Faculty of Engineering Technology (CTW)
Investigation into a new simple
DBD-plasma actuation model
Report Internship ISAS/JAXA
s1115103
Thijs Bouwhuis
Contents
1
2
3
4
5
6
7
General Introduction . . . . . . . .
Introduction to the research . . . .
Research method . . . . . . . . . .
3.1
Body force distribution . .
3.2
Parametric study . . . . . .
3.3
Normalization of results . .
Results . . . . . . . . . . . . . . . .
Alternative body force distribution
Conclusion and recommendations .
Bibliography . . . . . . . . . . . .
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1
2
4
4
5
6
9
15
17
18
1
General Introduction
In this report, one can read the results of the study conducted at the Institute of Space and Astronautical Sciences, ISAS, located in Japan. This study is part of my internship, a key element in
the educational program Master Mechanical Engineering, in which skills and knowledge acquired in
previous years have to be used in a real life situation at a graduate level.
In this context, I have visited Japan from October first 2015 till Februari fourth 2016. In this period,
I have worked as an intern at the Institute of Space and Astronautical Sciences, ISAS, located near
Sagamihara, Kanagawa in Japan. ISAS is part of the Japan Aerospace Exploration Agency, JAXA,
founded in 2003 as a fusion of the three major aerospace agencies of Japan: Institute of Space and Astronautical Science (ISAS), National Aerospace Laboratory (NAL) and National Space Development
Agency of Japan (NASDA).
At ISAS I worked at the department of Space Flight Systems. More specific, I was part of the
Fujii-Oyama lab.This lab worked on three topics: Acoustics of rocket jets, a mars airplane and flow
separation control. The last subject, flow separation control, was my topic of research for four months.
I am very grateful to Fujii-sensei and Oyama-sensei for accepting me in their research group. I also
like to thanks Professor Hoeijmakers, for introducing me at ISAS/JAXA and without whom I would
not have been able to experience this great opportunity. Words of gratitude and thanks go to Mr.
Nonomura-san, Ms. Yakeno-san and Mr. Abe-san for their great help during my research and many
fruitful discussions. Thanks also goes to Ms. Tamura-san, for she helped me a lot with arranging all
the practical conditions making this internship possible. Finally, words of thanks go to all members
of the Fujii-Oyama lab since I had an educative and great time, Arigatou gozaimasu.
Thijs Bouwhuis
1
2
Introduction to the research
Active flow separation control using dielectric barrier discharge plasma actuators (hereafter: PA)
has been studied intensively during the last decade, with the aim of improving performance and/or
efficiency of a wide variety of fluid machinery. Flow separation occurs when a boundary layer flow
is unable to overcome an adverse pressure gradient and the velocity of the fluid with respect to the
boundary falls to zero. The flow becomes detached. This phenomenon can occur with a flow around an
airfoil at a high angle of attack. The angle at which the flow starts to separate from the airfoil is called
the stall angle. As a result of the detached flow, a lot of undesirable effects occur. Among others: a
decrease in lift and an increase in drag. By controlling this flow separation, large improvements in
efficiency and performance of airfoils can be achieved.
Flow separation can be controlled by using a PA. The PA consists of two electrodes with a dielectric
material in between. When a high voltage O(103 V ), high frequency AC is applied between the two
electrodes a plasma is created. This plasma induces a wall jet which can be utilized in flow separation
control. A schematic representation of the PA can be observed in figure 1.
Figure 1: Schematic representation of the Dielectric Barier Discharge Plasma Actuator (PA)
The PA has different mechanisms influencing the flow separation. First, the momentum in the boundary layer increases by the wall jet behaviour of the PA. Secondly relative large vortices introduce
additional momentum to the flow and thirdly small scale vortices, turbulence, is introduced which can
transition the laminar flow to turbulent flow.
A conventional numerical method for the PA, which gives results in good agreement with experimental
results [2], solves the flow equations and an additional two equations on an additional grid. It is known
as the Suzen model [1]. The resulting body force field, shown is figure 2, of this model is coupled to
the flow equations.
Much experimental and numerical research is focused on the relation between the operational parameters of the PA (voltage, base- and burst frequencies etc.) and the performance of the PA. However,
the dominant parameters of the body force which determine the induced flow have not been identified
so far.
The main focus of this study will be to identify the dominant parameters of the body force field,
determining the induced flow. To this end the body force field of the Suzen model will be reduced in
complexity and a Gaussian distributed body force will be studied. A future objective is to propose
an new simple PA model, based on the dominant parameters, for which the induced flow is similar to
that of existing models but without additional equations to be solved on an extra grid.
