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This paper deals theoretically with the two-dimensional flow driven by a symmetrical Joukowski profile oscillating harmonically along its axis of symmetry in an unbounded viscous fluid otherwise at rest. Slow steady secondary flow which occurs outside the boundary layer around the profile is studied in detail. It is found that a uniform flow is induced at infinity owing to the asymmetry of the profile in the direction of oscillation.

The basic relations governing viscous flow of liquids with variable transport properties is considered. A dimensional analysis of the problem is given, and the basic equations of generalizedNewtonian flow through a pipe are derived. A perturbation solution is given for the special case of aNewtonian liquid with pressure and temperature dependent viscosity and expressions for the velocity components, pressure distribution and flow rate are obtained for both isothermal and adiabatic wall conditions. It is shown that pressure and viscous heating effects may result in apparently nonNewtonian behaviour.

This paper deals theoretically with the two-dimensional flow driven by a circular cylinder which performs unharmonically translational oscillation in an unbounded viscous incompressible fluid otherwise at rest. Slow steady secondary flow occurs outside the oscillatory boundary layer around the cylinder, provided that the non-dimensional amplitude of the oscillation e and the boundary layer thickness are both extremely small. It is found that induced steady flow of O(e) is symmetric with respect to the direction of the oscillation and vanishes at infinity; however, in O(e 2 ) asymmetric steady streaming is induced in the main field and remains at infinity as a uniform flow in the direction of oscillation, due to the effect of the asymmetry in the motion of the circular cylinder.

Numerical solutions for laminar incompressible fluid flows past an abruptly started elliptic cylinder at 45° incidence are presented. Various finite-difference schemes for the stream-function/vorticity formulation are used and their merits briefly discussed. Almost steady-state solutions are obtained forRe= 15 and 30, whereas forRe= 200 a Kármán vortex street develops. The transient period from the start to the steady or quasi-steady state is investigated in terms of patterns of streamlines and lines of constant vorticity and drag, lift and moment coefficients.

We obtain the solutions, under the Oseen and Boussinesq approximations, for the flow field disturbance due to a line singularity in an otherwise uniform, horizontal, inviscid, incompressible flow of a vertically stratified fluid. The results obtained show no upstream influence for those singularities across which [xdtri ]2ψ + (N/U)2ψ is continuous. Doublet and vortex singularities are examples of these. Uniform flows past doublets and vortices are considered for a range of internal Froude numbers, including the calculation of the pressure distributions and drag for the doublet. An application of the vortex solution to flows in the β-plane is discussed.

The dynamic equation with the generalized Maxwell model is solved numerically in moderate Reynolds number range. The results of calculation reveal the effect of elasticity in cross flow around a circular cylinder. The electrochemical method is used for the measurement of Sherwood number in cross flow of dilute polymer solutions. Comparison of calculated values with the experimental results shows good agreement.

The stability of flow of a viscous incompressible fluid contained between a stationary outer sphere and rotating inner sphere is studied theoretically and experimentally. Previous theoretical results concerning the basic laminar flow (part 1) are compared with experimental results. Small and large Reynolds number results are compared with Stokes-flow and boundary-layer solutions. The effect of the radius ratio of the two spheres is demonstrated. A linearized theory of stability for the laminar flow is formulated in terms of toroidal and poloidal potentials; the differential equations governing these potentials are integrated numerically. It is found that the flow is subcritically unstable and that the observed instability occurs at a Reynolds number close to the critical value of the energy stability theory. Observations of other flow transitions, at higher values of the Reynolds number, are also described. The character of the stability of the spherical annulus flow is found to be strongly dependent on the radius ratio.

When air blows over a closed channel of water a steady surface velocity develops, as part of a flow extending to the channel bed. With concentrated solutions of surfactants an entirely different flow is produced. The surface velocity us increases with the downwind distance x as us∝x13, and the flow is confined to a boundary layer adjacent to the surface. The boundary layer flow is a new type, but is related to that generated by a solid surface moving through a quiescent fluid. The characteristics of this flow are determined experimentally and theoretically, and the importance of surface tension gradients to its genesis and maintenance is developed.

