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

UNSTEADY TWO-DIMENSIONAL FLOW RECONSTRUCTION AND
FORCE COEFFICIENT ESTIMATION AROUND ARBITRARY SHAPES VIACONFORMAL MAPPING AIDED DEEP NEURAL NETWORKS

Ali Girayhan Özbay
Turbulence Simulation Group
Department of Aeronautics
Imperial College London
London SW7 2AZ, UK
aligirayhan.ozbay14@imperial.ac.uk

Sylvain Laizet
Turbulence Simulation Group
Department of Aeronautics
Imperial College London
London SW7 2AZ, UK
s.laizet@imperial.ac.uk

ABSTRACTIn many practical fluid dynamics experiments, measuring
variables such as velocity and pressure is possible only at alimited number of sensor locations. However, knowledge ofthe full fields is necessary to understand the dynamics of manyflows. Deep learning reconstruction of full flow fields fromsparse measurements as a way of overcoming this limitation
has recently garnered significant research interest, referred toas the flow reconstruction (FR) task.
We extend existing FR models by enabling such models
to make predictions on flows around arbitrary 2D geometries
without the need for re-training. This geometry flexibility isachieved through an innovative mapping approach, whereby
multiple fluid domains are mapped to an annulus.
Using this mapping approach, we explore the performance of a novel FR model trained on 64 geometries andtested on a further 16 different geometries. We demonstratethat the model trained using the mapping approach reconstructs the flow fields well even on geometries not present in
the training data.

Figure 1: A random geometry (left) and its preimage inthe w domain (right). Blue and green contours depict thenorm and argument in the w domain, respectively.

only one work in previous literature investigates this avenue;
Chen et al. (2021) used graph convolutional neural networks(GCNNs) to reconstruct laminar, steady pressure and velocity
fields around random geometries at a low Reynolds number(Re = 10). An alternate to GCNNs is the usage of a mapping,ensuring all geometries are mapped to a single shape. In 2D,
this can be achieved through a Schwarz-Christoffel (S-C) conformal mapping; each k-connected region can be mapped to adisc with k holes, as explored by Crowdy (2020).
In this work, we explore doubly connected S-C mappingsto generate datasets for training of FR models on large numbers of geometries, targeting reconstruction in space and time.
Figure 1 depicts an example mapping for a geometry used inthis study. Using this approach, we extend the capabilities
demonstrated by Chen et al. (2021), investigating unsteady reconstruction at a much higher Reynolds number Re = 300.
As a further remark, for readers who are interested in the
mapping strategy, our previous paper on the subject (Özbay &Laizet (2022)) covers the subject in greater detail. The present
work builds upon the results in the results in the previous paper, extending them by adding an investigation of lift and dragprediction.

INTRODUCTION
Flow reconstruction (FR) involves the prediction of dense
fields such as velocity based on sparse measurements. Though
the usage of various statistical methods for this purpose has
a long history in works such as Bonnet et al. (1994); Callaham et al. (2019). Recent advancements in deep learning (DL)rekindled an interest in flow reconstruction as seen in recent
works such as Erichson et al. (2020); Dubois et al. (2022).
One drawback even in the recent DL based approaches isthe necessity to train on the same geometry on which inference
is to be performed. This inhibits the usefulness of FR models
in practical scenarios, for instance in physical wind tunnel testing driven shape optimization. Overcoming this requires FR
models that are capable of handling arbitrary geometries. Augmenting the training dataset with different geometries is not asolution for existing models such as the Shallow Decoder (SD)by Erichson et al. (2020), as our experimentation revealed thattraining the SD on a dataset with different geometries leads
to a model that erroneously predicts fields that are averaged
across all geometries in the dataset.
Sidestepping this issue is possible by representing theflow field in a way that removes the necessity to predict thephysical geometry of the object in the fluid flow. However,

DATA AND EXPERIMENTAL SETUP
Our dataset consists of 80 geometries Gi , i ∈ [0, 79], randomly generated using Bezier curves using the code provided
by Viquerat et al. (2021). The control points of the Bezier
curves were chosen randomly in a square domain with charac-

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
temporal flow reconstruction from sparse sensors, when usedin conjunction with the spatial model.
All models were implemented using Tensorflow (Abadiet al. (2016)). Training was conducted using the Adam optimizer by Kingma & Ba (2017) with an initial learning rate of
10−3 . The learning rate was dynamically tapered as validation
loss plateaued. Number of training epochs was determined using the early stopping approach based on validation loss levels.

