Modeling and simulation in supersonic three-temperature carbon dioxide turbulent channel flow

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Modeling and simulation in supersonic three-temperature carbon

dioxide turbulent channel ﬂow

Guiyu Caoa,b, Yipeng Shic,∗, Kun Xud, Shiyi Chena,b,∗

aAcademy for Advanced Interdisciplinary Studies, Southern University of Science and Technology,

Shenzhen, Guangdong 518055, People’s Republic of China

bGuangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering

Applications, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s

Republic of China

cDepartment of Aeronautics & Astronautics Engineering, Peking University, Beijing 100871, People’s

Republic of China

dDepartment of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay,

Kowloon, Hong Kong

Abstract

This paper pioneers the direct numerical simulation (DNS) and physical analysis in su-

personic three-temperature carbon dioxide (CO2) turbulent channel ﬂow. CO2is a linear

and symmetric triatomic molecular, with the thermal non-equilibrium three-temperature

eﬀects arising from the interactions among translational, rotational and vibrational modes

under room temperature. Thus, the rotational and vibrational modes of CO2are addressed.

Thermal non-equilibrium eﬀect of CO2has been modeled in an extended three-temperature

BGK-type model, with the calibrated translational, rotational and vibrational relaxation

time. To solve the extended BGK-type equation accurately and robustly, non-equilibrium

high-accuracy gas-kinetic scheme is proposed within the well-established two-stage fourth-

order framework. Compared with the one-temperature supersonic turbulent channel ﬂow,

supersonic three-temperature CO2turbulence enlarges the ensemble heat transfer of the

wall by approximate 20%, and slightly decreases the ensemble frictional force. The ensemble

density and temperature ﬁelds are greatly aﬀected, and there is little change in Van Dri-

est transformation of streamwise velocity. The thermal non-equilibrium three-temperature

eﬀects of CO2also suppress the peak of normalized root-mean-square of density and tem-

perature, normalized turbulent intensities and Reynolds stress. The vibrational modes of

CO2behave quite diﬀerently with rotational and translational modes. Compared with the

vibrational temperature ﬁelds, the rotational temperature ﬁelds have the higher similarity

with translational temperature ﬁelds, especially in temperature amplitude. Current thermal

non-equilibrium models, high-accuracy DNS and physical analysis in supersonic CO2tur-

bulent ﬂow can act as the benchmark for the long-term applicability of compressible CO2

turbulence.

Keywords: Carbon dioxide ﬂow, Vibrational modes, Three-temperature eﬀects,

Supersonic turbulent channel ﬂows

Preprint submitted to Elsevier October 5, 2022

arXiv:2210.01621v1 [physics.flu-dyn] 4 Oct 2022

1. Introduction

Mars exploration programs are currently experiencing a revival, such as amazing Mars

robotic helicopter ”Ingenuity” operating on Mars [1]. Mars’s atmosphere consists of 95.32%

carbon dioxide (CO2). For accurate predictions of surface drag and heat ﬂux on Martian

vehicles, it is necessary to take the peculiarities of CO2into account. Diﬀerent with the

dominant diatomic gases nitrogen (N2) and oxygen (O2) on earth, CO2is a linear and

symmetric triatomic molecular, which has three vibrational modes [2]. Triatomic molecular

CO2is equipped with the inherent thermal non-equilibrium multi-temperature eﬀects arising

from the interactions among the translational, rotational and vibrational modes [3, 4].

Carbon dioxide is widely studied in the applications of physical chemistry and ﬂuid

dynamics, i.e., CO2-N2gas laser system [5], environmental green-house problems [6], and

Mars entry vehicles [7, 8, 9]. Complex molecular structure and multiple internal energy

relaxation mechanisms in CO2 signiﬁcantly aﬀect its physical and chemical properties [10].

In ﬂuids community, of special interest is the evaluation of bulk viscosity in CO2. Stokes’

viscosity relation does not hold for CO2[11], as the bulk viscosity can be thousands of

times larger than the shear viscosity. Many research works are engaged in experimental and

theoretical studies in the bulk viscosity of CO2[12, 13]. Recent experiments [14, 15] show

that the vibrational modes contribute dominantly to the bulk viscosity of CO2, and the

bulk viscosity from rotational modes is only one half of its shear viscosity approximately

[15]. Thus, the vibrational modes should be modelled carefully when simulating CO2ﬂows.

In view of the importance of multiple internal energy modes, the multiple internal energy

relaxation mechanisms of CO2[3, 4] have been modeled and analyzed, which conﬁrm that

equilibrium one-temperature gas ﬂow description is not valid for CO2ﬂows even under room

temperature (i.e., 300K). To the author’s knowledge, the thermal non-equilibrium physical

models considering the multi-temperature eﬀects of CO2and its applications in turbulent

ﬂows are seldom reported. For accurate predictions of CO2turbulence, it is necessary to

take the three-temperature eﬀects of CO2into account.

