Incompressible Pipe Flow Research Articles

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 (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity.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.

Prediction of pipeline restart using different rheological models of gelled crude oil

Abstract Shutdown of a waxy crude oil pipeline is unavoidable due to maintenance or emergency. It is critical to select the rheological models of gelled crude oil when investigating the pipeline restart process. Three crude oil rheological models are summarized based on previous researches in this paper, which are the viscoelastic model of a viscous type (Model 1), viscoelastic model of an elastic type (Model 2), and pure viscous thixotropic model without yield stress (Model 3). The same rheological data was fitted by the three models respectively. The critical state that pipeline can restart successfully is dominated by the slow creep of the gelled crude oil that can be regarded as an incompressible pipe flow, and this is verified by the pipe restart experiments under constant pressure conditions in this paper. To discuss the effects of the rheological models on calculation of pipeline restart separately, a simplified one-dimensional mathematical model with the pump boundary condition is established. The different calculated flow rate indicates that rheological models affect the judgement of the pipeline restart even though they are fitted from the same rheological data.

On an analytic solution for general unsteady/transient turbulent pipe flow and starting turbulent flow

Abstract The literature already offers analytic solutions for steady-state laminar (Hagen–Poiseuille), transient laminar (Szymanski) and steady-state turbulent (Pai) incompressible pipe flows. However, no reference could be found for the most general case that encompasses all of them: the unsteady turbulent incompressible pipe flow. This study presents a General Analytic Solution (GAS) for the Reynolds–averaged Navier–Stokes equations corresponding to unsteady turbulent incompressible circular pipe flow, which is composed of the transient response of the initial mean velocity field to external interactions, the unsteady response to time-dependent pressure gradient, and the unsteady component due to the turbulence’s Reynolds stress. The particular cases of laminar, turbulent, steady-state and transient incompressible pipe flows emanate directly from such solution. A general method is proposed to obtain an analytical solution for any particular pipe flow problem for which some relevant function could be provided. That general method is exemplified in a test case, a much simplified version of the starting flow initiated in a circular pipe, after suddenly applying a constant pressure gradient. The time evolution for such starting flow, with a growing stage and a shrinking stage, is presented graphically and discussed.

True bond graph formulation for one-dimensional incompressible pipe flows: modeling and analytical benchmarks

In this paper, a bond graph methodology described in a previous contribution by Balino (Simul Model Pract Theory 17:35–49, 3) is used to model incompressible one-dimensional pipe flows with rigid walls. As the volumetric flow is independent of position, the balance equation corresponding to the inertial port can be simplified and readily integrated. The volumetric flow and a nodal vector of entropy are defined as bond graph state variables. The state equations and the coupling between the inertial and entropy ports are modeled with true bond graph elements. The methodology was implemented using constant piecewise shape functions and linear piecewise weight functions for the entropy port. Different problems are simulated: advective heating due to constant wall heat flux and due to constant wall temperature, advective heating due to viscous dissipation and combined advection–diffusion. The numerical results show an excellent agreement with the corresponding analytical solutions, both for transient and steady-state flow. A representation with single-bond elements is obtained by lumping the temperature matrix. Simulations were run with the lumping approximation, showing no significant deterioration of the numerical solutions.

Efficient parallel implementation of incompressible pipe flow algorithm based on SIMPLE

SummaryParallel semi‐implicit method for pressure‐linked equations(SIMPLE) algorithm is used to solve the 3‐D incompressible pipe flow problem. In this paper, we proposed a novel parallel SIMPLE algorithm that uses the alternate tiling technique. Firstly, a parallel SIMPLE algorithm based on domain decomposition method was established, and the implementation of domain partition and data exchange was presented. Then, we presented serial finite difference stencil algorithm based on alternate tiling. Furthermore, an iteration space parallel two‐way finite difference stencil algorithm based on alternate tiling was proposed, introducing the sequence of iterative space tiles as the sequence of execution and using time skewing technique to partition the iteration space, thus to improve the data locality of algorithm. The cache misses and the cost of communication and synchronization are reduced by reordering the tiles of iteration space. Finally, the effectiveness of the two parallel SIMPLE algorithms were compared. The results showed that the parallel SIMPLE algorithm that uses the two‐way finite difference stencil algorithm based on alternate tiling has good data locality, performance, and scalability in the Deepcomp7000 cluster computing environment. Copyright © 2013 John Wiley & Sons, Ltd.

Solenoidal bases for numerical studies of transition in pipe flow

In this paper, solenoidal basis functions are employed in numerical studies of transition in incompressible pipe flow. The bases are formulated in terms of Legendre polynomials, which are more favorable both for the functional form of the basis functions and for the inner product integrals arising in the Galerkin projection scheme. The projection is performed onto the dual solenoidal basis set to eliminate the pressure gradient term from the governing equations. This simplifies the numerical approach to the problem and the resulting scheme is used to study the nonlinear stability of pipe flow. Some preliminary results are compared with those found in the literature.

Analysis of pipe flow transition. Part II. Energy transfer

The present article describes the results from a study of nonlinear mechanisms at work during the process of transition to turbulence in pipe flows. Using an accurate hybrid finite-difference code for the simulation of unsteady incompressible pipe flow, we have performed a direct numerical simulation designed to model experiments performed by Han, Tumin and Wygnanski [12]. Based on these numerical data, we have conducted a meticulous investigation of the dynamic interactions of the structures and flow modes that can be observed during this process. Based on this study, we can paint a detailed picture of the dynamical interactions of flow structures during both the linear and nonlinear stages of pipe flow transition. While this picture does have some similarities to earlier proposed mechanisms, we find that even for the simple cases considered here the structure of the pertinent interactions is much richer than suggested by these earlier models.

