prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the infall rate and gas density in the magnetized model compare to non-magnetized accretion?

Read the following text and answer briefly.

Context 1: \section{Introduction}
The averaged quantities can be obtained in two different ways in
magnetohydrodynamics  The first way is to solve 3D MHD equations
and then average the results  The second way is to solve some
system of equations on averages  Combination of numerical
simulations and averaged theory brings phenomenology that can
describe observations or experimental data 

The problem of spherically symmetric accretion takes its origin
from Bondi's work \citep{bondi}  He presented idealized
hydrodynamic solution with accretion rate $\dot{M}_B $ However,
magnetic field $\vec{B}$ always exists in the real systems  Even
small seed $\vec{B}$ amplifies in spherical infall and becomes
dynamically important \citep{schwa} 

Magnetic field inhibits accretion \citep{schwa}  None of many
theories has reasonably calculated the magnetic field evolution
and how it influences dynamics  These theories have some common
pitfalls  First of all, the direction of magnetic field is usually
defined  Secondly, the magnetic field strength is prescribed by
thermal equipartition assumption  In third, dynamical effect of
magnetic field is calculated with conventional magnetic energy and
pressure  All these inaccuracies can be eliminated 

In Section 2\ref{section_method} I develop a model that abandons
equipartition prescription, calculates the magnetic field
direction and strength and employs the correct equations of
magnetized fluid dynamics  In Section 3\ref{results} I show this
accretion pattern to be in qualitative agreement with Sgr A*
spectrum models  I discuss my assumptions in Section 4
\ref{discussion} 

\section{Analytical method}\label{section_method}
 Reasonable turbulence evolution model is the key difference of my
 method  I build an averaged turbulence theory that corresponds to
numerical simulations  I start with the model of isotropic
turbulence that is consistent with simulations of collisional MHD
in three regimes  Those regimes are decaying hydrodynamic
turbulence, decaying MHD turbulence and dynamo action  I introduce
effective isotropization of magnetic field in 3D model 
Isotropization is taken to have a timescale of the order of
dissipation timescale that is a fraction $\gamma\sim1$ of the
Alfven wave crossing time $\tau_{\rm diss}=\gamma r/v_A $

Common misconception exists about the dynamical influence of
magnetic field  Neither magnetic energy nor magnetic pressure can
represent $\vec{B}$ in dynamics  Correct averaged Euler and energy
equations were derived in \citep{scharlemann} for radial magnetic
field  Magnetic force $\vec{F}_M=[\vec{j}\times\vec{B}]$ can be
averaged over the solid angle with proper combination of
$\vec{\nabla}\cdot\vec{B}=0 $ I extend the derivation to random
magnetic field without preferred direction  Dynamical effect of
magnetic helicity \citep{biskamp03} is also investigated  I
neglect radiative and mechanical transport processes 

The derived set of equations requires some modifications and
boundary conditions to be applicable to the real astrophysical
systems  I add external energy input to turbulence to balance
dissipative processes in the outer flow  The outer turbulence is
taken to be isotropic and has magnetization $\sigma\sim1 $
Transonic smooth solution is chosen as possessing the highest
accretion rate as in \citep{bondi} 

\begin{figure}\label{fig1}
  \includegraphics[height= 5\textheight]{velocities}
  \caption{Normalized to Keplerian speed characteristic velocities of magnetized flow  Horizontal lines correspond to self-similar solution $v\sim r^{-1/2} $}
\end{figure}

\section{Results \& Application to Sgr A*}\label{results}

\begin{figure}\label{fig2}
  \includegraphics[height= 5\textheight]{magnetization}
  \caption{Plot of magnetization $\sigma=(E_M+E_K)/E_{Th}$ with radius }
\end{figure}
The results of my calculations confirm some known facts about
spherical magnetized accretion, agree with the results of
numerical simulations and have some previously unidentified
features 

Initially isotropic magnetic field exhibits strong anisotropy with
larger radial field $B_r $ Perpendicular magnetic field
$B_\perp\ll B_r$ is dynamically unimportant in the inner accretion
region Fig\ref{fig1}  Because magnetic field dissipates, infall
onto the black hole can proceed \citep{schwa} 

Turbulence is supported by external driving in the outer flow
regions, but internal driving due to freezing-in amplification
takes over in the inner flow Fig\ref{fig2}  Magnetization of the
flow increases in the inner region with decreasing radius
consistently with simulations \cite{igumen06}  Density profile
appears to be $\rho\sim r^{-1 25}$ that is different from
traditional ADAF scaling $\rho\sim r^{-1 5}$ \citep{narayan}  Thus
the idea of self-similar behavior is not supported 

Compared to non-magnetized accretion, infall rate is 2-5 times
smaller depending on outer magnetization  In turn, gas density is
2-5 times smaller in the region close to the black hole, where
synchrotron radiation emerges \citep{narayan}  Sgr A* produces
relatively weak synchrotron \citep{narayan}  So, either gas
density $n$ or electron temperature $T_e$ or magnetic field $B$
are small in the inner flow or combination of factors works  Thus
low gas density in magnetized model is in qualitative agreement
with the results of modelling the spectrum 

Flow is convectively stable on average in the model of moving
blobs, where dissipation heat is released homogeneously in volume 
Moving blobs are in radial and perpendicular pressure
equilibriums  They are governed by the same equations as the
medium 

\section{Discussion \& Conclusion}\label{discussion}
The presented accretion study self-consistently treats turbulence
in the averaged model  This model introduces many weak assumptions
instead of few strong ones 

I take dissipation rate to be that of collisional MHD simulations 
But flow in question is rather in collisionless regime 
Observations of collisionless flares in solar corona
\citep{noglik} gives dissipation rate $20$ times smaller than in
collisional simulations \citep{biskamp03}  However, flares in
solar corona may represent a large-scale reconnection event rather
than developed turbulence  It is unclear which dissipation rate is
more realistic for accretion 

Magnetic field presents another caveat  Magnetic field lines
should close, or $\vec{\nabla}\cdot\vec{B}=0$ should hold  Radial
field is much larger than perpendicular in the inner region 
Therefore, characteristic radial scale of the flow is much larger
than perpendicular  If radial turbulence scale is larger than
radius, freezing-in condition does not hold anymore  Matter can
freely slip along radial field lines into the black hole  If
matter slips already at the sonic point, the accretion rate should
be higher than calculated 

Some other assumptions are more likely to be valid  Diffusion
should be weak because of high Mach number that approaches unity
at large radius  Magnetic helicity was found to play very small
dynamical role  Only when the initial turbulence is highly
helical, magnetic helicity conservation may lead to smaller
accretion rate  Neglect of radiative cooling is justified a
posteriori  Line cooling time is about $20$ times larger that
inflow time from outer boundary 

The study is the extension of basic theory, but realistic
analytical models should include more physics  The work is
underway 
\begin{theacknowledgments}
I thank my advisor Prof  Ramesh Narayan for fruitful discussions 
\end{theacknowledgments}

\bibliographystyle{aipproc}


Context 2: Paper Info

Title: Is In-hospital Meta-information Useful for Abstractive Discharge Summary Generation?
Publish Date: 10 Mar 2023
Author List: Mamoru Komachi (from Tokyo Metropolitan University), Takashi Okumura (from Kitami Institute of Technology), Hiromasa Horiguchi (from National Hospital Organization), Yuji Matsumoto

