close

Вход

Забыли?

вход по аккаунту

?

О ТОЧНЫХ РЕШЕНИЯХ a-- - МОДЕЛИ ТУРБУЛЕНТНОСТИ.

код для вставкиСкачать
2015
ɇȺɍɑɇɕɃ ȼȿɋɌɇɂɄ ɆȽɌɍ ȽȺ
ʋ 220
ɍȾɄ 514.7
Ɉ ɌɈɑɇɕɏ Ɋȿɒȿɇɂəɏ k − ε –ɆɈȾȿɅɂ ɌɍɊȻɍɅȿɇɌɇɈɋɌɂ
ɇ.Ƚ. ɏɈɊɖɄɈȼȺ
ȼ ɫɬɚɬɶɟ ɦɟɬɨɞ ɩɨɢɫɤɚ ɢɧɜɚɪɢɚɧɬɧɵɯ ɪɟɲɟɧɢɣ ɩɪɢɦɟɧɹɟɬɫɹ ɤ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ, ɨɩɢɫɵɜɚɸɳɢɯ k − ε –
ɦɨɞɟɥɶ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ. ɇɚɣɞɟɧɵ ɬɨɱɧɵɟ ɪɟɲɟɧɢɹ ɷɬɨɣ ɫɢɫɬɟɦɵ. Ɇɟɬɨɞɢɤɚ ɩɨɫɬɪɨɟɧɢɹ ɬɨɱɧɵɯ ɪɟɲɟɧɢɣ,
ɢɫɩɨɥɶɡɨɜɚɧɧɚɹ ɜ ɞɚɧɧɨɣ ɪɚɛɨɬɟ, ɩɪɢɦɟɧɢɦɚ ɢ ɤ ɞɪɭɝɢɦ ɦɨɞɟɥɹɦ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɥɨɤɚɥɶɧɵɟ ɫɢɦɦɟɬɪɢɢ,
ɢɧɜɚɪɢɚɧɬɧɵɟ ɪɟɲɟɧɢɹ, k − ε –ɦɨɞɟɥɶ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ.
1. ȼȼȿȾȿɇɂȿ
Ɋɚɡɧɨɨɛɪɚɡɧɵɟ ɩɪɨɰɟɫɫɵ ɜ ɮɢɡɢɤɟ, ɬɟɯɧɢɤɟ, ɛɢɨɥɨɝɢɢ, ɷɤɨɧɨɦɢɤɟ ɢ ɞɪɭɝɢɯ ɨɛɥɚɫɬɹɯ
ɱɟɥɨɜɟɱɟɫɤɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɱɚɫɬɨ ɦɨɞɟɥɢɪɭɸɬɫɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɫɢɫɬɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. Ʉɚɤ ɩɪɚɜɢɥɨ, ɪɟɲɢɬɶ ɩɨɥɧɨɫɬɶɸ ɬɚɤɭɸ ɫɢɫɬɟɦɭ ɧɟ ɭɞɚɟɬɫɹ.
Ɉɞɧɚɤɨ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɸɬ ɢ ɱɚɫɬɧɵɟ ɪɟɲɟɧɢɹ. ɉɨɷɬɨɦɭ ɡɚɞɚɱɚ ɨɬɵɫɤɚɧɢɹ ɬɨɱɧɵɯ ɪɟɲɟɧɢɣ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢɥɢ ɫɢɫɬɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɫɟɝɞɚ ɹɜɥɹɟɬɫɹ
ɚɤɬɭɚɥɶɧɨɣ.
Ɇɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɦɧɨɝɢɟ ɢɡɜɟɫɬɧɵɟ ɦɟɬɨɞɵ ɩɨɫɬɪɨɟɧɢɹ ɬɨɱɧɵɯ ɪɟɲɟɧɢɣ ɹɜɥɹɸɬɫɹ
ɫɥɟɞɫɬɜɢɟɦ ɧɚɥɢɱɢɹ ɬɨɣ ɢɥɢ ɢɧɨɣ ɫɢɦɦɟɬɪɢɢ ɭ ɢɫɫɥɟɞɭɟɦɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɢɥɢ ɫɢɫɬɟɦɵ. ɇɚɩɪɢɦɟɪ, ɡɧɚɹ ɫɢɦɦɟɬɪɢɸ ɭɪɚɜɧɟɧɢɹ, ɦɨɠɧɨ ɩɨɩɵɬɚɬɶɫɹ ɧɚɣɬɢ ɢɧɜɚɪɢɚɧɬɧɨɟ
(ɚɜɬɨɦɨɞɟɥɶɧɨɟ) ɪɟɲɟɧɢɟ (ɫɦ., ɧɚɩɪɢɦɟɪ, [1], [6]). ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɢɧɜɚɪɢɚɧɬɧɵɟ ɪɟɲɟɧɢɹ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɨɩɢɫɵɜɚɸɬ ɩɪɨɰɟɫɫɵ, ɢɧɬɟɪɟɫɭɸɳɢɟ ɢɫɫɥɟɞɨɜɚɬɟɥɹ, ɜ ɞɪɭɝɢɯ
ɫɥɭɱɚɹɯ ɬɚɤɢɟ ɪɟɲɟɧɢɹ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɟɡɧɵɦɢ ɩɪɢ ɬɟɫɬɢɪɨɜɚɧɢɢ ɩɪɨɝɪɚɦɦ, ɢɫɩɨɥɶɡɭɟɦɵɯ ɞɥɹ
ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ.
ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɦɵ ɫɬɪɨɢɦ ɦɧɨɝɨɩɚɪɚɦɟɬɪɢɱɟɫɤɢɟ ɫɟɦɟɣɫɬɜɚ ɢɧɜɚɪɢɚɧɬɧɵɯ ɪɟɲɟɧɢɣ
ɞɢɫɫɢɩɚɬɢɜɧɨɣ ɞɜɭɯɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ( k − ε –ɦɨɞɟɥɶ).
2. ɈɋɇɈȼɇɕȿ ɉɈɇəɌɂə ɌȿɈɊɂɂ ɋɂɆɆȿɌɊɂɃ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ
ȼ ɑȺɋɌɇɕɏ ɉɊɈɂɁȼɈȾɇɕɏ
ɉɭɫɬɶ E = {F = 0 } — ɫɢɫɬɟɦɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ
(ɞɚɥɟɟ "ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ" ɢɥɢ ɩɪɨɫɬɨ "ɭɪɚɜɧɟɧɢɟ") ɜ ɜɟɤɬɨɪɧɨɦ ɪɚɫɫɥɨɟɧɢɢ
π : E n + m → M n . ȼ ɪɚɦɤɚɯ ɚɥɝɟɛɪɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɥɸɛɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɩɨɞɦɧɨɝɨɨɛɪɚɡɢɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɞɠɟɬɨɜ k -ɝɨ ɩɨɪɹɞɤɚ
J k (π ) ɪɚɫɫɥɨɟɧɢɹ π , ɝɞɟ k — ɦɚɤɫɢɦɚɥɶɧɵɣ ɩɨɪɹɞɨɤ ɭɪɚɜɧɟɧɢɣ, ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ, n —
ɱɢɫɥɨ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ, ɚ m — ɱɢɫɥɨ ɧɟɢɡɜɟɫɬɧɵɯ ɮɭɧɤɰɢɣ (ɡɚɜɢɫɢɦɵɯ
ɩɟɪɟɦɟɧɧɵɯ). ɇɚ ɦɧɨɝɨɨɛɪɚɡɢɢ ɛɟɫɤɨɧɟɱɧɵɯ ɞɠɟɬɨɜ
J ∞ (π )
ɢɦɟɟɬɫɹ n -ɦɟɪɧɨɟ
ɢɧɬɟɝɪɢɪɭɟɦɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ C (ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ʉɚɪɬɚɧɚ), ɡɚɞɚɜɚɟɦɨɟ ɨɩɟɪɚɬɨɪɚɦɢ
ɩɨɥɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ D1 ,…, Dn , ɤɨɬɨɪɨɟ ɞɨɩɭɫɤɚɟɬ ɨɝɪɚɧɢɱɟɧɢɟ C ɧɚ ɛɟɫɤɨɧɟɱɧɨ
ɩɪɨɞɨɥɠɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ E ∞ ⊂ J ∞ (π ) . ɉɨɞ ɫɢɦɦɟɬɪɢɹɦɢ ɭɪɚɜɧɟɧɢɹ E ɩɨɧɢɦɚɸɬ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ (ɤɨɧɟɱɧɵɟ ɢɥɢ ɢɧɮɢɧɢɬɟɡɢɦɚɥɶɧɵɟ) ɛɟɫɤɨɧɟɱɧɨ ɩɪɨɞɨɥɠɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ E ∞
, ɤɨɬɨɪɵɟ ɫɨɯɪɚɧɹɸɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ʉɚɪɬɚɧɚ ɧɚ E ∞ . Ɍɚɤɨɣ ɩɨɞɯɨɞ ɤ ɨɩɪɟɞɟɥɟɧɢɸ
ɫɢɦɦɟɬɪɢɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɪɢɜɟɥ ɤ ɫɨɡɞɚɧɢɸ ɷɮɮɟɤɬɢɜɧɨɝɨ ɚɥɝɨɪɢɬɦɚ
ɜɵɱɢɫɥɟɧɢɹ ɚɥɝɟɛɪɵ ɜɵɫɲɢɯ ɢɧɮɢɧɢɬɟɡɢɦɚɥɶɧɵɯ ɫɢɦɦɟɬɪɢɣ ɤɨɧɤɪɟɬɧɵɯ ɭɪɚɜɧɟɧɢɣ
([1], [2]).
