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Summary

    Velocity boundary condition generating synthetic turbulence-alike
    time-series for LES and DES turbulent flow computations.

    To this end, two synthetic turbulence generators can be chosen:
    - Digital-filter method-based generator (DFM)

    \verbatim
    Klein, M., Sadiki, A., and Janicka, J.
        A digital filter based generation of inflow data for spatially
        developing direct numerical or large eddy simulations,
        Journal of Computational Physics (2003) 186(2):652-665.
        doi:10.1016/S0021-9991(03)00090-1
    \endverbatim

    - Forward-stepwise method-based generator (FSM)

    \verbatim
    Xie, Z.-T., and Castro, I.
        Efficient generation of inflow conditions for large eddy simulation of
        street-scale flows, Flow, Turbulence and Combustion (2008) 81(3):449-470
        doi:10.1007/s10494-008-9151-5
    \endverbatim

    In DFM or FSM, a random number set (mostly white noise), and a group
    of target statistics (mostly mean flow, Reynolds stress tensor profiles and
    length-scale sets) are fused into a new number set (stochastic time-series,
    yet consisting of the statistics) by a chain of mathematical operations
    whose characteristics are designated by the target statistics, so that the
    realised statistics of the new sets could match the target.

    Random number sets ---->-|
                             |
                         DFM or FSM ---> New stochastic time-series consisting
                             |           turbulence statistics
    Turbulence statistics ->-|

    The main difference between DFM and FSM is that the latter replaces the
    streamwise convolution summation in DFM by a simpler and a quantitatively
    justified equivalent procedure in order to reduce computational costs.
    Accordingly, the latter potentially brings resource advantages for
    computations involving relatively large length-scale sets and small
    time-steps.

Resolved bugs (If applicable)

Verified for serial, scotch-parallel (4), hierar.-parallel (1 2 2), hierar.-parallel (1 2 4), serial-restart, and parallel-restart in terms of input Reynolds stress tensor components through channel395DFSEM tutorial (one-cell domain).

Checked for various possible (commonly encountered) wrong inputs, e.g. arbitrary Reynolds stress tensor components.

Details of new models (If applicable)

The model input:

1. Spatial-variant Reynolds stress symmetric tensor (6-components)
2. Spatial-variant mean velocity profile
3. Spatial-invariant (for now) integral-length scale tensor (9-components)

The model output: Stochastic time-series involving the statistics of the model input sets.

The model computation has four subsequent steps:

1. Generation of random-number sets obeying the standard normal probability distribution function
2. Analytical computation of digital-filter coefficients as a function of integral-length scales in either Gaussian or exponential form
3. Convolution summation between random-number sets and digital-filter coefficients
4. Embedment of Reynolds stress tensor and mean velocity input into the digital-filtered random-number sets via elementwise multiplication and summation

Fidelity:

Preliminary statistically-stationary results from a channel-height profile on the patch (one-cell domain channel395DFSEM case: hierar.-parallel (1 2 4)):

Preliminary not-statistically developed (0.6 sec run, ongoing) with non-optimal input results from full channel395DFSEM case:

Performance:

Preliminary comparisons with DFSEM suggests that the current model is ~1.8x faster for the channel395DFSEM tutorial.

Risks

1. Model is itself not divergence-free (yet convertible); therefore, should not be preferred for aeroacoustic applications as is. Nonetheless, the mass flow rate correction reduces the inlet pressure fluctuations to the level of Poletto et al.'s DFSEM (quantified and verified by Bercin in comparison to Moser et al'. DNS data for pressure fluctuations and correlations).
2. For now, Taylor's frozen turbulence hypothesis is applied in the streamwise direction.
3. For now, bilinear interpolation is not fully functional.
4. Code duplications with DFSEM exist for template funcs.
5. For now, integral-length scale set (9-components) is spatial-invariant across patch.
6. Further verification is ongoing through high-order statistics from Moser et al.'s DNS data, e.g. correlations, kinetic energy budget, enstrophy and so on.