An Empirical V-notch Weir Equation and Standard Procedure to Accurately Estimate Drainage Discharge

HighlightsAn empirical flow equation was developed for a metal-edge sharp-crest V-notch weir.A top-down approach was used to determine the height of the V-notch apex.A combination of the weighing method and a flow meter was used to develop the stage-discharge equation.A standard procedure was presented to accurately estimate the flow rate.Abstract. A reliable empirical flow equation for V-notch weirs will provide flow estimates that can be used to calculate nutrient loads leaving fields with subsurface drainage. The objective of this study was to develop such an equation for an AgriDrain metal-edge sharp-crest 45° V-notch weir. In this undertaking, we measured flow rate with a combination of the weighing method for low flow and a turbine flow meter for high flow. The head of water (H) was measured inside a 25-cm AgriDrain control structure with a three-step method. First, we measured the water level (a) and height of the control structure (b). Second, we measured the height of the V-notch apex (c). Third, we calculated head using this equation: H= (b-a) – (b-c). Based on the flow meter readings (Q) and H measurements, we developed the following stage-discharge equation: Q = 0.749H 2.25, with Q in liters per minute and H in centimeters. This equation is valid for an H less than the height of the V-notch (i.e., flow through the V-notch) with unsubmerged flow. Based on field experience, we provide a standard procedure for accurate estimation of drainage discharge. In conclusion, the stage-discharge equation developed in this study can provide reliable flow estimates for subsurface drainage studies. Keywords: Flow rate, Metal-edge weir, Sharp-crest weir, Subsurface drainage, Tile drainage, Weir placement.

On the limiting Stokes wave of extreme height in arbitrary water depth

Both Schwartz (J. Fluid Mech., vol. 62 (3), 1974, pp. 553–578) and Cokelet (Phil. Trans. R. Soc. Lond., vol. 286 (1335), 1977, pp. 183–230) failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Padé approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of$120^{\circ }$included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.

Numerical Simulation of Free Surface and Flow Field Turbulence in a Circular Channel with the Side Weir in Subcritical Flow

When flow surface is higher than of a side weir crest, the overflow spilt over the crest and divert into a side channel. These structures are extensively used in urban sewage disposal networks, water supply systems, and drainage and flood diversion networks. This study simulates stream free surface, discharge over a sharp-crest side weir, and discharge coefficient of a side weir in a circular channel using FLOW-3D software. Numerical model results were compared with the experimental ones and the comparison proved an acceptable consistency between the numerical and experimental results. RNG k-ε turbulence model was used for simulating flow turbulence. The volume of fluid (VOF) method was used in this CFD analysis for predicting changes of flow free surface. Then, the numerical simulation results were examined for discharge coefficient of the side weir and flow free surface for different discharge passing through the main channel. The changes of dividing stream surface from main channel bed toward stream free surface were examined. The concluding section assessed the effect of shape of a circular channel on the pattern and intensity of a secondary flow in the main channel and the impacts of the discharge passing through the circular channel on height of stagnation point and shear stress pattern in the main channel bed.

Period tripling and energy dissipation of breaking standing waves

We examine the dynamics of two-dimensional steep and breaking standing waves generated by Faraday-wave resonance. Jiang et al. (1996) found a steep wave with a double-peaked crest in experiments and a sharp-crested steep wave in computations. Both waveforms are strongly asymmetric in time and feature large superharmonics. We show experimentally that increasing the forcing amplitude further leads to breaking waves in three recurrent modes (period tripling): sharp crest with breaking, dimpled or flat crest with breaking, and round crest without breaking. Interesting steep waveforms and period-tripled breaking are related directly to the nonlinear interaction between the fundamental mode and the second temporal harmonic. Unfortunately, these higher-amplitude phenomena cannot be numerically modelled since the computations fail for breaking or nearly breaking waves. Based on the periodicity of Faraday waves, we directly estimate the dissipation due to wave breaking by integrating the support force as a function of the container displacement. We find that the breaking events (spray, air entrainment, and plunging) approximately double the wave dissipation.

Free-surface flows past a surface-piercing object of finite length

Steady two-dimensional flows past a parabolic obstacle lying on the free surface in water of finite depth are considered. The fluid is treated as inviscid and incompressible and the flow is assumed to be irrotational. Gravity is included in the free-surface condition. The problem is solved numerically by using boundary integral equation techniques. It is shown that there are solutions for which the flow is supercritical both upstream and downstream and others for which the flow is subcritical both upstream and downstream. These flows have continuous tangents at both ends of the obstacle at which separation occurs. For supercritical flows, there are up to three solutions corresponding to the same value of the Froude number when the obstacle is concave and up to two solutions when the obstacle is convex. For subcritical flows, there are solutions with waves behind the obstacle. As the Froude number decreases, these waves become steeper and the numerical calculations suggest that they, ultimately, reach limiting configurations with a sharp crest forming a 120° angle.

On the forming of sharp corners at a free surface

By applying the technique for time-dependent irrotational flows proposed in the preceding paper, a new class of exact free-surface flows is derived. In these, the free surface has the form of a variable hyperbola, whose axes rotate in space. The angle γ between the asymptotes, and the angle δ of orientation of the axes, are found explicitly as functions of the time. The solutions fall into three groups. First there are those in which γ diminishes smoothly from 90° (a rectangular hyperbola) to zero (a slender hyperbola) while the angle of orientation 8 increases towards a finite limit. Secondly there are solutions in which γ diminishes to a positive minimum, and then returns again to 90°. Thirdly γ may begin from small values, increase to less than 45° and return again towards zero. In each case the total angle δ max remains finite. An exceptional but very interesting solution, in which the vertex angle γ remains constant at 45° and the free surface rotates with uniform angular velocity δ, is described in terms of elementary functions of the time t . It is suggested that these flows, which are generalizations of the symmetrical Dirichlet hyperbolae, are relevant to the flow near the tip of a breaking gravity wave. Since for large values of t the angle δ tends to its limit like t ~ 1 , the flows may be matched asymptotically to the parabolic arch of a plunging breaker. In other cases, the tip of the wave can curl over and appear to form a vortex. By the inclusion of terms cubic in the space coordinates it is also possible to represent a sharp crest pointing upwards and tending towards a cusp.