Я.А. Коркодинов ОБЗОР СЕМЕЙСТВА k–ε МОДЕЛЕЙ ДЛЯ

ɍȾɄ 621
ə.Ⱥ. Ʉɨɪɤɨɞɢɧɨɜ
I.A. Korkodinov
ɉɟɪɦɫɤɢɣ ɧɚɰɢɨɧɚɥɶɧɵɣ ɢɫɫɥɟɞɨɜɚɬɟɥɶɫɤɢɣ ɩɨɥɢɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ
Perm National Research Politechnic University
ɈȻɁɈɊ ɋȿɆȿɃɋɌȼȺ k-İ ɆɈȾȿɅȿɃ
ȾɅə ɆɈȾȿɅɂɊɈȼȺɇɂə ɌɍɊȻɍɅȿɇɌɇɈɋɌɂ
THE REVIEW OF SET OF k-İ MODELS
FOR MODELING TURBULENCE
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɨɫɧɨɜɧɵɟ ɭɪɚɜɧɟɧɢɹ ɦɟɯɚɧɢɤɢ ɠɢɞɤɨɫɬɢ ɢ ɝɚɡɚ, ɬɚɤɢɟ ɤɚɤ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɢ ɭɪɚɜɧɟɧɢɹ ɇɚɜɶɟ – ɋɬɨɤɫɚ. Ɉɩɢɫɵɜɚɟɬɫɹ ɩɪɢɦɟɧɟɧɢɟ ɞɟɤɨɦɩɨɡɢɰɢɢ Ɋɟɣɧɨɥɶɞɫɚ
ɞɥɹ ɪɟɲɟɧɢɹ ɩɪɨɛɥɟɦ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɢ ɜɨɡɧɢɤɚɸɳɚɹ ɩɪɢ ɷɬɨɦ ɩɪɨɛɥɟɦɚ ɡɚɦɵɤɚɧɢɹ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ. Ɋɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɝɢɩɨɬɟɡɚ Ȼɭɫɫɢɧɟɫɤɚ, ɩɨɡɜɨɥɹɸɳɚɹ ɜɜɟɫɬɢ ɩɨɧɹɬɢɟ ɬɭɪɛɭɥɟɧɬɧɨɣ ɜɹɡɤɨɫɬɢ. Ɉɩɢɫɵɜɚɸɬɫɹ ɨɫɧɨɜɧɵɟ k–İ ɦɨɞɟɥɢ, ɢɯ ɩɪɟɢɦɭɳɟɫɬɜɚ ɢ ɧɟɞɨɫɬɚɬɤɢ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ, ɭɪɚɜɧɟɧɢɹ ɇɚɜɶɟ – ɋɬɨɤɫɚ, ɞɟɤɨɦɩɨɡɢɰɢɹ
Ɋɟɣɧɨɥɶɞɫɚ, ɝɢɩɨɬɟɡɚ Ȼɭɫɫɢɧɟɫɤɚ, k–İ ɦɨɞɟɥɢ.
The governing equations of fluid and gas mechanics are considered such as continuity equation
and Navier – Stokes equations. The application of Reynolds decomposition for closure problem of turbulence is described. Boussinesq hypothesis which allows introducing the conception of turbulent viscosity is considered. The most widely-known k–İ models are described as well as their advantages and
disadvantages.
Keywords: continuity equation, Navie – Stokes equations, Reynolds decomposition, Boussinesq
hypothesis, k–İ models.
ȼ ɧɚɲɢ ɞɧɢ ɩɚɤɟɬɵ ɱɢɫɥɟɧɧɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɨɬɤɪɵɜɚɸɬ ɨɝɪɨɦɧɵɟ
ɜɨɡɦɨɠɧɨɫɬɢ ɞɥɹ ɢɧɠɟɧɟɪɨɜ ɢ ɢɫɫɥɟɞɨɜɚɬɟɥɟɣ ɢɡ ɫɚɦɵɯ ɪɚɡɧɵɯ ɨɛɥɚɫɬɟɣ.
ɉɭɬɟɦ ɩɪɨɫɬɨɝɨ ɧɚɠɚɬɢɹ ɪɹɞɚ ɤɧɨɩɨɤ ɢ ɜɜɨɞɚ ɧɟɨɛɯɨɞɢɦɵɯ ɜɯɨɞɧɵɯ ɞɚɧɧɵɯ
ɢɫɫɥɟɞɨɜɚɬɟɥɶ ɦɨɠɟɬ ɩɨɥɭɱɚɬɶ ɪɟɲɟɧɢɹ ɞɥɹ ɤɨɦɩɥɟɤɫɧɵɯ ɦɟɠɞɢɫɰɢɩɥɢɧɚɪɧɵɯ ɡɚɞɚɱ. Ɇɟɠɞɭ ɬɟɦ ɜɫɟ ɷɬɢ ɩɚɤɟɬɵ ɨɫɧɨɜɚɧɵ ɧɚ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɡɚɤɨɧɚɯ
ɦɟɯɚɧɢɤɢ, ɢ ɤɚɠɞɵɣ ɜɯɨɞɧɨɣ ɩɚɪɚɦɟɬɪ ɹɜɥɹɟɬɫɹ ɤɪɢɬɢɱɟɫɤɢ ɜɚɠɧɵɦ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɞɨɫɬɨɜɟɪɧɨɝɨ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɪɟɲɟɧɢɹ.
ȼ ɧɚɲɟɣ ɫɬɚɬɶɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɦɟɯɚɧɢɤɢ ɠɢɞɤɨɫɬɢ
ɢ ɝɚɡɚ. Ɋɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɇɚɜɶɟ – ɋɬɨɤɫɚ ɢ ɞɚɟɬɫɹ ɜɨɡɦɨɠɧɨɟ ɮɢɡɢɱɟɫɤɨɟ ɨɛɴɹɫɧɟɧɢɟ ɨɫɧɨɜɧɵɯ ɟɝɨ ɤɨɦɩɨɧɟɧɬ. Ɍɚɤɠɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɨɫɧɨɜɧɵɟ ɦɨɞɟɥɢ, ɩɪɢɧɹɬɵɟ ɞɥɹ ɨɩɢɫɚɧɢɹ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɢ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɟɦɵɟ
ɜ ɩɚɤɟɬɚɯ ɱɢɫɥɟɧɧɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ. ɋɬɚɜɢɬɫɹ ɡɚɞɚɱɚ ɪɚɫɫɦɨɬɪɟɬɶ ɬɟɨɪɟɬɢɱɟɫɤɢɣ ɛɚɡɢɫ, ɧɟɨɛɯɨɞɢɦɵɣ ɞɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɩɪɨɛɥɟɦ
ɢɡ ɨɛɥɚɫɬɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ.
