Multiphase Flows

Model of Droplet Impingement Based on Least-Squares Solution of Proper Orthogonal Decomposition Basis Matrices

[+] Author and Article Information
M. S. Hanchak

 University of Dayton Research Institute Dayton, OH 45469Michael.Hanchak@udri.udayton.edu

L. W. Byrd

 Air Force Research Laboratory, Wright-Patterson AFB, OH 45433Larry.Byrd@wpafb.af.mil

A. M. Briones

 University of Dayton Research Institute, Dayton, OH 45469brioneam@notes.udayton.edu

J. S. Ervin

 University of Dayton Research Institute, Dayton, OH 45469Jamie.Ervin@udri.udayton.edu

S. A. Putnam

 Universal Technology Corporation, Dayton, OH 45432Shawn.Putnam@wpafb.af.mil

J. Fluids Eng 134(4), 041301 (Mar 27, 2012) (8 pages) doi:10.1115/1.4006226 History: Received October 05, 2011; Revised February 21, 2012; Published March 27, 2012; Online March 27, 2012

A reduced order model of the surface profiles of droplets impinging on a flat surface is presented based on axisymmetric, transient computational fluid dynamics (CFD). The free surfaces resulting from the volume-of-fluid simulations were interpolated in polar coordinates and arranged as rectangular matrices (time versus space). Proper orthogonal decomposition was then used to expand the data into sets of temporal and spatial basis vectors, which were truncated beyond diminishing singular values. The reduced model is a general linear combination of constant matrices and dimensionless parameters that, when combined, recreate the temporal and spatial basis vectors for each case. The constant matrices were determined with a least-squares solution to the overdetermined linear combinations. To predict a new case, the initial Reynolds, Weber, and Ohnesorge numbers were combined with the calculated constant matrices to determine the new basis vectors, which were used to create the new free surface profile. A new case predicted by the model was validated using a CFD simulation. The single maximum error between the CFD profile and the general linear model was approximately 9% of the initial droplet diameter. The root-mean-squared error for the entire droplet motion was approximately 2% of the initial droplet diameter.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

The reduced SVD is represented visually in terms of matrices

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Figure 2

This is a sequence of images taken from the CFD model of a droplet impinging on a flat surface. The droplet diameter is 54 μm. The time ranges from 0 to 293.3 μs. Adapted with permission from [1]. Copyright 2010 American Chemical Society.

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Figure 3

Instantaneous images of the droplet-gas interface are shown for the case of Weber number 1.4. The spatial and temporal scales are in units of μm and μs, respectively. All data is axisymmetric about the left vertical axis.

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Figure 4

This series of plots illustrates the discretization of the raw profile data. The lower-right plot shows the profile reconstructed from the reduced-order POD (red circles) versus the raw interpolated data (green circles).

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Figure 5

This is a plot of the singular values of the SVD

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Figure 8

Model predictions (red circles) compared to the original profiles (black lines). Each column represents a separate impingement case. The rows advance in time from top to bottom and the spatial units are μm.

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Figure 9

The reduced spatial basis vectors plotted together

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Figure 6

The rms error between the reduced POD reconstruction and the original data set diminishes with the order of the reduction

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Figure 7

The first five basis vectors from U (temporal) and V (spatial) are plotted against the number of elements of each. These are from case 3 in Table 1.

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Figure 10

A three-dimensional rendering of the droplet from case 3 (Table 1) at the fifth time step is shown. The vertical line is the axis of symmetry.



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