Ealuaton o Vaous Types o Wall Bounday Condtons o the Boltzmann Equaton Chstophe D. Wlson a, Ramesh K. Agawal a, and Felx G. Tcheemssne b a Depatment o Mechancal Engneeng and Mateals Scence Washngton Unesty n St. Lous, Mssou 6330, USA b Mechancs Depatment Computng Cente o Academy o Scence, Moscow, Russa Abstact. Ths pape pesents the ealuaton o seeal sold wall bounday condtons when used n the numecal soluton o the Boltzmann equaton usng the nte-deence/nte-olume methods. Fe sold wall bounday condtons ae consdeed: (a) adsopton, (b) specula electon, (c) duse electon, (d) Maxwellan electon, and (e) adsopte Maxwellan electon. The bounday condtons ae appled on a two-dmensonal dscetzed elocty space mesh. Methods o applyng the same bounday condtons on a thee-dmensonal elocty space gd ae also pesented. The bounday condtons ae mplemented o the numecal soluton o the hypesonc aeed low oe a lat plate usng a thee-dmensonal genealzed Boltzmann equaton (GBE) sole. The deates that contbute to heat tanse and skn cton at the sold bounday ae calculated and compaed. Recommendatons o uthe ealuaton o the bounday condtons ae made. Keywods: Boltzmann Equaton, Bounday Condtons, Hypesonc Flow, Flat Plate PACS: 5.0.+y, 05.20.Dd INTRODUCTION Applcaton o appopate bounday condtons n the computatonal doman s cucal to obtanng accuate solutons o any poblem beng soled analytcally o numecally. Appopate types o bounday condtons must be mplemented on aous boundaes o the computatonal doman. The aous types o bounday condtons, and whee they ae appled n the computatonal doman, ae dscussed n the ollowng sectons. The man emphass s placed on the sold wall bounday condtons. Fe types o bounday condtons: (a) adsopte, (b) specula electon, (c) duse electon, (d) Maxwellan electon, and (e) adsopte Maxwellan electon ae consdeed, mplemented, and ealuated o the accuacy n computng the skn cton and heat tanse on the sold wall. INFLOW BOUNDARY CONDITION The nlow bounday condton s assumed to be a Dchlet bounday condton. It s assumed that the nlow bounday s at equlbum. A Maxwellan dstbuton uncton centeed at the mean elocty o the ncomng low s used at all the gd ponts o the nlow bounday. Ths condton s mantaned at each tme step. An example o the nlow bounday condton usng the Maxwellan dstbuton uncton s shown on the let sde (L) o Fgue. SOLID WALL BOUNDARY CONDITIONS Fe sold wall bounday condtons ae consdeed o applcaton to the poblem o low aound mmesed bodes adsopton, specula electon, duse electon, Maxwellan electon, and adsopte Maxwellan electon. These bounday condtons ange om elately smple to mplement to dcult to mplement, and om
physcally unealstc to equng tunng o the accommodaton coecent to match the empcal data. The omulaton o each o these bounday condtons and the mplementaton ae dscussed n the ollowng sectons. Adsopton The adsopte bounday condton s analogous to the no slp bounday condton o scous walls n contnuum soles. Adsopton needs to be accompaned by a de-adsopte phase n ode o the conseaton o mass to be satsed at the sold wall bounday. An appoach o modelng adsopton/de-adsopton at the wall s to apply a Maxwellan dstbuton centeed at zeo elocty usng the pobablty denstes om the physcal space gd pont at the pecedng tme step. Ths appoach to the adsopte bounday condton, as appled n a two-dmensonal elocty space s llustated n the ght hand sde (R) o Fgue. The nlow bounday condton s also depcted wth the adsopte bounday condton o eeence. The adsopte bounday condton s ndependent o the angle o ncdence o the suace elate to the ncomng low. Applcaton o ths method to a thee-dmensonal elocty space smply noles utlzng the Maxwellan dstbuton that s a uncton o the thee coodnate decton eloctes. FIGURE. Inlow bounday condton (L) and adsopte bounday condton (R) Specula Relecton The specula electon bounday condton mples that molecules elect o o the sold wall wth the angles o ncdence and electon beng equal. A undamental assumpton behnd specula electon s a completely smooth suace. Makng that assumpton s not completely ealstc when smulatng molecula nteactons wth a eal suace. The magntude o the elocty ate the collson s the same as the elocty beoe the collson. Howee, the component o the elocty ecto nomal to the suace changes sgn. The nomal ecto, nˆ o the sold wall needs to be calculated at each physcal space node. Then the elected elocty, elocty, ξ can be elated to the ncdent ξ, usng Equaton (). The pobablty o each pont n the elected elocty space s calculated usng weghted aea ntepolaton o a two-dmensonal elocty space, usng Equaton (2), o weghted olume ntepolaton o a thee-dmensonal elocty space. The elatonshp between the aeas, A, and the pobabltes,, ae shown n Fgue 2. The ntepolated pobablty alue s then tanseed to the elected elocty space gd pont usng Equaton (3). The esults o applyng the specula electon bounday condton n a two-dmensonal elocty space o two angles o ncdence (90 and 0 ) ae shown n Fgues 3. ξ = ξ 2nˆ ( ξ nˆ ) o ˆ n 0 ( ) = ( Am, n m, n + Am, n+ m, n+ + Am+, n m+, n + Am+, n+ m+, n+ ) A ( ξ ) ( ξ ) ξ ξ () = (3) (2)
