MN — Ship Resistance and Flow · Ch. 2

1. Overview — the equations that govern the flow

Chapter 2 derives the equations that govern the viscous flow of water around a ship moving at steady forward speed, and states the boundary conditions they must satisfy. The chapter builds, step by step, a closed system of equations for four unknowns: the pressure $p$ and the three velocity components $u$, $v$, $w$. Two pieces make up that system — the continuity equation (conservation of mass) and the three Navier–Stokes equations (Newton's second law applied to a fluid particle). The chapter closes with two short but important ideas: the boundary condition modified by surface tension, and the split of pressure into a hydrostatic and a hydrodynamic part.

Reading guide. This is the most formula-heavy chapter of the book. Every equation below is given with the meaning of each symbol. One key assumption runs through the whole chapter: water is treated as an incompressible fluid — its density $\rho$ is taken as constant. The author warns the reader that the Navier–Stokes derivation is lengthy; a reader who only wants the result may jump straight to the final equations (2.13a)–(2.13c).

The closed system in one glance. Four unknowns — $p$, $u$, $v$, $w$ — and four equations: one continuity equation (2.1) plus three Navier–Stokes equations (2.13a–c). Once boundary conditions are added, the flow field is, in principle, fully determined.

**Symbols used throughout Chapter 2.**
Symbol	Quantity	Note
$x,\ y,\ z$	Cartesian coordinates	$x$ sternward, $y$ to starboard, $z$ upward; origin at midship and undisturbed water level
$u,\ v,\ w$	velocity components	along $x$, $y$, $z$ respectively
$p$	pressure	split later into $p_{hs}$ (hydrostatic) and $p_{hd}$ (hydrodynamic)
$V$	ship's steady forward speed	the inflow speed in the ship-fixed frame
$\rho$	density	assumed constant (incompressible water)
$\mu,\ \nu$	dynamic / kinematic viscosity	$\nu = \mu/\rho$
$g$	acceleration of gravity	acts along the negative $z$-axis only
$\sigma_{ij}$	viscous stress tensor	$i$ = face normal, $j$ = stress direction; symmetric, 6 independent components
$\gamma,\ \Delta p_\gamma$	surface tension / its pressure jump	$\Delta p_\gamma = \gamma(1/r_1 + 1/r_2)$
$\zeta(x,y)$	free-surface (wave) height	used in the kinematic 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 2 — Governing Equations.

Edital: Anexo 2-A, Área I, item 5 (Resistências do Navio e métodos para sua obtenção — escoamento inviscido/viscoso, equações de governo) · Anexo 2-B, Área I, item 3 (Larsson & Raven), Chapter 2.

2. The global coordinate system (§2.1)

The book deals with the flow around ships at steady forward speed, denoted $V$. Unsteadiness from ship motions, waves and manoeuvring is neglected. The nomenclature follows the recommendation of the ITTC.

Fig. 2-1 Fig. 2.1 — Global coordinate system

shows the adopted global Cartesian system:

$x$ — directed sternward (toward the stern).
$y$ — directed to starboard (the ship's right-hand side).
$z$ — directed vertically upward.
Origin — at midship and at the undisturbed water level.

The coordinate system moves with the ship. Equivalently, we picture a ship held at a fixed position inside a uniform inflow coming from ahead. In this frame, the entire flow field is steady in a time-averaged sense — the turbulent fluctuations are filtered out (see Section 9.7). In other words, the mean velocity and pressure fields and the wave pattern are functions of the spatial coordinates but not of time.

Why "unsteady form" anyway? Although the flow is steady on average, turbulent fluctuations do occur. So the equations are derived in their unsteady (time-dependent) form, keeping the $\partial/\partial t$ terms, for later use with turbulence modelling in Section 9.7.

3. The continuity equation (§2.2)

The continuity equation expresses conservation of mass. It is derived by looking at an infinitesimal fluid element of volume $dx\,dy\,dz$ and balancing the mass that flows in and out across its faces.

Fig. 2-2 Fig. 2.2 — Mass flow in the x-direction

shows the mass flows across the two faces whose normals point along $x$.

Only the $u$-component of velocity can carry mass through those two $x$-faces. The mass inflow through the left face is $\rho\,u\,dy\,dz$; the outflow through the right face is $\left[\rho u + \dfrac{\partial(\rho u)}{\partial x}\,dx\right]dy\,dz$. The net outflow in the $x$-direction is therefore $\dfrac{\partial(\rho u)}{\partial x}\,dx\,dy\,dz$. The same reasoning in the $y$- and $z$-directions gives the net outflows $\dfrac{\partial(\rho v)}{\partial y}\,dx\,dy\,dz$ and $\dfrac{\partial(\rho w)}{\partial z}\,dx\,dy\,dz$.

