MN Ch. 6 — Ship Resistance and Flow: The Flow Around the Hull and the Viscous Resistance

1. Overview — viscous resistance and the boundary layer (§6 intro, §6.1)

Chapter 5 dealt with wave resistance; this chapter turns to the other major component, the viscous resistance. The two are linked, but there are good reasons to treat them apart — so here the waves and every other free-surface effect are neglected. Two consequences follow at once: the Froude number becomes irrelevant (the only similarity parameter left is the Reynolds number), and the hydrostatic pressure disappears, leaving only the hydrodynamic pressure (the index "hd" and the word "hydrodynamic" are dropped throughout).

1.1 Body-fitted coordinate system (§6.1)

To solve the boundary-layer equations it is far more convenient to use a coordinate system fitted to the hull than the global Cartesian system of Fig. 2.1. Body-fitted systems are curvilinear and generally non-orthogonal, but inside boundary layers orthogonal systems may be used. Following the ITTC nomenclature, the coordinates $x$, $y$, $z$ are reused here with a local meaning (the small risk of confusion is accepted):

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 6 — The Flow Around the Hull and the Viscous Resistance.

Edital: Anexo 2-A, Área I, item 5 (escoamento viscoso em torno do casco; camada-limite; resistência viscosa; placa plana e linhas de fricção; fator de forma; separação; camada-limite do navio; rugosidade e redução de arrasto) e item 1 (resistência de fricção e de pressão viscosa) · Anexo 2-B, Área I, item 3 (Larsson & Raven), Chapter 6.

2. The boundary layer (§6.2)

2.1 Physical description (§6.2.1)

The no-slip condition states that the relative velocity between a solid surface and the fluid is zero — the fluid "sticks" to the wall by molecular action. Moving away from the surface, the velocity rises across a thin region: the boundary layer, which covers the whole hull and generally grows downstream (Fig. 6.1). At the front it is laminar; the velocity rises smoothly, without fluctuations, toward the edge.

Momentum parallel to the surface is exchanged between layers by molecular motion: a molecule jumping outward carries lower momentum and slows the faster layer, and vice versa. Treating the fluid as a continuum, this effect is modelled as a stress. For a 2-D boundary layer it reduces to

If the body is long enough, the laminar flow becomes unstable at the point of neutral stability, entering a transitional region of turbulent bursts that grow more frequent until the layer is turbulent. Its innermost part, the viscous sublayer, is still essentially laminar but interrupted by bursts. Outside it, the turbulent layer holds eddies of many sizes — the smallest (Kolmogoroff scale) about 0.1 mm at ship and model Reynolds numbers alike, the largest of the order of the boundary-layer thickness (0.1 m at model scale, 1–10 m at full scale).

The macroscopic effect of the eddies mirrors the microscopic effect of molecules and is modelled by a new concept, the turbulent (eddy) viscosity $\mu_t$. In a turbulent 2-D boundary layer the shear stress uses an effective viscosity:

2.2 First-order boundary-layer theory (§6.2.2)

The boundary-layer concept, due to Prandtl (1904), splits the flow into a near-wall/wake region where viscosity dominates and an outer region where it may be neglected (Fig. 6.1). Note why viscosity may be dropped outside: not because $\mu$ is smaller, but because the velocity gradients (rates of strain) are negligible there. The outer flow is therefore, somewhat inconsistently, called inviscid.

2.3 Local boundary-layer quantities (§6.2.3)

The most important local quantity is the thickness $\delta$. Since the velocity merges smoothly with the outer flow, $\delta$ is defined where $u$ reaches a stated percentage of the edge velocity $U_e$ (both 99% and 99.5% are used — the percentage must be given).