teristic length Lm . Each geometry was placed in the center of
a 40Lm /3 × 40Lm /3 square domain, on the edges of which a
uniform (u, v) = (1.0, 0.0) velocity Dirichlet BC was imposed.
The flow around each Gi was computed at Re = uLm /ν =300 with the PyFR solver by Witherden et al. (2014). 600
snapshots between times τ ∗ = 3.333 and τ ∗ = 23.333 wererecorded per geometry, where τ ∗ = uτ/Lm and τ is the physical time.
Subsequently, denoting the fluid domain around each Gi
as Fi , the forward mappings fi and inverse mappings gi between annuli Ai and original domains Fi were computed using
a set of Python bindings written for a modified version of the
DSCPACK code by Hu (1998). Each Ai has an outer radius of
1.0 and a geometry-specific inner radius ri . This process maps
the boundary of the geometry Gi to the inner ring of the annulus, while the outer boundary of the domain Fi is mapped tothe outer ring.
Using w and z to refer to complex coordinates in Ai and
Fi respectively, 64 × 256 grid points, equispaced in the radial
and angular directions respectively, with coordinates wA,i were
mapped to the original domains to obtain zA,i = f (wA,i ). The
velocity fields ut,i , vt,i for snapshots t ∈ [0, 600] of each Gi
were used to obtain vorticity fields ωt,i . Finally, ut,i , vt,i , ωt,i
plus the pressure fields pt,i were interpolated to zA,i to obtainũt,i , ṽt,i , ω̃t,i and p̃t,i , which constitute the target values of our’Annular Sampling’ dataset. This strategy is compared against
a ’Cartesian Sampling’ dataset, composed of ground truth values interpolated to a regular 128 × 128 Cartesian grid within
the domain.
The corresponding inputs are vectors st,i of sparse pressure and velocity measurements. The former are placed on
object surfaces and the latter are arranged in a grid behindthe objects. Based on this general template, two setups withvarying sensor quantities were considered; a large setup with
50 pressure and 25 velocity (50P+25V) sensors, and a small
12P+4V setup.
Two classes of experiments were conducted using thisdataset; spatial multi-geometry flow reconstruction (SMGFR)
whereby the model is expected to reconstruct a target field
˜·t,i given si,t , and spatiotemporal MGFR (STMGFR) which issimilar to SMGFR the target field is k timesteps in the future
relative to the sensor input - e.g. reconstruct ˜·t+k,i given st,i .

RESULTS
Below, we present the results from a series of SMGFRand STMGFR experiments. First, we begin with the investigation of spatial pressure and velocity reconstruction, and subsequently present results for instantaneous drag and lift coefficient estimation using the reconstructed fields. Next, we focus on reconstruction of vorticity both in space and time, andstatistically demonstrate that the prediction of vorticity in our
setting presents a substantially greater challenge than the prediction of velocity and pressure.

Spatial pressure and velocity reconstruction
We start with the investigation of spatial pressure and velocity reconstruction, to directly compare the quality of ourreconstruction methodology with the results by Chen et al.
(2021). For the sake of brevity, only the results using the SDUNet spatial model coupled with the FNO-based denoising autoencoder are presented in this section, using the large sensor
setup. Table 1 shows absolute and percentage validation error levels, averaged over the validation snapshots. Figure 2
depicts the ground truth and predicted fields for a random validation snapshot from the annular sampling dataset, alongside
the corresponding percentage error field. Highest percentage
errors are concentrated near the zero ground truth contours ofeach field, despite the absence of high concentrations of absolute errors near these areas, as the denominator in the percentage error calculation approaches zero, as further discussed inÖzbay & Laizet (2022).

Table 1: Mean absolute error (MAE) and mean abso-lute percentage error (MAPE) levels for pressure andvelocity SMGFR experiments. MAPE figures were filtered to exclude grid points exhibiting percentage errors
above 200% since such high levels of percentage error
are almost exclusively caused by ground truth values approaching zero, as seen in Figure 2.

