^{1}

^{1}

The fifth-order monotonicity-preserving (MP5) scheme is an accurate and low dissipative numerical method. As a finite-volume method, MP5 adopts the Roe-flux scheme for solving the numerical flux in the compressible Euler equation. However, due to the deficiency of the MP limiter and Roe-flux in maintaining positive density and pressure, the calculation could fail in cases of extreme flow involving small values of density and pressure. In this study, to overcome such a limitation but still to achieve a high-accuracy of MP5, we propose a hybrid flux method: the Roe-flux is used in the global computational domain, but the first-order Lax-Friedrich (LF)-flux is adopted only for trouble grids. The numerical results of shock-tube and complicated interaction problems indicate that the present scheme is more accurate at discontinuities and local extrema compared to the previous scheme, maintaining positive density and pressure values. For two-dimensional applications, a supersonic jet is explored with different Mach numbers and temperature conditions. As a result, small vortices induced by the shear layer can be clearly captured by the proposed scheme. Furthermore, a simulation was successfully conducted without blow-up of calculation even in the extreme jet flow condition.

The usage of an efficient high-order shock-capturing scheme is important in the large eddy simulation (LES)/direct numerical simulation (DNS) particularly in supersonic flow conditions because of the requirements to capture a sharp discontinuity and simultaneously to resolve small structures of turbulence. In addition, supersonic flow sometimes causes very low density and pressure values that have the potential to induce failure of calculation. Therefore, the numerical robustness is an additional requirement for a stable simulation.

The well-known essentially nonoscillatory (ENO) [

On the other hand, following several classical limiting strategies [

However, it was found that the calculation using MP5 scheme could blow up in specific situations when the flow involves very low density or pressure. On the region of local minimum value, nonphysical negative density or pressure can be generated during simulation. He et al. [

In this study, we try to develop a new efficient MP scheme to solve the negative density and pressure problem and simultaneously to achieve the high-accuracy. The key strategy of present scheme is the hybridization of Roe- and the first-order LF-fluxes depending on local density and pressure state. Initially, the solution in the whole flow-field is obtained using the original MP5 scheme; however, only for the local trouble region detected by criteria, the first-order LF-flux is applied to modify the numerical flux. The first-order never introduces the nonphysical oscillation; therefore, the additional TVD condition used in MP5-R is not necessary.

The order of accuracy test is conducted to assess a global level of error and convergence in a different number of grids. After that, the shock-tube and complicated interaction problems with normal/extreme conditions are carried out to evaluate the performances of schemes in various situations. In normal conditions, we compare the accuracy between MP5-R and MP5-Hyb scheme. However, in the extreme conditions involving low density and pressure, the robustness of scheme can be investigated as well. For the application, supersonic jet flow simulations are considered, which require the same numerical properties of high-accuracy and robustness. Initially, the position of initial vortex induced by a cold jet with Mach 1.56 was investigated to verify numerical solutions and to compare the resolving performance for small flow structures. In addition, a Mach 3.0 jet with an extremely high temperature of 2,100K was considered to ensure the robustness of scheme in such tough condition. Note that the solutions of original MP5 just before blow-up of calculation are presented as well in tests involving low density and pressure.

Consider the methods for one-dimensional linear advection equation with speed of wave

For the moment, assume that positive wave speed

For accurate calculations, the high-order interpolation for the numerical flux

The procedure of MP scheme is separated in two parts: (i) high-order interpolation and (ii) application of MP limiter (or constraints) for shock-capturing. The former procedure enhances the order of accuracy by obtaining the cell-interface flux using neighboring grids. Let us consider the cell-averaged variable on an arbitrary grid as

The minmod function is nothing but selecting the minimum if signs of terms are same; otherwise, it gives zero. The definition of others and details can be found in [

The error characteristics of the scheme can be investigated by Fourier analysis (or modified wavenumber analysis). The relation determining the spatial-accuracy can be obtained by combining (

Then Fourier transform is adapted to estimate spectral properties for wide range of wavenumbers. Since we can simply obtain

The wavenumber (

Error characteristics: (a) dispersion; (b) dissipation.

