Three dimensional vortex simulation of entrainment augmentation due to streamwise structures

The transport element method is extended to three dimensions to study the evolution of the scalar field in a flow with high concentration of vorticity. The numerical scheme is based on following Lagrangian computational elements used to transport, along with vorticity, the local gradients of the scalar field. The formulation of the numerical scheme is first presented as a direct generalization of the three dimensional vortex element method. The numerical algorithms required to implement this scheme are then developed. In particular, problem associated with severe distortion of the flow map due to the growth of perturbation are s h m to cause difficulties including loss of numerical accuracy and resolution. Means to overcome this problem are discussed in the paper and are sham to yield accurate solutions. The method is applied to study the evolution of an initially two-dimensional shear layer perturbed in the streamwise and the spanwise directions. Attention is focused of the role of the spanwise instability in enhancing the rate of scalar entrainment into the large scale structures that form as the streamwise instability develops. It is shown that the total entrainment increases by almost 25% over the Wimensional case. Two mechanisms, associated with the onset of the three-dimensional instability, are responsible for this increase; the intensification of vorticity within the core of the large eddy due to spanwise stretching delays its collapse, and the generation of transverse entrainment currents towards the eddy core due to the formation of streamwise vortex Structures within the core and along the braids between neighboring cores. Preferential entrainment is detected along the spanwise direction due to the streamwise vorticity. Copyright01989 by A.F. Ghoniem. mblished by the American Institute of Aeronautics and Astronautics, Inc., with permission. Grauduate student, member, AIAA. Associate professor, associate fellow, AIM. d I I ~ ~ C T I O N Experimental investigations of turbulent shear layers suggest that their evolution can be test analyzed in terms of the coherent vortical structures which form as a result of perturbing the initial flow (e.g., Brown and Roshko 111. mvies and Yule [2 I , Hussain 131, Yule [ 4 I , Roshko 151). The dominant effect of these structures in the early developrent of the layer and during mixing transition has been emphasized, and their presence has also k e n detected or inferred in the far field region (Dimtakis et al. [61, Wygnanski et al. [71) . lclo types of structures, distinguished by the orientation of vorticity with respect to the main flow, have been identified: spanwise and stremise structures. The fomtion of spanwise structures occurs as a result of the "rollup" of the vorticity layer which leads to the concentration of vorticity in organized, primarily two-dimensional eddies. The merger of these eddies has been s h to be the primacy mechanism controlling the early growth of the vorticity layer in the cross-stream direction (Winant and Browand (81. Ho and Huerre 191). Rlrther downstream, secondary stremise structures are observed. The fomtion of these structures is accompanied by spanwlse deformation of the cores of the otherwise two-dimensional eddies and by the generation of stremise vorticity (Berm1 and ROshko 1101, Jimenez 1111, Jimenez et al. 1121, Breidenthal (131). The behavior and interaction of the deformed vortices in "transition" regions have not yet been explained based on experimental results, and the visualizations of vorticity or scalar fields are often blurred by the various mechanisms which affect their motion. The desire to control these mechanisms in such a way as to promote or retard turbulent behavior is the main motivation behind this study. Linear stability analysis of shear layers has teen another important tool in the study of their evolution. In particular, it has been shown that the initial rollup occurs as a result of the Kelvin-Helmholtz instability of the vorticity layer (Michalke [141). The behavior of the spamise eddies is associated with the growth of the fundamental d e , while their merger is found to be dominated by subharmonic modes [8,91. It also states that the appearance of the spanwise undulations of the eddies and stremise scalar streaks is associated with three dimensional instabilities of the flow (Pierrehumhert and Widnall ( 1 5 1 ) . Results of these analyses are particularly useful in predicting the shape, wavelength and growth rate of a given mode. However. when large amplitude deviations from initially assumed states are reached, the results often cease to be applicable, and extensions into non-linear regimes become extremely difficult.