Seite 1

Introduction |

Computational Fluid Dynamics |

04.07.2016

Computational Fluid Dynamics 1 Theory, Numerics, Modelling Martin Pietsch Computational Biomechanics

Summer Term 2016

Seite 2

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

Fluid phase system Physical laws:

State variables: Density ρ (1d)

Mass conservation

Velocity u~ (3d)

Momentum conservation

Pressure p (1d)

Energy conservation

Energy e (1d)

Equation of state

Temperature T (1d)

Example for the equations of state: p = ρ Rs T

and

e = cν T

Seite 3

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

Reynolds transport theorem: d dt

Z 

Z f (x, t) dΩ = Ω(t)

 ∂f (x, t) + ∇ · (f u~ ) dΩ ∂t

Ω(t)

For the proof:1 For A, Y quadratic matrices, if d Y (t) = A(t) · Y (t) dt

y ∈ Ω(t)

holds, so does: d det Y (t) = trA(t) · det Y (t) dt 1

Skript 1994, Prof. Dr. J. Lorenz, RWTH-Aachen

φ(·, t) x ∈ Ω0

Seite 4

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

Mass conservation: Look at the mass m inside of an arbitrary volume Ω(t)  Z Z  d dm ∂ρ rtt ! = + ∇ · (ρ u~ ) dΩ = 0 ρ dΩ = dt dt ∂t Ω(t)

Ω(t)

Continuity equation: ∂ρ + ∇ · (ρ u~ ) = 0 ∂t Reynolds transport theorem:  Z Z  d ∂f f (x, t) dΩ = (x, t) + ∇ · (f u~ ) dΩ dt ∂t Ω(t)

Ω(t)

Seite 5

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

Momentum conservation: Look at the momentum p~ inside of an arbitrary volume Ω(t)  Z Z  d~ p d ∂ρ~ u rtt = + ∇ · (ρ u~ u~ ) dΩ = F~ ρ~ u dΩ = dt dt ∂t Ω(t)

Ω(t)

Force: Z F = FΩ + F∂Ω = Ω(t)

ρ f~ dΩ +

Z σ n~ dS ∂Ω(t)

Reynolds transport theorem:  Z Z  d ∂f f (x, t) dΩ = (x, t) + ∇ · (f u~ ) dΩ dt ∂t Ω(t)

Ω(t)

Seite 6

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

Momentum conservation: Handling the boundary force: Z Z σ n~ dS = ∇ · σ dΩ ∂Ω(t)

Ω(t)

  ∇ · ~σ1 with ∇ · σ = ∇ · ~σ2  ∇ · ~σ3

Therefore we get:  Z  Z n o ∂ρ~ u + ∇ · (ρ u~ u~ ) dΩ = ρ f~ + ∇ · σ dΩ ∂t Ω(t)

Ω(t)

Momentum equation: ∂ρ~ u + ∇ · (ρ u~ u~ ) = ρ f~ + ∇ · σ ∂t

Seite 7

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

Momentum equation: ∂ρ~ u + ∇ · (ρ u~ u~ ) = ρ f~ + ∇ · σ ∂t Look on the left side: ∂ρ~ u ∂~ u ∂ρ + ∇ · (ρ u~ u~ ) = ρ + u~ + u~ ∇ · (ρ u~ ) + (ρ u~ ) ∇ · u~ ∂t ∂t ∂t   ∂ρ ∂~ u =ρ + u~ + ∇ · (ρ u~ ) + (ρ u~ · ∇) u~ ∂t ∂t ∂~ u + (ρ u~ · ∇) u~ =ρ ∂t Reminder continuity equation: ∂ρ + ∇ · (ρ u~ ) = 0 ∂t

Seite 8

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

Energy equation: d dt

Z 

1 ρ |~ u |2 + ρe 2



Z n o ρf~ · u~ + ρ Q dΩ dΩ = Ω(t)

Ω(t)

Z +



∂Ω(t)

According to: volume force: energy source:

Ω(t) ρ

f~ · u~ dΩ

R

Q dΩ
σ n~ · u~ dS

R

R

Ω(t) ρ

surface force: ∂Ω(t) R heat flux: ∂Ω(t) κ ∇ T · n~ dS


σ n~ · u~ + κ ∇ T · n~ dS

Seite 9

Theoretical background |

Computational Fluid Dynamics |

04.07.2016

System equations: 1

mass conservation ∂ρ + ∇ · (ρ u~ ) = 0 ∂t

2

momentum conservation ρ

3

energy conservation ρ

4

∂~ u + (ρ u~ · ∇) u~ = ρ f~ + ∇ · σ ∂t



∂e = ρ Q + ∇ · (κ ∇ T ) + ∇ · σ u~ − ∇ · σ u~ ∂t

equation of state (e.g. ideal gas equation)

Seite 10

Navier-Stokes euqation |

Computational Fluid Dynamics |

04.07.2016

The stress tensor σ: σ = −p · 1 + τ

with τ is the viscous stress tensor

The viscosity term: 1

General viscous stress tensor: τ = F (D(t, x), t)

2

Strain rate tensor: D :=

i ∂ 1h = (∇ u~) + (∇ u~)T ∂t 2

Seite 11

Navier-Stokes euqation |

Computational Fluid Dynamics |

04.07.2016

Behaviour of the viscous stress tensor: τ = F (D(t, x), t) Time-dependent

Time-independent

increase with time printer ink, synovial fluid

shear thickening corn starch in water

decrease with time gelatin gels, yogurt

shear thinning ketchup, blood generalized newonian fluids water, blood plasma

Newtonian fluid:  h i 2 τ = µ · (∇ u~) + (∇ u~)T − µ∇ · u~ 3 with the dynamic viscosity µ

1

Seite 12

Navier-Stokes euqation |

Computational Fluid Dynamics |

04.07.2016

Incompressible fluid assumption: 0=

dρ ∂ (x, t) = ρ(x, t) + ∇ρ(x, t) · u~ dt ∂t

Continuity equation: ∂ρ ∂ρ + ∇ρ · u~ + ρ∇ · u~ + ∇ · (ρ u~ ) = ∂t ∂t = ρ∇ · u~ = 0 It follows: ∇ · u~ = 0 (divergency free velocity field) Viscous stress tensor: (Newtonian fluid)  h i 2  T  ~ 1 τ = µ · (∇ u~) + (∇ u~) − µ∇  ·u 3 

Seite 13

Navier-Stokes euqation |

Computational Fluid Dynamics |

04.07.2016

Incompressible 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 ρ

(1)

∂~ u + ρ (~ u · ∇) u~ = ρ f~ − ∇p + µ∇ · τ ∂t

Note: often, the kinematic viscosity ν :=

µ ρ

is used if ρ = const

(2)

Seite 14

Turbulence modelling |

Computational Fluid Dynamics |

Application to biofluid systems 1

Human air system Fluid-particle interaction Fluid-structure interaction Blood-air barrier

2

Human blood system Oxygen transportation Fluid-structure interaction Transport of medicine

3

...

04.07.2016