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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 330
Pressure Fluctuation on Finite Flat Plate above Wing in Sinusoidal
Gust
K.Nakatake, K.Ohashi, J.Ando
(Kyushu University, Japan)
ABSTRACT
It is generally said that the amplitude of pressure fluctuation induced by a sphere varying own volume with time is
proportional to the second derivative of the volume variation. Pressure fluctuation induced by a cavitating wing is a
similar example.
In this paper, as a modeled cavitation phenomenon, we treat the pressure fluctuation on a finite flat plate induced by
a wing advancing in uniform flow with a sinusoidal gust and varying its thickness with time. The wing is represented by a
simple surface panel method “SQCM” which can treat the unsteady motion.
In calculation for a finite flat plate, we use four kinds of calculation methods: the first one uses a mirror image
method, the second one does the solid boundary factor method, the third one does the source distribution method and the
last one does QCM. Comparing these four kinds of results for 2-D and 3-D cases, we discuss the availability of the four
methods and investigate the relation between the amplitude of pressure fluctuation and the second derivative of the wing
volume.
1. INTRODUCTION
Pressure fluctuation on a hull surface induced by a propeller causes ship hull vibration. Pressure on the hull surface
fluctuates largely because the propeller is working in the hull wake. If cavitation occurs, the amplitude of pressure
fluctuation becomes larger. Many researchers, such as Huse (1968), Vorus (1974), Hoshino (1980), Wang (1981), Breslin
et al. (1982), Ikehata & Funaki (1986), Kehr et al. (1996) etc. studied pressure fluctuation induced by a propeller.
As a modeled cavitation phenomenon, we treat the pressure fluctuation on a finite flat plate induced by a wing
advancing in uniform flow with a sinusoidal gust and varying thickness with time. In calculating pressure fluctuation, we
need to model a propeller and a ship hull. There are a lot of studies on propeller characteristics and we can obtain the
highly accurate results (ITTC, 1993). As to modeling of ship stern, there are a few studies using panel method, but ship
stern is usually treated as an infinite flat plate, and a solid boundary factor (2.0) is used to express the effect of the flat
plate (Huse, 1968). In order to represent hydrodynamically the finite flat plate above a wing, we adopt four kinds of
methods: the first one is the mirror image method, the second one is the so called solid boundary factor method, the third
one is the source distribution method (SDM) and the forth one is QCM (Quasi-Continuous vortex lattice Method, Lan,
1974). SDM treats the flat plate as a mere solid boundary and QCM does it as a wing. By using a simple surface panel
method “SQCM” (Ando et al., 1998), we calculate the characteristics of an unsteady wing varying its volume in a
sinusoidal gust and pressure fluctuation on the flat plate above a 2-D wing and a 3-D wing in a sinusoidal gust.
By comparing the obtained results, we discuss the availability of the solid boundary factor method, the mirror image
method, SDM and QCM and investigate the relation between the amplitude of pressure fluctuation and the second
derivative of the wing volume with respect to time.
2. CALCULATION METHODS FOR PRESSURE FLUCTUATION
Let us consider the problem of pressure fluctuation acting on a finite flat plate, induced by a wing advancing in
uniform flow with a sinusoidal gust. The unsteady wing is well represented by a simple surface panel method “SQCM”
and its formulations are described in the reference (Ando et al., 1998). Therefore we outline the main equations in this
paper. Fig. 1 shows the schematic diagrams to represent the 2-D finite flat plate by QCM and by the mirror image method
when a wing is advancing in uniform flow with a sinusoidal gust and changing its thickness with time. We take the
coordinate system fixed to the wing and adopt four kinds of
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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 331
calculation methods to represent the finite flat plate. The first one is the mirror image method and the second one is the
solid boundary factor method, which means to multiply the pressure induced by the unsteady mono-wing by the factor
2.0. The third one is the source distribution method (SDM) and the fourth one is QCM (Lan, 1974). Here we describe the
formulations for SDM and QCM, since other methods are included partly in these formulations.
Fig. 1 Coordinate System and Schematic Diagram for Wing and Flat Plate
Firstly, the induced velocity due to the bound vortex on the camber surface and the shed vortex at time tL is
expressed as follows.
