On swirl development in a square cross-sectioned, S-shaped duct

The flow in a uniform square cross-sectioned, S-shaped duct was investigated experimentally, at Reynolds number (Re) = 4.73 × 10⁴ and 1.47 × 10⁵, using three S-ducts of different curvature and turning angle. The hydraulic diameter (D) for each S-duct is 150 mm. Besides studying the square cross-sectioned S-duct flow at moderately higher Re than current literature, the S-ducts’ geometry used in this study also have larger curvatures and higher turning angles than those reported in the literature. With surface pressure measurement and smoke wire flow visualization, flow separation at the inside wall of the first bend was detected. Using surface oil flow visualization on the bottom wall of the S-duct and cross-wires measurement at the duct exit, it is shown here that the swirl developed in the first bend was partly attenuated in the second bend due to the formation of swirl of opposite direction. The swirl of an opposite sign results in the formation of a clear dividing or separation line on the bottom wall (and top wall) of the duct. Additional flow features include the formation of streamwise vortices on the outer-wall of the second bend. These streamwise vortices can either be a pair of counter-rotating vortices or a single vortex. The formation mechanism of these streamwise vortices is explained using the Squire and Winter [J Aeronaut Sci 18(4):271–277, 1951] formula and it is shown that the said mechanism is applicable to both Re in the present study.     

Prediction of developing turbulent flow in 90°‐curved ducts using linear and non‐linear low‐Re k–ε models

This paper reports the outcome of applying two different low‐Reynolds‐number eddy‐viscosity models to resolve the complex three‐dimensional motion that arises in turbulent flows in ducts with 90^° bends. For the modelling of turbulence, the Launder and Sharma low‐Re k–ε model and a recently produced variant of the cubic non‐linear low‐Re k–ε model have been employed. In this paper, developing turbulent flow through two different 90^° bends is examined: a square bend, and a rectangular bend with an aspect ratio of 6. The numerical results indicate that for the bend of square cross‐section the curvature induces a strong secondary flow, while for the rectangular cross‐section the secondary motion is confined to the corner regions. For both curved ducts, the secondary motion persists downstream of the bend and eventually slowly disappears. For the bend of square cross‐section, comparisons indicate that both turbulence models can produce reasonable predictions. For the bend of rectangular cross‐section, for which a wider range of data is available, while both turbulence models produce satisfactory predictions of the mean flow field, the non‐linear k–ε model returns superior predictions of the turbulence field and also of the pressure and friction coefficients. Copyright © 2006 John Wiley & Sons, Ltd.     

Heat transfer to non-Newtonian flows over a cylinder in cross flow

Over a range of 102<Re^*<5800, 6.5<Pr^*<79, and 0.6<n<1, circumferential wall temperatures for water and aqueous polymer (purely viscous) solution flows over a smooth cylinder were measured experimentally. The cylinder was heated by passing direct electric current through it. Aqueous solutions of Carbopol 934 and EZ1 were used as power-law non-Newtonian fluids. The peripherally averaged heat transfer coefficient for purely viscous non-Newtonian fluids, at any fixed flow rate, decreases with increasing polymer concentration. A new correlation is proposed for predicting the peripherally averaged Nusselt number for power-law fluid flows over a heated cylinder in cross flow.     

Turbulent pipe flow of a drag-reducing rigid “rod-like” polymer solution

Fully developed turbulent pipe flow of an aqueous solution of a rigid “rod-like” polymer, scleroglucan, at concentrations of 0.005% (w/w) and 0.01% (w/w) has been investigated experimentally. Fanning friction factors were determined from pressure-drop measurements for the Newtonian solvent (water) and the polymer solutions and so levels of drag reduction for the latter. Mean axial velocity u and complete Reynolds normal stress data, i.e. u′, v′ and w′, were measured by means of a laser Doppler anemometer at three different Reynolds numbers for each fluid. The measurements indicate that the effectiveness of scleroglucan as a drag-reducing agent is only mildly dependent on Reynolds number. The turbulence structure essentially resembles that of flexible polymer solutions which also lead to low levels of drag reduction.     