5
Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɦɟɯɚɧɢɤɢ ɠɢɞɤɨɫɬɢ. ɋɨɝɥɚɫɧɨ ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɨɣ
ɦɨɞɟɥɢ ɜɟɳɟɫɬɜɚ [1] ɠɢɞɤɨɫɬɶ ɢ ɝɚɡ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɫɩɥɨɲɧɭɸ ɬɟɤɭɱɭɸ
ɢɡɨɬɪɨɩɧɭɸ ɧɶɸɬɨɧɨɜɫɤɭɸ ɫɪɟɞɭ ɫ ɧɟɩɪɟɪɵɜɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɦɚɫɫɵ
ɢ ɞɪɭɝɢɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ. ȼɫɩɨɦɧɢɦ ɧɟɫɤɨɥɶɤɨ ɨɫɧɨɜɧɵɯ ɩɨɧɹɬɢɣ, ɩɪɢɦɟɧɹɟɦɵɯ ɜ ɦɟɯɚɧɢɤɟ ɠɢɞɤɨɫɬɢ ɢ ɝɚɡɚ (ɞɚɥɟɟ ɛɭɞɟɦ ɝɨɜɨɪɢɬɶ ɨ ɦɟɯɚɧɢɤɟ
ɠɢɞɤɨɫɬɢ, ɩɨɞɪɚɡɭɦɟɜɚɹ, ɱɬɨ ɦɨɞɟɥɢ, ɨɩɢɫɵɜɚɸɳɢɟ ɩɨɜɟɞɟɧɢɟ ɠɢɞɤɨɫɬɢ,
ɬɚɤɠɟ ɩɪɢɝɨɞɧɵ ɞɥɹ ɨɩɢɫɚɧɢɹ ɩɨɜɟɞɟɧɢɹ ɝɚɡɚ). Ɍɟɤɭɱɟɫɬɶ ɫɪɟɞɵ – ɫɜɨɣɫɬɜɨ
ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɞɟɮɨɪɦɢɪɭɟɦɨɫɬɢ ɫɪɟɞɵ, ɬ.ɟ. ɫɩɨɫɨɛɧɨɫɬɶ ɢɡɦɟɧɹɬɶ ɫɜɨɸ
ɮɨɪɦɭ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɤɨɥɶ ɭɝɨɞɧɨ ɦɚɥɵɯ ɫɢɥ, ɟɫɥɢ ɠɢɞɤɨɫɬɶ ɧɟ ɫɞɟɪɠɢɜɚɟɬɫɹ ɤɚɤɢɦɢ-ɥɢɛɨ ɫɬɟɧɤɚɦɢ. ɋɩɥɨɲɧɨɫɬɶ ɢɥɢ ɧɟɪɚɡɪɵɜɧɨɫɬɶ ɫɪɟɞɵ – ɫɩɨɫɨɛɧɨɫɬɶ ɡɚɩɨɥɧɹɬɶ ɜɟɫɶ ɨɛɴɟɦ, ɡɚɧɢɦɚɟɦɵɣ ɦɚɬɟɪɢɚɥɨɦ ɬɟɥɚ, ɛɟɡ ɜɫɹɤɢɯ ɩɭɫɬɨɬ,
ɨɛɳɧɨɫɬɶ ɫɜɨɣɫɬɜ ɥɸɛɨɣ ɱɚɫɬɢ ɫɪɟɞɵ ɢ ɫɪɟɞɵ ɜ ɰɟɥɨɦ. ɂɡɨɬɪɨɩɧɨɫɬɶ ɫɪɟɞɵ –
ɧɟɡɚɜɢɫɢɦɨɫɬɶ ɜɫɟɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ ɢ ɫɜɨɣɫɬɜ ɫɪɟɞɵ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ.
ɇɶɸɬɨɧɨɜɫɤɚɹ ɫɪɟɞɚ – ɫɪɟɞɚ, ɜ ɤɨɬɨɪɨɣ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ ɝɪɚɞɢɟɧɬɭ ɫɤɨɪɨɫɬɢ (ɢɥɢ ɫɤɨɪɨɫɬɢ ɭɝɥɨɜɵɯ ɞɟɮɨɪɦɚɰɢɣ).
Ʉɪɨɦɟ ɩɨɥɹ ɫɤɨɪɨɫɬɟɣ ɬɚɤɠɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɫɤɚɥɹɪɧɵɟ ɜɟɥɢɱɢɧɵ: ɩɥɨɬɧɨɫɬɶ U, ɤɝ/ɦ3 ; ɬɟɦɩɟɪɚɬɭɪɚ Ɍ , Ʉ ; ɬɟɧɡɨɪɵ ɧɚɩɪɹɠɟɧɢɣ ɢ ɫɤɨɪɨɫɬɟɣ ɞɟɮɨɪɦɚɰɢɣ. Ʉɚɠɞɚɹ ɢɡ ɞɚɧɧɵɯ ɜɟɥɢɱɢɧ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɤɨɨɪɞɢɧɚɬ ɢ ɜɪɟɦɟɧɢ.
Ɉɞɧɢɦ ɢɡ ɨɫɧɨɜɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɦɟɯɚɧɢɤɟ ɠɢɞɤɨɫɬɢ ɢ ɝɚɡɚ ɹɜɥɹɟɬɫɹ
ɭɪɚɜɧɟɧɢɟ ɇɚɜɶɟ – ɋɬɨɤɫɚ
&
du
&
& &
* 1 &
u ˜ ’ u ’p Q'u f .
(1)
U
dt
Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɞɪɨɛɧɨ ɨɩɢɫɵɜɚɟɬ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɠɢɞɤɨɫɬɢ ɩɨ
&
du
ɜɪɟɦɟɧɢ
ɫ ɩɨɦɨɳɶɸ ɱɟɬɵɪɟɯ ɤɨɦɩɨɧɟɧɬ.
dt
&
&
ɉɟɪɜɚɹ ɢɡ ɧɢɯ, u ˜ ' u , ɩɨɤɚɡɵɜɚɟɬ, ɤɚɤ ɞɢɜɟɪɝɟɧɰɢɹ ɜɥɢɹɟɬ ɧɚ ɫɤɨɪɨɫɬɶ. ȿɟ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɦɨɠɧɨ ɧɚɝɥɹɞɧɨ ɨɛɴɹɫɧɢɬɶ ɧɚ ɩɪɢɦɟɪɟ ɬɟɱɟɧɢɹ
ɪɟɤɢ [2]. Ɍɚɤ, ɤɨɝɞɚ ɪɟɤɚ ɫɭɠɚɟɬɫɹ, ɫɤɨɪɨɫɬɶ ɜɨɞɵ ɜ ɧɟɣ ɜɨɡɪɚɫɬɚɟɬ, ɢ ɧɚɨɛɨɪɨɬ, ɤɨɝɞɚ ɪɟɤɚ ɪɚɫɲɢɪɹɟɬɫɹ, ɫɤɨɪɨɫɬɶ ɜɨɞɵ ɭɦɟɧɶɲɚɟɬɫɹ.
1 &
ȼɬɨɪɚɹ ɤɨɦɩɨɧɟɧɬɚ, ’p , ɩɨɤɚɡɵɜɚɟɬ, ɤɚɤ ɜɥɢɹɟɬ ɧɚ ɞɜɢɠɟɧɢɟ ɢɡɦɟU
ɧɟɧɢɟ ɞɚɜɥɟɧɢɹ, ɨɫɨɛɟɧɧɨ ɧɚ ɧɚɩɪɚɜɥɟɧɧɨɫɬɶ ɞɜɢɠɟɧɢɹ ɨɬ ɨɛɥɚɫɬɟɣ ɫ ɛɨɥɟɟ
ɜɵɫɨɤɢɦ ɞɚɜɥɟɧɢɟɦ. Ɍɚɤɠɟ ɹɫɧɨ, ɱɬɨ ɱɟɦ ɛɨɥɶɲɟ ɩɥɨɬɧɨɫɬɶ ɠɢɞɤɨɫɬɢ, ɬɟɦ
ɬɪɭɞɧɟɟ ɟɣ ɨɫɭɳɟɫɬɜɥɹɬɶ ɩɟɪɟɦɟɳɟɧɢɟ.
&
ɋɥɟɞɭɸɳɚɹ ɤɨɦɩɨɧɟɧɬɚ Q'u , ɝɞɟ Q – ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ ɜɹɡɤɨɫɬɶ, ɩɨɤɚɡɵɜɚɟɬ ɜɥɢɹɧɢɟ, ɨɤɚɡɵɜɚɟɦɨɟ ɧɚ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ ɫɨɫɟɞɧɢɯ ɱɚɫɬɢɰ. ɑɟɦ
ɛɨɥɶɲɟ ɜɹɡɤɨɫɬɶ, ɬɟɦ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɛɨɥɶɲɟ ɜɟɥɢɱɢɧɚ ɞɚɧɧɨɝɨ ɜɥɢɹɧɢɹ.
6
&
ɂ ɱɟɬɜɟɪɬɚɹ ɤɨɦɩɨɧɟɧɬɚ, f , ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɜɥɢɹɧɢɟ, ɨɤɚɡɵɜɚɟɦɨɟ ɧɚ
ɞɚɧɧɭɸ ɠɢɞɤɨɫɬɶ ɫɨ ɫɬɨɪɨɧɵ ɥɸɛɨɣ ɞɪɭɝɨɣ ɫɢɥɵ.
Ⱦɪɭɝɢɦ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ
&
wU
div(Uu ) 0.
wt
Ⱦɥɹ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ U const ɢ ɭɪɚɜɧɟɧɢɟ ɩɪɢɨɛɪɟɬɚɟɬ ɜɢɞ
&
div(Uu ) 0.
(2)
(3)
ɍɪɚɜɧɟɧɢɹ (2), (3) ɫɩɪɚɜɟɞɥɢɜɵ ɤɚɤ ɞɥɹ ɢɞɟɚɥɶɧɨɣ, ɬɚɤ ɢ ɞɥɹ ɪɟɚɥɶɧɨɣ
ɠɢɞɤɨɫɬɢ [3].