In this report an outline of the research method is given: first a body force distribution is determined
based on the Suzen model. This body force distribution is used in a numerical simulation for two
2
dimensional flow, in order to carry out a parametric study on the parameters of the body force. After
that an outline of the normalization is given, which is used to present the results afterwards. Some
additional, more advanced, work is presented in which the body force distribution is somewhat adjusted.Finally some concluding remarks are given.
Additionally one appendix is added: An abstract of the most important topics of the research, submitted to 11th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements.
3
3
Research method
A parametric study is carried out using numerical simulations for two dimensional flow. The flow
field is described by the Navier-Stokes equations, in which the PA forcing is implemented as a source
term. The number of grid points in the computational grid is 442 × 203. Near the body force
field, the grid is uniformly distributed and grid spacing is small, as illustrated in figure 2. The
computational domain is taken sufficiently large to ensure the far field boundaries do not affect the
induced flow. The grid coarsens further away from the body force field. The computational Mach
number is taken as M a = 0.2 (incompressible upper limit) and the computational Reynolds number is
taken as Re = 63000. A computational Reynolds number Re = 23000 is used as a validation later on.
Sixth-order compact difference schemes [3] were used to discretize the spatial derivatives and ADI-SGS
methods [4] were used for time integration. This is an inhouse CFD code of ISAS/JAXA. No further
commands are made on these schemes and computational methods, since it is not in the scope of this
research. The flow field which is numerically simulated, is the induced flow of a PA in quiescent air. In
the simulation the PA is represented as a body force distribution, on which the next subsection (3.1)
will elaborate. In order to find the dominant parameters of this body force distribution, a parametric
study is set up. How this parametric study is designed will be explained in subsection (3.2). Because
we’re dealing with an induced flow in quiescent air, there is no free stream velocity which can be used
as a reference to normalize the results. To deal with this problem a new normalization is derived in
subsection (3.3).
3.1
Body force distribution
A conventional way to represent a PA in a numerical flow simulation, is by using the Suzen-Huang
model. This model incorporates the effects of the PA on the external flow into the Navier Stokes
computations as a body force vector. Two additional equations are solved in order to compute this
body force vector. One equation for the electric field due to the applied AC voltage at the electrode
of the PA and one equation for the charge density representing the ionized air. This numerical model
is calibrated against an experiment having a PA driven flow in quiescent air. The numerical model
shows good agreement with the experimental results. The body force field distribution is shown is
figure 2. The components of the force in x- andp
z-direction are shown in figures 2(a) and 2(b). The
amplitude of the force is in each point as: (F = fx2 + fz2 ).
Spatial integration of the body force field over the whole flow domain yields the total induced momentum per second, which is denoted as Cµ . In this research we do not use this Suzen-Huang model, but
a more simple spatial Gaussian distributed body force field. This is a temporal constant body force,
(a) x-component of Suzen model (b) z-component of Suzen model (c) Amplitude of Suzen model body
body force
body force
force
Figure 2: The commonly used Suzen-Huang model
4
in general computed by:
(x − x0 )2 (z − z0 )2
f (x, z)x = a exp −
+
2σx2
2σz2
(3.1)
This model can be implemented in the Navier Stokes computations as a body force vector. As
described in the introduction a gaussian body force model will be implemented in the governing
equations. Equation (3.1) is first simplified by setting the aspect ratio to unity (σx = σz = σ) and
setting x0 = 0 and z0 = 0, thus obtaining the following body force distribution:
2
x + z2
f (x, z)x = a exp −
(3.2)
2σ 2
In here the parameters a and σ have to be determined and will be based on the amplitude of the
body force of the suzen model (figure 2). In the amplitude plot (figure 2(c)) one can observe a very
high peak in the amplitude on the left (upstream) side, which will be ommited in the modeling of the
Gaussian profile. Instead we focus on the larger region right of this high peak. Here we determine
the maximum amplitude, which is directly implemented in the Gaussian distribution as the amplitude
(a). The standard deviation (σ) is determined from the Cµ value of the Suzen-Huang model, to ensure
the Gaussian model has the same Cµ value as the Suzen-Huang model.
Note: When determining the Half Width at Half Maximum (HWHM) as a characteristic length of
the Suzen-Huang model and ensuring the Gaussian model has got the same characteristic length, by
setting the standard deviation as σ = √HWHM
, we end up with a Gaussian model which does not
2 · log 2
significantly differ from the one determined above.
