2. Fluid-Flow Equations Governing Equations • Conservation equations for: ‒ mass ‒ momentum ‒ energy ‒ (other constituents) • Alternative forms: ‒ integral (control-volume) equations ‒ differential equations Integral (Control-Volume) Approach Consider the budget of any physical quantity in any control volume V V TIME DERIVATIVE ADVECTIVE + DIFFUSIVE FLUX SOURCE + = of amount in 푉 through boundary of 푉 in 푉 → Finite-volume method for CFD Mass Conservation (Continuity) Mass conservation: mass is neither created nor destroyed d u (mass) = net inward mass flux d푡 V un A d (mass) + net outward mass flux = 0 d푡 d mass + ෍ mass flux = 0 d푡 faces Mass in a cell: ρ푉 Mass flux through a face: 퐶 = ρu • A Mass Conservation - Differential Equation t Conservation statement: d z mass + net outward mass flux = 0 w n d푡 y b e d s ρ푉 + (ρ푢퐴) − (ρ푢퐴) + (ρ푣퐴) − (ρ푣퐴) + (ρ푤퐴) − (ρ푤퐴) = 0 x d푡 푒 푤 푛 푠 푡 푏 d ρΔ푥Δ푦Δ푧 + [(ρ푢) − ρ푢) Δ푦Δ푧 + [(ρ푣) − ρ푣) Δ푧Δ푥 + [(ρ푤) − ρ푤) Δ푥Δ푦 = 0 d푡 푒 푤 푛 푠 푡 푏 Divide by volume: 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) + + + = 0 d푡 Δ푥 Δ푦 Δ푧 Shrink to a point: 휕ρ 휕(ρ푢) 휕(ρ푣) 휕(ρ푤) 휕ρ + + + = 0 + ∇ • (ρu) = 0 휕푡 휕푥 휕푦 휕푧 휕푡 Continuity in Incompressible Flow t z w n b y s e d x (volume) + net outward volume flux = 0 d푡 휕푢 휕푣 휕푤 + + = 0 ∇ • u = 0 휕푥 휕푦 휕푧 Momentum Equation Momentum Principle: force = rate of change of momentum F If steady: force = (momentum flux)out – (momentum flux)in If unsteady: force = d/d푡(momentum inside control volume) + (momentum flux)out – (momentum flux)in Momentum Equation Rate of change of momentum = force d (momentum) + net outward momentum flux = force d푡 u V un A d mass × u + ෍ (mass flux × u) = F d푡 faces Momentum of fluid in a cell = mass × u = (ρ푉)u Momentum flux through a face = mass flux × u = (ρu • A)u Fluid Forces force Surface forces (proportional to area): stress = area • pressure y 휕푢 • viscous force: τ = μ 휕푦 U • reactions from boundaries force Body forces (proportional to volume): force density = z volume • gravity: −ρe푧 g axis 2 R R 2 • centrifugal force: ρΩ R r u −2ρΩ ∧ u • Coriolis force: In inertial frame In rotating frame Differential Equation Conservation statement: t d z (momentum) + net momentum flux = force d푡 w n y b e d s ρ푉푢 + (ρ푢퐴) 푢 − (ρ푢퐴) 푢 + (ρ푣퐴) 푢 − (ρ푣퐴) 푢 + (ρ푤퐴) 푢 − (ρ푤퐴) 푢 d푡 푒 푒 푤 푤 푛 푛 푠 푠 푡 푡 푏 푏 x = 푝푤퐴푤 − 푝푒퐴푒 + viscous and other forces d ρΔ푥Δ푦Δ푧 푢 + [(ρ푢) 푢 − ρ푢) 푢 Δ푦Δ푧 + [(ρ푣) 푢 − ρ푣) 푢 Δ푧Δ푥 + [(ρ푤) 푢 − ρ푤) 푢 Δ푥Δ푦 d푡 푒 푒 푤 푤 푛 푛 푠 푠 푡 푡 푏 푏 = 푝푤 − 푝푒 Δ푦Δ푧 + viscous and other forces Divide by volume: d(ρ푢) (ρ푢푢)푒 − (ρ푢푢)푤 (ρ푣푢)푛 − (ρ푣푢)푠 (ρ푤푢)푡 − (ρ푤푢)푏 푝푒 − 푝푤 viscous and + + + = − + d푡 Δ푥 Δ푦 Δ푧 Δ푥 other forces d(ρ푢) Δ(ρ푢푢) Δ(ρ푣푢) Δ(ρ푤푢) Δ푝 viscous and + + + = − + d푡 Δ푥 Δ푦 Δ푧 Δ푥 other forces Shrink to a point: 휕(ρ푢) 휕(ρ푢푢) 휕(ρ푣푢) 