MOTIONS, FORCES AND MODE TRANSITIONS IN VORTEX-INDUCED VIBRATIONS AT LOW MASS-DAMPING

These experiments, involving the transverse oscillations of an elastically mounted rigid cylinder at very low mass and damping, have shown that there exist two distinct types of response in such systems, depending on whether one has a low combined mass-damping parameter (low m^*ζ), or a high mass-damping (highm^*ζ ). For our low m^*ζ, we find three modes of response, which are denoted as an initial amplitude branch, an upper branch and a lower branch. For the classical Feng-type response, at highm^*ζ , there exist only two response branches, namely the initial and lower branches. The peak amplitude of these vibrating systems is principally dependent on the mass-damping (m^*ζ), whereas the regime of synchronization (measured by the range of velocity U^*) is dependent primarily on the mass ratio, m^*ζ. At low (m^*ζ), the transition between initial and upper response branches involves a hysteresis, which contrasts with the intermittent switching of modes found, using the Hilbert transform, for the transition between upper–lower branches. A 180° jump in phase angle φ is found only when the flow jumps between the upper–lower branches of response. The good collapse of peak-amplitude data, over a wide range of mass ratios (m^*=1–20), when plotted against (m^*+C_A) ζ in the “Griffin” plot, demonstrates that the use of a combined parameter is valid down to at least (m^*+C_A)ζ 0·006. This is two orders of magnitude below the “limit” that had previously been stipulated in the literature, (m^*+C_A) ζ>0·4. Using the actual oscillating frequency (f) rather than the still-water natural frequency (f_N), to form a normalized velocity (U^*/f^*), also called “true” reduced velocity in recent studies, we find an excellent collapse of data for a set of response amplitude plots, over a wide range of mass ratiosm^* . Such a collapse of response plots cannot be predicted a priori, and appears to be the first time such a collapse of data sets has been made in free vibration. The response branches match very well the Williamson–Roshko (Williamson & Roshko 1988) map of vortex wake patterns from forced vibration studies. Visualization of the modes indicates that the initial branch is associated with the 2S mode of vortex formation, while the Lower branch corresponds with the 2P mode. Simultaneous measurements of lift and drag have been made with the displacement, and show a large amplification of maximum, mean and fluctuating forces on the body, which is not unexpected. It is possible to simply estimate the lift force and phase using the displacement amplitude and frequency. This approach is reasonable only for very low m^*.     

Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models

The steady planar sink flow through wedges of angle π/α with α≥1/2 of the upper convected Maxwell (UCM) and Oldroyd-B fluids is considered. The local asymptotic structure near the wedge apex is shown to comprise an outer core flow region together with thin elastic boundary layers at the wedge walls. A class of similarity solutions is described for the outer core flow in which the streamlines are straight lines giving stress and velocity singularities of O(r⁻²) and O(r⁻¹), respectively, where r1 is the distance from the wedge apex. These solutions are matched to wall boundary layer equations which recover viscometric behaviour and are subsequently also solved using a similarity solution. The boundary layers are shown to be of thickness O(r²), their size being independent of the wedge angle. The parametric solution of this structure is determined numerically in terms of the volume flux Q and the pressure coefficient p₀, both of which are assumed furnished by the flow away from the wedge apex in the r=O(1) region. The solutions as described are sufficiently general to accommodate a wide variety of external flows from the far-field r=O(1) region. Recirculating regions are implicitly assumed to be absent.     

Experimental investigation on turbulent flow in a square-sectioned 90-degree bend

The development of steady, turbulent flow in a 90° section of a curved square duct was studied at a Reynolds number of 4 × 10⁴ by hot-wire anemometer. The curved duct has a cross-section measuring 80 × 80 mm and a curvature radius ratio of 4 and is connected with a long, straight duct at its both ends. The longitudinal and lateral components of mean and fluctuating velocities, and the Reynolds stresses were measured by the method of rotating a probe with an inclined hot-wire. The velocity fields of the primary and secondary flows, and the Reynolds stress distributions in the cross-section were illustrated in the form of contour map. The development of the primary flow was found to be connected with a strong pressure gradient near the outer and inner wall and a secondary flow induced in the cross-section of the bend by a pressure difference between the outer and inner wall and a centrifugal force acting on the fluid; the fluid is accelerated near the inner wall and decelerated near the outer wall between the bend angle ϕ ≅ 0° and ϕ ≅ 30°, but an increase and decrease of the fluid velocity are reversed between ϕ ≅ 30° and ϕ ≅ 90°. The fluctuating velocity correlations, i.e. the Reynolds stresses follow a complicated progress according to the complex development of the primary flow. The results obtained can be available to verify various types of turbulence models and to develop new models. Received: 10 May 1999/Accepted: 15 March 2000     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

