Some observations concerning "laminarization" in heated vertical tubes

https://doi.org/10.1016/j.ijheatmasstransfer.2020.120101Get rights and content

Highlights

  • Use of direct numerical simulation to study laminarization of heated gas flow.

  • Hypothesize that streamwise vortices provide least resistance for momentum transfer.

  • The number of vortices decreases as the flow "laminarizes".

  • A modified wall Reynolds number provides a criterion for laminarization.

Abstract

For low "turbulent" Reynolds numbers in strongly-heated vertical gas flow, fundamental results of existing direct numerical simulation (DNS) databases have been examined to deduce differences between cases which "revert" to turbulent and those that yield integral parameters which correspond to laminar flow (henceforth laminarizing or laminarized). Objectives are (1) to examine the streamwise evolution of near-wall flow structures and (2) to determine which, if any, proposed laminarization parameters could be used to discriminate between turbulent and laminarizing flows resulting. Views of streamlines in r-θ cross sections showed evidence of a ring of irregular-shaped streamwise vortices near the wall at all locations. The transverse advection of these vortices is hypothesized to provide a path of least resistance to the transport of streamwise momentum in the wall-normal direction. It is demonstrated that a steady streamwise laminar vortex can provide apparent turbulent momentum transport (as a consequence of the Reynolds decomposition) such that a reasonable mean velocity profile is predicted near the wall. For the present cases, the values of the non-dimensional radius (yc,w+ = Reτ,w) and of the modified wall Reynolds number did discriminate whether turbulent or laminarizing flow resulted.

Introduction

Experiments to determine heat transfer characteristics of internal flows frequently employ a gas in a resistively heated vertical tube with an unheated entry for flow development [39]. At low “turbulent” Reynolds numbers with “high” heating rates, results – presented as graphs of local Nusselt number versus local Reynolds number – show two characteristic shapes as in Fig. 1(a) and (b); a comparable discrimination is shown by Bankston [5] in terms of the Stanton number. Since viscosity increases with temperature, the bulk Reynolds number (Reb = GD/μb) decreases as the gas is heated, i.e., successive streamwise values move from right to left in the figures. Fig. 1(b) was discussed in a recent note by McEligot et al. [51]; the shape is hypothesized to occur when the viscous layer is thicker than the thermal boundary layer at the heated entry so that a laminar Leveque solution is reasonable. This situation is called “laminarization” [52] or “deterioration” [39]. In this note, we generally define the wall region as by Bradshaw [12] with the region where molecular transport dominates called the "linear" layer and the region where it is noticeable called the viscous layer (to y+ ≈ 30 in unheated, high Re flows). However, when discussing the work of other investigators we may use their terminology.

The trends demonstrated in Fig. 1(a) occur when the flow can be considered to remain turbulent. Shown for comparison is the popular Dittus-Bölter correlation [21], [22]Nub0.021Reb0.8Prb0.4for fully-developed turbulent flow of gases in tubes with constant properties. For a given experimental run the high values in the figures represent results in the thermal entry where the temperature ratio Tw{x}/Tb{x} increases with streamwise distance (and Reb decreases). Further downstream the increase in Tb leads to Tw{x}/Tb{x} decreasing. Typical empirical correlations describing heat transfer with variable gas properties are ones like those of McEligot et al. [56]Nub0.021Reb0.8Prb0.4(TwTb)1/2for the downstream region and by Gnielinski [25]. Beyond the thermal entry the effect of the variable gas properties (represented by Tw/Tb) is to reduce Nub below the Dittus-Bölter prediction. Then, as Tw/Tb decreases towards unity, Nub converges towards the correlation as seen in Fig. 1(a). For further background on effects of gas property variation on internal convective heat transfer, literature reviews in the papers by McEligot [50], Shehata and McEligot [76], Bae et al. [2], Lee et al. [39], Chu et al. [19], Valentin et al. [82] and others may be helpful.

Comparing Fig. 1(a) and (b), one sees that – at a given Reynolds number in this region – a wide range in Nu (and therefore h) is possible so that predicted wall temperatures can be highly uncertain. A general aim of the present note is to reduce this uncertainty by examining proposed criteria that might discriminate between the likelihood of remaining turbulent or laminarizing. For that purpose we will employ direct numerical simulations (DNS) of the turbulent and laminarizing experiments of Shehata [75], [76] and variations of his conditions.

The temporal three-dimensional DNS results provide much more information than just the integral parameters to address the above aim. For example, since the majority of the convective thermal resistance occurs in the viscous layers for low-Reynolds-number wall flows, it is appropriate to investigate the flow behavior there directly – as can be done with the DNS results.

Useful reviews of studies of near-wall structure of turbulent boundary layer and duct flows have been provided by Kovasznay [35], Cantwell [17], Blackwelder [8], Robinson [67], Jimenez [30], [31], Marusic and Monty [48], Tuckerman, Chantry and Barkley [81] and others. All treat incompressible flow with constant properties in contrast to the current cases with their strongly varying gas properties. The review by Robinson is still a valuable overview to various ideas on the near-wall structure of turbulent boundary layers and the one by Jimenez [31] provides a detailed update with concentration on the coherent structures of all scales.

From his DNS Chu [19] has presented a couple realizations of instantaneous streamwise cross-sections (rx plane) for Shehata's Run 445 in his Fig. 14: one at the thermal entry and the other showing the five diameters at the end of the tube. The latter shows so-called "streaks," one "low-speed" and one "high-speed" [30], [67] which are large relative to the diameter and persist along the entire sub-figure, i.e., 25–30 D. Bae [2] also provides instantaneous cross-sections in the rx plane. He presents distributions of ρ and u at 22.5 < (x/D) < 25 for four runs in his Fig. 22. In his Fig. 23 he also shows instantaneous isosurfaces near the wall for a constant value of the streamwise vorticity along the entire length of the tube, giving a clear demonstration of laminarizaton.

Consequently, the key objectives for the present study are (1) to examine the streamwise evolution of near-wall flow structures and (2) to determine which, if any, proposed laminarization parameters could be used to discriminate between turbulent and laminarizing flows resulting. These two objectives are not necessarily directly related.

Section snippets

Direct numerical simulations

As noted in our previous paper [51], experiments such as those by Lee [38], Shehata [75] and others do not exactly replicate the desired idealizations of a step change in the wall heat flux nor the uniform wall heat flux in the heated section. Direct numerical simulations can match these idealizations more closely; in addition they can give more details on the turbulent flow and its temporal behavior than experiments usually can. So for closer representation of these thermal boundary

"Flow visualization" of near-wall structure

Additional insight into the evolution of Runs 618 U, 635 U and 445 U can be obtained by examining the streamwise variation of the near-wall structure of their instantaneous fields, i.e., a computational version of photographic flow visualization. The DNS of these three runs involves heat transfer to a gas with significant variation of the fluid properties, except in the isothermal turbulence generating inlet. On the other hand, in this present Section 3 the references cited generally involve

Potential criteria

In the past, stability analyses have been applied to the problem of transition from laminar to turbulent boundary layers to account for various stabilizing/destabilizing phenomena [72]. For example, favorable streamwise pressure gradients associated with flow acceleration have been found to delay transition of a boundary layer, a heated wall is believed to destabilize a gas flow [47] and vertical free convection can produce strong destabilizing effects [59]. Narasimha and Sreenivasan mention

Discussion

For low "turbulent" Reynolds numbers in strongly-heated vertical gas flow, fundamental results of DNS have been examined to deduce differences between cases which "revert" to turbulent and those that yield integral parameters which correspond to laminar flow (called laminarizing or laminarized). Key objectives are (1) to examine the streamwise evolution of near-wall flow structures and (2) to determine which, if any, proposed laminarization parameters could be used to discriminate between

CRediT authorship contribution statement

Donald M. McEligot: Conceptualization, Methodology, Formal analysis, Data curation, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration, Funding acquisition. Xu Chu: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization. Joong Hun Bae: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We thank Prof. J.-i. Lee of KAIST for providing his tabulated data and his continued friendly help over the years. This study was partly supported by the Korea/U.S. International NERI (Nuclear Energy Research Initiative) program under the auspices of the Korea Institute of Science and Technology Evaluation and Planning (KISTEP) and also by the Creative Research Initiative (CRI) program of KISTEP, by the Forschungsinstitut für Kernenergie und Energiewandlung, by the Ministerium für Wissenschaft,

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