Lecture 12

PLANING VESSELS – I

Prepared by: Ravi Kota (ravi.kota@ntnu.no)

CONTENT:

• Introduction,

• 2.5D (2D+t) theory for Steady Lift Forces

• SAVITSKY’S FORMULA

• Free-Surface Profile behind Transom

• Dynamic Stability

• Porpoising

• Wave-Induced Motions

• Maneuvering

INTRODUCTION

- By definition,

1.0U

Fng L

= >≈⋅

,

It is more common to use “Breadth-Froude-number”, FnB for planing analysis instead of length.

- For steady flow, hull weight during planing is predominantly supported by hydrodynamic lift rather than buoyancy.

- Planing vessels are usually characterized by Chines – a sharp corner around the bilge of the planing hull cross-section, generally running through the length of the hull. Chines are essential to cause flow-separation which produces the steady lift forces and trim on the hull.

o Double-Chine Hulls – Upper-chine line to maximize waterplane area with increased beam at low speeds, while lower-chine line causes flow-separation at planing speeds.

o Stepped Planing Hulls – A notch in the keel line of the hull profile to cause separation followed by reattachment of the flow of the longitudinal flow ahead of the transom. The intent is to reduce the skin-friction resistance on the hull by creating a partially-ventilated section, just aft of the notch. The notch or the step is located as much aft as possible so that the lift forces generated forward are not adversely affected.

- Trim-tabs and Interceptors are often used to control the trim angle of the hull. Interceptors should be within the boundary layer of the flow.

2.5D (2D + t) THEORY FOR STEADY VERTICAL FORCES IN PLANING

The analysis of steady-vertical forces on a prismatic planing hull is analogous to the analysis of the water-entry problem of a 2-D wedge.

To understand this, consider an Earth-fixed plane through which the planing hull is passing with a steady forward speed, as shown in the figure.

It can be inferred that for an observer on this plane, the effect of the hull passing through would seem like a wedge dropping into calm water, with a steady vertical velocity V = U·τ, as shown in the figure.

Stepped Planing Hull

Flow separation here, to minimize wetted area

SIMPLIFIED LIFT CALCULATIONS FROM 2.5D THEORY

From the theory of water-impact of a 2D wedge,

Vertical force, ( )3 33 ,df a Vdt

= ⋅

where a33 is the 2D, infinite frequency heave added-mass of the wedge,

In the body-fixed system, then, with –

X = U·t, t being the time,

and V = U·τ,

we can write, ( )3 33df U a Udx

τ= ⋅ ⋅

Earth-fixed Plane

τ, trim angle

Mean Water Line

U, steady forward speed

V = U·τ

Water-Impact of a 2D Wedge – view of planing hull from Earth Fixed Plane

t = t0

t = t1

t = t2

Then, the total lift force on the hull is the sectional force, f3, integrated over the length of the hull –

i.e. ( )33

t

bow

X

X

dL U a U dxdx

τ= ⋅ ⋅∫

Noting that X is measured from the bow, for a prismatic hull,

( )33 0d

dxa = , (AFTER THE CHINES ARE WET since the wetted beam of

the section is constant after flow-separation occurs at the chine)

So that, 233( )tL U a Xτ=

From this formulation, the following important conclusions can be made, as essential for lift to be generated on the hull –

1) The hull must have a transom stern, so that a33(Xt) ≠ 0,

2) The vessel must have a positive trim angle,

3) Flow separation from the chines is essential for trim. (Due to flow-separation at the chines, the center of the lift force distribution along the hull length is located forward of lcg, and thus produces a trimming moment.)

4) From the expression for f3 it can be seen that hull sectional pressures (and hence forces) become negative when τ, the local trim angle is negative. This occurs when (τ = dz/dx), the slope of the hull surface is negative, i.e. when the keel line and the buttock lines are convex. Negative pressures may lead to cavitation and hence dynamic instabilities.

5) Also, note that

233 ( ) ( )

2a C beam

πβ ρ= ⋅ , where β is the local deadrise angle,

From this, we can infer two conditions which may adversely affect the lift force on the hull –

Negative local trim angle here, danger of cavitation.

o da33/dx < 0 if d(beam)/dx < 0. This must therefore be avoided.

o From Wagner’s wedge-impact theory, it is known that the added-mass increases with decreasing β. Generally, for planing hulls the deadrise angle decreases from bow to stern. This is beneficial since this ensures that d(a33) /dx > 0. If the hull longitudinal sides are warped (warping is out of plane twisting or bending of a plate), then it should be noted that the warping be in the direction that prevents d(a33) /dx < 0 and hence possibility of cavitation is avoided.

SAVITSKY’S FORMULAE FOR PRISMATIC HULLFORMS

0.60 0

2.51.1 0.5

0 deg 2

2 2

0.0065

(0.012 0.0055 )

10.75

5.21 / 2.39

L L L

WL W

B

p

W B W

C C C

CFn

l

B Fn

β β

λτ λ

λ λ

= − ⋅

= ⋅ + ⋅

= −⋅ +

In these formulae,

C L0 and CLβ are lift coefficients defined as the lift forces normalized by (1/2·ρ·U2·B2),

( )

2k c

W

L L

Bλ += , where Lk and Lc are keel- and chine- wetted lengths

respectively, and

lp is the center of pressure of the lift force distribution on the hull – measured from Transom.