40
ɇ.Ƚ. ɏɨɪɶɤɨɜɚ
ȼ
ɚɥɝɟɛɪɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ
ɬɟɨɪɢɢ
ɜɵɫɲɚɹ
ɨɬɨɠɞɟɫɬɜɥɹɟɬɫɹ ɫ ɷɜɨɥɸɰɢɨɧɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ
j
ɗϕ = ¦ Dσ (ϕ )
σ,j
ɢɧɮɢɧɢɬɟɡɢɦɚɥɶɧɚɹ
ɫɢɦɦɟɬɪɢɹ
∂
,
∂uσj
(1)
ɤɨɬɨɪɨɟ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɜɨɟɣ ɩɪɨɢɡɜɨɞɹɳɟɣ ɮɭɧɤɰɢɟɣ ϕ = (ϕ 1 ,…, ϕ m ) ,
ϕ i ∈ C ∞ ( J ∞ (π )) . ȼ ɮɨɪɦɭɥɟ (1), ɨɩɪɟɞɟɥɹɸɳɟɣ ɷɜɨɥɸɰɢɨɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɜ
ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ, ɢɫɩɨɥɶɡɨɜɚɧɵ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ: x1 ,…, xn , u1 ,…, u m , uσj ( u∅j = u j ) —
ɤɚɧɨɧɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɛɟɫɤɨɧɟɱɧɵɯ ɞɠɟɬɨɜ J ∞ (π ) , Di = ∂ + ¦ j ,σ uσj i ∂ j —
∂xi
∂uσ
ɨɩɟɪɚɬɨɪ ɩɨɥɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɩɨ ɩɟɪɟɦɟɧɧɨɣ xi , Dσ — ɤɨɦɩɨɡɢɰɢɹ ɨɩɟɪɚɬɨɪɨɜ
ɩɨɥɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɦɭɥɶɬɢɢɧɞɟɤɫɭ σ , ɱɟɪɬɚ ɨɛɨɡɧɚɱɚɟɬ ɨɝɪɚɧɢɱɟɧɢɟ
ɨɩɟɪɚɬɨɪɚ ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɩɪɨɞɨɥɠɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ E ∞ (ɩɨɞɪɨɛɧɟɟ ɫɦ., ɧɚɩɪɢɦɟɪ,
[1], [6]).
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɚɥɝɟɛɪɵ ɫɢɦɦɟɬɪɢɣ ɭɪɚɜɧɟɧɢɹ E : F = 0 ɧɚɞɨ ɪɟɲɢɬɶ ɫɢɫɬɟɦɭ
l F (ϕ ) = 0,
(2)
ɝɞɟ l F — ɨɩɟɪɚɬɨɪ ɭɧɢɜɟɪɫɚɥɶɧɨɣ ɥɢɧɟɚɪɢɡɚɰɢɢ, ɨɬɜɟɱɚɸɳɢɣ ɮɭɧɤɰɢɢ F = ( F1 ,…, Fr ) ,
Fi ∈ C ∞ ( J k (π )) , l F (ϕ ) = ɗϕ ( F ) .
Ɇɧɨɠɟɫɬɜɨ ɜɫɟɯ ɜɵɫɲɢɯ ɢɧɮɢɧɢɬɟɡɢɦɚɥɶɧɵɯ ɫɢɦɦɟɬɪɢɣ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɹɜɥɹɟɬɫɹ
ɚɥɝɟɛɪɨɣ Ʌɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɤɨɛɤɢ əɤɨɛɢ { ϕ ,ψ } = ɗϕ (ψ ) − ɗψ (ϕ ) . Ʉɨɦɩɨɧɟɧɬɵ ɫɤɨɛɤɢ əɤɨɛɢ
ɜɵɱɢɫɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
j
j
§
α ∂ψ
α ∂ϕ ·
{ ϕ ,ψ } = ¦ ¨ Dσ (ϕ ) α − Dσ (ψ ) α ¸ .
∂uσ
∂uσ ¹
σ ,α ©
j
Ɋɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ E = {F = 0 } , ɢɧɜɚɪɢɚɧɬɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɥɟɞɭɸɳɟɣ ɫɢɫɬɟɦɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ:
ɫɢɦɦɟɬɪɢɢ
ϕ,
­ F1 = 0,…, Fr = 0,
®
ϕ = 0.
¯
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ E = {F = 0 } , ɢɧɜɚɪɢɚɧɬɧɨɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɦɦɟɬɪɢɣ ϕ ɢ ψ , ɛɭɞɟɬ
ɬɚɤɠɟ ɢɧɜɚɪɢɚɧɬɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɤɨɛɤɢ { ϕ ,ψ } . ɉɨɷɬɨɦɭ ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ,
ɢɧɜɚɪɢɚɧɬɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɩɨɞɚɥɝɟɛɪɵ g ɚɥɝɟɛɪɵ ɫɢɦɦɟɬɪɢɣ. ȿɫɥɢ ϕ1 , ... ϕ s —
ɨɛɪɚɡɭɸɳɢɟ ɷɬɨɣ ɩɨɞɚɥɝɟɛɪɵ, ɨɬɵɫɤɚɧɢɟ ɢɧɜɚɪɢɚɧɬɧɵɯ ɪɟɲɟɧɢɣ ɫɜɨɞɢɬɫɹ ɤ ɪɟɲɟɧɢɸ
ɩɟɪɟɨɩɪɟɞɟɥɟɧɧɨɣ ɫɢɫɬɟɦɵ
­ F1 = 0,…, Fr = 0,
®
¯ϕ1 = 0,…, ϕ s = 0.
