One-dimensional turbulence investigation of variable density effects due to heat transfer in a low Mach number internal air flow

Highlights

• We introduce an extension of the spatial ODT formulation (S-ODT), in order to study a variable density internal air flow.

• S-ODT results are compared to the results of a traditional temporal ODT formulation (T-ODT) and to reference DNS data.

• For bulk quantities, both the T-ODT and S-ODT formulations have a similar performance in comparison to DNSs.

• For the evaluation of radial distributions, the S-ODT formulation clearly outperforms the T-ODT formulation.

Abstract

A novel spatial formulation of the One-Dimensional Turbulence (ODT) model is applied to a vertical pipe-flow with heat transfer, analogous to the Direct Numerical Simulation (DNS) performed by Bae et al. [Phys. Fluids 18, (075102) (2006)]. The framework presented here is an extension for radially confined domains of the cylindrical ODT spatial formulation for low Mach number flows with variable density. The variable density simulations for air (Prandtl number $Pr=0.71$) are performed at an initial bulk Reynolds number $R{e}_{b,0,DNS}=6000$ and Grashof number $G{r}_{0,DNS}=6.78\times {10}^6$. ODT results are presented for both the spatial formulation introduced in this work and the standard temporal formulation for cylindrical flows introduced by Lignell et al. [Theor. Comput. Fluid Dyn. 32, 4 (2018), pp. 495–20]. Streamwise bulk profiles and radial profiles at specific streamwise positions for the temporal and spatial formulations are in good agreement with the DNS results from Bae et al. For the present application, the spatial formulation yields physically better results in comparison to the temporal formulation. Overall, the findings in the original work of Bae et al. were corroborated with ODT. Although the framework proposed in this work is not a compressible framework and has some clear limitations regarding conservation properties, we suggest its use for future studies in the low Mach number variable density regime.

Introduction

Some understanding of the physics associated with the thermal boundary layers in wall-bounded flows with adverse pressure gradients has been achieved parallel to the increase in the number of heat transfer applications for pipes (Van Driest, 1956, McEligot, Coon, Perkins, 1970, Huang, Coleman, Bradshaw, 1995). Although it is tempting to think about compressible flow whenever analyzing variable density effects, there is a large number of everyday applications involving variable density, which do not necessarily deal with the intrincate phenomena happening at elevated velocity (high Mach number), e.g. heating or cooling, as well as some geophysical applications (Davidson, 2013). In comparison, however, to the research on incompressible pipe-flow in Direct Numerical Simulations (DNSs) (Satake, Kunugi, Himeno, 2000, Wu, Moin, 2008, Chin, Monty, Ooi, 2014, Ahn, Lee, Lee, Kang, Sung, 2015), only few studies have tackled pipe-flow problems with variable properties (Bae, Yoo, Choi, 2005, Xu, Lee, Pletcher, 2005, Bae, Yoo, Choi, McEligot, 2006, Ghosh, Foysi, Friedrich, 2010, Modesti, Pirozzoli, 2019). Bae et al. discussed the dissimilar behavior between the velocity and temperature profiles in a low Mach number internal air flow with large temperature gradients (Bae et al., 2006). Turbulence statistics from compressible and incompressible pipe and channel flow were also studied up to a second-order degree in the work done by Ghosh et al. (2010). Nonetheless, and due to practical reasons, the detailed numerical simulations with variable properties remain restricted to relatively low Reynolds numbers (e.g. up to $R{e}_b=31500,$ as in Modesti and Pirozzoli (2019)).

In order to simulate flows at large Reynolds numbers, a large family of turbulence models have been formulated. In the field of map-based stochastic turbulence models, Alan Kerstein proposed in 1999 the idea of One-Dimensional Turbulence (ODT) (Kerstein, 1999). Much of the research effort in ODT has been devoted to free shear and wall-bounded flows. As first formal applications in this context, Dreeben and Kerstein investigated the problem of stationary buoyant turbulent flow in a vertical slot using an incompressible formulation and a Boussinesq approach (Dreeben and Kerstein, 2000). Likewise, Echekki et al. (2001) presented a framework for the evaluation of stationary spatially developing jets on the basis of temporally evolving jets, where heat and mass transfer were evaluated in the context of a jet diffusion flame. Ashurst and Kerstein were the first ones to present a framework for variable density treatment in ODT. They also introduced an extension for spatially developing free shear flows with variable density, which did not require a temporal-to-spatial transformation (Ashurst and Kerstein, 2005). Recently, a novel cylindrical extension of ODT was formulated by Lignell et al. (2018). As in Ashurst and Kerstein (2005), the spatial formulation from Lignell et al. (2018) is limited to free shear flows, such as jets, which have a mathematical description portraying a parabolic behaviour in nature, in the light of the low Mach number approximation. Therefore, in order to partially overcome the constraints associated with the traditional spatial ODT formulation, we seek an extension of the formulation for wall-bounded flows with variable properties within the low Mach number approximation.

This paper is structured as follows: in Section 2 we discuss the low Mach number treatment of the deterministic ODT part for the temporal and spatial formulations, as well as the stochastic turbulent advection treatment. Section 3 discusses the initial conditions of the simulation setup and presents the results for bulk quantities, velocity and temperature distributions, as well as their turbulent fluctuations. In Section 4, we summarize and conclude our findings. As a complement to the work, theoretical considerations regarding the chosen form of the shear stress tensor for the cylindrical formulation are given in Appendix A. Details of the numerical procedure are given in Appendix C, while a discussion about statistical quantities in ODT for the variable density formulation are given in Appendix D.