## Introduction

Wall-bounded turbulent flows of organic fluids have drawn increasing attention in recent years, in part due to the application of organic Rankine cycle (ORC) systems for waste heat recovery. With wall-bounded flows present in the majority of ORC system components, an improved understanding of the detailed fluid mechanics within real-gas boundary layers is fundamental to (a) the validation of Computational Fluid Dynamics (CFD) codes, and therefore (b) the maximisation of component-level performance.

The majority of research within this area (Colonna et al., 2006; Colonna et al., 2008; Wheeler and Ong, 2014; Persico et al., 2015) has analysed blade-bounded flows in a turbine or nozzle, based upon the Reynolds Averaged Navier Stokes (RANS) simulation method. However further work is required for validation, to confirm that real-gas effects do not disturb the fundamental assumptions contained within RANS simulations.

Real-gas effects distinguish organic fluids from perfect gases such as air. The perfect-gas Equation of State (EoS),

Work by Colonna & Guardone (Colonna and Guardone, 2006) has shown that as the gas molecular complexity increases, its heat capacity,

This work studies the real-gas effect on the turbulent boundary layer in a fully developed channel flow by means of Direct Numerical Simulation (DNS), in which no turbulence model is required due to flows being resolved down to the Kolmogorov microscale. DNS helps to get rid of the uncertainty caused by any turbulent modelling assumption, and produces the realization of a real-gas flow.

Kim, Moin & Moser (Kim et al., 1987) firstly applied the DNS method on a incompressible fully developed channel flow at Reynolds number of 3,300. They studied the turbulence statistics near the wall, and found that their results showed a good agreement with the experimental data.

Huang, Coleman & Bradshaw (Huang et al., 1995) studied compressible flow within a channel, and analysed the compressibility-associated terms in Reynolds averaged energy equations. They found that the averaged property profiles matched the corresponding incompressible curves well, by scaling by mean density,

Patel et al (Patel et al., 2015) studied a turbulent channel with variable properties, in different cases of the relation between dynamic viscosity and temperature. They found that the normal Reynolds stress anisotropy and turbulence-to-mean time scales were influenced by that relation.

Sciacovelli, Cinnella, & Gloerfelt (Sciacovelli et al., 2017) studied a dense gas, PP_{11} (C_{14}F_{24}), in a turbulent channel flow. They focused on one operating point inside the region

The study within this work focuses on two commonly-used organic fluids, and compares three different working conditions of each fluid to give insight into the impact of real-gas effects. The main properties of these two organic fluids are show in Table 1. Refrigerant R1233zd(E), trans-1-chloro-3,3,3-trifluoropropene (CF_{3}CH = CHCl), is accepted as a high-efficiency and environmental-friendly fluid for ORC systems, and it can be a substitute of refrigerant R245fa. Meanwhile, MDM, formally known as octamethyltrisiloxane (C_{8}H_{24}O_{2}Si_{3}), is a representative of the group of siloxane fluids, since they also can be a potential fluid of ORC systems.

## Methodology

### Numerical method

A fully developed channel flow is simulated in the domain shown in Figure 1, where the boundaries on X and Z direction are periodic boundaries and the boundary on Y direction is no-slip isothermal wall boundary - mean flow is in the X direction. The scale of the domain is *h* is the characteristic length of the channel.

This work is based on the solution of the three-dimensional Navier-Stokes (NS) equations (Equations 2–4), which are closed by the equation of state *h*, globally averaged density,

where:

Non-dimensional parameters can be declared as:

For a real gas, all the thermodynamic parameters are two-variable functions of both temperature and density. The global density and wall temperature (

For the fully developed channel flow there is a periodic boundary instead of inlet and outlet boundaries, so the pressure drop along the stream is balanced by an artificial added body force

The DNS solver is developed based on a compressible DNS code from X. Li (Li et al., 2001) with an addition of a real-gas model. All the thermal properties in this model are two-variable functions of the local density and temperature, and they are calculated by the state-of-art Helmholtz energy EoS in REFPROP (Lemmon et al., 2007). Organic fluid cases are calculated by this real-gas model while the air case is calculated by a perfect-gas model. The NS equations are solved after non-dimensionalisation. For spectral discretisation, the convective flux derivatives are calculated by seventh-order upwind finite-difference scheme, and the viscous flux derivatives are calculated by sixth-order central scheme. For time advancement, it is solved by a third-order Runge-Kutta method. The validation of the simulation method is shown in the appendix.

The number of nodes in this work is ^{6}. The spatial resolution is chosen as DNS requirements. Zonta (Zonta et al., 2012) advised that

##### Table 2.

Case | Air | R1 | R2 | R3 | M1 | M2 | M3 |
---|---|---|---|---|---|---|---|

12.6 | 9.9 | 11.1 | 14.3 | 9.4 | 9.4 | 10.6 | |

0.8 | 1.0 | 1.1 | 1.9 | 1.1 | 1.1 | 1.3 | |

6.3 | 5.0 | 5.6 | 7.1 | 4.7 | 4.7 | 5.3 |

Figure 2a and 2b show two-point correlation of velocity u and density

This work is solved by the High Performance Computer in Imperial College London. Each case uses 512 CPUs, and it takes 10–20 days to get converged. After that, the statistic is taken every

### Case set-up

There are no standard inlet and outlet boundaries in this domain for the periodic boundary condition, so the global averaged density *T _{s}* = 110.56 K is the effective temperature.

