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DEVELOPMENT AND VALIDATION OF MODEL FOR AEROSOLS TRANSPORTATION IN BOUNDARY LAYERS
A.S. Petrosyan, K.V. Karelsky, I.Smirnov
Space Research Institute
Russian Academy of Sciences
Objectives
• To understand the structure and governing mechanisms of the boundary layer processes specifically, the influence of the thermal and orografyc inhomogeneity
• To analyze the structure of wind flows over complex terrain in close proximity to the surface
• To analyze the structure of wind flows over complex terrain in the important case non stationary conditions of dust lifting and transportation
Traditional boundary layer model
• No slip conditions for wind velocity on thesolid surface
• Prandtl hypothesis: fast velocity gradientsbrings the balance of viscosity and inertiaforces effects
• The viscous dumping is almost impossibleto neglect near surface even with highReynolds number
Details of our boundary layer model
• To abandon no-slip conditions, horizontal velocity must be computing parameter
• To reduce order of the used equations
• To ensure high gradients of the flow near surface in consequence of Prandtl hypothesis
• To provide viscous dissipation by means of the scheme viscosity
Model assumptions
• Volume concentration of the rigid particles is moderately high
• Viscosity and heat condition of the fluid and solid phases do not take effect at impulse and energy transfer in macroscopic scales
• Characteristic of the time scales of atmosphere motions are in excess of the interphase relaxation time
Governing Equations
0z
v)pe(
y
v)pe(
x
v)pe(
t
e
0z
)vp(
y
)vv(
x
)vv(
t
)v(
0z
)vv(
y
)vp(
x
)vv(
t
)v(
0z
)vv(
y
)vv(
x
)vp(
t
)v(
0z
)v(
y
)v(
x
)v(
t
zyx
2zzyzxz
zy2yyxy
zxyx2xx
zyx)
2
vvv(e
2z
2y
2x
.volumeunitperenergyernalint
,volumeunitperenergytotale
,velocityv,v,v
,pressurep
,density
zyx
Model physics Model of the effective perfect gas for atmosphere with solid particles
phaseeachofdensity
phaseeachandmixtureforcapacitiesthermalc,c,c
phasefluidandmixturefortstanconsgasR,R
1xx,x
cxcxc,RxR
RTp,cT
i
21
1
21i
i
221111
Advantages of the atmosphere model
Equations for dusty atmosphere are similar perfect fluid equations
with changed specific heat ratio and sound speed Cs
1c/)Rc(
s11
1s1s C
xC
pC
Possibility to describe uniformly atmosphere flows over complex terrain
in the conditions of impurity lifting and deposition
Integral form of the equations
0dtdxdyy
v)pe(dtdxdy
x
u)pe(dtdxdyt
e
0dtdxdyy
)vp(dtdxdy
x
uvdtdxdy
t
v
0dtdxdyy
uvdtdxdy
x
)up(dtdxdy
t
u
0dtdxdyy
vdtdxdy
x
udtdxdyt
LLL
L
2
LL
LL
2
L
LLL
L-arbitrary volume
Integral form of the equations after using Gauss theorem
0vdtdx)pe(udtdy)pe(edxdy
0dtdx)vp(uvdtdyvdxdy
0uvdtdxudtdy)up(udxdy
0vdtdxudtdydxdy
L
L
2
L
2
L
Computational algorithm |)CD|cosvR|AB|cosvR|BC|vR|AD|vR|CD|sinuR|AB|sinuR(
SRR )3()3(
1)1()1(
1)2()2(
1)4()4(
1)3()3(
1)1()1(
1ABCD
11
|)CD|cosvR|AB|cosvR|BC|vR|AD|vR|CD|sinuR|AB|sinuR(S
RR )3()3(2
)1()1(2
)2()2(2
)4()4(2
)3()3(2
)1()1(2
ABCD22
)cos|CD|vuRcos|AB|vuR
|BC|vuR|AD|vuRsin|CD|)uRP(sin|AB|)uRP((S
uRuR
)3()3()3(1
)1()1()1(1
)2()2()2(1
)4()4()4(1
2)3()3(1
)3(2)1()1(1
)1(
ABCD11
)sin|CD|vuRsin|AB|vuR|AD|)vRP(
cos|CD|)vRP(|BC|)vRP(cos|AB|)vRP((S
vRvR
)3()3()3(1
)1()1()1(1
2)4()4(1
)4(
2)3()3(1
)3(2)2()2(1
)2(2)1()1(1
)1(
ABCD11
)cos|CD|v)PE(cos|AB|v)PE(
|BC|v)EE(|AD|v)PE(sin|CD|u)PE(sin|AB|v)PE((S
EE
)3()3()3()1()1()1(
)2()2()2()4()4()4()3()3()3()1()1()1(
ABCD
Details of Godunov method
• Divergent difference scheme
• Provides for the viscosity by local tangential gaps at every step of the grid
• Such tangential gaps do not manifest themselves on the outer flow scale and make available dissipation energy
Digitization procedure
• Computational domain is sectored on curved trapezium grid by two set of lines
• Transformed computational space consists of rectangles
• If each grid hydrodynamic quantities are replaced by certain average constant at present instants of time
• Beginning with initial constant in each grid we make computations step by step at time intervals of interest to us
