Velocity in the plane of spanwise symmetry

For elliptic sections, by equation (7.68), the profile of the vertical component of velocity in the plane of spanwise symmetry,

$$v_n\!\!\raisebox{1ex}{\scalebox{1.414}[0.7071]{$\circ$}}(X,0) = \frac{X(1-4X^2)}{8(3+\mbox{$\mathcal S$}^{-2})},$$ (7.77)

is independent of $\mathcal S$; only the magnitude changes, as can be described by the magnitude of the vorticity at $X=Z=0$ (see §7.6.1). This is not so for rectangular sections, as is clear by inspection of figures 7.2 and 7.8.

Figure 7.8: Buoyancy-induced velocity, $v_n$, in the plane of spanwise symmetry, $Z=0$, of a duct or cavity of rectangular section.

In figure 7.8, (7.25) is used, rather than (7.28) or (7.59), since it accounts equally for the effect of the two end-walls, as appropriate on the plane of spanwise symmetry. The figure also confirms that the flow in the spanwise plane of symmetry is practically independent of $\mathcal S$ for $\mbox{$\mathcal S$}\geq 1.7$. The location of the maximum is closer to the hot wall for smaller values of $\mathcal S$. This location is a simple scalar measure of the effect of $\mathcal S$ on the velocity profile in the plane of spanwise symmetry.

According to Theorem 3 (p. ), if a flow has zero gradient in some direction, then the component of velocity in that direction is constant along vortex-lines. The flow given by $v_n^{\Box}$, (7.25) or (7.28), falls under the hypothesis of this theorem, having no gradient in the vertical direction. If there exists an isolated maximum, then, of $v_n^{\Box}$ in a section, there can be no projection in the section of a vortex-line through this point, otherwise the other points on the projection of the vortex-line would have the same velocity as at the maximum and the maximum would not be isolated. Thus, isolated maxima of $v_n^{\Box}$ can only occur at points where the horizontal components of vorticity vanish.

The buoyancy-induced vorticity in a rectangular section is, from (7.28) and (7.71):

$${\mbox{\boldmath$\omega$}_n^{\Box}=}\hspace{120mm}$$

$${ \left\{X\mbox{$\mathcal S$}Z - \frac{2\mbox{$\mathcal S$}}{\upi... ... S$}]} \sin[(2k+1)\upi Z]\right\}\mbox{\boldmath$\hat{\imath}$} }\hspace{120mm}$$

$${ + \left\{\frac{\mbox{$\mathcal S$}^2(1-4Z^2)}{8} - \frac{2\mbox... ...)\upi /2\mbox{$\mathcal S$}]} \right\}\mbox{\boldmath$\hat{k}$} }\hspace{120mm}$$

from which it is clear that $\mbox{\boldmath$\hat{\imath}$}\cdot\mbox{\boldmath$\omega$}_n^{\Box}=0$ if $Z=0$, so that the maximum of $v_n^{\Box}$ must lie in the plane of spanwise symmetry. Thus, the problem of finding the maximum reduces to finding $X_{max}$ such that

$$\mbox{\boldmath$\hat{k}$}\cdot\mbox{\boldmath$\omega$}_n^{\B... ...=0, \qquad\mbox{and}\quad X_{max}\in(0,\mbox{$\frac{1}{2}$}).$$ (7.79)

The limiting behaviour of $X_{max}$ for large and small values of $\mathcal S$ can be obtained from the Jones-Furry (7.24) and Hele-Shaw (7.27) limiting vorticity profiles as:

$$\lim_{\mbox{$\mathcal S$}\rightarrow\infty} X_{max} \equiv X_{max}^{\Vert} = \frac{\sqrt{3}}{6}$$ (7.80)

$$\lim_{\mbox{$\mathcal S$}\rightarrow 0} X_{max} \equiv X_{max}^{=} = \mbox{$\frac{1}{2}$}.$$ (7.81)

