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 = −ρ Δ푝 = −ρΔ푧 휕푧 h(x) p = p atm + g(h-z) • Validity: x ‒ always true in stationary fluid ‒ good approximation if vertical acceleration ≪ • Consequence: ‒ pressure is determined everywhere from the depth below the free-surface: 푝 = 푝atm + ρ(ℎ − 푧) Boussinesq Approximation for Density • Application: variable-density environmental flows: ‒ atmosphere (temperature); ‒ oceans (salinity). D푤 휕푝 휕푝 ρ = − − ρ + ⋯ = − − ρ − (ρ − ρ ) + ⋯ 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 ) 푝∗ = 푝 + ρ 푧 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) + ⋯ D푡 휕푧 Density-determining scalar θ (e.g. temperature or salinity): ρ − ρ0 = −α(θ − θ0) ρ0 D푤 휕푝∗ ρ = − + ρ α(θ − θ ) + ⋯ 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) = − (1ℎ2) + (휏 − 휏 ) 휕푡 휕푥 휕푥 2 ρ surface bed • Compressible-flow analogy: ‒ discontinuity: hydraulic jump shock ‒ wave speed: 푐 = ℎ ↔ 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.