Transcript Large Eddy Simulation - Consulting & Performing Industrial

Lecture 12 - Large Eddy Simulation
Applied Computational Fluid Dynamics
Instructor: André Bakker
© André Bakker (2002)
© Fluent Inc. (2002)
Outline
• Brief summary of turbulence models.
• Introduction to large eddy simulation (LES).
• Examples.
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–
Bluff body jet.
HEV static mixer.
Pitched blade impeller.
Rushton turbine.
High efficiency impeller.
Modeling turbulence
• Turbulence is a 3D transient phenomenon.
– Fluctuations cover a wide range of time and length scales.
• Turbulence models range from approximate to highly rigorous:
– Steady-state isotropic models.
– Transient 3D models of entire spectrum.
• Models are incorporated into the Navier-Stokes equations using a
variety of methods.
The turbulence spectrum
• Many scales of turbulent eddies exist:
– Large eddies contain most of the turbulent kinetic energy.
• Scale sizes are on the order of the flow passages.
– Energy cascades from large to small eddies.
– Small eddies dissipate the energy they receive from larger eddies in
the spectrum.
• Difficulty in turbulence modeling is trying to accurately capture the
contributions of all scales in the spectrum.
Direct numerical simulation (DNS)
• Navier-Stokes equations are solved on a fine grid using a small
time-step.
• Goal is to capture the smallest turbulence scales.
– Large scales are captured as well.
• Result is accurate, 3D, transient behavior.
• Great for simple flows, but computationally intensive.
– Not suited to industrial applications with CPU resources available
today.
The cost of DNS
• The number of grid points per dimension needed to resolve the
small scales is:
 k
3/ 4
N1D ~ Ret , Re t 

• The number of grid points needed for a 3D DNS simulation is:
N3D ~ Re9t / 4
• The overall cost, including time step, of the computational effort is
proportional to Ret3.
RANS turbulence models (1)
• Velocities are described by an equilibrium (vo) and fluctuating (v’)
contribution:
vi = voi + vi’.
• Momentum equations are rewritten, then time-averaged
(Reynolds Averaged Navier-Stokes equations).
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–
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–
Averaging eliminates terms with v’ as a factor.
Terms with vi’vj’ remain.
These Reynolds stresses are computed with a turbulence model.
Impact on transport equations is through the effective viscosity:
eff ~ t + o (1 and 2 equation models).
RANS turbulence models (2)
• Many flavors exist, such as:
– k-e: Robust, popular 2-equation model using constants taken from
simple, high Re flows.
• isotropic turbulence effects:eff is a scalar.
– RSM: 5-equation (2D) or 7-equation (3D) model.
• non-isotropic turbulence effects makes this suitable for highly swirling
flows.
Large eddy simulation (LES)
• LES is midway between DNS and RANS in terms of:
– Rigor.
– Computational requirement.
• Spectrum of turbulent eddies in the Navier-Stokes equations is
“filtered”:
– The filter is a function of the grid size.
– Small eddies are removed, and modeled using a subgrid-scale
(SGS) model.
– Large eddies are retained, and solved for directly using a transient
calculation.
Filtered variables
• A variable, f(x’), is filtered using a filter function, G.
~
f (x)   f (x' )G(x, x' )dx'
D
• G is a function of the cell volume.
Thus:
1/ V
G( x, x' ) 
0
for x'
otherwise
1
f (x)   f (x' )dx',
V
~
x'V
Filtered transport equations
• The filtered continuity and momentum equations use filtered
variables:
~
 u j

0
t
x j
and:
~~
~
u i u i u j
~
p t ij s ij




t
x j
x i x j x j
tij is the filtered stress tensor.
sij are the subgrid-scale Reynolds stresses.
Subgrid-scale (SGS) modeling
• SGS Reynolds stresses are modeled by:
1
s


ij
s ij 3 s kk  2 t Sij
s
where t is the subgrid-scale eddy viscosity and Sij is the rate of
strain tensor.
• Two common models are:
Smagorinsky SGS model
t   L
2
2 S ij S ij
1


L  min   d, C s V 3 


RNG SGS model

  2s  tot


 t   1  H
 C 
3

 


