Dynamic Mode Decomposition of Hydrofoil Cavitation
Jiahao Jia, Juanjuan Qiao and Tingrui Liu
*
College of Mechanical & Electronic Engineering, Shandong University of Science & Technology, Qingdao 266590, China
Keywords: Dynamic Mode Decomposition, DMD.
Abstract: This study performs dynamic mode decomposition (DMD) for the NACA66 cavitation process, and the
obtained modes have stable linear characteristics. The diagrams of different modes and the frequency energies
of the corresponding modes are also analyzed. We found that these different modes capture the flow
characteristics at different frequencies. The mean mode (Mode1) represents the basic flow structure and plays
a dominant role in the cavitation process. Mode2 denotes the cavitated region, and Mode3 and Mode4
represent the cavitated stretch off. The higher-order modes represent the alternating shedding of cavitation
and some high-frequency characteristics in the cavitation. The research in this study is essential for our
understanding of the three-dimensional characteristics of the flow field during cavitation and the three-
dimensional dynamical mode characteristics.
1 INTRODUCTION
During the operation of hydraulic machinery, when
the partial pressure is less than the saturated vapor
pressure of water, the form of water will change from
liquid to vapor, called cavitation. With the decrease
of the cavitation number, the cavitation phenomena
are successively manifested as cavitation inception,
sheet cavitation, cloud cavitation, and supercavitation.
Cavitation has obvious unsteady characteristics, and
it is easy to cause adverse effects on the hydrofoil,
such as pressure fluctuation, vibration, noise, and
erosion (Asnaghi, 2018; Dang, 2019). Hydrofoils are
the most fragile and most prone to unsteady cavitation
among turbomachinery, propeller, and other devices.
Therefore, it is crucial to analyze hydrofoil cavitation
and understand the physical mechanisms involved in
this destabilizing phenomenon.
Scholars and experts in related fields have done a
lot of research work to analyze the physical principles
of cavitation and use various methods to predict and
analyze the cavitation phenomenon. It contains two
data-driven methods for research, POD (proper
orthogonal decomposition) and DMD (dynamic
mode decomposition) (Taira, 2020; Taira, 2017).
Both ways use previous experimental or simulated
data to analyze and predict the development of
cavitation.
Liu (Liu, 2019) used POD and DMD methods for
the coherent structure of ALE15 hydrofoil cavitation.
They found that DMD is more advantageous than
POD for decomposing complex flows into uncoupled
coherent structures. Chen (Chen, 2012) proposed an
improved DMD method and tested it in low Reynolds
number in-cylinder fluid flow. To achieve a balance
between the number of modes and computational
efficiency, Jovanović (Jovanovi ć, 2014) proposed
sparsity to facilitate Sparsity-promoting dynamic
mode decomposition DMD, which performed well in
Poiseuille flow, supersonic flow, and two-cylinder
jet. Kou and Zhang (Kou, 2017) proposed an
additional criterion DMD (DMDc), which uses an
improved criterion to align the flow modes and
performs well in airfoil flutter. Grilli (Grilli, 2012)
used the DMD method to study the unstable behavior
of the shock-turbulent boundary layer and obtained
that the low-frequency mode was related to the
separation and impact of the bubble.
Although predecessors have conducted extensive
and in-depth research on cavitation, it can be seen that
dynamic mode decomposition is a relatively new
analysis and research method by combing the latest
recent research, so it has high research value. In this
study, the unsteady cavitation process of NACA66
was analyzed by DMD. The contour of different
modes are shown, and the frequency energy of
different modes is analyzed. We found that these
modes capture the flow characteristics at different
frequencies. The research in this study is of great
significance for us to understand the three-
dimensional characteristics of the flow field and the
three-dimensional dynamic mode characteristics in
the cavitation process.
102
Jia, J., Liu, T. and Qiao, J.
Dynamic Mode Decomposition of Hydrofoil Cavitation.
DOI: 10.5220/0012150100003562
In Proceedings of the 1st International Conference on Data Processing, Control and Simulation (ICDPCS 2023), pages 102-108
ISBN: 978-989-758-675-0
Copyright
c
2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
2 NUMERICAL SIMULATION
AND METHODOLOGY
2.1 Numerical Simulation
The numerical simulation of the NACA66 hydrofoil
is carried out, and the computational domain is a
square tunnel with a length of 1000mm and a width
of 192mm, as shown in Figure 1. The leading edge of
the hydrofoil is 2c away from the velocity inlet, and
the angle of attack(AOA) is 6°. The turbulence model
adopts SST k-omega, and the cavitation model
chooses Schnerr-Sauer. Then the inlet boundary
condition is set to velocity inlet, and the velocity is set
to 5.33m/s. The outlet boundary condition is set to the
pressure outlet, and the pressure is set to 19000Pa.