휕(ρ푤푢) 휕푝 + + + = − + μ∇2푢 + other forces 휕푡 휕푥 휕푦 휕푧 휕푥 General Scalar Time derivative + net outward flux = source ϕ = concentration (amount per unit mass) ρ푉ϕ Amount in a cell: (mass concentration) u Flux through a face: V un ‒ advection: (ρu • A)ϕ (mass flux concentration) A 휕ϕ ‒ diffusion: −Γ 퐴 휕푛 (diffusivity gradient area) Source: 푆 = 푠푉 (source density volume) d 휕ϕ mass × ϕ + ෍ (mass flux × ϕ − Γ 퐴) = 푠 푉 d푡 휕푛 faces 휕(ρϕ) 휕 휕ϕ 휕 휕ϕ 휕 휕ϕ + ρ푢ϕ − Γ + ρ푣ϕ − Γ + ρ푤ϕ − Γ = 푠 휕푡 휕푥 휕푥 휕푦 휕푦 휕푧 휕푧 Momentum Components as General Scalars Momentum equation: d 휕푢 mass × 푢 + ෍ mass flux × 푢 = ෍ ( μ 퐴) + other forces d푡 휕푛 faces faces viscous forces d 휕푢 mass × 푢 + ෍ mass flux × 푢 − μ 퐴 = other forces d푡 휕푛 faces General scalar-transport equation: d 휕ϕ mass × ϕ + ෍ mass flux × ϕ − Γ 퐴 = 푆 d푡 휕푛 faces • Velocity components 푢, 푣, 푤 satisfy individual scalar-transport equations: ‒ concentration, ϕ velocity component, 푢, 푣, 푤 ‒ diffusivity, viscosity, μ ‒ source, 푆 non-viscous forces • Differences: ‒ momentum equations are non-linear ‒ momentum equations are coupled ‒ the velocity field also has to be mass-consistent Differential Equations For Fluid Flow Forms of the equations in primitive variables may be: • Conservative ‒ can be integrated directly to give “net flux = source” • Non-conservative ‒ can’t be integrated directly Other forms of the equations include those for: • Derived variables ‒ e.g. velocity potential Example d (푦2) = (푥) conservative d푥 d푦 2푦 = (푥) non-conservative d푥 Same equation! ... but only the first can be integrated directly Rate of Change Following the Flow ϕ ≡ ϕ(푡, x) Total derivative dϕ 휕ϕ 휕ϕ d푥 휕ϕ d푦 휕ϕ d푧 ≡ + + + (following any path x(푡) (x(t), y(t), z(t)) d푡 휕푡 휕푥 d푡 휕푦 d푡 휕푧 d푡 Material derivative d푥 Dϕ 휕ϕ 휕ϕ 휕ϕ 휕ϕ = u ≡ + 푢 + 푣 + 푤 (following the flow): d푡 D푡 휕푡 휕푥 휕푦 휕푧 D 휕 휕 휕 휕 ≡ + 푢 + 푣 + 푤 D푡 휕푡 휕푥 휕푦 휕푧 D 휕 ≡ + u • ∇ D푡 휕푡 Non-Conservative Flow Equations conservative form non-conservative form 휕 휕 휕 휕 Dϕ (ρϕ) + (ρ푢ϕ) + (ρ푣ϕ) + (ρ푤ϕ) → ρ 휕푡 휕푥 휕푦 휕푧 D푡 (mass conservation) D푢 휕푝 ρ = − + μ∇2푢 e.g. momentum equation: D푡 휕푥 mass × acceleration forces 휕(ρϕ) 휕(ρ푢ϕ) 휕(ρ푣ϕ) 휕(ρ푤ϕ) Proof: + + + 휕푡 휕푥 휕푦 휕푧 휕ρ 휕ϕ 휕(ρ푢) 휕ϕ 휕(ρ푣) 휕ϕ 휕(ρ푤) 휕ϕ = ϕ + ρ + ϕ + ρ푢 + ϕ + ρ푣 + ϕ + ρ푤 휕푡 휕푡 휕푥 휕푥 휕푦 휕푦 휕푧 휕푧 휕ρ 휕(ρ푢) 휕(ρ푣) 휕(ρ푤) 휕ϕ 휕ϕ 휕ϕ 휕ϕ = + + + ϕ + ρ + 푢 + 푣 + 푤 휕푡 휕푥 휕푦 휕푧 휕푡 휕푥 휕푦 휕푧 =0 by continuity =Dϕ/D푡 by definition Dϕ = ρ D푡 Example Q1 (Equation Manipulation) In 2-d flow, the continuity and x-momentum equations can be written in conservative form as 휕ρ 휕 휕 휕 휕 휕 휕푝 + (ρ푢) + (ρ푣) = 0 (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 휕푥 (a) Show that these can be written in the equivalent non-conservative forms: Dρ 휕푢 휕푣 D푢 휕푝 + ρ( + ) = 0 ρ = − + μ∇2푢 D푡 휕푥 휕푦 D푡 휕푥 (b) Define carefully what is