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The drag and lift characteristics of flow around a circular cylinder with a slit
Journal Pre-proof The drag and lift characteristics of flow around a circular cylinder with a slit Li-Chieh Hsu, Chien-Lin Chen
PII: DOI: Reference:
S0997-7546(19)30383-8 https://doi.org/10.1016/j.euromechflu.2020.02.009 EJMFLU 103602
To appear in:
European Journal of Mechanics / B Fluids
Received date : 2 July 2019 Revised date : 22 January 2020 Accepted date : 25 February 2020 Please cite this article as: L.-C. Hsu and C.-L. Chen, The drag and lift characteristics of flow around a circular cylinder with a slit, European Journal of Mechanics / B Fluids (2020), doi: https://doi.org/10.1016/j.euromechflu.2020.02.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Masson SAS.
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The Drag and Lift Characteristics of Flow around a Circular Cylinder with a Slit
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Li-Chieh Hsu*, Chien-Lin Chen
Department of Mechanical Engineering, National Yunlin University of Science and Technology 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan
ABSTRACT
Flow past a circular cylinder with a slit results in local perturbation, which affects the flow pattern, vortex shedding and the aerodynamic forces. This study uses spectral element method to simulate flow past a cylinder with a normal slit or an inclined slit at Reynolds numbers of Re=100 to 500. For a cylinder with a normal slit, the drag and amplitude of the oscillating lift both increase as the width of the slit increases. There are two modes for a cylinder with an inclined slit: an injection mode and a blowing/suction mode. The drag also increases as the angle of inclination of the slit increases and is greater than that for a cylinder that has no slit for flow in the blowing/suction mode, except at low Reynolds numbers, such as Re=100. Increasing the width of slit or the angle of inclination of the slit increases the frequency of vortex shedding. However, a wider slit or larger angle of inclination also results in additional drag. There is a similarity between the external surface and the slit in that the aerodynamic forces due to pressure or shear stress vary with the angle of inclination of the slit. Keywords: Cylinder, Normal Slit, Inclined slit, Vortex shedding, Drag and lift
Nomenclature α : 𝛼
angle of inclination of slit : Critical angle of inclination for slit
CQ:
Coefficient of flow rate for a slit
Cd:
Coefficient of drag
Cl:
Coefficient of lift
Cp :
Coefficient of surface pressure for a cylinder
D:
Diameter of the cylinder
𝑄:
Volumetric flux through slit
S:
Width of slit
St:
St
𝑈 :
Flow velocity in the far field
Subscripts curve:
Drag or lift forces on the external surface
curve_pre:
Drag or lift forces due to pressure on the external surface
curve_shear: Drag or lift forces due to shear stress on the external surface
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slit:
Drag or lift forces on the walls of the slit
slit_pre:
Drag or lift forces due to pressure on the walls of the slit
slit_shear:
Drag or lift forces due to shear stress on the walls of the slit
total:
Sum of drag or lift forces on each surface
, Strouhal number
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𝑉 x, t : Velocity at the outlet of slit
1. Introduction
Passive control schemes to control flow are much easier and less costly to implement than active schemes. Using a slit as the communicating channel to the separation flow and wake behind a cylinder is a noteworthy method of flow control. Igarashi [1] experimentally studied the effect of the inclination of a slit on the flow pattern, drag and the frequency of vortex shedding for a cylinder using angles of inclination for the slit from 0° to 90°, between Reynolds numbers 13800 and 52000 and using two slit width ratios: (S/D) 0.08 and 0.185. Two flow patterns exist. For an angle of inclination α 40°, the flow passes through the slit to the wake region and produces an increase in the base pressure, which is called self injection. This results in lower drag and a higher frequency of vortex shedding than for a cylinder that has no slit. For angles of inclination 60° α 90°, the flow exhibits a periodic blowing/suction mode at the exits of the slit. The *
Corresponding author. Tel.: +886-5-5342601 ext. 4151, Fax.: +886-5-5342062 E-mail address: [email protected]
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frequency of vortex shedding for a slotted cylinder is greater than that for a cylinder that has no slit and this frequency increases as the angle of inclination of the slit increases. For angles of inclination α 60°, the drag increases as the angle of inclination increases and is greater than that for a cylinder that has no slit. Olsen and Rajagopalan [2] used circular cylinders with a slit and a concave notch at the rear surface of the cylinder to study vortex shedding for flow with a Reynolds number of Re = 60 to 2300. The experimental results of this study show that the values for the drag and the frequency of vortex shedding for a cylinder with a slit that is normal to the direction of flow are greater than those for a cylinder that has no slit. A cylinder that has a slit and a concave notch at the rear surface produces the greatest drag and a cylinder with a horizontal slit produces the least drag. Ma and Kuo [3] used Ansys/Fluent software to simulate a two-dimensional flow over a circular cylinder with a normal slit at Re=200. The evolution of the boundary layer and the wake were determined using a modified phase-averaged method. It was found that the periodic shedding vortices create a periodic pressure difference between the top and the bottom exits of the slit. The periodic blowing/suction at each exit of the slit creates a perturbation in the boundary layer that significantly delays the separation of the boundary layer along the rear surface of a cylinder with a slit and reduces the length of vortex formation. The upstream induction due to the shedding vortex street leads to primary lock-on of the wake behind a cylinder with a slit. Yang et al. [4] studied the structure of the near wake for a cylinder with a normal slit, S/D=0.15, at Re=1500~7200 using Large Eddy Simulation and experiments. Their results show that because of the periodic blowing/suction motion, the boundary layer passes through the slit and fundamentally changes the structure of wake. The frequency of vortex shedding or the Strouhal number for a slotted cylinder is greater than that for a cylinder that has no slit. Kuo et al [5] studied a cylinder with a normal slit to determine the variation in the separation of the boundary layer along the upper-rear surface and vortex formation in the near-wake at a Reynolds number of 1000. The slit ratio (S/D) was 0.3. The results of these experiments show that the alternate blowing/suction at the exit of the slit creates perturbation in the boundary layer, near the shoulder of a cylinder with a slit. This produces a significant change in the characteristics of the boundary-layer flow along the upper-rear surface of the cylinder, which entails a delay in flow separation and flow reattachment. The history of circulation shows that the shedding behind a cylinder with a slit is much more regular than that behind a baseline cylinder. However, a cylinder with horizontal slit is quite different to a cylinder with a normal slit. Baek and Karniadakis [6] studied a fixed and a free cylinder with a slit that was parallel to the main direction of flow at Re=500 to 1200 using a spectral element method. The results of the experiments show the amplitudes of the oscillating drag and the lift coefficients are significantly decreased when there is a horizontal slit. For a fixed cylinder, the oscillation is regular for a slit ratio(S/D) of less than 0.14. If the slit ratio is greater than 0.14, the oscillation is irregular. For a free cylinder, the 0.16 and the oscillation is suppressed completely if 0.16. oscillation is irregular if Sheng and Chen [7] used a finite volume method to simulate a cylinder with a horizontal slit at low Reynolds numbers from 60 to 250. The results show that at Re=150, the frequency of vortex shedding and the power spectral density in the FFT diagram are reduced when the slit width ratio is increased from S/D=0.1 to 0.3 in increments of 0.05. For Re=250 and slit width ratios S/D = 0.20 and 0.25, the Strouhal number is zero because there is no fluctuation in the lift coefficient, which means that vortex shedding is suppressed. The drag coefficient decreases when the slit width ratio increases because some of the flow goes through the slit from the front of the cylinder to the lower pressure or wake region. Ordia et al. [8] used two-color dyes and a cylinder with a horizontal slit to study the mechanism for the formation of the vortex behind bluff bodies at Re=200-2300 and to determine the potential of a cylinder with a parallel slit as an improved vortex generator. The results of these experiments show that at Re < 1000, the Strouhal number increases when the slit width is increased, except for S/D=0.4. The variation in the Strouhal number with the slit width is different to the relationship that was determined by Sheng and Chen [7]. Notably, these experiments [8] were conducted in a circular pipe, so the wall confinement effect is relevant. For Re > 1000, the slit width does not have a significant effect on the frequency of vortex shedding. However, for S/D=0.4, the flow behavior is similar to that for dual bluff bodies with a much lower Strouhal number than that for a cylinder that has no slit. For S/D < 0.4, the flow behavior resembles that for a single body. The vortex behind a cylinder for which S/D < 0.4 is shorter and narrower than that behind a cylinder that has no slit. For S/D=0.4, the vortex is longer and wider than that behind a cylinder that has no slit. Gao et al. [9] used particle image velocimetry (PIV) to study a cylinder with a horizontal slit in a flow at Re 2.67 10 . The results of these experiments show that a slit reduces the drag and eliminates the fluctuation in the amplitude of the lift. The diagram for the mean pressure around the surface shows that the difference in pressure between the windward and leeward stagnation points is reduced as the slit ratio (S/D) increases. For a slit ratio S/D of 0.05, the RMS value for the lift coefficients is a minimum and the lift is suppressed by up to 81.78%. The frequency spectrum for the lift time history diagram shows that the frequency increases as the slit width increases. When the S/D ratio increases, the region in which the vortex is formed is pushed further downstream and the jet vortices from the slit become stronger. For S/D < 0.1 the interaction and competition between the slit jets and the separation flows results in two recirculation regions. For S/D > 0.1 the flow behaves like a side-by-side system and the recirculation bubble becomes longer when the 2
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2. Numerical Method
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slit ratio is increased. The experimentally obtained values for the Strouhal number for a cylinder with a horizontal slit [8, 9] do not agree with the numerical results [7]. Although there is some disagreement between the numerical and experimental results for a cylinder with a horizontal slit, a normal slit increases the frequency and the strength of vortex shedding, but a sufficiently large horizontal slit suppresses vortex shedding and the drag and the amplitude of the oscillations in the lift are reduced [6]. A normal slit reduces the length of vortex formation [3], but a horizontal slit prolongs it. The Strouhal number for vortex shedding is increased for a normal slit [4] and decreased for a sufficiently large horizontal slit [9]. Therefore, a cylinder with normal slit can enhance the flow rate measurement because it has a stronger and more stable and regular signal [10-12]. It remains to be determined whether these measured signals are representative of the signal that is measured for a cylinder that has no slit and a real flow rate. To study this problem, Peng et al. [13] experimentally studied the effect of the slit width (S/D= 0 to 0.3) on the quality of the measurement for a cylinder with a slit. The results show that for S/D=0.1 to 0.15, the measurement has the best signal quality, with the minimum error for seven different slit widths. Although the effect of the slit ratio on a horizontal cylinder with a slit has been studied, there have been no systematic studies that involve a cylinder with a normal slit. This study determines the effect of a normal slit ratio and the angle of inclination for the slit on the flow pattern, the drag, the lift, the vortex shedding frequency and the mechanism for the formation of the flow patterns.
The incompressible Navier-Stokes equations are solved as:
u 1 u u p 2u t u 0
(1)
(2) A time splitting scheme [14] is used to treat the nonlinear convective terms explicitly and the pressure and velocity diffusion terms are solved implicitly as follows:
uˆ u n t
2
r 0
r
( u u ) nr
(3)
(4)
ˆ u n1 uˆ 2u n1 t
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ˆ 1 uˆ uˆ p n1 t
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The scheme uses intermediate time step values for velocity, 𝑢⃑ and 𝑢⃑, where and ˆ are between the nth and n+1st time steps. The nonlinear terms are advanced by a third-order Adams-Bashforth scheme with coefficients, 𝛽 . The pressure term is treated separately as an elliptical problem by taking the divergence on both sides of Eq.(4). Due to the divergence free condition, the surface term is zero. Hence,
2 p n 1
t
uˆ
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The pressure and the velocity calculations are types of Helmholtz problems, expressed in the following general form:
( 2 2 ) g
on
(7)
The variational form of Eq.(7) is:
dx 2 dx gdx
3
(8)
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which is discretized using the spectral element method. The solution domain is divided into macro elements. All variables are substituted by discrete approximations as: N
M
k hk pq hp (r )hq (s)
(9)
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p0 q0
where the h functions are the Lagrangian interpolants, which use the orthogonal set of Legendre polynomials for a high degree N (or M ), and r, s are the local coordinates of each element, k. Performing a Gauss-Lobatto quadrature and summing the contributions of adjacent elements gives the global matrix equation as:
( A 2 B ) Bg
(10)
where A is the discrete Laplacian operator and B is the mass matrix. The matrix equation is solved using a preconditioned conjugate gradient. Details can be found in previous studies [15-17].