Figure

Fig  1 Example of part of a discharge summary which is a dummy we created 
Fig  2  Overview of our proposed method A new feature embedding layer encoding hospital, physician, disease, and length of stay is added to the standard transformer architecture The figure shows an example of hospital embedding 
Statistics of our data for experiment 
of summarization models with different meta-information The best results are highlighted in bold Each score is the average of three models with different seeds The BS and BR indicate BERTScore and BLEURT, respectively 
Statistics on the number of cases handled by physicians C/P denotes Cases/Physician, which indicates how many cases an individual physician has Method of Grouping Physician IDs A most naive method of mapping physician IDs to features is without any grouping process The data contains 4,846 physicians, so |M | was set to 4,846 However it caused our model's training to be unstable This might be due to the many physician IDs appearing for the first time in the test time Table

abstract

During the patient's hospitalization, the physician must record daily observations of the patient and summarize them into a brief document called "discharge summary" when the patient is discharged  Automated generation of discharge summary can greatly relieve the physicians' burden, and has been addressed recently in the research community 
Most previous studies of discharge summary generation using the sequenceto-sequence architecture focus on only inpatient notes for input  However, electric health records (EHR) also have rich structured metadata (e g , hospital, physician, disease, length of stay, etc ) that might be useful  This paper investigates the effectiveness of medical meta-information for summarization tasks 
We obtain four types of meta-information from the EHR systems and encode each meta-information into a sequence-to-sequence model  Using Japanese EHRs, meta-information encoded models increased ROUGE-1 by up to 4 45 points and BERTScore by 3 77 points over the vanilla Longformer  Also, we found that the encoded meta-information improves the precisions of its related terms in the outputs 
Our results showed the benefit of the use of medical meta-information 

INTRODUCTION

Clinical notes are written daily by physicians from their consults and are used for their own decision-making or coordination of treatment  They contain a large amount of important data for machine learning, such as conditions, laboratory tests, diagnoses, procedures, and treatments  While invaluable to physicians and researchers, the paperwork is burdensome for physicians ,  
Discharge summaries, a subset of these, also play a crucial role in patient care, and are used to share information between hospitals and physicians (see an example in Figure )  It is created by the physician as a summary of notes during hospitalization at the time of the patient's discharge, which is known to be very time-consuming 
Researchers have begun to apply automatic summarization techniques to address this problem -   Previous studies used extractive or abstractive summarization methods, but most of them focused on only progress notes for inputs  Properly summarizing an admission of a patient is a quite complex task, and requires various meta-information such as the patient's age, gender, vital signs, laboratory values and background to specific diseases 
Therefore, discharge summary generation needs more medical meta-information, than similar but narrower tasks such as radiology report generation  However, what kind of meta-information is important for summarization has not been investigated, even though it is critical not only for future research on medical summarization but also for the policy of data collection infrastructure 
In this paper, we first reveal the effects of meta-information on neural abstractive summarization on admissions  Our model is based on an encoder-decoder transformer with an additional feature embedding layer in the encoder (Figure )  Hospital, physician, disease, and length of stay are used as meta-information, and each feature is embedded in the vector space 
For experiments, we collect progress notes, discharge summaries and coded information from the electronic health record system, which are managed by a largest multi-hospital organization in Japan  Our main contributions are as follows: • We found that a transformer encoding meta-information generates higher quality summaries than the vanilla one, and clarified the benefit of using meta-information for medical summarization tasks 
• We found that a model encoding disease information can produce proper disease and symptom words following the source  In addition, we found that the model using physician and hospital information can generate symbols that are commonly written in the summary  • We are the first to apply the abstractive summarization method to generate Japanese discharge summaries 
In the studies of summarization of medical documents, it is common to retrieve key information such as disease, examination result, or medication from EHRs -   Other researchs more similar to our study targeted to help physicians get the point of medical documents quickly by generating a few key sentences -  
Studies generating contextualized summaries can be categorized by the type of model inputs and architectures  Some studies produced a whole discharge summary using structured data for input - The sensitivity of the gram stain for bacterial meningitis is about 60%, and the sensitivity of the culture is not high either 
Also, the glucose in the cerebrospinal fluid would have been slightly lower  Although no definitive diagnosis could be made, bacterial meningitis was the most suspicious disease  The causative organism was assumed to be MRSA, and vancomycin and meropenem (meningitis dose) were used to cover a wide range of enteric bacteria 
a whole discharge summary from free-form inpatient records -   The free-form data is more challenging since it is noisier than structured data  In inputting of the free-form data, extractive summarization methods, which extract sentences from the source, are commonly used , -   On the other hands, an encoder-decoder model was used for abstractive summarization , , with a limited number of studies 
The various issues in the abstractive generation of discharge summary would be studied in the future  Studies using medical meta-information have long been conducted on a lot of tasks -   In abstractive summarization on discharge summary,  developed a model incorporating similarity of progress notes and information of the record author 
They presented an idea of integrating meta-information into the abstractive summarization model on medical documents, but did not reveal how meta-information would affect the quality of the summaries  Our method is based on the encoder-decoder transformer model  The transformer model is known for its high performance and has been widely used in recent studies, thus it is suitable for our purpose 
As shown in Figure , the standard input to a transformer's encoder is created by a token sequence T = [t 0 , t 1 ,    , t i ] and position sequence P = [p 0 , p 1 ,    , p i ], where i is the maximum input length  The token and position sequences are converted into token embeddings E T and positional embeddings E P by looking up the vocabulary tables 
The sum of E T and E P is input into the model  In this paper, we attempt to encode meta-information to feature embeddings  We follow the segment embeddings of BERT and the language embeddings of XLM , which provide additional information to the model  It is not a new idea but is suitable for our validation 
Our method is formulated as follows: Let M be feature type, M ∈ {Vanilla, Hospital, Physician, Disease, Length of stay}, since we set five types of features  Feature embeddings E M is created by looking up the feature table where m j is featue value (e g , pysician ID, disease code, etc ) and |M | is the maximum number of differences in a feature 
In our study, |M | is set to four different values depending on features  Specifically, they are as follows  a) Hospital: As shown in Table , the data includes five hospital records  They were obtained mechanically from the EHR system  b) Physician: Physicians are also managed by IDs in the EHR systems  We hashed the physician IDs into 485 groups containing 10 people each 
Specifically, as a naive strategy, we shuffled and listed the cases within each hospital, and hashed them into groups in the order of appearance of the physician IDs  So each group has the information about the relevance of the hospitals  The reason for employing a grouping strategy is described in Appendix A 
c) Disease: Two types of disease information exist in our EHRs: disease names and disease codes called ICD-10   We did not use any disease names in the inputs for our experiment  Instead, we encoded diseases with the first three letters of the ICD-10 code, because they represent well the higher level concept 
The initial three letters of the ICD-10 codes are arranged in the order of an alphabetic letter, a digit, and a digit, so there are a total of 2,600 ways to encode a disease  In our data, some ICD-10 codes were missing, although all disease names were systematically obtained from the EHR system  For such cases, we converted the disease names into ICD-10 codes using MeCab with the J-MeDic (MANBYO 201905) dictionary 
Also, diseases can be divided into primary and secondary diseases, but we only deal with the primary diseases  d) Length of stay: The length of stay can be obtained mechanically from the EHR system and the maximum value was set to 1,000 days  We set |M | for vanilla, hospital, physician, disease, and length of stay to 1, 5, 485, 2,600, and 1,000, respectively  
The vanilla embedding is prepared for the baseline in our experiment and to equalize the total number of parameters with the other models  The input to our model is the sum of E T , E P and E M   We also prepare an extra model with all features for our experiments  This takes all four feature embeddings (hospital, physician, disease, and length of stay) added to the encoder 

Datasets and Metrics

We evaluated our proposed method on a subset of data from National Hospital Organization (NHO), the largest multiinstitutional organization in Japan  The statistics of our data are shown in Table  , which includes 24,630 cases collected from five hospitals  Each case includes a discharge summary and progress notes for the days of stay 
The data are randomly split into 22,630, 1,000, and 1,000 for train, validation, and test, respectively  Summarization performances are reported in ROUGE-1, ROUGE-2, ROUGE-L and BERTScore in terms of F1  In addition, we also employed BLEURT , which models human judgment 

