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FULLY LAGRANGIAN APPROACH

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University of Brighton, UK, January 2012
A FULLY LAGRANGIAN APPROACH
FOR MODELING DISPERSE (COLLISIONLESS AND
PRESSURELESS) MEDIA.
ADVANTAGES AND PROSPECTS
A.N. Osiptsov
Head of Laboratory of Multiphase Flow
Institute of Mechanics, Lomonosov MSU
Professor of Dept. Math&Mech MSU
osiptsov@imec.msu.ru
20.09.2014
OUTLINE
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Instead of introduction
Examples of collisionless media and motivation
Problems in modeling: caustics, “folds”, “wrinkles”,
density singularities, etc.
Fully Lagrangian approach (FLA)
Examples of solutions found using the FLA (multivalued density fields, density singularities, collimated
particle beams, etc. )
Combined FLA and Lagrangian “viscous-vortex”
method.
Example: impulse two-phase jet
LOMONOSOV MOSCOW STATE
UNIVERSITY
ONE OF THE FOUNDERS OF MSU (1755)
Mikhail Lomonosov
MAIN BUILDING AND
INSTITUTE OF MECHANICS
LABORATORY OF MECHANICS OF MULTIPHASE MEDIA
Institute of Mechanics Lomonosov Moscow State University
Alexander Osiptsov
Head of Lab, Prof.
Dr. Sci.
Natalia Lebedeva
Ph.D., RS
Eugene Asmolov
Ph.D., SRS
Yurii Nevskii
Ph.D., RS
Sergei Boronin
Ph.D., SRS
Ninel Pashchenko
Ph.D., SRS
Irina Golubkina
Ph.D., RS
Katya Popushina
JRS
Vlad Izmodenov
Prof., Dr. Sci., SRS
Oyuna Rybdylova
JRS
OBJECTIVE: DEVELOPMENT OF GENERAL APPROACHES TO
MATHEMATICAL MODELING OF MULTIPHASE SYSTEMS
CURRENT AREAS OF INTEREST
- TWO-FLUID AND KINETIC-CONTINUUM MODELS
- PROBLEMS OF CLOSURE OF MACROSCOPIC MODELS
(COLLECTIVE EFFECTS, EFFECTS OF SAFFMAN, VIRTUAL MASS,
”HISTORY” FORCES, ETC.)
- FULLY LAGRANGIAN APPROACH
- BOUNDARY LAYER APPROXIMATION
- HYPERSONIC DUSTY FLOWS AND HEAT TRANSFER
- LINEAR STABILITY AND TRANSITION PROBLEMS IN MULTIPHASE FLOWS
- MECHANISMS OF FORMATION OF MESOSCALE NONUNIFORMITIES
(IN SEDIMENTING SUSPENSION, IN VORTEX FLOWS, ETC.)
- INERTIAL FOCUSING OF PARTICLES IN HIGH-SPEED FLOWS
- ASTROPHYSICAL AND GEOPHYSICAL APPLICATIONS
MOTIVATION FOR MULTIPHASE FLOW RESEARCH
- AERODYNAMIC APPLICATIONS (NOZZLE FLOWS WITH CONDENSED
PARTICLES, VEHICLE MOTION THROUGH DUSTY AND AEROSOL CLOUDS,
ETC.)
- ASTROPHYSICAL APPLICATIONS (SPACE DUST, COMET ATMOSPHERES,
MARTIAN PROJECTS, ETC.)
- ENVIRONMENTAL PROBLEMS (DUST STORMS, DUST RISE BEHIND
MOVING FRONTS, POLLUTION SPREADING, ETC. )
- CLEANING, COATING, DIRECT-WRITE, AND PARTICLE INTRUSION
TECHNOLOGIES
- POWER ENGINEERING, HEAT EXCHANGERS, COMBUSTION CHAMBERS,
ETC.
- CHEMICAL ENGINEERING, BIOTECHNOLOGIES, GAS AND OIL INDUSTRY
EASY TO CONTINUE ...
