2. FLUID-FLOW EQUATIONS SPRING 2021 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid dynamics is governed by conservation equations for: • mass; • momentum; • energy; • (for a non-homogenous fluid, any other constituents). Equations for these can be expressed mathematically as: • integral (control-volume) equations; • differential equations. This course focuses on the control-volume approach (the basis of the finite-volume method) because it relates naturally to physical quantities, is intrinsically conservative and is easier to apply in modern, unstructured-mesh CFD with complex geometries. However, the equivalent differential equations are easier to write down, manipulate and, in a few cases, solve analytically. The aims of this section are: (i) to derive differential equations for fluid flow; (ii) to demonstrate equivalence of integral and differential forms; (iii) to show that, although there are many different physical V quantities, all satisfy a single generic equation: the scalar- transport or advection-diffusion equation: TIME DERIVATIVE ADVECTIVE + DIFFUSIVE FLUX SOURCE ( ) + ( ) = ( ) (1) of amount in 푉 through boundary of 푉 in 푉 The finite-volume method is a natural discretisation of this. CFD 2 – 1 David Apsley 2.2 Conservative Differential Equations 2.2.1 Mass Conservation (Continuity) Mass is neither created nor destroyed, so: rate of change of mass in cell = net inward mass flux With the more conventional flux direction (positive outward): time derivative of mass in cell + net outward mass flux = 0 d (2) (mass) + ∑ (mass flux) = 0 d푡 faces For a cell volume 푉 and a typical face area 퐴: u mass of fluid in the cell: ρ푉 V un mass flux through one face: 퐶 = ρu • A A t z n A differential equation for mass conservation can be derived by w considering the small cartesian control volume shown left b y s e x If density and velocity are averages over cell volume or cell face, respectively: d(ρ푉) + ⏟(ρ 푢퐴 ) − ( ρ 푢퐴 ) + ( ρ 푣퐴 ) − ( ρ푣퐴 ) + ( ρ푤퐴 ) − (ρ 푤퐴 ) = 0 ⏟d 푡 푒 푤 푛 푠 푡 푏 net outward mass flux time derivative of mass Writing volume 푉 = Δ푥Δ푦Δ푧 and areas 퐴푤 = 퐴푒 = Δ푦Δ푧 etc: d(ρΔ푥Δ푦Δ푧) + [(ρ푢) − (ρ푢) ]Δ푦Δ푧 + [(ρ푣) − (ρ푣) ]Δ푧Δ푥 + [(ρ푤) − (ρ푤) ]Δ푥Δ푦 = 0 d푡 푒 푤 푛 푠 푡 푏 Dividing by the volume, 푥푦푧: dρ (ρ푢) − (ρ푢) (ρ푣) − (ρ푣) (ρ푤) − (ρ푤) + 푒 푤 + 푛 푠 + 푡 푏 = 0 d푡 Δ푥 Δ푦 Δ푧 dρ Δ(ρ푢) Δ(ρ푣) Δ(ρ푤) i.e. + + + = 0 d푡 Δ푥 Δ푦 Δ푧 Taking the limit as Δ푥, Δ푦, Δ푧 → 0: ∂ρ ∂(ρ푢) ∂(ρ푣) ∂(ρ푤) + + + = 0 (3) ∂푡 ∂푥 ∂푦 ∂푧 This analysis is analogous to the finite-volume procedure, but there the control volume does not shrink to a point (finite-volume, not infinitesimal-volume) and cells can be any shape. CFD 2 – 2 David Apsley (*** Advanced / optional ***) For an arbitrary volume 푉 with closed surface 휕푉: d ∫ρ d푉 + ∮ ρu • dA = 0 (4) d푡 ⏟푉 ⏟∂ 푉 mass in cell net mass flux For a fixed control volume, take d/d푡 under the integral sign and apply the divergence theorem to turn the surface integral into a volume integral: ∂ρ ∫ { + ∇ • (ρu)} d푉 = 0 푉 ∂푡 u Since 푉 is arbitrary, the integrand must be identically zero. Hence, V un ∂ρ A + ∇ • (ρu) = 0 (5) ∂푡 Incompressible Flow For incompressible flow, volume as well as mass is conserved, so that: ⏟(푢퐴 ) 푒 − ( 푢퐴 ) 푤 + ( 푣퐴 )푛 − ( 푣퐴 ) 푠 + (푤퐴 ) 푡 − (푤퐴 ) 푏 = 