1. Overview — why a model can stand in for the ship
Most experiments in ship hydrodynamics are run with scale models: small replicas of the ship are towed in a water basin while forces and motions are measured. The interesting case here is the towing-tank experiment, where the model is towed by a carriage and the force — and sometimes the flow around the hull — is recorded at several speeds. The model is obviously made geometrically similar to the ship (a geosim), but two questions are far less obvious: at what speed must the model be run, and how do we scale up the measured forces and velocities to full scale? Chapter 3 answers both by deriving the similarity laws.
| Symbol | Quantity | Note |
|---|---|---|
| $L$ | reference length | usually the length between perpendiculars $L_{pp}$ |
| $U_\infty$ (= $V$) | reference velocity | normally the ship speed $V$ |
| $u,\ v,\ w$ | velocity components | made dimensionless by $U_\infty$ |
| $p_{hd}$ | hydrodynamic pressure | used to define the dimensionless pressure $\overline{p}$ |
| $\rho$ | density | assumed constant |
| $\nu$ | kinematic viscosity | $\nu=\mu/\rho$; enters $Rn$ |
| $g$ | gravity | enters $Fn$ |
| $\gamma$ | surface tension | enters $Wn$ |
| $p_a,\ p_v$ | atmospheric / vapour pressure | $p_a$ enters $En$ and $\sigma$; $p_v$ enters only $\sigma$ |
| $\zeta,\ r_1,\ r_2$ | wave height / principal radii of curvature | free-surface condition |
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 3 — Similarity.
Edital: Anexo 2-A, Área I, item 5 (Resistências do Navio e métodos para sua obtenção — leis de semelhança, ensaio de modelo, número de Froude/Reynolds) · Anexo 2-B, Área I, item 3 (Larsson & Raven), Chapter 3.
2. Types of similarity (§3.1)
Three kinds of similarity must be distinguished. They build on one another: geometric similarity is the precondition, and matching the right dimensionless numbers then delivers the kinematic and dynamic similarity.
- Geometric similarity — ship and model have the same shape. Necessary in principle, but not down to the smallest detail: it is virtually impossible to scale the surface roughness exactly. As Section 6 will show, roughness has no effect if it is small enough. The requirement is easily met for the model; for the full-scale ship roughness does matter, but it is handled empirically (Section 8).
- Kinematic similarity — all velocities in the flow (including their components!) are scaled by the same factor. As a result the streamlines around the hull are geometrically similar at model and full scale.
- Dynamic similarity — all forces of the flow (including components!) are scaled by the same factor. The force vectors therefore have the same direction at both scales.
3. Proof of similarity (§3.2)
To derive the similarity laws, the quantities in the governing equations and their boundary conditions are made dimensionless. The idea is to find under which conditions the equations become independent of scale. If that can be achieved, the solution in nondimensional variables is unique — meaning both kinematic and dynamic similarity hold between any scales. Dimensional results are then recovered simply by converting back.
3.1 The nondimensional quantities
The following dimensionless variables are introduced (a bar denotes a dimensionless quantity):
$$ \overline{x}=\frac{x}{L},\ \overline{y}=\frac{y}{L},\ \overline{z}=\frac{z}{L},\ \overline{\zeta}=\frac{\zeta}{L},\ \overline{r_1}=\frac{r_1}{L},\ \overline{r_2}=\frac{r_2}{L} $$
$$ \overline{u}=\frac{u}{U_\infty},\ \overline{v}=\frac{v}{U_\infty},\ \overline{w}=\frac{w}{U_\infty},\ \overline{p}=\frac{p_{hd}}{\rho U_\infty^2},\ \overline{t}=\frac{t\,U_\infty}{L} $$
$L$ — a reference length, usually the length between perpendiculars $L_{pp}$.
$U_\infty$ — a reference velocity, normally the ship speed $V$.
Lengths are divided by $L$, velocities by $U_\infty$, time by $L/U_\infty$. Crucially, the pressure is made dimensionless with the hydrodynamic pressure $p_{hd}$ (not the total $p$), using the dynamic-head scale $\rho U_\infty^2$. This is the link back to §2.5: dropping the hydrostatic part removes the stand-alone gravity term from the field equations.