Aircraft and helicopter motion in dusty atmosphere
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Sahara dust over Atlantic ocean
Prospective missions to Mars
Landing unit
Global dust storm
Dust devils
Heat fluxes
Dunes on Mars
Condensation in near-sonic flows
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F/A-18 Hornet
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F/A-18
Dust rise behind moving shock
waves and fronts
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Dust explosions in coal mines
Explosions in pneumatic
conveying systems for powder
materials
Front of a dust storm
Surface cleaning, cold-spray coating, particle intrusion,
etc. using high-speed two-phase jets
В«NEDDLE-FREEВ» DELIVERY OF
POWDER DRUGS
“COLD-SPRAY” COATING
OTHER EXAMPLES OF COLLISIONLESS MEDIA
MEDIA CONSISTING OF INDIVIDUAL NON-COLLIDING
ELEMENTS (PARTICLES)
HYPERSONIC GAS FLOWS, LARGE SCALE MASS DISTRIBUTION IN
THE UNIVERSE, TRAFFIC JAMS, HIGH-SPEED DEFORMATION OF SOLID
MEDIA, “COLD” PLASMA, GRANULAR FLOWS, DISPERSED PHASE IN
MULTIPHASE FLOWS, ETC.
Typical dynamic behavior of collisionless medium:
Flock of birds
MATHEMATICAL MODELING OF
COLLISIONLESS MEDIA
Different Levels of Description of Dispersed Phase in FluidParticle Mixture
Р°) Individual particles
Carrier phase: N-S Eqs. +
b) Kinetic level
c) Continuum level
Concentration and mean
particle velocity
N-S Eqs. +
Governing Equations
f
(1)
t
пЂ«c
ns (t, r ) пЂЅ
пѓІ
f
r
f
m
f
(1)
(1)
пЂ«
(1)
(f / m )
c
пЂ«
d c d пЃі dT s , V s пЂЅ
d Vs
пЂЅ fs ,
dt
ns
t
f
(1)
( q / csm )
 Ts
пѓІ
csm
f
(1)
dT s
dt
пЂ« div( n s V s ) пЂЅ 0
пЂЅ0
c d c d пЃі dT s / n s
пЂЅ qs
Kinetic equation for oneparticle distribution
function for collisionless
medium
Pressureless-continuum
equations
Basic continuum model of a collisionless
and pressureless medium
fs = fs(t, x, y, z, V, V, ns……)
Basic Two-Continuum Model of ”Dusty Gas”
CARRIER PHASE
DISPERSED PHASE
- Correction factors in the force and heat
flux on the particle (Carlson, Hoglund, 1964 )
ADDITIONAL SIMILARITY
PARAMETERS
``EFFECTIVE-GASВґВґ MODEL
Simple model?
No, very difficult…
High compressibility of “pressureless” medium
and singularities in the density field
Particles, moving in velocity field generated by an array of point vortices
Low inertia particles
пЃґв†’0: Vs=V, div Vs =0
пЃґ = Оµ в‰ 0: singularities
LIGHT-WEIGHT PARTICLES
HEAVY PARTICLES
Non-orthogonal stagnation points: particle motion with
penetration and fold formation (inertial particles)
пЃІ = 1.0
m = 10.0
пЃ¬ = 1.6 (j1 = 51.34п‚°)
пЃЈ2 = 1.2 (j2 = 120.96п‚°)
b = 0.5
EXAMPLES OF FLOWS WITH SINGULARITIES IN PARTICLE NUMBER
DENSITY OF POINT NON-INTERACTING PARTICLES
Osiptsov, Fluid Dyn. 1980, 1985
1-D flows with the formation of
“folds” and caustics
IMPORTANT:
at the points of integrable singularities
the mean distance between particles
is finite and the model of
non-colliding particles remains valid
Flows past blunt bodies in the regime of
absence particle inertial deposition
Accumulation of Stokes
particles on the wall
Caustics in Stochastic Flows
Wilkinson et al., Phys. Fluids, 2007
STANDARD EULERIAN APPROACH
IS INVALID….
WHAT TO DO?
HOW TO CALCULATE THESE FLOWS?
IT IS NATURAL TO USE LAGRANGIAN
APPROACH, BUT HOW TO CALCULATE THE
PARTICLE CONCENTEATION?
