Development of a modified Runge-Kutta scheme with TVD limiters for the ideal 1-D MHD equations

Harada, Shigeki; Augustinus, Justin; Hoffmann, Klaus A.; Agarwal, Ramesh K.

doi:10.2514/6.1997-2090

Cited by 12 publications

(5 citation statements)

References 6 publications

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“…for low m R approximation Second-order central difference approximations have been utilized for the convective and diffusion terms according to the procedure outlined in Ref. 13. The post-processing stage is the last stage of computation and consists of correcting the last calculated unknown vector.…”

Section: Methodsmentioning

confidence: 99%

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Effect of Applied Magnetic Field on Shock Boundary Layer Interaction

Khan

Martin

2012

50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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The governing magneto-hydrodynamic (MHD) equations contain classical fluid dynamics equations along with coupled Maxwell's magnetic induction equations. These equations model both advection and diffusion effects of electromagnetic field. However, available literature indicates that some previous investigations neglect the diffusion of magnetic field and considered only ideal MHD equations for modeling a typical MHD problem. In this work, the effects of magnetic field diffusion term also known as viscous magnetic term have been investigated over flow structure. Low magnetic Reynolds number approximation and ideal full MHD set of equations have been considered and solved using a four-stage modified Runge-Kutta scheme augmented with the Davis-Yee symmetric Total Variation Diminishing model in post-processing stage. Results obtained from viscous and ideal flow computations without applied magnetic field have been found in close agreement. However, results obtained from viscous MHD and ideal MHD computations substantially disagree from each other which indicate that the effect of magnetic diffusion term on overall flow structure is significant. I = identity tensor J = Jacobian of transformation

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Section: Methodsmentioning

confidence: 99%

“…The TVD model is based on the eigenstructure of the convective flux Jacobian matrices. The eigenstructure and numerical method for general ideal MHD equations are provided in Ref 13…”

mentioning

confidence: 99%

Effect of Applied Magnetic Field on Shock Boundary Layer Interaction

Khan

Martin

2012

50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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“…At each iteration level, the solution is enhanced with a Total Variation Diminishing (TVD) model in a post-process stage to stabilize it after modified Fourth-Order Runge-Kutta was applied to solve problems [15]. The modified Runge-Kutta scheme is then presented, followed by a description of the Davis-Yee symmetric TVD model in the methodology.…”

Section: Introductionmentioning

confidence: 99%

The Influence of Limiters on Davis-Yee and Harten-Yee TVD Schemes

Alimuddin

Ismail²,

Mohammed³

et al. 2022

CFDL

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TVD schemes have many selections of limiters, but the recommendation of limiters for specific cases is not available in the literature. This study focuses on incorporating two flux limiters as the extension of the TVD schemes proposed by Harten-Yee and Davis-Yee and extends the test case on external flows, blunt-body. The method used in this study is Harten-Yee Upwind TVD and Davis-Yee Symmetric TVD scheme with different limiter functions to simulate cases for two-dimensional compressible flow. The results show that all the limiter functions can capture shock waves when the flow passes through the geometry at Mach number . The flow features such as bow shock, oblique shock, shock wave reflection, interaction, and expansion wave can all be captured in the case of the bump in a channel and wedge. While in the case of the external supersonic flow passing through the blunt-body, the presence of a bow shock was captured. We discovered that Davis-Yee limiter number 2 performs significantly better than other proposed Davis-Yee and Harten-Yee limiters for the case in this study. Therefore, the Davis-Yee Upwind TVD method is recommended to be applied for the identical case on the expansion of this study

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“…Due to the mathematical properties of this hyperbolic system of the ideal eight-wave MHD equations mentioned earlier, the introduction of TVD scheme to solve this system has attracted extensive attentions [10][11][12][13]. However, it is difficult since it requires the determination of the eigenvalues and eigenvectors for the system of equations, which has been proved to be not strictly with non-genuine nonlinearity (non-convexity) of some of the local wave fields.…”

mentioning

confidence: 97%

“…Dai and Woodward developed an approximate Riemann solver for multi-dimensional ideal MHD equations [10]. Then they introduced the piecwise parabolic method and a simple Riemann solver for MHD equations [11]. In ref.…”

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confidence: 99%

Simulation of three-dimensional nonideal MHD flow at high magnetic Reynolds number

Hao

2010

Sci. China Ser. E-Technol. Sci.

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A conservative TVD scheme is adopted to solve the equations governing the three-dimensional flow of a nonideal compressible conducting fluid in a magnetic field. The eight-wave equations for magnetohydrodynamics (MHD) are proved to be a non-strict hyperbolic system, therefore it is difficult to develop its eigenstructure. Powell developed a new set of equations which cannot be numerically simulated by conservative TVD scheme directly due to its non-conservative form. A conservative TVD scheme augmented with a new set of eigenvectors is proposed in the paper. To validate this scheme, 1-D MHD shock tube, unsteady MHD Rayleigh problem and steady MHD Hartmann problem for different flow conditions are simulated. The simulated results are in good agreement with the existing analytical results. So this scheme can be used to effectively simulate high-conductivity fluids such as cosmic MHD problem and hypersonic MHD flow over a blunt body, etc. magnetohydrodynamics (MHD), numerical simulation, TVD, MHD Rayleigh flow, MHD Hartmann flow Citation:Lu H Y, Lee C H. Simulation of three-dimensional nonideal MHD flow at highThe physical mechanisms in space and near-space physics are involved in the interaction of aerodynamics and electromagnetic dynamics. For instance, investigations have indicated that the strength of local magnetic field on the solar surface is of the order of kilogauss, and the elements of gas in the vicinity of solar surface are almost fully ionized. Therefore, the physical activities on solar surface are controlled by the interactions of strong magnetic field and plasma [1-3]. Renewed scientific interest has arisen all over the world for the potential application of MHD processes for the advancement of flight vehicles. In the early nineties of the last century, Russians proposed a new concept of combined propulsion system which was composed of a turbine and an MHD enhanced scramjet for hypersonic cruised aircraft. Later, they established the AJAX program [4]. The most innovative and key issue about this program was the utilization of MHD system to develop an MHD bypass scramjet technology, and to control the inlet of the propulsion system. According to the AJAX program, the inlet control can be furnished by an MHD device through the modification of the external flow upstream of the scramjet inlet. It helps in improving the performance of engine at off-design conditions [3][4][5][6]. The governing equations for MHD include the conservation laws of fluid dynamics augmented by terms including magnetic effects and Maxwell's equation of electrodynamics. It forms an eight-wave model demonstrating the strong interaction between the flow field and electromagnetic field. This system of equations is not strictly hyperbolic since eigenvalues of the system may coincide, causing the system to be nonconvex [7,8]. Due to these mathematical characteristics, the algorithms implemented on Euler equations do not guarantee the solution of the eight-wave equations.

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Development of a modified Runge-Kutta scheme with TVD limiters for the ideal 1-D MHD equations

Cited by 12 publications

References 6 publications

Effect of Applied Magnetic Field on Shock Boundary Layer Interaction

Effect of Applied Magnetic Field on Shock Boundary Layer Interaction

The Influence of Limiters on Davis-Yee and Harten-Yee TVD Schemes

Simulation of three-dimensional nonideal MHD flow at high magnetic Reynolds number

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