PNA Ch. 9 — Controllability

1. Introduction and Scope

Controllability covers every aspect of regulating a ship's trajectory, speed and orientation, both at sea and in the restricted waters where positioning and station keeping matter. It embraces starting, steering a steady course, turning, slowing, stopping, backing and, for submarines, diving. The study is conventionally split into three distinct functions.

Performance depends on water depth, channel restrictions and hydrodynamic interference from nearby vessels. Coursekeeping and maneuvering are particularly sensitive to trim, and for conventional ships the two tend to work against each other: an easy-turning ship is often hard to hold on course, while a good course-keeper is often hard to turn. A workable compromise is nearly always attainable.

Producing a ship with good controllability involves three tasks: (a) establishing realistic criteria for coursekeeping, maneuvering and speed changing; (b) designing the hull, control surfaces, appendages, steering gear and control systems to meet them, and predicting the result; and (c) conducting full-scale trials to compare measured performance with the criteria and predictions.

1.1 The control loop

For surface ships, coursekeeping, speed changing and maneuvering involve forces, moments and motions in the horizontal plane; for submarines the third dimension enters. The problem is best understood as a closed-loop directional control system (Segel, 1960).

That control force induces an angle of attack and an angular velocity, generating the hydrodynamic forces and moments that change heading and path. External disturbances (wind, current, waves) act at the same time, and feeding the actual heading and path back to the display closes the loop. In practice the full instantaneous path is rarely known: usually only heading, and sometimes rate of turn, is continuously determinate, while position is available occasionally (radar, visual cross bearings, Loran C, GPS). The last two elements of the loop, the ship and the steering gear and rudder, are of greatest concern to the naval architect. A second loop, the speed control loop, sets the speed along the path; the only element common to both loops is the conning officer.

1.2 Axes fixed in the earth

The dynamics are described with Newton's equations of motion referred first to axes fixed in the earth.

Equation (1). $X_0,Y_0$ = total forces along $x_0,y_0$; $\Delta$ = mass of the ship; $N$ = total moment about an axis through the CG parallel to $z_0$; $I_z$ = mass moment of inertia about $z_0$; the double dots denote second time derivatives.

1.3 Axes fixed in the ship

The motion is more conveniently expressed on axes $x,y$ fixed in the moving ship, origin at the CG. The $x$-axis lies along the centreplane (positive forward, its direction being the heading), $z$ is positive downward and $y$ positive to starboard. With $u$ and $v$ the components of the velocity $V$ along $x$ and $y$, transforming Equation (1) yields the equations of motion in the horizontal plane, assuming zero roll, pitch and heave.

Equation (5). The terms $-\Delta\,v\dot{\psi}$ (in $X$) and $+\Delta\,u\dot{\psi}$ (in $Y$) are the centrifugal-force terms: they appear only when moving axes are used and vanish with earth-fixed axes. Surge, sway and heave are the translations along $x,y,z$; roll, pitch and yaw the rotations about them. The origin is usually placed at midlength rather than at the CG, and throughout the chapter the LCG is assumed to coincide with the LCB and the midship location.

1.4 Forces acting during a maneuver

The left-hand side of the equations of motion is built from four types of force and moment: (a) hydrodynamic forces on hull and appendages from ship velocity and acceleration, rudder deflection and propeller rotation, which fall into damping forces (from velocity through the water) and added-mass forces (from accelerations); (b) inertial reaction (d'Alembert) forces from ship acceleration; (c) environmental forces from wind, waves and current; and (d) external forces from tugs or thrusters.

The effect of a rudder is indirect: moving it produces a moment that changes the heading, so the ship takes an angle of attack, which generates hull hydrodynamic forces that shift the CG laterally against the inertial reactions. With the rudder held fixed, a steady turn evolves once the hydrodynamic and inertial forces and moments balance. Current is normally folded into the hydrodynamic forces through relative velocity; wind forces scale roughly with above-water area and the square of relative wind speed; and steady, slowly varying second-order wave (drift) forces generally matter more for controllability than the first-order forces of seakeeping. The simplest case, a calm open sea with no wind, waves, current or external forces, is treated first.

2. Motion Stability and Linear Equations

2.1 Definitions of motion stability

Path keeping is closely tied to course stability. A body is stable in a state of equilibrium if, after a momentary disturbance and once released from it, it tends to return to the equilibrium it held before. For path keeping the disturbance is typically a wave or a gust, and ideally the ship resumes its original path after the disturbance passes, with no helm. The kinds of motion stability are classified by which attributes of the initial straight-line, constant-speed equilibrium the final path retains, forming an ascending hierarchy (

Achieving straight-line stability (Case I) is the designer's usual goal for hand-steered ships; the higher cases require increasing degrees of automatic control.

2.2 Controls-fixed versus controls-working stability

Stability has meaning with rudders fixed at zero, free to swing, or manually or automatically worked; in marine usage the unqualified term means controls-fixed stability. Three facts follow. In the vertical plane a surface ship in a calm sea has positional stability with controls fixed (Case IV), thanks to hydrostatic forces. In the horizontal plane, however, a self-propelled ship with stern propulsion cannot have positional or directional stability with controls fixed, because the stabilizing buoyancy changes are absent there. The only motion stability possible in the horizontal plane with controls fixed is straight-line stability, and many ships lack even that; wherever "controls-fixed stability" appears later it means controls-fixed straight-line stability, which is desirable but not mandatory. Each kind of controls-fixed stability carries a numerical index whose sign marks stable or unstable and whose magnitude gives the degree.

2.3 Linearity and the Taylor expansion

The force components $X,Y$ and the moment $N$ are functions of the ship's velocities and accelerations, expressed functionally in Equation (6):

To obtain a numerical index these functional forms are reduced through the Taylor expansion (

For two variables the linearization is a sum of three linear terms (Equation 7c). Applying this to the $Y$-force (Equation 8) and using the initial equilibrium (subscript 1 = straight-line, constant speed), symmetry about the $xz$-plane makes $v_1=0$ and the equilibrium $Y$ zero, so only $u_1\neq 0$ (equal to $V$). This reduces $Y$ to a few linear terms; the surge and yaw equations follow the same pattern. Cross-coupled derivatives such as $Y_{\dot{r}}$ and $Y_r$ have small nonzero values because bow and stern shapes differ, whereas $Y_{\dot{u}}$ and $Y_u$ vanish by symmetry.

2.4 Force and moment derivatives (SNAME notation)

In the simplified derivative notation of SNAME (Nomenclature, 1952), a symbol such as $Y_v \equiv \partial Y/\partial v$ or $N_r \equiv \partial N/\partial r$ denotes the slope of the force or moment with respect to a motion variable. For motions restricted to the horizontal plane, $\dot{\psi}\equiv r$ and $\ddot{\psi}\equiv\dot{r}$. A primed symbol denotes the nondimensional form.