2.1 Grid independence
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A rectangular computational domain is shown in Fig. 1. The diameter of the cylinder (D) is 1 unit. The domain extends to a distance of 20D in upstream from the center of the cylinder and a distance of 30D downstream. The upper and lower boundaries are both 20D from the center of the cylinder. Uniform flow Dirichlet boundary conditions (u=U0, v=0) are used for the inflow, the upper and the lower boundaries. The outflow boundary condition is used for the boundary that is farthest downstream. To determine grid independence, the number of elements (K) and the order of the Legendre polynomial (N) are serially increased to determine the convergence of the surface pressure coefficient for a regular cylinder in a flow of Re=100. Figure 2 (b) shows that the pressure coefficient converges when N 10. For this number of elements and order of the Legendre polynomial, the surface pressure coefficient for a higher Reynolds number Re=140 is in agreement with the result of Park [18], so the following simulation of a cylinder with a slit for flow at Re=100 to 200 uses a similar number of elements (K) and N=11 and the same computational domain that is described previously. However, for flow at higher Reynolds number, Re=500, more degree of freedoms are considered to simulate at this flow regime. The convergence test is implemented from order of N=11 to 15. The solution of N=15 is used as the reference solution. Figures 2 (c) and (d) indicates for reaching convergence, the order of the Legendre polynomial is not lower than N=13 for Re=500. Hence the following simulation of a cylinder with a slit for flow at Re=500 uses a similar number of elements (K) and N=15.
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Fig.1 The computational domain: (a) number of elements, k=334 and (b) a magnified view of the elements and grids
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Fig.2 The coefficient of surface pressure for a cylinder with a different order of the Legendre polynomial (N) and number of elements, k=334: (a) at Re=100, (b) a comparison with the surface pressure coefficient for the cylinder that is used in this study with K=334 and N=11 and the results for other studies, for Re=140 (c) at Re=500 (d) the L2 error with different degrees of freedom (DoF)
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2.2 Validation Using the same computational domain and number of elements, the case of flow at Re=200 past a cylinder with a normal slit is used for validation. This scenario is shown in Fig. 3(a). Fig. 3(b) shows a drag-lift polar diagram, which shows that the result for this numerical method agrees well with that of Ma and Kuo [3]. The amplitudes and mean values for the drag and lift are listed in Table 1. The drag and lift are calculated using Eqs. (11) and (12). The range of integration includes the surface of the cylinder and the walls of the slit. The coefficients of drag and lift are defined by Eqs. (13) and (14). The second case that is used for validation is a flow of Re=500 past a cylinder with a horizontal slit. Figure 4 shows that the values for the periodic drag and lift coefficients for this numerical simulation are in good agreement with those of Baek and Kaniadarkis [6], but the period of time for this simulation is different from that for the previous study. There is a 0.77% difference in the amplitude of the oscillations in the lift and a 1.7% difference in the drag coefficient between this simulation and that study.
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𝐩 ∙ 𝒏𝒙
𝐹𝐿
μ ∇𝐮
∇𝐮𝑇 ∙ 𝒏𝒙 ds
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𝐹𝐷
𝐩 ∙ 𝒏𝒚
μ ∇𝐮
∇𝐮𝑇 ∙ 𝒏𝒚 ds
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𝐹𝐷 1 𝜌𝑈 𝐷 2
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𝐶𝑙
𝐹𝐿 1 𝜌𝑈 𝐷 2
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𝑉𝑠 x, t ∙ dx
𝑄 𝑡
𝑄 𝑈 ∙𝑆
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𝐶
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Fig. 3 Flow past a cylinder with a normal slit S/D=0.15 at Re=200: (a) vorticity contour of present result and (b) the comparison of drag-lift polar diagram of presnt result and that of Ma and Kuo [3].
Table 1 Comparison of the drag and lift coefficients for a cylinder with a slit for which S/D=0.15, 𝛼=90˚ at Re=200 Ma and Kuo [3]
Present
Cl
±0.775
±0.78
Cd
1.419±0.0555
1.419±0.0553
6
15
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Fig. 4 Comparison of the drag and lift coefficient history for flow at Re=500 past a cylinder with a horizontal slit for present results and those of Baek and Kaniadarkis [6]: the upper curve represents Cd, the lower curve represents Cl and time
3. Results and discussion
t𝑈 /𝐷
To determine the effects of the width and inclination of the slit on the flow patterns and aerodynamic forces, the following study initially focuses on the flow of Re=100 and extends the theory to the flows with a higher Reynolds number. 3.1 The effect of the slit ratio
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The ratio of the width of slit to the diameter of the cylinder is the slit ratio (S/D). Simulations of flow past a cylinder with a normal slit using different slit ratios from 0.08 to 0.2 were conducted to determine the effect of the width of the slit on flow patterns and aerodynamic forces. When vortices are shed from behind the cylinder, there is a pressure difference between the two exits of the slit, so the flow passes through the slit from one end to the other. However, the external flow is similar to that for a cylinder that has no slit, as shown in Fig. 5(a). The internal flow inside the slit exhibits a periodic blowing/suction flow pattern, with small vortices near the exits, as shown in Figs. 5(b) and 5(c). A difference in the slit ratio results in a difference in the size of the local vortices near the inlet and outlet of the slit, as shown in Fig. 6. The non-dimensional flow rate coefficient, CQ , is measured for flow inside the slit. A positive value means that flow exits the slit and a negative value means that flow is sucked into the slit. The flow rate through the slit is increased when the slit ratio is increased, as shown in Fig. 7. The frequency of this periodic blowing/suction flow inside the silt is the same as that for vortex shedding behind the cylinder, as shown in Fig. 8. The frequency of vortex shedding increases when the slit ratio increases and is greater than that for a cylinder that has no slit, as shown in Fig. 7(b). However, there is a very minor phase lag between the slit flow rate and the lift coefficient, as shown in Fig. 9(a). The magnitude of this phase lag increases when the slit ratio is increased, as shown in Fig. 9(b). To determine the mechanism that causes a variation in the aerodynamic forces due to the slit, the following analyses of drag and lift respectively use the contributions from the external surface and the walls of slit. The drag due to the external surface is called the external drag. The drag due to the walls of the slit is called the slit drag. The external lift and slit lift are similarly defined. Each aerodynamic force that exerts on a surface is divided into pressure and a shear stress. Drag A larger slit ratio results in greater drag, but the drag coefficients for all slotted cylinders are smaller than that for a cylinder that has no slit, as shown in Fig. 10(a). A wider slit produces a decrease in the external drag but an increase in the slit drag, as shown in Fig. 11. Figure 12(a) shows that the shear drag is the principal cause of variation in the external drag, in terms of the magnitude of the change. The shear drag of the external flow decreases linearly because the effective surface becomes smaller when the slit ratio is increased. In terms of the slit flow, the pressure drag increases linearly when the slit ratio is increased, as shown in Fig. 12(b). A wider slit results in an increase in the flow rate through the slit and produces a greater imbalance in the pressure that is exerted on the two walls of the slit channel and a greater local vortex near the inlet or the outlet of the slit, as shown in Figs. 6, 7 and Fig. 10(b). However, shear stress inside slit has no effect on drag because the direction of the shear stress is perpendicular to the direction of the drag or the bulk flow. 7
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Summing every contribution to the drag, variation in the total drag is principally due to the slit pressure drag. This conclusion is different from those of previous studies [1, 19] which determine that for a cylinder with a normal slit, the drag is reduced when the slit ratio is increased. However, the results of this study show that a larger slit ratio results in greater drag when 0.1 and less drag when 0.1. The main reason for this difference is that this study combines the drag on the external surface of the cylinder and on the walls of the slit that results from pressure and shear stress into the total drag, but previous studies [1,19] only consider the drag that results from the pressure that exerts on the external surface of the cylinder. When the slit ratio is increased, the effect of the slit pressure drag is augmented and results in greater drag, as shown in Figs. 10(a), 11(b) and Fig. 12(b).
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Fig. 6 Streamlines and pressure distribution at Re=100 for different slit ratios: (a) S/D=0.08, (b) S/D=0.1, (c) S/D=0.15 and (d) S/D=0.2
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Fig. 7 Slit flow rate and vortex shedding: (a) the coefficient of flow rate inside the slit and (b) the frequency of vortex shedding for a cylinder with t𝑈 /𝐷
.
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Fig. 8 Lift coefficients of slit and total at S/D=0.1 and Re=100: (a) lift due to the walls of the slit and total lift
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(b) a magnified view; time
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Fig. 9 Phase lag between the oscillation in the slit flow rate coefficient and the lift coefficient : (a) the normalized slit flow rate coefficient and the lift coefficient and (b) the phase lag for different slit ratios
Lift
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The mean lift coefficient is zero for each case because the geometry is symmetrical, as shown in Figs. 13(a) and (b). When the slit ratio is increased, the amplitude of the external lift is reduced, but the amplitude of the lift due to the slit is increased. However, the amplitude of the oscillations in the total lift coefficient increases when the slit ratio is increased, as shown in Fig. 14(a). Because the effective surface in the y-direction is shortened, the amplitude of the oscillations in the lift force due to the external pressure decreases when the slit width is increased, as shown in Fig. 14(b). The two discontinuous sections of the surface of the cylinder that are divided by the slit limit the growth of a boundary layer and generate a greater shear stress. The effect of this shear stress on the amplitude of the oscillations in the lift is increased when the width of the slit is increased, as shown in Fig. 14(b). In terms of flow inside the slit, the amplitude of the lift due to shear stress increases when the slit width is increased because there is a greater flow rate through a wider slit, as shown in Figs. 7, 13(b) and 14(c). However, the pressure inside the slit has no effect on lift because of the direction in which it acts. Therefore, variations in the lift amplitude for different sizes of slit are mainly due to the shear stress that acts on the slit. A larger slit ratio results in a thinner boundary layer because there is a discontinuous external surface and there is greater flow through the slit so there is greater shear stress on the external surface and on the walls of the slit. Therefore, the amplitude of the oscillations in the lift coefficient increases when the slit ratio is increased, as shown in Fig. 14(a), so changes in the amplitude of the oscillations in the lift coefficient are mainly due to the effect of shear stress. This conclusion involving the lift coefficient is also different from the conclusions of previous studies [1, 19] which only consider the lift due to the pressure that exerts on the external surface.
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Fig. 10 Drag of a cylinder with different slit ratios: (a) time mean total drag coefficient and (b) pressure coefficient along the surface of the cylinder
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In summary, for a cylinder with a normal slit, shear drag dominates external drag and pressure drag dominates slit drag. Total drag is dominated by pressure drag of slit. Lift amplitude is dominated by shear stress. Hence, to determine more precise characteristics of aerodynamic forces, the contribution from slit and effect of shear stress are not negligible.
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Time mean drag coefficient for different slit ratios: (a) external drag and (b) slit drag
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Fig. 11
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Fig. 12 Mean drag coefficients at Re=100: (a) external drag due to pressure and shear stress and (b) slit drag due to pressure and shear stress
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Fig. 13 Lift coefficients for different slit ratios: (a) lift coefficient due to the external surface and (b) lift coefficient due to the slit
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Fig. 14 Lift amplitude: (a) total lift (b) lift amplitude due to pressure and shear stress on the external surface and (c) lift amplitude due to pressure and shear
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stress on the walls of the slit
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3.2 Inclined slit The angle of inclination for the slit is measured from α 0° at the leeward base point to α 180° at the windward stagnation point. The flow past a cylinder with a slit that is inclined at angles from 0° to 90° is simulated to determine the effect of the inclination on the variation in the flow pattern and the aerodynamic forces. The overall flow pattern is shown in Fig. 15. Generally, if the angle of inclination of the slit, α 45°, some flow goes through the slit in one direction from the upstream inlet to the wake region, which is called the injection mode and is shown in Figs. 16 (a) to (d). If α 60°, flow goes periodically back and forth through the slit. The flow alternately exits the slit and is sucked back into the slit. This flow pattern is the blowing/suction mode and is shown in Figs. 16 (e) and (f) and Fig. 5. In the injection mode, if the angle of inclination, α 15°, the flow exits the slit into the wake and sprays up and down periodically to form a part of a vortex. For an angle of inclination, α 30°, the flow exits the slit in an upward direction. After a half period, there is a saddle point at the exit of the slit. For α 45°, the flow exits the slit steadily in an upward direction and is not affected by the shedding vortex. It is arguable that there is a transition between these modes for 45° α 60°.
