Numerical Investigation of Laminar Flow over a Rotating Circular Cylinder

Transcription

1 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 32 Numerical Investigation of Laminar Flow over a Rotating Circular Cylinder Ressan Faris Al-Maliky Department of Mechanical Engineering Kufa University, Ira Corresponding author; address: Ressan_Faris@yahoo.com Abstract-- This is the work deals with a numerical study of laminar flow of two - dimensional, incompressible, and steady state over rotating circular cylinder. The solution of the flow is presented for dimensionless rotation rate varying from (1 6) (in the steps of 1) at each value of Reynolds number based on diameter of cylinder is (2, 4, 8, and 1). Navier Stokes and continuity euations were solved numerically by using finite volume techniue is conducted with FLUENT version (6.2) package program was used in present work. Stream lines or function and vorticity contours and pressure, lift, and skin friction coefficients results are presented along curve length of cylinder at each value of rotation rate and Reynolds number. The results of lift coefficient and stream lines and vorticity contours were compared with other previously published research that presented support the validity of results. Results have shown approximately increase values of pressure, and skin friction coefficients with increasing of rotation rate at known Reynolds number. Index Term-- Rotating cylinder, laminar flow, skin friction pressure lift coefficients I. INTRODUCTION In (CFD) computational fluid dynamics, laminar flow past a rotating cylinder is interesting problem; it's applications in many fields such as rockets, projectiles, aeronautics, and marine ships. The pressure gradient can be explained simply by Bernoulli's principle, in which pressure and velocity are inversely proportional. The phenomena of a rotating cylinder's lift is know as the Magnus effect, named after a 19th century German engineer, and is related to the circulation around an a flow field. (Rayleigh) studied the lift of a rotating cylinder for an inviscid (frictionless) fluid, and related lift to the circulation of a rotating cylinder by the following formula: L = ρ.u.γ in which the circulation, Γ is given by: Γ = 2.π.ω.R 2 therefore, L = ρ.u.(2.π.ω.r 2 ) The relationship between lift and circulation is known as Kutta Joukowsky relationship and applies to all shapes, particularly to the aerodynamic shapes such as an airplane wing. In a laminar fluid, like air, the cylinder is subjected to both pressure and viscous forces, and the explanation is more complex. Studies (Smith, 1979) indicate that the circulation does not result from the common explanation of the air set into an opposing rotation by the friction of a no slip wall, as this only occurs in a very thin boundary layer next to the surface. But this motion of the fluid in the boundary layer does affect the manner in which the flow separates from the cylinder. Boundary layer separation is moved back on the side of the cylinder that is moving with the fluid, and is moved forward on the side opposing the main stream. The wake then shifts to the side moving against the main stream causing the flow to be deflected on that side, and the resulting change in free stream flow creates a force on the spinning cylinder [1]. In the present work, the asymmetrical flow was considered, a laminar fluid which is generated by rotating a circular cylinder in a uniform stream of fluid. There are two basic parameters in the problem, namely, first the Reynolds number based on the diameter of cylinder, second rotation rate, which is a dimensionless measure of the rotation rate. When =, the motion is symmetrical about the direction of translation and this situation has previously received a considerable amount of attention [2]. (Watson, 1995) pointed out that the pressure field given by Smith's asymptotic form is not single-valued and proposed that an additional term to Jeffery's Fourier series is necessary. However, he did not derive the force, since the outer flow which is governed by the Navier-Stokes euations was not obtained. The problem of flow past rotating cylinders was considered by (Sennitskii, 1973). The problem was studied using a boundary layer approach for the case of a large distance between the centers of cylinders. In the work of (Sennitskii, 1975) the first terms of an asymptotic expansion by inverse degree of the Reynolds number were obtained [3]. (Ingham, 1983) obtained numerical solutions of the two-dimensional steady incompressible Navier Stokes euations in terms of vorticity and stream function using finite differences for flow past a rotating circular cylinder for Reynolds numbers Re = 5 and 2 and dimensionless rotation rate velocity between and.5. Solving the same form of the governing euations, but expanding the range for, (Ingham & Tang, 199) showed numerical results for Re = 5 and 2 and 3. With a substantial increase in Re, (Badr et al., 199) studied the unsteady two-dimensional flow past a circular cylinder which translates and rotates starting impulsively from rest both numerically and experimentally for 13 Re 14 and.5 3. They solved the unsteady euations of motion in terms of vorticity and stream function. The agreement Numerical study of the steady-state uniform flow past a rotating cylinder 193 between numerical and experimental results was good except for the highest rotational velocity where they observed three-dimensional and turbulence effects. Choosing a moderate interval for Re, (Tang & Ingham 1991) followed with numerical solutions of the steady two-

