Laminar flow of some inelastic time-dependent fluids in pipes

The paper presents some modification of the previously published model of thixotropic fluids for the case of fluids with irreversible destruction of the structure. In the second part of the paper the laminar flow of this kind of fluids in pipes is discussed. A simple method of determining the initial value of the structural parameter at the inlet to the pipe is proposed. D pipe diameter, m - k ₁,k ₂ rheological parameter in eq. (5), Nsⁿ/m² - L pipe length, m - m ₁,m ₂ rheological parameter in eq. (5) - n ₁,n ₂ rheological parameter in eq. (6) - p pressure, Pa - v mean linear velocity in the pipe, m/s - rheological parameter in eq. (2), = 1 s - shear rate, s^–1 - structural parameter - ₀ initial value of structural parameter - ₁, ₂ mean value of natural time for breakdown and build-up of the structure, respectively, s - shear stress, Pa - fluid density, kg/m³ - mean value of the friction factor - De modified Deborah number defined by eq. (11) - Re _m generalized Reynolds number defined by eq. (16) - Re _n generalized Reynolds number defined by eq. (10) - Se structural numbers defined eq. (12)     

Memory effects during the flow of thixotropic fluids in pipes

Summary The paper presents the phenomenon of thioxotropy from the point of view of the theory of fluids with fading memory. In the first part of the paper the mechanism of thixotropy was discussed in order to justify the application of the concept of structural parameter (this parameter occurs in the previously presented rheological model of thixotropic materials). In the second part of the paper an equation was derived, which enables the prediction of the mean value of the friction factor during the flow of a thixotropic fluid in a pipe. According to the obtained equation the friction factor is a function of three dimensionless numbers: the generalized Reynolds number, a modified Deborah number and a new dimensionless number which may be called a structural number. The preliminary experimental results confirmed the applicability of the obtained equation.

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Zusammenfassung Die Veröffentlichung behandelt das Phänomen der Thixotropie vom Standpunkt der Theorie der Flüssigkeiten mit schwindendem Gedächtnis. Im ersten Teil wird der Mechanismus der Thixotropie untersucht und die Einführung eines sog. Strukturparameters begründet (dieser Parameter kommt bereits in dem früher behandelten rheologischen Modell eines thixotropen Körpers vor). Im zweiten Teil wird dann eine Formel abgeleitet, welche die Voraussage des mittleren Wertes des Widerstandskoeffizienten bei der Strömung einer thixotropen Flüssigkeit durch ein Rohr ermöglicht. Dieser Formel gemäß ist der Widerstandskoeffizient eine Funktion von drei dimensionslosen Zahlen: einer verallgemeinerten Reynolds-Zahl, einer modifizierten Deborah-Zahl und einer neuen dimensionslosen Zahl, die als Struktur-Kennzahl bezeichnet werden kann. Die vorläufigen Versuchsergebnisse bestätigen die Brauchbarkeit der abgeleiteten Formel.

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a rheological parameter in eq. [1], s^–1 - A rheological parameter in eq. [1]; function defined in eq. [15] - b rheological parameter in eq. [1] - B constant in eq. [15] - c rheological parameter in eq. [4] - c function defined in eq. [4] - C function defined in eq. [48] (see also eq. [43]) - D pipe diameter,m - K ₁,K₂ coefficients of proportionality in eq. [6] - k rheological parameter in eq. [12], Nsⁿ/m² - k ^* rheological parameter in eq. [1], Ns^m/m² - L pipe length, m - m rheological parameter in eq. [1] - n rheological parameter in eq. [12] - N number of particles in unit volume - p pressure, Pa - p ₀ pressure at the pipe entrance, Pa - r radial coordinate, m - R pipe radius, m - s rheological parameter in eq. [1] - t time, s - u _z axial local velocity in the pipe, m/s - v mean linear velocity in the pipe, m/s - z axial coordinate, m - rheological parameter in eq. [5], = 1 s - shear rate, s^–1 - nominal shear rate defined by eq. [39], s^–1 - structural parameter - substantial derivative of structural parameter, s^–1 - _e equilibrium structural parameter in eqs. [2] and [5] - _en nominal structural parameter - ₀ initial value of structural parameter - function of natural time - mean value of natural time, s - shear stress, Pa - ₀ shear stress field atZ = 0 (at pipe entrance) - _y0 equilibrium yield stress, Pa - shear stress field atz - fluid density, kg/m³ - v number of bonds in an average aggregate - mean value of the friction factor - De modified Deborah number defined by eq. [46] - Re generalized Reynolds number defined by eq. [45] - Se structural number defined by eq. [41a] With 4 figures and 1 table     

