title: Ð¢ÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ ÑÑ ÐµÐ¼ Ð°Ð¿Ð¿ÑÐ¾ÐºÑÐ¸Ð¼Ð°ÑÐ¸Ð¸ ÐºÐ¾Ð½Ð²ÐµÐºÑÐ¸Ð²Ð½ÑÑ Ð¿Ð¾ÑÐ¾ÐºÐ¾Ð², Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐµÐ¼ÑÑ Ð´Ð»Ñ ÑÐ°ÑÑÐµÑÐ° Ð²ÑÑÐ¾ÐºÐ¾ÑÐºÐ¾ÑÐ¾ÑÑÐ½ÑÑ ÑÐµÑÐµÐ½Ð¸Ð¹
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Testing of numerical schemes for convective flux evaluation used for supersonic flows calculations
Creators: name: Ð¨Ð¸ÑÐºÐ¸Ð½Ð°, ÐÑÐ¸Ð½Ð°
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2020
Subjects: ÑÐ²ÐµÑÑ Ð·Ð²ÑÐºÐ¾Ð²ÑÐµ ÑÐµÑÐµÐ½Ð¸Ñ
ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÑÑ ÐµÐ¼Ñ Ð°Ð¿Ð¿ÑÐ¾ÐºÑÐ¸Ð¼Ð°ÑÐ¸Ð¸ ÐºÐ¾Ð½Ð²ÐµÐºÑÐ¸Ð²Ð½ÑÑ Ð¿Ð¾ÑÐ¾ÐºÐ¾Ð²
ÑÑ ÐµÐ¼Ñ ÑÐ°ÑÑÐµÐ¿Ð»ÐµÐ½Ð¸Ñ Ð²ÐµÐºÑÐ¾ÑÐ° Ð¿Ð¾ÑÐ¾ÐºÐ¾Ð²
"ÐºÐ°ÑÐ±ÑÐ½ÐºÑÐ»"-Ð½ÐµÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÑ
supersonic flows
numerical simulation
numerical schemes for convective flux evaluation
flux vector splitting
carbuncle instability

description: Ð Ð°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¾ÑÐ¾Ð±ÐµÐ½Ð½Ð¾ÑÑÐµÐ¹ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ ÑÑÐµÐ¼ 
Ð°Ð¿Ð¿ÑÐ¾ÐºÑÐ¸Ð¼Ð°ÑÐ¸Ð¸ ÐºÐ¾Ð½Ð²ÐµÐºÑÐ¸Ð²Ð½ÑÑ Ð¿Ð¾ÑÐ¾ÐºÐ¾Ð² Ð´Ð»Ñ ÑÐ°ÑÑÐµÑÐ° Ð²ÑÑÐ¾ÐºÐ¾ÑÐºÐ¾ÑÐ¾ÑÑÐ½ÑÑ ÑÐµÑÐµÐ½Ð¸Ð¹.
ÐÐ°Ð´Ð°ÑÐ¸ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ:
1. ÐÐ±Ð·Ð¾Ñ ÑÑÑÐµÑÑÐ²ÑÑÑÐ¸Ñ ÑÑÐµÐ¼ Ð°Ð¿Ð¿ÑÐ¾ÐºÑÐ¸Ð¼Ð°ÑÐ¸Ð¸ ÐºÐ¾Ð½Ð²ÐµÐºÑÐ¸Ð²Ð½ÑÑ Ð¿Ð¾ÑÐ¾ÐºÐ¾Ð², 
Ð¾ÑÐ±Ð¾Ñ ÑÑÐµÐ¼ Ð¸Ð· ÑÐµÐ¼ÐµÐ¹ÑÑÐ²Ð° AUSM.
2. Ð ÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ Ð½Ð° Ð±Ð°Ð·Ðµ ÑÑÑÐµÑÑÐ²ÑÑÑÐµÐ³Ð¾ Ð½ÐµÑÑÑÑÐºÑÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð½Ð¾Ð³Ð¾ ÐºÐ¾Ð½ÐµÑÐ½Ð¾-
Ð¾Ð±ÑÐµÐ¼Ð½Ð¾Ð³Ð¾ ÐºÐ¾Ð´Ð° SINF/Flag-S Ð²ÑÐ±ÑÐ°Ð½Ð½ÑÑ ÑÑÐµÐ¼ ÑÐµÐ¼ÐµÐ¹ÑÑÐ²Ð° AUSM.