3.2 The dimensionless governing equations
Inserting these quantities into the $x$-component of the Navier–Stokes equations and dividing through by $U_\infty^2/L$ yields the dimensionless momentum equation. The same operation in $y$ and $z$ gives the other two:
$$ \frac{\partial \overline{u}}{\partial \overline{t}} + \overline{u}\frac{\partial \overline{u}}{\partial \overline{x}} + \overline{v}\frac{\partial \overline{u}}{\partial \overline{y}} + \overline{w}\frac{\partial \overline{u}}{\partial \overline{z}} = -\frac{\partial \overline{p}}{\partial \overline{x}} + \frac{1}{Rn}\left(\frac{\partial^2 \overline{u}}{\partial \overline{x}^2} + \frac{\partial^2 \overline{u}}{\partial \overline{y}^2} + \frac{\partial^2 \overline{u}}{\partial \overline{z}^2}\right) \tag{3.1a} $$
$$ \frac{\partial \overline{v}}{\partial \overline{t}} + \overline{u}\frac{\partial \overline{v}}{\partial \overline{x}} + \overline{v}\frac{\partial \overline{v}}{\partial \overline{y}} + \overline{w}\frac{\partial \overline{v}}{\partial \overline{z}} = -\frac{\partial \overline{p}}{\partial \overline{y}} + \frac{1}{Rn}\left(\frac{\partial^2 \overline{v}}{\partial \overline{x}^2} + \frac{\partial^2 \overline{v}}{\partial \overline{y}^2} + \frac{\partial^2 \overline{v}}{\partial \overline{z}^2}\right) \tag{3.1b} $$
$$ \frac{\partial \overline{w}}{\partial \overline{t}} + \overline{u}\frac{\partial \overline{w}}{\partial \overline{x}} + \overline{v}\frac{\partial \overline{w}}{\partial \overline{y}} + \overline{w}\frac{\partial \overline{w}}{\partial \overline{z}} = -\frac{\partial \overline{p}}{\partial \overline{z}} + \frac{1}{Rn}\left(\frac{\partial^2 \overline{w}}{\partial \overline{x}^2} + \frac{\partial^2 \overline{w}}{\partial \overline{y}^2} + \frac{\partial^2 \overline{w}}{\partial \overline{z}^2}\right) \tag{3.1c} $$
The continuity equation, made dimensionless and divided by $U_\infty/L$, keeps the same clean form:
$$ \frac{\partial \overline{u}}{\partial \overline{x}} + \frac{\partial \overline{v}}{\partial \overline{y}} + \frac{\partial \overline{w}}{\partial \overline{z}} = 0 \tag{3.2} $$
3.3 The dimensionless boundary conditions
The boundary conditions of §2.4 are made dimensionless in the same way.
Hull surface (no-slip, from eq. 2.14):
$$ \overline{u}=\overline{v}=\overline{w}=0 \tag{3.3} $$
Earth-fixed boundaries (from eq. 2.15):
$$ \overline{u}=1, \qquad \overline{v}=\overline{w}=0 \tag{3.4} $$
Free surface — kinematic condition (from eq. 2.20): the dimensionless vertical velocity equals the total derivative of the dimensionless wave height,
$$ \overline{w}=\frac{d\overline{\zeta}}{d\overline{t}} \tag{3.5} $$
Free surface — dynamic condition (from eq. 2.18, neglecting viscous stresses). Here $p$ includes both the hydrodynamic and the hydrostatic pressures, so the balance reads $p_{hd}+p_{hs}=p_a-\Delta p_\gamma$, i.e.
