NUMERICAL SIMULATION OF FLOW AROUND BLUFF BODIES BASED ON VIRTUAL BOUNDARY METHOD

Transcription

1 The Eighth Asia-Paciic Conerence on Wind Engineering, December 1 14, 213, Chennai, India NUMERICAL SIMULATION OF FLOW AROUND BLUFF BODIES BASED ON VIRTUAL BOUNDARY METHOD Qing Yang 1, Shuyang Cao 2 1 Ph.D candidate o State Key Laboratory o disaster reduction in Civil Engineering, TONGJI University, Shanghai, China, 811benson@tongji.edu.cn 2 Proessor o State Key Laboratory o disaster reduction in Civil Engineering, TONGJI University, Shanghai, China, shuyang@tongji.edu.cn ABSTRACT The Immersed Boundary Method or rigid boundary (Virtual Boundary Method) is selected and integrated with staggered grid and ractional time-step method to simulate the low around blu bodies. A bilinear interpolation procedure is combined to realize the data transmit between grid points and boundary points to orm the boundary. The selection o constant values in the orcing term is discussed based on the error o velocity at the boundary. Then the method is used to simulate the low around blu bodies. The qualitative description about this simulation is irstly presented by providing the current simulation s streamlines and pressure contours. The phenomena illustrated in these sketches are the same as the other available literatures. At the end o this phase, the virtual boundary method is applied to the preliminary simulation o low around two-dimensional bridge section to demonstrate Virtual Boundary Method can cope with complex geometry. Under this basis, the quantitative analysis o this simulation is carried out to investigate the inluence o additional boundary treatment used in virtual boundary method on the accuracy o the aero-dynamic coeicients. The easibility o Virtual Boundary Method is veriied by the good agreement between this simulation s results and other reerences outcomes. Keywords: Virtual Boundary Method, Blu bodies, Aero-dynamic coeicients, Additional boundary treatment Introduction Virtual boundary method is a new type o mathematic ormulation and numerical scheme based on the immersed boundary idea which was proposed by Peskin (1972) to simulate the interaction between heart valve and blood. The core conception o immersed boundary idea is that a set o shear orce is exerted on grid points nearby the boundary to build the body shape. Based on this conception, immersed boundary algorithm has two characteristics. First, the numerical simulation o low over body, which is conducted by this model, can be solved on the regular Cartesian coordinate. It can avoid the use o coordinate transormations and mapping techniques which require highly accurate solution and a lot o time to accomplish the Jacobians transormation. Second, when immersed boundary method is applied to simulate moving body in the low ield, it is unnecessary to regenerate the grid to grab the updated position o boundary at every time step which is commonly used in conventional CFD method and creates large computational overhead. These above two eatures o immersed boundary method indicate that the problems o complex geometry and moving boundary, which are diicult to deal with in traditional numerical simulation, could be easily analyzed by immersed boundary method. Originally the immersed boundary model was only adopted to study low patterns and luid-structure interaction o elastic boundary, such as the deormation o red blood cells in a shear low [Eggleton and Popel (1998)], low in elastic blood vessels [Vesier and Yoganathan (1992)], the swimming o sperm and bacteria [Fauci and McDonald (1994), Dillon Fauci and Gaver (1994)]. The classical immersed boundary algorithm model or elastic boundary Proc. o the 8th Asia-Paciic Conerence on Wind Engineering Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds) Copyright c 213 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: doi:1.385/