Approximate values, used in the numerical model for a and σ are 1290.04 and 4.27E − 4 respectivily.
This gaussian distribution will be referred to as the basic gaussian. A spatial body force distribution
of this basic gaussian can be observed in figure 3.
Figure 3: Basic gaussian, described in equation 3.2
3.2
Parametric study
A parametric study is performed by deviating from the basic gaussian, by variating the parameters
a and σ. The values for σ are chosen as follows (normalised with σ from basic gaussian): 0.50 , 0.75
1
1
, 1.00 , 1.50 , 2, 00 For a the following (normalised) values are selected: 2.00
2 = 0.25, 1.502 ≈ 0.44,
1
1
1
= 1.00, 0.75
2 ≈ 1.78, 0.502 = 4.00. By combining these parameters we obtain 25 cases. Since
1.002
2
Cµ ∝ aσ the normalised Cµ values can be easily obtained. The parameters are determined in such a
way, that we obtain same Cµ cases. The Cµ values for all cases are summerized in the table below:
5
HH
σ
HH
0.50
a
H
H
0.25
0.44
1.00
1.78
4.00
6.25E-2
1.11E-1
2.50E-1
4.44E-1
1.00
0.75
1.00
1.50
2.00
1.41E-1
2.50E-1
5.62E-1
1.00
2.25
2.50E-1
4.44E-1
1.00
1.78
4.00
5.62E-1
1.00
2.25
4.00
9.00
1.00
1.78
4.00
7.11
1.60E1
Table 1: Cµ of all 25 computational cases with the corresponding a and σ parameter. All values are
normalised using the basic gaussian.
3.3
Normalization of results
Since we are dealing with a numerical simulation of an PA induced flow in quiescent air, there is no
free stream velocity with which the results can be normalized. In this subsection a new normalization
is derived by nondimensionalizing the Navier-Stokes equations and additionaly, from a steady state
form of the Navier-Stokes equations an approximation for the velocity of the induced flow is derived.
Nondimensionalization
We start with the following dimensional form of the Navier-Stokes equations (3.3):
!
∂ui
∂ui
1 ∂p
µ ∂ 2 ui
+ uj
=−
+
+ fi
∂t
∂xj
ρ ∂xi
ρ ∂x2j
(3.3)
We now define the reference parameters for bodyforce(3.4), density(3.5), lengthscale(3.6) and viscocity(3.7) with the following dimensions:
fref
ρref
σref
µref
ML
−2 −2
=
=
M
L
T
T 2 L3
M
= M L−3
=
3
L
= [L]
M
=
= M L−1 T −1
LT
(3.4)
(3.5)
(3.6)
(3.7)
The reference bodyforce and lengthscale are based on the amplitude and body force distribution of the
plasma actuator. Using the reference parameters we define a reference velocity(3.8), lengthscale(3.9)
and timescale(3.10).
s
Uref =
fref σref
ρref
Lref = σref
Lref
Tref =
Uref
Using this Uref , Lref and Tref we define dimensionless variables as:
6
(3.8)
(3.9)
(3.10)
∂ui
∂ ũi Uref
=
∂t
∂ t̃ Tref
∂ ũi Uref
∂ui
= ũj
uj
∂xj
∂ x̃j Tref
∂ p̃ Uref
1 ∂p
=
ρ ∂xi
∂ x̃i Tref
!
!
∂ 2 ũi µref Uref
µ ∂ 2 ui
=
ρ ∂x2j
∂ x̃2j ρref L2ref
Uref
fi = f˜i
Tref
Here, the variables with a tilde on top are dimensionless. Substitution of these variables in the NavierT
Stokes equations (3.3) and multiplying with Uref
yields the following:
ref
!
