An Experimental Study on the Effect of the Side Hull Symmetry on the Resistance Performance of a Wave-Piercing Trimaran

Abstract

This paper presented the results of an experimental investigation into the resistance performance of a wave-piercing trimaran with three alternative side hull forms, including asymmetric inboard, asymmetric outboard, and symmetric at various stagger/separation positions. Model tests were carried out at the National Iranian Marine Laboratory (NIMALA) towing tank using a scale model of a trimaran at the Froude numbers from 0.225 to 0.60. Results showed that by moving the side hulls to the forward of the main hull transom, the total resistance coefficient of trimaran decreased. Findings, furthermore, demonstrated that the symmetry shape of the side hull had the best performance on total resistance among three side hull forms. Results of this study are useful for selecting the side hull configuration from the resistance viewpoint.

Appendices

Appendix 1: Theory of trimaran resistance

The total calm water resistance coefficient of a monohull ship model can be written as:

$${C}_{TM}({R}_{e},{F}_{r})={\left(1+k\right)C}_{FM}({R}_{e})+{C}_{WM}({F}_{r})$$

(1)

Insel and Molland (1991) proposed the total calm water resistance coefficient of a catamaran as follows. This formula is also applicable for other multihulls:

$${C}_{T}^{\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}=(1+\beta k){C}_{F}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}+\tau {C}_{W}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}$$

(2)

${C}_{T}^{\text{Model}}$ is the total resistance coefficient, ${C}_{F}^{\text{isol}.}$ and ${C}_{W}^{\text{isol}..}$ are the coefficient values of frictional and wave resistance in isolation, respectively. ${C}_{F}$ is calculated by the ITTC-1957 method using the below equation:

$${C}_{F}=\frac{0.075}{{\left({\text{log}}_{10}\mathrm{ }{R}_{e}-2\right)}^{2}}$$

(3)

The length of the main hull of a trimaran ship is normally different from that of the side hulls. This will result in a different ${R}_{e}$ for the main and side hulls and different frictional resistance coefficients. Individual lengths of the main and the side hulls are used in deriving different ${R}_{e}$ and the separate resistance coefficients. The total frictional resistance coefficient of a trimaran ship may be expressed as a function of the wetted surface area of each hull with respect to the total wetted surface area of all three hulls as follows:

$${C}_{F}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}={C}_{F}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}\left(\frac{{S}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}}{{S}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}+2{S}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}}\right)+{C}_{F}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}\left(\frac{{2S}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}}{{S}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}+2{S}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}}\right)$$

(4)

And wave resistance coefficient as:

$${C}_{W}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}={C}_{W}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}\left(\frac{{S}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}}{{S}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}+2{S}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}}\right)+{C}_{W}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}\left(\frac{{2S}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}}{{S}^{\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}+2{S}^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}}\right)$$

(5)

According to Hughes-Prohaska, $(1+k)$ is the form factor that is assumed to be the same for both the monohull and multihull in isolation. $\beta$ and $\tau$ are the viscous and wave resistance interference factors, respectively. For practical purposes, the form factor is assumed constant over the velocity range and between the model and ship. Although for the first time, those interference factors were derived by Insel and Molland (1991) for a catamaran, they have formed reasonable approximations for other multihulls (Yanuar et al., 2015a, b). Molland et al. (2011) developed empirical formula for $(1+k)$ for round bilge monohulls and $(1+\beta k)$ for catamarans based on model test results:

$$\left(1+k\right)=2.76{(L/{\nabla }^{1/3})}^{-0.4}$$

(6)

$$\left(1+\beta k\right)=3.03{(L/{\nabla }^{1/3})}^{-0.4}$$

(7)

$\tau$ is defined as:

$$\tau =\frac{{C}_{W}^{\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}}{{C}_{W}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}}=\frac{{C}_{T}^{\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}-{(1+\beta k)C}_{F}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}}{{C}_{T}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}-{(1+k)C}_{F}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}}$$

