Flow structure to first order

The flow field correction due to buoyancy, $\mathbf{u}_{1B}$, is topologically equivalent to that due to inertia, $\mathbf{u}_{1I}$, though the sense is reversed.

Three-dimensional flow structures are conveniently discussed in terms of their critical points (Perry & Fairlie 1974; Perry & Chong 1987; Chong, Perry & Cantwell 1990). Interior critical points are those at which the velocity vanishes. Critical points on the boundary are those at which the normal derivative of each component of velocity vanishes.

The two basic types of interior critical points are the saddle-node and the focus. A plane passes through each saddle-node which is tangential to the stream-lines at the point and in which the projections of the stream-lines converge to or diverge from the point--hence the `node' part of the name. The node is referred to as stable or unstable, accordingly as the stream-lines converge or diverge. There are two other planes through the point which are also tangential to stream-lines. The projections of the stream-lines on these planes are separated into four regions by the lines corresponding to the other two planes. The flow is inward along the line corresponding to a stable node and outward along the other line--hence the `saddle' part of the name; the situation is reversed for an unstable node.

A plane tangential to stream-lines also passes through each focus, in which the projections of the stream-lines spiral into or out of the focus accordingly as the focus is stable or unstable. The stream-lines outside this plane coil around the axis of the focus. A focus on the border between stability and instability is called a centre.

The three basic types of surface critical points are the saddle, node and focus.

For $\mathbf{u}_{1I}$ and $\mathbf{u}_{1B}$, the interior critical points consist of an axisymmetric saddle-node at $r=0$ and two congruent circles about the axis of symmetry along which each point is a centre in its meridian plane; these may be called vortex-rings, since $\mbox{\boldmath$\nabla$}\times\mathbf{u}_{1I}$ and $\mbox{\boldmath$\nabla$}\times\mathbf{u}_{1B}$ are both purely azimuthal, having only a $\phi$ and a $\upsilon$ component, respectively, so that the circles are indeed everywhere parallel to the vorticity. The boundary critical points are a separation point (stable node), for $\mathbf{u}_{1I}$, and a stagnation point (unstable node), for $\mathbf{u}_{1B}$, at each end of the axis of symmetry and a circle of stagnation points along $\theta=\pi/2$ for $\mathbf{u}_{I1}$ and separation points along $\eta=\pi/2$ for $\mathbf{u}_{1B}$.

Each point on the $z$-axis is a centre in the plane of constant $z$ for the creeping flow, $\mathbf{u}_0$. The boundary critical points of $\mathbf{u}_0$ are centres at $z=\pm1/2$.

Since the three points $\{x=y=0, z=\pm\mbox{$\frac{1}{2}$}, 0\}$ are critical for $\mathbf{u}_0$, $\mathbf{u}_{1I}$ and $\mathbf{u}_{1B}$, they are also critical points for the combined flow to first order in $\mbox{\textit{Gr}}$, $\mathbf{u}_0+\mbox{\textit{Gr}}(1+N)\mathbf{u}_{1}$. The addition of the inertial correction to the creeping flow causes the centres at $z=\pm1/2$ to become stable foci, meaning that the limiting surface stream-lines spiral in to the points. Similarly the centre in the plane $z=0$ at $r=0$ becomes an unstable focus. The addition of the buoyancy correction is not as obvious, since it is not orthogonal to the creeping flow, except in the plane $x=0$, where it is very similar to the inertial correction.

It seems that at low Grashof numbers the basic structure of the flow is defined by the unstable focus in the $xy$-plane at $r=0$ and the stable surface foci at $\sin\theta=0$. The same conclusion was reached by Hiller et al. (1989) from their visualization studies of a cubic enclosure. This distribution of critical points over the boundary is consistent with the `hairy-sphere theorem' (Perry & Chong 1987), which states that for surfaces topologically equivalent to the sphere, the number of nodes and foci must exceed the number of saddles by two, the Euler number of the surface (Hilbert & Cohn-Vossen 1952, p. 295). It will be noticed that the entire flow field can be seen as two nested families of (topological) tori, separated by the $xy$-plane and surrounding the $z$-axis, plus the line segments on each half of the $z$-axis. The Euler number of the torus is zero (Griffiths, H. B. 1976, p. 110), so that the surfaces need not have any critical points.

The three-dimensional nature of the flow is illustrated in figure 8.11

Figure 8.11: The trace of a particle released at the point $r=0.45,\theta=\phi=0.05$, marked with a $\times$, into the flow $\mathbf{u}_0+\mbox{\textit{Gr}}(1+N)\mathbf{u}_1$ with $\mbox{\textit{Gr}}(1+N)\mbox{\textit{Pr}}=1000, \mbox{\textit{Pr}}=0.7, \mbox{\textit{Sc}}=0.6, N=0$ after a dimensionless time interval (units of $b^2/\nu\mbox{\textit{Gr}}(1+N)$) of (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6.

Figure 8.12: Continuation of previous figure. Elapsed times: of (g) 7, (h) 9, (i) 15, (j) 24, (k) 30, (l) 36.

by the locus of a particle released into a steady flow at $\mbox{\textit{Gr}}(1+N)\mbox{\textit{Pr}}=1000$, $\mbox{\textit{Pr}}=0.7$, $\mbox{\textit{Sc}}=0.6$ and $N=0.0$. The release point, $r=0.45,\theta=\phi=0.05$, is marked with an $\times$. The subfigures are a sequence of snapshots. The particle traces out a stream-line (since the flow is steady) which clearly lies on one of the surfaces homeomorphic to a torus described above. The boundary of the sphere is not shown for clarity, but the Cartesian axes are only shown within the domain. The particle tracks were obtained by second order Runge-Kutta integration of the velocity field. The resemblance of the tracks to those in a cuboid produced from the numerical solutions of the full Boussinesq equations of Mallinson and de Vahl Davis (1973, 1977) is startling. The reader is urged to compare figure 8.11 with their figures, some of which were reproduced by de Vahl Davis (1998) and Gebhart et al. (1988, p. 775).

The reason why the particle doesn't return to a starting point after a single `traverse' of the stream-surface is not numerical error. In discussing their computed stream-lines in a cuboid, Mallinson and de Vahl Davis (1977), concluded that the stream-lines must be closed, but pointed out that `multiple traverses on the same surface without streamline intersection are, however, possible and cannot be rejected a priori'. In fact, it may be demonstrated that it is entirely (kinematically and topologically) possible for a particle to travel an infinite distance in a steady flow field without returning to its starting point. Stream-lines can only cross at critical points, where their direction is undefined. It follows that stream-surfaces also only intersect at critical points. Consider the family of circles formed by the intersection of the surface of a torus and the family of cones coaxial with the torus and with its vertex at the torus's centre. This family of curves satisfies the kinematic and topological requirements of a set of stream-lines free from critical points; it is, for example, homeomorphic to the family of stream-lines on a typical stream-tube of the creeping flow, $\mathbf{u}_0$. Now cut the surface of the torus with a semi-plane bounded by its axis, twist one of its ends and then rejoin the surface and stream-lines. Unless a rational number of twists were applied to the end, the particle will never return to its starting point, even though it remains on the same finite surface (cf. the discussion of force-free magnetic field lines by Moffatt 1978, p. 30).

The addition of the correction for inertia to the creeping flow has a similar effect to the above cutting, twisting and rejoining operation.