Proceedings of Global Power and Propulsion Society ISSN-Nr: 2504-4400

GPPS Chania20 7th – 9th September, 2020

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GPPS-CH-2020-0068Direct Numerical Simulation of Real-gas Effects within Turbulent

Boundary Layers for Fully-developed Channel FlowsTao Chen, Bijie Yang, Miles Robertson and Ricardo Martinez-Botas

Turbo Group, Department of Mechanical Engineering, Imperial College LondonLondon SW7 2AX, UK

ABSTRACT

Real-gas effects have a significant impact on compressibleturbulent flows of dense gases, especially when flow propertiesare in proximity of the saturation line and/or the thermodynamiccritical point. Understanding of these effects is key for the analysisand improvement of performance for many industrial components,including expanders and heat exchangers in organic Rankine cyclesystems.

This work analyzes the real-gas effect on the turbulent boundarylayer of fully developed channel flow of two organic gases,R1233zd(E) and MDM - two candidate working fluids for ORCsystems. Compressible direct numerical simulations (DNS) withreal-gas equations of state are used in this research. Three cases areset up for each organic vapour, representing thermodynamic statesfar from, close to and inside the supercritical region, and these casesrefer to weak, normal and strong real-gas effect in each fluid.

The results within this work show that the real-gas effectcan significantly influence the profile of averaged thermodynamicproperties, relative to an air baseline case. This effect has a reverseimpact on the distribution of averaged temperature and density.As the real-gas effect gets stronger, the averaged centre-to-walltemperature ratio decreases but the density drop increases. In astrong real-gas effect case, the dynamic viscosity at the channelcenter point can be lower than at channel wall. This phenomenoncan not be found in a perfect gas flow.

The real-gas effect increases the normal Reynolds stress in thewall-normal direction by 7-20% and in the spanwise direction by10-21%, which is caused by its impact on the viscosity profile.It also increases the Reynolds shear stress by 5-8%. The real-gaseffect increases the turbulence kinetic energy dissipation in theviscous sublayer and buffer sublayer (y∗ < 30) but not in the outerlayer. The turbulent viscosity hypthesis is checked in these twofluids, and the result shows that the standard two-function RANSmodel (k− ε and k−ω) with a constant Cµ =0.09 is still suitable inthe outer layer (y∗ > 70), with an error in ±10%.

NOMENCLATUREc Sonic velocity [m/s]e Inner heat energy [J/kg]fi Body force [m/s2]k Turbulent kinetic energy [J/kg]m Mass flow rate [kg/s]p Pressure [Pa]qi Heat conduction [J/(m2s)]s Specific entropy [J/kg·K]t Time [s]u Velocity in x direction [m/s]v Velocity in x direction [m/s]w Velocity in x direction [m/s]xi Position [m]Cµ Turbulent model constantCp Isobaric specific heat capacity [J/(kg·K)]Cv Isochoric specific heat capacity [J/(kg·K)]L Channel length scale [m]M Molecular weight [kg/kmol]N Number of nodes [-]Pk Turbulent kinetic energy generation rate [W]R Specific gas constant [J/(kg·K)]T Temperature [K]V Specific volume [m3/kg]Z Compressibility factor [-]

Ec Eckert number [-]Ma Mach number [-]Re Reynolds number [-]Pr Prandtl number [-]

Greeksγ Specific heat ratio [-]ε Turbulent kinetic energy dissipation rate [W]η Smallest length scale [m]ηk Kolmogorov length scale [m]ηθ Length scale in temperature field [m]

λ Heat conductivity [W/(m·K)]µ Dynamic viscosity [Pa·s]ν Kinematic viscosity [m2/s]ρ Density [kg/m3]τ Viscous stress [Pa]Γ Fundamental derivative [-]

Subscripts0 Stagnation conditionscr Critical conditionpg Perfect gas

t Turbulent parameterw Channel wallB Bulk paramerC Channel central layerG Global parameterτ Viscous parameter

Accentsx Non-dimensional parameterx Reynolds-averaged mean parameterx′ Reynolds-averaged fluctuationx Favre-averaged mean parameterx′′ Favre-averaged fluctuation

INTRODUCTIONWall-bounded turbulent flows of organic fluids have drawn

increasing attention in recent years, in part due to the applicationof organic Rankine cycle (ORC) systems for waste heat recovery.With wall-bounded flows present in the majority of ORC systemcomponents, an improved understanding of the detailed fluidmechanics within real-gas boundary layers is fundamental to (a)the validation of Computational Fluid Dynamics (CFD) codes, andtherefore (b) the maximisation of component-level performance.

The majority of research within this area [1] [2] [3] [4]has analysed blade-bounded flows in a turbine or nozzle, basedupon the Reynolds Averaged Navier Stokes (RANS) simulationmethod. However further work is required for validation, to confirmthat real-gas effects do not disturb the fundamental assumptionscontained within RANS simulations.