(1)
where,
Nγ: number of divisions of camber surface
L: number of shed vortices
In Eq. (1), and are the induced velocities due to the bound vortex and the shed vortex with unit strength,
respectively, and Γ(tL) is a circulation around the wing at time tL, and γv(ξv, tL) is the strength of the bound vortex on each
panel.
Next the induced velocity due to the source panel on the wing surface is expressed as
(2)
where,
Nσ: number of divisions of one side of the wing
In Eq. (2), expresses the induced velocity due to a line source with unit strength, and σj(tL) does the source
strength on each panel at time tL. SDM uses the similar equation with strength σp to represent the finite plate.
Adding these velocity components to the relative inflow velocity we have for a mono-wing
(3)
and the boundary condition to be satisfied on the wing and camber surfaces is the solid boundary condition,
(4)
where,
unit normal vector at each panel
and the unsteady Kutta condition (Ando et al., 1998).
Next QCM treats the finite plate as a thin wing and gives the following expression for the induced velocity
(5)
where,
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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 332
lp: length of plate
: number of divisions of plate.
In Eq. (5), and are the induced velocities due to the bound vortex on the flat plate and the shed vortex with
unit strength, respectively, and Γp(tL) is a circulation around the flat plate and is the strength of bound vortex
on each panel on the flat plate.
As to divisions of the finite flat plate, we divide it into three parts i.e. the front part, the wing part and the aft part,
and adopt cosine divisions for each part. Summing up all velocity components, we have total velocity in case of
presence of the finite plate as
(6)
The boundary conditions on the flat plate are also the solid boundary condition and the unsteady Kutta condition.
The unknown strengths are γ, σ and γp or σp, but we solve the boundary conditions for the wing and those for the flat
plate, iteratively. In this case, we need usually several times for enough convergence.
Lastly, the unsteady pressure p(t) is expressed in the wing-fixed coordinate system as follows.
(7)
where,
In Eq. (7), ρ, p0, and are water density, the ambient pressure and the velocity potential in the wing-fixed coordinate
system, respectively. can be obtained analytically and is evaluated by two-points upstream difference scheme.
We define the unsteady pressure coefficient Cp as
(8)
and denote Cp for the mono-wing case by
3. RESULTS OF 2-D PROBLEM
Let us consider the case where a wing with NACA0012 section is set under a finite flat plate in uniform flow VI with
a sinusoidal gust of vertical velocity We take the vertical distance d=0.5c, (c: chord length), the
angle of attack α=0°, the reduced frequency k(=ωc/2VI)=1.0, the amplitude υ0=0.1VI and the time increment ∆t=πc/72VI.
ω expresses the circular frequency of the gust and the finite flat plate has length lp of 5c or 10c and the center of the plate
is set so as to coincide with the midchord of the wing. In this case, we calculate the pressure fluctuation on the flat plate
induced by the wing in uniform flow with a sinusoidal gust by the four methods.
Firstly we calculate it by QCM and SDM representing the flat plate and then compare these results with those by the
mirror image method and the solid boundary factor method.
Number of panel divisions of the wing surface is 60 along the perimeter and number of discrete vortices on the
camber is 29 and those of the flat plate are 49 (29 for wing part, 10 for fore and aft parts) for QCM and number of source
panels is 10 per chord length c for SDM.
