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An Elementary Treatise on Theoretical Mechanics by J.H. Jeans, M.A., F.R.S

By J.H. Jeans, M.A., F.R.S

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Convergence of Computed Solutions We would expect the convergence of the solution for both the three-point implicit and trapezoidal time differencing schemes to be second order in the time step, and this is confirmed in Fig. 5. Plotted there is the dimensionless frequency (the inverse of the period) of the single oscillation computed using both second-order accurate schemes for the three values of time steps in the figures above. The frequency is plotted as a function of the square of the time step, and the nearly straight lines for each set of three points confirm second order accuracy for both schemes.

The dimensionless frequencies determined a t the Reynolds numbers Red = 60, Red = 120, and Red = 180 are used to determine the coefficients in a three-term expansion in inverse square root of the Reynolds number, resulting in which is the dashed curve plotted in Fig. 7 The best three-term curve fit to the experimental data for this flow has been determined by Williamson & Brown [36] to be which is the solid curve plotted in Fig. 50 per cent over the range of Reynolds number for which the present results have been computed (60 Red 5 180).

The total energy per unit volume). The equation of state for a calorically perfect gas then gives where y = c,/c, is the ratio of specific heats. The stresses appearing in the viscous flux vectors are given by and the Cartesian components of the heat flux vector are given by aT q, = -k-, ax and q, = -k- dT ay The two-dimensional form of the equations is written here and throughout. While the computations presented here are for two-dimensional flows only, the methods used are readily extendible to the three-dimensional case.