^{1,a)}, Bashar R. Qawasmeh

^{1}, Matthew Barone

^{2}, Bart G. van Bloemen Waanders

^{2}and Lin Zhou

^{1,b)}

### Abstract

The aim of this work is to develop nonlinear low-dimensional models to describe vortex dynamics in spatially developing shear layers with periodicity in time. By allowing a free variable to dynamically describe downstream thickness spreading, we are able to obtain basis functions in a scaled reference frame and construct effective models with only a few modes in the new space. To apply this modified version of proper orthogonal decomposition (POD)/Galerkin projection, we first scale the flow along *y* dynamically to match a template function as it is developing downstream. In the scaled space, the first POD mode can capture more than 80% energy for each frequency. However, to construct a Galerkin model, the second POD mode plays a critical role and needs to be included. Finally, a reconstruction equation for the scaling variable *g* is derived to relate the scaled space to physical space, where downstream spreading of shear thickness occurs. Using only two POD modes at each frequency, our models capture the basic dynamics of shear layers, such as vortex roll-up (from a one-frequency model) and vortex-merging (from a two-frequency model). When arbitrary excitation at different harmonics is added to the model, we can clearly observe the promoting or delaying/eliminating vortex merging events as a result of *mode competition*, which is commonly demonstrated in experiments and numerical simulations of shear layers.

We thank Professor Clancy Rowley for constructive discussion. M.W. and B.Q. also gratefully acknowledge the support from Sandia-University Research Program (SURP). Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

I. INTRODUCTION

II. SIMULATION OF SPATIALLY DEVELOPING SHEAR LAYERS

III. PARABOLIZATION OF GOVERNING EQUATIONS

IV. LOW-DIMENSIONAL MODELS

A. Scaling the flow dynamically

B. Equations in scaled space

C. Equation for scaling variable

D. Galerkin projection

V. RESULTS AND DISCUSSIONS

A. Basic model without artificial excitation

B. Forced model with artificial excitations

VI. CONCLUSIONS

### Key Topics

- Navier Stokes equations
- 28.0
- Rotating flows
- 28.0
- Vortex dynamics
- 16.0
- Numerical modeling
- 13.0
- Shear flows
- 12.0

## Figures

Schematic of the two-dimensional free shear layer simulation.

Schematic of the two-dimensional free shear layer simulation.

Snapshots of the shear layer at different times: (a) *t*=50.4, (b) *t*=60, (c) *t*=69.6, (d) *t*=79.2, (e) *t*=88.8, and (f) . Contours show vorticity .

Snapshots of the shear layer at different times: (a) *t*=50.4, (b) *t*=60, (c) *t*=69.6, (d) *t*=79.2, (e) *t*=88.8, and (f) . Contours show vorticity .

The thickness growth along *x* direction while the shear layer is developing; sample vortex structure is shown for comparison.

The thickness growth along *x* direction while the shear layer is developing; sample vortex structure is shown for comparison.

for POD modes at : (a) and (b) . The thin solid line represents the real value, the thin dashed line represents the imaginary value, and the thick solid line represents the absolute value.

for POD modes at : (a) and (b) . The thin solid line represents the real value, the thin dashed line represents the imaginary value, and the thick solid line represents the absolute value.

of the instability mode for . The thin solid line represents the real value, the thin dashed line represents the imaginary value, and the thick solid line represents the absolute value.

of the instability mode for . The thin solid line represents the real value, the thin dashed line represents the imaginary value, and the thick solid line represents the absolute value.

for POD modes at : (a) and (b) . The thin solid line represents the real value, the thin dashed line represents the imaginary value, and the thick solid line represents the absolute value.

of the instability mode for . The thin solid line represents the real value, the thin dashed line represents the imaginary value, and the thick solid line represents the absolute value.

Comparison of (a) direct projection from DNS data to (b) 2-mode model results and (c) 4-mode model results: _____, real part of mode coefficient and ; _ _ _ _, real part of mode coefficient and ; and _ _ _, shear-layer thickness .

Comparison of (a) direct projection from DNS data to (b) 2-mode model results and (c) 4-mode model results: _____, real part of mode coefficient and ; _ _ _ _, real part of mode coefficient and ; and _ _ _, shear-layer thickness .

Relation between the shear layer thickness variation and the sudden change of the phase difference of the first two POD modes: (a) from the projection of full simulation, (b) from the solution of 2-mode model, (c) from the solution of 4-mode model. Notations are _______ phase difference between and , phase difference between and , and _ _ _ the shear layer thickness .

Relation between the shear layer thickness variation and the sudden change of the phase difference of the first two POD modes: (a) from the projection of full simulation, (b) from the solution of 2-mode model, (c) from the solution of 4-mode model. Notations are _______ phase difference between and , phase difference between and , and _ _ _ the shear layer thickness .

Comparison of flow fields at time visualized from: (a) DNS data, (b) projection of DNS data onto 2 POD modes at , (c) 2-mode model, (d) projection of DNS data onto all 4 modes, and (e) 4-mode model. Contours show vorticity .

Comparison of flow fields at time visualized from: (a) DNS data, (b) projection of DNS data onto 2 POD modes at , (c) 2-mode model, (d) projection of DNS data onto all 4 modes, and (e) 4-mode model. Contours show vorticity .

Comparison of forcing at frequency with mode (1,1): different amplitudes of forcing are applied at 0 (a), 0.5 (b), 1.0 (c), and 1.5 (d); _______, real part of mode coefficient and ; _ _ _ _, real part of mode coefficient and ; _ _ _, shear-layer thickness .

Comparison of forcing at frequency with mode (1,1): different amplitudes of forcing are applied at 0 (a), 0.5 (b), 1.0 (c), and 1.5 (d); _______, real part of mode coefficient and ; _ _ _ _, real part of mode coefficient and ; _ _ _, shear-layer thickness .

Comparison of forcing at frequency with mode (2,1): different amplitudes of forcing are applied at 0 (a), 1.0 (b), 2.0 (c), and 3.0 (d); _______, real part of mode coefficient and ; _ _ _ _, real part of mode coefficient and ; _ _ _, shear-layer thickness .

Comparison of forcing at frequency with mode (2,1): different amplitudes of forcing are applied at 0 (a), 1.0 (b), 2.0 (c), and 3.0 (d); _______, real part of mode coefficient and ; _ _ _ _, real part of mode coefficient and ; _ _ _, shear-layer thickness .

Vorticity field snapshots (top) (at , 648, 672, 696, 720, 744, 768, 792, and 816) and average thickness variations (bottom) from: (a) DNS projection on 4 modes, (b) 4-mode model, (c) 4-mode model with excitation at *k*=1, and (d) 4-mode model with excitation at *k*=2. Contours show vorticity .

Vorticity field snapshots (top) (at , 648, 672, 696, 720, 744, 768, 792, and 816) and average thickness variations (bottom) from: (a) DNS projection on 4 modes, (b) 4-mode model, (c) 4-mode model with excitation at *k*=1, and (d) 4-mode model with excitation at *k*=2. Contours show vorticity .

## Tables

Energy captured by different POD modes.

Energy captured by different POD modes.

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