Substituting the linearized derivatives into Equation (5) gives the linear equations of motion with moving axes, controls fixed, Equation (10):

Each term of the first two equations has the dimensions of a force and each term of the third has the dimensions of a moment. To nondimensionalize, the force equations are divided by $\tfrac{\rho}{2}L^2V^2$ and the moment equations by $\tfrac{\rho}{2}L^3V^2$ (analogous to the resistance-coefficient nondimensionalizers of Chapter V). Neglecting the surge equation and using $u_1/V\approx 1$ for small perturbations, Equation (10) becomes the nondimensional pair, Equation (11):

Because $Y'_{\dot{v}}$ enters as an addition to the mass term, $(\Delta' - Y'_{\dot{v}})$ is the virtual mass coefficient, identical to added mass; $Y'_{\dot{v}}$ is always negative, since $Y$ opposes a positive $\dot{v}$. Likewise $(I'_z - N'_{\dot{r}})$ is the virtual moment of inertia coefficient, and $Y'_{\dot{r}}$, $N'_{\dot{v}}$ are the coupled virtual inertia coefficients, which would vanish if the hull and appendages were symmetric about the $yz$-plane.

2.5 Control forces and moments

Every term of Equations (10) and (11) assumes the rudder held at zero. With the rudder working, the right-hand side carries the control force and moment. The linearized side force acting at the CG is $Y_\delta\,\delta_R$ and the linearized moment about the $z$-axis is $N_\delta\,\delta_R$, where $\delta_R$ is the rudder-deflection angle measured from the ship's $xz$-plane to the rudder plane (positive deflection turning to port for stern rudders). The rudder's side force creates a turning moment, the ship takes an angle of attack, and the well-designed hull, acting as a foil, generates a hull moment $N_v v$ that greatly augments the rudder moment (

where $n'_z = I'_z - N'_{\dot{r}} \approx 2I'_z$ and $\Delta'_y = \Delta' - Y'_{\dot{v}} \approx 2\Delta'$. Numerical predictions need coefficient values, obtained chiefly from captive model tests plus theoretical and empirical estimation.

3. Coursekeeping and Controls-Fixed Stability

3.1 The stability indexes

Solving the linear sway and yaw equations simultaneously for $v'$ and $r'$ gives a second-order differential equation whose standard solutions are Equation (13):

Here $e=2.718$; $V_1,V_2,R_1,R_2$ are constants of integration; and $\sigma_1,\sigma_2$ are the stability indexes, with dimensions of $1/t$. If both $\sigma$ are negative, $v'$ and $r'$ decay to zero and the path resumes a straight line (Case I). If either is positive the motions grow, no straight path is resumed, and the ship may settle into a steady turn with the rudder at zero.

Substituting Equation (13) into the equations of motion gives a quadratic in $\sigma$, Equation (14), whose roots are Equation (14a):

For controls-fixed stability both roots must be negative. In practice $\sigma_1$ alone is usually quoted for surface ships: it is algebraically less negative than $\sigma_2$, so its term dominates after the disturbance ends, making it a good measure of the degree of stability.

3.2 The stability criterion

Both roots are negative only if two conditions hold: $C/A>0$ (otherwise one root is always positive) and $B/A>0$ (if $B/A<0$ with $C/A>0$, both roots are positive). So stability requires $A$, $B$ and $C$ to share the same sign. Examining the derivatives shows why $A$ and $B$ are always large and positive. The inertial derivatives $Y_{\dot{v}}$ and $N_{\dot{r}}$ are always negative and large: for ellipsoids $Y_{\dot{v}}$ ranges from about $-0.9\,\Delta$ toward $-1.0\,\Delta$ as $L/B$ grows, and $N_{\dot{r}}$ from about $-0.7\,I_z$ ($L/B=5$) to $-1.0\,I_z$ ($L/B=\infty$), so $A$ is a large positive quantity.

The damping derivative $Y_v$ is always negative (bow and stern both give lift opposing $v$), with the bow dominating, so its centre of action lies forward of midlength and $N_v$ is usually negative for ships without fins or rudders. Adding a stern rudder raises $Y_\delta$ and reduces the negative magnitude of $N_v$ (rarely making it positive). For angular velocity $r$, bow and stern add to give a large negative $N_r$, while they oppose for $Y_r$ (small, sign uncertain). Since $B$ is likewise large and positive, the stability condition collapses to $C>0$, so $C$ is the discriminant of dynamic stability, Equations (14b)-(14e):

The last form reads as a relationship between the lever arms of the yawing and swaying forces. These inequalities tell whether the ship is stable, not the quantitative degree, which comes from $\sigma$. A related index $T\approx -1/\sigma_1$ of the K-T pair can be developed from ordinary trials for comparing vessels, and Clark's (1982) regression methods estimate stability from major dimensions early in design.

In the horizontal plane the stability indexes are not speed dependent at low and moderate Froude numbers, where $C_T$ and the nondimensional derivatives stay roughly constant. A ship stable at low speed is therefore stable at higher speed (up to a limiting $F_n$), and an unstable one stays unstable.

3.3 The Dieudonné spiral maneuver

The Dieudonné spiral (Dieudonné, 1953) is a definitive trial for directional stability. The ship is steadied on a straight course at a preset speed; the power plant is then left untouched. The rudder is turned to about 15 deg and held until the yaw rate is constant for roughly a minute, then decreased in steps of about 5 deg from large starboard through zero to large port and back, each step held to a new steady yaw rate. Plotting steady yawing rate against rudder angle characterizes the ship (

For the unstable ship the loop's height and width measure the degree of instability (a larger loop is more unstable), and the slope of the yaw-rate curve at zero rudder measures the degree of stability or instability. Linear theory cannot predict the loop of an unstable ship; the nonlinear theory is essential. The existence of a loop means that at zero rudder the yaw rate is not necessarily zero, the ship may keep turning with the rudder amidships, which is itself the signature of controls-fixed instability. Abkowitz (1964) drew the analogy with stability in heel: a righting-moment-versus-heel curve with positive slope at the origin is stable (ship A), negative slope unstable (ship B); on the unstable branch the ship can even turn against its rudder before swinging suddenly to the opposite stable branch.

For a stable ship the rudder angle at zero yaw rate is the equilibrium (neutral) rudder angle $\delta_{R1}$, usually with a nonzero $v_1$; for an unstable ship the neutral angle is approximated by the rudder angle at half the loop height. Adequate settling time at each rudder angle is essential: too short an interval yields a spurious sloped loop for a ship that is really stable (Strom-Tejsen, 1965). For submarines, where the spiral cannot be run in the vertical plane, the meander test is used: stern planes are deflected briefly and returned, a decaying oscillation indicating directional stability and a growing one instability. Directional instability is not necessarily bad, large slow ships can be handled well despite it; what matters is the degree relative to type, size and speed.

3.4 Bech reverse spiral and pullout

The Bech (reverse spiral) test (Bech, 1968) steers the ship at a constant rate of turn and measures the mean rudder angle required, repeated over a range of yaw rates (for example 0.5 deg/s port to 0.5 deg/s starboard). For a stable vessel the result resembles the direct spiral; for an unstable one a definite relationship is obtained within the hysteresis loop, since the test is no longer controls-fixed. It needs a calibrated rate-gyro and an accurate rudder-angle indicator, and points may be taken in any order. The pullout test (Burcher, 1972) puts the ship into a turn and then returns the rudder to midships: if stable, the rate of turn decays to zero for both sides; if moderately unstable, it falls to a residual value. It is run to both sides to reveal asymmetry.

4. Stability and Control

4.1 The five elements of path keeping

Controls-fixed stability is only one element of path keeping. Because path keeping involves continual path correction, its elements merge with path changing, and the ability at both depends on five elements of the control loop: (a) the magnitude and frequency of the disturbing yawing moments and sway forces; (b) the ship's controls-fixed response to disturbances; (c) how quickly path error is detected and corrective action begun; (d) the rate at which correction becomes rudder movement (steering-gear rate); and (e) the magnitude of the control force and moment applied by the rudder.