Architectures and Hyperparameters

Due to our hardware constraints we need a model that is computationally efficient, so we employed the Longformer instead of the conventional transformer  Longformer can    In our model, number of layers, window size, dilation, input sequence length, output sequence length, batch size, learning rate and number of warmup steps are 8, 256, 1, 1024, 256, 4, 3e-5 and 1K, respectively 
Other hyperparameters are the same as in the original Longformer, except for the maximum number of epochs is not fixed and the best epoch  It is selected for each training using the validation data based on ROUGE-1  Also, the original Longformer imports pretrained-BART parameters to initial values, but we do not use pre-trained Japanese BART in this study 
We used three GeForce RTX 2080 TI for our experiments  Our vocabulary for preparing input to Longformer is taken from UTH-BERT , which is pre-trained on the Japanese clinical records  Since the vocabulary of UTH-BERT is trained by WordPiece , we also tokenize our data with WordPiece  However, the vocabulary does not include white space and line breaks, which cannot be handled, so we add those two tokens to the vocabulary, resulting in a total size of 25,002 
The vocabulary has all tokens in full characters, so we normalized full-wdith characters by converting all alphanumeric and symbolic characters to half-width for byte fallback  , we found that all the models with encoded medical meta-information perform better in ROUGE-1, ROUGE-L and BLEURT than the vanilla Longformer 
However, in BERTScore, only hospital and disease models outperform the vanilla  Specifically, disease information is most effective, improving ROUGE-1, ROUGE-2, ROUGE-L, BERTScore and BLEURT by 4 45, 0 73, 3 12, 3 77 and 0 21 points over the vanilla model, respectively  This seems to be because disease information and the ICD-10 ontology efficiently cluster groups with similar representations 
In contrast, in ROUGE-2 and ROUGE-L, the model with physician embedding is inferior to the vanilla model  This seems to be a negative effect of grouping physicians without any consideration of their relevance  It would be better to cluster them by department, physician attributes, similarity of progress notes, etc  Regarding low ROUGE-2 scores in all models, a previous study using the English data set also reported a low ROUGE-2 score of about 5%, which may indicate an inherent difficulty in discharge summary generation 
In BERTScore, the models with the physician and the length of stay did not reach the performance of the vanilla model, suggesting that the system's outputs are semantically inferior  The model with all features performed the lowest of all models in BERTScore  The reason for the low score of the model with all features seems to be that its number of parameters in feature embedding was four times larger than that of the model with the individual feature, and the amount of training data was insufficient 
In BLEURT, all models with meta-information outperform vanilla, which suggests that they are more natural to humans  To analyze the influence of encoded meta-information on the outputs, we evaluate the precisions of the generated text  Specifically, we measure the probability that the generated words are included in the gold summary to investigate if the proper words are generated 
Some previous studies on faithfulness, which also analyze the output of summarization, have employed words or entities -   In this study, we focused on words, not entities, because we wanted to visualize expressions that are not only nouns  The words were segmented by MeCab with the J-MeDic  For each segmented word, the numeral and symbol labels were assigned as parts of speech by MeCab, the morphological analyzer, while the disease and symptom were assigned by the J-Medic dictionary 
The results, shown in Figure , indicate that the encoded disease information leads to generate more proper disease and symptom words  This indicates that the meta-information successfully learns disease-related expressions  The encoded hospital or physician information also improved the precision of symbols generation 
This suggests that different hospitals and physicians have different description habits (e g , bullet points such as "•", "*" and "-", punctuation such as "。" and " ", etc ), which can be grouped by meta-information  In this paper, we conducted a discharge summary generation experiment by adding four types of information to Longformer and verified the impact of the meta-information 
The results showed that all four types of information exceeded the performance of the vanilla Longformer model, with the highest performance achieved by encoding disease information  We found that meta-information is useful for abstractive summarization on discharge summaries  Our limitations are that we used Japanese EHR, the limited number of tested features and not performing human evaluations 
As for the efficacy of the meta-information, we believe that our results are applicable to non-Japanese, but it is left as Fig    The precisions of words in the generated summaries  The vertical axis shows the probability that the words exist in the gold summary  a future work  Other meta-information may be worth verifying such as the patient's gender, age, race, religion and used EHR system, etc 
It is hard to collect a large amount of medical information and process it into meta-information, so we may need to develop a robust and flexible research infrastructure to conduct a more large scale cross-sectional study in the future  In the discharge summary generation task, which demands a high level of expertise, the human evaluation requires a lot of physicians' efforts and it is a very high cost which is unrealistic 
This is a general issue in tasks dealing with medical documents, and this study also could not perform human evaluations  On this research, informed consent and patient privacy are ensured in the following manner  Notices about their policy and the EHR data usage are posted at the hospitals  The patients who disagree with the policies can request opt-out and are excluded from the archive 
In case of minors and their parents, followed the same manner  In the case of minors and their parents are same  To conduct a research on the archive, researchers must submit their research proposals to the institutional review board  After the proposal is approved, the data is anonymized to build a dataset for analysis 
The data is accessible only in a secured room at the NHO headquarters, and only statistics are brought out of the secured room, for protection of patients' privacy  In the present research, the analysis was conducted under the IRB approval (IRB Approval No : Wako3 2019-22) of the Institute of Physical and Chemical Research (RIKEN), Japan, which has a collaboration agreement with the National Hospital Organization 
This data is not publicly available due to privacy restrictions  shows the detailed number of cases handled by physicians  In all hospitals, there is a large difference between the median and the maximum of cases/physician  This indicates that a few physicians handle a large number of cases and many physicians handle fewer cases 
It is impossible to avoid physician IDs first seen at test time without some process that averages the number of cases a physician holds 
Context 3: Paper Info

Title: Specific-heat ratio effects on the interaction between shock wave and heavy-cylindrical bubble: based on discrete Boltzmann method
Publish Date: May 29, 2023
Author List: Yanbiao Gan (from School of Liberal Arts and Sciences, Hebei Key Laboratory of Trans-Media Aerial Underwater Vehicle, North China Institute of Aerospace Engineering), Yudong Zhang (from School of Mechanics and Safety Engineering, Zhengzhou University)

Figure

Figure 1: Research orientation and tasks of DBM 
Figure 2: Sketch of D2V16 model The numbers in the figure represent the index i in Eq  (3) 
Figure 3: The computational configuration of the shock-bubble interaction 
In the figure, results from odd rows are experimental, and the even rows indicate DBM simulation results The typical wave patterns and bubble's main characteristic structures are marked out in the figures Numbers in the pictures represent the time in µs Schlieren images of DBM results are calculated from the density gradient formula, i e , |∇ρ|/|∇ρ| max , with |∇ρ| = (∂ ρ/∂ x) 2 + (∂ ρ/∂ y)2  At t = 0µs, the incident shock wave impacts the upstream interface, and subsequently generates a transmitted shock (TS) propagating downstream in the bubble and a reflected shock wave moving upward in ambient gas The incident shock wave travels downstream contin-
The definitions and the corresponding physical meanings of the common TNE quantities in DBM, where the operator ∑ ix,iy indicates integrating over all the fluid units and multiply the unit area dxdy From a certain perspective, the TNE strength is increasing; While from a different perspective, the TNE strength, on the other hand, may be decreasing It is one of the concrete manifestations of the complexity of non-equilibrium flow behavior 
Figure 4: Snapshots of schlieren images of the interaction between a shock wave and a heavy-cylindrical bubble The odd rows represent experimental results from Ref  [31] with permission, and the even rows are DBM simulation results The typical wave patterns and the bubble's main characteristic structure are marked out in the figures Numbers in the picture represent the time in µs 
Figure 5: The temporal variations of length and width of the bubble The symbols represent DBM results and the lines are experimental The definition of the length and the width of the bubble can be seen in the illustration Experimental results are obtained from Fig  12, in Ref  [31] with permission 
Figure 6: Density contours and particle tracer images at three different moments (i e , t = 0 07,t = 0 11, and t = 0 16) with various specific-heat ratios The odd rows represent density contours, and the even rows are tracer particle images 
Figure 9: Vorticity contours at t = 0 134, with various specific-heat ratios The arrows in the vorticity image point out the apparent difference between case γ = 1 4 and case γ = 1 09 
Figure 11: Density contours (first row) and mixing degree M (second row) at several typical moments 
Figure 13: (a) Temporal evolution of D * 3,1 and D * 4,2  (b) Temporal evolution of D * 2 and D * 3  Lines with different colors represent the cases with various specificheat ratios 
The development of schemes for checking TNE state, extracting TNE information and describing corresponding TNE effects in DBM 

abstract

Specific-heat ratio effects on the interaction between a planar shock wave and a two-dimensional heavy-cylindrical bubble are studied by the discrete Boltzmann method  Snapshots of schlieren images and evolutions of characteristic scales, being consistent with experiments, are obtained  The specific-heat ratio effects on some relevant dynamic behaviors such as the bubble shape, deformation process, average motion, vortex motion, mixing degree of the fluid system are carefully studied, as well as the related Thermodynamic Non-Equilibriums (TNE) behaviors including the TNE strength, entropy production rate of the system 
Specifically, it is found that the influence of specific-heat ratio on the entropy production contributed by non-organized energy flux (NOEF) is more significant than that caused by non-organized momentum flux (NOMF)  Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary 
The effects of specific-heat ratio on various TNE quantities show interesting differences  These differences consistently show the complexity of TNE flows which is still far from clear understanding 