3.1 Pressure fluctuation due to wing in a gust
Fig. 2 shows the instantaneous pressure distribution (time step 280) calculated by QCM and SDM on the flat plate of
two length 5c and 10c and Fig. 3 shows vortex distribution γp and source distribution σp at the same time step. From
Fig. 2 we find that QCM gives nearly the same pressure distribution for lp=5c, 10c, while SDM does different
distributions for two kinds of lp though σp is nearly same. γp distribution of QCM shows a little different distributions
between 5c and 10c, but γp satisfies the unsteady Kutta condition in addition to the solid boundary condition and shows a
little rise near the leading edge as a usual thin wing. Fig. 4 shows a comparison of pressure distributions obtained for
lp=5c at two time steps 245, 280 by QCM and SDM and Fig. 5 shows the corresponding γp and σp distributions. From
these Figures, we notice that QCM results give more stable and reasonable pressure distributions than SDM results even if
σp shows smoother distribution. This seems to be due to the stable flow produced by the unsteady Kutta condition. Fig. 6
shows a comparison of amplitude of pressure fluctuation ∆Cp between QCM and SDM. We find QCM and SDM give
fairly different distributions of ∆Cp except the central part. Especially SDM gives larger ∆Cp than QCM in the fore part
and does lower ∆Cp after the central part than QCM, and then gives large ∆Cp near the trailing edge. From these results,
we think that QCM gives more stable and realistic pressure fluctuation for 2-D problem than SDM. Fig. 7 and Fig. 8 show
the contributions of the unsteady component and the velocity component to the total ∆Cp in case of QCM and
SDM, respectively. We must notice that these curves are not additive since each pressure fluctuation has phase difference.
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From these Figures, we understand that QCM gives more reasonable behavior near the leading edge of the flat plate than

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Fig. 4 Pressure Distribution on Flat Plate
Fig. 2 Pressure Distribution on Flat Plate
Fig. 6 Amplitude of Pressure Fluctuation on Flat Plate
on flat Plate
PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST
Fig. 5 Singularity Distribution on Flat Plate
Fig. 3 Singularity Distribution on Flat Plate
Fig. 7 Component of Amplitude of Pressure Fluctuation
333

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on flat Plate
Fig. 10 Amplitude of Pressure Fluctuation on Flat Plate
Fig. 12 Comparison of Pressure Distribution of 2-D Wing
Fig. 8 Component of Amplitude of Pressure Fluctuation
on flat Plate
PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST
Fig. 13 Time History of Lift Coefficient
Fig. 9 Pressure Distribution on Flat Plate
Fig. 11 Component of Amplitude of Pressure Fluctuation
334

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Fluctuation
Pressure Fluctuation
Fig. 14 Comparison of Solid Boundary Factor
Fig. 16 Effects of Frequency kt on Amplitude of Pressure
Fig. 18 Effects of Phase Difference ψ on Amplitude of
PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST
Pressure Fluctuation
Fig. 15 Variation of Wing Section
Fig. 17 Effects of th on Amplitude of Pressure Fluctuation
Fig. 19 Effects of Phase Difference ψ on Amplitude of
335

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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 336
SDM. We think that the Kutta condition stabilizes the flow over the flat plate.
Next we compare QCM results with those of the mirror image method and the solid boundary factor method. Fig. 9
shows a comparison of pressure distributions at time step 280 on the flat plate obtained by the three methods. The mirror
image method gives nearly the same results as but different distribution from QCM result near the aft part of the
plate. Fig. 10 shows a comparison of amplitude ∆Cp and Fig. 11 does components of ∆Cp composed of the flat plate, the
wing and the interaction components. From these Figures, we understand that the difference between QCM and other two
methods is caused by the flat plate component due to the finite length effect. Next we investigate the effect of the flat
plate on the lift of 2-D wing and check also the solid boundary factor 2.0. We show the pressure distributions on the wing
surface in Fig. 12, and the lift coefficient CL in Fig. 13, and the solid boundary factor Sb at the center of the flat plate in
Fig. 14. We understand that the finite flat plate of CL gives similar effect on the wing characteristics to the mirror image,
because QCM results are almost same as those of the mirror image method As to the solid boundary factor, the value of
the mirror image method converges to 2.0 with increase of the distance d, while the value of QCM shows a different
tendency and seems to converge to 1.0. Accordingly, we think that the solid boundary factor 2.0 is not always applicable
to the finite flat plate.
