numerical investigation of unstable hydrodynamic force on
TRANSCRIPT
Numerical investigation of unstable hydrodynamic force on the oscillating finite height
rectangular prism 1,2La Ode Ahmad Barata
1Graduate School of Natural Science and Technology
Kanazawa University Kanazawa-shi, 920-1192, Japan
2Mechanical Engineering Department Halu Oleo University
Kendari 93232, Indonesia [email protected];
Takahiro Kiwata Research Center for Sustainable
Energy and Technology Kanazawa University
Kanazawa-shi 920-1198, Japan
Takaaki Kono Division of Mechanical Science and
Engineering Kanazawa University
Kanazawa-shi 920-1198, Japan
Shunichi Mizukami Sugino Machine Limited 2410 Hongo, Uozu City,
Toyama Prefecture 937-8511, Japan
Abstract— Unstable hydrodynamic force acting around the slender rectangular prism induce the prism to oscillate with harmonic oscillation. This phenomenon is studied numerically using Large Eddy Simulation turbulence model by employing finite volume method discretization. The numerical results of vibration characteristics are verified by experimental data result. The oscillation frequency characteristic is analyzed using the FFT method. The locked-in phenomenon of wake frequency to frequency prism oscillation is solely found on the infinite prism with small side ratio at maximum response amplitude (Vr
= 3.0). It is not found in the finite prism for all cross-section models over all range reduced velocity conditions. However, the prism with a side ratio of 0.5, the normalized oscillation frequency tends to increase by increasing reduced velocity, where it inverses to the experimental result. In case of vibration characteristic, the free vibration was initialized by forced vibration as an initial condition in a specified range of flow time. At low reduced velocity, after employing forced oscillation, both the fluid force component and body motion amplitude act as the positive damping force with small-scale body oscillation is still recognized over the flow time. However, at high reduced velocity (maximum response amplitude), fluctuation separated shear layer on the side surface induce fluid force behave as the negative damping force. Hence, the uniform harmonic oscillation is introduced after specified flow time. In the prism with a side ratio of 0.5, small-scale fluctuation force still appears over the time flow due to the fluctuation of spanwise vortices behind the prism in which it is not found on the prism with small side ratio. Therefore, small-scale vortex excitation potentially exists in the elastically cantilevered of finite prism model with the side ratio of 0.5.
Keywords: numerical investigation, unstable oscillation, FFT method, fluid force, frequency.
I. INTRODUCTION
The rectangular prism with a sharp edge, separated shear flow occurs at the leading edge, and flow behaviors on side surfaces and behind the prism are considerably influenced by depth of prism (side ratio). In nature, separated shear layers at leading edge roll-up and reattach on the surfaces while rolled up of symmetrical or antisymmetric vortices behind the prism shed that is known as the Karman vortex. Consequently, Reynold number, depth to height ratio (side ratio), and the interaction mode of flow wake structure take an essential role on the flow behavior around the prism.
Reference [1] investigated instability of aerodynamic behavior of an oscillating rectangular cylinder with the
critical section. The fluctuating flow wake around an oscillating prism is considerably different from the static model. The vortex formation around an oscillating prism affect wake formation that influences drag and lift coefficient fluctuations. Fluctuations of dynamic forces over discrete reduced velocity appear over the range, and unstable oscillation behavior is identified in lower reduced velocity below resonant reduced velocity which known as low-speed galloping. Oscillation behaviors may differ for reduced velocity above reduced resonant velocity and critical side ratio which oscillation behavior is attributed by the vortex induced oscillation. In this case, unstable behavior does not relate to vortex-induced oscillation as well as high-speed galloping [2][3].
To support vibration characteristic analysis on the elastically wall mounted prism in the water channel, we carry out numerical simulation model to verify experimental data result and provide sufficient description about unstable oscillation behavior around of the prism below critical side ratio as well as its interaction during oscillation condition as mentioned in [4]. Finally, this study is intended to support sufficient description regarding the study of harvesting energy from low-speed galloping behavior using the magnetostrictive effect in the water flow.
