View of the layer ahead of the water-air surface. Ice is covered with a supercooled water film. The axis is parallel to the direction of the supercooled water flow and the axis is normal to it. is the temperature at the water-air surface. (a) is the situation in absence of airflow. A linear air temperature distribution was assumed. (b) is the situation in presence of airflow. and are undisturbed velocity and temperature distributions. and are the thickness of the water layer and that of the thermal boundary layer, respectively. is the gravitational acceleration and is the angle with respect to the horizontal. The flowing supercooled water layer, not to scale, is much thinner than the thickness of the thermal boundary layer.
For , , , and , (a) air temperature distribution , and distributions of , , and at the dimensionless wave number of . (b) Perturbed part of air temperature gradient at the water-air surface: in the absence of airflow ; in the presence of airflow and are the real and imaginary parts of . Here corresponds to the wavelength of 4.1 mm when .
For , and , (a) dimensionless amplification rate vs dimensionless wave number ; (b) dimensionless phase velocity vs dimensionless wave number . Solid and dashed lines indicate the presence and absence of airflow, respectively. (c) The behavior of the real and imaginary parts of the perturbed temperature gradient at the water-air surface with respect to , in the presence of airflow (solid lines) and in the absence of airflow (dashed lines). Here corresponds to the wavelength of when .
(a) The wavelength vs at . (b) The wavelength vs at . (c) The phase velocity vs at . Solid lines indicate the absence of airflow (Refs. 4 and 6 ); dashed lines and dashed-dotted lines indicate the presence of airflow for and .
(a) and (b) are illustrations of the time evolution of an initial disturbance of the ice-water interface from to . The solid arrows in the water film and the dashed arrows in the air boundary layer show the direction of the supercooled water flow and airflow, respectively. The arrows attached and are the maximum point of heat flux at the ice-water interface and water-air surface, respectively. (a) represents the disturbance of in the absence of airflow. (b) represents the disturbance of in the presence of airflow. (c) represents the phase shift of the water-air surface, , total heat flux at the ice-water interface, , and heat flux at the water-air surface, , relative to the ice-water interface with respect to .
For and , the dependence of modified local Grashof number, Gr, ice growth rate, , temperature at water-air surface, , thickness of thermal boundary layer, , wavelength of ripple, , and dimensionless translation velocity of ripple, , on air temperature far away, , and position from the bottom of a gutter, .
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