MODEL ARCHITECTURE AND TRAINING
For spatial reconstruction, the performance of four architectures was evaluated:
1. Shallow Decoder (SD): A 3 layer multilayer perceptron(MLP) with 40 units and ReLU activations in the intermediate layers, as used in Erichson et al. (2020).
2. SD-Large: A larger SD with 4 layers, 2048 units in eachlayer, leaky ReLU activations and batch normalization.
3. SD-UNet: SD model with 512 and 2048 units in the intermediate layers, followed by a reshape operation to a 2D
grid and a U-Net (Ronneberger et al. (2015)) model.
4. SD-FNO: An SD model identical to the one in the SDUNet, followed by four FNO (Li et al. (2020)) layers.
In addition, the training of a further FNO model was explored, using the predictions of the above four spatial models
as inputs and ground truth snapshots k = 0, k = 20 and k = 80
timesteps in the future relative to the inputs as the targets. In
the k = 0 configuration, this FNO model serves as a denoising
autoencoder. The k = 20 and k = 80 configurations explore
time-marching the full fields into the future, enabling spatio-

Annular sampling

Cartesian sampling

MAE

MAPE

MAE

MAPE

p

0.01182.43%

0.01333.32%

u0.02648.26%

0.0332

11.56%

v

0.01229.40%

0.016415.61%The error levels clearly display that annular sampling iscapable of substantially improving the quality of reconstructions, bringing error levels below 3% and 10% for pressureand velocity respectively, and reducing absolute error levels
by up to 25% in the case of v-velocity.

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

1
0
1
2
0
2
1
0
1
0

20

40

60

80 100Figure 2: Ground truth (left), predicted (middle) and percentage error (right) plots for pressure (top), u-velocity (middle)

and v-velocity (bottom) reconstruction for a validation snapshot.
Lift and drag coefficient estimation Reconstructions of pressure and velocity, when used together, canbe utilized to estimate the instantaneous lift and drag coefficients. To achieve this, we adopt a strategy on the computationof body forces. Table 2 compares the errors in lift and dragcoefficient derived from the Annular and Cartesian sampling
methods relative to those obtained from ground truth data. Figure 3 depicts the time evolution of predicted and ground truth
lift and drag coefficients for a validation geometry when using
annulus sampling.

the need for interpolation. This enables the estimation of liftand drag with an accuracy level comparable to the results by
Chen et al. (2021), who reported percentage errors on the orderof 3-4%. The similar levels of performance with our method
is despite the order-of-magnitude larger Reynolds number andthe presence of unsteady flow in the present study.

Spatial vorticity reconstruction
To push the boundaries in multi-geometry flow reconstruction, in addition to the pressure and velocity results inthe previous chapter, we detail the results of vorticity-based
SMGFR tasks. Reconstruction of vorticity from pressure andvelocity sensors presents a substantially greater challenge, as
the reconstruction relationship relating pressure and velocity
sensors to vorticity is more difficult to model, as explored further on. This creates larger differences between the best andworst performing configurations, and paints a clearer picture
regarding the effect of the three variables controlled for in
our study: sensor setups, model architectures and samplingstrategies. Table 3 summarizes the percentage errors fromthe SMGFR investigations with all configurations. In this table, in addition to the standard mean absolute percentage error(MAPE), we further provide the high vorticity MAPE (HVM)
metric, which is MAPE filtered such that grid points whichcorrespond to ground truth vorticity values with magnitudes
below 1% of the maximum vorticity in each snapshot are removed. This metric better represents the models’ performance

Table 2: Mean absolute error (MAE) and mean absolute

percentage error (MAPE) levels for pressure and velocity SMGFR experiments
Annular sampling

Cartesian sampling

MAE

MAPE

MAE

MAPE

CL

0.0253

4.97%

0.096628.18%

CD

0.02148.57%

0.068429.67%

Unlike the Cartesian sampling method, the annular sampling method allows for the computation of the forces without

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

1.2
1.0

based SMGFR experiments.
Small sensor setup

Sampling

Ann.

Cart.

Ann.

Cart.

0.6

MAPE

44.29%

59.88%

47.35%

57.40%

HVM

34.28%

46.14%

34.72%

50.89%

MAPE

43.80%

57.52%

47.20%

59.77%

HVM

31.85%

53.36%

34.06%

48.59%

MAPE

39.92%

47.64%

45.03%

51.56%

HVM

31.37%

39.88%

33.17%

44.47%

MAPE

40.83%

46.56%

44.04%

51.81%

HVM

31.78%

39.34%

33.77%

45.81%

10

20
Time (s)SD-FNO SD-UNet SD-Large

0.4
30

0.85
0.80
Cd

Large sensor setup

0.8

SD

Cl

Table 3: Percentage error metrics from the vorticity-

Ground truth
Predicted

0.75
0.70
0.65

Ground truth
Predicted
1020Time (s)sensor readings. Table 4 summarizes the percentage error metrics obtained from this setup with varying values of the temporal gap k, while Figure 5 depicts the results obtained from the
STMGFR procedure for the same snapshot in Figure 2.
The results clearly display that our setup is capable ofpredicting vorticity snapshots in the future from current sensor measurements with minimal penalties incurred in terms ofaccuracy.