Finally, the no-slip boundary conditions are set on
both side walls, and the upper and lower walls are set
with slip walls.
Figure 1: Computing domain and mesh arrangement.
2.2 Methodology
Figure 2: Matrix of cavitation velocity data.
This chapter introduces the basic principles and steps
of the Dynamic Mode Decomposition DMD method.
As shown in Figure 2, successive snapshots of the
flow field variables with the same time interval Δ𝑡
are preprocessed to form column vectors, and then the
column vectors of each time snapshot are combined
to form a matrix 𝑉
.
𝑉
=
𝑣
, 𝑣
, 𝑣
, ⋯, 𝑣
(1)
Among them,
𝑣
represents the i-th flow field
variable snapshot, and N is the number of selected
flow field variable snapshots. Our goal is to build a
linear dynamic system A. Among them,
⃗
= 𝐴𝑣⃗
.
Linear system A transforms present data
𝑣
into
future data
𝑣
. Therefore, the linear dynamic system
A satisfies the following conditions.
𝑉
=
𝐴
𝑉
(2)
Where 𝑉
represents the future moment of
𝑉
. A linear dynamical system can be represented
Dynamic Mode Decomposition of Hydrofoil Cavitation
103
using the pseudo-inverse matrix (𝑉
)
of 𝑉
.
𝐴
= 𝑉
(𝑉
)
(3)
Linear dynamic system Least squares fit the
present state and future state, and we call this exact
DMD.
As shown in Equation 4, perform singular value
(SVD) decomposition on the data at the present
moment.
𝑉
= 𝑈Σϒ
*
(4)
Generally speaking, due to the high
dimensionality of 𝑉
, this results in excessive
computation. So we use the low-rank structure for
SVD. It can be represented as follows:
𝑉
= 𝑈
Σ
ϒ
*
(5)
So the linear dynamic system 𝐴
×
can be
expressed as Equation 6 by replacing the pseudo-
inverse of 𝑉
𝐴
×
= 𝑉
(𝑉
)
= 𝑉
ϒ
Σ
𝑈
*
(6)
Although the linear dynamic system 𝐴
×
has
been calculated, its dimension is still too large, and
the following transformation is applied to reduce its
dimension:
𝐴
×
= 𝑈
*
𝐴
𝑈
= 𝑈
*
(𝑉
ϒ
Σ
𝑈
*
)𝑈
= 𝑈
*
𝑉
ϒ
Σ
(7)
Now the low-dimensional linear dynamic system
𝐴
~
can be defined as:
𝐴
~
=
𝐴
×
∈ ℝ
×
, 𝑟≪𝑛.
(8)
𝐴
~
is a low-dimensional linear dynamic system,
which is simple to calculate the eigenvalue and
eigenvector:
𝐴
~
𝑊 = 𝑊Λ
(9)
Where 𝑊 is the eigenvectors and Λ is the eigen
values
In the previous steps, the feature vector 𝑊 is
calculated in the low-dimensional space, and the 𝑊
obtained by the low-dimensional calculation can be
returned to the original high-dimensional space by the
following algorithm.
Φ = 𝑉
¡
Σ
-
𝑊
(10)
Among them, Φ is the dynamic modal
decomposition mode (DMD modes) of the original
space, and the eigenvalue Λ has not changed.
Through the obtained linear dynamic system, the
eigenvector Φ and eigenvalue Λ are obtained by
calculation, and then the variable 𝑣 is calculated
using the eigenvector and eigenvalue to achieve the
purpose of reconstructing and predicting the flow
field by using the decomposition mode.
𝑣(𝑡)=Φ𝑒
Ω
𝑏 = 𝜙
𝑒
𝑏
(11)
3 DYNAMIC MODE
DECOMPOSITION(DMD)
As shown in Figure 3, the middle three cavitation
periods were selected for analysis, with times of
0.1938s-0.8750s and a cavitation frequency of
4.404Hz. The DMD decomposition is performed
using velocity sequences in the flow field with a
sampling interval of 0.0025s. Each cavitation cycle
contains 90 snapshots of the flow field, for a total of
270 snapshots.
Figure 3: The periodicity of cavitation.
ICDPCS 2023 - The International Conference on Data Processing, Control and Simulation
104
Figure 4: Eigenvalues in the complex plane.
Figure 5: Dynamic mode energy at different frequencies.
Figure 4 shows the distribution of eigenvalues
corresponding to different modes in the complex
plane, with the horizontal coordinates representing
the real part and the vertical coordinates representing
the imaginary part. It can be seen that the eigenvalue
distribution is symmetric along the real axis of the
complex plane, indicating that the eigenvalues appear
conjugately. The black dashed line represents the unit
circle, and the relative positions between the
eigenvalue points and the unit circle reveal the
respective properties of the corresponding modes.
