# Fair Comparison: Quantifying Variance in Results
for Fine-grained Visual Categorization

Matthew Gwilliam1,2 Adam Teuscher1 Connor Anderson1 Ryan Farrell1
1Brigham Young University, 2University of Maryland
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For the task of image classification, researchers work arduously to develop
the next state-of-the-art (SOTA) model, each bench-marking their own
performance against that of their predecessors and of their peers.
Unfortunately, the metric used most frequently to describe a model’s
performance, average categorization accuracy, is often used in isolation. As
the number of classes increases, such as in fine-grained visual categorization
(FGVC), the amount of information conveyed by average accuracy alone dwindles.
While its most glaring weakness is its failure to describe the model’s
performance on a class-by-class basis, average accuracy also fails to describe
how performance may vary from one trained model of the same architecture, on
the same dataset, to another (both averaged across all categories and at the
per-class level). We first demonstrate the magnitude of these variations
across models and across class distributions based on attributes of the data,
comparing results on different visual domains and different per-class image
distributions, including long-tailed distributions and few-shot subsets. We
then analyze the impact various FGVC methods have on overall and per-class
variance. From this analysis, we both highlight the importance of reporting
and comparing methods based on information beyond overall accuracy, as well as
point out techniques that mitigate variance in FGVC results.

##  1 Introduction

Figure 1: Results from 10 trained networks in per-class accuracy plot, sorted
according to the average per-class accuracies. Deviation is standard
deviation, and the per-class deviation was averaged across all the classes.

Convolutional neural networks and their variants have taken over the image
classification task, both at the general level [34] and for fine-grained
domains [3, 49]. This has advanced the state-of-the-art in fine-grained
recognition [4, 9, 13, 23, 25, 45, 50] to previously inconceivable levels, but
with great research comes a need for great reporting, and current standards
are insufficient. If a given model yielded the same results every time it was
trained, and these results were the same for every class, then overall average
accuracy would be not only sufficient, but also comprehensive, and additional
metrics would be mostly trivial. Unfortunately, this is not the case, and as
FGVC transitions from theory to real-world application, it is critical that
both researchers and practitioners have enough information at their disposal
to fairly compare different models.

One of the primary limitations in comparing FGVC methods is that models with
the exact same dataset, training procedure, and architecture (identically-
trained models) have non-trivial differences (variance) in performance.
Variance is an important concern about a model’s performance that is distinct
from accuracy – the level of variance can vary independent of overall
accuracy. We investigate a few main types, two of which are represented in
Figure 1. The first, which we measure as overall deviation in Section 3 and
Section 4, is the wide distribution of per-class accuracies: for some classes,
the trained model is able to classify nearly all the images correctly, while
for a few classes it accurately classifies less than 20%. The second type of
variance, which we measure as per-class deviation in Section 3 and Section 4
is the distribution of accuracies for a given class between trained instances
of a model. The third type, which is not represented in the figure, is
variance in overall accuracy. Ironically, the variance between results from
different trained instances is due primarily to what they have in common:
architecture, training procedure, and dataset.

The architecture is what makes variance possible. Deep neural networks have a
series of layers, with millions of parameters in total, typically initialized
at random. Since there are so many learned weights, the possible solution
space is enormous, and due to the way these weights are learned and the way
they work together, even a reduced space consisting only of “good” solutions
would still be incredibly large. With the introduction of any randomness in
the training pipeline (explained in the following paragraph), the learned
weights will differ even when not randomly initialized (such as with
pretrained networks) and trained models become like snowflakes, in that it is
highly unlikely any two would ever have the exact same weights. Two trained
models with non-identical weights would logically make predictions that, while
largely similar, have some differences (variance).

In terms of training procedure, while every new state-of-the-art (SOTA)
approach to FGVC is different, many share some similarities in their
pipelines. The order of training images presented during training is often
shuffled, with transformations applied that randomly flip, crop, and modify
the color of the images in each batch. While the addition of randomness to the
pipeline through these methods helps prevent overfitting, it allows for
variance between instances of a model that started off with identical weights.

Figure 2: Blue Jays, from Flickr. Differences in pose make it difficult to
discern (without prior domain knowledge) whether these are all images of the
same species of bird.

The dataset is a major factor in determining the level of variance observed in
the evaluation results, as we prove in Section 3. It is important to remember
that the training images in a dataset are a random sample taken from the space
spanned by the selected visual domain, and a model’s performance can be highly
dependent on how representative this random sample is of the full visual
domain. If there are only a few images for a given class in the training set,
it is unlikely that the spectrum of possible appearances for that class is
adequately represented. Figure 2 illustrates how this can be difficult – with
only 5 images, it would be impossible to represent each of the 6 poses shown,
and even with more than 20 images it would logically be challenging to achieve
adequate, well-proportioned representation of the Blue Jay. With a limited
training set, performance on the test data will likely be both (i) worse for
the underrepresented classes and (ii) less consistent overall. Additionally,
some classes in a dataset are more challenging to classify than others. For
example, with the 2011 Caltech-USCD Birds (CUB) dataset [44], many models
report very high accuracy for highly-distinctive birds like the green
violetear but lower accuracy for birds that are hard to differentiate, such as
the common and elegant terns, shown in Figure 3.

Variance can be seriously problematic in a few different ways. The wideness of
the distribution of per-class accuracy scores presents challenges for a
multitude of practical use cases. Consider the distribution in Figure 1 as an
example, and note while some per-class accuracies are near 100%, other are
close to 0%. An x-ray classifier with such a distribution would often fail to
classify certain types of breaks and fractures. Such a bird classifier may
never be sufficiently accurate for the endangered species in the domain.
Ensemble methods can turn some of the per-class variance into a strength, but
due to resource-consumption, such an approach may be impractical for many
applications, making variance in performance between trained instances of the
same model an important consideration. For some use cases, when two models
have similar average accuracies, it may be better to use the model with less
variance between trained instances. However, since data that describes the
variance is often not reported, it is difficult for FGVC practitioners to make
such decisions in an informed manner.

Variance also makes fair reporting and comparison of results quite
challenging. Consider two fine-grained researchers, A and B, who develop two
different methods for the classification task. Researcher A, in order to make
sure their results are reproducible, writes a paper where they report an
overall accuracy that comes from the average of 10 trials. Researcher B runs
10 trials as well, and unfortunately, the mean accuracy of their results is
lower than that of Researcher A. However, 2 of their trials achieve higher
accuracies than Researcher A’s reported accuracy. Perhaps unaware that
Resarcher A was reporting an average over several trials, Researcher B reports
the maximum from the 10 trials as their overall accuracy. Other readers, not
knowing which methodology was used to get the overall accuracy figure (since
it is often not reported or hard to find), assume that in terms of overall
accuracy the method of Researcher B outperforms the method of Researcher A,
when in reality, Researcher A’s method is better on average. This and other
potential issues can be observed in the experiments we run using real methods
and real data in Section 3.