Only element (b) depends on controls-fixed stability. A deficiency in one element can be offset by improvement in another, for instance a well-designed automatic control in (c) can correct controls-fixed instability in (b). Enlarging the rudder (e) or speeding the rudder rate (d) does not necessarily cure path-keeping or path-changing deficiencies; the best design minimizes the deficiency in each element.

4.2 Definitive maneuvers

Definitive maneuvers demonstrate elements (b), (d) and (e) while excluding element (c), establishing basic stability and control characteristics independent of helmsman or autopilot: the direct or reversed spiral determines stability, the zigzag determines control, and the turning circle denotes turning qualities.

4.3 The zigzag maneuver

The zigzag, also called the Kempf overshoot or "Z" maneuver (Kempf, 1944), proceeds as follows. The ship is steadied as in the spiral test; the rudder is thrown at maximum rate to a preset angle (say 20 deg) and held until a preset heading change (say 20 deg) is reached; then it is thrown at maximum rate to the opposite checking angle and held until the execute heading change on the opposite side is reached, completing the overshoot test. For a full zigzag the cycle repeats through several executes, of which the first overshoot is the most important.

The zigzag results are speed dependent: time to execute falls with speed, while overshoot yaw angle and width of path grow with speed. The nondimensional time to execute (in ship lengths) rises with speed because the rudder-deflection rate is essentially speed-independent, so at low speed the rudder exerts full influence for longer. For submarines the overshoot is run in both planes, where overshoot pitch angle and overshoot change of depth also matter.

4.4 The K and T indexes (Nomoto)

Nomoto (1957) showed the linear equations can be rewritten as decoupled second-order equations with time constants $T'_1,T'_2,T'_3,T'_4$ and system gain $K'$, Equation (15), whose roots relate to the time constants by $\sigma_1=-1/T'_1$ and $\sigma_2=-1/T'_2$:

Neglecting the small cross-coupling terms and eliminating sway, turning depends only on the yaw rate $r$, giving the simplified yaw equation (16) and, on dividing by the yaw damping coefficient, the first-order Nomoto equation (17):

In words, $T'$ is the yaw inertia coefficient divided by the yaw damping coefficient, and $K'$ is the turning moment coefficient divided by the yaw damping coefficient. In dimensional form $T\dot{r}+r=K\delta_R$, with $T'=T(V/L)$ and $K'=K(L/V)$. For a rudder put over suddenly to $\delta_0$ and held, the yaw-rate solution is Equation (19):

The yaw rate rises exponentially to the steady value $K\delta_0$ (equivalently $K'V\delta_0/L$). A larger $K$ gives greater steady-state turning ability; a smaller $T$ gives quicker initial response to the helm, good for both course changing and course checking. $T$ has no effect on the steady turning rate but a small $T$ shortens the time to reach a steady turn. $T'$ is a reciprocal measure of course stability (stability improves as $T'$ decreases), a negative $T'$ signals an unstable dynamic character, and $T'\approx -1/\sigma_1$ ties it directly to the straight-line stability index.

Neglecting sway, steady turning at constant rudder gives the steady turning diameter $D_0$, Equations (20)-(22):

so a larger $K'$ lets a smaller rudder angle achieve a given turning diameter. Ranking maneuvering qualities by $T'$ and $K'$ (larger being better for turning and responsiveness), a highly maneuverable ship, quick to the rudder and turning tightly at the cost of low course stability, has small $T'$ and large $K'$. The large ratio $K'/T'$, the Norrbin parameter $P = K'/2T'$ (Nomoto and Norrbin, 1969), indicates good maneuverability but is not a good indicator of coursekeeping; a large $K'/T'$ suggests good overall controllability only if the stability is no greater than necessary.

Nomoto's further guidance (Fig. 19 in the book) relates the coefficients to rudder area $A_R$ and displaced volume $\nabla$, giving Equation (23):

where $c_1$ is a constant of proportionality nearly independent of ship type and rudder angle. Since a large $K'/T'$ is favourable, the ship's length and rudder area strongly affect controllability; once overall dimensions are set, both coursekeeping and turning improve by increasing rudder size or effectiveness. The indexes $T'$ and $K'$ can be computed from Equation (15) if the hydrodynamic and mass coefficients are known, or derived from standard trials or free-running model maneuvers, giving those trials direct physical meaning.

5. Analysis of Turning Ability

Almost every maneuver a ship performs — except some stopping — involves turning. When the rudder is deflected and held, the ship's response splits into an early transient part (with surge, sway and yaw accelerations) and a later steady part, where the rate of turn and the forward speed are constant and the track is a circle.

5.1 The Turning Path and the Turning Circle

The turning path is described by four numerical measures (Fig. 20). All but the last are tied to how far the ship's heading has swung, not to the tangent of the curve. They are measured from the moment the rudder order is given, the point called "execute".

Most merchant ships reach a tactical diameter of 2 to 4 ship lengths at full rudder, and many turn in 2 lengths or less. These tight turns are strongly nonlinear, so the linear estimates below apply best to stable ships turning in about 4 lengths or more (and to torpedoes or less-than-full rudder).

5.2 The Pivot Point and the Drift Angle

To an observer on deck the ship seems to swing about a pivot point, usually a little abaft the bow. There the water flows parallel to the ship's x-axis, so a fixed vertical fin placed at that point would see no angle of attack. Its distance forward of the center of gravity is:

where $x_c$ is the pivot-point distance from the CG, $R$ the turning radius, and $\beta$ the drift angle (the angle between the ship's x-axis and its actual path). Because tight (small-$R$) turns come with large drift angles and wide turns with small ones, the product $R\sin\beta$ changes little from ship to ship. For most ships the pivot point sits between the bow and about $1/5\,L$ abaft the bow (Mandel, 1953).

Empirically the drift angle (in degrees) follows one of two relationships with $L/R$:

The first gives a pivot-point distance $x_c$ of about 0.4 to 0.5 $L$ (depending on $L/R$); the second gives $x_c \approx 0.3\,L$. During the very first part of the turn there is also a temporary pivoting point near the bow: the bow initially runs along the straight extension of the approach path while the stern swings outward.

5.3 The Three Phases of a Turn

With the rudder deflected to a fixed angle, the path develops in three distinct phases (Fig. 21).

In the first phase the transverse acceleration $\dot v$ is negative (to port) even though the turn will finish to starboard, because a stern rudder pushes the stern to port to start a starboard swing. The linearized first-phase equations keep only the acceleration and rudder terms:

Here $\Delta$ is the mass term, $I_z$ the yaw moment of inertia, the $Y$ and $N$ terms the sway-force and yaw-moment derivatives, $\delta_R$ the rudder angle, and dotted symbols the accelerations. In phase 2 the path first strays slightly to port before the $Y_v v$ force enforces the starboard turn; this port offset is negligible in practice because phase 1 is short.

5.4 Steady Turning Radius

In the steady phase the accelerations vanish and only velocities remain, giving the steady-turn equations:

With the nondimensional turn rate $r' \equiv \dot\psi = rL/V$ and the steady radius $R = V/r$, so that $r' = L/R$, these solve for the radius and drift angle:

Primed symbols are the nondimensional hydrodynamic derivatives, $\delta_R$ and $\beta$ are in radians, and a positive $R$ means a starboard turn. In words, the steady radius is proportional to ship length $L$ and inversely proportional to the rudder angle $\delta_R$, while the drift angle is directly proportional to $\delta_R$.