Transitionally rough zero pressure gradient turbulent boundary layers

Near-wall measurements are performed to study the effects of surface roughness and viscous shear stresses on the transitionally rough regime (5 < k ⁺ < 70) of a zero pressure gradient turbulent boundary layer. The x-dependence is known from the eleven consecutive measurements in the streamwise direction, which allows for the computation of the streamwise gradients in the boundary layer equations. Thus, the skin friction is computed from the integrated boundary layer equation with errors of 3 and 5% for smooth and rough, respectively. It is found that roughness destroys the viscous layer near the wall, thus, reducing the contribution of the viscous stress in the wall region. As a result, the contribution in the wall shear stress due to form drag increases, while the viscous stress decreases. This yields Reynolds number invariance in the skin friction as k ⁺ increases into the fully rough regime. Furthermore, the roughness at the wall reduces the high peak of the streamwise component of the Reynolds stress in the near-wall region. However, for the Reynolds wall-normal and shear stress components, its contribution is not significantly altered for sand grain roughness.     

An improved EMMS model for turbulent flow in pipe and its solution

Based on the existing energy-minimization multi-scale (EMMS) model for turbulent flow in pipe, an improved version is proposed, in which not only a new radial velocity distribution is introduced but also the quantification of total dissipation over the cross-section of pipe is improved for the dominant mechanism of fully turbulent flow in pipe. Then four dynamic equality constraints and some other constraints are constructed but there are five parameters involved, leading to one free variable left. Through the compromise in competition between dominant mechanisms for laminar and fully turbulent flow in pipe respectively, the above four constructed dynamic equality constraints can be closed. Finally, the cases for turbulent flow in pipe with low, moderate and high Reynolds number are simulated by the improved EMMS model. The numerical results show that the model can obtain reasonable results which agree well with the data computed by the direct numerical simulation and those obtained by experiment. This illustrates that the improved EMMS model for turbulent flow in pipe is reasonable and the compromise in competition between dominant mechanisms is indeed a universal governing principle hidden in complex systems. Especially, one more EMMS model for a complex system is offered, promoting the further development of mesoscience.     

Simulations of dispersed turbulent multiphase flow

Direct numerical simulations of homogeneous isotropic turbulence are used to investigate the effects of turbulence on the transport of particles in gas flows or bubbles in liquid flows. The inertia associated with the bubbles or the particles leads to locally strong concentrations of these in regions of instantaneously strong vorticity for bubbles or strain-rate for particles. This alters the average settling rates and other processes. If the mass-loading of the dispersed phase is significant a random “turbulent” flow is generated by the particle settling. A simple demonstration of this is given, showing the statistically axisymmetric character of this flow and how it can modify an ambient turbulent flow.     

The turbulent/non-turbulent interface in an adverse pressure gradient turbulent boundary layer

The turbulent/non-turbulent interface (TNTI) in an adverse pressure gradient (APG, β = 1.45) turbulent boundary layer (TBL) is explored here by using direct numerical simulation (DNS) data; β is the Clauser pressure gradient parameter. For comparison, the DNS data for a zero pressure gradient (ZPG) TBL is included. The interface is extracted with an approach based on enstrophy criteria. Depending on the enstrophy, the outer boundary layer flow can be classified into the free stream, boundary layer wake, and intermittent flow regimes. The fractal dimension of the interface is obtained by using the box-counting algorithm, and was found to be constant over a long range of box sizes. The TNTI shows a monofractal behavior. The geometric complexity of a TNTI can be determined in terms of the genus, which is defined as the number of handles in a geometric object. We examine the volume and projection area of the genus of the TNTI to analyze the entrainment process. The geometric complexity of the APG TBL interface and the local entrainment are greater than those of the ZPG TBL, as is evident in the increases in the genus near the interface. The local entrainment velocity is dominantly affected by the viscous diffusion at the interface.     