Introduction

The applications of shock-accelerated inhomogeneous flows (SAIFs) are of significant value in biomedicine, energy utilization, and astrophysics fields, including but not limited to scenarios such as the impact of shock waves on kidney stones, the interaction between shock waves with foams, the impacting of detonation wave with burning flames in supersonic combustion systems, the formation of supernova remnants, etc  
Shock-bubble interaction (SBI) is one of the most fundamental problems in the research of SAIFs  Its applications and academic research are interdisciplinary  Generally, there are two kinds of problems encountered in SBI research: (i) The geometry of shock waves, the shape of material interfaces, and the structure of container are complex in the actual scene 
They will result in various wave patterns and significantly affect the flow morphology and bubble's evolution  (ii) There usually exist multi-physics coupling problem in the engineering application of SBI  Such as the supersonic combustion machines  When the shock waves passing through the reactants, it may lead to phase transition and chemical reactions, making the flow morphology more complex and inducing small structure (or fast-changing pattern)  
In an underwater explosion experiment, the interaction between shock waves and bubbles may refer to the cavitation and annihilation effects  The other scene is the inertial confinement fusion (ICF), in which the laser ablation, electron heat conduction, self-generated electromagnetic field, radiation, and many other factors may complicate the investigation of hydrodynamic instabilities  
Commonly, research on SBI mainly includes three methods: theoretical derivation, experiment, and numerical simulation  As a fundamental research method, theoretical research can provide a clear understanding of physical processes  In 1960, Rudinger et al  developed a theory that permits computing the response of bubbles to accelerations  
In order to describe the formation and evolution processes of vortex structure quantitatively, many scholars have developed circulation models   However, theoretical works provide limited information  Meanwhile, in the late stage of SBI evolution, the bubble deformation and flow morphology dominated by the developed Richtmyer-Meshkov instability (RMI) and Kelvin-Helmholtz instability (KHI) are difficult to be predicted accu-rately 
As the research method closest to engineering application, the experimental results are often regarded as standard results to verify the rationality and accuracy of theoretical and numerical works  To study the SBI process accurately, the scholars have made a series of improvements to experimental equipment or technique, including the generation techniques of different types of shock waves, interface formation methods, schlieren facilities, and image recognition techniques  
Among these, two of important and valuable works are performed by Ding et al   Based on the soap film technique, they formed kinds of initial interfaces with different curvatures through the wire-restriction method and captured the wave patterns and interface evolution with high-speed schlieren photography  
Other works, such as evolutions of a spherical gas interface under reshock conditions , developments of a membrane-less SF 6 gas cylinder under reshock conditions , and interactions of a cylindrical converging shock wave with an initially perturbed gaseous interface , are also performed by many other scholars 
However, we know that the experimental studies mainly depend on the experimental platform  When studying some complex and demanding condition problems, it takes much work to build the experimental platform  In this situation, numerical simulation research becomes an option  Generally, there are three kinds of physical modeling methods (or models) for SBI numerical research, i e , the macroscopic, mesoscopic, and microscopic modeling methods 
Most of the existing numerical researches on SBI are related to the macroscopic modeling methods (such as the Euler and Navier-Stokes (NS) models) based on the continuous hypothesis (or equilibrium and nearequilibrium hypothesis)   For example, presented the computational results on the evolution of the shock-accelerated heavy bubbles through the multi-fluid Eulerian equation  
There also exist a few SBI works based on the mesoscopic modeling method, such as the Direct Simulation Monte Carlo method   The microscopic modeling methods such as the Molecular dynamics (MD) simulation, is capable of capturing much more flow behaviors but restricted to smaller spatiotemporal scales because of its huge computing costs 
In the numerical research on SBI, three points need to be concerned  (i) Investigation of kinetic modeling that describes the non-continuity/non-equilibrium flows  Most of the current researches are based on macroscopic models  However, there exist abundant small structure (and fast-changing patterns) behaviors and effects such as the shock wave, boundary layer, material defects, etc 
For cases with small structures, the mean free path of molecules cannot be ignored compared to the characteristic length, i e , the non-continuity (discreteness) of the system is pronounced, which challenge the rationality and physical function of the macroscopic models based on the continuity hypothesis 
For cases with fast-changing patterns, the system dose not have enough time to relax to the thermodynamic equilibrium state, i e , the system may significantly deviate from the thermodynamic equilibrium state  Therefore, the rational-ity and physical function of the macroscopic models based on the hypothesis of thermodynamic equilibrium (or near thermodynamic equilibrium) will be challenged 
(ii) Improvement of method that describes the evolution characteristics of bubbles and flows morphology  Most of the studies describe bubble characteristics and flows morphology from a macroscopic view  The mesoscopic characteristics such as the kinetic effects which help understand the kinetic process, are rarely to be studied 
(iii) Further studies of effects of specific-heat ratio on SBI process  The specific-heat ratio is an essential index for studying the compressibility of the gas  Research from Igra et al  has shown that the differences in the specific-heat ratio of bubbles would cause various wave patterns and pressure distribution inside the bubbles during the interaction process  
Besides, many works on hydrodynamic instability have also demonstrated the importance of investigating the specific-heat ratio effect   Among these, Chen et al  investigated the specific-heat ratio effects on temperature gradient and the TNE characteristics of compressible Rayleigh-Taylor (RT) system  
For the above three points, in this work we apply the recently proposed discrete Boltzmann method (DBM)   The Lattice Boltzmann Method (LBM) research has two complementary branches   One aims to work as a kind of new scheme for numerical solving various partial differential equation(s)  The other aims to work as a kind of new method for constructing kinetic model to bridge the macro and micro descriptions 
The two branches have different goals and consequently have different rules  The current DBM is developed from the second branch of LBM and focusing more on the Thermodynamic Non-Equilibrium (TNE) behaviors that the macro modeling generally ignore  It breaks through the continuity and near-equilibrium assumptions of traditional fluid modeling, discards the lattice gas image of standard LBM, and adds various methods based on phase space for checking, exhibiting, describing and analyzing the non-equilibrium state and resulting effects 
More information extraction technologies and analysis methods for complex field are introduced with time  The numerical simulation includes three parts, as shown in Fig    (1) Physical modelling, (2) Algorithm design, (3) Numerical experiments and analysis of complex physical fields  The research of equation algorithm corresponds to the part (2) of the above three parts 
The DBM aims at parts (1) and (3) of the three mentioned above  It belongs to a physical model construction method rather than a numerical solution for the equations  The tasks of DBM are to: (i) Ensure the rationality of the physical model (theoretical model) and balance the simplicity for the problem to be studied; (ii) Try to extract more valuable physical information from massive data and complex physical fields 
Based on the coarse-grained modeling method of nonequilibrium statistical physics, the DBM aims to solve the following dilemma: (i) The traditional hydrodynamic modelings are based on the continuous hypothesis (or near-equilibrium hypothesis)  They only concern the evolution of three conserved kinetic moments of the distribution function, i e  the density, momentum and energy, so their physical functions are insufficient 
(ii) The situation that the MD can be used is restricted to too small spatial-temporal scales  The physical requirement for the modeling is that except for the Hydrodynamic Non-Equilibriums (HNE), the most related TNE are also needed to be captured  Theoretically, the Boltzmann equation is suitable for all-regime flows, including the continuum regime, slip regime, transition regime, and free molecule flow regime 
Based on the Chapman-Enskog (CE) multiscale analysis , through retaining various orders of Kn number (or considering different orders of TNE effects), the Boltzmann equation can be reduced to the various orders of hydrodynamic equations  They can be used to describe the hydrodynamic behaviors, i e , the conservations of mass, momentum and energy , in corresponding flow regimes 
Because what the traditional hydrodynamic equations describe are only the conservation laws of mass, momentum and energy  Consequently, it should be pointed out that, the information lost in the traditional hydrodynamic equations increases sharply with increasing the Kn number  With increasing the Kn number, to ensure the describing capability not to decrease significantly, the more appropriate hydrodynamic equations should be the Extended Hydrodynamic Equations (EHEs) which include not only the evolution equations of conserved kinetic moments but also the most relevant nonconserved kinetic moments of distribution function 
For convenience of description we refer the modeling method that derives EHEs from the fundamental kinetic equation to Kinetic Macroscopic Modeling (KMM) method  It is clear that, the complex process of CE expansion is necessary and the simulation is still based on the macroscopic equations in KMM  As a comparison, the DBM is a kind of Kinetic Direct Modeling (KDM) method 
In DBM modeling, the CE analysis is only used to quickly determine which kinetic moments should keep values unchanged, the final EHEs are not needed, and the simulation is not based on the complicated EHEs  As the TNE degree of the flow to be described rises gradually, the complexity of the derivation process and difficulty of numerical simulation in the KMM method increase sharply 
However, in the DBM method, to describe flows in a one-order more deeper depth of TNE, only two more related kinetic moments need to be added  Since without needing to derive and solve the EHEs, as the TNE degree deepens, the complexity of the DBM approach increases much slower than that of KMM method 
The core step in DBM modeling is to provide a feasible scheme for detecting, describing, presenting, and analyzing TNE effects and behaviors beyond traditional macroscopic modeling  Based on the non-equilibrium statistical physics, we can use the non-conservative moments of ( f − f eq ) to describe how and how much the system deviates from the thermodynamic equilibrium state and to check corresponding effects due to deviating from the thermodynamic equilibrium 
The non-conservative moments of ( f − f eq ) open a phase space, and this space and its subspaces provide an intuitive geometric correspondence for describing complex TNE system properties  The development of schemes for checking TNE state, extracting TNE information and describing corresponding TNE effects in DBM are seen in Table  
Actually, this set of TNE describing methods has been applied in many kinds of complex fluid systems such as hydrodynamic instability system , combustion and detonation systems , multiphase flow system , plasma system , etc  Besides the scheme for detecting, describing, presenting, and analyzing TNE effects and behaviors, the DBM incorporates other methods for analyzing the complex physical field 
One of them is the tracer particle method  The introduction of the tracer particle method makes the gradually blurred interface appear clearly   The rest of the paper is structured as follows  Section 2 shows Year Scheme for investigating TNE effects and behaviors Before 2012 Two classes of LBMs did not show a significant difference in physical function 
2012 Use the non-conservative moments of ( f − f eq ) to check and describe TNE   This is the starting point of current DBM approach  2015 Open TNE phase space based on non-conservative moments of ( f − f eq ) and define a TNE strength using the distance from a state point to the origin  This is the starting point of the phase space description method  
2018 Extend the distance concepts in phase space to describe the TNE difference/similarity of TNE states and kinetic processes   2021 Further extend the phase space description methodology to any set of system characteristics   the modeling method  Then, the numerical simulations and results are presented in Section 3, which includes two subsections 
Section 4 concludes the paper  Other complementary information is given in the Appendix 