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɥɚɦɢɧɚɪɧɨɣ ɠɢɞɤɨɫɬɢ ɦɵ ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɢɡ ɱɟɬɵɪɟɯ ɭɪɚɜɧɟɧɢɣ: ɬɪɢ ɭɪɚɜɧɟɧɢɹ ɇɚɜɶɟ – ɋɬɨɤɫɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɢ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɞɥɹ ɱɟɬɵɪɟɯ ɧɟɢɡɜɟɫɬɧɵɯ: ɬɪɢ ɤɨɦɩɨɧɟɧɬɚ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɢ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɟ ɞɚɜɥɟɧɢɟ.
Ⱦɟɤɨɦɩɨɡɢɰɢɹ Ɋɟɣɧɨɥɶɞɫɚ. Ɍɭɪɛɭɥɟɧɬɧɚɹ ɠɢɞɤɨɫɬɶ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ
ɤɨɥɟɛɚɧɢɹɦɢ ɫɤɨɪɨɫɬɢ ɜɨ ɜɫɟɯ ɧɚɩɪɚɜɥɟɧɢɹɯ ɢ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ [4]. Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɣ ɇɚɜɶɟ – ɋɬɨɤɫɚ ɞɥɹ ɬɭɪɛɭɥɟɧɬɧɨɣ
ɠɢɞɤɨɫɬɢ ɡɚɬɪɭɞɧɟɧɨ, ɬɚɤ ɤɚɤ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɹ ɷɥɥɢɩɬɢɱɟɫɤɢɟ, ɧɟɥɢɧɟɣɧɵɟ ɢ ɫɨɞɟɪɠɚɬ ɩɨ ɞɜɟ ɧɟɢɡɜɟɫɬɧɵɯ ɜɟɥɢɱɢɧɵ. ɀɢɞɤɨɫɬɶ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɯɚɨɬɢɱɟɫɤɚɹ, ɞɢɮɮɭɡɢɨɧɧɚɹ, ɞɢɫɫɢɩɚɬɢɜɧɚɹ ɢ ɩɪɟɪɵɜɢɫɬɚɹ.
ɋɭɳɟɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɩɭɬɟɣ ɪɟɲɟɧɢɹ ɞɚɧɧɨɣ ɩɪɨɛɥɟɦɵ. Ɉɞɧɢɦ ɢɡ ɧɢɯ ɹɜɥɹɟɬɫɹ ɞɟɤɨɦɩɨɡɢɰɢɹ Ɋɟɣɧɨɥɶɞɫɚ, ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɣ ɩɪɨɢɡɜɨɥɶɧɭɸ ɜɟɥɢɱɢɧɭ xi
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɤɚɤ ɫɭɦɦɭ ɟɟ ɫɪɟɞɧɟɝɨ ɡɧɚɱɟɧɢɹ xi ɢ ɨɬɤɥɨɧɟɧɢɹ xic [5]:
xi
xi x'i .
Ɍɚɤɚɹ ɞɟɤɨɦɩɨɡɢɰɢɹ ɛɭɞɟɬ ɞɚɜɚɬɶ ɧɚɛɨɪ ɭɪɚɜɧɟɧɢɣ, ɨɩɢɫɵɜɚɸɳɢɯ ɧɟɤɨɬɨɪɨɟ ɫɪɟɞɧɟɟ ɩɨɥɟ ɠɢɞɤɨɫɬɢ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɵ ɩɨɥɭɱɢɦ ɭɫɪɟɞɧɟɧɧɵɟ ɩɨ Ɋɟɣɧɨɥɶɞɫɭ ɭɪɚɜɧɟɧɢɹ ɇɚɜɶɟ – ɋɬɨɤɫɚ, ɤɨɬɨɪɵɟ ɬɚɤɠɟ ɧɚɡɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ
Ɋɟɣɧɨɥɶɞɫɚ, ɚ ɬɚɤɠɟ ɭɫɪɟɞɧɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ.
ɍɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɜ ɤɨɦɩɨɧɟɧɬɚɯ ɞɥɹ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ
ɢɦɟɟɬ ɜɢɞ
wui
wxi
0.
(4)
0.
(5)
Ɍɨɝɞɚ ɞɥɹ ɭɫɪɟɞɧɟɧɧɨɣ ɫɤɨɪɨɫɬɢ ui
wui
wxi
7
ȼɵɱɢɬɚɹ (5) ɢɡ (4), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɞɥɹ ɨɬɤɥɨɧɟɧɢɹ
wu'i
wxi
0.
ɂɫɩɨɥɶɡɭɹ (4), ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ (1) ɜ ɤɨɦɩɨɧɟɧɬɚɯ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ:
U
wVij
wui
w
Uui u j Ugi ,
wt wx j
wx j
(6)
ɝɞɟ Vij – ɧɚɩɪɹɠɟɧɢɹ ɜ ɠɢɞɤɨɫɬɢ, ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
Vij
§ wu wu j
pGij P ¨ i ¨ wx j wxi
©
·
¸¸ .
¹
(7)
ɋɨɨɬɧɨɲɟɧɢɹ (7) ɹɜɥɹɸɬɫɹ ɨɩɪɟɞɟɥɹɸɳɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ ɞɥɹ ɧɶɸɬɨɧɨɜɫɤɨɣ ɠɢɞɤɨɫɬɢ. Gij ɧɚɡɵɜɚɟɬɫɹ ɞɟɥɶɬɨɣ Ʉɪɨɧɟɤɟɪɚ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ
­°Gij
®
°̄Gij
1, i
j,
0, i z j.
ɂɫɩɨɥɶɡɭɹ ɞɟɤɨɦɩɨɡɢɰɢɸ Ɋɟɣɧɨɥɶɞɫɚ, ɭɪɚɜɧɟɧɢɟ (6) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ [5]:
§ wu
wu
U¨ i u j i
¨ wt
wx j
©
·
w
Vij Uu'i u j' .
¸¸ Ugi wx j
¹
(8)
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɢɡɜɟɫɬɧɨ ɤɚɤ ɭɪɚɜɧɟɧɢɟ Ɋɟɣɧɨɥɶɞɫɚ. Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ
ɞɨɫɬɚɬɨɱɧɨ ɩɨɯɨɠɟ ɧɚ ɭɪɚɜɧɟɧɢɟ (6) ɢ ɨɬɥɢɱɚɟɬɫɹ ɥɢɲɶ ɞɨɩɨɥɧɢɬɟɥɶɧɵɦ ɫɥɚɝɚɟɦɵɦ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ Uu'i u j' . ɗɬɨ ɫɥɚɝɚɟɦɨɟ ɧɚɡɵɜɚɟɬɫɹ ɧɚɩɪɹɠɟɧɢɹɦɢ Ɋɟɣɧɨɥɶɞɫɚ ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɢɦɦɟɬɪɢɱɧɵɣ ɬɟɧɡɨɪ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, ɫɨɫɬɨɹɳɢɣ ɢɡ ɲɟɫɬɢ ɧɟɡɚɜɢɫɢɦɵɯ ɤɨɦɩɨɧɟɧɬ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɬɭɪɛɭɥɟɧɬɧɨɣ ɠɢɞɤɨɫɬɢ ɢɦɟɸɬɫɹ ɜɫɟ ɬɟ ɠɟ ɱɟɬɵɪɟ ɭɪɚɜɧɟɧɢɹ ɢ ɭɠɟ ɞɟɫɹɬɶ
ɧɟɢɡɜɟɫɬɧɵɯ: ɬɪɢ ɤɨɦɩɨɧɟɧɬɚ ɫɤɨɪɨɫɬɢ, ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɟ ɞɚɜɥɟɧɢɟ ɢ ɲɟɫɬɶ
ɧɚɩɪɹɠɟɧɢɣ Ɋɟɣɧɨɥɶɞɫɚ. Ⱦɚɧɧɚɹ ɩɪɨɛɥɟɦɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɩɪɨɛɥɟɦɵ ɡɚɦɵɤɚɧɢɹ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ.