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Fig. 15 Vorticity at Re=100, S/D=0.1 and an angle of inclination for the slit, 𝛼=45°
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Fig. 16 Flow patterns for a cylinder with an inclined slit: (a) 𝛼
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There are two flow patterns, but there is a critical angle of inclination at which the flow transforms from the 56.33° injection mode to the suction/blowing mode. The results of several trials show that the critical angle is 𝛼 for flow at Re=100. If the angle of inclination for the slit is less than the critical angle, the flow is in the injection mode. If the inclination of the slit is greater than critical angle, the flow is in the suction/blowing mode. If the angle of inclination of the slit α 𝛼 , the flow rate in all cases is positive so there is injection flow only, as shown in Fig. 18(a). However, for α 𝛼 , the flow rate oscillates periodically between a positive and a negative value, so flow is in the blowing/suction mode, as shown in Fig. 18(b). As shown in Fig. 18(b), for α 60°, the blowing motion is stronger than the suction motion because the positive value is greater than the negative value. However, for α 75°, the suction motion is stronger. For α 90°, the blowing rate is equal to the suction rate. For any angle of inclination for the slit, the slit flow rate is periodic. Notably, if α 𝛼 , there is an alternating injection flow and a zero flow rate inside the slit, as shown in Figs. 17 and 18(b). Figures 18(a) and (b) show that the time mean flow rate of slit is reduced as the angle of inclination for the slit is increased. However, the amplitude of the oscillations in the slit flow rate increases as the inclination of the slit is increased. The non-dimensional frequency of the oscillations in the slit flow at zero degrees of slit inclination is double that for other angles of inclination, as shown in Figs. 18(a) and (c). The reason is the frequency of slit flow at α 0° is double that of vortex shedding. That is, during a cycle of vortex shedding, the slit flow has been injected twice. For other angles of inclination for slit, the frequency of slit flow is very close to that of vortex shedding. When α 10°, the frequency of the slit flow increases as the angle of inclination for the slit is increased, as shown in Fig.18 (d). When α 14
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𝛼 , the frequency of the slit flow is almost as high as that for α 90°. The frequency of the slit flow is lower than that of vortex shedding when the slit flow is in the injection mode. However, for slit flow in the blowing/suction mode, the frequency of the slit flow is higher than that of vortex shedding, except for α 60°, as shown in Fig. 18(d). Only at α 90° does the frequency of slit flow coincide with that of vortex shedding for a cylinder. (a)
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Fig. 17 Streamline and pressure distribution for a slit that is inclined at the critical angle, 𝛼
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Fig. 18 Flow rate of the slit and the Strouhal number for different angles of inclination for the slit, S/D=0.1: (a) flow rate in the injection mode, (b) flow rate in the blowing/suction mode, (c) Strouhal numbers based on the frequency of slit flow rate and the frequency of vortex shedding for a cylinder and (d) a magnified view of (c); time t𝑈 /𝐷 15
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Drag
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An increase in the angle of inclination for the slit results in greater external drag but a decrease in the drag due to the slit, as shown in Figs. 19 (a) and (b). The total drag is the sum of the external drag and the drag due to the slit. The drag due to the slit has a significant effect on the total drag at small angles of inclination for the slit, as shown in Fig. 19(c), but the slit has less effect on the total drag is at large angles of inclination for the slit, as shown in Fig. 19(d). As shown in Fig. 20(a), the pressure drag due to the external surface and the wall of the slit both increase with high similarity as the angle of inclination for the slit is increased. This is because the effective area in the x-direction on which pressure is exerted is increased when the angle of inclination for the slit is increased. For larger angles of inclination, the corners at the exit of slit generate local vortices, which also increase the pressure drag due to the slit. The variation in the pressure drag due to the external surface was demonstrated by Igarashi [1]. The shear drag due to the external surface and the wall of the slit both decrease in the same manner as the angle of inclination for the slit is increased, as shown in Fig. 20(b). The effective area in the x-direction on which shear stress is exerted decreases so the external shear drag is decreased when the angle of inclination for the slit is increased. In terms of the slit, the flow rate and the horizontal component of the shear stress are smaller at higher angles of inclination for the slit so there is a lower shear drag due to the slit. In terms of the magnitude of the force, the pressure drag has a dominant effect on the external drag and increases when the angle of inclination for the slit is increased, as shown in Fig. 20(a). The shear stress has a dominant effect on the drag inside the slit and decreases when the angle of inclination for the slit is increased, as shown in Fig. 20(b). Therefore, the total drag does not vary regularly with the angle of inclination for the slit, as shown in Fig. 21(a).
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Fig. 19 Drag coefficient for different angles of inclination for the slit: (a) external drag, (b) slit drag, (c) drag coefficient history for an angle of inclination for the slit of α 0° and, (d) ) drag coefficient history for an angle of inclination for the slit of α 90°; time t𝑈 ⁄𝐷
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Figure 20 shows that the magnitude of variation in drag due to shear stress on the walls of slit from α 0° to 90° is same with the drag due to pressure on external surface. That means the variation of shear stress inside slit is significant enough to change the regular variation of drag on the external surface as flow at low Reynolds number. For Re=100, the drag on a cylinder with any inclined slit is less than that on a cylinder that has no slit. For α 𝛼 , the drag has a maximum value that is close to that for a solid cylinder, because the flow rate is periodically zero. The drag due to the shear stress on the external surface exists a lowest value as 𝛼 75° shown in Fig. 20(b). It is because a strong suction flow at upper inlet of slit delays the separation point to result in the increase of shear stress on the rear surface of cylinder as the comparison of Figs. 16(c), (e) and (f). In fact, the suction flow results in shear stress in the opposite direction of drag on the rear upper surface. Besides, Fig.18(b) shows the suction flow is stronger than the blowing flow as 𝛼 75°. The blowing flow at lower exit of slit does not affect the separation much because the location of blowing is in the front of separation point. The periodic oscillation of drag due to the shear stress on the external surface at different angles of inclination of slit are shown in Fig. 21(b) which shows the case of 𝛼 75° has the smallest shear drag on external surface. The lower drags due to the shear stress on external surface and slit both reduce the total drag, especially, for 𝛼 75° as shown in Fig.21 (a).
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Fig. 20 Variation in the pressure drag and shear drag with the angle of inclination for the slit : (a) pressure drag due to the external surface and the slit
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Fig. 21 Total drag at different angles of inclination for the slit : the dashed line shows the total drag for a solid cylinder and the solid line shows the drag for a slotted cylinder
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Lift
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For a cylinder with an inclined slit, the total lift is the sum of the lift on the external surface and the lift on the wall of the slit, as shown in Fig. 22 (a). The Strouhal number for a cylinder with an inclined silt is greater than that for a cylinder that has no slit and increases as the angle of inclination for the slit is increased, as shown in Fig.22 (b). The lift coefficient oscillates periodically but the following analyses are condensed by using a time mean lift coefficient. The external lift that is caused by pressure depends on the projection of the external surface in the y-direction and the external lift that is caused by shear stress depends on the projection of the external surface in the x-direction. When the angle of inclination of the slit is increased, the projection in the y-direction is decreased and the projection in the x-direction is increased. In the injection mode, at α ≤ 35°, when the slit angle is increased, the lift due to pressure on the external surface and on the wall of the slit decreases from zero to a negative value because the effective surface in the y-direction is decreased, as shown in Fig. 23 (a). In the periodic blowing/suction mode, at 𝛼 35°, when the angle of inclination for the slit is increased, the lift due to pressure on the external surface and the wall of the slit recovers from a minimum back to zero because at larger angles of inclination for the slit, the flow pattern is more symmetrical with respect to the main direction of flow. The change of effective area due to the slit is related to the pressure loss on the surface. The region of surface of cylinder can be divided by the separation point into higher pressure region and lower pressure region as shown in Fig. 2(a). The location on the surface in front of separation point is higher pressure region. The location on the surface behind the separation point is lower pressure region. For low angle of inclination of slit, for example, 𝛼 30°, there is an loss of effective area for lift force in higher pressure region on the lower surface of the cylinder due to the inlet of slit. However, the other loss of effective area for pressure exists in lower pressure region because the outlet of slit on the upper surface of the cylinder locates behind the separation point. The lower surface experiences more loss of pressure force than that of upper surface. This unbalanced pressure results in downward force or negative lift as shown in Fig. 23(a). For high angle of inclination of slit, for example, 𝛼 60°, the loss of pressure in higher pressure region for lower surface of the cylinder is diminished because the surface pressure is smaller at that location. The other exit of slit is still in lower pressure region of upper surface. Therefore, smaller unbalanced pressure between the lower surface and upper surface results in lesser negative lift as shown in Fig. 23(a). In terms of lift due to the shear stress, Fig. 23 (b) shows that for α ≤ 35°, as the angle of inclination for the slit is increased, the external shear lift increases from zero to a positive value because the effective surface is increased. The shear lift due to the slit is also increased because the upward component of the shear stress becomes larger. For α 35°, as the angle of inclination for the slit is increased, the shear lift on the external surface decreases from a maximum value at α 35° to zero at α 90° because the flow pattern is more symmetrical with respect to the main direction of flow. The flow rate inside slit actually affects the shear stress of slit. Figure 23(b) shows the trend of variation of lift forces due to shear stress on the walls of the slit at different slit angles associates with the flow rate inside slit in y-direction.
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Fig. 22 Lift and Strouhal number for a cylinder with an inclined slit: (a) the history of the lift coefficient for α Strouhal number for vortex shedding for a cylinder with a slit, S/D=0.1, at different angles of inclination
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60° and time
t𝑈 ⁄𝐷 and (b) the
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However, the shear stress has an considerable effect on external lift as the comparison of Figs. 23 (a) and (b). The unbalanced pressure between lower surface and upper surface and shear stress on external surface results in the maximum exists around 𝛼 75° and the minimum around 𝛼 35° for external lift as shown in Fig. 23(c). Furthermore, as 𝛼 75°, the stronger suction flow at upper inlet of slit delays the separation point and results in the increase of shear stress on the rear surface of cylinder. The greater shear stress contributes some upward components of force or lift and enhances the total lift at 𝛼 75°. However, the shear stress has a dominant effect on lift due to slit, as shown in Fig. 23 (d). Figures 23(a) and (b) show that regardless of whether lift is due to pressure or to shear stress, the angle of inclination for the slit has a similar effect on the variation in the lift on the external surface and on the slit. The same is true of the drag on the external surface and the slit due to pressure or to shear stress, as shown in Fig. 20.
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Fig.23 Lift coefficient at different angles of inclination for the slit: (a) lift due to pressure on the external surface and on the walls of the slit, (b) ) lift due to shear stress on the external surface and the walls of the slit, (c) lift due to the external surface only and (d) lift due to the walls of the slit only
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3.3 Higher Reynolds Number Regimes
Flow patterns
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Flow at Re=200 and 500 is used to determine the effect of higher Reynolds numbers on the variation in the flow pattern and the aerodynamic forces for a slotted cylinder. A comparison of Figs 24(a) to (c) shows that for a cylinder with a normal slit, the distance between the third vortex and the center of the cylinder decreases from more than 4 to less than 4. Moreover, the distance of third vortex to the center of cylinder is shorten to 3 as S/D=0.2. The effect of the slit ratio is more obvious for a Reynolds number of 500. Figs. 24(g) to (i), show that the distance of the first three vortices to the center of a cylinder with a normal slit is much less than that for the cylinder that has no slit. For x 8 at Re=500, more vortices are shed by a cylinder with a normal slit than by a cylinder that has no slit. For Re=200 or 500, the larger slit ratio results in shed vortices closer to the cylinder. For Re=200, the distance between the first three vortices and the center of cylinder is not changed obviously as shown in Figs. 24 (a) and (d) to (f) shows. However, Fig. 25 (b) also shows that at Re=200, the frequency of vortex shedding for a cylinder with a slit at any angle of inclination is higher than that for a cylinder which has no slit. The same is true for a flow at Re=100. For a cylinder with an inclined slit at Re=500, for which flow is in the injection mode at α 0°, the presence of the slit increases the distance between the third vortex and the center of cylinder from 5 to 6 and the frequency of vortex shedding is decreased, as is seen by comparing Figs. 24(g) and (j). However, at a flow of Re=500 in the blowing/suction mode at α 60°, the presence of the slit decreases the distance between the third vortex and the center of cylinder and vortex shedding has a higher frequency, as a comparison of Figs. 24(g) and (l) shows. The variation in the frequency of vortex shedding for different angles of inclination for the slit are shown in Fig. 25(b). The reason that for Re=200, the flow pattern in the wake of a cylinder with an inclined slit is not changed obviously is very low Reynolds number of the flow inside slit. The Reynolds numbers of flow inside slit are Re=1 to 3 for bulk flow at Re=200 and Re=10 to 18 for bulk flow at Re=500. For all Reynolds numbers, the frequency of vortex shedding for a slotted cylinder increases as the slit ratio or the angle of inclination for the slit increases, as shown in Fig. 25. The frequency of vortex shedding for a cylinder with a normal slit for any slit ratio is greater than the frequency for a cylinder that has no slit. For inclined slit, at Re=100 and 200, the Strouhal number for a slotted cylinder with a slit at any angle of inclination is greater than that for a non-slotted cylinder, as shown in Fig. 25(b). For Re=500, only when α 30° the Strouhal number for a slotted cylinder is greater than that for a non-slotted cylinder. An increase in the angle of inclination for the slit produces a more significant increase in the Strouhal number at higher Reynolds numbers and at greater angles of inclination for the slit.
