MN — Ship Resistance and Flow · Ch. 8 — Experimental Resistance Prediction and Flow Measurement

1. Overview — predicting resistance from model tests (§8 intro)

Towing ship models to measure resistance has a long history (see §1.2). Since William Froude's work in the middle of the 19th century the basic testing technique has not changed — a model is towed and its resistance measured — but the scaling procedures (model data to full scale) and the measuring equipment have improved a great deal. Most leading towing tanks have run for many years and built up a wealth of knowledge on testing and on interpreting the results.

Why the tank still rules. Even though the numerical methods of §9 are now accurate enough to replace some systematic model tests in design optimisation, the towing tank still gives the most accurate predictions of resistance and power for the majority of ships.

**What this chapter covers.**
§	Topic	Core idea
8.1	Experimental facilities	Towing tanks, water tunnels and circulating water channels; their sizes and instrumentation
8.2	Model resistance tests	How the resistance test is run; model size compromise; turbulence stimulation
8.3	Prediction of effective power	The two extrapolation methods — Froude and ITTC-78 — and the form factor
8.4	Model flow measurements	Pitot/LDV/PIV velocity fields, wave gauges, tuft, paint, alignment and wave-pattern tests

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 8 — Experimental Resistance Prediction and Flow Measurement.

Edital: Anexo 2-A, Área I, item 5 (previsão experimental da resistência ao avanço: ensaios em tanque de reboque, extrapolação modelo→navio pelos métodos de Froude e ITTC-78, fator de forma, medição de escoamento) · Anexo 2-B, Área I, item 3 (Larsson & Raven), Chapter 8.

2. Experimental facilities (§8.1)

In towing-tank testing a model is towed at constant speed in still water. Commercial tanks use mechanically or electrically driven towing carriages running on rails along the model basin. The models are 4 to 10 m or more in length, and the tanks perform resistance tests, propulsion tests and many other experiments. The carriage carries computer systems for data processing — allowing direct inspection and a first analysis on the spot — and platforms from which the test and the flow features are observed; photographs and video are usually taken for later study.

Fig. 8-1 Fig. 8.1 — Model basin and towing carriage

**Typical tank dimensions (Fig. 8.1).**
Tank type	Length	Width	Depth	Model length
Large commercial	~250 m	~10 m	~5 m	4–10+ m
Shallow-water	—	> 20 m	variable, up to ~3 m	—
High-speed craft	extra long	~4 m (narrow)	~4 m	—
Educational / research	small	—	—	1–3 m

High-speed craft need extra-long tanks; these are often relatively narrow (~4 m × 4 m) and the carriage must reach speeds well above the ~10 m/sec maximum used in ordinary tanks. Some tanks have wave makers for seakeeping tests. Besides the commercial tanks there are many small tanks at educational and research establishments using 1–3 m models.

Water tunnels and circulating water channels. Water tunnels are not used for measuring resistance but are very good for measuring and visualising the flow around the hull. A special case is the circulating water channel, whose measuring section has a free water surface, so resistance and wave tests can be done there; usually only small models fit, though large channels exist (e.g. the Technical University of Berlin's, 11 m long and 5 m wide).

In ordinary resistance tests the measurement of carriage speed and resistance force is often combined with sinkage and trim measurements. For flow work the tanks keep pitot tubes for wake measurements and underwater observation for tuft tests; some have laser-Doppler velocimetry (LDV), and particle-image velocimetry (PIV) is in increasing use. These extra experiments are described in §8.4.

3. Model resistance tests (§8.2)

3.1 General (§8.2.1)

The resistance test is usually the first test done with a model. It serves three purposes: to predict the full-scale resistance; to feed the prediction of the full-scale required power; and — by comparing with data for similar vessels or with empirical/statistical methods — to measure the quality of the design from a resistance point of view.

The model is attached to the carriage, ballasted and trimmed to the required displacement and waterline, and connected to the resistance dynamometer. (A clear statement of whether moulded or total displacement is meant should be included.) The model is free to take up any sinkage or trim dictated by the water forces, but yawing is prevented by guides. On a run the carriage is driven at the desired constant speed and speed and resistance are recorded; the so-called towing force, the sinkage at the forward and aft perpendiculars, and photo/video of the waves near the model are usually taken too.

The "jump" in Reynolds number. Going from model to ship the Reynolds number jump is very large. For a 125-m ship and a 5-m model running at 25 and 5 knots, $Rn$ (proportional to $V\!\cdot\!L$) is in the ratio 1 : 125. (This is not always realised on experiment plots, which use a base of $\log Rn$ and so greatly reduce the apparent extrapolation.) The model resistance must therefore be measured extremely accurately: constant carriage speed and an accurate dynamometer are basic, and the models must be made to close tolerances, correctly finished, ballasted and trimmed.

