S. C. R. Dennis

Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100

Finite-difference solutions of the equations of motion for steady incompressible flow around a circular cylinder have been obtained for a range of Reynolds numbers from R = 5 to R = 100. The object is to extend the Reynolds number range for reliable data on the steady flow, particularly with regard to the growth of the wake. The wake length is found to increase approximately linearly with R over the whole range from the value, just below R = 7, at which it first appears. Calculated values of the drag coefficient, the angle of separation, and the pressure and vorticity distributions over the cylinder surface are presented. The development of these properties with Reynolds number is consistent, but it does not seem possible to predict with any certainty their tendency as R → ∞. The first attempt to obtain the present results was made by integrating the time-dependent equations, but the approach to steady flow was so slow at higher Reynolds numbers that the method was abandoned.

Laminar boundary layer on an impulsively started rotating sphere

The problem of determining the development with time of the flow of a viscous incompressible fluid outside a rotating sphere is considered. The sphere is started impulsively from rest to rotate with constant angular velocity about a diameter. The motion is governed by a coupled set of three nonlinear time‐dependent partial differential equations which are solved by first employing the semi‐analytical method of series truncation to reduce the number of independent variables by one and then solving numerically a finite set of partial differential equations in one space variable and the time. The calculations have been carried out on the assumption that the Reynolds number is very large. The physical properties of the flow are calculated as functions of the time and compared with existing solutions for large and small times. A radial jet is found to develop with time near the equator of the sphere as a consequence of the collision of the boundary layers.

Flow past an impulsively started circular cylinder

An accurate method is described for integrating the Navier-Stokes equations numerically for the time-dependent flow past an impulsively started circular cylinder. Results of integrations over the range of Reynolds numbers, based on the diameter of the cylinder, from 5 to ∞ are presented and compared with previous numerical, theoretical and experimental results. In particular, the growth of the length of the separated wake behind the cylinder has been calculated for R = 40, 100 and 200 and is found to be in very good agreement with the results of recent experimental measurements. The calculated pressure distribution over the surface of the cylinder for R = 500 is also found to be in reasonable agreement with experimental measurements for the case R = 560. For Reynolds numbers up to 100 the equations were integrated until most of the features of the flow showed a close approximation to steady-state conditions. The results obtained are in good agreement with previous calculations of the steady flow past a circular cylinder. For R > 100 the integrations were continued until the implicit method of integration broke down by reason of its failure to converge. A secondary vortex appeared on the surface of the cylinder in the case R = 500, but for higher Reynolds numbers, including the case R = ∞, the procedure broke down before the appearance of a secondary vortex. In all cases the flow was assumed to remain symmetrical.

The steady flow due to a rotating sphere at low and moderate Reynolds numbers

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates ( r , θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations. The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.

Calculation of the steady flow past a sphere at low and moderate Reynolds numbers

The steady axially symmetric incompressible flow past a sphere is investigated for Reynolds numbers, based on the sphere diameter, in the range 0·1 to 40. The formulation is a semi-analytical one whereby the flow variables are expanded as series of Legendre functions, hence reducing the equations of motion to ordinary differential equations. The ordinary differential equations are solved by numerical methods. Only a finite number of these equations can be solved, corresponding to an approximation obtained by truncating the Legendre series at some stage. More terms of the series are required as R increases and the present calculations were terminated at R = 40. The calculated drag coefficient is compared with the results of previous investigations and with experimental data. The Reynolds number at which separation first occurs is estimated as 20·5.

Unsteady flow past a rotating circular cylinder at Reynolds numbers 10 3 and 10 4

The unsteady flow past a circular cylinder which starts translating and rotating impulsively from rest in a viscous fluid is investigated both theoretically and experimentally. The theoretical study is based on numerical solutions of the two-dimensional unsteady Navier-Stokes equations, while the experimental investigation is based on visualisation of the flow using very fine suspended particles. The object of the study is to examine the effect of increase of rotation on the flow structure. There is excellent agreement between the numerical and experimental results for all speed ratios considered, except in the case of the highest rotation rate. Here three-dimensional effects become more pronounced in the experiments and the laminar flow breaks down, while the calculated flow starts to approach a steady state. For lower rotation rates a periodic structure of vortex evolution and shedding develops in the calculations which is repeated exactly as time advances. Another feature of the calculations is the discrepancy in the lift and drag forces at high Reynolds numbers resulting from solving the boundary-layer limit of the equations of motion rather than the full Navier-Stokes equations. Typical results are given for selected values of the Reynolds number and rotation rate.

Steady Laminar Forced Convection from a Circular Cylinder at Low Reynolds Numbers

Comparison of experimental data for the heat transfer between a circular cylinder and a steady stream of viscous, incompressible fluid at low Reynolds numbers with the only available theoretical solution, that based on Oseen type linearization of the heat transfer equation, indicates considerable discrepancies. This question can only be resolved by obtaining solutions of the heat transfer equation based on the correct velocity distribution in the field of flow according to the full Navier‐Stokes equations. This problem is considered for a range of Reynolds numbers from 0.01 up to 40. Solutions of the equation for forced heat convection are obtained using numerical methods. The mean and local Nusselt numbers are calculated and compared with available experimental data and correlations. The results are found to compare favorably. At the upper end of the Reynolds number range the solutions are examined to see if their trend checks with boundary‐layer results at high Reynolds numbers and with some recent pred...

Compact h4 finite-difference approximations to operators of Navier-Stokes type

A method of obtaining compact finite-difference approximations of h4 accuracy to operators of Navier-Stokes type is considered. The basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time. Four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods. Comparisons with previous results and with the results of h2-accurate computations are made.

Finite-difference methods for calculating steady incompressible flows in three dimensions

Abstract A new stable numerical method is described for solving the Navier-Stokes equations for the steady motion of an incompressible fluid in three dimensions. The basic governing equations are expressed in terms of three equations for the velocity components together with three equations for the vorticity components. This gives six simultaneous coupled second-order partial differential equations to be solved. A finite-difference scheme with second-order accuracy is described in which the associated matrices are diagonally dominant. Numerical results are presented for the flow inside a cubical box due to the motion of one of its sides moving parallel to itself for Reynolds numbers up to 100. Several methods of approximation are considered and the effect of different discretizations of the boundary conditions is also investigated. The main method employed is stable for Reynolds numbers greater than 100 but a finer grid size would be required in order to obtain accurate results.

Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers

Abstract Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.