Design of S-shaped diffusers in incompressible flow

The hodograph method, in conjunction with a numerical form of the Schwarz-Christoffel transformation, is applied to the determination of the shape of an S-shaped diffuser subject to certain prescribed characteristics in incompressible flow. It is shown how the resulting diffuser of infinite length can be modified to one of finite length by limiting the upstream and downstream velocities to within a small percentage of their normal asymptotic values     

Numerical simulation of swirling flow in diffusers

A reduced form of Navier–Stokes equations is developed which does not have the usual minimum axial step size restriction. The equations are able to predict accurately turbulent swirling flow in diffusers. An efficient single sweep implicit scheme is developed in conjunction with a variable grid size domain-conforming co-ordinate system. The present scheme indicates good agreement with experimental results for (1) turbulent pipe flow, (2) turbulent diffuser flow, (3) turbulent swirling diffuser flow. The strong coupling between the swirl and the axial velocity profiles outside of the boundary layer region is demonstrated.     

Erosion in conical diffusers in particulate-laden cavitating flow

This paper highlights the serious damage that can occur in diffusing sections of pipework in which a cavitating particulate-laden fluid is flowing. The combined effects of particle erosion and cavitation are shown to remove considerably more material than would be expected from summing the effects of the individual mechanisms. It is demonstrated that, to be sure of avoiding this accelerated surface erosion, the transition from a smaller flow section to a larger one needs to be an abrupt expansion. If pressure recovery is important, a possible design solution is proposed. In the case of swirling flow, the expansion again needs to be abrupt. Evidence was also obtained which showed that, by allowing air to be entrained into the low pressure region in the flow, the cavitation and the erosion can be substantially reduced.     

LES of supersonic turbulent flow in nozzles and diffusers

Large-eddy simulations of supersonic nozzle and diffuser flows with circular cross-sections using high-order compact schemes and an explicit filtering version of the approximate deconvolution method are presented in this paper. Two flow cases each for nozzle and diffuser having different outlet to inlet area ratios are presented. The effect of the geometry variations on the Reynolds stresses as well as on the production and pressure-strain terms in their transport equations is demonstrated. A Green’s function analysis of the Poisson equation for pressure fluctuations using LES data is presented and the results show similar trends as found in previous analyses using DNS data. The effects of geometry changes on the rapid and slow parts of pressure-strain correlations is also demonstrated.     

Numerical simulation of axisymmetric turbulent flow in combustors and diffusers

Numerical studies of turbulent flow in an axisymmetric 45° expansion combustor and bifurcated diffuser are presented. The Navier-Stokes equations incorporating a k–? model were solved in a non-orthogonal curvillinear co-ordinate system. A zonal grid method, wherein the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. However, treatment of the zonal boundaries required special handling. Boundary overlap and interpolating techniques were used and an adjustment of the flow variables was required to assure conservation of mass flux. Three finite differencing methods—hybrid, quadratic upwind and skew upwind—were used to represent the convection terms. Results were compared with existing experimental data. In general, good agreement between predicted and measured values was obtained.     

Shallow flow modelling using curvilinear depth-averaged stream function and vorticity transport equations

Most receiving water, such as lakes and open reservoirs, have large plan dimensions with respect to their depth. In such cases, the flow may be nearly two-dimensional and the depth-averaged Reynolds equations are appropriate. This paper presents a new version of the governing equations in curvilinear depth-averaged stream function and vorticity transport (ψ, ω) form appropriate for non-orthogonal computational meshes. The equations are discretized using finite differences and solved using successive over-relaxation for the depth-averaged stream function equation and an alternating direction implicit scheme for the vorticity transport equation. Results from the numerical model are validated against data from flow past a backward facing step and jet-forced flow in a circular reservoir. The results indicate that the (ψ, ω) form of the shallow water equations may be useful for applications where the free surface can either be assumed horizontal, or is know a priori.     

Convective flow with uniform vorticity in a vertical layer

The free convective circulation of liquid in plane vertical slits of circular and square cross section with a longitudinal horizontal temperature gradient at the boundaries was investigated experimentally. It was found that under such heating conditions there is a uniform-vorticity flow with a region of quasirigid rotation, which has the shape of a disk in a circular slit and the shape of a cross in a square slit; in each longitudinal section of this zone the liquid moves along concentric trajectories with constant angular velocity. Dimensionless numbers for the problem were established by tests with various liquids and cavities of different dimensions. In dimensionless form, the angular velocity of the vortex and the temperature gradient in it depend linearly on the temperature difference at the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–165, May–June, 1984.     