3.2 Pressure fluctuation due to thickness-varying wing in a gust
Let us consider the case where a thickness-varying wing is in uniform flow with a sinusoidal gust. We assume that
the wing thickness tw(t) varies with time by the following expression,
(9)
In Eq. (9), t0 is the original wing thickness, th the amplitude of thickness variation, ωt the circular frequency of
thickness variation and ψ the phase difference from the sinusoidal gust. Then the second derivative of wing volume with
respect to time becomes as
(10)
In Eq. (10), c1 is a constant known from the expression of NACA wing section. Then the amplitude of of the
volume variation becomes
At first, we consider the case where the wing is in uniform flow and is varying only upper surface. (see Fig. 15)
Introducing kt(=ωtc/2VI) instead of ωt, we show in Fig. 16 the relations between the amplitude of pressure fluctuation ∆Cp
at the center of the flat plate and kt, when kt changes from 0.5 to 3.0 and from 0.001 to 0.05. We find by three methods
that ∆Cp changes almost linearly with kt for kt=1.0~3.0 and ∆Cp does quite differently for kt<0.05. For small kt, the limit
of ∆Cp will be the value of the quasi-steady case. Next we show some results of ∆Cp when the wing is in a gust and is
changing the thickness with phase difference from the sinusoidal gust. In cases of kt=1.0, ψ=0 and th=0.01t0, 0.05t0, 0.1t0,
we show the relation of ∆Cp and th in Fig. 17 and get the linear relation between ∆Cp and th. By setting the phase
difference ψ as −π, −π/2, −π/4, π/2, π between thickness variation and the gust, we show the relation between ∆Cp and ψ
in Fig. 18. The phase lead of thickness variation (ψ=−π/4) gives a maximum value of ∆Cp, because maximum lift and
maximum thickness work together for maximum ∆Cp. Fig. 19 shows the effects of phase difference on ∆Cp. Therefore we
understand that occurrence of cavitation with phase difference may affect the amplitude of the pressure fluctuation on the
hull.
4. RESULTS OF 3-D PROBLEM
Corresponding to the 2-D problem, we perform calculations for the 3-D problem. Let us consider a case where a 3-D
wing with a span s=3c and NACA0012 section is set in uniform flow with a sinusoidal gust and a finite flat plate with a
breadth 2s and a length 5c is set at a vertical distance d=0.5c above the wing. (see Fig. 20). We take lp=5c, 6c, k=1.0,
υ0=0.1VI, and the same conditions for the gust and thickness variation tw (t) and phase difference ψ. Fig. 21 shows
instantaneous pressure distributions obtained by QCM and SDM on the midspan line (y=0.0) on the flat plate at time step
148 and Fig. 22 does the pressure distributions on the midchord line (x=0.5c) at the same time step. These distributions
seem to be plausible around the fore and wing parts. Though SDM gives some edge effect to Cp, but degree of the effect
is not large compared with 2-D case. Figs. 23 and 24 show a comparison of Cp obtained by the four methods in both x, y
directions. It is interesting to notice that the four methods give similar distributions except the aft part and SDM can give
reasonable Cp above the wing. We think that this may be due to the weaker edge effect compared with 2-D flat plate. We
show a comparison of the amplitude distribution of the pressure fluctuation ∆Cp in Fig. 25 and Fig. 26. Around the wing
part, all methods give nearly the same values, but do different tendencies in the aft part as the 2-D case. Fig. 27 shows the
relations between ∆Cp and kt in case of the thickness-varying wing. We find again linearity between them in the range of
kt=1.0~3.0. Then we show in Fig. 28 a comparison of the solid boundary factors at the center of the plate obtained by
QCM and the mirror image method and finally in Fig. 29 a comparison of the phase difference effects
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Wing and Flat Plate
Fig. 24 Pressure Distribution on Flat Plate
Fig. 22 Pressure Distribution on Flat Plate
Fig. 20 Coordinate System and Schematic Diagram for
PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST
Fig. 23 Pressure Distribution on Flat Plate
Fig. 21 Pressure Distribution on Flat Plate
Fig. 25 Amplitude of Pressure Fluctuation on Flat Plate
337

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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 338
on ∆Cp by the four methods. From these results we understand that the solid boundary factor method is not always
applicable to the finite flat plate and SDM is also applicable to estimation of pressure fluctuation on the flat plate near the
wing. Since these results are obtained only by numerical calculations, we must evaluate them by corresponding
experiments. These calculation methods are easily extended to the case of a pitching or heaving wing in uniform flow.