FULLY LAGRANGIAN APPROACH
Continuity Equation in Lagrangian Form
d
dt
пѓІ
V (t )
n s ( t , r ) dxdydz пЂЅ
d
dt
пѓІ
n s ( t , r0 ) | J | dx 0 dy 0 dz 0 пЂЅ 0
V (0)
n s ( t , r0 ) | J |пЂЅ n s (0, r0 ),
J ij   x i /  x j 0
| J | пЂЅ det( J ij )
BASIC IDEAS OF THE FULLY LAGRANGIAN APPROACH
A.N. Osiptsov, 1984, 1988, 2000, 2008
System of ordinary differential equations (ODE)
on a fixed particle trajectory :
Joseph-Louis Lagrange
1736-1813

3D - 24
2D - 12
  x 1  x 10
пѓ§
J   x 1  x 20
пѓ§
 x x
30
пѓЁ 1
 x 2 x 10
 x 2  x 20
 x 2  x 30
 x 3  x 10 
пѓ·
 x 3  x 20
пѓ·
 x 3  x 30 
“FLA” IN CURVILINEAR ORTHOGONAL COORDINATES
n s (  10 ,  20 ,  30 , t ) H 1 H 2 H 3 det ||   i /   j 0 ||
пЂЅ n s 0 ( пЃё 10 , пЃё 20 , пЃё 30 , 0) H 10 H 20 H 30
Hi – Lamé
coefficients
i
t
пЂЅ
v si
,
t
t
t
Hi
 J ij
 ij
 v si
пЂЅ
пЂЅ
пЃ— ij
Hi

 j0
пЂЅ

k
пЂ­
vi
Hi
2
пѓ¦
H k
H i 
пЂ­ v si
пѓ§ v sk
пѓ· пЂ« f si
H iH k пѓЁ
i
i 
v sk

k
H
 k
j
J kj ,
J ij пЂЅ
i
 j0
пѓ¬
v sk пѓ¦
H k
H i  
пѓЇ
  f si
пЂ­ v si

пѓ§ v sk
пѓ·пѓЅ пЂ«
 i
 i  
пѓЇ
пѓ® k H iH k пѓЁ
  j0
DEVELOPMENT OF THE FULLY LAGRANGIAN APPROACH
Taking into account flow
nonuniformities and non-stationary
effects in interphase force
Simplification for steday-state flows reduction of the number of unknowns
3D - 18, 2D - 8
ODE system on a fixed trajectory:
DEVELOPMENT OF THE FULL LAGRANGIAN APPROACH FOR
KINETIC DESCRIPTION OF A SYSTEM OF POLYDISPERSE
EVAPORATING
PARTICLES
ПР�МЕР: 1-D ПОЛ�Д�СПЕРСНАЯ
С�СТЕМА
С ФАЗОВЫМ� ПЕРЕХОДАМ�
Kinetic equation
(Eulerian form):
Lagrangian form:
System of ODE on a “trajectory”:
SIMPLIFICATIONS FOR STEADY-STATE
FLOWS
r ( r0 , t пЂ« dt ) пЂЅ r ( r0 пЂ­ d l 0 , t ),
r
 l0
xi
x0
пЂЅ
( r0 , t ) пЂЅ
v si
V s 0 n1
пЂ­
1 r
Vs 0 t
n 2 xi
n1  y 0
пЂ­
пЂЅ
d l 0 пЂЅ V s 0 dt
Vs
V s0
n3 xi
n1  z 0
( x1 пЂЅ x , x 2 пЂЅ y , x 3 пЂЅ z ),
n i пЂЅ cos( V s 0 , e i ),
,
( i пЂЅ 1, 2 ,3 )
us
x / y0
x / z0
n s ( r0 , t ) det v s
y / y0
 y /  z 0  n s 0 ( r0 , 0 ) u s 0
ws
z / y0
z / z0
AS A RESULT:
THE NUMBER OF ADDITIONAL EQUATIONS
ON A PARTICLE TRAJECTORY IS REDUCED
IMPORTANT PARTICULAR CASES
Flows with appearance
of new particles

t
(n
s
n sur d пЃ“ пЂЅ n sur 0 d пЃ“ 0 ,
Evolution of material
lines (II) and surfaces (I)
(I )
J пЂЅ det
(I )
z 0 пЂЅ s ( x0 , y0 )
( II )
y 0 пЂЅ g ( x 0 ),
n l dl пЂЅ n l 0 dl 0
z 0 пЂЅ h ( x0 )
n sur ( x 0 , y 0 , t ) J пЂ« J 1 пЂ« J 2 пЂЅ n sur 0 ( x 0 , y 0 , 0 ) 1 пЂ« s ' x 0 пЂ« s ' y 0
2
x / x0
y / x0
x / y0
y / y0
( II )
( r0 , t ) det || J ij || пЂ© пЂЅ j n det || J ij ||
,
J 1 пЂЅ det
 x
n l ( x 0 , t ) пѓ§пѓ§
 x0
2
2
2
2
z / x0
y / x0
z / y0
y / y0
пѓ¶
 y
пѓ· пЂ«пѓ§
пѓ·
 x
пѓё
пѓЁ 0
2
пѓ¶
 z
пѓ· пЂ«пѓ§
пѓ·
 x
пѓё
пѓЁ 0
пѓ¶
пѓ·
пѓ·
пѓё
,
J 2 пЂЅ det
2
x / x0
z / x0
x / y0
z / y0
2
пЂЅ n l 0 ( x 0 ,0 ) 1 пЂ« g ' x0 пЂ« h ' x0
2
2
EXAMPLES OF APPLICATION OF
FULLY LAGRANGIAN APPLROACH
Aerodynamic focusing of inertial particles behind the point of
interaction of shock waves in steady 2D flows
