3. Approximations and Simplified Equations Common Approximations

• Reduction of dimension: ‒ steady-state ‒ 2-dimensional

• Neglect of some fluid property: ‒ incompressible ‒ inviscid

• Simplified forces: ‒ hydrostatic ‒ Boussinesq approximation for density

• Averaging: ‒ depth-averaging (shallow-water flows) ‒ Reynolds averaging (turbulent flows) Time-Dependent vs Steady-State

• Use time-dependent calculations for: ‒ time-dependent problems (waves, pumps, turbines, …) ‒ flows with a natural instability (vortex shedding, ... ) ‒ time-marching to steady state (high-Mach or high-Froude flows)

• Time-dependent calculations: ‒ “parabolic” or “hyperbolic” equations ‒ 1st-order in time; solved by forward-marching in time

• Steady-state calculations: ‒ “elliptic” equations ‒ 2nd-order in space: implicit, iterative solution methods Marine Current Turbine Vortex Shedding From a Cylinder 2- or 3-Dimensional

• Determined by geometry and boundary conditions – but flow instabilities can still lead to 3-dimensionality

• Significant computational implications

• 2-d flows difficult to achieve in the laboratory

• Axisymmetric flows also “2-d” Incompressible Flow

• Definition: a flow is incompressible if flow-induced pressure and temperature changes don’t cause significant density changes

• Usually requires velocity ≪ speed of sound (Ma ≪ 1 )

• “Incompressible” does not necessarily mean “uniform density” ‒ Environmental flows driven by density differences in atmosphere or ocean can still be regarded as incompressible Compressible vs Incompressible CFD Compressible flow: • density changes along a streamline due to large pressure or temperature changes;

• requires an equation for internal energy 푒 (or enthalpy ℎ): change in energy = heat input + work done on fluid

• mass equation → density  energy equation → temperature 푇 equation of state → pressure 푝 (e.g. ideal gas law, 푝 = ρ푅푇)

Incompressible flow: • density constant along a streamline; volume conserved

• mechanical energy equation: change in kinetic energy = work done on fluid can be derived from momentum equation: no separate energy equation

• a pressure equation arises from the requirement that solutions of the momentum equation also be mass-consistent Compressible vs Incompressible CFD

Compressible CFD: • requires an energy equation • pressure determined by equation of state

Incompressible CFD: • does not require a separate energy equation • pressure arises from mass-consistency Viscous vs Inviscid y U

• Viscous (Navier-Stokes) equations: viscous ‒ dynamic (“no-slip”) boundary condition: (푢, 푣, 푤) = 0

y • Inviscid (Euler) equations: U ‒ no 2nd-order derivatives (one less b.c.) inviscid ‒ kinematic (“slip-wall”) boundary condition: 푢푛 = 0

• Inviscid approximation OK only if boundary layer is: ‒ thin (high Re) ‒ attached (no flow separation)

• Inviscid approximation implies no drag, no heat transfer, no sediment transport, … and no flow separation. Potential Flow • Approximation: inviscid, incompressible

• Velocity derived from a velocity potential ϕ: 휕ϕ 휕ϕ 휕ϕ 푢 = , 푣 = , 푤 = 휕푥 휕푦 휕푧 휕2ϕ 휕2ϕ 휕2ϕ + + = 0 휕푢 휕푣 휕푤 휕푥2 휕푦2 휕푧2 + + = 0  휕푥 휕푦 휕푧 Laplace’s Equation u = ∇ϕ ∇2ϕ = 0 ∇ • u = 0  • Consequences: ‒ entire flow field determined by a single scalar ‒ very common equation; plenty of good solvers around ‒ ignores boundary-layer effects: no drag or flow separation Hydrostatic Approximation

z • Approximation: patm ‒ pressure forces balance weight: 휕푝 h-z = −ρ𝑔 Δ푝 = −ρ𝑔Δ푧 휕푧 h(x) p = p atm +  g(h-z)

• Validity: x ‒ always true in stationary fluid ‒ good approximation if vertical acceleration ≪ 𝑔

• Consequence: ‒ pressure is determined everywhere from the depth below the free-surface:

푝 = 푝atm + ρ𝑔(ℎ − 푧) Boussinesq Approximation for Density

• Application: variable-density environmental flows: ‒ atmosphere (temperature); ‒ oceans (salinity). D푤 휕푝 휕푝 ρ = − − ρ𝑔 + ⋯ = − − ρ 𝑔 − (ρ − ρ )𝑔 + ⋯ D푡 휕푧 휕푧 0 0

• Approximation: ‒ retain density changes in buoyancy force ‒ neglect density changes in inertial term (mass  acceleration) D푤 휕푝∗ ρ = − − (ρ − ρ )𝑔 푝∗ = 푝 + ρ 𝑔푧 0 D푡 휕푧 0 0

• Comments: ‒ sometimes needed in theoretical work (to linearise equations); ‒ usually unnecessary in general-purpose CFD (included in iteration). Density-Determining Scalar

D푤 휕푝∗ Vertical momentum equation: ρ0 = − − (ρ − ρ0)𝑔 + ⋯ D푡 휕푧

Density-determining scalar θ (e.g. temperature or salinity):

ρ − ρ0 = −α(θ − θ0) ρ0

D푤 휕푝∗ ρ = − + ρ α(θ − θ )𝑔 + ⋯ 0 D푡 휕푧 0 0 buoyancy force Atmospheric Boundary Layer

stable boundary layer

 u

mixing depth

convective boundary layer

 u Fresh-Water Outfall Shallow-Water Equations

• Application: open-channel hydraulics z • Approximation: depth-averaged h(x,t) ‒ horizontal velocities 푢, 푣 u ‒ water depth ℎ 휕ℎ 휕 mass + 푢ℎ = 0 휕푡 휕푥 x 휕 휕 휕 1 (momentum) (푢ℎ) + (푢2ℎ) = − (1𝑔ℎ2) + (휏 − 휏 ) 휕푡 휕푥 휕푥 2 ρ surface bed

• Compressible-flow analogy: ‒ discontinuity: hydraulic jump  shock ‒ wave speed: 푐 = 𝑔ℎ ↔ 푐 = γ푝/ρ ‒ ratio of current to wave speed: Froude number Fr  Mach number Ma Turbulent Flow

Instantaneous Average Reynolds Averaging

Application: turbulent flows

푢 = 푢lj + 푢′ mean fluctuation

Averaging produces an additional effective stress in the mean flow equations:

′ ′ 휏turb = −ρ푢 푣 Reynolds stress

Common modelling practice: 휕푢ത 휏 = μ μ = eddy viscosity turb 푡 휕푦 푡