Development and Validation of MicrobEx: an Open-Source Package for
Microbiology Culture Concept Extraction.

Garrett Eickelberg1

Department of Preventitive Medicine (Health & Biomedical Informatics)

Feinberg School of Medicine,

750 N Lake Shore, Chicago, IL 60611, USA.

Yuan Luo2

Department of Preventitive Medicine (Health & Biomedical Informatics)

Feinberg School of Medicine

750 N Lake Shore, Chicago, IL 60611, USA.

Electronic address<EMAIL_ADDRESS>
L Nelson Sanchez-Pinto3

Department of Preventitive Medicine (Health & Biomedical Informatics)

Feinberg School of Medicine

750 N Lake Shore, Chicago, IL 60611, USA;

Department of Pediatrics (Critical Care)

225 E. Chicago Avenue, Chicago, IL 60611, USA.

Electronic address: lazaro.sanchez-pintonorthwestern.edu.

Phone: 1.800.543.7362

Keywords: Concept extraction, Information extraction, Electronic health
records, Natural language processing, Microbiology report

Word Counts (excluding references, headings, tables, abstract, and figures):

2000

###### Abstract

Microbiology culture reports contain critical information for important
clinical and public health applications. However, microbiology reports often
have complex, semi-structured, free-text data that present a barrier for
secondary use. Here we present the development and validation of an open-
source package designed to ingest free-text microbiology reports, determine
whether the culture is positive, and return a list of SNOMED-CT mapped
bacteria. Our rule-based natural language processing algorithm was developed
using microbiology reports from two different electronic health record systems
in a large healthcare organization, and then externally validated on the
reports of two other institutions with manually-extracted results as
benchmark. Our algorithm achieved F-1 scores $>$0.95 on all classification
tasks across both validation sets. Our concept extraction Python package,
MicrobEx, is designed to be reused and adapted to individual institutions as
an upstream process for other clinical applications, such as machine learning
studies, clinical decision support, and disease surveillance systems.

##  1 Introduction

Microbiology culture reports are relied upon for myriad healthcare
applications ranging from guiding clinical treatment decisions to global
disease surveillance. In a clinical setting, microbiology culture reports are
helpful in answering if an infection is present and what organisms are driving
that infection [1]. Outside of the clinical setting, microbiology data are
used to monitor disease outbreaks, improve healthcare operations (e.g. monitor
nosocomial infection rates), and are leveraged in a variety of observational
studies [2, 3, 4, 5]. Thus, the data within microbiology reports impacts
clinical treatment and public policy decisions, and are therefore critical for
secondary use [6, 2].

Unlike many other structured laboratory test results, microbiology culture
reports are often complex, semi-structured reports that pose unique challenges
for large-scale secondary use applications. Samples sent to a microbiology
laboratory routinely undergo numerous tests, such as gram stains and
antibiotic susceptibility tests, each of which have different turnaround
times, can produce more than a single result, and need to be linked to the
original accession number [1, 2]. Additionally, results from each test can
include both quantitative and qualitative data, and need to be reported as
they become available to facilitate treatment decisions [1, 2]. Unfortunately,
although there are efforts to standardize reporting and analysis of clinical
microbiology data, the suitability of existing microbiology reports for
secondary use are hindered by reporting variability and analysis practices [7,
8, 9]. Finally, microbiology reports contain varying amounts of protected
health information as defined by the Health Insurance Portability and
Accountability Act, thus limiting the flexibility of this data for data
sharing projects. Therefore, there is critical need for informatic tools that
can navigate microbiology report data challenges and extract information to
facilitate their secondary use. The goal of this study was to develop,
validate, and release an open-source microbiology concept extraction
(MicrobEx) system to facilitate secondary use of microbiology reports.

##  2 Materials & Methods

###  2.1 Datasets

The derivation datasets for this study were extracted from the Northwestern
Medicine (NM) Enterprise Data Warehouse (EDW). The regular expressions and
logic flow of our extraction system were developed using 216,372 raw free-text
microbiology reports extracted from critical care patients treated at one of
10 Northwestern Medicine intensive care units between 1/1/2010-1/1/2020. To
define microbiology reports, we queried the NMEDW and manually curated 235
unique procedures associated with microbiology culture orders. The collection
of microbiology reports had highly heterogeneous formatting and lacked
consistent template features such as concept-value pairs and table structures.
Additionally, our corpora contained full microbiology reports, as well as
individual microbiology components such as gram stains and antibiotic
susceptibility reports. To address these challenges, rules were crafted to
separate reports into sections wherever possible. For cultures with multiple
report entries tied to the same accession number, only the notes with the
latest report update time were selected for downstream processing and
analysis. Testing and validation of our extraction system was performed on two
external datasets with 119,789 expertly annotated free-text microbiology
reports from University of Chicago (validation 1) and Ann & Robert H. Lurie
Children’s Hospital (validation 2). The validation sets of microbiological
culture results were part of prior study and details have been previously
published [6]. The reports from both hospitals were annotated by the same
senior clinical research coordinator. All four datasets included microbiologic
cultures reports from blood, urine, respiratory, and cerebral spine fluid
samples.

###  2.2 Algorithm Overview

A summary of our algorithm workflow is presented in Figure 1. Our concept
extraction algorithm uses a comprehensive set of rules, as well as context,
keyword, and morphologic features that capture overall bacterial infection
status and identify bacterial species present in a microbiology report.
Rulesets and regular expressions were developed through an iterative process
based on document structural and context features in addition to clinical
criteria and domain knowledge. For bacterial species captures, we wrote
regular expressions to capture the genus and species for bacteria present in a
dictionary of clinically relevant organisms collated from knowledgebases [10,
1]. Organisms captured were mapped to Observational Health Data Sciences and
Informatics (OHDSI) and Systemized Nomenclature of Medicine (SNOMED) IDs via a
dictionary included in the source code. The mapping dictionary for
microorganism to OHDSI and SNOMED IDs was constructed by passing the collated
microorganism list into Usagi software indexed on SNOMED vocabulary and
restricted to class ‘ORGANISM’ and domain ‘OBSERVATION’[11]. During each
iteration, concept extraction performance was reviewed manually using a
variety of different pattern occurrence-based audits on our training data
sets. Customized regular expressions were created to capture remaining complex
patterns. Each regular expression was developed with generalizability in mind
to maximize dissemination and reusability. For all false positive and negative
cases, we reviewed the associated case context, assigned a reason for
misclassification. We addressed the cases by either refining existing rules or
implementing new ones. This iteration process was repeated until all remaining
uncaptured cases were caused by report noise, uncommon misspellings, or lack
of report clarity [12].

###  2.3 Validation

Figure 2. includes example reports annotated with extracted concepts, species,
and estimated bacterial culture positive status. Both species extraction and
binary bacteria positive culture status (yes/no) were evaluated as outcomes
for validation of our algorithm and compared to the manually annotated results
in the validation sets. For species extraction, we compared species captured
across all report sections by our algorithm and the expert annotation. We
encoded our binary outcome as positive if MicrobEx captured all the species
identified by the expert. Similarly, positive bacteria culture status was
assigned to all report sections and were compared to the expert annotation at
the report level using a maximum function.