2 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 33 dimensional incompressible euations of motion for Re = 6 and 1 and 1. They employed a scheme that avoids the difficulties regarding the boundary conditions far from the cylinder. Considering a moderate constant Re = 1, (Chew, Cheng & Luo, 1995) further expanded the interval for the dimensionless rotation rate, such that 6. They used a vorticity stream function formulation of the incompressible Navier Stokes euations. The numerical method consisted of a hybrid vortex scheme, where the time integration is split into two fractional steps, namely, pure diffusion and convection. They separated the domain into two regions: the region close to the cylinder where viscous effects are important and the outer region where viscous effects are neglected and potential flow is assumed. Using the expression for the boundary-layer thickness for flow past a flat plate, they estimated the thickness of the inner region. Their results indicated a critical value for about 2 where vortex shedding ceases and the lift and the drag coefficients tend to asymptotic values. (Nair, Sengupta & Chauhan, 1998) expanded their choices for the Reynolds number by selecting a moderate Re = 2 with =.5 and 1 and two relatively high values of Re = 1 and Re = 38, with = 3 and = 2, respectively. They performed the numerical study of flow past a translating and rotating circular cylinder solving the two-dimensional unsteady Navier Stokes euations in terms of vorticity and stream function using a third-order upwind scheme. (Kang, Choi & Lee, 1999) followed with the numerical solution of the unsteady governing euations in the primitive variables velocity and pressure for flows with Re = 6, 1 and 16 with 2.5. Their results showed that vortex shedding vanishes when increases beyond a critical value which follows a logarithmic dependence on the Reynolds number (e.g., the critical dimensionless rotation rate = 1.9 for Re = 16). (Chou, 2) worked in the area of high Reynolds numbers by presenting a numerical study that included computations falling into two categories: 3 with Re = 13 and 2 with Re = 14. Chou solved the unsteady two dimensional incompressible Navier Stokes euations written in terms of vorticity and stream function. In contrast, the work of (Mittal & Kumar, 23) performed a comprehensive numerical investigation by fixing a moderate value of Re = 2 while considering a wide interval for the dimensionless rotation rate of 5. They used the finite-element method to solve the unsteady incompressible Navier Stokes euations in two-dimensions for the primitive variables velocity and pressure [4]. (Dennis, 22) investigated the steady asymmetrical flow past an elliptical cylinder using the method of series truncation to solve the Navier-Stokes euations with the Oseen approximation throughout the flow. He found that by considering the asymptotic nature of the decay of vorticity at large distances that for asymmetrical flows it is not sufficient merely that the vorticity shall vanish far from the cylinder but it must decay rapidly enough [2]. (Kang et al., 1999) pointed out, the simulations may be started with arbitrary initial conditions. They performed a numerical study with different initial conditions, including the impulsive start-up, for Re = 1 and = 1. and the same fully developed response of the flow motion was eventually reached in all cases [4]. II. GOVERNING EQUATION and BOUNDARY CONDITIONS The applied system consists of a two dimensional infinite long circular cylinder Fig. (1), having diameter D and is rotating in a counter clockwise direction with a constant angular velocity ω. It is exposed to a constant free stream velocity of U at the inlet. The governing partial differential euations are the form of continuity and Navier Stokes or momentum euations in two dimensions for the incompressible, steady state, and laminar flow around a rotating circular cylinder [5] as below: Continuity euation: u v (1) x y x - momentum euation: 2 2 u u 1 P u u u v ν (2) 2 2 x y ρ x x y y - momentum euation: 2 2 v v 1 P v v u v (3) 2 2 x y ρ y x y The boundary conditions for the flow across a rotating circular cylinder see Fig. (1), can be written as: at the inlet boundary: u = U, v = at the exit boundary: p = On the surface of the cylinder: u = -ω D sin(θ)/2, v = -ω D cos(θ)/2, where θ 36. The boundary conditions on the surface of the cylinder can be implemented by considering wall motion: moving wall and motion: rotational for a particular rotational rate in FLUENT. The above governing euations (1, 2, & 3) when solved using the above boundary conditions yield the primitive variables, i.e., velocity u,v, and pressure p are calculated numerically. III. AERODYNAMICS CHARACTERISTICS To describe the problems must be define Reynolds number as ρ.u D Re (4) μ and dimensionless rotation rate: ω.d (5) 2U Three relevant parameters computed from the velocity and pressure fields are the pressure, skin friction, and lift coefficients, which represent dimensionless expressions of the forces that the fluid produces on the circular cylinder, these are defined, respectively, as follows [4]:

