Wake of a cruciform appendage on a generic submarine at 10\(^\circ \) yaw

Abstract

The present model geometry is a recent iteration of the Joubert (Defence Science and Technology, Tech. Rep. TR-1920, 2006) generic conventional submarine design and is known as the “BB2”. Wind-tunnel testing of the model at 10\(^\circ \) yaw, by China-clay visualisation and by ensemble-averaged measurements using high-resolution stereoscopic particle image velocimetry, shows a similar wake flow at the model-length Reynolds numbers \(Re_\mathrm{L} = 4 \times 10^6\) and \(8 \times 10^6\). The most significant flow feature is on the model upper hull. It is a system of three co-rotating vortices produced by a cruciform appendage which consists of a vertical fin (or sail in American terminology) and two horizontal hydroplanes. Circulation is strongest from the fin tip followed by the windward hydroplane, then the leeward hydroplane. Vortex tracking shows a down-wash of the fin-tip vortex, where the wind-ward- and lee-ward-hydroplane vortices spiral in the rotation direction of the fin-tip vortex. The interpreted flow includes a U-shaped vortex line around the leeward hydroplane, where this vortex line connects the fin-tip vortex and a surface vortex on the leeward side of the fin.

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Acknowledgements

Model manufacturing and SPIV traverse automation by QinetiQ and the financial support by DST Maritime Division are acknowledged. Our thanks go to fellow DST colleagues and the anonymous referees for providing helpful feedback on this work.

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Appendix 1: Bias error in experimental technique

Appendix 1: Bias error in experimental technique

For SPIV, the bias error \(\langle u \rangle _\mathrm{b}/U_{\infty }\) is approximated by the uncertainties in the length-scale conversion \((\delta _\mathrm{l})\), the laser timing \((\delta _t)\), the in-plane displacement \((\delta _{yz})\) and the out-of-plane displacement \((\delta _x)\), viz.,

$$\begin{aligned} \frac{\langle u \rangle _\mathrm{b}}{U_{\infty }} \simeq \sqrt{ \left( \frac{\delta _\mathrm{l}}{r_\mathrm{m} / (R_\mathrm{l} r_\mathrm{s})} \right) ^{2} + \left( \frac{\delta _t}{\Delta t} \right) ^{2} + \left( \frac{\delta _{yz}}{\langle \mathbf {s}_{yz} \rangle } \right) ^{2} + \left( \frac{\delta _x}{\langle \mathbf {s}_x \rangle } \right) ^{2}. } \end{aligned}$$
(15)

The terms on the right side of Eq. 15 are described as follows.

  • For the uncertainty in length-scale conversion, \(\delta _\mathrm{l} / (r_\mathrm{m} / (R_\mathrm{l} \, r_\mathrm{s})) \simeq 4 \,\text {pixels} / (137\,\text {mm} / (10 \times 5.5\,\mu \text {m/pixel})) \simeq {\pm }\,0.002\), where \(r_s\) is the spatial resolution of the camera sensor or charge-coupled device and \(R_l\) is the reproduction ratio. The reference length \(r_\mathrm{m}\) is the maximum radius of the bare hull.

  • For the uncertainty in laser timing due to electronics jitter, \(\delta _t / \Delta t \simeq {\pm }\,0.001\), where \(\Delta t \simeq 11\,\upmu \)s to \(22\,\upmu \)s is the time interval applied between laser pulses.

  • For the uncertainty in the in-plane (yz) displacement, \(\delta _{yz}/\langle \mathbf {s}_{yz} \rangle \,\simeq \)\(0.125\,\text {pixels} / 7\,\text {pixels} \simeq {\pm }\,0.018\), where \(\langle \mathbf {s}_{yz} \rangle \simeq 0.6U_{\infty } \times \Delta t / (R_l \, r_s\)) is the resolvable particle-image displacement based on a maximum swirl velocity of \(\simeq 0.6U_{\infty }\). The uncertainty \(\delta _{yz} = f \{ \langle \mathbf {s}_{yz} \rangle \}\) is effectively a lower limit on resolvable displacement as a function of \(\langle \mathbf {s}_{yz} \rangle \)—it is taken from the digital particle-image-velocimetry analysis (fig. 6b) of [35]; the uncertainty is for a seeding density of \(\simeq \) 6 particles per \(32 \times 32\) pixels interrogation window.

  • For a symmetrical arrangement of SPIV cameras at oblique angles of 45\(^\circ \) from the laser sheet, the ratio between the in-plane and the out-of-plane particle-image displacement uncertainties is unity based on thin-lens ray-tracing analysis [27, 36]. Assuming that the field of view is not affected by optical aberrations or vignetting, and the measurement uncertainty is approximately homogeneous and isotropic, the uncertainty in the out-of-plane (x) displacement may be written as

    $$\begin{aligned} \frac{\delta _{x}}{\langle \mathbf {s}_x \rangle } \simeq \frac{\delta _{yz}}{\langle \mathbf {s}_{yz} \rangle }. \end{aligned}$$
    (16)

To summarise, the in-plane and the out-of-plane displacement uncertainties are approximately the same \((\delta _{x}/\langle \mathbf {s}_x \rangle \simeq \delta _{yz}/\langle \mathbf {s}_{yz} \rangle \simeq {\pm }\,0.018)\), and are larger than the uncertainties in the length-scale conversion \(( \delta _l / (r_\mathrm{m} / (R_l \, r_s)) \simeq {\pm }\,0.002)\) and the laser timing \((\delta _t / \Delta t \simeq {\pm }\,0.001)\). Overall, by substituting Eq. 16 into Eq. 15, this gives the bias error \(\langle u \rangle _\mathrm{b}/U_{\infty } \simeq {\pm }\,0.026\).

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Lee, SK., Manovski, P. & Kumar, C. Wake of a cruciform appendage on a generic submarine at 10\(^\circ \) yaw. J Mar Sci Technol 25, 787–799 (2020). https://doi.org/10.1007/s00773-019-00680-x

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Keywords

  • Generic submarine
  • Wind-tunnel testing
  • Flow visualisation
  • Particle image velocimetry