Fluid Mechanics and Marine Hydrodynamics I: Fluid Statics, Conservation Laws, Similarity, Viscous Flow and the Boundary Layer

A fluid deforms continuously under shear. Its density ρ (fresh water about 1000 kg/m³, sea water about 1025 kg/m³), its dynamic viscosity μ, defined by Newton’s law of viscosity τ = μ du/dy, and the kinematic viscosity ν = μ/ρ are the properties that matter most here; surface tension and vapour pressure matter for small scales and for cavitation. In a fluid at rest the pressure is the same in every direction and increases with depth as dp/dz = −ρg, so p = p₀ + ρgh in a liquid of constant density.

The hydrostatic force on a plane surface of area A whose centroid is at depth h_c is F = ρg h_c A, acting at the centre of pressure, which lies below the centroid at depth h_cp = h_c + I_G/(h_c A), I_G being the second moment of the area about its own horizontal centroidal axis. For a vertical rectangle of depth d with its top at the surface, h_c = d/2 and the centre of pressure is at 2d/3. On a curved surface the horizontal component equals the force on the vertical projection and the vertical component equals the weight of fluid above the surface.

A floating body displaces its own weight of fluid (Archimedes), and its buoyancy acts at the centre of buoyancy B, the centroid of the displaced volume. It is stable in small heel when the metacentre M lies above the centre of gravity G: GM = KB + BM − KG > 0, with BM = I/∇, I the second moment of the waterplane about the heel axis and ∇ the displaced volume. A fully submerged body has no waterplane, so BM = 0 and it is stable only if G is below B. The naval architecture chapter develops this into the whole of ship stability.

For a control volume, conservation of mass says that the rate of increase of mass inside equals the net inflow; in steady flow through a duct this is ρ₁A₁V₁ = ρ₂A₂V₂, and for an incompressible fluid A₁V₁ = A₂V₂. In differential form it is ∂ρ/∂t + ∇·(ρV) = 0, reducing to ∇·V = 0 for incompressible flow. The momentum equation for a steady control volume, ΣF = Σ(ṁV)_out − Σ(ṁV)_in, gives the force on a pipe bend, a nozzle or a vane and the thrust of a jet or a propeller; the energy equation adds heat and shaft work to the balance of pressure, kinetic and potential energy, and its mechanical-energy form includes the head lost to friction.

For an inviscid fluid, Newton’s law per unit volume is Euler’s equation ρ DV/Dt = −∇p + ρg. Integrated along a streamline in steady flow it becomes Bernoulli’s equation, p/ρ + V²/2 + gz = constant, which holds along each streamline; if the flow is also irrotational the constant is the same everywhere. Its four conditions — steady, inviscid, incompressible, along a streamline — are what an examiner tests when the equation is misapplied. The generalised, unsteady form for potential flow is taken up in the next chapter.

The Buckingham π theorem: a relation among n variables involving k fundamental dimensions can be written as a relation among n − k dimensionless groups. Ship resistance R depends on ρ, V, L, μ and g — six variables and three dimensions (M, L, T) — so R/(½ρV²L²) = f(Re, Fn), a relation among three groups. The groups that matter in marine hydrodynamics are the Reynolds number Re = VL/ν (inertia to viscous forces), the Froude number Fn = V/√(gL) (inertia to gravity, which governs the wave pattern), the Weber number (inertia to surface tension), the Euler number (pressure to inertia) and the cavitation number σ = (p − p_v)/(½ρV²).

Dynamic similarity between model and prototype needs geometric similarity and equal values of every relevant dimensionless group. For a ship model this is impossible: equal Fn requires V_m = V_s/√λ, while equal Re in water of similar viscosity requires V_m = λV_s. The way out is Froude’s hypothesis, which treats the resistance as a frictional part depending on Re and a residuary part depending on Fn; the model is run at the corresponding speed (equal Fn) and the frictional part is corrected between the two Reynolds numbers, as the resistance section of the naval architecture chapter shows.

The Navier–Stokes equations add the viscous term to Euler’s: ρ DV/Dt = −∇p + ρg + μ∇²V for an incompressible Newtonian fluid. They have exact solutions for parallel flows. Couette flow between a fixed plate and one moving at U, gap h, with no pressure gradient, has the linear profile u = Uy/h and shear stress τ = μU/h everywhere. Plane Poiseuille flow between fixed plates driven by a pressure gradient has the parabolic profile u = (1/2μ)(−dp/dx)y(h − y), with u_max = 1.5 ū. In a circular pipe (Hagen–Poiseuille) u_max = 2ū, Q = πR⁴(−dp/dx)/(8μ), and the Darcy friction factor is f = 64/Re.

Pipe flow is laminar below a Reynolds number of about 2000 and becomes turbulent above a few thousand, with random fluctuations that transport momentum far more effectively than viscosity — modelled as a Reynolds stress −ρu′v′ and an eddy viscosity. The turbulent profile is much flatter (roughly a 1/7-power law), with a thin viscous sublayer at the wall. In either regime the head lost to friction is given by the Darcy–Weisbach equation h_f = f(L/D)(V²/2g); in turbulent flow f depends on Re and the relative roughness ε/D, read from the Moody chart. Minor losses at bends, valves and fittings are K V²/2g each.

At high Reynolds number the viscous effects are confined to a thin boundary layer next to the surface. Prandtl’s order-of-magnitude argument reduces Navier–Stokes to the boundary-layer equations u∂u/∂x + v∂u/∂y = −(1/ρ)dp/dx + ν∂²u/∂y², with ∂p/∂y = 0 across the layer, so the pressure is impressed by the outer potential flow. For a flat plate with zero pressure gradient, Blasius’s similarity solution gives the thickness δ ≈ 5.0x/√Reₓ, the displacement thickness δ* = 1.72x/√Reₓ, the momentum thickness θ = 0.664x/√Reₓ, the local skin-friction coefficient cf = 0.664/√Reₓ and the plate drag coefficient C_D = 1.328/√Re_L. The shape factor H = δ*/θ is about 2.59 for the laminar layer.

The displacement thickness δ* = ∫(1 − u/U)dy is the distance the outer flow is pushed away by the velocity deficit; the momentum thickness θ = ∫(u/U)(1 − u/U)dy measures the momentum lost, and the drag per unit width of a plate is ρU²θ at its trailing edge. On a flat plate the layer becomes turbulent at Reₓ of roughly 5 × 10⁵, after which it grows faster, δ ≈ 0.37x/Reₓ^1/5, with C_D ≈ 0.074/Re_L^1/5 and H ≈ 1.3. A ship at full scale has a turbulent boundary layer over nearly its whole length, which is why its frictional resistance is computed from turbulent friction lines.

Separation occurs in an adverse pressure gradient (dp/dx > 0), where the slow fluid near the wall is decelerated until it reverses; the criterion is (∂u/∂y) at the wall = 0, that is zero wall shear. Downstream the flow is a recirculating wake and a large pressure drag appears. A turbulent layer, with more momentum near the wall, resists separation longer than a laminar one — which is why the drag of a sphere falls sharply when its boundary layer turns turbulent. Boundary-layer control delays separation by suction (removing the slow fluid), blowing or tangential injection (re-energising it), slots and slats, moving surfaces, and vortex generators that mix high-momentum fluid down to the wall.