OCR Text |
Show the material surface on which the scalar gradients were oncentrated by the ini tial and boundary conditions. The scalar field evolution is coupled to the veloCity field throuerh the local strain rate cr(s) in (4). which is determined from the deformation rate at each point on this material surface. The material surface evolves from the Biot-Savart indu ed velocities resulting from the vorticity in the now field calculation as f ' x - X'd' , u( x, t) = w( x ,l) x I '1-" x x' X - X (5) Vorticity concentrations that result from the advection-diffusion balance in the vorticity transport equation, namely aW 1 ., -+u· Vw--V-w= w· Vu at Re (6) allow a similar local integral treatment in the flow field calculation that underlies the solution of (4). This leads to a set of analogous local integral equations for the moments r, for j = {I, ... , oo} of the vorticity. For simplicity. only the zeroth integral nloment is tracked. which satisfies a similar equation as in (4a). namely (7) where fo(s) is the zeroth moment of the local vorticity profile. Since the integral moment equation in (7) requires the circulation associated with material pOints to ren1ain constant with time. it can be solved via a vortex-in-cell implementation for given initial and boundary conditions. At every tin1e I. the vorticity field from (7) produces the velocity field via (5). which lhen is us d to move each point s on the material surface to the next tinle step. The relativ displacements of neighborin rf pOints on this surface then give the strain rate 0 (.\ ) at each pOint on the material surface. which determines the evolution of the scalar erradient moments G/s ) at that point via (4). In this manner. the time evolution of the integral moments for the scalar gradient field at each point on the evolving material surface are computed. The distribution of local integral moments G .(s) over J the compu tational surface is then used to construct the scalar field sex .t) throughout any region of interest. A class of profile shapes with k independent parameters is used to represent the local scalar gradient profile along the lay r-normal direction n. Tracking the first k integral nloments G/s) via (4) then d termines the local gradient profile. This allows the scalar gradient field Vc;,(x,r) to be evaluated at each point on a grid chos n for the sealar field construction. The divergence V· Vs(x .t) of the resulting scalar gradient field is then numerically evaluated on this grid. and then used as the (known) right hand side in the definition of the Laplacian . nan1ely (8) The Poisson equation in (8) is then solved to give the cons erved scalar field c;,(x,t). which i th n differentiated to p:iv th scalar en rf!Y dissipation rate field Vc;, ·Vc;,(x,t). The joint scalar and s alar dissipation fields are then used to onstruct the chemical |