3.2 Model size (§8.2.2)

The choice of model size is a compromise. A larger model can be made more accurately and gives larger forces to measure — both improving accuracy — but it is more expensive to build and handle and needs larger facilities and instruments.

**Limits that constrain the model size.**
Constraint	Rule of thumb	Why
Wall / bottom interference	Model length not much more than the water depth or half the basin width	Avoid interference with the wave resistance
Blockage	Midship cross-section < about 1/200 of the basin's	Avoid a return flow around the model
Critical depth Froude number	Keep $V/\sqrt{gh} < \approx 0.7$ of the critical value (=1)	Above ~0.7 the resistance differs from deep water
Propeller / laminar flow	Account for propeller size; avoid laminar regions	Self-propulsion follows; laminar flow falsifies resistance

There is still no full agreement on assessing the interference effect; where wave making is small, larger models are used and a blockage correction is applied (Comstock & Hancock, 1942; Emerson, 1959; Hughes, 1957, 1961; Kim, 1962; Telfer, 1953). If self-propelled experiments are to follow, the propeller size also constrains the scale. A final consideration is the need to avoid significant laminar flow on the model — the subject of the next subsection.

3.3 Turbulence stimulation (§8.2.3)

The flow over the model must be made fully turbulent, because the flow around the full-scale ship is turbulent. Laminar flow can usually be spotted from the shape of the resistance curve (Fig. 8.2):

Fig. 8-2 Fig. 8.2 — Effects of laminar flow at low speed

At low Froude number, where wave resistance is vanishingly small, the $C_T$ curve should run roughly parallel to the skin-friction curve $C_F$ — the path ABC. A curve that falls away or even goes horizontal (ABD or ABE) is at once suspect of partial laminar flow, which gives too small a resistance.

Tripping the flow. The practical cure is to deliberately "trip" the laminar flow with roughness near the bow. Trip wires about 1 mm diameter placed around the hull at a station 5% of the length from the forward perpendicular are now standard. For appendages (and the hull) sand strips or studs are also used; a typical stud is ~2.5 mm high and ~3 mm in diameter, set in a row parallel to the leading edge.

The stimulator adds its own parasitic drag. Placed too close to the stem the laminar flow may re-establish if the pressure gradient is favourable; placed in the usual position (5% $L_{pp}$ aft of the stem) it leaves any laminar flow undisturbed over the first part of the length, where the resistance is then less than the desired turbulent value. It is usual to assume that this defect balances the parasitic drag of the wire or studs.

4. Prediction of effective power (§8.3)

The effective power $P_E$ is the power needed to drive the ship at a given speed without propulsive losses — the resistance times the speed, $R_T\,V$. To get the full-scale (index $S$) resistance from the model (index $M$) test, essentially only two methods have been used: Froude's method and the method recommended by the ITTC in 1978. The latter is used by most tanks today, but Froude's is still met. Both rest on the principles of §3.3:

All force coefficients are obtained by dividing the forces by $\tfrac{1}{2}\rho V^2 S$, where $V$ is the speed and $S$ the wetted surface.

4.1 Froude's method (§8.3.1)

In Froude's method the friction is supposed equal to that of an equivalent flat plate (same wetted surface, same length, towed at the same speed — so the same $Rn$), and the residuary resistance is everything else. The residuary thus contains not only the wave resistance but also the form effect on friction and on pressure. In modern terms:

Fig. 8-3 Fig. 8.3 — Graphical view of Froude's method

$$ C_T(Rn, Fn) = C_{F0}(Rn) + C_R(Fn) \tag{8.1} $$ $C_{F0}$ is the plate friction coefficient and $C_R$ the residuary coefficient. In practice $C_{F0}$ is replaced by $C_F$ from the ITTC-57 line, eq. (6.38). Because that line contains ~12% form effect (which should belong to $C_R$), the procedure is not exactly as Froude proposed.