Avoid

The presence of Froude number in Savitsky’s formulation indicates two effects arising from gravity – hydrostatic pressure forces and wave-generation.

We neglect the wave-making effect noting it to be relatively small for increasingly large Froude numbers, hydrostatic forces are assessed. The intent is to understand how hydrostatic pressure forces are manifested in Savitsky’s formulation.

For this purpose, we determine the submerged volume of a planing hull, as shown in the figure.

x1 = location at which the chine intersects the mean water line,

Sectional Area till Chine Wetted station x1, 2 2tan ( )

( )tan( )

xA x

τβ

=

Volume of the wedge till x1, 1 2

31

0

1 tan ( )1 ( )

3 tan( )

x

Vol A x dx xτβ

= =∫

Breadth of the section at x1, 12 tan( )

tan( )

xB

τβ

=

Volume of the hull from x1 to transom below chines, 212 ( )(0.25)( tan( ))kVol L x B β= −

2x·tanτ tanβ

x tan(τ) β

x

vol. 2 vol. 3 τ

x = x1 vol. 1

For 0 < x < x1

Volume between a plane through chines and MWL, 213 (0.5)( ) tan( )kVol L x B τ= −

Total Hydrostatic force, thus, ( 1 2 3)HSF g Vol Vol Volρ= + +

Or, in terms of a non-dimensional parameter, 2 2 2 3

2:

0.5HS

LHSB

F VolC

U B Fn Bρ= =

Then, to account for suction pressure at transom occurring from flow-separation, we introduce a correction. This correction is done by reducing the keel-wetted length, Lk, by 0.5B.

Doing so gives a good correlation of the Hydrostatic + Suction Pressure (CLHS) to Savitsky’s CLβ, as shown in Figure 9.13 (pp. 355).

This approach is very approximate but gives an idea of the relative magnitude of the hydrostatic and suction forces, against hydrodynamic forces, per Savitsky’s formulae.

EXAMPLE OF PREDICTING RUNNING ATTITUDE & RESISTANCE OF A PLANING HULL – USING SAVITSKY’S FORMULA

Given,

M = 27,000 kg

lcg = 8.84m

B = 4.27m

β = 10◦

U = 20.58 m/s

To determine the trim angle and transom draft (i.e. running attitude) and resistance thus being able to estimate horsepower (EHP) requirements, in calm water.

STEP 1: Determine λw.

At equilibrium,

lp = lcg,

2 2

10.75

5.21 / 2.39W B W

lcg

B Fnλ λ= −

⋅ +

With given values of lcg, and FnB, we get λw = 3.43.

STEP 2: Determine τ.

By definition, CLβ = FLβ / (1/2·ρ·U2·B2),

Since FLβ = Mg, at equilibrium, we get

2 2

0.067(0.5)L

MgC

U Bβ ρ= =

Then, using Savitsky’s formula for CLβ and CL0, and with the known value of β in degrees, we get τ, thus,

0.60 00.0065L L LC C Cβ β= − ⋅

2.5

1.1 0.50 deg 2

(0.012 0.0055 )WL W

B

CFn

λτ λ= ⋅ + ⋅

τ = 2.21 deg.

FLβ N

Rv T

lcg

τ

Mg

STEP 3: To determine wetted length, Xs = Lk – Lc.

From Wagner’s solution for a wedge impact on calm water, we have the beam of the local wetted section as

( )2 tan

Vtc t

πβ

=

In body-fixed system, if ts is the time at which the chine is wet, we can write,

Xs = U ts and so, c(ts) = B/2,

2 2 tan

tan

s

s

XB U

U

B X

π τβ

π τβ

⋅∴ =

⋅⇒ =

So, with λw = (Lk – Lc )/2B and Xs = Lk – Lc known, we get

Lc = 11.5m, and

Lk = 17.7m.

Thus, transom draft, Ds = Lk sin(τ) = 0.68m

STEP 4: To obtain resistance, RT and EHP.

Wetted Area, S = 63.3 m2 (includes spray-root wetted area. See discussion on pp. 361 for this)

Viscous Resistance, ( )21

2V F FR U S C Cρ= + Δ ,

ITTC Formula,2

10

0.075

(log 2)FCRn

=−

Using an increase for Average Hull Roughness (AHR) = 150μm, based on the following empirical formula (see pp. 31, Eqn 2.86),

( )1/ 33 1/ 310 44 / 10 0.125FC AHR L Rn−⎡ ⎤Δ = − ⋅ +⎣ ⎦

Gives, RV = 29,603 N. Note that to calculate Rn and (AHR/L), L = Lk is used.

To this, we add the resistance due to Lift-induced pressure drag,

RP = (FLβ )(τ) to get a total hull resistance, RT = 39,285N

This will imply that the power required (EHP) = RT U = 820 kW = 1,115 hp

Stepped Planing Hull

How can we predict the location of longitudinal flow re-attachment aft of the step?