Real-gas effects distinguish organic fluids from perfect gasessuch as air. The perfect-gas Equation of State (EoS), pV = RT , isinapplicable in these fluids when they work close to the saturationcurve and/or critical point. In this case, the thermodynamicproperties such as pressure, viscosity, and heat capacity, willdepend not only on temperature, but also on density. As a resultof this, many perfect-gas-based simplifications must be checkedbefore their application. Meanwhile, most frequently-used organicfluids, including refrigerants, siloxanes and alkanes, have a largemolecular weight and a high-level molecular complexity, also calleddense gases [5] [6].

Work by Colonna & Guardone [7] has shown that as thegas molecular complexity increases, its heat capacity, Cv/R, willincrease, and the fundamental derivative of gas dynamics, Γ

(defined in Eq. 1 [8]) will decrease. Perfect gases always have a

constant Γ = (γ + 1)/2 of more than unity, while dense gas canhave a variable and lower Γ. For most of the frequently-usedorganic fluids, the minimum Γ can be lower than 1, where the speedof sound will decrease with pressure in an isentropic process [8].There are also a group of theoretical BZT fluids (named after Bethe[9], Zel’dovich [10] and Thompson [8]), which contain a regionof negative Γ. In that region an ’inversion’ of typical supersonicbehaviour is observed, with flows achieving compression via fans,as opposed to shock waves [8]. To summarise, real-gas anddense-gas effects can have a significant influence on the mechanicsof an organic fluid - it is the purpose of this work to study theseeffects in detail.

Γ≡ c4

2V 3

(∂ 2V∂ p2

)s

(1)

This work studies the real-gas effect on the turbulent boundarylayer in a fully developed channel flow by means of DirectNumerical Simulation (DNS), in which no turbulence model isrequired due to flows being resolved down to the Kolmogorovmicroscale. DNS helps to get rid of the uncertainty caused by anyturbulent modelling assumption, and produces the realization of areal-gas flow.

Kim, Moin & Moser [11] firstly applied the DNS method on aincompressible fully developed channel flow at Reynolds number of3300. They studied the turbulence statistics near the wall, and foundthat their results showed a good agreement with the experimentaldata.

Huang, Coleman & Bradshaw [12] studied compressible flowwithin a channel, and analysed the compressibility-associatedterms in Reynolds averaged energy equations. They foundthat the averaged property profiles matched the correspondingincompressible curves well, by scaling by mean density,ρ/ρw. They also proposed the model for Reynolds analogy,mean-Favre-averaged fluctuations, and pressure-dilatation terms.

Patel et al [13] studied a turbulent channel with variableproperties, in different cases of the relation between dynamicviscosity and temperature. They found that the normalReynolds stress anisotropy and turbulence-to-mean time scaleswere influenced by that relation.

Sciacovelli, Cinnella, & Gloerfelt [14] studied a dense gas,PP11 (C14F24), in a turbulent channel flow. They focused onone operating point inside the region Γ < 1, and found the meantemperature is nearly constant and the mean viscosity decreasesfrom the wall to the centreline, showing a liquid-like behavior.Their results showed that dense gas channel flows can tend towardsthe behaviour of an incompressible channel flow with variableproperties, due to their heat capacity.

The study within this work focuses on two commonly-usedorganic fluids, and compares three different working conditions ofeach fluid to give insight into the impact of real-gas effects. Themain properties of these two organic fluids are show in Tab. 1.Refrigerant R1233zd(E), trans-1-chloro-3,3,3-trifluoropropene(CF3CH=CHCl), is accepted as a high-efficiency and

2

environmental-friendly fluid for ORC systems, and it can bea substitute of refrigerant R245fa. Meanwhile, MDM, formallyknown as octamethyltrisiloxane (C8H24O2Si3), is a representativeof the group of siloxane fluids, since they also can be a potentialfluid of ORC systems.

Table 1 KEY PROPERTIES OF R1233zd(E) AND MDM [15].

Property R1233zd(E) MDMTcr (K) 439.6 564.1

pcr (kPa) 3624 1415ρcr (kg/m3) 480.2 256.7M (kg/kmol) 130.5 236.5

METHODOLOGYNumerical Method

A fully developed channel flow is simulated in the domainshown in Fig. 1, where the boundaries on X and Z direction areperiodic boundaries and the boundary on Y direction is no-slipisothermal wall boundary - mean flow is in the X direction. Thescale of the domain is Lx×Ly×Lz = 2πh×2h×πh, where h is thecharacteristic length of the channel.

Fig.1 CALCULATION DOMAIN OF THE CHANNEL FLOWSIMULATION.

This work is based on the solution of the three-dimensionalNavier-Stokes (NS) equations (Eq.2-4), which are closed bythe equation of state p = p(ρ,T ). These equations arenon-dimensionalised by half-channel height, h, globally averageddensity, ρG ≡ 1

h∫ h

0 ρdy, and global averaged velocity, uG ≡1

hρG

∫ h0 ρudy, before being calculated.