µref Uref Tref ∂ 2 ũi
∂ ũi
∂ ũi
∂ p̃
+ ũj
=−
+
+ f˜i
(3.11)
∂ x̃j
∂ x̃i ρref L2ref Uref ∂ x̃2j
∂ t̃
Here we find an expression for the Reynoldsnumber in front of the second derivative term as:
µref Tref
µref Uref Tref
µref
µref
µref
1
q
=
=
=
=q
=
fref σref
ρref Lref Uref
Re
ρref L2ref Uref
ρref L2ref
3
f
fref ρref σref
ρref σref
ρref
Inverting yields the Reynoldsnumber we will use from now on:
Ref =
1 q
3
fref ρref σref
µref
(3.12)
Using this Reynoldsnumber, we nondimensionalized the Navier-Stokes equations (3.3) and obtained
the following equation (note: from this point on we ommit the tilde signs on the variables, but keep
in mind that they are dimensionless variables):
∂ui
∂ui
∂p
1 ∂ 2 ui
+ uj
+ fi
=−
+
∂t
∂xj
∂xi Ref ∂x2j
(3.13)
Steady state
Starting from equation (3.13) we define the following:
ui = ūi + u0i
,
pi = p̄i + p0i
(3.14)
Here ū & p̄ are time mean values and u0i & p0i are fluctuations. Substitution in equation (3.13) and taken
the time average yields, after ommiting (near)zero terms, the steady state equation for momentum:
ūj
∂ ūi
∂xj
=−
∂ p̄
1 ∂ 2 ūi
+
+ f¯i
∂xi Ref ∂x2j
7
(3.15)
Assuming a 2D flow, so there are no fluctuations in the y-direction (i = 2). With this we obtain two
equations, for x- and z- direction respectively:
∂ ū
∂ ū
∂ p̄
1
+ w̄
=−
+
∂x
∂z
∂x Ref
∂ 2 ū ∂ 2 ū
+ 2
∂x2
∂z
+ fx
2
∂ w̄
∂ w̄ ∂ 2 w̄
∂ w̄
∂ p̄
1
ū
+
+ fz
+ w̄
=−
+
∂x
∂z
∂z Ref ∂x2
∂z 2
ū
(3.16)
(3.17)
Analytical derivation of induced flow
From the steady state momentum equation in streamwise direction (equation (3.16), we attempt to find
so expression for the induced flow. For low Reynoldsnumber, the momentum equation in streamwise
direction (3.16) reduces to:
1 ∂ 2 ū
= −fx
Ref ∂z 2
(3.18)
If we now assume, as a most simplyfied model, a constant body force in the wall normal direction we
can solve equation 3.18 by twice integrating with respect to z:
1
ū(z) = −Ref fx z 2 + c1 z + c2
2
(3.19)
Applying the boundary conditions
z=0 ,
ū = 0
(3.20)
z=1 ,
ū = Uref
(3.21)
yields the following relationship for ū:
Ref
fx (1 − z)z + Uref z
2
= 0, one can state:
ū(z) =
In absence of a free stream, Uref
ū(z) ∝ Ref
8
(3.22)
(3.23)
4
Results
Umax,induced [m/s]
The results of the flow computation are quasi steady, as can be seen in figure 4. The results for
0 < Tf t < 2 are not utilized. For further analyses instantaneous flow fields (for Tf t > 2) are used
unless stated otherwise.
The maximum induced velocity is presented in figure 5. Here we see for an increase in induced velocity
for both increasing amplitude a and increasing standard deviation σ of the body force distribution.
Both of these results are as expected, since the momentum added to the flow increases in both cases.
Rearranging the results yields figure 6. We observe for low Cµ that Uinduced,max is constant. On the
other hand, for high Cµ , Uinduced,max decreases with increasing σ.
The velocity profiles of five Cµ = 1, are shown in figure 7. The shape of the five profiles show
similarities, only the magnitude of the velocity varies with Ref . Normalization of figure 5 using Uref
and arranging the results by Ref yields figure 8. It can be observed that all results collapse into one
curve, depending on Ref only. On this curve two regions can be distinguished. Below Ref ≈ 100, all
results are on the line UUmax
∝ Ref denoting a Stokes flow regime. This was also derived in equation
ref
3.23. Above Ref ≈ 100 the results start deviating from this straight line and tend to Umax ∝ Uref .
Cµ
From figure 8 it is derived for low Ref : Umax ∝ µref
. For high Ref , assuming asymptotic behaviour,
q
1
the following relation is derived: Umax ∝ σref
. To check if the assumption of asymptotic behaviour
is legitimate, additional computations at a high Ref number are done. These additional results are
plotted in figure 9 In order to validate the results shown in figure 10 the simulation was recomputed on
a different computational Reynolds number. All previous results were computation at a computational
Reynolds number of Re = 63000, the validation is done at Re = 23000. It can be seen, that the results
for the new computational Reynolds number collapse with the origional data of the parametric study.
10
Cµ = 1/16
Cµ = 1/4
Cµ = 1
Cµ = 4
Cµ = 16
1
0.1
0.01
0.001
0
1
2
3
4
5
6
7
Tft [-]
Figure 4: The maximum induced velocity as a function of flow trough time Tf t .