$$ \overline{p}\,\rho U_\infty^2 - \rho g\,\overline{\zeta}L = p_a - \frac{\gamma}{L}\left(\frac{1}{\overline{r_1}}+\frac{1}{\overline{r_2}}\right) \tag{3.7} $$
where the hydrostatic surface pressure (eq. 2.22) and the surface-tension expression (eq. 2.19) have been inserted. Solving for the dimensionless pressure $\overline{p}$:
$$ \overline{p} = \left(\frac{p_a}{\rho U_\infty^2}\right) + \left(\frac{gL}{U_\infty^2}\,\overline{\zeta}\right) - \left(\frac{\gamma}{\rho U_\infty^2 L}\right)\left(\frac{1}{\overline{r_1}}+\frac{1}{\overline{r_2}}\right) \tag{3.8} $$
Infinity (from eq. 2.21):
$$ \overline{u}=1, \qquad \overline{v}=\overline{w}=0, \qquad \overline{p}=0 \tag{3.9} $$
Finally, geometrically similar bodies and boundaries are represented by functions $f_i(\overline{x},\overline{y},\overline{z})=0$ that are independent of scale:
$$ f_i(\overline{x},\overline{y},\overline{z})=0 \tag{3.10} $$
1st bracket → the Euler number $En=p_a/(\rho U_\infty^2)$;
2nd bracket → contains $gL/U_\infty^2 = 1/Fn^2$, the inverse square of the Froude number;
3rd bracket → contains $\gamma/(\rho U_\infty^2 L)=1/Wn$, the inverse Weber number.
Together with $1/Rn$ from the momentum equations (3.1a–c), these are the only parameters that appear anywhere in the problem.
3.4 The four similarity numbers
The complete problem is defined by the field equations (3.1a–c) and (3.2), the solid-surface conditions (3.3)–(3.4), the free-surface conditions (3.5) and (3.8), and the infinity condition (3.9). The only parameters appearing are the following four.
Reynolds number — appears in (3.1a–c) as $1/Rn$:
$$ Rn = \frac{U_\infty L}{\nu} \tag{3.11} $$
Euler number — appears in (3.8):
$$ En = \frac{p_a}{\rho U_\infty^2} \tag{3.12} $$
Froude number — appears in (3.8) as $1/Fn^2$:
$$ Fn = \frac{U_\infty}{\sqrt{gL}} \tag{3.13} $$
Weber number — appears in (3.8) as $1/Wn$:
$$ Wn = \frac{\rho U_\infty^2 L}{\gamma} \tag{3.14} $$
| Number | Definition | Physical meaning — what it scales | Where it appears |
|---|---|---|---|
| Reynolds $Rn$ | $U_\infty L/\nu$ | effect of viscosity; ratio of inertial to viscous forces | momentum eqs. (3.1a–c) |
| Euler $En$ | $p_a/(\rho U_\infty^2)$ | absolute pressure level; governs onset of cavitation | free-surface eq. (3.8) |
| Froude $Fn$ | $U_\infty/\sqrt{gL}$ | effect of gravity on the free surface; wave making | free-surface eq. (3.8) |
| Weber $Wn$ | $\rho U_\infty^2 L/\gamma$ | effect of surface tension; spray, breaking, capillary waves | free-surface eq. (3.8) |
| Cavitation $\sigma$ | $\dfrac{p_a-p_v}{\frac{1}{2}\rho U_\infty^2}$ | "margin" to cavitation; interchangeable with $En$ | derived from $En$ |
The Euler number matters because $p_a$ sets the absolute pressure level: raising the atmospheric pressure at the water surface raises the pressure everywhere by the same amount. If the pressure anywhere falls below the vapour pressure $p_v$, cavitation occurs. A measure of the margin to cavitation is $p_a-p_v$, which leads to the cavitation number:
$$ \sigma = \frac{p_a - p_v}{\frac{1}{2}\,\rho U_\infty^2} \tag{3.15} $$
Because the Euler and cavitation numbers differ only by a constant, they are interchangeable as similarity parameters.
"The constancy of the Reynolds number, the Euler (cavitation) number, the Froude number, and the Weber number is a necessary and sufficient condition for flow similarity between geosim bodies at different scales." (Larsson & Raven, §3.2)
4. Consequences of the similarity requirements (§3.3)
4.1 Summary of requirements (§3.3.1)
In theory, towing-tank testing of a ship model should satisfy all of the following:
- Geometric similarity — model and ship must have the same shape, except for surface roughness.