In the past few decades, the gas-kinetic scheme (GKS) based on the Bhatnagar-Gross-

Krook (BGK) model [16, 17] has been developed systematically for the computations from

low speed ﬂows to hypersonic ones [18, 19]. Based on the time-dependent ﬂux solver, in-

cluding generalized Riemann problem solver and GKS [20, 21], a reliable two-stage fourth-

order framework was provided for developing the high-order GKS (HGKS) into fourth-order

accuracy. With the advantage of ﬁnite volume GKS and HGKS, they have been natu-

rally implemented as a direct numerical simulation (DNS) tool in simulating turbulent ﬂows

[22, 23, 24], especially for compressible turbulence [25, 26]. Aiming to conduct the large-scale

DNS, a parallel in-house computational platform of HGKS has been developed in uniform

grids and curvilinear grids [27, 28], with high eﬃciency, fourth-order accuracy and super ro-

bustness. In addition, with the discretization of particle velocity space, a uniﬁed gas-kinetic

∗Corresponding author

Email addresses: caogy@sustech.edu.cn (Guiyu Cao), syp@mech.pku.edu.cn (Yipeng Shi),

makxu@ust.hk (Kun Xu), chensy@sustc.edu.cn (Shiyi Chen)

scheme (UGKS) [29, 30] and uniﬁed gas-kinetic wave particle method (UGKWP) [31, 32, 33]

have been developed for multi-scale physical transport problems. The well-developed HGKS

and multi-scale UGKS/UGKWP provides the solid foundation for thermal non-equilibrium

multi-temperature modeling and simulation in CO2ﬂows. The multi-scale modeling and nu-

merical framework can be applied in multi-scale CO2ﬂows, i.e., the Mar’s re-entry vehicles

from rareﬁed to continuum regimes. As a starter, current study focuses on the supersonic

CO2turbulence in the continuum regime.

In this paper, the vibrational modes of CO2are addressed, and the translational, rota-

tional and vibrational relaxation time of CO2are calibrated. The three-temperature eﬀects

of CO2are modeled in an extended three-temperature BGK-type model within the well-

established kinetic framework [34, 35, 36]. To achieve high-order accuracy in space and time

for simulating the supersonic CO2turbulence, the non-equilibrium high-accuracy GKS has

been constructed with the second-order kinetic ﬂux, ﬁfth-order WENO-Z reconstruction [37],

and two-stage fourth-order time discretization [21]. One-temperature supersonic turbulent

channel ﬂow [38, 39] is simulated ﬁrstly to validate the numerical set-up with bulk Mach

number Ma = 3 and bulk Reynolds number Re = 4880. Considering the translational,

rotational, and vibrational speciﬁc heats at constant volume, one-temperature supersonic

turbulent channel ﬂow of thermally perfect gas has been studied [40]. With implement-

ing the non-equilibrium high-accuracy GKS in the large-scale parallel in-house platform

[27, 28], for the ﬁrst time, the DNS in supersonic three-temperature CO2turbulent chan-

nel ﬂow is conducted. Compared with the one-temperature supersonic turbulent channel

ﬂow, the three-temperature eﬀects of CO2are analyzed. Numerical simulation conﬁrms

the thermal non-equilibrium three-temperature performance of CO2. Both the maximum

ensemble temperature and normalized r.m.s. temperature sort from high to low is transla-

tional temperature, rotational temperature, and vibrational temperature. Compared with

the vibrational temperature ﬁelds, the rotational temperature ﬁelds has the higher similarity

with translational temperature ﬁelds both in temperature amplitude and its structure.

For physical modeling and numerical simulation in supersonic three-temperature CO2

turbulent channel ﬂow, this paper is organized as follows. Section 2 addresses the inter-

nal energy modes of CO2. Extended thermal non-equilibrium three-temperature BGK-type

model and corresponding non-equilibrium high-accuracy GKS for CO2are included in Sec-

tion 3. Numerical examples and discussions are presented in Section 4. The last section is

the conclusion and remarks.

2. Internal energy modes of carbon dioxide

Thermal non-equilibrium three-temperature eﬀects of CO2mainly arise from the inter-

actions among internal energy modes [4]. This section addresses the vibrational modes of

CO2, and focuses on the translational, rotational and vibrational relaxation time for the

extended three-temperature BGK-type model.

2.1. Rotational and vibrational modes

Carbon dioxide is a linear and symmetric triatomic molecular, with rotational and vi-

brational internal energy modes. The characteristic rotational temperature is deﬁned as

θr=h2

P/(8π2kBI), where hPis the Planck constant, kBthe Boltzmann constant, Ithe

molecular moment of inertia. For carbon dioxide, θr= 0.56Kcan be obtained [2], while

the θrfor N2and O2is 2.88Kand 2.08K, respectively. Under room temperature, it is well

known that the rotational degrees of freedom (d.o.f.) are assumed to be excited completely

for N2and O2. Since CO2is with the smaller characteristic rotational temperature, the

rotational d.o.f. of CO2are regarded as complete excitation in current study (i.e., CO2gas

temperature above 300K).

Carbon dioxide is equipped with three vibrational modes as one symmetric stretching

mode ν1, one double degenerated bending mode ν2, and one asymmetric stretching mode ν3.

