Mean-velocity field calculations

Write a program to solve by means of RFM the equation of motion and using the velocity field, calculate viscous dissipation and solve for the energy equation. Neglect inertial and convective effects. Consider T0=200°C, Ti=150°C, /x=24000 Pa-s, k=0.267 W/mK, i o=0.1 m, i i=0.13 m, k=0.769, cc=0.496 rad/s. Compare the numerical results with the analytical solution. Hint The couette flow is constant along the angular direction, hence, it is no necessary to use the whole domain. 

The ability of MTEN calculations to predict accurately the mean velocity field and turbulence kinetic-energy distribution is demonstrated by Fig. 17 from Mellor and Herring s contribution to the Stanford conference. Their use of a fine computational mesh near the wall is reflected in their accurate prediction of the inner regions. 

Figures 12.3 and 12.3c show mean velocity (Fig. 12.36) and mean temperature (Fig. 12.3c) fields under bluff-body stabilized combustion of stoichiometric methane-air mixture at inlet velocity 10 m/s, and ABC of Eq. (12.19) at the combustor outlet. Functions Wj, Wij, and W2j in Eq. (12.1) were obtained by solving the problem of laminar flame propagation with the detailed reaction mechanism [31] of Ci-C2-hydrocarbon oxidation (35 species, 280 reactions) including CH4 oxidation chemistry. The PDF of Eq. (12.4) was used in this calculation.

Figure 12.5 Calculated mean temperature fields in combustors with a set of similar open-edge V-gutter flame holders of height H = 3 cm and apex angle of 60°. The isoterms divide the entire temperature interval from the initial temperature To to combustion temperature Tc into 10 uniform parts and correspond to t = 27.5 ms. The combustor is 1 m long and the distance between the planes of flame holders is 0.05 m. Flame holders are shifted in longitudinal direction by OH (no shift) (a), IH (6), 2H (c), 3H (d), and 5H (e). Combustion of stoichiometric methane-air mixture at the mean inlet velocity Ui = 20 m/s, po = 0.1 MPa, To = 293 K, ko = 0.24 J/kg, /o = 4 mm. The lower and upper boundaries of the computational domain are the symmetry planes...

The calculating procedure is based on sub-division of the Arctic Basin into grids (Eijk. This is realized by means of a quasi-linearization method (Nitu et al., 2000a). All differential equations of the SSMAE are substituted in each box E by easily integrable ordinary differential equations with constant coefficients. Water motion and turbulent mixing are realized in conformity with current velocity fields which are defined on the same coordinate grid as the E (Krapivin et al., 1998). 

We conclude that over the continuum scale the determining parameters are the wind speed Uh and turbulence initial parameters of the cloud/plume when it reaches the top of the canopy or, equivalently, the virtual source at the level of the canopy. Using suitable fast approximate models for the flow field over urban areas (e.g. RIMPUFF, FLOWSTAR), the variation of the mean velocity and turbulence above the canopy can be calculated. The FLOWSTAR code (Carruthers et al., 1988 [105]) has been extended to predict how (Uc) varies within the canopy. Dispersion downwind of the canopy can also be estimated using cloud/plume profiles, denoted by Gc,w,GA,w which are shown in Figures 2.20 and 2.22. 

To complete the specification of the problem for 9, we must specify a particular velocity field u. In the case of Re <creeping-flow solutions of Chaps. 7 and 8, and it is again convenient to focus our attention on the case of a sphere in a uniform streaming flow, in which a first approximation to the velocity field is given by the Stokes solution, Eq. (7-158), from which we can calculate the velocity components by means of (7-102). 

Even if we computed the microscopic velocity field in the x, y and z directions, we only considered the macroscopic averages in the y and z directions, as no heterogeneity is considered in the x-direction. The macroscopic fluxes were calculated by averaging the microscopic velocities over 50 time steps, over five sites in the z-direction, and over half-cross sections of the micropore matrix and of the crack, multiplied by the respective mean water contents. 

Based on the solution of the velocity field, the force and torque acting on the aggregates can be calculated and the hydrodynamic equivalent diameters for translation (xh,t) and rotation (xi -) can be derived. They scale with the aggregate mass via a power law that reflects the stmctural properties of the aggregate. That means, while for hep aggregates the aggregate mass (N) is proportional to the third power of Xh, there is a fractal-tike relationship for DLCA aggregates with a hydrodynamic dimension 4i close to the fiactal dimension (Fig. 4.17, left cf. discussion to Kirkwood-Riseman theory on pp. 164). This fractal relationship... 

Reasonable agreement is obtained between measured and calculated flow fields. Rotor creates a jet stream in the radial direction towards the cylindric wall. Two main flows circulates back to the impeller, one through the top side and second from the lower side of the stator. The comparisons of the measured and computed mean velocity components for radial direction as a function of cell height is showed in figure 10. 

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