Smooth Pipe Flow Research Articles

Developed log-wake law and turbulent behaviour of flow along a stepped spillway

Abstract The log-wake law for turbulent current has been developed and tested with laboratory data on turbulent flow in smooth pipes. However, flow with turbulence and vortices in a stepped spillway have not been described. Therefore, in this study, a log-wake law has been developed for use in stepped spillway systems. It can be divided into three parts. The first part, a logarithmic equation, describes the effect of shear stress between the flow layers with a von Kármán constant of 0.41. The second part, a third-degree polynomial, describes the effect of the shear stress on the wall. The last part, a fourth-degree polynomial, describes the effect of changing the flow pressure distribution, similar to the wall-free shear stress. Calibration tests (68 datasets) are used with a flow rate between 0.0233 and 3.285 m3/s, a spillway slope of 14–30°, and a step height of 0.0380–0.610 m. The developed log-wake law characterized the flow in a stepped spillway well. The limitation of the equation is a maximum flow velocity of 4 m/s; the accuracy of this equation decreases as the step height increases.

Experimental and Numerical Analysis of the Effect of Rheological Models on Measurements of Shear-Thinning Fluid Flow in Smooth Pipes

The aim of this research is to investigate the effects of rheological models of shear-thinning fluids and their estimated parameters on the predictions of laminar, transitional, and turbulent flow. The investigation was carried out through experimental and computational fluid dynamics (CFD) studies in horizontal pipes (diameters of 19.1 mm and 76.2 mm). Six turbulent models using Reynolds averaged Navier–Stokes equations in CFD_ANSYS Fluent 19.0 were examined in a 3D simulation followed by comparison studies between numerical and experimental results. Regarding results of laminar regions in power-law rheology models, Metzner and Reed presented the best fit for the pressure loss and transitional velocity. For the turbulent region, correlations observed by Wilson and Thomas as well as Dodge and Matzner had good agreement with the experimental results. For Herschel–Bulkley fluids, pressure losses and transitional regions based on a yielded region were examined and compared to the experimental results and the modified Slatter Reynolds number, where the results provided good estimation. For both pipe diameters, the Slatter model was the best fit for pressure losses of Herschel–Bulkley fluids in the turbulent regime. Furthermore, when comparing k-omega and k-epsilon turbulence models to the power-law behaviour, numerical studies delivered the most accurate results with fluids that have a higher behaviour index. However, the error percentage significantly increased at a higher shear rate in the Herschel–Bulkley fluids with a greater yield stress effect. Moreover, the modified Herschel–Bulkley viscosity function by Papanastasiou was implemented in the current CFD study. This function was numerically stabilized, devoid of discontinuity at a low strain rate, and more effective in transitional regions.

A New Approach to Correlate Heat Transfer Data for Turbulent Flow in Smooth Pipes Based on Generalization of Reynolds Number

A New Approach to Correlate Heat Transfer Data for Turbulent Flow in Smooth Pipes Based on Generalization of Reynolds Number

Numerical determination of condensation pressure drop of various refrigerants in smooth and micro-fin tubes via ANN method

Abstract In the current work, the pressure drop of the refrigerant flow in smooth and micro-fin pipes has been modeled with artificial neural networks as one of the powerful machine learning algorithms. Experimental analyses have been evaluated in two groups for the numerical model such as operation parameters/physical properties and dimensionless numbers used in two-phase flows. Feed forward back propagation multi-layer perceptron networks have been developed evaluating the practically obtained dataset having 673 data points covering the flow of R22, R134a, R410a, R502, R507a, R32 and R125 in four different pipes. The outputs acquired from the artificial neural network have been evaluated with the target ones, and the performance factors have been estimated and the prediction accuracy of the network models has been resourced comprehensively. The results revealed that the neural networks could predict the pressure drop of the refrigerant flow in smooth and micro-fin pipes between 10% deviation bands.