9
3
Uinduced,max [m/s]
2.5
2
a=0.25
a=0.44
a=1.00
a=1.77
a=4.00
1.5
1
0.5
0
0e+00
2e-05
4e-05
6e-05
8e-05
1e-04
σ [m]
Figure 5: Maximum induced velocity as a function of σ for a number of amplitudes. The amplitude
a is normalized with the maximum amplitude of the Suzen model.
1.6
Uinduced,max [m/s]
1.4
1.2
1
Cµ=0.25
Cµ=0.44
Cµ=0.56
Cµ=1.00
Cµ=1.78
Cµ=2.25
Cµ=4.00
0.8
0.6
0.4
0.2
0
0e+00
2e-05
4e-05
6e-05
8e-05
1e-04
σ [m]
Figure 6: Maximum induced velocity results, arranged by Cµ (normalized with Suzen model Cµ ) of
the body force field
10
5
Ref = 141.2
Ref = 122.3
Ref = 99.8
Ref = 86.5
Ref = 70.6
z / σref [-]
4
3
2
1
0
-0.2
0
0.2
0.4
0.6
0.8
Uinduced/Uref [-]
Figure 7: Normalized velocity profiles, for Cµ = 1 at x = xumax
Umax / Uref [-]
10
a=0.25
a=0.44
a=1.00
a=1.78
a=4.00
1
0.1
0.01
10
100
1000
Ref [-]
Figure 8: Normalized maximum induced velocity arranged by Ref . All data of figure 5 collapse in
one curve
11
Figure 9: Maximum induced velocity for high Ref computations, plotted together with the origional
results. It can be observed that the results tending towards a horizontal asymptote.
Umax / Uref [-]
10
Restdin=63000
Restdin=23000
1
0.1
0.01
10
100
1000
Ref [-]
Figure 10: Normalized maximum induced velocity arranged by Ref for two computational Reynolds
number. Restdin represents the computational Reynolds number.
12
∝ Ref for high
A momentum balance is made, in order to clarify the deviation from the line UUmax
ref
Reynolds number in figure 8. To that end, a control volume is required. This is schematically depicted
in figure 11. In this momentum balance analysis, time averaged results are used (time average taken
for 2 < Tf t < 7). For the momentum balance we have an input of momentum into the control volume
from the body force and output of momentum as dissipation at the wall and induced momentum
(pout − pin ). Note: The pressure term in the momentum balance is neglected, since the flow velocity
is very low. The results of the momentum balance analysis can be observed in figure 12. It can be
observed that for Ref < 100 the wall dissipation equals Cµ , so we have a dissipation dominated flow,
a stokes flow. This corresponds to the stokes flow regime observed in figure 8. For higher Ref , we
observe a relative decrease in wall dissipation and an increasing induced momentum.
Figure 11: The control volume for the momentum balance. Their is momentum convection into the
c.v. (pin ) and a source of momentum, the body force field. The quantity of momentum added to the
c.v. by the body force is denoted with Cµ . Momentum is convecting out of the c.v. and momentum
is dissipated at the wall. Quiescent air is assumed at the top of the control volume.
13
1.2
wall dsp.
ind. mom.
p s-1 /Cµ [-]
1
0.8
0.6
0.4
0.2
0
10
100
1000
10000
Ref [-]
Figure 12: Momentum per second normalized with the body forces Cµ as a function of Ref .
14
5
Alternative body force distribution
Up to this point, all body force distributions had the maximum amplitude at the wall. This section
briefly discusses an alternative body force distribution discribed in equation 5.1 forz0 6= 0 and depicted
in figure 13. With this new body force distribution additional simulations are computed with the same
settings as the origional parametric study. The results will be compared. Simulations are done for
z0
σ = 0, 1, 2, 3, 4 and 5. Parameters a and σ are similar to the values of the basic gaussian.
Figure 13: Body force distribution for z0 6= 0
For the case for which z0 6= 0, the normalization is adjusted. Uref is adapted to take an varying
Cµ value into account. Lref is adapted to account for two lengthscales: The height from the wall,
computed as a ’gravity point’ (eq. 5.4) of the distribution and a characteristic length of the body force
distribution itself. Half the harmonic mean of the two separate lengthscales becomes the new Lref .
With these new reference parameters the Navier-Stokes equations are non-dimensionalised and a new
Reynoldsnumber is obtained, equation 5.5.
The original results (figure 8) of the parametric study are also re-normalized with the new normalization and the results are shown in figure 14. It is shown that the results for both z0 = 0 and z0 6= 0
collapse into one, Ref,new -dependent curve.