###  2.4 Performance Benchmark

In order to benchmark our algorithm’s performance against a well-established
clinical natural language processing (NLP) tool, we applied MetaMap[13] to
both validation sets and built a rule-based decision workflow to predict
positive bacterial culture status and capture bacterial species.

###  2.5 Dataset customization

To identify and address dataset-specific patterns capable of causing
misclassifications, we audited our workflow as described in the _use guide_
prior to final validation. The generalizable regular expressions we added
during the audits were both appended into the codebase prior to our validation
studies. The detailed code and Python package installation instructions have
been made available at: https://github.com/geickelb/rbmce. See the _use guide_
section for Regular expression examples and a description on how to deploy and
customize our package to a new dataset.

##  3 Results

###  3.1 Validation

Table 1. summarizes the distribution of positive bacterial culture status in
the four datasets. The ratio of positive to negative cases across our training
set predictions is consistent with that seen in the two curated validation
sets.

| Positive bacterial culture | Negative bacterial culture
---|---|---
Derivation set 1 | 14,376 (20.7%) | 55,065 (79.3%)
Derivation set 2 | 23,549 (16%) | 123,382 (84%)
Validation set 1 | 2,185 (14.5%) | 12,915 (85.5%)
Validation set 2 | 7,391 (14.7%) | 42,957 (85.3%)
Table 1: Bacterial culture positive status distribution.

Table 2. summarizes the validation results across both species and positive
bacterial culture status classification tasks. The algorithm had excellent and
consistent performance, with validation sets 1 and 2 having F1-scores of 0.99
and 0.96 for positive culture classification and species capture,
respectively. To estimate the improvements made by introducing customized
regular expressions from the data audits, each validation set was reanalyzed
using a codebase with the associated regular expressions deactivated. From
this, we estimate that culture positivity classification increased from 0.93
to 0.96 and 0.69 to 0.96 for validation sets 1 and 2, respectively. The
addition of customized regular expressions was found to cause little-to-no
effect on species capturing across both validation sets.

| True Negative | False Positive | False Negative | True Positive | Precision | Recall | NPV | F-1
---|---|---|---|---|---|---|---|---
Validation set 1 |  |  |  |  |  |  |  |
Species capture | 12,463 (82.54%) | 2 (0.01%) | 209 (1.38%) | 2,426 (16.07%) | 0.998 | 0.921 | 0.984 | 0.958
Positive culture status | 12,909 (85.48%) | 7 (0.05%) | 22 (0.15%) | 2,162 (14.32%) | 0.995 | 0.990 | 0.998 | 0.992
Validation set 2 |  |  |  |  |  |  |  |
Species capture | 42,391 (84.20%) | 4 (0.01%) | 68 (0.14%) | 7,885 (15.66%) | 0.999 | 0.991 | 0.999 | 0.995
Positive culture status | 42,950 (85.31%) | 7 (0.01%) | 606 (1.20%) | 6,785 (13.48%) | 0.998 | 0.918 | 0.986 | 0.956

Table 2: Infection classification and species capture performance across
Validation sets.

Supplemental Table 1. presents the results from our customized MetaMap based
benchmarking algorithm against both validation sets. Across both positive
culture classification and species capture, MicrobEx matched or surpassed the
benchmark algorithm performance. These results suggest that our task-specific
classifier can outperform more general-use clinical NLP tools like MetaMap.

###  3.2 Error Analysis

In the error analysis we identified a collection of five patterns in which our
concept extraction workflow had the majority of errors. Figure 2. presents
annotated visual examples of the classification hierarchical logic for the
different patterns observed, with examples for both correct classifications as
well as misclassifications. Examples 5 and 6 depict the two most common types
of false positive patterns and examples 7 and 8 present the most common
patterns found in false negatives in the validation sets. We can summarize
these patterns as a combination of multiple positive and negative organisms
where the negative regex capture supersedes the positive captures, and the use
of the term “contaminant” leading to a false negative classification.

##  4 Discussion

In this study, we developed and validated an open-source, rule-based framework
to extract and map clinical concepts from microbiology reports to standardized
terminologies to facilitate secondary use of microbiology reports. Our main
finding is that our algorithm can reliably estimate binary bacterial culture
status, extract bacterial species, and map these to SNOMED organism
observations when applied to semi-structured, free-text microbiology reports
from different institutions with relatively low customization.

Top performing rule-based concept extraction applications commonly employ a
well-established clinical NLP tool that can map mentions to a corresponding
medical concept(s) for broad medical corpora, such as cTAKES[14] and
MetaMap[13]. Like the well-established tools, MicrobEx performs concept
matching by leveraging existing microbiology knowledgebases as described in
Materials & Methods. In contrast to these tools however, MicrobEx uses custom
rules and regular expressions tailored to microbiology reports for dependency
recognition and modifier detection. MicrobEx’s higher performance on bacterial
positive culture status prediction suggests that for this classification task,
MicrobEx’s more tailored approach provides advantages over an out-of-the-box
approach using a well-established NLP tool. To further improve MicrobEx’s
prediction performance, additional institution-specific customized rules could
be added. Figure 2. depicts four representative examples of cases
misclassified for positive culture status that could be addressed with
institution-specific custom rules.

To our best knowledge, three previously published studies have applied
clinical concept extraction methods to microbiology notes [15, 16, 17]. Jones
et al. [16] applied a set of crafted rules to blood culture reports from the
Salt Lake City Healthcare system to extract organism information, antibiotic
susceptibilities, and infer if methicillin-resistant staphylococcus aureus
(MRSA) was present. An evaluation was performed against approximately 10,000
expertly annotated reports to measure successful identification of MRSA.
Matheny et al. and Yim et al. [17, 15] used hybrid and rule-based systems to
capture combinations of microorganisms species and antibiotic susceptibilities
from blood and multiple sample types, respectively. Our algorithm is notably
different from the previously published systems in the following ways: (1) we
estimate positive bacterial culture status, (2) our algorithm was designed to
work with a variety of disparate microbiology report formats from different
institutions, (3) we performed external validation on two expertly annotated
microbiology datasets, and (4) our software is entirely open-source and
available as a python package that can be further adapted to the reports of
other institutions as described in the _supplemental use guide_ and supported
by our results.

