Steady Pipe Flow Research Articles

Numerical study of steady pipe flows and head loss coefficients

The study concerns steady pipe flows within 90o-bend and T-junctions, the head loss coefficients and the physical mechanisms responsible for the energy loss. Numerical viscous flow modelling based on SimpleFOAM solver were implemented in the study. Although secondary flow acts as the mechanism of energy loss in both the bend and in the branch flow of T-junction, the head loss coefficient in the former more critically depends on Reynolds (Re) number. This is supported by the comparison of the streamline fields and the radially varying profiles of the pressures from both cases. The location of flow separation prior to the secondary flow in the bend changes with Re number unlike the fixed location at the junction in the branch flow. The study confirms a practical implication related to the dependence of the head loss coefficients for pipe with 90o-bend and T-junction on flow velocity in an analysis of pipe network.

A time-consistent stabilized finite element method for fluids with applications to hemodynamics

Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov–Galerkin stabilization (SUPG) term, which is weighted by tau _{text{SUPG}}. The conventional formulation of tau _{text{SUPG}} includes a constant that depends on the time step size, producing an overall method that becomes exceedingly less accurate as the time step size approaches zero. In practice, such method inconsistency introduces significant error in the solution, especially in cardiovascular simulations, where small time step sizes may be required to resolve multiple scales of the blood flow. To overcome this issue, we propose a consistent method that is based on a new definition of tau _{text{SUPG}}. This method, which can be easily implemented on top of an existing streamline upwind Petrov–Galerkin and pressure stabilizing Petrov–Galerkin method, involves the replacement of the time step size in tau _{text{SUPG}} with a physical time scale. This time scale is calculated in a simple operation once every time step for the entire computational domain from the ratio of the L2-norm of the acceleration and the velocity. The proposed method is compared against the conventional method using four cases: a steady pipe flow, a blood flow through vascular anatomy, an external flow over a square obstacle, and a fluid–structure interaction case involving an oscillatory flexible beam. These numerical experiments, which are performed using linear interpolation functions, show that the proposed formulation eliminates the inconsistency issue associated with the conventional formulation in all cases. While the proposed method is slightly more costly than the conventional method, it significantly reduces the error, particularly at small time step sizes. For the pipe flow where an exact solution is available, we show the conventional method can over-predict the pressure drop by a factor of three. This large error is almost completely eliminated by the proposed formulation, dropping to approximately 1% for all time step sizes and Reynolds numbers considered.

Spatiotemporal Intermittency in Pulsatile Pipe Flow.

Despite its importance in cardiovascular diseases and engineering applications, turbulence in pulsatile pipe flow remains little comprehended. Important advances have been made in the recent years in understanding the transition to turbulence in such flows, but the question remains of how turbulence behaves once triggered. In this paper, we explore the spatiotemporal intermittency of turbulence in pulsatile pipe flows at fixed Reynolds and Womersley numbers (, ) and different pulsation amplitudes. Direct numerical simulations (DNS) were performed according to two strategies. First, we performed DNS starting from a statistically steady pipe flow. Second, we performed DNS starting from the laminar Sexl–Womersley flow and disturbed with the optimal helical perturbation according to a non-modal stability analysis. Our results show that the optimal perturbation is unable to sustain turbulence after the first pulsation period. Spatiotemporally intermittent turbulence only survives for multiple periods if puffs are triggered. We find that puffs in pulsatile pipe flow do not only take advantage of the self-sustaining lift-up mechanism, but also of the intermittent stability of the mean velocity profile.

The effect of pulsation frequency on transition in pulsatile pipe flow

Pulsatile flows are common in nature and in applications, but their stability and transition to turbulence are still poorly understood. Even in the simple case of pipe flow subject to harmonic pulsation, there is no consensus among experimental studies on whether pulsation delays or enhances transition. We here report direct numerical simulations of pulsatile pipe flow at low pulsation amplitude$A\leqslant 0.4$. We use a spatially localized impulsive disturbance to generate a single turbulent puff and track its dynamics as it travels downstream. The computed relaminarization statistics are in quantitative agreement with the experiments of Xuet al. (J. Fluid Mech., vol. 831, 2017, pp. 418–432) and support the conclusion that increasing the pulsation amplitude and lowering the frequency enhance the stability of the flow. In the high-frequency regime, the behaviour of steady pipe flow is recovered. In addition, we show that, when the pipe length does not permit the observation of a full cycle, a reduction of the transition threshold is observed. We obtain an equation quantifying this effect and compare it favourably with the measurements of Stettler & Hussain (J. Fluid Mech., vol. 170, 1986, pp. 169–197). Our results resolve previous discrepancies, which are due to different pipe lengths, perturbation methods and criteria chosen to quantify transition in experiments.

