Dean instability and secondary flow structure in curved rectangular ducts

The pressure driven, fully developed turbulent flow of incompressible viscous fluid (water) in 120° curved ducts of rectangular cross-section is investigated experimentally and numerically. Three different types of curved duct (A-CL, B-SL and C-IL) with continuously varying curvature conform to blade profile as the inner and outer curvature walls to simplify and guide the impeller design of pumps. After validating the numerical method against Particle Image Velocimetry (PIV) measurements, the flow development in the ducts is analyzed in detail by Computational Fluid Dynamics (CFD) for a wide range of Reynolds numbers (Re = 2.4 × 10⁴–1.4 × 10⁵) and aspect ratios (Ar > 1.0, =1.0 and <1.0). The results clearly depict the existence of multiple Dean vortices along the duct: while the axial velocity profile is more related to an inner Dean vortex (called split base vortex), the wall pressure is more influenced by the Dean vortex attached to the inner curvature wall (called ICW Dean vortex). The induced multiple Dean vortices and the secondary flow patterns in the duct cannot be faithfully predicted by using traditional techniques. Therefore, a new criterion based on the vortex core velocities is proposed. With this approach, the effects of Re, Cr and Ar on the Dean instabilities in curved ducts are carefully studied. Decreasing Re promotes the generation of Dean vortices closer to the duct inlet, a trend that is as opposed to laminar flow. In addition, a new pair of vortices called entrainment Dean vortex occurs near the outlet of the curved duct with Ar = 1.0, which has not been previously reported in the literature.     

An exact analytical solution for creeping Dean flow of Bingham plastics through curved rectangular ducts

In this paper, an exact analytical solution for creeping flow of Bingham plastic fluid passing through curved rectangular ducts is presented for the first time. The closed form of axial velocity distribution, flow resistance ratio, and wall shear stress are derived using bounded Fourier transformation. An extensive investigation on mutual effects of Hedstrom number, curvature ratio, and aspect ratio is conducted. The results indicate that a drag reduction is caused in the flow field by increasing the Hedstrom number. It is shown that unlike the Newtonian creeping Dean flow, the critical aspect ratio (an aspect ratio in which the flow resistance ratio is independent from curvature ratio) does not exist at large enough Hedstrom numbers. Analytical solution also indicated that as Hedstrom number is increased, the value of Poiseuille number is enhanced, and unlike the Newtonian flows, the value of Poiseuille number is not zero at edges of cross section.     

Fluid flow in a rotating curved rectangular duct

The fluid flowing in a rotating curved duct is subjected to both the Coriolis force due to a rotation and the centrifugal force due to a curvature. In this paper, the combined effects of the two forces on the flows in rotating curved rectangular ducts are examined numerically. According to the aspect ratio of the cross-section, the rectangular ducts are divided into three types: η>1, η=1, η<1, where η is the aspect ratio. The variations of the flow structures with the force ratio F (the ratio of the Corislis force to the centrifugal force) are studied in detail and many hitherto unknown flow patterns are found. The effects of the force ratio and the aspect ratio of the cross-section on the friction factor are also examined. Present results show both the characteristics of the secondary flow, axial flow and the natures of the friction factor.     

Creeping flow of Newtonian fluids in curved rectangular channels

The object of this paper is to present accurate numerical data concerning the creeping flow in curved annular channels with rectangular cross sections of which the outer wall is rotating with constant angular velocity. Dimensionless expressions for velocity profiles, flow rates and friction factors are obtained analytically for both the “drag” and “pressure” flow contributions. Numerical data were obtained on a digital computer and are presented in tabular and graphical form. The results of the theoretical analysis are also expressed in terms of the flow rate correction factors widely used in calculating the pumping efficiency of screw-pumps, agitators and extruders. This enables to estimate the effect of flight curvature on the pumping capacity.     

Pressure effects on electroosmotic flow of power-law fluids in rectangular microchannels

In this paper, the fully developed electroosmotic flow of power-law fluids in rectangular microchannels in the presence of pressure gradient is analyzed. The electrical potential and momentum equations are numerically solved through a finite difference procedure for a non-uniform grid. A complete parametric study reveals that the pressure effects are more pronounced at higher values of the channel aspect ratio and smaller values of the flow behavior index. The Poiseuille number is found to be an increasing function of the channel aspect ratio for pressure assisted flow and a decreasing function of this parameter for pressure opposed flow. It is also observed that the Poiseuille number is increased by increasing the zeta potential. Furthermore, the results show that an increase in the flow behavior index results in a lower flow rate ratio, defined to be the ratio of the flow rate to that of a Newtonian fluid at the same conditions. Moreover, whereas the flow rate ratio in the presence of an opposed pressure gradient is smaller than that of a favorable pressure force for shear thinnings, the opposite is true for shear-thickening fluids.     

