Based on a direct numerical simulation (DNS) of a temporally evolving mixing layer, we present a detailed study of the turbulent/non-turbulent (T/NT) interface that is defined using the two most common procedures in the literature, namely either a vorticity or a scalar criterion. The different detection approaches are examined qualitatively and quantitatively in terms of the interface position, conditional statistics and orientation of streamlines and vortex lines at the interface. Computing the probability density function (p.d.f.) of the mean location of the T/NT interface from vorticity and scalar allows a detailed comparison of the two methods, where we observe a very good agreement. Furthermore, conditional mean profiles of various quantities are evaluated. In particular, the position p.d.f.s for both criteria coincide and are found to follow a Gaussian distribution. The terms of the governing equations for vorticity and passive scalar are conditioned on the distance to the interface and analysed. At the interface, vortex stretching is negligible and the displacement of the vorticity interface is found to be determined by diffusion, analogous to the scalar interface. In addition, the orientation of vortex lines at the vorticity and the scalar based T/NT interface are analyzed. For both interfaces, vorticity lines are perpendicular to the normal vector of the interface, i.e. parallel to the interface isosurface.
We examine finite Reynolds number contributions to the inertial range solution of the third order structure functions D 3,0 and D 1,2 stemming from the unsteady and viscous terms. Under the assumption that the second order correlations f and g are self-similar under a coordinate change, we are able to rewrite the exact second order equations as a function of a normalized scale r only. We close the resulting system of equations using a power law and an eddy-viscosity ansatz. If we further assume K41 scaling, we find the same Reynolds number dependence as previously in the literature. We proceed to extrapolate towards higher Reynolds numbers to examine the unsteady and viscous terms in more detail. We find that the intersection between the two terms, where their contribution to the solution of the structure function equations is relatively small, scales with the Taylor scale λ.
We examine balances of structure function equations up to the seventh order N = 7 for longitudinal, mixed and transverse components. Similarly, we examine the traces of the structure function equations, which are of interest because they contain invariant scaling parameters. The trace equations are found to be qualitatively similar to the individual component's equations. In the even-order equations, the source terms proportional to the correlation between velocity increments and the pseudo-dissipation tensor are significant, while for odd N, source terms proportional to the correlation of velocity increments and pressure gradients are dominant. Regarding the component equations, one finds under the inertial range assumptions as many equations as unknown structure functions for even N, i.e. can solve for them as function of the source terms. On the other hand, there are more structure functions than equations for odd N under the inertial range assumptions. Similarly, there are not enough linearly independent equations in the viscous range r → 0 for orders N > 3.
Kolmogorov introduced dissipative scales based on the mean dissipation ε and the viscosity ν, namely the Kolmogorov length η = (ν 3 / ε ) 1/4 and the velocity u η = (ν ε ) 1/4 . However, the existence of smaller scales has been discussed in the literature based on phenomenological intermittency models. Here, we introduce exact dissipative scales for the even-order longitudinal structure functions. The derivation is based on exact relations between even-order moments of the longitudinal velocity gradient (∂u 1 /∂x 1 ) 2m and the dissipation ε m . We then find a new length scale η C,m = (ν 3 / ε m/2 2/m ) 1/4 and u C,m = (ν ε m/2 2/m ) 1/4 , i.e. the dissipative scales depend rather on the moments of the dissipation ε m/2 and thus the full probability density function (p.d.f.) P(ε) instead of powers of the mean ε m/2 . The results presented here are exact for longitudinal even-ordered structure functions under the assumptions of (local) isotropy, (local) homogeneity and incompressibility, and we find them to hold empirically also for the mixed and transverse as well as odd orders. We use direct numerical simulations (DNS) with Reynolds numbers from Re λ = 88 up to Re λ = 754 to compare the different scalings. We find that indeed P(ε) or, more precisely, the scaling of ε m/2 / ε m/2 as a function of the Reynolds number is a key parameter, as it determines the ratio η C,m /η as well as the scaling of the moments of the velocity gradient p.d.f. As η C,m is smaller than η, this leads to a modification of the estimate of grid points required for DNS.
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