Unsteady finite-depth effects during resistance tests on a ship model in a towing tank

Abstract

This paper covers an extension of the study of Doctors et al. (J Ship Res 52(4):263–273, 2008) on oscillations in wave resistance during the constant-velocity phase of a towing-tank resistance test on a ship model to the case of relatively shallow water. We demonstrate here that the unsteady effects are very prominent and that it is essentially impossible to achieve a steady-state resistance curve in a towing tank of typical proportions for a water-depth-to-model-length ratio of 0.25. This statement is particularly true in the speed region near a depth Froude number of unity. However, on the positive side, we show here that an application of unsteady linearized wave-resistance theory provides an excellent prediction of the measured total resistance, when one accounts for the form factor in the usual manner. Finally, a simple application of the results to the planning and analysis of towing-tank tests is presented.

Appendix: Formulation for unsteady wave resistance

The formulation is identical to that published by Doctors et al. [2]. We repeat it here for the convenience of the reader. The final result for the wave resistance is:

$$ R_W = R_{W,1} + R_{W,2} $$

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$$ R_{W,1} = {\frac{\rho\dot{U}} {2\pi}} \sum_{i=0}^\infty \epsilon \sum_{j=-\infty}^\infty \int \limits_{S_0} {\frac{\partial {b}} {\partial {x}}} \hbox {d}S \int \limits_{S_0} {\frac{\partial {b'}}{\partial {x'}}} \hbox{d} S' (-1)^j \left[\begin{array}{ll} {\frac{1} {r}} - {\frac{1} {r'}} \end{array}\right] $$

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$$ R_{W,2} = {\frac{2\rho g} {\pi w}} \int \limits_0^t U(\tau) \hbox{d} \tau \int \limits_0^\infty k_x^2 \hbox{d} k_x \sum_{i=0}^\infty \epsilon ({{\mathcal{U}}}^2 + {{\mathcal{V}}}^2) \times \cos [\omega (t - \tau)] \cos \{ k_x [s(t) - s (\tau)] \}. \\ $$

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The first term in Eq. 16 relates to the inertia of the water and is simply proportional to its density ρ. This term equals the product of the acceleration and the thin-ship infinite-Froude-number added mass of the vessel. The integrals in Eq. 18 are effected over the centerplane area of the vessel S ₀. The geometry of the vessel enters the equation through the appearance of its local beam b(x,z). The symbol ′ is used to distinguish the source variables from the field variables. The two radial distances from the source point to the field point are

$$ r = \sqrt{(x - x')^2 + (iw)^2 + (z - z' - 2jd)^2} $$

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$$ r' = \sqrt{(x - x')^2 + (iw)^2 + (z + z' + 2jd)^2}. $$

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It can be demonstrated that, as the Froude number approaches zero, the second term in Eq. 16 approaches twice the negative of the image-sink distribution. Thus, at such low speeds, one can model the hydrodynamics of the vessel using just a simpler, added-mass, concept.

The second term in Eq. 16 relates to wave effects. Here, s(t) is the distance traveled from the start of the motion. We also have

$$ \omega = \sqrt {gk\;\tanh(kd)}. $$

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The wave numbers k _x and k _y are related to the circular wave number k and the wave angle θ through the relationship:

$$ k_x + \hbox{i} k_y = k \exp (\hbox{i} \theta). $$

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The summation factor $\epsilon$ and the transverse wave number k _y are given by the formulas:

$$ \epsilon = \left\{\begin{array}{ll} {\frac {1}{2}} & \hbox{for }i = 0 \\ 1 & \hbox{for }i\geq 1\\ \end{array}\right., $$

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$$ k_y = 2 \pi i /w. $$

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The index i of the summation in these equations has been dropped for the sake of simplicity.

Finally, the finite-depth wave functions in Eq. 18 are given by the formulas

$$ {{\mathcal{U}}} = {\frac{P^+ + \exp (-2kd) P^-} {1 + \exp (-2kd)}} $$

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$$ {{\mathcal{V}}} = {\frac{Q^+ + \exp (-2kd) Q^-} {1 + \exp (-2kd)}}, $$

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in which the Michell deep-water wave functions P ^± and Q ^± are

$$ P^\pm + \hbox{i} Q^\pm = \int \limits_{S_0} b(x,z) \exp (\hbox{i} k_x x \pm kz) \hbox{d}S. $$

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