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