Foppl’s stability analysis (1913) of the equilibrium positions of a line vortex pair in the downstream neighbourhood of a circular cylinder in the two-dimensional potential flow of an incompressible fluid moving with uniform speed at infinity has been corrected recently by Smith (1973). Smith shows that it is only for small symmetric displacements from any one pair of points on the locus of possible rest positions which is known as the Foppl curve there is neutral spatial stability, all other displacements being unstable. This correction now makes the results of the stability analysis consistent with the work of Dolaptchiev and Sendov (1959) who determined the trajectories followed by such a vortex pair when constrained so as to move in a wholly symmetric motion. Dolaptchiev and Sendov obtained the analytical solution of the differential equations for the completely symmetric motion, but for motion other than symmetric no simple solutions of this sort appear to exist although the vortex trajectories for any case can be deduced numerically without difficulty.

Conventional boundary-layer theory cannot be applied when the fluid velocity outside the layer changes direction, and the leading edge of a finite body changes ends. In this paper an approximate method for examining the details of the boundary layer during a single flow reversal (occurring at t = 0) is described. It is based on the expectation that (a) long before reversal (t < -t1), there will be a quasi-steady boundary layer appropriate to flow in one direction; (b) long after reversal (t > t2) there will be a quasi-steady boundary layer appropriate to flow in the opposite direction, and (c) in between there will be a period of pure diffusion. The method is applied to a simple heat-transfer problem, in which a fluid of thermal diffusivity D flows with uniform velocity U = At over the plane y = 0; the strip 0 < x < L of the plane is maintained at temperature T1, while the restof the plane and the fluid far away have temperature T0. The approximate solution is compared with an exact solution of the boundary-layer equation, and is shown to give an accurate prediction of the heat transfer as a function of time. The boundary-layer approximation itself breaks down in regions of length O(D2/3A−1/3) near the ends of the heated strip, as usual; it also breaks down in the neighbourhood of the point x = 1/2At2, t > 0, at which the influence of the new leading edge is first felt after flow reversal. In a solution of the full equation, this region is examined in detail, and boundary-layer theory is shown to be sufficiently accurate for the calculation of heat transfer.

A new correlation for two-phase wall friction is proposed derived from analytical considerations and based upon an annular flow model. In spite of this basis in annular flow the correlation is consistent for all flow patterns in vertical two-phase flow.

The problem considered here is to study the MHD effects on the steady flow of an incompressibe viscous conducting fluid through two concentric porous non-conducting infinite circular cylinders, rotating in various ways with uniform angular velocities in presence of a radial magnetic field. It is supposed that the rate of suction at the inner cylinder is equal to the rate of injection at the outer.

An approximate method is derived for obtaining the flow in magnetic boundary layers with aligned fields which were discussed by Stewartson (1965). On comparing with exact, computed solutions the method is found to be accurate when applied to forward stagnation point flow for the whole, allowable range of the magnetic field strength parameter. It is then applied to the layer that develops from the forward stagnation point on a circular cylinder placed in a uniform stream and values obtained for the point at which it separates from the surface.

The purpose of this work was to examine the high speed flow of viscoplastic fluid past a circular cylinder. Included in this work were measurements of the time-averaged wake length and the pressure and temperature distributions over the cylinder surface. The time-averaged wake length was very characteristic of viscoplasticity: The viscoplastic flow showed quite longer wake length than theNewtonian flow under certain conditions. It was found that there were the regions of stationary fluids adjacent to a cylinder and that they obstructed the heat transfer between the cylinder and fluid very much.

Potential flow theory predicts: (1) the added mass of a cylinder near a plane boundary is independent of the direction of the flow and more than twice as large as that of the same cylinder far away from the boundary; (2) a possible failure mechanism of alternating positive and negative lift force, which is proportional to horizontal velocity squared, for pipelines resting on the bottom boundary and subjected to waves and currents; and (3) the force from convective acceleration of the ambient flow can be as large as 30% of total acceleration force for waves. Laminar boundary-layer theory suggests that a phase shift exists between maximum velocity and maximum drag force for cylinders subjected to waves. Laboratory wave force experimentation confirms the aforementioned theoretical findings 1 and 2.