Model construction

Based on the Bhatnagar-Gross-Krook (BGK) singlerelaxation model, a two-fluid DBM with a flexible specific-heat ratio is presented in this part  From the origin Boltzmann equation to a DBM, four fundamental steps are needed: (i) Simplification and modification of the Boltzmann equation according to the research requirement 
(ii) Discretization of the particle velocity space under the condition that the reserved kinetic moments keep their values unchanged  (iii) Checking the TNE state and extracting TNE information  (iv) The selection/design of the boundary conditions 

Simplification and modification of the Boltzmann equation

As we know, the collision term in the original Boltzmann contains high dimensional distribution functions  Therefore, the direct solution to it needs too much computing consumption  The most common method to simplify the collision operator is to introduce a local equilibrium distribution function ( f eq ) and write the complex collision operator in a linearized form, i e , the original BGK collision operator − 1 τ ( f − f eq ), where τ is the relaxation time  
The original BGK operator describes the situation where the system is always in the quasi-equilibrium state  Namely, it characterizes only the situation where the Kn number of the system is small enough and f ≈ f eq   The currently used BGK operator for non-equilibrium flows in the field is a modified version incorporating the meanfield theory description  
Based on the above considerations, the simplified Boltzmann equation describing the SBI process is where the two-dimensional equilibrium distribution function is ) where ρ, T , v, u, I, R, and η are the mass density, temperature, particle velocity vector, flow velocity vector, the number of the extra degrees of freedom including molecular rotation and vibration inside the molecules, gas constant, and a free parameter that describes the energy of the extra degrees of freedom, respectively 
The specific-heat ratio is flexible by adjusting parameter I, i e , γ = (D + I + 2)/(D + I), where D = 2 represents the two-dimensional space 

Discretization of the particle velocity space and determination of f σ ,eq i

The continuous Boltzmann equation should be discretized for simulating  Specifically, the continuous velocity space can be replaced by a limited number of particle velocities  So that the values of continuous kinetic moments can be obtained from the summation form of kinetic moments  In this process, it requires the reserved kinetic moments, which are used to characterize the system behaviors, keep their values unchanged after discretizing the velocity space 
Namely, f the reserved kinetic moments  According to the CE analysis, f can be expressed by f eq   Therefore, in the process of discretization, the reserved kinetic moments of f eq should keep their values unchanged, i e , where i represents the kind of discrete velocities and α (α = x or y) is the direction in cartesian coordinate 
To simulate the interaction between two different fluids, a two-fluid DBM should be constructed  Based on the singlerelaxation model, the discrete two-fluid Boltzmann equation can be written as : where σ represents the types of material particle and f σ ,eq i = f σ ,eq i (ρ σ , u, T)



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Context 3:  In two-fluid DBM, the macroscopic quantities of the mixture and each component are
where ρ σ and u σ are the mass density and flow velocity of the component σ , respectively  ρ and u represent the mass density and flow velocity of the mixture, respectively  There exist two kinds of temperature (internal energy) definitions in two-fluid DBM because the definition of temperature (internal energy) depends on the flow velocity to be chosen as a reference 
The first definition is by choosing the velocity of the mixture to be a reference, i e , So that the expressions of temperature of component σ and mixture are where We can also choose the flow velocity of component as a reference, i e , , where u σ is the flow velocity of component σ   The corresponding definitions of temperature for component σ and the mixture are
where ∆E * I is It is clear to see that these two definitions of temperature for mixture are the same, but for temperature of component σ are different  We choose the first definition in this manuscript  To solve the Eq  ( ), it is necessary to determine the values of f σ ,eq i   Its values depend on the reserved kinetic moments which characterize the main system behaviors 
In DBM modeling, the CE multiscale analysis is used to determine quickly the reserved kinetic moments  Specifically, when constructing a DBM which only the first order term of Kn number is retained (i e , only the first order TNE effects are retained), seven kinetic moments should be reserved, i e , the M 0 , M 1 , M 2,0 , M 2 , M 3,1 , M 3 , M 4,2  
Two more kinetic moments ( M 4 and M 5,3 ) are needed when the second order TNE is considered   However, it should be noted that the function of CE analysis in DBM modeling is only to determine the kinetic moments that need to be preserved  Whether or not to derive the hydrodynamic equations does not affect the DBM simulation 
The kinetic moments used in our physical modeling are shown in the Appendix B  Their expressions can be obtained by integrating v and η with continuous-form f eq   For better understanding, the Appendix C gives the two-fluid hydrodynamic equations recovered from the Boltzmann equation  The kinetic moments in Appendix B can be written in matrix form, i e , C • f σ ,eq = fσ,eq , (
where C is the matrix of discrete velocity and feq represents the kinetic moments  A proper discrete velocity model is needed to confirm the values of f σ ,eq i   The f σ ,eq can be obtained by solving the inverse matrix, i e , f σ ,eq = C −1 • fσ,eq , where C −1 is the inverse matrix of C  It is very convenient to obtain the inverse matrix of C through some mathematical softwares such as Mathematica, etc 
The D2V16 model is chosen in this paper, its sketches can be seen in Fig    The specific values of D2V16 are given in the following equations: where "cyc" indicates cyclic permutation and c is an adjustable parameter of the discrete velocity model  The sketch of η in D2V16 is η i = η 0 for i = 1 − 4, and η i = 0 for i = 5 − 16 