Therefore, variance presents a major stumbling block for the fair reporting,
comparison, and practical application of different FGVC models. In this paper,
we draw attention to the variations in performance based on many different
factors. We break down and reveal trends describing how variations in overall
accuracy, per-class accuracy, and range of per-class accuracies can be
influenced by visual domain (dataset), shape of a long-tailed distribution,
optimizer, model architecture, loss function, and number of training images
per class. We both demonstrate the prevalence of variance and uncover insights
on what drives it.

Figure 3: From left to right: green violetear, common tern and elegant tern.
In CUB, the violetear is visually distinct, while the common and elegant terns
are easily confused.

##  2 Background and Related Work

One recent and striking example of the variance in model performance caused by
random initialization is the lottery ticket hypothesis [11]. The hypothesis
states that for a dense, randomly initialized network there exists a sub-
network with fewer parameters that can attain the same or better accuracy in
the same or less amount of time when trained independently on the same task.
These sub-networks are dubbed “winning tickets”, because it is the unique
combination of initial weights and sub-network structure that allow them to
learn effectively. The authors observed that when the sub-networks were re-
initialized with different random parameters, they failed to achieve the same
level of performance. This precise dependence on the random initialization is
evidence of the large variation that can occur when training neural models.

As discussed throughout Section 1, variance is an important subject in FGVC,
where classification algorithms seek to identify the precise category/class
within a particular domain (e.g. genus/species of a bird or make/model/year of
a car). FGVC datasets such as those mentioned earlier frequently have limited
numbers of images per category. Each image therefore contributes
proportionally more toward overall accuracies (and variations therein). Near
state-of-the-art approaches for FGVC include methods that use pooling [6, 24],
attention mechanisms [12, 50] or regularization [9].

Accuracy is the standard metric for comparing results across fine-grained
classification models. However, not all deep learning disciplines are able to
use such a simple metric. Machine translation (MT), a subfield of natural
language processing, is one such discipline. Similar to FGVC, MT has seen
exciting advances in the past decade with the development of Neural MT methods
[2, 36, 43]. Unlike FGVC, however, translation is an inherently subjective
task, where there is no single right answer, and preferences vary between
different human evaluators [40].

As a result, MT practitioners have had to be resourceful in crafting their
evaluation metrics. The most popular automatic metric is BLEU score [31],
which can be calculated using frameworks such as SacreBLEU [33].
Unfortunately, simple comparison of BLEU scores is inferior to human
evaluation. To try to bridge the gap between human and machine evaluation of
machine translation, recent work has focused on producing additional metrics
that measure things like linguistic richness [42] or level of formality [29]
in translated text. We take a similarly creative approach in our proposed
evaluation of FGVC systems.

The most significant resource that we are aware of that addresses the fair
reporting and comparison gap in FGVC is paperswithcode.com [1], which
facilitates the comparison of common metrics like top 1 and top 5 accuracy,
with links to both the accompanying paper and code. Our work also bears some
similarity to general machine learning benchmark suites, such as the UCI
Machine Learning Repository [8, 26] and the Penn machine learning benchmark
[30]. Instead of focusing merely on accuracy, we propose additional metrics
and ways to evaluate the performance of classifiers.

##  3 Variance by Data

Figure 4: Per-class accuracy for each dataset, aggregated across 10 training
runs. The standard deviation (gray band) and minimum/maximum performance (red-
colored curves) for each class are shown along with the mean (black curve).
The overall mean and standard deviation across classes are also shown (dotted
lines). Bottom right: Overall accuracy for each trial.

The experiments in this section provide evidence of the different types of
variance that we discussed in Section 1. The analysis here examines how
variations in the image data lead to variance in the results of trained
ResNet-50 models. In Section 3.1, we investigate variance that occurs in the
results of classifiers trained and evaluated on different datasets. In Section
3.2, we demonstrate how the variance in long-tailed distributions shows
similar trends to variance across datasets. In Section 3.3, we conduct few-
shot experiments to investigate trends unique to higher-variance scenarios.
These experiments provide many examples of both intuitive and counter-
intuitive trends and types of variance that occur in FGVC results. Together,
they make a compelling case for the need to consider variance metrics in
addition to accuracy metrics, a need we further emphasize in Section 4.

Dataset | Classes | # Images | Accuracy
---|---|---|---
Aircraft | 100 | 10000 | 84.61
Birds | 200 | 11788 | 81.46
Cars | 196 | 16185 | 89.86
Dogs | 120 | 28580 | 82.22
NABirds | 555 | 48000 | 78.58
Table 1: Baseline performance (averaged over 10 trials) of our ResNet-50s on
different FGVC datasets. This is not intended to be competitive with state-of-
the-art results, but instead gives benchmarks for our models. From this table
onward, “Birds” refers to CUB.

foo

Training Procedure. Unless otherwise specified, we use the same network
architecture for each of these experiments – a ResNet-50 pre-trained on
ImageNet [48]. We use the implementation available in PyTorch [32]. We
optimize cross-entropy loss using the Adam optimizer [20], starting with a
learning rate of 0.0001. We train for 50 epochs, decreasing the learning rate
by a factor of 10 at 15, 25, and 35 epochs. During training, we apply random
horizontal flips and take random re-sized crops (224 x 224). During
evaluation, our networks take center crops. To place our method in context,
the accuracy scores of this architecture for different popular fine-grained
datasets (FGVC datasets) are provided in Table 1.

Terminology. We use the term per-class accuracy to refer to the classification
accuracy of the subset of images belonging to one particular class rather than
the whole dataset. For CUB, which has 200 classes, this means that there are
200 corresponding per-class accuracies: 1 for each class. Overall deviation
refers to the standard deviation of all per-class accuracy scores for a
particular experiment. For example, if 10 models are trained and evaluated on
CUB, which has 200 classes, then the overall deviation is the standard
deviation of the 2000 resulting per-class accuracies Per-class deviation is
the standard deviation of per-class accuracy across multiple training runs,
averaged over all classes. For the experiments here, where each data point is
the result of 10 training runs, the per-class deviation is computed by taking
the standard deviation of the 10 accuracy scores for a given class, for all
classes, and averaging those standard deviations. Whereas overall standard
deviation describes the range of accuracy values and their relationship to the
mean, per-class standard deviation describes the differences in performance
between the models for each individual class.

###  3.1 Variance by Visual Domain

Figure 5: Overall standard deviations, average per-class standard deviations,
and overall average accuracies for different datasets. Datasets are sorted by
overall accuracy.

We train and test 10 networks on each of the FGVC Aircraft [27], Caltech-UCSD
Birds, Stanford Cars [21], Stanford Dogs [19], and North American Birds [41]
datasets, which we refer to as Aircraft, Birds, Cars, Dogs, and NABirds,
respectively. More information on the datasets can be found in Table 1. The
results are summarized in Figure 4 and Figure 5.

Clear patterns emerge in Figure 4. For each dataset, the model tends to do
fairly well, correctly labeling more than 75% of all test images. Notice that
the distribution of per-class accuracies vary from 50% to 100%, with most
accuracies closer to the maximum than the minimum. Critically, most per-class
accuracies are above the mean. While it may seem beneficial that the average
class is labeled more accurately than the overall accuracy conveys, this is
actually a major issue. What this means, and Figure 4 represents this as well,
is that the model performs quite poorly for a non-trivial subset of classes.
Taking Birds as an example, for 25 species, the model fails to accurately
label more than 60% of the images.