3 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 34 p - p C p (6) 1 2 ρ.u 2 τ C f (7) 1 2 ρ.u 2 L Cl (8) 1 2 ρ.u 2 Where pressure forces act normal to the surface of the cylinder and the shear stress acts tangential to the surface of the cylinder. IV. RESULTS and DISCUSSION The numerical solution of the governing system of partial differential euations is carried out through the computational fluid dynamics package FLUENT version (6.2). This computer program applies a control-volume method to integrate the euations of motion, constructing a set of discrete algebraic euations with conservative properties. The segregated numerical scheme, which solves the discretized governing euations seuentially [4]. The computational grid for the problem under consideration is generated by using a commercial grid generator GAMBIT and the numerical calculations are performed in the full computational domain using FLUENT program for varying conditions of Reynolds number and rotation rate. In particular, the O-type shown in Fig. (2), grid structure is created here and it consists of non-uniform uadrilateral elements 346 having a total of 353 nodes or grid points in the full computational domain. The grid near the surface of the cylinder is sufficiently fine to resolve the boundary layer around the cylinder [5]. The results of lift, skin friction, and pressure coefficients have been represented from FLUENT (6.2) at each values of Re of range (2, 4, 8, and 1) and varying from 1 6 in the steps of 1 with angle in polar coordinate (angular direction), as shown in Fig. (1), θ in degrees units. Also, lines of stream function at same Re & ranges. While validations of the results of lift coefficient are compared with other numerical results as shown in Tab. (1) and give a good approach and convergence. Lift coefficient of rotating cylinder with at each values of Re is represented in Tab. (2) shows C l is increase in negative direction, if change direction of rotating to clockwise then lift coefficient is positive value, observe in Tab. (2), note at each value of Re (or in each column Re is constant) wherever increase rotation rate, will increase lift coefficient clearly, but when compare between first column and second until fourth each value of lift coefficient approximately convergent or similar for same rotation rate i.e., effect of Re is not significant on lift coefficient, in additional to lift coefficient is greater than at no-rotate a absolutely according to Magnus effect. Fig. (3), shows the streamline patterns for the various pairs of Re and, the rotation of the cylinder is counterclockwise while the streaming flow is from left to right considered in this investigation. Notice that the stagnation point lies above the cylinder, in the region where the direction of the free stream opposes the motion induced by the rotating cylinder. As the dimensionless rotation rate at the surface of the cylinder increases, for a fixed Re, the region of close streamlines around the cylinder extends far from the wall and, as a conseuence, the stagnation point moves upwards. For the lowest = 3, the region of close streamlines becomes narrow and the stagnation point lies near the upper surface of the cylinder. The contours of positive and negative vorticity are presented in Fig. (5), the positive vorticity is generated mostly in the lower half of the surface of the cylinder while the negative vorticity is generated mostly in the upper half. For the dimensionless rotation rates of = 3 and 4, a zone of relatively high vorticity stretches out beyond the region neighboring the rotating cylinder for θ 9, resembling "tongues" of vorticity. Increasing, the rotating cylinder drags the vorticity so the tongues disappear and the contours of positive and negative vorticity appear wrapped around each other within a narrow region close to the surface. Based on the velocity and pressure fields obtained from the simulations for the various Re and considered [4]. While Figs. (3, 5) shown stream lines and vorticity contours are represented to purpose of comparison with Figs. (4, 6) in other numerical results (J. C. Padrino, et al) in same conditions of the flow in which give same behavior and convergence in shape approximately excepting some small differences due to different in number of mesh nodes, iteration loop to arrive convergence values, and levels of contour. In additional to lift coefficient is greater than it's value at no-rotating cylinder. Figs. (7, 8, 9, and 1) are shown pressure coefficients of flow past rotating cylinder along curve length or angle from front as in Fig. (1) at various values of Re,, in which observe wherever increasing rotation rate, pressure coefficient will increase without looking to Reynolds number. Each curve differ than one to other while maximum values of pressure coefficient in four figures are -43, -4.7, , and appear at (225 24) degree & = 6, i.e., cylinder front approximately at Reynolds number are: 2, 4, 8, and 1 respectively. Maximum value of pressure coefficient are convergent approximately, this lead to vary in Re from 2 to 1 has small effect or not significant on pressure coefficient in same time value of rotation rate change from 1 6. Further published researches don t deal with skin friction of flow past rotating cylinder and don t meet paper discuss prediction to estimate skin friction numerically past rotating cylinder just past stationary cylinder such as (E. Achenbach, 1968) [7] investigated skin friction at θ 36 over stationary cylinder at Re with smooth surface experimentally and defined three states of the flow: the subcritical, critical, and supercritical then specified separation angle in each region. In this present Figs. (11, 12, 13, and 14) shown skin friction coefficient along curve length or angle from point on cylinder surface to origin in Fig. (1) at known Reynolds number & rotation rate, observe in these figures mostly (not along curve length) increase values of skin friction with increasing of rotation rate of each case i.e., Re is known and constant except at Re = 2, skin friction at