ɋɬɚɧɞɚɪɬɧɚɹ ɦɨɞɟɥɶ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ k İ . ɑɬɨɛɵ ɡɚɦɤɧɭɬɶ ɬɭɪɛɭɥɟɧɬɧɨɫɬɶ, ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɫɜɹɡɶ ɦɟɠɞɭ ɧɚɩɪɹɠɟɧɢɹɦɢ ɩɨ Ɋɟɣɧɨɥɶɞɫɭ ɢ ɩɚɪɚɦɟɬɪɚɦɢ ɨɫɪɟɞɧɟɧɧɨɝɨ ɬɟɱɟɧɢɹ. ɗɬɭ ɫɜɹɡɶ ɨɩɪɟɞɟɥɹɸɬ ɫ ɩɨɦɨɳɶɸ
ɪɚɡɥɢɱɧɵɯ ɦɨɞɟɥɟɣ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ [6]. ȼ ɷɬɢɯ ɦɨɞɟɥɹɯ ɩɪɢɧɢɦɚɸɬɫɹ ɨɩɪɟɞɟɥɟɧɧɵɟ ɞɨɩɭɳɟɧɢɹ, ɧɚ ɨɫɧɨɜɟ ɤɨɬɨɪɵɯ ɜɜɨɞɢɬɫɹ ɧɟɞɨɫɬɚɸɳɟɟ ɱɢɫɥɨ ɭɪɚɜ8
ɧɟɧɢɣ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɧɚɣɬɢ ɜɫɟ ɧɟɢɡɜɟɫɬɧɵɟ. Ɉɞɧɢɦ ɢɡ ɞɨɩɭɳɟɧɢɣ ɹɜɥɹɟɬɫɹ
ɜɜɟɞɟɧɢɟ ɬɭɪɛɭɥɟɧɬɧɨɣ ɜɹɡɤɨɫɬɢ, ɤɨɬɨɪɨɟ ɜɩɟɪɜɵɟ ɨɫɭɳɟɫɬɜɢɥ Ȼɭɫɫɢɧɟɫɤ.
Ɍɭɪɛɭɥɟɧɬɧɭɸ ɞɢɧɚɦɢɱɟɫɤɭɸ ɜɹɡɤɨɫɬɶ Pt ɨɧ ɜɜɟɥ ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɞɢɧɚɦɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɶɸ
§ wu wu j
Pt ¨ i ¨ wx j wxi
©
Uu'i u 'j
·
¸¸ .
¹
(9)
Ⱦɚɥɟɟ ɩɟɪɟɣɞɟɦ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɤ ɩɨɥɭɱɟɧɢɸ ɫɬɚɧɞɚɪɬɧɨɣ k H ɦɨɞɟɥɢ ɢɡ ɞɜɭɯ ɭɪɚɜɧɟɧɢɣ, ɤɨɬɨɪɚɹ ɫɟɝɨɞɧɹ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɫɬɚɧɞɚɪɬɧɚɹ ɦɨɞɟɥɶ ɞɥɹ ɨɩɢɫɚɧɢɹ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɢ ɪɟɲɟɧɢɹ ɢɧɠɟɧɟɪɧɵɯ ɡɚɞɚɱ. ȼ ɞɚɧɧɨɣ
ɦɨɞɟɥɢ ɜɜɨɞɹɬɫɹ ɞɜɚ ɜɚɠɧɵɯ ɩɨɧɹɬɢɹ – ɝɟɧɟɪɚɰɢɹ Ɋ ɢ ɞɢɫɫɢɩɚɰɢɹ İ. Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɝɟɧɟɪɚɰɢɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ Ɋ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɩɨɪɨɠɞɟɧɢɢ ɧɨɜɵɯ
ɜɢɯɪɟɣ ɢ ɩɭɥɶɫɚɰɢɣ, ɤɨɬɨɪɵɟ ɢ ɨɛɪɚɡɭɸɬ ɬɭɪɛɭɥɟɧɬɧɨɫɬɶ [7]. Ⱦɢɫɫɢɩɚɰɢɹ İ,
ɧɚɩɪɨɬɢɜ, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɫɫɟɢɜɚɧɢɟ ɛɨɥɶɲɢɯ ɜɢɯɪɟɣ ɧɚ ɛɨɥɟɟ ɦɚɥɵɟ,
ɩɪɢɜɨɞɢɬ ɤ ɭɫɪɟɞɧɟɧɢɸ ɬɟɱɟɧɢɹ ɢ ɭɦɟɧɶɲɟɧɢɸ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ. Ⱦɜɚ ɭɪɚɜɧɟɧɢɹ ɩɟɪɟɧɨɫɚ ɩɨɡɜɨɥɹɸɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɭɪɛɭɥɟɧɬɧɨɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
ɢ ɜɪɟɦɟɧɢ. Ⱦɚɧɧɚɹ ɦɨɞɟɥɶ ɹɜɥɹɟɬɫɹ ɩɨɥɭɷɦɩɢɪɢɱɟɫɤɨɣ ɢ ɨɩɢɪɚɟɬɫɹ ɧɚ ɮɟɧɨɦɟɧɨɥɨɝɢɱɟɫɤɢɣ ɩɨɯɨɞ ɢ ɪɟɡɭɥɶɬɚɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɨɩɵɬɧɵɦ ɩɭɬɟɦ.
ȼɵɩɨɥɧɢɜ ɧɟɤɨɬɨɪɵɟ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢ ɭɦɧɨɠɢɜ ɧɚ u j ,
(8) ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɤ ɫɥɟɞɭɸɳɟɦɭ ɜɢɞɭ [8]:
w t u'i u j' uk w k u'i u j'
1
u j' w i p u'i w j p 2Qw k u'i w k u j' U
(10)
w k uk' u'i u j' u j' uk' w k ui u'i uk' w k u j Q’ ui u j .
2
Ɉɩɪɟɞɟɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɤɚɤ k
ɢ ɩɨɞɫɬɚɜɢɦ ɟɟ ɜ (10), ɩɪɢɧɢɦɚɹ i
w t k uk w k k
0,5u'i u j'
j:
1
1
w i ui ' p Qw k ui ' w k ui ' w k uk ' ui ' ui ' ui ' uk 'w k ui Q’ 2 k . (11)
U
2
ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɩɪɚɜɨɣ ɱɚɫɬɢ (11), ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ [8], ɹɜɥɹɟɬɫɹ ɞɢɫɫɢɩɚɰɢɟɣ
H Qw k u'i w k u'i ,
(12)
ɬɨɝɞɚ ɤɚɤ ɱɟɬɜɟɪɬɨɟ ɫɥɚɝɚɟɦɨɟ ɩɪɚɜɨɣ ɱɚɫɬɢ ɜɵɪɚɠɟɧɢɹ (1.1), ɜɤɥɸɱɚɹ ɦɢɧɭɫ,
ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ [8], ɹɜɥɹɟɬɫɹ ɝɟɧɟɪɚɰɢɟɣ Ɋ:
P
ui'uk'w k ui .
(13)
9
Ⱦɚɥɟɟ ɞɟɥɚɟɦ ɞɨɩɭɳɟɧɢɟ [8], ɱɬɨ
§1
·
1
w j ¨ u j' u'i u'i u j p ¸ | w j (QT w j k ) .
U
©2
¹
(14)
ɍɱɢɬɵɜɚɹ (12)–(14), ɭɪɚɜɧɟɧɢɟ (11) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
wt k u j w j k
§§
Q
P H w j ¨¨ ¨ Q T
Vk
©©
·
·
¸ w j k ¸¸ .
¹
¹
(15)
ɍɪɚɜɧɟɧɢɟ (15) ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɞɥɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ k . Vk –
ɩɚɪɚɦɟɬɪ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɣ ɧɭɠɧɭɸ ɪɚɡɦɟɪɧɨɫɬɶ ɞɥɹ ɫɥɚɝɚɟɦɨɝɨ ɫ QT . Ɉɛɵɱɧɨ ɩɪɢɧɢɦɚɟɬɫɹ Vk 1. ɍɪɚɜɧɟɧɢɟ ɞɥɹ ɞɢɫɫɢɩɚɰɢɢ H ɚɧɚɥɢɬɢɱɟɫɤɢ ɧɟ ɜɵɜɨɞɢɬɫɹ ɢ ɩɪɨɫɬɨ ɡɚɩɢɫɵɜɚɟɬɫɹ ɩɨ ɚɧɚɥɨɝɢɢ ɫ (15):
wt H u j w j H
§§
C1H' P C2H' H
Q
w j ¨¨Q T
¨
VH
T
©©
·
·
¸ w j H ¸¸ ,
¹
¹
(16)
ɝɞɟ T k /H ɨɛɟɫɩɟɱɢɜɚɟɬ ɧɭɠɧɭɸ ɪɚɡɦɟɪɧɨɫɬɶ, ɚ ɤɨɧɫɬɚɧɬɵ C1H' , C2 H' , VH ɜɜɨɞɹɬɫɹ, ɩɨɫɤɨɥɶɤɭ ɮɨɪɦɚ ɭɪɚɜɧɟɧɢɹ (16) ɥɢɲɶ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɧɨ ɧɟ ɜɵɜɨɞɢɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɢ.