This demonstrates the importance of using metrics besides accuracy. While
top-1 accuracy conveys how the model performs on average, it fails to
communicate the majority of the information shown in Figure 4. Overall
accuracy communicates little about the minimum and maximum accuracy, and
nothing about the distribution of accuracies between those two values. Overall
deviation, on the other hand, expresses much of that information with a single
number. While it does not communicate in concrete terms the maximum or
minimum, when combined with overall accuracy it can be used to infer much
about the shape of the per-class accuracy curve, including how far from the
mean individual per-class accuracies would be. Indeed, as expected from the
graphs in Figure 4, overall deviations are fairly high for each dataset,
ranging from just under 10% for Cars to nearly 20% for NABirds.

Another finding, from Figure 5, is that there seems to be a correlation
between overall standard deviation and per-class standard deviation, meaning
that when the per-class accuracies are distributed closer to the mean, they
also tend to vary less between trained instances of a given model.
Additionally, accuracy seems to have an inverse correlation with standard
deviation, where the datasets with better overall accuracy also have 1)
tighter distributions and 2) less disagreement between models.

Some aspects of the variance seem to contradict intuition. For example, as
Figure 4 shows, the accuracy for any given class can vary significantly across
different training runs, despite the fact that they all start from the same
pre-trained weights. With NABirds, for instance, there were classes for which
one ResNet-50 would label images 35% more accurately than another ResNet-50.
Another surprising finding is that while it has neither the highest nor lowest
number of classes, and also has neither the highest nor lowest number of
images per class, the ResNet-50s we trained achieved their highest accuracies
and lowest deviations (both overall and per-class) on Cars, which suggests the
dataset is inherently easier than some of the others. Dogs, for example, has
more images representing each class and approximately 60% as many categories,
but the yet our ResNet-50s achieve both less consistent and less accurate
results on Dogs than on Cars.

###  3.2 Variance with Long-tailed Distribution

Figure 6: Standard deviations and accuracy results for different long-tailed
subsets of the Birds dataset. The subsets take a linearly decreasing number of
images from each class, from the high number (left of dash) to the low number
(right of dash). Total number of images is shown beneath the dash. Figure 7:
Left. Follows the convention described in Figure 8. Accuracy for few-shot
subsets of the Birds dataset. Right. Standard deviation and accuracy for the
same few-shot subsets.

Long-tailed distributions were simulated by taking subsets of the Birds
dataset, where for a given distribution, the subset was initialized randomly
and then kept the same for all trials. These subsets involved a “high” number,
which represents the maximum number of images per class for the subset, and a
“low” number, which represents the minimum number of images per class. Classes
were then randomly assigned numbers of images so that all numbers from low to
high number were used evenly. Additionally, numbers were adjusted when
possible so that the total number of images would be multiples of 100. Thus,
the 20-5 (high number is 20, low number is 5) distribution had the lowest
number of total images, at 2500, and 30-15 had the most, at 4500.

From Figure 6, note that while accuracy tends to improve as the number of
images increases, that result is strongly dependent on the distribution of
images, to the point where the 20-15 distribution is almost 4% better than the
30-5 distribution in terms of average accuracy, even though they both have
3500 images. The deviations follow the same trends, but in reverse- they tend
to decrease as accuracy increases. Figure 6 thus reinforces a key trend from
Figure 5 – the correlation between accuracy, overall deviation, and per-class
deviation. There is only one minor deviation from this tendency in the whole
chart, when per-class deviation increases slightly from the 30-5 subset to the
20-10 subset while accuracy increases and overall standard deviation
decreases.

Interestingly, both the per-class and overall deviations in Figure 6 decrease
consistently not as the overall number of images increases, but more so as the
minimum number increases. This is likely due to the fact that underrepresented
classes tend to be classified both less accurately (affecting overall
deviation) and less consistently between training runs (affecting per-class
deviation).

###  3.3 Variance In Few-Shot

10 subsets of Birds were taken, with the number of training images limited
from 1 up to 10 images per class, inclusive. We trained 10 models on each
subset and averaged the results. From the graph on the left in Figure 7, we
can see that the variance in overall results changes dramatically from 1 to 10
images per class, with the distributions getting much tighter as the number of
training images increases. Note that for these few shot trials, overall
accuracy results can vary by multiple percentage points between trials.

The graph on the right of Figure 6 is even more interesting, as it contradicts
some of the trends observed in Section 3.1 and in Section 3.2. We see that
overall and per-class deviations are both very high, as could be expected in a
lower accuracy setting. However, variance does not decrease monotonically as
accuracy increases. In fact, The variance actually increases with accuracy
until the jump from 4 to 5 images per class. This calls into question whether
variance is as straight-forward as much of this section has suggested it may
be (varying inversely with accuracy), or if it can change independent of
accuracy, and thus can be manipulated directly, similarly to how researchers
currently strive to manipulate (increase) accuracy. Section 4 answers this
question, by making it clear that variance exhibits a significant degree of
independence from accuracy.

##  4 Variance by Method

Whereas Section 3 focuses on variance resulting from different attributes of
the dataset, this section is dedicated to assessing the impact of variance in
various methods for fine-grained visual categorization. In Section 4.1, we
examine popular baseline optimizers and model architectures to highlight the
importance of considering variance in both designing and reporting the results
of FGVC pipelines. In Section 4.2 we analyze the effect different loss
functions have on variance. Finally, in Section 4.3 we present our findings on
variance in current state-of-the-art methods. From our discussions on loss
functions and state-of-the-art algorithms we identify techniques that seem to
affect variance independent of accuracy, and identify necessary future work.

###  4.1 Variance by Optimizer and Architecture

Figure 8: Overall accuracy for the Birds and Cars datasets trained with
different optimizers. Colored lines show the accuracy results from each of the
10 trials. Black lines show the mean across the 10 trials. The shaded region
represents 3 standard deviations above and below this mean.

For optimizers, we compare Adagrad [10], Adadelta [47], Adam (used in the
other experiments), RMSprop [14], and stochastic gradient descent (SGD). For
models, we compare DenseNet-161 [17], ResNet-152 [15], VGG19 [35] with batch
normalization, and Inception v3 [37].

Figure 8 shows the results of the experiments with optimizers. The performance
of Adadelta and Adam is similar. Nevertheless, it is worth noting that
Adadelta has a lower standard deviation and a higher mean for both Birds and
Cars, while Adam has the higher max for Cars. Also worth noting is that Adam’s
maximum accuracy is higher than Adadelta’s average accuracy for both datasets.
This could result in the reporting problem described in Section 1 – a
researcher using Adam could make it appear as though Adam has the “better”
result by reporting only the maximum (on Birds) or the average of the best 5
trials (on Cars) instead of all 10.

Figure 9: Overall model accuracy for different models trained and tested on
the Birds and Cars datasets. Uses format explained in caption for Figure 8.