0 net outward VOLUME flux Substituting for face areas, dividing by volume and proceeding to the limit as above produces ∂푢 ∂푣 ∂푤 + + = 0 (6) ∂푥 ∂푦 ∂푧 This is usually taken as the continuity equation in incompressible flow. 2.2.2 Momentum Newton’s Second Law: rate of change of momentum = force time derivative of momentum in cell + net outward momentum flux = force d (mass × u) + ∑ (mass flux × u) = F d푡 (7) faces For a cell volume 푉 and a typical face area 퐴: momentum of fluid in the cell = mass u = (ρ푉)u momentum flux through a face = mass flux × u = (ρu • A)u Momentum and force are vectors, giving (in principle) 3 equations. CFD 2 – 3 David Apsley Fluid Forces There are two main types: • surface forces (proportional to area; act on control-volume faces) • body forces (proportional to volume) (i) Surface forces are usually expressed in terms of stress: force stress = or force = stress × area area The main surface forces are: • pressure 푝: acts normal to a surface; • viscous stresses τ: frictional forces arising from relative motion; y • reaction forces from boundaries. For a simple shear flow there is only one non-zero stress component: ∂푢 τ ≡ τ = μ U 12 ∂푦 but, in general, τ푖푗 is a symmetric tensor with a more complex expression for its components. 1 In incompressible flow , 22 12 ∂푢푖 ∂푢푗 τ푖푗 = μ( + ) y 21 ∂푥푗 ∂푥푖 11 x 11 21 (ii) Body forces are often expressed as forces per unit volume, or force densities. 12 22 The main body forces are: z ● gravity: the force per unit volume is g ρg = −ρ푔e푧 (For constant-density fluids, pressure and weight can be combined as a piezometric pressure 푝∗ = 푝 + ρ푔푧; gravity then no longer appears explicitly in the flow equations.) • centrifugal and Coriolis forces (apparent forces in a rotating reference frame): axis 2 R R centrifugal force: 2 ρΩ R r u Coriolis force: 2 −2ρΩ ∧ u In inertial frame In rotating frame 1 There is a slightly extended expression in compressible flow; see the recommended textbooks. 2 The symbol ˄ here means vector product; Ω is the angular velocity vector, its direction that of the rotation axis. CFD 2 – 4 David Apsley Because the centrifugal force can be written as the gradient of some quantity (in this case, 1 ρΩ2푅2) it can, like gravity, be absorbed into a modified pressure; see the Examples. 2 t Differential Equation For Momentum z Consider a fixed cartesian control volume with sides Δ푥, Δ푦, Δ푧. w n Follow the same process as for mass conservation. y b e s For the 푥-component of momentum: x d (ρ푉푢) + ⏟(ρ 푢퐴 ) 푢 − (ρ 푢퐴 ) 푢 + ( ρ푣퐴 ) 푢 − ( ρ 푣퐴 ) 푢 + (ρ 푤퐴 ) 푢 − ( ρ 푤퐴 ) 푢 d⏟푡 푒 푒 푤 푤 푛 푛 푠 푠 푡 푡 푏 푏 net outward momentum flux time derivative of momentum = ⏟(푝 푤 퐴 푤 − 푝 푒퐴 푒 ) + viscous and other forces pressure force in 푥 direction Substituting cell dimensions: d (ρΔ푥Δ푦Δ푧 푢) + [(ρ푢푢) − (ρ푢푢) ]Δ푦Δ푧 + [(ρ푣푢) − (ρ푣푢) ]Δ푧Δ푥 + [(ρ푤푢) − (ρ푤푢) ]Δ푥Δ푦 d푡 푒 푤 푛 푠 푡 푏 = (푝푤 − 푝푒)Δ푦Δ푧 + viscous and other forces Dividing by volume Δ푥Δ푦Δ푧 (and changing the order of 푝푒 and 푝푤): d(ρ푢) (ρ푢푢) − (ρ푢푢) (ρ푣푢) − (ρ푣푢) (ρ푤푢) − (ρ푤푢) + 푒 푤 + 푛 푠 + 푡 푏 d푡 Δ푥 Δ푦 Δ푧 (푝 − 푝 ) = − 푒 푤 + viscous and other forces Δ푥 In the limit as Δ푥, Δ푦, Δ푧 → 0: ∂(ρ푢) ∂(ρ푢푢) ∂(ρ푣푢) ∂(ρ푤푢) ∂푝 + + + = − + μ∇2푢 + other forces (8) ∂푡 ∂푥 ∂푦 ∂푧 ∂푥 Notes. (1) The viscous term