A numerical study of a variable-density low-speed turbulent mixing layer.

*(English)*Zbl 1421.76109Summary: Direct numerical simulations of a temporally developing, low-speed, variable-density, turbulent, plane mixing layer are performed. The Navier-Stokes equations in the low-Mach-number approximation are solved using a novel algorithm based on an extended version of the velocity-vorticity formulation used by J. Kim et al. [ibid. 177, 133–166 (1987; Zbl 0616.76071)] for incompressible flows. Four cases with density ratios \(s=1,2,4\) and 8 are considered. The simulations are run with a Prandtl number of 0.7, and achieve a \(Re_\lambda\) up to 150 during the self-similar evolution of the mixing layer. It is found that the growth rate of the mixing layer decreases with increasing density ratio, in agreement with theoretical models of this phenomenon. Comparison with high-speed data shows that the reduction of the growth rates with increasing density ratio has a weak dependence with the Mach number. In addition, the shifting of the mixing layer to the low-density stream has been characterized by analysing one-point statistics within the self-similar interval. This shifting has been quantified, and related to the growth rate of the mixing layer under the assumption that the shape of the mean velocity and density profiles do not change with the density ratio. This leads to a predictive model for the reduction of the growth rate of the momentum thickness, which agrees reasonably well with the available data. Finally, the effect of the density ratio on the turbulent structure has been analysed using flow visualizations and spectra. It is found that with increasing density ratio the longest scales in the high-density side are gradually inhibited. A gradual reduction of the energy in small scales with increasing density ratio is also observed.

##### MSC:

76F25 | Turbulent transport, mixing |

76F10 | Shear flows and turbulence |

76F65 | Direct numerical and large eddy simulation of turbulence |

##### Software:

chebop
Full Text:
DOI

##### References:

[1] | Ashurst, W. T.; Kerstein, A. R., One-dimensional turbulence: variable-density formulation and application to mixing layers, Phys. Fluids, 17, (2005) · Zbl 1187.76028 |

[2] | Bell, J. H.; Mehta, R. D., Development of a two-stream mixing layer from tripped and untripped boundary layers, AIAA J., 28, 12, 2034-2042, (1990) |

[3] | Bogdanoff, D. W., Compressibility effects in turbulent shear layers, AIAA J., 21, 6, 926-927, (1983) |

[4] | Bretonnet, L.; Cazalbou, J.-B.; Chassaing, P.; Braza, M., Deflection, drift, and advective growth in variable-density, laminar mixing layers, Phys. Fluids, 19, 10, (2007) · Zbl 1182.76083 |

[5] | Brown, G. L.1974The entrainment and large structure in turbulent mixing layers. In Proceedings of the 5th Australasian Conference on Hydraulics and Fluid Mechanics, pp. 352-359. |

[6] | Brown, G. L.; Roshko, A., On density effects and large structure in turbulent mixing layers, J. Fluid Mech., 64, 4, 775-816, (1974) · Zbl 1416.76061 |

[7] | Carlier, J.; Sodjavi, K., Turbulent mixing and entrainment in a stratified horizontal plane shear layer: joint velocity-temperature analysis of experimental data, J. Fluid Mech., 806, 542-579, (2016) |

[8] | Chassaing, P.; Antonia, R. A.; Anselmet, F.; Joly, L.; Sarkar, S., Variable Density Fluid Turbulence, (2002), Springer |

[9] | Clemens, N. T.; Mungal, M. G., Two-and three-dimensional effects in the supersonic mixing layer, AIAA J., 30, 4, 973-981, (1992) |

[10] | Cook, A. W.; Riley, J. J., Direct numerical simulation of a turbulent reactive plume on a parallel computer, J. Comput. Phys., 129, 2, 263-283, (1996) · Zbl 0890.76049 |

[11] | Dimotakis, P. E., Two-dimensional shear-layer entrainment, AIAA J., 24, 11, 1791-1796, (1986) |

[12] | Dimotakis, P. E., Turbulent free shear layer mixing and combustion, High Speed Flight Propulsion Systems, 137, 265-340, (1991) |

[13] | Dimotakis, P. E., Turbulent mixing, Annu. Rev. Fluid Mech., 37, 1, 329-356, (2005) · Zbl 1117.76029 |

[14] | Driscoll, T. A.; Bornemann, F.; Trefethen, L. N., The chebop system for automatic solution of differential equations, BIT Num. Math., 48, 4, 701-723, (2008) · Zbl 1162.65370 |

[15] | Flores, O.; Jiménez, J., Hierarchy of minimal flow units in the logarithmic layer, Phys. Fluids, 22, 7, (2010) |

[16] | Fontane, J.; Joly, L., The stability of the variable-density Kelvin-Helmholtz billow, J. Fluid Mech., 612, 237-260, (2008) · Zbl 1151.76475 |

[17] | Gatski, T. B.; Bonnet, J.-P., Compressibility, Turbulence and High Speed Flow, (2013), Academic |

[18] | Hall, J. L.; Dimotakis, P. E.; Rosemann, H., Experiments in nonreacting compressible shear layers, AIAA J., 31, 12, 2247-2254, (1993) |

[19] | Higuera, F. J.; Moser, R. D., Effect of chemical heat release in a temporally evolving mixing layer, CTR Report, 19-40, (1994) |

[20] | Hoyas, S.; Jiménez, J., Scaling of the velocity fluctuations in turbulent channels up to Re_𝜏 = 2003, Phys. Fluids, 18, 1, (2006) |

[21] | Jahanbakhshi, R.; Madnia, C. K., Entrainment in a compressible turbulent shear layer, J. Fluid Mech., 797, 564-603, (2016) |