3. ÐÑÐ±Ð¾Ñ ÑÐµÑÑÐ¾Ð²ÑÑ Ð·Ð°Ð´Ð°Ñ, Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ ÑÐ¾Ð¿Ð¾ÑÑÐ°Ð²Ð¸ÑÐµÐ»ÑÐ½ÑÑ ÑÐ°ÑÑÐµÑÐ¾Ð².
4. Ð¢ÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¿Ð¾Ð´Ð°Ð²Ð»ÐµÐ½Ð¸Ñ Â«ÐºÐ°ÑÐ±ÑÐ½ÐºÑÐ»Â»-Ð½ÐµÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸ Ð´Ð»Ñ 
ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ ÑÑÐµÐ¼.
Ð Ð°ÑÑÐµÑÑ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð»Ð¸ÑÑ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÑÐ²ÐµÑÑÐ·Ð²ÑÐºÐ¾Ð²Ð¾Ð¹ Ð²ÐµÑÐ²Ð¸ ÐºÐ¾Ð´Ð° SINF/Flag-S, 
ÑÐ°Ð·ÑÐ°Ð±Ð°ÑÑÐ²Ð°ÐµÐ¼Ð¾Ð³Ð¾ Ð² ÐÐÐÐ Ð¡ÐÐ±ÐÐ£, ÐºÐ¾ÑÐ¾ÑÑÐ¹ Ð±ÑÐ» Ð´Ð¾Ð¿Ð¾Ð»Ð½ÐµÐ½ Ð²Ð¾Ð·Ð¼Ð¾Ð¶Ð½Ð¾ÑÑÑÑ 
ÑÐ°ÑÑÐµÑÐ° Ð²ÐµÐºÑÐ¾ÑÐ° Ð¿Ð¾ÑÐ¾ÐºÐ¾Ð² Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð²ÑÐ±ÑÐ°Ð½Ð½ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¾Ð±Ð·Ð¾ÑÐ° Ð»Ð¸ÑÐµÑÐ°ÑÑÑÑ ÑÑÐµÐ¼ 
(AUSM, AUSM+, AUSM+-up, AUSMD Ð¸ SLAU). Ð ÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½ Ð¼ÐµÑÐ¾Ð´Ð° Ð§ÐµÐ½Ð° (Chen et
al. 2018) Ð´Ð»Ñ Ð¿Ð¾Ð´Ð°Ð²Ð»ÐµÐ½Ð¸Ñ ÐºÐ°ÑÐ±ÑÐ½ÐºÑÐ» Ð½ÐµÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐ¹ Ð±ÑÐ» ÑÐ°ÐºÐ¶Ðµ
Ð¾Ð±Ð¾Ð±ÑÐµÐ½ Ð½Ð° ÑÑÐµÑÐ¼ÐµÑÐ½ÑÐ¹ ÑÐ»ÑÑÐ°Ð¹.
