Effect of Large Scale 3-D Structures on the Flow Around a Heated Cylinder at Low Reynolds Number

Abstract

This work presents results of flow around a heated circular cylinder in mixed convection regime and demonstrates that Prandtl number and angle of attack of the incoming flow have a large influence on the characterisation of the flow transition from 2-D to 3-D. Previous studies show that heat transfer can enhance the formation of large 3-D structures in the wake of the cylinder for Reynolds numbers between 75 and 127 and a Richardson number larger than 0.35. This transitional mode is generally identified as “mode E”. In this work, we compare the results for water-based flow (large Prandtl number) with the ones for air-based flows (low Prandtl number). The comparison is carried out at two Reynolds numbers (100 and 150) and at a fixed Richardson number of 1. It shows that at the low Reynolds number of 100 the low Prandtl number flow does not enter into transition. This is caused by the impairment of the baroclinic vorticity production provoked by the spanwise temperature gradient. At low Prandtl number temperature gradients are less steep. For an air-based flow at Reynolds number 150, several Richardson numbers have been simulated. In this situation, the flow enters into transition and exhibits the characteristics of “mode E”, with the development of Λ-shaped structures in the near wake and mushroom-like structures in the far wake. It is also observed that the transition is delayed at Richardson number of 0.5. Simulations are also carried to investigate the effect of the angle of attack on the incoming flow on the development of large coherent structures. When the angle of attack is positive, the development of the wake tends to return to a more bi-dimensional configuration, where large scale coherent structures are impaired. In contrast, when the angle of attack is negative, large scale tri-dimensional structures dominate the flow in the wake, but with a very chaotic behaviour and the regular pattern of zero angle of attack is destroyed. The different behaviour of the flow with the variation of the angle of attack is also related to the baroclinic vorticity production, where new terms appear in the equations, leading to a positive effect of the vorticity production in case of a negative angle of attack and the opposite for a positive angle of attack.

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Acknowledgements

The authors are grateful for financial support by the Hartree Centre and the UK Engineering and Physical Sciences Research Council (EPSRC) under grants EP/L000261/1, EP/N016602/1, EP/K038427/1 and EP/N030028/1.

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Correspondence to Stefano Rolfo.

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Appendices

Appendix A: Mesh Convergence Study for a 2-D Flow

The 2-D flow around a heated cylinder in forced and mixed convection is used to assess mesh convergence. Table 3 details the averaged force coefficients CL and CD, the heat transfer coefficient Nu, their r.m.s. variation and the Strouhal number of the lift coefficient for the regime of pure forced convection. Results are all consistent and in good agreement with the experimental data available, showing that the coarser mesh resolution is good enough to properly resolve the flow. Figures 26 and 27 present the variation of the lift and drag coefficients respectively as function of the Richardson number for several 2-D meshes. The lift coefficient is in excellent agreement for all meshes, whereas the drag coefficient has an increasing error with the increase of the Richardson number, up to 5% at Ri = 2. In contrast, M062 and M106 are in perfect agreement for all Richardson numbers. Indeed, for the largest Richardson numbers, the coarse mesh shows also some spurious oscillations, mainly due to the interaction of the wake plume with the top boundary.

Fig. 26
figure26

Comparison of drag coefficient CL versus Richardson number at Re = 150 for different mesh resolutions

Fig. 27
figure27

Comparison of drag coefficient CD versus Richardson number at Re = 150 for different mesh resolutions.

Table 3 Averaged force coefficient CL and CD, heat transfer coefficient Nu, their r.m.s variation and Strouhal number for the dominant mode for the lift coefficient spectrum

Therefore, only M062 and M106 will be extruded in the spanwise direction to study 3-D flow.

Appendix B: Boussinesq approximation versus variation of physical properties

In this section, the Boussinesq approximation is verified and compared with results obtained from the variation of density and viscosity according to the following laws:

  • density variation using the perfect gas law for standard air

    $$ \rho= \dfrac {p_{0}}{R^{*}T} $$
    (7)

    where the reference pressure p0 = 101.325 kPa and the gas constant R = 287.058 Jkg− 1K1

  • Sutherland’s law for variation of the viscosity

    $$ \mu= \mu_{0}\left( \frac{T}{T_{0}}\right)^{3/2}\frac{T_{0}+T}{T+S} $$
    (8)

    where the subscript 0 stands for air properties at T0 − 273.15 K and S = 110.4 K

  • Thermal conductivity as a function of the dynamic viscosity following

    $$ \lambda= \frac{C_{p}\mu}{Pr} $$
    (9)

    where μ is varying according to Eq. 8. Cp and the Prandtl number are kept constant since the maximum variation between incoming and wall temperature is in the region of 18 K for the largest Richardson number, Ri = 2.

For these calculations with full variation of the physical property the dilatable solver of Code_Saturne has been employed. The formulation takes into account not only the spatial variation of the physical properties, but also the time derivative of the density in the continuity equation. The comparison between Boussinesq approximation and variation of physical properties as a function of the Richardson number for lift and drag coefficients is presented in Figs. 28 and 29 respectively. Lift coefficient shows a relatively good agreement between the all Richardson numbers, whereas some differences can be seen for the drag coefficient for Ri > 1. However, the difference in the drag coefficient at Ri = 2 is of the order of 3%. Therefore, the Boussinesq approximation can be considered acceptable for all Richardson numbers considered in this work.

Fig. 28
figure28

Comparison of lift coefficient CL versus Richardson number at Re = 150 for simulations using Boussinesq approximation and variation of physical properties

Fig. 29
figure29

Comparison of drag coefficient CD versus Richardson number at Re = 150 for simulations using Boussinesq approximation and variation of physical properties

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Rolfo, S., Kopsidas, K., Rahman, S.A. et al. Effect of Large Scale 3-D Structures on the Flow Around a Heated Cylinder at Low Reynolds Number. Flow Turbulence Combust 101, 553–577 (2018). https://doi.org/10.1007/s10494-018-9970-y

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Keywords

  • Heated cylinder
  • Mixed convection
  • Overhead lines
  • Boussinesq approximation
  • Flow transition