2
x + (z − z0 )2
f (x, z) = fref exp −
2σ 2
s
Cµ
Uref,new =
ρ∞ σ
1
Lref,new = 1
1
σ + z0,g
RR
z f dxdz
z0,g = RR
f dxdz
s
Cµ ρ∞ L2ref,new
1
Ref,new =
µ∞
σ
15
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
Umax / Uref [-]
1
z0 = 0
z0 > 0
0.1
0.01
10
100
1000
Ref [-]
Figure 14: The results of the origional parametric study are plotted together with the results for
z0 6= 0. For the normalization equations 5.2 and 5.5 are used.
16
6
Conclusion and recommendations
The flow field resulting from a Gaussian body force field with z0 = 0, can be normalized using the
proposed Uref (eq. 3.8) and Ref (eq. 3.12). This normalization yields a relation of Uinduced,max as a
function of Ref . For low Ref , Cµ is the dominant parameter and for high Ref , Cµ and σref are the
dominant parameters. Introducing an extra parameter (z0 6= 0) requires an adapted normalization,
which yields a relation between Uinduced,max and Ref,new (eq. 5.5). This relationship shows that the
effect of height above the wall decreases when the height from the wall becomes larger (prescribed by
Lref,new , eq. 5.3).
Some recommendations can be made on both this work and some future work. In this work, the
momentum balance analysis could have been done more thoroughly. There is no balance between
input (Cµ ) and output (wall dissipation, induced momentum). This might improve by implementing
the pressure term, although the flow velocities are quite low, and by not assuming quiescent air at the
top of the control volume.
For the body force field with z0 6= 0 a relation is found for Uinduced,max and Ref,new , but to achieve
this a reference length is set (eq. 5.3) which holds no particular physical meaning. A more thorough
theoretical background is desirable.
For future work, a more simple plasma actuation model can be proposed based on the dominant
parameters for the induced flow. With this it might be possible to have an plasma actuator model,
which can be directly inplemented even in a course grid flow simulation.
17
7
Bibliography
[1] Suzen, Y.B., Huang, P.G.,Jacob, J.D. and Ashpis, D.E., Numerical simulations of plasma based
flow control application, AIAA 2005-4633, (2005).
[2] Aono, H., Sekimoto, S., Sato, M., Yakeno, A., Nonomura, T., Fujii, K., Computational and
experimental analysis of flow structures induced by a plasma actuator with burst modulations in
quiescent air, Mechanical Engineering Journal, 2.4 (2015), 15-00233.
[3] Lele, S.K., Compact finite difference scheme with spectral-like resolution, Journal of Computational
Physics, Vol. 103, (1992), pp.16-22.
[4] Hishida, H. and Nonomura, T., ADI-SGS scheme on ideal magnetohydrodynamics, Journal of
Computational Physics,Vol. 228, (2009), pp. 3182-3188.
18
1
Investigation into a new simple DBD-plasma actuation model
T. Bouwhuis1,2 , Y. Abe3 , A. Yakeno4 , T. Nonomura4 , H.W.M. Hoeijmakers1 and K. Fujii
5
1 University of Twente, Faculty of Engineering Technology, Drienerlolaan 5, 7522 NB Enschede, The Netherlands 2 [email protected]
3 University of Tokyo, Japan 4 ISAS/JAXA, Sagamihara, Kanagawa, Japan 5 Tokyo University of Science, Japan
Abstract
The dominant factors of a body force field, representing a plasma actuator, are identified by means of a parametric numerical
study. Two dimensional flow simulations have been performed for a plasma actuator operating in quiescent air. Because of the
absence of a free stream the induced velocity is normalized with a proposed reference velocity, based on parameters of the
body force field. The normalized maximum induced velocity depends on the Reynolds number.
1. BACKGROUND
Active flow control using dielectric barrier discharge plasma actuators (hereafter: PA) has been studied intensively, with the
aim of improving performance and/or efficiency of a wide variety of fluid machinery. The PA consists of two electrodes with
a dielectric material in between. When a high voltage O(103 V ), high frequency AC is applied between the two electrodes a
plasma is created. This plasma induces a wall jet which can be utilized in flow separation control.
A conventional numerical method for the PA, which gives results in good agreement with experimental results [2], solves the
flow equations and an additional two equations on an additional grid. It is known as the Suzen model [1]. The resulting body
force field, shown is figure 1(a), of this model is coupled to the flow equations. Much experimental and numerical research
is focused on the relation between the operational parameters of the PA (voltage, base- and burst frequencies etc.) and the
performance of the PA. However, the dominant parameters of the body force which determine the induced flow have not been
identified so far.