Analysis of pipe flow transition. Part I. Direct numerical simulation

We have developed an accurate hybrid finite-difference code for the simulation of unsteady incompressible pipe flow. The numerical scheme uses compact finite differences of at least eighth-order accuracy for the axial coordinate, and Chebyshev and Fourier polynomials for the radial and azimuthal coordinates, respectively. Boundary conditions for the incompressible flow are enforced using an influence-matrix technique, and the Poisson equation for pressure is solved using a fast direct method. The code has been used to simulate and analyze the spatial transition process in developed laminar pipe flow at a Reynolds number of Re=2350. Results of the simulation are compared to experimental data given by Han, Tumin and Wygnanski [18].

AN ACCURATE METHOD FOR THE SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE HEATED PIPE FLOW

A numerical scheme using Fourier expansions in the streamwise and azimuthal directions and Jacobi polynomials in the radial direction for the direct numerical simulation of three-dimensional incompressible pipe flow with heating is presented. The proposed basis and test functions for the thermal field in conjunction with those for the velocity field set forth by Leonard and Wary offer an accurate representation of flow variables in transitional flow. A simple test to a linear stability problem demonstrates that this method yields very accurate results with relatively few radial modes and is well suited for the simulation of nonisothermal pipe flow transition.

Spatial instability of flow in a semiinfinite cylinder with fluid injection through its porous walls

In this article, the stability of incompressible pipe flows induced by wall injection is investigated. A local linear spatial theory is applied with a special treatment near the axis due to the choice of the cylindrical coordinates. It shows that the flow is stable near the headwall until it reaches a critical axial position. The flow becomes unstable downstream for a range of frequencies which increases with the distance to the headwall. Unstable two-dimensional and also three-dimensional modes are exhibited. Comparison with the available experimental results confirms the existence of amplified instability waves and the presence of at least some of the predicted modes.

Flashing Flow Discharge of Initially Subcooled Liquid in Pipes

The release rates of hot fluids stored under pressure during a pipe break or valve failure are a pertinent consideration in source term modeling as part of the environmental health and safety issue. This technical brief presents an analytical treatment for flashing discharge of a hot liquid in a pipe from a high pressure reservoir. Specifically, the liquid is initially subcooled relative to stagnation condition but superheated relative to ambient pressure. The effect of inlet subcooling, pipe length-to-diameter ratio, and physical property groups of the fluid can be readily assessed by the present generalized model. A criterion related to the amount of ''limiting'' subcooling is found where the discharge rate can be evaluated using a form similar to the incompressible pipe flow formula. Unlike other previously published works, the present solution yields a smooth transition from saturated inlet to subcooled conditions and reduces exactly to the nozzle flow limit at zero pipe length.

Correlation of Preston tube data with laminar skin friction

Preston-tube data have been obtained on a sharp ten-degree cone in the NASA Ames Eleven-Foot Transonic Wind Tunnel. Data were obtained over a Mach number range of 0.30 to 0.95 and unit Reynolds numbers of 9.84, 13.1, and 16.4 million per meter. The portions of these data, that were obtained within laminar boundary layers, have been correlated with the corresponding values of theoretical skin friction. The rms scatter of skin-friction coefficient about the correlation is of the order of one percent, which is comparable to the reported accuracy for calibrations of Preston-tubes in incompressible pipe-flows. In contrast to previous works on Preston-tubelskin-friction correlations, which are based on the physical height of the probe's face, this very satisfactory correlation for compressible boundary-layer flows is achieved by accounting for the effects of a variable effective height of the probe. The coefficients, which appear in the correlation, are dependent on the particular tunnel environment. The general procedure can be used to define correlations for other wind tunnels. Nomenclature

The stability of unsteady axisymmetric incompressible pipe flow close to a piston. Part 2. Experimental investigation and comparison with computation

Flow visualization has been used quantitatively to determine the flow relative to a piston and a free surface started from rest. The discharge of water from a cylindrical reservoir was investigated. Flow with a free surface started from rest was found to have a critical Reynolds number (based on tube diameter and surface speed) of about 450 above which a ring vortex was produced just below the surface.Measurements at Reynolds numbers of 525 and 1200 were compared with computations made by the methods described in Part 1. The computed drift of tracer particles agreed well with observed values. The largest discrepancies occurred in the radial component of the drift in the early stages of the motion and amounted to 2½% of the tube diameter.

The Influence of Bends on Fluid Transients Propagated in Incompressible Pipe Flow

The influence of a bend on the fluid transients propagated by a rapid valve closure in a long pipeline is examined by means of a short dimensional analysis and a comprehensive series of experimental tests designed to record the effects of varying in turn: the pipeline diameter to thickness ratio (7.35 to 48), pipe elasticity to fluid bulk modulus ratio (0.27 to 93.6), the restraint applied to the pipeline, the bend radius of curvature to pipe bore ratio (0 to 5), the included angle of the bend (0° to 135°) and the initial flow velocity.From these tests the influence of a bend is found to be solely dependent on its geometry. Empirical formulae are proposed which express the reflection and transmission of a transient at a bend in terms of the bend radius of curvature to pipe bore ratio and included angle. The formulae are tested by varying both the angle and curvature of a series of bends mounted in turn in a steel pipeline and found to be accurate within the limits of measurement possible.A method designed to predict the effect of a multiple bend made up of a series of bends the individual reflection and transmission coefficients of which were known or predictable is tested and found to be accurate to within ±7 per cent.