The onset of turbulence in pulsating flows in smooth pipes

Experimental study examines the structure of nominally laminar flows subjected to sinusoidal variation of flow rate in smooth pipes. An experimental setup employed for this purpose provides high stability of flow rate variation pattern and is able to reproduce flow reversal during a fraction of pulsation period. The Reynolds number based on the highest cross-section average velocity over the pulsation period is 1200. SIV technique is used to measure the instantaneous vector fields of velocity in two regimes of pulsating flow. While there is no flow reversal in the first regime, it occupies approximately 0.4 of the pulsation period in the second regime. The increase in the amplitude of high-frequency fluctuation of velocity is documented in both regimes near the pipe wall within the Stokes layer. Evolution of the shape of velocity profiles and fluctuation amplitude profiles depending on the pulsation phase is presented. Our analysis suggests that forced sinusoidal pulsation of flow rate at the considered parameters results in the onset of instability and signs of transition to turbulence in the near-wall flow in the smooth pipe. We describe possible mechanism of turbulization of the near-wall flow under the impact of forced flow rate pulsation.

Thermo-hydraulic performance of tube with decaying swirl flow generators

• Investigation of experimental and numerical study for decaying swirl flow in a pipe are presented. • Pressure drop and heat transfer were evaluated and analyzed for different swirl angle. • Due to the swirl intensity, heat transfer enhances with increasing swirl helical angle. • Performance evaluation criteria is the highest at the lowest swirl angle. • Variation Local Nusselt number along pipe is higher at larger swirl angle. In this study, the influences of swirl generators fixed at the pipe inlet and having different swirl angles (0°, 22.5°, 41°and 50°) on the heat transfer and fluid flow characteristics in a pipe flow were experimentally and numerically studied under a constant heat flux condition in the range of Reynolds ( Re ) number 2400–23000. The axial type passive swirl generator aims to increase the turbulence effect and gives tangential acceleration while increasing the total flow length in the pipe. This study is novel with the aspects of; (i) the number of helical air passage channels employed in the investigated swirl generators and, (ii) 360° rotation of each helical channel at different swirl angles, i.e., lengths of considered swirl generators are different. Experiments were carried out in a smooth aluminum pipe having an inner diameter of 15 mm and a length of 675 mm with and without swirl generators for turbulent airflows. Empirical correlations for the local and mean Nusselt ( Nu ) numbers, and friction factor ( f ) were developed as a function of Re number and swirl angle as well as smooth pipe flow. The obtained experimental results for the heat transfer and friction factor at different swirl angles were compared with the smooth pipe results. The findings showed that the maximum heat transfer rate is achieved at a swirl angle of θ = 50°, and the heat transfer rate increases as the swirl angle and Re number are increased. The decaying swirl flow leads to around 1.1–1.41 times enhancement of Nu and 2.3–6.76 times increase of f compared to the smooth pipe. The maximum value of performance evaluation criteria ( PEC ), around 0.83, is achieved by the decaying swirl flow when the θ = 0° and Re = 2400. The experimental and numerical Nu and f values for swirl flow are in good agreement within the mean deviation ± 7.1% and ± 8.4%.

Secondary Flow in Smooth and Rough Turbulent Circular Pipes: Turbulence Kinetic Energy Budgets

Direct Numerical Simulations have been performed for turbulent flow in circular pipes with smooth and corrugated walls. The numerical method, based on second-order finite discretization together with the immersed boundary technique, was validated and applied to various types of flows. The analysis is focused on the turbulence kinetic energy and its budget. Large differences have been found in the near-wall region at low Reynolds number. The change in the near-wall turbulent structures is responsible for increase of drag and turbulence kinetic energy. To investigatselinae the effects of wall corrugations, the velocity fields have been decomposed so as to isolate coherent and incoherent motions. For corrugated walls, we find that coherent motions are strongest for walls covered with square bars aligned with the flow direction. In particular, the coherent contribution is substantial when the bars are spaced apart by a distance larger than their height. Detailed analysis of the turbulence kinetic energy budget shows for this set-up a very different behavior than for the other types of corrugations.

Integral measures of the zero pressure gradient boundary layer over the Reynolds number range ≤Rτ<∞

