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Investigation on the relative performance of various low-Reynolds number turbulence models for buoyancy-driven flow in a tall cavity

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Abstract

The present study deals with the numerical investigation of turbulent buoyancy driven flow in a differentially heated rectangular cavity with adiabatic horizontal walls. The aspect ratio of cavity is 5 and the Rayleigh number based on the cavity height is 4.56 × 1010. The computations have been carried out using the finite volume method on a staggered grid and SIMPLEC algorithm for pressure–velocity coupling. The low-Reynolds number \(k-\epsilon\) model proposed by Yang and Shih (YS), low-Reynolds number \(k-\omega\) model proposed by Wilcox, and \(k-\omega\) shear stress transport (SST) model of Menter have been applied for turbulence closure. The performance comparison of different models have been carried out using the experimental, LES and various RANS results available in the literature. The computation of turbulent natural convection flow is numerically challenging due to complex flow involving laminar, transition and turbulent regions, coupling of velocity with the energy equation, and some other problems reported in literature e.g. grid dependency of solution, numerical stability problem, etc. The flux Richardson number is calculated to get an estimate of relative importance of buoyancy and shear force in different regions of flow. The shearing and swirling zones have been identified in the entire flow domain using the \(\lambda _{2}\) criterion. Based on the comparison of mean flow, heat transfer and turbulence characteristics with the available results, it has been found that YS model performs better. The better performance obtained from YS model may be due to peculiarity of model that takes into account the Kolmogorov time scale near the wall and the conventional time scale \((k/\epsilon )\) away from the wall.

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Abbreviations

\(b\) :

Width of the cavity (m)

\(c_{p}\) :

Specific heat at constant pressure (J kg−1 K−1)

\(d\) :

Non-dimensional distance from the nearest wall

\(g\) :

Acceleration due to gravity (m s−2)

\(H\) :

Heigth of the cavity (m)

\(k\) :

Turbulent kinetic energy (m2 s−2)

\(k_f\) :

Thermal conductivity of fluid (W/m2K)

\(Nu_{local}\) :

Nusselt number \(h_{y}H/k_f\)

\(\overline{p_{0}}\) :

Ambient pressure (Pa)

\(\overline{p}\) :

Static pressure (Pa)

\(P\) :

Non-dimensional static pressure

\(Pr\) :

Prandtl number, \(\nu /\alpha\)

\(Pr_{t}\) :

Turbulent Prandtl number

\(Ra\) :

Rayleigh number, \(g\beta \Delta TH^{3} /\nu \alpha\)

\(Re_{t}\) :

Turbulent Reynolds number, \({k^{2}}/{\nu \epsilon } \, \hbox {or} \, k/\nu \omega\)

\(Re_{y}\) :

Non-dimensional distance, \({\sqrt{k}y}/{\nu }\)

\(T\) :

Dimensional temperature (K)

\(T^{+}\) :

Dimensionless temperature \(({T_{w}-T})/{T_{\tau }}\)

\(T_{\tau }\) :

Friction temperature, \({q}/{\rho c_{p}u_{\tau }}\) (K)

\(T_{c}\) :

Temperature of cold wall of the cavity (K)

\(T_{h}\) :

Temperature of hot wall of the cavity (K)

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

Reynolds shear stress (m2 s−2)

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

Dimensional mean velocities in \(x,\,y\)-directions, respectively (m s−1)

\(U,\,V\) :

Non-dimensional velocities in \(X,\,Y\)-directions, respectively

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

Fluctuating components of velocities in \(x\) and \(y\)-directions, respectively (m s−1)

\(U_{0}\) :

Reference velocity, \(\sqrt {g\beta {\Delta }TH}\) (m s−1)

\(u_{\tau }\) :

Friction velocity, \(\sqrt{{\tau _{w}}/{\rho }}\) (m s−1)

\(v^{+}\) :

Non-dimensional velocity, \({\overline{v}}/{u_{\tau }}\)

\(X,\,Y\) :

Non-dimensional coordinates

\(x,\,y\) :

Dimensional coordinates (m)

\(x^{+}\) :

Non-dimensional distance, \({xu_{\tau }}/{\nu }\)

\(y^{+}\) :

Non-dimensional distance, \({yu_{\tau }}/{\nu }\)

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

Laminar and turbulent thermal diffusivities, respectively (m2 s−1)

\(\beta\) :

Coefficient of thermal expansion (K−1)

\(\chi\) :

von Karman constant

\({\Delta }T\) :

Temperature difference between hot and cold wall of the cavity \(( T_{h}-T_{c})\) (K)

\(\epsilon\) :

Rate of dissipation of turbulent kinetic energy (m2 s−3)

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

Laminar and turbulent kinematic viscosities, respectively (m2 s−1)

\(\omega\) :

Rate of specific dissipation (s−1)

\(\omega _{xy}\) :

Vorticity in \(x-y\) plane (s−1)

\(\rho\) :

Density (kg m−3)

\(\tau _{w}\) :

Wall shear stress (Pa)

\(\theta\) :

Non-dimensional temperature

\(n\) :

Non-dimesional quantity

\(p\) :

Corresponds to the first near-wall grid point

\(w\) :

Corresponds to wall

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, 721302, West Bengal, India

    Sushil Kumar Rathore & Manab Kumar Das

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  1. Sushil Kumar Rathore
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  2. Manab Kumar Das
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Correspondence to Manab Kumar Das.

Appendix

Appendix

Production by shear (\(G_{n}\)):

$$G_{n}=\nu _{t,n}\sqrt{\frac{\text {Pr}}{\text {Ra}}}\left[ \left( \frac{\partial U_{i}}{\partial X_{j}}+\frac{\partial U_{j}}{\partial X_{i}}\right) \right] \frac{\partial U_{i}}{\partial X_{j}}$$
(6.1)

Buoyancy production term:

$$G_{k}=g\beta \overline{v^{\prime }T^{\prime }}$$
(6.2)

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Rathore, S.K., Das, M.K. Investigation on the relative performance of various low-Reynolds number turbulence models for buoyancy-driven flow in a tall cavity. Heat Mass Transfer 52, 437–457 (2016). https://doi.org/10.1007/s00231-015-1557-8

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