Figure 9 shows the variance in overall accuracy for different models trained
and evaluated on Birds and Cars. This offers more evidence to support the
importance of reporting an average over more than 5 trials, without cherry-
picking. With DenseNet-161, for example, while its average accuracy for Birds
is higher than the average of the 10 Inception-v3 networks, it is lower than
the overall accuracy from 1 of the Inception-v3 networks. In a similar
fashion, the 5 lowest accuracies for ResNet-152 on Cars overlap with the
highest 5 for Inception-v3.

Surprisingly, the networks with the highest accuracy scores do not always have
the most consistent scores. While the best network by overall accuracy, the
DenseNet-161, has the lowest standard deviation, the worst network, the
VGG-19, seems to be on par with ResNet-152 and Inception-v3 in terms of
consistency, despite having a significantly worse overall accuracy. This
highlights the need for comparing results across multiple runs, and is an
additional consideration to take into account when selecting a network type,
especially when resources are limited and ensemble methods are not feasible.

###  4.2 Variance by Loss Function

Figure 10: Overall model accuracy for models trained with different loss functions and tested on the Birds and Cars datasets. Uses format explained in caption for Figure 8. |  | Deviation
---|---|---
Loss Name | Accuracy | Overall | Per-class
CUB
baseline | 81.46% | 0.1539 | 0.03962
labelsmoothing | 81.79% | 0.1555 | 0.03942
maxentropy | 82.90% | 0.1493 | 0.03556
Cars
baseline | 89.89% | 0.0987 | 0.02632
labelsmoothing | 90.26% | 0.1007 | 0.02500
maxentropy | 90.96% | 0.0978 | 0.02418
Table 2: Presents accuracy and variance results for 10 ResNet-50s each trained
with 3 different loss functions, for CUB and Cars.

Recent work has challenged the notion that one-hot targets with cross-entropy
are the best way to model the objective for FGVC. We specifically compare 2
different approaches to standard one-hot cross-entropy: label-smoothing [28,
38], which treats the target class as most probable while assigning a uniform,
non-zero probability to all other classes, and maximum entropy regularization
[9], where regular cross entropy is computed and the entropy of each
prediction vector is subtracted from it to calculate the overall loss.

The results are somewhat unexpected. While Figure 10 shows that maxentropy
outpeforms labelsmoothing which outperforms baseline crossentropy, as
expected, note that the accuracies for label smoothing are distributed over a
wider range than those of the baseline. In a related vein, Table 2 shows that
overall deviation (which describes the distribution of per-class accuracies
around the mean) is larger for labelsmoothing than the baseline on both
datasets, which is quite different from the trend we observed in Section 3.
The existence of a method with higher accuracy and also higher variance
provides evidence for the idea that variance can be targeted and solved as its
own goal, independent of accuracy. Future work in loss functions could
potentially focus on lowering the variance.

###  4.3 Variance in the State-of-the-Art

Figure 11: Overall model accuracy for different models trained and tested on
CUB. Uses format explained in caption for Figure 8 Figure 12: Accuracy and
deviations for the SOTA models.

For state-of-the-art methods, we consider several different high-performing
approaches: WS-DAN [16], Fast MPN-COV [22], DCL [5], S3N [7], Densenet161
[18], Densenet161 with maximum entropy regularization, EfficientNet [39],
BilinearCNN, and ResNeXt50 [46]. Unlike in previous sections, we only trained
5 of each of these, and we evaluate on CUB only. Additionally, we used larger
images – 448x448.

One of the most surprising aspects of Figure 11 is what it shows about the
deviation in overall accuracy, particularly in how it does not decrease as
accuracy increases. High performing methods such as DCL and WSDAN show large
variations in overall accuracies between the 5 trials. On the other hand, the
lowest performing method, BCNN, and the second highest, S3N, both have very
tight overall accuracy distributions. It is clear that there is something
about the design of the models that lends itself to more consistent
performance, and insofar as it is not always practical to use an ensemble,
there is a real possibility one could get an unlucky draw with models such as
DCL and WSDAN. It is therefore important for researchers to consider the
consistency of their models, and important for the community to be aware of
how methods stack up in that regard.

The deviations in Figure 12 also do not follow the patterns from the previous
sections. The one trend that is mostly preserved is the correlation between
the two deviations, though even that is violated with WSDAN, where overall
deviation decreases while per-class deviation increases. One of the more
interesting violations of the accuracy-deviation correlation is maximum
entropy, which is comparable in variance to enet, which it outperforms by more
than 0.5%. The transition from BCNN to densenet is similar; while accuracy
increases considerably, the deviations also both increase substantially.

Both figures indicate that Bilinear CNN is very unique in terms of its
variance-accuracy relationship. In particular, despite having the lowest
accuracy, it has a low per-class deviation. Additionally, there is little
variance in the overall accuracy results of BCNN across its 10 trials.
Unsurprisingly, it does not have the lowest overall deviation – methods whose
accuracies are closer to 100% will naturally have lower overall deviations.
These results prove that variance is not completely dependent on accuracy, and
some methods, such as BCNN, are more consistent than methods that are
otherwise considered “better,” such as DCL. Studying these differences could
be the key to minimizing variance in the highest performing SOTA methods.

##  5 Conclusion

As incremental improvements to FGVC techniques increase the applicability of
automated image classification to real world problems, it becomes increasingly
important to cultivate a deep understanding of how these methods perform, both
in absolute and relative terms. The types of variance demonstrated in this
paper reveal an unexplored side of the FGVC problem. Specifically, this work
investigates variance by demonstrating its prevalence and highlighting the
need for change. We give examples of how to use deviation to quantify
different types of variance using standard deviation calculations. We
demonstrate when intuitive assumptions about the relationship between accuracy
and variance apply, and when they fail. We promote the adoption of variance as
a problem, and we argue that minimizing variance is a goal worthy of
consideration as researchers continue to develop increasingly accurate FGVC
methods.

## Acknowledgments

This work was supported by the National Science Foundation under Grant No.
IIS1651832. We gratefully acknowledge the support of NVIDIA Corporation for
their donation of multiple GPUs that were used in this research.

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# Transient growth in a flat plate boundary layer under a stream with uniform
shear

Shyam Sunder Gopalakrishnan  Laboratoire de Physique des Lasers, Atomes et
Molécules, CNRS UMR 8523, Université Lille 1 - 59655 Villeneuve d’Ascq Cedex,
France  Faculté des Sciences, Université libre de Bruxelles (ULB), CP. 231,
1050 Brussels, Belgium  Alakesh Chandra Mandal<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Aerospace Engineering, Indian Institute of
Technology, Kanpur 208016, India

###### Abstract

One of the simplest problems involving external vorticity in boundary layer
flows is the flow over a semi-infinite plate under a stream of uniform shear.
We study the transient growth phenomenon in this flow to investigate the role
of freestream shear on energy amplification, and analyse the differences with
the Blasius flow. The initial optimal disturbance which triggers the maximum
growth is found to be streamwise vortices, as in other shear flows. Compared
to the Blasius boundary layer, higher optimum energy and larger spanwise
wavelength of streamwise vortices have been observed. We provide scaling laws
for the maximum optimal amplification, which is found to increase
exponentially with the freestream shear gradient.