Consider the one-dimensional compressible Euler equation

The original fifth-order monotonicity-preserving (MP5) scheme proposed by Suresh and Huynh [

The procedure of MP5 scheme can be summarized as follows: (i) calculate the characteristic variable shown in (

The notation

To overcome the aforementioned limitation of original scheme, MP5-R [

The MP5-R scheme based on the LF-flux can be summarized as follows: (i) Calculate the characteristic variable shown in (

As introduced in the previous section, MP5 and MP5-R schemes have different numerical characteristics by adopting a different flux scheme. The former is quite accurate due to using Roe-flux; however, it is not robust in low density or pressure conditions. In contrast, the latter is robust; however, the accuracy is degenerated from the original MP5 scheme due to dissipative LF-flux.

However, in practice, the numerical problem owing to low density or pressure does not appear in the entire computational domain. Therefore, on the same MP scheme framework, we suggest the efficient method as follows: use the LF-flux only for trouble grids to modify the negative density and pressure; otherwise, apply the Roe-flux to obtain accurate solution. By implementing such a hybrid-concept, we try to achieve both of the respective advantages of the MP5 and MP5-R schemes.

Let us explain the detailed methodology of the hybrid flux method. If we know the solution at

Indeed, we do not need to recalculate numerical fluxes for all grids because solutions are normal except the trouble grids. Therefore, we call the solutions at the

Here, the candidates for the conservative and flux terms are changed so that

Let us explain the positivity of density and pressure [

We can see that

Equation (

For all these reasons, on a local trouble grid,

Therefore, we can calculate a new local residue as

Figure

Schematic for hybrid flux method.

In general, the multistage Runge-Kutta method is used for the time-integration. In this case, the modification procedures for negative density or pressure should be applied in every internal stages of Runge-Kutta method. Figure

Flow chart for hybrid flux method with

Implementation in a multidimensional case is quite straightforward. The procedures for flux modification are adopted in direction

In this section, several problems related to the normal and extreme flow conditions have been investigated for accuracy and robustness of the present method. Classical test problems in one-dimensional cases and supersonic jet flow simulations in asymmetrical two-dimensional domain are performed, which is the main application of the present scheme. For all the following computations, a third-order TVD Runge-Kutta time-integration method [

A typical way to verify the order of accuracy and convergence of numerical scheme is to simulate the advection of wave and compare the standard errors of solutions in different number of grid situations. For the compressible Euler equation, on the domain

_{2}

Number of grids | MP5-R (He et al.) | MP5-Hyb (Present) | ||
---|---|---|---|---|

| Slope | | Slope | |

15 | 2.04338E-04 | - | 9.29329E-05 | - |

30 | 6.53404E-06 | 4.97 | 2.96709E-06 | 4.97 |

60 | 2.04595E-07 | 5.00 | 9.32070E-08 | 4.99 |

120 | 6.38685E-09 | 5.00 | 2.91671E-09 | 5.00 |

240 | 2.00031E-10 | 4.99 | 9.18235E-11 | 4.99 |

Shock-tube problem demonstrates the dynamics of two different state of gas generating a shock-wave, contact-discontinuity, and expansion fan. Through this test, we can compare the accuracy of discontinuous solution. We consider two different initial conditions of normal and extreme cases, and all numerical solutions were compared with the analytical solution.

This is a normal condition case of shock-tube problem. On the domain

Sod shock-tube problem on 80 grids (density).

Sod shock-tube problem on coarse and fine grids (density): (a) coarse, N=40; (b) fine, N=120.

The LeBlanc problem [

LeBlanc problem on 400 grids: (a) density; (b) pressure.

LeBlanc problem on coarse and fine grids (density): (a) coarse, N=200; (b) fine, N=600.

In this subsection, we consider two different interaction problems. One is shock and wave interaction, and the other is two shock waves interaction. In these problems, there is no analytical solution because the interaction behaviors are complicated and highly nonlinear. Therefore, we produce a reference solution using the WENO scheme [

The simulation for shock-entropy wave interaction problem [

A shock-wave of Mach=3.0 interacts with the linear entropy waves. A zero-gradient boundary condition was imposed on both sides, and total 200 grids were used. The results were compared at the simulation time of t=1.8. For the comparison, the solution obtained by WENO scheme with 2,000 grids was used as a reference solution. As can be seen in Figure

Shock-entropy wave interaction problem on 200 grids (density).

