Simultaneous effect of rotation and natural convection on the flow about a liquid sphere (original) (raw)

2004, Heat and Mass Transfer

The problem of mixed convection around a liquid sphere that experiences a rotation about its axis parallel to the free stream is studied numerically using a finite-difference technique. The coupled boundary-layer energy and momentum equations are numerically solved over a wide range of Grashof number that represents the cases of aiding and opposing free convection and for wide range of the spin parameter Ta/Re 2 . The surface of the sphere also rotates as a result of the shear stress exerted from the external flow of air. The effect of both parameters on the velocity components as well as the temperature within the thermal boundary-layer is presented. Results show that increasing the aiding free convection and the spin parameter cause increases in the shear stress and the local heat transfer coefficient. Notation a sphere radius, m g gravitational acceleration, m/s 2 Gr Grashof number, Gr = 16gbqa 4 /kt 2 h local heat transfer coefficient, W/m 2°C k thermal conductivity of fluid, W/m°C m number of steps of the numerical mesh network in the x-direction n number of steps of the numerical mesh network in the z-direction Nu Nusselt number, 2ah k ¼ À2 @T @Z j o Pe Peclet number = Re à Pr Pr Prandtl number, m/a r radial coordinate measured from the sphere's center, m Re Reynolds number, 2U ¥ a/m t temperature,°C t w wall temperature,°C t ¥ free stream temperature,°C T dimensionless temperature, T = k(t w -t ¥ )/(aq) Ta Taylor number, Ta = 4W 2 a 4 /m 2 T w dimensionless wall temperature, T w = k(t w -t ¥ )/ (aq) u meridional (x-direction) component of velocity, m/s U dimensionless meridional component of velocity, u/U ¥ u à velocity component in x-direction for the potential flow outside the external boundary layer, -( ¶w/ ¶r)/ (r sin h) = U ¥ sin h [1 + a 3 /(2r 3 )], m/s U à dimensionless potential velocity component in the x-direction for external flow, u*/U ¥ U ¥ free stream velocity in the exterior flow, m/s v azimuthal velocity component, m/s v o circumferential velocity at the sphere's surface, v o = Wr o , m/s V dimensionless azimuthal velocity component, V = v/Wa V o dimensionless azimuthal velocity component at the sphere's surface, V o = r o /a w radial (z-direction) velocity component, m/s w à radial (z-direction) velocity component for potential flow outside the external boundary layer, ( ¶w/ ¶h)/(r 2 sin h) = -U ¥ cos h [1 -a 3 /r 3 ], m/s W dimensionless radial velocity component, w/U ¥