††preprint: AIP/123-QED

##  I  Introduction

The effects of freestream vorticity and/or turbulence on a laminar boundary
layer has been of academic interest for many a decade as it finds its
relevance in diverse engineering applications Klebanoff (1971); Kendall
(1998); Westin  _et al._ (1998); Saric, Reed, and Kerschen (2002); Schrader
_et al._ (2010); Manu, Mathew, and Dey (2010). In flows which are subject to a
high freestream disturbance environment, transition processes may bypass the
linear mechanism altogether, and can involve nonlinearity directly (Morkovin,
1969). It is by now well-established that the problem of receptivity to
freestream turbulence is directly coupled to the transient growth of non-
orthogonal modes Klebanoff (1971); Jonáš, Mazur, and Uruba (2000); Saric,
Reed, and Kerschen (2002); Henningson (2006). Here we study the transient
growth in a flat plate boundary layer subject to a uniform freestream shear,
which is one of the simplest problems involving external vorticity Van Dyke
(1969).

Under the linear approximation, Kovasznay Kovasznay (1953) has shown that a
small amplitude unsteady disturbance in the freestream can be decomposed into
an acoustic, vortical, and entropy components. The pressure fluctuation
associated with any acoustic disturbance can cause instability. On the other
hand, vortical disturbances do not create a pressure gradient at the boundary
layer edge and so do not provoke flow instability. However, vortical
disturbances can interact with surface irregularities leading to instability
Crouch (1994). Freestream turbulence and wake in the freestream often contains
a large amount of vortical disturbances, and undergo bypass transition
Ovchinnikov, Piomelli, and Choudhari (2006); Pan  _et al._ (2008). A simple
flow with freestream vorticity is the flow past a semi-infinite plate in a
stream with uniform shear, which arose as a practical problem in supersonic
flow past a blunt body Ferri and Libby (1954); Van Dyke (1969). Several
investigators have studied the role of freestream shear on the boundary layer
Li (1956); Glauert (1957); Murray (1961); Devan (1965); Koch, Ludford, and
Seebass (1971); Dey and Nath (1984); Bera and Dey (2005); Legner (2014);
Balamurugan and Mandal (2017a). The existence of a streamwise pressure
gradient due to interaction between the displacement thickness and external
vorticity was first pointed out by Li Li (1956). This was later confirmed
using the method of matched asymptotic expansions by Murray Murray (1961) with
the interaction between the displacement thickness and external vorticity
stemming from the second-order boundary layer effect. It was further extended
by Toomre & Rott Toomre and Rott (1964) wherein it was shown that the
interaction pressure gradients were correctly given only while the assumption
of an infinite, uniform shear flow is valid, and is strongly influenced by the
boundedness of the external shear. A detailed review on this flow along with
other relevant references can be found in this excellent article Van Dyke
(1969).

In the aforementioned boundary layer flows subject to freestream vortical
disturbances, transition scenarios occur on a short time scale, and may bypass
the linear mechanism Tumin and Reshotko (2001); Fransson  _et al._ (2004);
Mans, De Lange, and Van Steenhoven (2007); Duriez, Aider, and Wesfreid (2009);
Balamurugan and Mandal (2017b); Vavaliaris, Beneitez, and Henningson (2020);
Rigas, Sipp, and Colonius (2021). The modal stability theory which describes
the asymptotic fate of infinitesimal disturbances Drazin and Reid (2004),
fails to capture the short-term characteristics. On the other hand, non-modal
stability theory describes the stability over a finite-time horizon Schmid
(2007). The transient growth of energy is also referred to as non-modal since
it is caused due to the superposition of several eigenmodes. Mathematically,
the transient growth is due to the non-normal nature of the stability operator
Farrell (1988a, b); Reddy and Henningson (1993); Butler and Farrell (1992),
which results in non-orthogonal eigenfunctions leading to an algebraic growth
for a short time. This growth may be sufficient enough to trigger nonlinearity
in flow and hence, transition at a subcritical Reynolds number Schmid (2000,
2007). Transient growth phenomenon has been extensively studied for many shear
flows like the Blasius boundary layer Ellingsen and Palm (1975); Landahl
(1980); Hultgren and Gustavsson (1981); Andersson, Berggren, and Henningson
(1999); Gustavsson (1991); Butler and Farrell (1992); Corbett and Bottaro
(2000); Luchini (2000); Fransson  _et al._ (2004); Hœpffner, Brandt, and
Henningson (2005); Zuccher, Bottaro, and Luchini (2006); Vavaliaris, Beneitez,
and Henningson (2020).

The linear stability of a boundary layer flow over a flat plate subject to a
uniform freestream shear was studied by Bera & DeyBera and Dey (2005). They
showed that the freestream shear results in larger stability domains in
comparison with the Blasius boundary layer flow. However, the non-normal
nature of the governing operator may result in significant transient growth,
which forms the objective of this study. Recently, the effect of mean flow
shear in flat plate boundary layers was experimentally investigated in a low-
speed wind tunnel Balamurugan and Mandal (2017a). By using a non-uniform
parallel rod grid, they were able to control the freestream turbulence
resulting in a mean velocity profile with shear. They showed that the energy
amplification is higher compared to mean velocity profiles without shear,
thereby undergoing bypass transition earlier. In addition, they further
observed algebraic growth of distrubance energy within the boundary layer. In
the present work we perform a transient growth analysis of such a flow with
uniform freestream shear, which has a second-order boundary layer effect. As
observed in Balamurugan and Mandal (2017a), we find that the transient
amplification of the disturbances scale with the imposed freestream shear.

To this end, we organise the manuscript as follows. The governing equations
along with the mean flow velocity profiles are discussed in II. The tools used
in non-modal stability analyses are also briefly outlined for completeness.
Following this, we present the results outlining the differences in energy
amplification in flows with and without shear (III). We also provide scaling
laws for the transient growth as a function of the imposed freestream shear,
which is followed by some concluding remarks in IV.

