PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m = 1,2,... Nabla operator = i x + j y + k x = (x, y, z), v = (v x, v y, v z ) [ u = i u x + j u y + k u = v = v x v = det x + v y y + v z i j k x y u x, u y, u = ] T gradient v z y v y v x v z x divergence curl i z k j v y v x v y v z v y x v x y x u = ( u) = 2 u = 2 u x + 2 u 2 y + 2 u 2 2 Laplacian
Tensorial quantities in fluid dynamics elocity gradient v = [ v x, v y, v z ] = v x x v x y v x v y x v y y v y v z x v z y v z 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 v Remark. The trace (sum of diagonal elements) of v equals v. Deformation rate tensor (symmetric part of v) D(v) = 1 2 ( v + vt ) = 1 2 1 2 v x x ( vx y + v y x ( vx + v z x ) ) 1 2 1 2 ( ) vy x + v x y v y y ( vy + v z y ) 1 2 1 2 ( vz x + v x ( vz y + v y v z ) ) Spin tensor S(v) = v D(v) (skew-symmetric part of v)
ector multiplication rules Scalar product of two vectors a,b R 3, a b = a T b = [a 1 a 2 a 3 ] b 1 b 2 b 3 = a 1b 1 + a 2 b 2 + a 3 b 3 R Example. u v u = v x x + v u y y + v u z convective derivative Dyadic product of two vectors a,b R 3, a b = ab T = a 1 a 2 [b 1 b 2 b 3 ] = a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 R3 3 a 3 a 3 b 1 a 3 b 2 a 3 b 3
Elementary tensor calculus 1. αt = {αt ij }, T = {t ij } R 3 3, α R 2. T 1 + T 2 = {t 1 ij + t2 ij }, T 1, T 2 R 3 3, a R 3 t 11 t 12 t 13 3. a T = [a 1, a 2, a 3 ] t 21 t 22 t 23 = 3 a i [t i1, t i2, t i3 ] i=1 }{{} t 31 t 32 t 33 i-th row t 11 t 12 t 13 a 1 4. T a = t 21 t 22 t 23 a 2 = 3 t 1j t 2j a j (j-th column) j=1 t 31 t 32 t 33 a 3 t 3j t 5. T 1 T 2 1 11 t 1 12 t 1 13 t = t 1 21 t 1 22 t 1 2 11 t 2 12 t 2 { 13 23 t 2 t 1 31 t 1 32 t 1 21 t 2 22 t 2 3 } 23 = t 1 ik t2 kj 33 t 2 31 t 2 32 t 2 k=1 33 6. T 1 : T 2 = tr (T 1 (T 2 ) T ) = 3 3 t 1 ik t2 ik i=1 k=1
Divergence theorem of Gauß Let Ω R 3 and n be the outward unit normal to the boundary Γ = Ω\Ω. Then Ω f dx = Γ f n ds for any differentiable function f(x) Example. A sphere: Ω = {x R 3 : x < 1}, Γ = {x R 3 : x = 1} where x = x x = x 2 + y 2 + z 2 is the Euclidean norm of x Consider f(x) = x so that f 3 in Ω and n = x x volume integral: surface integral: Ω Γ f dx = 3 f n ds = Γ Ω dx = 3 Ω = 3 x x x ds = Γ [ ] 4 3 π13 x ds = Γ on Γ = 4π ds = 4π
Governing equations of fluid dynamics Physical principles 1. Mass is conserved 2. Newton s second law 3. Energy is conserved Mathematical equations continuity equation momentum equations energy equation It is important to understand the meaning and significance of each equation in order to develop a good numerical method and properly interpret the results Description of fluid motion z v Eulerian monitor the flow characteristics Lagrangian in a fixed control volume track individual fluid particles as i k j ܼ Ý ¼ Þ ¼µ y they move through the flow field x ܽ Ý ½ Þ ½µ
Trajectory of a fluid particle Description of fluid motion x i z k j v ܽ Ý ½ Þ ½µ ¼µ Þ ¼ Ý Ü¼ y x = x(x 0, t) x = x(x 0, y 0, z 0, t) y = y(x 0, y 0, z 0, t) z = z(x 0, y 0, z 0, t) dx dt = v x(x, y, z, t), x t0 = x 0 dy dt = v y(x, y, z, t), y t0 = y 0 dz dt = v z(x, y, z, t), z t0 = z 0 Definition. A streamline is a curve which is tangent to the velocity vector v = (v x, v y, v z ) at every point. It is given by the relation dx v x = dy v y = dz v z y v y(x) dy dx = v y v x x Streamlines can be visualized by injecting tracer particles into the flow field.