Checking the TNE state and extracting TNE information

Many physical quantities can characterize the degree of TNE in a fluid system, such as relaxation time, Kn number, viscosity, heat conduction, the gradients of macroscopic quantity, etc  They are all helpful to characterize the TNE strength and describe the TNE behaviors of a fluid system from their perspectives 
But it is not enough only relying on these quantities  Besides the above physical quantities describing the TNE behaviors, in DBM modeling, we can also use the non-conservative moments of ( f − f eq ) to characterize the TNE state and extract TNE information from the fluid system  Fundamentally, four TNE quantities can be defined in a firstorder DBM, i e , ∆ σ * 2 , ∆ σ * 3,1 , ∆ σ * 3 , and ∆ σ * 4,2  
Their definitions can be seen in Table , where v * i = v i − u represents the central velocity and u is the macro flow velocity of the mixture  Physically, ∆ σ * 2 = ∆ σ * 2,αβ e α e β and ∆ σ * 3,1 = ∆ σ * 3,1 e α represent the viscous stress tensor (or non-organized momentum flux, NOMF) and heat flux tensor (or non-organized energy flux, NOEF), respectively 
The e α (e β ) is the unit vector in the α (β ) direction  The later two higher-order TNE quantities contain more condensed information  Specifically, and it indicates the flux information of ∆ σ * 2   To describe the TNE strength of the whole fluid system, some TNE quantities contained more condensed information are also defined, i e ,
Other TNE quantities can be defined based on specific requirements  All the independent components of TNE characteristic quantities open a highdimensional phase space, and this space and its subspaces provide an intuitive image for characterizing the TNE state and understanding TNE behaviors   It should be emphasized that: (i) The TNE strength/intensity/degree is the most basic parameter of non-equilibrium flow description; And any definition of non-equilibrium strength/intensity/degree depends on the research perspective 
(ii) The physical meaning of D * m,n is the TNE strength of this perspective  (iii) From a certain perspective, the TNE strength is increasing; While from a different perspective, the TNE strength, on the other hand, may be decreasing  It is normal, one of the concrete manifestations of the complexity of non-equilibrium flow behavior 
Strictly speaking, those TNE intensity and effect descriptions that do not account for the research perspective are not correct  Do not explain the research perspective, the corresponding is not dependent on the research perspective 

Numerical simulations and results

In this section, we first validate the DBM code by comparing the DBM results with experimental results  Then, the effects of specific-heat ratio on the dynamic process and TNE behaviors on SBI are investigated 

Comparison with experimental results

In the following part, we use a first-order two-fluid DBM to simulate the interaction between a planar shock wave with a 2-D heavy-cylindrical bubbles, and compare the DBM results with the experimental results from Ref    The computational configuration can be seen in Fig    In a flow field which is filled with Air, there is a static bubble composed of 26% Air and 74% SF 6  
A shock with Ma = 1 2 would pass through the bubble from left to right  The initial conditions of ambient gas are ρ 0 = 1 29kg/m 3 , T 0 = 293K, p 0 = 101 3kPa  Ignoring the pressure difference between interior gas and ambient gas, the initial parameters of the bubble are ρ bubble = 4 859kg/m 3 , p bubble = 101 3kPa,
and T 0 = 293K  For simulating, these actual physical quantities should be transferred to dimensionless parameters  This process can refer to the Appendix A  The dimensionless conditions of macroscopic quantities of the fluid field in initial time are (ρ, T, u x , u y ) bubble = (4 0347, 1 0, 0 0, 0 0), (ρ, T, u x , u y ) 1 = (1 3416,
1 128, 0 3616, 0 0), (ρ, T, u x , u y ) 0 = (1 0, 1 0, 0 0, 0 0), where the subscript "0" ("1") represents downstream (upstream) region  In two-fluid DBM code, the distribution function f Air is used to describe the ambient gas, i e , Air  The f bubble characters the bubble which is a mixture that composed of Air and SF 6  
The grid number is N x × N y = 800 × 400, where the N x and N y are grid number in x and y direction, respectively  This grid size has passed the mesh convergence test  The below results also show that it is sufficient to meet the requirements of the following research problem  Other parameters used for the simulation are: c = 1 0, η Air = η bubble = 10 0,
I Air = 3, I bubble = 15, ∆x = ∆y = 1 2 × 10 −4 and ∆t = 1 × 10 −6   The viscosity effect is feeble compared to the shock compression effect, so it does not significantly affect the deformation of the bubble  Therefore, in this part, the relaxation time τ is set sufficiently small  The inflow (outflow) boundary condition is used in the left (right) boundary, and the periodic boundary is adopted in the y direction 
The first-order forward difference scheme is used to calculate the temporal derivative, and the second-order nonoscillatory nonfree dissipative scheme is adopted to solve the spatial derivative in Eq  ( )   Two quantitative comparisons between experimental results and DBM simulations are shown in the following part, including snapshots of schlieren images and evolutions of characteristic scales for the bubble 
The first is shown in Fig  non-organized momentum flux (NOMF) uously to form a diffracted shock (DS)  As TS propagates, it will split into three branches due to the considerable pressure perturbations caused by the gradual decay of the DS strength   Afterward, as shown in the subfigure at about t = 128µs, two high pressure regions (ROH) generate because of the interaction of these branches 
Subsequently, at about t = 148µs, the two ROHs meet, causing the shock focusing  On the one hand, at about t = 168µs, the shock focusing causes the generation of downstream-propagating second transmitted shock (STS) and upward-moving rarefaction wave  On the other hand, it will produce high pressure region inside the bubble, which later leads to a jet structure, as shown at about t = 288µs 
At about t = 428µs, due to the deposited vorticity, there will produce a pair of counter-rotating vortexes at the pole region of the bubble  The further development of the vortex pair and the effect of viscosity decrease the amplitude of the jet  Finally, the jet structure disappears  The second quantitative comparison is the interface structure described by the length and width of the bubble, as shown in Fig   
The experimental data are extracted from Fig  , in Ref    Quantitative agreements between DBM simulation and experimental results are seen  For the profile of bubble width, there are mainly two stages  At an early time (t < 150µs), it decreases to a minimum value because of the shock compression effect 
After the shock wave passes through the bubble (t > 150µs), the developed vortex pair caused by the deposited vorticity gradually dominates the growth of bubble width  Different from width evolution, the temporal variation of length experiences three stages  In the early stages (t < 150µs), it decreases quickly due to the shock compression effect 
Then, the jet structure emerges, which results in a growth in length (150µs < t < 250µs)  Because the upstream interface moves faster than the downstream interface, the bubble length would decrease at 250µs < t < 500µs  In the third stage (t > 500µs), the vortex pair forms and then leads to a continuous development of bubble length 
Both the length and width experience oscillations in the later stages due to complex wave patterns  The quantitative agreements between DBM simulation and experimental results indicate the following two facts: (i) the order of TNE considered in the current DBM is sufficient, (ii) the choosing of discrete velocities and spatial-temporal steps and simulation parameters like the relaxation times is suitable for characterizing the deformation of bubble, wave patterns, main characteristics of flow morphology 