A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B. J. Cantwell, “A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)]. More recently, the same profile was shown to accurately approximate velocity profiles in channel flow, the zero pressure gradient boundary layer, and the boundary layer in an adverse pressure gradient [M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows,” AIAA Paper No. 2021-0061, 2021 and M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers,” J. Fluid Mech. (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity. At a low Reynolds number, the profile approaches the laminar channel/pipe flow limit. The primary measure of the Reynolds number used in this work is the friction Reynolds number Rτ=uτδ/ν. It is a little unusual to use Rτ for the boundary layer since it requires that the velocity profile be cutoff using an arbitrarily defined overall boundary layer thickness, δ. Because of the slow approach of the velocity to the free stream, different conventions used to define the thickness lead to different values of Rτ assigned to a given flow. It will be shown in this paper that, through its connection to channel/pipe flow, the universal velocity profile can be used to define a practically useful, unambiguous, measure of overall boundary layer thickness, called here the equivalent channel half height, δh. For Rτ>≈5000, the universal velocity profile defines a Reynolds number independent shape function that can be used to generate explicit expressions for the infinite Reynolds number behavior of all the usual integral boundary layer measures; displacement thickness, momentum thickness, energy thickness, overall boundary layer thickness, and skin friction. The friction coefficient Cf(Rδ2) generated by the universal velocity profile accurately approximates data over a wide range of momentum thickness Reynolds numbers collected by Nagib et al. [“Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress?,” AIAA Paper No. 2004-2392, 2004]. The universal velocity profile is used to integrate the von Kaŕmań boundary layer integral equation [T. von Kármán, “Uber laminaire und turbulente reibung,” Z. Angew. Math. Mech. 1, 233–252 (1921)] in order to generate the various thicknesses and friction velocity as functions of the spatial Reynolds number, Rx=uex/ν.

A new universal model for friction factor in smooth pipes

Friction factor models for turbulent flow in smooth pipes express friction factor λ as a function of the bulk Reynolds number ReD and may be broadly grouped into two categories: power-law models and log-law models. While the former stem from the spectral scaling arguments applied to eddy momentum transfer close to the wall, the latter are derived from the mean velocity log law and are known to be consistent with the attached eddy model of wall turbulence structure. Interestingly, none of these models individually describes the entire range of Reynolds numbers (Re) accessed to date, without requiring adjustment of coefficients and/or exponents, i.e., these models are not universal. In this work, we present a new semi-empirical universal model that combines, without introducing any additional empirical coefficients, the essence of both power-law and log-law models. Due to this, our model successfully describes the variation of friction factor over the entire range of Reynolds numbers (more than four decades in ReD) at once. The physical basis for our model is the observation that at finite Reynolds numbers, the flow appears to be a small perturbation of the so-called ultimate regime of smooth-pipe turbulence, as far as friction is concerned; in the ultimate regime, λ→0 asymptotically as ReD→∞. The new model has significant potential toward accurate estimation of friction factor or flow rate in smooth pipe flows.

Turbulence characteristics of turbulent pipe flow with porous wall

We report on the experiments of turbulent pipe flow with a porous wall. The pipe wall consists of a cylindrical porous medium in which there are many radial fine through-holes connected with a plenum chamber. We have investigated the dependence of friction factor f on Reynolds number Re up to Re~105 in porous pipe flow with a chamber, porous pipe flow without a chamber, and a smooth pipe flow. In the non-porous case, turbulent flow exhibits the Blasius law f = 0.3164Re-1/4 in the range of 3 × 103 ≲ Re ≲ 105. In porous pipe flow with chamber, on the other hand, we have found the scaling f~Re0 at higher Re ≳ 2 × 104, implying the Taylor dissipation law. In porous pipe flow without chamber, the friction drag increases compared to the non-porous case, but f decreases with increasing Re at Re ≲ 105. The significant difference in porous pipe flows with and without chamber suggests that the scaling f~Re0 in porous pipe flow with chamber is attributed to the fluid motion through the porous wall.

Asymptotic properties of mixing length closures for turbulent pipe flow

Turbulence is investigated as the Reynolds number approaches infinity for the flow of an incompressible fluid through straight pipes with a circular cross section under the assumptions that the continuum hypothesis holds, the pipe wall is smooth, and the mixing length closure constructed by Cantwell [“A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)] is sufficiently accurate to allow the extrapolation to Reynolds numbers beyond the range of the Princeton superpipe data used as a foundation for the closure model. Two sets of scales are introduced to set up two sets of dimensionless equations and two Reynolds numbers for the near wall region (Re, inner scaling) and the center part of the pipe flow (Rτ, outer scaling). It is shown analytically that the turbulent flow asymptotically approaches a two-layer structure: The core of the pipe flow becomes uniform with constant mean velocity and zero shear stress in the outer scaling and the near wall region (inner scaling) with the mean velocity satisfying the law of the wall and non-zero shear stress. The core part of the flow pushes, as Rτ → ∞, the near wall layer to the boundary restricting it to a cylindrical subdomain with zero volume.