∂ρ

∂ t+

∂ρu j

∂x j= 0 (2)

∂ρui

∂ t+

∂ρu jui

∂x j=− ∂ p

∂xi+

∂τi, j

∂x j+ρ fi (3)

∂ρ(e+ 1

2 u2i)

∂ t+

∂ρu j(e+ 1

2 u2i)

∂x j

=−∂u j p∂x j

+∂uiτi, j

∂x j−

∂q j

∂x j+ρu j f j (4)

where:

τi j = µ

(∂ui

∂x j+

∂u j

∂xi

)− 2

3µ∇ ·ukδi j (5)

qi =−λ∂T∂xi

(6)

Non-dimensional parameters can be declared as:

t ≡ tuG

h(7)

xi ≡xi

h(8)

ui ≡ui

uG(9)

ρ ≡ ρ

ρG(10)

T ≡ TTw

(11)

p≡ pρGu2

G(12)

e≡ eu2

G(13)

µ ≡ µ

ReGµG(14)

λ ≡ Twλ

ρGu3Gh

=Cp,GTwλ

ReGPrGu2GλG

(15)

For a real gas, all the thermodynamic parameters are two-variablefunctions of both temperature and density. The global densityand wall temperature (ρG,Tw) are taken to determinate theglobal dynamic viscosity, µG, and heat capacity Cp,G during thenon-dimensionalisation. The global Reynolds number is definedby ReG ≡ ρGuGh

µGand the global Prandtl number is defined by

PrG ≡Cp,GµG

λG. Subsequently, expression of the non-dimensional NS

equations is the same as the original expression.

For the fully developed channel flow there is a periodic boundaryinstead of inlet and outlet boundaries, so the pressure drop alongthe stream is balanced by an artificial added body force fi, which isequivalent to the mean pressure gradient in physical flow [12]. Thebody force fi is non-zero only for the mean flow direction (i = 1),and it is uniform in the domain. The body force is calculated in eachtime step by keeping the global averaged mass flow rate invariant inthe calculation.

The DNS solver is developed based on a compressible DNScode from X. Li [16] with an addition of a real-gas model. Allthe thermal properties in this model are two-variable functions ofthe local density and temperature, and they are calculated by thestate-of-art Helmholtz energy EoS in REFPROP [15]. Organicfluid cases are calculated by this real-gas model while the aircase is calculated by a perfect-gas model. The NS equations aresolved after non-dimensionalisation. For spectral discretisation, theconvective flux derivatives are calculated by seventh-order upwind

3

finite-difference scheme, and the viscous flux derivatives arecalculated by sixth-order central scheme. For time advancement,it is solved by a third-order Runge-Kutta method. The validation ofthe simulation method is shown in the appendix.

The number of nodes in this work is Nx×Ny×Nz = 128×257×128, and the overall number of nodes is 4.2 × 106. The spatialresolution is chosen as DNS requirements. Zonta [17] advisedthat (∆x/η)max ≈ 12, (∆y/η)max ≈ 2 and (∆z/η)max ≈ 6, whileLee [18] advised that (∆x/η)max ≈ 12.4, (∆y/η)max ≈ 3.0 and(∆z/η)max ≈ 7.9, where η is the smallest length scale in the flow. η

is equal to the Kolmogorov length scale ηk when Prandtl number islower than 1, while it equal to ηθ ≈ ηk/

√Pr when Prandtl number

is higher than 1 [19]. The spatial resolution of cases in this workis shown in Tab. 2, and it can fulfill the DNS requirement. The y+

of the first layer from wall boundary is below 0.5, and the first 10layers are within y+ = 5.

Figure 2 (a) and (b) show two-point correlation of velocity u anddensity ρ between points in distance ∆x ∈ [0,Lx/2] at y+ ≈ 10, andFig. 2 (c) and (d) show the two-point correlation between points indistance ∆z ∈ [0,Lz/2]. It shows that all the two-point correlationsare close to zero at Lx/2 and Lz/2, which means that the domain sizeLx and Lz are large enough for the channel flow cases with periodicboundary.

This work is solved by the High Performance Computer inImperial College London. Each case uses 512 CPUs, and it takes10-20 days to get converged. After that, the statistic is taken every∆t = 2 within a time range ∆t = 1000.

Table 2 DNS CASES SPATIAL RESOLUTION.

Case Air R1 R2 R3 M1 M2 M3(∆x/η)max 12.6 9.9 11.1 14.3 9.4 9.4 10.6(∆y/η)max 0.8 1.0 1.1 1.9 1.1 1.1 1.3(∆z/η)max 6.3 5.0 5.6 7.1 4.7 4.7 5.3

Case Set-UpThere are no standard inlet and outlet boundaries in this domain

for the periodic boundary condition, so the global averaged densityρG and mass flow rate mG = uGhρG are set as an alternative. Theglobal averaged velocity is given as uG =MaGcG, where MaG is theglobal Mach number. cG is a reference global sound speed definedby wall temperature and global averaged density cG = crg(ρG,Tw)

for organic fluids, and only wall temperature cG = cpg(Tw) forair. The dynamic viscosity µG is defined in the same fashion -µG = µrg(ρG,Tw) for organic fluids and µG = µpg(Tw) for air. µrg

is calculated by the real-gas model, and µpg is calculated by theSutherland viscosity law [20] shown in Eq. 16, where Ts = 110.56K is the effective temperature.

