# AI-Supported Assessment of Load Safety

Julius Schöning<EMAIL_ADDRESS>Osnabrück University of Applied
SciencesFaculty of Engineering and Computer ScienceOsnabrückGermanyDE-49076
and  Niklas Kruse<EMAIL_ADDRESS>Osnabrück University of
Applied SciencesFaculty of Engineering and Computer
ScienceOsnabrückGermanyDE-49076

(202x)

###### Abstract.

Load safety assessment and compliance is an essential step in the corporate
process of every logistics service provider. In 2020, a total of 11,371 police
checks of trucks were carried out, during which 9.6% (1091) violations against
the load safety regulations were detected. For a logistic service provider,
every load safety violation results in height fines and damage to reputation.
An assessment of load safety supported by artificial intelligence (AI) will
reduce the risk of accidents by unsecured loads and fines during safety
assessments. This work shows how photos of the load, taken by the truck driver
or the loadmaster after the loading process, can be used to assess load
safety. By a trained two-stage artificial neural network (ANN), these photos
are classified into three different classes I) cargo loaded safely, II) cargo
loaded unsafely, and III) unusable image. By applying several architectures of
convolutional neural networks (CNN), it can be shown that it is possible to
distinguish between unusable and usable images for cargo safety assessment.
This distinction is quite crucial since the truck driver and the loadmaster
sometimes provide photos without the essential image features like the case
structure of the truck and the whole cargo. A human operator or another ANN
will then assess the load safety within the second stage.

load safety assessment, artificial intelligence (AI), image-based methods,
semi-automatic safety assessments

††copyright: acmcopyright††journalyear: 202x††doi:
10.1145/123123123.121212††conference: arXiv ’2x; date; arXiv††booktitle: arXiv
’2x, date, arXiv††price: 00.00††isbn: 978-1-4503-XXXX-X/2x/xx

##  1\. Introduction

The assessment of load safety is an important step in the business process of
any logistic service provider. In 2020, 11,371 traffic checks were carried
out, during which 1091 violations against load safety were detected (BLM2020,
). Load safety violations are accountable for almost one-tenth of all
violations and often result in high fines (BundesamtJustiz2023, ). Both the
driver and the loadmaster are liable in the event of a violation under current
case law. In addition to penalties imposed by authorities, civil claims may
also arise if outside parties are adversely affected (Sander2023, ). In this
respect, it is of enormous importance for the logistics service provider to
inspect the load safety, reducing their liability risk properly.

In order to prove that they are in line with the legal compliances of load
safety, logistic companies often take photos of the cargo in the trucks as an
internal safety check before the trucks leave the warehouses. Some logistic
companies inspect the photos by highly qualified quality management employees
and allow the truck’s departure only after successful control. However, due to
a large number of loads, this kind of follow-up inspection by a quality
management employee is very time-consuming and may not always be possible due
to the low image quality of the photos due to the circumstances of image
acquisition in the warehouse by the driver or the loadmaster.

This work investigates to what extent artificial intelligence (AI) can be used
to improve or facilitate the work of the quality management employee in
assessing load safety based on photos. Therefore the photos should be
classified into three different classes I) cargo loaded safely, II) cargo
loaded unsafely, and III) unusable image. By applying several architectures of
convolutional neural networks (CNN), it can be shown that it is possible to
distinguish between unusable and usable images. This distinction is quite
crucial since the truck driver and the loadmaster sometimes provide photos
without the essential image features like the case structure of the truck and
the whole cargo. The load safety will then be assessed by a human operator or
another artificial neural network (ANN). With such an AI-based safety
assessment system in place, the legal certainty of a logistics company will
increase, and the bottleneck, the quality management department, will vanish.

##  2\. State of the Art

Starting with the big picture, this section introduces a centralized system
for load safety assessment, then gives an overview of ANN for image
classification. Going even deeper, this section will also describe standard
techniques to improve and prepare the datasets for ANN by data augmentation.

###  2.1. Centralized System for Load Safety Assessment

Digitalizing logistics, processes, and data of trucks, cargo, traffic flow
etc., will be managed in a centralized platform. This platform could be a
cloud run by a cloud service provider or a cloud run on-premise, i.e., on an
in-house IT solution. Having a centralized platform available, this platform
could be used to verify cargo safety.

Systems for cargo safety verification are essential since not all of a
company’s drivers are permanent employees, yet the loadmaster is legally
liable for violations by these drivers (Sander2023, ). The cargo is first
loaded by a subcontractor and then inspected by a permanent company employee.
First, any dangerous goods are checked to see if they were loaded safely, and
then the rest of the load is examined. If both checks are successful, the
employee takes a photo of the load and sends it via a centralized platform to
quality management. Using a centralized platform that not only collects photos
of the load but also uses AI to examine the safety of the load could
significantly reduce the workload in both quality management and the
warehouse.

###  2.2. Artificial Neural Networks for Image Classification

ANN can be used for a wide variety of tasks like classifying objects in images
(Abiodun2018, ), recognizing speech and words (Schmidhuber2015, ), creating
CAD-models from images (Schoening2015, ), and controlling mechatronic systems
(Schoening2022, ), to only name a few. In general, ANN can recognize different
features of a dataset (Choi2020, ; Schmidhuber2015, ). Features on taken
images could be, e.g., the edges of a piece of cargo, the framework of the
truck, the floor of the truck, and the colors of labels. During the supervised
training of an ANN, the ANN selected the used features on its own. There are
different architectures of ANN. For image classification tasks like the load
safety assessment, CNN performs well. CNN are more efficient compared to other
architectures for recognizing and learning the relevant features of the image
itself (OShea2015, ).

CNN in the field of image classification have evolved rapidly in the past, so
that many models exist today, using the convolution technique (Khan2020, ).
Since these architectures are freely available, it seems reasonable to
identify architectures as state-of-the-art and then use them for load safety
assessment. CNN architectures such as VGG(Simonyan2014, ), AlexNet
(Krizhevsky2017, ), GoogleNet (Szegedy2017, ), and ResNet (He2016, ) are
commonly used. However, in many application cases, the size of the input image
for the CNN architecture is often not optimally chosen. Allowing an optimally
designed CNN architecture, Richter et al. (Richter2021, ) provide a Python
toolbox for analyzing the architecture of CNN by receptive fields. An overview
table of the minimum and maximum feasible input image resolution on Tensorflow
and Torch built-in CNN implementation is given in (Richter2022, ;
Schoening2023, )

In the training of ANN, the dataset is of particular importance since it was
shown in the literature (AbdElrahman2013, ; Kaur2019, ) that better results
can be achieved if the data of the individual classes are balanced. If this is
not the case, there is the problem that individual classes are under or
overrepresented, and the overfitting effect occurs.

###  2.3. Data Augmentation

In object classification, data augmentation describes the artificial extension
of a dataset by using image processing techniques (Khalifa2021, ; Taylor2018,
). Data augmentation should be used when an overfitting effect appears within
the training of the ANN (Khalifa2021, ). Overfitting usually occurs because
there is not enough data in the dataset, so the ANN does not have the
opportunity to learn in a generalized manner. The collection of new data can
be very costly (Taylor2018, ) because, on the one hand, the data must first be
reassigned to specific classes, and, on the other hand, new data can only be
found with considerable effort in a short time.

The group of classical data augmentation techniques is divided into subgroups
of geometric and photometric processing (Khalifa2021, ; Taylor2018, ).
Geometric data augmentation changes only the spatial properties of the image
and only, to a limited extent, the actual representation. The most
straightforward and commonly used image processing technique is to make a copy
of the image using a flip on the x or y-axis. The application of geometric
image processing techniques must be performed in a very targeted manner
(Khalifa2021, ). By randomly cropping an original image, relevant features may
be lost, negatively affecting learning. Thus, to achieve the desired effect,
it must be carefully investigated beforehand whether the processed images
still have a proper relation to the original domain (Khalifa2021, ;
Shorten2019, ).