II. NUMERICAL METHOD
A. Governing Equations
The governing equations of incompressible viscous flowfield are given by mass conservation and momentum equation of filtered Navier-Stokes equation as follow
and filtered Navier-Stokes equation as
Since the turbulent in the subgrid-scale is unknown, then the Boussinesq hypothesis is employed. Then subgrid-scale of the stress tensor is calculated as follow
(1)
(2)
2018 International Conference on Electrical Engineering and Computer Science (ICEECS)
105
where ���, �, ��, � are average velocity vector, stress tensor, kinematic viscosity, and pressure respectively. The effect of turbulent is lumped into a turbulent viscosity for which dissipation of kinetic energy at sub-grid scale (SGS) is considered as molecular diffusion. Hence, for subgrid-scale eddy viscosity, Wall Adapting Local Eddy- Viscosity (WALE) model is used in this study. In this case, Eddy viscosity SGS WALE model considers Eddy viscosity as the product of characteristic length and velocity scale. Hence, eddy viscosity can be written as equation (4).
where
�� = min (��, ����/�),Cw = 0.235, κ = 0.4187 where ��,κ,d, Cw, V, δij, are length scale, Karman’s constant energy, minimum wall spacing, WALE constant, cell volume, and Kronecker delta (δij = 0 if i ≠ j, otherwise 1 if i = j) respectively.
B. Computational Model
In this study, the discretization method used is a finitevolume method; with the convection term is the bounded center difference. The second precision implicit method and the pressure-velocity coupling algorithm PISO are employed for time integral and momentum balance.
The computational model used in this study, i.e., the finite rectangular prism with cross-section height of prism (H) normal to flow stream is 20 mm, depth (D) of 4 and 10 mm and span length (L) of 200mm. A three-dimensional O-grid domain is developed with the diameter of 40H and span length of 2L. The total number of grid points is 4.41 × 106 with minimum grid width is 5 × 10-3H as depicted in figure 1 with Reynold number of 22000. Because of the prism oscillate only transversely, a dynamic mesh is used, and at the boundary, Neumann condition is applied at outflow for which gradient velocity component (u v w) and the pressure are zero.
No slip condition at the prism and wall surface are applied then the symmetry plane in which frictionless wall is adopted at the upper and bottom surface of the domain. By considering figure 2, unstable oscillation characteristic is analyzed using one degree of freedom with damped harmonic oscillation following the equation (6).
The free vibration was initialized by force vibration with velocity displacement dy/dt = A.ωc where amplitude A=0.1H. H denotes the height of prism normal to flow stream and ωc is frequency characteristic.
where fluid damping and spring stiffness are denoted by
2���
���, and � = 4��� �
�
����
�
respectively. Equation of
motion (6) subsequently is discretized in time using Adams-Bashforth method multistep solution. For the convergence and stability, Adams Bashforth predictor-corrector was verified as in reference [5]. Therefore, the second order accuracy is employed in this study.
III. NUMERICAL RESULTS AND DISCUSSION
A. Unstable Oscillation behavior on low-speed galloping
As foregone mentioned that initial condition of velocitydisplacement (forced oscillation) is given by dy/dt = Aωc, for which A = 0.1H. The forced oscillation is taken after T0, and it is followed by free vibration test where displacement of the prism and their derivatives are considered at T. Prior to the free vibration test, the initial condition of the flow behaviors around fixed prism was taken at non-dimensional time condition T0 (=Ut/H) is 0.0055 - 497, where displacement, velocity, and acceleration condition are equal to zero.
,
yFkyycym
Fig. 1. Three dimensional of O-grid computational domain
(4)
(5)
(5a)
(5b)
(6)
Fig. 2. Schematic vibration model with one DOF
H=
20m
m
U
fc Cn
y
m
D
m
x
y
z
Fig. 3. Non-dimensional body displacement characteristics respect to reduced velocity for both models
0.00
0.05
0.10
0.15
0.20
0.25
0 1 2 3 4 5 6
hrm
s
Vr
D/H Exp. Cal.