1
3

 s   0.157V 


2
2Sij Sij
1/ 3
Prediction Methods
h = l/Re3/4
l
Direct numerical simulation (DNS)
Large eddy simulation (LES)
Reynolds averaged Navier-Stokes equations (RANS)
LES - what does it take?
• Requires 3-D transient modeling.
• Requires spatial and temporal resolution of scales in “inertial
subrange”.
Taylor scale
L   c     d
log E
Inertial
subrange
Kolmogorov scale
Dissipating
eddies
-5/3
Energycontaining
eddies
Kd  1/ L
Kc  1 /  c
K
Kd  1/  d  Re / L
3/ 4
Bluff-body jet
• Cold flow in a bluff-body coaxial burner.
• A range of length and time scales exists.
20 m/s
62 m/s
bluff body
R = 1.8 mm
Bluff body jet flow pattern
Time averaged flow pattern from LES simulations.
Bluff body jet axial velocities
• Transient, 3D solution done with LES (420k cells).
• Steady-state, 2D axisymmetric solutions done with k-e and RSM
(30k cells).
• Comparison with experimental data:
Vaxial (x/D=0.8)
Vaxial (x/D=1.5)
Experiment
33.5 m/s
12.7 m/s
LES (time average)
35.2
13.2
RSM
26.4
7.5
k-e
25.5
7.7
Bluff body jet conclusion
• Time-averaged results better predicted the axial flow pattern in a
bluff-body jet example.
HEV static mixer
• Circular or square cross-section pipe with sets of tabs mounted
on the walls.
• Flow around tabs is unsteady, with counter-rotating longitudinal
vortices, and hairpin vortices.
Source: “Kenics Static Mixers” brochure, 1996.
Previous models
• Assumptions:
– Eight-fold symmetry.
– Steady state flow with RANS model.
• Results:
– Longitudinal vortices observed.
• Disadvantages:
– Hairpin vortices not observed.
– Under-prediction of mixing near center.
– No material exchange between areas surrounding tabs.
Geometries studied
• Two models studied.
– Square duct.
• 0.1x0.1x1 m3.
• Air at 30 m/s.
• Re ~ 200k.
– Cylindrical pipe.
• D = 0.05 m.
• Water at 0.12 m/s.
• Re ~ 5000.
• Both models:
– 500k cells.
– Unstructured grid.
Square duct results: RNG k-e
Longitudinal vortices are symmetric and stable.
Between 3rd and 4th tabs
2-D downstream of last tabs
Square duct results: longitudinal vortices - 1
Between third and fourth sets of tabs
T = 0.1429 s
T = 0.1486 s
Square duct results: longitudinal vortices - 2
2-D downstream of last set of tabs
T = 0.1429 s
T = 0.1486 s
Cylindrical pipe: LES hairpin vortices - 1
T = 6.40 s
T = 6.43 s
Cylindrical pipe: LES hairpin vortices - 2
T = 6.46 s
T = 6.53 s
Hairpin vortex animation
Cylindrical pipe: LES longitudinal vortices - 1
Cylindrical. At tip of last set of tabs.
Cylindrical pipe: LES longitudinal vortices - 2
Cylindrical. At tip of last set of tabs.
Longitudinal vortices animation
Ratio between effective viscosity and molecular viscosity
Effective viscosity comparison
160
Square Duct
140
Subgrid LES
k-e RNG
k-e Standard
120
100
80
60
40
20
Cylindrical
0
Re = 5000
Re = 2E5
HEV mixer conclusion
• LES predicts unsteady vortex system including transient hairpin
vortices, as also seen in experiments.
• Interaction between vortices causes material exchange between
tabs, and between the center and tabs.
• Practical issues:
– Calculation time for 1500 time steps on the order of a week on 350
MHz P2 PC.
– 500k node model requires 0.5 GB of RAM.
– Data files ~ 50MB each (compressed).
Stirred tank flow instabilities
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Experimental work suggests that large-scale, time-dependent
structures, with periods much longer than the time of an
impeller revolution, are involved in many of the fundamental
hydrodynamic processes in stirred vessels.
Local velocity data histograms may be bi-modal or tri-modal
(Bakker and Van den Akker).
The gas holdup distribution may be asymmetric and oscillating
(Bakker and Van den Akker).
In solids suspension processes, solids can be swept from one
side of the vessel to the other in a relatively slow oscillating
pattern, even in dilute suspensions.
Digital particle image velocimetry experiments have shown
large scale asymmetries with periods up to several minutes
(Myers, Bakker and Ward).
Stirred tank modeling options
Hypothetical.
Research. Large scale turbulence
and unsteady structures.
Impeller-Baffle interaction.
Time dependence.
Impeller Design. When
velocity data is not available.
Daily design. General flow
fields. How many impellers
are needed. Instructional.
Stirred tank modeling
• The sliding mesh model was used to set up the transient motions
of the impeller in the tank.
• Two turbulence model approaches were evaluated:
– Reynolds-Averaged Navier-Stokes turbulence model, i.e., standard
k-e, RNG k-e, Reynolds Stress Model.
– Large eddy simulation or LES.
• Three impeller styles were tested:
– Pitched blade turbine (PBT).