On the basis of a detailed analysis of irrotational compressible flow a model problem describing subsonic stream fields is formulated.

A flow model has been developed to study the flow development after reattachment with supersonic external streams. Special attention is given to the pressure difference across the viscous layer, and it is suggested that such a flow redevelopment can be treated as a relaxation of this pressure difference. Upon correlating the pressure difference with a slope parameter of the velocity profile, the system of equations governing the flow would produce a saddle point singularity corresponding to the fully rehabilitated asymptotic flow condition. A method of calculation for this flowfield, in conjunction with the matching of the upstream flow, has been derived and is discussed. Samples of calculations are also presented. Reasonably good agreement with experimental data has also been observed.

The paper presents the results of some calculations aimed at studying the characteristics of turbulent boundary layers in unsteady flow. The calculations were performed using the method of Singleton and Nash, which is based on the turbulent kinetic energy equation. Emphasis is placed on the approach to zero wall stress, and the results show the extent to which onset of this condition can be delayed by the effects of time dependence. The delay is not associated merely with alleviation of the pressure gradients, and quasi-steady methods cannot readily be used to calculate this type of flow.

A new method of solving numerically the Navier-Stokes equations for time dependent flow is presented. The velocity components and pressure are used as the main variables, but in order to ensure continuity on a global scale, the Stokes stream function is also employed. The solution process is formulated in cylindrical polar co-ordinates and both implicit and explicit versions are discussed. Time dependent development in the entrance region of a cylindrical pipe is investigated using the method and several interesting flow features are predicted.

A method is described for the solution of transient, incompressible viscous flow in two dimensions. The dependent variables, stream function and vorticity, were approximated over each triangular element using linear interpolation functions. This approximation reduces the problem to a set of matrix equations whose term involving derivatives of time is the mass matrix. The lumping of this matrix together with the application of Euler integration scheme produces an efficient method of solution. Once the nodal values of the stream function are known the velocities and pressure can be computed. As an application a study of the vortex street development behind a rectangular obstruction is described. The flow has been impulsively accelerated to a constant speed in a channel of finite width. The Reynolds number range investigated is between 20 and 100.

This paper is devoted to the theoretical analysis of steady and unsteady boundary layer problems for non-Newtonian power-law fluid flow using a new variational principle ofHamilton's type. The standard method of variational calculus in the form of partial integration is a basic tool for obtaining approximative solutions. The main characteristic of the variational principle developed here is that all basic rules of variational calculus are preserved. The results are found to be in good agreement with those obtained by other authors. Several examples of practical importance, such as steady flow around a flat plate, a wedge and a circular cylinder as well as impulsive motion of a flat plate and a circular cylinder are considered in detail.

The Navier-Stokes equation of motion for two-dimensional, viscous, steady-state incompressible flow past an infinitely long circular cylinder was solved by numerical techniques for Reynolds numbers between 0.1 and 50. From the streamfunction and vorticity fields the pressure at the cylinder surface, the pressure drag, and the frictional drag were computed, and from the latter two the total drag on the cylinder was derived. The values found for the drag compared well with the best theoretical and experimental values reported in literature, suggesting that our flow fields were sufficiently accurate. These flow fields were used to determine the hydrodynamic interaction between simple columnar ice crystals idealized as circular cylinders of finite length L′, of radius aL′, and of Reynolds number NRe,L (67.1≤L′≤2440 µm; 23.5≤ aL′≤146.4 µm;0.2≤NRe,L

Finite-difference solutions of the Navier-Stokes equations are given in stream function-vorticity formulation for a series of Reynolds numbers and for varied positions of the cylinder in the channel. For flows in a doubly connected region, the value of the stream function on the body cannot always be given in advance. A new iteration scheme to solve such a problem is proposed for getting the proper solution by imposing a condition that the pressure is single-valued. The force and the moment of force acting on the cylinder are computed and their dependence on the Reynolds number as well as on the position of the cylinder is studied. The numerical errors caused by the existence of corners on the boundary are checked for Stokes flow by comparison with exact solutions, and their dependence on the mesh length is shown numerically.