Fig. 26 Amplitude of Pressure Fluctuation on Flat
Fig. 27 Effects of Frequency kt on Amplitude of Pressure
Fluctuation
Fig. 28 Comparison of Solid Boundary Factor Fig. 29 Effects of Phase Difference ψ on Amplitude of
Pressure Fluctuation
5. CONCLUSION
We applied QCM, SDM, the mirror image method and the solid boundary factor method to the problem of amplitude
of pressure fluctuation induced on the finite flat plate set above a wing in uniform flow with a sinusoidal gust.
From the obtained results, we conclude as follows.
• The mirror image method, the solid boundary factor method, QCM give nearly the same amplitude of pressure
fluctuation on a finite flat plate in the upstream and upper regions of the wing in 2-D and 3-D problem. SDM is
similar only in 3-D problem.
• Only QCM seems to give reasonable amplitude distribution on a finite flat plate in lengthwise direction.
• The amplitude of pressure fluctuation varies linearly with the second derivative of wing volume with respect to
time in a certain range of circular frequency of gust.
ACKNOWLEDGEMENT
The authors would like to deeply thank Mrs. Yasuko Yamasaki for her typing of this manuscript.
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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 339
REFERENCES
J.Ando, S.Maita, K.Nakatake, “A New Surface Panel Method to Predict Steady and Unsteady Characteristics of Marine Propeller,” Proc. of 22nd ONR
Symposium on Naval Hydrodynamics, Washington, 1998.
J.P.Breslin, R.J.Van Houten, J.E.Kerwin and C.-A.Johnsson, “Theoretical and Experimental Propeller -Induced Hull Pressures Arising from Intermittent
Blade Cavitation Loading and Thickness,” Trans SNAME., Vol. 90, pp. 111–151, 1982.
T.Hoshino, “Pressure Fluctuation Induced by a Spherical Bubble Moving with Varying Radius,” Trans. of the West-Japan Society of Naval Architects,
Vol. 58, pp. 221–234, 1979.
T.Hoshino, “Estimation of Unsteady Cavitation on Propeller Blades as a Base for Predicting Propeller Induced Pressure Fluctuations,” Journal of the
Society of Naval Architects of Japan, Vol. 148, pp. 33–44, 1980.
E.Huse, “The Magnitude and Distribution of Propeller-Induced Surface Forces on a Single-Screw Ship Model, ” Norwegian Ship Model Experiment
Tank Publication, No. 100, 1968.
M.Ikehata & H.Funaki, “Analytical Characteristics of Oscillating Pressure Distribution above a Propeller,” Journal of the Society of Naval Architects of
Japan, Vol. 159, pp. 71–81, 1986.
20th ITTC Propulsor Committee Report, 1993
Y.-Z.Kehr, C.-Y.Hsin and Y.-C.Sun, “Calculations of Pressure Fluctuations on the Ship Hull Induced by Intermittently Cavitating Propeller,” Proc. of
21st ONR Symposium on Naval Hydrodynamics, pp. 882–897, 1996.
C.E.Lan, “A Quasi-Vortex-Lattice Method in Thin Wing Theory,” Journal of Aircraft, Vol. 11, No. 9, pp. 518–527, 1974.
W.S.Vorus, “A Method for Analyzing the Propeller-Induced Vibratory Forces Acting on the Surface of a Ship Stern,” Trans. SNAME, Vol. 82, pp. 186–
210, 1974.
G.Wang, “The Influence of Solid Boundaries and Free Surface on Propeller Induced Pressure Fluctuations,” Norwegian Maritime Research, No. 2, pp.
34–46, 1981.
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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 340
DISCUSSION
T.Hoshino
Mitsubishi Heavy Industries, Ltd.
Japan
The authors should be congratulated for their efforts to make clear the solid boundary effect of a finite flat plate on
the pressure fluctuation induced by a wing. I have the following questions.
(1) Fig. 6 shows that there is very large difference of amplitude of pressure fluctuation between QCM and SDM,
especially in the aft part of the plat. This difference would be due to the consideration of trailing vortices
shedding from the trailing edge of the plat in QCM. Does this difference become small, if the aft part of the
flat plat was lengthened?