The steps to extrapolate the measured model resistance $R_{TM}$ to the full-scale $R_{TS}$:

Run the model at the same Froude number as the ship ($V_M = V_S\sqrt{L_M/L_S}$). Measure the total coefficient $C_{TM}$.
Compute the model friction by the ITTC-57 formula:
$$ C_{FM} = \frac{0.075}{\left({}^{10}\!\log Rn_M - 2\right)^2} \tag{8.2} $$where $Rn_M$ is the model Reynolds number (or use another line, §6.3.4).
Compute the model residuary: $\;C_{RM} = C_{TM} - C_{FM}\;$ (8.3).
Since $Fn$ is equal, the residuary is the same at both scales: $\;C_{RS} = C_{RM}\;$ (8.4).
Compute the ship friction:
$$ C_{FS} = \frac{0.075}{\left({}^{10}\!\log Rn_S - 2\right)^2} \tag{8.5} $$with $Rn_S$ for the ship.
Compute the ship total coefficient, adding a roughness allowance:
$$ C_{TS} = C_{FS} + C_{RS} + \Delta C_F \tag{8.6} $$where the roughness allowance is taken constant, $\Delta C_F = 0.0004$.
The full-scale resistance, then the effective power:
$$ R_S = C_{TS}\cdot\tfrac{1}{2}\rho_S V_S^2 S_S \quad(\mathrm{N}) \tag{8.7} \qquad P_E = R_S\cdot V_S \tag{8.8} $$

A graphical representation is given in Fig. 8.3.

4.2 The ITTC-78 method (§8.3.2)

In the ITTC-78 method — the one most used today — the decomposition is into a viscous resistance (including the form effect on friction and on pressure) and a wave resistance. Through the form factor a better division is obtained and the components scale better. The assumption is:

Fig. 8-4 Fig. 8.4 — Graphical view of the ITTC-78 method

$$ C_T(Rn, Fn) = (1+k)\,C_{F0}(Rn) + C_W(Fn) \tag{8.9} $$ $C_{F0}$ is the plate friction, $(1+k)$ the form factor, and $C_W$ the wave coefficient. Again $C_{F0}$ is replaced by $C_F$ from the ITTC-57 line (6.38).

The steps:

Run the model at the same $Fn$; measure $C_{TM}$.
Compute the model friction (ITTC-57):
$$ C_{FM} = \frac{0.075}{\left({}^{10}\!\log Rn_M - 2\right)^2} \tag{8.10} $$
Determine the form factor $k$ (usually by Prohaska's method, §8.3.3 / §4.3 below).
Compute the model wave coefficient: $\;C_{WM} = C_{TM} - (1+k)\,C_{FM}\;$ (8.11).
Assume $C_{WS} = C_{WM}$. Note this wave resistance is smaller than the residuary of Froude's method.
Compute the ship friction:
$$ C_{FS} = \frac{0.075}{\left({}^{10}\!\log Rn_S - 2\right)^2} \tag{8.12} $$
Compute the roughness allowance (Bowden):
$$ \Delta C_F = \left(105\left(\frac{k_{MAA}}{L}\right)^{1/3} - 0.64\right)\cdot 10^{-3} \tag{8.13} $$where $k_{MAA}$ is the roughness in microns ($10^{-6}$ m) by the MAA method (§6.8.2); ITTC recommends a typical $k_{MAA} = 150$ microns.
Determine the air-resistance coefficient:
$$ C_{AA} = 0.001\cdot\frac{A_T}{S} \tag{8.14} $$with $A_T$ the frontal area above the water (cf. eq. 7.22).
Compute the ship total coefficient:
$$ C_{TS} = (1+k)\,C_{FS} + C_{WS} + \Delta C_F + C_{AA} \tag{8.15} $$
The total resistance and effective power:
$$ R_S = C_{TS}\cdot\tfrac{1}{2}\rho_S V_S^2 S_S \tag{8.16} \qquad P_E = R_S\cdot V_S \tag{8.17} $$

A graphical representation is given in Fig. 8.4. Considerably more steps are required for extrapolating self-propulsion tests — described in the Propulsion volume of the PNA series.

Froude vs ITTC-78 in one line. Both test the model at the ship's Froude number. Froude keeps a single residuary coefficient equal across scales and scales only the bare plate friction. ITTC-78 introduces the form factor, keeps the smaller wave coefficient equal, and scales the whole viscous part $(1+k)C_F$ — a more physical split.