Or, more generally, what is the length of the hollow behind the transom of a planing hull?

A key parameter to analyze the condition of dry-transom or flow-separation at step is the

transom-draft Froude number, D

UFn

g D=

⋅, where D is the draft at the transom relative

to calm-water line, and thus including the rise and trim of the hull.

For flow-separation, FnD > 2.5.

For stepped hulls, we use FnDs i.e draft at the step where flow-separation is of interest, rather than at the transom.

We determine the free-surface profile in the centerplane of the hull, just aft of the transom (or step).

Assumptions – 2D Potential Flow in the X-Z plane as shown.

Us is the steady flow velocity at the transom.

Velocity Potential at the transom is given as,

cos( )nUs X Ar nθΦ = ⋅ +

- The second term is due to flow-separation at the stern. - In reference to the following figure, we assume r is small, - Potential satisfies 2D-Laplace equation,

Us

Z

X

Ds

τ

Mean Water Line

U

- Note that sin(-) term does not arise due to body-boundary condition at θ = 0; - X = -r·cos(θ),

Then, from the Dynamic Free Surface Boundary condition – i.e. satisfying Steady Bernoulli’s equation on the free-surface profile, we have,

2 21 1

2 2ap gz V p Uρ ρ ρ+ + = +

Noting that z is very small (close to the MWL), and with u and w being the velocities in X and Z directions due to the Arncos(nθ) term, we can write,

( ) ( )( )2 21 1

2 2g D Z Us u w Uρ ρ ρ− − + + + =

Where Z is the free-surface profile coordinate in the axes system shown. To the lowest order (zeroth order in u and w), then, we have

22Us gD U= + To the next order, we would get, on Z = 0,

0Us uρ ⋅ ⋅ = i.e., substituting for u, close to the stern, (r is small so that θ = π on the free-surface),

( )( )| cos | 0nu Ar nr θ πθ π θ

==∂= =∂

( )cos 0

1 3 5, , , .

2 2 2

n

n etc

π⇒ =

⇒ =

But here, n = ½ is not permitted. The reason is that the radial velocity due to the perturbation potential, Arncos(nθ), which is ( )1 cosnnAr nθ− ,will yield a singularity for

Centerline Free Surface Profile

θ

r

w

Us

Z

X

HULL

0r → , in this case. This singularity cannot be permitted since we are enforcing a smooth

tangential flow at the transom by Kutta condition. Therefore, the lowest order of 32n = .

Now, applying the Kinematic Free-Surface condition, we have, at Z = 0,

( ) 32

121 3cos |

2n

nw Ar n Arr

θ π

θθ ==

⎡ ⎤⎢ ⎥⎣ ⎦

∂= =∂

Then, the free-surface profile can be approximated as

dZ w

dX Us=

3

2AZ XUs

⇒ =

Comparing this against Savistky’s empirical formula for free-surface profile (Eq. 9.19, pp.355), shows that the least-squares fit of the latter agrees well with the simplified analytical result above.

2D planing surface Vs 2D Foil in infinite Fluid The assumption here is that the Froude number (FnB) is very high so that gravity effects can be neglected in the hydrodynamic pressures on the planing surface. Also, the situation that is being considered is when the vessel is supported by lift forces on a small area at the stern. This condition can justify assuming the planing surface to be of high-aspect ratio. For high-aspect ratio surfaces, 2D foil theory can be applied.

For linearization, project the planing surface, at a trim angle, on a horizontal line segment of length equal to keel wetted length, and transfer body boundary conditions to the line.

High frequency or high-speed assumption leads to the free-surface approximation of φ = 0.

Now take the mirror-image of the flow about the mean-water line. By the free-surface condition of φ = 0 imposed, the flow in the mirrored domain would be ANTI-SYMMETRIC.

The combined Body + Image flow would now be equivalent to the 2D flow over a flat-plate in an Infinite fluid Domain, at an angle of attack of τ.

The fact that the flow leaves smoothly at the transom, will translate to the Kutta condition at the trailing edge of the flat-plate foil.

Lift on a 2D flat-plate in infinite fluid, at an angle of attack is given as – 2LC π α= ⋅

In the case of a planing surface, α τ= , and the fluid pressures are only from the semi-infinite domain.

2

&

1

2

L

L

C

L U S C

π τ

ρ

∴ = ⋅

= ⋅

From 2D Foil theory, the pressure drag on a flat-plate at an angle of attack is Zero. This occurs because of leading edge suction pressures which exactly cancel out the anticipated Drag component of the normal pressure force. In the case of a flat-plate planing on a free-surface with a trim angle, the flow is not the same. A jet flow occurs near the Nose.

This jet flow in the opposite to the direction of incoming stream, acts as a pressure drag on the planing surface. Thus, for a 2D flat plate in Infinite Fluid, D = 0 But, for a 2D planing surface, D ≠ 0.

--- ooo ---

Smooth Flow separation here

Jet Flow here

U