Steady magnetohydrodynamic flow past a circular cylinder

The steady two-dimensional, viscous, electrically conducting flow around a circular cylinder is investigated. The flow and magnetic field are uniform and parallel at large distances from the cylinder. The equations and boundary conditions are derived for arbitrary values of R, R_mand β, where R is the Reynolds number, R_m the magnetic Reynolds number and β, the ratio of the square of the Alfvén speed to the square of the main stream speed. Because of the large number of parameters involved, the numerical solution is restricted to R = 40, R_m = 1 and infinity and 0 ? β ? 4. Also the cylinder is taken to be a perfect conductor, this avoids having to compute the magnetic field within the cylinder. The numerical computations for the non-magnetic case, i.e. β = 0, are presented and are found to be in good agreement with existing results. The effect of increasing the strength of the magnetic field (i.e. increasing β) on the drag coefficient, the size and position of the standing vortex and the increasing effect of the upstream propagation of disturbances are examined.     

Steady viscous flow past a tapered cylinder

The three-dimensional nature of the viscous flow past a linearly tapered circular cylinder is examined at low Reynolds numbers. The numerical solution of the unsteady Navier–Stokes equations converges to a steady state. The primary flow in planes perpendicular to the cylinder axis is practically indistinguishable from the two-dimensional flow past a uniform cylinder. A secondary spanwise flow is observed in the stagnation zone going from the wide end towards the narrow end, whereas a secondary motion on the rear side goes in the opposite direction. In spite of this secondary flow, the length of the separation zone varies linearly with the local Reynolds number.     

Steady flow past an elliptic cylinder inclined to the stream

The properties of steady two-dimensional flow past an elliptic cylinder inclined to the oncoming stream are investigated for small to moderate values of the Reynolds number for which good accuracy can be assured. The solutions are based on a numerical method of solution of the Navier-Stokes equations for incompressible fluids which ensures that all the correct conditions of the problem are satisfied. In particular, the solution is carried out in such a way that the vorticity decays rapidly enough at large distances from the cylinder for the lift and drag on the cylinder to be finite. Results are presented for the variation of lift, drag and streamline patterns with inclination and Reynolds number. Two elliptic cylinders (based on their minor-to-major axes ratio) are considered. For an elliptic cylinder with minor-to-major axes ratio 1:5, results are obtained for Reynolds numbers up to 40 and inclination varying from zero to 90°. Streamline plots for these results show a development of the solution from asymmetric flow at zero inclination (with no separation), through asymmetric flows with increasing inclination (with either no separation, separation with a single recirculating region, or separation with two recirculatory regions) to the symmetric flow at 90° incidence (with two counter rotating vortices). Of interest are asymmetric steady-state results which contain two recirculatory regions trailing the cylinder, one attached and one unattached to the cylinder. Results are also obtained for a second elliptic cylinder with minor-to-major axes ratio 1:10 at Reynolds numbers 15 and 30, inclination 45°. These results are found to be in good agreement with corresponding unsteady results taken to long times (which are tending to a steady state).     

Steady non–Newtonian flow past a circular cylinder: a numerical study

Summary. The steady and incompressible flow of non–Newtonian fluids past a circular cylinder is investigated for power law indices n between 0.2 and 1.4, blockage ratios of 0.037, 0.082 and 0.164, and the Reynolds numbers Re of 1, 20 and 40, using a stream function/vorticity formulation. The governing field equations have been solved by using a second-order accurate finite difference method to determine the drag coefficient, wake length, separation angle and flow patterns, and to investigate their dependence on power law index, blockage ratio and Reynolds number. The results reported here provide fundamental knowledge on the dependence of engineering flow parameters on blockage ratio and power law index, and further show that the effects on stream line and iso-vorticity patterns which result from an increase in the blockage ratio are similar to those which result from a decrease in the power law index.     

Steady circulatory flow of a second grade fluid about a rotating porous cylinder

An exact solution for the circulatory flow of an incompressible second grade fluid about a rotating porous cylinder is given. The solution is expressed in terms of the confluent hypergeometric functions and it is valid for all values of the cross-Reynolds number, the elastic number and the ratio of the circulation at infinity to that on the surface of the cylinder. The velocity, the vorticity and the torque exerted by the fluid on the cylinder are calculated. It is shown that there are some discrepancies between the results obtained by the exact solution and those obtained by the perturbation solution which is valid for small values of the elastic number.     