A technique is presented for the numerical solution of the unsteady Navier-Stokes equation for laminar flow through an orifice within a pipe. The solution is accomplished through the rearrangement of the equation of motion into a vorticity transport equation (VTE) and a definition-of-vorticity equation (DVE) which are solved by an implicyt numerical method. An initial series of studies was performed to analyze flow development at upstream and downstream infinity for the case of constantly increasing flow until a Reynolds number of 5 was reached followed by a period of constant flow until steady flow was approached. The solution during this series of studies never failed to produce convergent results, although a damped instability was observed when very large time increments were used. Results are presented for the unsteady development of flow far upstream of the orifice for non-dimentional flow accelerations of 1, 10, and 100. These demonstrate that the penetration of vorticity decreases with increasing acceleration, and that the fraction of volume flow adjacent to the wall increases with increasing acceleration. By allowing this solution to proceed during the period of constant flow, it was found that the stream function distribution approached a stedy conditon more rapidly than the vorticity distribution. Results are also presented for the asymptotic solution for steady flow through an orifice at a Reynolds number of 5 for an orifice-to-pipe dia ratio of 0·5. These results compared very favorably with those from a recent steady flow solution.

The article gives a solution to the plane problem of the motion of a deformed contour in a flow of an ideal incompressible liquid with a constant vorticity. An explicit expression is obtained for the hydrodynamic force when the velocity of the external flow depends linearly on the coordinates. In the case of a contour of small dimensions, this expression is valid also for an arbitrary external flow.

This paper deals with the flow generated by a stretching filament issuing from an orifice into a still fluid. The filament is assumed to have a velocity proportional to the axial distance. The solution is valid at large axial distances when the boundary layer is very much thicker than the cylinder.

The situation of viscous flow past a split plate is treated within the framework of the Oseen equations. The problem is reduced to an integral equation defined on disjoint intervals. The solution of the integral equation is determined asymptotically for slow flow. The drag force is calculated and compared with certain single plate results.

Possibility of occurrence of secondary flow in boundary layer around a circular cylinder in rotatory oscillation is studied by means of linear theory. The secondary flow can occur above a certain critical Reynolds number, and the condition of occurrence of the secondary flow is shown by curves in the Reynolds and wave number diagrams with a nondimensional thickness of the boundary layer as a parameter. Satisfactory agreement of the results with experimental ones by Taneda et al. is obtained.

The total and local heat transfer from a smooth circular cylinder to the cross flow of air has been measured over the Reynolds number range 3 × 104

The theory initiated by Lighthill (1952) to describe the sound radiated by turbulence embedded in an uniform fluid at rest is here extended to the case where the turbulence exists on the edge of a uniformly moving stream. An exact analogy is developed between the distant real sound field and that which would be radiated by a particular quadrupole distribution adjacent to a vortex sheet positioned in the linearly disturbed flow. The equivalent sources in this analogy are quadrupoles identical in strength with those in Lighthill's model, but the quadrupoles are now shown to convect with the fluid-particle velocity. There is no amplifying effect of shear. The particular case of a plane shear layer is worked out in detail for sound waves of scale large in comparison with the shear-layer thickness.A downstream zone of silence is predicted as is the formation of highly directional beams associated with the interference of sound radiated directly and sound reflected from the fluid interface. A distinct structure results in which the variation of sound with flow velocity, density and angle is not easily accounted for by simple power-law scaling. Finally a comparison is made with some features of jet noise; the modelling of the high frequency jet noise problem by a single shear layer yields some features consistent with experiment.

Theoretical test for the accuracy of the M 2 -expansion solutions to subsonic flow problems has been made by taking the simplest and basic case of the flow past a circular cylinder without circulation. Imai's equation of two-dimensional compressible flow is used for this purpose and the velocity q is assumed to bear the characteristics of the M 2 -expansion solutions of this particular flow problem. It is shown then that, under a certain condition, all q 's satisfying the above equation approximately do not differ much as to their magnitudes. This property which is called the `appropriateness' of the equation is utilized to check the accuracy of the M 2 -expansion solutions, leading to an error estimate in terms of the free stream Mach number M and the adiabatic index γ.

