Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation

We analyze the general applicability of a recent explicit expression of the Colebrook–White equation for turbulent flow friction factor calculation. This explicit expression, which is based on the Lambert W function, is characterized by an exponential term which imposes restrictions on its use. These constraints have been expressed in terms of pipe roughness (ε/D) and the Reynolds number R that are required for friction factor calculation. These constraints were determined as 8.0666?ln(R)+(ε/D)R<721.97 and 8.0666?ln(R)+(ε/D)R<5731.83, respectively, for machines using single precision and double precision computations. Using the Lambert W function, an explicit equation relating R and ε/D was derived at the limiting case which allowed for a graphical representation of the applicability of the explicit form of the Colebrook–White equation in the R versus ε/D space. Before computing friction factors using the explicit Colebrook–White equation, a quick check must be performed to see if the desired combination of R and ε/D values satisfies the applicable constraint mentioned above.     

Validation of Forchheimer's Law for Flow through Porous Media with Converging Boundaries

Modifications to the Forchheimer equation, which describes the flow behavior through coarse porous media, to account for the convergence of streamlines are made and experimentally verified. The applicability of a resistance law relating friction factor and Reynolds number using the square root of intrinsic permeability as the characteristic length is examined for flow with converging boundaries, and theoretical curves similar to Moody diagram for pipe flow are developed. The effect of convergence of streamlines on the linear and nonlinear parameters is demonstrated.     

Timescale Behavior of the Wall Shear Stress in Unsteady Laminar Pipe Flows

Based on two-dimensional (2D) flow model simulations, the effects of the radial structure of the flow (e.g., the nonuniformity of the velocity profile) on the pipe wall shear stress, τw, are determined in terms of bulk parameters such as to allow improved 1D modeling of unsteady contribution of τw. An unsteady generalization, for both laminar and turbulent flows, of the quasi-stationary relationship between τw and the friction slope, J, decomposes the additional unsteady contribution into an instantaneous energy dissipation term and an inertial term (that is, based on the local average acceleration-deceleration effects). The relative importance of these two effects is investigated in a transient laminar flow and an analysis of the range of applicability of this kind of approach of representing unsteady friction is presented. Finally, the relation between the additional inertial term and Boussinesq momentum coefficient, is clarified. Although laminar pipe flows are a special case in engineering practice, solutions in this flow regime can provide some insight into the behavior of the transient wall shear stress, and serve as a preliminary step to the solutions of unsteady turbulent pipe flows.     

Effect of Convergence on Nonlinear Flow in Porous Media

The behavior of flow through porous media has been the subject of study for a long time. The relationship relating friction factor and Reynolds number using the square root of intrinsic permeability as the characteristic length is examined for flow in porous media with converging boundaries. An experimental investigation of the effect of convergence of streamlines on the linear and nonlinear parameters for different radial flow lines in a converging permeameter for different ratios of radii of the test section is also studied. In the present case, crushed rocks of sizes 11.64 and 4.73 mm were used as media and water as fluid, to develop curves relating friction factor and Reynolds number for different radial flow lines with different ratios of radii of the test section of the permeameter.     

Efficient Treatment of the Vardy–Brown Unsteady Shear in Pipe Transients

An accurate, simple, and efficient approximation to the Vardy–Brown unsteady friction equation is derived and shown to be easily implemented within a one-dimensional characteristics solution for unsteady pipe flow. For comparison, the exact Vardy–Brown unsteady friction equation is used to model shear stresses in transient turbulent pipe flows and the resulting waterhammer equations are solved by the method of characteristics. The approximate Vardy–Brown model is more computationally efficient (i.e., requires one-sixth the execution time and much less memory storage) than the exact Vardy–Brown model. Both models are compared with measured data from different research groups and with numerical data produced by a two-dimensional turbulence waterhammer model. The results show that the exact Vardy–Brown model and the approximate Vardy–Brown model are in good agreement with both laboratory and numerical experiments over a wide range of Reynolds number and wave frequencies. The proposed approximate model only requires the storage of flow variables from a single time step while the exact Vardy–Brown model requires the storage of flow variables at all previous time steps and the two-dimensional model requires the storage of flow variables at all radial nodes.     