The initial condition is a turbulent flow field, in which the global averaged density

The global Mach number *T _{w}* = 288 K. The two thermodynamic conditions chosen for the real-gas cases were wall temperature and global density. From R1 to R3, the set-up point moves closer to the supercritical region, shown by the temperature-entropy diagram in Figure 3. The red and blue lines in Figure 3 are the averaged thermodynamic property distribution in each real-gas case.

## Results and discussion

### Averaged velocity and thermodynamic properties

The variables are averaged on each y level, since the x and z directions are periodic. For analysis, the wall shear stress is defined as the averaged value

Some important global results of each case are listed in Table 4. Bulk Reynolds number and bulk Mach number are defined as

##### Table 4.

The centre-line Reynolds number and Mach number are defined as

The friction Reynolds number is defined as

Reynolds averaged velocity and distribution versus non-dimensional wall distance

The averaged temperature, density and pressure along the wall-normal direction are shown in Figure 5, where the abscissa is the non-dimensional wall distance

For each organic fluid,

When averaging and integrating the energy equation, the heat transfer as a function of wall distance can be represented by Equation 18 [12]. In the expression,

By assuming the velocity distribution function and the thermal conductivity ratio the same in all these cases, the non-dimensional temperature gradient is approximately proportional to the global Prandtl number multiplied by global Eckert number

The density is higher at the channel wall than the channel centre (shown in Figure 5c, and the value of the wall-to-centre density ratio

Since the pressure is nearly constant (

The pressure ratio has a minimum value at

After integrating Equation 22 along wall distance y, the expression of for pressure ratio shown in Equation 23 is found. The pressure ratio is associated to

The distribution of averaged dynamic viscosity ratio

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

### Reynolds stress

The distributions of Reynolds stresses

The peak values of

The impact of real-gas effect on the normal Reynolds stress in spanwise and wall-normal directions (

The Reynolds shear stress

### Turbulence energy

The Reynolds averaged turbulence energy equation for fully developed channel flow is shown in Equation 25. The first term is turbulence energy production

##### (25)

The distribution of turbulence energy production

The turbulence energy production-to-dissipation ratio

### Turbulence viscosity hypothesis

The evaluation of the turbulence viscosity hypothesis for the standard two-function RANS model (*k* and dissipation *k* and

The ratio

With the data for R1233zd(E) and MDM collected so tightly around the constant

## Conclusions

This work studied the fully developed compressible turbulent channel flow of two organic fluids, R1233zd(E) and MDM, by means of Direct Numerical Simulation. Three cases for each fluid are set up at the same global Reynolds number (*Re _{G}* = 4,880) and global Mach number (

*Ma*= 2.25), and they are compared with an air baseline case. Three cases are set up for each organic vapour, representing thermodynamic states far from, close to and inside the supercritical region, and these cases refer to weak, normal and strong real-gas effect in each fluid.

_{G}The turbulent statistic parameters are analyzed and compared. The results show that the real-gas effect influences the averaged thermodynamic property profile. The averaged centre-to-wall temperature ratio is lower in the organic fluid with larger molecular weight, and this ratio will decrease as real-gas effect increases. The averaged wall-to-centre density ratio is also lower in the organic fluid with larger molecular weight, but increases as real-gas effect increases, which is reversed from the impact on averaged centre-to-wall temperature ratio.

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

The real-gas effect does not influence the normal Reynolds stress in the streamwise direction, but it has an obvious impact on the other two normal Reynolds stress and the Reynolds shear stress. The peak values of the wall-normal Reynolds stress

The turbulence kinetic energy dissipation

## Nomenclature

*c*

Sonic velocity (m/s)

*e*

Inner heat energy (J/kg)

Body force (m/s^{2})

*k*

Turbulent kinetic energy (J/kg)

Mass flow rate (kg/s)

*p*

Pressure (Pa)

Heat conduction (J/(m^{2}s))

*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)

Position (m)

Turbulent model constant

Isobaric specific heat capacity (J/(kg·K))

Isochoric specific heat capacity (J/(kg·K))

*L*

Channel length scale (m)

*M*

Molecular weight (kg/kmol)

*N*

Number of nodes (−)

Turbulent kinetic energy generation rate (W)

*R*

Specific gas constant (J/(kg·K))

*T*

Temperature (K)

*V*

Specific volume (m^{3}/kg)

*Z*

Compressibility factor (−)

Ec

Eckert number (−)

Ma

Mach number (−)

Re

Reynolds number (−)

Pr

Prandtl number (−)

### Greeks

Specific heat ratio (−)

Turbulent kinetic energy dissipation rate (W)

Smallest length scale (m)

Kolmogorov length scale (m)

Length scale in temperature field (m)

Heat conductivity (W/(m·K))

Dynamic viscosity (Pa·s)

Kinematic viscosity (m^{2}/s)

Density (kg/m^{3})

Viscous stress (Pa)

Fundamental derivative (−)