On the squeeze flow of a power-law fluid between rigid spheres

The lubrication solution for the squeeze flow of a power-law fluid between two rigid spherical particles has been investigated. It is shown that the radial pressure distribution converges to zero within the gap between the particles for any value of the flow index, n, provided that the gap separation distance is sufficiently small. However, in the case of the viscous force, it is useful to consider that there are two contributions. The first is developed in the inner region of the gap and corresponds to the lubrication limit. The second is due to an integration of the pressure in the adjacent outer region of the gap. The relative contribution to the force in this outer region increases as n decreases and the separation distance increases. In particular, for flow indices in the range n>1/3, the contribution in the outer region is negligible if the separation distance is sufficiently small. For n1/3, this is the dominant term and an accurate prediction of the viscous force is possible only for discrete liquid bridges.Based on “zero” pressure and lubrication criteria for the upper limits of integration, two closed-form solutions have been derived for the viscous force. Both are accurate for n>0.5 and are in close agreement with a previously published asymptotic solution in the range n>0.6. For smaller values of n, the asymptotic solution over-estimates the viscous force and predicts a singularity when n approaches 1/3. The two closed-form solutions show continuous and monotonic behaviour for all values of n. Moreover, the solution satisfying the lubrication limit is valid in the range n<1/3 provided that it is restricted to liquid bridges.     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

Near-wall flow in a three-dimensional boundary layer on the endwall of a 30° bend

 Turbulence measurements are reported on the three-dimensional turbulent boundary layer along the centerline of the flat endwall in a 30° bend. Profiles of mean velocities and Reynolds stresses were obtained down to y ⁺≈2 for the mean flow and y ⁺≈8 for the turbulent stresses. Mean velocity data collapsed well on a simple law-of-the-wall based on the magnitude of the resultant velocity. The turbulence intensity and turbulent shear stress magnitude both increased with increased three-dimensionality. The ratio of these two quantities, the a ₁ structure parameter, decreased in the central regions of the boundary layer and showed profile similarity for y ⁺<50. The shear stress vector angle lagged behind the velocity gradient vector angle in the outer region of the boundary layer, however there was an indication that the shear stress vector tends to lead the velocity gradient vector close to the wall. Received: 16 July 1996/Accepted: 14 July 1997     

Strain strengthening effects in Witwatersrand quartzite

A series of uniaxial compression specimens were tested over a range of applied ram displacement rates of 8.9 × 10⁻⁴ to 8.9 mm/sec to elucidate the effects of loading rate on the uniaxial compressive fracture stress of Witwatersrand quartzite. It was demonstrated that even within standard loading rate ranges, considerable scatter in the fracture strength (under uniaxial compression) existed in this particular quartzite rock. Nevertheless, a definite trend of increasing fracture resistance with increasing monotonic loading rate was evident inasmuch that increasing the loading rate (strain rate) by four orders of magnitude increase the fracture strength by almost 2.8 times. Prior fatigue loading also produced a significant strain strengthening as the uniaxial compressive fracture stress tended to increase in a sigmoidal fashion with increasing number of fatigue cycles prior to testing. Indeed, the fracture strength of quartzite was almost doubled in value after 10⁻ cycles. Plane strain fracture toughness tests utilising three point bend specimens were conducted and an average of K_lc = 1.7 MPa√m was realized. In both the uniaxial compression tests and the fracture toughness tests, failure occurred by crack extension predominantly by a transgranular flat cleavage-like mode through pure quartzite (silica) regions. However, crack extension was also observed to occur in an intergranular “ductile-like” mode through areas associated with inclusions prevalent in the quartzite.     

