MN — Ship Resistance and Flow · Ch. 7 — Other Resistance Components

1. Overview — the additional resistance components (§7 intro)

Chapter 4 introduced the two main resistance components — the wave resistance and the viscous resistance — and then listed the additional ones. Having treated the two main components in depth (Chapters 5 and 6), this chapter turns to those additional components. Note that blockage effects were already covered in §5.9–5.12, so they are not repeated here.

The four additional components (from §4.2). They are: (1) induced resistance; (2) appendage resistance; (3) air and wind resistance; and (4) added resistance in a seaway. The last is closely tied to seakeeping theory and is left to the Seakeeping volume — so this chapter concentrates on the first three, which matter for a wide range of vessels, from sailing yachts and slender high-speed ships to bluff tankers.

**What this chapter covers.**
§	Component	Physical origin	Governing parameter
7.1	Induced resistance	Lift generation in an asymmetric flow (trailing vortices); an inviscid phenomenon	Aspect ratio (governed by the Chapter 5 equations)
7.2	Appendage resistance	Friction and viscous pressure losses around shafts, brackets, keels, rudders, fins	Reynolds number
7.3	Air and wind resistance	Pressure deficiency in separated regions on the above-water hull and superstructure	Apparent wind (Rn-independent — sharp-edged separation)

Fonte: LARSSON, Lars; RAVEN, Hoyte C. Ship Resistance and Flow. The Principles of Naval Architecture Series. J. R. Paulling (ed.). Jersey City: SNAME, 2010. Chapter 7 — Other Resistance Components.

Edital: Anexo 2-A, Área I, item 1 (resistência induzida, de apêndices, do ar/vento) e item 5 (componentes adicionais da resistência ao avanço: apêndices, ar e vento, sustentação/arrasto induzido) · Anexo 2-B, Área I, item 3 (Larsson & Raven), Chapter 7.

2. Induced resistance (§7.1)

The induced resistance appears whenever lift is generated in an asymmetric flow. It matters for wing-like bodies — keels, rudders, hydrofoils, bilge keels, stabilizers, twin-skeg sterns, catamaran hulls, trimaran outriggers — and for hulls themselves under certain conditions (a sailing yacht or a ship sailing at non-zero leeway). A key point: induced resistance is an inviscid phenomenon, governed by the equations of Chapter 5, not by viscosity.

2.1 Lift generation (§7.1.1)

"Lift" is a term borrowed from aerodynamics, where it is normally directed upward to balance an aircraft's weight. In hydrodynamics the lift is mostly horizontal, acting to move the body sideward relative to the direction of motion (for a heeling sailing yacht it may be inclined with the boat).

A wing is an obstacle to the flow (Fig. 7.1). Because the fluid cannot penetrate it, it must follow the contour and leaves the trailing edge in a direction different from the undisturbed flow far ahead — so the wing's main effect is to change the flow direction. The wing exerts a force to turn the flow; by reaction the flow exerts an equal and opposite force on the wing — the lift. The streamlines generally turn downward, so the centre of curvature is below the wing. Recalling the link between streamline curvature and lateral pressure gradient (§5.2), the pressure on the lower side must be higher than undisturbed, while the upper side develops a suction.

Kutta condition. For the deflection (and hence the lift) to be effective, the trailing edge must be sharp and the flow must leave it smoothly on both sides. This is a key element of inviscid flow theory, applied either as an equal-pressure condition (equal pressure on the two sides of the trailing edge) or as a flow-direction condition (the flow lies in the bisector plane just after the trailing edge).