m,n+ m+,n+ A m+,n A m+,\n+ A m,n ( ) ξ A m,n+ m,n m+,n FIGURE 2. Weghted aea aeage o ncdent molecule pobablty undegong specula electon FIGURE 3. Specula electon bounday condton o ncdence angles o 90 (L) and 0 (R) The sold wall bounday suace can be sualzed as a plane wthn the thee-dmensonal elocty space that ntesects the ogn and s pependcula to the suace nomal. In these two-dmensonal epesentatons, the plane collapses to a lne that les wthn the u plane. Agan, the lne uns dectly though the ogn o elocty space. These gues llustate that the dstbuton uncton contaned n the elocty space doman that s located n the decton opposte the suace nomal s elected acoss the plane (o lne) that denes the suace bounday nto the doman o elocty space that s located n the decton o the suace nomal. Fo the suace pependcula to the ncomng low, almost all o the dstbuton uncton s elected nto the upwnd pat o the elocty space doman. At an ncdence angle o 0 (a suace paallel to the ncomng low) only hal o the dstbuton uncton s elected, essentally epoducng the nlow dstbuton uncton. The specula electon bounday condton s equalent to an nscd wall n a contnuum low sole. Duse Relecton The duse electon bounday condton s a bette appoxmaton o the seemngly andom nteacton o molecules wth a ough suace. The duse electon can be modeled by usng unomly dstbuted set o andom numbes to ay the elocty o the elected patcle. The molecula speed alues can be geneated usng the cumulate dstbuton uncton, as pesented by Shen [], shown n Equaton (4). In Equaton (4), ξ s the elected elocty, k s the Boltzmann constant, m s the molecula mass, T s the tempeatue o the molecule undegong the electon, and an s a andom acton unomly dstbuted between zeo and one. Fo a twodmensonal elocty space, the scatteng decton, θ, s detemned om a unomly dstbuted andom numbe between zeo and π, as shown n Equaton (5). Fo a thee-dmensonal elocty space, the scatteng decton n the plane paallel to the suace, φ, also needs to be consdeed. The pe-collson pobabltes ae ncementally dstbuted to the elected elocty space gd usng a smla weghted aea (o olume) appoach as o the specula electon. A total pobablty o each elected elocty space gd pont s detemned by summng all o the ncemental contbutons.
2 k ξ = ln( an ) whee 0 an < and β = 2 T β m (4) θ = π an whee 0 an < and φ = 2 π an whee 0 an < (5) FIGURE 4. Duse electon bounday condton o ncdence angles o 90 (L) and 0 (R) The suace s dened n the same manne as o the specula electon. Theeoe, the scatteng decton s dened by an angle elate to the suace nomal ecto. Fo the case whee the suace s pependcula to the ncomng low, the ente dstbuton uncton s elected and scatteed nto the pat o the elocty space doman that les n the decton o the suace nomal. Unlke the specula electon, the dstbuton uncton o the duse electon has the hghest pobablty o a elocty located at the ogn o elocty space. The total pobablty densty o the nlow dstbuton uncton and the dusely elected dstbuton uncton ae equal. When the angle o ncdence eaches 0, only hal o the nlow dstbuton uncton s elected at the bounday. Agan, the total pobablty densty o the nlow dstbuton s the same as the dusely elected dstbuton uncton. Maxwellan Relecton The Maxwellan bounday condton s a combnaton o specula and duse electon. It s assumed that a cetan pecentage o ncdent molecules undego specula electon. The emanng popoton o molecules undegoes duse electon. The pecentage o molecules that undego duse electon s called the accommodaton coecent, α. The esultng pobablty dstbuton s detemned usng Equaton (6). The accommodaton coecent, α, can be adjusted to empcally match expemental data to accuately elect deences n suace popetes. The esults o applyng the Maxwellan electon bounday condton n a twodmensonal elocty space ae shown n Fgue 5. The alue o the accommodaton coecent, α, s equal to 0.5. Maxwellan Duse ( α ) Specula = α + (6) FIGURE 5. Maxwellan electon bounday condton (α = 0.5) o ncdence angles o 90 (L) and 0 (R)