Because, in the absence of mass sources, the total net transport of mass out of the element must be zero, the sum of the three contributions vanishes:

$$ \frac{\partial(\rho u)}{\partial x}\,dx\,dy\,dz + \frac{\partial(\rho v)}{\partial y}\,dx\,dy\,dz + \frac{\partial(\rho w)}{\partial z}\,dx\,dy\,dz = 0 $$

With the density $\rho$ constant (incompressible water), $\rho$ and the volume factor $dx\,dy\,dz$ divide out, leaving the continuity equation for incompressible flow:

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \tag{2.1} $$

Meaning of the symbols (eq. 2.1).
$u,\ v,\ w$ — velocity components in the $x$, $y$, $z$ directions.
$\partial u/\partial x$, etc. — the rate at which each velocity component changes along its own direction.
The equation says the volume of an incompressible fluid element is preserved: whatever flows in must flow out. The sum of the three diagonal velocity gradients (the divergence of the velocity field) is zero everywhere.

4. The Navier–Stokes equations (§2.3)

The Navier–Stokes equations apply Newton's second law to the same infinitesimal fluid element. The derivation is long; the author notes that a reader who only wants the answer may jump directly to the final result, equations (2.13a)–(2.13c). The steps below show where each term comes from.

4.1 Newton's second law for a fluid particle

We start from Newton's second law written for the element of mass $dm$:

$$ \overrightarrow{dF} = dm \cdot \vec{a} \tag{2.2} $$

where $\overrightarrow{dF}$ is the total force on the element, $dm$ its mass and $\vec{a}$ its acceleration. In fluid mechanics three kinds of force act on the particle: pressure forces $\overrightarrow{dF_p}$, body forces $\overrightarrow{dF_b}$ (here, gravity), and viscous forces $\overrightarrow{dF_v}$. Dividing (2.2) by $dm$ and inserting the three forces:

$$ \vec{a} = \frac{\overrightarrow{dF_p}}{dm} + \frac{\overrightarrow{dF_b}}{dm} + \frac{\overrightarrow{dF_v}}{dm} \tag{2.3} $$

4.2 The acceleration — the material derivative

The acceleration is the total (material) derivative of the velocity vector $\vec{u}$, whose components are $u(x,y,z,t)$, $v(x,y,z,t)$, $w(x,y,z,t)$:

$$ \vec{a} = \frac{d\vec{u}}{dt} \tag{2.4} $$

Applying the chain rule (the velocity changes both with time and because the particle moves to a new position), the three components of acceleration become:

$$ a_x = \frac{du}{dt} = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} \tag{2.5a} $$

$$ a_y = \frac{dv}{dt} = \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} \tag{2.5b} $$

$$ a_z = \frac{dw}{dt} = \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} \tag{2.5c} $$

The two parts of the acceleration. The first term, $\partial u/\partial t$, is the local (unsteady) acceleration — the velocity changing at a fixed point. The other three terms, $u\,\partial u/\partial x + v\,\partial u/\partial y + w\,\partial u/\partial z$, form the convective acceleration — the particle speeding up or slowing down because it has moved into a region of different velocity. Together they are the material derivative of velocity.

4.3 The pressure force

Consider the $x$-component of the pressure force, $dF_{px}$.

Fig. 2-3 Fig. 2.3 — Pressure on the x-faces

shows that the pressure force on the left surface is $p\,dy\,dz$, while on the right surface it has grown to $\left(p + \dfrac{\partial p}{\partial x}\,dx\right)dy\,dz$ because of the pressure gradient. Adding the two (with sign), the resulting force points in the negative $x$-direction:

$$ dF_{px} = -\frac{\partial p}{\partial x}\,dx\,dy\,dz $$

Dividing by the mass $dm = \rho\,dx\,dy\,dz$ gives the pressure term per unit mass for each of the three directions:

$$ \frac{dF_{px}}{dm} = -\frac{1}{\rho}\frac{\partial p}{\partial x} \tag{2.6a} $$

$$ \frac{dF_{py}}{dm} = -\frac{1}{\rho}\frac{\partial p}{\partial y} \tag{2.6b} $$

$$ \frac{dF_{pz}}{dm} = -\frac{1}{\rho}\frac{\partial p}{\partial z} \tag{2.6c} $$

Here $p$ is the pressure and $\rho$ the (constant) density. The minus sign means the fluid is pushed from high toward low pressure.