ȼ ɩɚɤɟɬɟ Ansys Fluent ɭɪɚɜɧɟɧɢɹ cɬɚɧɞɚɪɬɧɨɣ k H ɦɨɞɟɥɢ ɩɪɢɦɟɧɹɸɬɫɹ
ɜ ɧɟɫɤɨɥɶɤɨ ɢɧɨɦ, ɦɨɞɟɪɧɢɡɢɪɨɜɚɧɧɨɦ ɜɢɞɟ. ȿɝɨ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɢɡ (15)
ɢ (16) ɩɭɬɟɦ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ, ɢ ɨɧ ɬɚɤɠɟ ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɡɞɚɬɟɥɹɦɢ ɦɨɞɟɥɢ [9], www.ansys.com:
­w
Pt · wk º
w
w ª§
° Uk Ukui «¨ P » Gk Gb UH YM Sk ,
¸
wxi
wx j ¬«©
V k ¹ wx j ¼»
° wt
(17)
®
Pt · wH º
°w
w
w ª§
H
H2
° wt UH wx UHui wx «¨ P V ¸ wx » C1H k Gk C3HGb C2 HU k SH .
i
j «
j »
H ¹
©
¼
¯
ȼ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ Gk ɩɪɟɞɫɬɚɜɥɹɟɬ ɬɭɪɛɭɥɟɧɬɧɭɸ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ, ɨɛɪɚɡɨɜɚɧɧɭɸ ɨɬ ɫɪɟɞɧɢɯ ɝɪɚɞɢɟɧɬɨɜ ɫɤɨɪɨɫɬɢ. ɉɪɢɧɢɦɚɹ ɝɢɩɨɬɟɡɭ Ȼɭɫɫɢɧɟɫɤɚ, ɟɟ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɩɨ ɮɨɪɦɭɥɟ
Gk
Pt S 2 ,
ɝɞɟ
Pt
10
UɋP
k2
,
H
(18)
U – ɩɥɨɬɧɨɫɬɶ ɝɚɡɚ, CP
S
S – ɢɧɜɚɪɢɚɧɬ ɬɟɧɡɨɪɚ ɞɟɮɨɪɦɚɰɢɣ,
const,
2Sij Sij ; Gb – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɜɵɬɚɥɤɢɜɚɸɳɟɣ ɫɢɥɵ,
Egi
Gb
Pt wT
,
Prt wxi
ɝɞɟ Prt – ɬɭɪɛɭɥɟɧɬɧɚɹ ɩɨɫɬɨɹɧɧɚɹ ɉɪɚɧɞɬɥɹ ɞɥɹ ɷɧɟɪɝɢɢ, gi – ɤɨɦɩɨɧɟɧɬɚ
ɜɟɤɬɨɪɚ ɝɪɚɜɢɬɚɰɢɢ ɜ i-ɦ ɧɚɩɪɚɜɥɟɧɢɢ; E – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ
ɪɚɫɲɢɪɟɧɢɹ,
1 § wU ·
E ¨
¸ ,
U © wT ¹ p
ɝɞɟ Ɍ – ɬɟɦɩɟɪɚɬɭɪɚ.
C3H – ɤɨɧɫɬɚɧɬɚ, ɨɩɪɟɞɟɥɹɸɳɚɹ ɫɬɟɩɟɧɶ ɜɨɡɞɟɣɫɬɜɢɹ ɜɵɬɚɥɤɢɜɚɸɳɟɣ
ɫɢɥɵ ɧɚ H, ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
C3H
tanh
Q'
,
u'
ɝɞɟ Q' – ɤɨɦɩɨɧɟɧɬɚ ɫɤɨɪɨɫɬɢ ɠɢɞɤɨɫɬɢ, ɩɚɪɚɥɥɟɥɶɧɚɹ ɫɤɨɪɨɫɬɢ ɝɪɚɜɢɬɚɰɢɢ
ɢ u' – ɤɨɦɩɨɧɟɧɬɚ ɫɤɨɪɨɫɬɢ ɠɢɞɤɨɫɬɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚɹ ɫɤɨɪɨɫɬɢ ɝɪɚɜɢɬɚɰɢɢ.
C3H 1 ɞɥɹ ɫɥɨɟɜ ɠɢɞɤɨɫɬɢ, ɞɥɹ ɤɨɬɨɪɵɯ ɧɚɩɪɚɜɥɟɧɢɟ ɫɤɨɪɨɫɬɢ ɠɢɞɤɨɫɬɢ ɩɚɪɚɥɥɟɥɶɧɨ ɜɟɤɬɨɪɭ ɝɪɚɜɢɬɚɰɢɢ, C3H 0 ɞɥɹ ɫɥɨɟɜ ɠɢɞɤɨɫɬɢ, ɞɥɹ ɤɨɬɨɪɵɯ ɧɚɩɪɚɜɥɟɧɢɟ ɫɤɨɪɨɫɬɢ ɠɢɞɤɨɫɬɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɜɟɤɬɨɪɭ ɝɪɚɜɢɬɚɰɢɢ.
YM – ɜɤɥɚɞ ɩɟɪɟɦɟɧɧɨɝɨ ɪɚɫɲɢɪɟɧɢɹ ɩɪɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɫɠɚɬɢɹ ɜ ɨɛɳɭɸ ɫɤɨɪɨɫɬɶ ɞɢɫɫɢɩɚɰɢɢ. Ⱦɚɧɧɭɸ ɜɟɥɢɱɢɧɭ ɫɥɟɞɭɟɬ ɭɱɢɬɵɜɚɬɶ ɩɪɢ ɛɨɥɶɲɨɦ ɱɢɫɥɟ Ɇɚɯɚ. ȿɟ ɨɛɹɡɚɬɟɥɶɧɨ ɭɱɢɬɵɜɚɬɶ, ɤɨɝɞɚ ɦɨɞɟɥɢɪɭɟɬɫɹ ɫɠɢɦɚɟɦɵɣ
ɢɞɟɚɥɶɧɵɣ ɝɚɡ.
YM
2UHM t2 ,
ɝɞɟ M t – ɱɢɫɥɨ Ɇɚɯɚ ɞɥɹ ɬɭɪɛɭɥɟɧɬɧɨɣ ɠɢɞɤɨɫɬɢ,
Mt
ɝɞɟ ɚ – ɫɤɨɪɨɫɬɶ ɡɜɭɤɚ, a
k
,
a2
JRT .
Ɉɫɬɚɥɶɧɵɟ ɤɨɧɫɬɚɧɬɵ ɨɩɪɟɞɟɥɟɧɵ ɢɡ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɞɥɹ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɬɭɪɛɭɥɟɧɬɧɵɯ ɠɢɞɤɨɫɬɟɣ ɢ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɟ ɡɧɚɱɟɧɢɹ: ɋ1H 1, 44,
ɋ2 H 1,92, ɋP
0,09, Vk
1, 44, VH 1,3.
