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1 Seite 1 Introduction Computational Fluid Dynamics Computational Fluid Dynamics 2 Turbulence effects and Particle transport Martin Pietsch Computational Biomechanics Summer Term 2016

2 Seite 2 Review Computational Fluid Dynamics System equations: 1 mass conservation ρ t 2 momentum conservation 3 energy conservation + (ρ u ) = 0 ρ u t + (ρ u ) u = ρ f + σ ρ e t = ρ Q + (κ T ) + (σ u ) ( σ ) u 4 equation of state (e.g. ideal gas equation)

3 Seite 3 Review Computational Fluid Dynamics Incompressible fluid / flow assumption: Incompressible flow: 0 = dρ (x, t) dt = ρ(x, t) + ρ(x, t) u t Incompressible fluid: ρ (x, t) = ρ(x, t) = 0 t Continuity equation: 0,flow ass. {}}{ ρ ρ + (ρ u ) = + ρ u +ρ u = ρ u = 0 t }{{} t }{{} 0,fluid ass. 0,fluid ass. t It follows: u = 0 3 t 2 (divergency free velocity field) t 1

4 Seite 4 Review Computational Fluid Dynamics Incompressible flow/fluid + isothermal assumption: From T = const. with d dt ρ = 0 follows: 1 Pressure is given with p ρ (equation of state) 2 Energy is a function of ρ and u the energy conservation contains no extra information For a newtonian fluid we get the Navier-Stokes equations as Navier-Stokes equations u = 0 (1) u t + ( u ) u = f 1 p + ν τ (2) ρ Note: often, the kinematic viscosity ν := µ ρ is used

5 Seite 5 Dimensionless description Computational Fluid Dynamics Dimensionless Navier-Stokes: Navier-Stokes momentum equation u t + ( u ) u = f 1 ρ p + µ ρ τ Define characteristic time T, length L and velocity U with L = U T : τ = t T v = u U ξ = x L Dimensionless representation of the momentum equation: v τ + ( v ) v = L U 2 f 1 ρu 2 p + µ ρul τ dimensionless forcedensity κ := L f (look for Froude number) U 2 pressure rescaling p := (NOTE: only for inc. fluid) p ρu 2

6 Seite 6 Dimensionless description Computational Fluid Dynamics Diffusion term & Reynolds number: v µ + ( v ) v = κ p + τ ρul τ Definition of the Reynolds number: Re := inertia forces viscous forces = ρul µ inertia force: F in = ρl3 U T (momentum transfer) viscous force: F vis = µl 2 U L ( velocity diffusion ) Dimensionless Navier-Stokes equations v τ v = 0 (3) 1 + ( v ) v = κ p + τ (4) Re

7 Seite 7 Pressure equation Computational Fluid Dynamics Pressure equation: v τ + ( v ) v = κ p + 1 Re τ Divergency free velocity field implies ( ) ( v + ( v ) v = κ p + 1 ) τ Re τ with τ v = 0, we get the Poissin-Pressure equation: ( p = κ ( v ) v + 1 ) Re τ

8 Seite 8 Turbulent flow Computational Fluid Dynamics Turbulent flow: If Re << 1, the diffusion time scale is much smaller as the time scale for momentum transportation velocity field perturbations smooth out quickly velocity field tends to be laminar If Re >> 1, momentum transportation is the main effect for the fluid flow description velocity field perturbations increase quickly velocity field tends to be turbulent Example: (flow in pipe) Reynolds number: Re = ρ d vz µ Observation: Julius Rotta (at 1950) Re krit r v, µ, ρ z d

9 Seite 9 Turbulent flow Computational Fluid Dynamics Energy cascade: 1 energy injection range (small viscous effects) 2 inertial subrange 3 dissipation range (large viscous effects) log(e) log(k) visualization after the model of Lewis Fry Richardson e := energy, k := wave number

10 Seite 10 Turbulent flow Computational Fluid Dynamics Kolmogorov scales: The smallest scales that influences the turbulent flow by dissipation effects. Note: To retain energy conservation at the numerical domain, one have to resolve also the dissipative scales in the Navier-Stokes equation! The scales are given as: (ɛ is the average dissipation rate) ( ) 1 ( ) 1 µ 3 4 µ length : η = ɛ ρ 3 vel : u η = ρ ɛ 4 time : τ η = ( µ ρ ɛ ) 1 2 with Re η = η u η µ ρ = 1

11 Seite 11 Turbulence models Computational Fluid Dynamics Resolution problem: Approximation of the dissipation rate (from large scales): ɛ kinetic energy time U2 T = U3 L Therefore we get the relation: ( ) L µ 3 1 η = L ɛ ρ 3 4 L ( U 3 ρ 3 L µ 3 ) 1 4 = Re 3 4 Example: (L 10 3 m, v 1 m s Re , ρ 1.3 kg m 3, µ 17.1 µpa s) η m

12 Seite 12 Turbulence models Computational Fluid Dynamics Resolution problem: Approximation of the dissipation rate (from large scales): ɛ kinetic energy time U2 T = U3 L Therefore we get the relation: ( ) L µ 3 1 η = L ɛ ρ 3 4 L ( U 3 ρ 3 L µ 3 ) 1 4 = Re 3 4 Example: (L 10 3 m, v 0.1 m s Re 35 η m, ρ 1060 kg m 3, µ 3 mpa s)

13 Seite 13 Turbulence models Computational Fluid Dynamics Simulation approaches: Direct numerical simulation (DNS): Assumption that the flow inside of a volume element is purely laminar and no dissipation effect occurs. (Note: If this is not true, the energy conservation results in a different flow field.) Eddy dissipation modelling on small scales: Reynolds-Averaged Navier Stokes (RANS) Large-Eddy Simulation... v = v + v and p = p + p with the mean value of and the fluctuating part.

14 Seite 14 Turbulence models Computational Fluid Dynamics RANS: Special cases: temporal or spatial averaging N In general: f ( x, t) = lim f ( x, t) N n Fluctuating part: f = 0 Reynolds equations: v t v = 0 + ( v ) v = f p + 1 Re τ ( v ) v }{{} correlation property v v v xv x v x v y v xv = v y v x v y v y v y v z v zv x v z v y v zv z

15 Seite 15 Turbulence models Computational Fluid Dynamics RANS models: Zero equation models ν T = ξ 2 v (mixing length ξ) One equation models (example: Spalart and Allmaras) ( ) ν T νt + v ν T = ν T + S ν t σ T Two equation models (k ɛ, k ω, SST) k = 1 2 tr v v (mean of the fluctuating kinetic energy) dissipation rate ɛ eddy frequency ω 1 k ɛ: good on free flow fields with no walls 2 k ω: near wall approximation is good 3 SST brings the advantage of booth together

16 Seite 16 Turbulence models Computational Fluid Dynamics Large-Eddy simulations (LES): spatial averaging method v( x, t) := v( x, t) G( x, x, ) dv with 1 step-function { 1, if x x < /2 G := 3 0, else V 2 gauss-filter { β x x } G := A( ) exp

17 Seite 17 Turbulence models Computational Fluid Dynamics Large-Eddy simulations (LES): LES equation: v t v = 0 + ( v ) v = f p + 1 Re τ τ S with τ S := v v v v. Detailed look: τ S = v v v v + v v v v + v v }{{}}{{}}{{} L C τ SR Leonard-strain: creation of small eddys through large eddys Cross-stress: interaction of the different scales Subgrid-scale Reynolds stress tensor

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