Numerical simulation of axisymmetric jets using differential eddy viscosity models

Fluid, gas and plasma mechanics


Larina E. V.1*, Kryukov I. A.2**, Ivanov I. E.3***

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, 101-1, prospekt Vernadskogo, Moscow, 119526, Russia
3. Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russia



The work deals with numerical simulation using semi empirical turbulence models which include a differential equation for turbulent viscosity in two-dimensional case. k‑ε‑µt model [5] and k‑ε‑µt(Λ) model [6] are used to describe the mean turbulent parameters. The one-parameter models of eddy viscosity (νt −92 [7] and Spalart, Allmaras [8]) are used in one of the jet problems. Averaged flow and turbulence parameters are implemented in Cartesian and cylindrical coordinate systems.

Simulation of axisymmetric subsonic jet in quiescent air in initial and transition regions with mean velocity u=87m/s have been conducted using k-ε-μt model. Initial region length has been determined with reasonable accuracy. Difference between calculated and experimental velocities is less than 5%.

Verification of turbulence models in the flat plate boundary layer problem with M = 0.5 have been performed. Skin friction coefficient and velocity in logarithmic coordinates are compared with analytical solutions.

Underexpanded supersonic turbulent jet simulation at M = 2 in the nozzle exit [13] have been conducted. Calculated static pressure and Mach number distributions along the jet axis are compared with experimental data. The k-ε-μt model reproduced 7 jet barrels. The positions of the maxima and minima are closest to the experiment in comparison with other models. The maxima and minima of the pressure oscillations and Mach in all barrels underestimated. The νt‑92 model reproduced 12 jet barrels that is fully consistent with the experiment. The positions of the maxima and minima are distinctly shifted in comparison with experiment. The k-ε-µt(Λ) model reproduced the flow similarly with k-ε-μt model (7 barrels), but predicted flow core length is bigger. Spalart, Allmaras model reproduced only 5 barrels and predicted flow core length is smallest in comparison with other models.

Numerical simulation of supersonic axisymmetric jet from converging nozzle in quiescent air [15] has been performed. A comparison with the Pitot pressure and velocity distribution along the axis is shown. The ratio of the chamber total pressure to ambient static pressure equals to NPR=2.5 and 4. In the case of k-ε-µt model at NPR=2.5 the initial region length is in satisfactory agreement with experiment, but the number of oscillations is underestimated. The k-ε-µt(Λ) model overestimates the initial region length. Oscillations of the Pitot pressure and velocity have bigger magnitude for all jet barrels in comparison with k-ε-µt model. In the case of k-ε-μt model at NPR=4 the Pitot pressure in the first barrel is bigger than experimental value. The difference in amplitude decreases downstream. The peaks begin to keep up and become less appreciable. Pitot pressure produced by k‑ε‑μt(Λ) model is underestimated in the first barrel. The subsequent barrels predicted by the model are much better than the first.

Numerical simulation of overexpanded cold jet using the k‑ε‑μt model in quiescent air at Mach 3.005 in the nozzle exit [16] has been performed. The calculated axis Pitot pressure is underestimated, but the position of the local maxima and minima and local minima values are determined with good accuracy. The biggest difference along the axis between the calculation and experiment is in the two initial jet barrels. The largest deviation from the experiment in all cross-sections is found near the axis.


three-equation turbulence model, supersonic axisymmetric under- and over-expanded jets, boundary layer


  1. Kudimov N.F., Safronov A.V., Tretiyakova O.N. Trudy MAI, 2013, no. 69:

  2. Kudimov N.F., Safronov A.V., Tretiyakova O.N. Trudy MAI,. 2013, no.70:

  3. Ivanov I.E., Kryukov I.A. Matematicheskoe modelirovanie,1996, vol.8, no. 6, pp. 47-55.

  4. Glushko G.S., Ivanov I.E., Kryukov I.A. Математические модели и компьютерное моделирование, 2010, vol. 2, no. 4, pp. 407–422.

  5. Ivanov I.E., Kryukov I.A., Larina E.V. Fluid Dynamics, 2014, vol. 49, no. 5, pp. 694-702.

  6. Yoshizawa A., Abe H., Matsuo Y., Fujiwara H., Mizobuchi Y. Physics of fluids, 2012, vol. 24, no.7, URL:

  7. Gulyaev A.N., Kozlov V.E., Secundov A.N. Fluid Dynamics, vol.28, no.4, 1993, pp. 485-494.

  8. Spalart P.R., Allmaras S.R. A One-Equation Turbulence Model for Aerodynamic Flows. Recherche Aerospatiale. no. 1, 1994, pp. 5-21.

  9. Launder B.E., Spalding D.B. The numerical computation of turbulent flows // Computer Meth. Appl. Mech. Engn., 1974, Vol. 3, no. 2, pp. 269-289.

  10. Sarkar S., Erlebacher G., Hussaini M.Y., Kreiss H.O. The analysis and modelling of dilatational terms in compressible turbulence // J. Fluid Mech., 1991, Vol. 227, pp. 473–493.

  11. Abe K., Kondoh T., Nagano Y. On Reynolds-stress expressions and near-wall scaling parameters for predicting wall and homogeneous turbulent shear flows // Int. J. Heat Fluid Flow 18, 266, 1997.

  12. Abramovich G.N. Teoriya turbulentnykh strui (The Theory of Turbulent Jets), Izd-vo fiziko-matematicheskoi literatury, 1963, 671 p.

  13. Seiner J.M., Norum T.D. Experiments of shock associated noise on supersonic jets // AIAA Pap. 79-1526, 1979.

  14. Glushko G.S., Ivanov I.E., Kryukov I.A. Fiziko-khimicheskaya kinetika v gazovoi dinamike, 2010, vol. 9, URL:

  15. Saddington A. J., Lawson N. J., Knowles K. An experimental and numerical investigation of under expanded turbulent jets. Aaeronaut. Journal. 2004. V. 108. № 1081. P. 145–152.

  16. Zapryagaev V.I., Kudryavtsev A.N., Lokotko A.V. An experimental and numerical study of a supersonic jet shockwave structure, West East High Speed Flow Fields 2002, CIMNE, Barcelona, Spain, 2002, 6 pp.

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