5.5 Radius and the Hydrodynamic Derivatives — the Stability Link

Rewriting the radius (Equation 26a) exposes a clean result: its numerator is exactly the controls-fixed stability criterion $C$, and the denominator is always positive.

So a stable ship (positive numerator) with a stern rudder produces a starboard turn from a negative rudder angle, and vice versa — it turns "with" its rudder. An unstable ship has a negative numerator, so $R$ takes the same sign as $\delta_R$ and the ship in effect turns against its rudder. Because Equation 26 only describes the slope of $R$ versus $\delta_R$ at zero, it cannot predict the actual turning radius of an unstable ship. A useful corollary: since yaw damping $N'_r$ is usually much larger than $N'_v$, increasing the side-force derivative $|Y'_v|$ usually increases the steady radius — the $Y_v v$ force starts the turn but a bigger $Y_v$ does not necessarily tighten it.

5.6 Heel in a Turn

The rudder also cross-couples into roll, and the resulting heel can be large. Its size is estimated from the vertical spacing of the acting forces. Gathering the first-phase terms and setting them to zero gives the balance used for the heeling moment:

For a surface ship the heel changes sign between phase 1 and phase 3 (Fig. 24). In the first phase of a starboard turn the ship heels slightly to starboard (into the turn); in the steady phase it heels to port (outward), because the inward $Y_v v$ force at the CG must overcome the rudder force to hold the turn.

A potentially dangerous situation exists just prior to the completion of the first large heel to port. A helmsman, fearing too large a heel, might return the rudder quickly to amidships; this would eliminate the rudder force and the heel to port would be aggravated rather than alleviated. The only safe action is to immediately, but slowly and cautiously, reduce the rudder angle and at the same time reduce speed as quickly as possible. — Principles of Naval Architecture (SNAME), Vol. III, Sec. 6.5

A submerged submarine behaves differently: it heels inboard (starboard heel for a starboard turn) through all phases, because its force positions sit higher relative to the CG (the bridge fairwater acts as a high lifting surface). The large first-phase inboard heel is the snap roll. The ratio of snap roll to steady heel can be as high as about 3.5 for a submarine with a large fairwater and about 5 for one without. The fairwater plays a double role: it raises the roll excitation and also raises the roll damping that curbs the snap-roll overshoot.

5.7 Speed Loss in a Turn

Speed falls as soon as an appreciable drift angle appears, and the loss depends mostly on the tightness of the circle (Davidson, 1944). Figure 25 relates the ratio (steady-turn speed / approach speed) to the turning diameter in ship lengths, with block coefficient as the parameter; the scatter could not be tied to rudder angle, approach speed or rudder area.

Despite this loss, tightening the tactical diameter to about 2 ship lengths pays off operationally. A 122 m (400 ft), 20-knot ship at TD/L = 2.0 completes a full course reversal and nearly regains approach speed in the roughly 1¾ min it takes to pass the original execute point heading the opposite way; the same ship at TD/L = 4.5 needs about 2½ min and far more sea room. One caution for calculations: the speed used in the steady-phase heel estimate must be the reduced turn speed, not the approach speed. Transient turning and complex maneuvers cannot be predicted by linear theory — they need nonlinear captive-model or free-running model tests.

6. Accelerating, Stopping and Backing

Getting a ship up to speed, slowing it and reversing it matter most near land, other vessels and fixed structures. The hull–propeller interactions in these maneuvers are complex and change moment to moment, so when motion-equation coefficients are missing, naval architects fall back on empirical calculations.

6.1 Four Maneuvers Defined

Performance is scored by the time and distance from start to finish. Analyses often assume a straight track, but on a single-screw or unirotating ship the propeller swings the stern to one side (to port for a right-handed propeller). When the ship curves, the along-track projections — head reach and side reach — become the more useful indexes.

6.2 Accelerating

The accelerating force at any speed is simply the net thrust available minus the resistance. Dividing by the virtual mass gives the acceleration:

where $R_t$ is the ship resistance, $T_{net}$ the net thrust, $t$ the thrust deduction, $m$ the ship mass and $a_x$ the longitudinal added mass. Time, speed and distance follow by integration, $t = \int dV/\dot u$ and $S = \int V\,dt$. Two simplifications keep the sums workable: propeller thrust is taken to build up instantly (the ship takes far longer to speed up than the propeller does), and the thrust deduction is treated as constant.

6.3 Stopping Distances and the "80-50" Rule

Stopping is about avoiding collisions, rammings and groundings. The key index is the head reach — the distance still made good in the original direction while coming to rest. Operators treat head reach from harbor speed (12 knots) as the real measure of backing power; head reach from full speed matters little, because at high speed a hazard is dodged more easily by turning than by stopping.

Two forces do the stopping work: the ship's own resistance (which dissipates kinetic energy but falls off fast as speed drops) and the astern backing thrust, both opposed by the ship's mass plus longitudinal added mass. Chase et al. (1957) model resistance as $R = kV^n$ (strictly valid for $n = 2$) and group the variables into three dimensionless ratios — a dynamic potential (giving head reach), a dynamic impulse (giving time to stop) and the ratio of ahead resistance to astern thrust $R_0/T_1$. At slow speeds, where resistance is small, thrust dominates and — because thrust goes roughly as RPM² — head reach and stopping time obey an inverse-square law in RPM.

For early-design estimates the astern thrust at dead-in-the-water is taken as $T_1 = 5.5\,Q_1/P$ ($Q_1$ = astern torque, $P$ = propeller pitch), and the time to close the ahead and open the astern throttle is about 20 s for a modern automated vessel. Historically, merchant-ship turbines were designed for the "80-50" backing power: astern torque equal to 80% of rated ahead torque, delivered at astern RPM equal to 50% of rated ahead RPM.

6.4 Stopping with Freedom to Turn

In a real single-screw crash astern the trajectory is usually unpredictable, because directional control is lost once the propeller reverses (the 1955 Esso Lima / Esso Paterson trials). At high speed with sea room, a hard-over turn beats a straight stop for clearing a hazard: the advance in a turn is far shorter than the head reach in stopping, and steering is retained. Below about 6 knots the head reach and deflections shrink and turning loses its general advantage.

An intermediate technique is the rudder cycling maneuver (Esso Bernicia, 1969): four partial turns to alternating sides about the base heading, with the engine reduced in steps then reversed. It sheds speed through the hull's inertial reactions in the turns and offers a more predictable track and a shorter head reach than a plain crash astern. Still, where sea room allows, a simple hard-over turn is better than either; and from approach speeds below about 8 knots the direct crash astern is generally best. If impact is unavoidable, crash astern at least reduces the striking energy.

6.5 Machinery Limits and Displacement

Machinery strongly shapes stopping ability. For light, high-powered ships the head reach in a crash stop drops sharply as the propeller RPM can be reversed faster; for very large, low-powered ships like big tankers the effect is small. A caution in design: direct-drive diesel ships can only reverse a limited number of times before exhausting their compressed-air supply, risking a temporary loss of reversing ability in tight waters.

Simulation captures what hand calculation cannot. Two robust results stand out. First, head reach and time to stop vary almost directly with displacement (Fig. 71) — bigger ships stop over longer distances. Second, doubling the absorbed astern horsepower (from ~30% to ~60% of maximum ahead) cuts head reach only 20 to 25%, because astern thrust grows only as the 2/3 power of shaft power and the RPM-reversal lag dilutes the gain. Propeller cavitation can also cut astern thrust once astern RPM exceeds about 70% of maximum ahead.