µpg

µw=( T

Tw

)3/2 Tw +Ts

T +Ts(16)

The initial condition is a turbulent flow field, in which the globalaveraged density ρG and mass flow rate mG are set. During the

calculating, ρG and mG will not change. After each time step, ρG iscalculated and compared with its value in the previous step. If thereis a gap between them (caused by numerical error), the density atall points will plus or minus a same small density source term ∆ρ

so as to keep ρG invariant after this step. mG is also calculated ineach step and compared with its value in the previous step. If theyare different, the body force will be changed so as to maintain aninvariant mG.

The global Mach number MaG ≡ uGcG

and global Reynolds

number ReG ≡ ρGuGhµG

are set at 2.25 and 4880 for all seven cases.The air case follows the perfect-gas Equation of State, requiringonly one thermodynamic condition - the wall temperature, Tw =288 K. The two thermodynamic conditions chosen for the real-gascases were wall temperature and global density. From R1 to R3,the set-up point moves closer to the supercritical region, shown bythe temperature-entropy diagram in Fig. 3. The red and blue linesin Fig. 3 are the averaged thermodynamic property distribution ineach real-gas case. Cv/R for each case is list in Tab. 3, and thevalue for MDM is the largest whilst air is the smallest. As the set-uppoint closer to the supercritical region Cv/R increases, although thischange is small in comparison to the difference in values betweenR1233zd(E) and MDM. The fundamental derivative Γ shows thatthe minimum value is at R2 and M2 for the two organic fluids,respectively.

Table 3 CALCULATING CONDITIONS FOR CASE SET-UP (R= R1233zd(E), M = MDM).

Case Air R1 R2 R3 M1 M2 M3Tw/Tcr - 0.86 0.98 1.03 0.86 0.98 1.03ρG/ρcr - 0.02 0.27 1.00 0.02 0.27 1.00

ReG 4880 4880 4880 4880 4880 4880 4880MaG 2.25 2.25 2.25 2.25 2.25 2.25 2.25

Cν ,G/R 2.5 13.4 15.7 17.3 53.1 58.3 61.9Γ 1.2 1.01 0.82 1.90 0.97 0.59 1.62Z 1 0.96 0.67 0.32 0.96 0.68 0.33

RESULTS AND ANALYSISAveraged Velocity and Thermodynamic Properties

The variables are averaged on each y level, since the x and zdirections are periodic. For analysis, the wall shear stress is definedas the averaged value τw ≡ [µ(∂u/∂y)]w and wall friction velocityis defined as uτ ≡

√τw/ρw. Huang et al [12] proposed a semi-local

friction velocity u∗τ ≡√

τw/ρ as well, which takes compressibilityeffects into consideration.

Some important global results of each case are listed in Tab. 4.Bulk Reynolds number and bulk Mach number are defined as ReB ≡ρGuGh/µw and MaB ≡ uG/cw [14], and they are different from ReG

and MaG in real gases (µw 6= µG).The centre-line Reynolds number and Mach number are defined

as ReC ≡ ρCuCh/µC and MaC ≡ uC/cC.The values show that ReB

decreases as the set-up point is closer to the supercritical region(from R1 to R3 and from M1 to M3), while ReC increases. Bothorganic fluids have smaller ReB and larger ReC relative to the

4

0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Air

R1

R2

R3

M1

M2

M3

(a)

0 0.5 1 1.5 2 2.5 3

-0.2

0

0.2

0.4

0.6

0.8

1

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(d)

Fig.2 TWO-POINT CORRELATION OF VELOCITY U AND DENSITY IN STREAMWISE AND SPANWISE. CORRELATION INSTREAMWISE (a),(b); CORRELATION IN SPANWISE (c),(d).

air baseline. The Mach number MaB firstly increases and thendecreases as the set-up point is closer to the supercritical region(from R1 to R3 and from M1 to M3), while the centreline Machnumber shows minimal change across cases.

The friction Reynolds number is defined as Reτ ≡ ρwuτ h/µw.The friction Reynolds number is lower in the organic fluid casesthan air, while lowest in the supercritical cases (R3 and M3). Thesemi-local friction Reynold number Re∗τ ≡ ρu∗τ h/µ has a highervalue for the two organic gases than air, and Re∗τ is highest in thesupercritical cases R3 and M3 for each fluid.

Reynolds averaged velocity and distribution versusnon-dimensional wall distance y+ ≡ ρwuτ y/µw is shown inFig. 4(a) The position y+ = 0 is the wall, while the maximumy+ is the central layer of the channel. It shows that the organicfluid cases are closer to the log law line u+ = 0.25 ln(y+) + 5.5in the outer layer than air. In each fluid, as the case moves closerto the supercritical region, the velocity distribution increases indeviation from the log law line. After the application of a Van

Driest transformation (Eq. 17) [21] good agreement to the log lawin the outer layer is observed (shown in Fig 4(b)).

uV D =∫ u

0

√ρ

ρwdu (17)

The averaged temperature, density and pressure along thewall-normal direction are shown in Fig. 5, where the abscissa isthe non-dimensional wall distance y∗ ≡ ρu∗τ y/µ . Figure 5(a) showsthat the temperature in the central region of the channel is higherthan wall temperature, and centre-to-wall temperature ratio TC/Tw

is lower in the organic fluid cases than in the air case (shown in Tab.4).

