| dc.description.abstract | The problem of flow around a circular cylinder has long attracted the attention of many research workers because of its many practical applications and exciting mathematical features. This thesis contains the results of numerical studies of some non-linear problems based on the Navier-Stokes equations in fluid mechanics. The flow of a viscous incompressible fluid around a circular cylinder has been examined under various initial and boundary conditions. In the introductory chapter, a brief survey of the theoretical and experimental investigations on some of these problems by other authors is given. This is followed by a detailed summary of the work done in this thesis.
Some authors, Thom, Kawaguti, Keller and Takami have proposed finite-difference schemes for solving the problem of steady flow past a circular cylinder. In Chapter 2, a mathematical formulation along with a numerical method is presented to study the transient problem, in order to examine the existence of limiting steady-state solutions or of the Karman vortex street. This numerical method is subsequently used to get the solutions of non-stationary problems based on the full Navier-Stokes equations under various initial and boundary conditions. The equations of motion have been used in terms of polar coordinates. The stream function and vorticity function are introduced as dependent variables. Since these functions vary most rapidly near the surface of the cylinder, a very fine mesh in this region of the flow field has to be used, and this is achieved by using an exponential transformation. The time derivative is replaced by its forward difference equivalent, and the space derivatives by their central difference equivalents. Thus, the equations of motion are reduced to finite difference equations which are then solved to get transient solutions with the help of a high-speed digital computer. The stability criterion for the difference equations of motion is discussed for choosing a suitable time step for computation when the Reynolds number and the mesh size are specified. Over-relaxation techniques have been applied for accelerating the rate of convergence of the iterative method.
Chapter 3 gives the solution of the transient problem on the assumption that the fluid starts moving with a constant velocity at the initial time at infinity. The investigation is carried out for Reynolds numbers up to 60. Kawaguti and Jain obtained the numerical solutions of the problem at Reynolds numbers up to 50 by assuming the symmetry condition of the flow. The flow in the wake of the cylinder may not satisfy the symmetry condition after the critical Reynolds number at which the wake becomes unstable. Therefore, the symmetry is removed to estimate the value of the critical Reynolds number. The numerical results have been obtained by computing the flow in the entire two-dimensional region. The limiting steady-state solutions are obtained up to the Reynolds number 60. The corresponding values of the flow characteristics are plotted. At R = 60, the length of the standing vortex is 8.35, and the total drag, 1.3786. The flow pattern, equi-vorticity lines, pressure distribution on the surface, vorticity distribution on the surface, are plotted and are found to be in good agreement with the other theoretical and experimental results.
Greenspan and Schultz have proposed a numerical method for computing flows at high Reynolds numbers. The question of whether limiting steady-state solutions exist at high Reynolds number has been examined systematically by Jain and Sankara Rao. Chapter 4 deals with the numerical solutions of the problem at high Reynolds numbers. The process of vortex shedding by computing the unsteady flow around a circular cylinder at R = 100 and 200 is shown. This is achieved by introducing artificial vorticity within the wake at one instant of time. The suppression process of vortex shedding is also shown at R = 100 and 200 by applying artificial boundary conditions along the line of symmetry in the wake of the cylinder. The effect of artificial boundary conditions amounts to the use of a splitter plate in the wake of the cylinder. It is known experimentally that the use of a splitter plate in the wake results in the suppression of vortex shedding. The dependence of the flow pattern on time, vorticity and pressure distribution on the surface of the cylinder, and the dependence of the drag on the Reynolds number and time, are shown with the help of figures.
In Chapter 5, the effect of time-dependent fluctuations in the magnitude of the oncoming stream velocity on the flow past a circular cylinder at Reynolds numbers 20, 30 and 40 has been numerically studied. The boundary conditions at infinity are taken as (in non-dimensional units):
u=1+?cos??t, ?v=0,u = 1 + \delta \cos \Omega t,\; v = 0,u=1+?cos?t,v=0,
where (u, v) are components of velocity along the coordinate axes. It is found that the drag coefficient has reduced, while the angle of separation and the length of the standing vortex has increased when compared with the case of uniform flow at infinity. The limiting steady cycle exists for R = 30 and 40. Various flow parameters have been tabulated for a comparison of the numerical results. Streamlines, equi-vorticity lines, total drag, pressure distribution, vorticity distribution and other flow characteristics are plotted and compared with the corresponding values of the uniform flow.
In Chapter 6, the flow of a viscous incompressible fluid past a circular cylinder, when the cylinder is held stationary and the fluctuating disturbances are superimposed upon the direction of the velocity of the external stream, has been numerically investigated. Two problems have been solved at R = 40. In Problem A, the periodicity of the external stream velocity is generated by
A/rsin??(1+?cos??).A/r \sin \theta (1 + \epsilon \cos \theta).A/rsin?(1+?cos?).
In Problem B, the flow past a row of circular bars parallel to the circular cylinder is considered, so that
rsin??+?1sin?(?2).r \sin \theta + \epsilon_1 \sin(\theta_2).rsin?+?1sin(?2).
It is found that the directional fluctuations exert a strong influence on the flow pattern in the shear layer behind the cylinder. The value of the total drag is observed to increase for low-frequency fluctuations with relatively increasing positive values of the amplitude ?, and to decrease for the same low frequency at relatively high negative values of ??. The flow pattern in the wake changes from the flow with alternate shedding of vortices to a steady one in which the separation streamline reattaches to the line of symmetry giving rise to two closed recirculatory regions attached to the cylinder. The greatest effects on the values of flow characteristics like the total drag, the angle of separation and the length of the standing vortex have been observed for low-frequency space fluctuations. The length of the standing vortex grows with the drag coefficient. Figures are drawn to show the flow patterns and the variations of the flow characteristics. Results are compared with the case of the uniform flow.
Since the results for the flow past a bluff body are of great importance at high Reynolds numbers R, an attempt is made to modify the numerical scheme described in Chapter 2 to work at high R, and is discussed in Chapter 7. An appropriate transformation has been used, by which the mesh size in the vicinity of the body becomes very fine. This has enabled us to obtain numerical results of the problem at high R = 2000, which are compared with the available experimental results and found to be in reasonable agreement. Accurate analytical solutions are not available in the literature for high Reynolds numbers. Fluctuations in the numerical values of the drag are observed. This phenomenon is explained by the shedding of vortices which causes the pressure to fluctuate in the downstream vicinity of the cylinder. The minimum total drag is found to be 1.17 during the first 100 time steps of calculations with an amplitude of total drag fluctuations of about 0.06. The calculated value of the drag due to friction is about 10% of the total drag. Figures have been drawn to show the flow pattern and variations of flow characteristics.
In the final chapter, the dynamical behaviour of a system of parallel line vortices in an inviscid fluid is studied numerically. The initial configuration of the system is assumed to be such that the points of intersection of the line vortices with a plane normal to the vorticity form a regular polygon. The numerical experiments show that the vortex polygon which is unstable according to linearised theory is rearranged due to non-linear interactions among the line vortices in such a way as to produce a more or less uniform distribution of vortices inside the fluid with an approximately constant mean separation. The average angular velocity of the rotation of the vortex lines about the instantaneous centroid of the vortex system remains approximately constant. These results agree with the conjecture of Raja Gopal. The stability of this system against small disturbances is also studied. It is shown that a polygon of more than seven vortices is always unstable whether the system is bounded or unbounded by a coaxial cylinder. | |