A space fractional constitutive equation model for non-Newtonian fluid flow

Non-Newtonian fluid flow can be driven by spatially nonlocal velocity, the dynamics of which can be described by promising fractional derivative models. This study reports a left-side, Caputo type, space fractional-order constitutive equation (FCE) using a nonlocal, fractional velocity gradient and then interprets physical properties of non-Newtonian fluids for steady pipe flow. Results show that the generalized FCE model contains previous non-Newtonian fluid flow models as end-members by simply adjusting the order of the fractional index, and a preliminary test shows that the FCE model conveniently captures the observed growth of shear stress for various velocity gradients. Further analysis of the velocity profile, frictional head loss, and Reynolds number using the FCE model also leads to analytical tools and criterion that can extend standard models in quantifying the complex dynamics of non-Newtonian fluid flow with a wide range of spatially nonlocal velocities. Model extension using the general two-side fractional derivative is also briefly discussed.

Retrospective cost adaptive Reynolds-averaged Navier–Stokes k–ω model for data-driven unsteady turbulent simulations

This paper presents a data-driven computational model for simulating unsteady turbulent flows, where sparse measurement data is available. The model uses the retrospective cost adaptation (RCA) algorithm to automatically adjust the closure coefficients of the Reynolds-averaged Navier–Stokes (RANS) k–ω turbulence equations to improve agreement between the simulated flow and the measurements. The RCA-RANS k–ω model is verified for steady flow using a pipe-flow test case and for unsteady flow using a surface-mounted-cube test case. Measurements used for adaptation of the verification cases are obtained from baseline simulations with known closure coefficients. These verification test cases demonstrate that the RCA-RANS k–ω model can successfully adapt the closure coefficients to improve agreement between the simulated flow field and a set of sparse flow-field measurements. Furthermore, the RCA-RANS k–ω model improves agreement between the simulated flow and the baseline flow at locations at which measurements do not exist. The RCA-RANS k–ω model is also validated with experimental data from 2 test cases: steady pipe flow, and unsteady flow past a square cylinder. In both test cases, the adaptation improves agreement with experimental data in comparison to the results from a non-adaptive RANS k–ω model that uses the standard values of the k–ω closure coefficients. For the steady pipe flow, adaptation is driven by mean stream-wise velocity measurements at 24 locations along the pipe radius. The RCA-RANS k–ω model reduces the average velocity error at these locations by over 35%. For the unsteady flow over a square cylinder, adaptation is driven by time-varying surface pressure measurements at 2 locations on the square cylinder. The RCA-RANS k–ω model reduces the average surface-pressure error at these locations by 88.8%.

Transition to turbulence in pulsating pipe flow

Fluid flows in nature and applications are frequently subject to periodic velocity modulations. Surprisingly, even for the generic case of flow through a straight pipe, there is little consensus regarding the influence of pulsation on the transition threshold to turbulence: while most studies predict a monotonically increasing threshold with pulsation frequency (i.e. Womersley number, $\unicode[STIX]{x1D6FC}$), others observe a decreasing threshold for identical parameters and only observe an increasing threshold at low $\unicode[STIX]{x1D6FC}$. In the present study we apply recent advances in the understanding of transition in steady shear flows to pulsating pipe flow. For moderate pulsation amplitudes we find that the first instability encountered is subcritical (i.e. requiring finite amplitude disturbances) and gives rise to localized patches of turbulence (‘puffs’) analogous to steady pipe flow. By monitoring the impact of pulsation on the lifetime of turbulence we map the onset of turbulence in parameter space. Transition in pulsatile flow can be separated into three regimes. At small Womersley numbers the dynamics is dominated by the decay turbulence suffers during the slower part of the cycle and hence transition is delayed significantly. As shown in this regime thresholds closely agree with estimates based on a quasi-steady flow assumption only taking puff decay rates into account. The transition point predicted in the zero $\unicode[STIX]{x1D6FC}$ limit equals to the critical point for steady pipe flow offset by the oscillation Reynolds number (i.e. the dimensionless oscillation amplitude). In the high frequency limit on the other hand, puff lifetimes are identical to those in steady pipe flow and hence the transition threshold appears to be unaffected by flow pulsation. In the intermediate frequency regime the transition threshold sharply drops (with increasing $\unicode[STIX]{x1D6FC}$) from the decay dominated (quasi-steady) threshold to the steady pipe flow level.