11
RNG k İ ɦɨɞɟɥɶ. RNG ɦɨɞɟɥɶ ɛɵɥɚ ɩɨɥɭɱɟɧɚ ɩɪɢ ɩɨɦɨɳɢ ɬɟɨɪɢɢ ɪɟɧɨɪɦɚɥɢɡɨɜɚɧɧɵɯ ɝɪɭɩɩ [10]. Ɉɧɚ ɢɦɟɟɬ ɫɯɨɠɭɸ ɮɨɪɦɭ ɫɨ ɫɬɚɧɞɚɪɬɧɨɣ k H
ɦɨɞɟɥɶɸ, ɧɨ ɜɤɥɸɱɚɟɬ ɫɥɟɞɭɸɳɢɟ ɭɥɭɱɲɟɧɢɹ:
– ɢɦɟɟɬ ɞɨɩɨɥɧɢɬɟɥɶɧɵɣ ɱɥɟɧ ɜ ɭɪɚɜɧɟɧɢɢ ɞɥɹ H, ɤɨɬɨɪɵɣ ɭɥɭɱɲɚɟɬ
ɬɨɱɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ ɞɥɹ ɠɢɞɤɨɫɬɟɣ ɫ ɜɵɫɨɤɢɦɢ ɫɤɨɪɨɫɬɹɦɢ ɞɟɮɨɪɦɚɰɢɣ;
– ɜ ɦɨɞɟɥɢ ɭɱɬɟɧɨ ɜɥɢɹɧɢɟ ɡɚɜɢɯɪɟɧɧɨɫɬɢ ɧɚ ɬɭɪɛɭɥɟɧɬɧɨɫɬɶ, ɱɬɨ ɭɜɟɥɢɱɢɜɚɟɬ ɬɨɱɧɨɫɬɶ ɞɥɹ ɜɵɫɨɤɨɡɚɜɢɯɪɟɧɧɵɯ ɠɢɞɤɨɫɬɟɣ;
– ɞɚɧɧɚɹ ɬɟɨɪɢɹ ɩɪɟɞɥɚɝɚɟɬ ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɮɨɪɦɭɥɵ ɞɥɹ ɬɭɪɛɭɥɟɧɬɧɵɯ
ɱɢɫɟɥ ɉɪɚɧɞɬɥɹ, ɬɨɝɞɚ ɤɚɤ ɫɬɚɧɞɚɪɬɧɚɹ ɦɨɞɟɥɶ ɢɫɩɨɥɶɡɭɟɬ ɡɚɞɚɧɧɵɟ ɩɨɥɶɡɨɜɚɬɟɥɟɦ ɩɨɫɬɨɹɧɧɵɟ ɡɧɚɱɟɧɢɹ;
– RNG ɦɨɞɟɥɶ ɩɪɟɞɥɚɝɚɟɬ ɚɧɚɥɢɬɢɱɟɫɤɢ ɩɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ ɞɥɹ ɷɮɮɟɤɬɢɜɧɨɣ ɜɹɡɤɨɫɬɢ, ɤɨɬɨɪɚɹ ɩɪɟɞɧɚɡɧɚɱɟɧɚ ɞɥɹ ɠɢɞɤɨɫɬɟɣ ɫ ɧɢɡɤɢɦɢ ɱɢɫɥɚɦɢ Ɋɟɣɧɨɥɶɞɫɚ. Ɍɟɦ ɧɟ ɦɟɧɟɟ ɷɮɮɟɤɬɢɜɧɨɟ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɷɬɨɣ ɨɩɰɢɢ ɡɚɜɢɫɢɬ
ɨɬ ɩɪɚɜɢɥɶɧɨɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ ɩɪɢɫɬɟɧɨɱɧɨɣ ɨɛɥɚɫɬɢ.
Ⱦɚɧɧɵɟ ɭɥɭɱɲɟɧɢɹ ɞɟɥɚɸɬ RNG ɦɨɞɟɥɶ ɛɨɥɟɟ ɬɨɱɧɨɣ ɢ ɧɚɞɟɠɧɨɣ, ɩɨɡɜɨɥɹɹ ɷɮɮɟɤɬɢɜɧɨ ɩɪɢɦɟɧɹɬɶ ɟɟ ɞɥɹ ɛɨɥɟɟ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ ɠɢɞɤɨɫɬɟɣ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫɨ ɫɬɚɧɞɚɪɬɧɨɣ k H ɦɨɞɟɥɶɸ.
ɍɪɚɜɧɟɧɢɹ RNG ɦɨɞɟɥɢ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
­w
w
w §
wk ·
° Uk ¨¨ ak P eff
¸ Gk Gb UH YM Sk ,
Ukui wxi
wx j ©
wx j ¸¹
° wt
°
°w
w
w §
wH ·
H
UHui ¨¨ aH P eff
¸¸ C1H (Gk C3H Gb ) ® UH k
wxi
wx j ©
wx j ¹
° wt
°
2
°C U H R S .
H
H
° 2H k
¯
(19)
Ⱦɚɥɟɟ ɪɚɫɫɦɨɬɪɢɦ ɡɧɚɱɟɧɢɟ ɜɟɥɢɱɢɧ, ɧɨɜɵɯ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ (17). ak , aH –
ɨɛɪɚɬɧɵɟ ɷɮɮɟɤɬɢɜɧɵɟ ɱɢɫɥɚ ɉɪɚɧɞɬɥɹ ɞɥɹ k ɢ H ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ȼ (19) P eff
ɨɡɧɚɱɚɟɬ ɷɮɮɟɤɬɢɜɧɭɸ ɜɹɡɤɨɫɬɶ. Ⱦɚɧɧɚɹ ɜɹɡɤɨɫɬɶ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɪɚɜɧɚ Pt ɢɡ
ɫɬɚɧɞɚɪɬɧɨɣ k H ɦɨɞɟɥɢ ɞɥɹ ɜɵɫɨɤɢɯ ɱɢɫɟɥ Ɋɟɣɧɨɥɶɞɫɚ. Ⱦɥɹ ɧɢɡɤɢɯ ɱɢɫɟɥ
Ɋɟɣɧɨɥɶɞɫɚ ɫɨɡɞɚɬɟɥɹɦɢ ɦɨɞɟɥɢ [10] ɩɪɟɞɥɚɝɚɟɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɡɜɨɥɹɸɳɟɟ ɛɨɥɟɟ ɬɨɱɧɨ ɜɵɱɢɫɥɢɬɶ P eff .
Ƚɥɚɜɧɨɟ ɨɬɥɢɱɢɟ RNG ɦɨɞɟɥɢ ɨɬ ɫɬɚɧɞɚɪɬɧɨɣ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɞɨɩɨɥɧɢɬɟɥɶɧɨɦ ɱɥɟɧɟ ɜ ɭɪɚɜɧɟɧɢɢ ɞɥɹ H. RH ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
RH
12
CP UK3 (1 K / K0 )H 2 H 2
,
k
1 EK3
(20)
ɝɞɟ K Sk /H, K0 4,38, E 0,012. Ɂɧɚɱɟɧɢɟ RH ɦɨɠɟɬ ɫɬɚɬɶ ɛɨɥɟɟ ɨɱɟɜɢɞɧɵɦ, ɟɫɥɢ ɡɚɩɢɫɚɬɶ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ (20) ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
w
w
UH UHui wt
wxi
ɝɞɟ C2*H
C2 H w
wx j
CP K3 1 K / K0 1 EK3
§
wH ·
H
H2
*
¨¨ aH P eff
¸¸ C1H Gk C3H Gb C2 HU ,
wx j ¹
k
k
©
.
ȼ ɫɥɭɱɚɟ ɤɨɝɞɚ K K0 , RH ɜɧɨɫɢɬ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɜɤɥɚɞ, C2*H ɫɬɚɧɨɜɢɬɫɹ
ɛɨɥɶɲɟ, ɱɟɦ C2 H . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɠɢɞɤɨɫɬɟɣ ɫɨ ɫɥɚɛɵɦɢ ɢɥɢ ɭɦɟɪɟɧɧɵɦɢ ɫɤɨɪɨɫɬɹɦɢ ɞɟɮɨɪɦɚɰɢɣ RNG ɦɨɞɟɥɶ ɞɚɟɬ ɪɟɡɭɥɶɬɚɬɵ, ɫɯɨɠɢɟ ɫ ɩɨɥɭɱɟɧɧɵɦɢ ɩɪɢ ɩɨɦɨɳɢ ɫɬɚɧɞɚɪɬɧɨɣ k H ɦɨɞɟɥɢ.