4 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 35 = 5 has maximum value greater than it's at = 6 happen at point (15, 1.156) i.e., at back of cylinder, while another figure it's maximum values at point: (315, 1.112), (3,.667), and (3,.596) at = 6, Re = 4, 8, and 1 respectively. Commonly each curve of skin friction has five vertex or apex in various location, as behavior of curve half is repeat after θ = 18, but reversely in shape and differ in values of another half i.e., shape of skin friction coefficient curve in upper surface same as lower surface. V. CONCLUSIONS 1. comparison of the numerical results with other published researches for lift coefficient and stream lines and vorticity contours therefore give good agreement. 2. increase lift coefficient with increasing rotation rate and effect of Reynolds number is not significant or small change on lift coefficient. 3. increasing rotation rate, pressure coefficient will increase without look to Reynolds number i.e., variation in Reynolds number from 2 to 1 has small effect or not significant on pressure coefficient in same time value of rotation rate change from mostly increase values of skin friction coefficient with increasing of rotation rate at Reynolds number is eual 4, 8, and 1 except at Re = in each case skin friction coefficient has five vertex in various location, as behavior of curve half is repeat after θ = 18, but shape of skin friction coefficient in upper surface same as lower surface VI. SCOPE of the STUDY and LIMITATIONS Rotating circular cylinder application play important rule in missiles and projectiles where it's rotation add lift force as well as original lift force when existence cross wind that lead to increase in range, stability, and performance just as decrease drag force in aerodynamics fields. In automotive design, good aerodynamic consideration aims for the least drag to achieve efficiency, and also to optimize negative lift particularly in motor sport. Similarly such effort has been proven to tremendously save the fuel cost in the aviation industry. Other engineering applications of cylinder like structures such as air flow past a group of buildings or bundle of pipes in a chemical plant, where reuire reduce wind force on their side. In heat transfer, coolant flow past tubes in a heat exchanger, sea water flow past columns of a marine structure, twin chimney stacks. The flow around a rotating cylinder involves complex transport phenomenon because of many factors such as the effect of cylinder rotation on the production of lift force and moment. There are two parameters that influence this flow problem: The Reynolds number, and the rotation rate of the cylinder is non-dimensionalized uantities, the first Reynolds No. is limited flow model i.e., laminar or turbulent, while the second represented relative velocity of uniform flow and rotational cylinder. In this paper low Reynolds No. is considered to generate laminar, steady, incompressible flow, no slip without average roughness surface of cylinder. All these factors are limited this work and any cases outside this field are not satisfying assumptions of model. REFERENCES [1] John Middendorf, "CFD Modeling of Wind Tunnel Flow over Rotating Cylinder", Computation Fluid Dynamics, Professors Tracie Barber/Eddie Leonardi, May 3, 23. [2] D. B, Ingham and T. Tang, "A Numerical Investigation into the Steady Flow Past a Rotating Circular Cylinder at Low and Intermediate Reynolds Numbers", Reprinted from Journal Of Computational Physics Vol. 87, No.1, New York and London, March 199. [3] Surattana Sungnul and Nikolay Moshkin, "Numerical Simulation of Steady Viscous Flow past Two Rotating Circular Cylinders", Suranaree J. Sci. Technol. 13(3): , May 3, 26. [4] J. C. Padrino and D. D. Joseph, "Numerical study of the steady-state uniform flow past a rotating cylinder", J. Fluid Mech. (26), Vol. 557, pp , 26 Cambridge University Press. [5] Varun Sharma and Amit Kumar Dhiman, "Heat Transfer from a Rotating Circular Cylinder in the Steady Regime: Effects of Prandtl Number", Indian Institute of Technology Roorkee, Roorkee , dhimuamit@rediffmail.com, India. [6] Sanjay Mittal, S. & Bhaskar Kumar, "Flow Past a Rotating Cylinder", J. Fluid Mech. Vol. 476, , Cambridge University Press, United Kingdom, 23. [7] E. Achenbach, "Distribution of Local Pressure and Skin Friction around a Circular Cylinder in Cross-Flow up to Re = 5 16", J. Fluid Mech., Vol.34, pp , NOMENCLATURE Latin Description symbols L Lift force (N) rotation rate Re Reynolds number r radius in polar coordinate U free velocity of fluid (m/s) pressure as the radial coordinate r goes to p infinity (N/m 2 ) p local pressure (N/m 2 ) C p pressure coefficient C f skin friction coefficient D diameter of the cylinder R Radius of the cylinder C l lift coefficient U co free velocity of air (m/s) velocity component in x, y direction u, v respectively (m/s) Cartesian coordinate in horizontal direction x (m) y Cartesian coordinate in vertical direction (m) Greek Description symbols τ shear stress (N/m 2 ) α angle of attack Γ circulation (m 2 /s) ρ density of the air (kg/m 3 ) θ angle in polar coordinate (degree) ω angular velocity (rad/s) µ dynamic viscosity of the fluid (kg/m.s) v kinematic viscosity of the fluid (m 2 /s) symbols CFD Abbreviations Computational Fluid Dynamic

