modeling complex direct flows numerical simulationsgtryggva/cfd-course/2011-lecture-36.pdf · in...

Computational Fluid Dynamics!

Modeling Complex Flows!

Grétar Tryggvason!Spring 2011!

http://www.nd.edu/~gtryggva/CFD-Course/!Computational Fluid Dynamics!

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!

Streamwise velocity!

Flow direction!

Periodic streamwise and spanwise boundaries!

Wall!


Streamwise vorticity!


Streamwise vorticity!

Turbulent shear stress!

Turbulent eddies generate a nearly uniform velocity profile!

Channel Flow!


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 =

"u"x

"u"y

"u"z

"v"x

"v"y

"v"z

"w"x

"w"y

"w"z

#

$

% % % % % %

&

'

( ( ( ( ( (

S = 12!u +!Tu( ) = 12

2"u"x

"u"y

+ "v"x

"u"z

+ "w"x

"v"x

+ "u"y

2"v"y

"v"z

+ "w"y

"w"x

+ "u"z

"w"y

+ "v"z

2"w"z

#

$

% % % % % %

&

'

( ( ( ( ( (

! = 12"u -"Tu( ) = 12

0 #u#y

$ #v#x

#u#z

$ #w#x

#v#x

$ #u#y

0 #v#z

$ #w#y

#w#x

$ #u#z

#w#y

$ #v#z

0

%

&

' ' ' ' ' '

(

)

* * * * * *


It can be shown that the second eigenvalue of !

S2 + !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.!

Other quantities have also been used, such as the second invariant of the velocity gradient:!

Q = !ui!x j

!u j

!xi

!2


!2 = "0.3

!2 = "0.2


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 !


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 = l02 2S ijS ij( )1/ 2

S ij = 12

!Ui

!x j

+!U j

!xi

"

# $ $

%

& ' '

Since the viscosity increases, the size of the smallest flow scales increases and lower resolution is needed !


Multiphase Flow!


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!

Examples:! 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!


Mixed!

Stratified!

Dispersed!

Slugs!

Flow in pipes!


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!


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 εp describes how much of the region is occupied by phase p. Obviously:!

�

εp =1∑While the averaging is similar to turbulent flows, here we must account for the different phases!

�

α p =1 inside phase p0 otherwise

⎧ ⎨ ⎩

The void fraction is found by!

�

εp = 1V

α p dvV∫


�

ˆ φ p = 1εpV

α pφ dvV∫Averages are found by!

Where the volume V goes to zero in some way!

The velocity is found by!

�

ˆ u p = 1εpV

upα p dvV∫The averages can also be interpreted as time or ensemble averages!

The effective density of phase p is!

�

ˆ ρ = εpρp


�

α pρp dvV∫The total mass of phase p in a control volume is!

And the mass conservation equation can be averaged to yield!

�

∂∂tεpρp + ∇ ⋅ εpρpup( ) = ˙ m p

Here!

Since a mass that leaves one phase must add to another phase!�

˙ m p = 0∑


The conservation of momentum equation becomes!

�

∂∂t

εpρpup( ) + ∇ ⋅ εpρpupup( ) = −εp∇pp

+∇ ⋅ εpµpDp( ) + εpρpg + ∇ ⋅ εpρp < uu >( ) + Fint

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 non-zero in laminar flow!

interfacial forces!

Reynolds stresses!


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!

�

Fp = 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!

�

dudt

= Fp

If the particles have no influence on the fluid: One way coupling!If the particles exert a force on the fluid: Two way coupling!


where! kD =

34

CDεrρq

ur − up( )dr

�

CD = CD Re( )

is obtained from experimental correlations, such as!

and!

�

CD = 24Re

1+ 0.15Re0.687( ) Re <103

For solid particles! Re based on slip velocity!

�

Fp = kD u−up( ) + g ρD − ρρ

+ Fother

Usually the force is written:!

Other forces due to added mass, pressure, lift, etc!Gravity!

buoyancy!

Drag force!


For turbulent flow, set particle velocity!

�

up + u'

Random velocity fluctuations from !

This allows particles to cross streamlines as they do in turbulent flow!

Particles can accumulate here!�

kp = u'u'∑

The force allows us to find the particle velocity by integrating:!

�

dupdt

= Fp and trajectories by!

�

dx pdt

= up


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!


Similar approach can be taken for the temperature and the size of a particle (heat and mass transfer)!

�

mpcpdTpdt

= hAp (Tf −Tp ) + εpApσ (T∞4 −Tp

4 )

�

dmp

dt= ˙ m p

For dilute flows this does work reasonably well — if the initial or inlet conditions are knows!

Mass transfer due to evaporation, for example!


Turbulent in the continuous phase!

�

DkDt

=+ <U ⋅Fp >

�

<U ⋅Fp >= τρ

< uf (uf − up ) >= τρ(< uf uf > − < uf up >)

This term can lead to both reduction and increase in the turbulence in the liquid!

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 kp. Called k ε kp models.! The k equation is!


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!


Flow! Gravity!

Bubbly flow in a vertical channel!

Need to know!• The bubble distribution!• The velocity profile and the flow rate!

x

y

�

∂∂y

= 0

Assume that the flow is independent of y, so!

�

∂pl∂y is given!

but!

S.P. Antal, R.T. Lahey and J.E. Flaherty. Intʼl. J. Multiphase Flow 17 (1991), 635-652.!


�

ε ∂ε∂x

Ur2

51−ε( ) = −εCLUr

∂ul∂x

−ε Cw1 + Cw2Rb

s⎛ ⎝ ⎜

⎞ ⎠ ⎟ Ur

2

Rb�

(1−ε) dpldy

+ 1−ε( )ρlgy = (1−ε)µl∂2vl∂x 2

+ 38

εRb

CDρlUr Ur

�

ε x( ) = 1L

εdx0

L

∫ , ul 0( ) = ul H( ) = 0

Simple two-fluid model for laminar multiphase flow

Comparison with a two-fluid model!

�

ε dpdy

+ ερggy = − 38

εRb

CDρlUr Ur

�

dpgdy

= dpldy

= dpdy

Bubble vertical momentum!Liquid vertical momentum!Bubble horizontal momentum!

Lift! Wall repulsion!(away from wall or zero)!

�

CD = 24Re1+ 0.1Re0.75( )

�

Re=2Rbρl Ur

µm

�

µm = µ l

1−ε

Computational Fluid Dynamics!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 17 (1991), 635-652.!


For more information about computing multiphase flow, see:!

Modeling of multiphase flows is still a very immature area. Interpret the results with care!!

modeling complex direct flows numerical simulationsgtryggva/cfd-course/2011-lecture-36.pdf · in...

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