The main focus of this study will be to identify the dominant parameters of the body force field, determining the induced
flow. To this end the body force field of the Suzen model will be reduced in complexity and a Gaussian distributed body force
will be studied. A future objective is to propose an new simple PA model, based on the dominant parameters, for which the
induced flow is similar to that of existing models but without additional equations to be solved on an extra grid.
2. METHOD
A parametric study is carried out using numerical simulations for two dimensional flow. The flow field is described by
the Navier-Stokes equations, in which the PA forcing is implemented as a source term. The number of grid points in the
computational grid is 442 × 203. Near the body force field, the grid is uniformly distributed and grid spacing is small, as
illustrated in figure 1(a). The computational domain is taken sufficiently large to ensure the far field boundaries do not affect
the induced flow. The grid coarsens further away from the body force field. Sixth-order compact difference schemes [3] were
used to discretize the spatial derivatives and ADI-SGS methods [4] were used for time integration.
The Gaussian body force model will have the force pointing in x-direction only, in contrast to the Suzen model which
features
pa multidirectional force field (x- and z -direction) with a complex distribution of the body force field. The amplitude
(F = fx2 + fz2 ) of the force field produced by the Suzen model is plotted in figure 1(a). The Gaussian body force field used
in the present study is prescribed by equation 1 and shown in figure 1(b). In here the parameters fref and σ are based on the
local maximum and characteristic length of the Suzen model (fig: 1(a)). Note that the body force strength is independent of
time. Spatial integration of the body force field over the whole flow domain yields the total induced momentum per second,
which is denoted as Cµ . The Cµ of the Gaussian body force model is in good agreement with the Cµ of the Suzen model
(fig. 1(a)).
(a) Amplitude of Suzen model body force
(b) Body force distribution for z0 = 0
(c) Body force distribution for z0 6= 0
Figure 1: Body force fields, all presented on the same scale. A length indicator is shown in figure 1(b)
2
A parametric study has been carried out varying the parameters fref
2
x + z2
and σ , in the Gaussian model (eq. 1). Because the Cµ from a Gaussian
f (x, z) = fref exp −
(1)
function can be determined exactly, the parameters fref and σ are
2σ 2
s
determined such that the set of studies contains body force models
fref σ
with similar Cµ , but different fref and σ . The total number of flow
Uref =
(2)
ρ∞
computations is 25.
Because of the absence of a free stream velocity, there is not a proper
Lref = σ
(3)
q
reference velocity that can be used to normalize the results. Therefore
1
fref ρ∞ σ 3
(4)
Ref =
a set of 4 reference parameters is chosen: fref (maximum value in the
µ∞
body force field), σ (standard deviation of body force), µ∞ (dynamic
2
viscosity determined from flow computation as µref = M aCFD /ReCFD )
x + (z − z0 )2
f (x, z) = fref exp −
(5)
and ρ∞ (constant). Using these reference parameters, a reference velocity
2σ 2
s
is defined in equation 2.
Lref
Cµ
By using a reference length (eq. 3) and time Tref = Uref , the Navier
Uref,new =
(6)
ρ∞ σ
Stokes equation is non dimensionalized. This nondimensionalization
1
yields a Reynolds number defined in equation 4. These equations (2,
(7)
Lref,new = 1
1
3 & 4) are used to normalize the parametric study.
σ + z0,g
RR
Besides the parametric study, variations of the height z0 of the center of
z f dxdz
the body force from the wall is studied. Using the formulation for the
(8)
z0,g = RR
f dxdz
body force field given in equation 5.
s
An example of such a body force field is depicted in figure 1(c). By
Cµ ρ∞ L2ref,new
1
Re
=
(9)
f,new
introducing one extra parameter, an additional length-scale is obtained.
µ∞
σ
Also, both the shape and Cµ change. Therefore the reference velocity
and reference length need to be redefined as defined in equations 6 and 7. In eq. 7, z0,g is the ’gravity point’, the weighted
height from the wall, of the body force field. It is defined in equation 8. Normalization of the Navier-Stokes equations with
the new reference dimensions yields a new Reynoldsnumber defined in equation 9.
3.