For each organic fluid, TC/Tw decreases as the point moves closerto the supercritical region (R1 to R3 and M1 to M3). TC/Tw

correlated positively to PrEc in these cases (Prandtl number Pr ≡Cpµ/λ , Eckert number Ec ≡ c2/CpT ), which means the largerPrEc the larger Tc/Tw will be.

5

1700 1750 1800 1850 1900 1950 2000 2050

Entropy [J/(kg K)]

360

380

400

420

440

460

480

Te

mp

era

ture

[K

]

500

1000

1500

2000

250030

00

35004000

45005000

Two-phase RegionR1

R1

R2

R2

R3

R3

Isobar [kPa]

(a) R1233zd(E)

600 650 700 750 800

Entropy [J/(kg K)]

480

500

520

540

560

580

600

Te

mp

era

ture

[K

]

500

1000

15002000

2500

3000

Two-phase Region

M1

M1

M2

M2

M3

M3

Isobar [kPa]

(b) MDM

Fig.3 TEMPERATURE-ENTROPY DIAGRAM OF R1233zd(E)AND MDM. DOTS REFER TO SET-UP CASES, WHILST LINESREFER TO MEAN THERMAL PROPERTY DISTRIBUTION.

When averaging and integrating the energy equation, the heattransfer as a function of wall distance can be represented by Eq. 18[12]. In the expression, Fu(y) is a velocity distribution functionwhich equal to 0 at y = 0 and equal to 1 at y = 1. Sincethe heat conductivity fluctuation is small compared with averagedconductivity, the gradient of averaged temperature can be derivedto Eq. 19.

By assuming the velocity distribution function and the thermalconductivity ratio the same in all these cases, the non-dimensionaltemperature gradient is approximately proportional to the globalPrandtl number multiplied by global Eckert number PrGEcG whosevalues are shown in Tab. 4. For a perfect gas, EcG = γR/Cp = γ−1.The organic fluids have higher molecular complexity, leading to alarger Cp/R and therefore a smaller Ec. For each organic fluid, Prand Ec are variable, and the closer to the super critical region thesmaller the value of PrEc.

λ∂ T∂ y

= τw

∫ 1

yρ udy

qq+ qt

= τwFu(y)q

q+ qt(18)

100

101

102

0

5

10

15

20

25

30

Air

R1

R2

R3

M1

M2

M3

(a)

100

101

102

0

5

10

15

20

25

(b)

Fig.4 DISTRIBUTION OF X-DIRECTION AVERAGEDVELOCITY ALONG WALL DISTANCE. BLACK DASHEDLINES REFER TO u+ = y+ IN THE VISCOUS SUBLAYERAND u+ = 0.25ln(y+)+5.5 IN THE OUTER LAYER.

∂ T∂ y≈ τwFu(y)

λ

qq+ qt

(19)

∂ T∂ y

∝1

λ=

ReGPrGu2GλG

Cp,GTwλ∝ PrGEcG (20)

The density is higher at the channel wall than the channel centre(shown in Fig. 5(c), and the value of the wall-to-centre density ratioρw/ρC is shown in Tab. 4. The largest density ratio also appears inthe air case, which is the same for the temperature ratio mentionedabove. However, for each organic fluid, the density ratio increasesas the set-up point moves closer to the supercritical region, whilethe temperature ratio has a reverse trend.

Since the pressure is nearly constant (p/ pw > 0.985 shown inFig. 5(d)), the density gradient is approximately equal to thetemperature gradient multiplied by −

(∂ ρ

∂ T

)p (Eq. 21). The value

6

100

101

102

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Air

R1

R2

R3

M1

M2

M3

(a) Averaged temperature

100

101

102

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

(b) Averaged temperature (zoomed)

100

101

102

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

(c) Averaged density

100

101

102

0.985

0.99

0.995

1

(d) Averaged pressure

Fig.5 DISTRIBUTION OF AVERAGED TEMPERATURE, DENSITY AND PRESSURE ALONG THE WALL-NORMAL DIRECTION.

Table 4 DNS RESULTS FOR EACH CASE

Case Air R1 R2 R3 M1 M2 M3ReB 4880.0 4881.1 4530.4 2683.1 4880.7 4836.4 4175.0ReC 3512.5 4824.6 5219.2 5648.0 5394.9 5519.9 5632.4MaB 2.25 2.26 2.52 1.26 2.25 2.32 2.09MaC 1.92 2.40 2.34 2.21 2.53 2.50 2.51Reτ,C 395.9 306.9 306.2 205.7 287.6 287.9 262.9Re∗

τ,C 183.4 243.4 266.0 300.7 270.1 277.4 287.8TC/Tw 1.875 1.155 1.069 1.045 1.038 1.014 1.004

PrGEcG 0.28 0.055 0.025 0.018 0.013 0.005 0.002PrG 0.7 0.81 1.02 4.77 0.73 0.67 1.41EcG 0.4 0.069 0.025 0.004 0.017 0.007 0.001

Cp,G/R 3.5 14.7 21.7 81.3 54.3 64.5 139.4ρw/ρc 1.89 1.18 1.33 1.52 1.05 1.07 1.14

−(

∂ ρ

∂ T

)p,G 1 1.1 3.9 31.4 1.1 3.9 33.3

pw 0.26 0.22 0.28 0.27 0.21 0.31 0.43µc/µw 1.57 1.16 1.00 0.56 1.04 1.01 0.86

of −(

∂ ρ

∂ T

)p is shown in Tab. 4. It is equal to 1 in perfect gas,

but it increases significantly as the set-up point moves closer tothe supercritical region in real gases(from R1 to R3, and M1 toM3). The effect of the increase in −

(∂ ρ

∂ T

)p is larger than that of

the decrease in temperature gradient, so the density drop is larger incases closer to the supercritical region.