##  3\. Image-Based Load Safety Classification

For assessing load safety, the ANN must be able to classify the images into
three different classes I) cargo loaded safely, II) cargo loaded unsafely, and
III) unusable image. This classification task could be solved in two ways.
First, a multi-class classification directly classifies whether the images
show either safe cargo, unsafe cargo, or an unusable image. An alternative way
is to divide the classification task into a decision tree of two binary
classifications. Thus the first branch of the decision tree assesses whether
the input image is usable or unusable. In the case of a usable image, a
distinction between cargo loaded safely or unsafely is made, respectively.

By visual inspection of the used dataset, the class III) unusable image is
easily distinguishable from the classes I) cargo loaded safely and II)
unsafely, even for a non-trained human observer. It can therefore be assumed
that using a decision tree with two binary classifications might also be
easier for an ANN. The recognition of unsafely and safely loaded cargo seems
more complex at first sight. However, it is currently carried out by humans in
the quality management department of a logistics company; thus, an ANN might
be able to perform this task too. Nevertheless, solving only the first
classification step to identify unusable images will already have considerable
value for any logistic service provider.

###  3.1. Structure of the Dataset

The used dataset is composed of 5712 images. These images were taken during
the shipping process of a logistic company and always showed the truck’s rear
view. These images were taken by employees of the logistic company who check
the safety of the load while still at the loading center and ultimately create
and transmit a photo for quality assurance. This real-world dataset is taken
since, in the context of AI, the transferability of the academic datasets to
real-world industrial deployment is often quite challenging (Wuest2016, ;
Lee2020, ). Using a dataset as close as possible to the target domain can
simplify the usability of the results (Chen2021, ). The advantage of the
dataset is that it does not come from an outside source but was used for
evaluation at a logistic company and manually classified into the relevant
classes. At the same time, however, this classification could lead to errors
in the initial classification, which, given the dataset size, could be
attributed to human error or different standards in the assessment
(Northcutt2021, ). Nevertheless, this should facilitate the subsequent
transfer of AI findings to real-life applications in the field.

With some exceptions, the images have a resolution of 3456x4608 pixels and RGB
color space. In advance, the images were divided into the three classes
mentioned above.

The I) cargo loaded safely class is formed by all images that show suitable
load safety and represent the optimum result from the logistics company’s
point of view. There are 1813 images in this class.

The II) cargo loaded unsafely class contains the images that were previously
described as safe in the warehouse and before the departure of the load but
would be considered unsafe in a subsequent inspection. Thus, images in this
class do not satisfy legal requirements and could carry significant liability
risks (BundesamtJustiz2023, ; Sander2023, ). Examples of this class represent
the least desirable case, as they could give rise to liability risks. In this
respect, this class must be reliably detected to allow a logistics company to
defend itself against fines. This class consists of 2355 images.

The III) unusable image class is characterized by the fact that they are not
suitable for checking cargo security by quality management. In other words, no
visible features in the images can provide information about the safety of the
goods because they are too blurred, too dark, or do not show the entire cargo
hold, for example. There are 1544 images in this class.

###  3.2. Applied Data Augmentation

Data augmentation aims to prevent possible overfitting and thus improve the
training effect, whereby one reason for overfitting is a too-small dataset
(Khalifa2021, ; Shorten2019, ). However, the number of datasets collected
within the logistics company is limited and cannot be easily expanded with
original data due to the significant effort involved in the new survey.
Initial pre-run tests found that overfitting occurs very early in the training
process with the given data.

The preliminary test with the InceptionV3 architecture showed, as illustrated
in Fig. 1 that the overfitting effect occurred approximated at epoch 50. From
this epoch, the model lost its generalized competence and emphasized the need
for data augmentation. The remaining networks showed comparable effects in the
same preliminary tests before they reached the desired metrics in terms of
accuracy.

For the implementation of data augmentation, photometric and geometric methods
are used. By not altering the classes of the images, only the random flip on
the y axes, slight random rotation, random brightness modification, and random
color adjustments are used for data augmentation.

$1$$10$$20$$30$$40$$50$$60$$70$$80$$90$$100$$110$$120$$130$$140$$150$$160$$170$$180$$190$$0$$0.2$$0.4$$0.6$$0.8$$1$epochslosstraining
lossvalidation loss Figure 1. Training and validation loss gradient of the
InceptionV3 architecture, overfitting occurs approximated at epoch 50.

###  3.3. Architecture Design of the Convolutional Neural Network

Three CNN architectures are used for the experiment, which can be assigned to
two categories. On the one hand, two deep ANN were used for the InceptionV3
(Szegedy2015, ) and the ResNet101 (He2016, ). On the other hand, one shallow
architecture, called LogisticNet based on the AlexNet architecture
(Krizhevsky2017, ), is used.

In the standardized design, InceptionV3 and the ResNet101 have an input
resolution of 299x299 (Szegedy2015, ; Szegedy2017, ), and 244x244 (He2016, )
pixels, respectively. The resolution can significantly impact the model’s
performance, so a higher resolution can also lead to better classification
accuracy (Kannojia2018, ). The resolution to be chosen can be further
influenced by the receptive field of an ANN (Richter2021, ; Richter2022, ).
Since the receptive field can be calculated, for some state-of-the-art models,
there are defined maximum and minimum values that the input image should have
to utilize the existing model optimally. For the ResNet101 this resolution is
a maximum of 971x971 pixels; for the InceptionV3, it is 1311x1311 pixels. Due
to hardware limitations and to allow batch sizes larger than two, the
resolution here was capped at 800x800 pixels.

The LogisticNet, shown in Fig. 2 is based on the AlexNet, which itself is
purely sequential (Simonyan2014, ). For this reason, the input resolution was
limited to 227x227 pixels (Krizhevsky2017, ). LogisticNet intends to find out
whether a simple model is also capable of solving the task, so a comprehensive
redesign of LogisticNet was omitted.

However, the LogisticNet has some adaptations compared to the AlexNet. In its
first convolution layer, the AlexNet has a large kernel size of 11x11
(Krizhevsky2017, ). Newer models tend to use smaller kernels here (Khan2020, ;
Novakovic2017, ), as they act more efficiently and are better at extracting
small features from the dataset (Camgoezlue2021, ). The remaining convolution
layers are oriented based on the original AlexNet architecture, but the last
dense layer reduces the number of classes to two classes instead of 1000. As
with the other models, the softmax activation function was chosen as the
activation function of the last dense layer.

InputConv2D (K_size=3x3)Conv2D (K_size=11x11)Batch
NormalizationMaxPool2D(P_Size=3)Conv2D (K_size=5x5)Batch
NormalizationMaxPool2D(P_Size=3)Conv2D (K_size=3x3)Batch NormalizationConv2D
(K_size=3x3)Batch NormalizationConv2D (K_size=3x3)Batch
NormalizationMaxPool2D(P_Size=3)FlattenDense (4096)DropOut (0.5)Dense
(4096)DropOut (0.5)Dense (2)Output Figure 2. LogisticNet architecture based on
the AlexNet (Krizhevsky2017, ) architecture.

##  4\. Evaluation

All three architectures are trained throughout 300 epochs. Since it is
expected that they would show the effects of overfitting at a certain point,
an epoch had to be found at which the training was terminated before
overfitting occurred, and the model can no longer recognize general features.

The ResNet101 achieved a validation accuracy of 87% at the resolution of
244x244 after 300 epochs and reached its highest validation accuracy at epoch
242 with a validation accuracy of 93%. On the resolution of 800x800px, the
validation accuracy was 94% after 300 epochs and reached its highest point
after 292 epochs with 94.5%. As plotted in Fig. 3 the loss curves of both
models, the overfitting effect is noticeable. At the 244x244px resolution, the
ResNet101 reached the point of overfitting after 90 epochs, while at the
800x800px resolution, the overfitting did not occur until epoch 120. For the
final evaluation, ResNet101 was trained at a resolution of 244x244px over 87
epochs and on 800x800px over 117 epochs avoiding the overfitting on the
training dataset.

$1$$20$$40$$60$$80$$100$$120$$140$$160$$180$$200$$220$$240$$260$$280$$0$$0.2$$0.4$$0.6$$0.8$$1$epochslosstraining
lossvalidation loss (a) ResNet101

$1$$20$$40$$60$$80$$100$$120$$140$$160$$180$$200$$220$$240$$260$$280$$0$$0.2$$0.4$$0.6$$0.8$$1$epochslosstraining
lossvalidation loss (b) InceptionV3

Figure 3. Loss curves of LABEL:sub@sub:ResNet18 ResNet101 and
LABEL:sub@sub:Inception InceptionV3 for the input resolution 800x800px.