4.4 The body force — gravity

The only body force considered is gravity. Because $z$ points vertically upward, gravity has no component in $x$ and $y$, and in $z$ it equals $-g\,dm$, where $g$ is the acceleration of gravity. Per unit mass:

$$ \frac{dF_{bx}}{dm} = 0 \tag{2.7a} $$

$$ \frac{dF_{by}}{dm} = 0 \tag{2.7b} $$

$$ \frac{dF_{bz}}{dm} = -g \tag{2.7c} $$

4.5 The viscous force — stress tensor and Newton's hypothesis

The viscous force $\overrightarrow{dF_v}$ is harder to obtain because it comes from stresses acting on the faces of the element, both normal and tangential.

Fig. 2-4 Fig. 2.4 — Viscous stresses in the x-direction

shows the stresses acting in the $x$-direction on the six faces. Each stress $\sigma_{ij}$ carries two indices: the first ($i$) names the surface on which it acts (identified by the direction of its normal), the second ($j$) its own direction. Since both indices take three values, the viscous stress tensor $\sigma_{ij}$ has nine components.

Adding the contributions from opposing faces (with sign), the total viscous force in the $x$-direction is:

$$ dF_{vx} = \left[\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{yx}}{\partial y} + \frac{\partial \sigma_{zx}}{\partial z}\right]dx\,dy\,dz \tag{2.8} $$

so the viscous term per unit mass, for the three directions, is:

$$ \frac{dF_{vx}}{dm} = \frac{1}{\rho}\left[\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{yx}}{\partial y} + \frac{\partial \sigma_{zx}}{\partial z}\right] \tag{2.9a} $$

$$ \frac{dF_{vy}}{dm} = \frac{1}{\rho}\left[\frac{\partial \sigma_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} + \frac{\partial \sigma_{zy}}{\partial z}\right] \tag{2.9b} $$

$$ \frac{dF_{vz}}{dm} = \frac{1}{\rho}\left[\frac{\partial \sigma_{xz}}{\partial x} + \frac{\partial \sigma_{yz}}{\partial y} + \frac{\partial \sigma_{zz}}{\partial z}\right] \tag{2.9c} $$

The stresses themselves are found from a hypothesis, very well proven over the years. In his Principia (1687), Newton postulated a linear relationship between the shear stress and the normal velocity gradient (for flow around a rotating cylinder). In 1845, Stokes generalised this to general three-dimensional flows. For most flows of engineering interest the viscous stress tensor $\sigma_{ij}$ is proportional to a rate-of-strain tensor $S_{ij}$:

$$ S_{ij} = \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \qquad\qquad \sigma_{ij} = \mu\,S_{ij} \tag{2.10} $$

Meaning of the symbols (eq. 2.10).
$i,\ j$ — each takes the value 1, 2 or 3; $u_1, u_2, u_3$ mean $u, v, w$ and $x_1, x_2, x_3$ mean $x, y, z$.
$S_{ij}$ — the rate-of-strain tensor, built from velocity gradients; it measures how fast the fluid element is being deformed.
$\mu$ — the dynamic viscosity, the constant of proportionality (a property of the fluid).
The tensor is symmetric ($\sigma_{ij}=\sigma_{ji}$), so of the nine components only six are independent.

The six independent stress components are:

$$ \sigma_{xx} = 2\mu\frac{\partial u}{\partial x} \tag{2.11a} $$

$$ \sigma_{xy} = \sigma_{yx} = \mu\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right) \tag{2.11b} $$

$$ \sigma_{zx} = \sigma_{xz} = \mu\left(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}\right) \tag{2.11c} $$

$$ \sigma_{yy} = 2\mu\frac{\partial v}{\partial y} \tag{2.11d} $$

$$ \sigma_{yz} = \sigma_{zy} = \mu\left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}\right) \tag{2.11e} $$

$$ \sigma_{zz} = 2\mu\frac{\partial w}{\partial z} \tag{2.11f} $$

Introducing the $x$-components (2.11a)–(2.11c) into the $x$-equation (2.9a) and rearranging gives, at first, six terms. The last three of them vanish because of continuity: the gradient in $x$ of $\left(\partial u/\partial x + \partial v/\partial y + \partial w/\partial z\right)$ is the gradient of zero (by eq. 2.1). Defining the kinematic viscosity $\nu = \mu/\rho$, the viscous $x$-component simplifies to:

$$ \frac{dF_{vx}}{dm} = \nu\left[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right] \tag{2.12a} $$

$$ \frac{dF_{vy}}{dm} = \nu\left[\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right] \tag{2.12b} $$

$$ \frac{dF_{vz}}{dm} = \nu\left[\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right] \tag{2.12c} $$

