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Turbulent low-Reynolds-number k–ε model effect on buoyancy-driven free convection flow past a vertical cylinder

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Abstract

The main objective of the present analysis is to characterize the transient buoyancy-motivated free convection turbulent flow and heat transfer characteristic features of an incompressible viscous fluid past a vertical cylinder with low-Reynolds-number (LRN) k–ε turbulence model in a two-dimensional coordinate system numerically. The Reynolds averaged Navier–Stokes equations (RANS) such as continuity, momentum, and energy are considered in terms of cylindrical coordinate system. The extra stress tensors obtained from the RANS model are closed using the eddy diffusive model. The local value of turbulent kinematic viscosity (\({\nu }_{t}\)) is determined by utilizing the kinetic energy \((k)\) and dissipation rate \((\epsilon )\) equations. The resulting system of partial differential equations (PDEs) with high nonlinearity, governing the turbulent boundary layer flow are solved using the implicit Crank–Nicolson technique. The discretized set of dimensionless tridiagonal algebraic equations are simplified by utilizing Thomas algorithm. Also, the simulated results are expressed in terms of graphs to analyse the average velocity, temperature, kinetic energy, dissipation rate, and also average momentum and heat transfer rates for the varying values of turbulent Prandtl (\({Pr}_{t}\)), Grashof \({(Gr}_{t})\) and Reynolds (\({Re}_{t}\)) numbers. It is noted that the average velocity, kinetic energy, dissipation rate of kinetic energy fields suppressed, and temperature field enhanced with increasing \({Re}_{t}\). Also, the rising turbulent Prandtl parameter decreased the average velocity, temperature, turbulent kinetic energy, and dissipation rate profiles. Further, the increasing turbulent Grashof number decreased the kinetic energy and dissipation rate profiles. Further, the obtained results from the present turbulent investigation are compared with the existing results and observed an excellent agreement.

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Abbreviations

\(x\) :

Axial coordinate

\(r\) :

Radial axis perpendicular to the cylinder

\({r}_{0}\) :

Radius of the cylinder

\(X\) :

Non-dimensional axial coordinate

\(R\) :

Non-dimensional radial coordinate

\(u, v\) :

Velocity components along \(x\) and \(r\)  paths

\(\overline{u }, \overline{v }\) :

Average velocities along \(x\) and \(r\) directions

\({u}^{^{\prime}}, {v}^{^{\prime}}\) :

Fluctuating velocities along \(x\) and \(r\) directions

\(\overline{{u }^{^{\prime}}{v}^{^{\prime}}}\) :

Shear stress of the turbulent flow

\(\overline{{u }{^{\prime}}{\theta }{^{\prime}}}\) :

Longitudinal turbulent heat flux

\(\overline{{v }{^{\prime}}{\theta }{^{\prime}}}\) :

Transverse turbulent heat flux

\({\tau }_{w}\) :

Wall shear stress of turbulent flow

\(U, V\) :

Dimensionless velocities in \(X\) and \(R\) directions

\({\theta }{^{\prime}}\) :

Fluctuating temperature

\(\overline{\theta }\) :

Average thermal filed

\(T\) :

Dimensionless temperature

\({t}{^{\prime}}\) :

Dimensional time

\(t\) :

Non-dimensional time

\(g\) :

Acceleration due to gravity

\({Gr}_{t}\) :

Turbulent thermal Grashof parameter

\({Pr}_{t}\) :

Turbulent Prandtl number

\(Pr\) :

Laminar Prandtl number

\({C}_{1}, {C}_{2}\) :

Empirical constants

\({C}_{k}\) :

Arbitrary Prandtl number-dependent coefficient

\({f}_{1, }{ f}_{2}, {f}_{\mu }\) :

Near wall treatment functions

\(k\) :

Average turbulence energy

\(\epsilon\) :

Average dissipation rate

\(K\) :

Dimensionless kinetic energy

\(E\) :

Dimensionless dissipation rate

\({Re}_{t}\) :

Reynolds number

\({C}_{\mu }\) :

Proportional constant

\({K}_{T}\) :

Thermal conductivity

\({\overline{u} }_{\infty }\) :

Free stream velocity

\({\alpha }_{t}\) :

Turbulent heat diffusivity

\({\sigma }_{T}, {\sigma }_{k}\) :

Turbulent Prandtl numbers for \(\overline{\theta }\) and \(k\)

\({\sigma }_{\epsilon }\) :

Dissipation Prandtl number

\(\mu\) :

Laminar flow's viscosity

\({{\mu }_{t} }\) :

Turbulent flow's viscosity

\(\nu , {\nu }_{t}\) :

Laminar and turbulent kinematic viscosities, respectively

\(\rho\) :

Density

\(a, b\) :

Grid levels, respectively, in the directions of \(X \& R\)

\(w\) :

Surface condition

\(\infty\) :

Free stream condition

\(c\) :

Time step level

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Authors and Affiliations

  1. Laboratory on Computational Fluid Dynamics, Department of Mathematics, Central University of Karnataka, Kalaburagi, 585367, India

    S. P. Suresha & G. Janardhana Reddy

  2. Department of Mathematics, Government Degree College, Sindhanur, 584128, India

    Hussain Basha

Authors
  1. S. P. Suresha
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  2. G. Janardhana Reddy
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  3. Hussain Basha
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Correspondence to G. Janardhana Reddy.

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Suresha, S.P., Reddy, G.J. & Basha, H. Turbulent low-Reynolds-number k–ε model effect on buoyancy-driven free convection flow past a vertical cylinder. Indian J Phys 98, 659–677 (2024). https://doi.org/10.1007/s12648-023-02797-7

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