−dρ

dy≈−

(∂ ρ

∂ T

)p

dTdy

(21)

The pressure ratio has a minimum value at y∗ ≈ 70 for allthe cases (shown in Fig. 5 (d)). As the set-up point movescloser to the supercritical region, the maximum pressure drop firstlyincreases and then decreases in R1233zd(E) (from R1 to R3), whileit increases monotonically in MDM (from M1 to M3). This trend ofmaximum pressure drop is nearly negatively related to the averagedpressure at wall pw (shown in Tab. 3), and this relation can beexplained by the Reynolds averaged momentum equation in the ydirection (Eq. 22) in the following paragraph.

7

100

101

102

0.4

0.6

0.8

1

1.2

1.4

1.6

Air

R1

R2

R3

M1

M2

M3

Fig.6 DISTRIBUTION OF AVERAGED DYNAMIC VISCOSITYALONG THE WALL-NORMAL DIRECTION.

After integrating Eq. 22 along wall distance y, the expression offor pressure ratio shown in Eq. 23 is found. The pressure ratio is

associated to 1/ pw and∫ y

0d(ρ v′′v′′)

dy dy. If the later part differs slightlybetween cases, it can be concluded that the larger the wall pressurepw is the smaller the pressure drop will be.

0 =−d pdy

+ddy

(τ2,2− ρ v′′v′′)≈−d pdy− d(ρ v′′v′′)

dy(22)

ppw

=− 1pw

∫ y

0

d(ρ v′′v′′)dy

dy+1 (23)

The distribution of averaged dynamic viscosity ratio µ/µw isshown in Fig. 6. It shows that the viscosity increases as y∗ increasesin the Air baseline, R1 and M1, while it decreases in Case R3and M3. Sciacovelli et al [14] proposed that the former trend isa gas-like behaviour while the latter is a liquid-like behaviour, andthat liquid-like behaviours can be observed in dense-gases.

From the result of Fig. 6, it shows that the liquid-like behaviouronly exists in the supercritical cases (R3 and M3) of the twoorganic fluids. By implication there is a transition line from gas-likebehaviour to liquid-like behaviour in each organic fluid, which is( ∂ µ

∂T )p = 0 shown in Fig. 7. The slope of the transition line is largerthan that of the isobar lines. When fluid works below the transitionline, the dynamic viscosity increases with wall distance; otherwisewhen fluid works above that line, the dynamic viscosity decreasesas wall distance is increased.

Reynolds StressThe distributions of Reynolds stresses u′′u′′, v′′v′′, w′′w′′ and

u′′v′′ normalized by u∗2τ are shown in Fig. 8. The normal Reynoldsstress in the streamwise direction, u′′u′′, shows a slight deviation(less than 4%) between cases (shown in Fig. 8(a)), while the normalReynolds stress in spanwise and wall-normal directions (v′′v′′ andw′′w′′) show larger values of deviation (Fig. 8(b) and Fig. 8(c)).

1700 1750 1800 1850 1900 1950 2000 2050

Entropy [J/(kg K)]

360

380

400

420

440

460

480

Te

mpe

ratu

re [K

]

500

1000

1500

2000

2500

3000

35004000

Two-phase Region

Transition Line

Isobar [kPa]

(a) R1233zd(E)

600 650 700 750 800

Entropy [J/(kg K)]

480

500

520

540

560

580

600

Tem

pe

ratu

re [

K]

500

15002000

2500

3000

Two-phase Region

Transition Line

Isobar [kPa]

(b) MDM

Fig.7 TRANSITION LINE FOR R1233zd(E) and MDM.

The peak values of v′′v′′ in real-gas cases are larger than thatin air case by 7-20%. For each fluid, when the real-gas effectincreases(from R1 to R3, or M1 to M3), the peak value willincrease. Meanwhile, this impact is stronger in R1233zd(E) thanin MDM. Real-gas effect also influence the peak values of w′′w′′,which are larger by 10-21% in real-gas cases. The trend of thisimpact is the same as that of the real-gas effect on v′′v′′.

The impact of real-gas effect on the normal Reynolds stress inspanwise and wall-normal directions (v′′v′′ and w′′w′′) is associatedwith the viscosity profile and bulk Reynolds number ReB. Patelet al. [13] proposed that if the centre-to-wall viscosity ratio lessthan 1, the normal Reynolds stress in spanwise and wall-normaldirections will increase, and Sciacovelli et al [14] proposed that theincrease of bulk Reynolds number ReB will lead to a larger peakvalue of normal Reynolds stress. Both these two effects influencethe normal Reynolds stress in the cases of this work, while theviscosity ratio effect seems more predominant in this work, sincesupercritical case R3 (with the lowest µc/µw and lowest ReB) hasthe largest maximum v′′v′′ and w′′w′′ among all cases.