(8)

The following equations were applied to derive the ${C}_{Ts}$ value for the ship.

$${C}_{Ts}={C}_{W}^{\mathrm{S}\mathrm{h}\mathrm{i}\mathrm{p}}+{(1+\beta k)C}_{FS}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}$$

(9)

$${C}_{W}^{\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}={C}_{W}^{\mathrm{S}\mathrm{h}\mathrm{i}\mathrm{p}}$$

(10)

$${C}_{Ts}={C}_{T}^{\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}-{\left(1+\beta k\right)C}_{F}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}+\left(1+\beta k\right){C}_{FS}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}$$

(11)

$${C}_{Ts}={C}_{T}^{\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}-(1+\beta k){(C}_{Fm}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..}-{C}_{Fs}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}..})$$

(12)

The residuary resistance coefficient C_R of the tested model was derived by subtracting the frictional resistance from the total measured resistance. The frictional resistances for the central and the side hulls were firstly calculated individually using the “ITTC 1957 model-ship correlation line”. Then, C_R was in the below form:

$${C}_{R}=\frac{{R}_{T}-{R}_{FC}-{2R}_{FS}}{\frac{1}{2}\rho {V}^{2}({S}_{C}+{2S}_{S})}-{C}_{A}$$

(13)

where ${R}_{T}$ is the total resistance of the model, ${R}_{FC}$ is the calculated frictional resistance for the central hull, ${R}_{FS}$ is the calculated frictional resistance for each side hull, ${S}_{C}$ is the wetted surface area of the central hull, ${S}_{S}$ is the wetted surface area of a side hull, and ${C}_{A}$ is the correlation factor equal to 0.0004 (Lewis 1988). For slender ship hulls, the form resistance is very small; so, the measured residuary resistance is dominated by the wave-making resistance.

It is also important to identify the components of the wave-making resistance of a trimaran ship. Firstly, the wave-making resistance calculations are separately carried out for the central and the side hulls. The resulted wave-making resistance coefficients ${C}_{w0}$, excluding the wave interference between the hulls, is:

$${C}_{w0}=\frac{{R}_{WC}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}.}+{2R}_{WS}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}.}}{\frac{1}{2}\rho {V}^{2}({S}_{C}+{2S}_{S})}$$

(14)

where ${R}_{WC}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}.}$ is the wave-making resistance of the central hull alone, and ${R}_{WS}^{\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}.}$ is the wave-making resistance of a side hull alone. Then, the wave-making resistance coefficient ${C}_{wi}$, due to the wave interaction effects of the three hulls will be:

$${C}_{wi}={C}_{w}-{C}_{w0}$$

(15)

A positive ${C}_{wi}$ represents added wave-making resistance due to the wave interference between the central and the side hulls. A negative ${C}_{wi}$ means a reduction in total wave-making resistance due to wave cancelation effects between the hulls.

Appendix 2: The uncertainty analysis

For the results of the model experiments, an uncertainty analysis was carried out according to the 7.5–02-02–02 ITTC procedures and guidelines. The relative standard uncertainty components (wetted surface area and representative length of hull model) of resistance related to the hull geometry can be approximately estimated by the following equations, respectively (ITTC, 2014):

$${u}_{11}^{\text{'}}\left({R}_{T}\right)={u}^{\text{'}}(S)\frac{2}{3}\mathrm{ }{u}^{\text{'}}(\Delta )$$

(16)

$${u}_{12}^{\text{'}}\left({R}_{T}\right)=\frac{{C}_{F}}{{C}_{T}}.\mathrm{ }{\frac{0.87}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{R}_{e}-2}u}^{\text{'}}\left(L\right)\frac{{C}_{F}}{{C}_{T}}.\mathrm{ }{\frac{0.29}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{R}_{e}-2}u}^{\text{'}}\left(\Delta \right)$$

(17)

