# Modeling Complex Flows! Direct Numerical Simulations! Computational Fluid Dynamics!

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1 Modeling Complex Flows! Grétar Tryggvason! Spring 2011! Direct Numerical Simulations! In direct numerical simulations the full unsteady Navier-Stokes equations are solved on a sufficiently fine grid so that all length and time scales are fully resolved. The size of the problem is therefore very limited. The goal of such simulations is to provide both insight and quantitative data for turbulence modeling! Channel Flow! Wall! Flow direction! Streamwise velocity! Periodic streamwise and spanwise boundaries! Streamwise vorticity! Channel Flow! Turbulent shear stress! Turbulent eddies generate a nearly uniform velocity profile! Streamwise vorticity!

2 Turbulence are intrinsically linked to vorticity, yet laminar flows can also be vortical so looking at the vorticity is not sufficient to understand what is going on in a turbulent flows. Several attempts have been made to define properties of the turbulent flows that identifies vortices as opposed to simply vortical flows.! One of the most successful method is the lambda-2 method of Hussain.! Visualizing turbulence! #"u "x "v!u = "x "w \$ "x "u "u & "y "z "v "v "y "z "w "w "y "z ' ) = 1 2 S = 1 2!u +!T u ) = 1 2! = 1 2 "u - "T u # 2 "u "x "v "x + "u "y "w "x + "u \$ "z ' 0 ' '#v #x \$ #u ' #y '#w ' #x \$ #u & #z "u "y + "v "x 2 "v "y "w "y + "v "z #u #y \$ #v #x 0 #w #y \$ #v #z "u "z + "w & "x "v "z + "w "y 2 "w "z ' #u #z \$ #w * #x #v #z \$ #w * * #y * * 0 * ) It can be shown that the second eigenvalue of! S 2 +! 2 define vortex structures! Referece: J. Jeong and F. Hussain, "On the identification of a vortex," Journal of Fluid Mechanics, Vol. 285, 69-94, 1995.!! 2! 2 = "0.2 Other quantities have also been used, such as the second invariant of the velocity gradient:! Q =!u i!x j!u j!x i! 2 = "0.3 Large Eddy Simulations! Unsteady simulations where the large scale motion is resolved but the small scale motion is modeled. Frequently simple models are used for the small scale motion. Most recently some success has been achieved by intrinsic large eddy simulations where no modeling is used but monotonicity is enforced by the methods described in the lectures on hyperbolic methods!

3 In the simplest case, the Smagorinsky eddy viscosity is used in simulation of unsteady flow, thus resulting in a viscosity that depends of the flow.!! T = l 2 0 2S ij S ij ) 1/ 2 S ij = 1 "!U i +!U j 2 \$ #!x j!x ' i & Multiphase Flow! Since the viscosity increases, the size of the smallest flow scales increases and lower resolution is needed! Examples:! Spray drying! Pollution control! Pneumatic transport! Slurry transport! Fluidized beds! Spray forming! Plasma spray coating! Abrasive water jet cutting! Pulverized coal fired furnaces! Solid propellant rockets! Fire suppression and control! Disperse flow! Solid-liquid: Slurries, quicksand, sediment transport! Solid-air: dust, fluidized bed, erosion! Liquid-air: sprays, rain! Air-liquid: bubbly flows! Single component Multicomponent! Single water flow air flow! phase Nitrogen flow emulsions! Multiphase Steam-water flow air-water flow! Freon-Freon slurry flow! vapor flow! Flow in pipes! Stratified! Slugs! Mixed! Dispersed! This figure shows schematically one of several different configurations of a circulating fluidized bed loop used in engineering practice. The particles flow downward through the aerated standpipe, and enter the bottom of a fast fluidized bed riser. The particles are centrifugally separated from the gas in a train of cyclones. In this diagram, the particles separated in the primary cyclone are returned to the standpipe while the fate of the particles removed from the secondary cyclone is not shown.! From: Computational Methods for Multiphase Flow, Edited by A.Prosperetti and G.Tryggvason!

4 Need model equations to predict flow rates, pressure drop, slip velocities, and void fraction! Mixture models: one averaged phase! Two-fluid models: two interpenetrating continuum! Although commercial codes will let you model relatively complex multiphase flows, it is really only in the limit of dispersed and dilute flows where we can expect reasonable accuracy! To treat systems like this, the two-fluid model is usually used. The continuous phase is almost always used in an Eularian way where the continuity, momentum, and energy equations are solved on a fixed grid.! The void fraction ε p describes how much of the region is occupied by phase p. Obviously:! ε p =1 While the averaging is similar to turbulent flows, here we must account for the different phases! 1 inside phase p = 0 otherwise The void fraction is found by! ε p = 1 V dv V Averages are found by! ˆ φ p = 1 ε p V φ dv V Where the volume V goes to zero in some way! The velocity is found by! u ˆ p = 1 ε p V dv V The averages can also be interpreted as time or ensemble averages! The effective density of phase p is! ˆ ρ = ε p The total mass of phase p in a control volume is! dv V And the mass conservation equation can be averaged to yield! Here! t ε p + ε p ) = m p m p = 0 Since a mass that leaves one phase must add to another phase! The conservation of momentum equation becomes! t ε p ) + ε p ) = ε p p p + ε p µ p D p ) + ε p g + ε p < uu > ) + F int Reynolds stresses! interfacial forces! In addition to the Reynolds stresses, it is now necessary to model the interfacial forces. The kinetic energy is often neglected, even though the fluctuations are nonzero in laminar flow!

