In numerical simulations of fluid flows, just as in experiments, we often question the accuracy of the results, and we construct error bars that reflect the numerical accuracy of the solution. In many cases, however, there exists a much larger error associated with the fact that the physical parameters, the geometry, and the operating conditions of the simulated flow are not precisely known. With the computational fluid dynamics field reaching now some degree of maturity, we naturally pose the more general question of how to model uncertainty and stochastic input mathematically, and how to develop new algorithms that will yield simulation results that reflect accurately the propagation of uncertainty. To this end, the Monte Carlo approach can be employed but it is computationally expensive and it is only used as the last resort.
In this grant we develop a new approach similar to high-order finite element methods but instead decomposing the random domain. Specifically, we extend the pioneering ideas of Norbert Wiener in generalized Fourier series -- the so-called polynomial chaos expansion -- and apply it locally to each random element. The resulting system of governing equations forms a set of coupled modified flow equations, which are deterministic and thus can be solved with standard numerical methods. Comparisons with the Monte Carlo method show that the new method is faster by a factor of 100 to 1000 on the average. We propose to document systematically this method and use it to study in detail important problems in high-speed flows and in modeling blood flow in the human arterial tree. The proposed approach will affect fundamentally the way we design new experiments and the type of questions that we can address, while the interaction between simulation and experiment will become more meaningful and more dynamic. This, in turn, will find its way into the design of flow systems equipment and will provide a rigorous reliability framework.
We plan to involve graduate and undergraduate students in the current research and we will develop a specific initiative to attract pre-college female students to mathematics and computational science.
