Grid-Free Simulation of the Spatially Growing Turbulent Mixing Layer

Abstract

The spatially developing, unforced, turbulent mixing layer is simulated via a grid-free vortex method. Vortex filaments composed of straight tubes are used as the computational element with new vortex tubes produced as the filaments stretch. A loop removal algorithm serves as a de facto subgrid model limiting growth in the number of elements to practical levels. The computations are high resolution and well resolve the mixing layer from its unforced inception as a laminar flow through transition to a self-similar turbulent state. Mean velocity statistics including growth rate and Reynolds stresses agree well with experimental values. The vortical composition of the transition region is found to develop in one or another of the modes that have been documented in previous experiments and computations: roller/rib vortices, the chain-link fence structure in a diamond shaped pattern, and somewhat oblique roller/rib configuration with partial pairing. Evidently, small perturbative effects that are intrinsic to the numerical scheme influence which transitional mode appears locally in the simulations. The computations offer a clear view of the downstream dissolution of the identifiable structure into turbulence in the late transition and the salient aspects of the process are noted. Nomenclature h = maximum vortex tube length L = downstream boundary l = similarity length scale N = number of vortex tubes R x = boundary layer Reynolds number s i = axial vector along the ith vortex tube U in = inlet velocity U p x; t = potential velocity U t , U b = velocities at top and bottom of shear layer x, y, z = streamwise, vertical, and spanwise coordinates x i = center of ith vortex tube x 1 i , x 2 i = beginning and end of ith vortex tube i = circulation of ith vortex tube " = vertical offset of downstream vortex sheet = similarity variable = momentum thickness = smoothing parameter = smoothing function

Spanwise domain effects on the evolution of the plane turbulent mixing layer

“…All simulations show that the pre-transition primary vortices undergo localised interactions, which are particularly evident in Case D1 (Figure 7a) at x/θ i ≈ 220 and in Case D3 at x/θ i ≈ 180. The vortex structure visualised in these images is remarkably similar to that observed in other simulations of the pre-transition region of the mixing layer (Bernard 2008;Lesieur et al 1997), where the pre-transition structure takes a 'chain-link fence' appearance. There is no particular evidence for a 'streaky' streamwise structure which has been observed in experimental studies (Bernal and Roshko 1986;Bell and Mehta 1992), and has also been observed to co-exist with the chain-link fence structure in numerical simulations (Bernard 2008).…”

“…The value of k 1 for Case D3 falls within the range of mixing layer thickness growth rate constants reported in the literature (Brown and Roshko 1974;Bernard 2008). When the flow development becomes inhibited by the spanwise domain extent in both cases D1 and D2, the momentum thickness growth rate decreases.…”

“…The secondary streamwise structure observed in numerical simulations of the plane mixing layer appears to be a function of the imposed inflow conditions. Simulations with three-dimensional white-noise fluctuations produce oblique rollers that undergo pre-transition localised pairing interactions or helical pairings (Lesieur et al 1997;Bernard 2008), with the pre-transition vortex structure taking the appearance of a chain-link fence. In contrast, highly two-dimensional disturbances produce thin rib vortices in the interconnecting braid region between primary rollers (Lesieur et al 1997).…”

Initial condition effects on large scale structure in numerical simulations of plane mixing layers

“…This localised pairing can be seen in the light colours in the plan view taking a corrugated appearance in this region of the flow. This type of localised pairing has been observed in many other simulations of the plane mixing later with a pseudo-random white noise fluctuation environment, 22,24,26 and is sometimes referred to as giving the vortex structure a chain-link fence appearance. Once the streamwise structure has formed, a further interaction between the primary vortices precipitates the transition to turbulence in the mixing layer.…”

“…23 Flow visualisation of the high Reynolds number simulated flow shows that streamwise vorticity develops in the mixing layer, and that the streamwise vorticity persists far into the turbulent region. 22,23,26,27 Evidence for the presence of statistically stationary streamwise vorticity, however, appears to be lacking in all of these spatially-developing mixing layer computations.…”