The displacement thickness $\delta_1$ measures the volume-flux deficit (Fig. 6.2):

The momentum thickness $\theta$ is used for estimating resistance (Fig. 6.3):

Behind the plate, $\theta$ stays constant — so a velocity profile measured anywhere downstream yields the plate friction. Eq. (6.6) holds for all 2-D bodies if the profile is measured far enough behind for the pressure to be undisturbed; then it gives the total viscous resistance (friction + form + roughness), not friction alone. This is the basis of wake-survey resistance measurement.

Beyond thicknesses, the friction coefficients are defined as

3. The flat plate (§6.3)

The simplest boundary layer develops on an infinitely thin flat plate, parallel to the flow, leading (and trailing) edge at right angles to it. Its key simplification: the pressure is constant and undisturbed along the plate (first-order theory: the plate does not disturb the pressure, and the displacement-thickness effect is neglected). So $U_e \approx U_\infty$.

3.1 Laminar boundary layer — the Blasius solution (§6.3.1)

The laminar flat-plate layer is one of the few viscous flows with an analytical solution, found by Blasius: all velocity profiles coincide when scaled in the normal direction (a "similar" solution):

So $\delta_1\approx\tfrac13\delta$ and $\theta\approx\tfrac18\delta$ for the laminar layer. The total friction coefficient is exactly twice the local one at any distance from the leading edge. All laminar thicknesses grow as $x^{1/2}$.

3.2 Transition (§6.3.2)

Under ideal conditions transition completes at a critical Reynolds number $Rn_{crit}\approx 3\times 10^6$, where

That high value needs a very low upstream turbulence level. Defining the turbulence level $T$ from the RMS fluctuations,

Transition also depends on roughness; the critical roughness height (Feindt) is $U_\infty k_s/\nu = 120$ (eq. 6.18), about the same requirement as a hydraulically smooth surface. In model testing, because models run at much lower $Rn$, transition occurs much further aft — so turbulence is deliberately stimulated near the bow to force premature transition (§8). For shapes other than the flat plate, the longitudinal pressure gradient strongly influences where transition occurs.

3.3 Turbulent boundary layer (§6.3.3)

In the inner part of a turbulent layer the velocity profile depends only on the wall shear stress $\tau_w$ and the fluid properties — by dimensional reasoning $u/u_\tau = f(y\,u_\tau/\nu)$ (eq. 6.19), where the friction velocity is

In the outer part the velocity-defect depends on $\delta$: $(U_\infty - u)/u_\tau = f_1(y/\delta)$ (eq. 6.25). Millikan showed the only way for the two to merge smoothly is a logarithmic intermediate region. The result is the four-region wall-wake law (Fig. 6.4):

The four-region law is accurate but complex, so a simpler empirical power law is very often used (poor near the wall, fine away from it):

Here $\delta_1\approx\tfrac18\delta$ and $\theta\approx\tfrac{1}{10}\delta$ ($n=7$). Crucially, turbulent thicknesses grow as $x^{4/5}$ while laminar ones grow as $x^{1/2}$ — the turbulent layer grows faster. Eqs. (6.31)–(6.34) give quick estimates even for more complex bodies.

3.4 Flat-plate friction and extrapolation lines (§6.3.4)

The total turbulent friction follows by integrating (6.34), but the historical "plank" friction lines matter because the friction of an "equivalent flat plate" has anchored model-test procedure since William Froude's day.

William Froude (1872, 1874) towed varnished planks (0.6–15 m, 0.5–4 m/s) at Torquay and found resistance per unit area lower for a long plank than a short one — anticipating the boundary layer before Prandtl. He gave $R_F = f\,S\,V^{n}$ (eq. 6.35), with $n$ falling from 2.0 (short plank) to 1.83 (15 m plank) for smooth varnish. R. E. Froude (1888) extended the curves to 366 m without further data.

