Nonlinear interactions in turbulence with strong irrotational straining

Abstract

The rate of growth of the nonlinear terms in the vorticity equation are analysed for a turbulent flow with r.m.s. velocity u₀ and integral length scale L subjected to a strong uniform irrotational plane strain S, where (u₀/L)/S=ε[Lt ]1. The rapid distortion theory (RDT) solution is the zeroth-order term of the perturbation series solution in terms of ε. We use the asymptotic form of the convolution integrals for the leading-order nonlinear terms when β= exp(−St)[Lt ]1 to determine at what time t and beyond what wavenumber k (normalized on L) the perturbation series in ε fails, and hence derive the following conditions for the validity of RDT in these flows. (a) The magnitude of the nonlinear terms of order ε depends sensitively on the amplitude of eddies with large length scales in the direction x₂ of negative strain. (b) If the integral of the velocity component u₂ is zero the leading-order nonlinear terms increase and decrease in the same way as the linear terms, even those that decrease exponentially. In this case RDT calculations of vorticity spectra become invalid at a time t_NL∼L/u₀k⁻³ independent of ε and the power law of the initial energy spectrum, but the calculation of the r.m.s. velocity components by RDT remains accurate until t= T_NL∼L/u₀, when the maximum amplification of r.m.s. vorticity is ω/S∼εexp(ε⁻¹)[Gt ]1. (c) If this special condition does not apply, the leading-order nonlinear terms increase faster than the linear terms by a factor O(β⁻¹). RDT calculations of the vorticity spectrum then fail at a shorter time t_NL∼(1/S) ln(ε⁻¹k⁻³); in this case T_NL∼(1/S) ln(ε⁻¹) and the maximum amplification of r.m.s. vorticity is ω/S∼1. (d) Viscous effects dominate when t[Gt ](1/S) ln(k⁻¹(Re/ε)^1/2). In the first case RDT fails immediately in this range, while in the second case RDT usually fails before viscosity becomes important. The general analytical result (a) is confirmed by numerical evaluation of the integrals for a particular form of eddy, while (a), (b), (c) are explained physically by considering the deformation of differently oriented vortex rings. The results are compared with small-scale turbulence approaching bluff bodies where ε[Lt ]1 and β[Lt ] 1.

These results also explain dynamically why the intermediate eigenvector of the strain S aligns with the vorticity vector, why the greatest increase in enstrophy production occurs in regions where S has a positive intermediate eigenvalue; and why large-scale strain S of a small-scale vorticity can amplify the small-scale strain rates to a level greater than S – one of the essential characteristics of high-Reynolds-number turbulence.