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Dynamic mode decomposition of separated flow over a finite blunt plate: time-resolved particle image velocimetry measurements

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Abstract

Dynamic mode decomposition (DMD) analysis was performed on a large number of realizations of the separated flow around a finite blunt plate, which were determined by using planar time-resolved particle image velocimetry (TR-PIV). Three plates with different chord-to-thickness ratios corresponding to globally different flow patterns were particularly selected for comparison: L/D = 3.0, 6.0 and 9.0. The main attention was placed on dynamic variations in the dominant events and their interactive influences on the global fluid flow in terms of the DMD analysis. Toward this end, a real-time data transfer from the high-speed camera to the arrayed disks was built to enable continuous sampling of the spatiotemporally varying flows at the frequency of 250 Hz for a long run. The spectra of the wall-normal velocity fluctuation, the energy spectra of the DMD modes, and their spatial patterns convincingly determined the energetic unsteady events, i.e., St = 0.051 (Karman vortex street), 0.109 (harmonic event of Karman vortex street) and 0.197 (leading-edge vortex) in the shortest system L/D = 3.0, St = 0.159 (Karman vortex street) and 0.242 (leading-edge vortex) in the system L/D = 6.0, and St = 0.156 (Karman vortex street) and 0.241 (leading-edge vortex) in the longest system L/D = 9.0. In the shortest system L/D = 3.0, the first DMD mode pattern demonstrated intensified entrainment of the massive fluid above and below the whole plate by the Karman vortex street. The phase-dependent variation in the low-order flow field elucidated that this motion was sustained by the consecutive mechanisms of the convective leading-edge vortices near the upper and lower trailing edges, and the large-scale vortical structures occurring immediately behind the trailing edge, whereas the leading-edge vortices were entrained and decayed into the near wake. For the system L/D = 6.0, the closely approximated energy spectra at St = 0.159 and 0.242 indicated the balanced dominance of dual unsteady events in the measurement region. The Karman vortex street was found to induce considerable localized movement of the fluid near the trailing edges of the plate. However, the leading-edge vortices near the trailing edge were found to detach away from the plate and fully decay around 0.5D behind the trailing edge, where a well-ordered origination of the downstream large-scale vortical structures (the Karman vortex street) was established and might be locally energized by the decayed leading-edge vortex. In the longest system L/D = 9.0, the phase-dependent variations in the low-order flow disclosed a rapid decay of the leading-edge vortices beyond the reattachment zone, reaching the fully diffused state near the trailing edges. Accordingly, no clear signature of the interaction between the Karman vortex street and the leading-edge vortex could be found in the dynamic process of the leading-edge vortex.

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Abbreviations

A :

Matrix comprised of mode coefficients \(a_{i} (t)\)

B :

System matrix

C :

Companion matrix

\(\tilde{C}\) :

Full companion matrix

D :

Thickness of the blunt plate (m)

G vv :

Power spectra of wall-normal fluctuation (dB)

L :

Chord of the blunt plate (m)

M :

Finite number of POD modes

R :

Temporal correlation matrix

Re D :

Reynolds number based on the thickness of the blunt plate

S :

Infinite Vandermonde matrix

\(\tilde{S}\) :

Finite Vandermonde matrix

St :

Strouhal number (St = fD/U 0)

K :

Koopman operator

\(U\) :

Discrete velocity matrix

\(U_{0}\) :

Free-stream velocity (m s−1)

\(a_{i} (t)\) :

Mode coefficients of POD

\(c_{i}\) :

\(c = \left( {c_{1} , \ldots ,c_{N - 1} ,c_{N} } \right)\)

d :

Diameter of glass beads (m)

e :

\(e = \left( {0, \ldots ,0,1} \right)^{\text{T}}\)

f :

Frequency (Hz)

f():

Map comparing a manifold to itself

g :

Velocity gradient tensor

g():

Vector-valued observable

r :

Residual vector

t :

Time (s)

T :

Period of the referenced unsteady event (s)

u :

Time-averaged streamwise velocity (m s−1)

\(\bar{u}\) :

Time-mean velocity (m s−1)

u′:

Fluctuation velocity (m s−1)

u k :

Instantaneous velocity (m s−1)

\(v_{j}\) :

Koopman mode

x :

Streamwise coordinate (m)

\(\Delta{x}\) :

Distance between two adjacent points along the streamwise direction

x 0 :

Streamwise coordinate of the reference point (m)

y :

Wall-normal coordinate (m)

y 0 :

Wall-normal coordinate of the reference point (m)

\(\varLambda\) :

Diagonal matrix with eigenvalues

\(\lambda_{i}\) :

Eigenvalues of the Koopman operator

\(\mu\) :

Eigenvalues mapping onto the complex plane

\(\rho\) :

Density of the glass beads (kg m−3)

\(\rho_{vv}\) :

Cross-correlation coefficient of the wall-normal velocity fluctuation

\(\sigma_{k} \left( x \right)\) :

Spatial eigenfunctions

Φ :

Dynamic modes

\(\left\| \varPhi \right\|\) :

Energy norm of dynamic mode

\(\varphi_{i}\) :

Koopman eigenfunctions

CMOS:

Complementary metal-oxide semiconductor

DMD:

Dynamic mode decomposition

FFT:

Fast Fourier transformation

POD:

Proper orthogonal decomposition

RAID:

Redundant arrays of inexpensive disks

SSD:

Solid-state disk

TR-PIV:

Time-resolved particle image velocimetry

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Acknowledgments

The authors gratefully acknowledge financial support for this study from the National Natural Science Foundation of China (Grant Nos. 51176108 and 11372189).

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Authors and Affiliations

  1. Key Lab of Education Ministry for Power Machinery and Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

    Yingzheng Liu & Qingshan Zhang

  2. Gas Turbine Research Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

    Yingzheng Liu & Qingshan Zhang

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  1. Yingzheng Liu
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Correspondence to Yingzheng Liu.

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Liu, Y., Zhang, Q. Dynamic mode decomposition of separated flow over a finite blunt plate: time-resolved particle image velocimetry measurements. Exp Fluids 56, 148 (2015). https://doi.org/10.1007/s00348-015-2021-8

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