Viscous shock anomaly in a variable-viscosity Burgers flow with an active scalar

• Published in: Fluid dynamics research 2015-12, Vol.47 (6), p.1-22
• Main Authors: Talbot, B, Mammeri, Y, Bedjaoui, N
• Format: Article
• Language: English
• Subjects: active scalar; Advection-diffusion equation; Burgers' flow; Energy dissipation; Mathematical models; Propagation; Scalars; Sine waves; Small scale; spectral methods; variable-viscosity; Viscosity; viscous shock
• ISSN: 0169-5983; 1873-7005
• DOI: 10.1088/0169-5983/47/6/065502

This paper examines the effect of viscosity variability on the formation of shocklets (small transient shocks) through the inhomogeneity in composition of the propagating medium. For this purpose, both analytical estimates and numerical spectral method are applied to a Burgers' equation-where viscosity is a space-time function depending on a coupled advection-diffusion equation for the local mass fraction. The coefficient of viscosity thus behaves as an active scalar. The inhomogeneous shocklet is modeled by a fixed sine wave for the initial velocity profile while different sine waves of higher frequency are used for the initial embedded distribution of scalar. The initial kinematic viscosity ratio ranges from 1 to 4. It is found that, surprisingly, for all conditions at R > 1, i.e., for waves becoming more and more viscous on average, there was (1) a steeper maximum gradient in the shock transition zone, and consequently (2) velocity spectra extended toward the finest small scales, and (3) an enhanced energy dissipation rate is observed at the time of peak energy dissipation. These results will be useful to the understanding of small-scale dynamics for one-dimensional shocklets propagating in multi-component gas mixtures where noticeable active scalar effects are present.