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Fig.24 Variation in the flow pattern for different slit ratios and angles of inclination for the slit at higher Reynolds numbers: Re=200: (a) non-slotted cylinder, (b) S/D=0.1, α S/D=0.1, α
90°, (c) S/D=0.2, α 90°, (i) S/D=0.2, α
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90°, (j) S/D=0.1, α
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are -20 to 20 for Re=200 and -30 to 30 for Re=500.
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The effect of the slit ratio on the drag and lift
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For a cylinder with a normal slit at flows with a higher Reynolds number, there is a significant increase in drag as the slit ratio increases. This increase is much greater than that for a cylinder that has no slit, as shown in Fig. 26(a). At Re=200 or 500, as the slit ratio increases, the pressure drag due to the external surface increases significantly and mainly causes the great increase in the total drag as the analysis in Fig. 26(c) and (e). This phenomenon is different from that in the lower Reynolds number. At Re=100, the increase in total drag as the slit ratio increases is mainly caused by the increase in pressure drag due to the slit as shown in Fig.12. The amplitude of the oscillation in the lift increases as the slit ratio increases for Re=100 and 200, as shown in Fig. 26 (b). However, for Re=500, the amplitude of lift has a maximum value for S/D=0.04 and decreases with slit ratio as 0.04. This phenomenon also can be found in the Figs. 24(g) to (i) which show that the oscillation of vortices are more constrained as the slit ratio increases. For higher Reynolds number, the external pressure dominates the variation of amplitude of lift as the analysis shown in Fig. 26(d) and (f). In summary, at higher Reynolds number, the external pressure is the critical factor for the variation of drag and lift of a cylinder with different slit ratios.
Fig. 25 Strouhal number for different slit ratios and angles of inclination for the slit at Re=100 to 500: (a) variation of St with the slit ratio and (b)
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variation in St. with the angle of inclination for the slit
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Fig. 26 Drag and lift coefficients for a cylinder with a normal slit for different slit ratios at Re=100 to 500: (a) total drag (b) amplitude of total lift (c) decomposition of drag at Re=200 (d) decomposition of lift amplitude at Re=200 (e) decomposition of drag at Re=500 (f) decomposition of lift amplitude at Re=500
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The effect of the inclination of the slit on the drag and lift
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The drag varies irregularly with the angle of inclination for the slit at Re=100, but the drag increases as the angle of inclination of the slit is increased at Re=200 and Re=500, as shown in Figs. 27 (a) and (b). For all Reynolds numbers, the external drag increases as the angle of inclination for the slit is increased, but the drag due to the slit decreases as the angle of inclination for the slit is increased, as shown in Figs. 27(c) and (d). However, the external drag has a dominant effect on the total drag at higher Reynolds numbers so for α 45° and value for Re=200 and 500, the total drag on a cylinder with an inclined slit is greater than the drag on a cylinder that has no slit. For α 45°, the total drag on a cylinder with an inclined slit is less than the drag on a cylinder that has no slit, as shown in Figs 27 (a) and (b). This is because that at smaller angle of inclination for the slit, there is an increase in flow through the slit from the windward side of the cylinder to the wake region, so there is a stronger injection flow and a lesser difference in the pressure between the windward and the leeward surfaces. This produces less external drag and total drag but more slit drag at smaller angles of inclination for the slit. Similarly to the effect of lift at Re=100, the locations of two exits of slit in the higher pressure region and lower pressure region, respectively, result in the unbalanced pressure which associates with external lift as shown in Fig. 28(a). The slit flow rate also affects the lift due to shear stress on the walls of the slit as shown in Fig. 28(b). The pressure dominates the external lift, whereas shear stress dominates the lift due to slit. As 𝛼 30°, the change in the effective surface area means that external lift decreases but lift due to the slit increases as the angle of inclination for the slit is increased, as shown in Figs 28(a) and (b).
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Fig. 27 Variation in the drag with the angle of inclination for the slit: (a) total drag at Re=100 to 200, (b) total drag at Re=100 to 500, (c) external drag and (d) slit drag
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However, at 𝛼 30°, both types of lift recovers to zero as the angle of inclination for the slit is increased because the flow is more symmetrical with respect to the direction of the main flow. The variation in the lift with the angle of inclination for the slit at Re=200 and Re= 500 is similar to that at Re=100. For higher Reynolds numbers, the variation in the total lift with the angle of inclination for the slit is also irregular, as it is at Re=100. The maximum of lift exists at α 75° and the minimum at 𝛼 30°~40° for Re=500. Because the external pressure force dominates the total lift at higher Reynolds numbers, Fig. 29 may show the reason that lift coefficient is higher at α 75° and lower at α 30° for Re=500. The lower surface of cylinder is exerted by larger pressure than that on the upper surface as inclination angle of slit is at α 75°. The unbalanced pressure between lower surface and upper surface results in larger lift. However, for α 30° , the pressure on the upper surface of cylinder is larger than that on the lower surface. This results in lower or even negative lift. For α 0° or α 90°, the pressure distributions for upper and lower surfaces are much more equal to each other. This results in lift coefficient is closer to zero. Figure 29 also shows the external drag is greater as the inclination of slit is larger. However, among that, the largest drag exists at α 75° which is even larger than that of α 90°. This trend of drag with different angle of inclination for slit is similar to the result of Igarashi [1] which shows the maximum drag exist at α 65° and the drag is lower than that of non-slotted cylinder as α 45° for flow of Re=45000.
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Fig. 28 Comparison of lift for different Reynolds regimes for different angles of inclination for the slit: (a) total lift, (b) external lift, (c) slit lift, (d) pressure
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Fig.29 Coefficient of surface pressure for a cylinder with different angles of inclination of slit for flow at Re=500
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Critical Angle
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Figures. 17 and 18(b) show the characteristic of slit flow at critical angle is an alternating injection flow which associates with a periodic zero flow rate such as 𝛼 56.33° for Re=100. Similarly, there are critical angles for higher Reynolds number flow. Bases on this characteristic, the critical angles are determined as 𝛼 49.26° for Re=200 and 𝛼 48.21° for Re=500. Figure 30 (a) to (c) show the slit flow is in injection mode which enters the lower inlet of slit and ejects out from the upper exit. The slit flow goes upward inside slit and causes the positive vorticity on the right side of the wall and negative vorticity in the left side of the wall. Figures 30 (d) to (g) shows no any flow goes through the slit. Figures 30 (i) and(h) starts another cycle of injection. For flow modes at higher Reynolds number regimes are divided by critical angle of slit. If the angle of inclination is less than the critical angle, the flow pattern is in the injection mode. If the angle of inclination is greater than the critical angle, the flow pattern is in the blowing/suction mode.
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Fig. 30 A cycle of vorticity of cylinder with a slit, S/D=0.1, 𝛼 one is negative vorticity.
48.21°, at flow of Re=500: the red color indicates positive vorticity and blue
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4. Conclusion For a cylinder with a normal slit, shear stress has a dominant effect on external drag and pressure has a dominant effect on slit drag. The slit pressure drag has a dominant effect on the total drag and increases as the slit ratio is increased. However, shear stress has a dominant effect on the lift amplitude and increases as the slit ratio is increased. Nevertheless, for flow at higher Reynolds numbers, drag and amplitude of lift are dominated by the pressure on the external surface. For a cylinder with an inclined slit, there are two modes for the flow: an injection mode and a blowing/suction mode. If the angle of inclination is less than the critical angle, the flow pattern is in the injection mode. If the angle of inclination is greater than the critical angle, the flow pattern is in the blowing/suction mode. For all Reynolds numbers, the drag increases as the angle of inclination for the slit increases. The pressure on the external surface dominates the variation of drag due to the change of angle of inclination for the slit. The drag is greater than that for a cylinder that has no slit when flow is in the blowing/suction mode, except for flow at a low Reynolds number such as Re=100. For flow at any Reynolds number, external lift is equally affected by pressure and shear stress. However, shear stress has a dominant effect on the lift due to the slit. There is a similarity between external surface and slit in which aerodynamic forces due to the pressure or shear stress is varied with the angle of inclination for slit. For a cylinder with a normal slit, the distance between the shed vortex and the cylinder is decreased and the frequency of vortex shedding is increased. This effect is more significant for a cylinder with a wider slit or at a higher Reynolds number. Any inclined slit also increases the frequency of vortex shedding, except for an angle of inclination for the slit of less than 30° for Re=500. This effect is more significant for a cylinder with a slit that is inclined at a greater angle or for a flow at a higher Reynolds number. A normal slit gives a greater increase in the frequency of vortex shedding than an inclined slit as a comparison of Figs. 25 (a) and (b) shows. However, wider slit produces greater drag, especially at flows with a higher Reynolds number. An inclined slit produces less drag for angles of inclination for the slit of less than 45° and for flows at lower Reynolds numbers, such as Re 100. Further, the results provide the aerodynamic data for the design of a heat sink with perforated pin fins by selecting an appropriate width of slit and the inclination angle of the slit to produce the favorable frequency of vortex shedding and lower drag for the purpose of heat transfer enhancement and lower power consumption on the electronics cooling. 26
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Acknowledgement The authors are grateful to Ministry of Science and Technology, R.O.C. for its financial support under grant No. 106-2221-E-224-028.
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References [1] T. Igarashi, “Flow characteristics and a circular cylinder with a slit. 1st report, flow control and flow patterns,” Bulletin of JSME, Vol. 21 (154), pp.656–664, 1978. [2] J. F. Olsen, S. Rajagopalan, “Vortex shedding behind modified circular cylinders,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 86, pp.55-63, 2000. [3] H.-L. Ma, C.-H. Kuo, Control of boundary layer flow and lock-on of wake behind a circular cylinder with a normal slit, European Journal of Mechanics B/Fluids, Vol.59, pp. 99-114, 2016. [4] J. W. Yang, Y. He, F. Bao, “Large eddy stimulation and experimental measurement of near wake structures of a slotted circular cylinder,” ACTA Aerodynamica Sinica, Vol. 32(3), pp. 308-314, 2014. [5] C.-H. Kuo, H.-W. Lin, C.-T. Chai, F. Cheng, “Flow characteristics around circular cylinders with a normal slit,” Defect and Diffusion Forum , Vol. 379, pp 48-57, 2017. [6] H. Baek and G.E. Karniadakis, “Suppressing vortex-induced vibrations via passive means,” Journal of Fluids and Structures, Vol. 25, pp.848–866, 2009. [7] W. J. Sheng and W. Chen, “Features of flow past a circular cylinder with a slit,” Scientia Iranica B, Vol. 23(5), pp. 2097-2112, 2016. [8] L. Ordia, A. Venugopal, A. Agrawal, S. V. Prabhu, “Vortex shedding characteristics of a cylinder with a parallel slit placed in a circular pipe,” Journal of Visualization, Vol. 20, pp. 263–275, 2017. [9] D.-L. Gao, W.-L. Chen, H. Li, H. Hu, “Flow around a circular cylinder with slit,” Experimental Thermal and Fluid Science, Vol.82, pp. 287–301, 2017 [10] C.O. Popiel, “Vortex shedding from a circular cylinder with a slit and concave rear surface,” Applied Scientific Research, Vol. 51, pp.209-215, 1993. [11] J. T. Turner, C. O. Popiel and D. I. Robinson, “Evolution of an improved vortex generator,” Flow Measurement Instrument, Vol. 4(4), pp.249-258 1993. [12] T. Igarashi, “Flow resistance and Strouhal Number of a vortex shedder in a circular pipe,” JSME International Journal, Series B, Vol.42 (4), pp. 586-595, 1999. [13] B.H. Peng, J.J. Miau, F. Bao, L.D. Weng, C.C. Chao and C.C. Hsu, “Performance of vortex shedding from a circular cylinder with a slit normal to the stream,” Flow Measurement and Instrumentation, vol. 25, pp.54–62, 2012. [14] S. A. Orszag, and L. C. Kells, Transition to turbulence in plane poiseuille and plane couette flow, Journal of Fluid Mechanics, Vol. 96, pp. 159-205, 1980. [15] L.-C. Hsu and J.-Z. Ye, “Numerical Study of Flow Patterns in a Tandem Array Cylinder System,” International Journal of Applied Mechanics, Vol. 7(3), pp.1550034 (35 pages), 2015. [16] L.-C. Hsu and G.-J. Gao, “Simulation of vortex shedding behind a flat plate with vorticity based adaptive spectral element method,” Mathematical Problems in Engineering, Vol. 2014, Article ID 959615, 17 pages, 2014. [17] L.-C. Hsu, J.-Z. Ye, C.-H. Hsu, “Simulation of flow past a cylinder with adaptive spectral element method,” Journal of Mechanics, Vol. 33(2), pp. 235-245, 2017. [18] J. Park, K. Kwon and H. Choi, Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160, KSME International Journal, Vol. 12(6), pp. 1200-1205, 1988. [19] T. Igarashi, “Flow characteristics and a circular cylinder with a slit. 2nd report, effect of boundary sunction,” Bulletin of JSME, Vol. 25 (207), pp.1389–1397, 1982.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
PII: DOI: Reference:
S0997-7546(19)30383-8 https://doi.org/10.1016/j.euromechflu.2020.02.009 EJMFLU 103602
To appear in:
European Journal of Mechanics / B Fluids
Received date : 2 July 2019 Revised date : 22 January 2020 Accepted date : 25 February 2020 Please cite this article as: L.-C. Hsu and C.-L. Chen, The drag and lift characteristics of flow around a circular cylinder with a slit, European Journal of Mechanics / B Fluids (2020), doi: https://doi.org/10.1016/j.euromechflu.2020.02.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Masson SAS.