##  II Problem formulation and methodology

In the following, $x$, $y$ and $z$ denote the streamwise, wall-normal and
spanwise directions, respectively. A simple model representative of freestream
vorticity is a steady incompressible laminar flow over a semi-infinite plate
placed in a stream with uniform shear, with the freestream velocity given by
$U_{s}=U_{0}+\kappa y$. Here $U_{0}$ and $\kappa$ are constants, with the
uniform freestream Blasius flow corresponding to $\kappa=0$. Since the
freestream vorticity introduces an additional second-order boundary layer
effect, this flow has been studied by several authors Li (1956); Glauert
(1957); Murray (1961); Koch, Ludford, and Seebass (1971); Dey and Nath (1984).
For the flow under consideration, the imposed freestream shear is $dU_{s}/dy$,
the non-dimensional vorticity number being $N_{1}(=\kappa\nu/U_{0}^{2})\ll$
1\. The stream function, $\psi$, may be expressed as the perturbation about
the Blasius flow,

$\psi(x,y)=\sqrt{U_{0}x\nu}\big{[}f_{0}(\eta)+N_{1}\sqrt{(Re)}f_{1}(\eta)+....\big{]}$
(1)

where $\eta=y\sqrt{U_{0}/\nu x}$, $f_{0}(\eta)$ is the non-dimensional Blasius
stream function, and $f_{1}(\eta)$ is the non-dimensional perturbed stream
function. The governing similarity equations for $f_{0}(\eta)$ and
$f_{1}(\eta)$ that follows from the boundary layer momentum equations (Murray,
1961) are

$f^{\prime\prime\prime}_{0}+f_{0}f^{\prime\prime}_{0}=0,$ (2)

$f^{\prime\prime\prime}_{1}+f_{0}f^{\prime\prime}_{1}-f^{\prime}_{0}f^{\prime}_{1}+2f^{\prime\prime}_{0}f_{1}=A.$
(3)

Here a prime denotes the derivative with respect to $\eta$, and the constant
$A$ is due to the imposed pressure. The boundary condition for the above set
of equations are as follows.

$f_{0}(0)=f^{\prime}_{0}(0)=0,\qquad f^{\prime}_{0}(\infty)\rightarrow 1$ (4)
$f_{1}(0)=f^{\prime}_{1}(0)=0,\qquad f^{\prime}_{1}(\infty)\rightarrow\eta$
(5)

Figure 1: Boundary layer velocity profiles. - - - - Blasius; —– uniform shear,
for $N$ = 0.01.

The base flow used in the transient growth analysis is given by
$U=f^{\prime}(\eta)=f^{\prime}_{0}(\eta)+Nf^{\prime}_{1}(\eta)$, where
$N=N_{1}Re/1.73$. Figure 1 shows the mean velocity profiles with and without
the uniform freestream shear. It can be noted that the principal effect of the
external vorticity is to result in an increasing displacement thickness Murray
(1961). This results in a modified pressure field outside the boundary layer
region, thereby affecting the skin friction, boundary layer separation, and
stability region. Indeed, the linear stability analysis has shown that the
freestream shear stabilizes the flow in comparison with the Blasius flow
without any shear Bera and Dey (2005).

We now briefly outline the tools used in the non-modal stability analysis for
completeness. For the transient growth analysis on the above base flow, which
we denote as $\bf{U}$ = $(U,0,0)$, we add a three-dimensional disturbance with
the velocities given by $\mathbf{\tilde{u}}=(\tilde{u},\tilde{v},\tilde{w})$.
The velocities are normalized by $U_{0}$, and distances by the displacement
thickness of the Blasius boundary layer. Pressure has been normalized by $\rho
U_{0}^{2}$. Under the parallel flow assumption, the linearized disturbance
equations and the continuity equation reads as

$\frac{\partial\tilde{u}}{\partial t}+U\frac{\partial\tilde{u}}{\partial
x}+vU^{\prime}=-\frac{\partial p}{\partial
x}+\frac{1}{Re}\nabla^{2}\tilde{u},$ (6) $\frac{\partial\tilde{v}}{\partial
t}+U\frac{\partial\tilde{v}}{\partial x}=-\frac{\partial p}{\partial
y}+\frac{1}{Re}\nabla^{2}\tilde{v},$ (7) $\frac{\partial\tilde{w}}{\partial
t}+U\frac{\partial\tilde{w}}{\partial x}=-\frac{\partial p}{\partial
z}+\frac{1}{Re}\nabla^{2}\tilde{w},$ (8)

$\frac{\partial\tilde{u}}{\partial x}+\frac{\partial\tilde{v}}{\partial
y}+\frac{\partial\tilde{w}}{\partial z}=0.$ (9)

Considering the normal vorticity

$\tilde{\Omega}=\frac{\partial\tilde{u}}{\partial
z}-\frac{\partial\tilde{w}}{\partial x},$

and eliminating the perturbation pressure, we obtain the following two
equations (Schmid and Henningson, 2001)

$\Big{[}\Big{(}\frac{\partial}{\partial t}+U\frac{\partial}{\partial
x}\Big{)}\nabla^{2}-U^{\prime\prime}\frac{\partial}{\partial
x}-\frac{1}{Re}\nabla^{4}\Big{]}\tilde{v}=0,$ (10)
$\Big{[}\frac{\partial}{\partial t}+U\frac{\partial}{\partial
x}-\frac{1}{Re}\nabla^{2}\Big{]}\tilde{\Omega}=-U^{\prime}\frac{\partial\tilde{v}}{\partial
z}.$ (11)

The boundary conditions are $\tilde{v}=\tilde{v^{\prime}}=\tilde{\Omega}=0$.
The solutions of these equations can be sought as follows (Schmid and
Henningson, 2001)

$\tilde{v}(x,y,z,t)=v(y,t)e^{i(\alpha x+\beta z)},$ (12)
$\tilde{\Omega}(x,y,z,t)=\Omega(y,t)e^{i(\alpha x+\beta z)}.$ (13)

Here we consider the homogeneous nature of the flow in the streamwise and the
spanwise coordinate directions, with $\alpha$ and $\beta$ denoting the
corresponding wave numbers. By substituting these velocity and vorticity in
(10) and (11), we obtain

$\Big{[}\big{(}\frac{\partial}{\partial t}+i\alpha
U\big{)}\big{(}D^{2}-k^{2}\big{)}-i\alpha
U^{\prime\prime}-\frac{1}{Re}\big{(}D^{2}-k^{2}\big{)}^{2}\Big{]}v=0,$ (14)
$\Big{[}\big{(}\frac{\partial}{\partial t}+i\alpha
U\big{)}-\frac{1}{Re}\big{(}D^{2}-k^{2}\big{)}\Big{]}\Omega=i\beta
U^{\prime}v.$ (15)

along with the boundary conditions $v=Dv=\Omega=0$ at the wall and in the far
field. Here, $D=\frac{d}{dy}$ and $k=\sqrt{\alpha^{2}+\beta^{2}}$.

Using the continuity equation, the horizontal velocity components, $u$ and
$w$, can be recovered from the normal velocity and the normal vorticity by the
following equations.

$u=\frac{i}{k^{2}}\big{(}\alpha Dv-\beta\Omega\big{)},$ (16)
$w=\frac{i}{k^{2}}\big{(}\beta Dv+\alpha\Omega\big{)}.$ (17)

The equations (14) and (15) can also be written in vector notation (Schmid and
Henningson, 2001)

$\mathbf{M}\frac{\partial\mathbf{q}}{\partial
t}=\mathbf{L}\mathbf{q}\qquad\mathrm{or}\qquad\frac{\partial\mathbf{q}}{\partial
t}=\mathbf{M}^{-1}\mathbf{L}\mathbf{q}=\mathbf{L_{1}\mathbf{q}},$ (18)

where the vectors $\bf q$, $\bf M$, and $\bf L$ are

$\bf{q}=\big{(}\begin{array}[]{c}v\\\ \Omega\end{array}\big{)}$,
$\bf{M}=\left(\begin{array}[]{cc}k^{2}-D^{2}&0\\\ 0&1\end{array}\right)$ and
$\bf{L}=\left(\begin{array}[]{cc}L_{OS}&0\\\ i\beta
U^{\prime}&L_{SQ}\end{array}\right)$.