Poiseuille flow of a thixo-elastic fluid: analytical and semi-analytical solutions

In this study, the steady pipe flow of a thixo-elastic fluid is considered. Thixo-elasticity is modeled using the modified Delaware thixotropic model (MDTM). In MDTM model, the total stress tensor is decomposed into a viscous part and an elastic part and each stress component is properly modeled using the concept of single structural parameter which represent the state of thixotropic microstructure in space and time. Analytical solutions are attained for three special cases of MDTM model and a general semi-analytical solution is devised as well. The effect of various thixo-elastic parameters is studied on the profiles of axial velocity and structural parameter. Additionally, volumetric flow rate is presented for all the considered cases.

A New Coupled Explicit Finite Difference Model for Wellbore Flow and Heat Transfer in CO2 Geological Storage

The wellbore pressure and temperature are very significant parameters for CO2 geological storage projects, which determine the safety and effectiveness of projects. To realize the coupling calculation between wellbore pressure and temperature conveniently, this work takes the one dimensional steady flow with homogeneous fluid in wellbores as the analysis object and divides the wellbore into finite micro-segments. Then a new coupled explicit finite difference model (EFDM) for wellbore flow and heat transfer is proposed. It is novel in three aspects: (i) the EFDM about wellbore pressure is derived from the equation of continuity for one-dimensional steady pipe flow, the equation of motion and the equation of state of the fluid within a certain micro-segment. (ii) The EFDM about temperature reflecting the wellbore heat transfer between wellbore and formation is developed according to the equation of energy balance for the same micro-segment. (iii) To realize the coupling calculation of them among the integrated wellbore, the specific method of the coupling iterative calculation between the wellbore pressure and temperature is presented. At last, a comparison among the simulation results of EFDM, other models and the log data from engineering is implemented, which demonstrated that the prediction results of EFDM are more consistent with the log data than the results of other models. Therefore, the reliability of the new coupled EFDM is verified. Generally, the prediction of wellbore pressure and temperature distribution and the comprehension about the internal essence of wellbore flow and heat transfer will be effectively promoted by this work.

A simplified approach for the computation of steady two-phase flow in inverted siphons

Hydraulic, sanitary, and sulfide control conditions of inverted siphons, particularly in large wastewater systems, can be substantially improved by continuous air injection in the base of the inclined rising branch. This paper presents a simplified approach that was developed for the two-phase flow of the rising branch using the energy equation for a steady pipe flow, based on the average fluid fraction, observed slippage between phases, and isothermal assumption. As in a conventional siphon design, open channel steady uniform flow is assumed in inlet and outlet chambers, corresponding to the wastewater hydraulic characteristics in the upstream and downstream sewers, and the descending branch operates in steady uniform single-phase pipe flow. The proposed approach is tested and compared with data obtained in an experimental siphon setup with two plastic barrels of different diameters operating separately as in a single-barrel siphon. Although the formulations developed are very simple, the results show a good adjustment for the set of the parameters used and conditions tested and are promising mainly for sanitary siphons with relatively moderate heights of the ascending branch.

Transition phenomena and velocity distribution in constant-deceleration pipe flow

The critical Reynolds number for transition to turbulence in a constant-acceleration pipe flow is well known to be significantly higher than the value for steady pipe flow. This is a consequence of the significant suppression of amplification of disturbances entering the pipe by the acceleration in spite of the increase of inertial force due to increase in instantaneous Reynolds number. In contrast, relatively little is known of the transition phenomena in a constant-deceleration pipe flow. Such a flow system is investigated by an experimental method in which the cross-sectional mean velocity is initially kept constant and subsequently decreased linearly to zero. The characteristics of the transition phenomena will be determined by two factors – deceleration and viscosity. Deceleration will amplify the disturbances, while a decrease in the instantaneous Reynolds number is expected to suppress the amplification of the disturbances due to viscosity. The study uses hot-wire experiments to clarify which of the two effects predominates in constant-deceleration pipe flow.