2.2 Vortices and induced resistance (§7.1.2)

Consider a yacht keel at an angle of attack equal to the hull's leeway angle (Fig. 7.2). Nothing prevents the flow on the high-pressure (leeward) side from escaping below the tip to the low-pressure side, so a cross-flow around the tip is generated. The pressure side acquires a downward velocity component, the suction side an upward one, and the effect grows toward the tip. Where the two flows meet at the trailing edge they move in slightly different directions; the fluid elements just behind it feel opposite stresses on the two sides and start to rotate. Longitudinal trailing vortices are generated — none at the root (the flow is along the hull bottom there), increasingly strong toward the tip. The vortex sheet is unstable and rolls up into one concentrated vortex behind the keel. Because this vortex system contains rotational energy, it corresponds to an increase in resistance.

A 3-D wing can be represented by its vortex system (Fig. 7.3): the wing itself becomes a bound vortex (along the $y$-axis) and free trailing vortices stream behind it. The system is a superposition of horseshoe vortices of different span; along each filament the vortex strength is constant (Helmholtz's first theorem). The bound vorticity is maximum at the centreplane and drops to zero at the tips. By Helmholtz's second theorem a vortex cannot end in the fluid, so the trailing vortices close downstream through the starting vortex — usually neglected in steady flow.

The trailing vortex system produces a downwash ($w$) inside the sheet and an upwash outside it. The downwash at the wing, superimposed on the undisturbed velocity $U_\infty$, tilts the effective flow and defines the induced angle of attack $\alpha_i$:

$$ \alpha_i = \operatorname{atan}\frac{w}{U_\infty} \tag{7.1} \qquad\Longrightarrow\qquad \alpha_i \approx \frac{w}{U_\infty} \tag{7.2} $$ Because the downwash $w$ is always small, the arctangent reduces to the ratio itself ($\alpha_i$ in radians).

By the Kutta–Joukowski theorem, the lift on a bound vortex of strength $\Gamma$ acts at right angles to the local approaching flow $U$ (not to $U_\infty$) with magnitude $\rho U \Gamma$. The lift is thus tilted backward by the angle $\alpha_i$ relative to the $z$-axis. Resolving at right angles to, and parallel with, the undisturbed flow $U_\infty$ gives lift and induced drag per unit span:

$$ L' = \rho U \Gamma \cos\alpha_i \approx \rho U_\infty \Gamma \tag{7.3} \qquad D_i' = \rho U \Gamma \sin\alpha_i \approx \rho U_\infty \Gamma\,\alpha_i \tag{7.4} $$ The prime denotes force per unit span. The induced drag $D_i'$ is proportional to the induced angle $\alpha_i$ — another way (besides the energy argument) of relating trailing vorticity to drag.

2.3 The elliptical load distribution (§7.1.3)

Equations (7.1)–(7.4) explain the physics but are not handy for calculation, because circulation and induced angle are not easily obtained. The case of great practical importance is the elliptical load distribution (where the circulation $\Gamma$ is an elliptical function of the spanwise coordinate $y$): the integrated effect of the trailing vortices can then be obtained analytically. With the coefficients defined on the projected wing area $S_p$,

$$ C_L = \frac{L}{\tfrac{1}{2}\rho U_\infty^2 S_p} \qquad C_{Di} = \frac{D_i}{\tfrac{1}{2}\rho U_\infty^2 S_p} \tag{7.5} $$ the following simple expressions follow (Kuethe & Chow, 1986; Anderson, 1991): $$ C_L = \frac{2\pi}{1+\dfrac{2}{AR}}\,\alpha \tag{7.6} \qquad C_{Di} = \frac{C_L^2}{\pi\,AR} \tag{7.7} $$

In (7.6) the numerator $2\pi$ is for symmetric wings in inviscid flow with $\alpha$ in radians. For $\alpha$ in degrees the constant becomes 0.11; a more realistic value, allowing for the small viscous effect on lift, is 0.10.

The aspect ratio $AR$ of an arbitrary planform is

$$ AR = \frac{b^2}{S_p} \tag{7.8} $$ where $b$ is the span. For a trapezoidal wing this is the span divided by the mean chord. Note: $S_p$ is the projected wing area, distinct from $S$, the total wetted surface used for most other coefficients in the book.

