# Instability analysis of natural convection

### From Thermal-FluidsPedia

Natural convection in an enclosure has been a classic problem in fluid mechanics and heat transfer due to its fundamental and practical significance. It has attracted extensive experimental, analytical and numerical studies. The majority of the earlier studies have focused on the steady-state solutions, particularly for natural convection in a rectangular cavity with differentially heated sidewalls. In this case, the basic flow is a simple, unicellular circulation. When the horizontal temperature gradient increases, however, the base flow undergoes instability, and eventually becomes unsteady and turbulent. Computation of flow instabilities is one of the most fascinating and challenging aspects of the present computational fluid dynamics. The aim of instability analysis is to predict the values of the parameters at which instabilities occur, the nature of the instability (steady or unsteady, supercritical or subcritical), the type of solution that results from the instability, the modification of the base flow solution that results from the non-linear interactions of the unstable modes and the subsequent route to chaos (Gadoin, et al., 2001).

The instability analysis of natural convection in a vertical fluid layer (Bergholz, 1978), for example, shows that if the value of the Prandtl number is in the low to moderate range (Pr < 12.7), there is a transition from stationary to travelling-wave instability if the vertical temperature stratification exceeds a certain magnitude. However, if the Prandtl number is large, the transition with increasing stratification, is from travelling-wave to stationary instability. These predictions of the instability analysis are in good agreement with experimental observations for both stationary and travelling-wave instabilities (e.g., Elder, 1965).

Xin and Le Quere (2001) developed a general methodology for investigating the stability of a two-dimensional base solution with respect to both two- and three-dimensional perturbations by combining the Arnoldi’s method, the preconditioned Newton’s iteration and the preconditioned continuation method. They applied this methodology to investigate in detail the stability of natural convection in a differentially heated square cavity with conducting horizontal walls: 2D base solutions with respect to both 2D and 3D perturbations for a large range of Prandtl number.

Their results show that for 2D perturbations, the critical Rayleigh number corresponding to the onset of unsteadiness increases with the Prandtl number. The following table lists the critical Rayleigh numbers and angular frequencies under 2D perturbations for Pr = 7 and 0.015, respectively.

**Table Critical Rayleigh numbers and angular frequencies under 2D perturbations**

Pr | Mode 1 | Mode 2 | Mode 3 | |||
---|---|---|---|---|---|---|

Ra_{c}
| ω_{c}
| Ra_{c}
| ω_{c}
| Ra_{c}
| ω_{c}
| |

7 | 5,573,177±1 | 2.466713 | 5,694,340±1 | 2.681573 | 6,305,615±1 | 2.883617 |

0.015 | 40,524±1 | 0.712348 | - | - | - | - |

The numerical results for 3D perturbations show that (1) for Pr=1, the base solutions are more unstable to 3D perturbations, and the most unstable 3D mode is connected to the third most unstable 2D mode; (2) for larger Prandtl numbers (3, 7 and 20), the most dangerous perturbations are two-dimensional; (3) for smaller Prandtl numbers (0.71, 0.1, and 0.015) the base solutions are also more unstable to 3D perturbations, but the most unstable 3D mode corresponds to a stationary mode, instead of connecting to the most unstable 2D modes. Moreover, the critical Rayleigh numbers for Pr=0.015 and 0.1 are approximately 40 times smaller than those found for 2D perturbations.

As far as a differentially heated square cavity with conducting horizontal walls is concerned, the study by Xin and Le Quere (2001) suggests that for large Prandtl number two-dimensional flows, two-dimensional studies make sense at least up to the onset of unsteadiness, while for other Prandtl numbers, it is necessary to perform three-dimensional investigations. The need to do so is even stronger for small Prandtl numbers because the two-dimensional base flows are unstable to three-dimensional perturbations at very low Rayleigh numbers.

## References

Bergholz, R.F., 1978, “Instability of Steady Natural Convection in a Vertical Fluid Layer,” J. Fluid Mech., Vol. 84, pp. 743-768.

Elder, J.W., 1965, “Laminar Free Convection in a Vertical Slot,” J. Fluid Mech., Vol. 23, pp. 77-98.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, *Advanced Heat and Mass Transfer*, Global Digital Press, Columbia, MO.

Gadoin, E., Le Quere, P., and Daube, O. 2001, “A General Methodology for Investigating Flow Instabilities in Complex Geometries: Application to Natural Convection in Enclosures,” Int. J. Numer. Meth. Fluids, Vol. 37, pp. 175-208.

Xin, S., and Le Quere, P., 2001, “Linear Stability Analyses of Natural Convection Flows in a Differentially Heated Square Cavity with Conducting Horizontal Walls,” Physics of Fluids, Vol. 13, pp. 2529-2542.