I.V. Golubkina, A.N. Osiptsov, Fluid Dynamics, 2007, No. 4
1.Symmetric regular shock wave reflection
2.Symmetric Mach shock wave reflection
1
3.Asymmetric regular shock wave reflection regime
2
3
Aerodynamic focusing of inertial particles behind the point of
interaction of shock waves in steady 2D flows
1. Symmetric regular shock
wave reflection regime
2. Symmetric Mach shock
wave reflection regime
3. Asymmetric regular shock wave
reflection regime
Parametric Numerical Study of the Particle Focusing Effect
Surface in the space of governing
parameters corresponding to
optimal regimes of particle
focusing (Оґв†’в€ћ):
Parameters characterizing the
rate of particle focusing
d ,пЃ¤ пЂЅ
D
d
Governing parameters:
M 0 , j 0 ,пЃ№ 0 , Re
s0
38
Interaction of Bow Shock with an
Oblique Shock Wave in Dusty Gas
Osiptsov A.N. et al., Fluid Dyn., 2011, N1
Edney B.E., “Anomalous
heat transfer and
pressure distributions
on blunt bodies at
hypersonic speeds in
the presence of an
impinging shock”, FFA
Rept. 115, 1968.
Borovoy V. Ya., Chinilov
A. Yu., Gusev V. N., et
al. “Interference
between a cylindrical
bow shock and a plane
oblique shock” // AIAA
J. 1997. V. 35.
39
Gas Flow in III and IV Shock Wave Interaction Regimes
Carrier-phase parameters are found from the solution of the Navier-Stokes equations
on a nonunform Eulerian grid
V. Ya. Borovoy et al.
“Interference between
a cylindrical bow
shock and a plane
oblique shock”, 1997
пЃ§ пЂЅ 1 .4 , M 0 пЂЅ 6 ,
Re пЂЅ 2 . 25 пѓ— 10 ,
5
Pr пЂЅ 0 . 7 , T w пЂЅ 1
40
Particle Trajectories and Concentration Fields for III Regime
b
Re
пЂЅ 2 пѓ— 10
s0
пЂЅ 450
пЂ­2
4 . 08 пѓ— 10
315
пЂ­2
0 . 125
2
180
45
41
Particle Trajectories and Concentration Fields for IV Regime
b
Re
пЂЅ 5 пѓ— 10
s0
пЂЅ 900
пЂ­3
2 пѓ— 10
450
пЂ­2
8 пѓ— 10
225
пЂ­2
2
45
42
PARTICLE ACCUMULATION ZONES IN TORNADO-LIKE FLOWS
Carrier phase
• viscous
п‚· incompressible
Flow:
• Axisymmetric
• Steady-state
• Far from the axis – flow from
a free vortex filament:
z п‚® п‚Ґ:
*
v пЂЅ v пЂЅ 0, v пЂЅ
*
r
p пЂЅ p пЂ­ пЃІпЃ‡
*
*
п‚Ґ
*
z
2
*
j
2r
пЃ‡
r
*
,
*2
Scales:
L – length
 L – velocity
Similarity parameter:
Re пЂЅ пЃ‡пЃІ m
пЃІпЃ‡
2
2
L
– pressure
ZONES OF PARTICLE ACCUMULATION IN TORNADO-LIKE FLOWS
N.A. Lebedeva, A.N. Osiptsov, Fluid Dynamics, 2009
z
0.0
0.4
1.5
0.6
1
0.8
0.5
0
z0
0.9
0.5
1
1.5
r
0.0
20
0.2
15
0.3
10
0.5
0.7
5
0.9
0
0
5
10
r
TWO-PHASE FLOW STRUCTURE IN A LOCALIZED
LARGE-VORTICITY REGION (Kelvin “cat’s eye” flow)
particles
3
g
2
1
y0
x
пЃ№ пЂЅ ln ( A ch( y ) пЂ« C cos( x ) пЂ©
St, Fr
-1
-2
-3
0
2
4
6
Carrie phase:
uпЂЅ
vпЂЅ
C sh ( y )
C ch ( y ) пЂ« A cos( x )
A sin ( x )
C ch ( y ) пЂ« A cos( x )
8
10
12
Dispersed phase:
dv s
пЂЅ
(v пЂ­ vs )
dt
ns J пЂЅ 1
St
пЂ­
Nondimensional
parameters:
1
Fr
2
j
C, A пЂЅ
C пЂ­1
2
KELVIN «CAT’S EYE»: FAST SEDIMENTATION
St = 1.0, Fr = 1.0, C = 2.0, vs0 = 0
KELVIN «CAT’S EYE: FAST SEDIMENTATION
5
1.0
1.0
1.0
1.0
y
