(98ax) Inverse Approach in CFD Models Using Optimized Lattice Boltzmann Method

Authors: 
Jhon, M. S., Carnegie Mellon University
Biegler, L. T., Carnegie Mellon University



Uncertainties in the output of a complicated phenomena such as turbulent flow simulation are not only caused by uncertainties in its input but may also result from heuristically motivated artificial model parameters. These additional equations in modeling this flow, constituting the so-called turbulence models with higher degrees of freedom (i.e., two equation models), typically involve certain parameters  which can be justified only heuristically from empirical information and experimental data. The adjustment of these turbulence parameters is often not obvious, guided principally by “what worked before,” even if the model is now used in a different application. Thus, an important question in the CFD community now is how the results of the overall simulation depend on the turbulence parameters, where if the results are strongly affected by these parameters, the uncertainty of the simulation is considered to be high.

Recently, mesoscale methodologies based on kinetic theory approach via Boltzmann transport equation, namely lattice Boltzmann method (LBM) is increasingly competing with conventional computational fluid mechanics (CFD) methodologies in understanding complex phenomena of turbulence. LBM approaches to the asymptotically correct hydrodynamic [1,2] and local thermo-hydrodynamic [3,4] limits, and turbulence modeling methodologies can be efficiently incorporated into LBM via its advantages including highly efficient massively parallel computation ability, ease in handling complex geometries, and a “rule based” approach which offers significant flexibility.

Thus, in this work, to gain deeper understanding into the heuristic approach of turbulence modeling, using LBM we perform the parameter estimation of the closure parameters which predict the experimental data, and asses the sensitivity of the results of such a turbulence simulation, a derivative-based sensitivity analysis is carried out via automatic differentiation to investigate the closure parameters. Automatic differentiation (AD) [5] is a chain-rule-based technique for evaluating the sensitivity derivatives with respect to their input variables. In contrast to the approximation of derivatives by difference methods, AD does not incur any truncation error and is computationally efficient. For our benchmark example of turbulent flow simulation, the derivatives are computed via AD by specifying the closure parameters as the independent variables and the velocity components u and vof the velocity in directions x and y as the dependent variables.

In summary, this sensitivity analysis methodology will be used to adjust the approximation models within the system in order to improve its physical simulation or to obtain desired numerical solution properties such as stability or convergence. Furthermore, this analysis will help us to assess the robustness of the LBM, and can lead to deeper insight into the area of turbulence simulation that has never been done before and will lead to the best turbulence models by simply adopting an inverse approach.

  1. H. Chen, S. Chen, and W. Matthaeus, “Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method,” Phys. Rev. E, 45, 5339 (1992).
  2. L.S. Luo and S.S. Girimaji, “Theory of the lattice Boltzmann method: Two-fluid model for binary mixtures,” Phys. Rev. E, 67, 036302 (2003).H. Chen, C. Teixeira, and K. Molvig, “Realization of fluid boundary conditions via discrete Boltzmann dynamics,” Int. J. Mod. Phys., 9(8), 1281 (1998).
  3.  H. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi, and V. Yahkot, “Extended Boltzmann kinetic equation for turbulent flows,” Science, 301, 633 (2003).
  4. H. Chen, C. Teixeira, and K. Molvig, “Realization of fluid boundary conditions via discrete Boltzmann dynamics,” Int. J. Mod. Phys., 9(8), 1281 (1998).
  5. Rall, Louis B. (1981). Automatic Differentiation: Techniques and Applications. Lecture Notes in Computer Science. 120. Springer, Heidelberg.