4.3 Determination of the form factor (§8.3.3)

The form factor $(1+k)$ was defined in eq. (6.39): the ratio of the (smooth) viscous resistance to the resistance of the equivalent flat plate. It absorbs the form effect on the viscous resistance — increased flow speed and pressure gradients from the body's thickness, the boundary-layer and wake displacement effect on the stern pressure, and possible separation. Given for 2-D flows, it holds also for 3-D. Strictly the real plate friction $C_{F0}$ should appear, but in practice the ITTC-57 line is used; as it already carries ~12% form effect the approach is not quite logical, yet it gives correct results provided the form factor is used with the friction line it is intended for. Three ways to find $k$:

**Three ways to determine the form factor.**
Method	How	Note
Empirical formula (Watanabe)	Eq. (8.18) from the main hull parameters	Usable only with the ITTC-57 line
Low-speed test	Run at $Fn \approx 0.15$ where $C_W \approx 0$; read $k$ from (8.9) (Fig. 8.5)	Laminar-flow problems and tiny forces hurt accuracy
Prohaska's method (1966)	Plot $C_T/C_F$ vs $Fn^4/C_F$; the intercept is $(1+k)$ (Fig. 8.6)	Most widely used; works to somewhat higher speed

The most popular empirical formula is attributed to Watanabe:

$$ k = -0.095 + 25.6\cdot\frac{C_B}{\left(\dfrac{L}{B}\right)^2\sqrt{\dfrac{B}{T}}} \tag{8.18} $$ $C_B$ is the block coefficient, $L$ the length between perpendiculars, $B$ the beam, $T$ the draft. Usable only with the ITTC-57 line.

Fig. 8-5 Fig. 8.5 — Form factor at low speed

The low-speed determination runs the model at, say, $Fn \approx 0.15$, where the wave resistance is negligible, so $k$ follows from (8.9) with $C_W = 0$ and $C_{F0}$ from the ITTC-57 line (Fig. 8.5). Unfortunately laminar-flow problems may occur and the forces at such low speeds are very small, making accurate measurement difficult.

The most widely used is Prohaska's method (1966), which assumes the wave coefficient is proportional to the fourth power of the Froude number (supported by Wigley's calculations, §5.4.5). Then:

$$ C_T = (1+k)\,C_F + k_1\,Fn^4 \tag{8.19} $$ Dividing by $C_F$: $$ \frac{C_T}{C_F} = (1+k) + k_1\,\frac{Fn^4}{C_F} \tag{8.20} $$ a straight line (Fig. 8.6). If the wave-resistance assumption holds, the points fall on the line with $(1+k)$ as the intercept on the vertical axis. Normally this is true in the lower Froude-number range; the line is fitted to those points and the form factor read at the axis.

4.4 Discussion (§8.3.4)

ITTC-78 is generally preferred and considered more physically correct, but the form factor is not always easy to find. Prohaska's method assumes that, in the low-speed regime considered, there are no clear wave-interference effects; a bulbous bow near the surface can make local waves already at low speed (high local $Fn$), giving non-smooth resistance, and an immersed transom dry only at higher speeds can cause inaccuracies. If the experimental determination fails, empirical formulas help, but accuracy is limited; a less accurate form factor does not immediately cause gross errors.

Limits of the form-factor idea. Assuming the viscous resistance is proportional to the equivalent flat plate's friction holds well for slender ships without separation. For full forms with a model-scale bubble separation that may be absent at full scale, the proportionality is doubtful. For vessels with sharp edges the separation is $Rn$-independent, so a direct scaling of that contribution may be better — as discussed for appendage resistance in §7.2.1 (Fig. 7.18): when the appendage resistance is large it should be scaled separately.

A further approximation is the basic assumption that wave making and viscous flow are independent ($Fn$ vs $Rn$). In reality the stern wave is reduced by viscous effects, less so at full scale, so a small increase of $C_W$ from model to ship is expected; and the viscous resistance can itself be $Fn$-dependent (the wavy surface and altered pressure distribution affect the boundary layer and wetted surface, and can even cause separation). Hence a form factor from a low-speed test may not suit higher speeds (computational study: Raven et al., 2008). Even so, these methods remain the best established; the correlation allowances, fixed by regression of trial data, hide a variety of corrections beyond roughness, so experience still matters for an accurate full-scale prediction.

"Fourth power" and "sixth order" are consistent. Prohaska takes the wave coefficient proportional to $Fn^4$ (eqs. 8.19–8.20). Since the wave force is $R_W = C_W\cdot\tfrac{1}{2}\rho V^2 S$, it scales as $Fn^4\cdot V^2 \propto V^6$ — a sixth-order function of the speed. The two statements in the source describe the same assumption from different angles (coefficient vs force).

5. Model flow measurements (§8.4)

Besides resistance tests, many other tests are run in towing tanks to determine the properties of the flow around the hull. First the techniques for measuring velocities and wave elevations, then the usual tests relevant to resistance and flow.