ɉɪɢ ɛɨɥɶɲɢɯ ɫɤɨɪɨɫɬɹɯ ɞɟɮɨɪɦɚɰɢɣ, K ! K0 , RH ɜɧɨɫɢɬ ɨɬɪɢɰɚɬɟɥɶɧɵɣ
ɜɤɥɚɞ, C2*H ɫɬɚɧɨɜɢɬɫɹ ɦɟɧɶɲɟ, ɱɟɦ C2 H , ɫɧɢɠɚɟɬɫɹ k , ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɮɮɟɤɬɢɜɧɚɹ ɜɹɡɤɨɫɬɶ. ȼ ɪɟɡɭɥɶɬɚɬɟ, ɜ ɠɢɞɤɨɫɬɹɯ ɫ ɛɨɥɶɲɢɦɢ ɫɤɨɪɨɫɬɹɦɢ ɞɟɮɨɪɦɚɰɢɣ RNG ɦɨɞɟɥɶ ɞɚɟɬ ɦɟɧɶɲɭɸ ɬɭɪɛɭɥɟɧɬɧɭɸ ɜɹɡɤɨɫɬɶ, ɱɟɦ ɫɬɚɧɞɚɪɬɧɚɹ k H ɦɨɞɟɥɶ. Ʉɨɧɫɬɚɧɬɵ C1H ɢ C2H ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɟ ɡɧɚɱɟɧɢɹ:
C1H 1, 42, C2H
1,68 .
Ɋɟɚɥɶɧɚɹ k İ ɦɨɞɟɥɶ. ɉɨ ɫɪɚɜɧɟɧɢɸ ɫɨ ɫɬɚɧɞɚɪɬɧɨɣ k H ɦɨɞɟɥɶɸ
ɞɚɧɧɚɹ ɦɨɞɟɥɶ ɢɦɟɟɬ ɞɜɚ ɫɭɳɟɫɬɜɟɧɧɨ ɜɚɠɧɵɯ ɨɬɥɢɱɢɹ [11]:
– ɪɟɚɥɶɧɚɹ k H ɦɨɞɟɥɶ ɫɨɞɟɪɠɢɬ ɚɥɶɬɟɪɧɚɬɢɜɧɭɸ ɮɨɪɦɭɥɢɪɨɜɤɭ ɞɥɹ
ɬɭɪɛɭɥɟɧɬɧɨɣ ɜɹɡɤɨɫɬɢ;
– ɦɨɞɢɮɢɰɢɪɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɟɪɟɧɨɫɚ ɞɥɹ ɫɤɨɪɨɫɬɢ ɞɢɫɫɢɩɚɰɢɢ H
ɛɵɥɨ ɩɨɥɭɱɟɧɨ ɢɡ ɬɨɱɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɩɟɪɟɧɨɫɚ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɡɚɜɢɯɪɟɧɧɨɫɬɢ.
Ⱦɚɧɧɚɹ ɦɨɞɟɥɶ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɬɨɱɧɵɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɝɪɚɧɢɱɟɧɢɹɦ ɩɨ
ɧɚɩɪɹɠɟɧɢɹɦ Ɋɟɣɧɨɥɶɞɫɚ, ɜɵɬɟɤɚɸɳɢɦ ɢɡ ɮɢɡɢɤɢ ɬɭɪɛɭɥɟɧɬɧɨɣ ɠɢɞɤɨɫɬɢ.
ɇɟɪɚɜɟɧɫɬɜɨ ɒɜɚɪɰɚ ɢ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɡɧɚɤ ɧɚɩɪɹɠɟɧɢɣ ɩɨ Ɋɟɣɧɨɥɶɞɫɭ
ɧɚɤɥɚɞɵɜɚɸɬ ɧɟɤɨɬɨɪɵɟ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɋP . Ɂɧɚɱɟɧɢɹ ɞɚɧɧɨɣ ɤɨɧɫɬɚɧɬɵ ɦɟɧɹɸɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɜɨɣɫɬɜ ɠɢɞɤɨɫɬɢ ɢ ɦɟɫɬɨɩɨɥɨɠɟɧɢɹ, ɨɧɢ ɞɨɫɬɚɬɨɱɧɨ ɬɨɱɧɨ ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ ɩɭɬɟɦ ɞɥɹ ɪɚɡɧɵɯ ɭɫɥɨɜɢɣ.
Ⱦɪɭɝɚɹ ɩɪɨɛɥɟɦɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɫɤɨɪɨɫɬɢ ɞɢɫɫɢɩɚɰɢɢ H ɧɟ ɜɫɟɝɞɚ ɪɚɛɨɬɚɟɬ ɞɨɫɬɚɬɨɱɧɨ ɯɨɪɨɲɨ. ɇɚɩɪɢɦɟɪ, ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɚ
ɚɧɨɦɚɥɢɹ ɤɪɭɝɥɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɪɭɢ. ɋɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ
ɞɥɹ ɤɪɭɝɥɨɝɨ ɫɟɱɟɧɢɹ ɨɩɢɫɵɜɚɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɯɨɪɨɲɨ, ɬɨɝɞɚ ɤɚɤ ɞɥɹ ɚɫɫɢɦɟɬɪɢɱɧɨɝɨ ɪɟɡɭɥɶɬɚɬɵ ɩɨɥɭɱɚɸɬɫɹ ɧɟɭɞɨɜɥɟɬɜɨɪɢɬɟɥɶɧɵɦɢ.
13
Ɋɟɚɥɶɧɚɹ k H ɦɨɞɟɥɶ ɭɱɢɬɵɜɚɟɬ ɷɬɢ ɧɟɞɨɫɬɚɬɤɢ ɫ ɩɨɦɨɳɶɸ ɫɥɟɞɭɸɳɢɯ
ɭɥɭɱɲɟɧɢɣ: ɜɨ-ɩɟɪɜɵɯ, ɩɪɟɞɥɚɝɚɟɬɫɹ ɧɨɜɚɹ ɮɨɪɦɭɥɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɭɪɛɭɥɟɧɬɧɨɣ ɜɹɡɤɨɫɬɢ, ɩɟɪɜɨɧɚɱɚɥɶɧɨ ɩɪɟɞɥɨɠɟɧɧɚɹ ɟɳɟ Ɋɟɣɧɨɥɶɞɫɨɦ; ɜɨɜɬɨɪɵɯ, ɢɫɩɨɥɶɡɭɟɬɫɹ ɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɞɢɫɫɢɩɚɰɢɢ H, ɨɫɧɨɜɚɧɧɨɟ ɧɚ ɞɢɧɚɦɢɱɟɫɤɨɦ ɭɪɚɜɧɟɧɢɢ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɡɚɜɢɯɪɟɧɧɨɫɬɢ.
Ɉɝɪɚɧɢɱɟɧɢɟɦ ɹɜɥɹɟɬɫɹ ɬɨ, ɱɬɨ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɧɟɮɢɡɢɱɧɵɟ ɬɭɪɛɭɥɟɧɬɧɵɟ ɜɹɡɤɨɫɬɢ ɜ ɫɢɬɭɚɰɢɹɯ, ɤɨɝɞɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɛɥɚɫɬɶ ɫɨɞɟɪɠɢɬ ɤɚɤ ɡɨɧɵ
ɫ ɬɭɪɛɭɥɟɧɬɧɨɫɬɶɸ, ɬɚɤ ɢ ɡɨɧɵ ɫɨ ɫɬɚɰɢɨɧɚɪɧɨɣ ɠɢɞɤɨɫɬɶɸ.
ɍɪɚɜɧɟɧɢɹ ɪɟɚɥɶɧɨɣ k H ɦɨɞɟɥɢ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
­w
Pt · wk º
w
w ª§
Uku j ° Uk «¨ P » Gk Gb UH YM Sk ,
¸
wx j
wx j ¬«©
V k ¹ wx j ¼»
° wt
°
Pt · wH º
°w
w
w ª§
UHu j «¨ P ¸
» UC1S H ® UH wx j
wx j ¬«©
VH ¹ wx j ¼»
° wt
°
H2
H
°UC
C1H C3H Gb SH .
2
°
k
k QH
¯
ɉɪɢ ɷɬɨɦ
C1
ª
K º
max «0, 43;
.