Flow models and reference frames Eulerian Lagrangian S S integral fixed C of a finite size moving C of a finite size d fixed infinitesimal C ds d moving infinitesimal C ds differential Good news: all flow models lead to the same equations
Eulerian vs. Lagrangian viewpoint Definition. Substantial time derivative d dt is the rate of change for a moving fluid particle. Local time derivative is the rate of change at a fixed point. Let u = u(x, t), where x = x(x 0, t). The chain rule yields du dt = u + u dx x dt + u dy y dt + u dz dt = u + v u substantial derivative = local derivative + convective derivative Reynolds transport theorem d dt t u(x, t) d = u(x, t) t d + u(x, t)v n ds S S t rate of change in a moving volume = rate of change in a fixed volume + convective transfer through the surface
Derivation of the governing equations Modeling philosophy 1. Choose a physical principle conservation of mass conservation of momentum conservation of energy 2. Apply it to a suitable flow model Eulerian/Lagrangian approach for a finite/infinitesimal C 3. Extract integral relations or PDEs which embody the physical principle Generic conservation law Z Z Z u d + f n ds = S ds n f S q d f = vu d u flux function Divergence theorem yields Z Z Z u d + f d = Partial differential equation u + f = q in q d
Derivation of the continuity equation Physical principle: conservation of mass dm dt = d ρ ρ d = dt t t d + ρv n ds = 0 S S t accumulation of mass inside C = net influx through the surface Divergence theorem yields [ ] ρ + (ρv) Lagrangian representation d = 0 Continuity equation ρ + (ρv) = 0 (ρv) = v ρ + ρ v dρ dt + ρ v = 0 Incompressible flows: dρ dt = v = 0 (constant density)
Conservation of momentum Physical principle: f = ma (Newton s second law) d g ds h n total force f = ρg d + h ds, where h = σ n body forces g gravitational, electromagnetic,... surface forces h pressure + viscous stress Stress tensor σ = pi + τ momentum flux For a newtonian fluid viscous stress is proportional to velocity gradients: τ = (λ v)i + 2µD(v), where D(v) = 1 2 ( v + vt ), λ 2 3 µ Normal stress: stretching τ xx = λ v + 2µ v x x τ yy = λ v + 2µ v y y τ zz = λ v + 2µ v z Ý ÜÜ Ü Shear stress: deformation vy τ xy = τ yx = µ + v x x y τ xz = τ zx = µ ` v x + v z x v τ yz = τ zy = µ z + v y y Ý ÝÜ Ü
Derivation of the momentum equations Newton s law for a moving volume d (ρv) ρv d = d + (ρv v) n ds dt t t S S t = ρg d + σ n ds t S S t Transformation of surface integrals [ ] (ρv) + (ρv v) d = [ σ + ρg] d, σ = pi + τ (ρv) Momentum equations + (ρv v) = p + τ + ρg [ ] [ ] (ρv) v ρ + (ρv v) = ρ + v v + v + (ρv) = ρ dv dt }{{}}{{} substantial derivative continuity equation
Conservation of energy Physical principle: δe = s + w (first law of thermodynamics) d g ds δe accumulation of internal energy n h s heat transmitted to the fluid particle w rate of work done by external forces Heating: s = ρq d f q ds Fourier s law of heat conduction q f q T κ internal heat sources diffusive heat transfer absolute temperature thermal conductivity f q = κ T the heat flux is proportional to the local temperature gradient Work done per unit time = total force velocity w = f v = ρg v d + v (σ n) ds, σ = pi + τ
