Unsteady free convection from a heated sphere at high Grashof number (Q797429)

The authors consider a Boussinesq fluid in which variable fluid properties are ignored except in the buoyancy-driven unsteady laminar motion. A constant Prandtl number of 0.72 is chosen. At time \(t=0\), the surface temperature of the sphere is raised to and subsequently maintained at a constant temperature \(T_ w>T_{\infty}\). Dimensionless boundary-layer coordinates are adopted employing \(y=(r- a)a\epsilon^{{1\over2}}\), where \(\epsilon^ 2=Gr^{-1}\), Gr is the Grashof number, a the radius of the sphere, and r the radial coordinate. The solution for the velocity components u,v, and temperature \(\theta\) is (i) developed into a series in powers of \(t^{{1\over2}}\) in order to yield the initial condition at \(t=0.25\) for (ii) a difference solution where (iii) a stagnation point representation is employed in a neighborhood of the lower pole \(x=0\). Here, (ii) is an adaptation of a method by \textit{M. G. Hall} [Ing. Arch. 38, 97-106 (1969; Zbl 0175.241)]. For the system of ordinary differential equations in the case of (i), a collocation method with Chebychev polynomials is employed. The numerical method for (iii) is a special case of the one for (ii). For (ii), the solution is advanced in time at fixed x before proceeding to the next x- station where the solution then is advanced in the y-direction. The object of the paper is the evolution of the solution towards a steady state and the verification of the singularity at the upper pole \(x=\pi\) of the sphere at a time \(t_ s\). Numerical results are presented for the velocity \(v_{\infty}(x,y_{\infty},t)\) at the outer edge \(y_{\infty}\) of the boundary layer, the heat transfer coefficient as a function of x and t, and the thickness of the thermal boundary layer. Up to \(x=3\pi /4\), the boundary layer is thin and steady state at \(t=6\). For \(x>3\pi /4\), steady state is approached at a considerably later time not investigated in the paper. Authors obtain \(t_ s=2.922\) and state that the overall understanding of this unsteady free convection problem is now almost complete except the gap due to the employment of boundary layer approximations in a neighborhood of \(x=\pi\) an \(t\geq t_ s\).