An accurate and robust AUSM-family scheme on two-dimensional triangular grids

Phongthanapanich, Sutthisak

doi:10.1007/s00193-019-00892-5

Cited by 13 publications

(20 citation statements)

References 30 publications

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“…Generally, the HLL scheme and its variants are often used to stabilize approximate Riemann solvers; 25,29,32 however, the author found the schemes to be too dissipative and may deteriorate the accuracy of the contact discontinuity and the shear wave profiles. Instead, in this article, the AUSMV + scheme 18 is modified and then applied to construct an accurate hybrid Roe scheme, namely the Roe + scheme. In the modified AUSMV+ scheme, the numerical flux function at the cell interface is written in a general form 37 as

\begin{align} {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+}} & = m_{1 false/ 2}^{+} {boldF}_{L}^{normalAUSM {normalV}^{normal+}} + m_{1 false/ 2}^{prefix-} {boldF}_{R}^{normalAUSM {normalV}^{normal+}} + {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normalp false)}, \\ = {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normalc false)} + {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normald false)} + {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normalp false)}, \end{align}

where

F_{1 / 2}^{AUSM V^{+}, (c)} = \frac{1}{2} m_{1 / 2} false(F_{L}^{AUSM V^{+}} +

…”

Section: Governing Equations Finite Volume Discretization and The Rmentioning

confidence: 99%

“…The split Mach numbers (

M_{L / R}^{(4, δ) \pm}

) and the split pressures (

p_{L / R}^{(5, ϵ) \pm}

) are defined as follows 18

M_{L / R}^{(4, δ) \pm} (M) = \{\begin{matrix} prefix\pm \frac{1}{4} {false(M prefix\pm 1 false)}^{2} prefix\pm δ {false(M^{2} prefix- 1 false)}^{2} & false| M false| < 1 \\ \frac{1}{2} false(M prefix\pm false| M false| false) & otherwise \end{matrix},

p_{L / R}^{(5, ϵ) \pm} (M) = \{\begin{matrix} \frac{1}{4} {false(M prefix\pm 1 false)}^{2} false(2 \mp M false) prefix\pm ϵ M {false(M^{2} prefix- 1 false)}^{2} & false| M false| < 1 \\ \frac{1}{2} false(1 prefix\pm s i g n false(M false) false) & otherwise \end{matrix},

where

δ = 1 / 8

and

ϵ = 3 / 16

.…”

Section: Governing Equations Finite Volume Discretization and The Rmentioning

confidence: 99%

“…In this section, the linearized analysis 9,18 is applied to evaluate the numerical perturbation sensitivity of the Roe + scheme according to the odd‐even decoupling numerical instability mechanism. Consider a uniform flow with normalized values of

ρ_{0} = 1

, u 0 ≠ 0, v 0 = 0, and p 0 = 1 with their corresponding perturbation quantities

(\hat{ρ}, \hat{u}, \hat{p})

.…”

Section: Linearized Analysismentioning

confidence: 99%

“…Also, there is a circulation behind the detached shock wave near the stagnation line, which perturbs the solution and may induce some oscillations, as shown by the streamlines in Figure 1 (right). Similarly, AUSM family of schemes also admit numerical shock anomalies for this problem 15‐18 …”

Section: Introductionmentioning

confidence: 96%

“…The ADC and SWM techniques were also adopted to cure numerical shock instability in the HLLEM Riemann solver 34 . Recently, the new hybrid idea has been successfully implemented to improve both stability and accuracy of the AUSMDV scheme 35 with a less amount of numerical dissipation 18 . The scheme successfully provided a carbuncle‐free solution for several problems on triangular grids.…”

Section: Introductionmentioning

confidence: 99%

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A stable hybrid Roe scheme on triangular grids

Phongthanapanich

2020

Numerical Methods in Fluids

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Numerical shock instability is a common problem for shock‐capturing methods that try to resolve contact and shear waves with minimal diffusion. Most flux‐difference splitting and the AUSM family of schemes produce the carbuncle phenomenon on both structured and unstructured grids. The original Roe scheme is well known to generate shock anomalies and can lead to nonentropic weak solutions to the Euler equations. A simple and robust approach for healing these numerical instabilities is to apply the hybrid technique incorporated with an efficient weighting switch function to control the amount of dissipation in the vicinity of shock waves. This article proposes a simple, robust, and accurate hybrid Roe scheme (Roe+ scheme) by hybridizing the Roe scheme and the modified AUSMV+ scheme. A new normalized pressure/density‐based weighting switch function is proposed and applied to the scheme to minimize the numerical dissipation and maintain the robustness of the hybridization. The linearized discrete analysis is performed to evaluate the proposed scheme according to the perturbation damping mechanism of an odd–even decoupling problem. The resulting recursive equations indicate that the hybridized mechanism damps all perturbations effectively. Finally, several numerical examples demonstrated that the Roe+ scheme provides an accurate, robust, and carbuncle‐free solution on both structured and unstructured triangular grids.