2 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) problems was summarized by Peskin (22). And then it was successully employed to biology research to modeling arthropod iliorm hair motion in the air by Heys et al. (28). Ater previous comprehensive researches on the elastic border, the immersed boundary algorithm was applied by some researchers to study the rigid boundary problems. However, when initial immersed boundary model reerred in above papers, which is suitable to elastic boundary, was used to solve rigid boundary problems, it existed problem, including sever stability constraints and spurious elastic eects, because o the discrepancy o constitutive laws between elastic border and rigid surace. Under this background, virtual boundary method was introduced by Goldstein et al. (1993) to simulate low over rigid bodies. In this method, orcing term take advantage o eedback loop o the luid velocity on the surace to achieve the required rigid boundary conditions: the luid velocity is equivalent to the surace velocity. In Saiki s paper [Saiki and Biringen (1996)], virtual boundary method was combined with high-order inite dierence and bilinear interpolation to simulate stationary, rotating, oscillating cylinders in uniorm low at low Reynolds numbers (Re<4). Recently virtual boundary method was applied by Deng et al. (25) to show the three dimensional low structures o two spheres in tandem arrangement. In the last papers about virtual boundary method, the simulations were ocused on the simple body shape: circular cylinder or other theoretical researches. Other blu body shapes were not studied in the numerical simulations. The local continuity or the mesh containing the immersed border, which has been discussed in detail in other immersed boundary models [Li et al. (24), Kim et al. (21)], was not presented in virtual boundary method. In the present work, virtual boundary method is combined with staggered grid and inite dierence to make simulation. Firstly the mechanism o orming border is discussed in detail to demonstrate the unction o eedback loop in the model. Secondly the simulation o the low over stationary blu bodies, including square cylinder, circular cylinder and other blu shape, is perormed. The emphasis o second stage in this paper is to veriy the easibility o virtual boundary method under regular Cartesian grid. The advantage o immersed boundary idea or coping with moving body isn t presented in this work. Finally the local conservation is considered in the simulation to make comparison under traditional Cartesian grid. Numerical Method Governing Equation Basic equations o virtual boundary method can be expressed as below: u 2 ρ + ( u ) u = P+ μ u+ F t (1) U = (2) In this equation, F is orcing term to orm the rigid immersed boundary by eedback loop. Its expression is deined by the reerence o Goldstein et al. (1993): t ' F = α [ u V] dt + β [ u V] (3) Here α and β are the negative constants, which are critical actors o the eedback loop; u is the luid velocity on the boundary; V is the velocity o body border. The above equation is the eedback to velocity dierence u-v and acts to enorce the classic no-slip boundary condition (u=v) on the rigid immersed boundary. The irst term o the equation is inclined to annihilate any velocity dierence between u and V. It can be 583

3 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) viewed as the restoring orce to bring the luid velocity close to the interace velocity. The second term is considered as the energy dissipation caused by surrounding luid cell. In order to illustrate distribution o the boundary points and virtual boundary method, Fig.1 and Fig.2 are presented as below: Fig.1 Distribution o boundary points Fig.2 Virtual boundary illustration In Fig.1, the hollow points are boundary points, which can also be called Lagrangian points. The other grid points are regarded as Euler mesh points. The dotted line stands or the embedded border. The orcing terms are exerted on these boundary points to orm the desired body shape. Fig.2 indicates that ater orming the boundary, the luid ield is divided into two parts: inner low ield and outer low ield. The low inside the body is independent rom the outer luid. As the time increase, the inner luid velocities tend to decay. The outer low ield is undisturbed by internal low ield. Fractional Step Method Fractional step method is used to discretize the governing equations. This method is described as ollows: The irst step: the initial luid components including u and v are set to calculate the estimated velocities u* and v*. * n 2 2 u u u u 1 u u = u v + + F (4) Δt x y γ x y * n 2 2 v v v v 1 v v = u v + + F (5) Δt x y γ x y The second step: Poisson equation is solved to get the correction pressure. Pi+ 1 2Pi, j + Pi 1 Pj+ 1 2Pi, j + Pj 1 + = D 2 2 i, j (6) Δx Δy ( ) ( ) u* v* Di, j=δ t + x y (7) The third step: the updated velocities are gotten by using equation (8)~(9) n+ 1 P u = u* Δt x (8) n+ 1 P v = v* Δt y (9) 584

4 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) Interpolation Procedures In virtual boundary method, the ideal condition is that boundary points (Lagrangian points) coincide with luid grid points (Euler mesh points). The rontier luid velocity can be directly gotten by solving governing equation. The orcing term can be easily calculated to impose on the boundary. But body shape is generally complex. The embedded border can cut through the underlying Cartesian grid in an arbitrary manner. In particular, when the staggered mesh is adopted to avoid the numerical oscillation caused by collocated grid, all luid components can t coincide with the grid points consistently. Considering the above conditions, the interpolation procedure is necessary to distribute the eect o orcing term to the Euler mesh points nearby the boundary and extrapolate the luid velocity on the border rom surrounding Euler grid points. It s a data transer process. In this process, the bilinear interpolation is well understood and widely applied in immersed boundary model to realize the data transer [Saiki and Biringen (1996), Li et al (24)]. Compared with spectral interpolation utilized by Goldstein et al. (1993), the bilinear interpolation is more accurate and convenient to be applied in the Virtual Boundary Method. The bilinear interpolation expression, as illustrated in Fig.3, is summarized as below: Fig.3 Bilinear interpolation illustration In Fig.3, P is the boundary point (x s, y s ); (i, j), (i+1, j), (i, j+1), (i+1, j+1) are the our surrounding Euler velocity grid points. i+ 1, j+ 1 ( ) ( ) U x = D x U (1) s i, j s i, j i, j ( ) ( ) ( ) Di, j xs = d xs xi d ys y (11) j In equation (1), the luid velocities o virtual boundary points (x s, y s ) are interpolated rom the surrounding grid points by bilinear unction D i, j (see equation (11)) to calculate the orcing term on the border. And then the impaction o immersed border is extrapolated back to the Euler mesh points by the equation (12). It is put into the governing equations to solve luid velocities at new time-step. Nb 1 i, j i, j s n s N b n = 1 ( ) ( ) F = D x F X (12) Where N b is the number o boundary points which aects the (i, j)th Euler grid point; F n (X s ) is the orcing term o the border. Additional Boundary Treatment Additional boundary treatment is the characteristic o immersed model. It is employed to reinorce local mass conservative law o the embedded border. Li et al. (24) postulated 585