Why continuity simplifies the viscous term. The full expansion of (2.9a) contains three extra cross-derivative terms, $\partial^2 u/\partial x^2 + \partial^2 v/\partial x\,\partial y + \partial^2 w/\partial x\,\partial z$. This equals $\dfrac{\partial}{\partial x}\!\left(\dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}+\dfrac{\partial w}{\partial z}\right)$, which is the $x$-derivative of the continuity expression (2.1) — and that is zero. So those terms drop, leaving only the clean Laplacian of each velocity component.

4.6 The final Navier–Stokes equations

Substituting the acceleration (2.5a–c), the pressure force (2.6a–c), the body force (2.7a–c) and the viscous force (2.12a–c) into the three components of (2.3) gives the Navier–Stokes equations for incompressible flow, assuming gravity is the only body force and acts along the negative $z$-axis:

$$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + \nu\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) \tag{2.13a} $$

$$ \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial y} + \nu\left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) \tag{2.13b} $$

$$ \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial z} - g + \nu\left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) \tag{2.13c} $$

How to read the Navier–Stokes equations (2.13a–c).
Left-hand side — the acceleration of a fluid particle in the $x$, $y$, $z$ directions (local term $\partial/\partial t$ plus the convective terms).
Right-hand side — the forces per unit mass: the first term is the pressure gradient, the last (in brackets, times $\nu$) is the viscous force. An intermediate term $-g$ appears only in the $z$-equation — that is the effect of gravity.
$\rho$ — density (constant); $p$ — pressure; $\nu = \mu/\rho$ — kinematic viscosity; $g$ — gravity.
These equations are written in component form; other forms are used in later sections.

5. Boundary conditions (§2.4)

Mathematically, the Navier–Stokes equations are second-order, elliptic partial differential equations. Elliptic equations require conditions on all boundaries of the computational domain. The boundaries for the unknowns $u$, $v$, $w$, $p$ are of three kinds: solid surfaces, water surfaces, and "infinity".

5.1 Solid surfaces — the no-slip condition (§2.4.1)

At a solid–liquid interface, interaction occurs at the molecular level: molecules cross between phases and mix in a very thin layer, so the tangential velocity is transferred from one side to the other. Velocity differences are smoothed out, and essentially all experience in fluid mechanics shows the difference is zero — the liquid sticks to the submerged solid surface. This is the "no-slip" condition.

Because the coordinate system moves with the hull, the no-slip condition on the hull surface is simply:

$$ u = v = w = 0 \tag{2.14} $$

On other solid surfaces fixed to the earth — the seabed, beaches, canal banks — the condition is instead:

$$ u = V, \qquad v = w = 0 \tag{2.15} $$

because, in the ship-fixed frame, those surfaces move backward at the speed $V$ relative to the hull. Here $V$ is the steady forward speed of the ship.

A caveat the author flags. Recent research (Watanabe, Udagawa & Udagawa, 1999) suggests that for extremely hydrophobic (water-repellent) surfaces the no-slip condition may not hold — but this is not yet well proven, so the book assumes no-slip throughout.

5.2 Water surface — the free surface (§2.4.2)

The same molecular argument applies to a liquid–gas interface such as the water surface, which the book calls the free surface. Because of molecular interchange between water and air, both attain the same speed at the interface, and there must be an equilibrium of forces across it. Using a local Cartesian system $s,t,n$ with $n$ normal to the surface, the tangential equilibrium reads:

$$ \sigma_{(ns)w} = \sigma_{(ns)a}, \qquad \sigma_{(nt)w} = \sigma_{(nt)a} \tag{2.16} $$

where indices $w$ and $a$ refer to water and air. The normal force equilibrium, including the effect of surface tension $\Delta p_\gamma$ (positive for a concave surface), is:

$$ (\sigma_{nn} - p)_w = (\sigma_{nn} - p)_a + \Delta p_\gamma \tag{2.17} $$

These are the dynamic boundary conditions on the surface. The viscous stresses are normally very small and are mostly neglected; dropping them gives the inviscid dynamic boundary condition:

$$ p = p_a - \Delta p_\gamma \tag{2.18} $$

(the water-pressure index has been dropped). The pressure jump due to surface tension is obtained from (White, 1994, p. 28):

$$ \Delta p_\gamma = \gamma\left(\frac{1}{r_1} + \frac{1}{r_2}\right) \tag{2.19} $$

