a Similarity Model of Nonlinear Convection
The nonlinear equations for a steady axisymmetric convective flow of an incompressible fluid over a horizontal plate are analyzed using a similarity relation which has been used in the study of geophysical vortices. The principal assumption of the present model is that the components of the fluid velocity field decrease inversely with distance from a point singularity located at the intersection of the axis of symmetry and the plate. Although the tangential velocity component around the vertical axis is assumed to be zero, the model can be generalized to include unsteady swirling motion. The inverse distance scaling for the velocity field transforms the Navier-Stokes equations into a set of nonlinear ordinary differential equations in the polar angle. No slip and impermeability conditions are imposed along the flat plate. Along the axis we exclude fluid sources and require that stresses be finite. In closing the model, it is convenient to specify to solenoidal term of the vorticity equation explicitly as a function of the polar angle and distance from the point singularity. After computing the velocity field and assuming a relevant equation of state, consideration of the energy equation then leads to a computation of the heating required to maintain the flow. Numerical results indicate that axial jets can form in association with updrafts but not with downdrafts. On the other hand, wall jets can be found in downdrafts but not in updrafts. In addition, the theory predicts that no steady state solution exists when the magnitude of the solenoidal field in the updraft exceeds a critical finite value. No such theoretical limit exists for a downdraft. These results are interpreted through a consideration of the vorticity balance. Their possible relevance for real atmospheric convective phenomena such as the downburst and tornado formation is indicated.
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- Physics: Fluid and Plasma