The InceptionV3 achieved a validation accuracy of 95% after 300 epochs in the
native resolution of 299x299px and a value of 97% in the high resolution of
800x800px. It reached its high point with an accuracy of 96% at epoch 277 in
the low resolution and 96% after 280 epochs in the high resolution. The
previously observed overfitting effect is also seen in InceptionV3. Here, the
overfitting occurred in the test with the low resolution after epoch 58. In
the higher resolution test, the overfitting could be observed from epoch 85.
In the following test, the InceptionV3 network was trained in low resolution
over 55 epochs and high resolution over 82 epochs.

The LogisticNet was only tested in a resolution of 227x227px, so only one
version of the model must be evaluated. After a test duration of 300 epochs,
the LogisticNet achieved a validation accuracy of 88%. The LogisticNet reached
the highest point of validation accuracy after 254 epochs with a value of 89%.
The overfitting effect was visible in the LogisticNet from epoch 49. The
LogisticNet was trained over 46 epochs in a second run and then evaluated.

Table 1. Metrics of the distinction between usable, class I) as well as II),
and unusable images class III) by the different ANN architectures. ResNet101
---
Resolution: 244x244px | Resolution: 800x800px
Recall: 94% | Precision: 91% | Recall: 92% | Precision: 89%
InceptionV3
Resolution: 299x299px | Resolution: 800x800px
Recall: 94% | Precision: 92% | Recall: 93% | Precision: 90%
LogisticNet
Recall: 98% | Precision: 95%

##  5\. Result

After finding the optimal count of training epochs for all networks, they were
trained again, and the average results are shown in Tab. 1. On distinguishing
between usable and unusable images, all CNN architecture achieves precision
and recall above 90%.

By applying, i.e., training and evaluating, the ResNet101 and the InceptionV3
architecture on the second task of the decision tree to distinguish between
classes I) cargo loaded safely and II) cargo loaded unsafely, the archived
precision and recall drastically decrease, as summarized in Tab. 2.

As shown in the confusion matrix in Fig. 4, the InceptionV3 network, as the
best architecture, was able to recognize 72 out of 100 images of class I)
cargo loaded safely but only 37 out of 100 of the II) class, correctly. In the
high resolution of 800x800px, different results were seen. Here 61 out of 100
images of class I) were recognized, and 51 out of 100 of class II) were
correctly classified.

The ResNet101 could not achieve the same good results in this task’s high and
low resolution. In class I) only 45 out of 100 images were correctly
classified, and in class II) 62 of 100 images were correctly recognized.
Almost identical values were achieved in the high resolution, but the value of
correctly classified results in class II) increased by three percent.

Table 2. Metrics of the distinction between classes I) cargo loaded safely and
II) cargo loaded unsafely by ResNet101 and InceptionV3; MCC is the Matthews
correlation coefficient where a 1.00 would represent a perfect classification.
ResNet101
---
Resolution: 244x244px | Resolution: 800x800px
F1-Score: 0.57 | MCC: 0.18 | F1-Score: 0.59 | MCC: 0.20
InceptionV3
Resolution: 299x299px | Resolution: 800x800px
F1-Score: 0.43 | MCC: 0.09 | F1-Score: 0.52 | MCC: 0.13

(a) 299x299px  (b) 800x800px

Figure 4. Confusion Matrix distinguishing between classes I) cargo loaded
safely and II) cargo loaded unsafely by Inception V3; resolution
LABEL:sub@sub:low 299x299px and LABEL:sub@sub:high 800x800px.

##  6\. Conclusion

Based on the result of the previous sections, AI-Supported assessment of load
safety is feasible. It is valuable for any logistic service provider since a
CNN can distinguish between usable and unusable images for cargo safety
assessments. By implementing this function in the centralized logistic
platform, the photos from the drivers and the loadmaster can automatically
check if these photos are taken correctly for cargo safety assessment. This
check will reduce the number of unusable images already during the acquisition
of the images in the warehouse. Nevertheless, recognizing safe and unsafe
loaded cargo could not be solved satisfactorily in this work. As seen in Tab.
2 and Fig. 4, the CNN architectures used could not sufficiently predict the
images into the respective classes. One, but not the only, reason for this
result could be the used dataset. By manual inspection of the results, an
intersection between the classes I) cargo loaded safely and II) cargo loaded
unsafely was notable. These intersections occur due to some boundary cases,
where one cloud assesses the cargo shown on the image as both safe or unsafe
loaded cargo. Overcoming this issue, the intersection of the classes must be
removed. By removing these intersections, the used CNN architectures might
generalize better between classes I) and II). In parallel, a hybrid dataset, a
mixture of computer-generated and real-world images, will increase the number
of training images and provide a basis for further work. In conclusion, AI can
already support the assessment of load safety quite well but still need to be
supervised by a human operator.

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# Zabreiko’s lemma in 2-normed space and its applications.

Akshay S. RANE 111Department of Mathematics, Institute of Chemical Technology,
Nathalal Parekh Marg, Matunga, Mumbai 400 019, India, email :
<EMAIL_ADDRESS>
###### Abstract

We prove the Zabreiko’s lemma in 2-Banach spaces. As an application we shall
prove a version of the closed graph theorem and open mapping theorem.

Key Words : 2-normed space, Zabreiko’s Lemma, closed graph theorem, open
mapping theorem.

AMS subject classification : 46B20, 46C05, 46C15,46B99,46C99

##  1 Introduction

n-normed spaces can be found in [1], [5], [6], [3], and [7]. Open mapping
theorem and closed graph theorem are fundamental theorems in functional
analysis. In a usual normed space, Zabreiko [11] proved a lemma due to which
the major theorems of functional analysis, like open mapping theorem and
closed graph theorem followed as corollaries. In case of 2-normed spaces [10]
and [9] proved open mapping and closed Graph theorem in the usual way as in
usual normed spaces. In this paper we would like to extend the Zabreiko’s
lemma in $2-$normed spaces and the open mapping theorem and a version of the
closed graph theorem follows as a consequence.

##  2 Methods

We shall now introduce definitions and results which shall be used in this
paper.

######  Definition 2.1.

Let $X$ be a vector space over $\mathbb{R}$ of dimension $>1.$ An 2-norm is a
real valued function from $X\times X$ satisfying the following properties.

  1. 1.

$\|x_{1},x_{2}\|=0$ if and only if $x_{1},x_{2}$ are linearly dependent.

  2. 2.

$\|x_{1},x_{2}\|=\|x_{2},x_{1}\|$.

  3. 3.

$\|\alpha x_{1},x_{2}\|=|\alpha|\|x_{1},x_{2}\|$ for $\alpha\in\mathbb{R}$.

  4. 4.

$\|x_{1}+x_{1}^{\prime},x_{2}\|\leq\|x_{1},x_{2}\|+\|x_{1}^{\prime},x_{2}\|$

Then $(X,\|.,.\|)$ is called as a 2-normed space.

######  Definition 2.2.

A real valued function $p$ on $X\times X$ satisfying properties $2$, $3$ and
$4$ is called 2-seminorm.

We incorporate various definition which can be found in [10], [2] and [8].

######  Definition 2.3.

If $p$ is a 2-seminorm, then for $e\in X,$ $p(x,e)$ is a semi-norm on $X$.

######  Definition 2.4.

A sequence $x_{n}$ in $X$ is said to be convergent to $a\in X,$ if

$\lim_{n\rightarrow\infty}\|x_{n}-a,y\|=0,$

for every $y\in X.$

######  Definition 2.5.

A sequence $x_{n}$ in $X$ is said to be Cauchy if

$\lim_{n,m\rightarrow\infty}\|x_{n}-x_{m},y\|=0,$

for every $y\in X.$

######  Definition 2.6.

A 2-normed space $X$ is said to be an $2-$Banach space if every Cauchy
sequence in $X$ converges.