meant by the statement that a flow is incompressible. To what does the continuity equation reduce in incompressible flow? (c) Write down conservative forms of the 3-d equations for mass and x-momentum. (d) Write down the 푧-momentum equation, including the gravitational force. (e) Show that, for constant-density flows, pressure and gravity can be combined in the momentum equations via the piezometric pressure 푝 + ρ푧. axis (f) In a rotating reference frame there are additional apparent forces (per unit volume): 2 R R 2 centrifugal force: ρΩ R r Coriolis force: −2ρΩ ∧ u where Ω is the angular velocity of the reference frame, u is the fluid velocity in that frame, r is the position vector and R is its projection perpendicular to the axis of rotation. By writing the centrifugal force as the gradient of some quantity show that it can be subsumed into a modified pressure. Also, find the components of the Coriolis force if rotation is about the 푧 axis. In 2-d flow, the continuity and x-momentum equations can be written in conservative form as 휕ρ 휕 휕 휕 휕 휕 휕푝 + (ρ푢) + (ρ푣) = 0 (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 휕푥 (a) Show that these can be written in the equivalent non-conservative forms: Dρ 휕푢 휕푣 D푢 휕푝 + ρ( + ) = 0 ρ = − + μ∇2푢 D푡 휕푥 휕푦 D푡 휕푥 Continuity: Momentum: 휕ρ 휕 휕 휕 휕 휕 휕푝 + (ρ푢) + (ρ푣) = 0 (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 휕푥 휕ρ 휕푢 휕ρ 휕ρ 휕푢 휕ρ 휕푣 LHS = 푢 + ρ + 푢 + ρ + 푣 + ρ = 0 휕푡 휕푡 휕푡 휕푥 휕푥 휕푦 휕푦 휕(ρ푢) 휕푢 + 푢 + ρ푢 휕푥 휕푥 휕(ρ푣) 휕푢 + 푢 + ρ푣 휕ρ 휕ρ 휕ρ 휕푢 휕푣 휕푦 휕푦 + 푢 + 푣 + ρ + = 0 휕푡 휕푥 휕푦 휕푥 휕푦 휕ρ 휕(ρ푢) 휕(ρ푣) 휕푢 휕푢 휕푢 LHS = + + 푢 + ρ + 푢 + 푣 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 Dρ 휕푢 휕푣 =0 by mass conservation =D푢ΤD푡 + ρ + = 0 D푡 휕푥 휕푦 D푢 휕푝 ρ = − + μ∇2푢 D푡 휕푥 (b) Define carefully what is meant by the statement that a flow is incompressible. To what does the continuity equation reduce in incompressible flow? Incompressible: flow-induced changes to pressure (or temperature) do not cause significant changes in density Dρ = 0 D푡 Dρ 휕푢 휕푣 + ρ + = 0 D푡 휕푥 휕푦 휕푢 휕푣 + = 0 휕푥 휕푦 휕ρ 휕 휕 2-d continuity: + (ρ푢) + (ρ푣) = 0 휕푡 휕푥 휕푦 휕 휕 휕 휕푝 휕 휕 2-d x-momentum: (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 ∇2≡ + 휕푡 휕푥 휕푦 휕푥 휕푥2 휕푦2 (c) Write down conservative forms of the 3-d equations for mass and x-momentum. 휕ρ 휕 휕 휕 3-d continuity: + (ρ푢) + (ρ푣) + (ρ푤) = 0 휕푡 휕푥 휕푦 휕푧 휕 휕 휕 휕 휕푝 휕 휕 휕 3-d x-momentum: (ρ푢) + (ρ푢푢) + (ρ푣푢) + (ρ푤푢) = − + μ∇2푢 ∇2≡ + + 휕푡 휕푥 휕푦 휕푧 휕푥 휕푥2 휕푦2 휕푧2 (d) Write down the 푧-momentum equation, including the gravitational force. 휕 휕 휕 휕 휕푝 3-d z-momentum: ρ푤 + ρ푢푤 + ρ푣푤 + ρ푤푤 = − − ρ + μ∇2푤 휕푡 휕푥 휕푦 휕푧 휕푧 (e) Show that, for constant-density flows, pressure and gravity can be combined in the momentum equations via the piezometric pressure 푝 + ρ푧.