0.2
0.5
(3)
106
In figure 3, the non-dimensional response amplitudes of prisms are presented with respect to reduced velocity. The response amplitude for two different side ratios of the prisms increases when reduced velocity increases. In the onset vibration, a prism with a side ratio of 0.2 vibrates at lower reduced velocity rather than prism with a side ratio of 0.5. Based on those experimental results, we calculated response amplitude numerically with respect to reduced velocity from the experimental result. We find that numerical calculation result on response amplitudes for both models agree with the experimental results. In the case of the oscillation frequency, normalized body oscillation frequency in experimental result shows an imperfect synchronization (f/fc < 1) for all range reduced velocity as depicted in figure 5. The deviation increases when response amplitude increases. In denser fluid such as water, imperfectly of such frequency synchronous is acceptable due to viscous and added mass effect (mass inertia).
In the numerical calculation results, the normalized frequencies for two finite rectangular models have good agreement with the experimental results over reduced velocity. However, it is not found on infinite models with a side ratio of 0.5. Both prism models, the normalized frequency of body oscillation grow linearly with a reduced velocity, which it indicates that strong interaction in the wake take place mainly at high reduced velocity. In the some researches indicate that strong resonance takes place at the critical side ratio for rectangular prism in the uniform flow i.e. around 0.6 with resonant reduced velocity is 8.0 as inreferences[2][3][6][7]. Those previous investigations findthat the outstanding features of wake flow characteristic ofaround the prism with side ratio around critical side ratio aredistinctly different in case of vortex street pattern for whichdrag and lift fluctuation behaviors change abnormaly. Figure5 also shows that the synchronization between bodyoscillation and wake frequency also can be found only for theinfinite model (D = 0.2H) at the maximum response (Vr=3.0)in which wake frequency known as Karman frequencymatches with oscillation structure frequency. Hence,harmonic oscillation can be found in figure 4(a) and 7(c). Thewake fluctuation of the finite and infinite prism is found infigure 5 in which the effect of flow separation at the tip of themodel is pronounced at high reduced velocity. Otherwise, thisdisparity is not found at the prism with a side ratio of 0.5which wake frequency for all reduced velocity range is notcaptured-in to body motion frequency.
B. Fluid force components on the unstable oscillation
Figure 6 shows the effect of unstable hydrodynamic forceon the drag coefficient behavior. In this case, we also present the comparative numerical result of drag from the two-dimensional model. In case of the stationary prisms in which reduced velocity is equal to zero, coefficient drag for finite height prisms does not agree with infinite height prism for both models as described in the several investigations both experiment and numeric such as in references [7][8][9]. In this case, the drag coefficient of the two-dimensional and infinite models show the different pattern with a finite model, which it indicates that the flow wake structure is distinctly different. It is followed by the alteration of alternating vortices behavior in the wake as described widely in the above-cited references mentioned. The interaction of regular vortex shedding with a separated shear layer at the prism tip
enhances momentum recovery around the prism. When the prism is imposed by the oscillation, the amplification of flow interaction takes place in the wake. As a result, the drag coefficient is reduced.
Fig. 4. Experimental result of body oscillation characteristic over flowtime. (a) D/H = 0.2, (b) D/H = 0.5
-0.4
0
0.4
3.5 4 4.5 5
η
t [s]
-0.4
0
0.4
3.5 4 4.5 5
η
t [s]
(a)
Vr=1.5 Vr =3.0
Vr=3.0 Vr =5.0
(b)
Fig. 6. Mean drag coefficient of prisms respect to reduced velocity (Vr)
0.0
1.0
2.0
3.0
4.0
0 1 2 3 4 5 6
CD
Vr
D/H 0.5InfiniteFinite
0.2
2D
Fig. 5. Normalized frequency of body motion and wake for both models.
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6
f/f c
Vr
D/H Exp. Cal. Wake
0.20.5
0.50.2
Finite
Infinite
107
Figure 7 shows the time histories of non-dimensional body motion and lift coefficient fluctuation for both side ratios. In the onset of vibration in which forced oscillation is employed at the designed time, prism oscillation experience positive damping. However, after T = 150, prism motion is still recognized with small-scale fluctuations. In this condition, we recognize onset vibration in a reduced velocity of 1.5 and 3.0 for the prism with the side ratio of 0.2 and 0.5 respectively.