Bifurcation phenomena of generalized newtonian fluids in curved rectangular ducts

The bifurcation phenomenon in flow through a curved rectangular duct is investigated in this study. The non-linear equations of motion governing the steady, fully developed laminar flow of an incompressible generalized Newtonian fluid have been solved numerically. Extensive results have been generated in an effort to map the regions of multiple solution in the parameter space of Dean number, Dn, aspect ratio, γ, power-law index, n, and radius of curvature, r. For a Newtonian fluid (n = 1), at a fixed curvature (r = 100), the transition between a symmetric 2-cell and a symmetric 4-cell solution appears to follow a tilted cusp. The extent of the stable, symmetric 2-cell solution surface is critically influenced by the length scale γ. In the non-Newtonian case, at a fixed aspect ratio (γ = 1) and a fixed curvature (r = 100), the flow transition follows that of a fold catastrophe. The influence of the curvature is reasonably well accounted for in Dn. The bifurcation set determined in the Dn-γ space remains qualitatively the same at any value of n or r. These parameters merely shift and/or stretch the equilibrium surface determined by Dn and γ.     

Instability investigation of creeping viscoelastic flow in a curved duct with rectangular cross-section

In this paper, instability in the creeping viscoelastic flow inside a curved rectangular duct is investigated numerically for the first time. Using the Criminale–Eriksen–Filbey (CEF) model as the constitutive equation, the governing equations are solved by a second order of finite difference method based on the artificial compressibility algorithm in a staggered mesh. The effects of normal stress differences on the flow stability are investigated. The numerical results obtained indicate that the increase of the negative second normal stress difference of viscoelastic fluid causes stability in the creeping flow in curved ducts, however, the increase of the first normal stress difference intensifies the instability. Furthermore, at the special value of Ψ₂/Ψ₁=?0.5, the interaction of the two normal stress differences results in a stable flow field.     

A criterion for detection of the onset of Dean instability in Newtonian fluids

Dean instability for Newtonian fluids in laminar secondary flow in 180° curved channels was studied experimentally and numerically. The numerical study used Fluent CFD code to solve the Navier–Stokes equations, focusing on flow development conditions and the parameters influencing Dean instability. An accurate criterion based on the radial gradient of the axial velocity was defined that allows detection of the instability threshold, and this criterion is used to optimize the grid geometry. The effects on Dean instability of the curvature ratio (from 5.5 to 20) and aspect ratio (from 0.5 to 12) are studied. In particular, we show that the critical value of the Dean number decreases with the increasing duct curvature ratio. The variation of the critical Dean number with duct aspect ratio is less regular.In the experimental study, flows were visualized in several tangential positions of a 180° curved channel with aspect ratio 8 and curvature ratio 10. The flow is hydrodynamically developed at the entrance to the curved channel. The critical Dean number is detected and the development of secondary flow vortices by additional counter-rotating vortex pairs is observed. A diagram of different critical Dean numbers is established.     

Experimental and numerical investigations of pressure drop in a rectangular duct with modified power law fluids

Numerical solutions are presented for fully developed laminar flow for a modified power law fluid (MPL) in a rectangular duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian behavior at low shear rates, through a transition region, to power law behavior at higher shear rates. The analysis identified a dimensionless shear rate parameter which, for a given set of operating conditions, specifies where in the shear rate range a particular system is operating, i.e. in the Newtonian, transition, or power law regions. The numerical results of the friction factor times Reynolds number for the Newtonian and power law region are compared with previously published results showing agreement within 0.05% in the Newtonian region, and 0.9% and 5.1% in the power law region. Rheological flow curves were measured for three CMC-7H4 solutions and were found to be well represented by the MPL constitutive equation. The friction factor times Reynolds number values were measured in the transition region for which previous measurements were unavailable. Good agreement was found between experiment and calculation thus confirming the validity of the analysis.     

Magnetohydrodynamic flow in a rectangular duct

The magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct, with an external magnetic field applied transverse to the flow, has been investigated. One of the duct's boundaries which is perpendicular to the magnetic field is taken partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 100. All the infinite series obtained are transformed to infinite integrals first and then to finite integrals which contain modified Bessel functions of the second kind. In this way, the difficulties associated with the computation of infinite integrals with oscillating integrands and slowly converging infinite series, the convergence of which is further affected for large values of M, have been avoided. It is found that, as M increases, boundary layers are formed near the non-conducting boundaries and in the interface region, and a stagnant region is developed in front of the conducting boundary for velocity field. The maximm value of magnetic field takes place on the conducting part. These behaviours are shown on some graphs.     

The generation of steady flow in a rectangular duct

The generation of the steady rectilinear flow of an Oldroyd B fluid in a rectangular duct is considered. The Laplace transform is used to separate out the time dependence from the governing equations and a Fourier series is used to solve the resulting Poisson problem in the space variables. It is seen that the presence of elasticity in the fluid gives rise to velocity overshoot. The solution of Waters and King for the planar channel is recovered at large aspect ratios.     