######  Definition 2.7.

An open ball in a 2-normed space is $B_{e_{2}}(a,r):=\left\\{x\in
X:\|x-a,e_{2}\|<r\right\\}$ and a closed ball is $B_{e_{2}}[a,r]:=\left\\{x\in
X:\|x-a,e_{2}\|\leq r\right\\}$

######  Definition 2.8.

A subset $U$ of $X$ is said to be open in $X,$ if there exist $r>0$ and
$e_{2}\in X$ such that $B_{e_{2}}(a,r)\subset X$.

######  Definition 2.9.

A linear operator $T:X\rightarrow Y$ is said to be $e$ bounded if there exist
$M_{e}>0$ such that

$\|Tx,Te\|\leq M_{e}\|x,e\|,$

where $X$ and $Y$ are $2-$normed spaces. The operator $T$ is said to be
bounded if $T$ is $e$ bounded for each $x\in X.$

##  3 Results

We shall now prove the following lemma which will be used to prove the
Zabreiko’s Lemma.

######  Lemma 3.1.

Let $p$ be a seminorm on a $2-$normed space $(X,\|.,.\|)$ and for $\alpha>0$
consider

$V_{\alpha}=\left\\{x\in X:p(x,e)\leq\alpha,\text{for}\;e\in X\right\\}$

Suppose there is $a\in X$ $r>0$ and $e_{2}\in X$, such that

$B_{e_{2}}[a,r]\subset\overline{V_{\alpha}}.$

Then for every $\delta>0$,

$B_{e_{2}}[0,\delta r]\subset\overline{V_{\delta\alpha}}.$

###### Proof.

We shall prove the result for $\delta=1$. Let $x\in B_{e_{2}}[0,r].$ Then

$\|(x+a)-a,e_{2}\|\leq r,\;\|(-x+a)-a,e_{2}\|\leq r.$

This implies that $x+a,-x+a\in B_{e_{2}}[a,r]\subset\overline{V_{\alpha}}$.
There is a sequence $u_{m}$ and $v_{m}$ in $V_{\alpha}$ such that
$u_{m}\rightarrow x+a$ and $v_{m}\rightarrow-x+a$ with $p(u_{m},e)\leq\alpha,$
and $p(v_{m},e)\leq\alpha.$ Hence

$\frac{u_{m}-v_{m}}{2}\rightarrow x$

with

$p(\frac{u_{m}-v_{m}}{2},e)\leq\alpha.$

So $x\in\overline{V_{\alpha}}$ and we are done. Now consider $\delta>0$ and

$\|x,e_{2}\|\leq\delta r$

Let $y=\frac{x}{\delta}.$ Then $\|y,e_{2}\|\leq r.$ This implies
$y\in\overline{V_{\alpha}}$. So that there is a sequence $y_{m}$ in
$V_{\alpha}$ so that $y_{m}\rightarrow y$ and $p(y_{m},e)\leq\alpha.$ And we
have

$\delta y_{m}\rightarrow\delta y,\;p(\delta y_{m},e)\leq\delta\alpha.$

This implies $x\in\overline{V}_{\delta\alpha}.$ ∎

We shall now prove the Zabreiko’s theorem in $2-$normed spaces. The Baire
Category theorem which we shall use to prove Zabreiko’s Lemma for $2-$normed
spaces is proved in [9].

######  Theorem 3.2.

Let $p$ be a countabely subadditive seminorm on a $2-$Banach space. Then there
is $e_{2}\in X$ and a constant $M>0$ such that

$p(x,e)\leq M\|x,e_{2}\|,$

for any $e\in X.$ In particular,

$p(x,e_{2})\leq M\|x,e_{2}\|.$

###### Proof.

For $n\in\mathbb{N}$, let

$V_{n}=\left\\{x\in X:p(x,e)\leq n,e\in X\right\\}$

Then

$X=\bigcup_{n=1}^{\infty}V_{n}=\bigcup_{n=1}^{\infty}\overline{V}_{n}$

As a consequence

$\bigcap_{n=1}^{\infty}\overline{V}_{n}^{c}=\phi.$

By the Baire Category theorem atleast one of $\overline{V}_{n}^{c}$ will not
be dense in $X.$ As a result

$\overline{\overline{V}_{m}^{c}}\subset X.$

That is there is $a\in X$ such that $a$ does not belong to
$\overline{\overline{V}_{m}^{c}}.$ There exist $r>0$ and $e_{2}\in X$ such
that

$B_{e_{2}}[a,r]\cap\overline{V}_{m}^{c}=\phi$

This means

$B_{e_{2}}[a,r]\subset\overline{V}_{m}$

Let $x\in X$. Let $\epsilon>0.$ Define

$\epsilon_{0}=\frac{\|x,e_{2}\|}{r}$

and

$\epsilon=m\sum_{k=1}^{\infty}\frac{\epsilon}{m2^{k}}=m\sum_{k=1}^{\infty}\epsilon_{k}.$

Since $\|x,e_{2}\|=r\epsilon_{0},$ implies $x\in B_{e_{2}}[0,\epsilon_{0}r]$.
From the above lemma

$B_{e_{2}}[0,\epsilon_{0}r]\subset\overline{V}_{\epsilon_{0}m}$

$x\in\overline{V}_{\epsilon_{0}m}$, for $\epsilon_{1}r$ and any $\tilde{e}\in
X$

$B_{\tilde{e}}[x,\epsilon_{1}r]\bigcap V_{\epsilon_{0}m}\neq\phi.$

There exist $x_{1}\in B_{\tilde{e}}[x,\epsilon_{1}r]\bigcap
V_{\epsilon_{0}m}.$ This implies

$\|x-x_{1},\tilde{e}\|\leq\epsilon_{1}r,\;\;p(x_{1},e)\leq\epsilon_{0}m,$
(3.1)

for any $\tilde{e},e\in X.$ In particular for $\tilde{e}=e_{2}$

$\|x-x_{1},e_{2}\|\leq\epsilon_{1}r.$

Let $u_{1}=x-x_{1}$ so that $x=x_{1}+u_{1}$ and
$\|u_{1},e_{2}\|\leq\epsilon_{1}r$. So that $u_{1}\in
B_{e_{2}}[0,\epsilon_{1}r]$ From the above lemma once again

$B_{e_{2}}[0,\epsilon_{1}r]\subset\overline{V}_{\epsilon_{1}m}$

$u_{1}\in\overline{V}_{\epsilon_{1}m}$, for $\epsilon_{2}r$ and any
$\tilde{e}\in X$

$B_{\tilde{e}}[u_{1},\epsilon_{2}r]\bigcap V_{\epsilon_{1}m}\neq\phi$

There exist $x_{2}\in B_{\tilde{e}}[u_{1},\epsilon_{2}r]\bigcap
V_{\epsilon_{1}m}$ This implies

$\|u_{1}-x_{2},\tilde{e}\|\leq\epsilon_{2}r,\;\;p(x_{2},e)\leq\epsilon_{1}m,$
(3.2)

for any $\tilde{e},e\in X.$ In particular for $\tilde{e}=e_{2}$

$\|u_{1}-x_{2},e_{2}\|\leq\epsilon_{2}r$

Let $u_{2}=u_{1}-x_{2}$ so that $x=x_{1}+u_{1}=x_{1}+x_{2}+u_{2}$ and
$\|u_{2},e_{2}\|\leq\epsilon_{2}r.$ Now using previous lemma

$u_{2}\in B_{e_{2}}[0,\epsilon_{2}r]\subset\overline{V}_{\epsilon_{2}m}$

We get $x_{3}$ such that

$\|u_{2}-x_{3},\tilde{e}\|\leq\epsilon_{3}r,\;\;p(x_{3},e)\leq\epsilon_{2}m,$
(3.3)

for any $\tilde{e},e\in X.$ Continuing this way we get, for each $k$,
$x_{k},u_{k}\in X$ such that