is given without proof (but see the optional notes below). ∂2 ∂2 ∂2 ∇2 is the Laplacian operator + + . ∂푥2 ∂푦2 ∂푧2 (2) The pressure force per unit volume in any direction is minus the pressure gradient in that direction. (3) The 푦 and 푧-momentum equations can be obtained by inspection / pattern-matching. CFD 2 – 5 David Apsley (*** Advanced / optional ***) With surface forces determined by stress tensor σ푖푗 and body forces determined by force density 푓푖, the control-volume equation for the component of momentum may be written d ∫ρ푢푖 d푉 + ∮ ρ푢푖푢푗 d퐴푗 = ∮ σ푖푗 d퐴푗 + ∫푓푖 d푉 d푡 ⏟푉 ⏟∂ 푉 ⏟∂ 푉 ⏟푉 (9) momentum in cell net momentum flux surface forces body forces For fixed 푉, take d/d푡 inside integrals and apply the divergence theorem to surface integrals: ∂(ρ푢푖) ∂(ρ푢푖푢푗) ∂σ푖푗 ∫ { + − − 푓푖} d푉 = 0 푉 ∂푡 ∂푥푗 ∂푥푗 As 푉 is arbitrary, the integrand vanishes identically. Hence, for arbitrary forces: ∂(ρ푢푖) ∂(ρ푢푖푢푗) ∂σ푖푗 + = + 푓푖 (10) ∂푡 ∂푥푗 ∂푥푗 The stress tensor has pressure and viscous parts: σ푖푗 = −푝δ푖푗 + τ푖푗 (11) ∂(ρ푢푖) ∂(ρ푢푖푢푗) ∂푝 ∂τ푖푗 + = − + + 푓푖 (12) ∂푡 ∂푥푗 ∂푥푖 ∂푥푗 For a Newtonian fluid, the viscous stress tensor (including compressible part) is given by ∂푢푖 ∂푢푗 2 ∂푢푘 τ푖푗 = μ( + − δ푖푗 ) ∂푥푗 ∂푥푖 3 ∂푥푘 If the fluid is incompressible and viscosity is uniform then the viscous term simplifies: ∂(ρ푢 푢 ) ∂(ρ푢푖) 푖 푗 ∂푝 2 + = − + μ∇ 푢푖 + 푓푖 ∂푡 ∂푥푗 ∂푥푖 CFD 2 – 6 David Apsley 2.2.3 General Scalar A similar equation may be derived for any physical quantity that is advected and diffused in a fluid flow. Examples include salt, sediment and chemical pollutants. For each such quantity an equation is solved for the concentration (amount per unit mass of fluid) ϕ. Diffusion causes net transport from regions of high concentration to regions of low concentration. For many scalars this rate of transport is proportional to area and concentration gradient and may be quantified by Fick’s diffusion law: rate of diffusion = −diffusivity × gradient × area ∂ϕ = −Γ 퐴 ∂푛 This is often referred to as gradient diffusion. An example is heat conduction. u For an arbitrary control volume: V un amount in cell: ρ푉ϕ (mass concentration) A advective flux: (ρu • A)ϕ (mass flux concentration) diffusive flux: ∂ϕ (– diffusivity gradient area) −Γ 퐴 ∂푛 source: 푆 = 푠푉 (source density volume) Balancing the time derivative of the amount in the cell, the net flux through the boundary and rate of production yields the general scalar-transport (or advection-diffusion) equation: time derivative + net outward flux = source d ∂ϕ (mass × ϕ) + ∑ (mass flux × ϕ − Γ 퐴) = 푆 d푡 ∂푛 (13) faces (Conservative) differential equation: ∂(ρϕ) ∂ ∂ϕ ∂ ∂ϕ ∂ ∂ϕ + (ρ푢ϕ − Γ ) + (ρ푣ϕ − Γ ) + (ρ푤ϕ − Γ ) = 푠 (14) ∂푡 ∂푥 ∂푥 ∂푦 ∂푦 ∂푧 ∂푧 (*** Advanced / optional ***) The integral equation may be expressed more mathematically as: d ∫ρϕ d푉 + ∮ (ρuϕ − Γ∇ϕ) • dA = ∫푠 d푉 (15) d푡 푉 ∂푉 푉 For a fixed control volume, taking the time derivative under the integral sign and using the divergence theorem gives a corresponding conservative differential equation: ∂(ρϕ) + ∇ • (ρuϕ − Γ∇ϕ) = 푠 (16) ∂푡 CFD 2 – 7 David Apsley 2.2.4 Momentum Components as Transported Scalars In the momentum equation, if the viscous force τ퐴 = μ(∂푢/ ∂푛)퐴 is transferred to the LHS it looks like a diffusive flux.