The calculation of mechanical energy loss for incompressible steady pipe flow of homogeneous fluid

The calculation of the mechanical energy loss is one of the fundamental problems in the field of Hydraulics and Engineering Fluid Mechanics. However, for a non-uniform flow the relation between the mechanical energy loss in a volume of fluid and the kinematical and dynamical characteristics of the flow field is not clearly established. In this paper a new mechanical energy equation for the incompressible steady non-uniform pipe flow of homogeneous fluid is derived, which includes the variation of the mean turbulent kinetic energy, and the formula for the calculation of the mechanical energy transformation loss for the non-uniform flow between two cross sections is obtained based on this equation. This formula can be simplified to the Darcy-Weisbach formula for the uniform flow as widely used in Hydraulics. Furthermore, the contributions of the mechanical energy loss relative to the time averaged velocity gradient and the dissipation of the turbulent kinetic energy in the turbulent uniform pipe flow are discussed, and the contributions of the mechanical energy loss in the viscous sublayer, the buffer layer and the region above the buffer layer for the turbulent uniform flow are also analyzed.

A Unified Formula for Determination of Wellhead Pressure and Bottom-hole Pressure

The number of gas injection or exploitation projects is on fast increase with growing energy needs and environmental problems. Of these projects, the calculation and the control of wellhead pressure or bottom-hole pressure in wellbores are usually key steps. However, methods used to predict wellhead and bottom-hole pressure are multifarious, making the program writing and practical application inconvenient. In this study, a unified formula derived from the differential equations of one-dimensional steady pipe flow and applicable in both injection wells and production wells was presented for calculation of both wellhead pressure and bottom-hole pressure. The recalculation with a traditional numerical method of the same equations corroborated well the accuracy of the unified formula, and the positive- reverse checking calculation verified its reliability. Basic data from Shenhua CCS project were used for studying the influence of injected gas temperature, geothermal gradient and flow rate on wellbore pressure by applying the unified formula. The results can provide a technical guidance for practical operation.

A fast explicit finite difference method for determination of wellhead injection pressure

A fast explicit finite difference method (FEFDM), derived from the differential equations of one-dimensional steady pipe flow, was presented for calculation of wellhead injection pressure. Recalculation with a traditional numerical method of the same equations corroborates well the reliability and rate of FEFDM. Moreover, a flow rate estimate method was developed for the project whose injection rate has not been clearly determined. A wellhead pressure regime determined by this method was successfully applied to the trial injection operations in Shihezi formation of Shenhua CCS Project, which is a good practice verification of FEFDM. At last, this method was used to evaluate the effect of friction and acceleration terms on the flow equation on the wellhead pressure. The result shows that for deep wellbore, the friction term can be omitted when flow rate is low and in a wide range of velocity the acceleration term can always be deleted. It is also shown that with flow rate increasing, the friction term can no longer be neglected.

Direct numerical simulation of fully developed turbulent and oscillatory pipe flows at

Shear flow turbulence in oscillatory fluid motions is of theoretical interest and practical relevance, since the onset of turbulence can drastically change the transport properties and mixing efficiency. To supplement former theoretical and experimental investigations on the transition to turbulence in Sexl–Womersley (SW) flows, we perform three-dimensional direct numerical simulations (DNS) of oscillatory pipe flows at three Womersley numbers (Wo∈{26, 13, 5}) and one constant Reynolds number () based on the friction velocity and the pipe diameter. For this, the incompressible Navier–Stokes equations are solved in cylindrical coordinates using a fourth-order-accurate finite-volume method on staggered grids, motivated by Schumann’s volume balance procedure. We generate a well-correlated high-Reynolds-number initial flow field for the oscillatory flows by means of a DNS of a statistically steady pipe flow at . To underline the reliability of the DNS results for the oscillatory pipe flows, we validate the finite-volume method, the spatial resolution of the computational grid and the length of the computational domain by comparing the results for the statistically steady pipe flow with experimental data obtained by laser Doppler anemometry (LDA). Comparing the statistical moments of the velocity components up to the fourth order shows good agreement with the corresponding LDA data. When started from the turbulent initial velocity field, the oscillatory flows relaminarise or reach a conditionally or fully turbulent state, depending on Wo. The peak flow rates decrease with increasing Wo, while the relaxation phase for the initially steady flow converging to a purely oscillating flow increases with increasing Wo. For the highest Wo considered, the flow completely relaminarises and we do not find any instabilities close to the wall, as expected from former stability analyses. On the other hand, we confirm the existence of turbulent bursts and increasing turbulence intensity during the deceleration phase of the flow and relaminarisation in the acceleration phase for Wo=13 in agreement with experimental results in the literature. By analysing selected terms of the transport equations for the mean and turbulent kinetic energy, we demonstrate the transport of turbulent kinetic energy from the axial to the radial and azimuthal velocity components during flow deceleration.

