneous measurementsfrom multiple orthogonal planes are used. Future work willconsider reconstruction from input data consistent with stereoscopic and tomographic PIV (i.e., 2D-3C and 3D-3C) as wellas wall-based measurements.

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