Effects of specific-heat ratio on SBI

The major of current works on SBI research have not focused on specific-heat ratio effects  In this part, the simulation parameters are fine-adjusted based on the parameters in Section 3 1 to highlight the influence of specific-heat ratio  Through adjusting the extra degree of freedom I, five cases with various specific-heat ratios of the bubble are simulated, i e , γ = 1 4,
1 28, 1 18, 1 12, and 1 09  Two kinds of analysis methods, including tracer particle method and two-fluid model, are used to characterize qualitatively the macroscopic behaviors such as the shape, deformation process, mixing degree, etc  The related TNE behaviors are also studied 

Effects of specific-heat ratio on jet shape, deformation process, and average motion

We first observe the specific-heat ratio effect on the bubble shape from the view of density contour and images of particle tracer visually  As shown in Fig  , pictures with three typical moments are plotted, i e , t = 0 07,t = 0 11, and t = 0 16  The odd rows represent density contours and the even rows are tracer particle images 
It can be seen that the specific-heat ratio significantly affects the length and shape of the jet structure  The smaller the specific-heat ratio is, the stouter the jet structure can be seen  The reason is that the specific-heat ratio significantly changes the propagation speed of shock waves and wave patterns inside the bubble 
The specific-heat ratio also influences the vortex structure in early stage but contributes little effects to it in later stage  In the later stage, for cases with different specific-heat ratios, the differences in vortex pairs are almost invisible  Then, the effects of specific-heat ratio on deformation process are analyzed 
Shown in Fig  are the evolutions of characteristic scales which used to describe the bubble size, i e , width and length  It can be seen that the smaller the specific-heat ratio of bubble, the smaller the bubble width and length  For the fluid with smaller specific-heat ratio, it is easier to be compressed 
Therefore, the characteristic scales of bubbles with smaller specific-heat ratio tend to be compressed smaller  It can also be seen that the case with the largest specific-heat ratio reaches the minimum characteristic scales firstly  The reason is that the shock wave propagates faster in case with larger specific-heat ratio 
Through the method of tracer, information on the average motion of the bubble is easy to obtain  Shown in Fig  are the average position and average velocity of the bubble, with different specific-heat ratios  It is found that, in the shock compression stage (t < 0 03), the effect of specific-heat ratio contributes little to the average motion 
However, when the shock wave passes through the bubble (t > 0 03), a larger specific-heat ratio speeds up the average motion of the bubbles  The reason is that the bubbles with smaller specific-heat ratio need more energy to compress their size, so their translational energy is smaller 

Effects of specific-heat ratio on vortex motion

Vorticity is one of the most important physical quantities in describing the vortex motion  In the 2-D case, the vorticity can be calculated by the following equation: The positive (negative) value of ω represents the positive (negative) direction along the z axis  Vorticity contours at t = 0 134, with various specific-heat ratios, are shown in Fig   
The discernable difference between cases with various specific-heat ratios can be observed  The arrows in the vorticity images point out the obvious differences around the interface between case γ = 1 4 and case γ = 1 09  That is to say, there exists influences of specific-heat ratio on the rotational motion of the bubble 
The strength of vorticity is described by circulation Γ, where Γ = ∑ ω∆x∆y



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Question: How does the specific-heat ratio affect the average motion of the bubble?

Read the following text and answer briefly.

Context 3:  Γ + = ∑ ω| ω>0 ∆x∆y is the positive circulation and Γ − = ∑ ω| ω<0 ∆x∆y represents the negative circulation  Figure shows the temporal evolution of circulations on SBI process  It can be seen that the values of Γ are equal to zero all the time because the values of Γ + and Γ − are the same 
But they point in the opposite direction  In the shock compression stage (t < 0 03), the specific-heat ratio effect contributes little to the circulation of the bubble  When the shock wave sweeps through the bubble (t > 0 03), the specific-heat ratio affects the value of circulation obviously  The cases with a smaller specific-heat ratio experiences a larger range of amplitude of change, which is caused by its good compressibility 

Effects of specific-heat ratio on mixing degree

The mixing process is a fundamental research content on SBI  In two-fluid DBM, the mixing degree at each fluid unit can be defined as M = 4 • M A M B , where M σ represents the mass fraction of component σ   The higher the value of M, the higher the mixing amplitude  Images of density (first cow) and mixing degree M (second row) at several typical moments are shown in Fig   
As can be seen, the mass mixing occurs in the region where two media contact  In addition, the mixing degree M g described the whole fluid field can be defined where the symbol "−" indicates integrating the M σ over the whole fluid field and then dividing the grid size N x • N y   Shown in Fig  is the temporal evolution of the global mixing degree M g  
As can be seen, temporal profiles of the global mixing degree show two stages: t < 0 03 and t > 0 03  When t < 0 03, there is almost no difference between cases with various specific-heat ratios  But for t > 0 03, the stronger the specific-heat ratio effect, the larger the mixing degree  Actually, there are mainly two indicators that measure the global mixing degree: the amplitude of mixing and the area of the mixing zone between two fluids 
At the stage t > 0 03, the shock compression dominates the mix by enhancing the mixing amplitude and increasing the area of the mixing zone simultaneously  In this stage, the specific-heat ratio effect contributes little to the mix  However, when the shock passes through the bubble, the deformation of interface and the evolution of vortex core both significantly raise the area of the mixing zone 
As can be seen in Fig  , the smaller specific-heat ratio of bubble, the stronger global mixing degree of fluid field  Intuitively, for the fluid with smaller specific-heat ratio, it is easier to deform and compress, which is beneficial for the fluid mixing  It can also be explained by the diffusion formula, i e ,  
The specific-heat ratio affects both the temperature T and the gradient of density simultaneously  Therefore, these two aspects comprehensively influence the material diffusion between the two fluids  Due to the complex reflected shock wave, the global mixing degree shows a tendency for oscillating growth 

Effects of specific-heat ratio on TNE behaviors

The investigation of TNE behaviors is of great importance for understanding the kinetics process on SBI  These TNE quantities describe the fluid system deviating from the thermodynamic state from their own perspectives  The effects of specific-heat ratio on global TNE strength, i e , D * 2 , D * 3 , D * 3,1 , and D * 4,2 , are shown in Fig   
It can be seen that the effects of specific-heat ratios on various TNE quantities are different  Theoretically, the influence of specific-heat ratio on the non-equilibrium effect is reflected in two aspects: transport coefficient and macroscopic quantity gradient  For example, on the one hand, the specific-heat ratio reduces heat conductivity, while on the other hand, it enhances the temperature gradient 
Therefore, the effect of specific heat ratio on NOEF is the comprehensive result of the competition between the two aspects  As shown in Fig  , the smaller the specific-heat ratio, the stronger strength of D * 3,1   It indicates that the specific-heat ratio increase the strength of D * 3,1 by raising the heat conductivity  
For the strength of D * 3 , as shown in Fig  (b), it is seen that it decreases as the specific-heat ratio becomes small  The reason is that a smaller specific-heat ratio decreases the temperature gradient  Effects of specific-heat ratio on D * 4,2 show two-stage  In the shock compression stage (t < 0 03), the smaller specificheat ratio, the larger the strength of D * 4,2  
But the situation is reversed at the stage t > 0 03  For strength of D * 2 , the specificheat effects are more significant in later stage 

