Efforts are currently underway to extend the current LBM for higher Mach number jet flows. Recently, Sun and Hsu 25 used an LBM technique to study a shock tube problem and obtained good results compared to the Reimann solution.
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Shan et al. Recently, Li et al. They obtained good results and showed that their methodology is valid up to a Mach number of 0. The pipe diameter was chosen to match the jet diameter of the experiments of Tanna. The addition of a nozzle in the computational domain was intended to eliminate the need for an artificial forcing mechanism to trip the flow. As discussed later, however, computational resources did not allow a very fine grid within the pipe.
The computational domain is partitioned into several variable resolution VR regions, in order to tailor the grid as needed to resolve the flow details, and reduce computational costs. This methodology is similar to grid stretching techniques typically employed in traditional CFD. Successive VR regions are concentric and cylindrical. The outer most bounding region is rectangular. These tones were as high as 15 dB and appeared to bias the overall spectral makeup.
Providing a steamwise spacing of at least two jet diameters between VR regions actually eliminated these tones in the lower frequency portion. However, the VR tones are still somewhat present in the high frequency range. Hence, each VR region represents one grid resolution level and the VRs cascade outwards from the fine resolution region toward the coarse resolution region.
The voxel cell size between each successive VR region differs by a factor of 2. The ratio needed to resolve the duct boundary layers is at least one order of magnitude less without the implementation of a wall model. This was deemed prohibitively expensive.
Although the adopted cell size did not resolve the boundary layer details, it was sufficiently small to supply physical jet inflow conditions without the need for artificial forcing techniques. These values, which may seem large at first, were chosen to minimize reflections of the propagating sound waves and convected jet plume flow structures back into the near-field.
Non-reflecting type boundary conditions are applied at the inlet lateral and outlet boundaries. Virtual probes are placed on the jet axis centerline and nozzle lip-line along the streamwise direction for the eventual calculations of turbulent spectral content. Additional probes are placed sufficiently far away from the non-linear turbulent region.
These probes are used to compute far-field acoustics. The far-field acoustic measurement probes are placed 15 r o above the centerline jet axis. We consider no mean density or temperature gradients in the nozzle compared to ambient conditions. In order to reasonably resolve the turbulent scales with the available voxel resolution, the kinematic viscosity had to be adjusted. The modeling of the subgrid-scale stress tensor still raises some fundamental issues as discussed by Bogey and Bailly.
An alternative to the use of an explicit eddy-viscosity model is the use of spatial filtering for modeling the effects of the subgrid-scales. Using this alternative, the turbulent energy is only dissipated when it is transferred from the larger scales to the smaller scales discretized by the mesh grid. The unsteady, Favre-filtered, compressible, non-dimensional LES equations were solved.
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Transformation from curvilinear coordinates to a uniform grid was done in computational space. The non-dissipative sixth-order compact scheme developed by Lele 34 was used to compute the solution on the internal points. For the points on the boundaries, a third-order one-sided compact scheme was used.
The points next to the boundaries were computed using a fourth-order compact central differencing technique. For time advancement, the explicit fourth-order Runge—Kutta scheme was used.
In addition, a sponge zone 37 is attached to the end of the computational domain to dissipate the vortices present in the flow field before they hit the outflow boundary. This minimizes unwanted reflections from the outflow boundary. To excite or trip the flow, randomized perturbations in the form of induced velocities from a vortex ring proposed by Bogey and Bailly 38 are added to the velocity profile at a short distance approximately one jet radius downstream from the inflow boundary.
This is done to ensure the break up of the potential core within a reasonable distance. Beyond the streamwise location of 60 r o is the sponge zone. The physical domain length of 60 r o was chosen for two reasons. First, Uzun et al. Second, in order to capture the overall sound pressure levels OASPL adequately at shallow angles, a domain length of at least 55 r o is required based on the recommendations of Shur et al.
The physical extends to 60 r o. Every 3rd grid point is shown.
Noise prediction of a subsonic turbulent round jet using the lattice-Boltzmann method
The vortex ring used included up to 16 azimuthal jet modes. Bogey and Bailly 39 performed a simulation with all modes present and later removed the first four modes and found that the jet was quieter for the latter case, in better agreement with experimental results. Accordingly, the first four azimuthal modes of forcing were not included in the forcing.
Since there is no nozzle-lip in the simulation domain, the hyperbolic tangent velocity profile was used on the inflow boundary given by.
The parameter that controls the thickness of the shear layer is b. A higher value of b implies a thinner shear layer. For laboratory jets however, the measured value of b is usually an order of magnitude higher compared to that used in LES and DNS of jets.
Considering an isothermal jet, the Crocco—Busemann density is accordingly specified as a function of the jet Mach number and inlet velocity profile. The physical time scaling or time step for the LBM is 1. A total of time steps was required to achieve reasonably converged statistics for flow analysis. In terms of computational resources used, this test case took approximately two days of runtime using processors in parallel on a Dell Xeon cluster See Acknowledgments.
In comparison, the LES simulations took approximately 3. The mean streamwise velocity on the jet centerline is plotted in Fig. Here, U j is the jet exit velocity and U c x is the local mean streamwise velocity on the centerline. The LES data shows a slightly faster decay compared to the experiments of Bridges and Wernet but are nonetheless reasonable. When Fig. The experimental correlation of 0. In the present simulation, a potential core length of 12 r o and This value is within the range of core lengths, between 10 and 14 jet radii, typically observed in laboratory experiments.
The simulation results are also compared to recent experimental measurements of Laurendeau et al. Qualitatively, the trends of the axial turbulence intensities for this jet are consistent with those of an axisymmetric turbulent jet; the peak fluctuation for the centerline occurs earlier and is greater compared to the centerline peak fluctuation.
The peak value and the axial location of the computed lip-line turbulence intensity are in agreement with experimental observations. The lower values may be due to the fact that the simulated exit shear layer is still laminar or transitional at best.
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Recall that there is no artificial excitation used in the LBM jet. The Reynolds number of the simulated jet, , is much smaller than the Reynolds number of the measured jet, which was , due to the need for an increased viscosity. The experiments were performed using a converging nozzle rather than a straight pipe. Nonetheless, according to Laurendeau, the initial high turbulence intensities measured along the lip-line in the experiments could be due to the surrounding air not being seeded and a possible influence on how the boundary layer is tripped.
But nevertheless the high intensity at the nozzle exit appears to be very plausible. The centerline mean axial turbulence intensity on the other hand shows a better agreement compared to the experiments of Laurendeau et al. Again, note a slight under-prediction compared to the experiments from the exit to approximately 9 r o.