[22] | Jang, Y.; De Bruyn Kops, S. M., Pseudo-spectral numerical simulation of miscible fluids with a high density ratio, Comput. Fluids, 36, 2, 238-247, (2007) · Zbl 1177.76270 |

[23] | Kaneda, Y.; Ishihara, T., High-resolution direct numerical simulation of turbulence, J. Turbul., 7, N20, (2006) |

[24] | Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166, (1987) · Zbl 0616.76071 |

[25] | Knio, O. M.; Ghoniem, A. F., The three-dimensional structure of periodic vorticity layers under non-symmetric conditions, J. Fluid Mech., 243, 353-392, (1992) · Zbl 0825.76644 |

[26] | Lee, M. J.; Kim, J.; Moin, P., Structure of turbulence at high shear rate, J. Fluid Mech., 216, 561583, (1990) |

[27] | Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 1, 16-42, (1992) · Zbl 0759.65006 |

[28] | Lele, S. K., Compressibility effects on turbulence, Annu. Rev. Fluid Mech., 26, 1, 211-254, (1994) · Zbl 0802.76032 |

[29] | Mahle, I.; Foysi, H.; Sarkar, S.; Friedrich, R., On the turbulence structure in inert and reacting compressible mixing layers, J. Fluid Mech., 593, 171-180, (2007) · Zbl 1128.76027 |

[30] | Mcmullan, W., Coats, C. & Gao, S.2011Analysis of the variable density mixing layer using large eddy simulation. In 41st AIAA Fluid Dynamics Conference and Exhibit, pp. 2011-3424. AIAA. |

[31] | Mcmurtry, P. A.; Jou, W.-H.; Riley, J.; Metcalfe, R. W., Direct numerical simulations of a reacting mixing layer with chemical heat release, AIAA J., 24, 6, 962-970, (1986) |

[32] | Moin, P.; Mahesh, K., Direct numerical simulation: a tool in turbulence research, Annu. Rev. Fluid Mech., 30, 1, 539-578, (1998) · Zbl 1398.76073 |

[33] | Nicoud, F., Conservative high-order finite-difference schemes for low-Mach number flows, J. Comput. Phys., 158, 1, 71-97, (2000) · Zbl 0973.76068 |

[34] | O’Brien, J.; Urzay, J.; Ihme, M.; Moin, P.; Saghafian, A., Subgrid-scale backscatter in reacting and inert supersonic hydrogen – air turbulent mixing layers, J. Fluid Mech., 743, 554-584, (2014) |

[35] | Pantano, C.; Sarkar, S., A study of compressibility effects in the high-speed turbulent shear layer using direct simulation, J. Fluid Mech., 451, 329-371, (2002) · Zbl 1156.76403 |

[36] | Papamoschou, D.; Roshko, A., The compressible turbulent shear layer: an experimental study, J. Fluid Mech., 197, 453-477, (1988) |

[37] | Peters, N., Turbulent Combustion, (2000), Cambridge University Press · Zbl 0955.76002 |

[38] | Pickett, L. M.; Ghandhi, J. B., Passive scalar measurements in a planar mixing layer by PLIF of acetone, Exp. Fluids, 31, 3, 309-318, (2001) |

[39] | Ramshaw, J. D., Simple model for mixing at accelerated fluid interfaces with shear and compression, Phys. Rev. E, 61, 5339-5344, (2000) |

[40] | Reinaud, J.; Joly, L.; Chassaing, P., The baroclinic secondary instability of the two-dimensional shear layer, Phys. Fluids, 12, 10, 2489-2505, (2000) · Zbl 1184.76450 |

[41] | Rogers, M. M.; Moser, R. D., Direct simulation of a self-similar turbulent mixing layer, Phys. Fluids, 6, 2, 903-923, (1994) · Zbl 0825.76329 |

[42] | Sekimoto, A.; Dong, S.; Jiménez, J., Direct numerical simulation of statistically stationary and homogeneous shear turbulence and its relation to other shear flows, Phys. Fluids, 28, 3, (2016) |

[43] | Da Silva, C. B.; Pereira, J. C. F., Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets, Phys. Fluids, 20, 5, (2008) · Zbl 1182.76172 |

[44] | Soteriou, M. C.; Ghoniem, A. F., Effects of the free-stream density ratio on free and forced spatially developing shear layers, Phys. Fluids, 7, 8, 2036-2051, (1995) · Zbl 1032.76547 |

[45] | Spalart, P. R.; Moser, R. D.; Rogers, M. M., Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions, J. Comput. Phys., 96, 2, 297-324, (1991) · Zbl 0726.76074 |

[46] | Spencer, B. W. & Jones, B. G.1971 Statistical investigation of pressure and velocity fields in the turbulent two-stream mixing layer. AIAA Paper 71-613. |

[47] | Thorpe, S. A., The Turbulent Ocean, (2005), Cambridge University Press |

[48] | Turner, J. S., Buoyancy Effects in Fluids, (1979), Cambridge University Press · Zbl 0443.76091 |

[49] | Vreman, A. W.; Sandham, N. D.; Luo, K. H., Compressible mixing layer growth rate and turbulence characteristics, J. Fluid Mech., 320, 235-258, (1996) · Zbl 0875.76159 |

[50] | Wang, P.; Fröhlich, J.; Michelassi, V.; Rodi, W., Large-eddy simulation of variable-density turbulent axisymmetric jets, Intl J. Heat Fluid Flow, 29, 3, 654-664, (2008) |

[51] | Williams, F. A., Combustion Theory, (1985), Westview Press |

[52] | Wyngaard, J. C., Turbulence in the Atmosphere, (2010), Cambridge University Press · Zbl 1380.86001 |

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