6.6 Coasting and Propeller Drag

Coasting means letting the ship slow with no backing power, and the propeller state decides how fast. A windmilling propeller turns freely and makes no thrust, so only hull resistance slows the ship. A locked (stopped) propeller adds its own large drag on top of hull resistance, estimated from the standard drag formula:

where $A$ is the developed propeller area, $V = V_A = (1-w)V_0$ the flow speed at the disc, $w$ the wake fraction, and $C_D \approx 1.0$ for a locked propeller (Hewins et al., 1950). This locked-propeller drag is large — the ratio of propeller drag to hull resistance runs from about 1.5 (very large slow ships) to about 3.0 (large fast twin-screw liners) — so locking the propellers instead of letting them windmill can cut coasting distance by a factor of 2 to 4.

A neat consequence of the $n = 2$ resistance law: the coasting distance to a given fraction of the initial speed is independent of the initial speed. The table below (Chase et al., 1957) gives that distance in feet, where $\Delta$ is displacement, $V_0$ the initial speed in knots and $R_0$ the total resistance at $V_0$.

6.7 Backing

Around docks, backing is judged less by raw stopping and more by maneuverability: for clearing a slip, the astern speed reached after one ship length is often an adequate criterion. That speed is found by equating the work done by the accelerating force over the distance to the ship's kinetic energy:

where $S$ is the distance run, $X$ the instantaneous accelerating force, $V$ the astern speed reached, $\Delta$ the displacement and $X_{\dot u}$ the longitudinal added-mass derivative (so $\Delta - X_{\dot u}$ is the virtual mass, hull mass plus added mass).

6.8 Auxiliary Stopping Devices and Tugs

Because hull resistance scales with speed squared, it only helps at high speed — which is why water parachutes and brake flaps are largely useless from moderate speeds, exactly where unplanned stops happen (even 20× normal hull resistance barely shortens a slow-speed stop). A non-hydrodynamic fixed force, such as a rocket motor, can help more: up to 400 tons of retarding force (about the takeoff thrust of forty Boeing 707 engines) was studied for a 190,000-dwt tanker.

At harbor speeds tugboats become part of the stopping system, modeled as a constant added retarding force in power tie-up (same forward speed). But Figure 76 makes a sharp point: approach speed and the tanker's own astern RPM matter far more than the number of tugs (given at least 40 RPM astern). Zero tanker RPM with six tugs gives about the same head reach as 55 RPM astern with no tugs at all. Above roughly 6 knots tugs become impractical to tie up.

7. Effects of the Environment

The immediate environment — wind, current and waves — can strongly influence a ship's controllability. It may reduce coursekeeping stability, cause complete loss of the ability to hold a course, or simply add resistance that demands extra power. The importance of each effect depends heavily on the ratio of the disturbing velocity to the ship's own speed, so the same wind or current is far more troubling at low speed than at service speed.

7.1 Dynamic Behavior in Wind

Wind matters most when the ratio of wind velocity to ship speed is large; even a moderate wind can make a slow-advancing ship hard to control. The effect grows with the above-water (windage) area, with the distance from the centre of lateral area to the LCG, and with the aerodynamic drag coefficients. Ships of large windage — car carriers, containerships, LNG carriers — are critically influenced at low speed. Fig. 82 shows non-dimensional aerodynamic side-force coefficients for many ships plotted against wind direction; they scatter fairly evenly about an amplitude of roughly 1.0.

• $\rho_a$ — air mass density.
• $u_a,\,v_a$ — longitudinal and transverse components of the relative wind velocity.
• $A_{ax},\,A_{ay}$ — maximum longitudinal and transverse projections of the aerodynamic (above-water) area.
• $x_{ac}$ — longitudinal coordinate of the centre of transverse aerodynamic force, measured from midships.
• $X'_a,\,Y'_a,\,N'_a$ — the corresponding non-dimensional coefficients.

Referred to the ship's $x$ and $y$ body axes, the velocity components relative to the air are:

• $U_a$ — wind speed (velocity of the wind).
• $\psi_a$ — direction from which the wind arrives, relative to the earth-fixed axes.

Typical results for a Mariner-class cargo ship (Fig. 84) show that, for a given wind-to-ship velocity ratio, greater rudder angles are required when the wind is on the beam. Because most ships have a maximum rudder angle near 35 degrees, a ship is not generally controllable when the wind demands a rudder angle close to this limit. A full-load Mariner is uncontrollable at some headings when the beam-wind velocity is large ($U_a = 10V$), yet may remain controllable in a wind of the same magnitude coming from another direction. The directional-stability requirement can set a critical wind velocity that is even lower than the one implied by the 35-degree limit.

A stability analysis of an unsteered ship (eigenvalues of the equations of motion) reveals three regimes for wind from the bow ($U'_a > 0$), where $U'_a$ is the relative wind speed:

In the first two regimes the ship tends to keep its heading without control forces. For wind from the stern ($U'_a < 0$) the ship is always unstable: the imaginary parts of the critical roots are zero, so the motion diverges without oscillation, at a rate that grows monotonically with wind velocity. For an automatically steered ship in a wind from an arbitrary direction the characteristic equation is fifth order; the ship is stable in a not-too-strong wind from near the bow and most unstable in the same wind from the stern (180 deg). A well-designed automatic control system improves stability in wind more than even a highly experienced helmsman.

7.2 Current Effects

Current affects controllability differently from wind and is usually handled through the relative velocity between ship and water, especially in maneuvering simulation. By analogy with the wind relations, the relative velocities are:

• $U_c$ — current velocity (drift).
• $\psi_c$ — current direction (reciprocal of the "set"), relative to the earth-fixed axes.

Open-ocean surface currents are modest and nearly constant, so they pose no difficulty for open-sea controllability. Current becomes important in restricted waters, where operating speeds are low and the current is non-uniform — particularly when running downstream in a river or canal that carries significant current, and above all at bends, where spatial velocity gradients occur. There the low relative speed through the water ($u - v_c$) may be too small to develop adequate rudder and hull force. Ship-handling simulators are the practical tool for evaluating such effects, and detailed current surveys are often a necessary input (Miller, 1978, on river-tow safety).

7.3 Stability and Control in Waves

In rough seas a ship experiences wave-induced oscillatory motion in all six degrees of freedom. In linear theory the coupled transverse motions of sway, yaw and roll can be treated separately from pitch, heave and surge; this chapter is concerned mainly with yaw and sway (closely related even in calm water) plus roll, which is important for high-speed ships because turning causes heel and rolling in turn affects steering.

In head and bow seas the encounter frequency is high and course stability is usually large, so serious difficulty seldom occurs: the experienced helmsman ignores the high-frequency yawing and steers to the mean heading. Automatic controls tend to call for high-frequency rudder movements that add resistance with little effect on heading, so filtering or suitable control settings (adaptive autopilots) are desirable. Yaw-roll rudder coupling arises in turns, since the rudder produces heeling as well as yawing moments.