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\begin{align} {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+}} & = m_{1 false/ 2}^{+} {boldF}_{L}^{normalAUSM {normalV}^{normal+}} + m_{1 false/ 2}^{prefix-} {boldF}_{R}^{normalAUSM {normalV}^{normal+}} + {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normalp false)}, \\ = {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normalc false)} + {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normald false)} + {boldF}_{1 false/ 2}^{normalAUSM {normalV}^{normal+} normal, false(normalp false)}, \end{align}

where

F_{1 / 2}^{AUSM V^{+}, (c)} = \frac{1}{2} m_{1 / 2} false(F_{L}^{AUSM V^{+}} +

…”

Section: Governing Equations Finite Volume Discretization and The Rmentioning

confidence: 99%

“…The split Mach numbers (

M_{L / R}^{(4, δ) \pm}

) and the split pressures (

p_{L / R}^{(5, ϵ) \pm}

) are defined as follows 18

M_{L / R}^{(4, δ) \pm} (M) = \{\begin{matrix} prefix\pm \frac{1}{4} {false(M prefix\pm 1 false)}^{2} prefix\pm δ {false(M^{2} prefix- 1 false)}^{2} & false| M false| < 1 \\ \frac{1}{2} false(M prefix\pm false| M false| false) & otherwise \end{matrix},

p_{L / R}^{(5, ϵ) \pm} (M) = \{\begin{matrix} \frac{1}{4} {false(M prefix\pm 1 false)}^{2} false(2 \mp M false) prefix\pm ϵ M {false(M^{2} prefix- 1 false)}^{2} & false| M false| < 1 \\ \frac{1}{2} false(1 prefix\pm s i g n false(M false) false) & otherwise \end{matrix},

where

δ = 1 / 8

and

ϵ = 3 / 16

.…”

Section: Governing Equations Finite Volume Discretization and The Rmentioning

confidence: 99%

ρ_{0} = 1

, u 0 ≠ 0, v 0 = 0, and p 0 = 1 with their corresponding perturbation quantities

(\hat{ρ}, \hat{u}, \hat{p})

.…”

Section: Linearized Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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A stable hybrid Roe scheme on triangular grids

Phongthanapanich

2020

Numerical Methods in Fluids

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An improvement of the AUSMDV$$^{+}$$ scheme on unstructured grids

Phongthanapanich

Matthujak

2021

Shock Waves

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It is well known that many contact-and shear-preserving approximate Riemann solvers admit various forms of numerical shock anomalies and numerical shock instabilities such as the carbuncle phenomenon, sonic glitch, and post-shock wave oscillations. Recently, the AUSMDV + scheme was developed to improve the numerical stability and accuracy of the AUSMDV scheme. For the AUSMDV + scheme, the AUSMD + and the AUSMV + schemes were introduced as the accurate and the moderately dissipative schemes, respectively. In this paper, the AUSMDV2 + scheme is presented as an improved version of the AUSMDV + scheme. In the AUSMDV2 + scheme, a newly proposed mass flux function is used for both the AUSMD + and the AUSMV + schemes. Additionally, a new normalized weighting function is proposed to control the amount of dissipation. Many tests show that the proposed scheme's stability and accuracy are easily controlled by varying the constant parameter κ between 0.1 and 1.0. Furthermore, the dissipation mechanism of the scheme is studied by employing linearized discrete analysis on the odd-even decoupling problem. The resulting recursive equations show that the proposed scheme can effectively damp out all perturbations over time. Finally, the scheme's numerical stability and accuracy are examined in several test cases on structured and unstructured grids with the first-and second-order accurate methods. The results show that the proposed scheme is stable, accurate, and suitable for solving a wide range of high-speed compressible flow problems.

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