Mirror image (Fig. 7.2). In computing the aspect ratio — but not the area — the mirror image of the keel in the hull bottom must be included. If the keel were attached to an infinitely large horizontal flat plate, the span would be exactly doubled by the mirror image; for a modern flat-bottom hull this is a good approximation.

Finite-wing theory shows that the elliptical circulation distribution gives minimum induced drag and that it is produced by an untwisted wing of elliptical planform — preferable from a resistance standpoint. Most wings are not elliptical, so the spanwise distribution differs.

Fig. 7-4 Fig. 7.4 — Schrenk's approximation of the lift distribution

Schrenk (1940) showed empirically that for unswept, untwisted wings the lift at every spanwise position is halfway between the elliptical and the actual chord distribution for the same area and span (Fig. 7.4). At non-zero sweep the centroid moves outward for positive sweep and inward for negative sweep.

Fig. 7-5 Fig. 7.5 — Definition of planform parameters

The sweep angle is measured relative to the quarter-chord line (the line joining the points 25% of the distance from leading to trailing edge at every station). The taper ratio is 1 for a rectangular wing and 0 for a triangular one; the value that best matches an ellipse for a trapezoidal wing is about 0.45. Larger ratios overload the outer wing, smaller ratios underload it — but an unfavourable planform can be partly compensated by sweep.

The relation between taper ratio and sweep angle for the optimum (minimum induced resistance) distribution is given in Fig. 7.6: the larger the sweep angle, the smaller the taper for best performance.

For arbitrary distributions no simple expressions like (7.6)–(7.7) exist; instead one introduces an effective aspect ratio $AR_e$, always smaller than the geometrical one, defined from (7.7) as

$$ AR_e = \frac{C_L^2}{\pi\,C_{Di}} \tag{7.9} $$ where $C_L$ and $C_{Di}$ are now for the nonelliptical distribution. The sensitivity to deviations from the ellipse is small: for most shapes $AR_e$ differs from $AR$ by only a few percent.

Fig. 7.7 shows the very large influence of aspect ratio on $C_L$; these curves agree well with equation (7.6).

3. Appendage resistance (§7.2)

In Fig. 4.1 no appendage resistance was specified for displacement hulls — an approximation, because all hulls with exposed shafts gain some resistance from the shaft and its support. The component can be considerable for a planing hull: as the dynamic lift raises the hull and shrinks its wetted surface, the appendages stay submerged and take over more and more of the viscous resistance with speed. It is also large for the sailing yacht. Crucially, appendage resistance is of viscous origin — frictional and viscous-pressure losses around the appendage — so the flow and resistance are governed by the Reynolds number.

3.1 Streamlined bodies (§7.2.1)

Streamlined sections appear throughout fluid mechanics, often to create lift (the aircraft wing; its hydrodynamic counterpart the hydrofoil; rudders and keels), but also to reduce resistance on struts and brackets.

Fig. 7.8 shows the huge effect of streamlining a 2-D body: the streamlined body and a circular cylinder have about the same drag, although the streamlined body has a frontal area about 30 times larger. On the streamlined body the flow stays attached to the trailing edge with very little pressure resistance (the small residue comes from the boundary-layer displacement effect); on the cylinder massive separation destroys the pressure recovery, so the low base pressure causes a very large pressure resistance with friction almost negligible.

Foils are generally asymmetric (Fig. 7.9). The mean line lies midway between upper and lower surfaces; the chord length $C$ is the distance between the intersections of the mean line with the nose and the trailing edge. The local thickness $t$ is measured at right angles to the mean line, and its maximum $t_{max}$ (or $t_{max}/C$) is often used to classify the section. The nose radius is the radius of curvature at the nose — usefully approximated by a circular arc back to about 45° on either side of the intersection.