5 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) that i no treatment is applied, the local continuity o the low adjacent to the immersed boundary may not be satisied, especially at the beginning o the simulation when incorrect initial condition is used. The mass source introduced as a treatment measure [Kim et al. (21)] to ensure the local conservation is regarded as adverse to the global mass conservative within the entire calculation domain. Although the velocities can satisy the no-slip boundary condition, the pressure distribution near the boundary may be incorrect by using the mass source term. The imposition o zero normal gradient condition o pressure is another additional boundary treatment measure proposed by Li et al. (24) in order to obtain the realistic solution and increase the accuracy o the solution near the boundary. This alternative approach has been tested that it can alleviate the above problems. It is used to make sure that there is no low across the border. The local continuity o the boundary can be satisied automatically. a b Fig.4 Implementation o zero normal gradient pressure As demonstrated in Fig.4, the zero normal gradient condition o pressure can be imposed by setting P A =P B (Fig.4a) and P A =P E (Fig.4b) to achieve the desired parameter distribution. In this paper, simulations o low around square cylinder with and without zero normal gradient condition are tested. The analysis is stated in ollowing section below. Calculation domain The traditional equidistance Cartesian grid is applied in the simulation. The grid size DX=DY=.25. The time step size t= The two dimensional blu body with diameter D is centered inside calculation domain (height H). When the body is cylinder, the blockage ratio is ixed at 1/8. As or the 2D bridge deck, the blockage ratio is set at 1/2. Frictionless Boundary Condition is used on the lateral boundary: u/ y= P/ y= v=. The boundary condition o inlet is set as: u=1. v=. The Neumann boundary condition is applied on the outlet o the calculation grid. / =. v=. u=1. / =. / =. Fig.5 Calculation domain and boundary condition 586

6 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) Simulation Results Mechanism analysis o orming virtual border The equation (1) can be simpliied as ollows: dq t = α qdt+ β q dt (13) Where q=u-v is velocities dierences as discussed in the above sections. The irst term including α can be viewed as the spring connecting between the luid cell on the boundary and virtual border. This equation indicates that when the luid velocity u on the boundary deviates rom the setting value o border velocity V, the orcing term compels u back to V. In unsteady low ield, the value o α must large enough to trace the changing low ield. At the same time, the time restriction in this method can be expressed as below: 2 β ( β 2 αk) Δ t < (14) α Where k is a dependent constant o order one. The time step size is determined by constants: α β. Hence, the selection o constant value is needed to be discussed. The dierent combination o constants are applied to simulate the no-slip wall boundary. The proper constant value is selected based on the error o velocity at the boundary (l 2 -norm) to launch simulation. 1 nb l2 norm= u i i n (15) b Where n b is the number o boundary points, u i is the velocity o boundary points. a. =-16, =-6 b. =-16, =-6 c. =-16, =-6 d. =-1.6x1 6, =-6 Fig.6 l 2 -norm o boundary points with dierent constant combinations In Fig.6, as the constants increase, the time required to achieve desirable boundary condition (U=) lessen. It indicates the larger constant value could realize immersed boundary `` quickly, and the combination c and d could both act more eiciently in reinorcing boundary condition, in which the time required is almost same. Furthermore, the combination c tends to retain the entire calculation eiciency based on the time restriction. Ater the above analysis, the combination c is considered as the most appropriate selection or the simulation U.3.2 V T.1 5 T 1 15 Fig.7 Visualization o Fig.8 The variation o x- Fig.9 The variation o y-direction immersed boundary direction velocity u o velocity v o internal internal points with time points with time 587

7 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) As shown in the Fig.7, the streamline bypass the immersed boundary to veriy the success o simulation or physical border. Furthermore the inner luid ield is analyzed. The variations o velocities o random internal points with time are drawn in the Fig.8 and Fig.9. The variation o inner velocities is at low amplitude within the entire calculation time. And the values are very small in the range o 1-2 ~1-3. These phenomena indicate that inner low ield is completely isolated rom the outer low ield. Qualitative Description o simulation o low around blu bodies The simulation o low around two dimensional blu bodies conducted by virtual boundary method is carried out to veriy easibility o this program. Considering the vortex structure become three dimension in high Reynolds numbers and the blockage ratio (1/8) adopted in this paper [Guo et al. (23)], the range o Reynolds numbers o present two dimensional simulation is 1~5. Fig.1 is the streamline contour o low around square cylinder at Re=1~6. Fig.1a reveal that two steady symmetrical vortices appear in the downstream region. As Re increases, the length o separated bulb becomes longer (shown in Fig.1b). When Re is 6, the steady and closed near-wake becomes unstable. The transverse oscillation starts at the end o the near-wake and initiates a wave along the trail. The critical value o Re crit or this calculation is 6. It is the same as determined by Guo et al. (23). This phenomenon or square cylinder can also be visualized by vortex contour in Fig.11a (Re=6). When Re reaches 1, the ree shear layers begin to roll up and orm eddies as seen in Fig11.b. This eature is well known as the von Karman vortex street. As Re comes to the upper limit (Re=5), the low pattern change into turbulence low. The regularity disappears in the wake region (seen in Fig.11c). These above description about this simulation coincide with phenomena observed by Breuer et al. (2) and Liu et al. (28). a. Re=1 b. Re=3 c. Re=6 Fig.1 Streamline contour o Flow around square at dierent Re a. Re=6 b. Re=1 c. Re=5 Fig.11 Vortex contour o Flow around square at dierent Re Fig.12 is the streamline sketch o low around square with attack angle at dierent Re. The constant values in this simulation are the same as the square cylinder 5 ( α = β = 6 ). For circular cylinder, β is dierent rom the square cylinder ( β = 4 ). The attached, recirculating, symmetric bubble is revealed in the wake at Re=4 (Fig.13a). The vortex shedding rom the circular cylinder occurs at Re=5 (Fig.13b). The same situation was obtained in the Saiki et al. s (1996) simulation. 588

8 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) a. Re=1 b. Re=6 a. Re=4 b. Re=5 Fig.12 Flow around square with attack angle at dierent Re Fig.13 Flow around a circular cylinder at dierent Re The virtual boundary method is directly applied to simulate the low around twodimensional bridge section. Fig.14 is the preliminary simulation under Re=2 to demonstrate that virtual boundary method can cope with the complicated geometry. a. Pressure contours b. Velocity streamlines Fig.14 Flow over bridge deck (Re=2) Quantitative Analysis o simulation o low around blu bodies Considering the adoption o boundary treatment in virtual boundary program, two cases (case1 and case2), which are dierentiated by zero normal gradient condition, are proposed to analyze the inluence on the numerical accuracy o aero-dynamic coeicients or square cylinder Liu 3 Guo.4 Case2 Breuer 2.5 Case1.3 Case1 Case2 2.2 Cd Re Re Fig.15Variation law o Cd or square Fig.16 Data dierence variation o Cd or cylinder at dierent Re square cylinder depending on Re As shown in Fig.15, it is observed that although the drag coeicient Cd o square cylinder without zero normal gradient condition (case1) is smaller than others, the drag coeicient trend depending on Re o case1 is in line with other reerence s variation law [Breuer et al. (2), Guo et al. (23), Liu et al. (28)]. On the other hand, the case2 s drag coeicient has good agreement with other s results [Breuer et al. (2), Guo et al. (23), Liu et al. (28)]. The accuracy o numerical simulation is improved by using the boundary treatment. Fig.16 relects data dierence variation o Cd or square cylinder depending on Re. In this diagram, the data dierence means the discrepancy between the current simulation (case1 and case2) and other literatures. The discrepancy o case2 is smaller than the case1 within the whole calculation time zone. Although the conclusion can be summarized the boundary treatment measures is helpul to annihilate the deviation o simulation based on the above Dierence 589