We recognize several limitations of our study. First, for users of this
software, classifying positive culture status is the prediction task with the
largest potential error. Compared to species extraction, which is largely
string matching, estimating infection status requires significantly more
complex logic. The hierarchical logic involved with positive bacterial culture
status estimation is potentially susceptible to syntactic heterogeneity and
report complexity, as depicted in Figure 2. Additionally, we focused on
bacterial cultures for the development and validation of the algorithm given
the importance of antibiotic stewardship, antibiotic resistance, and bacterial
sepsis in hospitalized patients. While our algorithm captures other
microorganism species (including fungal and viral species), we did not
validate the performance on those. Finally, we included logic to extract
relevant quantitative and semi-quantitative concepts, however the performance
of this was variable due to syntactic heterogeneity. As a result, we continue
to provide quantitative captures as a feature of the MicrobEx algorithm,
however these were not included in our validation.

##  5 Conclusion

In this article we detail the development, validation, and use of our open-
source microbiology concept extractor (MicrobEx) algorithm and package. Our
workflow achieved excellent performance in two independent validation sets
with minimal customization. Our concept extraction Python package is designed
to be reused and adapted to individual institutions as an upstream process for
other clinical applications such as machine learning, clinical decision
support, and disease surveillance systems.

##  6 Acknowledgements

LNSP and YL are co-corresponding authors. This research is partly supported by
grants U01TR003528 & R01LM013337 from the National Institutes of Health (Luo),
grant 5T32LM01220304 from the National Library of Medicine (Eickelberg), and
grant R01HD105939 from the National Institute of Child Health & Human
Development (Sanchez-Pinto).

##  7 Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

Figure 1:  The MicrobEx algorithm structure. The input of our algorithm is a
whole or parsed section of a free-text microbiology culture reports. Within
the algorithm, a series of regular expression collections are applied to the
text input and the captures are associated with bacterial absence (negative),
bacterial presence (positive), microbiological species, potential bacterial
contamination, and uncertainty. Bacterial species captured are subsequently
mapped to both OHDSI and SNOMED concept IDs. Hierarchical decisions are
applied to the regular expression collection captures to categorize the
culture as positive or negative for bacteria. Figure 2:  Examples of annotated
reports for validation and error analysis on validation set reports. Colored
underlines correspond to parts of the report captured by the associated
regular expression collection. For bacterial culture positive status
classification, the concepts captured in each block are considered in a
hierarchical decision structure according to Figure 1. Examples 1 to 4
demonstrate algorithm annotation on cases found to be correctly classified as
positive and negative. Examples 5 to 8 depict four examples representative of
common misclassifications. Abbreviations: TP, true positive; TN, true
negative; FP, false positive; FN, false negative.

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# Re-picturing viscoelastic drag-reducing turbulence by introducing dynamics
of elasto-inertial turbulence

Wenhua Zhang1,2  Hongna Zhang1<EMAIL_ADDRESS>Yuke Li3  Fengchen Li1
<EMAIL_ADDRESS>1Key Laboratory of Efficient Utilization of Low and Medium
Grade Energy, MOE, School of Mechanical Engineering, Tianjin University,
Tianjin 300350, China 2Sino-French Institute of Nuclear Engineering and
Technology, Sun Yat-sen University, Zhuhai 519082, China 3Department of
Physics of Complex System, Weizmann Institute of Science, Rehovot 7610001
Israel

###### Abstract

Recently, the nature of viscoelastic drag-reducing turbulence (DRT),
especially maximum drag reduction (MDR) state, has become a focus of
controversy. It has long been regarded as polymers-modulated inertial
turbulence (IT), but is challenged by the newly proposed concept of elasto-
inertial turbulence (EIT). This study is to re-picture DRT in parallel plane
channels by introducing dynamics of EIT based on statistical and budget
analysis for a series of flow regimes from the onset of DR to EIT. Energy
conversion between velocity fluctuations and polymers as well as polymeric
pressure redistribution effect are of particular concern, based on which a new
energy self-sustaining process (SSP) of DRT is re-pictured. The numerical
results indicate that at low Reynolds number (Re), the flow enters laminar
regime before EIT-related SSP is formed with the increase of elasticity,
whereas, at moderate Re, EIT-related SSP can get involved and survive from
being relaminarized. This somehow explains the reason why relaminarization is
observed for small Re while the flow directly enters MDR and EIT at moderate
Re. Moreover, with the proposed energy picture, the newly discovered
phenomenon that the streamwise velocity fluctuations lag behind those in wall-
normal direction can be well explained. The re-pictured SSP certainly justify
that IT nature is gradually replaced by that of EIT in DRT with the increase
of elasticity.

##  1 Introduction

Adding small quantities of high-molecular weight polymers to Newtonian
turbulence can significantly reduce the flow drag. Since its discovery by Toms
(1948), extensive researches on its kinematics and dynamics have been carried
out (see, e.g., Lumley 1969; White & Mungal 2008; Graham 2014; Xi 2019), and
important concomitant phenomena were found. Especially, the MDR phenomenon
(Virk 1971), turbulence will not be completely removed, by polymers has long
been a challenge for understanding the essence of viscoelastic drag reducing
turbulence (DRT). So far, DRT was widely regarded as a perturbation of
Newtonian inertial turbulence (IT) by polymers. Starting from this point,
various theories of DR induced by polymers including the classical viscous
theory (Lumley 1969) and elastic theory (de Gennes 1990) were proposed. MDR
state was considered to be a form of “hibernating” turbulence, which is
inherently part of Newtonian turbulence but becomes unmasked by polymer
elasticity (Xi & Graham 2010; Zhu & Xi 2021). However, the mechanism of DRT
has not been unified yet and the proposed theories cannot explain all the
accompanying phenomena.

Recently, the knowledge about DRT, especially MDR, has been significantly
advanced benefiting from the investigations on elastic turbulence (ET,
Groisman & Steinberg 2000) and elasto-inertial turbulence (EIT, Samanta et al.
2013). Spectra analysis in DRT done by Watanabe & Gotoh 2010 and 2014 implied
the possible connection between DRT and ET. Different from pure IT and ET, EIT
is induced by elastic nonlinearity and the maintenance of disturbance requires
the participation of fluid inertia. One of its distinctive features is the
emergence of trains of spanwise oriented vortical structures with alternating
sign that appear on elongated sheets of highly stretched polymers (Dubief et
al. 2013; Shekar et al. 2019; Sid et al. 2018; Gillissen 2019; Choueiri et al.
2018). Therefore, EIT is essentially regarded as two-dimensional (2D)
turbulence (Sid et al. 2018; Gillissen 2019). Today, the connection between
DRT and EIT has become a hotspot in the field of visceoasltic turbulence.
Samanta et al. (2013) and Dubief et al. (2013) found that the DRT displays
features of EIT after the flow enters MDR state. Choueiri et al. (2018)
reported that the MDR asymptotic limit can be exceeded by carefully
controlling concentration of the dilute polymer solution, and further
increasing the polymer concentration will lead the flow to a saturated EIT
regime with the flow drag corresponding to that in MDR state. Their
observations directly support that the MDR state is essentially an EIT regime,
and there exists a “coexistence phase” of EIT and IT immediately before MDR.
In our recent work, we developed a characterization method of IT and EIT
related dynamics by decomposing the drag coefficients, and proposed a concept
that EIT-related dynamics start to play in DRT long before entering MDR and
DRT phenomenon is possibly the result of IT and EIT-related dynamics
interaction (Zhang et al. 2021b). Those work identifies there does exist a
link between DRT and EIT, although it is figured from the perspectives of
phenomenological comparisons.