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The Drag and Lift Characteristics of Flow around a Circular Cylinder with a Slit
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Li-Chieh Hsu*, Chien-Lin Chen
Department of Mechanical Engineering, National Yunlin University of Science and Technology 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan
ABSTRACT
Flow past a circular cylinder with a slit results in local perturbation, which affects the flow pattern, vortex shedding and the aerodynamic forces. This study uses spectral element method to simulate flow past a cylinder with a normal slit or an inclined slit at Reynolds numbers of Re=100 to 500. For a cylinder with a normal slit, the drag and amplitude of the oscillating lift both increase as the width of the slit increases. There are two modes for a cylinder with an inclined slit: an injection mode and a blowing/suction mode. The drag also increases as the angle of inclination of the slit increases and is greater than that for a cylinder that has no slit for flow in the blowing/suction mode, except at low Reynolds numbers, such as Re=100. Increasing the width of slit or the angle of inclination of the slit increases the frequency of vortex shedding. However, a wider slit or larger angle of inclination also results in additional drag. There is a similarity between the external surface and the slit in that the aerodynamic forces due to pressure or shear stress vary with the angle of inclination of the slit. Keywords: Cylinder, Normal Slit, Inclined slit, Vortex shedding, Drag and lift
Nomenclature α : 𝛼
angle of inclination of slit : Critical angle of inclination for slit
CQ:
Coefficient of flow rate for a slit
Cd:
Coefficient of drag
Cl:
Coefficient of lift
Cp :
Coefficient of surface pressure for a cylinder
D:
Diameter of the cylinder
𝑄:
Volumetric flux through slit
S:
Width of slit
St:
St
𝑈 :
Flow velocity in the far field
Subscripts curve:
Drag or lift forces on the external surface
curve_pre:
Drag or lift forces due to pressure on the external surface
curve_shear: Drag or lift forces due to shear stress on the external surface
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slit:
Drag or lift forces on the walls of the slit
slit_pre:
Drag or lift forces due to pressure on the walls of the slit
slit_shear:
Drag or lift forces due to shear stress on the walls of the slit
total:
Sum of drag or lift forces on each surface
, Strouhal number
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𝑉 x, t : Velocity at the outlet of slit
1. Introduction
Passive control schemes to control flow are much easier and less costly to implement than active schemes. Using a slit as the communicating channel to the separation flow and wake behind a cylinder is a noteworthy method of flow control. Igarashi [1] experimentally studied the effect of the inclination of a slit on the flow pattern, drag and the frequency of vortex shedding for a cylinder using angles of inclination for the slit from 0° to 90°, between Reynolds numbers 13800 and 52000 and using two slit width ratios: (S/D) 0.08 and 0.185. Two flow patterns exist. For an angle of inclination α 40°, the flow passes through the slit to the wake region and produces an increase in the base pressure, which is called self injection. This results in lower drag and a higher frequency of vortex shedding than for a cylinder that has no slit. For angles of inclination 60° α 90°, the flow exhibits a periodic blowing/suction mode at the exits of the slit. The *
Corresponding author. Tel.: +886-5-5342601 ext. 4151, Fax.: +886-5-5342062 E-mail address: [email protected]
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frequency of vortex shedding for a slotted cylinder is greater than that for a cylinder that has no slit and this frequency increases as the angle of inclination of the slit increases. For angles of inclination α 60°, the drag increases as the angle of inclination increases and is greater than that for a cylinder that has no slit. Olsen and Rajagopalan [2] used circular cylinders with a slit and a concave notch at the rear surface of the cylinder to study vortex shedding for flow with a Reynolds number of Re = 60 to 2300. The experimental results of this study show that the values for the drag and the frequency of vortex shedding for a cylinder with a slit that is normal to the direction of flow are greater than those for a cylinder that has no slit. A cylinder that has a slit and a concave notch at the rear surface produces the greatest drag and a cylinder with a horizontal slit produces the least drag. Ma and Kuo [3] used Ansys/Fluent software to simulate a two-dimensional flow over a circular cylinder with a normal slit at Re=200. The evolution of the boundary layer and the wake were determined using a modified phase-averaged method. It was found that the periodic shedding vortices create a periodic pressure difference between the top and the bottom exits of the slit. The periodic blowing/suction at each exit of the slit creates a perturbation in the boundary layer that significantly delays the separation of the boundary layer along the rear surface of a cylinder with a slit and reduces the length of vortex formation. The upstream induction due to the shedding vortex street leads to primary lock-on of the wake behind a cylinder with a slit. Yang et al. [4] studied the structure of the near wake for a cylinder with a normal slit, S/D=0.15, at Re=1500~7200 using Large Eddy Simulation and experiments. Their results show that because of the periodic blowing/suction motion, the boundary layer passes through the slit and fundamentally changes the structure of wake. The frequency of vortex shedding or the Strouhal number for a slotted cylinder is greater than that for a cylinder that has no slit. Kuo et al [5] studied a cylinder with a normal slit to determine the variation in the separation of the boundary layer along the upper-rear surface and vortex formation in the near-wake at a Reynolds number of 1000. The slit ratio (S/D) was 0.3. The results of these experiments show that the alternate blowing/suction at the exit of the slit creates perturbation in the boundary layer, near the shoulder of a cylinder with a slit. This produces a significant change in the characteristics of the boundary-layer flow along the upper-rear surface of the cylinder, which entails a delay in flow separation and flow reattachment. The history of circulation shows that the shedding behind a cylinder with a slit is much more regular than that behind a baseline cylinder. However, a cylinder with horizontal slit is quite different to a cylinder with a normal slit. Baek and Karniadakis [6] studied a fixed and a free cylinder with a slit that was parallel to the main direction of flow at Re=500 to 1200 using a spectral element method. The results of the experiments show the amplitudes of the oscillating drag and the lift coefficients are significantly decreased when there is a horizontal slit. For a fixed cylinder, the oscillation is regular for a slit ratio(S/D) of less than 0.14. If the slit ratio is greater than 0.14, the oscillation is irregular. For a free cylinder, the 0.16 and the oscillation is suppressed completely if 0.16. oscillation is irregular if Sheng and Chen [7] used a finite volume method to simulate a cylinder with a horizontal slit at low Reynolds numbers from 60 to 250. The results show that at Re=150, the frequency of vortex shedding and the power spectral density in the FFT diagram are reduced when the slit width ratio is increased from S/D=0.1 to 0.3 in increments of 0.05. For Re=250 and slit width ratios S/D = 0.20 and 0.25, the Strouhal number is zero because there is no fluctuation in the lift coefficient, which means that vortex shedding is suppressed. The drag coefficient decreases when the slit width ratio increases because some of the flow goes through the slit from the front of the cylinder to the lower pressure or wake region. Ordia et al. [8] used two-color dyes and a cylinder with a horizontal slit to study the mechanism for the formation of the vortex behind bluff bodies at Re=200-2300 and to determine the potential of a cylinder with a parallel slit as an improved vortex generator. The results of these experiments show that at Re < 1000, the Strouhal number increases when the slit width is increased, except for S/D=0.4. The variation in the Strouhal number with the slit width is different to the relationship that was determined by Sheng and Chen [7]. Notably, these experiments [8] were conducted in a circular pipe, so the wall confinement effect is relevant. For Re > 1000, the slit width does not have a significant effect on the frequency of vortex shedding. However, for S/D=0.4, the flow behavior is similar to that for dual bluff bodies with a much lower Strouhal number than that for a cylinder that has no slit. For S/D < 0.4, the flow behavior resembles that for a single body. The vortex behind a cylinder for which S/D < 0.4 is shorter and narrower than that behind a cylinder that has no slit. For S/D=0.4, the vortex is longer and wider than that behind a cylinder that has no slit. Gao et al. [9] used particle image velocimetry (PIV) to study a cylinder with a horizontal slit in a flow at Re 2.67 10 . The results of these experiments show that a slit reduces the drag and eliminates the fluctuation in the amplitude of the lift. The diagram for the mean pressure around the surface shows that the difference in pressure between the windward and leeward stagnation points is reduced as the slit ratio (S/D) increases. For a slit ratio S/D of 0.05, the RMS value for the lift coefficients is a minimum and the lift is suppressed by up to 81.78%. The frequency spectrum for the lift time history diagram shows that the frequency increases as the slit width increases. When the S/D ratio increases, the region in which the vortex is formed is pushed further downstream and the jet vortices from the slit become stronger. For S/D < 0.1 the interaction and competition between the slit jets and the separation flows results in two recirculation regions. For S/D > 0.1 the flow behaves like a side-by-side system and the recirculation bubble becomes longer when the 2
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2. Numerical Method
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slit ratio is increased. The experimentally obtained values for the Strouhal number for a cylinder with a horizontal slit [8, 9] do not agree with the numerical results [7]. Although there is some disagreement between the numerical and experimental results for a cylinder with a horizontal slit, a normal slit increases the frequency and the strength of vortex shedding, but a sufficiently large horizontal slit suppresses vortex shedding and the drag and the amplitude of the oscillations in the lift are reduced [6]. A normal slit reduces the length of vortex formation [3], but a horizontal slit prolongs it. The Strouhal number for vortex shedding is increased for a normal slit [4] and decreased for a sufficiently large horizontal slit [9]. Therefore, a cylinder with normal slit can enhance the flow rate measurement because it has a stronger and more stable and regular signal [10-12]. It remains to be determined whether these measured signals are representative of the signal that is measured for a cylinder that has no slit and a real flow rate. To study this problem, Peng et al. [13] experimentally studied the effect of the slit width (S/D= 0 to 0.3) on the quality of the measurement for a cylinder with a slit. The results show that for S/D=0.1 to 0.15, the measurement has the best signal quality, with the minimum error for seven different slit widths. Although the effect of the slit ratio on a horizontal cylinder with a slit has been studied, there have been no systematic studies that involve a cylinder with a normal slit. This study determines the effect of a normal slit ratio and the angle of inclination for the slit on the flow pattern, the drag, the lift, the vortex shedding frequency and the mechanism for the formation of the flow patterns.
The incompressible Navier-Stokes equations are solved as:
u 1 u u p 2u t u 0
(1)
(2) A time splitting scheme [14] is used to treat the nonlinear convective terms explicitly and the pressure and velocity diffusion terms are solved implicitly as follows:
uˆ u n t
2
r 0
r
( u u ) nr
(3)
(4)
ˆ u n1 uˆ 2u n1 t
(5)
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The scheme uses intermediate time step values for velocity, 𝑢⃑ and 𝑢⃑, where and ˆ are between the nth and n+1st time steps. The nonlinear terms are advanced by a third-order Adams-Bashforth scheme with coefficients, 𝛽 . The pressure term is treated separately as an elliptical problem by taking the divergence on both sides of Eq.(4). Due to the divergence free condition, the surface term is zero. Hence,
2 p n 1
t
uˆ
(6)
The pressure and the velocity calculations are types of Helmholtz problems, expressed in the following general form:
( 2 2 ) g
on
(7)
The variational form of Eq.(7) is:
dx 2 dx gdx
3
(8)
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which is discretized using the spectral element method. The solution domain is divided into macro elements. All variables are substituted by discrete approximations as: N
M
k hk pq hp (r )hq (s)
(9)
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p0 q0
where the h functions are the Lagrangian interpolants, which use the orthogonal set of Legendre polynomials for a high degree N (or M ), and r, s are the local coordinates of each element, k. Performing a Gauss-Lobatto quadrature and summing the contributions of adjacent elements gives the global matrix equation as:
( A 2 B ) Bg
(10)
where A is the discrete Laplacian operator and B is the mass matrix. The matrix equation is solved using a preconditioned conjugate gradient. Details can be found in previous studies [15-17].
