The velocity-scalar cross spectrum of stretched spiral vortices

Abstract

The stretched-spiral vortex model is used to calculate the velocity-scalar cross spectrum for homogeneous, isotropic turbulence in the presence of a mean scalar gradient. The only nonzero component of the cospectrum is that contributed by the velocity component in the direction of the imposed scalar gradient while the quadrature spectrum is identically zero, in agreement with experiment. For the velocity field provided by the stretched-spiral vortex, the velocity-scalar spectrum can be divided into two additive components contributed by the velocity components along the vortex axis, and in the plane normal to this axis, respectively. For the axial velocity field, a new exact solution of the scalar convection-diffusion equation is found exhibiting scalar variation in the direction of the vortex tube axis. An asymptotic expression was found for the cospectrum contributed by this solution and the axial velocity, with the leading order term showing a k–5/3 range. This term is produced by the winding of the initial axial velocity field by the axisymmetric vortex core. The next order term gives a k–7/3 range, and arises from the lowest order effect of the nonaxisymmetric vorticity on the evolution of the axial velocity. Its coefficient can be of either sign or zero depending on the initial conditions. The contribution to the cospectrum from the velocity in the plane of the vortex is also calculated, but no universal high wave number asymptotic form is found. The integrals are evaluated numerically and it is found that the resulting cospectrum does not remain of one sign. Its form depends on the choice of the vortex core velocity profile and time cutoff in the spectral integrals. The one-dimensional cospectrum contributed by the axial velocity is compared with the experimental data of Mydlarski and Warhaft [J. Fluid Mech. 358, 135–175 (1998)].

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