Ɉ ɬɨɱɧɵɯ ɪɟɲɟɧɢɹɯ
3.
k − ε –ɦɨɞɟɥɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ
41
k − ε –ɆɈȾȿɅɖ ɌɍɊȻɍɅȿɇɌɇɈɋɌɂ
Ⱦɥɹ ɪɚɫɱɟɬɚ ɬɭɪɛɭɥɟɧɬɧɵɯ ɬɟɱɟɧɢɣ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ k − ε –ɦɨɞɟɥɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ
ɫɢɫɬɟɦɚ ɨɫɪɟɞɧɟɧɧɵɯ ɩɨ Ɋɟɣɧɨɥɶɞɫɭ ɭɪɚɜɧɟɧɢɣ ɇɚɜɶɟ-ɋɬɨɤɫɚ ɡɚɦɵɤɚɟɬɫɹ ɞɜɭɦɹ ɭɪɚɜɧɟɧɢɹɦɢ
ɞɥɹ ɬɭɪɛɭɥɟɧɬɧɨɣ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ k ɢ ɫɤɨɪɨɫɬɢ ɞɢɫɫɢɩɚɰɢɢ ɬɭɪɛɭɥɟɧɬɧɨɣ
ɷɧɟɪɝɢɢ ε . ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɢɫɩɨɥɶɡɭɸɬɫɹ ɪɚɡɥɢɱɧɵɟ ɜɚɪɢɚɧɬɵ k − ε –ɭɪɚɜɧɟɧɢɣ.
ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɫɢɫɬɟɦɚ, ɤɨɬɨɪɚɹ ɨɩɢɫɵɜɚɟɬ ɞɜɢɠɟɧɢɟ
ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ ɫ ɭɱɟɬɨɦ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɩɪɢ ɛɨɥɶɲɢɯ ɱɢɫɥɚɯ
Ɋɟɣɧɨɥɶɞɫɚ ([4], [3]):
∂u i
= 0;
∂xi
(3)
∂ ( ρ u j ) ∂ ( ρ u iu j )
∂p
∂ §¨
′ ′·
¸
+
=−
+
¨τ ij − ρ ui u j ¸ , j = 1, 2, 3;
©
¹
∂t
∂xi
∂x j ∂xi
(4)
cμ k 2 · ∂k ½°
∂k
∂k
∂ ­°§
′ ′ ∂u j
ν
+ui
=
+
−ε;
®¨
¾ − ui u j
¸
∂t
∂xi ∂xi °¯©
∂xi
σ k ε ¹ ∂xi °¿
cμ k 2 · ∂ε °½
ε ′ ′ ∂u j
ε2
∂ε
∂ε
∂ °­§
c
u
u
c
ν
,
+ui
=
+
−
−
®¨
¾
¸
ε2
k
∂t
∂xi ∂xi °¯©
σ ε ε ¹ ∂xi °¿ ε1 k i j ∂xi
(5)
(6)
ɝɞɟ
§ ∂u
∂u j ·
τ ij = ρν ¨ ∂x i + ∂x ¸ ,
i ¹
© j
cμ k 2 § ∂ u i ∂ u j · 2
+
¨
¸ − δ k,
ε ¨© ∂x j ∂xi ¸¹ 3 ij
ν , ρ , cμ , cε1 , cε 2 , σ k , σ ε − ɤɨɧɫɬɚɧɬɵ.
− ui ′u j′ =
ɗɬɨ ɫɢɫɬɟɦɚ ɧɚ ɲɟɫɬɶ ɧɟɢɡɜɟɫɬɧɵɯ ɮɭɧɤɰɢɣ u1 = u 1 , u 2 = u 2 , u 3 = u 3 , u 4 = p , u 5 = k ,
u 6 = ε , ɡɚɜɢɫɹɳɢɯ ɨɬ ɱɟɬɵɪɟɯ ɩɟɪɟɦɟɧɧɵɯ x1 , x2 , x3 , x4 = t . ɉɪɢ ɨɩɢɫɚɧɢɢ ɤɥɚɫɫɢɱɟɫɤɢɯ
ɫɢɦɦɟɬɪɢɣ ɢɫɩɨɥɶɡɭɸɬɫɹ ɤɚɧɨɧɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɞɠɟɬɨɜ [1].
4. ɂɇȼȺɊɂȺɇɌɇɕȿ Ɋȿɒȿɇɂə
Ⱥɥɝɟɛɪɚ ɤɥɚɫɫɢɱɟɫɤɢɯ ɫɢɦɦɟɬɪɢɣ k − ε –ɦɨɞɟɥɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ (3) – (6) ɛɵɥɚ
ɜɵɱɢɫɥɟɧɚ ɜ [3]. ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɦɵ ɢɳɟɦ ɪɟɲɟɧɢɹ, ɢɧɜɚɪɢɚɧɬɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɞɚɥɝɟɛɪɵ,
ɩɨɪɨɠɞɟɧɧɨɣ ɫɢɦɦɟɬɪɢɹɦɢ X 1 ( f ) , X 2 ( g ) , X 3 (h) , ɝɞɟ
fx1 , fk1 , f ε1 ),
X 1 ( f ) = ( fu11 − f , fu12 , fu13 , fp1 + ρ 2 , gk 2 , gε 2 ),
X 2 ( g ) = ( gu12 , gu 22 − g , gu 23 , gp2 + ρ gx
, hk , hε ),
X (h) = (hu1 , hu 2 , hu 3 − h, hp + ρ hx
3
3
3
3
f , g , h — ɧɟɤɨɬɨɪɵɟ ɮɭɧɤɰɢɢ ɩɟɪɟɦɟɧɧɨɣ t .