The present paper further attempts to build a mechanistic link by drawing an
energy picture for the self-sustaning process (SSP) in DRT considering the
role of EIT. Regarding EIT, Dubief et al. (2013) first proposed its SSP cycle,
later confirmed by Terrapon et al. 2015, in which they argue that the unstable
nature of the nonlinear advection of polymers plays an important role, and
once triggered, EIT is self-sustained and fed upon the velocity fluctuations
created by the elastic instability. In a recent work (Zhang et al., 2021a), we
also pictured the energy transfer process in EIT and argue that rather than
the nonlinear advection of polymers, the nonlinear part of elastic shear
stress (ESS, the so-called “stress loss”), originated from the polymers and
turbulence interaction, plays a dominant role in forming sheet-like structures
and supplying energy to turbulent structures so as to sustain EIT. The role of
inertia seems to lift the polymer sheets towards the channel center, pumping
energy from near-wall region to the bulk region. Although its SSP still needs
to be further uncovered, it is certain that EIT follows a very different
turbulence self-sustaining mechanism from that of IT.

The present paper aims at re-picturing the energy transfer process in DRT by
introducing EIT-related dynamics and revealing the potential link between DRT
and EIT. It is organized as follows: Section 2 discribes the methods used in
this study, including problem formulation and computations details, budget
equations as well as the decompositions of ESS and pressure fluctuations;
Section 3 presents and analyzes the obtained results including the statistics
and the energy and stress budgets, and then propose an energy picture for SSP
in DRT; Section 4 gives the conclusions and outlooks.

##  2 Methods

###  2.1 Problem formulation and computational details

This work employs datasets from our recent DNSs of DRT passing a parallel
plane channels with the Oldroyd-B model, which have been soundly validated
(Zhang et al. 2021a and 2021b). In the three-dimensional channels, _x_ , _y_
and _z_ are the streamwise, wall-normal and spanwise respectively, and the
corresponding velocity components are _u_ , _v_ and _w_ , abbreviated as
${u_{i}}$ (${i=1,2,3}$ represents _x_ , _y_ and _z_ directions). For brevity,
variables with superscript “*” are dimensionless variables based on outer
scale, and those without superscript are original ones. The outer scale takes
the channel half height ${h}$ as the reference length, the bulk mean velocity
${u_{\text{b}}=\frac{1}{h}\int_{0}^{\delta}{{{\bar{u}}}d{y^{*}}}}$(${\bar{u}}$
is the local mean streamwise velocity) as the reference velocity, and
${h/u_{\text{b}}}$ and ${\rho}u_{\text{b}}^{2}$ as the reference time and
stress, respectively. Neglecting the volume force, the dimensionless governing
equations based on the outer scale and Oldroyd-B model are:

$\frac{{\partial u_{i}^{*}}}{{\partial x_{i}^{*}}}=0,$ (1)

$\frac{{\partial u_{i}^{*}}}{{\partial{t^{*}}}}+u_{j}^{*}\frac{{\partial
u_{i}^{*}}}{{\partial x_{j}^{*}}}=-\frac{{\partial{p^{*}}}}{{\partial
x_{i}^{*}}}+\frac{2}{{{\mathop{\rm Re}\nolimits}}}\frac{\partial}{{\partial
x_{j}^{*}}}\left({\frac{{\partial u_{i}^{*}}}{{\partial
x_{j}^{*}}}}\right)+\frac{{\partial\tau_{ij}^{*}}}{{\partial x_{j}^{*}}},$ (2)

$\frac{{\partial{c_{ij}}}}{{\partial{t^{*}}}}+u_{k}^{*}\frac{{\partial{c_{ij}}}}{{\partial
x_{k}^{*}}}={c_{ik}}\frac{{\partial u_{j}^{*}}}{{\partial
x_{k}^{*}}}+{c_{kj}}\frac{{\partial u_{i}^{*}}}{{\partial
x_{k}^{*}}}-\frac{1}{{Wi}}\left({{c_{ij}}-{\delta_{ij}}}\right),$ (3)

where, ${t^{*}}$ is time; ${p^{*}}$ is pressure; ${\tau_{ij}^{*}}$ is the
elastic stress, and is calculated by
${\tau^{*}_{ij}=\frac{\beta(c_{ij}-\delta_{ij})}{\text{ReWi}}}$;
${c_{ij}^{*}}$ is the conformation tensor; ${\delta_{ij}^{*}}$ is the unit
tensor; ${\text{Re}}=\it{2hu_{b}/\nu}$ is Reynolds number using ${2h}$;
${\text{Wi}}=\it{\lambda u_{b}/\delta}$ is Weissenberg number;
${\beta=\eta/\nu}$ is the viscosity contribution ratio; $\eta$ and $\nu$ are
contributions of additives and solvent to zero shear viscosity of solution,
respectively; $\lambda$ is the relaxation time of polymer solution.

The datasets under a wide range of flow conditions are selected at
$\beta=1/9$, Re=2500–20000, and Wi=0–60 (Wi=0 denotes Newtonian IT) which
covers a series of flow regime including IT and DRT from the onset of DR to
EIT as well as the re-laminarized flow regime by polymers. The numerical
details can be found in the supplementary materials or our previous work
(Zhang et al. 2021a and 2021b).

###  2.2 Budget equations

To understand the SSP of DRT, necessary budget equations including
decompositions of ESS and pressure fluctuations involved are given below.

The Reynolds stress budget equation can be obtained by re-arranging equation
(2):

${\partial\overline{u_{i}^{\rm{{}^{\prime}*}}u_{j}^{\rm{{{}^{\prime}*}}}}}/{{\partial{t^{*}}}}={P_{ij}^{\rm{R}}}{\rm{+}}{\Phi_{ij}}+{\varepsilon_{ij}^{\rm{R}}}-{G_{ij}}+{T_{ij}^{\rm{R}}},$
(4)