A new explicit friction factor formula for laminar, transition and turbulent flows in smooth and rough pipes

Pressure drop due to friction during internal pipe flow is one of the most important considerations in designing fluid and heat flow systems. There are many separate correlations valid for different situations such as laminar or turbulent rough and smooth surfaces for the estimation of friction factor. The present study introduces the derivation of a single explicit equation which is valid over the entire range of Reynolds number for both smooth and rough surfaces. This equation combines the laminar and turbulent friction factor equations through relation developed for the transition region. The results of the proposed equation were compared with the existing experimental data and the data from the Colebrook equation for smooth and rough surfaces respectively. The results show very good agreement with the experimental data as well as the Colebrook equation over the entire range of Reynold number. The percentage error lies within ± 2% and ± 10% for laminar and turbulent flows over the rough and smooth surfaces respectively.

MR-based wall shear stress measurements in fully developed turbulent flow using the Clauser plot method

In arterial blood flow wall shear stress (WSS) quantifies the frictional force that flowing blood exerts on a vessel wall. WSS can be directly estimated from phase-contrast (PC) MR velocity measurements and has been suggested as a biomarker in cardio-vascular diseases. We present and investigate the application of the Clauser plot method for estimating WSS in fully developed turbulent stationary flow using PC velocity measurements. The Clauser plot method estimates WSS from the logarithmic region of boundary layer in fully developed turbulent stationary flow. The Clauser plot method was evaluated using 2D PC-MR phantom measurements at 3 T for different in-plane resolutions at various Reynolds numbers. WSS values derived from the Clauser plot were compared to results from Laser Doppler Velocimetry (LDV) measurements and theoretical results calculated using the friction factor formula for smooth pipe flow. For all Reynolds numbers, WSS values derived from the Clauser plot were in good agreement with results from LDV measurements and values using the friction factor formula (relative deviations ∼5%). Furthermore, Clauser plot derived results were almost independent of spatial resolution, in contrast to WSS results obtained with our in-house software tool for MR-based WSS quantification showing relative deviations of more than 100%. In fully developed turbulent flow, the Clauser plot method provides highly consistent WSS independent of the underlying spatial resolution. Therefore, it renders a valuable approach for MR-based WSS estimates in controllable flow settings. Although its direct in vivo applicability is severely limited because of the different flow character, it may serve as helpful approach for validation of MR-based WSS quantification algorithms prior to their clinical application.

Average shear rate in airlift bioreactors: searching for the true value.

The shear rate is an important bioreactor parameter that needs to be evaluated due to its impact on microorganism morphology and viability, and consequently on bioproduct formation. Airlift bioreactors, classified as low-shear devices, are used as an alternative to conventional stirred-tank reactors. Considerable efforts have been made to characterize the shear environments in airlift bioreactors, using the average shear rate ([Formula: see text]) as a key parameter. However, there is no agreement among the values obtained in different studies, which can differ even in orders of magnitude. The methodologies used to obtain [Formula: see text] in the different studies could be the reason for the lack of agreement among them. In this work, [Formula: see text] in a concentric tube airlift bioreactor was evaluated using computational fluid dynamics (CFD), as well as based on universal velocity profiles for liquid flows in smooth pipes and annuli. Good agreement was obtained between the CFD-based average shear rates and the values obtained from universal velocity profiles, indicating that CFD simulation is a valuable tool for [Formula: see text] prediction.

Further Thoughts on Turbulent Flow in a Pipe

Nikuradse’s 1932 paper on turbulent flow in a smooth pipe contains a wealth of information on flow resistance (friction factor) and profiles of velocity and eddy viscosity. The goal here is to study this information in detail with the objective of applying it to other turbulent-flow situations. In particular, reverse engineering supports a value of n = 2 for the exponent on the volumetric dissipation in the decay term of the equation of the dissipation theorem. Of equal importance, integration of Nikuradse’s profiles of eddy viscosity does not lead to his formula for the universal resistance law; instead the presence of the viscous sublayer has an overt effect on the result even though we had thought that such a region influenced only the form of mass transfer at high values of the Schmidt number. A formula is proposed for the decay of dissipation for turbulent flow in smooth pipes.