In maximum responses amplitude, figure 7(c),7(d), lift force fluctuation is still dominant over the flow time, and body oscillation experience the negative damping force for which energy is transferred from fluid to the bluff body structure. It is also observed that both prism and lift fluctuation oscillate with the higher frequency on the prism with small side ratio (D/H = 0.2) rather than side ratio of 0.5. The different behavior is observed at the prism with a side ratio of 0.5 in which the small-scale fluctuations of lift coefficient and body motion along designed time flow are
still recognized in the prism as exhibited in figure 7(d). This behavior is also observed in the experimental result as shown in figure 4(b). It is distinctly different with the prims in figure 7(c) in which after T = 25, both of them are relatively uniform. The behavior of lift fluctuation in the oscillating rectangular prism with small side ratio is observed in experimental data such as in [10]. It is noted that fluid force fluctuation behavior is concomitant with body motion but it different in magnitude.
C. Wake flow structure on the oscillating prism
Instantaneous spanwise vortices (ωz) behaviors of oscillating finite rectangular prisms below critical side ratio are presented in figure 8. These behaviors are also considerable different with fixed and infinite height prism as widely described previously. Along the span length of the prism (Z/H), vortices structure may differ in behavior as presented in figure 9 in which the effect of the tip end is pronounced. Detail visualization about these vortices
-0.1
0.0
0.1
-4.0
0.0
4.0
0 50 100 150 200
hCL
T
-0.1
0.0
0.1
-4.0
0.0
4.0
0 50 100 150 200
hCL
T
-2.0
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0 50 100 150 200 250
hCL
T
-2.0
0.0
2.0
-2.0
0.0
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0 50 100 150 200 250
hCL
T
CL η
(a) D/H = 0.2 Vr = 1.5
(b) D/H = 0.5 Vr = 3.0
CL η
CL η
CL η
Fig. 7. Time histories of prism oscillation and lift coefficient fluctuation at (a),(b) onset vibration, (c),(d) Maximum vibration
(d) D/H = 0.5 Vr = 5.0
(c) D/H = 0.2 Vr = 3.0
108
behavior along spanwise can be found in reference [4]. Both of prisms with a side ratio of 0.2 and 0.5, the separated shear layer is introduced at leading edge which commonly takes place in the prism with the sharp edge mainly in high Reynold number. In the case of the oscillating prism, the features of reattachment, reserved flow, and recirculation region depend on the side ratio and reduced velocity. In this case,
reattachment point takes place further downstream. In low reduced velocity (onset galloping) separated shear layer on the side surface roll-up and recirculated with small-scale intensity behind the prism. Flapped shear layer from the leading edge induced negative pressure on the side surface. At low reduced velocity, the intensity of the separated shear layer on the side surfaces both side ratio of 0.2 and 0.5 are not
sufficient to induce fluid damping force to become more negative, hence, base pressure is still low. After forced oscillation is employed, fluid force behave as a positive damping force for the defined range of flow time. However, small-scale response amplitude is still recognized over the flow time, and onset galloping is introduced. These phenomena are seen clearly in figure 8(a) and 8(b).