Velocity Pulse Model for Turbulent Diffusion from Flowing Water into a Sediment Bed

The “velocity pulse model” simulates the transfer of turbulence from flowing water into a sediment bed, and its effect on the diffusional mass transfer of a solute (e.g., oxygen, sulfate, or nitrate) in the sediment bed. In the “pulse model,” turbulence above the sediment surface is described by sinusoidal variations of vertical velocity in time. It is shown that vertical velocity components dampen quickly inside the sediment when the frequency of velocity fluctuations is high and viscous dissipation is strong. Viscous dissipation (ν) inside the sediment is related to the apparent viscosity depending on the structure of the sediment pore space, i.e., the porosity and grain diameter, as well as inertial effects when the flow is turbulent. A value ν/ν0 between 1 and 20 (ν0 is kinematic viscosity of water) has been considered. Turbulence penetration into the sediment is parametrized by the Reynolds number Re = UL/ν and the relative penetration velocity W/U, where U=amplitude of the velocity pulse; and W=penetration velocity; L = WT=wave length of the velocity pulse; and T is its period. Amplitudes of vertical velocity components inside the sediment and their autocorrelation functions are computed, and the results are used to estimate eddy viscosity inside the sediment pore system as a function of depth. Diffusivity in the sediment pore system is inferred by using turbulent or molecular Schmidt numbers. Turbulence penetration from flowing water can enhance the vertical diffusion coefficient in a sediment bed by an order of magnitude or more. Penetration depth of turbulence is higher for low frequency velocity pulses. Vertical diffusivity inside the pore system is shown to decrease more or less exponentially with depth below the sediment/water interface. Vertical diffusivities in a sediment bed estimated by the “velocity pulse model” can be used in pore water quality models to describe vertical transport from or into flowing surface water. The analysis has been conducted for a conservative material, but source and sink terms can be added to the vertical transport equation.     

Experimental Study and 3D Numerical Simulations for a Free-Overflow Spillway

The main objectives of the present work were to investigate the flow field over a spillway and to simulate the flow by means of a three-dimensional (3D) numerical model. Depending on the wall curvature, the boundary layer parameters decreased or increased with increasing distance along the spillway. The growth of the boundary layer along the spillway is better described as a function of Reynolds number than the normalized streamwise length. A simplified form of the 3D momentum equation can be used to obtain a rough estimate of the skin friction. The velocity profile in the boundary layer along the spillway is described by a velocity–defect relationship. Numerical models provide a cost-effective means of simulating spillway flows. In this study, the water surface profiles and the discharge coefficients for a laboratory spillway were predicted within an accuracy range of 1.5–2.9%. The simulations were sensitive to the choice of the wall function, grid spacing, and Reynolds number. A nonequilibrium wall function with a grid spacing equal to a distance of 30 wall units gave good results.     

Flow Resistance Law in Channels with Flexible Submerged Vegetation

In this paper, experimental data collected in a straight flume having a bed covered by grasslike vegetation have been used to analyze flow resistance for flexible submerged elements. At first, the measurements are used to test the applicability of Kouwen’s method. Then, a calibration of two coefficients appearing in the semilogarithmic flow resistance equation is carried out. Finally, applying the Π-theorem and the incomplete self-similarity condition, a flow resistance equation linking the friction factor with the shear Reynolds number, the depth-vegetation height ratio and the inflection degree is deduced.     

Turbulent/viscous interactions control Doppler/catheter pressure discrepancies in aortic stenosis. The role of the Reynolds number