(2) In the calculation of the pressure fluctuation induced by a propeller, the pressure fluctuation is calculated on
the plate fixed in space, not moving with propeller. On the other hand, the pressure fluctuation on the plat
moving with the wing is calculated in the present paper. In this case, the pressure fluctuation due to the
thickness effect can't be considered and only the loading effect, is considered. I think that this would be the
reason why there is large difference between QCM and SDM.
AUTHOR'S REPLY
Thank you for your raising questions.
(1) We think that this difference is due to both the shedding vortex (QCM) and the unstable flow (SDM) near the
trailing edge. In case of 2-D problem, lengthening of only aft plate does not improve the difference near the
leading edge of the plate. Lengthening of plates forward and afterward improves the difference in the fore
and upper parts of the plate (See Fig. 2), but still it can not improve that in the aft part of the plate. In case of
3-D problem, however, the effects of shedding vortex and the unstable flow seem to be small for the fore and
upper parts of the plate (see Fig. 21).
(2) In this calculation, only the loading effect is considered in the sense that the relative position of the flat plate
to the wing is unchanged. We think that large difference between QCM and SDM is due to the unstable flow
(SDM) at the trailing edge. Especially this effect is sensitive in the 2-D problem.
DISCUSSION
K.Kim
Naval Surface Warfare Center, USA
Cavitation-induced hull pressure has been a continuing subject for many researchers in this field. The authors tried
different numerical schemes to predict induced pressure on a flat plate above a pulsating wing in a sinusoidal gust. I have
some questions.
(1) It is not clear where the boundary condition expressed in Equation (4) was applied; on camber surface, on the
wing surface or on both camber and the wing surface?
(2) It appears that Fig. 9 (pressure distribution at time step 280) and Fig. 10 (amplitude of pressure fluctuation)
are identical. I am wondering if Fig. 10 should be replace by Cp figure.
(3) For Fig. 7, the authors stated that total Cp was not the sum of the amplitude of the components due to the
phase angle difference. In Fig. 11, however, total Cp appears to be sum of the components. Was phase angle
considered here or not?
(4) For 3-D case, the authors applied unsteady Kutta condition at downstream edge of the flat plate. How did the
authors treat the side edges in terms of boundary condition?
(5) In Figs. 16 and 27, Cp is expected to be sensitive to the time-step size for different reduced frequencies. Did
the authors use different time step size in these figures?
(6) Judging from the large discrepancies in predicted pressure distribution by QCM and SDM, it appears that the
validity of the SDM is questionable. I would like to suggest that the authors revisit the formulation and/or
numerical schemes to identify possible causes of the discrepancy.
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PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 341
AUTHOR'S REPLY
We thank the discusser for his minute discussions and reply as follows:
(1) The boundary condition (4) was applied to on both the camber and the wing surfaces.
(2) We considered always the phase difference in our calculation. We show the pressure fluctuations of
components with time at x/c=3.0 in Fig. A-1 in case of Fig. 7, and that in Fig. A-2 in case of Fig. 11. From
these Figures, we understand that in case of Fig. 7 the phase difference between unsteady and velocity
components is so large that two components cancel to each other, on the other hand, in case of Fig. 11, the
phase differences among three components are small, then the total amplitude is nearly equal to sum of three
components.
(3) We did not apply any boundary condition at the side edges, because its effect seems to be small.
(4) We tested several time-step sizes in the calculation and confirmed the used one is sufficient for the given
range of reduced frequency. Therefore we used the same time-step size.
(5) We think that the validity of SDM is not fine behind the unsteady wing especially in the 2-D case. The main
cause is that SDM can not satisfy the Kutta condition at the trailing edge of the flat plate. This condition
assures the smooth flow downwards at the trailing edge and we think that the flow field near and downstream
the trailing edge is not expressed by SDM in principle.
Fig. A-1 Component of Pressure Fluctuation or Flat Plate at x/c=3.0
the authoritative version for attribution.
Fig. A-2 Component of Pressure Fluctuation or Flat Plate at x/c=3.0