NACA sections. NACA (the precursor of NASA) tested systematically varied sections in the 1930s–40s. The most useful for hydrodynamics are the four-digit and six-series. In the four-digit NACA 0010, the last two digits are the thickness ratio $t_{max}/C$ and the first two the camber (zero for symmetric). In the six-series 63-010 / 65-010, the first digit marks the series, the second (×10) gives the chordwise position of minimum pressure in percent, and the two digits after the dash give the thickness ratio.

**Table 7.1 — Three NACA sections (half-thickness $t/2$ in percent of chord, all symmetric).**
$x/C$ (%)	0010	63-010	65-010
2.5	2.178	1.756	1.574
10	3.902	3.362	3.040
30	5.002	4.938	4.760
50	4.412	4.496	4.812
70	3.053	2.712	3.156
90	1.207	0.604	0.810
100	0.105	0	0

Extract of Table 7.1; the maximum half-thickness is ~5% near 30–40% chord (the full table is in the source). Suitable for struts, brackets, rudders and keels.

Fig. 7.10 gives the nose radii versus thickness ratio: the four-digit series has twice the radius of an ellipse, the six-series lies in between. As a general rule, the larger the nose radius, the less sensitive the section to leading-edge separation at an angle of attack — so the four-digit series is the most robust in this respect.

Fig. 7.11 plots the pressure coefficient $C_p$ along the chord with the positive axis downward (a convention used for wings, since the upper side is normally at negative pressure). The 0009 section has its minimum very far forward (~10% chord); the two six-series sections have their minima near midchord, as their designation implies. Why care about the minimum? As in §6.4.3, the location of transition is dictated by the pressure distribution — a negative gradient stabilises the flow — and transition tends to occur at the pressure minimum. Pushing it back means more laminar flow and less friction; this is exploited in the so-called laminar (six-series) sections (the 63-series has ~30% laminar flow, the extreme 67-series ~70%; in practice the 65-series is the most extreme commonly used).

Fig. 7-13 Fig. 7.13 — Drag comparison, four six-series sections

Comparing the drag of two 9%-thick profiles (Fig. 7.12, $Rn = 3\times10^6$): the 0009 has higher drag at small angles but a smooth rise; the 63-009 has low drag below ~2° (its larger laminar region) then a rapid increase between 2° and 3° as the pressure minimum jumps forward, ending up worse than the 0009. This low-drag interval is the drag bucket. For the six-series (Fig. 7.13), thicker profiles (21%) hold the bucket to higher angles (~5–6°) than the thin ones (~2–3°), and there is a region (~3–7°) where the thick profiles even have lower drag than the thin ones.

At zero angle of attack (Fig. 7.14) the drag increases with thickness for all series, and the more extreme the profile (the further back the pressure minimum), the lower the drag at this angle.

The lift is quite similar between four-digit and six-series, at least below ~7° (Fig. 7.15). The theoretical lift slope is the same for all symmetric sections:

$$ \frac{\partial C_L}{\partial \alpha} = 2\pi \;\text{per radian} \;\approx\; 0.11 \;\text{per degree} \;(\approx 0.10 \text{ in a viscous fluid}) $$ This follows from (7.6) with an infinite aspect ratio (2-D sections). Small differences between sections come from different boundary-layer developments.

Fig. 7-16 Fig. 7.16 — Different types of stall

Above the linear range the lift ceases to follow the straight line because of flow separation on the suction side — the wing stalls. The pattern depends on thickness (Fig. 7.16): (a) a thick profile stalls by trailing-edge separation, beginning at small angles and growing gradually; (b) a very thin profile stalls by a leading-edge separation bubble that grows with angle; (c) a moderate profile (9–12%, the type most used in hydrodynamics) separates from both ends, with a drastic lift drop when the two bubbles meet.

Fig. 7-17 Fig. 7.17 — Maximum lift coefficient vs thickness ratio

For a rudder, the maximum lift is the key parameter (the turning moment is the lift coefficient times the rudder area, and must exceed a safety minimum). Fig. 7.17 shows that the four-digit series has the largest maximum lift: for a 9% profile, $C_{Lmax} = 1.32$ (four-digit) versus 1.12 (63-series) and 1.05 (65-series). A 65-series rudder thus needs about 25% more area to generate the required maximum lift, increasing friction over a wide range of angles. Because rudders operate over a wide angle range, the four-digit series is preferable — and the 12% profile is the best for this purpose.