Measurement of pressure loss in the flow of polymer solutions through wavy channels

In order to examine the flow behavior of polymer solutions through porous media, the measurement of pressure loss and the experiment for flow visualization were carried out with wavy channels as one of the model channels of porous media. The test fluids used are aqueous solutions of polyacrylamide (PAA) with two different concentrations. The occurrence of the excess pressure loss, which was not due to the effect of the centrifugal force, was found for the PAA solutions. The relations between the friction factor ratio and the Deborah number were similar to that obtained for the flow through porous media. Furthermore, the results of the flow visualization suggest that the elongational property of the PAA solutions is connected with the occurrence of the excess pressure loss.     

The turbulent character and pressure loss produced by periodic symmetric ribs in a circular duct

The structural character and steady-state statistics of the turbulence inside a rib-wall circular duct is investigated by the large-eddy simulation (LES) methodology. The impetus of this study is to gain an understanding of the principle physics attributing to minimizing the pressure recovered (or maximizing the pressure loss) within the core flow. For a rib periodicity with height (h) to pitch (p) ratio p/h=5, the computational results show that the majority of turbulence produced due to the rib’s presence is concentrated near the rib crest leading edge. Pairs of counter-rotating streamwise vortices form soon after the leading edge that are quickly convected radially toward the core flow. The turbulent activity within the duct trough region is negligible compared to the turbulence levels of the core flow. At this rib periodicity, the separated shear layers from the trailing edge of each rib nearly reattach to the trough floor before reaching the next rib. The resultant irrecoverable pressure loss in the form a centerline frictional coefficient is verified by an ‘at-sea’ test on board a US Navy submarine. Based on the duct diameter, their Reynolds numbers are Re_D^LES=8×10³ and (Re_D^exp)_avg=4×10⁶, respectively.     

Correlation of turbulent flow rate-pressure drop data for non-newtonian solutions and slurries in pipes

Isothermal and non-isothermal flow rate-pressure drop data in turbulent flow through smooth pipes have been obtained for non-Newtonian fluids, including aqueous solutions of polymers and aqueous suspensions of titanium dioxide. It has been found that the friction factor, f, is a function of a new form of Reynolds number, Re_B, based on the parameters A, x and w of Bowen's correlation, viz.

$\text{τ}{}_{\mathrm{w}}\text{D}{}^{\mathrm{x}}\text{=A}\text{u}{}^{\mathrm{w}}$

where τ_w is the wall shear strees, ?u the mean velocity, D the pipe diameter; A, x and w are experimentally derived parameters which characterise the fluid.     

Partial-average-based equations of incompressible turbulent flow

This paper presents a detailed study of incompressible turbulent flow based on a newly developed statistical partial average scheme. As the ensemble average is taken on two groups of turbulent fluctuations separately, the partial average scheme is able to capture the first-order statistical moment of the fluctuation field, providing valuable information in addition to what have been known in the past from the conventional Reynolds average. The first-order statistical moment serves as the foundation in formulating theories of orthotropic turbulence and a momentum transfer chain in the modeling of second-order correlation terms, and eventually leads to a complete set of equations of incompressible turbulent flow. Without any empirical coefficients, the same set of the equations is used to simulate statistical mean behaviors and coherent structures of various benchmark turbulent flows. The simulated results are in good agreement with experimental data.     

Laminar/turbulent oscillating flow in circular pipes

A two-dimensional oscillating flow analysis was conducted simulating the gas flow inside Stirling engine heat exchangers. Both laminar and turbulent oscillating pipe flow were investigated numerically for Re_max = 1,920 (Va = 80), 10,800 (Va = 272), 19,300 (Va = 272), and 60,800 (Va = 126). The results are here compared with experimental results of previous investigators. Predictions of the flow regime on present oscillating flow conditions are also checked by comparing velocity amplitudes and phase difference with those from laminar theory and quasi-steady profile. A high Reynolds number k-ε turbulence model was used for turbulent oscillating pipe flow. Finally, the performance of the k-ε model was evaluated to explore the applicability of quasi-steady turbulent models to unsteady oscillating flow analysis.     