The Orr–Sommerfeld operator, $L_{OS}$, and the Squire operator, $L_{SQ}$, are

$L_{OS}=i\alpha U\big{(}D^{2}-k^{2}\big{)}-i\alpha
U^{\prime\prime}-\frac{1}{Re}\big{(}D^{2}-k^{2}\big{)}^{2},$ $L_{SQ}=i\alpha
U^{\prime\prime}-\frac{1}{Re}\big{(}D^{2}-k^{2}\big{)}.$

Assuming solutions of the form

$\mathbf{q}=\mathbf{\tilde{q}}\exp(-i\omega t)\qquad\qquad\mathbf{\omega\in
C}$

the initial value problem (eq. 13) becomes a generalized eigenvalue problem

$\mathbf{L}\mathbf{\tilde{q}}=-i\omega\mathbf{M}\mathbf{\tilde{q}}.$ (19)

The energy measure of the disturbance is

$E(t)=\frac{1}{2k^{2}}\int_{0}^{\infty}\big{(}\begin{array}[]{c}v\\\
\Omega\end{array}\big{)}^{H}\Big{(}\begin{array}[]{cc}k^{2}-D^{2}&0\\\
0&1\end{array}\Big{)}\big{(}\begin{array}[]{c}v\\\ \Omega\end{array}\big{)}\
dy,$ (20)

and the maximum possible amplification of the initial energy density (Schmid
and Henningson, 2001) is

$G(t)=\max_{\bf q_{0}\neq 0}\frac{\bf\|q(t)\|^{2}}{\bf\|q_{0}\|^{2}}$ (21)

where $\|\bf{q(t)}\|^{2}$ = $E(t)$.

Here $G(t)$ is the maximum possible energy amplification at a given time, and
includes an optimisation over all initial conditions. At different time
instants a different initial condition may yield the maximum possible energy
amplification. The interested reader is referred to these excellent references
Schmid and Henningson (2001); Schmid (2007) for further details.

##  III Results and Discussion

Figure 2:  Eigenvalue spectra for Blasius and uniform freestream shear flows
at $\alpha=0.2$ and $Re=500$.

Figure 3:  Maximum amplification, $G(t)$, for Blasius ($N$ = 0; solid line)
and uniform freestream shear ($N=0.01$; dashed line) flows at $Re=300$.
$\alpha=0$ and $\beta$ $\approx$ 0.65 for $N=0$; $\alpha=0$ and $\beta$
$\approx$ 0.60 for $N=0.01$.

Using a spectral code, the eigenfunctions and eigenvalues of the stability
operator, $\bf L_{1}$, are obtained. An arbitrary initial disturbance may be
obtained by using the eigenfunctions as the basis functions and $G(t)$, as
defined in (21), is evaluated and optimized. In the present study, we use a
spectral collocation method based on Chebyshev polynomials (Schmid and
Henningson, 2001). Figure 2 shows the eigenvalue spectra for the Blasius
boundary layer ($N=0$) and for a boundary layer flow with uniform freestream
shear ($N=0.01$). The eigenvalue spectra are qualitatively similar for both
the flows with a small shift between them. The straight line part in the
respective spectra correspond to the discretised modes of the continuous
spectrum (Schmid and Henningson, 2001). The phase velocity of the disturbance
is given by $c=\beta/\alpha=c_{r}+ic_{i}$. It can be noted that the phase
velocity is greater than one for the uniform shear flow. This is due to the
normalization by $U_{0}$ which is less than the total freestream velocity with
uniform shear.

(a)(b)

Figure 4:  Initial disturbance and its response in uniform freestream shear
flows at $t$ = $t_{max}$ for $Re=300$; $\alpha=0$, $\beta$ $\approx$ 0.60.
Vectors are of the spanwise and the wall normal disturbance velocities; line
contours denote the streamwsie fluctuating velocity. (a) Optimal initial
disturbance. (b) Optimal response.

(a)(b)(c)

Figure 5:  Contours of $G_{max}$($\alpha$, $\beta$, $Re$) in $\alpha$-$\beta$
plane for three different Reynolds numbers and $N=0.01$. Parameter settings:
(a) $Re=300$; (b) $Re=500$; (c) $Re=800$.

Figure 3 shows the evolution of the energy amplification, $G(t)$, at a
subcritical Reynolds number of $Re=300$, for $N=0$ and $N=0.01$. For the
Blasius boundary layer (Butler and Farrell, 1992), the optimum values of
$\alpha$ and $\beta$ are $0$ and $0.65$, respectively, whereas they are $0$
and $0.60$, respectively, for the present uniform shear case. These values of
$\alpha$ and $\beta$ are found to be optimum in the sense that an optimum
energy amplification occurs at these values, as discussed below. A lower
$\beta$ implies a higher spanwise wavelength than that for the Blasius flow.
We may note that this is higher than the plane Couette flow (Schmid and
Henningson, 2001) case $(\alpha=35/Re,\beta=1.6$; $Re$ appropriate to this
flow). We see that there is significant transient growth in a boundary layer
flow with uniform freestream shear as well. It can be seen that in the initial
phase, the growth of $G(t)$ is the same for both the uniform shear and Blasius
flow, and is unaffected by the freestream shear. However, a higher peak
transient amplification is observed for the freestream shear flow.

For $N=0.01$, the optimal initial disturbances which induce the maximum growth
in the boundary layer, and the optimal response are shown in Fig. 4. We see
that the optimal initial disturbances which trigger the maximum growth are the
streamwise vortices. These are similar to those in other shear flows
(Andersson, Berggren, and Henningson, 1999; Butler and Farrell, 1992; Corbett
and Bottaro, 2000). We can also note from the contour levels that the maxima
of the optimal response is orders of magnitude larger than that of the optimal
initial disturbance representative of the algebraic growth of energy. Thus the
emergence of strong streamwise velocity streaks in Fig. 4(b) may be explained
as the result of the initial streamwise vortex Henningson (2006).

Figure 6:  Temporal variation of $G(t)$ at three different Reynolds numbers.
Parameter settings: $N=0.01$, $\alpha=0$ and $\beta=0.60$.

For the boundary layer flow corresponding to $N=0.01$, $G_{max}$ is found to
be optimum at $\alpha=0$ and $\beta\approx 0.60$. This may be clearly seen in
Fig. 5, where the level curves of $G_{max}$ are shown in the $\alpha$-$\beta$
plane for three different Reynolds numbers. The peak value of $G_{max}$ in the
$\alpha$-$\beta$ plane corresponds to the optimal energy growth, $G_{opt}$.
One may also notice that the value of $G_{opt}$ increases with increasing
Reynolds number. The effect of increasing Reynolds number in the transient
growth amplification is shown in Fig. 6. For a particular value of $N$, indeed
$G_{opt}$ depends on the Reynolds number. We may also note that the contour
lines in the lower branch of each frame in Fig. 5 are not smooth. The kink on
the constant growth rate curve in the Orr–Sommerfeld solutions Bera and Dey
(2005) may be a reason for this effect.