2.1 Grid independence
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A rectangular computational domain is shown in Fig. 1. The diameter of the cylinder (D) is 1 unit. The domain extends to a distance of 20D in upstream from the center of the cylinder and a distance of 30D downstream. The upper and lower boundaries are both 20D from the center of the cylinder. Uniform flow Dirichlet boundary conditions (u=U0, v=0) are used for the inflow, the upper and the lower boundaries. The outflow boundary condition is used for the boundary that is farthest downstream. To determine grid independence, the number of elements (K) and the order of the Legendre polynomial (N) are serially increased to determine the convergence of the surface pressure coefficient for a regular cylinder in a flow of Re=100. Figure 2 (b) shows that the pressure coefficient converges when N 10. For this number of elements and order of the Legendre polynomial, the surface pressure coefficient for a higher Reynolds number Re=140 is in agreement with the result of Park [18], so the following simulation of a cylinder with a slit for flow at Re=100 to 200 uses a similar number of elements (K) and N=11 and the same computational domain that is described previously. However, for flow at higher Reynolds number, Re=500, more degree of freedoms are considered to simulate at this flow regime. The convergence test is implemented from order of N=11 to 15. The solution of N=15 is used as the reference solution. Figures 2 (c) and (d) indicates for reaching convergence, the order of the Legendre polynomial is not lower than N=13 for Re=500. Hence the following simulation of a cylinder with a slit for flow at Re=500 uses a similar number of elements (K) and N=15.
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1
0
-1
(b)
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0
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1
Fig.1 The computational domain: (a) number of elements, k=334 and (b) a magnified view of the elements and grids
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(b)
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Fig.2 The coefficient of surface pressure for a cylinder with a different order of the Legendre polynomial (N) and number of elements, k=334: (a) at Re=100, (b) a comparison with the surface pressure coefficient for the cylinder that is used in this study with K=334 and N=11 and the results for other studies, for Re=140 (c) at Re=500 (d) the L2 error with different degrees of freedom (DoF)
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2.2 Validation Using the same computational domain and number of elements, the case of flow at Re=200 past a cylinder with a normal slit is used for validation. This scenario is shown in Fig. 3(a). Fig. 3(b) shows a drag-lift polar diagram, which shows that the result for this numerical method agrees well with that of Ma and Kuo [3]. The amplitudes and mean values for the drag and lift are listed in Table 1. The drag and lift are calculated using Eqs. (11) and (12). The range of integration includes the surface of the cylinder and the walls of the slit. The coefficients of drag and lift are defined by Eqs. (13) and (14). The second case that is used for validation is a flow of Re=500 past a cylinder with a horizontal slit. Figure 4 shows that the values for the periodic drag and lift coefficients for this numerical simulation are in good agreement with those of Baek and Kaniadarkis [6], but the period of time for this simulation is different from that for the previous study. There is a 0.77% difference in the amplitude of the oscillations in the lift and a 1.7% difference in the drag coefficient between this simulation and that study.
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𝐩 ∙ 𝒏𝒙
𝐹𝐿
μ ∇𝐮
∇𝐮𝑇 ∙ 𝒏𝒙 ds
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𝐹𝐷
𝐩 ∙ 𝒏𝒚
μ ∇𝐮
∇𝐮𝑇 ∙ 𝒏𝒚 ds
12
𝐶𝑑
𝐹𝐷 1 𝜌𝑈 𝐷 2
13
𝐶𝑙
𝐹𝐿 1 𝜌𝑈 𝐷 2
14
𝑉𝑠 x, t ∙ dx
𝑄 𝑡
𝑄 𝑈 ∙𝑆
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𝐶
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11
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Fig. 3 Flow past a cylinder with a normal slit S/D=0.15 at Re=200: (a) vorticity contour of present result and (b) the comparison of drag-lift polar diagram of presnt result and that of Ma and Kuo [3].
Table 1 Comparison of the drag and lift coefficients for a cylinder with a slit for which S/D=0.15, 𝛼=90˚ at Re=200 Ma and Kuo [3]
Present
Cl
±0.775
±0.78
Cd
1.419±0.0555
1.419±0.0553
6
15
16
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Fig. 4 Comparison of the drag and lift coefficient history for flow at Re=500 past a cylinder with a horizontal slit for present results and those of Baek and Kaniadarkis [6]: the upper curve represents Cd, the lower curve represents Cl and time
3. Results and discussion
t𝑈 /𝐷
To determine the effects of the width and inclination of the slit on the flow patterns and aerodynamic forces, the following study initially focuses on the flow of Re=100 and extends the theory to the flows with a higher Reynolds number. 3.1 The effect of the slit ratio
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The ratio of the width of slit to the diameter of the cylinder is the slit ratio (S/D). Simulations of flow past a cylinder with a normal slit using different slit ratios from 0.08 to 0.2 were conducted to determine the effect of the width of the slit on flow patterns and aerodynamic forces. When vortices are shed from behind the cylinder, there is a pressure difference between the two exits of the slit, so the flow passes through the slit from one end to the other. However, the external flow is similar to that for a cylinder that has no slit, as shown in Fig. 5(a). The internal flow inside the slit exhibits a periodic blowing/suction flow pattern, with small vortices near the exits, as shown in Figs. 5(b) and 5(c). A difference in the slit ratio results in a difference in the size of the local vortices near the inlet and outlet of the slit, as shown in Fig. 6. The non-dimensional flow rate coefficient, CQ , is measured for flow inside the slit. A positive value means that flow exits the slit and a negative value means that flow is sucked into the slit. The flow rate through the slit is increased when the slit ratio is increased, as shown in Fig. 7. The frequency of this periodic blowing/suction flow inside the silt is the same as that for vortex shedding behind the cylinder, as shown in Fig. 8. The frequency of vortex shedding increases when the slit ratio increases and is greater than that for a cylinder that has no slit, as shown in Fig. 7(b). However, there is a very minor phase lag between the slit flow rate and the lift coefficient, as shown in Fig. 9(a). The magnitude of this phase lag increases when the slit ratio is increased, as shown in Fig. 9(b). To determine the mechanism that causes a variation in the aerodynamic forces due to the slit, the following analyses of drag and lift respectively use the contributions from the external surface and the walls of slit. The drag due to the external surface is called the external drag. The drag due to the walls of the slit is called the slit drag. The external lift and slit lift are similarly defined. Each aerodynamic force that exerts on a surface is divided into pressure and a shear stress. Drag A larger slit ratio results in greater drag, but the drag coefficients for all slotted cylinders are smaller than that for a cylinder that has no slit, as shown in Fig. 10(a). A wider slit produces a decrease in the external drag but an increase in the slit drag, as shown in Fig. 11. Figure 12(a) shows that the shear drag is the principal cause of variation in the external drag, in terms of the magnitude of the change. The shear drag of the external flow decreases linearly because the effective surface becomes smaller when the slit ratio is increased. In terms of the slit flow, the pressure drag increases linearly when the slit ratio is increased, as shown in Fig. 12(b). A wider slit results in an increase in the flow rate through the slit and produces a greater imbalance in the pressure that is exerted on the two walls of the slit channel and a greater local vortex near the inlet or the outlet of the slit, as shown in Figs. 6, 7 and Fig. 10(b). However, shear stress inside slit has no effect on drag because the direction of the shear stress is perpendicular to the direction of the drag or the bulk flow. 7
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Summing every contribution to the drag, variation in the total drag is principally due to the slit pressure drag. This conclusion is different from those of previous studies [1, 19] which determine that for a cylinder with a normal slit, the drag is reduced when the slit ratio is increased. However, the results of this study show that a larger slit ratio results in greater drag when 0.1 and less drag when 0.1. The main reason for this difference is that this study combines the drag on the external surface of the cylinder and on the walls of the slit that results from pressure and shear stress into the total drag, but previous studies [1,19] only consider the drag that results from the pressure that exerts on the external surface of the cylinder. When the slit ratio is increased, the effect of the slit pressure drag is augmented and results in greater drag, as shown in Figs. 10(a), 11(b) and Fig. 12(b).
(a)
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Fig. 5 A cylinder with a normal slit for which S/D=0.1 and Re=100: (a) vorticity contours, streamlines and pressure distribution at (b) 3/8 T and (c) 7/8 T
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0.6
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0
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Fig. 6 Streamlines and pressure distribution at Re=100 for different slit ratios: (a) S/D=0.08, (b) S/D=0.1, (c) S/D=0.15 and (d) S/D=0.2
(a)
(b)
Fig. 7 Slit flow rate and vortex shedding: (a) the coefficient of flow rate inside the slit and (b) the frequency of vortex shedding for a cylinder with t𝑈 /𝐷
.
(a)
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different slit ratios, time
(b)
Fig. 8 Lift coefficients of slit and total at S/D=0.1 and Re=100: (a) lift due to the walls of the slit and total lift
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(b) a magnified view; time
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(b)
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Fig. 9 Phase lag between the oscillation in the slit flow rate coefficient and the lift coefficient : (a) the normalized slit flow rate coefficient and the lift coefficient and (b) the phase lag for different slit ratios
Lift
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The mean lift coefficient is zero for each case because the geometry is symmetrical, as shown in Figs. 13(a) and (b). When the slit ratio is increased, the amplitude of the external lift is reduced, but the amplitude of the lift due to the slit is increased. However, the amplitude of the oscillations in the total lift coefficient increases when the slit ratio is increased, as shown in Fig. 14(a). Because the effective surface in the y-direction is shortened, the amplitude of the oscillations in the lift force due to the external pressure decreases when the slit width is increased, as shown in Fig. 14(b). The two discontinuous sections of the surface of the cylinder that are divided by the slit limit the growth of a boundary layer and generate a greater shear stress. The effect of this shear stress on the amplitude of the oscillations in the lift is increased when the width of the slit is increased, as shown in Fig. 14(b). In terms of flow inside the slit, the amplitude of the lift due to shear stress increases when the slit width is increased because there is a greater flow rate through a wider slit, as shown in Figs. 7, 13(b) and 14(c). However, the pressure inside the slit has no effect on lift because of the direction in which it acts. Therefore, variations in the lift amplitude for different sizes of slit are mainly due to the shear stress that acts on the slit. A larger slit ratio results in a thinner boundary layer because there is a discontinuous external surface and there is greater flow through the slit so there is greater shear stress on the external surface and on the walls of the slit. Therefore, the amplitude of the oscillations in the lift coefficient increases when the slit ratio is increased, as shown in Fig. 14(a), so changes in the amplitude of the oscillations in the lift coefficient are mainly due to the effect of shear stress. This conclusion involving the lift coefficient is also different from the conclusions of previous studies [1, 19] which only consider the lift due to the pressure that exerts on the external surface.
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Fig. 10 Drag of a cylinder with different slit ratios: (a) time mean total drag coefficient and (b) pressure coefficient along the surface of the cylinder
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In summary, for a cylinder with a normal slit, shear drag dominates external drag and pressure drag dominates slit drag. Total drag is dominated by pressure drag of slit. Lift amplitude is dominated by shear stress. Hence, to determine more precise characteristics of aerodynamic forces, the contribution from slit and effect of shear stress are not negligible.
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Time mean drag coefficient for different slit ratios: (a) external drag and (b) slit drag
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Fig. 11
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Fig. 12 Mean drag coefficients at Re=100: (a) external drag due to pressure and shear stress and (b) slit drag due to pressure and shear stress
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Fig. 13 Lift coefficients for different slit ratios: (a) lift coefficient due to the external surface and (b) lift coefficient due to the slit
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Fig. 14 Lift amplitude: (a) total lift (b) lift amplitude due to pressure and shear stress on the external surface and (c) lift amplitude due to pressure and shear
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stress on the walls of the slit
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3.2 Inclined slit The angle of inclination for the slit is measured from α 0° at the leeward base point to α 180° at the windward stagnation point. The flow past a cylinder with a slit that is inclined at angles from 0° to 90° is simulated to determine the effect of the inclination on the variation in the flow pattern and the aerodynamic forces. The overall flow pattern is shown in Fig. 15. Generally, if the angle of inclination of the slit, α 45°, some flow goes through the slit in one direction from the upstream inlet to the wake region, which is called the injection mode and is shown in Figs. 16 (a) to (d). If α 60°, flow goes periodically back and forth through the slit. The flow alternately exits the slit and is sucked back into the slit. This flow pattern is the blowing/suction mode and is shown in Figs. 16 (e) and (f) and Fig. 5. In the injection mode, if the angle of inclination, α 15°, the flow exits the slit into the wake and sprays up and down periodically to form a part of a vortex. For an angle of inclination, α 30°, the flow exits the slit in an upward direction. After a half period, there is a saddle point at the exit of the slit. For α 45°, the flow exits the slit steadily in an upward direction and is not affected by the shedding vortex. It is arguable that there is a transition between these modes for 45° α 60°.