Shock-entropy wave interaction problem on coarse and fine grids (density): (upper) coarse, N=150; (lower) fine, N=250.

We considered the two-blast-wave interaction problem [

On the domain ~~.~~

Figure

Two-blast wave interaction problem on 400 grids: (a) density; (b) pressure.

Figure

Two-blast wave interaction problem on coarse and fine grids (density): (a) coarse, N=200; (b) fine, N=400.

The main purpose of developing the present scheme is to solve the supersonic jet flow that requires the numerical capabilities we have investigated because (1) the supersonic jet flow contains the shock-wave and shear layer to be solved using the method of high-accuracy for a good quality of solution, and (2) low density and pressure appear at the core of vortices induced by high-speed jet flows.

First, we assess that MP schemes successfully simulate the supersonic jet flow by comparing with experiment data, and we compare the detailed flow structures. After that, we evaluated the robustness of present scheme in a high-speed supersonic jet case which generates the vortices with very low density and pressure. The governing equation is the two-dimensional axisymmetric compressible Euler equation,

We tried to validate and evaluate the performance of MP schemes for the supersonic jet flow. Therefore, the pulse jet problem [

Boundary conditions for axisymmetric supersonic jet.

Initial structure of the supersonic jet.

Distance of the initial vortex position: (a) axial coordinate; (b) radial coordinate.

Figure

Instantaneous density contour at

The present scheme was applied for a high-Mach number and high-temperature supersonic jet simulation to investigate the robustness of scheme. When jet flow speed is high, low density and pressure can be observed at the core of initial vortex and small vortices induced by shear layer. This can lead to negative density or pressure value, and thereby, the calculation may fail unless an adequate numerical treatment is adopted. The computational domain has been changed a little from the previous problem. The domain size in x-direction was doubled owing to higher jet flow speed, and a stretching grid was generated to make the actual computing environment (

Figure

Instantaneous pressure contours when the calculation of MP5 fails at the initial vortex (

Instantaneous pressure contours when the calculation of MP5 fails at the shear layer (

Instantaneous density and pressure contours of fully developed jet calculated using MP5-Hyb (

In this study, MP5-Hyb scheme was proposed for robust and accurate simulation in supersonic flow conditions particularly involving low density/pressure values. The original MP5 scheme based on the Roe-flux is an accurate shock-capturing scheme; however, the calculation could fail when very low density/pressure appears. MP5-R scheme was proposed to overcome this limitation by adopting the LF-flux and additional TVD condition. However, the LF-flux scheme is more dissipative than the Roe-flux scheme; therefore, the overall accuracy is degenerated from the original MP5.

To overcome such limitations but still achieve a high-accuracy of MP5, we have proposed a hybrid flux method. The strategy is that the Roe-flux is used in the global computational domain, but the first-order LF-flux is adopted only for trouble grids resulting in negative density or pressure. The first-order LF never introduces numerical oscillation, and it has a positivity-preserving capability itself; therefore, the additional treatment such as TVD condition is not necessary.

The results of shock-tube and complicated interaction problems proved that the MP5-Hyb scheme has the robustness without resulting in calculation failure. In addition, it is beneficial to capture the shock waves and local extrema. The MP5-Hyb scheme was applied to simulate supersonic jet flows in different conditions. As a result, the MP5-Hyb scheme has better accuracy than the MP5-R scheme for small-scale structures of vortices. In addition, the MP5-Hyb scheme can maintain a stable calculation even with the extreme jet flow condition whereas the MP5 scheme failed. Therefore, we conclude that MP5-Hyb scheme can be an efficient tool to simulate the supersonic flow involving low density or pressure.

Left eigenvector of Jacobian matrix

All data can be accessed in the Numerical Validations section of this article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by Space Core Technology Development Program through the National Research Foundation of Korea (NRF), which is funded by the Ministry of Science, ICT & Future Planning