Figure 7:  Contours of $k^{2}G_{max}(\alpha,\beta,Re)/\beta^{2}Re^{2}$ in $k$
and $\alpha$-$Re$ plane. Solid line: $Re=500$; dashed line: $Re=300$.
Parameter settings: $N=0.01$.

Figure 8: Curves of maximum optimal amplification for different $Re$ with
$G(t)$ rescaled by $Re^{2}$ and $t$ by $Re$. Parameter settings: $N=0.01$,
$\alpha=0$, $\beta=0.60$.

(a)(b)(c)

Figure 9:  Contours of $G_{max}(\alpha,\,\beta,\,Re$) in the $\alpha$-$\beta$
plane at $Re=300$ for different values of $N$. Parameter settings: (a)
$N=0.01$, (b) $N=0.03$, (c) $N=0.05$.

Since an estimation of $G_{max}$($\alpha$, $\beta$, $Re$) in the three-
dimensional parameter space ($\alpha$, $\beta$, $Re$) is computationally
expensive, Reddy & Henningson Reddy and Henningson (1993) derived a scaling
relation in the two parameter $(k,\alpha Re)$ space for $G_{max}$($\alpha$,
$\beta$, $Re$) given by

$G_{max}(\alpha,\beta,Re)\approx\frac{\beta^{2}Re^{2}}{k^{2}}H_{2}(k,\alpha
Re),$ (22)

for some function $H_{2}$. This scaling relation holds good for Couette and
Poiseuille flows more accurately, for $\alpha Re\rightarrow 0$ Reddy and
Henningson (1993). As shown in Fig. 7, where the level curves of
$k^{2}G_{max}(\alpha,\,\beta,\,Re)/\beta^{2}Re^{2}$ are shown for two Reynolds
numbers, this scaling relation holds good even for a boundary layer developing
under a freestream with uniform shear. This quadratic scaling of energy is
clearly seen in Fig. 8, where $G(t)$ and $t$ are scaled with $Re^{2}$ and
$Re$, respectively. For $N=0.01$, the optimum value of $G_{max}$ in the
$\alpha$-$\beta$ plane scales as $1.69\times 10^{-3}Re^{2}$, whereas for the
Blasius boundary layer Butler and Farrell (1992); Schmid and Henningson
(2001), the same scales as $1.50\times 10^{-3}Re^{2}$.

(a)(b)(c)

Figure 10:  Maximum amplification, $G(t)$, for different values of $N$.
Parameter settings: (a) $Re=300,\alpha=0.0,\beta=0.60$. (b)
$Re=500,\alpha=0.0,\beta=0.65$, (c) $Re=1000,\alpha=0.0,\beta=0.65$.

(a)(b)(c)(d)

Figure 11:  Rescaled $G(t)$ curves are shown in panels (a,b,c) for the same
parameter settings as in Fig.10. The scaling law holds well for all the
different cases, which is summarized in panel (d).

Figure 12:  Maximum optimal amplification, $G_{max}$, rescaled by
$e^{12N}Re^{2}$, as a function of the freestream shear $N$ for the different
cases considered in the present study.

So far, we have discussed the results for $N=0.01$. We now vary the freestream
shear at a fixed value of the Reynolds number. Figure 9 shows the contours of
$G_{max}(\alpha,\,\beta,\,Re$) for $N=0.01$, $0.03$ and $0.05$ at $Re=300$.
The optimum value of $G_{max}(\alpha,\,\beta,\,Re$) in the $\alpha$-$\beta$
plane is found to increase with increasing $N$, at a given Reynolds number.
With increasing $N$, the peak value of $G_{max}(\alpha,\,\beta,\,Re$) in the
$\alpha$-$\beta$ plane occurs at a lower value of $\beta$, but at $\alpha=0$.
Specifically, the optimum growth for $N=0.03$ occurs at $\alpha=0$ and $\beta$
$\approx$ $0.44$, and at $\alpha=0$ and $\beta=0.36$ for $N=0.05$.

Figure 10 summarizes the effect of increasing freestream shear on the
transient energy growth at three different values of the Reynolds number. It
can be seen that the maximum amplification increases with increasing $N$.
Though the linear stability analysis has shown that the freestream shear
stabilises the flow Bera and Dey (2005), such an increase in amplfication can
lead to a bypass transition scenario Balamurugan and Mandal (2017a). This is
in congruence with the experimental observations reported in Balamurugan and
Mandal (2017a) wherein they found the wall normal distribution of the
normalized $u_{\textrm{rms}}$ profile to follow the non-modal theory of
Luchini Luchini (2000). In addition, it can be seen that the peak transient
amplification is delayed with increasing $N$ and $Re$.

The rescaled $G(t)$ curves are shown in Fig. 11, where $G(t)$ is scaled as
$g_{i}Re^{2}$, and $t$ by $(1-6N)\beta/Re$. Here $g_{i}$ denotes the ratio of
$G_{max}$ for the corresponding value of $N$ normalized by $G_{max}$ at $N=0$,
which is the Blasius flow. The scaling law holds well for all the different
cases considered in the present study, which is evident from Fig. 11(d). As
remarked earlier, the time instant at which the maximum optimal growth is
attained increases with increase in $N$, as evidenced by the scaling for $t$.
This is further elucidated in Fig.12 where $G_{max}$ as a function of $N$ is
shown. The maximum amplification is found to scale as $e^{12N}Re^{2}$
indicating an exponential increase with increasing value of the freestream
shear. These results strongly indicate that a bypass transition scenario is
likely to occur as the external freestream vorticity is increased.

##  IV Concluding remarks

We studied the transient growth in a flat plate boundary layer under a uniform
freestream shear, which is one the simplest problems involving external
vorticity. The freestream vorticity introduces an additional second-order
boundary layer effect. Though many aspects of the transient growth phenomenon
were found to be similar with the Blasius flow, a larger optimum disturbance
growth and a longer spanwise wavelength, in comparison with the Blasius flow
is found. The maximum optimal amplification is found to increase with the
freestream shear, with $G_{max}$ scaling exponentially with the freestream
shear gradient. The present study is of interest in understanding the bypass
transition mechanism in boundary layer flows subject to freestream vorticity,
and should motivate further experimental research.

## Acknowledgements

We are grateful to Prof. Jyothirmoy Dey for introducing us to this interesting
boundary layer flow, and the Department of Aerospace Engineering (Low Speed
Wind Tunnel Lab) at the Indian Institute of Science, where the study began. S.
S. G. acknowledges the support from the French National Research Agency (LABEX
CEMPI, Grant No. ANR-11- LABX-0007) as well as the French Ministry of Higher
Education and Research, Hauts de France council and European Regional
Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER
Photonics for Society P4S). A. C. M. thankfully acknowledges the financial
support provided by IIT Kanpur.

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