where,
${P_{ij}^{\rm{R}}={-\left({\overline{u_{i}^{{}^{\prime}*}u_{k}^{{}^{\prime}*}}\frac{{\partial\bar{u}_{j}^{*}}}{{\partial
x_{k}^{*}}}+\overline{u_{j}^{{}^{\prime}*}u_{k}^{{}^{\prime}*}}\frac{{\partial\bar{u}_{i}^{*}}}{{\partial
x_{k}^{*}}}}\right)}}$ is production rate;
${\Phi_{ij}={\overline{{{p}^{{}^{\prime}*}}\left({\frac{{\partial
u_{j}^{{}^{\prime}*}}}{{\partial x_{i}^{*}}}+\frac{{\partial
u_{i}^{{}^{\prime}*}}}{{\partial x_{j}^{*}}}}\right)}}}$ is pressure-strain
term; ${\varepsilon_{ij}={-\frac{4}{{{\mathop{\rm
Re}\nolimits}}}\overline{\frac{{\partial u_{i}^{{}^{\prime}*}}}{{\partial
x_{k}^{*}}}\frac{{\partial u_{j}^{{}^{\prime}*}}}{{\partial x_{k}^{*}}}}}}$ is
dissipation rate;
${G_{ij}={\left({\overline{\tau_{jk}^{\rm{{{}^{\prime}*}}}\frac{{\partial
u_{i}^{\rm{{{}^{\prime}*}}}}}{{\partial
x_{k}^{\rm{{*}}}}}}+\overline{\tau_{ik}^{\rm{{{}^{\prime}*}}}\frac{{\partial
u_{j}^{\rm{{{}^{\prime}*}}}}}{{\partial x_{k}^{\rm{{*}}}}}}}\right)}}$ is
elastic stress-strain term; $\frac{\partial}{\partial
y^{*}}\left(-\partial\overline{{u}_{i}^{{}^{\prime}*}{u}_{j}^{{}^{\prime}*}{v}^{{}^{\prime}*}}+\partial\overline{{\tau}_{j2}^{{}^{\prime}*}{u}_{i}^{{}^{\prime}*}}+\partial\overline{{\tau}_{i2}^{{}^{\prime}*}{u}_{j}^{{}^{\prime}*}}+\frac{2}{\operatorname{Re}}\frac{\partial}{\partial
y^{*}}\overline{{u}_{i}^{{}^{\prime}*}{u}_{j}^{{}^{\prime}*}}\right)-\left(\overline{\frac{\partial{{{{p}}}^{{}^{\prime}*}}{u}_{j}^{{}^{\prime}*}}{\partial
x_{i}^{*}}}+\overline{\frac{\partial{{{{p}}}^{{}^{\prime}*}}{u}_{i}^{{}^{\prime}*}}{\partial
x_{j}^{*}}}\right)$ is transport rate.

Pressure redistribution term $\Phi_{ij}$ is an important link in turbulent
SSP, responsible for the generation of TKE in the wall-normal direction.
Following Ptasinski et al., (2003) and Terrapon et al. (2015), the fluctuating
pressure $p^{\prime}$ can be decomposed into rapid, slow and polymer
contributions to investigate the modulation effect of polymers, as
${{{p^{\prime}}^{*}}\left(x\right){\rm{=}}{p^{\prime}}_{R}^{*}\left(x\right)+{p^{\prime}}_{s}^{*}\left(x\right)+{p^{\prime}}_{p}^{*}\left(x\right)}$,
where:

${\frac{{{\partial^{2}}p_{\rm{R}}^{{}^{\prime}*}}}{{{\partial^{2}}x_{i}^{*2}}}=-2\frac{{d{{\bar{u}}^{*}}}}{{d{y^{*}}}}\frac{{\partial
v^{\prime}}}{{\partial{x^{*}}}}},{\frac{{{\partial^{2}}p_{\rm{S}}^{{}^{\prime}*}}}{{{\partial^{2}}x_{i}^{*2}}}=-\frac{{\partial
u_{i}^{{}^{\prime}*}}}{{\partial x_{j}^{*}}}\frac{{\partial
u_{j}^{{}^{\prime}*}}}{{\partial
x_{i}^{*}}}+\frac{{{d^{2}}\overline{{{v}^{{}^{\prime}*2}}}}}{{d{y^{*2}}}}},{\frac{{{\partial^{2}}p_{\rm{P}}^{{}^{\prime}*}}}{{{\partial^{2}}x_{i}^{*2}}}=\frac{{{\partial^{2}}\tau_{ij}^{{}^{\prime}*}}}{{\partial
x_{i}^{*}\partial x_{j}^{*}}}},$ (5)

and the first two are identified as inertial contribution.

The elastic stress budget equation can be obtained by arranging equation (3):

${{\partial\bar{\tau}_{ij}^{*}}}/{{\partial{t^{*}}}}={P_{ij}^{\rm{E}}}+{G_{ij}}+{T_{ij}^{\rm{E}}}{\rm{+}}{V_{ij}}+{\varepsilon_{ij}^{\rm{E}}},$
(6)

where,
${P_{ij}^{\rm{E}}}={\left({\bar{\tau}_{ik}^{*}\frac{{\partial\bar{u}_{j}^{*}}}{{\partial
x_{k}^{*}}}+\bar{\tau}_{kj}^{*}\frac{{\partial\bar{u}_{i}^{*}}}{{\partial
x_{k}^{*}}}}\right)}$ is production rate; ${G_{ij}}$ is elastic stress-strain
term; ${T_{ij}^{\rm{E}}}={-\overline{\frac{{\partial
v^{{}^{\prime}*}\tau_{ij}^{{}^{\prime}*}}}{{\partial y^{*}}}}}$ is turbulent
transport rate; ${V_{ij}^{*}={\frac{{2\beta}}{{{\mathop{\rm
Re}\nolimits}Wi}}\left({\frac{{\partial\bar{u}_{j}^{*}}}{{\partial
x_{i}^{*}}}+\frac{{\partial\bar{u}_{i}^{*}}}{{\partial x_{j}^{*}}}}\right)}}$
is viscous effect term;
${\varepsilon_{ij}^{\rm{E}}={-\frac{1}{{Wi}}\bar{\tau}_{ij}^{*}}}$ is elastic
disspation rate.

As the excitation of EIT is independent of the viscous effect term in (6)
which is a linear term, the mean ESS can be decomposed into linear part
$\bar{\tau}_{E1}^{*}$ and nonlinear part $\bar{\tau}_{E2}^{*}$ (see Zhang et
al. 2021a and 2021b for more information):

$\bar{\tau}_{E}^{*}{\rm{=}}{\bar{\tau}_{E1}^{*}}+{\bar{\tau}_{E2}^{*}},$ (7)

where, ${\bar{\tau}_{E1}^{*}={\frac{\beta}{{{\mathop{\rm
Re}\nolimits}}}\frac{{\partial\bar{u}^{*}}}{{\partial y^{*}}}}}$ ;
${\bar{\tau}_{E2}^{*}={Wi\left({\bar{\tau}_{yy}^{*}\frac{{\partial\bar{u}^{*}}}{{\partial
y^{*}}}{\rm{+}}\left({\overline{\tau_{1k}^{\rm{{}^{\prime}*}}\frac{{\partial
v^{{}^{\prime}*}}}{{\partial
x_{k}^{*}}}}+\overline{\tau_{k2}^{{}^{\prime}*}\frac{{\partial
u^{{}^{\prime}*}}}{{\partial x_{k}^{*}}}}}\right)-\overline{\frac{{\partial
u_{k}^{{}^{\prime}*}\tau_{E}^{{}^{\prime}*}}}{{\partial
x_{k}^{*}}}}}\right)}}$ is caused by the interaction between polymers and
turbulence, that is, the so-called ”stress loss”.