$\|u_{k-1}-x_{k},\tilde{e}\|\leq\epsilon_{k}r,\;\;p(x_{k},e)\leq\epsilon_{k-1}m,$
(3.4)

for any $\tilde{e},e\in X.$ As a result

$x=x_{1}+u_{1}=x_{1}+x_{2}+u_{2}=x_{1}+x_{2}+\ldots+x_{k}+u_{k}$

Thus for any $\tilde{e}\in X$ we have

$\|x-\sum_{k=1}^{n}x_{k},\tilde{e}\|=\|u_{n},\tilde{e}\|\leq\epsilon_{m}r=\frac{\epsilon
r}{m2^{n}}\rightarrow 0$

as $n\rightarrow\infty.$ We see that

$x=\sum_{k=1}^{\infty}x_{k}.$

By the countable subadditivity of $p$ we have

$p(x,e)\leq\sum_{k=1}^{\infty}p(x_{k},e)\leq
m\sum_{k=1}^{\infty}\epsilon_{k-1}=m\epsilon_{0}+\epsilon.$

The above statement is true for any $\epsilon.$ This implies for $e\in X$ we
have

$p(x,e)\leq\frac{m}{r}\|x,e_{2}\|.$

In particular for $e=e_{2}$

$p(x,e_{2})\leq\frac{m}{r}\|x,e_{2}\|.$

∎

######  Remark 3.3.

Let $T:X\rightarrow Y$ be linear operator. Then for $x,e\in X$

$p(x,e)=\|Tx,Te\|$

is an example of a $2-$seminorm on $X.$

######  Definition 3.4.

A linear operator $T:X\rightarrow Y$ is said to be closed if for a sequence
$x_{n}\rightarrow x$ in $X$, $Tx_{n}\rightarrow y$ in $Y$ then $y=Tx.$

We require the following result.

######  Theorem 3.5.

Let $X$ be a $2-$Banach space. Then if
$\displaystyle\sum_{n=1}^{\infty}\|x_{n},e\|<\infty$ for each $e\in X.$ Then
$\displaystyle\sum_{n=1}^{\infty}x_{n}<\infty.$

###### Proof.

Let

$s_{n}=\sum_{k=1}^{n}x_{k}$

For any $e\in X$ and $n>m$, consider

$\|s_{n}-s_{m},e\|=\|\sum_{k=m+1}^{n}x_{k},e\|\leq\sum_{k=m+1}^{n}\|x_{k},e\|$

Since $\displaystyle\sum_{k=m+1}^{n}\|x_{k},e\|\rightarrow 0$ as
$n,m\rightarrow\infty$ This implies $s_{n}$ is a Cauchy sequence in $2-$Banach
space and $\lim s_{n}$ exists. As a result
$\displaystyle\sum_{k=1}^{\infty}x_{k}$ converges. ∎

In the following results, we apply the Zabreiko’s Lemma to prove a version of
the closed graph and open mapping theorem.

######  Theorem 3.6.

Let $X$ and $Y$ be $2-$Banach spaces and $T:X\rightarrow Y$ be a closed linear
operator. Then $T$ is e bounded.

###### Proof.

We show that for $e\in X$

$p(x,e)=\|Tx,Te\|$

is a countabely subadditive 2 norm on $X.$ Let
$x=\displaystyle\sum_{n=1}^{\infty}x_{n}$ with
$\displaystyle\sum_{n=1}^{\infty}\|T(x_{n}),Te\|<\infty.$ Since $Y$ is a
$2-$Banach space by the above result $\displaystyle
y=\sum_{n=1}^{\infty}Tx_{n}<\infty.$ Define
$s_{n}=\displaystyle\sum_{k=1}^{n}x_{k}$. Note that $s_{n}\rightarrow x$ and
$T(s_{n})\rightarrow y.$ As $T$ is closed
$y=T(x)=\displaystyle\sum_{n=1}^{\infty}Tx_{n}$.

$p(x,e)=\|Tx,Te\|\leq\sum_{n=1}^{\infty}\|Tx_{n},Te\|=\sum_{n=1}^{\infty}p(x_{n},e)$

By the Zabreiko’s lemma there is an $M.0$ and $e_{2}\in X$ such that

$\|Tx,Te_{2}\|=p(x,e_{2})\leq M\|x,e_{2}\|,$

which proves the result. ∎

######  Theorem 3.7.

Let $X$ and $Y$ be $2-$Banach spaces. Let $T:X\rightarrow Y$ be continuous and
surjective. Then $T$ is an open map.

###### Proof.

Define

$p(y,e^{\prime})=\inf\left\\{\|x,e\|:T(x)=y,\;T(e)=e^{\prime}\right\\}$

Since $T$ is surjective and $\|.,.\|$ is non negative, $p(y,e^{\prime})$ is
well defined. it is easy to see that $p(kx,e)=|k|p(x,e).$ We shall show that
$p$ is countabely subadditive seminorm on $Y$. Let
$y=\sum_{k=1}^{\infty}y_{k}$ with

$\sum_{k=1}^{\infty}p(y_{k},e^{\prime})<\infty.$

Let $\epsilon>0$. For $\frac{\epsilon}{2^{k}}$ there is $x_{k}\in X$ such that
$T(x_{k})=y_{k}$ and

$\|x_{k},e\|<p(y_{k},e^{\prime})+\frac{\epsilon}{2^{k}}$

This implies

$\sum_{k=1}^{\infty}\|x_{k},e\|<\sum_{k=1}^{\infty}p(y_{k},e^{\prime})+\epsilon$

As a result

$\sum_{k=1}^{\infty}\|x_{k},e\|<\infty$

and as $X$ is a 2-Banach space $x=\sum_{k=1}^{\infty}x_{k}$ converges. $T$
closed

$T(x)=\sum_{n=1}^{\infty}T(x_{n})=\sum_{n=1}^{\infty}y_{n}=y$

As a result

$p(y,e^{\prime})=\|y,e^{\prime}\|\leq\|x,e\|\leq\sum_{k=1}^{\infty}\|x_{k},e\|<\sum_{k=1}^{\infty}p(y_{k},e^{\prime})+\epsilon$

By Zabreiko’s Lemma, there exist $e_{2}\in Y$, $M>0$ such that
$p(y,\tilde{e})<M\|y,e_{2}\|.$ There exist $x\in X$ such that
$T(x)=y,T(e)=e^{\prime}$ and

$\|x,e\|\leq M\|y,e_{2}\|.$

Consider an open set $E$ in X and let $x_{0}\in X$. There exits $\delta>0$ and
$e\in X$ such that $B_{e}(x_{0},\delta)\subset E.$ Let $y\in Y$,
$\|y-T(x_{0}),e_{2}\|<\frac{\delta}{M}.$ We shall show that $y\in T(E).$ Since
$y-T(x_{0})\in Y$ there exist $x\in X$ such that $T(x)=y-T(x_{0})$ and

$\|x,e\|\leq M\|y-T(x_{0}),e_{2}\|<\delta$

Thus

$\|x+x_{0}-x_{0},e\|=\|x\|<\delta$

This implies $x+x_{0}\in B_{e}(x_{0},\delta)\subset E$ So that
$y=T(x)+T(x_{0})\in T(E)$ ∎

##  4 Conclusions

We prove the Zabreiko’s Lemma in 2-normed spaces, but for n-normed s]paces can
be generalized in a similar manner. However using the Zabreiko’s Lemma we were
able to prove that a closed linear operator is e-bounded. It would be interest
to show when the operator is bounded or continuous. Also question could be
asked is whether analog of uniform boundedness principle could be obtained
from the Zabreiko’s lemma.

##  5 List of Abbreviations

  1. 1.

$B_{e_{2}}(a,r):=\left\\{x\in X:\|x-a,e_{2}\|<r\right\\}.$

  2. 2.

$B_{e_{2}}[a,r]:=\left\\{x\in X:\|x-a,e_{2}\|\leq r\right\\}.$

  3. 3.

$\overline{V}:$ closure of the set $V.$

##  6 Declarations

Availability of data and materials: Not applicable
Competing interests : Not Applicable
Funding : Not Applicable
Authors’ contributions : Not Applicable.
Acknowledgements : UGC FRP program, India
Authors’ information :
UGC Assistant Professor,
Department o Mathematics,
Institute of Chemical technology, Mumbai.

