Multiply branching fluid flows are modelled in two contexts. The first (type I) is for one-to-many branching. Computations are described for flow through a, channel, with fully developed motion upstream, which branches abruptly into a number of subchannels downstream. The differences in pressure between the upstream end of the channel and the downstream ends of the subchannels are substantial. Comparisons with recent analytical predictions show fair agreement for Reynolds numbers in the low tens and above. The second context (type II) has successive generations of bifurcation in a network. Modelling, computations and analysis include the effects of many bifurcations.

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).

In this part-review part-new work, studies on branching tube flows are described. These are based on modelling for increased flow rates as well as on direct numerical simulations and are motivated by applications to the cardiovascular system, lung airways and cerebral arteriovenous malformations. Small pressure differentials acting across a multiple branching are considered first, followed by substantial pressure differentials in a side branching, multiple branching or basic three-dimensional branching. All cases include a comparison of results between the modelling and the direct simulations. Wall shear, pressure variation, influence lengths, and separation or its suppression are examined, showing in particular sudden spatial adjustment of the pressure between mother and daughter tubes, nonunique flow patterns and a linear increase of flow rate with increasing number of daughters, dependent on the specific conditions. The agreement between modelling and direct simulations is generally close at moderate flow rates, suggesting their combined use in the biomedical applications.

The planar flow of incompressible fluid past a blunt obstacle mounted on a flat (horizontal) fixed solid surface of infinite extent is examined in the presence of an incident linear velocity profile, modelling the fluid behaviour close to a small surface roughness for instance. The motion is taken to be steady and laminar. The obstacle is blunt in the sense that its typical surface slopes are not small, a feature which here always induces flow separation both upstream and downstream of the obstacle. Computations and nonlinear theory are applied, together with comparisons. The direct computations of the Navier-Stokes equations, using for example a higher order upwind-difference scheme, deal with a moderate range of Reynolds numbers up to 200, based on the obstacle height and the incident uniform shear. In addition the accuracy is necessarily limited as the Reynolds number increases. The theory is for large Reynolds numbers and is based on viscous-inviscid reasoning, back-pressure effects from the obstacle and slender-layer separation locally, among other influences. The comparisons nevertheless yield encouragingly close agreement, for the present computed cases of a vertical flap or a rectangular block. This is both quantitatively, in terms of the upstream separation and downstream reattachment positions in particular, and generally, in terms of the separating flow structure, even at the notably moderate Reynolds numbers covered accurately by the computations.

A numerical method is developed for investigating the two-dimensional unsteady viscous flow over an inclined elliptic cylinder placed in a uniform stream of infinite extent. The direction of the free stream is normal to the cylinder axis and the flow field unsteadiness arises from two effects, the first is due to the flow field development following the start of the motion and the second is due to vortex shedding in the wake region. The time-dependent flow is governed by the full conservation equations of mass and momentum with no boundary layer approximations. The parameters involved are the cylinder axis ratio, Reynolds number and the angle of attack. The investigation covers a Reynolds number range up to 5000. The minor-major axis ratio of the elliptic cylinder ranges between 0.5 and 0.6, and the angle of attack ranges between 0 degrees and 90 degrees. A series truncation method based on Fourier series is used to reduce the governing Navier-Stokes equations to two coupled infinite sets of second-order differential equations, These equations are approximated by retaining only a finite number of terms and are then solved by approximating the derivatives using central differences. The results reveal an unusual phenomenon of negative lift occurring shortly after the start of motion. Various comparisons are made with previous theoretical and experimental results, including flow visualizations, to validate the solution methodology. Copyright (C) 2001 John Wiley & Sons, Ltd.

The temporal development of two-dimensional viscous incompressible flow induced by an impulsively started circular cylinder which performs time-dependent rotational oscillations about its axis and translates at right angles to this axis is investigated. The investigation is based on the solutions of the unsteady Navier–Stokes equations. A series expansion for small times is developed. The Navier–Stokes equations are also integrated by a spectral–finite difference method for moderate values of time for both moderate and high Reynolds numbers. The numerical method is checked with the results of the analytical solution. The effects of the Reynolds number and of the forcing Strouhal number S on the laminar asymmetric flow structure in the near-wake region are studied. The lift and drag coefficients are also extracted from numerical results. An interesting phenomenon has been observed both in the flow patterns and in the behaviour of drag coefficients for S = π/2 at Reynolds number R = 500 and is discussed. For comparison purposes the start-up flow is determined numerically at a low Reynolds number and is found to be in good agreement with previous experimental predictions.

