OCR Text |
Show layers, we define local integral moments GJ of all orders j for the scalar gradient profile across each layer as G = j ' - +~ ¥n J - J~ n n dn. ) - 0, 1. 2 •... (3) The exact set of transport equations for these scalar gradient moments can be readily dertved from (2), giving ordinary difTerential equations (ODE's) for the time-evolution of the local moments as dGo(s) = ° dt (4a) d ~t(S) = -a(s)G\ (s) (4b) dG2 (s) 2 --=-2a(s)G (s)---G (s) de 2 ReSc 0 (4c) where the coordinate S identifies the location along the layer. These local integral moment equations are exact. The infinite set of ODE's for the integral moments Gj for j = {I ..... 00} are equivalent to the partial difTerential equation (POE) in (2). However, owing to the linearity of the one-dimensional strained diffusion equation in (2). the local integral moment equations have the property that they are exactly closed at any level of truncation of the infinite set. In other words, the set of equations for the integral moments Gj for j = {I, .... k} Involves only the moments Gj of orderk and lower. As a result. the time evolution of the local moments up to any deSired ordercan be determined from the local integral equations without any closure approximation. The LIM model determines the time-evolution of the integral moments everywhere on the material surface on which the scalar gradients were concentrated by the iniUal and boundary 4.0 .0 4.0 o o o o o (a) (b) 40 B.O Fig. 1. LIM model results for confinement effects In two-dimensional and axtsymmetric bluff body flows. shoWing the more pronounced effects of confinement In two-dimensional flows, and the differences In reCirculation zone structure In twodimensional and axisymmetric flows. Shown are Instantaneous shapes of the time-evolvtng computational surface for: (a) two-dimensional bluff body with low confinement, (b) axlsymmetrtc bluff body With low confinement, (e) two-dimensional bluff body with high confinement, and (d) axl -symmetrlc bluff body with high confinement. Note that the prtmary recirculation zone length changes significantly With confinement In the twodimensional case, but not In the axisymmetric case, In agreement with experimental results. Note also the fundamentally different vortlcal structure of the reclrculatlon zones In the twodimensional cases and the axisymmetric cases, which Is also In agreement with expert mental results. ( c) 5 0 10 0 15 0 20 0 |