##  3 Results and analysis

###  3.1 Statistical properties

At first, we present some results for the flow statistics. Figure 1
illustrates friction coefficient $f$, global TKE production rate $P_{\rm k}$
by RSS, global TEE production rate $P_{\rm e}$ by NESS $\tau_{\rm E2}$ as well
as the global energy transfer rate $G^{*}$ between EE and TKE. The datasets
qualitatively reproduces the interesting observations in Choueiri’s
experiments, which can be read from the drag coefficients $f$ as shown in
Figure 1a. At a medium Re (e.g., 6000 or 20000), the viscoelastic turbulence
experiences the regimes of the onset of DR, LDR, HDR, MDR and EIT with the
increase of Wi. Correspondingly, evolutions of ${P_{\rm{k}}^{*}}$,
${P_{\rm{e}}^{*}}$ and ${G^{*}}$ share similar trends for moderate Re. Since
the onset of DR, ${P_{\rm{k}}^{*}}$ starts to decrease, while
${P_{\rm{e}}^{*}}$ starts to increase. When the flow enters MDR and EIT
regime, ${P_{\rm{k}}^{*}}$ becomes negligible, whereas ${P_{\rm{e}}^{*}}$
becomes dominant and saturated. During this process, $G^{*}$ firstly decreases
to a valley and then starts to increase again with Wi. More strikingly, the
value of $G^{*}$ is negative below the critical Wi of MDR indicating globally
the energy is transferred from TKE to TEE, but positive above the critical Wi
of MDR indicating the globally inversed energy transfer. Moreover, the
evolution of $G^{*}$ after the valley point for two Re collapse to each other
showing a power$-$law relationship with Wi implying a scaling for $G^{*}$
independent of Re, which needs further investigations in the future. At a
small Re (e.g., 2500), it experiences the regimes of the onset of DR, LDR,
HDR, laminar flow regime and EIT ($Wi>15$). The existence of the laminarized
flow regime for moderate Wi ($\approx 6-15$ in the current work) which
distinguishes the origins of DRT in two different stages. Evolutions of
${P_{\rm{k}}^{*}}$, ${P_{\rm{e}}^{*}}$ and ${G^{*}}$ show a different behavior
for low Re (e.g., 2500) comparing with that for moderate Re. Before
relaminarization, ${P_{\rm{k}}^{*}}$ is dominant and the value of $G^{*}$ is
always negative indicating the turbulence is always IT dominated which is
inhibited by polymers; when the flow enters laminar regime for Wi $\approx
6-15$, ${P_{\rm{k}}^{*}}$, ${P_{\rm{e}}^{*}}$ and ${G^{*}}$ become eliminated
in laminar regime. In contrast, during the process of entering EIT,
${P_{\rm{e}}^{*}}$ is dominant and the value of $G^{*}$ is positive for
$Wi>15$ indicating the turbulence is sustained by polymers. The above results
demonstrate a global picture for the elastic effect on the viscoelastic
turbulence: (1) at small Wi (LDR for medium Re or before relaminarization for
small Re), the flow is IT-dominated and in these cases the traditional TKE
production by Reynolds stress is suppressed by polymers and thus SSP of IT is
modulated; (2) increasing Wi above a critical $Wi_{\rm c}$, a new TKE
generation term induced by elastic effect appears, which grows with Wi and
changes the SSP cycle of DRT; (3) further increase Wi, the turbulence is
sustained from by $P_{\rm k}^{*}$ to $P_{\rm e}^{*}$, and the nature of
turbulence changes from polymer-modulated IT to EIT.

Figure 1:  Evolution of statistics with Wi: (a) friction coefficient $f$; (b)
global TKE production
${P_{\rm{k}}^{*}{\rm{=}}-\int_{0}^{2}{\overline{{{u^{\prime}}^{*}}{{v^{\prime}}^{*}}}\frac{{\partial{{\bar{u}}^{*}}}}{{\partial{y^{*}}}}d{y^{*}}}}$
and EE production rate
${P_{\rm{e}}^{*}{\rm{=}}-\int_{0}^{2}{\bar{\tau}_{E2}^{\rm{+}}\frac{{\partial{{\bar{u}}^{*}}}}{{\partial{y^{*}}}}d{y^{*}}}}$;
(c) global energy transformation rate
${G^{*}{\rm{=}}\int_{0}^{2}{\overline{\tau_{ij}^{{}^{\prime}*}\frac{{\partial
u_{i}^{{}^{\prime}*}}}{{\partial x_{j}^{*}}}}d{y^{*}}}}$. $G^{*}>0$ represents
the energy is globally transferred from polymers to the flow structures,
otherwise from the flow structures to the polymers.

The characteristics of Reynolds stress attract the most attention in the
analysis of IT and DRT. However, although there exist obvious $u^{\prime}$ and
$v^{\prime}$ (especially $u^{\prime}$) in EIT,
$\overline{{{u^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ is extremely small, implying
somehow the decoupling of $u^{\prime}$ and $v^{\prime}$. From our recent
demonstrated SSP cycle of EIT (Zhang et al. 2021b), the TKE production in
different directions and the embedded TKE transfer amongst them clearly differ
from that of IT, which is possible to delay or decouple the generation of
$u^{\prime}$ from $v^{\prime}$. Figure 2 presents the relative cross-
correlation $C_{uv}(\xi)$ between $u^{\prime}$ and $v^{\prime}$ over $z$ and
$t$ where
$C_{uv}(\xi)={\left\langle{u^{\prime}\left({{\bf{x}},t}\right)v^{\prime}\left({{\bf{x}}+{\bf{r}},t}\right)}\right\rangle/\left\langle{u^{\prime}\left({{\bf{x}},t}\right)v^{\prime}\left({{\bf{x}},t}\right)}\right\rangle}$.
Strikingly, unlike that in IT, there exists an obvious lag effect between
$u^{\prime}$ and $v^{\prime}$ in HDR and EIT: $u^{\prime}$ lags behind
$v^{\prime}$ ($v^{\prime}$ induces the generation of $u^{\prime}$ in other
words). Moreover, in EIT, the lagging distance $\xi_{\rm{max}}$ can reach
several times of the half channel height $h$. For example, at Re = 1000, Wi =
60, $\xi$ can reach more than $5h$. In addition, $\xi_{\rm{max}}$ gradually
increases with an approximately linear relationship with Wi and
$\xi_{\rm{max}}$ also increases with Re due to the inertia enhancement. The
above results once again imply a different picture of TKE transfer process
among velocity components in IT and DRT.

Figure 2:  Distributions of $C_{uv}(\xi)$: (a) in $\xi-y$ plane, (b) at
$y^{*}\approx 0.5$. Inset: $\xi_{\rm{max}}$ vs. Wi at different Re.
$\xi_{\rm{max}}$ corresponds to the $\xi$ where $C_{uv}(\xi)$ reaches the
maximum.