## References

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# The onset of turbulence in particle-laden pipe flows

Willian Hogendoorn  Delft University of Technology, Multiphase Systems
(3ME-P&E), Leeghwaterstraat 39, 2628 CB Delft, The Netherlands
Bidhan Chandra  Delft University of Technology, Multiphase Systems (3ME-P&E),
Leeghwaterstraat 39, 2628 CB Delft, The Netherlands
Indian Institute of Science Education & Research, Chemical Engineering, Bhopal
462066, India  Christian Poelma<EMAIL_ADDRESS>Delft University of
Technology, Multiphase Systems (3ME-P&E), Leeghwaterstraat 39, 2628 CB Delft,
The Netherlands

###### Abstract

We propose a scaling law for the onset of turbulence in pipe flow of neutrally
buoyant suspensions. This scaling law, based on a large set of experimental
data, relates the amplitude of the particle-induced perturbations ($\epsilon$)
to the critical suspension Reynolds number, $Re_{s,c}$. Here $\epsilon$ is a
function of the particle-to-pipe diameter ratio and the volume fraction of the
suspended particles, $\epsilon=(d/D)^{1/2}\phi^{1/6}$. $Re_{s,c}$ is found to
scale as $\epsilon^{-1}$. Furthermore, the perturbation amplitude allows a
distinction between classical, intermediate and particle-induced transition.

††preprint: APS/123-QED

The recurrent question when and how pipe flow transitions to turbulence long
predates the seminal pipe flow experiments by Osborne Reynolds in the early
1880s [1]. Since pipe flow is linearly stable, finite amplitude perturbations
are thus responsible for this onset of turbulence [2, 3]. For increasing
Reynolds number ($Re=UD/\nu$; with $U$ the bulk velocity, $D$ the pipe
diameter and $\nu$ the kinematic viscosity), smaller perturbation amplitudes
are sufficient to initiate this transition [4]. Initially, turbulence is found
to be transient: localized patches of turbulence are embedded in a laminar
flow. These turbulent puffs are known to have an increasing lifetime for
increasing Reynolds number [5, 6]. Beyond a Reynolds number of 2040, puffs
grow and split, eventually leading to sustained turbulence [7].

Particle-laden flows are known to exhibit significantly different transition
behavior [8, 9, 10, 11]. This is in particular evident from the critical
Reynolds number, $Re_{c}$, which is known to strongly depend on the particle
volume fraction ($\phi$) and the particle-to-pipe diameter ratio ($d/D$) of
the suspended particles. For increasing volume fraction and a large enough
$d/D$, critical (suspension) Reynolds numbers as low as 600 are reported [8].
Furthermore, beyond a critical volume fraction, which depends on the particle-
to-pipe diameter ratio, a different regime is observed: the transition is
smooth without the presence of turbulent puffs [9, 10]. The exact onset of
turbulence is of major importance for a variety of applications, as this onset
is accompanied by a significant - sometimes intermittent - drag increase.
Precise control over the flow rate (by means of a driving pressure) is
essential for process control, whether it is in additive manufacturing, food
processing, slurry transport, or dredging. However, despite considerable
efforts, a definite scaling indicating the onset of the drag increase in the
transitional regime is still absent. We propose a scaling for $Re_{c}$ in
neutrally buoyant suspensions, which is supported by a large set of
experimental data.

The first detailed study reporting a prominent effect of particles on laminar-
turbulent transition was performed by Matas  _et al._ [8]. They determined
$Re_{c}$ for a wide range of $d/D$ and $\phi$, using the low frequency
component in the pressure spectrum as an indicator for the presence of
turbulent puffs, characteristic for the onset of turbulence. This $Re_{c}$ was
based on the corrected viscosity using Kriegers’ viscosity model [12], to
account for the presence of particles. A scaling in terms of $\phi D/d$ as
function of this viscosity corrected critical Reynolds number ($Re_{s,c}$) was
proposed to collapse all results on a single master curve. However, the ratio
$D/d$ is used in both axes of this master curve, suggesting that $Re_{s,c}$ is
a function of $\phi$ only.

Lashgari  _et al._ [13] studied the influence of neutrally buoyant particles
($d/h$ = 0.1, with $h$ the channel height) numerically for a channel flow
configuration. They introduced a distinction based on the dominant term in the
stress budget. Three different regimes are identified: a laminar regime for
low volume fractions and low Reynolds numbers, a turbulent regime for low
volume fractions and high Reynolds numbers, and an inertial shear-thickening
regime for $\phi\gtrapprox$ 0.15. This distinction is only feasible in
numerical studies as it requires very detailed flow information.

A different transition mechanism, without the presence of turbulent puffs in
the transition region, was found by Hogendoorn and Poelma [9] for higher
volume fractions ($\phi\geq$ 0.175). This particle-induced transition behavior
is characterised by a smooth transition curve, which collapses on $64/Re$ for
low $Re_{s}$ after viscosity correction. The onset of turbulence was
identified using a 10% deviation from the law of Hagen-Poiseuille. Agrawal
_et al._ [10] independently reported this particle-induced transition for
higher volume fractions as well.

This smooth, particle-induced transition was also found for lower volume
fractions ($\phi$ = 0.05) in combination with a larger particle-to-pipe
diameter ratio ($d/D$ = 0.17) by Leskovec  _et al._ [11]. The authors proposed
a scaling to distinguish between classical and particle-induced transition
based on a model of viscous dissipation and particle agitation.

Hogendoorn  _et al._ [14] showed that for large $d/D$ even very dilute systems
($\phi$ = 0.0025) exhibit this particle-induced transition. Based on
instantaneous velocity measurements they showed that particles introduce
perturbations and formulated a model predicting that these disturbances are
proportional to $d/D$ and $U$.

Based on these previous studies, it appears that the onset of turbulence in
particle-laden flows is dependent on at least two parameters: the particle-to-
pipe diameter ratio and the volume fraction. We propose a scaling for this
onset in neutrally buoyant suspensions, based on a wide range of experimental
data, both new and from aforementioned studies. Furthermore, this scaling can
be used to predict which transition behavior will be observed for a given
system.

Experiments are performed in two different pipe-flow facilities. The first
experimental setup is the same as the setup described in Hogendoorn and Poelma
[9]. In short, this setup consist of a precision glass pipe with a diameter
($D$) of 10.00$\pm$0.01 mm. The flow is gravity driven using an overflow
reservoir to prevent perturbations from the pumps. The height of this
reservoir can be adjusted to control $Re_{s}$. At the inlet a settling chamber
and a smooth contraction are used to maintain (single-phase) laminar flows for
Reynolds numbers exceeding 4000. An orifice, similar to the one used by
Wygnanski and Champagne [15], is used to ensure a controlled transition at a
fixed Reynolds number of 2000 for single-phase flows. The total pipe length
($L$) after the orifice is 310$D$. The pressure drop ($\Delta p$) is measured
(Validyne DP15) from 125$D$ to 250$D$, ensuring sufficient development length.
$Re_{s}$ is determined with an uncertainty smaller than 0.5%, by collecting
and weighing an amount of suspension from the outflow during a given time. A
set of peristaltic pumps are used to feed the outflow back to the overflow
reservoir.

The second experimental setup is similar to the one described above; for
brevity the differences will be addressed only. This setup consist of a PMMA
pipe with an inner diameter of 19.98$\pm$0.06 mm. Using a settling chamber
with a smooth contraction in combination with smooth pipe connectors a laminar
flow is maintained for single-phase Reynolds numbers up to 5000. In this setup
the flow is either perturbed using an orifice or perturbed using an active
perturbation mechanism, which is a zero-net mass flux injector (adapted from
Draad  _et al._ [16]) at the beginning of the pipe (positioned 10$D$ after the
inlet chamber). The perturbation method is no longer significant beyond a
certain critical volume fraction as was reported by Matas  _et al._ [8] and
Agrawal  _et al._ [10]. We also confirmed this for our experiments. The
average pressure drop is obtained from 125$D$ to 200$D$ after this active
perturbation. An inline Coriolis mass flow meter (KROHNE Optimass 7050c) is
used to measure the flow rate with a maximum uncertainty of $\pm$ 0.1%. A
progressive cavity pump (Monopump, AxFlow B.V.) is used to transport the
suspension back to the feeding reservoir. For $d/D$ = 0.088, the overflow was
removed to be able to drive very viscous flows. We confirmed that this did not
influence the (single-phase) transition, as this was still dominated by the
orifice perturbation.