Under seagoing conditions, especially in long overtaking waves at high speed, several new factors enter:

• (a) Low wave-encounter frequencies allow large roll and yaw moments to build up.
• (b) High-speed ships generally have relatively low static transverse stability (low $GM$).
• (c) Static stability changes significantly in waves — with crests at bow and stern the midship is in a trough, developing less righting moment, hence large roll — which affects yaw.
• (d) The rudder's large effect on roll as well as yaw makes the design of the automatic control system critical.
• (e) High-speed hulls have a fore-and-aft asymmetry that itself changes as the ship rolls.

A non-zero heel at non-zero speed shifts the local sectional-area centroid transversely (like the camber line of a wing), introducing a hydrodynamic yaw moment and side force; bulbous bows make this more pronounced. Significant coupling among yaw, sway, roll and rudder action is therefore possible, and automatic control can help overcome unfavourable coupling.

7.4 Coursekeeping in Astern Seas

Much more serious difficulty occurs in quartering and following seas. Wahab and Swaan (1964) studied coursekeeping and broaching — turning broadside to the waves — concentrating on the limiting case where ship speed equals wave velocity (zero frequency of encounter). They concluded that the difficulty in steering and the danger of broaching arise from dynamic course instability: "all unsteered ships appear to be unstable somewhere on the downward slope of a wave." Increasing the ship's smooth-water course stability reduces the danger, and a superior control system helps overcome the instability.

The period and frequency of encounter govern how the ship experiences the seaway:

• $L_w$ — wave length; $V_w$ — wave velocity; $V$ — ship speed.
• $\chi$ — angle from the ship's velocity vector to the direction of wave advance.

For seas from ahead ($90^\circ < \chi < 270^\circ$), $\cos\chi < 0$ and the encounter frequency is high. In astern seas the term $(V_w - V\cos\chi)$ can become very small, giving a very low encounter frequency — which is why coursekeeping in astern seas is usually more difficult than in head seas. Three situations arise in astern seas: (a) overtaking seas, where the waves overtake the ship at low encounter frequency; (b) the semistatic case, where the encounter frequency is zero and a ship poised on a downslope stays there; and (c) following seas, where the ship overtakes the waves. In cases (a) and (c), but not (b), the sway forces and yaw moments oscillate with time — a destabilizing yaw moment when the stern is on a crest and the bow in a trough, reversing to a stabilizing moment half a period later.

Wahab and Swaan computed the static-equilibrium heading deviation as a function of position on the wave: the required deviation is greater with the bow in a trough than with the bow in a crest, and the bow-in-trough positions are unstable with controls fixed while bow-in-crest positions may be stable. No matter how much smooth-water stability a ship has, it becomes unstable in long waves when its CG lies a certain distance ahead of a crest; conversely, a ship unstable in smooth water may become stable in long waves with the bow near the crest. Their conclusions for a destroyer with automatic controls working:

• (a) Great danger of broaching persists when the bow is in the trough of an astern wave of length $\geq 1.5L$ travelling at about wave speed.
• (b) The likelihood of broaching increases with wave height.
• (c) The danger is reduced by increased fin area aft (better smooth-water controls-fixed stability).
• (d) An autopilot with a large control constant $k_1$ can shrink the instability regions.
• (e) Sensitivity to yaw rate does not much reduce the instability regions, and increasing the control time lag does not much enlarge them.

Horizontal-plane motions (yaw, sway, surge) have no natural frequency of their own (except a very low autopilot-related one), so encounter-frequency effects differ from those on pitch, heave and roll. At high encounter frequency the steered and unsteered ship differ little (small ratio of rudder force to wave excitation); at low encounter frequency the unsteered yaw becomes extremely large, rudder effectiveness increases greatly, and steering is clearly advantageous. Large time constants help prevent violent rudder activity in following seas, whereas a large yaw-rate gain greatly increases rudder motion. Increasing rudder size improves both stability and turning in following seas at wave speed, but has very little effect at high encounter frequency (low speed, e.g. $F_n < 0.30$, quartering sea).

8. Vessel–Waterway Interactions

8.1 General

Safe operation in restricted waterways depends on the ship, its pilot, the local environment and informational factors. Direct analysis is very complex, and decisions have relied mainly on rules of thumb, comparison with successful practice, and "seaman's eye." Interest has grown with the ever-increasing size of ships, the carriage of hazardous cargoes and the social cost of accidents. Two definitions frame the discussion:

• Shallow water (for maneuvering) — water in which the ratio of water depth to ship draft is 3 or less; above this ratio shallow-water effects fall off rapidly.
• Restricted waters — narrow channels or canals, waterways with vertical or overhanging banks, or areas with piers and breakwaters that substantially change maneuvering characteristics. Most restricted waters include shallow water, and many include current and tide.

The hydrodynamic effects are grouped as: (a) water depth relative to draft; (b) channel width and topography relative to beam; (c) large changes in depth or width relative to ship size; (d) interaction between two ships; and (e) combinations of these.

8.2 Shallow-Water Effects

Yeh (1964) reduced shallow-water, current-induced forces at zero ship speed to curves of side force and yawing moment versus flow angle and depth-to-draft ratio.

From the Esso Osaka trials the checking and counter-turning ability first decreased as depth fell to an intermediate value (50 per cent bottom clearance) and then increased again at the shallowest depth (20 per cent under-keel clearance), reflecting an apparent reversal in controls-fixed course stability — stability first drops, then rises as the water becomes very shallow. Stopping distance was largely independent of water depth, though the heading deviation while stopping grew from 18° to 50° to 88° going from deep to medium to shallow water. It is worth noting that 20 per cent under-keel clearance is not itself very shallow: ships often operate at about 10 per cent in low water and only about 5 per cent at a berth.

Hirano and Takashina (1987) estimate the linear derivatives in shallow water through an effective ship aspect ratio $k_e$:

• $k$ — effective aspect ratio in deep water.
• $T$ — mean draft; $D_w$ — water depth.
• $\lambda$ — an empirical parameter, evaluated from tests. Proposed values (VLCC, LNG carrier, passenger car carrier): $\lambda = 2.3$ for $Y'_v$, $\lambda = 1.7$ for $N'_v$, and $0.7$ for both $Y'_r$ and $N'_r$.

8.3 Effects of Narrow Channels

In deep, wide water the flow passes around the sides and under the bottom of the hull. In shallow water the under-hull flow is restricted, so more flow is forced along the sides; this changes the side forces and moments and hence the hydrodynamic derivatives ($Y_v$, $N_v$, $Y_r$). In a narrow, shallow canal the derivatives are altered even more severely.

The magnitudes of $Y_{y_0}$ and $N_{y_0}$ increase as the canal width decreases, implying a sensitivity to position that is absent in the open ocean. There is also a heading sensitivity: a yaw angle $\psi$ creates a moment $N$ that tends to increase $\psi$, so the derivative $N_\psi$ is always positive — a destabilizing derivative. Because of the nature of $N_\psi$, no ship can possess controls-fixed positional stability with respect to the canal centreline; any ship on the centreline is in unstable equilibrium. The only way to hold the path on the centreline is by controls — manual, or automatic with a continuous distance-from-bank signal added to the control law.

