Flow In Rough Pipes Research Articles

Virtual Nikuradse

In this paper we derive an accurate composite friction factor versus Reynolds number correlation formula for laminar, transition and turbulent flow in smooth and rough pipes. The correlation is given as a rational fraction of rational fractions of power laws which is systematically generated by smoothly connecting linear splines in log-log coordinates with a logistic dose function algorithm. We convert Nikuradse's (1933) (J. Nikuradse, 1933 Stromungsgesetz in rauhren rohren, vDI Forschungshefte 361. (English translation: Laws of flow in rough pipes). Technical report, NACA Technical Memorandum 1292. National Advisory Commission for Aeronautics (1950), Washington, DC.) data for six values of roughness into a single correlation formula relating the friction factor to the Reynolds number for all values of roughness. Correlation formulas differ from curve fitting in that they predict as well as describe. Our correlation formula describes the experimental data of Nikuradse's (1932, 1933) (J. Nikuradse, Laws of turbulent flow in smooth pipes (English translation), NASA (1932) TT F-10: 359 (1966).) and McKeon et al. (2004) (B.J. McKeon, C.J. Swanson, M.V. Zaragola, R.J. Donnelly, and J.A. Smits, Friction factors for smooth pipe flow, J. Fluid Mech. 511 (2004), 41–44.) but it also predicts the values of friction factor versus Reynolds number for the continuum of sand-grain roughness between and beyond those given in experiments. Of particular interest is the connection of Nikuradse's (1933) data for flow in artificial rough pipes to the data for flow in smooth pipes presented by Nikuradse (1932) and McKeon et al. (2004) and for flow in effectively smooth pipes. This kind of correlation seeks the most accurate representation of the data independent of any input from theories arising from the researchers ideas about the underlying fluid mechanics. As such, these correlations provide an objective metric against which observations and other theoretical correlations may be applied. Our main hypothesis is that the data for flow in rough pipes terminates on the data for smooth and effectively smooth pipes at a definite Reynolds number R σ(σ); if λ = f(Re, σ) is the friction factor in a pipe of roughness parameter σ then λ = f(R σ(σ), σ) is the friction factor at the connection point. An analytic formula giving R σ(σ) is obtained here for the first time.

Intermittency and rough-pipe turbulence

Recently, by analyzing the measurement data of Nikuradze [NACA Tech. Memo No. 1292 (1950)], it has been proposed [N. Goldenfeld, Phys. Rev. Lett. 96, 044503 (2006)] that the friction factor, f , of rough-pipe flow obeys a scaling law in the turbulent regime. Here, we provide a phenomenological scaling argument to explain this law and demonstrate how intermittency modifies the scaling form, thereby relating f to the intermittency exponent, eta . By statistically analyzing the measurement data of f , we infer a satisfactory estimate for eta ( approximately 0.02) , the inclusion of which is shown to improve the data-collapse curve. This provides empirical evidence for intermittency other than the direct measurement of velocity fluctuations.

Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow

AbstractThe friction factor data from transitional rough test pipes, from the measurements of Sletfjerding and Gudmundsson (2003, “Friction Factor Directly From Roughness Measurements,” J. Energy Resour. Technol. 125, pp. 126–130), have been analyzed in terms of directly measurable roughness parameters, Ra the arithmetic mean roughness, RZ the mean peak to valley heights roughness, Rq the root mean square (rms) roughness, and Rq∕H rms textured roughness (H, the Hurst exponent is a texture parameter), in addition to h the equivalent sand grain roughness. The proposed friction factor λ, in terms of new scaling parameter, viz., the roughness Reynolds number Reϕ=Re∕ϕ (where ϕ is a nondimensional roughness scale), is a universal relation for all kinds of surface roughness. This means that Prandtl’s smooth pipe friction factor relation would suffice provided that the traditional Reynolds number Re is replaced by the roughness Reynolds number Reϕ. This universality is very well supported by the extensive rough pipe data of Sletfjerding and Gudmundsson, Shockling’s (2005, “Turbulence Flow in Rough Pipe,” MS thesis, Princeton University) machined honed pipe surface roughness data, and Nikuradse’s (1933, Laws of Flow in Rough Pipe, VDI, Forchungsheft No. 361) sand grain roughness data. The predictions for the roughness function ΔU+, and the roughness scale ϕ for inflectional roughness compare very well with the data of the above mentioned researchers. When surface roughness is present, there is no universality of scaling of the friction factor λ with respect to the traditional Reynolds number Re, and different expressions are needed for various types of roughnesses, as suggested, for example, with inflectional roughness, monotonic roughness, etc. In traditional variables, the proposed friction factor prediction for inflectional roughness in the pipes, is supported very well by the experimental data of Sletfjerding and Gudmundsson, Shockling, and Nikuradse. In the present work, the predictions of friction factor as implicit relations, as well as approximate explicit relations, have also been proposed for various roughness scales.