3
3
3
3
42
ɇ.Ƚ. ɏɨɪɶɤɨɜɚ
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɫɢɦɦɟɬɪɢɢ ɩɪɢ ɩɨɞɯɨɞɹɳɟɦ ɜɵɛɨɪɟ ɮɭɧɤɰɢɣ f , g , h ɜɤɥɸɱɚɸɬ ɜ
ɫɟɛɹ ɬɪɚɧɫɥɹɰɢɢ ɩɨ ɩɟɪɟɦɟɧɧɵɦ xi ɢ ɝɚɥɢɥɟɟɜɭ ɫɢɦɦɟɬɪɢɸ. ɋɢɦɦɟɬɪɢɢ X 1 ( f ) , X 2 ( g ) , X 3 (h)
ɩɨɩɚɪɧɨ ɤɨɦɦɭɬɢɪɭɸɬ:
{X 1 ( f ), X 2 ( g ) } = {X 2 ( g ), X 3 ( h) } = {X 1 ( f ), X 3 ( h) } = 0.
ɉɨɤɚɠɟɦ ɧɚ ɩɪɨɫɬɨɦ ɩɪɢɦɟɪɟ, ɤɚɤ ɦɨɠɧɨ ɫ ɩɨɦɨɳɶɸ ɫɢɦɦɟɬɪɢɣ ɫɜɟɫɬɢ ɢɫɯɨɞɧɭɸ
ɫɢɫɬɟɦɭ ɤ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ ɫ ɦɟɧɶɲɢɦ ɱɢɫɥɨɦ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ. Ⱦɥɹ ɨɬɵɫɤɚɧɢɹ
ɪɟɲɟɧɢɣ ɫɢɫɬɟɦɵ (3) – (6), ɢɧɜɚɪɢɚɧɬɧɵɯ, ɧɚɩɪɢɦɟɪ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɦɦɟɬɪɢɢ X 1 ( f ) , ɫɧɚɱɚɥɚ
ɪɟɲɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ X 1 ( f ) = 0 :
f
∂ 1
f u − f = 0 Ÿ u 1 = x1 + A(t , x2 , x3 );
∂x1
f
2
∂
f u = 0 Ÿ u 2 = B(t , x2 , x3 );
∂x1
∂ 3
f u = 0 Ÿ u 3 = C (t , x2 , x3 );
∂x1
ρ f 2
∂p
+ ρ f
fx1 = 0 Ÿ p = −
x1 + D(t , x2 , x3 );
∂x1
2f
f ∂k = 0 Ÿ k = E (t , x2 , x3 );
∂x1
f ∂ε = 0 Ÿ ε = F (t , x2 , x3 ).
∂x1
Ɂɚɬɟɦ ɩɨɥɭɱɟɧɧɵɟ ɮɭɧɤɰɢɢ u 1 , u 2 , u 3 , p , k , ε ɩɨɞɫɬɚɜɢɦ ɜ ɫɢɫɬɟɦɭ (3) – (6), ɜ
ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɞɚɧɧɚɹ ɫɢɫɬɟɦɚ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɧɚ ɲɟɫɬɶ ɮɭɧɤɰɢɣ ɬɪɟɯ
ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ t , x2 , x3 .
Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɮɭɧɤɰɢɢ u 1 , u 2 , u 3 , p , k , ε , ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ
X 1 ( f ) = X 2 ( g ) = X 3 ( h) = 0,
ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
u =
1
f
g
h
x1 + α (t ), u 2 = x2 + β (t ), u 3 = x3 + γ (t );
f
g
h
§ ·
g
h ¸
ρ¨ f
p = − ¨¨ x12 + x22 + x32 ¸¸ + p(t );
2 ¨© f
g
h ¸¹
k = k (t ), ε = ε (t ).
(7)
(8)
(9)
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ (7) – (8) ɜ ɭɪɚɜɧɟɧɢɹ (3) – (4) ɩɨɥɭɱɚɟɦ, ɱɬɨ
α = a1 f −1 , β = a2 g −1 , γ = a3 h −1 , fgh = A,
ɝɞɟ A, ai , — ɩɪɨɢɡɜɨɥɶɧɵɟ ɤɨɧɫɬɚɧɬɵ, ɢ
(10)
Ɉ ɬɨɱɧɵɯ ɪɟɲɟɧɢɹɯ
k − ε –ɦɨɞɟɥɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ
1
u =
ȼɵɱɢɫɥɢɦ ɫɭɦɦɭ −ui ′u j′
43
+a
+a
+a
hx
fx
gx
1
1
, u2 = 2 2 , u3 = 3 3 .