Schoenherr (1932) collected the plank data and fitted the ATTC line:

Hughes (1952/1954) towed planks and pontoons up to 78 m ($Rn$ to $3\times10^8$), extrapolated to infinite aspect ratio, and obtained a minimum-turbulent-resistance curve for plane smooth 2-D surfaces, the Hughes line:

Since Hughes' formula could not estimate the total ship viscous resistance, the ITTC (Madrid, 1957) made a correction folding in a typical form effect — the ITTC-57 model-ship correlation line:

4. Two-dimensional bodies (§6.4)

The flat plate misses three effects present in the full 3-D case: longitudinal pressure gradients, lateral streamline convergence, and lateral pressure gradients. They are introduced one at a time. This section adds the longitudinal pressure gradient — recall the normal pressure is always constant in first-order theory. A "2-D" body is infinitely long in $z$ with constant cross-section; streamlines cannot bend in $z$.

4.1 Pressure distribution (§6.4.1)

The boundary layer disturbs the inviscid pressure (Fig. 5.3) via its displacement effect. Near the stern the viscous streamlines are displaced outward (lower velocity inside the layer), as in Fig. 6.7:

Adding the displacement thickness to the hull (Fig. 6.8) and running an inviscid calculation on this thickened body gives the real viscous pressure along the body. Because the thickened body has no aft end (the wake extends to infinity), there is no stagnation pressure there: the straighter streamlines give a smaller pressure at the stern than the inviscid flow (Fig. 6.9). This lower stern pressure is the form effect on the pressure (viscous pressure resistance) of Fig. 4.1.

4.2 General effects of the longitudinal pressure variation (§6.4.2)

In inviscid flow, pressure and velocity are linked by Bernoulli (Chapter 5): high pressure ↔ low velocity (zero at stagnation points), low pressure ↔ high velocity (maxima at the shoulders). In viscous flow the link uses the edge velocity $U_e$. A fluid element in the boundary layer is acted on by viscous forces and, when pressure is not constant, by longitudinal pressure forces too.

There are no simple flat-plate-type relations for general gradients, but if the gradients are not too large the flat-plate relations still give reasonable estimates of thickness, friction and profile.

4.3 Transition and separation (§6.4.3–6.4.4)

Transition. A falling pressure (acceleration) stabilises the flow and delays transition; a rising pressure does the opposite. This is exploited to make low-drag laminar airfoil sections (§7). The pressure-gradient effect is usually stronger than the $Rn$ effect, so over a range of $Rn$ transition occurs near the first pressure minimum — a good estimate of its location, at least at model scale.

Separation. If the pressure rise at the aft end is too rapid, the decelerating force can stop the longitudinal motion of fluid right at the wall; the streamlines leave the surface and a zone of reversed flow forms — separation (Fig. 6.10). The separation zone has very small axial mean velocity but large fluctuations (unlike an attached turbulent layer). The separation itself then reduces the pressure gradient: the straighter outside streamlines build up even less stern pressure (Fig. 6.9).

4.4 Form effects and form factor (§6.4.5)

Because the pressure gradient changes the edge velocity, the friction on the body differs from a flat plate. It is normally larger: the body displaces the streamlines, packing them closer over the body than at infinity, raising $U_e$ and hence the friction — the form effect on the friction of Fig. 4.1. The two form effects (on pressure and on friction) are lumped into a form factor $1+k$:

5. Axisymmetric bodies (§6.5)

From a boundary-layer viewpoint, the new feature of an axisymmetric body (generated by a plane curve rotating about an axis in its plane) is that the width of the surface over which the layer develops varies along the body. The near-surface streamlines diverge from the forward stagnation point, are most spread at the largest diameter, and converge to the aft stagnation point (Fig. 6.11).

The lateral distance between streamlines is proportional to the local radius. Where the radius increases (diverging streamlines), continuity reduces the boundary-layer growth; where it decreases (converging), growth is enhanced. So the initial thickness growth is smaller than a 2-D body at the fore end and larger at the tail.

6. Three-dimensional bodies (§6.6)

The final step adds lateral pressure gradients (in a plane parallel to the surface, at right angles to the edge flow), completing the picture: longitudinal pressure variation (2-D body) + lateral streamline convergence/divergence (axisymmetric) + lateral pressure gradient (3-D).