BACKGROUND: Despite good correlation between Doppler and catheter pressure drops in numerous reports, it is well known that Doppler tends to apparently overestimate pressure drops obtained by cardiac catheterization. Neither (1) simplification of the Bernoulli equation nor (2) pressure recovery effects can explain this dilemma when taken alone. This study addressed the hypothesis that a Reynolds number-based approach, which characterizes (1) and (2), provides a first step toward better agreement of catheter and Doppler assessments of pressure drops. METHODS AND RESULTS: Doppler and catheter pressure drops were studied in an in vitro model designed to isolate the proposed Reynolds number effect and in a sheep model with varying degrees of stenosis. Doppler pressure drops in vitro correlated with the directly measured pressure drop for individual valves (r = .935, .960, .985, .984, .989, and .975) but with markedly different slopes and intercepts. A Bland-Altman type plot showed no useful pattern of discrepancy. The Reynolds number was successful in collapsing the data into the profile proposed in the hypothesis. Parallel results were found in the animal model. CONCLUSIONS: Apparent overestimation of net pressure drop by Doppler is due to pressure recovery effects, and these effects are countered by both viscous effects and inertial/turbulent effects. Only by reconciliation of discrepancies by use of a quantity such as Reynolds number that embodies the relative importance of competing factors can the noninvasive and invasive methods be connected. This study shows that a Reynolds number-based approach accomplishes this goal both in the idealized in vitro setting and in a biological system.     

Numerical Sensitivity Study of Unsteady Friction in Simple Systems with External Flows

This paper investigates the importance of unsteady friction effects when performing water hammer analyses for pipe systems with external fluxes due to demands, leaks, and other system elements. The transient energy equation for a system containing an orifice-type external flow is derived from the two-dimensional, axial momentum equation. A quasi-two-dimensional flow model is used to evaluate the relative energy contribution of total friction, unsteady friction, and the external flow, in a 1,500?m pipeline, with orifice flows ranging from steady-state flows of 2–70% of the mean pipe flow, and a Reynolds number of 600,000. It is found that for initial lateral flows larger than around 30% of the mean flow, unsteady friction effects can probably be neglected, whereas for external flows smaller than this, unsteady friction should generally be considered. Overall, the relative role of unsteady friction is found to diminish as the external flux increases, implying that unsteady friction is not critical for systems with large external flows. These results imply that unsteady friction may have a significant impact on the validity of transient leak detection techniques that have been derived assuming quasi-steady friction. To demonstrate this point, an existing transient leak detection method, originally derived under quasi-steady conditions, is tested with unsteady friction included.     

Power Law Velocity Profile in Fully Developed Turbulent Pipe and Channel Flows

The power law velocity profile has been analyzed in terms of the envelope of the friction factor which gives the friction factor log law. The power law index α and prefactor C are shown as the function of the friction Reynolds number, as well as the function of the alternate variable the nondimensional friction velocity. The fully developed turbulent superpipe flow data of McKeon et al. and fully developed channel flow data of Zanoun et al. have been analyzed and the power law index α and prefactor C data have been estimated, first as a function of the friction Reynolds number and second as function of the nondimensional friction velocity. Based on analysis, several correlations have been proposed that have been supported by the data.     

1D Modeling of Mud∕Debris Unsteady Flows

The National Weather Service 1D dynamic flood routing model FLDWAV is enhanced to include capability of modeling mud∕debris unsteady flows by including an additional friction slope term in the momentum equation of the Saint-Venant equations. Three techniques are incorporated into the model to determine the mud∕debris related friction slope term due to the internal viscous dissipation of non-Newtonian fluids and granular sliding friction of coarse-grained debris surges. These techniques are tested and some computational results are compared with observed field data and experimental data.     

Reservoir Routing using Steady and Unsteady Flow through Rockfill Dams

In this paper a two-dimensional (2D) model for flow through rockfill dams is presented and its results have been compared to 1D model. The model is based on the 2D continuity equation. In the model, an exponential relationship between Reynolds number (R) and Darcy-Weisbach coefficient (f) is suggested and combined with the continuity equation. Coefficients of this relationship are estimated by using real data and a nonlinear optimization technique. Introducing inflow hydrograph to the reservoir and rockfill characteristics as input data and utilizing the above model the outflow hydrograph can be determined. The model has been calibrated and verified using real data. The results of the numerical solution have been shown to be more reliable than the 1D model. To demonstrate the model sensitivity to different parameters, a parametric sensitivity analysis has been conducted. Finally, a comparison between the steady- and unsteady-state results is introduced.     

