Dynamique et modélisation de la Turbulence - UPMCchibbaro/FILES/M2-MF2A-CM4.pdf · Dynamique et...
Transcript of Dynamique et modélisation de la Turbulence - UPMCchibbaro/FILES/M2-MF2A-CM4.pdf · Dynamique et...
Dynamique et modélisation de la Turbulence
Cours 4
SB Pope, «Turbulent flows» Cambridge University Press
mercredi 2 novembre 2016
4 axioms of O. ReynoldsThe averaging operator (denoted by a bar) must fulfill th
following constraints:1. Linearity
2. Constant preservation
3. Commutation with derivatives in x,t
4. Be a projector
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Reynolds decomposition• Turbulent variable splitting (velocity, pressure, …)
• Properties:
« mean flow » « turbulent » or « fluctuating » flow
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Pdf-related parameters
• Skewness factor
• Flatness factor
Rem: Gaussian process, S=0 and F=3
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S>0
S<0
large δ
small δ
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Governing equations in physical space
Framework:
• Incompressible flow
• Single phase flow
• Single species flow
• Newtonian flow
•--> hyp: the Navier-Stokes equations hold (rem: Clay prize)
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Mean flow equations
Applying the filtering operator to NS eqs., one obtains
Reynolds decomposition:
Coupling term between mean flow and turbulencemercredi 2 novembre 2016
Fluctuating field equationsSubstracting averaged momentum eq. from the original one, one has
Rem: the nonlinear term can be further decomposed as follows
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Vitesse moyenne Température moyenne
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Reynolds stress budget
Noticing that one obtains
Transfer term (production)
Turbulent diffusion Pressure power
Viscous diffusion dissipation
Mean flow advection
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Mean kinetic energy budgetdefining the averaged momentum equation yields
dissipation transfer term
Turbulent diffusionViscous diffusion
Pressure diffusion
External force power
Advection along Mean flow streamlines
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Turbulent kinetic energy budgetDenoting and
Mean flow advection Transfer term (production)
Turbulent diffusion Pressure diffusion
Viscous diffusion dissipation
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Control volume budget
Diffusive processes
Diffusive processesDiffusive processes
Diffusive processes
production
dissipation
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Description level hierarchyMicroscopic level:• Boltzmann equation• Discrete description: particule transport & collision
Macroscopic level:• Navier-Stokes equations• Continuum mechanics: velocity, pressure, viscosity, fluxes
Statistical description of turbulence• Reynolds-averaged Navier-Stokes equations•New fluxes induced by turbulent motion
Local statistical average
Reynolds statistical average
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