**1. Introduction**

In order to improve the heat transfer in technical devices, pipes with a rough surface are used. In addition, there are many other possibilities to passively improve the heat transfer, for example various internals or the use of nanofluids, an overview of these methods can be found in the paper of Ajarostaghi et al. [1]. The roughness ensures that the boundary layer of the flow is disturbed, transport processes normal to the wall are enhanced, and, additionally, the heat transfer surface is increased. Due to larger frictional and pressure forces, the pressure loss is increased [2]. Most published studies of internally ribbed pipes are based on experimental methods. Due to the increasing computing power, it is now possible to perform detailed flow simulations. This enables an improved virtual product development to reduce the costs of expensive experiments and speed up the development.

In the past, numerous experimental studies have been performed on internally ribbed pipes. With the aim of improving heat transfer and pressure loss, helically ribbed pipes were investigated by Webb et al. [2], Gee andWebb [3], Withers [4,5], Han et al. [6], and Nakayama et al. [7]. These authors have established correlations for the heat transfer and the friction coefficient based on their experiments, which are valid for various physical and geometrical parameters. By applying measured data to a linear model, Ravigururajan and Bergles [8] developed general correlations for pressure loss and heat transfer in single-phase turbulent flow using the research data of Webb [9], Gee and Webb [3], Withers [4,5], and Kumar and Judd [10]. In addition, to develop correlations for heat transfer and friction

**Citation:** Kügele, S.; Mathlouthi, G.O.; Renze, P.; Grützner, T. Numerical Simulation of Flow and Heat Transfer of a Discontinuous Single Started Helically Ribbed Pipe. *Energies* **2022**, *15*, 7096. https:// doi.org/10.3390/en15197096

Academic Editors: Artur Bartosik and Dariusz Asendrych

Received: 30 August 2022 Accepted: 22 September 2022 Published: 27 September 2022

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in pipes with internal helical ribs, Zdaniuk et al. [11] used an artificial neural network approach based on experimental data.

For the investigation of internally structured pipes, the importance of simulations increases due to the ever increasing computing resources and the resulting reduced computing times. In the past, several studies have been performed in this area, and a selection of them are briefly described in the following.

A comparison between the experimental data of Ravigururajan and Bergles [8] with the results of Reynolds-averaged Navier–Stokes (RANS) simulations for a single started helically ribbed pipe can be found in Hossainpour and Hassanzadeh [12]. They show that it is possible to provide reasonably good results in the range of the Reynolds number from 25,000 to 80,000. In this study, no comparison of the local values such as the local heat transfer has been made. A comparison between RANS and a large-eddy simulation (LES) was performed by Vijapurapu and Cui [13]. Cauwenberge et al. [14] demonstrate that RANS simulations are unable to predict certain secondary flow phenomena that have an effect on heat transport and pressure loss. In the work of Wang [15], it has been shown by means of an LES simulation that the performance of helically corrugated pipes is superior to the transverse corrugated pipe. A multiple-started helically ribbed pipe is investigated in Akermann et al. [16] using an LES method, where the simulation setup is validated with experimental data for the Nusselt number and pressure drop for Reynolds numbers of 8000 and 16,000 and Prandtl numbers of 5, 7 and 9. Based on the previously mentioned experimental data from Mayo et al. [17], two studies have been carried out to validate a simulation with an LES model with the simpler continuous geometry. The first study was done by Cauwenberg et al. [18] and the second by Campet et al. [19]. Both studies show a good agreement between the simulation and measurement for the mean values of velocity and temperature. The authors were able to perform a successful validation for a continuous ribbed pipe geometry using an LES. The literature referenced above only deals with continuous structures at internal pipe walls. Measurements of three-dimensional structures were performed at the von Karman Institute, where Mayo et al. [17] determined experimentally the heat transfer and flow properties in a pipe with a helically structured rib. Further experiments for the identical pipe and a comparison with a modified version were performed in the publications of Virgillio et al. [20,21]. The flow and heat transfer between the ribs are investigated in these measurements and used for the comparison in the present study.

The aim of this study is to explain and further analyze the findings from these heat transfer measurements at such complex structures with the help of detailed flow simulations. Three-dimensional wall structures lead to very complex flow phenomena that have been rarely investigated so far. The experiments from the literature can usually only provide global values such as pressure loss and heat transfer, and local values can only be determined with complex measurement methods. With the current simulation method, it should be possible in the future to view the global results in connection with the local physical processes without complex measurement methods.

That is why in the present work an LES of a discontinuous single-started helically ribbed pipe is performed and validated with experimental data for the velocity and heat transfer between the ribs. The results demonstrate that it is possible to reproduce the measurement data even for complex geometries. The difficulty here is especially in the interruptions of the helix, so the local results are more difficult to compare than with a continuous helix. The aim of this study is to speed up the development process of new complex geometries and the associated correlations for the calculation of heat transfer and pressure loss.