On corner flows of Oldroyd-B fluids

Exact series solutions for planar creeping flows of Oldroyd-B fluids in the neighbourhood of sharp corners are presented and discussed. Both reentrant and non-reentrant sectors are considered. For reentrant sectors it is shown that more than one type of series solution can exist formally, one type exhibiting Newtonian-like asymptotic behaviour at the corner, away from walls, and another type exhibiting the same kind of asymptotics as an Upper Convected Maxwell (UCM) fluid. The solutions which are Newtonian-like away from walls are shown to develop non-integrable stress singularities at the walls when the no-slip velocity boundary condition is imposed. These mathematical solutions are therefore inadmissible from the physical viewpoint under no-slip conditions. An inadmissible solution, with stress singularities which are not everywhere integrable, is identified among the solutions of UCM-type. For a 270° reentrant sector the radial behaviour of the normal stress is everywhere r^−0.613. In the viscometric region near a wall, the radial normal stress σ^rr behaves like (rε)^−0.613, where ε is the angle made with the wall. In addition σ^rθ is infinite (not integrable) at the wall even when r is non-zero. Another UCM-type solution has a normal stress behaviour away from walls which is r^−0.985 for 270° sector. Again, this solution has a non-integrable stress singularity and is therefore inadmissible. Finally, for non-reentrant sectors it is shown that the flow is always Newtonian-like away from walls.     

Experimental flow study over a blunt-nosed axisymmetric body at incidence

The flowfield over a blunt-nosed cylinder was examined experimentally at a low subsonic speed for Re=1.88×10⁵ and angles of attack up to 40°. Velocity measurements were carried out (employing a seven-hole Pitot tube) as well as wall static pressure and wall shear-stress measurements. Surface flow visualization was applied using liquid crystals and a mixture of oil–TiO₂. For all the examined cases no flow asymmetries were found. For high angles of attack (20° and above) a separation “bubble” appears at the leeside of the nose area (streamwise flow separation). The basic feature of the circumferential pressure distribution at the after body area for these angles of attack is a plateau close to the suction peak and a fast recovery next to it. One streamwise vortex on each side of the symmetry plane is formed as well as a separation bubble about 90° far from this plane, where the cross-flow primary separation line is located. Each cross-flow primary separation line starts at the leeside nose area and moves towards the windward side along the cylindrical after body. The space between the two primary separation lines close to the wall is characterized by high flow fluctuations on the leeside, compared to the low fluctuations of the windward side.     

Direct numerical simulation of turbulent heat transfer in annuli: Effect of heat flux ratio

Fully developed turbulent flow and heat transfer in a concentric annular duct is investigated for the first time by using a direct numerical simulation (DNS) with isoflux conditions imposed at both walls. The Reynolds number based on the half-width between inner and outer walls, δ=(r₂-r₁)/2, and the laminar maximum velocity is Re_δ=3500. A Prandtl number Pr=0.71 and a radius ratio r^*=0.1 were retained. The main objective of this work is to examine the effect of the heat flux density ratio, q^*=q₁/q₂, on different thermal statistics (mean temperature profiles, root mean square (rms) of temperature fluctuations, turbulent heat fluxes, heat transfer, etc.). To validate the present DNS calculations, predictions of the flow and thermal fields with q^*=1 are compared to results recently reported in the archival literature. A good agreement with available DNS data is shown. The effect of heat flux ratio q^* on turbulent thermal statistics in annular duct with arbitrarily prescribed heat flux is discussed then. This investigation highlights that heat flux ratio has a marked influence on the thermal field. When q^* varies from 0 to 0.01, the rms of temperature fluctuations and the turbulent heat fluxes are more intense near the outer wall while changes in q^* from 1 to 100, lead to opposite trends.     

Heat transfer in subsonic high-temperature gas flow through three-dimensional curved channels

The heat transfer in subsonic high-temperature (T 2000°K) gas flow through a curved channel of rectangular cross section has been experimentally investigated. The local heat flow into the channel wall was measured by the modified gradient method, which consists in subdividing the walls by means of thin annular thermally-insulating partitions, measuring the temperature in the upper (gas) and lower (coolant) sections of the modules thus formed, and using these measurements to determine the local heat flux q_w on the assumption that the thermal field in the module is homogeneous. The soundness of this method has been demonstrated theoretically and experimentally and the expediency of using it in the intensive wall cooling regime has been confirmed. The method is employed to find the local heat flux fields over the- entire surface of the channel. The integrated fluxes q_w coincide to within 5% with the independently determined total increase in the enthalpy of the water in the cooling channels. A distinguishing feature of the investigation is the high relative curvature of the bend in the channel, which leads to the formation of a zone of intense separation on the convex (inner) wall. Three types of channel are examined. These differ with respect to the section beyond the bend which is either long or short or short with a contraction. A close correlation between the characteristics of the q_w fields and the hydrodynamic effects is detected and explained. These effects comprise: separation and reattachment of the flow, secondary effects in the bend, the formation of an unclosed separation zone in the short outlet section, the localization of this zone when the outlet section includes a contraction, and specific gas dynamic effects near the intersection of the surfaces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 93–99, May–June, 1989.     

