Turbulent Flow In Smooth Pipes 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.

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

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.

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.

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.

Optimal structure of tree-like branching networks for fluid flow

Tree-like branching networks are very common flow or transportation systems from natural evolution. In this study, the optimal structures of tree-like branching networks for minimum flow resistance are analyzed for both laminar and turbulent flow in both smooth and rough pipes. It is found that the dimensionless effective flow resistance under the volume constraint for different flows is sensitive to the geometrical parameters of the structure. The flow resistance of the tree-like branching networks reaches a minimum when the diameter ratio β∗ satisfies β∗=Nk, where, N is the bifurcation number N=2,3,4,… and k is a constant. For laminar flow, k=−1/3, which is in agreement with the existing Murray’s law; for turbulent flow in smooth pipes, k=−3/7; for turbulent flow in rough pipes, k=−7/17. These results serve as design guidelines of efficient transport and flow systems.

Friction factor correlations for laminar, transition and turbulent flow in smooth pipes

In this paper we derive an accurate composite friction factor vs. Reynolds number correlation formula for laminar, transition and turbulent flow in smooth 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 curve algorithm. 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 correlation is as accurate, or more accurate, than other correlations in the range of Reynolds numbers in which the correlations overlap. However, our formula is not restricted to certain ranges of Reynolds numbers but instead applies uniformly to all smooth pipe flow data for which data is available. The properties of the classical logistic dose response curve are reviewed and extended to problems described by multiple branches of power laws. This extended method of fitting which leads to rational fractions of power laws is applied to data of Marusic and Perry (1995) [1] for the velocity profile in a boundary layer on a flat plate with an adverse pressure gradient, to data of Nikuradse (1932) [2] and McKeon et al. (2004) [3] on friction factors for flow in smooth pipes and to the data of Nikuradse [4] for effectively smooth pipes.

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.

Modified log-wake laws for turbulent flow of the outer and inner regions in smooth pipes

Based on theoretical analysis two modified log-wake laws for turbulent flow in smooth pipes are obtained, one is applicable to the outer region and other one to inner region, the new law for the outer region fits the velocity profile measured in smooth pipes by Zagarola very well and the effect of Reynolds number can be taken into consideration, the velocity profile for inner region satisfies the wall boundary conditions, equals zero at the pipe wall and smoothly joins up with the velocity profile in outer region, the adopted eddy viscosity model is consistent with Laufer's, Nunner's and Reichardt's experimental data.

Modified log-wake law for turbulent flow in smooth pipes

A modified log-wake law for turbulent flow in smooth pipes is developed and tested with laboratory data. The law consists of three terms: a log term, a sine-square term and a cubic term. The log term reflects the restriction of the wall, the sine-square term expresses the contribution of the pressure gradient, and the cubic term makes the standard log-wake law satisfy the axial symmetrical condition. The last two terms define the modified wake law. The proposed velocity profile model not only improves the standard log-wake law near the pipe axis but also provides a better eddy viscosity model for turbulent mixing studies. An explicit friction factor is also presented for practical applications. The velocity profile model and the friction factor equation agree very well with Nikuradse' and other recent data. The eddy viscosity model is consistent with Laufer's, Nunner's and Reichardt's experimental data. Finally, an equivalent polynomial version of the modified log-wake law is presented.

A higher-order asymptotic theory for fully developed turbulent flow in smooth pipes

A complete second-order asymptotic theory for fully developed turbulent flow in smooth pipes at high turbulent Reynolds numbers is presented in the paper. The theory is based on Prandtl's mixing-length hypothesis involving a fourth-order polynomial representation for the mixing length and taking into account its dependence on the Reynolds number. Two main contributions with respect to the existing literature have been achieved:

Evaluation of von Karman's Constant from Integral Flow Parameters

Theory and data for turbulent flow in smooth pipes are employed to evaluate von Karman's constant from integral flow parameters. A set of equations for mean velocity and centerline velocity are der...

An empirical model of velocity profiles for turbulent flow in smooth pipes

An empirical model of velocity profiles for turbulent flow in smooth pipes

Location of the eddy viscosity maximum in smooth pipe flow

Nikuradse found from his measurements of turbulent flow in smooth pipes that the eddy viscosity maximum occurs midway between the pipe wall and the center line. It is shown that a velocity-defect law for a similar flow leads to the same result.

An analysis of heat transfer to turbulent flow of drag reducing fluids

Reichardt's analysis for heat transfer to turbulent flow in smooth pipe is extended to drag reducing viscoelastic fluids. Velocity profile expression by Deissler is modified to account for changes brought about by viscoelastic drag reducing fluids. The final correlation compares very favourably over a wide range of Prandtl (or Schmidt) numbers, with experimental data.

Developing turbulent flow in smooth pipes

A numerical and experimental investigation of steady incompressible developing turbulent flow in smooth pipes is presented. Finite difference techniques are used to solve simultaneously the vorticity transport and stream function equations utilising a modified form of the Van Driest effective viscosity model. The numerical solutions are verified experimentally using air as a working fluid at pipe Reynolds 1 × 105, 2 × 105 and 3 × 105.

A simple expression for the velocity distribution in turbulent flow in smooth pipes

A simple expression for the velocity distribution in turbulent flow in smooth pipes

The Flow and Pressure Losses in Smooth Pipe Bends of Constant Cross Section

SummaryIncompressible turbulent flow in smooth pipe bends of circular-arc curvature and constant cross section is discussed. The effects of bend angle, radius ratio, duct cross-sectional shape and Reynolds number are considered. Particular attention is paid to the influence of the tangents on the flow and losses in a bend. In addition consideration is given to the definition and measurement of the bend loss.A correlation of experimental pressure loss data is made. It is shown that large variations in experimental results, which cannot be ascribed to the effects of geometrical parameters or Reynolds number, are accounted for by the effects of the downstream tangent and position at which the pressure measurements were made. The inadequacy of existing experimental information is an obstacle to the presentation of comprehensive data on the effect of geometrical parameters on bend losses.

Pressure Losses in Smooth Pipe Bends

The results of extensive experimental studies to determine the pressure losses for turbulent flow in smooth pipe bends of circular cross section are presented in this paper. To make the data usable in practical design problems, the results are discussed in relation to those found by previous investigators, and empirical formulas for the bend-loss coefficient are given. The general correlation of the test data appears to be as good as our present test information will permit.

Study of Turbulent Flow in Smooth Pipes

亂流に関するPrandtlの變動の假説を用ひて滑かなる管内の混合距離、管内の速度、速度變化交換量等の間の理論的関係式(6)(7)(8)(9)を求め、Nikuradseの實驗値より混合距離を表はす實驗式(10)を見出し、これより管内の速度、速度變化、交換量等を表はす式(11)(13)(19)(19α)(20)を得た。これ等の諸式はレイノルド數が100×103以上なる場合には實驗値とよく一致する。尚上記の理論的関係式を用ひて管内の速度分布を與へるPrandtlの一般式(1).ポテンシャル式(2)及びKarmanの理論式(3)を調べこれ等の式より求めた速度變化、混合距離及び交換量の値が實驗値と甚しく相違する事を示した。