The NACA profiles were tested at $Rn = 3$, $6$ and $9\times10^6$ — typical for pleasure-craft keels and rudders. Full-scale ships have much larger appendage Reynolds numbers and models much smaller, so the appendage scale effects differ from the hull's — a problem in model testing. Fig. 7.18 (after Hoerner, 1965) shows section drag at zero lift for 6%, 12% and 25% thicknesses, with laminar/turbulent flat-plate lines and a smooth cylinder for comparison:

Section drag versus Reynolds number (Fig. 7.18)

Below a critical $Rn \approx 10^5$: the layer is laminar and separates before transition. Pressure (separation) drag grows with thickness; the extreme is the circular cylinder, whose drag coefficient is roughly constant here (dominated by a Rn-independent pressure deficit). The 6% profile nearly follows the flat-plate line.
Above the critical $Rn$: transition occurs before separation; the turbulent layer mixes high-speed fluid inward, delays separation and cuts the pressure resistance. For the cylinder, separation moves from ~80° back to ~120°, narrowing the wake and reducing drag by about 75%.
Range $10^6$–$10^7$: the drag coefficient is roughly constant — transition moving forward (more turbulent layer, more drag) is just offset by the normal drop of drag with $Rn$, which reappears above $10^7$ where transition is nearly fixed.

3.2 Bluff bodies (§7.2.2)

The large drag drop at the critical $Rn$ for the circular cylinder is explained by Fig. 7.19, the pressure versus angular position (zero at the forward stagnation point):

**Pressure distribution on a circular cylinder (Fig. 7.19).**
Case	Behaviour	Consequence for drag
Inviscid	From $C_p = 1.0$ at the stagnation point down to $-3.0$ at 90°, then a symmetric rise back to $1.0$ at 180°	Perfectly balanced — no pressure drag (d'Alembert)
Supercritical	Minimum ~$-2.5$ near 90°; a plateau aft of 120° at a slightly negative $C_p$ to 180°	No balancing of the stagnation pressure — a small pressure drag
Subcritical	Minimum near 70°, then roughly constant at $C_p \approx -1.0$ over the whole back side	Large negative back-side pressure — a large drag

The plotted curves are mean values; in reality the flow fluctuates periodically, generating side-to-side lift forces (vortex shedding).

Fig. 7-20 Fig. 7.20 — Effect of roughness on drag

A way to reduce the drag below the critical $Rn$ is to disturb the boundary layer — most practically by surface roughness, which trips premature transition and lowers the critical $Rn$. Fig. 7.20 shows a circular mast covered with sand grains: the larger the grain, the larger the effect on the critical $Rn$, but the larger the minimum drag as well. This trade-off — lower drag and narrower wake — is of interest for sailing-yacht masts, with the added benefit of improved flow over the sail.

Practical corrections. Propeller shafts are not at right angles to the flow; the axial component's effect is Rn-dependent (no simple formula — a procedure is given by Kirkman & Kloetzli, 1980). Some appendages, notably the rudder, sit in the propeller race where the speed exceeds the ship speed (≈ +10% is reasonable); others are partly immersed in the hull boundary layer where the speed is reduced (estimate with the flat-plate formulas of §6.3.3).

4. Air and wind resistance (§7.3)

A ship moving on a smooth sea in still air feels a resistance from driving its above-water part through the air; it depends on the ship speed and on the area and shape of the upper structure. When a wind blows, the resistance depends also on the wind speed and its relative direction (and the wind raises waves, whose added resistance is treated in the Seakeeping volume — only the air effect is dealt with here).