NUMERICAL STUDIES OF TRANSITIONAL TURBULENT PULSATILE FLOW IN PIPES WITH RING–TYPE CONSTRICTIONS

Pulsatile flows in the vicinity of mechanical ring-type constrictions in pipes were studied for transitional turbulent flow with a Reynolds number (Re) of the order of 10⁴. The Womersley number (Nw) is in the range 30–50, with a corresponding Strouhal number (St) range of 0·0143–0·0398. The pulsatile flows considered are a pure sinusoidal flow, a physiological flow and an experimental pulsatile flow profile for mechanical aortic valve flow simulations. Transitional laminar and turbulent flow characteristics in an alternating manner within the pulsatile flow fields were studied numerically. It was observed that fluid accelerations tend to suppress the development of flow disturbances. All the instantaneous maximum values of turbulent kinetic energy, turbulent viscosity and turbulent shear stress are smaller during the acceleration phase than during the deceleration period. Various parametric equations have been formulated through numerical experimentation to better describe the relationships between the instantaneous flow rate (Q), the pressure loss (ΔP), the maximum velocity (V_max), the maximum vorticity (ζ_max), the maximum wall vorticity (ζ_w,max), the maximum shear stress (τ_max) and the maximum wall shear stress (τ_w,max) for turbulent pulsatile flow in the vicinity of constrictions in the vascular tube. An elliptic relationship has been found to exist between the instantaneous flow rate and the instantaneous pressure gradient. Other linear and quadratic relations between various flow parameters were also obtained.     

OPERATOR–SPLITTING COMPUTATION OF TURBULENT FLOW IN AN AXISYMMETRIC 180° NARROWING BEND USING SEVERAL k—ϵ MODELS AND WALL FUNCTIONS

This paper describes a finite element implementation of an operator-splitting algorithm for solving transient/steady turbulent flows and presents solutions for the turbulent flow in an axisymmetric 180° narrowing bend, a benchmark problem dealt with at the 1994 WUA-CFD annual meeting. Three k–ϵ based models are used: the standard linear k–ϵ model, a non-linear k–ϵ model and an RNG k–ϵ model. Flow separation after the bend, as observed in the experiment, is predicted by the RNG model and by both the linear and non-linear k–ϵε models with van Driest mixing length wall functions. Good agreement with experimental data of pressure distribution on bending walls is obtained by the present numerical simulation. Results show that there is very little difference between the linear and non-linear k–ϵε models in terms of predicted velocity fields and that the non-linearities mainly affect the distribution of turbulent normal stress and pressure, in analogy to the effect of second-order viscoelastic fluid models on laminar flow. Both the linear and non-linear k–ϵε models fail to predict any flow separation if logarithmic wall functions are used.     

Numerical simulation of fluid flow and heat transfer characteristics in channel with V corrugated plates

A detailed numerical study is carried out to investigate fluid flow and heat transfer characteristics in a channel with heated V corrugated upper and lower plates. The parameters studied include the Reynolds number (Re = 2,000–5,500), angles of V corrugated plates (θ = 20°, 40°, 60°), and constant heat fluxs (q″ = 580, 830, 1,090 W/m²). Numerical results have been validated using the experimented data reported by Naphon, and a good agreement has been found. The angles of V corrugated plates (θ) and the Reynolds number are demonstrated to significantly affect the fluid flow and the heat transfer rate. Increasing the angles of V corrugated plates can make the heat transfer performance become better. The increasing Reynolds number leads to a more complex fluid flow and heat transfer rate. The numerical calculations with a non-equilibrium wall function have a better accuracy than with a standard wall function for solving high Reynolds numbers or complex flow problems.