Nonlinear interactions in turbulence with strong irrotational straining

The rate of growth of the nonlinear terms in the vorticity equation are analysed for a turbulent flow with r.m.s. velocity u 0 and integral length scale L subjected to a strong uniform irrotational plane strain S , where ( u 0 / 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 x 2 of negative strain. ( b ) If the integral of the velocity component u 2 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 / u 0 k −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 / u 0 , 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 −1 k −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.