# Analysis of ENO scheme slope limiters

## Keywords:

supersonic flow, multicomponent gas, ENO scheme, limiters, Navier- Stokes equations.## Abstract

The interaction of the three-dimensional supersonic turbulent air fl ow with the transversely injected hydrogen jet is numerically simulated by solving the Reynold s-averaged NavierStokes equations using the ENO (essentially non-oscillatory) scheme of the third order of accuracy. Since the choice of the limiter functions significantly a ffects the accuracy of the problem, preliminary test problems are solved to validate numerical method and choose optimal slope limiter. Analysis of the different variati ons of the limiter functions for developed algorithm was done to define optimal function produced the smallest solution spreading. Then, the effect of the different variations of the limiter functions for developed algorithm on the mixing layer is studied since the exact calculation of the mass concentration spreading is important issue in combustion pr oblem modeling. Also, by numerical experiments the effect of the slope limiters on the shock-wave structure formation is studied. It was shown that choice of some limiters can result in excessive expansion of the mixing layer, that is important issue in numeri cal modeling of scramjet engine. As result, the optimal limiter function which produces the sm allest spread of solution for the spatial problem was defined. Also the mechanism o f the formation of vortices in front of the injected jet and behind that is studied.## References

[1] Adams N.A., Shariff K. A High-Resolution Hybrid Compact-ENO Scheme for Shock-Turbulence Interaction Problems // Journal of Computational Physics, Volume 127, Issue 1, August 1996, pp. 27–51

[2] Sun De-chuan, Hu Chun-bo, Cai Ti-min Computation of Supersonic Turbulent Flowfield with Transverse Injection // Applied Mathematics and Mechanics. English Edition. Vol.23, No 1, Jan 2002. pp. 107-113

[3] Amano R.S., Sun D. Numerical Simulation of Supersonic Flowfield with Secondary Injection // The 24th Congress of ICAS, September 2004, Yokohama, pp. 1–8

[4] Bruel P., Naimanova A. Zh. Computation of the normal injection of a hydrogen jet into a supersonic air flow // Thermophysics and Aeromechanics, Vol. 17 issue 4 December 2010. pp. 531–542

[5] A.KulikovskiiA.G., Pogorelov N.V., Semenov A.Yu.Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Moscow:Fizmatlit, 2001. 656 p.

[6] Harten A. High resolution schemes for hyperbolic conservation laws // J. Comp. Phys., Vol. 49, 1983, pp. 357-393

[7] Harten A., Engquist B., Osher S., Chakravarthy S. Uniformly high-order accurate essentially non-oscillatory schemes III // J. Comput. Phys., Vol. 71, 1987, pp. 231-303

[8] Shu C., Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes // J. Comp. Phys. 1988. Vol. 77. pp. 439–471.

[9] Shu C., Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes, II // J. Comp. Phys. 1989. Vol. 83. pp. 32–78.

[10] Berger M.J., Aftosmis M.J., Murman S.E. Analysis of slope limiters on irregular grids // In 43rd AIAA Aerospace Sciences Meeting, Reno, NV , 2005. Paper AIAA 2005-0490. pp.22

[11] Poinsot T.J., Lele S.K. Boundary conditions for direct simulation of compressible viscous flows // J. Comp. Phys. 1992. Vol. 101. pp. 104–129.

[12] Sweby, P.K. High resolution schemes using flux-limiters for hyperbolic conservation laws // SIAM J. Num. Anal. 1984. Vol. 21, No 5. pp. 995–1011

[13] Danaila I., Joly P., Kaber S.M., Postel M. Introduction to scientific computing: twelveprojects solved with MATLAB // Springer, 2007. 308 p.

[14] Harten A. The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes // Math. Comp. 1978. Vol. 32. pp. 363–389 .

[2] Sun De-chuan, Hu Chun-bo, Cai Ti-min Computation of Supersonic Turbulent Flowfield with Transverse Injection // Applied Mathematics and Mechanics. English Edition. Vol.23, No 1, Jan 2002. pp. 107-113

[3] Amano R.S., Sun D. Numerical Simulation of Supersonic Flowfield with Secondary Injection // The 24th Congress of ICAS, September 2004, Yokohama, pp. 1–8

[4] Bruel P., Naimanova A. Zh. Computation of the normal injection of a hydrogen jet into a supersonic air flow // Thermophysics and Aeromechanics, Vol. 17 issue 4 December 2010. pp. 531–542

[5] A.KulikovskiiA.G., Pogorelov N.V., Semenov A.Yu.Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Moscow:Fizmatlit, 2001. 656 p.

[6] Harten A. High resolution schemes for hyperbolic conservation laws // J. Comp. Phys., Vol. 49, 1983, pp. 357-393

[7] Harten A., Engquist B., Osher S., Chakravarthy S. Uniformly high-order accurate essentially non-oscillatory schemes III // J. Comput. Phys., Vol. 71, 1987, pp. 231-303

[8] Shu C., Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes // J. Comp. Phys. 1988. Vol. 77. pp. 439–471.

[9] Shu C., Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes, II // J. Comp. Phys. 1989. Vol. 83. pp. 32–78.

[10] Berger M.J., Aftosmis M.J., Murman S.E. Analysis of slope limiters on irregular grids // In 43rd AIAA Aerospace Sciences Meeting, Reno, NV , 2005. Paper AIAA 2005-0490. pp.22

[11] Poinsot T.J., Lele S.K. Boundary conditions for direct simulation of compressible viscous flows // J. Comp. Phys. 1992. Vol. 101. pp. 104–129.

[12] Sweby, P.K. High resolution schemes using flux-limiters for hyperbolic conservation laws // SIAM J. Num. Anal. 1984. Vol. 21, No 5. pp. 995–1011

[13] Danaila I., Joly P., Kaber S.M., Postel M. Introduction to scientific computing: twelveprojects solved with MATLAB // Springer, 2007. 308 p.

[14] Harten A. The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes // Math. Comp. 1978. Vol. 32. pp. 363–389 .

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*Journal of Mathematics, Mechanics and Computer Science*,

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Mathematical modeling