The characteristic vibrational temperature reads θv=hP˜νcL/kB, where ˜νis the character-

istic wavenumber, and cLthe speed of light in the vacuum. In experimental studies of CO2,

infrared spectrum gives the wavelength for corresponding vibrational modes ν2and ν3, and

Raman spectroscopy provides the wavelength for ν1[2]. Table 1 presents the characteristic

wavenumber, wavelength (λ= 1/˜ν), characteristic vibrational temperature and correspond-

ing degeneracy of CO2. The double degenerated bending modes ν2are most likely to be

activated with characteristic vibrational temperature θv= 959.66K, which is much lower

than that of N2with θv= 3521Kand O2with θv= 2256K. Thus, the vibrational modes of

N2and O2is usually considered in high-temperature applications, i.e., re-entry vehicles ex-

periencing the temperature above 800K[41]. However, the excitation of vibrational modes

of CO2requires to be modeled and simulated even under the room temperature [4, 15].

Vibrational mode Wavenumber(˜ν)/cm−1Wavelength(λ)/µm θv/KDegeneracy

ν11388 7.20 1997.02 1

ν2667 14.99 959.66 2

ν32349 4.26 3379.69 1

Table 1: Parameters for three vibrational modes of CO2.

With the assumption that there is a unique vibrational temperature at each point in the

ﬂow ﬁelds, the translational internal energy Et, rotational internal energy Er, and vibrational

internal energy Evper unit mass of CO2read

Et=Nt

2RTt,(1)

Er=Nr

2RTr,(2)

Ev=R

i=1

θv,i

eθv,i/Tv−1,(3)

with translational d.o.f. as Nt, rotational d.o.f. as Nr, and vibrational d.o.f. as Nv, where

Tt,Trand Tvrepresent translational temperature, rotational temperature and vibrational

temperature, respectively. In Eq.(3), θv,i is the characteristic vibrational temperature of

vibrational mode νi, and giis the corresponding degeneracy of mode νias shown in Table

1. The total speciﬁc heat at constant volume CVcan be obtained by the sum of three

components as

CV=CV,t +CV,r +CV,v,(4)

with

CV,t =Nt

2R, (5)

CV,r =Nr

2R, (6)

CV,v =

i=1

(θv,i/Tv)2

eθv,i/Tv+e−θv,i/Tv−2R, (7)

where CV,t,CV,r and CV,v denotes the componential speciﬁc heat at constant volume for

translational internal energy, rotational internal energy and vibrational internal energy, re-

spectively. In current study, the supersonic CO2turbulence is considered above 300K, thus,

translational d.o.f. as Nt= 3 and rotational d.o.f. as Nr= 2 are adopted. Vibrational d.o.f.

as Nvcan be obtained by the deﬁnition as CV,v =NvR/2, and the speciﬁc heat ratio γis

the function of vibrational d.o.f. as

Nv= 2

i=1

(θv,i/Tv)2

eθv,i/Tv+e−θv,i/Tv−2,(8)

γ=CP

= 1 + 2

5 + Nv

.(9)

In Eq.(9), CPis the total speciﬁc heat at constant pressure. In the high-temperature limit,

note that the vibrational d.o.f. as Nvapproaches to classical deﬁnition with 2 P3

i=1 giθv,i/Tv

eθvib,i/Tv−1.

For compressible wall-bounded turbulence, the Prandtl number P r plays a key role in

determining the statistical turbulent quantities [27], especially for density and temperature

ﬁelds. For CO2, the P r can be calculated as P r =µCP/κ, where shear viscosity µand

thermal conductivity κdepends on the translational temperature Ttas subsequent Eq.(10)

and Eq.(12), and CPrelies on the vibrational temperature as Eq.(9). Figure 1 presents the

comparisons on γand P r between CO2and air. For air, the speciﬁc heat ratio γalmost

keeps as 1.4 up to 800K, and P r is ﬁxed at approximate 0.7 between 300Kand 800K.

In terms of CO2, experimental measurements on γand P r show the strong temperature-

dependent behavior [42]. We ﬁnd the numerical proﬁle of γin CO2with Eq.(9) agrees well

with the experimental one. The numerical proﬁle of P r is calculated with the assumption

of Tt=Tv, and this assumption dose not hold for experimental measurement. Thus, it is

reasonable to ﬁnd the discrepancy in P r between the numerical proﬁle and experimental

result. In following simulation, without the assumption of Tt=Tvas shown in Figure 1,

the P r of CO2depends on the practical translational and vibrational temperatures. Notice

that the γand P r must be computed locally in each time step for each computational grid

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Modelingandsimulationinsupersonicthree-temperaturecarbondioxideturbulentchannelowGuiyuCaoa,b,YipengShic,,KunXud,ShiyiChena,b,aAcademyforAdvancedInterdisciplinaryStudies,SouthernUniversityofScienceandTechnology,Shenzhen,Guangdong518055,People'sRepublicofChinabGuangdong-HongKong-MacaoJointLaboratory...

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Modeling and simulation in supersonic three-temperature carbon dioxide turbulent channel flow

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