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Show Nakahashi and Diewert (1986)]. The formulation results in a tri-diagonal system of equations along each coordinate line for the final grid point positions. Wi-iSi-i - (Wi-i + W, + TijSi + WiSl + i = -TjSt (28) where physically, T^^ is the product of a torsional spring constant and its reference position and represents the contribution from grid straightness and orthogonality considerations (see above cited references). The weighting function, Wi, is defined as a function of a selected normalized flow property, /, such that Wi = l + Aff where A and B are constants related to the specified m a x i m u m and minimum grid spacings (see Harvey, 1993 and references therein). As in m a n y of the studies on solution-adaptive gridding, a linear combination of gradients and curvature of selected dependent variables is used here as weighting functions. Finding a flow-field variable which will consistently represent solution error in all regions of the solution domain is difficult, if not impossible. S o m e variables change very rapidly in certain regions, but remain fairly smooth in other regions where another variable could be undergoing severe change. In the present study, an algorithm developed earlier by Harvey et. al. (1993) is used to automatically choose which variables are to represent the weighting function at each grid point location. The scheme computes gradients and/or curvature of each user specified variable and then, after normalizing each of these, chooses the largest in magnitude to represent the weighting function at each point. This method has proven superior to conventional weighting function selection processes, especially where large changes in one dependent variable exist in certain regions of the flow, but for another variable these large changes exist in different regions. The normalized flow variable, /, at the zth point along the current coordinate line is computed as fi = U - f MIN JhlAX ~ JMIN where /* is a linear combination of the gradients and curvature of the dependent flow variables, fa. The values f\fiN and /MAX are the minimum and maximum values of ft along the current adaptation line. d24>7 -lT?J+'v (27) dfr ds max k d(j>7 ds ds2 max k d2^ ds'' for all specified 0*'s, ie. species mass fraction, temperature, density etc. Hence, the gradients of all selected flow variables are computed, then the m a x i m um is chosen to represent ft at each point along the current adaptation line. Numerical Results The methods outlined above are tested on the confined laminar methane-air diffusion flame studied experimentally by Mitchell (1975). A s mentioned in the introduction, this problem was also studied numerically by Smooke et al. (1989) and more recently by X u et al. (i993). The experimental burner setup (Mitchell, 1975) consists of two concentric tubes connected to a burner plate as shown in Fig. 1. Methane fuel flows through the inner tube and dry air is introduced through the annular (outer) tube. Inlet conditions used as well as the burner dimensions are specified in the figure. T h e outer tube enclosure was assumed to at a constant wall temperature of 300K. -A r= 0.635 cm) i ___ - i - - - *" air T= 300 K U= 9.88 cm/s CH. T= 300 K U= 4.5cm/s L= 30 cm Fig. 1: Burner geometry Implementation Details A n 89 x 41 computational grid was used for the calculations. Initially, a converged cold mixing solution was obtained. The reference density in the gravity term of Eq. (6) was set to that of air in order to reduce the magnitude of the R H S term and the associated numerical stiffness. A attempt was m a d e at "lighting" the resulting methane/air plume with the detailed mechanism by introducing an artificial energy source on the R H S of the energy equation without success. It became necessary to obtain a starting solution for C O 2 using a single-step global reaction mechanism: CH4 + 202 <=> C02 + 2H20 The single-step reaction was assumed to proceed in the forward direction, independent of temperature (ar = Er = 0 of Eq. 18a) with an arbitrary pre-exponential reaction rate of AT = 6.3 x IO14 G |