6.1 Cross-flow and limiting streamlines (§6.6.1)

A lateral pressure gradient $\partial p/\partial z$ bends the external flow sideways (Fig. 6.13). Since the pressure is constant across the layer in $y$, $\partial p/\partial z$ is the same at every $y$ on a normal — so $u^2/r$ must be the same at all $y$. But $u$ falls toward the wall, so the local radius of curvature $r$ falls too (in proportion to $u^2$): the slower fluid turns more sharply. A cross-flow develops toward the lower pressure, going to zero at the wall (no-slip).

If $\partial p/\partial z$ changes sign (an inflexion point), the innermost streamlines react at once while the higher-momentum outer flow lags, giving an S-shaped cross-flow profile over some distance. The cross-flow angle is

The limiting flow direction at the surface is thus the direction of the wall shear stress. Lines traced along the friction direction are the limiting streamlines — the limit of the boundary-layer streamlines on the surface. In tank tests they are revealed by girthwise strips of wet paint dragged by the friction of the moving hull (see Fig. 8.10).

6.2 Three-dimensional separation (§6.6.2)

3-D flow can separate two ways (Fig. 6.14):

7. The boundary layer around ships (§6.7)

Table 6.1 summarises the build-up. The ship case is treated with all wave effects neglected: the undisturbed free surface is a symmetry plane, so the real underwater hull plus its mirror image is studied — a double model. The example is the KVLCC2, a very bluff VLCC (block coefficient $C_B = 0.85$) much used for CFD validation, so its three-dimensionality is strong and easily seen. The discussion uses computed results (inviscid by a potential-flow panel method, viscous by a RANS method).

7.1 Pressure distribution and boundary-layer development (§6.7.1)

The boundary layer is driven by the pressure field. The inviscid pressure around the KVLCC2 (Fig. 6.15, = Fig. 5.5) shows two high-pressure zones (bow, stern) and low-pressure zones at the two shoulders and at the bilges near bow and stern.

In the real viscous flow (Fig. 6.16, model $Rn = 4.6\times10^6$) the displacement effect is tiny on the forebody but larger aft: near the transom the high inviscid pressure is reduced by 0.1–0.2 in $C_p$, and the bilge pressure minimum is about 0.05 higher. The distribution is similar but smoothed.

Turning to the lateral pressure gradients (tied to lateral streamline curvature, Fig. 6.17): the upper streamlines are nearly horizontal (little cross-flow), but those in the bilge region curve down under the bottom and back, giving an inflexion point on the bilge where the lateral pressure gradient changes sign. Above the line of inflexion points cross-flow goes downward; below it, upward — both toward the line, so the flow converges there. This is exactly the vortex-sheet case (Fig. 6.14): a bilge vortex forms.

7.2 Cross-sections through the boundary layer (§6.7.2)

At about half draft the outer flow converges (close to the external streamlines) while the near-wall flow diverges — a distinct two-layer structure with a stepwise velocity change (Löfdahl & Larsson, 1984). Iso-velocity contours of the axial velocity $u/U_\infty$ at three stations (Fig. 6.19) show the redistribution: the outer layer thickens where the external streamlines converge (note the bulge of the 0.9 contour), while the inner layer thins where the limiting streamlines diverge.

At the propeller plane an "island" of low-velocity fluid (0.3–0.4 contours) sits in the centre — created by the stern bilge vortex hitting the disk, producing the characteristic "hooks" in the velocity contours (Fig. 6.20). Predicting these hooks was a major CFD challenge for two decades; the contour shape strongly affects propeller operation (§11.4.1). First-order boundary-layer theory is adequate over most of the hull but fails near the stern, where the layer is too thick — the less-approximate methods of §9.7–9.8 are needed.

