Velocity and Temperature Boundary- Layer Modeling Using Averaged Molecule cluster Transport Equations

Transcription

1 IUVSTA 011 Leiseiler, May 16 th 011 Velocity ad Temperature Boudary- Layer Modelig Usig Averaged Molecule cluster Trasport Equatios R. Groll Uiversity of Breme, Am Fallturm, D-859 Breme R. Groll 1

2 Micro chael flo cotiuum approch IUVSTA 011 Leiseiler, May 16 th 011 Micro- ud macroscopic approach Characteristic layer thicess L Molecular umber desity iside trassoic ozzle R. Groll

3 Micro chael flo slip velocity IUVSTA 011 Leiseiler, May 16 th 011 Poiseuille flo of high Kudse umbers x 1 x u Gas -u Wall L s Modificatio of parabola ear the all Slip Velocity Slip Legth R. Groll

4 Statistical modelig Filterig IUVSTA 011 Leiseiler, May 16 th 011 Molecule umber desity ad statistical filters Molecule umber N Cotrol volume V0 Number desity N lim V0 0 V 0 Molecule velocity Mea velocity i i fb(, t x, ) d i Number-desity eighted Velocity deviatio u 1 i i i '' i i i x R. Groll 4

5 Statistical modelig Velocity correlatios IUVSTA 011 Leiseiler, May 16 th 011 Cotiuity of molecular diffusio 0 0 t x t x t x x ' D x Velocity correlatio ' D x ' i D i x x i Aalogy of macroscopic values m m T B u md Sc=1 i i pm m E d m R. Groll 5

6 Statistical modelig Mometum trasport IUVSTA 011 Leiseiler, May 16 th 011 Number-desity eighted averagig Molecule velocity ξ E E 0 0 t x t x Mometum trasport equatio t x t x x 0 i i i i i i Usteady D i x x xi x Covectio Source D 1 Diffusio x xi x x x Coversio i R. Groll 6

7 Statistical modelig Eergy trasport IUVSTA 011 Leiseiler, May 16 th 011 Turbulece aalogy 1 D 1 i i i i D i xi x x x xi x x i Eergy trasport equatio i 0 D i t x t x x x x x Terms of ietic dissipatio ad other eergetic productio Depedig o velocity ad umber desity gradiets covectio diffusio compressio dissipatio i 1 D D D x xi xi xi x coversio Eergy Diffusio ithout desity gradiets ors ith Pr=κ/(κ +) = 1/γ < 1. R. Groll 7

8 Statistical modelig thermodyamic cosistace IUVSTA 011 Leiseiler, May 16 th 011 Thermodyamic cosistece of trasport equatios de Tds pdv e Ts pv Costitutive equatio Variatio eergy / itrisic eergy Etropy icrease i 1 D i Ts i D D x x xi xi x xi x x x dissipatio coversio diffusio Compressio Eergy icrease Expasio Etropy icrease u e eu Ts e t x x covectio 1 compressio Oly for γ>1 R. Groll 8

9 Boudary Coditios Wall shear stress IUVSTA 011 Leiseiler, May 16 th 011 Boudary coditios Well-o all shear stress L L L 1 Lp0 0 K c Lp0 L Modeled ifluece of large mea-free-paths o all shear stress c K 1 L 1 / / L L L L L 1 / / / L 1 K R. Groll 9

10 Boudary Coditios Velocity profiles IUVSTA 011 Leiseiler, May 16 th 011 Computatioal resuts Pressure gradiet coditio: dp 0 dx K No-dimesioal values: u p u p i 0 i Hp0 p0 K 0 L x 0 0 x L K=0.05 K=1.00 R. Groll 10

11 Coclusios Scale-trasitioig Aalogy IUVSTA 011 Leiseiler, May 16 th 011 Coclusios Cotiuity, mometum ad erergy are calculated by trasport equatios of umber desity ad the first ad secod statistical momets of the molecular velocity. ( statistical modellig) (,, ) i i fb t x d E u 1 E e Aalogy of macroscopic values Algebraic model describes scalar diffusio coefficiet ( diffusio modellig) m E d m For high Kudse umbers : Diffusio is described by formulatio of dilute gases ( slip modellig) L 1 K / 1 R. Groll 11

12 IUVSTA 011 Leiseiler, May 16 th 011 May thas for your attetio! R. Groll 1