Hydrodynamic stability of multiple steady-states of vertical buoyancy-induced flows in cold pure water

Abstract

Numerical solutions of the hydrodynamic stability equations which model disturbances in steady-state, laminar flows generated by natural convection of cold, pure water adjacent to a vertical, planar, isothermal surface have been obtained. The results yield neutral stability curves representing the relative hydrodynamic stability of the two previously predicted multiple, steady-state base flows. These two steady states are called the upper- and lower-branch solutions (the upper-branch solution has the higher heat transfer rate (- { b ′(0)) at the vertical isothermal surface). New results presented here include the first hydrodynamic stability analysis for cold water which includes the range of ambient temperature ( T α) 5.6896–8.0586°C (if the plate temperature T 0 is 0°C), where inside buoyancy force reversals exert a strong influence upon the flow. This range corresponds to values of the density extremum parameter R=( Tm–T ∞/(T 0–T ∞) in the interval (0.29181, 0.50) where Tm is the density extremum temperature. The results show that for the flows corresponding to the upper- and lower-branch solutions, the critical Grashof number systematically decreases as the heat transfer rate decreases. Namely, an increase in the magnitude of inside buoyancy force reversals, which are associated with the locations of the two points of inflection in the vertical velocity components of the base flow, always cause the flows to be significantly more unstable. The flows corresponding to the upper-branch solutions were, in general, found to be more stable than the flows corresponding to the lower-branch solutions. This agrees with previously reported configurational stability results. The present results also indicate that oscillation between the upper and the lower steady-states is possible (inasmuch as it is consistent with hydrodynamic stability theory) in the range 0.29181 ⩽ R < 0.34, but not for R ⩾ 0.34.