– Rushton turbine (RT).
– High efficiency impeller (HE-3).
Stirred tank models - LES grid size
• Grid size used was on the order of 800k cells.
• This corresponded to an average grid size of 2E-3m.
• From RANS simulations it was determined that:
Integral length scale ~ 1E-2 m
Taylor length scale ~ 1.3E-3 m
Kolmogorov scale ~ 6E-5 m.
Pitched blade turbine
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•
Flat bottom vessel with four baffles:
– T=0.292m.
– Z/T=1.
Pitched-blade turbine (PBT):
– Four blades at 45°.
– D/T=0.35.
– W/D=0.2.
– C/T=0.46.
– 60 RPM.
Water.
•
Reynolds number ~ 1E4.
•
Experimental DPIV Data
Unsteady RANS (RSM) flow field
Pitched Blade Turbine - Velocity Magnitude
Tracer dispersion (RANS)
Pitched Blade Turbine
Unsteady LES flow field
Pitched Blade Turbine - Velocity Magnitude
Unsteady LES flow field
Iso-surface of Vorticity Magnitude
Tracer dispersion (LES)
Pitched Blade Turbine
Time series of axial velocity - 1
(a) x = 0.185m y = -0.04m, z = -0.04m
(b) x = 0.185m y = 0.04m z = 0.04m
PBT from 168.13306 (2500 time steps) to 178.12756s
(3500 time steps) after start-up from a zero-velocity field.
Time series of axial velocity - 2
(c) x=0.25m y=0.05m z=0.05m
(d) x=0.05m y=0.05m z=0.05m
Rushton turbine model
• Flat-bottom tank with four flat baffles:
– T=0.2 m.
– Z/T=1.
• Impeller:
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Six blades.
D/T=1/3.
W/D=0.2.
C/T=1/3.
290 RPM.
• Water.
• Reynolds number ~ 2E4.
2-D simulation
Rushton turbine - trailing vortices
Iso-Surface of Vorticity Magnitude (550 s-1)
Colored by velocity magnitude
Rushton turbine - trailing vortices
Iso-Surface of Vorticity Magnitude (550 s-1)
Colored by velocity magnitude
Rushton turbine - trailing vortices
Iso-Surface of Vorticity Magnitude (550 s-1)
Colored by velocity magnitude
Rushton turbine - vorticity
Iso-Surface of Vorticity Magnitude (80 s-1)
Colored by velocity magnitude
Rushton turbine - vorticity
Iso-Surface of Vorticity Magnitude (80 s-1)
Colored by velocity magnitude
Rushton turbine - vortices at surface
Iso-Surface of Vorticity Magnitude (80 s-1)
Colored by velocity magnitude
Rushton turbine - trailing vortices
Iso-Surface of Vorticity Magnitude (550 and 80 s-1)
Colored by velocity magnitude
Rushton turbine - axial velocity
Iso-surface of axial velocity of 0.1m/s. The velocity is
directed upwards in the regions enclosed by the isosurface. The surface is colored by strain rate on a scale
of 0 to 100 1/s.
Hydrofoil impeller (HE-3)
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•
Flat bottom vessel with four
baffles:
– T=0.292m.
– Z/T=1.
– Water.
HE-3:
– Three blades.
– D/T=0.39.
– C/T=0.33.
– 60 RPM.
– Reynolds ~ 1.3E4.
Reference: Myers K.J., Ward R.W.,
Bakker A. (1997) A Digital Particle
Image Velocimetry Investigation of
Flow Field Instabilities of Axial Flow
Impellers, Journal of Fluids
Engineering, Vol. 119, No. 3, page 623632.
(m/s)
Experimental PIV data measured at Chemineer Inc.
Animation has approximately one snapshot every
five revolutions. Plays approximately 12 times
faster than real time.
Time dependent velocity magnitude
(m/s)
2-D Fix
3-D LES (14.5 revs.)
3-D MRF
Velocity on vorticity iso-surfaces
(m/s)
Iso-Surface of Vorticity
Magnitude (30 s-1)
Iso-Surface of Vorticity
Magnitude (15 s-1)
15.5 revs.
Vorticity is: xV
Shear rate is: V
Velocity on vorticity iso-surfaces
(m/s)
Iso-Surface of Vorticity Magnitude (5 s-1)
3.9 revs.
Flow at the surface
(m/s)
HE-3 “oilflow” lines at liquid
surface (8.8 revolutions)
“Oilflow” lines are pathlines constrained to
the surface from which they are released.
HE-3 “oilflow” lines at liquid surface (12.3 revolutions)
(m/s)
(m/s)
HE-3 “oilflow” at vessel wall (18 revolutions)
(m/s)
HE-3 - iso-surfaces of vorticity
Iso-surface of vorticity magnitude for values increasing
from 2 1/s to 30 1/s. The tank rotates while the iso-value
is increased gradually. This animation is for the flow field
at one instant in time only.
Summary stirred vessels
• LES is a transient turbulence model that falls midway between
RANS and DNS models.
• The differences between predicted mixing patterns with RANS
and LES are clear.
• The predicted flow patterns for the HE-3 and PBT compared well
with digital particle image velocimetry data reported in the
literature, and exhibited the long time scale instabilities seen in
the experiments.
• The results of these studies open the way to a renewed
interpretation of many previously unexplained hydrodynamic
phenomena that are observed in stirred vessels.
• However, 2-D fix, 3-D fix, and MRF models are much faster
computationally and still have their place, especially in design.