In contrast, at the reduced velocity of 3.0 where maximum response amplitude of prism with small side ratio take place, separated shear layer from leading edge is drawn back close to the side surface and irregular vortices behind the prism are recognized proximity to the prism. It is implied that flow around the surface of prism mainly on the side surface becomes more negative rather than the former condition. In
x/h-0.1 0 0.1
x/h-0.1 0 0.1
x/h-0.1 0 0.1
0.4
0.6
Fig. 9. Detailed view of instantaneous vortices and mean-streamline of side surface of prism with side ratio of 0.2 at reduced velocity of 2.5
(a) Z/H = 2.5
(b) Z/H = 5.0
(c) Z/H = 9.8
Z Vor
50038927816756
-56-167-278-389-500
(a)
(b)
Fig. 8. Instantaneous spanwise vortices (��) near free end (Z/H=9.8) for finite rectangular prism with side ratio (a) D/H = 0.2 and (b) D/H = 0.5
Vr = 1.5
Vr = 2.0
Vr = 3.0
Vr = 3.0
Vr = 4.0
Vr = 5.0
y/h
-1
-0.5
0
0.5
1
x/h-0.5 0 0.5 1 1.5 2 2.5 3
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x/h
y/h
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-1
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109
this case, fluid damping force completely works as negative damping where amplitude grows exponentially. As a result, the prism oscillation and lift fluctuation are relatively uniform as shown in figure 7(c). Meanwhile, in the prism model with D = 0.5H, flapped shear layer is also drawn back close to side surface prism, and fluctuation of spanwise vortices (ωz) is recognized behind the prism. However, small-scale fluctuations of body motion and lift coefficient still appear in several points in the flow time which indicate that small-scale vortex induced oscillation is introduced. It is different for the model with small side ratio. We recognize that the prism oscillation of side ratio of 0.5 is potential as the transition to vortex induced oscillation in the elastically mounted finite rectangular prism. However, those instabilities of rectangular prisms with side ratio below critical section are induced prominently by the intensity of separation bubble at side surfaces. Figure 9 shows the instantaneous spanwise vortices and mean-streamline behavior on the side surface of the prism with a small side ratio of 0.2 at a reduced velocity of 2.5. Flapped shear layer behavior from leading edge on the side surface is distinctly different with vortices behavior near the free end. In this case, the circulation and reattachment point region drawback. The intensity of the separation bubble at the leading edge near the tip is relatively higher than vortices in the mid-span area and near the base. It triggers recirculation becomes faster due to the fluid force more negative which enhance strong flow interaction in the wake. The interaction of spanwise vortices (ωz) at the free end is considered as a cause of the distinctly different behavior of wake flow near the free end as found in [11] as well.
IV. CONCLUSIONS
Unstable of hydrodynamic force phenomena on the elastically wall mounted prism with the side ratio of 0.2 and 0.5 in the uniform flow has been studied numerically using three-dimensional turbulence model of Large Eddy Simulation. The numerical results of vibration characteristics are compared with the experimental result. In case of response amplitude of elastically mounted rectangular prism, the numerical results are good agreement with experimental data result. The normalized frequencies for both models of side ratios agree with the experimental result. However, a locked-in phenomenon is not found over flow time condition. The locked-in phenomenon appears on the infinite model only at maximum response amplitude (Vr = 3.0). The prism with side ratio around critical side ratio, the normalized frequency of the prism oscillation tends to increase by increasing reduced velocity, where it inverses to the experimental result. We also found that the drag and lift coefficient show different characteristic in elastically mounted finite height prism over reduced velocity compared with two-dimensional and stationary cases. In stationary cases, we find underestimated value of drag coefficient compared with the finite model as depicted by other investigations.
In low reduced velocity (onset galloping), the fluid force acts as a positive damping force after forced oscillation is employed. However, small-scale body oscillation frequency is still recognized over flow time. In contrast, at high reduced velocity, fluctuation separated shear layer on the side surface becomes more negative and the fluid force behaves as a negative damping force, then uniform body oscillation is introduced with exponential growing at specified flow time.
We also find that body motion of prism is completely induced by intensity of separation bubble in the wake mainly on the small side ratio. However, a rectangular prism with critical side ratio, small-scale lift, and oscillation motion fluctuation still appear over time flow due to the fluctuation of spanwise vortices (ωz) behind the prism and interaction of shear layer form the tip end. Therefore, small-scale vortex excitation is potential in the elastically cantilevered finite height prism with a side ratio of 0.5.
ACKNOWLEDGMENT
The authors are thankful for Ph.D. research fellowship between Kanazawa University Japan and Directorate General of Higher Education of Indonesia through KU-DIKTI program funded by Indonesia Endowment Fund for Education (LPDP).
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