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Fig. 15 Vorticity at Re=100, S/D=0.1 and an angle of inclination for the slit, 𝛼=45°
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Fig. 16 Flow patterns for a cylinder with an inclined slit: (a) 𝛼
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There are two flow patterns, but there is a critical angle of inclination at which the flow transforms from the 56.33° injection mode to the suction/blowing mode. The results of several trials show that the critical angle is 𝛼 for flow at Re=100. If the angle of inclination for the slit is less than the critical angle, the flow is in the injection mode. If the inclination of the slit is greater than critical angle, the flow is in the suction/blowing mode. If the angle of inclination of the slit α 𝛼 , the flow rate in all cases is positive so there is injection flow only, as shown in Fig. 18(a). However, for α 𝛼 , the flow rate oscillates periodically between a positive and a negative value, so flow is in the blowing/suction mode, as shown in Fig. 18(b). As shown in Fig. 18(b), for α 60°, the blowing motion is stronger than the suction motion because the positive value is greater than the negative value. However, for α 75°, the suction motion is stronger. For α 90°, the blowing rate is equal to the suction rate. For any angle of inclination for the slit, the slit flow rate is periodic. Notably, if α 𝛼 , there is an alternating injection flow and a zero flow rate inside the slit, as shown in Figs. 17 and 18(b). Figures 18(a) and (b) show that the time mean flow rate of slit is reduced as the angle of inclination for the slit is increased. However, the amplitude of the oscillations in the slit flow rate increases as the inclination of the slit is increased. The non-dimensional frequency of the oscillations in the slit flow at zero degrees of slit inclination is double that for other angles of inclination, as shown in Figs. 18(a) and (c). The reason is the frequency of slit flow at α 0° is double that of vortex shedding. That is, during a cycle of vortex shedding, the slit flow has been injected twice. For other angles of inclination for slit, the frequency of slit flow is very close to that of vortex shedding. When α 10°, the frequency of the slit flow increases as the angle of inclination for the slit is increased, as shown in Fig.18 (d). When α 14
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𝛼 , the frequency of the slit flow is almost as high as that for α 90°. The frequency of the slit flow is lower than that of vortex shedding when the slit flow is in the injection mode. However, for slit flow in the blowing/suction mode, the frequency of the slit flow is higher than that of vortex shedding, except for α 60°, as shown in Fig. 18(d). Only at α 90° does the frequency of slit flow coincide with that of vortex shedding for a cylinder. (a)
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Fig. 18 Flow rate of the slit and the Strouhal number for different angles of inclination for the slit, S/D=0.1: (a) flow rate in the injection mode, (b) flow rate in the blowing/suction mode, (c) Strouhal numbers based on the frequency of slit flow rate and the frequency of vortex shedding for a cylinder and (d) a magnified view of (c); time t𝑈 /𝐷 15
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Drag
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An increase in the angle of inclination for the slit results in greater external drag but a decrease in the drag due to the slit, as shown in Figs. 19 (a) and (b). The total drag is the sum of the external drag and the drag due to the slit. The drag due to the slit has a significant effect on the total drag at small angles of inclination for the slit, as shown in Fig. 19(c), but the slit has less effect on the total drag is at large angles of inclination for the slit, as shown in Fig. 19(d). As shown in Fig. 20(a), the pressure drag due to the external surface and the wall of the slit both increase with high similarity as the angle of inclination for the slit is increased. This is because the effective area in the x-direction on which pressure is exerted is increased when the angle of inclination for the slit is increased. For larger angles of inclination, the corners at the exit of slit generate local vortices, which also increase the pressure drag due to the slit. The variation in the pressure drag due to the external surface was demonstrated by Igarashi [1]. The shear drag due to the external surface and the wall of the slit both decrease in the same manner as the angle of inclination for the slit is increased, as shown in Fig. 20(b). The effective area in the x-direction on which shear stress is exerted decreases so the external shear drag is decreased when the angle of inclination for the slit is increased. In terms of the slit, the flow rate and the horizontal component of the shear stress are smaller at higher angles of inclination for the slit so there is a lower shear drag due to the slit. In terms of the magnitude of the force, the pressure drag has a dominant effect on the external drag and increases when the angle of inclination for the slit is increased, as shown in Fig. 20(a). The shear stress has a dominant effect on the drag inside the slit and decreases when the angle of inclination for the slit is increased, as shown in Fig. 20(b). Therefore, the total drag does not vary regularly with the angle of inclination for the slit, as shown in Fig. 21(a).
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Fig. 19 Drag coefficient for different angles of inclination for the slit: (a) external drag, (b) slit drag, (c) drag coefficient history for an angle of inclination for the slit of α 0° and, (d) ) drag coefficient history for an angle of inclination for the slit of α 90°; time t𝑈 ⁄𝐷
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Figure 20 shows that the magnitude of variation in drag due to shear stress on the walls of slit from α 0° to 90° is same with the drag due to pressure on external surface. That means the variation of shear stress inside slit is significant enough to change the regular variation of drag on the external surface as flow at low Reynolds number. For Re=100, the drag on a cylinder with any inclined slit is less than that on a cylinder that has no slit. For α 𝛼 , the drag has a maximum value that is close to that for a solid cylinder, because the flow rate is periodically zero. The drag due to the shear stress on the external surface exists a lowest value as 𝛼 75° shown in Fig. 20(b). It is because a strong suction flow at upper inlet of slit delays the separation point to result in the increase of shear stress on the rear surface of cylinder as the comparison of Figs. 16(c), (e) and (f). In fact, the suction flow results in shear stress in the opposite direction of drag on the rear upper surface. Besides, Fig.18(b) shows the suction flow is stronger than the blowing flow as 𝛼 75°. The blowing flow at lower exit of slit does not affect the separation much because the location of blowing is in the front of separation point. The periodic oscillation of drag due to the shear stress on the external surface at different angles of inclination of slit are shown in Fig. 21(b) which shows the case of 𝛼 75° has the smallest shear drag on external surface. The lower drags due to the shear stress on external surface and slit both reduce the total drag, especially, for 𝛼 75° as shown in Fig.21 (a).
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Fig. 20 Variation in the pressure drag and shear drag with the angle of inclination for the slit : (a) pressure drag due to the external surface and the slit
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Fig. 21 Total drag at different angles of inclination for the slit : the dashed line shows the total drag for a solid cylinder and the solid line shows the drag for a slotted cylinder
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Lift
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For a cylinder with an inclined slit, the total lift is the sum of the lift on the external surface and the lift on the wall of the slit, as shown in Fig. 22 (a). The Strouhal number for a cylinder with an inclined silt is greater than that for a cylinder that has no slit and increases as the angle of inclination for the slit is increased, as shown in Fig.22 (b). The lift coefficient oscillates periodically but the following analyses are condensed by using a time mean lift coefficient. The external lift that is caused by pressure depends on the projection of the external surface in the y-direction and the external lift that is caused by shear stress depends on the projection of the external surface in the x-direction. When the angle of inclination of the slit is increased, the projection in the y-direction is decreased and the projection in the x-direction is increased. In the injection mode, at α ≤ 35°, when the slit angle is increased, the lift due to pressure on the external surface and on the wall of the slit decreases from zero to a negative value because the effective surface in the y-direction is decreased, as shown in Fig. 23 (a). In the periodic blowing/suction mode, at 𝛼 35°, when the angle of inclination for the slit is increased, the lift due to pressure on the external surface and the wall of the slit recovers from a minimum back to zero because at larger angles of inclination for the slit, the flow pattern is more symmetrical with respect to the main direction of flow. The change of effective area due to the slit is related to the pressure loss on the surface. The region of surface of cylinder can be divided by the separation point into higher pressure region and lower pressure region as shown in Fig. 2(a). The location on the surface in front of separation point is higher pressure region. The location on the surface behind the separation point is lower pressure region. For low angle of inclination of slit, for example, 𝛼 30°, there is an loss of effective area for lift force in higher pressure region on the lower surface of the cylinder due to the inlet of slit. However, the other loss of effective area for pressure exists in lower pressure region because the outlet of slit on the upper surface of the cylinder locates behind the separation point. The lower surface experiences more loss of pressure force than that of upper surface. This unbalanced pressure results in downward force or negative lift as shown in Fig. 23(a). For high angle of inclination of slit, for example, 𝛼 60°, the loss of pressure in higher pressure region for lower surface of the cylinder is diminished because the surface pressure is smaller at that location. The other exit of slit is still in lower pressure region of upper surface. Therefore, smaller unbalanced pressure between the lower surface and upper surface results in lesser negative lift as shown in Fig. 23(a). In terms of lift due to the shear stress, Fig. 23 (b) shows that for α ≤ 35°, as the angle of inclination for the slit is increased, the external shear lift increases from zero to a positive value because the effective surface is increased. The shear lift due to the slit is also increased because the upward component of the shear stress becomes larger. For α 35°, as the angle of inclination for the slit is increased, the shear lift on the external surface decreases from a maximum value at α 35° to zero at α 90° because the flow pattern is more symmetrical with respect to the main direction of flow. The flow rate inside slit actually affects the shear stress of slit. Figure 23(b) shows the trend of variation of lift forces due to shear stress on the walls of the slit at different slit angles associates with the flow rate inside slit in y-direction.
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Fig. 22 Lift and Strouhal number for a cylinder with an inclined slit: (a) the history of the lift coefficient for α Strouhal number for vortex shedding for a cylinder with a slit, S/D=0.1, at different angles of inclination
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However, the shear stress has an considerable effect on external lift as the comparison of Figs. 23 (a) and (b). The unbalanced pressure between lower surface and upper surface and shear stress on external surface results in the maximum exists around 𝛼 75° and the minimum around 𝛼 35° for external lift as shown in Fig. 23(c). Furthermore, as 𝛼 75°, the stronger suction flow at upper inlet of slit delays the separation point and results in the increase of shear stress on the rear surface of cylinder. The greater shear stress contributes some upward components of force or lift and enhances the total lift at 𝛼 75°. However, the shear stress has a dominant effect on lift due to slit, as shown in Fig. 23 (d). Figures 23(a) and (b) show that regardless of whether lift is due to pressure or to shear stress, the angle of inclination for the slit has a similar effect on the variation in the lift on the external surface and on the slit. The same is true of the drag on the external surface and the slit due to pressure or to shear stress, as shown in Fig. 20.
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Fig.23 Lift coefficient at different angles of inclination for the slit: (a) lift due to pressure on the external surface and on the walls of the slit, (b) ) lift due to shear stress on the external surface and the walls of the slit, (c) lift due to the external surface only and (d) lift due to the walls of the slit only
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3.3 Higher Reynolds Number Regimes
Flow patterns
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Flow at Re=200 and 500 is used to determine the effect of higher Reynolds numbers on the variation in the flow pattern and the aerodynamic forces for a slotted cylinder. A comparison of Figs 24(a) to (c) shows that for a cylinder with a normal slit, the distance between the third vortex and the center of the cylinder decreases from more than 4 to less than 4. Moreover, the distance of third vortex to the center of cylinder is shorten to 3 as S/D=0.2. The effect of the slit ratio is more obvious for a Reynolds number of 500. Figs. 24(g) to (i), show that the distance of the first three vortices to the center of a cylinder with a normal slit is much less than that for the cylinder that has no slit. For x 8 at Re=500, more vortices are shed by a cylinder with a normal slit than by a cylinder that has no slit. For Re=200 or 500, the larger slit ratio results in shed vortices closer to the cylinder. For Re=200, the distance between the first three vortices and the center of cylinder is not changed obviously as shown in Figs. 24 (a) and (d) to (f) shows. However, Fig. 25 (b) also shows that at Re=200, the frequency of vortex shedding for a cylinder with a slit at any angle of inclination is higher than that for a cylinder which has no slit. The same is true for a flow at Re=100. For a cylinder with an inclined slit at Re=500, for which flow is in the injection mode at α 0°, the presence of the slit increases the distance between the third vortex and the center of cylinder from 5 to 6 and the frequency of vortex shedding is decreased, as is seen by comparing Figs. 24(g) and (j). However, at a flow of Re=500 in the blowing/suction mode at α 60°, the presence of the slit decreases the distance between the third vortex and the center of cylinder and vortex shedding has a higher frequency, as a comparison of Figs. 24(g) and (l) shows. The variation in the frequency of vortex shedding for different angles of inclination for the slit are shown in Fig. 25(b). The reason that for Re=200, the flow pattern in the wake of a cylinder with an inclined slit is not changed obviously is very low Reynolds number of the flow inside slit. The Reynolds numbers of flow inside slit are Re=1 to 3 for bulk flow at Re=200 and Re=10 to 18 for bulk flow at Re=500. For all Reynolds numbers, the frequency of vortex shedding for a slotted cylinder increases as the slit ratio or the angle of inclination for the slit increases, as shown in Fig. 25. The frequency of vortex shedding for a cylinder with a normal slit for any slit ratio is greater than the frequency for a cylinder that has no slit. For inclined slit, at Re=100 and 200, the Strouhal number for a slotted cylinder with a slit at any angle of inclination is greater than that for a non-slotted cylinder, as shown in Fig. 25(b). For Re=500, only when α 30° the Strouhal number for a slotted cylinder is greater than that for a non-slotted cylinder. An increase in the angle of inclination for the slit produces a more significant increase in the Strouhal number at higher Reynolds numbers and at greater angles of inclination for the slit.
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Fig.24 Variation in the flow pattern for different slit ratios and angles of inclination for the slit at higher Reynolds numbers: Re=200: (a) non-slotted cylinder, (b) S/D=0.1, α S/D=0.1, α
90°, (c) S/D=0.2, α 90°, (i) S/D=0.2, α
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are -20 to 20 for Re=200 and -30 to 30 for Re=500.
