The investigators will study improved computational algorithms for model reduction of complex fluid flows. This research will have two main thrusts, new closure models and a parametric modeling framework. The investigators will investigate closure for models that are based on the proper orthogonal decomposition. This component of the research will leverage current developments in large eddy simulation along with two-level algorithms for efficient computational implementation. The fact that the closure models follow from both mathematical and physical arguments will allow for a methodology that generalizes to physical settings other than fluid flow. A new globalization methodology that combines more accurate local models with an interpolation framework to evaluate the models in parameter space will also be studied. The local model improvements are provided using flow sensitivity analysis and basis functions are constructed using a multidimensional Hermite interpolation method. The investigators will emphasize problems in geophysical fluid dynamics as well as flows with free convection.
This research will expand the applicability of computationally efficient simulations to a wider class of fluid flows. The availability of dramatically faster simulations is important to many engineering applications including optimization and control of fluid flows, data assimilation and uncertainty quantification. Therefore, the research conducted in this project has direct application to a wide range of problems that include data assimilation in climate and weather modeling as well as simulation, optimization, and control of energy efficient buildings.
Physical models of complex systems often lead to simulations that require a large computational effort. Simulating the full-order discretization of the model can provide insights into modeling assumptions and can be used to accurately predict the behavior of physical systems at specific values of the model parameters. However, reduced-order models (ROMs) are required for a wide range of practical problems where either ``real-time’’ simulation is required (for applications such as data assimilation and model predictive control) or where a low dimensional problem size is required (for applications such as dynamical systems theory and most control algorithms). Reduced-order models are also key ingredients in a number of algorithms for important problems such as uncertainty quantification, design optimization, and parameter estimation. This research project has delivered numerous algorithmic and theoretical advances in reduced-order modeling methods for complex fluid flows. The most common method for producing ROMs for these flows combines a low-dimensional basis for the flow computated using the proper orthogonal decomposition (POD) with a low-dimensional differential equation for basis coefficients derived using a Galerkin projection of the model equations onto the POD basis. A number of improvements to this process includes developing better methods for creating the low-dimensional bases and including models of the influence of the discarded (truncated) basis functions on the retained basis functions. These include incorporating parametric derivatives of the solutions in creating the ROMs to produce models that are more accurate as parameters vary as well as closure models based upon large-eddy simulation. For low Reynolds number flows, we developed strategies for incorporating parametric derivatives of the flow into the ROMs. Using derivatives of the flow to compute derivatives of the POD basis with respect to important problem parameters allowed us to propose several strategies for improving the accuracy of ROMs both in terms of reproducing the dominant flow features across a range of parameters and capturing important dynamical properties such as how the Strouhal number changes with Reynolds number. The developed algorithms are very efficient though require parametric derivatives of the simulation. Software was developed to calculate this derivative information by applying operator overloading to Matlab programs was developed to aid this study and is freely available at (https://github.com/jborggaard/AD_Deriv). Applying the traditional POD/Galerkin approach to mildly turbulent flows requires additional terms to stabilize the model. The most common approaches either modify terms in the coefficient model or add additional dissipative terms. Our approach was to develop and test a physically motivated modeling strategy based on the success of large eddy simulation (LES). Specifically, we demonstrated success with both the dynamic subgridscale model (DSM) as well as the variational multiscale model (VMS). This lead to superior ROMs without the need to "tune" model parameters. Truncation of the POD basis played the role of LES filtering and the two models (DSM and VMS) achieved more accurate ROMs for complex 3D mildly turbulent flows. Studies included flow past a circular cylinder at a Reynolds number of 1000 and Rayleigh-Benard convection at a Reynolds number of 6800.
