Reduced Order Models of the Navier-Stokes Equations of Fluid Flows

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Turbulence is the apparently random motion that occurs commonly in almost all fluid flows. It has been called one of the most challenging unsolved problems in Newtonian physics. The aggregate cost to our society resulting from our incomplete understanding of turbulence is significant. Consider, for example, the environmental and economic costs associated with sub-optimal performance of virtually every fluid-thermal system such as internal combustion engines and air-conditioners. It is usually not possible to directly and routinely simulate turbulent flow because its multi-scale characteristics require prohibitively high computational resources. Even when adequate computational resources are available, simulations often provide too little understanding of the solutions they produce. There are significant scientific and engineering benefits in developing and studying (reduced-order) models of turbulence that retain the needed physical fidelity while substantially reducing the size and cost of the computational model. The ultimate goal of reduced-order modeling of turbulence is to provide efficient and accurate solutions with minimal reliance on auxiliary empirical models. Empirical models are inherently undesirable because they degrade simulation accuracy and reliability.The research objective of this project is to develop a reduced-order modeling approach to turbulent fluid flows that is free of empirical closure models. Unlike traditional approaches, the new methodology does not rely on empirical turbulence modeling or ad hoc modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new basis functions also provide stable reduced-order models. The approach is illustrated with three test cases: two-dimensional flow inside a square lid-driven cavity, two-dimensional mixing layer, and three-dimensional turbulent flow around the Ahmed body. Future work will extend this method to more complex flows including the effects of higher spatial dimensions and higher flow velocities (Reynolds numbers).