A Novel Model of Turbulent Convective Heat Transfer in Round Tubes at Supercritical Pressures

In the present study, a novel model was established to investigate the enhanced heat transfer to turbulent pipe flow of supercritical pressure fluids. The governing equations for the steady turbulent compressible pipe flow were simplified into the one-dimensional nondimensionalized forms based on the boundary layer theory. A conventional mixing length turbulence model for constant-property pipe flows was modified by introducing the effect of density fluctuations into the equations of turbulent transport, and the modified turbulence model was applicable to both constant-property and variable-property pipe flows. With the suggested model, which was a combination of the nondimensional governing equations and the modified turbulence model, the numerical calculations were carried out for the turbulent convective heat transfer of water in round tubes at supercritical pressures. The results showed that the present model can provide a relatively precise prediction about the effect of pressure, mass flux, and wall heat flux on heat transfer for supercritical fluid flows and greatly reduce the calculation workload. The modified turbulence model showed a much better agreement with the experimental results than the original turbulence model.

Effect of Initial Constant Acceleration on the Transition to Turbulence in Transient Circular Pipe Flow

Experimental investigation is carried out on the transition to turbulence in a transient circular pipe flow. The flow is accelerated from rest at a constant acceleration until its cross-sectional mean velocity reaches a constant value. Accordingly, the history of the flow thus generated consists of the initial stage of constant acceleration and the following stage of constant cross-sectional mean velocity. The final Reynolds number based on the constant cross-sectional mean velocity and the pipe diameter is chosen to be much greater than the transition Reynolds number of a steady pipe flow of about 3000. The transition to turbulence is judged from the output signal of the axial velocity component and its root-mean-square value measured with a hot-wire anemometer. A turbulent slug appears after the cross-sectional mean velocity of the flow reaches the predetermined constant value under every experimental condition. Turbulence production therefore is suppressed, while the flow is accelerated. The time lag for the appearance of the turbulent slug after the cross-sectional mean velocity of the flow reaches the constant value decreases with an increase in the constant acceleration value. An empirical equation is proposed for estimating the time lag. The propagation velocity of the leading edge of the turbulent slug is independent of the constant acceleration value under the present experimental conditions.

Transport Phenomena of Solid Particles in Pulsatile Pipe Flow

The transportation mechanism of single solid particles in pulsating water flow in a vertical pipe was investigated by means of videography and numerical simulations. The trajectories of alumina particles were observed experimentally by stereo videography. The particle diameter was 3 mm or 5 mm, and the pipe diameter was 18 mm or 22 mm. The frequency of flow pulsation was less than or equal to 6.67 Hz. It was found that the critical minimum water flux at which the particle can be transported upward depended on the pulsating pattern. Two types of numerical simulations were conducted, namely, one-dimensional simulations for tracking the vertical motion of the solid particles and two-dimensional simulations of the pulsating pipe flows in an axisymmetric coordinate system. The computer simulations of axisymmetric pipe flows revealed that the time-averaged radial velocity profile of water in the pulsating flows was very different from that in steady pipe flows. The motion of the particles is discussed in detail for a better understanding of the physics of the transport phenomena.

Unsteady turbulent pipe flow due to water hammer using k–θ turbulence model

The behaviour of unsteady turbulent pipe flow resulting from water hammer is herein numerically studied. An accurate k–θ turbulence model for two-dimensional boundary layers under adverse and favorable pressure gradients (Wilcox 1994) was applied. The results of the numerical method are in good agreement with test data obtained from both steady and unsteady turbulent pipe flows. Also, the behaviour and the structure of turbulence in unsteady turbulent pipe flows caused by water hammer were investigated. A non-dimensional parameter P as the time scale ratio of turbulence diffusion in the radial direction to the pressure wave speed introduced by Ghidaoui et al. (2002) was used. The results show that for P ≈ 1, the turbulence structure is significantly changed in the first cycle of water hammer, which is important for water quality modeling, whereas these variations are small for P ≫ 1.

Effect of the pattern of initial perturbations on the steady pipe flow regime

The influence of the inlet flow formation mode on the steady flow regime in a circular pipe has been investigated experimentally. For a given inlet flow formation mode the Reynolds number Re* at which the transition from laminar to turbulent steady flow occurred was determined. With decrease in the Reynolds number the difference between the resistance coefficients for laminar and turbulent flows decreases. At a Reynolds number approximately equal to 1000 the resistance coefficients calculated from the Hagen-Poiseuille formula for laminar steady flow and from the Prandtl formula for turbulent steady flow are equal. Therefore, we may assume that at Re > 1000 steady pipe flow can only be laminar and in this case it is meaningless to speak of a transition from one steady pipe flow regime to the other.