The unsteady two-dimensional flow of a viscous incompressible fluid past an impulsively started oscillating and translating elliptic cylinder has been investigated. The governing Navier-Stokes equations expressed in terms of a stream-function/vorticity formulation are solved numerically for the early stages of the flow for moderate to high Reynolds numbers. A boundary-layer type transformation was adopted to scale out the singular nature in the vorticity at the start of the motion. Some comparisons of the flow patterns with the impulsively started translating case have been included to illustrate the effect of oscillation. Wherever possible, comparisons with existing numerical results have been made and the agreement was found to be good.

This paper treats the two-dimensional steady flow of a viscous incompressible fluid driven through a channel bounded by two walls which are the radii of a sector and two arcs (the ‘inlet’ and ‘outlet’), with the same centre as the sector, at which inflow and outflow conditions are imposed. The computed flows are related to both a laboratory experiment and recent calculations of the linearized ‘spatial’ modes of Jeffery–Hamel flows. The computations, at a few values of the angle between the walls of the sector and several values of the Reynolds number, show how the first bifurcation of the flow in a channel is related to spatial instability. They also show how the end effects due to conditions at the inlet and outlet of the channel are related to the spatial modes: in particular, Saint-Venant's principle breaks down when the flow is spatially unstable, there being a temporally stable steady flow for which small changes at the inlet or outlet create substantial effects all along the channel. The choice of a sector as the shape of the channel is to permit the exploitation of knowledge of the spatial modes of Jeffery–Hamel flows, although we regard the sector as an example of channels with walls of moderate curvature.

The development of the two-dimensional flow of a viscous incompressible fluid around a circular cylinder which suddenly starts to move with the velocity U = U0 + U1t + U2t2 is studied. Equations for the flow in terms of the stream function and vorticity in boundary-layer coordinates are presented. A perturbation series solution for small times is developed. The flow for longer times is computed numerically using an accurate implicit time-integration procedure. The numerical method is checked for small times by comparison with the results of the analytical solution. Reynolds numbers R in the range 200 to 104 (based on the diameter of the cylinder) are considered. One particularly interesting result is that for certain values of U1 and U2 at R = 500 and R = 103 it is found that two co-rotating vortices and three co-rotating vortices develop with time in each half of the wake in the two respective cases.

A method of solution for the two-dimensional Navier-Stokes equations for incompressible flow past a cylinder is given in which the equation of continuity is solved by a step-by-step integration procedure at each stage of an iterative process. Thus the formulation involves the solution of one first-order and one second-order equation for the velocity components, together with the vorticity transport equation. The equations are solved numerically by h(4)-accurate methods in the case of steady flow past a circular cylinder in the Reynolds number range 10-100. Results are in satisfactory agreement with recent h(4)-accurate calculations. An improved approximation to the boundary conditions at large distance is also considered.

In this paper the two-dimensional steady-state problem of laminar forced convective heat transfer from an isothermal cylinder, elliptic in cross section, inclined to a uniform stream is investigated. Numerical solutions of the Navier-Stokes and energy equations have been obtained for Reynolds numbers, Re, 5 and 20 for various values of the Prandtl number, Pr, and inclination angle, α. Particular attention is focussed on a solution process utilizing the asymptotic properties of the Navier-Stokes and energy equations. The average rate of heat transfer, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGobGaamyDaaaaaaa!36B5!\[\overline {Nu} \], was found to behave closely to the theoretical result % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGobGaamyDaaaacqWI8iIocaWGqbGaamyzamaaCaaaleqabaWaaSGb% aeaacaaIXaaabaGaaG4maaaaaaaaaa!3B58!\[\overline {Nu} \sim Pe^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \] for large Pe where Pe=RePr is the Peclet number.

Some methods are proposed for solving the Navier-Stokes equations for two-dimensional, incompressible, flow using the velocity-vorticity formulation. The main feature of the work is the solution of the equation of continuity using boundary-value techniques. This is possible because both of the velocity components are known at each boundary point. Some illustrative results are computed including some for heat convection inside a square cavity when one side is held at a constant temperature.

