**3. Geometry and Reference Cases**

The validation reference case is based on experimental data published in the literature. The experiments were conducted at the "TU - Heat and Mass Transfer Laboratory" at the von Karman Institute in Sint Genesius Rode, Belgium. The flow data are taken from Virgilio et al. [20] and the heat transfer data from Virgilio et al. [21]. For the velocity measurements, a low-speed water tunnel was used. The measurement setup is configured in a way that the pressure at the inlet can be assumed to be constant; for a full optical access, the pipe is made of acrylic glass. Two different helical turbulators are measured, defined as RIB-1 and RIB-2, illustrated in Figure 1. The RIB-1 is the continuous helicoidal turbulator studied by Mayo et al. [17]. These are made of acrylonitrile butadiene styrene using a 3D printer. The diameter D of the pipe examined here is 150 mm; in the experiment, the pipe length is 15 × D. The rib pitch *p* corresponds to 63 mm, rib height *e* is 5.4 mm, angle between the axial and helix-wise directions is 80°, and the rib width is 10.8 mm. RIB-2 is different in that the height of the semi-circular rib changes along the helical pattern direction, but the maximum height remains the same for both ribs. After an angular displacement of 11°, this height becomes zero in both directions of the helix. This results in six obstacles on one pitch of the helix, with a total of nine pitches. For RIB-1 and RIB-2, Reynolds numbers of 24,400 and 21,100 have been investigated in the simulation. The stereoscopicv particle image velocimetry (S-PIV) images are taken between the seventh and eighth pitch. Data are only available in the area between the ribs. Liquid Crystals Thermography (LCT) is used to measure the stationary heat transfer. To reduce the angular influence on the measurement inaccuracy, a lately developed calibration technique for the narrow-band LCT was successfully applied. Since the simulations are computationally expensive and RIB-1 has already been examined in detail by Campet et al. [19] and Cauwenberg et al. [18] with an LES, in this article, RIB-2 has been investigated with the use of LES.

**Figure 1.** Schematics of the continuous helicoidal turbulator RIB-1 (**left**) and RIB-2 (**right**) from Virgillio et al. [20].

#### **4. Numerical Methodology**

The accurate calculation of heat transfer in turbulent pipe flow requires a fully developed turbulent flow inside the pipe. Limitations of computational resources did not allow simulations of a long pipe to ensure a fully developed flow. By simulating a small section of a pipe with cyclic boundary conditions at the inlet and outlet, a fully developed turbulent flow could be achieved, if a sufficient number of flow cycles were carried out. Table 1 lists the boundary conditions used, and the schematic illustration is shown in Figure 2. For the calculation, it is necessary to insert a source term for momentum and the energy equation due to the cyclic boundary conditions. This setup is also applied in the studies by

Campet et al. [19], Kügele et al. [25] and Akermann et al. [16]. Second-order methods are used for all discretisations. A Courant number of 0.3 is used for the computation, and, after a quasi-steady state of the flow has set in, the results are averaged over seven cycles as in the study by Campet et al. [19].

**Figure 2.** Schematic representation of the boundary conditions used.



The length of the calculation area is 4 × D. For the cyclical boundary conditions, it must be ensured that inlet and outlet are exactly identical. The mesh without the boundary layers is created with cfMesh; snappyHexMesh is then used to generate the final mesh. The maximum cell size within the geometry is 0.002 mm. Near the wall, the grid resolution is reduced to about 0.0005 mm to be able to better represent the structure of the ribs and to keep the transition between layers and cells small. The number of cells adds up to 23.33 million, with 67% of the cells forming the layers. The total number of layers used is ten, the smallest layer has a height of 0.03116 mm, and the growth rate of the layers is set to 1.2. The influence of the refinement on the wall and the addition of layers can be seen on the inlet and on the ribs in Figure 3. With these techniques, a *y*<sup>+</sup> value of less than 0.2 can be achieved. The values for *x*<sup>+</sup> and *z*<sup>+</sup> correspond to 7, which is sufficient for LES as shown in Akermann et al. [16].

**Figure 3.** Display of the grid with layers at the inlet (**left**) and along the center (**right**) of the rib in RIB-2 geometry.

To ensure a sufficient grid resolution for the LES, the LSR (Length Scale Resolution) parameter is investigated. It corresponds to the ratio between the actually resolved energy level and the corresponding lower limit of the inertial subrange and is defined as:

$$LSR = \frac{\Delta}{60 \cdot \eta\_{\text{kol}}} \tag{6}$$

where Δ corresponds to the grid width. The Kolmogorov length scale is calculated with:

$$
\eta\_{kol} = (\frac{\nu^3}{\epsilon})^{1/4} \tag{7}
$$

$$
\epsilon = \frac{k^{3/2}}{D/6} \tag{8}
$$

and the turbulent kinetic energy *k* is determined using the simulation results. If the value is equal to 1, the turbulent scales are resolved up to the dissipation range. Therefore, a link between resolved energy levels and the local filter sizes can be created [26]. The evaluation in Figure 4 shows that the value is below 1 in all ranges. This ensures that the grid resolution is high enough.

**Figure 4.** Display of the grid with layers at the inlet of the RIB-2 geometry.