For general $\mathcal S$, $X_{max}$ may be found by bisection (Kahaner, Moler & Nash 1989, p. 240) on the line interval $\{\mathbf{X}: X\in[X_{max}^{\Vert},X_{max}^{=}], Z=0\}$; higher order methods, such as Newton-Raphson, being unsuitable since the vorticity is almost independent of $X$ except near the hot wall if $\mathcal S$ is small. For very small $\mathcal S$ the representation (7.54) of the velocity profile developed in §7.4.1 is more appropriate; so much more so, in fact, that only the Hele-Shaw form and the first term of the series are required to give three figure accuracy in $X_{max}$ for $\mbox{$\mathcal S$}<0.4$. This leads to the approximation for the vorticity field:

$$\mbox{\boldmath$\hat{k}$}\cdot\mbox{\boldmath$\omega$}_n^{\B... ...thcal S$}\rightarrow 0, X\;\mbox{near}\;\mbox{$\frac{1}{2}$}),$$ (7.82)

from which the asymptotic relation (7.81) may be extended to:

$$X_{max} \sim \frac{1}{2} + \frac{\mbox{$\mathcal S$}}{\upi }... ...hcal S$}\upi ^2}{16} \qquad(\mbox{$\mathcal S$}\rightarrow 0).$$ (7.83)

For large $\mathcal S$, (7.80) may be extended by taking the Jones-Furry form and the first term of the series in (7.25). The approximate spanwise vorticity in the plane $Z=0$ is then:

$$\mbox{\boldmath$\hat{k}$}\cdot\mbox{\boldmath$\omega$}_n^{\B... ...x{$\mathcal S$})}\qquad(\mbox{$\mathcal S$}\rightarrow\infty).$$ (7.84)

By expanding the cosine in a second order Taylor series about $X=\sqrt{3}/6$, and finding the root of the resulting quadratic equation in $X$, an explicit approximation for $X_{max}$ is obtained:

$$X_{max} \sim \frac{\sqrt{3}}{6} + \left\{ \frac{\upi ^2}{3}\cosh\upi \mbox{$\mathcal S$}-2\upi \sin\frac{\upi }{\sqrt{3}} \right.$$

$$+ \left.\left[ \left( 2\upi \sin\frac{\upi }{\sqrt{3}}- \frac{\upi ^2}{3}\cosh\upi \mbox{$\mathcal S$} \right)^2 \right. \right.$$

$$- \left. \left. 4\cos\frac{\upi }{\sqrt{3}} \left( 2\upi ^2\cos\fra... ...{\sqrt{3}} -\upi ^2\cosh\upi \mbox{$\mathcal S$} \right) \right]^{1/2} \right\}$$

$$\div 2 \left[ 2\upi ^2\cos\frac{\upi }{\sqrt{3}}-\upi ^2\cosh \upi \mbox{$\mathcal S$} \right] \qquad(\mbox{$\mathcal S$}\rightarrow\infty)$$ (7.85)

The various estimates; (7.80), (7.81), (7.83) and (7.85); of $X_{max}$, along with the root found by bisection of (7.78), are plotted in figure 7.9

Figure 7.9: Velocity maximum (or zero of vorticity) in a fully developed buoyant flow in a rectangular section.

where it can be seen that there is only a narrow transition region, $0.4<\mbox{$\mathcal S$}<0.9$, inside which neither the large (7.85) or small (7.83) $\mathcal S$ asymptotic relations are accurate. The maximum error in $X_{max}$ in using the more appropriate of these relations is less than 0.5% of the cavity width, and occurs near $\mbox{$\mathcal S$}=0.57$.

Figure 7.9 again confirms that for $\mbox{$\mathcal S$}>1.7$, the flow in the plane of spanwise symmetry is essentially two-dimensional.

The behaviour of the location of the point of maximum vertical velocity is similar to that of the point of maximum deflection in a vertical elastic plate subjected to a hydrostatic pressure variation (Timoshenko & Woinowsky-Krieger 1959, p. 125): the point moves further from the centre as the aspect ratio (height to width, in this case) increases. These two problems are not exactly analogous, however, as the plate deflection satisfies a fourth order equation, rather than the Poisson equation (7.20). The problems become identical if the plate has no resistance to bending, and so is a uniformly stretched membrane.