4.1 True and apparent wind (§7.3.1)

Fig. 7-21 Fig. 7.21 — Relation between true and apparent wind

The true wind $\vec{V}_{TW}$ is the wind due to natural causes that would exist had the ship been absent (zero true wind = still air). The apparent (relative) wind $\vec{V}_{AW}$ is the vector sum of the true wind and the wind generated by the ship's own motion through still air (Fig. 7.21):

$$ \vec{V}_{AW} = \vec{V}_{TW} - \vec{V} \tag{7.10} $$ where $\vec{V}$ is the ship velocity. (In the yachting literature "wind" means "wind velocity".)

Taking both components parallel to the water surface, the magnitude and direction of the apparent wind are

$$ V_{AW} = \left(V_{TW}^2 + V^2 + 2\,V\,V_{TW}\cos\beta_{TW}\right)^{1/2} \tag{7.11} $$ $$ \beta_{AW} = \operatorname{atan}\frac{V_{TW}\sin\beta_{TW}}{V + V_{TW}\cos\beta_{TW}} \tag{7.12} $$ where $\beta_{TW}$ is the angle between the true wind and the $x$-axis and $\beta_{AW}$ that of the apparent wind.

The true wind varies with height $z$ because of the atmospheric boundary layer (Blendermann, 1990):

$$ V_{TW}(z) = V_{TW}(10)\left(\frac{z}{10}\right)^{1/n} \tag{7.13} \qquad V_{TW}(10) = 0.836\,(Bft)^{3/2} \tag{7.14} $$ The reference value is taken at 10 m height and related to the Beaufort scale by (7.14). Use $n \approx 10$ for strong, stable winds (a fuller profile) and $n \approx 5$ for light, unstable airs (a less full profile).

Because $V$ is independent of $z$ but $V_{TW}$ rises with $z$, the apparent-wind direction $\beta_{AW}$ also depends on $z$ — the apparent-wind profile is twisted. Near the surface the ship speed dominates and the flow is more along the hull; higher up the true wind dominates. The twist matters for sailing-yacht sails; for ships it is mostly neglected, and the flow direction is taken at the mean height of the lateral projection of the above-water hull.

4.2 Forces and moments (§7.3.2)

For most ships the airflow force is taken as horizontal (exceptions: some high-speed hulls using aerodynamic lift). The components considered are forces $X$ and $Y$ (along $x$ and $y$) and moments $K$ and $N$ (roll about $x$, yaw about $z$). They are non-dimensionalised by a dynamic pressure and a representative area:

$$ q_a = \tfrac{1}{2}\,\rho_a\,V_{AW}^2 \tag{7.15} $$ where the index $a$ stands for air.

$$ C_X = \frac{X}{q_a\,A_T} \qquad C_Y = \frac{Y}{q_a\,A_L} \tag{7.16} $$ $$ C_K = \frac{K}{q_a\,A_L\,H_M} \qquad C_N = \frac{N}{q_a\,A_L\,L} \tag{7.17} $$ $A_L$ is the lateral projected area and $A_T$ the transverse projected area; the longitudinal force $X$ is referred to $A_T$ (the convention adopted here), the others to $A_L$. $L$ is the length overall and $H_M$ the mean height of the lateral projection: $$ H_M = \frac{A_L}{L} \tag{7.18} $$

The coefficients depend on the apparent-wind direction. At zero apparent-wind angle the projected area in the wind direction is small ($A_T$) but the force is along the hull; at non-zero angles the area is larger but the force is not along the hull. A maximum in the resistance force $X$ is therefore typically found at $20° \le \beta_{AW} \le 30°$ (and a corresponding extreme at $150° \le \beta_{AW} \le 160°$).

Reynolds independence. Wind forces are dominated by the pressure deficiency in separated regions on the leeward side; frictional forces are very small. Because separation is fixed by the sharp edges of the hull and superstructure, the separation lines are independent of the Reynolds number. Hence all force and moment coefficients are Rn-independent, and wind-tunnel coefficients may be used at full scale with good accuracy.