5 Euler/Euler approach! All phases are treated as interpenetrating continuum! The dispersed phase is averaged over each control volume! Each phase is governed by similar conservation equations! Modeling is needed for!!interaction between the phases!!turbulent dispersion of particles!!collision of particles with walls! A size distribution requires the solution of several sets of conservation equations! Numerical diffusion at phase boundaries may result in errors! This approach is best suited for high volume fraction of the dispersed phase! Euler/Lagrange approach! The fluid flow is found by solving the Reynolds-averaged Navier-Stokes equations with a turbulence model.! The dispersed phase is simulated by tracking a large number of representative particles.! A statistically reliable average behavior of the dispersed phase requires a large number of particles! The point particles must be much smaller than the grid spacing! Modeling is needed for!!collision of particles with walls!!particle/particle collisions and agglomeration!!droplet/bubble coalescence and breakup! A high particle concentration may cause convergence problems! If there is no mass transfer m=0 and F is the force that one phase exerts on the other! F p = 0 In principle the conservation equations can be solved for both the continuous and the dispersed phase Euler/Euler approach).! However, the dispersed phase is not all that continuous and an other approach is to explicitly tract representative) particles by solving! du = F p If the particles have no influence on the fluid: One way coupling! If the particles exert a force on the fluid: Two way coupling! Usually the force is written:! ) + g ρ D ρ F p = k D u Drag force! ρ Gravity! buoyancy! + F other Other forces due to added mass, pressure, lift, etc! where! k D = 3 4 C ε ρ u r ) and! C D = C D Re D r q d r ) is obtained from experimental correlations, such as! C D = 24 Re 1 + ) 0.15Re0.687 Re <10 3 For solid particles! Re based on slip velocity! The force allows us to find the particle velocity by integrating:! d = F p and trajectories by! For turbulent flow, set particle velocity! + u' dx p = Random velocity fluctuations from! k p = u'u' Usually a large number of particles is used to get a well converged particle distribution! Notice that almost all the interactions particles/flow) particle/particle, particle/wall) are highly empirical! This allows particles to cross streamlines as they do in turbulent flow! Particles can accumulate here!

6 Similar approach can be taken for the temperature and the size of a particle heat and mass transfer)! m p c p dt p dm p = m p = ha p T f T p ) + ε p A p σt 4 T p 4 ) Mass transfer due to evaporation, for example! For dilute flows this does work reasonably well if the initial or inlet conditions are knows! Turbulent in the continuous phase! Either ignore the contributions of the dispersed phase when computing the flow, or use a k-ε model! Solve for k and ε in the liquid and k p. Called k ε k p models.! The k equation is! Dk Dt = + < U F p > < U F p >= τ ρ < u f u f ) >= τ ρ < u f u f > < u f >) This term can lead to both reduction and increase in the turbulence in the liquid! The full two-fluid model suffers from several problems, in addition to uncertainties about the various closure assumptions:! The major one is that the full equations are ill-posed and one cannot expect a fully converged solution under grid refinement! One possible way around this is to use the drift flux approximation where the particle velocity is assumed to be a given function of the local conditions.! Modeling of Laminar Flow in a Vertical Channel! y Flow! Gravity! x S.P. Antal, R.T. Lahey and J.E. Flaherty. Intʼl. J. Multiphase Flow ), ! Bubbly flow in a vertical channel! Need to know! The bubble distribution! The velocity profile and the flow rate! Assume that the flow is independent of y, so! y = 0 p l y but! is given! ε ε x 2 Comparison with a two-fluid model! Simple two-fluid model for laminar multiphase flow ε dp dy + ερ g = 3 ε g y C D ρ l 8 R Bubble vertical momentum! b 1 ε) dp l dy + 1 ε)ρ 2 v lg y = 1 ε)µ l l x + 3 ε Liquid vertical C 2 D ρ l 8 R b momentum! u 1 ε) = εc L U l r 5 x ε C + C R b w1 w2 s L ε x) = 1 L εdx, u l 0) = u l H ) = 0 0 Lift! R b Wall repulsion! away from wall or zero)! C D = 24 Re Re0.75 ) Re = 2R bρ l µ m µ m = µ l 1 ε dp g dy = dp l dy = dp dy 2 Bubble horizontal momentum!

7 Comparison with a two-fluid model! Comparison with experimental results. Graph from: S.P. Antal, R.T. Lahey and J.E. Flaherty. Intʼl. J. Multiphase Flow ), ! Modeling of multiphase flows is still a very immature area. Interpret the results with care!! For more information about computing multiphase flow, see:!

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