Explicit Reformulation of the Colebrook−White Equation for Turbulent Flow Friction Factor Calculation

In this paper, we present an improvement of a mathematically equivalent representation of the Colebrook−White (CW) equation to compute friction factors for turbulent flow in rough pipes. This new form is simple and very well-suited for accurately estimating the friction factor, because no iterative calculations are necessary. Specifically, the friction factor is expressed as the sum of known simple functions and an unknown correction term. This correction term satisfies an auxiliary equation that can be accurately and easily solved with predictable error bounds over the complete range of pipe roughness and Reynolds number values encountered in practice. The simplest case, with the unknown correction term set to zero, resulted in friction factor estimates with errors of <1%. A simple linear approximation of the correction term resulted in a maximum error of 3.64 × 10-4%, whereas friction factor estimates from a continued-fractions-based approximation had a maximum error of 1.04 × 10-10%. These maximum erro...

Turbulent Flow Friction Factor Calculation Using a Mathematically Exact Alternative to the Colebrook–White Equation

We present a novel, mathematically equivalent representation of the Colebrook–White equation to compute friction factor for turbulent flow in rough pipes. This new form is simple, no iterative calculations are necessary, and is well suited for accurate friction factor estimation. A limiting case of this equation provided friction factor estimates with a maximum absolute error of 0.029 and a maximum percentage error of 1% over a 20×500 grid of e∕D and R values ( 10−6 ⩽e∕D⩽5× 10−2 ; 4× 103 <R< 108 ). This was more accurate than the best currently available noniterative approximation of the Colebrook–White equation (maximum absolute error of 0.058; maximum percentage error of 1.42%). The superior accuracy, however, was obtained at the expense of a 30% increase in computational effort over the noniterative approximation. The novel equation presented in this study is theoretical and eliminates the need for best fit parameters or complicated initial guesses that are a characteristic of various empirical approxi...

Evaluation of a universal transitional resistance diagram for pipes with honed surfaces

A method for evaluating a universal transitional resistance diagram for pipes that relates the pressure drop in the pipe to Reynolds number, as a function of relative surface roughness, is presented. The method assumes a universal wake function, a logarithmic overlap region and a power fit in the viscous and buffer layer. Estimates can be made of the friction factor–Reynolds number relationship for arbitrary relative roughness, based on a given surface geometry. The method is illustrated for a pipe with a honed surface finish and uses data of Shockling (“Turbulent flow in a rough pipe,” MSE dissertation, Princeton University, 2005). Honed roughness demonstrates an inflectional behavior in the transitionally rough regime, much like sand grain roughness [Laws of flow in rough pipes, VDI Forschungsh, 361 (1933), 1292 (NACA TM, 1950)], but the method proposed here can be applied to any given roughness behavior. It is suggested that the critical parameter that determines whether the resistance diagram shows inflectional characteristics is the ratio of roughness height to outer layer scale. Based on analysis of data from previous researchers it is suggested that if the relative surface roughness krms∕D&amp;lt;0.0025, where krms is the rms amplitude of the roughness and D is the pipe diameter, inflectional relationships should be observed.

Characterization of surface roughness effects on pressure drop in single-phase flow in minichannels

Roughness features on the walls of a channel wall affect the pressure drop of a fluid flowing through that channel. This roughness effect can be described by (i) flow area constriction and (ii) increase in the wall shear stress. Replotting the Moody’s friction factor chart with the constricted flow diameter results in a simplified plot and yields a single asymptotic value of friction factor for relative roughness values of ε∕D&amp;gt;0.03 in the fully developed turbulent region. After reviewing the literature, three new roughness parameters are proposed (maximum profile peak height Rp, mean spacing of profile irregularities RSm, and floor distance to mean line Fp). Three additional parameters are presented to consider the localized hydraulic diameter variation (maximum, minimum, and average) in future work. The roughness ε is then defined as Rp+Fp. This definition yields the same value of roughness as obtained from the sand-grain roughness [H. Darcy, Recherches Experimentales Relatives au Mouvement de L’Eau dans les Tuyaux (Mallet-Bachelier, Paris, France, 1857); J. T. Fanning, A Practical Treatise on Hydraulic and Water Supply Engineering (Van Nostrand, New York, 1877, revised ed. 1886); J. Nikuradse, “Laws of flow in rough pipes” [“Stromungsgesetze in Rauen Rohren,” VDI-Forschungsheft 361 (1933)]; Beilage zu “Forschung auf dem Gebiete des Ingenieurwesens,” Ausgabe B Band 4, English translation NACA Tech. Mem. 1292 (1937)]. Specific experiments are conducted using parallel sawtooth ridge elements, placed normal to the flow direction, in aligned and offset configurations in a 10.03mm wide rectangular channel with variable gap (resulting hydraulic diameters of 325μm–1819μm with Reynolds numbers ranging from 200 to 7200 for air and 200 to 5700 for water). The use of constricted flow diameter extends the applicability of the laminar friction factor equations to relative roughness values (sawtooth height) up to 14%. In the turbulent region, the aligned and offset roughness arrangements yield different results indicating a need to further characterize the surface features. The laminar to turbulent transition is also seen to occur at lower Reynolds numbers with an increase in the relative roughness.