7.3 Effects on viscous resistance (§6.7.3)

Both bow and stern bilge vortices raise the drag, but the stern vortex brings a trade-off: it rounds the velocity contours in the propeller plane, keeping the axial velocity at a given radius more constant, minimising angle-of-attack and load variations on the blades — better for noise, vibration and propulsive efficiency (§11). Bubble separation (large momentum loss, big drag rise) should be avoided. For both separation types the drag rise is mainly a pressure loss on the hull, with a small compensating friction loss. A more important friction effect: where the boundary layer is thin, the shear stress is large — friction is very high along the keel (diverging external streamlines) and higher on the forebody than the afterbody (Fig. 6.21).

7.4 Scale effects (§6.7.4)

The boundary layer depends on $Rn$ ($\delta \propto Rn^{-1/5}$), and a ship's $Rn$ is at least two orders larger than a model's — so the relative layer is much thinner at full scale. At $Rn = 2.0\times10^{10}$ (Fig. 6.22) the full-scale stern pressure is closer to the inviscid result than the model's (smaller displacement effect ⇒ less smoothing). The limiting streamlines diverge/converge less (Fig. 6.23), the bilge vortex is weaker, and the wake "hooks" disappear (Fig. 6.24) — smoother contours, plus the expected thinning. Only the inviscid results are scale-independent.

8. Roughness allowance (§6.8)

So far the surface was assumed hydraulically smooth — true for good ship models, but not for full-scale ships, where the roughness allowance (Fig. 4.1) can be considerable.

8.1 Roughness and fouling on ships (§6.8.1)

Hull plates are shot-blasted (fairly smooth, but not hydraulically smooth at ship speed); welding joints, paint layers, and deterioration add roughness; and fouling can increase it greatly. The self-polishing copolymer (SPC) antifouling paints get smoother with time (the faster flow over peaks abrades them), which also reduces fouling — extending the docking period from typically 1.5 years to about 5. (TBT, the SPC agent, was ~70% of commercial ship coatings in 1999; tin-based paints were phased out 2003–2008 by IMO agreement, prompting copper-oxide biocides and biocide-free silicone/fluoropolymer or structured surfaces.)

8.2 Characterization; hydraulically smooth surfaces (§6.8.2–6.8.3)

The classical reference is densely packed sand: the equivalent sand roughness $k_s$ is tabulated for various materials but rarely known for hulls.

In ship hydrodynamics the accepted measure is the Mean Apparent Amplitude (MAA): a needle tracks the contour along a 750 mm line, divided into 50 mm intervals; within each, the highest-peak-to-lowest-valley distance is recorded, and the roughness height $k$ is the mean of the 15 values. Before the SPC era $k$(MAA) was ~100 μm for newer ships and 600–700 μm for older ones (cleaned, not shot-blasted).

A surface is hydraulically smooth if its roughness elements are embedded in the viscous sublayer ($y^{+}\le 5$):

8.3 Roughness allowance prediction and Bowden's formula (§6.8.4–6.8.6)

If $k_s$ is known, the Prandtl-Schlichting diagram gives $\Delta C_F$ — but it rarely is, so MAA-based experiments are used. A floating element balance (Fig. 6.26) cuts a small element (~200×200 mm) flush with a tunnel surface and measures the shear force, yielding diagrams like Fig. 6.27 (Johansson, 1984). Pipe-flow pressure-drop tests are cheaper and good for large friction increases (fouling) at high $Rn$.

With few rough-surface data available, the ITTC (1978) adopted an empirical roughness allowance, Bowden's formula:

Fouling. A fouled surface gives much larger drag increases; geometry varies too much for general rules, but the diagram of Fig. 6.28 (densely packed barnacles ~5 mm high) may be used (Johansson, 1984; Leer-Andersen & Larsson, 2003).

9. Drag reduction (§6.9)

For ~60 years it has been known that friction can be cut below that of a hydraulically smooth surface, by acting on the turbulence. The main techniques: