Water flows around a concrete bridge pier that has a cross section in the shape of a Rankine oval, formed by superimposing a line source and line sink placed 2 m apart, and each having a strength of 8 m²/s, Fig. 7–36. If the uniform flow around the body is U = 10 m/s, determine the required

Question

Water flows around a concrete bridge pier that has a cross section in the shape of a Rankine oval, formed by superimposing a line source and line sink placed 2 m apart, and each having a strength of 8 m²/s, Fig. 7–36. If the uniform flow around the body is U = 10 m/s, determine the required dimensions b and h for the pier.

Accepted Answer

Fluid Description. We assume steady flow and assume water is an ideal fluid.

Analysis. The stagnation points are at x = ±b, and so using Eq. 7–59, we have

[latex] b=\left(\frac{q}{U \pi} a+a^2 ight)^{1 / 2} [/latex] (7-59)

[latex] \begin{aligned} & =\left[\frac{8 m ^2 / s }{(10 m / s ) \pi}(1 m )+(1 m )^2 ight]^{1 / 2} \ & =1.12 m \end{aligned} [/latex]

The stream function for the flow is defined by Eq. 7–55. Here it becomes

[latex] \psi=U y-\frac{q}{2 \pi} an ^{-1}\frac{2 a y}{x^2+y^2-a^2} [/latex] (7-55)

[latex] =10 y-\frac{4}{\pi} an ^{-1} \frac{2 y}{x^2+y^2-1} [/latex]

Setting ψ = 0 gives the boundary of the pier, since it contains the stagnation points.

[latex] \begin{gathered} \psi=10 y-\frac{4}{\pi} an ^{-1}\left(\frac{2 y}{x^2+y^2-1} ight)=0 \ an (2.5 \pi y)=\frac{2 y}{x^2+y^2-1} \end{gathered} [/latex]

To find the half width of the pier, set x = 0, y = h in this equation, Fig. 7–36. This gives

[latex] h=\frac{h^2-1}{2} an (2.5 \pi h) [/latex]

This same result can also be obtained by applying Eq. 7–60 with a = 1 m, U = 10 m/s, and q = 8 m²/s. To solve, we note that for a half body, its half width is defined by Eq. 7–49, q /2U = (8 m²/s) /[2(10 m/s)] = 0.4 m. And so if we start with a slightly smaller value, say h = 0.35 m, for a Rankine oval, then we can increase or decrease it until it satisfies the equation. We find that h = 0.321 m

[latex] h=\frac{h^2-a^2}{2 a} an \left(\frac{2 \pi U h}{q} ight) [/latex] (7-60)

[latex] h=\frac{q}{2 U} [/latex] (7-49)

Question 7.10: Water flows around a concrete bridge pier that has a cross s...

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