Transient turbulent friction in fully rough pipe flows

A weighting-function model of unsteady skin friction in fully rough-walled flows in one-dimensional ducts is derived using an idealized radial viscosity distribution. The model complements previous work by the authors for smooth-walled flows. It is assumed that, for sufficiently short-lived transients, the viscosity distribution in the cross-section may be regarded as constant and equal to that in a pre-existing steady flow. The eddy viscosity in an outer annulus is assumed to vary linearly from a minimum at the wall to a maximum at the edge of a central core of uniform viscosity. The resulting weighting-function model for short-lived transients is used to develop a simple formula predicting values of unsteady skin friction coefficients suitable for an instantaneous-acceleration model of unsteady skin friction in fully rough pipe flows.

Sedimentation of particles in developed turbulent flow in rough pipes

Sedimentation of particles on a pipe wall in developed turbulent flow is a rather common phenomenon in industrial processes. Various experimental investigations showed that the sedimentation velocity of particles is greatly affected by the roughness height and the friction factor. In this article, new expressions to calculate the friction factor and the sedimentation velocity of the particles in turbulent flow in rough pipes are suggested for the range of the Reynolds number, 10 4≤Re≤10 7 and the range of the dimensionless relaxation time, 10 −1≤ τ +≤10 2, respectively. Comparisons of the suggested equations in this work with the published experimental data from the literature show good agreement.

Friction and heat and mass transfer for turbulent pseudoplastic non‐newtonian fluid flows in rough pipes

AbstractFriction factor and heat and mass transfer coefficient correlations are developed for turbulent pseudoplastic (purely viscous or inelastic) non‐Newtonian fluids flowing through rough pipes.A correlation for friction factor which is applicable in the region of transition from the smooth to the fully‐rough regime is proposed. It is obtained by adding the two correlations for the limiting situations, i.e. smooth pipes and fully‐rough pipes. The predictions of the proposed correlation are compared with published experimental data and other previous correlations.A model for heat and mass transfer is developed on the basis of the Levich three‐zone model. It is found that the proposed model is capable of predicting satisfactorily heat and mass transfer coefficients for Newtonian and pseudoplastic non‐Newtonian fluid flow in rough pipes.

Mass transfer controlled corrosion of pipelines under two phase (gas—liquid) flow

AbstractRates of diffusion controlled corrosion of smooth and rough pipes were studied under single phase liquid flow and two phase (gas–liquid) flow using the diffusion controlled dissolution of copper in acidified FeCl3. The study was carried out using a recirculating batch tubular reactor through which acidified FeCl3 solution was circulated. The mass transfer coefficient of the dissolution of the copper pipe in the solution was obtained from FeCl3 concentration–time data. Variables studied were solution velocity, gas velocity, and peak to valley height of the rough pipe. For single phase fully developed flow, the mass transfer coefficients of the smooth uncorroded pipes were correlated for the conditions 3000 < ReL < 18 000 and 1688 < Sc < 1899 by the equation Sh = 0·054ScO·33ReL0·79, where ReL is the liquid Reynolds number, Sc is the Schmidt number, and Sh is the Sherwood number. For single phase flow in rough pipes, the data were correlated for the conditions 5 < e+ < 52·5, 3000 < ReL < 18 000, and ...

Effects of Pipe Roughness on Friction Pressures of Fracturing Fluids

Summary In this study, the effects of pipe roughness on friction pressures of uncrosslinked and crosslinked fracturing gels are reported from flow data gathered in smooth and rough pipes. Pipe-roughness effects are found to be very significant in the estimation of friction pressures of fracturing fluids. These effects are more pronounced with less viscous fluids than with more viscous fluids and at higher rates with both fluids. From the data gathered with fluids tested in this study, correlations to estimate friction pressures of fracturing gels in turbulent flow in rough pipes are also presented. Though not ideal, these correlations allow better estimates of friction pressures in rough tubular conductors than existing methods do.

So-called constants of turbulence in rough pipes

A direct analysis of Nikuradse's experiments on flow in rough pipes has shown that on the quadratic resistance interval the value of the second constant of turbulence depends on the relative roughness of the pipe. It is established that the universal logarithmic relation for the velocity profile in a pipe in the quadratic resistance regime is an approximation of the more accurate relation obtained by the authors, which takes into account the effect of the relative roughness of the pipe. Where the roughness is significant, the proposed expression for the skin-friction coefficient is more accurate than the well-known Prandtl-Nikuradse formula.