f
g
h
(11)
∂u j
ɞɥɹ ɮɭɧɤɰɢɣ u i , ɨɩɪɟɞɟɥɹɟɦɵɯ ɮɨɪɦɭɥɨɣ (11):
∂xi
3 § c k2 §
· ∂u
∂u j
∂u ∂u j · 2
−ui u
=¦¨ μ ¨ i+
− δ ij k ¸ j =
¸
¸ ∂xi
∂xi i, j =1 ¨© ε ¨© ∂x j ∂xi ¸¹ 3
¹
′
′
j
cμ k 2
2
§ ∂u i · 2 3 ∂ui k 2
=2
¦ ¨ ¸ − k ¦ = L,
ε i =1 © ∂xi ¹ 3 i =1 ∂xi ε
3
(12)
ɝɞɟ
§ § f · 2 § g · 2 § h · 2 ·
L = 2cμ ¨ ¨ ¸ + ¨ ¸ + ¨ ¸ ¸ .
¨© f ¹ © g ¹ © h ¹ ¸
©
¹
Ɍɟɩɟɪɶ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (5) – (6) ɫ ɭɱɟɬɨɦ (9), (12) ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
2
k
k = L − ε ;
ε
ε = cε Lk − cε
1
ε2
2
k
(13)
.
(14)
Ɋɚɫɫɦɨɬɪɢɦ ɫɧɚɱɚɥɚ ɫɥɭɱɚɣ L ≡ 0 , ɬɨ ɟɫɬɶ f = g = h = 0 ɢ u i = const (ɜ ɷɬɨɦ ɫɥɭɱɚɟ
ɫɢɦɦɟɬɪɢɢ X 1 ( f ) , X 2 ( g ) , X 3 (h) ɹɜɥɹɸɬɫɹ ɬɪɚɧɫɥɹɰɢɹɦɢ):
ε = −cε 2
k2
−ε .
k =
ɂɫɤɥɸɱɢɦ
ε2
ε ɢ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ k:
k 2
k = cε 2 2
k
ɢɥɢ
k
k
=
c
⇔ (ln k)⋅ − cε 2 (ln k )⋅ = 0,
ε
2
k
k
ɨɬɤɭɞɚ ɩɨɥɭɱɚɟɦ
k
k
cε 2
1− c
k ε2
= const Ÿ
= C 1t + C 2 (cε 2 ≠ 1).
1 − cε 2
44
ɇ.Ƚ. ɏɨɪɶɤɨɜɚ
ɂɬɚɤ,
cε 2
1
1 − cε 2
1 − cε 2
C1
, ε = − k =
(C1t + C2 )
,
k = (C1t + C2 )
1 − cε 2
ɝɞɟ C1 , C2 — ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɨɛɳɢɣ ɫɥɭɱɚɣ ɢ ɩɪɟɨɛɪɚɡɭɟɦ ɬɟɩɟɪɶ ɫɢɫɬɟɦɭ (13) – (14) ɤ ɨɞɧɨɦɭ
ɭɪɚɜɧɟɧɢɸ. ɍɦɧɨɠɢɜ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ ɧɚ ε ɢ k ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜɵɱɢɫɥɢɦ ɩɪɨɢɡɜɨɞɧɭɸ
⋅
2
2
2
§ k · ε k − kε (1 − cε1 ) L k + (cε 2 − 1)ε
§k·
=
= (1 − cε1 ) L¨ ¸ + (cε 2 − 1).
¨ ¸=
ε2
ε2
©ε ¹
©ε ¹
ȼɜɟɞɟɦ ɧɨɜɭɸ ɮɭɧɤɰɢɸ w =
k
ε
. Ɍɨɝɞɚ
w = −l (t ) w2 + a,
(15)
ɝɞɟ
l (t ) = (cε1 − 1) L ≥ 0, a = cε 2 − 1 > 0.
ɉɨɞɫɬɚɜɢɜ ɜɵɪɚɠɟɧɢɟ k = wε ɜ ɫɢɫɬɟɦɭ (13) – (14), ɭɛɟɠɞɚɟɦɫɹ, ɱɬɨ ɨɧɚ ɫɜɨɞɢɬɫɹ ɤ
ɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
ε = §¨© cε Lw − cε w−1 ·¸¹ ε ,
1
(16)
2
ɢɡ ɤɨɬɨɪɨɝɨ ɩɨ ɪɟɲɟɧɢɸ w ɭɪɚɜɧɟɧɢɹ (15) ɧɚɯɨɞɢɦ ɮɭɧɤɰɢɸ ε .