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The effect of the slit ratio on the drag and lift
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For a cylinder with a normal slit at flows with a higher Reynolds number, there is a significant increase in drag as the slit ratio increases. This increase is much greater than that for a cylinder that has no slit, as shown in Fig. 26(a). At Re=200 or 500, as the slit ratio increases, the pressure drag due to the external surface increases significantly and mainly causes the great increase in the total drag as the analysis in Fig. 26(c) and (e). This phenomenon is different from that in the lower Reynolds number. At Re=100, the increase in total drag as the slit ratio increases is mainly caused by the increase in pressure drag due to the slit as shown in Fig.12. The amplitude of the oscillation in the lift increases as the slit ratio increases for Re=100 and 200, as shown in Fig. 26 (b). However, for Re=500, the amplitude of lift has a maximum value for S/D=0.04 and decreases with slit ratio as 0.04. This phenomenon also can be found in the Figs. 24(g) to (i) which show that the oscillation of vortices are more constrained as the slit ratio increases. For higher Reynolds number, the external pressure dominates the variation of amplitude of lift as the analysis shown in Fig. 26(d) and (f). In summary, at higher Reynolds number, the external pressure is the critical factor for the variation of drag and lift of a cylinder with different slit ratios.
Fig. 25 Strouhal number for different slit ratios and angles of inclination for the slit at Re=100 to 500: (a) variation of St with the slit ratio and (b)
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variation in St. with the angle of inclination for the slit
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Fig. 26 Drag and lift coefficients for a cylinder with a normal slit for different slit ratios at Re=100 to 500: (a) total drag (b) amplitude of total lift (c) decomposition of drag at Re=200 (d) decomposition of lift amplitude at Re=200 (e) decomposition of drag at Re=500 (f) decomposition of lift amplitude at Re=500
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The effect of the inclination of the slit on the drag and lift
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The drag varies irregularly with the angle of inclination for the slit at Re=100, but the drag increases as the angle of inclination of the slit is increased at Re=200 and Re=500, as shown in Figs. 27 (a) and (b). For all Reynolds numbers, the external drag increases as the angle of inclination for the slit is increased, but the drag due to the slit decreases as the angle of inclination for the slit is increased, as shown in Figs. 27(c) and (d). However, the external drag has a dominant effect on the total drag at higher Reynolds numbers so for α 45° and value for Re=200 and 500, the total drag on a cylinder with an inclined slit is greater than the drag on a cylinder that has no slit. For α 45°, the total drag on a cylinder with an inclined slit is less than the drag on a cylinder that has no slit, as shown in Figs 27 (a) and (b). This is because that at smaller angle of inclination for the slit, there is an increase in flow through the slit from the windward side of the cylinder to the wake region, so there is a stronger injection flow and a lesser difference in the pressure between the windward and the leeward surfaces. This produces less external drag and total drag but more slit drag at smaller angles of inclination for the slit. Similarly to the effect of lift at Re=100, the locations of two exits of slit in the higher pressure region and lower pressure region, respectively, result in the unbalanced pressure which associates with external lift as shown in Fig. 28(a). The slit flow rate also affects the lift due to shear stress on the walls of the slit as shown in Fig. 28(b). The pressure dominates the external lift, whereas shear stress dominates the lift due to slit. As 𝛼 30°, the change in the effective surface area means that external lift decreases but lift due to the slit increases as the angle of inclination for the slit is increased, as shown in Figs 28(a) and (b).
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Fig. 27 Variation in the drag with the angle of inclination for the slit: (a) total drag at Re=100 to 200, (b) total drag at Re=100 to 500, (c) external drag and (d) slit drag
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However, at 𝛼 30°, both types of lift recovers to zero as the angle of inclination for the slit is increased because the flow is more symmetrical with respect to the direction of the main flow. The variation in the lift with the angle of inclination for the slit at Re=200 and Re= 500 is similar to that at Re=100. For higher Reynolds numbers, the variation in the total lift with the angle of inclination for the slit is also irregular, as it is at Re=100. The maximum of lift exists at α 75° and the minimum at 𝛼 30°~40° for Re=500. Because the external pressure force dominates the total lift at higher Reynolds numbers, Fig. 29 may show the reason that lift coefficient is higher at α 75° and lower at α 30° for Re=500. The lower surface of cylinder is exerted by larger pressure than that on the upper surface as inclination angle of slit is at α 75°. The unbalanced pressure between lower surface and upper surface results in larger lift. However, for α 30° , the pressure on the upper surface of cylinder is larger than that on the lower surface. This results in lower or even negative lift. For α 0° or α 90°, the pressure distributions for upper and lower surfaces are much more equal to each other. This results in lift coefficient is closer to zero. Figure 29 also shows the external drag is greater as the inclination of slit is larger. However, among that, the largest drag exists at α 75° which is even larger than that of α 90°. This trend of drag with different angle of inclination for slit is similar to the result of Igarashi [1] which shows the maximum drag exist at α 65° and the drag is lower than that of non-slotted cylinder as α 45° for flow of Re=45000.
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Fig. 28 Comparison of lift for different Reynolds regimes for different angles of inclination for the slit: (a) total lift, (b) external lift, (c) slit lift, (d) pressure
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Fig.29 Coefficient of surface pressure for a cylinder with different angles of inclination of slit for flow at Re=500
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Critical Angle
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Figures. 17 and 18(b) show the characteristic of slit flow at critical angle is an alternating injection flow which associates with a periodic zero flow rate such as 𝛼 56.33° for Re=100. Similarly, there are critical angles for higher Reynolds number flow. Bases on this characteristic, the critical angles are determined as 𝛼 49.26° for Re=200 and 𝛼 48.21° for Re=500. Figure 30 (a) to (c) show the slit flow is in injection mode which enters the lower inlet of slit and ejects out from the upper exit. The slit flow goes upward inside slit and causes the positive vorticity on the right side of the wall and negative vorticity in the left side of the wall. Figures 30 (d) to (g) shows no any flow goes through the slit. Figures 30 (i) and(h) starts another cycle of injection. For flow modes at higher Reynolds number regimes are divided by critical angle of slit. If the angle of inclination is less than the critical angle, the flow pattern is in the injection mode. If the angle of inclination is greater than the critical angle, the flow pattern is in the blowing/suction mode.
(c)
(d)
(e)
(h)
(i)
(j)
Fig. 30 A cycle of vorticity of cylinder with a slit, S/D=0.1, 𝛼 one is negative vorticity.
48.21°, at flow of Re=500: the red color indicates positive vorticity and blue
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4. Conclusion For a cylinder with a normal slit, shear stress has a dominant effect on external drag and pressure has a dominant effect on slit drag. The slit pressure drag has a dominant effect on the total drag and increases as the slit ratio is increased. However, shear stress has a dominant effect on the lift amplitude and increases as the slit ratio is increased. Nevertheless, for flow at higher Reynolds numbers, drag and amplitude of lift are dominated by the pressure on the external surface. For a cylinder with an inclined slit, there are two modes for the flow: an injection mode and a blowing/suction mode. If the angle of inclination is less than the critical angle, the flow pattern is in the injection mode. If the angle of inclination is greater than the critical angle, the flow pattern is in the blowing/suction mode. For all Reynolds numbers, the drag increases as the angle of inclination for the slit increases. The pressure on the external surface dominates the variation of drag due to the change of angle of inclination for the slit. The drag is greater than that for a cylinder that has no slit when flow is in the blowing/suction mode, except for flow at a low Reynolds number such as Re=100. For flow at any Reynolds number, external lift is equally affected by pressure and shear stress. However, shear stress has a dominant effect on the lift due to the slit. There is a similarity between external surface and slit in which aerodynamic forces due to the pressure or shear stress is varied with the angle of inclination for slit. For a cylinder with a normal slit, the distance between the shed vortex and the cylinder is decreased and the frequency of vortex shedding is increased. This effect is more significant for a cylinder with a wider slit or at a higher Reynolds number. Any inclined slit also increases the frequency of vortex shedding, except for an angle of inclination for the slit of less than 30° for Re=500. This effect is more significant for a cylinder with a slit that is inclined at a greater angle or for a flow at a higher Reynolds number. A normal slit gives a greater increase in the frequency of vortex shedding than an inclined slit as a comparison of Figs. 25 (a) and (b) shows. However, wider slit produces greater drag, especially at flows with a higher Reynolds number. An inclined slit produces less drag for angles of inclination for the slit of less than 45° and for flows at lower Reynolds numbers, such as Re 100. Further, the results provide the aerodynamic data for the design of a heat sink with perforated pin fins by selecting an appropriate width of slit and the inclination angle of the slit to produce the favorable frequency of vortex shedding and lower drag for the purpose of heat transfer enhancement and lower power consumption on the electronics cooling. 26
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Acknowledgement The authors are grateful to Ministry of Science and Technology, R.O.C. for its financial support under grant No. 106-2221-E-224-028.
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References [1] T. Igarashi, “Flow characteristics and a circular cylinder with a slit. 1st report, flow control and flow patterns,” Bulletin of JSME, Vol. 21 (154), pp.656–664, 1978. [2] J. F. Olsen, S. Rajagopalan, “Vortex shedding behind modified circular cylinders,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 86, pp.55-63, 2000. [3] H.-L. Ma, C.-H. Kuo, Control of boundary layer flow and lock-on of wake behind a circular cylinder with a normal slit, European Journal of Mechanics B/Fluids, Vol.59, pp. 99-114, 2016. [4] J. W. Yang, Y. He, F. Bao, “Large eddy stimulation and experimental measurement of near wake structures of a slotted circular cylinder,” ACTA Aerodynamica Sinica, Vol. 32(3), pp. 308-314, 2014. [5] C.-H. Kuo, H.-W. Lin, C.-T. Chai, F. Cheng, “Flow characteristics around circular cylinders with a normal slit,” Defect and Diffusion Forum , Vol. 379, pp 48-57, 2017. [6] H. Baek and G.E. Karniadakis, “Suppressing vortex-induced vibrations via passive means,” Journal of Fluids and Structures, Vol. 25, pp.848–866, 2009. [7] W. J. Sheng and W. Chen, “Features of flow past a circular cylinder with a slit,” Scientia Iranica B, Vol. 23(5), pp. 2097-2112, 2016. [8] L. Ordia, A. Venugopal, A. Agrawal, S. V. Prabhu, “Vortex shedding characteristics of a cylinder with a parallel slit placed in a circular pipe,” Journal of Visualization, Vol. 20, pp. 263–275, 2017. [9] D.-L. Gao, W.-L. Chen, H. Li, H. Hu, “Flow around a circular cylinder with slit,” Experimental Thermal and Fluid Science, Vol.82, pp. 287–301, 2017 [10] C.O. Popiel, “Vortex shedding from a circular cylinder with a slit and concave rear surface,” Applied Scientific Research, Vol. 51, pp.209-215, 1993. [11] J. T. Turner, C. O. Popiel and D. I. Robinson, “Evolution of an improved vortex generator,” Flow Measurement Instrument, Vol. 4(4), pp.249-258 1993. [12] T. Igarashi, “Flow resistance and Strouhal Number of a vortex shedder in a circular pipe,” JSME International Journal, Series B, Vol.42 (4), pp. 586-595, 1999. [13] B.H. Peng, J.J. Miau, F. Bao, L.D. Weng, C.C. Chao and C.C. Hsu, “Performance of vortex shedding from a circular cylinder with a slit normal to the stream,” Flow Measurement and Instrumentation, vol. 25, pp.54–62, 2012. [14] S. A. Orszag, and L. C. Kells, Transition to turbulence in plane poiseuille and plane couette flow, Journal of Fluid Mechanics, Vol. 96, pp. 159-205, 1980. [15] L.-C. Hsu and J.-Z. Ye, “Numerical Study of Flow Patterns in a Tandem Array Cylinder System,” International Journal of Applied Mechanics, Vol. 7(3), pp.1550034 (35 pages), 2015. [16] L.-C. Hsu and G.-J. Gao, “Simulation of vortex shedding behind a flat plate with vorticity based adaptive spectral element method,” Mathematical Problems in Engineering, Vol. 2014, Article ID 959615, 17 pages, 2014. [17] L.-C. Hsu, J.-Z. Ye, C.-H. Hsu, “Simulation of flow past a cylinder with adaptive spectral element method,” Journal of Mechanics, Vol. 33(2), pp. 235-245, 2017. [18] J. Park, K. Kwon and H. Choi, Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160, KSME International Journal, Vol. 12(6), pp. 1200-1205, 1988. [19] T. Igarashi, “Flow characteristics and a circular cylinder with a slit. 2nd report, effect of boundary sunction,” Bulletin of JSME, Vol. 25 (207), pp.1389–1397, 1982.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: