Unsteady Flow Around Cylinders with Cavities

Transcription

1 Unsteady Flow Around Cylinders with Cavities G. Iaccarino,P.Durbin 2,S.Talley Center for Turbulence Research, Stanford University, Stanford, CA Dept. Mechanical Engineering, Stanford University, Stanford, CA Reynolds-Averaged Navier-Stokes simulations of the flow around circular cylinder with V-shaped longitudinal cavities are carried out to study the effect of the cavity geometry on the flow characteristics. In particular the effect of the cavity depth on the unsteady aerodynamic forces is analyzed. It is found that the cavities reduce the overall drag and the amplitude of the lift fluctuations. Introduction Thisstudy ismotivated by the saguaro cactus andother tall arborescent(treelike) succulents that withstand high wind velocities in their natural habitat. These stationary desert plants experience Reynolds number ( Re) flows up to 6 and share acommon cylindrical shape modified with complex surface geometry. Because the shape of an object influences the surrounding airflow, natural selection may favor body morphologies that reduce forces exerted by wind gusts in their habitat. We hypothesize that the tall cacti morphology of longitudinal cavities and spines may function to reduce wind forces such as drag andfluctuating lift. In this paperweaddressthishypothesisbynumerical simulations. A concurrent experimental investigation is described in Talley et al.,2; some measured data are compared with the numerical predictions herein. There has been much speculation on the function of cavities and spines on cacti, and their significance is still open to speculation (Geller and Nobel, 984). Natural selection acts on the random mutations of existing structures (traits), resulting in improved structures, novel structures, and/or multiplefunctionality of existing structures. Therefore, one function of a trait does not necessarily preclude other functions, and many traits may contribute to a common function. Given that the shape of an object affects the flow, it is surprising that no studies have examined how cavities and spines on desert succulents influences airflow.

2 66 G. Iaccarino, P. Durbin, and S. Talley Because there are many species of tall arborescent succulents that vary in body size, depth and number of cavities, and spine arrangement, we will focus on one of the most studied of the tall arborescent succulents, the saguaro cactus, Carnegiea gigantea (Fig. ). Saguaros are long-lived and slow to mature. They take 3 to 5 years to reach reproductive maturity and live up to 5 years of age. Adult saguaros have one main cylindrical stem ranging from.3 to.8 m in diameter (Benson, 98) and over 8 to 5 m in height (Hodge, 99). Ten to 3 v-shaped cavities span the length of the stem (Hodge, 99). The number of cavities depends on the diameter of the stem, and new cavities can be added or deleted (Fig. a) to maintain a cavity depth ratio ( L/D - depth of the cavity divided by the diameter of the cylinder) of approximately. 7 ±. 5 (Geller and Nobel, 984). Apices of the cavity junctures are adorned with whorls of 5 to 3 spines 2.5 to 7.6 cm long (Benson, 98). In order for wind to be a selective agent on saguaros, high wind velocities must occur in saguaro habitats and they must affect their reproductive success. Within the distribution of saguaros, high wind velocities were recorded 5 m above the ground for a nine-year period (Nobel, 994). The maximum wind velocity recorded was 38 m/s,( Re = 6 ), and velocities exceeding 22 m/s ( Re = 7 5 ) occurred almost every month. Saguaro habitats contain less vegetation cover than other ecosystems and, consequently, have few if any other tall plants to shelter them from the wind (Fig. b). There is substantial circumstantial evidence that wind gusts exert enough force to topple saguaros, and thus, cause their premature mortality (Fig. c; Benson, 98; Alcock, 985; Pierson and Turner, 998). Information on the wind velocities required to topple large desert succulents is lacking. Consistent with the natural selection scenario, some saguaros are toppled by gusts, while many others remain standing. Considering most tall cacti live for 5 years and take over 3 to 5 years to reach reproduce maturity, strong gusts need only to occur every 3 to 5 years to be important in the natural selection of tall succulent morphology. 2 Numerical Method Numerical simulations of the flow around a cactus section are carried out by solving the Reynolds-Averaged Navier-Stokes equations in two dimensions. Two codes are used: INS2D (Rogers and Kwak, 99) and Fluent (Fluent, 999). INS2D is an upwind based, third-order accurate code for structured (multiblock) grids; the artificial compressibility approach is used for pressurevelocity coupling and the time integration is second-order accurate. Fluent is an unstructured-mesh solver based on second-order accurate spatial and time discretization; the SIMPLE technique is used for pressure-velocity coupling. Turbulence modeling isbased onthe v 2 f model (Durbin, 995; Iaccarino, 2).

3 3 Computational Grids Unsteady Flow Around Cylinders with Cavities 67 Cylinders with v-shaped cavities (with cavity ranging from L/D =. to L/D =. 5) are considered. Several meshes have been generated to assess the sensitivity of the solution. In Fig. 2, examples of the grids are reported. Simulations using the structured grids (Fig. 2a and 2b) have been performed using both Fluent and INS2D. The structured grid is generated as an O- type mesh wrapped around the cylinder. The cavities are slightly smoothed to improve the orthogonality of the grid lines at the cylinder surface. The height of the first cell is adjusted according to Re; the distance from the far field boundary is 25D as used in Rogers and Kwak, 99. The unstructured meshes are generated using a quadrilateral paving technique (Blacker et al., 99); this approach allows flexibility in clustering the grid cells in the wake region and close to the surface. In Table I, results are reported for the computations performed on different grids at a very low Reynolds number. The flow is unsteady and exhibits a periodic vortex shedding from the cylinder; only the averaged drag coefficient is reported. Grid convergence is achieved for the smooth cylinder L/D = using both the structured and the unstructured grids, and the corresponding values are extremely close. The results for the flow around the cylinders with cavities show that grid convergence is achieved only using the unstructured grids. An increase in cavity depth requires a finer resolution to capture accurately the in-cavity flow; in addition, the quality of the structured grid degrades as the cavity depth increases. It is worth noting that the results obtained using the finest structured grid (76 2) are in good agreement with the converged results for the unstructured mesh. In the following sections only results computed using the unstructured grids are reported. Grid Elements L/D L/D L/D L/D , , , , structured grids Elements L/D L/D L/D L/D , , , , unstructured grids Table I.Computed time-averaged C d for different computational grids Re = -Laminar Simulations 4 Laminar Simulations Flow simulations at low Reynolds number ( Re = and Re = 2) are carried out to evaluate the effect of cavity depth (and the accuracy of the predictions) without uncertainties related to the turbulence modeling. Twodimensional simulations have been performed with unstructured grids using

4 68 G. Iaccarino, P. Durbin, and S. Talley 6, to 42, elements (only the fine mesh results are presented but the results appear to be already insensitive ot the mesh for a grid size of 25, elements). The calculations are carried out using a timestep tu/d =. (corresponding to approximately 35 time steps per vortex shedding period) and for a total time of TU/D = 5. Simulations have been carried out using a smaller time step ( tu/d =. 65 and the lift and drag coefficient changed by less than.5%). The time history of drag and lift coefficients at Re = are reported in Fig. 3a and 3b respectively. The statistics (time averaged values and the Strouhal number, etc.) are computed over a period T av = 5D/U and are reported in Table II. L/D C d C l St.339 ±. ± ±. ± ±. ± ±.2 ± Re = L/D C d C l St.365 ±.37 ± ±.45 ± ±.57 ± ±.49 ±.74.7 Re =2 Table II. Statistics for low Reynolds number flow around cacti. St is the Strouhal number based on the frequency of the lift oscillations, f C l D/U The results indicate a small drag reduction ( %) associated with the presence of the cavities. The cavity depth L/D =. 5 is somewhat optimal. The change in the unsteady lift is also small, showing that the effect of the cavity is limited. The results presented for the smooth cylinder at Re = 2 are in good agreement with the numerical simulations and the experimental data reported in Rogers &Kwak(99).Itisworth noting that Re =9 representthe onset of three-dimensional flow in the wake of the cylinder. 5Turbulent Simulations Calculations at Re = 2, and Re =, (subcritical regime) are performed using the v 2 f turbulence model. The time step, the simulated time and the averaging time are the same as before; the time history of lift and drag are reported infig. 4. As compared to the results presented at low Re, the drag reduction is now larger ( 25%). The strength of the unsteady motion is also greatly reduced as reported in Table III.

5 L/D C d C l St.683 ±.64 ± ±.76 ± ±.83 ± ±.52 ± Re =2, Unsteady Flow Around Cylinders with Cavities 69 L/D C d C l St.644 ±.3 ± ±.2 ± ±.3 ± ±.79 ± Re =, Table III. Statistics for high Reynolds number flow around cacti From the results presented in Table III, it appears that the cavity depth has a relatively strong effect on the drag and a substantial dampening effect on the unsteady motion. The time averaged turbulent kinetic energy for the four geometries considered is reported in Fig. 5;the intensity very close to the cylinder decreases with the cavity depth, but higher values are observed in the near wake. The comparison ofthe computed C d with the experimental values for the smooth cylinder (Achenbach, 97) shows an overprediction of about 2%. The flow over the smooth cylinder in the subcritical regime is characterized by a laminar boundary layer separation; turbulence is generated in the separated shear layer and is sustained in the near wake. The smooth cylinder calculations ( L/D =)are carried out with the v 2 f turbulence model switched offfor θ 9 o in an effort toforce alaminar separation in the simulations. Itis well known that RANS turbulence models typically anticipate transition and, even with first part of the boundary layer forced to be laminar, in the present calculations the shear layer separates with very high level of turbulent kinetic energy. In addition, in the subcritical regime three-dimensional effects in the wake are substantial and not account for in the present calculations. The simulations with cavities are carried out with the turbulence model switched on from the stagnation point (θ = o )because itisexpected that transition occurs immediately after the first cavity. In addition, it is expected that three-dimensional effects are less substantial inthese cases (as observed in the experiments). A comparison of experimental and computed velocity profiles in the wake is reported in Fig. 6. The results for the smooth cylinder confirm that the calculation overestimate the drag (corresponding to the larger velocity defect in the wake); on the other hand, the data for the cylinder with cavities show aremarkable agreement. 6Conclusions A numerical study of the flow around cactus-like cylinders is presented; various cavity depth are considered to investigate their effect on the flow characteristics with particular emphasis on the aerodynamic forces. Preliminary Simulation are carried out at very low Reynolds numbers, namely and

6 7 G. Iaccarino, P. Durbin, and S. Talley 2. The effect of the cavity is limited and only a slight drag reduction is obtained. At higher Reynolds numbers (2, and, ) the effect of the cavities is more substantial with a considerable reduction of the drag and, perhaps more importantly, a strong damping of the oscillating lift. (c) Fig.. Addition of cavities (ribs) on an adult saguaro trunk Saguaro forest, and (c) Root system of asaguaro toppled by the wind.

7 Unsteady Flow Around Cylinders with Cavities (c) 7 (d) Fig. 2. Example of the computational Grids: (a-b) structured grids 24 elements; (c-d) unstructured grids 2, elements (a-c) L/D = (b-d) L/D = Cl Cd tu/d tu/d Fig. 3. Time history of drag and lift coefficients. Re =. Solid line: L/D = ; Dotted line: L/D =.7.

8 72 G. Iaccarino, P. Durbin, and S. Talley 2 2 C d.5 C l tu/d tu/d Fig. 4. Time history of drag and lift coefficients. Re = 2,. Solid line: L/D =; Dotted line: L/D =. 7. References. Achenbach, E. 97. Influence of surface roughness on the cross-flow around a circular cylinder. J. of Fluid Mech. 46, Alcock, J Sonaran Desert Spring. The University of Chicago, Chicago. 3. Benson, L. 98. The Cacti of Arizona. The University of Arizona Press, Tucson. 4. Blacker T.D., M.B. Stephenson & S. Canann 99 Analysis automation with paving: A new quadrilateral meshing technique Advances in Engineering Software, 56, Durbin, P.A. 995 Separated flow computations with the k - ɛ - v 2 model, AIAA J., Geller, G. N., and Nobel, P. S Cactus ribs: influence onpar interception and CO 2 uptake. Photosynthetica 8, Hodge, C. 99. All About Saguaros. Hugh Harelson-Publisheer, Phoenix. 8. Iaccarino, G. 2 Predictions of a turbulent separated flow using commercial CFD codes, J. Fluids Engineering, 23, Fluent Inc. 999 Fluent V5.3 User Manual.. Nobel, P. S Remarkable Agaves and Cacti. Oxford University Press, New York.. Pierson, E. A., and Turner, R. M An 85-year study of saguaro (Carnegiea gigantea) demography. Ecology. 79, Rogers, S. E. and Kwak, D. 99 An Upwind Differencing Scheme for the Time Accurate Incompressible Navier-Stokes Equations AIAA J., 28,

9 Unsteady Flow Around Cylinders with Cavities (c) (d) 73 Fig. 5. Time averaged turbulent kinetic energy: L/D = ; L/D =.35; (c) L/D =.7; (d) L/D =.5. u/vinl u/vinl y/d y/d 2 Fig. 6. Velocity profiles in the wake of cylinders. Solid line: simulations (Re =,); Circle: experiments (Re = 25,). smooth cylinder cylinder with L/D.7.

10 74 G. Iaccarino, P. Durbin, and S. Talley 3. Talley, S., Iaccarino, G., Mungal, G. and Mansur, N. N. 2. An Experimental and Computational Investigation of Flow Past Cacti, Annual Research Briefs, Center for Turbulence Research, 5 64