The Reynolds shear stress u′′v′′ is larger in two organic fluids by

8

100

101

102

0

1

2

3

4

5

6

7

8

9

10

Air

R1

R2

R3

M1

M2

M3

(a)

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(b)

100

101

102

0

0.5

1

1.5

(c)

100

101

102

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(d)

Fig.8 DISTRIBUTION OF REYNOLDS STRESSES.

(5-8%) in comparison to air, while the difference between organicfluid cases is not significant. The maximum value of Reynolds shearstress u′′v′′ is associated with the viscous shear stress. According tothe Reynolds averaged momentum equation in the y direction (Eq.24), the viscous shear stress and Reynolds shear stress balance thebody force together. Since the dynamic viscosity for real-gas casesis smaller in the outer layer, which leads to a smaller viscous shearstress, so the Reynolds shear stress becomes larger compared withair case.

0 =ddy

(τ1,2− ρ u′′v′′)+ρ f1 (24)

Turbulence EnergyThe Reynolds averaged turbulence energy equation for fully

developed channel flow is shown in Eq. 25. The first term isturbulence energy production Pk; the second term is the summaryof turbulent diffusion, velocity-pressure diffusion, and viscousdiffusion; the third term is energy dissipation εk; and the last three

terms are compressibility-related terms. The turbulence energyproduction term Pk and dissipation term εk are discussed in thiswork, which are defined in Eq. 26 and Eq. 27.

0 =−ρu′′1u′′2∂ u1

∂x2+

∂

∂x2

(− 1

2ρu′′i u′′i u′′2−ρ ′u′′2 +u′′i τ ′i,2

)− τ ′i,2

∂u′′i∂x2−u′′2

∂ p∂x2

+ p′∂u′′i∂xi

+u′′i∂τi,2

∂x2(25)

The distribution of turbulence energy production Pk is shownin Fig. 9(a). It shows that the fluid property do not significantlyinfluence non-dimensional turbulence energy production, with amaximum Pk difference of less than 5% across all cases. Thedistribution of turbulence energy dissipation εk is shown in Fig.9(b). The difference between the air and real-gas cases aresignificant in the viscous sublayer and buffer layer (y∗ < 30),with the magnitude decreasing in the outer layer (y∗ > 30). Thenon-dimensional dissipation at the wall is 5-13% higher in real-gas

9

cases than air. For each organic case, the non-dimensionaldissipation in the viscous sublayer and buffer layer increases as thereal-gas effect increases (from R1 to R3; M1 to M3).

The turbulence energy production-to-dissipation ratio −Pk/ε −1 is shown in Fig. 10. A clear peak is demonstrated at y∗ ≈ 10across all cases, followed by a reduction to nearly zero as y∗ ≈ 50is reached. In the region y∗ > 50, the air case remains stable beforereducing to -1, while those for real-gas cases will increase slowlybefore demonstrating the same decrease (albeit at higher values ofy∗). The position of the start point for the second drop is related toRe∗

τ,C, and the larger the Re∗τ,C the later the second drop will be.

Pk ≡−ρu′′i u′′2∂ ui

∂x2(26)

ε ≡−τ ′i,2∂u′′i∂x2

(27)

100

101

102

0

0.05

0.1

0.15

0.2

0.25

Air

R1

R2

R3

M1

M2

M3

(a)

100

101

102

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

(b)

Fig.9 DISTRIBUTION OF TURBULENCE ENERGY (a)PRODUCTION, (b) DISSIPATION.

100

101

102

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Air

R1

R2

R3

M1

M2

M3

Fig.10 DISTRIBUTION OF TURBULENCE ENERGYPRODUCTION - TO - DISSIPATION RATIO.

Turbulence Viscosity HypothesisThe evaluation of the turbulence viscosity hypothesis for the

standard two-function RANS model (k− ε and k−ω) is a focusof study in this work. The turbulence viscosity νt is defined asthe ratio of Reynolds shear stress to the gradient of the averagedvelocity, whose simplified expression in the full developed channelflow is shown in Eq. 28. In the standard k− ε model [22], theturbulence viscosity νt is calculated by the turbulence kinetic energyk and dissipation ε (Eq. 29), which are solved by equations for kand ε . The constant Cµ is equal to 0.09.

The ratio νtε/k2 is calculated for each case and shown in Fig.11. It shows that in the region y∗ < 70, the ratio is lower than 0.09for all cases, but approximately equal to 0.09 in the region y∗ > 70within deviation bands of ±10%.

With the data for R1233zd(E) and MDM collected so tightlyaround the constant Cµ line across all cases, there is little evidenceto suggest that real-gas effects have a significant impact on thisvalue. A conclusion can therefore be made that the assumptionCµ = 0.09 is still suitable in the standard two-function RANS model(k− ε and k−ω) within the outer layer region at y∗ > 70 for theorganic fluids within this study.

u′′v′′ = νtdudy

(28)

νt =Cµ

k2

ε(29)

CONCLUSIONSThis work studied the fully developed compressible turbulent

channel flow of two organic fluids, R1233zd(E) and MDM, bymeans of Direct Numerical Simulation. Three cases for each fluidare set up at the same global Reynolds number (ReG = 4880) andglobal Mach number (MaG = 2.25), and they are compared with anair baseline case. Three cases are set up for each organic vapour,representing thermodynamic states far from, close to and insidethe supercritical region, and these cases refer to weak, normal and

10

0 50 100 150 200 250

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

+10%

-10%

Air

R1

R2

R3

M1

M2

M3

0.09

Fig.11 DISTRIBUTION OF νtεk/k2t ACROSS ALL CASES.

strong real-gas effect in each fluid.

The turbulent statistic parameters are analyzed andcompared.The results show that the real-gas effect influencesthe averaged thermodynamic property profile. The averagedcentre-to-wall temperature ratio is lower in the organic fluid withlarger molecular weight, and this ratio will decrease as real-gaseffect increases. The averaged wall-to-centre density ratio isalso lower in the organic fluid with larger molecular weight, butincreases as real-gas effect increases, which is reversed from theimpact on averaged centre-to-wall temperature ratio.

The averaged centre-to-wall viscosity ratio is lower than 1, whichis a kind of ‘liquid-like behavior’, when the real-gas effect is strongenough (in case R1 and M1). There exists a transition line

(∂u∂T

)= 0

for each organic fluid. when working condition is above this line(strong real-gas effect), the viscosity at channel centre is lower thanat wall.

The real-gas effect does not influence the normal Reynolds stressin the streamwise direction, but it has an obvious impact on theother two normal Reynolds stress and the Reynolds shear stress.The peak values of the wall-normal Reynolds stress v′′v′′ in real-gascases are larger than that in air case by 7-20%, and the peak valuesof the spanwise Reynolds stress w′′w′′ in real-gas cases are largerby 10-21%. The real-gas effect influences the viscosity profile ,therefore, increases v′′v′′ and w′′w′′. The Reynolds shear stress u′′v′′

is also increase by the real-gas effect since its peak value is 5-8%larger in two real-gas fluids than in air.

The turbulence kinetic energy dissipation εk in the viscoussublayer and buffer sublayer (y∗ < 30) can be impacted by thereal-gas effect, which will increase as real-gas effect increases.The turbulence viscosity hypothesis within standard two-functionRANS model (k− ε and k−ω) is checked in these two organicfluids, and the result shows that it is still suitable in the outer layery∗ > 70 with a constant Cµ = 0.09 for all real-gas cases, with anerror in ±10%..

ACKNOWLEDGEMENTThe DNS code in this research is developed together with X. Li.

The research was funded in part by the China Scholarship Council(CSC).

APPENDIXValidation of the Simulation Method

The simulation method is validated by applying it on acompressible turbulent channel flow case of air, of which ReB =

4880 and MaB = 3.0. The domain size and grid are the same asthat used in this work. The result of this case is compared withthe DNS data of Coleman [23] and shown in Fig. 12. It shows thatthe mean velocity, root-mean-square of velocity and Reynolds shearstress distribution follow the result of Coleman, and the differenceis within an acceptable range.

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11

100

101

102

0

5

10

15

20

25

30

35

Coleman[23]

Code in this work

(a)

0 50 100 150 200

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 urms

vrms

wrms

urms

(Coleman[23])

vrms

(Coleman[23])

wrms

(Coleman[23])

(b)

0 0.2 0.4 0.6 0.8 1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Coleman[23]

Code in this work

(c)

Fig.12 VALIDATION OF THE METHOD (a) MEAN VELOCITY,(b) ROOT-MEAN-SQUARE OF VELOCITY AT DIFFERENTDIRECTION, (c) REYNOLDS SHEAR STRESS.

[12] P.G. Huang, G.N. Coleman and P. Bradshaw, 1995,”Compressible turbulent channel flows: DNS results andmodelling”, Journal of Fluid Mechanics, Vol. 305, pp. 185-218.

[13] A. Patel, J.W. Peeters, B.J. Boersma and R. Pecnik, 2015,”Semi-local scaling and turbulence modulation in variableproperty turbulent channel flows”, Physics of Fluids, Vol. 27,No. 9, 095101.

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[17] F. Zonta, C. Marchioli, and A. Soldati ,2012, ”Modulationof turbulence in forced convection by temperature- dependentviscosity”, Journal of Fluid Mechanics, vol 697, pp 150-174.

[18] J. Lee, S.Y. Jung, H.J. Sung, and T.A. Zaki, 2013,”Effect of wall heating on turbulent boundary layers withtemperature-dependent viscosity”, Journal of Fluid Mechanics,vol.726,pp 196-225.

[19] A.S. Monin, and A.M. Yaglom, 1975, ”Statistical FluidMechanics: Mechanism of Turbulence, book 2”, MIT Press.

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[23] G.N. Coleman, J. Kim, and R. D. Moser, 1995, ”A numericalstudy of turbulent supersonic isothermal-wall channel flow”,Journal of Fluid Mechanics, Vol 305, pp 159-184.

12