Comparison of organized motions in a constant and an adverse pressure gradient turbulent flow

The distribution of the statistical properties of coherent motions across a fully developed feed pipe is observed to change drastically as adverse pressure gradient is applied in a conical diffuser. These changes are associated with distortion of the turbulence structure and becomes more pronounced as the flow approaches detachment. A conceptual model based on present measurements effectively accounts for major turbulence characteristics in the diffuser.     

The effects of inlet sharpness on the pipe contraction pressure loss coefficient

Pipe contractions occur in heat exchangers at the entrance to tube bundles and in conjunction with enlargements, as ferrules which are often used for the close control of the flow distribution within the tube bundles. These contractions are normally sharp at the commencement of service but suffer erosion or corrosion over the lifetime of the plant. This significantly affects their pressure loss characteristics and upsets the flow distribution. Thus it is important to be able to predict the variation of contraction pressure loss coefficient with variations in the small-bore pipe inlet geometry, referred to as inlet sharpness. There are no known experimental data for the effects of inlet sharpness on the pipe contraction pressure loss coefficient, but there are data for intakes set flush in a plane wall which are used as approximations. Experimental data showing the variation of pressure loss coefficient with the inlet sharpness are presented and compared with the approximate data. The comparison shows significant differences.     

Two‐phase pressure drop caused by sudden flow area contraction/expansion in small circular pipes

Theoretical studies have been made to determine the pressure drops caused by abrupt flow area expansion/contraction in small circular pipes for two‐phase flow of air and water mixtures at room temperature and near atmospheric pressure. Two‐phase computational fluid dynamics (CFD) calculations, using Eulerian–Eulerian model (with the air phase being compressible for pipe contraction case) are employed to calculate the pressure drop across sudden expansion and contraction. The pressure drop is determined by extrapolating the computed pressure profiles upstream and downstream of the expansion/contraction. The larger and smaller tube diameters are 1.6 and 0.84 mm, respectively. Computations have been performed with single‐phase water and air, and two‐phase mixtures in a range of Reynolds number (considering all‐liquid flow) from 1000 to 12 000 and flow quality from 1.2 × 10^?3 to 1.6 × 10^?2. The numerical results are validated against experimental data from the literature and are found to be in good agreement. The expansion and contraction loss coefficients are found to be different for single‐phase flow of air and water, and they agreed reasonably well with the commonly used theoretical predictions. Based on the numerical results as well as experimental data, correlations are developed for two‐phase flow pressure drops caused by the flow area contraction as well as expansion. Copyright © 2010 John Wiley & Sons, Ltd.     

Heat transfer characteristics and Nusselt number correlation of turbulent pulsating pipe air flows

Heat transfer characteristics to turbulent pulsating pipe flows under a wide range of Reynolds number and pulsation frequency were experimentally investigated under uniform heat flux condition. Reynolds number was varied from 8462 to 48540 while the frequency of pulsation ranged from 1 to 29.5 Hz. The results showed that the relative mean Nusselt number is strongly affected by both pulsation frequency and Reynolds number. Enhancements in mean Nusselt number of up to 50% were obtained at medium pulsation frequency between 4.1 and 13.9 Hz for Reynolds number range of 8462 to 14581. An enhancement of up to 50% in mean Nusselt number was obtained at high pulsation frequency range between 13.9 and 29.5 Hz, specially as Reynolds number is close to 15000, while a reduction was observed at higher Reynolds number more than 21200. This reduction, at high Reynolds number, increased as pulsation frequency increased. Also, there was a reduction in mean Nusselt number of up to 20% that obtained at low pulsation frequency range between 1 and 4.1 Hz for Reynolds number range of 8462 to 48543. A significant reduction in mean Nusselt number of up to 40% was obtained at medium pulsation frequency between 4.1 and 13.9 Hz for Reynolds number range of 21208 to 48543. Empirical equations have been developed for the relative mean Nusselt number that related to Reynolds number and dimensionless frequency with about uncertainty of 10% rms.The support of both King Fahd University of Petroleum and Minerals and Cairo University for this research is acknowledged.     