The transient flow field caused by an infinitely long circular cylinder placed in an unbounded viscous fluid oscillating in a direction normal to the cylinder axis, which is at rest, is considered. The flow is assumed to be started suddenly from rest and to remain symmetrical about the direction of motion. The method of solution is based on an accurate procedure for integrating the unsteady Navier–Stokes equations numerically. The numerical method has been carried out for large values of time for both moderate and high Reynolds numbers. The effects of the Reynolds number and of the Strouhal number on the laminar symmetric wake evolution are studied and compared with previous numerical and experimental results. The time variation of the drag coefficients is also presented and compared with an inviscid flow solution for the same problem. The comparison between viscous and inviscid flow results shows a better agreement for higher values of Reynolds and a Strouhal numbers. The mean flow for large times is calculated and is found to be in good agreement with previous predictions based on boundary-layer theory.

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.

For flow past a cylinder it is known that the vorticity is only significant in a thin boundary-layer adjacent to the surface and within a parabolic wake far from the cylinder. To address this behaviour of the vorticity a numerical method is implemented whereby the flow field is decomposed into two regions: an inner region to deal with boundary-layer phenomena and an outer region to model wake phenomena. This method equally applies to any cylinder cross section. The equations of motion are solved in each region and are matched at the boundary. Numerical solutions have been carried out for the trial case of a circular cylinder and the agreement with existing results is good.

A mathematical model is proposed for the steady two-dimensional flow of a viscous incompressible fluid past a cylinder which incorporates the details of the structure of the vorticity in this case where its behaviour is known. The model is constructed to be consistent both with boundary-layer theory for sufficiently large Reynolds numbers and with the asymptotic solution at large distances from the cylinder. The governing Navier-Stokes equations are transformed to a set of equations which we refer to as the modified Navier-Stokes equations and are then solved numerically. Solutions have been obtained for the cases of flow past a stationary and a rotating circular cylinder and flow past an inclined elliptic cylinder. Good agreement is found with existing results.

An experimental and numerical investigation of the two-dimensional flow normal to a flat plate is described. In the experiments, the plate is started impulsively from rest in a channel for Reynolds numbers, based on the breadth of the plate, in the range 5 ≤ Re ≤ 20. Over this range of Re the flow remains symmetrical and stable and tends to a steady state but is shown to depend strongly on the ratio λ of the plate to channel breadth. The evolution of the experimental flow with time and Reynolds number is studied and the variation with λ in the range 0.05 ≤ λ ≤ 0.2 is investigated sufficiently to enable an estimate of properties of the flow as λ → 0 to be obtained for the steady-state flow. The numerical results are obtained for steady flow normal to a flat plate in an unbounded fluid for Reynolds numbers up to Re = 100. They supplement and extend results for this flow obtained for values of Re up to 20 by Hudson & Dennis (1985). The present solutions have been found using a vorticity-stream function formulation rather than the primitive-variable approach of Hudson & Dennis and provide an independent check on these results. A comparison of the theoretical results for Re ≤ 20 with the limit λ → 0 of the experimental results is, generally speaking, extremely satisfactory.

The problem of determining the two-dimensional steady motion of a viscous incompressible fluid which is injected radially over one small arc of a circle and ejected radially over another arc is considered and examples are given of both symmetrical and asymmetrical flows. The motion is governed by the Navier Stokes equations and the method of solution is based on the use of truncated Fourier series representations for the stream function and vorticity in the angular polar coordinate. The Navier-Stokes equations are reduced to ordinary differential equations in the radial variable and these sets of equations are solved using finite-difference methods, but with the boundary vorticity calculated using global integral conditions rather than local finite-difference approximations. One of the objects of the investigation is to relate this method to a previous study which did not use integral conditions and also to a recent study which uses an integro-differential method which is different in concept but which also uses integral conditions. A brief review of previous work on the problem is given. Comparisons of present and previous results are excellent.

It is shown that one condition only, derived from one of Green's identities, is necessary to determine the matching of inner and outer expansions in solving problems involving the initial flows of a viscous fluid. This condition has been named an integral condition, and when it has been satisfied, in the form of a set of conditions for the spectral components, the inner solution is obtained by a straightforward integration procedure and the outer solution is obtained merely by substituting the inner solution in an integral. Two examples of the method are considered, including that of flow past a suddenly started circular cylinder and flow due to a suddenly started rotating sphere.

In this paper we consider the asymptotic form, for large Dean number, of the solution which describes the fully developed laminar flow in a curved pipe of circular cross section. Although we have not been able to provide a complete solution we present strong evidence in favour of an asymptotic structure which is based upon an inviscid core flow enclosed by viscous boundary layers at the pipe wall.