Extensive model testing (David Taylor Research Center; Panama Canal studies) and digital simulation confirm this picture. In an Interoceanic Canal Study, Eda (1971, 1973) modelled a 250,000-dwt tanker (335 m, $C_B$ 0.83) and a 188 m cargo ship ($C_B$ 0.60): after a 2-degree yaw disturbance in a 158 m wide, 24 m deep canal at 6 knots, both ships were directionally unstable with rudder fixed (worse for the tanker), but both became directionally stable with an activated rudder (gains $K_1 = K_2 = 4$); the smaller cargo ship behaved much better. Design observations from this body of work include: hull forms full at the bow and sharp at the stern handle better in a canal; a rudder placed abaft the propellers, or one that deflects into the propeller race, is more effective; river-towboat pilots may order a rate of turn rather than a rudder angle through bends; and for a given ship and canal there may be a critical speed that causes the greatest difficulty. If handling is so poor that transit is impossible, a tugboat made fast astern by a towline both raises the propeller-race velocity (better rudder effect) and, through towline tension, improves stability.

8.4 Interaction Between Two Vessels

Passing close to another ship, like passing a channel boundary, creates forces and moments that do not exist in open water. The difference is that a channel boundary is long and of constant cross-section, so its interaction depends only on transverse distance $y_0$ and yaw $\psi$, whereas ship-to-ship interaction also depends on the longitudinal separation $x_0$, on the yaw angles, and on the relative sizes of the ships. It matters operationally for overtaking, meeting, collision avoidance, passing a moored ship and underway replenishment (UNREP).

The peak forces occur near the fully abeam position. At positions 3 (ship A) and 5 (ship B) both the interaction force and the interaction moment draw the ships together, so the rudder must overcome the interaction moment and introduce a yaw angle whose outboard force counteracts both the attraction and the rudder force. These positions immediately precede and follow the directly-abeam position (4), where the rudder must swing quickly from large port to large starboard deflection — the moment of greatest collision risk, worse in rough seas or heavy wind. If collision seems imminent, in position 3 ship A reduces speed and ship B increases; in position 5 ship B reduces and ship A increases. Theoretical methods (Tuck, Dand, Abkowitz) agree reasonably with the model data and support applications such as UNREP simulation and passing a moored ship.

8.5 Sinkage and Trim

Tuck (1966, 1967) showed that sinkage and trim in shallow and restricted water collapse onto a nearly universal non-dimensional curve, almost independent of ship form. Three terms are distinguished:

• Sinkage — the downward vertical displacement of the ship's centre of gravity.
• Positive trim — a bow-up rotation about the CG (trim can be negative, bow-down, at low speed in shallow water).
• Squat — the resultant of sinkage plus bow-up rotation.

Sinkage is plotted against a Froude number based on water depth, $F_n = V/\sqrt{gD_w}$. An approximate value of the vertical hydraulic force acting on a ship in a relatively narrow canal is:

• $F$ — vertical force, positive downward.
• $S_0$ — cross-sectional area of the canal; $S(x)$ — local cross-sectional area of the ship; $B(x)$ — local beam.
• $\rho$ — mass density of the canal water; $F_n$ — depth Froude number.

When comparing finite canal width with infinite width, Tuck's curves use an effective width $\bar{W} = (W/L)\sqrt{1 - F_n^2}$; the finite-width effect is much greater for sinkage than for trim. Agreement with experiment (Graff et al., 1964) is good at lower $F_n$ but deteriorates as $F_n \to 1$ — the critical speed $U \approx \sqrt{gD_w}$, close to the celerity of shallow-water waves. Sinkage dominates at subcritical speeds ($F_n < 1$) and trim dominates at supercritical speeds ($F_n > 1$); the large subcritical sinkage is always positive (downward), while the supercritical trim is positive (bow up).

Because sufficient bottom clearance is crucial, Eda (1971) gives contours of the speed limited in canals so that the ship clears the bottom, based on a semi-empirical relation from model tests (Yamaguchi, 1967, 1968):

• $F_{n_L}$ — limiting Froude number, $U/\sqrt{gL}$.
• $p$ — draft/ship length, $h/L$; $m$ — water depth/ship draft, $D_w/T$.
• $q = 1/(1+e)$, with $e = 0.24$ from the test results.

9. Hydrodynamics of Control Surfaces

A control surface develops a transverse control force from its orientation and movement relative to the water: a rudder for horizontal control of the ship, a diving plane for a submarine's vertical motion, or an activated fin to reduce rolling. A rudder placed at the stern creates a moment that rotates the ship, orienting the hull at an angle of attack; the forces and moments that then arise determine the maneuvering characteristics. This section treats the rudder principally, but most of the discussion applies to any control surface.

Source: Lewis, E. V. (ed.), Principles of Naval Architecture, 2nd rev., SNAME 1988–1989, Vol. III, Ch. IX, Section 14 (pp. 291–316). Equation numbers (121–127) and figure numbers (121–154) below are the book's own.

9.1 Geometry, Forces, and Moments

The simplest and most common control surface is the all-movable surface. Following aeronautical nomenclature, its dimensions are expressed as lying in three mutually orthogonal directions (

Because a rudder must develop lift in either of two opposed directions, its section shape is symmetrical about the centerplane. In nonviscous, two-dimensional (infinite-aspect-ratio) flow at angle of attack $\alpha$, the combination of forward velocity and $\alpha$ induces a circulation that produces a lift force normal to the free-stream velocity, with no drag (

The total resultant real-fluid force acts at the center of pressure (CP). It is resolved into a lift $L$ (normal to the direction of motion), a drag $D$ (parallel to it), and a $y$-component normal to the ship's axis — the last being the reason for having a rudder. For a rudder well isolated from the ship, the $y$-force and its yaw moment about the ship's $z$-axis are:

• $L$, $D$ — lift and drag of the rudder (always taken positive).
• $\beta_R$ — drift angle at the rudder (always taken positive).
• $x_R$ — distance from the ship's origin to the rudder CP; negative if aft of the origin, positive if forward.
• $Y_{rudder}$, $N_{rudder}$ — transverse force and yaw moment; $X_{rudder}$ — longitudinal force.

With these sign conventions, the sign of $Y_{rudder}$ follows the sign of the rudder angle $\delta_R$ (hence the $\pm$). Since $D\cos\beta_R$ is always larger than $L\sin\beta_R$, the term $X_{rudder}$ is always negative — that is, directed aft.

The component of the total rudder force normal to the centerplane, $F$, times the distance of the CP from the stock centerline gives the hydrodynamic torque carried by the rudder stock:

Here $d$ is the mean distance between the leading edge and the centerline of the rudder stock, and $CP_{\bar c}$ is the chordwise CP location. The sign of the stock moment depends on whether $d$ is greater or less than $CP_{\bar c}$ — not on the right-hand rule. The bending moment on the stock about the root section uses the spanwise CP; its maximum value, with the torque, sizes the stock, bearings and steering engine, and the stock diameter in turn fixes the rudder's root thickness.

To compare rudders of different size and speed, the forces and moments are made nondimensional by $(\rho/2)\,A_R\,U^2$ together with $\bar c$ or $b$, where $\rho$ is the mass density, $A_R$ the rudder area and $U$ the velocity:

• $C_L$, $C_D$ — lift and drag coefficients; $C_N$ — normal-force coefficient.
• $(C_M)_H$ — torque coefficient about the rudder stock; $C_{m\bar c/4}$ — moment coefficient about the quarter chord.
• $CP_{\bar c}$, $CP_s$ — chordwise and spanwise CP locations.

9.2 Flow Around a Ship's Rudder

A rudder works in a highly complicated medium. Three phenomena place definite limits on the maximum achievable performance.