We consider the problem of a stationary convective diffusion of a substance, dissolved in an incompressible fluid flow on the surface of a particle moving with constant speed in a shear flow field. We assume that the flow over the particle is inertia-free and that there is total absorption of the dissolved component on its surface. In the diffusing boundary layer approximation we determine the concentration field and obtain expressions for the total diffusing stream of a substance on the surface of a solid spherical particle and on the surface of a spherical drop (bubble).

An extensive wind-tunnel test of a round turbulent jet directed normally through a flat plate into a subsonic cross flow has been conducted. The results of the velocity field measurements are presented in a concise and usable form through the use of simple models to relate the velocity field to empirical values for the strength and location of the pair of contrarotating vortices associated with the jet.

The behaviour of the polynomial approximation to the boundary layer velocity profile is investigated. Various orders of polynomials and 4 different schemes of “reasonable” boundary conditions are examined for applicability as approximate solutions to the Blasius flow over a flat plate.

The conical flow solution for axisymmetric supersonic flow past cones has been found to be virtually independent of the ratio of specific heats when normalized in a certain way. A simple rational approximation to this flow is derived. The important singularities and the limiting behavior of the solution are also discussed.

Integral transformations are developed to construct three and five axisymmetric potentials for a needle or straight line segment. These potentials are applied to flow past a needle and one of the main results is for streaming flow past a line segment in which the fluid velocity vanishes on the boundary. This solution may also be regarded as a Stokes flow or an inviscid potential flow.

Separated flow past a two-dimensional smooth curved bluff body can be solved by the use of a free-streamline theory so long as the base pressure and the separation position are prescribed as empirical parameters. The purpose of the present paper is to eliminate one of these parameters by determining the separation position as a function of the base pressure with the help of the boundary layer theory. Calculations were performed for cases of a subcritical and a transcritical flow past a circular cylinder. Although some discrepancy is seen to exist between the theory and the experimental data, it is shown that solutions of Kawaguti, and of Bluston and Paulson can be included in the present theory as special examples. Furthermore, a conclusion of the latter paper is criticised from the viewpoint of the present theory.

A BOUNDARY value problem in a bounded region, whose solution approximates the solution of the problem is discussed. The effect of boundary conditions, specified on an additional distant boundary, is investigated. A numerical solution algorithm is described and the results of calculations for Re = 18 to 100 are presented.

The quasi-geostrophic flow over an obstacle placed on the lower of two horizontal planes in rapid rotation about the vertical z axis is considered. The flow field is calculated in the limit of vanishing viscosity after assuming the flow far upstream of the obstacle to be uniform, of magnitude V. The effects that occur in homogeneous flow are compared with those that occur in stratified flow.If the flow is homogeneous, there is a region of closed streamlines if \[ h_0R^{-1} > \min_r\left[r\left/\int_0^r xh(x)dx\right. \right], \] where the obstacle is assumed to be cylindrically symmetric and given by z = Hh0h(r/L), H is the distance between the planes and R is the Rossby number V/(fL). For any obstacle the right-hand side of (1) is greater than zero and hence h0 must be positive for a closed-streamline region to occur. It is argued, and illustrated by a particular example, that because (1) involves an integral of h(x) a representative flow pattern can be obtained for obstacles of less than critical height by considering the special case of a flat-topped obstacle, as is done by Ingersoll (1969).If the flow is stratified with constant Brunt-Väisälä frequency N, the condition for the existence of closed streamlines is shown to be \[ h_0R^{-1} > \min_r \left[B\int_0^{\infty}\int_0^{\infty}xt\cot h(Bt)h(x)J_0(tx)J_1(tr)\,dt\,dx \right]^{-1}, \] where B = NH/fL. In contrast to the homogeneous situation, the right-hand side of (2) can be zero and is so if the obstacle is somewhere vertical. Such obstacles will produce a closed-streamline region no matter now small their height and will hence not lead to patterns representative of smooth obstacles. This is because a stratified column of fluid cannot be stretched or compressed over an infinitesimal distance. Instead, the column bends markedly and the fluid flows around the obstacle. The critical conditions (1) and (2) for a number of specific obstacles are calculated and discussed.