Saline water (Na2SO4) or a water-glycerine mixture is used to obtain a
density-matched system with polystyrene particles (Synthos EPS; density $\rho$
= 1.032 kg/L). The volume fraction in the experimental facilities is
controlled based on the mass ratio of the working fluid and the particles.
Starting with a single-phase system, with a known initial mass, particles are
added in steps to obtain the desired volume fraction. Particles with diameters
of 0.30$\pm$0.034, 0.53$\pm$0.05, 1.31$\pm$0.07 and 1.75$\pm$0.12 mm are used.
All four particle types are used in the 10.00 mm facility and the 1.31 and
1.75 mm particles are also used in the 19.98 mm facility. This results in six
$d/D$ ratios (0.03, 0.053, 0.065, 0.088, 0.13, and 0.18); two pressure data
sets were re-used from previous studies ($d/D$ = 0.18, 0.053 [9, 14]).

The (critical) Reynolds numbers reported in this study are based on the
corrected viscosity ($\mu_{s}=\rho\nu_{s}$) of the suspension, determined
using Eilers’ viscosity model [17]:

$\frac{\mu_{s}}{\mu_{0}}=\bigg{(}1+1.25\frac{\phi}{1-\phi/0.64}\bigg{)}^{2}.$
(1)

Here $\mu_{0}$ is the viscosity of the continuous phase (i.e., saline water or
glycerol).

Fig. 1 shows the transition behavior for a range of different experiments
(i.e., various $d/D$) for a constant volume fraction, $\phi$ = 0.05. Here the
Darcy friction factor ($f\equiv\Delta p/(\frac{1}{2}\rho U^{2}L/D)$) is shown
as function of $Re_{s}$. The continuous line represents the Hagen-Poiseuille
law: $64/Re$, the solution for laminar flows. The dashed line shows Blasius’
equation. The single-phase case obtained in the 10.00 mm diameter setup, shown
for reference, displays a transition at $Re_{c,0}\approx$ 2000, resulting from
the perturbation in the beginning of the setup. For the particle-laden cases,
the influence of $d/D$ is clearly visible from the decrease of $Re_{s,c}$ for
increasing $d/D$. For $d/D<$ 0.065, a sharp transition is observed with a
clear local minimum in the transition region. This local minimum shifts to
lower $Re_{s,c}$ for increasing $d/D$, implying an earlier onset of
turbulence. A critical transition curve is shown for $d/D$ = 0.065, where the
local minimum is still (only just) present. Eventually, for $d/D>$ 0.065,
smooth transition curves are observed, characteristic for particle-induced
transition.

Notably, for the intermediate case ($d/D$ = 0.065), the friction factors in
the transition region are lower compared to the friction factors corresponding
to $d/D$ = 0.053. Related to this change in transition behavior is the non-
monotonically decreasing critical Reynolds number in this specific region.
This can likely be explained by the change in transition scenario and the
associated change in length scales. In a previous study, the integral length
scales corresponding to a smooth transition case were found to be smaller
(i.e., approximately 4$D$ in the transition region) and continuously present
[14]. This is in contrast to a sharp, intermittent transition, where turbulent
patches span about 20-30$D$ [15, 18].

Figure 1:  Friction factor, $f$, as function of suspension Reynolds number,
$Re_{s}$. The single-phase case is shown for reference. For the particle-laden
cases the concentration ($\phi$) is fixed at 0.05 to highlight the diameter-
ratio ($d/D$) effect.

All data are shown in the regime map ($\phi$ vs. $d/D$) in Fig. 2, where each
marker represents one of 51 complete transition curves (which consist of 11-39
measurements of $f$ vs. $Re_{s}$). The transition curves with a monotonically
decreasing friction factor for increasing $Re_{s}$ (i.e., $\partial f/\partial
Re<0$ for all values of $Re$) are indicated with square markers. The
transition curves with a local minimum are represented by the triangular
markers. Here the derivative is locally positive in the transition region. The
color of the markers indicates $Re_{s,c}$, determined using a threshold of
$70/Re$ [9]. This threshold was determined to be a sound balance between
robustness to measurement noise and accuracy. This figure confirms the
dependency of $Re_{s,c}$ on $d/D$ and $\phi$: an increase in either parameter
promotes transition. The dashed curve represents a constant Bagnold number
[19], which has previously been used to classify suspension behavior [13]. It
is defined as the ratio of the inertial to viscous stress:
$N=d^{2}\dot{\gamma}\lambda^{1/2}/\nu$, with $\dot{\gamma}$ the shear-rate and
$\lambda$ the linear concentration, $1/[(0.74/\phi)^{1/3}-1]$. The best
discrimination between transition mechanisms is found for $N$ = 7.2 (based on
a bulk shear-rate for $Re_{c,0}$ = 2000). It is evident that this is still not
satisfactory. Alternative values of $N$ will always only satisfy the
transition behavior at one $d/D$. We thus confirm the observation by Lashgari
_et al._ [13] that the Bagnold number by itself is not sufficient to predict
transition behavior. All experiments shown in Fig. 2 are well below $N$ = 40,
which suggest that all cases are in the viscous-dominated regime according to
Bagnolds’ theory. Another model, based on particle agitation versus laminar-
dissipation is proposed by Leskovec  _et al._ [11]. According to their model
the threshold between the two mechanisms is predicted by $\phi^{2}(d/D)^{2}Re$
and is indicated by the dashed-dotted curve. However, this model is based on a
limited range of experimental data, resulting in a less accurate prediction
for the transition mechanism at higher $d/D$. The solid curve, indicated by
$\epsilon$ = const., is based on our proposed model (introduced below) to
distinguish between intermediate and particle-induced transition.

Figure 2:  Regime map $\phi$ vs $d/D$, where each marker represents one
complete transition curve. The transition curves indicated with the triangular
markers have a local minimum, whereas the transition curves with a
monotonically decreasing friction factor are represented by a square marker.
The marker color represents the critical suspension Reynolds number,
$Re_{s,c}$.

Previously, it has been shown that the velocity fluctuations caused by the
finite-sized particles scale with $d/D$. Based on this, it is expected that
$Re_{s,c}$ is dependent on $d/D$ and $\phi$ [14]. The latter has been shown to
be a relevant scaling parameter before [8, 9, 10]. The exact respective
contributions of $d/D$ and $\phi$ on $Re_{s,c}$ are studied, based on the data
shown in Fig. 2. The effect of $d/D$ (for $\phi$ = 0.01, 0.05, and 0.1) and
$\phi$ (for $d/D$ = 0.065, 0.13, and 0.18) are shown in Fig. 3a and 3b,
respectively. In both figures a selection of three representative cases is
shown. From both figures it follows that $Re_{s,c}$ depends on $d/D$ and
$\phi$ with exponents of $-\frac{1}{2}$ and $-\frac{1}{6}$, respectively. The
exponents are determined using a regression analysis, where the found
exponents are rounded to the nearest common fraction. The dashed curves shown
are based on this fraction and the corresponding amplitude, resulting from the
regression analysis. While both exponents are obtained separately in this
approach, we also confirmed that a regression of both exponents
simultaneously, ($d/D)^{\alpha}\phi^{\beta}$, gave similar results. Note that
the square of the linear concentration in Bagnolds’ scaling also results in an
exponent of $\frac{1}{6}$ for the concentration in the dilute limit.

(a)

(b)

Figure 3: LABEL:sub@fig:3a $Re_{s,c}$ as function of $d/D$ for three
representative volume fractions and LABEL:sub@fig:3b $Re_{s,c}$ as function of
$\phi$ for three representative diameter ratios. The dashed curves are based
on a regression analysis, where the slope is rounded to the nearest common
fraction.