Numerical simulation of turbulent free-surface flow in curved channel

This paper presents first results of numerical simulation of turbulent free-surface flow. Simple implementation of surface capturing method is based on the variable density approach. The flow is treated as if there is only one fluid, but with variable material properties (density, viscosity). The switch in these values is done by a function resulting from the mass conservation principle. This approach simplifies the implementation of turbulence model. In this case the SST k−ω model was chosen in modification given by Hellsten.Numerical solution was carried out by finite-volume method with explicit Runge-Kutta time-integration. The artificial compressibility method was used for time-marching search for steady state solution. The whole model was tested on horizontally placed square-sectioned 90^∘ bend, which was partially filled by the water. The main goal of this study was to demonstrate the applicability of this model and solution method for capturing the water-air interface as well as for predicting the turbulent effects in both fluids.     

Numerical and experimental study of turbulent impinging twin-jet flow

The two dimensional impinging circular twin-jet flow with no-cross flow is studied numerically and experimentally. The theoretical predications are carried out through numerical procedure based on finite volume method to solve the governing mass, momentum, turbulent kinetic energy and turbulent kinetic energy dissipation rate. The parameters studied were jet Reynolds number (9.5 × 10⁴  Re  22.4 × 10⁴), nozzle to plate spacing (3  h/d  12), nozzle to nozzle centerline spacing (l/d = 3, 5 and 8) and jet angle (0°  θ  20°). It is concluded that the stagnation primary point moves away in the radial main flow direction by increasing the jet angle. This shift becomes stronger by increasing the nozzle to nozzle centerline spacing (l/d). A secondary stagnation point is set up between two jets. The value of pressure at this point decreases by decreasing Reynolds number and/or increasing the jet angle.

The sub atmospheric region occurs on the impingement plate. It increases strongly by increasing Reynolds number and decreases as the jet angle and/or a nozzle to plate spacing increases. The spreading of jet decreases by increasing nozzle to plate spacing. The intensity of re-circulation zone between two jets decreases by increasing of h/d and jet angle. The increase of turbulence kinetic energy occurs within high gradient velocity.     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

This paper presents a model for the evolution of a transversal concentration profile of a macromolecular solution (PEO) injected into a cylindrical pipe at turbulent flow conditions (R 40000). This model, based on the diffusion of a scalar quantity emitted by two diametrically opposed point sinks, proves to be in good agreement with the experimental data. C concentration - C _w wall concentration - C _i initial concentration before injection - x downstream distance from the slot - y normal distance from the wall - characteristic height of diffusion, i.e. the value ofg at whichC/C _w = 0.5 - n characteristic exponent - R radius of pipe - D diameter of pipe - a, b constants - L _m mixing length, i.e. the value ofx at whichC _w /C _i =Q _i /Q _T - Q _i flow rate of injection - Q _T flow rate - f, g annex functions - n ₀ maximal value ofn     

Direct numerical simulation of a turbulent Couette-Poiseuille flow,part 2: Large- and very-large-scale motions

A direct numerical simulation dataset of a fully developed turbulent Couette-Poiseuille flow is analyzed to investigate the spatial organization of streamwise velocity-fluctuating u-structures on large and very large scales. Instantaneous and statistical flow fields show that negative-u structures with a small scale on a stationary bottom wall grow throughout the centerline due to the continuous positive mean shear, and they penetrate to the opposite moving wall. The development of an initial vortical structure related to negative-u structures on the bottom wall into a large-scale hairpin vortex packet with new hairpin vortices, which are created upstream and close to the wall, is consistent with the auto-generation process in a Poiseuille flow (Zhou et al., J. Fluid Mech., vol. 387, 1999, pp. 353–396). Although the initial vortical structure associated with positive-u structures on the top wall also grows toward the bottom wall, the spatial development of the structure is less coherent with weak strength due to the reduced mean shear near the top wall, resulting in less turbulent energy on the top wall. The continuous growth of the structures from a wall to the opposite wall explains the enhanced wall-normal transport of the streamwise turbulent kinetic energy near the centerline. Finally, an inspection of the time-evolving instantaneous fields and conditional averaged flow fields for the streamwise growth of a very long structure near the centerline exhibits that a streamwise concatenation of adjacent large-scale u-structures creates a very-large-scale structure near the channel centerline.