Spiral motion of a cylinder in a power-law non-newtonian fluid

A study is made of steady-state flow in a power-law system in the presence of double shear caused by spiral motion of a circular cylinder.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 5, pp. 810–814, May, 1970.     

Analytical solutions of hydromagnetic boundary-layer flow of a non-Newtonian power-law fluid past a continuously moving surface

Summary The boundary-layer flow of a power-law non–Newtonian fluid over a continuously moving surface in the presence of a magnetic field B(x) applied perpendicular to the surface has been investigated. An analytical solution is obtained and compared with the numerical solution of the resulting non linear ordinary differential equation. The effects of the Stewart number (N) and the power law-index (n) on the velocity profiles and the skin-friction are studied.     

Stokes flow past a smooth cylinder

Summary Ranger's solution of the two-dimensional Stokes flow past a smooth body is analyzed in detail. The body can be made convex or concave depending on the choice of its two parameters and concavity is found to be necessary but not sufficient for separation to occur. A relationship exists between the formation of a Stokesian wake and the curvature at the concave end of the body. Particular attention is given to the case when the body degenerates to a circular arc. The two-dimensional Stokes flow past a circular arc is strikingly similar to the axisymmetric Stokes flow past a spherical cap.     

Uniform shear flow past a circular cylinder

Summary This paper solves the two-dimensional problem of the unsteady shear flow of a viscous incompressible fluid past a circular cylinder. The flow is calculated using two methods. The first takes the form of a double series solution where an expansion is carried out in powers of the time, t, and in powers of the parameter where R is the Reynolds number. This approximate analytical solution is valid for small times following the start of the motion and for large Reynolds numbers. The second method involves a spectral-finite difference procedure for numerically integrating the full Navier-Stokes equations expressed in terms of a stream function and vorticity. Our results demonstrate that for small times and moderately large R the two methods of solution are in good agreement. Results are presented for a wide range of Reynolds numbers. Comparisons with previous studies are also carried out and discussed.     

Stability of viscous flow past a circular cylinder

A spectral method which employs trigonometric functions and Chebyshev polynomials is used to compute the steady, incompressible laminar flow past a circular cylinder. Linear stability methods are used to formulate a pair of decoupled generalized eigenvalue problems for the growth of symmetric and asymmetric (about the dividing streamline) perturbations. We show that, while the symmetric disturbances are stable, the asymmetric perturbations become unstable at a Reynolds number about 40 with a Strouhal number about 0.12. The critical conditions are found to depend on the size of the computational domain in a manner similar to that observed in the laboratory.     

The initial oscillatory flow past a circular cylinder

This paper presents results obtained from an initial approximation for the flow around a circular cylinder in two-dimensional oscillating flow. The analysis is developed in terms of the scalar vorticity and stream function. An expansion in powers of time from the start of the motion is obtained using an exact analysis which extends the results of boundary-layer theory by taking into account corrections for finite Reynolds number. The time development of the physical properties of the flow are determined both by means of analytical expressions and by an accurate numerical procedure. The surface pressure, drag and surface vorticity are calculated and various estimates of the time of separation and the distance moved in this time are obtained. The phenomenon of steady streaming is not considered in this paper since the time of validity of the expansions is small. The agreement between the analytical and numerical results at small times is excellent.     

Shear flow past a square cylinder near a wall

A numerical study on the wake behind a square cylinder placed parallel to a wall has been made. The cylinder is considered to be within the boundary layer of the wall, so that the outside flow is taken to be due to uniform shear. Flow has been investigated in the laminar Reynolds number (based on the cylinder height) range. The interaction of wall boundary layer on the vortex shedding at Reynolds number up to 1400.0 has been investigated for cylinder to wall gap height 0.5 and 0.25 times the cylinder height. The gap flow between the cylinder and wall during a period of vortex shedding has been obtained. The governing unsteady Navier–Stokes equations are discretised through the finite volume method on staggered grid system. An algorithm SIMPLE has been used to compute the discretised equations iteratively. Our results show that at the gap height 0.5 times the cylinder height the vortex shedding occurs at a Strouhal number greater than for an isolated cylinder. Vortex shedding suppression occurs and wake becomes steady up to a certain value of Reynolds number at gap height 0.25 time the cylinder height. At higher Reynolds number the formation of a single row of negative vortices behind the cylinder when it is in close proximity to wall is noteworthy. Due to the shear, the drag experienced by the cylinder is found to decrease with the reduction of gap height.     