Blendermann (1990) measured the aerodynamic coefficients for many ship types and loading conditions (Fig. 7.22; ship data in Table 7.2). These results allow computing the aerodynamic loads of most ships with reasonable accuracy.

**Table 7.2 (extract) — Data for the ships of Fig. 7.22.**
Ship	LOA (m)	$A_L$ (m²)	$A_F$ (m²)
Car carrier	190.7	4257	654
Container vessel I	210.8	3751	802
Ferry	143.9	2126	325
Tanker, loaded	351.4	3402	1132
Tanker, ballast	351.4	7840	1804
Offshore supply vessel I	62.0	337	137

$A_F$ = front area, $A_L$ = lateral area. Note how the tanker in ballast presents far more lateral area (7840 m²) than when loaded (3402 m²). Full table in the source.

4.3 Boundary-layer correction (§7.3.2 cont.)

The wind-tunnel coefficients (measured without the atmospheric boundary layer) can be corrected to include it via the mean dynamic pressure over the ship's height. For a ship at rest ($V_{AW} = V_{TW}$), inserting (7.13) into (7.15) and integrating up to $H_M$ gives the average dynamic pressure $\bar q_a$ relative to the reference value $q_{10}$:

$$ q_a = \tfrac{1}{2}\rho_a\,V_{TW}^2(10)\left(\frac{z}{10}\right)^{2/n} \tag{7.19} \qquad \frac{\bar q_a}{q_{10}} = \frac{n}{n+2}\left(\frac{H_M}{10}\right)^{2/n} \tag{7.20} $$ $C_X$, $C_Y$ and $C_N$ are multiplied by $\bar q_a / q_{10}$; because the roll lever arm also shifts in the boundary layer, $C_K$ is multiplied instead by $\dfrac{n+1}{n+2}\dfrac{\bar q_a}{q_{10}}$.

The mean $C_X$ at zero apparent-wind angle for all hulls of Fig. 7.22 (except the drilling ship) is 0.6. In still air this gives the air-resistance force $R_{AA}$:

$$ R_{AA} = C_X\cdot\tfrac{1}{2}\rho_a V^2 A_T = 0.6\cdot\tfrac{1}{2}\rho_a V^2 A_T \tag{7.21} $$

Non-dimensionalising by the water density and the ship's wetted area $S$ (to match the hydrodynamic coefficients) gives the air-resistance coefficient $C_{AA}$:

$$ C_{AA} = \frac{R_{AA}}{\tfrac{1}{2}\rho V^2 S} = 0.72\times10^{-3}\cdot\frac{A_T}{S} \tag{7.22} $$ with air and water densities taken as $1.2$ and $10^3\ \mathrm{kg/m^3}$. $C_{AA}$ is used in extrapolating model tests to full scale (§8): models are towed without superstructure (the hull's own air resistance is neglected as streamlined), so $C_{AA}$ is neglected for the model but added for the ship — a small correction kept to one significant digit, the constant rounding to 0.001.

4.4 Indirect effects of the wind (§7.3.3)

When the wind is not along the hull, the side force makes the hull slide sideward until balanced by a hydrodynamic side force — the hull moves with a leeway angle. The two side forces usually do not act on the same line, so a yawing moment develops, balanced by the rudder; the ship then moves with a non-zero rudder angle.

Because both the hull and the rudder now generate lift, both produce induced drag. For the rudder, equations (7.6)–(7.7) give accurate values; for the hull the accuracy is poorer (its underwater part has a very small aspect ratio) but useful for a rough estimate. As in §7.1, the mirror image of the hull in the water surface must be included when computing the aspect ratio (the effective aspect ratio is twice the geometrical one if the free surface is a symmetry plane).

Order of magnitude (van Berlekom, 1981). In a thorough analysis of wind effects, the direct wind force on the above-water structure is of the same order as the added resistance due to waves; the effects of leeway and rudder are less important.