30

Figure 3: Time evolution of the predicted and groundtruth lift (top) and drag (bottom) coefficients for a validation geometry.

Table 4: Percentage error metrics for the vorticity-based

STMGFR task with varying temporal gaps.
in regions of interest within the flow, where the vortex shedding dynamics are observed.
The results show a clear trend regarding the relative importance of the three experimental variables; the samplingstrategy permits the improvements in performance, providing
an accuracy benefit exceeding 15 percentage points in the caseof the SD model with the large sensor setup. Relative to this,the improvements afforded by the choice of model architecture and sensor setup are modest. The best overall-performing
configuration is the SD-UNet coupled with annular sampling
and the large sensor setup. Figure 4 depicts vorticity predictions with this configuration; as also covered in Özbay &Laizet (2022), our method correctly replicates flow features ofinterest such as the intensity and placement of shed vortices
and concentrations of vorticity in boundary layers, while concentrations of high percentage error are once again observed
mostly in areas where the ground truth values approach zero.

k (∆τ ∗ )0 (0.0)20 (0.667)

80 (2.667)

MAPE

28.89%

31.02%

31.88%

HVM

17.86%

17.86%

21.97%

Difficulties in predicting vorticity
Our method permits predictions of pressure and velocity
with error levels largely in line with previous literature despite the greater challenge of tackling unsteady flows at muchhigher Reynolds numbers. However, much higher percentage
error levels are encountered in the previously unexplored taskof vorticity reconstruction around arbitrary geometries. An intuitive explanation for this is the fact that we use pressure andvelocity sensors to reconstruct a third, different target field.
To test this hypothesis, we constructed two metrics tocompare the difficulties of reconstructing pressure, u-velocity,
v-velocity and vorticity fields given the sensor inputs fromthe large sensor setup. The first, D, is based on the Spearman rank correlation coefficient (SRCC), which quantifies themonotonicity of the relationship between two quantities. The
second, M , is based on mutual information (MI), which compares the similarity (relative entropy) of the distributions of
two random variables. The scores are computed by constructing matrices containing the SRCC or MI between each sensor

Spatio-temporal vorticity reconstruction
Finally, we extend the results from vorticity-based
SMGFR efforts to STMGFR (reconstruction in both time andspace). The outputs from a spatial reconstruction model, chosen as the SD-UNet trained on the annular dataset for displaying the highest performance in SMGFR, are provided as inputs to an FNO model which is optimized to reconstruct the
ground truth snapshots a fixed number of timesteps k in the future relative to its input. This allows the combined SD-UNet+ FNO system to predict snapshots in the future, given current

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

5

0

5

0

20

40

60

80 100Figure 4: Ground truth (left), predicted (middle) and percentage error (right) plots for vorticity reconstructions of threevalidation snapshots.

input and each grid point in the target fields, and then subsequently computing the Frobenius norm of this matrix. Table 5shows both difficulty metrics for the four target fields investigated in this study.

models on this task is poor, and developed an approach forsampling the flow fields in a novel manner using SchwarzChristoffel mappings. Comparing the performance of models trained using this sampling method versus the traditional
Cartesian approach, we found that our method reduces validation errors substantially, enabling the reconstruction of pressure, velocity and vorticity with percentage errors below 3%,
10% and 30% respectively, as detailed in Özbay & Laizet(2022).

Table 5: Difficulty metrics D and M for the four target
fields investigated. Higher is easier.

Variable

p

u

v

ω

D

306.22298.99301.85242.82

M

302.72296.37269.81207.75Subsequently, we used the mapping strategy to accurately
predict instantaneous lift and drag coefficients, achieving percentage errors below 5% and 10% for lift and drag estimation respectively, compared to errors above 25% when using
Cartesian sampling. Finally, we extended the spatial-only reconstruction task to obtain flow fields in the future relative tothe sensor measurements, reconstructing vorticity fields in the
future with minimal error increases compared to contemporaneous predictions.

The difficulty metrics clearly support the hypothesis thata more difficult to learn relationship exists between the sensor
inputs and vorticity compared to pressure and velocity, explaning the greater error encountered in vorticity experiments.

In the future, we aim to extend this work by exploring
reconstruction in 3D with varying Reynolds numbers.

CONCLUSIONREFERENCES

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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the SD-UNet, while the bottom rows display the predictions from the FNO model for the vorticity field at timestep t + k.
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