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 Lsubjected 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 tand 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/ u0 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/ 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−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
Saligns with the vorticity vector, why the greatest increase in enstrophy production occurs in regions where Shas a positive intermediate eigenvalue; and why large-scale strain Sof 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.