Interdisciplinary Applied Mathematics

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15.5.2    Comparison with Navier-Stokes Solutions

A validation of the traditional LBM with direct solutions of the Navier-Stokes equations was presented in (Karniadakis et al., 1993) (LBM computations were performed by G. Zanetti). Specifically, the spectral element method described earlier in Section 14.1 was employed, and the incompressible Navier-Stokes equations were solved. The version of LBM used was similar to the method of (Higuera and Succi, 1989), which was made Galilean invariant by using a different choice of equilibrium distribution function. The resulting numerical scheme has some of the flavor of a finite-volume technique. It is, however, more microscopic in nature because it still involves the integration of a Boltzmann equation but in a limited phase

FIGURE 15.19. Pressure contours for the shear-driven flow past the shaded body. The top wall is moving, while the lower wall is stationary; Re = 110 (Karniadakis et al., 1993).

space with only a finite number of possible particle velocities. This new model successfully passed a series of consistency tests: tests of rotational symmetry, tests of the stress tensor, tests of Galilean invariance, sound, and shear waves. Specifically, we present the simulation of a sheared flow in a channel with a bluff body. This flow has simple boundary conditions: periodic in    the    flow direction    and no-slip    on    the    bottom    wall    and on    the

body. The flow is driven by a moving top wall. Moreover, the flow is steady in the range of Reynolds number considered.

The domain was mapped onto a parallelogram of 128 x 128 lattice sites, with a    top    wall    velocity    of U =    0.042, and    the    reduced density,    p,    of the

LBE fluid was chosen to be p = 0.3; the kinematic viscosity was adjusted so that the macroscopic Reynolds number defined using V0, the height of the channel, and the kinematic viscosity of the fluid was Re = 110. The same simulation was repeated using a higher resolution, i.e., 192 x 192 lattice sites, and a small discrepancy was found, less than 2% between the two runs. The results of the LBM simulations were rescaled appropriately for the comparison with the spectral element results. It is worth mentioning that there are no free parameters in this conversion process, except for an arbitrary constant added to the pressure. In the reference solution computed using the spectral element method the flow domain was subdivided into 28 elements, and to test for convergence, the same problem was run with three different spectral resolutions, 5 x 5, 7 x 7, and 9 x 9 modes per

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