K 5 »¼
¬
ȼ ɪɟɚɥɶɧɨɣ ɦɨɞɟɥɢ ɭɪɚɜɧɟɧɢɟ ɞɥɹ k ɬɚɤɨɟ ɠɟ, ɤɚɤ ɢ ɜ ɫɬɚɧɞɚɪɬɧɨɣ ɦɨɞɟɥɢ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ ɭɪɚɜɧɟɧɢɟ ɞɥɹ H ɨɬɥɢɱɚɟɬɫɹ ɫɭɳɟɫɬɜɟɧɧɨ. Ɉɞɧɨɣ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɱɟɪɬɨɣ ɹɜɥɹɟɬɫɹ ɬɨ, ɱɬɨ ɩɪɚɜɚɹ ɫɬɨɪɨɧɚ ɭɪɚɜɧɟɧɢɹ ɞɥɹ H ɧɟ ɫɨɞɟɪɠɢɬ Gk . ɋɱɢɬɚɟɬɫɹ, ɱɬɨ ɷɬɨ ɨɛɟɫɩɟɱɢɜɚɟɬ ɥɭɱɲɢɣ ɩɟɪɟɧɨɫ ɫɩɟɤɬɪɚɥɶɧɨɣ
ɷɧɟɪɝɢɢ. Ⱦɪɭɝɨɟ ɩɪɟɢɦɭɳɟɫɬɜɨ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɩɪɢ k 0 ɧɟ ɜɨɡɧɢɤɚɟɬ ɞɟɥɟɧɢɹ ɧɚ ɧɨɥɶ. ɏɨɪɨɲɢɟ ɪɟɡɭɥɶɬɚɬɵ ɞɥɹ ɷɬɨɣ ɦɨɞɟɥɢ ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɞɥɹ
ɜɢɯɪɟɜɵɯ ɨɞɧɨɪɨɞɧɵɯ ɠɢɞɤɨɫɬɟɣ ɫɨ ɫɞɜɢɝɨɜɵɦɢ ɧɚɩɪɹɠɟɧɢɹɦɢ, ɞɥɹ ɫɜɨɛɨɞɧɵɯ ɠɢɞɤɨɫɬɟɣ ɢ ɪɚɡɞɟɥɟɧɧɵɯ ɠɢɞɤɨɫɬɟɣ. Ⱦɥɹ ɜɫɟɯ ɷɬɢɯ ɫɥɭɱɚɟɜ ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɫɭɳɟɫɬɜɟɧɧɨ ɥɭɱɲɢɟ ɪɟɡɭɥɶɬɚɬɵ, ɱɟɦ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɫɬɚɧɞɚɪɬɧɨɣ
ɦɨɞɟɥɢ. Ɍɚɤɠɟ ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɯɨɪɨɲɢɟ ɪɟɡɭɥɶɬɚɬɵ ɞɥɹ ɚɫɫɢɦɟɬɪɢɱɧɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɪɭɢ.
ȼ ɪɟɚɥɶɧɨɣ ɦɨɞɟɥɢ ɧɚɯɨɠɞɟɧɢɟ ɬɭɪɛɭɥɟɧɬɧɨɣ ɜɹɡɤɨɫɬɢ ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɞɪɭɝɢɯ k H ɦɨɞɟɥɟɣ. ɏɨɬɹ ɫɚɦɚ ɜɹɡɤɨɫɬɶ ɬɚɤɠɟ ɧɚɯɨɞɢɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
(18), CP ɭɠɟ ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɧɫɬɚɧɬɨɣ ɢ ɧɚɯɨɞɢɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
CP
14
1
kU A0 AS
H
,
ɝɞɟ
U
:
Sij Sij :
ij ij ,
:ij
:
ij
:ij 2Hijk Zk ,
:ij Hijk Zk ,
ɝɞɟ :ij – ɬɟɧɡɨɪ ɫɪɟɞɧɢɯ ɫɤɨɪɨɫɬɟɣ ɜɪɚɳɟɧɢɹ; Zk – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ. Ʉɨɧɫɬɚɧɬɵ A0
M
6 cos M, ɝɞɟ
4,04, AS
1
cos 1
3
6W , W
Sij S jk S ki , S
S 3
Sij Sij , Sij
1 § wu j wui
¨
2 ¨© wxi wx j
·
¸¸ .
¹
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ CP ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ
ɢ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɢ ɞɟɮɨɪɦɚɰɢɣ, ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɢ ɩɟɪɟɦɟɧɧɵɯ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ k ɢ H . Ɉɫɬɚɥɶɧɵɟ ɤɨɧɫɬɚɧɬɵ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɟ ɡɧɚɱɟɧɢɹ:
C1H 1, 44, C2 1,9, Vk 1,0, VH 1, 2.
ȼɵɜɨɞɵ ɢ ɪɟɤɨɦɟɧɞɚɰɢɢ. ȼ ɬɟɨɪɢɢ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɟɳɟ ɧɟ ɪɚɡɪɚɛɨɬɚɧɚ
ɭɧɢɜɟɪɫɚɥɶɧɚɹ ɬɟɨɪɢɹ, ɩɨɡɜɨɥɹɸɳɚɹ ɨɞɢɧɚɤɨɜɨ ɭɫɩɟɲɧɨ ɧɚɯɨɞɢɬɶ ɪɟɲɟɧɢɹ
ɞɥɹ ɜɫɟɯ ɤɥɚɫɫɨɜ ɡɚɞɚɱ. k H ɦɨɞɟɥɢ ɹɜɥɹɸɬɫɹ ɧɚɢɛɨɥɟɟ ɲɢɪɨɤɨ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɦɢ ɢ ɩɪɢɦɟɧɢɦɵɦɢ ɜ ɱɢɫɥɟɧɧɵɯ ɩɚɤɟɬɚɯ. Ɋɟɡɭɥɶɬɚɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɫ ɢɯ
ɩɨɦɨɳɶɸ, ɩɨɥɭɱɢɥɢ ɩɪɚɤɬɢɱɟɫɤɨɟ ɩɨɞɬɜɟɪɠɞɟɧɢɟ ɞɥɹ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ
ɮɭɧɤɰɢɣ. Ɇɟɠɞɭ ɬɟɦ ɧɟ ɫɥɟɞɭɟɬ ɡɚɛɵɜɚɬɶ, ɱɬɨ ɞɚɧɧɵɟ ɦɨɞɟɥɢ ɢɦɟɸɬ ɤɚɤ
ɩɪɟɢɦɭɳɟɫɬɜɚ, ɬɚɤ ɢ ɧɟɞɨɫɬɚɬɤɢ, ɢ ɫɭɳɟɫɬɜɭɟɬ ɨɝɪɨɦɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɚɥɶɬɟɪɧɚɬɢɜɧɵɯ ɦɨɞɟɥɟɣ. ȼɵɛɨɪ ɦɨɞɟɥɢ ɞɨɥɠɟɧ ɨɫɭɳɟɫɬɜɥɹɬɶɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɭɫɥɨɜɢɣ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɢ, ɚ ɪɟɡɭɥɶɬɚɬɵ ɪɟɲɟɧɢɹ ɞɨɥɠɧɵ ɛɵɬɶ ɬɳɚɬɟɥɶɧɨ
ɩɪɨɜɟɪɟɧɵ.
ɋɩɢɫɨɤ ɥɢɬɟɪɚɬɭɪɵ
1. ɏɚɪɢɬɨɧɨɜ ȼ.ɉ. Ɏɭɧɞɚɦɟɧɬɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɦɟɯɚɧɢɤɢ ɠɢɞɤɨɫɬɢ
ɢ ɝɚɡɚ. – Ɇ.: ɆȽɌɍ ɢɦ. ɇ.ɗ. Ȼɚɭɦɚɧɚ, 2012. – 65 ɫ.
2. Dobek S. Fluid dynamics and the Navier – Stokes Equation, available at:
http://www.cs.umd.edu/~mount/Indep/Steven_Dobek/dobek-stable-fluid-final-2012.pdf.
3. Ɏɪɢɤ ɉ.Ƚ. Ɍɭɪɛɭɥɟɧɬɧɨɫɬɶ: ɦɨɞɟɥɢ ɢ ɩɨɞɯɨɞɵ. Ʉɭɪɫ ɥɟɤɰɢɣ. ɑ. I /
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ɉɨɥɭɱɟɧɨ 7.06.2013
Ʉɨɪɤɨɞɢɧɨɜ əɪɨɫɥɚɜ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ – ɚɫɩɢɪɚɧɬ, ɉɟɪɦɫɤɢɣ ɧɚɰɢɨɧɚɥɶɧɵɣ ɢɫɫɥɟɞɨɜɚɬɟɥɶɫɤɢɣ ɩɨɥɢɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ (614990, ɝ. ɉɟɪɦɶ,
Ʉɨɦɫɨɦɨɥɶɫɤɢɣ ɩɪ., 29, e-mail: svarogjk1989@rambler.ru).
Korkodinov Iaroslav Aleksandrovich – Postgraduate Student, Perm National Research Polytechnic University (614990, Perm, Komsomolsky av., 29,
e-mail: svarogjk1989@rambler.ru).
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