Since the Reynolds number in a typical resistance test is on the order of 10⁷, ${u}_{12}^{\text{'}}$ is relatively negligible to ${u}_{11}^{\text{'}}$. The combined standard uncertainty of resistance resulted from hull geometry can be estimated by Eq. (18) (ITTC, 2014):

$${u}_{1}^{\text{'}}\left(R\right)=\sqrt{{({u}_{11}^{\text{'}}\left({R}_{T}\right))}^{2}+{({u}_{12}^{\text{'}}\left({R}_{T}\right))}^{2}})\frac{2}{3}\mathrm{ }{u}^{\text{'}}(\Delta )$$

(18)

The uncertainty of carrier speed was also propagated into the resistance measurement as both dynamic pressure and Reynolds number (ITTC, 2014). These uncertainties were obtained as quantitatively by the following equations, respectively:

$${u}_{41}^{\text{'}}\left({R}_{T}\right)=2{u}^{\text{'}}(V)$$

(19)

$${u}_{42}^{\text{'}}\left({R}_{T}\right)=\frac{{C}_{F}}{{C}_{T}}.\mathrm{ }{\frac{0.87}{{log}_{10}{R}_{e}-2}u}^{\text{'}}(V)$$

(20)

where ${u}_{42}^{\text{'}}$ is usually much less than ${u}_{41}^{\text{'}}$ and negligible. Then, the combined standard uncertainty of resistance resulted from towing speed can be estimated as:

$${u}_{4}^{\text{'}}\left(R\right)=\sqrt{{({u}_{41}^{\text{'}}\left({R}_{T}\right))}^{2}+{({u}_{42}^{\text{'}}\left({R}_{T}\right))}^{2}})2\mathrm{ }{u}^{\text{'}}(V)$$

(21)

Moreover, the relative uncertainty of water viscosity resulted from temperature can be estimated:

$${u}_{3}^{\text{'}}\left({R}_{T}\right)=\frac{{C}_{F}}{{C}_{T}}.\mathrm{ }{\frac{0.87}{{log}_{10}{R}_{e}-2}u}^{\text{'}}(\nu )$$

(22)

The uncertainty component of resistance resulted from calibration of dynamometer was also estimated by standard error estimation (SEE) (ITTC, 2014):

$${u}_{2}^{\text{'}}\left({R}_{T}\right)={u}^{\text{'}}\left({R}_{T}\right)=\mathrm{S}\mathrm{E}\mathrm{E}$$

(23)

The standard uncertainty component from single test and repeat tests can be estimated by the following equations, respectively:

$${u}_{A}^{\text{'}}\left({R}_{T}\right)=\frac{Sdev}{{\widehat{R}}_{T}}$$

(24)

$${u}_{A}^{\text{'}}\left({R}_{T}\right)=\frac{Sdev/{R}_{T}}{\sqrt{N}}$$

(25)

Analyses of all significant uncertainty components related to the total resistance were combined to obtain the overall standard uncertainty by RSS method (ITTC, 2014):

$${u}_{C}^{\text{'}}\left({R}_{T}\right)=\sqrt{{({u}_{1}^{\text{'}})}^{2}+{({u}_{2}^{\text{'}})}^{2}{+({u}_{3}^{\text{'}})}^{2}+{({u}_{4}^{\text{'}})}^{2}{+({u}_{A}^{\text{'}})}^{2}}$$

(26)

Also, the expanded standard uncertainty of the resistance with a confidence level (t) was estimated by Eq. (27) (ASME, 2005; ITTC, 2014):

$${U}_{p}={k}_{p}{u}_{c}$$

(27)

An uncertainty analysis was finally carried out for results of symmetrical trimaran model in Tri-1 configuration. Figure 17 shows this analysis for carriage speed and total resistance coefficients. As shown, maximum uncertainty for this configuration of trimaran model occurs in ${F}_{r}$=0.25 which is about 2.6%.

Figure 17

Uncertainties of experimental results for symmetrical trimaran model in Tri-1 configuration