Derivation of the energy equation Total energy per unit mass: E = e + v 2 2 e specific internal energy due to random molecular motion v 2 2 specific kinetic energy due to translational motion Integral conservation law for a moving volume d (ρe) ρe d = d + ρe v n ds dt t t S S t = ρq d + κ T n ds t S S t + ρg v d + v (σ n) ds t S S t accumulation heating work done Transformation of surface integrals [ ] (ρe) + (ρev) d = [ (κ T) + ρq + (σ v) + ρg v] d, where (σ v) = (pv) + (τ v) = (pv) + v ( τ) + v : τ
Total energy equation (ρe) Different forms of the energy equation + (ρev) = (κ T) + ρq (pv) + v ( τ) + v : τ + ρg v (ρe) Momentum equations [ ] [ ] E ρ + (ρev) = ρ + v E + E + (ρv) }{{}}{{} substantial derivative continuity equation ρ dv dt ρ de dt = ρde dt + v ρdv dt = (ρe) Internal energy equation (ρe) = p + τ + ρg = ρ de dt (Lagrangian form) + (ρev) + v [ p + τ + ρg] + (ρev) = (κ T) + ρq p v + v : τ
Summary of the governing equations 1. Continuity equation / conservation of mass ρ + (ρv) = 0 2. Momentum equations / Newton s second law (ρv) + (ρv v) = p + τ + ρg 3. Energy equation / first law of thermodynamics (ρe) + (ρev) = (κ T) + ρq (pv) + v ( τ) + v : τ + ρg v E = e + v 2 2, (ρe) + (ρev) = (κ T) + ρq p v + v : τ This PDE system is referred to as the compressible Navier-Stokes equations
Conservation form of the governing equations Generic conservation law for a scalar quantity u + f = q, where f = f(u,x, t) is the flux function Conservative variables, fluxes and sources ρ ρv U = ρv, F = ρv v + pi τ ρe (ρe + p)v κ T τ v, Q = 0 ρg ρ(q + g v) Navier-Stokes equations in divergence form U + F = Q U R 5, F R 3 5, Q R 5 representing all equations in the same generic form simplifies the programming it suffices to develop discretization techniques for the generic conservation law
Constitutive relations ariables: ρ, v, e, p, τ, T Equations: continuity, momentum, energy The number of unknowns exceeds the number of equations. 1. Newtonian stress tensor τ = (λ v)i + 2µD(v), D(v) = 1 2 ( v + vt ), λ 2 3 µ 2. Thermodynamic relations, e.g. p = ρrt ideal gas law R specific gas constant e = c v T caloric equation of state c v specific heat at constant volume Now the system is closed: it contains five PDEs for five independent variables ρ, v, e and algebraic formulae for the computation of p, τ and T. It remains to specify appropriate initial and boundary conditions.
Initial and boundary conditions Initial conditions ρ t=0 = ρ 0 (x), v t=0 = v 0 (x), e t=0 = e 0 (x) in Ω Boundary conditions Inlet Γ in = {x Γ : v n < 0} Let Γ = Γ in Γ w Γ out Û ρ = ρ in, v = v in, e = e in Ò Å ÓÙØ prescribed density, energy and velocity Solid wall Γ w = {x Γ : v n = 0} v = 0 no-slip condition T = T w given temperature or ( T f n) = q κ prescribed heat flux Outlet Γ out = {x Γ : v n > 0} v n = v n or p + n τ n = 0 v s = v s or s τ n = 0 prescribed velocity vanishing stress The problem is well-posed if the solution exists, is unique and depends continuously on IC and BC. Insufficient or incorrect IC/BC may lead to wrong results (if any).