###  3.2 Budget analysis

This section re-evaluate the energy transfer process in DRT by budget analysis
of TKE, EE and stresses among different components as illustrated in Figure 3
considering whether the SSP cycle of EIT is formed. As well-known, in IT, RSS
absorbs energy from the mean motion and produces streamwise TKE, which is
transformed into wall-normal TKE through fast and slow pressure
redistribution, and regenerates RSS driven by wall-normal velocity
fluctuations. The above process forms a cycle to maintain the turbulence
against dissipation. As demonstrated in Zhang et al. (2021b), when the flow
enters EIT, it follows a very different SSP cycle comparing with that in IT.
The most striking differences lie in: (1) rather than by RSS, NESS absorbs
energy from the mean motion to sustain the turbulence; (2) the conversion of
wall-normal TKE into EE and streamwise EE into TKE are added; (3) the
interaction between polymers and turbulence suppress both RSS and NESS. In
addition, polymers also contributes to modify the energy redistribution term
in EIT which affects the generation of
$\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ in the SSP cycle of EIT
(Terrapon et al., 2015). EIT-related SSP cycle is possibly formed and survives
from being disspated once the above features appear. In the following, special
attention will be paid to the performance of terms related with these
features.

Common with the earlier studies (Xi, 2019), $P_{xx}^{\rm{R}}$ and
$\epsilon_{xx}^{\rm{R}}$ gradually decrease with the increase of Wi,
indicating the suppression of elasticity on the IT-related SSP in DRT.
Moreover, it is observed that (1) at a medium Re (e.g., 6000),
$P_{xx}^{\rm{E}}$ continuously grows with Wi and exceeds $P_{xx}^{\rm{R}}$ at
$Wi\approx 9$ in HDR stage, indicating that for cases of $\rm{Wi}>9$, more
energy is transferred from the mean flow to the polymers by NESS than that to
the turbulent structures and the turbulence is dominated by NESS; (2) at a
small Re (e.g., 2500), the flow is dominated by $P_{xx}^{\rm{R}}$ before the
relaminarization while by $P_{xx}^{\rm{E}}$ when EIT is excited. Interesting
to notice from profiles of $G_{xx}$ is that it experiences a morphological
change with the increase of Wi. For a small Wi (e.g., at Re=6000, $Wi<3$ and
Re=2500, $Wi<5$ before the re-laminarization), the energy is transferred from
$\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$ to the polymers in the bulk
region, which increases with Wi and introduces additional energy dissipation.
This effect modulates the IT-dynamics related SSP. For moderate and large Wi
(e.g., at Re=6000, $Wi>3$ and Re=2500, $Wi>15$ after the emergence of EIT
regime), the energy transfer direction turns over with the increase of Wi. The
negative energy transfer region (from the polymers to
$\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$) shifted from the near-wall
region to the bulk region, and enlarges with Wi. Meanwhile, the positive
energy transfer region (from $\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$
to the polymers) shifted from the bulk region to the near-wall region, and
shrinks with the Wi. Notable is that the region where polymers absorb energy
shifted upward with Wi, which is probably originated from the turbulence
streaks lifting as reported in Zhang et al. (2021b). The core area is mainly
characterized by the energy transfer of elastic energy to
$\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$, which indicates that EIT
dynamics can be self-sustained, i.e., EIT-related SSP involves. With the
increase of Wi, more elastic energy is transferred from polymers to TKE until
it completely develops into EIT.

Figure 3:  Budgets analysis for
$\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$,
$\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$,
$\overline{{{u^{\prime}}^{*}}{{v^{\prime}}^{*}}}$, $\overline{\tau_{xx}^{*}}$
and $\overline{\tau_{xy}^{*}}$ at: (a) Re = 6000, Wi = 0 (Newtonian IT); (b)
Re = 6000, Wi = 1; (c) Re = 6000, Wi = 4; (d) Re = 6000, Wi = 6; (e) Re =
6000, Wi = 10 (MDR); (f) Re = 6000, Wi = 30 (EIT); (g) Re = 2500, Wi = 0
(Newtonian IT); (h) Re = 2500, Wi = 5 (before relaminarization); (i) Re =
2500, Wi = 30 (EIT).

Comparing with $G_{xx}$, the energy transfer rate $G_{yy}$ between
$\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ and EE is negative for all
the covered flow regimes, indicating that the polymers always suppress the
normal-wise turbulence. With the increase of Wi, as
$\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ is suppressed by polymers,
less energy is transferred from
$\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ to polymers. Different from
$\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$, the generation of
$\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ is from pressure
redistribution. Figures 4 shows the evolutions of rapid, slow and polymeric
pressure redistribution terms under different flow conditions. In IT both
$\phi_{xx}^{\rm{R}}$ and $\phi_{xx}^{\rm{S}}$ behave negative in the
turbulence core area, i.e., TKE is redistributed to other directions from
$\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$ through pressure
fluctuations; the behaviors of both $\phi_{yy}^{\rm{R}}$ and
$\phi_{yy}^{\rm{S}}$ are opposite with that of $\phi_{xx}^{\rm{R}}$ and
$\phi_{xx}^{\rm{S}}$: near the wall (at $y^{*}<0.1$), $\phi_{yy}$ is negative,
indicating the energy is redistributed to that in spanwise forming turbulent
streaks and in the core area (at $y^{*}>0.1$), $\phi_{yy}$ is positive,
indicating the energy is redistributed to that in the wall-normal direction
and forms $\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$. In DRT, for small
Wi when the turbulence is IT dominated, both $\phi^{\rm{R}}$ and
$\phi^{\rm{S}}$ are far larger than $\phi^{\rm{P}}$, and dominate pressure
redistribution. With the increase of Wi, $\phi^{\rm{P}}$ is gradually excited
on one hand and $\phi^{\rm{R}}$ and $\phi^{\rm{S}}$ are dampened on the other
hand. When Wi is large enough, $\phi^{\rm{P}}$ catches up with those two
terms, and can affect the pressure redistribution. Strikingly, the evolutions
of $\phi^{\rm{P}}$ demonstrate a similar trend with that of $G_{xx}$ and
$P^{\rm E}_{xy}$ with the increase of Wi. For cases of small Wi, e.g., $Wi<3$
at Re = 6000 and $Wi<5$ at Re = 2500, $\phi_{yy}^{\rm{P}}$ is small and
negative, i.e., $\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ is
transferred to other directions through the polymeric pressure term. However,
for large Wi, e.g., $Wi>3$ at Re = 6000 and $Wi>30$ at Re = 2500, there is an
obvious positive polymer effect near the wall, i.e., TKE in other directions
is transferred to the wall-normal direction through the polymer pressure
effect. Polymers play a gain effect on the normal-wise TKE generation which
sustains the generation of $\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$
and turbulent SSP. Therefore, the existence of elastic effect introduces a new
pressure redistribution mechanism in DRT. It plays a modulation pressure
redistribution effect when Wi is small, but can reshape the pressure
redistribution effect when Wi is large enough, making the correlation between
${u^{\prime}}^{*}$ and ${v^{\prime}}^{*}$ in DRT more complex. When entering
EIT, the characteristics of pressure redistribution effect are obviously
different from that in IT, and the polymer contribution plays a role in
sustaining ${v^{\prime}}^{*}$ generation.