- No blockage — because contract conditions are normally for unrestricted water, the tank must be wide and deep enough to avoid blockage effects (discussed in Section 5).
- Equal Reynolds number $Rn=U_\infty L/\nu$ — for the effect of viscosity to be correct. Since $Rn$ sits in the Navier–Stokes equations, it affects all flows of hydrodynamic interest. (The one exception is the "inviscid flow" approximation, where viscosity is neglected and $Rn$ is irrelevant — see Section 5.)
- Equal Froude number $Fn=U_\infty/\sqrt{gL}$ — for the effect of gravity on the free surface to be correct. $Fn$ (and the remaining $Wn$, $\sigma$) appears only in the dynamic free-surface condition; with no free surface (as in most water tunnels without cavitation) none of these matters. A correct $Fn$ means correctly scaled waves, and hence a correctly scaled wave resistance.
- Equal Weber number $Wn=\rho U_\infty^2 L/\gamma$ — for the correct effect of surface tension: spray, wave breaking (water drops and air bubbles of small radius), and very short surface waves are then correctly scaled.
- Equal cavitation number $\sigma$ — for the same cavitation pattern. Cavitation hardly ever occurs around the hull, so it is not normally considered in towing-tank testing and is not pursued in this volume; it is, however, important in propeller design (treated in the propulsion volume).
4.2 The dilemma in model testing (§3.3.2)
Unfortunately it is impossible to satisfy all requirements at once. The model is, by definition, shorter than the ship, so its speed would have to be adjusted to give the correct $Rn$, $Fn$ and $Wn$ simultaneously — and that cannot be done:
- For the correct Froude number, the model speed must be lower than the ship's (speed scales as $\sqrt{L}$).
- For the correct Reynolds and Weber numbers, the model speed must be higher.
In principle the requirements could be met by changing the model fluid's physical properties ($\rho$, $g$, $\nu$ or $\gamma$), but no suitable fluid exists, and tests are run in fresh water, whose constants are very close to the salt water at full scale. In reality only one of the three (Reynolds, Froude, Weber) can be satisfied; the other two are sacrificed.
- breaking waves look different — much less "white water" than full scale;
- spray looks different at high speed — more coherent water films ejected sideways;
- capillary waves appear at low speed, with no full-scale counterpart;
- the wave pattern deviates if the model speed is around 0.23 m/s — a speed to be avoided when wave making matters.
That leaves the two parameters that really count — Reynolds and Froude — both important for the resistance and the flow. The tester faces a real dilemma: which to hold constant and which to sacrifice.
4.3 The practical solution — match the Froude number
The practical solution is to satisfy the Froude number and correct for the errors introduced by the wrong Reynolds number. This is exactly what William Froude proposed in 1868 (see §1.2.1). Testing at the right $Fn$ correctly scales gravity, the main driving force of the waves.
"The model shall be tested at the same Froude number as the ship." (Larsson & Raven, §3.3.2)
The other main resistance component — the viscous resistance caused by the fluid viscosity — is mainly governed by the Reynolds number. Froude proposed computing it from an empirical formula based on plank tests; even with today's more refined methods (Section 8), empirical formulas are still used, and all modern ones are functions of one parameter only — the Reynolds number (Section 6.3.4). There is a small Froude influence on the viscous resistance too, because the wetted surface changes slightly with speed as the wave profile along the hull changes.
4.4 Water tank versus wind tunnel
Knowing the scaling rules, testers exploit the strengths of each facility. Rudders and other appendages, and detailed boundary-layer measurements (even for ship models), are often done in wind tunnels, because equipment is easier to access there and robustness requirements are smaller in air.
Conversely, some aerodynamic problems are best solved in water. The classic example is testing automobiles in water tanks (Larsson et al., 1989): the car is towed along the bottom of the tank, which correctly reproduces the flow around the rotating wheels and under the car. In a wind tunnel there is always a boundary layer in the approaching flow — absent when a car moves through still air — that must be removed (not without problems), and the rotating wheels, which must not touch the floor, are hard to model.