RESULTS
The results of the flow computation are quasi steady so for further analyses, instantaneous flow fields are used unless stated
otherwise. The maximum induced velocity is presented in figure 2(a). Rearranging the results yields figure 2(b). We observe for
low Cµ that Uinduced,max is constant. On the other hand, for high Cµ , Uinduced,max decreases with increasing σ . The velocity
profiles of five Cµ = 1, are shown in figure 2(c). The shape of the five profiles show similarities, only the magnitude of the
velocity varies with Ref . Normalization of figure 2(a) using Uref and arranging the results by Ref yields figure 2(d). It can
be observed that all results collapse into one curve, depending on Ref only. On this curve two regions can be distinguished.
Below Ref ≈ 100, all results are on the line UUmax
∝ Ref denoting a Stokes flow regime. Above Ref ≈ 100 the results start
ref
µ
deviating from this straight line and tend to Umax ∝ Uref . From figure 2(d) it is derived for low Ref : Umax ∝ µCref
. For high
q
1
Ref , assuming asymptotic behaviour, the following relation is derived: Umax ∝ σref
For the case for which z0 6= 0, the normalization is adapted as given by equations 6, 7 and 9. The original results (figure
2(d)) of the parametric study are also re-normalized with the new normalization and the results are shown in figure 3. It is
shown that the results for both z0 = 0 and z0 6= 0 collapse into one, Ref,new -dependent curve.
4.
CONCLUSION
The flow field resulting from a Gaussian body force field with z0 = 0, can be normalized using the proposed Uref (eq. 2)
and Ref (eq. 4). This normalization yields a relation of Uinduced,max as a function of Ref . For low Ref , Cµ is the dominant
parameter and for high Ref , Cµ and σref are the dominant parameters. Introducing an extra parameter (z0 6= 0) requires an
adapted normalization, which yields a relation between Uinduced,max and Ref,new . This relationship shows that the effect of
height above the wall decreases when the height from the wall becomes larger (prescribed by Lref,new , eq. 7).
R EFERENCES
[1] Suzen, Y.B., Huang, P.G.,Jacob, J.D. and Ashpis, D.E., Numerical simulations of plasma based flow control application, AIAA 2005-4633, (2005).
[2] Aono, H., Sekimoto, S., Sato, M., Yakeno, A., Nonomura, T., Fujii, K., Computational and experimental analysis of flow structures induced by a
plasma actuator with burst modulations in quiescent air, Mechanical Engineering Journal, 2.4 (2015), 15-00233.
[3] Lele, S.K., Compact finite difference scheme with spectral-like resolution, Journal of Computational Physics, Vol. 103, (1992), pp.16-22.
[4] Hishida, H. and Nonomura, T., ADI-SGS scheme on ideal magnetohydrodynamics, Journal of Computational Physics,Vol. 228, (2009), pp. 3182-3188.
3
Uinduced,max [m/s]
2.5
2
1.6
a=0.25
a=0.44
a=1.00
a=1.77
a=4.00
1.5
1
1.2
1
0.8
0.6
0.4
0.5
0
0e+00
Cµ=0.25
Cµ=0.44
Cµ=0.56
Cµ=1.00
Cµ=1.78
Cµ=2.25
Cµ=4.00
1.4
Uinduced,max [m/s]
3
0.2
2e-05
4e-05
6e-05
8e-05
0
0e+00
1e-04
2e-05
σ [m]
4e-05
6e-05
8e-05
1e-04
σ [m]
(a) Maximum induced velocity as a function of σ for a number (b) Maximum induced velocity results, arranged by Cµ (normalized
of amplitudes. The amplitude a is normalized with the maximum with Suzen model Cµ ) of the body force field
amplitude of the Suzen model .
5
Umax / Uref [-]
4
z / σref [-]
10
Ref = 141.2
Ref = 122.3
Ref = 99.8
Ref = 86.5
Ref = 70.6
3
2
a=0.25
a=0.44
a=1.00
a=1.78
a=4.00
1
0.1
1
0
-0.2
0.01
0
0.2
0.4
0.6
0.8
10
Uinduced/Uref [-]
100
1000
Ref [-]
(c) Normalized velocity profiles, for Cµ = 1 at x = xumax
(d) Normalized maximum induced velocity arranged by Ref
Figure 2: Results of the parametric study. Figures (a) and (b) present the maximum induced velocity as a function of the
characteristic length σ of the body force field. Figures (c) and (d) show the normalization using the proposed reference velocity.
Umax / Uref [-]
1
z0 = 0
z0 > 0
0.1
0.01
10
100
1000
Ref [-]
Figure 3: The results of the origional parametric study are plotted together with the results for z0 6= 0. For the normalization
equations 6 and 9 are used.