Effects of specific-heat ratio on entropy production rate

and entropy production The concepts of entropy are commonly used in complex flows   In DBM, there are two kinds of en- tropy production rates, i e , ṠNOEF and ṠNOMF   They are key factors in compression science field  The former is induced by temperature gradient and the NOEF (∆ * 3,1 )  The latter is affected by velocity gradient and the NOMF (∆ * 2 ) 
The entropy production rates are defined by the following formulas : Integrating the ṠNOEF and ṠNOMF over time t, the entropy generations over this period of time are obtained, i e , S NOEF = t 0 ṠNOEF dt and S NOMF = t 0 ṠNOMF dt  Plotted in Fig  ) and 14(b) are the temporal evolution of ṠNOMF and ṠNOEF , respectively 
The evolution of entropy generation rate is related to two aspects: (i) the propagation of the shock wave, and (ii) the deformation of the bubble  The former generates a macroscopic quantity gradient, and the latter makes the contact interface wider, longer, and deformed  Depending on the location of the shock wavefront, there exist two critical moments in this SBI process: (i) at around t = 0 03, the shock wave just sweeps through the bubble, and (ii) at t = 0 06, the shock wave exits the flow field 
Therefore, the temporal evolution of the entropy production rate shows three stages, i e , t < 0 03, 0 03 < t < 0 06, and t > 0 06  At the stage t < 0 03, the shock compression stage, the shock effects compress the bubble  It generates the large macroscopic quantity gradients, resulting in a quick increase of ṠNOMF  
At around t = 0 03, the shock wave passed through the bubble  So the values of ṠNOMF decreases  The values of ṠNOMF would continue to decrease due to the gradually wider contact interface caused by the diffusion effect  At around t = 0 06, the shock wave comes out of the flow field so that the values of ṠNOMF drops rapidly 
In the third stage, i e , t > 0 06, because of the diffusive effect, the general trend of ṠNOMF is downward  However, it shows an oscillatory trend due to the influence of various reflected shock waves  The specific heat ratio indirectly changes the value of ṠNOMF by changing the velocity gradient  The smaller the specific-heat ratio, the larger ṠNOMF  
Different understanding can be seen in Fig  , where the temporal evolution of ṠNOEF is plotted  In the first stage (t < 0 03), cases with different specific-heat ratios show various trends  At the stage where the bubble deformation is not very large, i e , 0 03 < t < 0 06, values of ṠNOEF fluctuate near the average value 
In the third stage (t > 0 06), evolutions of ṠNOEF in cases with larger specific-heat ratios show an apparent growing tendency  Differently, the values of ṠNOEF in cases with smaller specific-heat ratios remain almost unchanged  The influence of specific heat ratio on the ṠNOEF , similar with the effect on NOEF, is also affected by the heat conductivity and the temperature gradient 
It can be seen that, except for the case of γ = 1 09, the larger the specific-heat ratio, the higher entropy production rate ṠNOEF   The temporal evolutions of ṠNOEF of case γ = 1 09 and case γ = 1 12 are very similar  Consequently, the specific-heat ratio increases the ṠNOEF by raising the temperature gradient 
Further understanding can be seen in Fig  , where the entropy productions over this period are plotted  For convenience, the sum and difference between S NOMF and S NOEF are also plotted in the figure  The variation range of S NOEF is larger than that of S NOMF   It indicates that the influence of specific-heat ratio on S NOEF is more significant than that on S NOMF  
Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary  Specifically, it can be seen that the entropy production contributed by NOMF increases with re- duced specific-heat ratio  But the entropy production caused by NOEF first reduces with decreasing specific-heat ratio and then approaches to a saturation value 
The S NOEF in case γ = 1 09 is almost the same with it in case γ = 1 12  When the specificheat ratio γ is smaller than a threshold value γ c (γ c ≈ 1 315), the entropy production induced by NOEF is more significant than that caused by NOMF  However, in the case of γ > γ c , the situation reverses  The temporal evolution of the total entropy production (S NOMF +S NOEF ) is similar to the S NOEF profile 
The difference between S NOMF and S NOEF increases with decreasing specific-heat ratio 

Conclusions

Specific-heat ratio effects on the interaction between a planar shock wave and a 2-D heavy-cylindrical bubble are studied by a two-fluid DBM which has a flexible specific-heat ratio and includes several schemes for analyzing the complex physical fields  Besides the HNE that NS easily describes, the DBM pays more attention to the related TNE that NS is not convenient to describe 
First, both the snapshots of schlieren images and evolutions of characteristic scales from DBM simulation are compared with those from experiment  The quantitative agreements between them indicate the following two facts: (i) the order of TNE considered in the current DBM is sufficient, (ii) the choosing of discrete velocities, spatial-temporal steps, and simulation parameters like the relaxation times are suitable for the following physical researches 
Then, five cases with various specific-heat ratios are simulated  Several analysis methods for complex physical fields, including the description scheme of TNE behaviors, tracer particle method, and two-fluid model, are used to characterize the effects of specific-heat ratio on the bubble shape, deformation process, average motion, vortex motion, mixing degree of the fluid system, TNE strength, and entropy production 
Specifically, for bubble shape, bubbles with different specific-heat ratios display various jet structures  The smaller the specific-heat ratio is, the stouter the jet structure can be seen  For the case with smaller specific-heat ratio, the fluid is easier to be compressed  So, the characteristic scales of bubbles with smaller specific-heat ratio tend to be compressed smaller 
For the bubble, the smaller the specific-heat ratio, the slower average motion  In the shock compression stage, the specific-heat ratio contributes little effects to the vortex motion  Differently, after the shock passes through the bubble, it significantly influences the vorticity around the interface and the corresponding amplitude of circulation due to the development of KHI 
The larger the difference in specific-heat ratio between the bubble and ambient gas, the higher the degree of material mixing  Effects of specific-heat ratio on various TNE quantities are different  These differences consistently show the complexity of TNE flows which is still far from a clear understanding 
In addition, it is found that the temporal evolution of the entropy production rates ṠNOMF and ṠNOEF both show three stages because of the influence of the shock wave location  The smaller the specific-heat ratio, the larger the velocity gradient, which indirectly enhances the strength of ṠNOMF   The specific-heat ratio increases the ṠNOEF by raising the temperature gradient 
The influence of specific-heat ratio on S NOEF is more significant than that on S NOMF   Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary  Specifically, the entropy production contributed by NOMF increases with reduced specific-heat ratio  But the entropy production caused by NOEF first reduces with decreasing specific-heat ratio and then approaches to a saturation value 
When the specific-heat ratio γ is smaller than a threshold value γ c (γ c ≈ 1 315), the entropy production induced by NOEF is more significant than that caused by NOMF  However, in the case of γ > γ c , the situation reverses  The fundamental research in this paper helps to understand the interaction mechanism between shock waves and bubbles in ICF, supersonic combustors, underwater explosions, etc 
The effects of viscosity and heat conduction on the interaction between shock waves and bubbles will be studied in the following work  where the subscript "m, n" means that the m-order tensor is contracted to n-order tensor  According to the CE multiscale analysis, the Boltzmann-BGK equation can be reduced to the hydrodynamic equations 
In the following part, the derivation process from Boltzmann-BGK equation to a two-fluid hydrodynamic equation are shown  More details can see the reference presented by Zhang et al    The discrete Boltzmann equation for component σ is (C 1) In Eq  C 1 there are two equilibrium distribution functions, i e , f σ ,seq = f σ ,seq (ρ σ , u σ , T σ ) and f σ ,eq = f σ ,eq (ρ σ , u, T ) 
For convenience, S σ i is defined as We perform the CE expansion around the f σ ,seq   That is, the distribution function f σ i can be expanded as where ε is a coefficient referring to Knudsen number  The partial derivatives of time and space can also be expanded to Substituting the above four equations into Eq 
(C 1), we can obtain (C 6) When retaining to ε terms, the following equation is obtained When retaining to ε 2 terms, we can obtain where M 2,αβ ( f σ ,(1) ) = ∑ i v iα v iβ f σ ,(1) i and M 3,1,α ( f σ ,(1) ) =   Substituting Eq  (C 14) into the above three equations, and replacing the time derivatives with the space derivatives, we obtain It should be noted that the ability to recover the corresponding level of macroscopic fluid mechanics equations is only part of the physical function of DBM 
The corresponding to the physical functions of DBM is the EHEs, which, in addition to the conserved moments evolution equations corresponding to the three conservation laws of mass, momentum and energy, also includes some of the most closely related nonconserved moments evolution equations  We refer the EHEs derivation based on kinetic equation to as KMM 
The necessity of the expanded part, the evolution equations of the relevant non-conserved moments, increases rapidly as increasing the degree of non-continuity/non-equilibrium  As the degree of non-continuity/non-equilibrium increases, the complexity will rapidly make KMM simulation studies, deriving and solving EHE, impossible 



ground_truth: The specific-heat ratio affects the average motion of the bubble. The bubbles with smaller specific-heat ratios have slower average motion.