The eigenvalues inside the unit circle indicate that this
mode is convergent, and the modes outside the circle
gradually increase in instability and eventually are
divergent. Concurrently, the modes located on the
unit circle are stable and do not change with time.
Almost all eigenvalues are within the unit circle
indicating good convergence of each mode after mode
decomposition. The modes on the unit circle represent
the linear characteristics of the cavitation process,
while the rest represent the nonlinear characteristics.
The cavitation process explored in this study has
evident periodicity. Most of the modes are located
near the unit circle, which verifies the stability and
simulation accuracy of the cavitation period in this
study.
Figure 5 shows the energy and frequency of each
mode obtained by DMD, where the 2-norm of the
modes defines the energy. According to the energy of
each mode, Mode1 is the most energetic mode with a
frequency of 0Hz, which represents the average flow
field characteristics. The energy of Mode2 decreases
Dynamic Mode Decomposition of Hydrofoil Cavitation
105
significantly, but the energy of Mode2 is still more
considerable than other modes, and its frequency is
4.414 Hz, which is consistent with the cavitation
frequency. Mode3-5 energy gradually decreases,
while the frequency of the modes gradually increases,
and the frequencies are multiples of the cavitation
frequency.
4 RESULT AND DISCUSSION
Figure 6 shows the modes obtained after DMD of the
axial velocity flow field data, and the longitudinal
section of the flow field is intercepted for display.
Where Re(phix) is the real part of the selected mode,
the magnitude of its absolute value represents the
magnitude of the regional energy, and positive or
negative represents the change of phase. Mode1 is the
average mode of the flow field, representing the flow
field's primary flow structure and plays a dominant
role in the whole cavitation process. The frequency of
Mode2 is consistent with the cavitation frequency,
which indicates that the critical periodically changing
structure in the flow field is captured. It can be seen
that the high-energy region of Mode2 is the same as
the cavitation region, indicating that Mode2 responds
to the cavitation region during the flow process. It can
be seen that the distribution of the high-energy
regions of Mode3 and Mode4 alternate along the
velocity direction, and the frequencies of these two
Modes are multiples of the cavitation frequency,
representing the stretching off in the cavitation
process. Mode6 and Mode8 have higher frequencies
and lower energies. The contour shows that the
energy of the high-energy region alternates along the
hydrofoil suction surface and the direction of
cavitation development. The energy of the high-
energy region also decreases gradually along the flow
direction. It represents the alternating shedding of
cavitation and the high-frequency characteristics in
some flow fields.
Figure 7 shows the 3D vortex structure(Q-
criteria=1000) of different Modes in the flow field.
The cavitation process of the hydrofoil is distributed
in three dimensions in the flow channel, which is a
typical flow with three-dimensional spatial
characteristics. Because the middle of the hydrofoil is
unaffected by shear forces on both sides of the wall,
cavitation is more intense. The selected longitudinal
section of the flow field can only show part of the
characteristics of the cavitation process. To explore
the flow field characteristics in more detail, need to
analyze the three-dimensional characteristics.
Figure 6: Different Modes in longitudinal plane of the flow field.
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106
Figure 7: 3D vortex structure(Q-criteria=1000) of different Modes in the flow field.
5 CONCLUSION
In this study, the unsteady cavitation process of
NACA66 was analyzed using DMD. It is found that
the eigenvalues corresponding to different modes are
symmetrically distributed along the real axis on the
complex plane, indicating that the eigenvalues are
conjugated. The eigenvalues are almost above the
unit circle, indicating that the modes obtained by
decomposition are convergent and have stable linear
characteristics, while the modes corresponding to the
eigenvalues deviating from the unit circle have
nonlinear characteristics. The cavitation studied in
this study has obvious periodicity. Most of the modal
eigenvalues are located on the side of the unit circle,
which verifies the accuracy of the simulation and
DMD decomposition. At the same time, the cloud
images of different modes are displayed, and the
frequency energy of different modes is analyzed.
These different modes capture the flow
characteristics at different frequencies. The average
mode represents the main flow structure of the flow
field and plays a dominant role in the entire flow
process. Mode2 represents the cavitation region,
Mode3 and Mode4 represent the pull-off of cavitation,
and higher-order modes represent the alternate
shedding of cavitation and some high-frequency
characteristics in the flow field. The research in this
study is of great significance for us to understand the
three-dimensional characteristics of the flow field and
the three-dimensional dynamic modal characteristics
in the cavitation process.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the support of the
National Natural Science Foundation of China (no.
51675315).
Dynamic Mode Decomposition of Hydrofoil Cavitation
107
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