CNS.1 Compressble Naver-Sokes Tme Averaged Insananeos flow conservaon prncples, compressble flow D M : L( ρ) = ρ + ( ρ ) = 0 x ρ D P : L( ρ ) = + ρ + pδ = 0 x D E : L( ρe) = ( ρe+ ρ / ) + ( ρh+ ρ / q) = 0 x Sae: p= ρrt defnons : h= e+ p/ ρ = c p T, e= ct 1 k 1 = μ, s δ s 3 x + k x x T μ h q = -κ = -, Pr = Prandl nmber x Pr x sae varable : q( x, ) = { ρ, ρ, ρe, p} T v
CNS. Compressble Naver-Sokes Tme Averaged Favre me-averagng elmnaes densy flcaons 1 1 + T mass-weghed me average : lm (, ) (, ) d 0 ρ xτ xτ τ= ρ T T hs ρ = ρ = ρ U + ρ Favre velocy resolon: = + comparng o convenonal me average: ρ = 0! Morkovn s hypohess(196) : densy flcaon effecs are small for M < 5 sae varable resolon : = + h = h + h ρ = ρ + ρ e= e + e p = P + p T= T+ T q = q + q h
CNS.3 Compressble Naver-Sokes Tme Averaged Favre-averaged conservaon prncples, compressble flow DM DP : L( ρ) = ρ + ( ρ ) = 0 x ρ : L( ρ ) = + ρ + Pδ + ρ = 0 x D E : L( ρe ) = ρe + ρ / + ρ / ρ h + ρ / + ρ / +q l + = 0 x + ρ h + ρ / + ρ Sae: P = ρrt defnons : Reynolds sress ensor : ρτ ρ rblence knec energy : ρk ρ / rblen hea flx: q = ρ h h lamnar hea flx : ql μ = - Pr x
CNS.4 Favre-averaged Compressble Naver-Sokes Eqaons Favre-averaged CNS conservaon law form L( k) formed va Favre-averagng L ( ρ ) wh: P = ρ R T, E = e + / + k, H h + + k
CNS.5 Favre-averaged Compressble Naver-Sokes Eqaons Favre-averaged Reynolds sress ranspor eqaon
CNS.6 Favre-averaged Compressble Naver-Sokes Eqaons FaCNS conservaon saemens reqre closre models D : - = k P ρ ρτ μ S ι k δ 3 x ρ δ k 3 μ c p - D : - = T μ h E ρ h q = Pr Pr x x μ k - + ρ / - μ + σ x D : E = ρε = μ s s x k, k, 3 υ s s 3 k, k, = υ ρωω+ ρ, ρ 3 ρε + ρε s d k ι,,,
CNS.7 Favre-averaged Compressble Naver-Sokes Eqaons FaCNS dsspaon he sm of solenod + dlaaon acon D E : ρε = ρε ρε : L( k) +, s d 4 = υ ρωω + υ ρ (,) 3 = υ (flcang vorcy + flcang velocy dvergence) Sarkar, Zeman dsspaon modfcaons, pressre D E =...+ ρ ( ε + ε ) =0 s d L( ε ) =...+ Cρε/ k =0 s ε s ε ξ f (M ) ε, M = ( k/a ) 1/ d s = rblence Mach nmber p M + (M ), α ρτ α ρε, α, α = 0.15,0.0, 3 3 M ρ kτ / ρε x
CNS.8 FaCNS Compressble Law of he Wall In log layer can neglec convecon, pressre, moleclar dffson DP : x D E: D E : D E : D E : where : τ / τ w ρw ρt = ρ T f β w w β β = β 0 1 ξ (M ) + β = β β ξ (M ) ξ 1 = f for -D 0 0 = consan (3/4 3/) perrbaon solon mehology ransforms y, hence d(.) d d(.) d(.) μ μ = ρ dy dy d w τ d
CNS.9 FaCNS Compressble Law of he Wall Near wall perrbaon solons employ frcon Mach nmber M τ T ρ 1 q w 1 k = 1-( γ -1) Pr M + w = + T ρ τ 3 w ρ τ w τ τ τ M / a τ τ w 1/4 ω 1 C + B A exp κ / 0 U U w τ A,B = f ( M, T, T,Pr w ) = U f ( A, B, / U ) 1 y k -ω law of he w all: ln τ + C κ υ w τ w w κ = (,,,,,Pr ) M w κ f β β α σ γ, ξ +... 0 0 τ 1 C = C + ( ) 1/4 ln ρ / ρ w κ w w ξ = Sarkar com pressbly correcon( 1)
CNS.10 FaCNS Compressble Law of he Wall Smlarly, for he k-ε rblen closre model 1 y law of he wall : ln τ + C κ υ ε τ ε w ( 1 ) κ = κ f C, C, σ, σ,γ, ξ,pr M +... ε ε ε k ε τ 1 C = C + ln ( ρ/ ρ ε ) κ w ε Predcons, -k- ω, --- k-ε-low Re adabac log layer, M=4.5, 10.3
CNS.11 FaCNS Bondary Layer Solons Comparave predcons, k-ω, k-ε-low Re compressble closre models Fg 5.4 Comped and measred skn frcon and velocy profle (x=1.18m), Mach 4,adabac-wall bondary layer wh adverse pressre graden:- k-ω model;--- Chen k-ε model; o Zwars. Fg 5.5 Comped and measred flow properes, Mach.65, heaed-wall bondary layer wh adverse pressre graden:- k-ω model;--- Chen k-ε model; o Fernando and Sms.
CNS.1 Smmary: Favre-averaged Compressble Naver-Sokes Favre-averagng of nsaaneos CNS conservaon PDE sysem elmnaes explc appearance of ρ generaes Re sress ensor τ = rblen hea flx vecor q = h nrodces oal nernal energy and enhalpy, E, H compressble law of he wall s INS-appearng Karman consan becomes fncon of frcon Mach nmber M τ Closre models reqred for hgher-order momens Re sress ensor = f ( S,., k) dsspaon of k conans solenodal and dlaaonal erms modfcaons correlaed wh rblence Mach nmber M ( k ) low Re regon correcons denfed for k - ω& k- εmodels + k-ω model beer agrees wh compressble law of he wall daa y 50 valdaon of boh models for spersonc BL daa