Meaning of the symbols (eq. 2.19).
$\gamma$ — the surface tension of the water surface.
$r_1,\ r_2$ — the two principal radii of curvature of the water surface (the maximum and minimum normal-curvature directions, at right angles to each other).
The more strongly curved the surface, the larger the pressure jump it can support.

There is also a kinematic condition on the surface: in the macroscopic sense, no flow passes through the surface (the interface is taken as sharp). A water particle moving along the surface must have a vertical velocity equal to the total time-derivative of the wave height $\zeta(x,y)$:

$$ w = \frac{d\zeta}{dt} \tag{2.20} $$

where $\zeta(x,y)$ is the equation of the free surface. The total derivative captures both the temporal change of wave height and its spatial change as the particle moves.

5.3 "Infinity" (§2.4.3)

Although the water has a limited extent, it is often convenient to treat the flow domain as infinite in some directions. The condition is then simply that all disturbances go to zero at infinity:

$$ u = V, \quad v = w = 0, \quad p = p_\infty \tag{2.21} $$

where $p_\infty$ is the undisturbed pressure. These are the mathematical boundary conditions. As Section 9 will show, a real computation must restrict the domain, so artificial numerical boundaries are introduced; there, the pressure and velocities — or their derivatives in one direction — must be specified.

6. Hydrodynamic and hydrostatic pressure (§2.5)

In a liquid at rest, pressure increases linearly downward: each element at a given depth carries the weight of all the fluid above it. This hydrostatic pressure $p_{hs}$ is, in the coordinate system adopted here ($z$ upward):

$$ p_{hs} = -\rho\,g\,z \tag{2.22} $$

(below the surface $z$ is negative, so $p_{hs}$ is positive and grows with depth). Once the liquid is disturbed, pressures related to the motion appear — the hydrodynamic pressure $p_{hd}$. The measurable pressure in a moving fluid is the sum of the two:

$$ p = p_{hs} + p_{hd} \tag{2.23} $$

$$ p_{hd} = p - p_{hs} = p + \rho\,g\,z \tag{2.24} $$

Now look at the pressure and gravity terms together in the $z$-component of Navier–Stokes (2.13c). They are $-\dfrac{1}{\rho}\dfrac{\partial p}{\partial z} - g$. Since $\rho g z$ has $z$-derivative $\rho g$, these two terms combine into a single gradient of $(p + \rho g z) = p_{hd}$:

$$ \frac{dF_p}{dm} + \frac{dF_b}{dm} = -\frac{1}{\rho}\frac{\partial p}{\partial z} - g = -\frac{1}{\rho}\frac{\partial(p + \rho g z)}{\partial z} = -\frac{1}{\rho}\frac{\partial p_{hd}}{\partial z} $$

So if the pressure $p$ in the $z$-component is replaced by $p_{hd}$, the gravity term may be dropped. And because $\partial p_{hd}/\partial x = \partial p/\partial x$ and $\partial p_{hd}/\partial y = \partial p/\partial y$ (from eq. 2.24, since $\rho g z$ does not depend on $x$ or $y$), $p$ may likewise be replaced by $p_{hd}$ in the $x$- and $y$-components. This gives an important conclusion:

"If the pressure in the Navier–Stokes equations is replaced by the hydrodynamic pressure, no gravity terms shall be included." (Larsson & Raven, §2.5)

Why this makes sense. Replacing $p$ by $p_{hd}$ amounts to removing the hydrostatic pressure from the equations. The motions of the flow — which the Navier–Stokes equations govern — are therefore independent of the hydrostatic pressure. This is in fact obvious: the hydrostatic pressure is exactly large enough to balance the weight of each fluid element, so it gives rise to no motion. The author notes this split into hydrodynamic and hydrostatic parts is not usually made in general fluid dynamics — for air the aerostatic pressure is tiny, and liquids are often studied in small systems where hydrodynamic effects dominate the hydrostatic ones.