Phenomenon	What happens	Effect on lift
Stall	As $\alpha$ increases, the separation point moves forward along the chord; the lift-curve slope falls; at a critical (stall) angle there is an abrupt discontinuity.	Beyond the stall angle, lift decreases with further $\alpha$.
Cavitation	Occurs when max negative pressure + atmospheric + hydrostatic pressure falls below vapor pressure; inception depends on nuclei and surface roughness ( ).	Slows the growth of lift with $\alpha$ (does not stop it); its inhibiting effect grows with speed.
Aeration (ventilation)	Air is drawn from the atmosphere into the suction side when the rudder is near the surface and the pressure difference exceeds the resistance to air drawing.	If that resistance is low, occurs at lesser angles / lower speeds than cavitation.

Typical curves of normal-force coefficient versus angle of attack carried through the stall point — in the free stream and in the propeller race — are shown in Fig. 248. At moderate speeds cavitation is less restrictive than stall; but at higher speeds its effect grows, and it can erode the rudder surface and cause rudder-induced vibration. Aeration is common in model turning tests, where the bottom of the model rudder may be seen clear of the water.

9.3 Scale Effects

Each phenomenon obeys a different law of similitude, so a free-running model run under Froude's Law cannot simultaneously reproduce the full-scale Reynolds number or Weber number. For an actual ship the rudder Reynolds number is roughly of the order of $10^{7}$; the model, run at the Froude-scaled speed, is much smaller:

Wind-tunnel results (Shiba 1960; NACA sections) show three trends for the Reynolds number effect, confirming that low-$Re$ model tests can be conservative (they underpredict the maximum lift of the actual ship):

• (a) The maximum lift coefficient increases with Reynolds number (the stall angle is delayed).
• (b) The lift-curve slope varies little with Reynolds number (or with section shape).
• (c) The drag coefficient decreases with increasing Reynolds number.

Surface roughness has an important effect on maximum lift. At low Reynolds numbers the model flow may be laminar — more susceptible to separation — causing premature stall on the model. The cavitation scale effect works in the opposite sense to stall: since atmospheric and vapor pressures have the same absolute value for model and ship, in coefficient form they are much larger for the model, so cavitation onset is delayed to a higher Froude number on the model than on the ship. This scale effect is not as severe as the stall scale effect.

Air drawing through the free surface depends on the Reynolds number and the Weber number $W$, plus angle of attack and geometry. Shiba (1960) found separation to be a necessary but not sufficient condition for air drawing; its occurrence in the model range also depends on $W$:

Even without aeration, proximity to the free surface degrades rudder performance through wave generation, which is properly scaled under Froude scaling. A ship's tactical diameter is nearly speed-independent below $F_n \approx 0.30$ but increases at higher speeds; as the tactical diameter grows, the inflow angle at the rudder $\beta_R$ decreases, so the angle of attack at the rudder (at constant deflection) increases — a likely cause of severe rudder stall in free-turning tests. For single-propeller, single-rudder ships two scale effects (propeller slip ratio and the thicker model boundary layer) tend to cancel; for multiple-screw single-rudder ships they do not, so model tests may overpredict the tactical diameter by 10–15%.

9.4 Effect of Aspect Ratio

An infinite-aspect-ratio surface has the same flow in all planes perpendicular to the span (strictly two-dimensional). For a finite aspect ratio, cross flow occurs over the root and tip from the high- to the low-pressure side, making the flow three-dimensional; this cross flow increases with decreasing span and decreases the lift for any given angle of attack.

Prandtl's theory (elliptical spanwise load distribution) relates the drag coefficient and the angle of attack (in radians) to aspect ratio for a constant lift coefficient (

Figure 131 shows that the lift-curve slope decreases sharply with decreasing aspect ratio (

9.5 Free-Stream Characteristics of Low-Aspect-Ratio Control Surfaces

The decade 1950–1960 produced extensive free-stream wind-tunnel testing of all-movable, low-aspect-ratio control surfaces, making the old Joessel formulas (1892) unnecessary. The most important work is that of Whicker and Fehlner (1958), which provides $C_L$, $C_D$, the moment coefficient about the quarter chord, the chordwise CP from the leading edge and the spanwise CP from the root, for both ahead and astern directions, at test Reynolds numbers of about 2–3 million (

Whicker and Fehlner also give semi-empirical equations for estimating the low-aspect-ratio, all-movable characteristics. The lift coefficient and its slope (per degree) are:

• $a$ — effective aspect ratio; $\Lambda$ — sweep angle of the quarter-chord line; $\alpha$ — angle of attack in degrees.
• $C_{Dc}$ — crossflow drag coefficient, dependent on both tip shape and taper ratio (Fig. 138).

The drag coefficient adds the minimum section drag to an induced-drag term:

The moment coefficient about the quarter chord, and the CP locations, complete the set:

9.6 Influence of Hull Shape on Effective Aspect Ratio

The groundboard idealization is not fully realized alongside a real hull: the gap between the root and the hull grows as the rudder is laid over (the hull above the rudder is rarely a plane normal to the stock). So the effective aspect ratio may be twice the geometric value at zero deflection but decreases as the rudder is laid over.

Harper and Simitses (1959) tested a surface of geometric aspect ratio 1.0 against a groundboard and against a conical half-body of revolution (body diameter at the quarter-chord $= 0.3\times$ the mean geometric chord; half-angle of closure $12^\circ$). The rudder against the body of revolution has the same lift-curve slope at the origin as against the groundboard, but beyond about $6^\circ$ the curves diverge, and its maximum lift is only about 80% of that against the groundboard. Applying Equation (122b), the effective aspect ratio varies roughly linearly from 2 at $0^\circ$ to 1.7 at $27^\circ$, then drops more rapidly:

9.7 Influence of Fixed Structure and Flapped Control Surfaces

The all-movable rudder has several important variants (

• A doubly all-movable rudder with even a small (unbalanced) flap has a much larger maximum lift than an all-movable rudder with no flap.
• Increasing flap size beyond 20% has little influence on the maximum lift.
• Lift gains come at the cost of large increases in hinge moments and somewhat increased drag; but at a fixed lift coefficient, the 20%-flap doubly all-movable rudder has less drag than a zero-flap rudder above $C_L = 0.6$, and comparable drag below.
• Disadvantages: increased hinge moments, mechanical complexity, and possible maintenance difficulties.

In practice the fixed structure may be a faired sternpost, a horn (full or partial depth,

The semibalanced skeg rudder incorporates balance area without a gap when laid over, but has a horizontal break between the top of the balance area and the lower side of the fixed portion (an unfavorable influence). Goodrich and Molland (1979) tested three such rudders (taper ratios 0.59 / 0.80 / 1.00, NACA 0020 sections) with skeg angles of $\pm 15^\circ$ to represent realistic limits of drift at the rudder in turning. Discontinuities in the lift and drag curves arise from early separation downstream of the skeg (confirmed visually): skeg separation starts at low rudder angles, all-movable-part separation at higher angles.

• Gaps precipitate stalling of the flap (a general aircraft-flap finding): sealing either the high- or the low-pressure side of a gap gives lift only slightly lower than sealing all gaps, but much higher than leaving all gaps open.
• The rudder-plus-skeg combination shows less lift increase with angle of attack than the all-movable rudder, but its stall angle is delayed about 12° and its maximum lift is only about 10% less (for the same lift, the skeg rudder has much higher drag). Reversing the rudder while turning adds the angles, giving a higher maximum lift for the skeg rudder; increasing the taper ratio improved lift at the larger angles.