Ɂɚɦɟɬɢɦ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ (15) ɫ ɩɨɦɨɳɶɸ ɩɨɞɫɬɚɧɨɜɤɢ w =
1 v
ɫɜɨɞɢɬɫɹ ɤ ɥɢɧɟɣɧɨɦɭ
lv
ɭɪɚɜɧɟɧɢɸ ɧɚ ɮɭɧɤɰɢɸ v :
= al 2 v.
lv − lv
(17)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɢɫɤ ɢɧɜɚɪɢɚɧɬɧɵɯ ɪɟɲɟɧɢɣ ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɩɨɞɚɥɝɟɛɪɵ
ɩɪɨɜɨɞɢɬɫɹ ɩɨ ɫɥɟɞɭɸɳɟɣ ɫɯɟɦɟ. ɋɧɚɱɚɥɚ ɜɵɛɢɪɚɟɦ ɮɭɧɤɰɢɢ f , g , h , ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ
ɭɫɥɨɜɢɸ (10), ɢ ɩɨ ɮɨɪɦɭɥɚɦ (11), (8) ɧɚɯɨɞɢɦ u i ɢ p . Ɂɚɬɟɦ ɜɵɱɢɫɥɹɟɦ ɮɭɧɤɰɢɸ l (t ) ɢ,
1 v
, ɩɨɞɫɬɚɜɥɹɟɦ
ɩɨɞɫɬɚɜɢɜ ɟɟ ɜ ɭɪɚɜɧɟɧɢɟ (17), ɧɚɯɨɞɢɦ ɟɝɨ ɪɟɲɟɧɢɟ v . Ɂɚɬɟɦ ɜɵɱɢɫɥɹɟɦ w =
lv
w ɜ (16) ɢ ɧɚɯɨɞɢɦ ɮɭɧɤɰɢɸ ε , ɚ ɡɚɬɟɦ ɮɭɧɤɰɢɸ k ɩɨ ɮɨɪɦɭɥɟ k = ε w .
ɇɚɩɪɢɦɟɪ, ɞɥɹ ɮɭɧɤɰɢɣ (ɜɵɛɪɚɧɵ ɫ ɭɱɟɬɨɦ (10)) f = b1e λ t , g = b2 e μ t , h = b3e − ( λ + μ ) t , ɝɞɟ λ ,
μ , bi — ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɨɥɭɱɚɟɦ:
u1 = λ x1 +
a
a1 − λt 2
a
e , u = μ x2 + 2 e − μt , u 3 = −(λ + μ ) x1 + 3 e( λ + μ ) t , p = p(t ).
b1
b2
b3
Ɉ ɬɨɱɧɵɯ ɪɟɲɟɧɢɹɯ
k − ε –ɦɨɞɟɥɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ
45
Ɏɭɧɤɰɢɹ l = const , l = 0 , ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (17) ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ
al t
v = C3e
+ C4 e −
al t
( C3 , C4 — ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ).
ɉɨɞɫɬɚɜɥɹɟɦ ɮɭɧɤɰɢɸ
a C3e 2
w=
l C3e 2
al t
− C4
al t
+ C4
ɜ ɭɪɚɜɧɟɧɢɟ (16) ɢ ɩɨɫɥɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ε :
ε = C5
§
¨¨
©
§
¨¨
©
C3e
al t
C3e
al t
+ C4 e −
cε
1
al t · cε1 −1
¸¸
¹
cε
2
− al t · cε 2 −1
¸¸
4
¹
−C e
,
ɢ ɞɥɹ k :
k = C5
§
¨¨
©
a C3e
l ¨§ C e
¨
©
3
1
al t
+ C4 e−
al t · cε1 −1
¸¸
¹
al t
− C4 e−
al t · cε 2 −1
¸¸
¹
1
( C5 — ɩɪɨɢɡɜɨɥɶɧɚɹ ɩɨɫɬɨɹɧɧɚɹ).
5. ɁȺɄɅɘɑȿɇɂȿ
ȼ ɧɚɫɬɨɹɳɢɣ ɦɨɦɟɧɬ ɢɦɟɟɬɫɹ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɦɨɞɟɥɟɣ ɞɥɹ ɪɚɫɱɺɬɚ ɬɭɪɛɭɥɟɧɬɧɵɯ
ɬɟɱɟɧɢɣ. Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɩɪɢɝɨɞɧɨɫɬɢ ɬɨɣ ɢɥɢ ɢɧɨɣ ɦɨɞɟɥɢ ɞɥɹ ɪɟɲɟɧɢɹ ɤɨɧɤɪɟɬɧɨɣ
ɡɚɞɚɱɢ ɢɧɨɝɞɚ ɬɪɟɛɭɟɬɫɹ ɩɪɨɜɟɞɟɧɢɟ ɞɨɪɨɝɨɫɬɨɹɳɢɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ. ɇɚɥɢɱɢɟ ɢɧɜɚɪɢɚɧɬɧɵɯ
ɪɟɲɟɧɢɣ ɦɨɠɟɬ ɩɨɦɨɱɶ ɫɞɟɥɚɬɶ ɜɵɛɨɪ ɚɞɟɤɜɚɬɧɨɣ ɦɨɞɟɥɢ [6]. Ɇɟɬɨɞɢɤɚ ɩɨɫɬɪɨɟɧɢɹ
ɬɨɱɧɵɯ ɪɟɲɟɧɢɣ, ɢɫɩɨɥɶɡɨɜɚɧɧɚɹ ɜ ɞɚɧɧɨɣ ɪɚɛɨɬɟ, ɩɪɢɦɟɧɢɦɚ ɢ ɤ ɞɪɭɝɢɦ ɦɨɞɟɥɹɦ
ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ.