Optimal vorticity regime in the flow tract of a hydroturbine

A study is made of the problem of maximizing the power taken from the shaft of the working rotor of a hydroturbine for a fixed available energy difference in the framework of a two-dimensional axisymmetric flow model. Necessary conditions of optimality of first and second order are derived and used to set up an algorithm for numerical solution of the problem. The results of calculations are given, and a comparison is made with optimal solutions obtained using two- and one-dimensional models of axisymmetric flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 68–73, May–June, 1984.     

Effects of swirl on flow separation and performance of wide angle diffusers

Experimental investigations have been carried out to determine whether the introduction of a circumferential velocity component can produce worthwhile improvements in the performance of, and eliminate flow separation in, wide angle conical diffusers. The swirl generator is a 24 flat-bladed, radial intake type. Systematic experimentation has been carried out for one diffuser configuration fitted with a tailpipe (16.5° and 4.4 area ratio) using varying strengths of inlet swirl and introducing the dissipated mechanical energy as the main criterion of diffuser performance. The best inlet swirl strength produced about 60% reduction of the total diffuser losses in swirl-free flow. The analysis of these results, together with information obtained from flow visualisation experiments, suggests that increasing the swirl beyond an observed threshold completely eliminated flow separation, but it also gave rise to a central zone of recirculating flow and hence additional dissipative losses. We conclude that the optimum improvement achievable in wide angle diffuser performance using swirl does not require the addition of more energy than it saves     

On patches of uniform vorticity in a plane of irrotational flow

We derive an evolution equation for the motions of patches of vorticity (vortex). Steady state solutions of this equation that include those of Kirchhoff and Moore & Saffman are established. The m-fold symmetric, m3, hypotrochoid is an exact steady solution of this equation when rotation and strain are present. When strain is absent but rotation is present, the m-fold symmetric, m2, hypotrochoid is a perturbation solution with a dispersion relation extending that of Lamb. The case of m=2 is exact and is the Kirchhoff elliptical vortex.     

Performance of a transverse vorticity probe in a turbulent channel flow

 The performance of a four hot-wire transverse vorticity probe is tested by comparing measurements in a fully developed turbulent channel flow with corresponding data obtained from direct numerical simulations (DNS) of the same flow. In the inner region, the probe performs poorly, the rms vorticities being consistently smaller than the DNS values. In the outer region of the flow, there is reasonable agreement between measured and DNS vorticity statistics, especially after correcting the measurements for the effect of spatial resolution. In this region, the imbalance indicated by the vorticity form of the streamwise momentum equation is approximately constant. The magnitude of the imbalance can be reduced to an acceptable level of accuracy by considering sources of error which affect the velocity–vorticity correlations. Received: 17 March 1997/Accepted: 17 November 1997     

Calculation of the initial segment and the stabilized flow segment in two-dimensional separation-free diffusers

The aerodynamic characteristics of a series of plane diffusere with straight walls are calculated for a broad range of divergence angles, Reynolds numbers, and the parameter which characterizes the initial flow nonuniformity.The fluid is assumed to be incompressible. The calculations are made for the case in which the boundary layer is fully turbulent, i. e., there is no laminar flow segment near the entrance section.The calculation of separation-free flow in diffuser channels is based on the use of boundary layer theory [1]. It has now become possible to carry out large-scale calculations for diffusers whose geometric and aerodynamic parameters vary over rather wide limits. This is the result both of the use of computers and of the fact that the modern approximate methods for calculating the turbulent boundary layer have been reduced to comparatively simple interpolation formulas [2].Usually, in the calculation of diffusers we examine only the initial flow segment, within the limits of which the boundary layers which form on the walls do not come together, i. e., there is a potential core. The laws governing diffuser flow in the absence of the potential core have been studied relatively little; the only known solutions are those of [3], which are valid at a very great distance from the entrance section.In this study we examine three characteristic flow zones: the initial segment, extending from the entrance section to the section at which the boundary layers come together; the stabilized-flow zone with closed boundary layers comprised of two characteristic segments-the transitional segment extending from the plane where the boundary layers join to the beginning of the radial flow segment; and, finally, the radial (self-similar) flow segment, characterized by constancy of all the dimensionless boundary layer characteristics along the flow. It is obvious that this division into characteristic zones is arbitrary: a consequence of the adopted flow idealization is a break in the curves expressing the aerodynamic characteristics as a function of the axial coordinate at the junction of the initial segment and the stabilized-flow segment. It is well known that a similar phenomenon occurs in the calculation of free turbulent jets based on arbitrary division of the jet into two segments-initial and primary segments [4].The computer calculation of the initial segments was performed by A. N. Smol'yaninova, and the stabilized-flow segments were calculated by I. N. Podol'nyi.     

