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.

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 $\langle u \rangle _\mathrm{b}/U_{\infty } \simeq {\pm }\,0.026$.