5 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 36 y U ω r θ x Fig. 1. laminar flow over rotating cylinder has diameter is D Table I Numerical lift coefficient of the steady state laminar flow past a rotating cylinder No. Re Present study J. C. Padrino & D. D. Mittal & Kumar [6] Joseph [4] Fig. 2. the O-type grid structure mesh

6 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 37 Table II lift coefficient with rotation rate & Reynolds No. lift coefficient C l Re = 2 Re = 4 Re = 8 Re = Fig. 3. Stream lines for various pairs of Re and [4].

7 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 38 (a) Re = 2, = 4 (b) Re = 2, = 5 (c) Re = 4, = 4 (d) Re = 4, = 5 (e) Re = 4, = 6 (f) Re = 1, = 3 Fig. 4. Stream lines for various pairs of Re and for present study.

8 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 39 Fig. 5. Vorticity contours for various pairs of Re and. The negative vorticity is shown as dashed lines. The rotation of the cylinder is counterclockwise while the streaming flow is from left to right [4]. (a) Re = 2, = 4 (b) Re = 2, = 5

9 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:3 4 (c) Re = 4, = 4 (d) Re = 4, = 5 (e) Re = 4, = 6 (f) Re = 1, = 3 Fig. 6. Vorticity contours for various pairs of Re and for present study 5 Cp Re = 2 = 1 = 2 = 3 = 4 = 5 = Fig. 7. Pressure coefficient vs. angular position at Re = 2 θ

10 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No: Cp Re = 4 = 1 = 2 = 3 = 4 = 5 = Fig. 8. pressure coefficient vs. angular position at Re = Cp Re = 8 = 1 = 2 = 3 = 4 = 5 = Fig. 9. pressure coefficient vs. angular position at Re = 8

11 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No: Cp Re = 1 = 1 = 2 = 3 = 4 = 5 = Fig. 1. Pressure coefficient vs. angular position at Re = Cf Re = 2 =1 = 2 = 3 = 4 = 5 = Fig. 11. Skin friction coefficient vs. angular position at Re = 2

12 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No: Cf Re = 4 =1 = 2 = 3 = 4 = 5 = Fig. 12. Skin friction coefficient vs. angular position at Re = 4 Cf Re = 8 =1 = 2 = 3 = 4 = 5 = Fig. 13. Skin friction coefficient vs. angular position at Re = 8

13 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No: Cf Re = 1 = 1 = 2 = 3 = 4 = 5 = Fig. 14. Skin friction coefficient vs. angular position at Re = 1