Numerical Analysis of Two-Dimensional Natural Convection in an Open Vertical Tube

Abstract

Numerical analysis of two-dimensional natural convection in an open vertical tube with volumetric heat generation involves solving the governing equations for fluid flow and heat transfer in such a configuration. This problem is typically addressed by considering the Navier-Stokes equations (for fluid flow), the energy equation (for heat transfer), and appropriate boundary conditions. The presence of volumetric heat generation adds complexity to the thermal analysis.Numerical analysis of natural convection in open ended ended two dimensional axisymmetric vertical tube has been carried out for a wide temperature range to find out the relation between non-dimensionalized number,Numerical analysis of two-dimensional natural convection in open ended vertical tube has been carried out for a wide range of heat generation rate to find out the appropriate non-dimensional numbers governing the process. Buoyancy induced flows in a tube with adiabatic wall boundary condition and uniform volumetric heat generation rate in fluids were analyzed by varying heat generation rate and the geometrical aspect ratio of the tubes with and without considering surface radiation. Curve fits have been provided for non-dimensionalized mass flow rate versus other relevant non-dimensional numbers based on heat generated and geometry of the tube. Non-dimensional correlation has been found out for natural convection flow in an open adiabatic tube with internal heat generation and neglecting surface radiation. One more non-dimensional correlation has been found out for natural convection flow in an open adiabatic tube with internal heat generation and considering surface radiation.

Introduction

Objective:

To numerically investigate high-temperature natural convection in an open vertical tube where buoyancy is caused by volumetric heat generation (e.g., from combustion), with the goal of developing correlations for non-dimensional mass flow rate (NDMF) based on relevant dimensionless parameters.

Key Motivations:

• High-temperature natural convection differs significantly from low-temperature cases due to strong temperature-dependent fluid properties.

• Existing low-temperature models and results cannot be directly applied.

• Literature lacks clarity on the governing non-dimensional numbers for such high-temperature flows.

Problem Definition:

• Simulates air induction and buoyancy-driven flow in a vertical, adiabatic tube where heat is generated internally (e.g., from combustion).

• Solves coupled continuity, momentum, energy, and radiation equations.

• The setup is simplified but relevant to practical systems like chulhas (traditional Indian stoves).

Literature Review Highlights:

• Existing studies include both Boussinesq (constant property) and variable property formulations.

• Phenomena like backflow and laminar-turbulent transition are considered.

• Variable property modeling is essential for accurate simulation of high-temperature flows.

Governing Equations:

• Axisymmetric 2D steady-state equations in cylindrical coordinates:

  • Continuity

  • Axial and radial momentum

  • Energy (with volumetric heat generation)

  • Radiation (surface only, with no gas absorption/scattering)

Radiation Modeling:

• Surface radiation is included since radiative heat flux dominates at high temperatures (due to T? dependence).

• Assumes a non-participating medium (no scattering, no absorption).

• Radiation intensity is calculated using simplified Radiative Transfer Equations (RTEs).

Non-Dimensionalization:

• Two methods were used to non-dimensionalize the governing equations.

• Key non-dimensional numbers identified:

  • Grashof number (Gr): for buoyancy-driven flow

  • Prandtl number (Pr): for fluid properties

  • Two derived dimensionless groups:

    • Nd1 = Gr / (1 + ΔT / Tref)

    • Nd2 = Nd1 / Aspect Ratio
      These represent the influence of temperature difference and geometry.

Numerical Simulation Setup:

• Tube dimensions: Length = 183.3 mm, Radius = 18.33 mm

• Volumetric heat generation: 1.5 × 10? W/m³

• Simulation includes:

  • Surface radiation

  • Variable property effects

  • Analysis of temperature and velocity profiles

Key Results:

• Temperature increases with tube height due to buoyancy-driven flow.

• When radiation is considered:

  • Peak temperatures shift away from the wall.

  • Maximum and average temperatures are reduced.

• These changes significantly affect the flow field and highlight the importance of including radiative heat transfer in high-temperature modeling.

Conclusion

1) A numerical investigation for a two-dimensional natural convection in open ended vertical tube has been performed with volumetric heat generation. A wide range of heat generation rate, radius and aspect ratio have been considered to find out the appropriate non-dimensional numbers governing the process and correlations between them. 2) Many sets of simulations were carried out varying dimensional quantities within range, and it was found that T max has a strong dependence on Qgen, for both the cases, considering and neglecting radiation 3) In the present work, laminar flow in an open vertical tube with volumetric generation was studied, firstly neglecting radiation and then taking radiation into account. In future turbulent flows could be analyzed both with and without inclusion of radiation. Also, criterion for transition from laminar to turbulent flows could be studied for the present case

References

[1] Biertumpfel R., and Beer H., 2002, “Natural convection heat transfer increase at the laminar-turbulent transition in the presence of instationary longitudinal vortices”, International Journal of Heat and Mass Transfer, Vol 46(2003), pp 3109-3117. [2] Fujii, T., Koyama, S. , and Buenconsejo, N.S., (Jr.), 1988, “Laminar free convection flow rate in a vertical tube”, Int.J.Heat Mass Transfer, Vol 31, N.4, pp 831-841 [3] Gebhart, B., Jaluria, Y., Mahajan, R.L., and Sammakia,B., 1988, “ Buoyancy-Induce Flows and Transport”, Hemisphere Publishing Washington, 1988). [4] Jones, W.P., and Launder, B.E. , 1972, \"The prediction of laminarization with a two-equation model of turbulence\", J. Heat Mass Transfer, Vol 15, pp 301–14. [5] Sparrow, E.M., Chrysler, G.M., and Azevedo, L.F., 1984, “Observed flow reversals and measured-predicted Nusselt numbers for natural convection in a one sided heated vertical channel”, J.Heat Transfer, Vol 106, pp 325-332. [6] White, F.M., 1991, “Viscous Fluid Flow”, 1991, ( McGraw Hill International, 1991)

Copyright

Copyright © 2025 Amit Kumar Sinha, Prakash Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.