Formulas for Friction Factor in Transitional Regimes

This note concerns variations of the friction factor in the two transitional regimes, one between laminar and turbulent flows and the other between fully smooth and fully rough turbulent flows. An interpolation approach is developed to derive a single explicit formula for computing the friction factor in all flow regimes. The results obtained for pipe flows give a better representation of Nikuradse’s experimental data, in comparison with other implicit formulas available in the literature. Certain modifications are also made for applying the obtained friction formula to open-channel flows.     

Surface Roughness for Shallow Overland Flow on Crushed Stone Surfaces

Results of laboratory experiments carried out to determine the effective surface roughness for shallow overland flow as a function of the runoff rate, roughness element height, and underlying soil condition are presented. This work was conducted to extend the understanding of the mechanics of shallow overland flows based on the relationships between the Reynolds number, Froude number, and surface resistance over a wide range of conditions. Results exhibit a strong dependence of the effective roughness on the ratio between the depth of flow and the height of the roughness element, and a strong inverse relationship was found between Manning's n (and the Darcy-Weisbach friction factor f) and the Froude number. Three distinct subcritical flow regimes were identified: (1) Submerged flow (R < 300); (2) sheet flow (R > 1,200); and (3) transitional (partially submerged) flow (300 < R < 1,200). Analysis and synthesis of laboratory data allowed relationships between the Reynolds and Froude numbers, and Manning's n, which are suitable for practical engineering applications, to be established in this study.     

Tensile Strength of Unsaturated Sand

A theory that accurately describes tensile strength of wet sand is presented. A closed form expression for tensile strength unifies tensile strength characteristics in all three water retention regimes: pendular, funicular, and capillary. Tensile strength characteristically increases as soil water content increases in the pendular regime, reaches a peak in the funicular regime, and reduces with a continuing water content increase in the capillary regime. Three parameters are employed in the theory: internal friction angle (at low normal stress) ?t, the inverse value of the air-entry pressure α, and the pore size spectrum parameter n. The magnitude of peak tensile strength is dominantly controlled by the α parameter. The saturation at which peak tensile strength occurs only depends on the pore size spectrum parameter n. The closed form expression accords well with experimental water retention and tensile strength data for different sands.     

Experimental Study of Particle Motion on a Smooth Bed under Shoaling Waves Using Particle Image Velocimetry

The motion of spherical particles (diameter 1.58 mm, specific gravity 2.5) on 2 and 3% plane slope was studied in a laboratory wave flume for shoaling wave conditions. The range of wave-height-to-water-depth ratio was 0.24相似文献   

Two-Phase Flow Characteristics of Stepped Spillways

An experimental study on a large model flume with fiber-optical instrumentation indicated that minimum Reynolds and Weber numbers of about 105 and 100, respectively, are required for viscosity and surface tension effects to become negligible compared to gravitational and inertial forces expressed by Froude similitude. Both the location of and the flow depth at the inception point of air entrainment can be expressed as functions of a so-called roughness Froude number containing the unit discharge, step height and chute angle. The depth-averaged air concentration is found to depend only on a normalized vertical distance from the spillway crest and the chute angle for chute slopes ranging from embankment to gravity dam spillways. Air concentration profiles can be expressed by an air bubble diffusion model. The pseudobottom air concentration allows the assessment of the cavitation risk of stepped chutes and is approximated by a regression function. Finally, a new velocity distribution function is presented consisting of a power law up to 80% of the characteristic nondimensional mixture depth, and a constant value above.     

7085铝合金的高温压缩流变应力及软化行为

对均匀化炉冷态7085铝合金进行高温压缩实验,研究该合金在变形温度为350~450℃、变形速率为0.001~0.1 s 1和应变量为0~0.6条件下的流变应力及软化行为。结果表明:流变应力在变形初期随着应变的增加而迅速增大,出现峰值后逐渐软化进入稳态流变;随着变形温度的升高和应变速率的降低,峰值流变应力降低。采用包含Zener-Hollomon参数的Arrhenius双曲正弦关系描述合金的流变行为。分析和建立了应变量与本构方程参数(激活能、应力指数和结构因子)的关系,研究发现本构方程参数随应变量的增加而减少。合金的流变行为差异与动态回复再结晶和第二相粒子相关。