Stability of the circular couette flow of a power-law fluid

The stability of the circular Couette flow of a non-Newtonian power-law fluid is analyzed in the small-gap approximation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 4, pp. 719–724, April, 1973.     

Steady and unsteady forced convection past an inclined elliptic cylinder

Summary Discussed in this paper is the two dimensional problem of forced convection past an inclined elliptic cylinder. Both the steady state and unsteady cases have been considered for moderate Reynolds numbers, R, in the range 40R70 with a Prandtl number Pr=1. The numerical results show that there is good agreement between the steady and limiting unsteady heat transfer quantities and Nu which are the average and local Nusselt numbers respectively. The isotherm plots show marked differences as R is increased due to vortex shedding which is known to take place. Lastly, the early development for high Reynolds number flow past a harmonically oscillating and translating thin elliptic cylinder is addressed to illustrate the effect of rotation on the convective heat transfer process. The main role of rotation is to cause fluctuations in the time variation of and also to distort the thermal tail region emanating from the rear tip of the ellipse.     

Magnetohydrodynamic flow for a non-Newtonian power-law fluid having a pressure gradient and fluid injection

Summary The nonlinear partial differential equation of motion for an incompressible, non-Newtonian power-law fluid flowing over flat plate under the influence of a magnetic field and a pressure gradient, and with or without fluid injection or ejection, is transformed to a nonlinear third-order ordinary differential equation by using a stream function and a similarity transformation.The necessary boundary conditions are developed for flow with and without fluid injection (or ejection), and a solution for four different power-law fluids, including a Newtonian fluid, is presented.The controlling equation includes, as special cases, the Falkner-Skan equation and the Blasius equation.     

Nonisothermal generalized couette flow of a fluid with a power-law rheological characteristic

The nonisothermal steady flow of a power-law fluid between two parallel plates is analyzed for different kinds of temperature boundary conditions and, moreover, with energy dissipation taken into account. It is assumed that the fluidity of the substance is a linear function of the temperature.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.22, No. 5, pp. 872–880, May, 1972.     

Wall effects on separation of flow past a circular cylinder

Abstract

The nature of the flow past a circular cylinder has been a classical problem, raising many questions concerning the various wake phenomena that have been observed. This study focused on the critical Reynolds number for the onset of the recirculation region for a flow restricted in a channel. The influence of the bounded walls is examined. The trend is that larger critical Reynolds numbers are accomplished with larger values of blockage ratio (defined as the ratio of cylinder diameter to the channel width). Furthermore, as the blockage ratio tends to zero, the trend seems to imply that the critical Reynolds number approaches the experimental value for flows in an unbounded domain.     

Slow viscoelastic fluid flow past a rotating sphere

The problem of axially symmetric flow of a particular type of non-Newtonian fluid past a rotating sphere due to a uniform stream at infinity is investigated. The presence of a region of reversed flow is found under certain conditions depending on the angular velocity of the sphere, the speed of the uniform stream and radius of the sphere. This region which is attached to the rear portion of the sphere is found to depend strongly on the viscoelasticity of the fluid. The vortex is seen to move towards the sphere as the viscoelastic parameter increases while the other parameters are kept fixed. As this viscoelastic parameter approaches a critical value, the vortex is found to disappear.     

Analysis of initial flow past a rotating and translating circular cylinder

Abstract

Matched asymptotic expansion is carried out for initial flow past an impulsively started rotating and translating circular cylinder. The study expands upon previous similar analysis to include the effect of rotation with specific emphasis on the evaluation of the lift, drag, moment coefficients and the pressure distribution along the cylinder surface. Three different approaches to evaluate the force, moment coefficients and surface pressure are shown to require different orders of asymptotic expansion as far as leading behaviors are concerned. Some indication of the present analysis about how to numerically implement these different approaches is also discussed.     

Steady forced flow of a micropolar fluid against a rotating disc

The steady forced flow of an incompressible micropolar fluid against a rotating disc is considered. The flow due to a rotating disc in an infinite fluid which is at rest and the axisymmetric stagnation flow on a flat plate are particular cases of this present problem. The equations of motion are solved numerically using the Gauss-Seidel iterative procedure and the Simpson's rule. The results are given in tabular form and compared with the known results for a Newtonian fluid.