Turbulent flow in a tube containing an offset rod

A numerical finite volume prediction method for arbitrary-shaped passages has been applied to the case of fully developed axial turbulent flow past a rod eccentrically placed in a circular tube. The numerical method was based on an orthogonal curvilinear mesh and employed an algebraic stress transport model to calculate the full three-dimensional velocity field directly from the governing partial differential equations. This study is one of a series of applications of this prediction method to a range of different non-circular passages that have been made in order to establish the capabilities and usefulness of this type of procedure. The present eccentric rod case was the subject of a comprehensive experimental investigation by Kacker¹ which has enabled a detailed comparison to be made between the present predictions and the measurements. This comparison included local distributions of axial velocity, wall shear stress and secondary velocities; and although found to be satisfactory overall, some differences in detail revealed possible shortcomings in the measurement of secondary flow. This, together with other previously reported cases, indicates, that, although the present method cannot be expected to replace experiment in providing turbulent passage flow data, it has an important role to play in interpreting and supplementing experiments.     

A numerical study of vorticity layers in a two-dimensional stratified shear flow

A numerical study is conducted to find out the conditions of occurrence of a secondary Kelvin-Helmholtz instability in the thin layers (referred to as baroclinic layers) that form in a stably-stratified shear layer. For this purpose, three high resolution calculations of a moderately stratified shear layer have been carried out, at a fixed Reynolds number. The wavelength of the initial perturbation is progressively increased, starting from the fundamental wavelength predicted by linear stability theory up to twice this fundamental wavelength. The baroclinic layer of the flow is shown to lengthen and destabilize progressively from one calculation to the other, eventually bearing a secondary Kelvin-Helmholtz instability. The structure and dynamics of the baroclinic layers of the three calculations are examined in the frame of a theoretical model proposed by Corcos and Sherman ([1]). An excellent agreement with the predictions of this model have been found. We next show that the stability of the layer is controlled by the large-scale Kelvin-Helmholtz vortex, via the strain field that it induces in the stagnation point region of the layer. A consequence of this study is that secondary Kelvin-Helmholtz instabilities are fostered by the pairing of primary Kelvin-Helmholtz vortices in a strongly-stratified shear layer.

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Sommario E stato condotto uno studio numerico per trovare le condizioni in cui insorge una instabilità secondaria di Kelvin-Helmholtz negli strati sottili che si formano in uno strato di scorrimento stabilmente stratificato. A questo scopo sono state effettuate tre simulazioni ad alta risoluzione a fissato numero di Reynolds e stratificazione bassa. La lunghezza d'onda della perturbazione iniziale è stata progressivamente aumentata dalla lunghezza fondamentale predetta dalla teoria lineare della stabilità fino a due volte questa stessa lunghezza. È stato osservato che da una simulazione all'altra lo strato baroclino del flusso si allunga e si destabilizza progressivamente, generando eventualmente un'instabilità di Kelvin-Helmholtz secondaria. Utilizzando il modello teorico proposto da Corcos e Sherman (1976), per le tre simulazioni sono state analizzate la struttura e la dinamica dello strato baroclino. È stato trovato un accordo eccellente con le predizioni di questo modello. È stato in seguito mostrato che la stabilità dello strato è controllato dai vortici di Kelvin-Helmholtz di larga scala attraverso il campo di deformazione che inducono nella regione del punto di ristagno dello strato. Una conseguenza di questo studio è che le instabilità secondarie di Kelvin-Helmholtz sono forzate dall'accoppiamento dei vortici primari in uno strato di scorrimento fortemente stratificato.

     

Steady axisymmetric flow with vorticity of an inviscid fluid in multistage turbomachines

The direct problem of steady axisymmetric flow of a gas with vorticity through a multistage turbomachine is formulated precisely and a generalized solution is constructed by a variational-difference method. The turbomachine is represented schematically by an annular channel in which there are fixed (₁) and rotating (₂) three-dimensional cascades and channels free of them (₀).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–15, October–December, 1981.