Viscous free-surface flows past cylinders

Free-surface flows of viscous liquid down an inclined plane and past cylinders of various cross-sections are investigated theoretically and experimentally. The cylinders are oriented with their axis perpendicular to the plane and are sufficiently tall that they are not overtopped. A lubrication model is applied to derive the steady governing equation for the flow depth, which is integrated numerically and analyzed asymptotically to calculate how the depth of a steady uniform flow is perturbed as it flows past the cylinder. Flows past cylinders that are narrow relative to the depth of the oncoming flow are only slightly perturbed, but for relatively wide cylinders, there forms a pond of nearly stationary fluid upstream of the cylinder and a dry region in which there is no fluid downstream of the cylinder. The structure of the flow in the regime of a relatively wide cylinder depends in detail upon the curvature of the cylinder at the upstream stagnation point. For flows past cylinders with circular and square cross-sections, the maximum flow depth occurs at the upstream stagnation point. Its numerical value may be predicted analytically on the basis of the asymptotic expressions and exhibits different dependencies upon the variables that characterize the motion. In addition, wedge-shaped obstructions are analyzed for which the flow depth increases along the wedge wall and the maximum flow depth occurs away from the upstream stagnation point. The results from new laboratory experiments of flows past circular cylinders are reported and these corroborate the theory, confirming the occurrence of both pond and dry regions. The investigation has direct relevance to the deflection of lava flows by barriers and buildings and the theory is employed to deduce simplified asymptotic expressions of the force exerted on the cylinders.

Figure 1

(a) Schematic for viscous flow down an inclined plane at an angle, $\beta$, to the horizontal, past a cylinder from a line source. (b) Plan view of the setup. The cylinder has radius $L$, the $X$ axis is in the direction of steepest descent and the $Y$ axis is in the cross-slope direction. (c) Side view of the flow showing the flow thickness far upstream, $H_{\infty }$, and the perturbation owing to the cylinder.

Figure 2

Contour plot of the thickness of the steady flow past a cylinder with dimensionless radius $r=1$ for three values of the flow parameter, $\mathcal{F}$. (a) $\mathcal{F}$ = 20 (diffusive terms dominate advective terms), (b) $\mathcal{F}$ = 1 (diffusive terms and advective terms are comparable and both are important), and (c) $\mathcal{F}$ = 0.025 (advective terms dominate the diffusive terms).

Figure 3

The maximum (blue solid line) and minimum (red solid line) depths, calculated from our numerical technique, for steady flow past a cylinder as functions of the parameter $\mathcal{F}$. The deepest point is always at the stagnation point on the upstream boundary at $\theta =\pi$, while the shallowest point is on the downstream boundary at $\theta =0$. The asymptotic predictions for the two extrema, calculated from Eq. (33), are plotted as black dotted lines, which show good agreement for $\mathcal{F}\gg 1$,

Figure 4

Comparison between the composite expansion Eq. (36) and the numerical results for $\mathcal{F}=15$. (a) The depth of liquid along the centerline of the steady flow past a cylinder as a function of the downstream distance, $x$, for $\mathcal{F}=15$. The composite expansion Eq. (36) shows good agreement with our numerical simulations. (b) Contours of the absolute difference between the numerical result and the composite expansion for the flow thickness. The error does not exceed 1.1% of the upstream flow depth.

Figure 5

Depth along the centerline for the steady flow past a cylinder as a function of distance, $x$, with $\mathcal{F}=0.025$. The depth from our numerical simulations (solid blue line) is compared to the “inner” approximation [black dashed line, Eq. (55)], which varies linearly with $x$, and is very close to the “interior” approximation (red dot-dashed line), the solution to Eq. (B2), which is plotted in Fig. 18.

Figure 6

Flow thickness on the upstream cylinder boundary, $h(1,\theta )$, as a function of polar angle $\theta$ for $\mathcal{F}=0.025$. The numerically calculated flow thickness (solid curve) is compared to the asymptotic approximation, $h(1,\theta )={\mathcal{F}}^{-1/4}G\left(\theta \right)$, Eq. (58) (dashed curve).

Figure 7

The maximum depth, for steady flow past a cylinder as a function of ${\mathcal{F}}^{-1}$. The deepest point is always at the stagnation point on the upstream boundary at $\theta =\pi$. The results from our numerical technique (solid black line) compare very closely to the leading-order prediction of Eq. (58) (red dot-dashed line) and there is particularly good agreement for $\mathcal{F}\ll 1$. The asymptotic expansion with the second term Eq. (61) is also plotted (blue dashed line).

Figure 8

The maximum thickness for steady flow past a cylinder as a function of $\mathcal{F}$. The numerical results are compared to our asymptotic predictions for the two regimes; $\mathcal{F}\ll 1$ [Eq. (61)] and $\mathcal{F}\gg 1$ [Eq. (34)]. We also include our empirical approximation Eq. (62) as a dotted red line.

Figure 9

(a) Schematic of the experimental setup. A constantly maintained head of syrup supplies constant flux. (b) Photograph of the steady syrup flow past the cylinder, illustrating the increase in flow depth upstream of the cylinder, the sharp decrease in flow depth around the cylinder and the dry region downstream.

Figure 10

Comparison of the experimental and theoretical results (calculated numerically) for the maximum and minimum flow thicknesses. Zero minimum flow thickness corresponds to a dry zone downstream of the cylinder in which there is no syrup.

Figure 11

Flow depth along the centerline as a function of dimensionless distance, $x$, upstream of the cylinder for three values of the flow parameter, $\mathcal{F}$. The experimental results (solid line) obtained from the laser line technique show very good agreement with our theoretical predictions (dashed lines), calculated using the numerical scheme described in Sec. 3.

Figure 12

Flow past a square with (a) $\mathcal{F}=10$ and (b) $\mathcal{F}=0.25$. The two panels have different axes to capture the region of interest. Note that the flow remains attached to the square in both cases and there is no dry region for these values of $\mathcal{F}$.

Figure 13

Flow depth as a function of position along the upstream boundary of the rectangle for $\mathcal{F}=0.025$. The asymptotic prediction is $h={\mathcal{F}}^{-2/5}G\left(y\right)$ where the function $G\left(y\right)$ is given by Eq. (74).

Figure 14

Maximum thickness for flow past a square cylinder as a function of the dimensionless flow parameter, $\mathcal{F}$. The asymptotic prediction is $h_{\max }={10}^{1/5}{\mathcal{F}}^{-2/5}$, Eq. (75) and is accurate for $\mathcal{F}\ll 1$. The empirical approximation is quite accurate for all $\mathcal{F}$ and is given by Eq. (76).

Figure 15

Flow past a rhombus with $\mathcal{F}=0.25$. The angle between the upstream boundary and the centerline is $\psi =\pi /4$. The rhombus sides have dimensionless length of unity. There is no dry region for $\mathcal{F}=0.25$.

Figure 16

Flow depth along the upstream wall for $\mathcal{F}=0.025$ and $\psi =\pi /4$. The agreement between the numerical results and asymptotic prediction declines near $x^{\prime }=1$ as discussed in the text.

Figure 17

Maximum flow depth as a function of the flow parameter, $\mathcal{F}$, for three wedge angles. The asymptotic prediction is given in Eq. (83).

Figure 18

Flow depth in the interior region of our asymptotic analysis, according to Eq. (B2).