A steady axisymmetric source-sink flow of an incompressible viscous fluid in a rotating circular cylinder is studied. The source-sink distribution is idealized as the line distribution along the concentric circles on the top and the bottom. The effect of the discontinuity of the angular velocity across the same circles is also examined. The method of analysis is the boundary layer analysis of the linearized basic equations. The attention is concentrated on the details of the flow structure in the shear layers, and several numberical results are shown.

The initial flow field of an incompressible, viscous fluid around a circular cylinder, set impulsively to move normal to its axis, is studied in detail. The nonlinear vorticity equation is solved by the method of matched asymptotic expansions. Analytic solutions for the stream function in terms of exponential and error functions for the inner flow field, and of circular functions for the outer, are obtained to the third order, from which a uniformly valid composite solution is found. Also presented are the vorticity, pressure, separation point and drag. These quantities agree with the numerical computations of Collins & Dennis. Extended solutions developed by Padé approximants indicate that higher than third-order approximations will yield only minor improvements.

The application of slender-body theory to the evaluation of the three-dimensional surface velocities induced by a boundary layer on an elliptic cylinder is considered. The method is applicable when the Reynolds number is sufficiently large so that the thin-boundary-layer approximation is valid. The resulting potential problem is reduced to a two-dimensional consideration of the flow over an expanding cylinder with porous boundary conditions. The limiting solutions for a flat plate of finite span and a nearly circular cross-section are obtained in a simple analytic form. In the former case, within the limitations of slender-body theory, the results are in exact agreement with the complete three-dimensional solution for this geometry.

The nearness of a cylinder to a wall has an important effect on the flow-induced forces exerted on that cylinder, particularly when the cylinder is relatively large compared to the cross section of the flow channel. This paper describes an investigation of wall interference effects that occur when crossflow-induced forces are exerted on circular cylinders with moderately large blockages (d/h = 0.2 to 0.33) at high Reynolds numbers (3.5 × 105 – 1.2 × 106). The results show that, within the range studied, the gap flow velocity is the correct flow velocity to compensate for wall interference effects. The data reported here represent the first available information on experimental cross-flow-induced forces at such high Reynolds numbers using water as a flow medium.

Second-order thermal boundary-layer problem is formulated for an inviscid plane or axisymmetric flow. The resulting energy equations which govern the effects of both longitudinal and transverse curvatures, external temperature gradient and vorticity are solved both exactly and approximately. Applications of the present theory are given for heat transfer from elliptical-rod bundles and that from a sphere.

The steady flow generated by an oscillating triangular cylinder immersed in a viscous, incompressible fluid is investigated experimentally. When an equilateral triangular cylinder is oscillated perpendicularly to one of its sides, the flow pattern in the vicinity of the cylinder is symmetric with respect to the axis of oscillation but asymmetric with respect to the axis perpendicular to it. The flow patterns are classified into two types on the basis of the values of two non-dimensional parameters, \(\textbf{\itshape r}_{0}\sqrt{\omega/\nu}\) and s / r 0 , where r 0 is half the side length of the cylinder section, ω the angular frequency, ν the kinematic viscosity and s the oscillation amplitude. On the other hand, when the triangular cylinder is oscillated parallel to one of its sides, the flow pattern is asymmetric with respect to the axis of oscillation but symmetric with respect to the axis perpendicular to it. The flow patterns are classified into three types.

Numerical solutions are obtained for viscous incompressible flow over an impulsively started cylinder and a 2:1 ellipse at an angle of attack of 20°. The Navier-Stokes equations with vorticity and stream function as variables are solved using a cmbination of finite differences and Green's function. The Reynolds numbers are 500 and 200 for the cylinder and ellipse respectively, and computations are continued up to a dimensionless time of 1.5. For the flow over an ellipse the lift force as well as the drag are initially very large and then rapidly decay. The rear stagnation point gradually moves down and separation occurs at t ∼ 1.1. Recent experimental studies by Taneda for flow at the same geometry indeed show large values of lift following the initial motion and gradual downward movement of the rear stagnation point.