Based on this, the onset of turbulence is modelled as
$Re_{s,c}\big{(}\frac{d}{D}\big{)}^{\frac{1}{2}}\phi^{\frac{1}{6}}=c$. Note
that dimensionless group,
$\big{(}\frac{d}{D}\big{)}^{\frac{1}{2}}\phi^{\frac{1}{6}}$, can also be
interpreted as a perturbation amplitude, $\epsilon$, resulting from the
suspended particles. This is analogous to finite perturbation amplitudes being
used in single-phase transition experiments [4, 20, 21]. Therefore we decided
to introduce the following perturbation amplitude:

$\epsilon=\Big{(}\frac{d}{D}\Big{)}^{\frac{1}{2}}\phi^{\frac{1}{6}}.$ (2)

This perturbation amplitude can also be rewritten to be proportional to
$N_{p}^{~{}\frac{1}{6}}\frac{d}{D}$, with $N_{p}$ the number of particles per
unit volume (i.e., $D^{3}$). We choose the former representation to separate
the parameters, as $N_{p}$ is a function of both $d/D$ and $\phi$. The
physical interpretation of $\epsilon$ is that the perturbation amplitude
increases with the number of particles per unit volume and for increasing
$d/D$.

In Fig. 4, all critical Reynolds numbers (i.e., the colors from Fig. 2) are
shown as function of $\epsilon$. As expected, for this scaling all $Re_{s,c}$
now collapse on one single curve given by: $Re_{s,c}=207~{}\epsilon^{-1}$. The
general interpretation of the exponent in case of single-phase flows is that
there is a balance between inertial and viscous forces [21]. For the particle-
laden cases the exponents for $d/D$ and $\phi$ are now responsible for this
balance. The pre-factor, also resulting from regression, is likely specific
for the current configuration: the flow of a suspension of neutrally buoyant,
spherical particles through a pipe. The relatively large horizontal errorbar,
shown for one experiment only, is based on a conservative error propagation
and predominantly originates from the polydispersity in particle diameter
(common for experimental studies). This new scaling is also validated using
data from literature [8, 10]. These $Re_{s,c}$ are indicated in the legend.
For $Re_{s,c}$ reproduced from Matas  _et al._ [8], three $Re_{s,c}$ for three
different $d/D$ (i.e., 0.056, 0.063 and 0.1) are shown, spanning a significant
range of the scaling. Note that $Re_{s,c}$ beyond the local minimum (see Fig.
3 in Matas  _et al._ [8]) are excluded from this analysis as these $Re_{s,c}$
are biased due to their measurement method, as is discussed in Hogendoorn and
Poelma [9]. Data for $Re_{s,c}$ taken from Agrawal  _et al._ [10] also support
our scaling.

Figure 4:  Critical suspension Reynolds number, $Re_{s,c}$, as function of
particle perturbation amplitude, $\epsilon$, for all experiments. All
$Re_{s,c}$ collapse on one single curve with slope -1. The marker on the
ordinate axis represents the critical Reynolds number for a single-phase flow,
$Re_{c,0}$.

For small $\epsilon$ (i.e., low $\phi$ and/or $d/D$), particles will not
affect $Re_{s,c}$. In this regime, the amplitude of the particle perturbations
is negligible with respect to the perturbation amplitude of the disturbance
mechanism. Therefore, the transition behavior will be described by the dashed
line in Fig. 4, indicating a fixed transition corresponding to the
perturbation amplitude of the used perturbation mechanisms ($Re_{c,0}$ = 2350
for this particular experiment). The presence of this plateau is confirmed for
experiments with a particle size ratio of $d/D$ = 0.088 (see also $Re_{s,c}$
for values of $\epsilon\rightarrow$ 0 in Fig. 3 in Matas  _et al._ [8]). The
marker on the ordinate axis represents the corresponding critical Reynolds
number for this single-phase case ($Re_{c,0}$), i.e., $\epsilon\rightarrow$ 0.

Using the scaling law, the conditions can be identified where the
perturbations of the particles are sufficiently damped by viscous effects, so
that transition is not triggered by the particles. For combinations of
$\epsilon$ and $Re_{s}$ above the solid line in Fig. 4, the friction factor
will deviate 10% or more from Poiseuille. Conversely, for combinations below
this line, the friction factor can safely be assumed to be $64/Re_{s}$.

The scaling shown in Fig. 4 covers a very wide range of situations, including
most engineering applications. As an indication, a 45% volume fraction of
particles with a size of 1/10 of the pipe diameter is characterized by
$\epsilon$ = 0.28. Larger values can be reached by increasing $d/D$ to unity
(leading to a theoretical limit of $\epsilon$ = 0.93). However, a different
scaling is then expected, as these extreme cases would represent stacked
particles with a diameter close to the pipe diameter. Obviously, this is no
longer a ‘flowing suspension’.

In the literature generally three different regimes are distinguished to
describe the underlying dynamics: classical, intermediate, and particle-
induced transition behavior [9, 10, 14]. Using the perturbation amplitude (Eq.
2), we can quantitatively distinguish between these different regimes; they
are indicated in the bottom of Fig. 4. The border between classical and
intermediate transition can be defined where $Re_{c}$ is found to deviate from
a typical transition Reynolds number for single-phase flow, $Re_{c,0}$ = 2000.
The corresponding critical perturbation amplitude is found to be $\epsilon$ =
0.103. The change between intermediate and particle-induced transition
behavior is indicated by the solid curve in Fig. 2, corresponding to
$\epsilon$ = 0.15. This value is determined by minimizing the error between
the number of square markers and triangular markers above and below the curve,
respectively. Given the limited number of data near the prediction curve, this
critical value for $\epsilon$ is bounded by 0.135 and 0.17. For simplicity we
report a value in the center of this range. Note that there is a smooth
transition between classical and particle-induced transition behavior for
increasing $\epsilon$. The intermittent nature of classical transition is
gradually replaced by continuous, particle-induced fluctuations, see also the
detailed characterization of one single case by Hogendoorn  _et al._ [14].

Our motivation to interpret $\epsilon$ as a perturbation amplitude follows
from similar approaches in single-phase flow experiments [20, 21]. Note that
for suspension flows the particle-induced perturbations are continuously
present along the length of the pipe. This is in contrast to single-phase
perturbation experiments, where the perturbation is temporally and spatially
bounded. For injection-based disturbances the perturbation amplitude is
typically defined as the ratio of the injection volume flux with respect to
the volume flux in the pipe. Similarly, for an orifice type perturbation the
orifice diameter can be expressed as a disturbance amplitude. The amplitude
required to trigger transition in single-phase flows scales with
$Re^{\gamma}$, where the exponent varies between -1 and -1.5, depending on the
type of perturbation [4, 22, 21]. This scaling is generally based on
experiments for $Re>$ 2000, while for particle-laden flows the range below
$Re_{s}$ = 2000 is especially important.

Further measurements with small $d/D$ and/or low $\phi$ (i.e., low $\epsilon$)
in the absence of a perturbation mechanism should reveal the flow stability
response for $Re_{s,c}>$ 2000\. This will also shed light on the universality
of the various perturbation parameters. On the edges of the investigated
parameter space (Fig. 2) various effects will come into play [23]. For
instance, the effect of the spatial distribution of particles on the
transition in the (semi-)dilute regime is still an open question. For
$d/D\rightarrow$ 0, our model predicts that particles will not affect the
transition, but this needs to be confirmed. Additionally, measurements for
higher volume fractions need to be performed to establish whether our scaling
law will hold in the inertial-shear thickening regime [13].

In summary, based on a large set of experimental data, we uncovered a scaling
law relating the amplitude of the particle-induced perturbations to the
critical suspension Reynolds number. The particle-induced perturbation
amplitude is a simple function of the particle-to-pipe diameter ratio and the
volume fraction: $\epsilon=(d/D)^{1/2}\phi^{1/6}$. The onset of turbulence in
neutrally buoyant suspensions is found to scale as
$Re_{s,c}\sim\epsilon^{-1}$. Data from literature also supports the validity
of this scaling. Furthermore, $\epsilon$ allows a prediction of the transition
scenario. For a variety of applications it will predict whether the transition
will be classical, intermediate, or particle-induced.

###### Acknowledgements.

We thank Vasudevan Krishnan for assisting in our experimental campaign. We
would like to thank Mr S. van Baal (Synthos EPS) for kindly providing
particles for this research. This work is funded by the ERC Consolidator Grant
No. 725183 “OpaqueFlows.” Underlying data are deposited in the 4TU.Centre for
Research Data, and will be accessible by: doi:10.4121/16586954.

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