Prediction of wake in a curved duct

Experimental data on the development of an aerofoil wake in a curved stream are compared with calculations based on the k-ε model of turbulence with standard constants and with the model constant C_μ dependent on the local curvature. The mean velocity profile is asymmetric, the half-width of the wake is more on the inner side of the curved duct than on the outer side, and the turbulent shear stress decreases rapidly on the outer side. The standard k-ε model is able to satisfactorily reproduce this behaviour. Making C_μ dependent on the local radius improves the agreement on the inner side but slightly worsens it on the outer side.     

The end correction for power-law fluids in the capillary rheometer

The finite element scheme developed by Nickell, Tanner and Caswell is used to compute the entry and exit losses for creeping flow of power-law fluids in a capillary rheometer. The predicted entry losses for a Newtonian fluid agree well with available experimental and theoretical results. The entry losses for inelastic power-law fluids increased with decreasing flow behaviour index and show an increasing deviation from available upper bound results as the flow behaviour index in the power-law decreases.The exit losses are found to be finite for inelastic power-law fluids and increase as the flow behaviour index decreases. The predicted die swell for Newtonian fluids agrees well with the available experimental data while the influence of shear thinning is to reduce the die swell.The end correction which is the sum of the entry and exit losses relative to twice the viscometric wall shear stress varies from 0.834 for n = 1 to 2.917 for n = 1/6. This figure reaches a very high value as n tends to zero. The experimental variation in the Couette correction factor in capillary rheometry is explained in terms of the shear thinning characteristics of the fluid. It is concluded that the exit flow is not viscometric, contrary to a common assumption.     

Laminar flow of power-law fluids past a rotating cylinder

In this work, the continuity and momentum equations have been solved numerically to investigate the flow of power-law fluids over a rotating cylinder. In particular, consideration has been given to the prediction of drag and lift coefficients as functions of the pertinent governing dimensionless parameters, namely, power-law index (1 ≥ n ≥ 0.2), dimensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (0.1 ≤ Re ≤ 40). Over the range of Reynolds number, the flow is known to be steady. Detailed streamline and vorticity contours adjacent to the rotating cylinder and surface pressure profiles provide further insights into the nature of flow. Finally, the paper is concluded by comparing the present numerical results with the scant experimental data on velocity profiles in the vicinity of a rotating cylinder available in the literature. The correspondence is seen to be excellent for Newtonian and inelastic fluids.     

Unsteady laminar flow developing in a curved duct

The flow developing in a tightly curved U-bend of square cross section has been investigated experimentally and via numerical simulation. Both long-time averages and time histories of the longitudinal (streamwise) component of velocity were measured using a laser-Doppler velocimeter. The Reynolds number investigated was Re = 1400. The data were obtained at different bend angles, θ, and were confined to the symmetry plane of the bend. At Re = 1400, the flow entering the bend is steady, but by θ = 90° it develops an oscillatory component of motion along the outer-radius wall. Autocorrelations and energy spectra derived from the time histories yield a base frequency of approximately 0.1 Hz for these oscillations. Flow-visualization studies showed that the proximity of the outer-radius wall served to damp the amplitude of the spanwise oscillations.

Numerical simulations of the flow were performed using both steady and unsteady version of the finite-difference elliptic calculation procedure of Humphrey et al. (1977). Although the unsteadiness observed experimentally does not arise spontaneously in the calculations, numerical experiments involving the imposition of a periodic time-dependent perturbation at the inlet plane suggest that the U-bend acts upon the incoming flow so as to damp the amplitude of the imposed oscillation while altering its frequency.

The oscillations observed experimentally, and numerically as a result of the periodic perturbation, have been linked to the formation of Goertler-type vortices of the outer-radius wall in the developing flow. The vortices, which develop as a result of the centrifugal instability of the flow on the outer-radius wall, undergo a further transition to an unsteady regime at higher flow rates.     

Three-dimensional compressible potential flow in a curved duct

Summary The three-dimensional problem of compressible potential flow in a bend of a circular duct is studied by means of a perturbation method. As perturbation parameter is used the quotient between the radius of the duct and the radius of curvature of the duct, and the solution includes terms of the first and second order of this parameter. Numerical calculations have been performed for the values 0.2 and 0.1 of the perturbation parameter, and the plotted result shows how the axial velocity distribution changes from the straight to the curved region of the duct.

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Übersicht Mit Hilfe einer Störungsmethode wird die dreidimensionale Strömung einer kompressiblen Flüssigkeit in einer gekrümmten kreisförmigen Rohrleitung untersucht. Als Störungsparameter wird das Verhältnis von Querschnittradius zum Krümmungsradius des Leitungsbogens verwendet. Die Lösung wird bis zu Gliedern von zweiter Ordnung angegeben. Numerische Berechnungen werden für die Werte 0,1 und 0,2 des Störungsparameters durchgeführt. In Diagrammen wird gezeigt, wie sich die axiale Geschwindigkeitsverteilung beim Übergang vom geraden zum gekrümmten Teil der Leitung verändert.