PDF Approach Hybrid Methodology Validation DEVELOPMENT OF A HYBRID EULERIAN-LAGRANGIAN METHOD FOR THE NUMERICAL MODELING OF THE DISPERSED PHASE IN TURBULENT GAS-PARTICLE FLOWS X. PIALAT 1 , 2 1 Departement Modèles pour l’Aérodynamique et l’Énergétique ONERA-CERT 2 Institut de Mécanique des Fluides de Toulouse UMR CNRS / INPT / UPS PhD Defense X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Introduction Gas-Particle Flows Applications pollutant dispersion, combustion chamber, . . . fluidized bed, Numerous Physical Issues fluid turbulence influence over the particles, backward influence of the particles over the fluid, inter-particle collisions, rebounds on walls, chemistry, combustion, evaporation, . . . Numerical Simulation Advantages parallel calculation improvement production of numerical "experiences" cheaper than experience X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Introduction Gas-Particle Flows Applications pollutant dispersion, combustion chamber, . . . fluidized bed, Numerous Physical Issues fluid turbulence influence over the particles, backward influence of the particles over the fluid, inter-particle collisions, rebounds on walls, chemistry, combustion, evaporation, . . . Numerical Simulation Advantages parallel calculation improvement production of numerical "experiences" cheaper than experience X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Particle phase numerical simulation Discrete Particle Simulation (DPS, Deterministic Lagrangian) statistical model accuracy of the description PDF Approach particle simulation Euler ”fluctuating Stochastic movement model” Lagrangian Moments Method ( q 2 p − q fp , R p , ij − q fp ) X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Particle phase numerical simulation Discrete Particle Simulation (DPS, Deterministic Lagrangian) statistical model accuracy of the description PDF Approach particle simulation Euler ”fluctuating Stochastic movement model” Lagrangian Moments Method direct simulations ( q 2 p − q fp , R p , ij − q fp ) high-cost simulations low numerical precision additional hypothesis low-cost simulations high numerical precision X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Motivation similarities with atmosphere re-entry flows Navier-Stokes cease to be valid near the obstacle, with two possible solutions : extend the domain of validity of the continuous model (Burnett equations. . . ) use of a hybrid method coupling Navier-Stokes simulations with particle Boltzmann resolution hybridation based on the kinetic derivation of the Navier-Stokes equations the fluxes used in Navier-Stokes can be divided into outgoing and ingoing half-fluxes the exchange of information between the two domains is done via these half-fluxes Le Tallec and Mallinger, JCP , 1997 X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Particle phase numerical simulation Discrete Particle Simulation (DPS, Deterministic Lagrangian) accuracy of the description PDF Approach Euler Stochastic Lagrangian Moments Method ( q 2 p − q fp , R p , ij − q fp ) X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Particle phase numerical simulation Discrete Particle Simulation (DPS, Deterministic Lagrangian) accuracy of the description PDF Approach Euler Stochastic Lagrangian Moments Method ( q 2 p − q fp , R p , ij − q fp ) X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Outline PDF Approach 1 Statistical modeling Stochastic lagrangian Eulerian approach Hybrid Methodology 2 Methodology Lagrangian condition Eulerian conditions Validation 3 Homogeneous Shear Flow Channel Flow X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Statistical modeling Stochastic lagrangian Eulerian approach Outline PDF Approach 1 Statistical modeling Stochastic lagrangian Eulerian approach Hybrid Methodology 2 Methodology Lagrangian condition Eulerian conditions Validation 3 Homogeneous Shear Flow Channel Flow X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Statistical modeling Stochastic lagrangian Eulerian approach Probability Density Function (PDF) Joint fluid-particle probability density function f fp [Simonin, VKI,1996], f fp ( t ; x , c p , c f ) dxdc p dc f is the probable number of particles at instant t with properties ( x p , u p , u f @ p ) ∈ ( x , c p , c f ) ”+” ( dx , dc p , dc f ) , obtained ideally by averaging over an infinity of realizations of the flow the evolution equation of this pdf is similar to the gas kinetic theory’s Boltzmann equation : ∂ f fp ∂ ∂ � < du p , k � � � ∂ t + c p , k f fp + | c f , c p > f fp ∂ x k ∂ c p , k dt � ∂ f fp ∂ � < du f @ p , k � � + | c f , c p > f fp = ∂ c f , k dt ∂ t coll X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Statistical modeling Stochastic lagrangian Eulerian approach Probability Density Function (PDF) Joint fluid-particle probability density function f fp [Simonin, VKI,1996], f fp ( t ; x , c p , c f ) dxdc p dc f is the probable number of particles at instant t with properties ( x p , u p , u f @ p ) ∈ ( x , c p , c f ) ”+” ( dx , dc p , dc f ) , obtained ideally by averaging over an infinity of realizations of the flow the evolution equation of this pdf is similar to the gas kinetic theory’s Boltzmann equation : ∂ f fp ∂ ∂ � < du p , k � � � ∂ t + c p , k f fp + | c f , c p > f fp ∂ x k ∂ c p , k dt � ∂ f fp ∂ � < du f @ p , k � � + | c f , c p > f fp = ∂ c f , k dt ∂ t coll X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Statistical modeling Stochastic lagrangian Eulerian approach Probability Density Function (PDF) Joint fluid-particle probability density function f fp [Simonin, VKI,1996], f fp ( t ; x , c p , c f ) dxdc p dc f is the probable number of particles at instant t with properties ( x p , u p , u f @ p ) ∈ ( x , c p , c f ) ”+” ( dx , dc p , dc f ) , obtained ideally by averaging over an infinity of realizations of the flow the evolution equation of this pdf is similar to the gas kinetic theory’s Boltzmann equation : ∂ f fp ∂ ∂ � < du p , k � � � ∂ t + c p , k f fp + | c f , c p > f fp ∂ x k ∂ c p , k dt � ∂ f fp ∂ � < du f @ p , k � � + | c f , c p > f fp = ∂ c f , k dt ∂ t coll X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Statistical modeling Stochastic lagrangian Eulerian approach Probability Density Function (PDF) Joint fluid-particle probability density function f fp [Simonin, VKI,1996], f fp ( t ; x , c p , c f ) dxdc p dc f is the probable number of particles at instant t with properties ( x p , u p , u f @ p ) ∈ ( x , c p , c f ) ”+” ( dx , dc p , dc f ) , obtained ideally by averaging over an infinity of realizations of the flow the evolution equation of this pdf is similar to the gas kinetic theory’s Boltzmann equation : ∂ f fp ∂ ∂ � < du p , k � � � ∂ t + c p , k f fp + | c f , c p > f fp ∂ x k ∂ c p , k dt � ∂ f fp ∂ � < du f @ p , k � � + | c f , c p > f fp = ∂ c f , k dt ∂ t coll X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
v r v r ( ) F d g PDF Approach Hybrid Methodology Validation Statistical modeling Stochastic lagrangian Eulerian approach Forces acting on a particle Small rigid spheres with d p ≤ η K Mass point approximation Inertial particles : τ p ≫ τ K v r Equation of motion for a particle : du p , k = − u p , k − u f @ p , k + g k τ p dt τ p = τ p ( | u p − u f @ p | ) is the particle relaxation time g is the gravity u f @ p is the ”seen” fluid velocity at the particle location : X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
PDF Approach Hybrid Methodology Validation Statistical modeling Stochastic lagrangian Eulerian approach Langevin scheme for the ”seen” fluid velocity Inspired by single-phase works [Pope,1983], the fluctuating velocity u ′′ f @ p , i = u f @ p , i − U f , i is predicted using a Langevin equation [Simonin, 1996] : f @ p , i = − ∂ R ff , ik G fp , ik − ∂ U f , i du ′′ u ′′ � � dt + f @ p , k dt + H fp δ W fp , i ∂ x k ∂ x k A = G fp − ∇ · U f is the drift tensor, which simplest model is given by the Simplified Langevin Model (SLM) : G fp , ij = − δ ij /τ t τ t f @ p = τ t f @ p , f H fp is the diffusion tensor, i.e. the tensor of amplitude of the Wiener process all these quantities are given data (and not calculated data as in fluid turbulence stochastic lagrangian modeling) X. PIALAT Hybrid Eulerian-Lagrangian Method (HELM)
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