Practical Aspects of the Implementation of Reduced-Order Models Based on Proper Orthogonal Decomposition
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This work presents a number of the practical aspects of developing reduced- order models (ROMs) based on proper orthogonal decomposition (POD). ROMS are derived and implemented for multiphase ﬂow, quasi-2D nozzle ﬂow and 2D inviscid channel ﬂow. Results are presented verifying the ROMs against existing full-order models (FOM). POD is a method for separating snapshots of a ﬂow ﬁeld that varies in both time and space into spatial basis functions and time coeﬃcients. The partial diﬀerential equations that govern ﬂuid ﬂow can then be pro jected onto these basis functions, generating a system of ordinary diﬀerential equations where the unknowns are the time coeﬃcients. This results in the reduction of the number of equations to be solved from hundreds of thousands or more to hundreds or less. A ROM is implemented for three-dimensional and non-isothermal multiphase ﬂows. The derivation of the ROM is presented. Results are compared against the FOM and show that the ROM agrees with the FOM. While implementing the ROM for multiphase ﬂow, moving discontinuities were found to be a ma jor challenge when they appeared in the void fraction around gas bubbles. A point-mode POD approach is proposed and shown to have promise. A simple test case for moving discontinuities, the ﬁrst order wave equation, is used to test an augmentation method for capturing the discontinuity exactly. This approach is shown to remove the unphysical oscillations that appear around the discontinuityin traditional approaches. A ROM for quasi-2D inviscid nozzle ﬂow is constructed and the results are com- pared to a FOM. This ROM is used to test two approaches, POD-Analytical and POD-Discretized. The stability of each approach is assessed and the results are used in the implementation of a ROM for the Navier-Stokes equations. A ROM for a Navier-Stokes solver is derived and implemented using the results of the nozzle ﬂow case. Results are compared to the FOM for channel ﬂow with a bump. The computational speed-up of the ROM is discussed. Two studies are presented with practical aspects of the implementation of POD- based ROMs. The ﬁrst shows the eﬀect of the snapshot sampling on the accuracy of the POD basis functions. The second shows that for multiphase ﬂow, the cross- coupling between ﬁeld variables should not be included when computing the POD basis functions.
Subjectcomputational fluid dynamics
proper orthogonal decomposition
Brenner, Thomas Andrew (2011). Practical Aspects of the Implementation of Reduced-Order Models Based on Proper Orthogonal Decomposition. Doctoral dissertation, Texas A&M University. Available electronically from