Reconsideration of the friction factor data and equations for smooth, rough and transition pipe flow

The key element in design of pipelines is the friction factor estimation. After the brief review of the experimental data and friction factor correlations for isothermal single phase flow, we have checked the validity of well-known correlations through statistical criteria. During this process it was statistically proved that some of the well-known and permanently cited friction factor equations can be improved. Moreover we have prepared, for practical engineering purposes, equations that cover the entire range of laminar, critical and turbulent pipe flow.

Numerical Modeling of Convective Heat Transfer for Turbulent Flow in “Bubbler” Cooling Channels

In the present study, a numerical model was developed to investigate the heat transfer effect of single-phase turbulent flow in a “bubbler” geometry, typical for cooling channels in the high-pressure die-casting process. A steady-state $$ k - \omega $$ model with fine near-wall resolution was compared with established empirical relations developed by Gnielinski et al. for pressure drop and Nusselt numbers in turbulent flow in smooth pipes and annular ducts. The empirical relations matched well with the simulations. At the “tip” of the bubblers, impingement of the flow to the channel wall was found to intensify heat transfer locally. Increasing the ratio between inner and outer pipe diameters caused the flow to separate, increasing the pressure drop at the bend and the intensification of heat transfer.

Application of copper oxide–thermal oil (CuO-HTO) nanofluid on convective heat transfer enhancement in inclined circular tube

The influence of using copper oxide–thermal oil on convective heat transfer and pressure drop in an upward flow in an inclined smooth tube is studied experimentally in this paper. The flow regime and wall temperature are laminar and constant, respectively. The effects of nanofluid, Graetz number, Prandtl number, negative inclination angle on convective heat transfer rate rise moderately with the augmentation of nanoparticles mass concentration. Both correlations are recommended to evaluate Nusselt number and Darcy friction factor in an upward flow under constant wall temperature and laminar flow in smooth pipe. The maximum deviations are 19% and 21%, respectively, which are acceptable for scientific research to be used in industrial applications. Accompaniment of heat transfer ratio with pumping power ratio is presented in this paper. If the increment of pressure drop is more than heat transfer enhancement, it will not be appropriate to use CuO–thermal oil, negative inclination angles and smooth tube. The figure of merit increases up to 1.58% which is calculated with 1.5% nanoparticle mass concentration and inclination angle of 30° at Prandtl number of 387. The results show that most of the values are more than unity, so the heat transfer enhancement is more than increment of pressure drop.

Refined dependences of power-law form for calculation of hydraulic resistance and velocity distribution

To solve the engineering problems of hydraulic engineering, water supply and sanitation, prevent emergency situations at hydraulic structures, regulate channel processes, efficient ecological monitoring of water bodies, and develop measures that exclude crisis ecological situations, more and more accurate estimates are required for the hydraulic characteristics of water flows. Inaccurate assumptions and approximate methods of integration can lead to significant errors that significantly affect the results of predicting flooding of riverine areas; this requires further refinement of the calculation methods. Until now, theoretical and experimental substantiations of the basic propositions of the theory of turbulence, hydraulic resistances, channel processes cannot be considered sufficiently complete. The compositions of problems associated with flows in wide channels, problems of turbulence, ax symmetric flows in smooth and rough pipes, are of scientific and practical interest. The article investigates the dependencies for the calculation of kinematic and dynamic flow characteristics in deformed and undeformable boundaries. New formulas are given for a refined description of the distribution of flow velocities and hydraulic resistance of turbulent flows in pipes and channels. Based on the known fundamental research, the calculation-theoretical methods are refined. The results obtained for flows in non-deformable channels can be useful for estimating not only flow in pipes, but also river flows in hard, undeformable boundaries and with minor channel deformations (in channels and rivers). For the first time, the analysis made it possible to obtain power-law approximations of the resistance laws for smooth and rough tubes and to establish the actual relationship between the exponents in the velocity profile and the law of resistance.

ИССЛЕДОВАНИЕ ПРОСТРАНСТВЕННОГО РАСПРЕДЕЛЕНИЯ ПЛОТНОСТИ СВЕРХКРИТИЧЕСКОЙ ВОДЫ ПРИ ВОСХОДЯЩЕМ ТЕЧЕНИИ В ВЕРТИКАЛЬНЫХ ОБОГРЕВАЕМЫХ ТРУБАХ

The data of CFD modeling of spatial distributions of density of supercritical water during its flow in vertical smooth pipes under conditions of mixed convection are presented. The effects of the influence of buoyancy forces on the indicated distributions are considered.