ɅɂɌȿɊȺɌɍɊȺ
1. Ȼɨɱɚɪɨɜ Ⱥ.ȼ., ȼɟɪɛɨɜɟɰɤɢɣ Ⱥ.Ɇ., ȼɢɧɨɝɪɚɞɨɜ Ⱥ.Ɇ., Ⱦɭɠɢɧ ɋ.ȼ., Ʉɪɚɫɢɥɶɳɢɤ ɂ.ɋ., Ɍɨɪɯɨɜ ɘ.ɇ.,
ɋɚɦɨɯɢɧ Ⱥ.ȼ., ɏɨɪɶɤɨɜɚ ɇ.Ƚ., ɑɟɬɜɟɪɢɤɨɜ ȼ.ɇ. ɋɢɦɦɟɬɪɢɢ ɢ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɭɪɚɜɧɟɧɢɣ
ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ. 2-ɟ ɢɡɞ. Ɇ.: ɂɡɞ-ɜɨ Ɏɚɤɬɨɪɢɚɥ-ɉɪɟɫɫ, 2005.
2. Khor’kova N.G., Verbovetsky A.M. On symmetry subalgebras and conservation laws for k-e
turbulence model and the Navier-Stokes equation. Amer. Math. Soc. Transl. Series 2. 1995. V. 167.
P. 61–90.
3. Jones W.P., Launder B.E. The prediction of laminarization with a two-equation model of turbulence. Internat.
J. Heat Mass Transfer 15 (1972), no. 2, p. 301–314.
4. Kollmann W. (ed.) Prediction method for turbulent flows, Hemisphere, Washington, 1980, 312 p.
5. Oberlack M. Symmetries and Invariant Solutions of Turbulent Flows and their Implications for Turbulence
Modelling. — Theories of Turbulence. International Centre for Mechanical Sciences. Vol. 442. 2002.
Ɋ. 301-366.
6. ȼɟɪɛɨɜɟɰɤɢɣ Ⱥ.Ɇ., ɏɨɪɶɤɨɜɚ ɇ.Ƚ., ɑɟɬɜɟɪɢɤɨɜ ȼ.ɇ. ɋɢɦɦɟɬɪɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ: ɭɱɟɛ.
ɩɨɫɨɛɢɟ.– Ɇ.: ɂɡɞ-ɜɨ ɆȽɌɍ ɢɦ. ɇ. ɗ. Ȼɚɭɦɚɧɚ, 2002.
46
ɇ.Ƚ. ɏɨɪɶɤɨɜɚ
ON EXACT SOLUTIONS OF
k − ε TURBULENCE MODEL
Khorkova N.G.
A method of constructing invariant solutions is applied to k − ε turbulence model. Exact solutions for k − ε
turbulence model are obtained. The method for construction of exact solutions used in this paper may be applied to other
models of turbulence.
Keywords: nonlinear differential equation, local infinitesimal symmetries, invariant solution,
turbulence model.
k −ε
REFERENCES
1.
2.
3.
4.
5.
6.
Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V.,
Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M. Symmetries and Conservation Laws for Differential
Equation of Mathemetical Phisics // Translations of Mathematical Monographs, vol. 182 – Providence, RI: AMS,
1999. 333 p.
Khor’kova N.G., Verbovetsky A.M. On symmetry subalgebras and conservation laws for k-e turbulence model
and the Navier-Stokes equation. Amer. Math. Soc. Transl. Series 2, 1995, v. 167, pp. 61–90.
Jones W.P., Launder B.E. The prediction of laminarization with a two-equation model of turbulence. Internat.
J. Heat Mass Transfer, 15 (1972), no. 2, pp. 301–314.
Kollmann W. (ed.) Prediction method for turbulent flows, Hemisphere, Washington, 1980, 312 p.
Oberlack M. Symmetries and Invariant Solutions of Turbulent Flows and their Implications for Turbulence
Modelling. Theories of Turbulence. International Centre for Mechanical Sciences. — Volume 442, 2002,
pp. 301-366.
Verbovetsky A.M., Khor’kova N.G., Chetverikov V.N. Simmetrii differenzialnych uravneniy. (Symmetries of
Differential Equations). Uchebnoe posobiye. Ɇoscow, MGTU, 2002, 36 p.
ɋɜɟɞɟɧɢɹ ɨɛ ɚɜɬɨɪɟ
ɏɨɪɶɤɨɜɚ ɇɢɧɚ Ƚɪɢɝɨɪɶɟɜɧɚ, ɨɤɨɧɱɢɥɚ ɆȽɍ ɢɦ. Ɇ.ȼ. Ʌɨɦɨɧɨɫɨɜɚ (1983), ɤɚɧɞɢɞɚɬ ɮɢɡɢɤɨɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ ɆȽɌɍ ɢɦ. ɇ. ɗ. Ȼɚɭɦɚɧɚ, ɚɜɬɨɪ 17 ɧɚɭɱɧɵɯ ɪɚɛɨɬ,
ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ - ɚɥɝɟɛɪɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɬɟɨɪɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ.
Документ
Категория
Без категории
Просмотров
3
Размер файла
328 Кб
Теги
турбулентность, точных, модель, решения
1/--страниц
Пожаловаться на содержимое документа