5.1 Measurement techniques (§8.4.1)

Fig. 8-7 Fig. 8.7 — Rake with two five-hole pitot tubes

**Devices for velocity fields (Fig. 8.7).**
Device	Principle	Limits
Five-hole pitot tube	Pressure differences between five holes in a spherical head give flow magnitude and direction (from calibration)	Large incidence angles, a nearby wall, strong gradients or turbulence reduce accuracy
LDV (Laser-Doppler Velocimetry)	Two laser beams cross in a small "measurement volume"; particles scatter light with a Doppler shift, the interference frequency gives the local speed and direction	Needs particles ("seeding"); from many particles, average speed and turbulence follow
PIV (Particle-Image Velocimetry)	A laser sheet lights particles; two photos a few ms apart show the particle shift, giving the in-plane velocity distribution; stereo gives the normal component	Pattern recognition matches particle groups; whole plane at once

Wave-elevation devices. The most used are capacitance or conductance probes — an electric signal depends on the immersion, read at a scanning frequency, giving wave elevation against time. They come as twin wires on a rod or as flexible tape on the model. With a substantial flow speed past the probe they are less suited; then servo-driven "finger probes" that touch and follow the surface are used (alternatives: stereo photography for a height map, or pictures of a vertical laser sheet illuminating the wave surface).

5.2 Wake field / flow field measurement (§8.4.2)

A conventional wake field measurement determines the velocity field at the propeller location, for a model without propeller — the "nominal" wake field (to distinguish it from the field with the working propeller). It is usually measured with a pitot-tube rake rotated around the shaft centre and set at various angles and radii to scan the propeller plane; all three components (axial, radial, tangential) are obtained, often over several runs. The nominal wake field gives first indications of the propeller inflow and so of the risk of cavitation.

The wake field can also be measured by LDV or PIV. LDV gives precise flow and turbulence around the hull and the inflow to a working propeller, but the whole system must ride on the carriage and many runs are needed, so it never became popular in tanks (it is far easier in circulating water channels); a special LDV use is the boundary layer and wake of full-scale ships, with the gear inside the hull working through Plexiglas windows. PIV measures a whole plane at once (fast), though several measurements are needed for a proper turbulent average, and the optical access can be a problem; PIV may become an alternative to the pitot-tube wake measurement for routine use (Fig. 8.8).

5.3 Tuft, paint and alignment tests (§8.4.3–8.4.5)

Fig. 8-9 Fig. 8.9 — Tuft test (afterbody)

Fig. 8-10 Fig. 8.10 — Paint test (forebody)

Besides quantitative flow measurements, more qualitative information is often collected to help understand and improve the model's performance:

Qualitative flow tests

Tuft test (§8.4.3). Short pieces of yarn on pins ~50 mm high align with the flow; an underwater camera shows the local flow direction and skewing within the first 50 mm, plus unsteadiness and possible separation — a rather qualitative result (Fig. 8.9).
Paint test (§8.4.4). More detailed and quantitative: special paint in girthwise strips is smeared by the flow into traces along the surface shear stress (skin friction) — the limiting streamlines (§6.6.1) — revealing skin-friction direction and separation regions. In the stern, low skin friction makes good traces hard (Fig. 8.10).
Appendage alignment test (§8.4.5). A paint test gives the shear-stress direction on the surface, but aligning shaft support struts needs the flow direction at a distance from the hull; for that, pitot tubes, small vanes on a rod at the strut position, or painted plates giving traces of the local flow are used.

5.4 Wave pattern measurement (§8.4.6)

Wave pattern measurements are most easily done with a wave gauge at a fixed position in the tank. The probes are read at a set frequency, giving wave elevation against time during the model's passage at constant speed; using the model speed, scanning frequency and a start signal this is converted to elevation against position — a longitudinal cut through the steady wave pattern. The useful part runs from the arrival of the model's wave disturbance until waves reflected from the tank wall reach the probes.

Such longitudinal wave cuts are used to compare model variations, to compare with computations, to generate far-field wave (wash) predictions (§5.11), or for wave-pattern analysis — deducing the wave spectrum (§5.4.6) and from it the wave resistance directly. In the 1960s much work went into wave-pattern analysis (Eggers, Sharma & Ward, 1967) hoping to use the resulting wave resistance to improve extrapolation, but this is hardly used in practice today.

Reach and alternatives. These measurements give cuts at a minimum distance of half the model beam; to measure elevations in the path of the vessel (e.g. aft of the stern) other techniques are needed — finger probes on a subcarriage, photographs of a laser-lit surface, or stereo photography from the ceiling. Simpler but less meaningful is the wave profile along the hull, usually just photographed against a grid of waterlines and sections painted on the model.