From the above budget analysis, there is a distinct SSP from IT in
viscoelastic DRT. On one hand, the existence of elastic effect suppresses the
self-sustaining mechanism related to inertial turbulence. On the other hand,
when Wi is large enough, a cycle of EIT-related SSP is introduced, which
finally replaces that of IT.

Figure 4:  Profiles of total pressure distribution effect on
$\overline{{{u^{\prime}}^{*}}{{u^{\prime}}^{*}}}$,
$\overline{{{v^{\prime}}^{*}}{{v^{\prime}}^{*}}}$ and the divided rapid, slow
and elastic contributions at: (a) Re = 6000, Wi = 0 (Newtonian IT); (b) Re =
6000, Wi = 1; (c) Re = 6000, Wi = 4; (d) Re = 6000, Wi = 6; (e) Re = 6000, Wi
= 10 (MDR); (f) Re = 6000, Wi = 30 (EIT); (g) Re = 2500, Wi = 0 (Newtonian
IT); (h) Re = 2500, Wi = 5 (before relaminarization); (i) Re = 2500, Wi = 30
(EIT).

###  3.3 Re-picturing viscoelastic drag-reducing turbulence

Based on the analysis, we re-picture the energy transfer process for
viscoelastic DRT from IT to EIT by combing TKE and stress budgets as shown in
Figure 5. The cycles of SSP for IT and EIT are illustrated by the cyan paths
and the blue paths, repectively. The energy process in DRT from onset of DR to
MDR, is illustrated by the dashed paths colored gradient from orange to blue.
Under LDR conditions, polymers passively respond to the SSP of IT. TKE is
partially transformed into EE through the interaction between polymers and
turbulence in both the streamwise and wall-normal directions. NESS is also
formed due to wall-normal polymer extension. NESS absorbs energy from mean
motion and turns into streamwise EE. The above process is a passive response
to the SSP of IT, which cannot support but intervent its SSP. Pressure
fluctuations induced by polymers distribute TKE from the wall-normal direction
to other directions such as the streamwise which is opposing with that of
inertial pressure distribution although with almost neglibile magnitude.
Moreover, there is an orange path showing the interaction between polymer and
turbulence, which inhibits both RSS and NESS and is another important factor
for polymers to intervent with the SSP of IT. When Wi is high enough, the
energy process of DRT gradually transits to that of EIT state. In this
process, the SSP of IT weakens and SSP cycle of EIT dominates the flow
replacing that of IT.

Figure 5:  Energy picture of viscoelastic DRT. — and —: SSP cycles in IT and
EIT. The —: the bypass of energy process caused by the passive response of
polymers on IT. - -: the path change from passive response on IT SSP to
actively participating in EIT SSP with the increase of Wi. —: energy source
and dissipation.

With this energy picture in mind, the different DR phenomena at low Re and
moderate Re can be well unified. As shown in Figures 3 and 4, the two
important links related with SSP cycle of EIT discribed by $G_{xx}$ and
$\Phi^{P}_{yy}$ are not formed untile $Wi=5$ for $\rm{Re}=2500$, but can be
well hold even at $\rm{Wi}=4$ for $\rm{Re}=6000$. Further increasing Wi, at
$\rm{Re}=2500$ the flow is laminarized since the weaker IT disappears before
EIT is excited, whereas, at $\rm{Re}=6000$ the flow finally enters EIT without
a period of relaminarization as EIT can be well sustained before IT
disappears.

We can also easily understand the reason why $u^{\prime}$ lags behind
$v^{\prime}$ in DRT observed in Figure 2. In IT, DRT and EIT, the formation of
$u^{\prime}$ are all inseparable with $v^{\prime}$ but with different routine.
In IT, the energy absorbed by RSS sustains the formation of $u^{\prime}$. The
formed TKE in streamwise is then redistributed to $v^{\prime}$ directly
through the pressure redistribution effect. The formation $u^{\prime}$ and the
energy redistribution to $v^{\prime}$ are direct processes and rapid terms.
However, in DRT and EIT regime, polymers modulate even change the formation
mechanism of $u^{\prime}$, as illustrated by the orange path and blue path in
Figure 5. Different from that in IT, $u^{\prime}$ is mainly derived from the
conversion of streamwise EE in EIT regime, and the generation of streamwise EE
is from the wall-normal EE caused by $v^{\prime}$. Due to the polymers memory
effect, the formation and development of this process are indirect and takes a
certain time, related to the fluid relaxation time, which results in the
phenomenon $u^{\prime}$ lags behind $v^{\prime}$. For DRT, with the increase
of Wi, the IT related SSP gradually weakens, while the EIT related SSP
enhances and the time memory effect increases, resulting in the phenomenon
that the lag distance increases.

##  4 Conclusions

In summary, we have re-pictured viscoelastic turbulent channel flow,
especially its TKE transfer process, based on statistical and budget analysis
throughout a wide range of flow state covering IT, LDR, HDR, MDR to EIT and
relaminarized flow regime. The evolutionary comparison of $P_{k}$, $P_{e}$ and
$G$ with Wi provide quantitative evidence that the nature of DRT gradually
changes from IT to EIT. The statistical results of $C_{uv}$ among IT, DRT and
EIT are in sharp contrast, which further implies that there exists a SSP
different from IT in DRT. For energy and stress budget, polymers do have the
passive feedback effect of modulating and interfering with the SSP of IT, but
more importantly, when Wi exceeds a certain critical value long before the
flow enters MDR state, polymers can actively excite EIT-related SSP. The two
obvious characteristics of this active behavior are that polymers pump energy
from mean motion to induce streamwise turbulent fluctuations, and that
polymers can induce the pressure distribution to provide energy for the
formation of wall-normal turbulent fluctuations. These evidences fully suggest
that the nature of DRT experienced the transition from pure IT, IT modulated
by polymers, participation of EIT and finally maybe pure EIT. The different
drag reduction phenomena at low (Re=2500) and moderate (Re=6000, 20000) Re are
unified through the proposed energy picture. The reason why the relaminarized
phenomenon doesn’t occur at moderate Re is that EIT dynamics gets involved in
the DRT earlier, so that the turbulence can be maintained and feeded by
elasticity instability. The proposed energy picture can well illustrate why
the existing DRT models cannot well predict HDR and MDR flows since EIT
dynamics are not involved, also provides a theoretical fundament for DRT
modelling.

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