The Influence of Surface Roughness on Resistance to Flow Through Packed Beds

Experiments were performed to determine the effect of roughness on flow through randomly packed beds of spheres. Three different packings were investigated, one of smooth spheres, and two others composed of spheres with roughness elements added to the surface. The relative roughness, defined as the height of the added elements divided by the diameter of the smooth spheres, was .012 and .026 for these two cases. The experiments covered a range of Reynolds numbers based on the sphere diameter from near unity where the flow is dominated by viscosity to 1600 where the flow is dominated by inertia. It was found that the pressure drop is substantially increased by the presence of surface roughness over the entire range of Reynolds numbers studied. The observed behavior is quite different from that which has been proposed previously by drawing analogy with flow in rough pipes, since the flow at low Reynolds number as well as high Reynolds number was affected by roughness.

Determination of turbulent pressure loss of non-newtonian oil flow in rough pipes

A detailed theoretical analysis is carried out of the turbulent flow of non-newtonian fluids in rough pipes in which approximate assumptions are of negligible importance. A new friction-factor equation (BNS-equation) is obtained which is valid for turbulent flow of “power-law type” non-newtonian fluids in the transition region between smooth and wholly rough wall turbulence.

General Algorithm for Rough Conduit Resistance

An analytical model for computing the frictional resistance to flow in rough pipes based on the geometric characteristics of the roughness elements on the boundary is developed. The the resistance coefficient is facilitated by charts which relate the projected area of the roughness elements to the velocity distribution. The results of computations are compared with experimental results for different types of artificial and natural roughness.

Sand Bed Friction Factors for Oscillatory Flows

Published data from the Carstens, Neilsen, and Altinbilek (4) oscillatory water tunnel tests are plotted as friction factor versus Reynolds number. These data, for three sand sizes and for both rippled and flat movable beds, are analyzed in a manner analogous to early treatment of flow in rough pipes that produced the Moody diagram. Laminar, transitional, and turbulent regimes are defined. The turbulent regime data are further analyzed to find the relative roughness, a combination of ripple height, grain size, and bottom particle displacement, which will predict the friction factor. The ripple heights for the 0.585mm sand increased with increasing Reynolds number while the ripple heights for the 0.297mm and 0.190mm sands decreased with increasing Reynolds number. Considering rippled beds only, the friction factor is proportional to the grain size divided by the square root of the Reynolds number. Grain roughness contributes 64% to 100% of the total bottom roughness, based on the relative roughness term.

Calculation of Variable Property Turbulent Friction and Heat Transfer in Rough Pipes

A numerical calculation method for turbulent flow in rough pipes is developed. A mixing length model is used in the turbulent core while a roughness element drag coefficient and a sub-layer Stanton number are used to characterize transport to the wall. Sample wall relations are developed for sandgrain roughness and transverse repeated rib roughness, and it is shown that large roughness heights require accounting for terms of order of roughness height divided by pipe radius. For gas flows with cooling, the effects of variable properties are investigated for smooth walls and both roughness patterns. For smooth walls, comparison with experiment is satisfactory; for rough walls experimental data is not available. Simple power law formulae representing variable property effects for fully rough flows are presented.

An eddy-viscosity approach to the calculation of velocity and heat-transfer properties of flow in rough pipes

An eddy-viscosity approach to the calculation of velocity and heat-transfer properties of flow in rough pipes

Turbulence Structure for Fully Developed Flow in Rough Pipes

This paper presents the results of an experimental study of fully developed turbulent flow in rough pipes. The investigation was undertaken in an attempt to better understand some of the changes in turbulence structure which accompany a change in wall roughness, and the experimental results include energy spectra of the fluctuating velocities in each coordinate direction, as well as microscales and macroscales of turbulence. The results indicate that the turbulence structure in the central region of the pipe is relatively independent of surface roughness, while the structure near the wall is very much dependent on the nature of the solid boundary. For hydraulically smooth flow, the principal effect of the wall is to radially suppress the extent of the larger eddies in the flow field. In the transition regime, the longitudinal extent of both large and small eddies is disrupted by the roughness elements, and the dissipation eddies appear to be stretched radially as they intermesh with the sand grains. The large eddies are almost completely obliterated by the roughness particles in the fully rough flow regime, while a tendency toward isotropy of the small eddies is noted. Thus, very definite changes in the flow structure occur as wall roughness is varied, and it is postulated that these changes are due principally to the fact that the roughness elements occupy the space where the three-dimensional streaky structure originates in smooth pipe flow.