The Maxwellan dstbuton, o an accommodaton coecent equal to 0.5, esults n hal o the specula electon bounday condton and hal o the duse electon bounday condton beng epesented n the pat o the elocty space doman located n the decton o the suace nomal. Fo an ncdence angle equal to 90, the duse electon component appeas to be o the same magntude as the nlow dstbuton uncton, but only hal o the dstbuton s pesent (the hal that s n the decton o the suace nomal). The specula electon component appeas to be compsed o a complete Maxwellan dstbuton, but the magntudes ae one hal o the nlow alues. The duse electon pobabltes become less domnant oe the specula electon pobabltes as the angle o ncdence eaches zeo. These obseatons may change a deent accommodaton coecent s selected. I ethe zeo o one ae selected, then the obseatons would be the same as o the specula electon o the duse electon, espectely. Adsopte Maxwellan Relecton The last bounday condton s a combnaton o adsopton and Maxwellan electon. A new coecent, β, s ntoduced to epesent the popoton o molecules that expeence adsopton. The emanng molecules ae assumed to expeence the Maxwellan electon, whch s dstbuted between specula and duse electons accodng to the selected accommodaton coecent, α. The pobablty dstbuton uncton s detemned usng Equaton (7). The esults o applyng the adsopte Maxwellan bounday condton n a two-dmensonal elocty space o aous ncdent angles ae shown n Fgue 6. The two coecents, α and β, ae equal to 0.5 and 0.33, espectely. As a esult, the pobablty dstbuton uncton s compsed o equal pats o adsopton, specula electon, and duse electon. ( ) ( β ) α + ( α ) Adsopte Maxwellan = β Adsopte + Duse (7) Specula FIGURE 6. Adsopte Maxwellan electon bounday condton (α = 0.5, β = 0.33) o ncdence angles o 90 and 0 It s clea by examnng Fgue 6 that the pobabltes assocated wth the specula electon component ae educed beyond that o the Maxwellan electon. The magntude o the duse electon component located at the ogn o the elocty space appeas unchanged. Ths appeaance s due to the act that the adsopte bounday condton, also located at the ogn o the elocty space, now compses pat o the elected bounday condton. It s mpotant to note that the behao noted n the pecedng dscusson wll undoubtedly change deent alues o α and β ae chosen. APPLICATION OF BOUNDARY CONDITIONS TO FLOW OVER A FLAT PLATE The sold wall bounday condtons dscussed n the pecedng sectons wee appled to smulaton o supesonc low oe a lat plate, at Mach 3, usng a dect Boltzmann sole based on the appoach o Tcheemssne [2]. These smulatons wee conducted n ode to make a pelmnay detemnaton egadng the eecteness o each type o bounday condton n smulatng the low nea a sold wall bounday. The equalent Knudsen numbe s 0.5 snce the lat plate extends between 4.0 and 6.0 n the X/λ decton. A wall tempeatue equal to the eesteam alue was assumed o the adsopte bounday condton. An example o the Mach numbe and densty contous s shown o the duse electon bounday condton n Fgue 7. The e bounday condtons can be compaed by calculatng the patal deates that ae popotonal to skn cton and heat tanse, as shown n Equaton (8). Fgue 8 pesents a compason o the skn cton and heat tanse popetes along the suace o the lat plate.
c ( U U ) ( Z λ) and q ( T T ) ( Z λ) (8) FIGURE 7. Mach numbe and densty contous o duse electon along a lat plate at Mach 3 FIGURE 8. Compason o deate o skn cton and heat tanse o low along a lat plate at Mach 3 As expected, the heat tanse and skn cton o the specula electon bounday condton s zeo acoss the ente suace o the lat plate. The duse electon bounday condton esults n heat tanse alues that ae not too dssmla om the specula electon alues. Howee, the skn cton alues ae hghe o duse electon. Fo the Maxwellan electon, the skn cton alls between the alues obtaned o specula electon and the duse electon. Inteestngly, the Maxwellan electon esults n absolute alues o heat tanse that ae hghe than those obtaned om ethe specula electon o duse electon. The adsopte bounday condton yelds the hghest skn cton alues and the hghest absolute heat tanse alues. When combned wth the Maxwellan electon, the adsopte bounday condton domnates the heat tanse pocess and esults n a pole that s almost the same as pue adsopton. The adsopte bounday condton also domnates the skn cton alues, but to a lesse extent. SUMMARY Fo smulatng low oe a lat plate, pue adsopton esults n the hghest alues o skn cton and heat tanse. Pue specula electon esults n the lowest alues o skn cton and heat tanse. Duse electon esults n skn cton alues that all between the adsopte and specula electon alues. The Maxwellan electon yelds some lexblty n tunng the skn cton and heat tanse alues acoss the plate. Howee, combnng the adsopte bounday wth the Maxwellan electon enables an een geate degee o lexblty. Ths lexblty wll be extaodnaly useul when attemptng to match numecal smulatons to expemental data. REFERENCES. C. Shen, Raeed Gas Dynamcs: Fundamentals, Smulatons & Mco Flows, Spnge-Velag, Beln, 2005. 2. F. G. Tcheemssne, Doklady Physcs 47, 872-875 (2002).