1.4
1.4
0.7
0.7
1.4
0
-5
пЃ°
2пЃ°
3пЃ°
St = 1.0, Fr = 1.0, C = 2.0, vs0 = 0
x
4пЃ°
Momentum exchange between the phases in
high-gradient flows: Stokes drag and Saffman lift force
FSaff
FStokes
• Stokes drag force (Klyachko
approximation for high Res):
FStokes
FSaff
• Saffman lifting force:
P.G. Saffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech. 1965. V. 22.
pp. 385-400. Corrigendum. J. Fluid Mech. 1968. V. 31. pp. 624-625.
DUST RISE BEHIND A MOVING SHOCK WAVE
(Osiptsov A.N. & WANG B.Y., Acta Mech. Sin. 2005)
Aerodynamic focusing of dispersed
particles in narrow channels due to the
action of Saffman force
COLLIMATED PARTICLE BEAMS IN
NANOTECHNOLOGIES В«DIRECT-WRITEВ»
I.S. Akhatov, A.N. Osiptsov, et al. 2009
Particle trajectories in a circular nozzle
X=1
X=3
X=6
X=10
Particle focusing behind a shock wave moving in a
microchannel (d ~ 100 Ојm)
O.D. Rybdylova, A.N. Osiptsov, Physics-Doklady, 2010
Examples of calculated particle trajectories behind a
moving shock wave
Modeling of Two-Phase Impulse Jet Using a Completely Meshless Method
Carrier Phase:
• Motion is induced by impulse jet injected in a quiescent fluid
during time interval пЃ„t;
• Initial vorticity in the jet corresponds to Poiseuille profile :  = 2y.
Dispersed Phase:
• Initial velocity slip is zero;
• Uniformly distributed in the injected fluid;
• Force on the particle – Stokes drag.
y
Scales:
L – length
2L
x
 – vorticity
U пЂЅ пЃ—L
T пЂЅпЃ—
пЂ­1
– velocity
– time
Nondimensional
parameters:
Re пЂЅ
b пЂЅ
LU
пЃ®
L
ls
ls – particle velocity
relaxation length
Motion of Viscous Vortices in the Impulse Jet
Re = 100, пЃ„t = 0.2
Motion of Impulse Two-Phase Jet
Re = 100, b = 1, пЃ„t = 0.2
Problems of Mixing: Evolution of a Contour of Particles
ADVANTAGES OF FULLY LAGRANGIAN APPROACH
- Calculation of all parameters of pressureless continuum (including
density) on a fixed particle trajectory from a system of ODE
- Possibility to calculate (with controlled accuracy) the flows with
“folds”, fragmentation of phase volume, and regions of multiple
intersections of particle trajectories
- Possibility to calculate steady-state and unsteady flows without change of
algorithm
- Possibility to calculate flows with integrable singularities of particle number
density
- Feasibility to calculate particle concentration fields in 3-D flows
Prospects for nearest future
пЃ®
Investigation of single-phase stratified flows
пЃ®
Investigation of the behavior of finite Lagrangian volumes,
surfaces, and filaments
пЃ®
Investigation of flows with phase transitions (evaporation,
condensation, etc.) and the variation of particle number
пЃ®
Combination of the Lagrangian viscous-vortex method
with the Fully Lagrangian approach for studying unsteady
viscous two-phase flows
пЃ®
Investigation of particle clustering in turbulent flows
SALVADOR DALI (1983)
THANK YOU FOR ATTENTION!
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