Ð ÐµÐ·ÑÐ»ÑÑÐ°ÑÑ ÑÐ°ÑÑÐµÑÐ¾Ð² Ð¿Ð¾ÐºÐ°Ð·Ð°Ð»Ð¸, ÑÑÐ¾ ÑÑÐµÐ¼Ñ ÑÐµÐ¼ÐµÐ¹ÑÑÐ²Ð° AUSM Ð² ÑÐµÐ»Ð¾Ð¼ Ð´Ð°ÑÑ 
ÑÑÐ¾Ð¶Ð¸Ðµ ÑÐ¾ ÑÑÐµÐ¼Ð°Ð¼Ð¸ Ð³Ð¾Ð´ÑÐ½Ð¾Ð²ÑÐºÐ¾Ð³Ð¾ ÑÐ¸Ð¿Ð° ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ. Ð¡ÑÐµÐ¼Ñ SLAU Ð¸ AUSM+
-up Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÑÑ Ð¿Ð¾Ð»ÑÑÐ¸ÑÑ Ð±Ð¾Ð»ÐµÐµ Ð³Ð»Ð°Ð´ÐºÐ¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð² Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ð´Ð¾Ð·Ð²ÑÐºÐ¾Ð²Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ, 
ÑÑÐµÐ¼Ð° AUSMD Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÐµÑ Ð¸Ð·Ð±ÐµÐ¶Ð°ÑÑ Ð²Ð¾Ð·Ð½Ð¸ÐºÐ½Ð¾Ð²ÐµÐ½Ð¸Ñ Ð¾ÑÑÐ¸Ð»Ð»ÑÑÐ¸Ð¹ Ð²Ð±Ð»Ð¸Ð·Ð¸ ÑÐ¾ÑÐµÐº 
Ð¸Ð·Ð»Ð¾Ð¼Ð° Ð¾Ð±ÑÐµÐºÐ°ÐµÐ¼Ð¾Ð¹ Ð¿Ð¾Ð²ÐµÑÑÐ½Ð¾ÑÑÐ¸. Ð¢ÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¿Ð¾Ð´Ð°Ð²Ð»ÐµÐ½Ð¸Ðµ ÐºÐ°ÑÐ±ÑÐ½ÐºÑÐ»-Ð½ÐµÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸ Ð½Ð° Ð¿ÑÐ¸Ð¼ÐµÑÐµ Ð³Ð¸Ð¿ÐµÑÐ·Ð²ÑÐºÐ¾Ð²Ð¾Ð³Ð¾ Ð¾Ð±ÑÐµÐºÐ°Ð½Ð¸Ñ ÑÐ¸Ð»Ð¸Ð½Ð´ÑÐ° (Ð² 2D Ð¸ 3D
Ð¿Ð¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐµ) Ð¿Ð¾ÐºÐ°Ð·Ð°Ð»Ð¾, ÑÑÐ¾ Ð´Ð»Ñ Ð¿Ð¾Ð´Ð°Ð²Ð»ÐµÐ½Ð¸Ñ Ð½ÐµÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸ Ð² ÑÐ»ÑÑÐ°Ðµ Ð´Ð²ÑÐ¼ÐµÑÐ½ÑÑ 
Ð·Ð°Ð´Ð°Ñ ÑÐ²Ð½Ð¾Ðµ Ð¿ÑÐµÐ¸Ð¼ÑÑÐµÑÑÐ²Ð¾ Ð¸Ð¼ÐµÑÑ ÑÑÐµÐ¼Ñ Ñ Ð´Ð¾Ð±Ð°Ð²Ð»ÐµÐ½Ð¸ÐµÐ¼ Ð¸ÑÐºÑÑÑÑÐ²ÐµÐ½Ð½Ð¾Ð¹ Ð²ÑÐ·ÐºÐ¾ÑÑÐ¸, 
Ð° Ð² ÑÐ»ÑÑÐ°Ðµ ÑÑÐµÑÐ¼ÐµÑÐ½ÑÑ Ð·Ð°Ð´Ð°Ñ â Ð¼ÐµÑÐ¾Ð´ Ð§ÐµÐ½Ð°.
description: The given work is devoted to studying the features of numerical schemes for 
convective flux evaluation in a supersonic flow calculation.
Research objectives:
1. Review of the existing numerical schemes for convective flux evaluation, 
selection of AUSM-family schemes.
2. Implementation of the selected AUSM-family schemes to the existing 
unstructured finite-volume SINF/Flag-S code.
3. Selection of test cases, performing comparative calculations. 
4. Testing of different approaches for curing the carbuncle instability in 
different schemes
The calculations were performed using the supersonic branch of SINF/Flag-S 
code developed at IAMM SPbPU, which was supplied with different flux schemes
evaluation selected during the literature review (AUSM, AUSM+
, AUSM+-up, AUSMD Ð¸ SLAU). Chen method (Chen et al. 2018) generalized by the author for the three-dimensional case was implemented to cure carbuncle instability.
The calculation results have shown that AUSM-family schemes give results 
similar to the Godunov-type schemes. SLAU and AUSM+-up schemes allow to give 
more smooth solutions in the subsonic flow region, AUSMD scheme allows to avoid oscillations in the vicinity of the fractures of a streamlined surface. Testing of methods
for curing carbuncle instability in the case of hypersonic flow past a cylinder (in 2D 
and 3D formulation) has shown that schemes with artificial viscosity have an obvious 
advantage to cure instability at 2D cases, so as Chen method at 3D cases.