9 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) investigation, the zero normal gradient condition can t be view the only sole actor to inluence the accuracy. Because rom the Fig.16, it can be obviously watched that as the Reynolds number increases, the discrepancy o two cases declines. The discrepancy may be caused by the coarse equidistance Cartesian grid in the vicinity o virtual boundary, which isn t good enough to capture the velocity dierence nearby the body surace. [Breuer et al. (2), Deng et al. (25)]..25 St.2.15 Suzuki Breuer Franke Case 1 Case2 Max(Cl)-Min(Cl) Breuer Case1 Case Re Re Fig.17 Strouhal number depending on Re Fig.18 Lit variation at dierent Re The variation o St at dierent Re is presented in Fig.17. In the range 6<Re<3, the case2 its other literature [Breuer et al. (2)] well, showing an increase in the St with increasing Reynolds numbers. The case1 show a little deviation in St. This phenomenon coincide with the above conclusion. In Fig.18, lit coeicient o this computation ( Max(Cl)-Min(Cl) ) are plotted. As shown in the igures, the numerical results o this paper have good agreement with previous literature [Breuer et al. (2)]. Conclusions In this paper, the proper constant value selection o orcing term in the simulation is discussed. Considering the time restriction, the appropriate combination is chosen to launch simulation. And then the mechanism o orming border is visualized. The streamline bypass the immersed boundary to veriy the success o simulation or physical border. The inner low ield is isolated rom the outer low ield and velocities decay with time. The qualitative description o this simulation is presented by providing sketches o low around blu bodies. Then the 2D bridge deck is simulated to show that Virtual Boundary Method can treat with the complicated geometry situation. Under this basis, the quantitative analysis is carried out to investigate the inluence o boundary treatment measures on the numerical accuracy o simulation. Although the boundary treatment measure is helpul to annihilate the deviation o simulation, the zero normal gradient condition can t be view the only sole actor to inluence the accuracy. Acknowledgement This work was unded by the Ministry o Education and SLDRCE9-B-4. Reerences Peskin C S. Flow patterns around heart valves [J]. Journal o Computational Physics, 1972, 2: Goldstein D, Handler R, and Sirovich L. Modeling a no-slip low boundary with an external orce ield [J]. Journal o Computational Physics, 1993, 15:

10 Proc. o the 8th Asia-Paciic Conerence on Wind Engineering (APCWE-VIII) Saiki E M and Biringen S. Numerical simulation o a cylinder in uniorm low: Application o a virtual boundary method [J]. Journal o Computational Physics, 1996, 123: Breuer M, Bernsdor J, Zeiser T and Durst F. Accurate computations o the laminar low past a square cylinder based on two dierent methods [J]. Int. Journal o Heat and Fluid Flow, 2,21: LIU Tiancheng, GE Yaojun, CAO Fengchan, LIU Gao. Simulation o low around square cylinder based on the lattice Boltzmann method [J]. Journal o Tongji University: Natural Science, 28, 36(8): DENG Jian, ZOU Jianeng, REN Anlu SHAO Xueming. Numerical simulation o low around two circular cylinders in cruciorm arrangement and two spheres in tandem arrangement by virtual boundary method [J]. Chinese Journal o Theoretical and Applied Mechanics, 25, 37(5): Guo W, Wang N, Shi B, et al. Lattice-BGK simulation o a two-dimensional channel low around a square cylinder [J]. Chinese Physics, 23, 12( 1) : 147. Dillon R, Fauci L J. A micro-scale model o bacterial and bioilm dynamics in porous media [J]. Biotechnol Bioengt, 2, 68: Eggleton C D, Popel A S. Large deormation o red blood cell ghosts in a simple shear low [J]. Phys Fluids, 1998, 1: Peskth C S. The immersed boundary method [J]. Acta Numerica, 22, 11: Heysa J J, Gedeonb T,.Knotta B C, Kimc Y. Modeling arthropod iliorm hair motion using the penalty immersed boundary method [J]. Journal o Biomechanics, 28, 41: Vesier C. C. and Yoganathan A. P. A computer method or simulation o cardiovascular low ields: Validation o approach [J]. J. Comput. Phys., 1992, 99: Fauci L. J. and Mcdonald A. Sperm motility in the presence o boundaries [J]. B. Math. Biol., 1994, 57: Kim J, Kim D, Choi H. An immersed-boundary inite-volume method or simulations o low in complex geometries [J]. Journal o Computational Physics 21; 171: Li C W, Wang L L. An immersed boundary inite dierence method or LES o low around blu shapes [J]. International Journal or numerical methods in luids, 24, 46: