Experimental Analysis and Numerical Simulation of Hydraulic Jump

Saint Venant equations are numerically solved to simulate the formation of hydraulic jump in a rectangular channel having a small bed slope. The MacCormack’s scheme is used for the solution by applying specified initial and boundary conditions until a steady state flow is reached. The location of the hydraulic jump is determined as a part of these computations. The artificial viscosity technique should be used in the computations to dampen the superior oscillations near the steep gradient of the simulated hydraulic jump. Twenty laboratory experiments were carried out for verification of the numerical model. The upstream Froude number for these experiments ranged from 2.17 to 7.0 in three different bed slopes 0, 0.02174, 0.0475. The simulated hydraulic jump profiles using the MacCormack’s scheme shows a good agreement with the experimental data. An empirical equation was developed to determine the location of hydraulic jump using regression analysis based on simulated data. Software based on Computational Fluid Dynamics (CFD) was also used to simulate two of these experiments. The results obtained from CFD analysis matched fairly with the experimental results.


Introduction
Hydraulic jump is a sudden transition to subcritical flow from a supercritical flow. The phenomenon is observed in canals below sluice gates, at the foot of spillways or when there is an abrupt change in slope from steep to flat. Hydraulic jumps are practically used as energy dissipaters below spillways to avoid scouring in the downstream, for mixing chemicals and aerate water for city water supplies or for removing air pockets from water supply lines for preventing air locking. It is essential to determine the various parameters like length and location of the jump, amount of energy dissipated etc. to design a hydraulic structure. The theory of jump was developed in earlier days mainly on the basis of extensive experimental and empirical work. Several researches were done by conducting extensive laboratory experiments and developing design charts and empirical relationships between various flow parameters [1][2][3][4][5][6][7][8][9][10][11]. The advent of high speed digital computers and developments in Computational fluid dynamics gave the opportunity to solve the governing equations of hydraulic jump numerically. An extensive amount of data has been reported in the literature on the hydraulic jump. To determine the jump location, computed the water surface profiles were computed for supercritical flow starting from the upstream end and the subcritical flow starting from the downstream end, and the jump is formed at a location IOP Conf. Series: Earth and Environmental Science 505 (2020) 012024 IOP Publishing doi: 10.1088/1755-1315/505/1/012024 2 where the specific forces on both sides of the jump are equal [12]. A strip-integral method was used to compute the jump length, water surface profile, and pressures at the bed [13]. The finite-difference method was used [14] and the finite-element method was used [15] to solve the St. Venant equations numerically until a steady state was reached. The location of the hydraulic jump is automatically computed as part of the solution. Boussinesq equations describing one-dimensional unsteady, rapidly varied flows were integrated numerically to simulate both the sub-and supercritical flows and the formation of a hydraulic jump in a rectangular channel having a small bottom slope. For this purpose the MacCormack (second-order accurate in space and time) and two-four (second-order accurate in time and fourth-order in space) explicit finite-difference schemes were used to solve the governing equations subject to specified end conditions until a steady state was reached. There was significant agreement between the experimental and numerical results. The fourth order accurate model was found to be more precise than the second order accurate model. But the simulations showed that the Boussinesq terms have little effect in determining the location of the hydraulic jump. Hydraulic jump was numerically simulated on a straight horizontal channel with supercritical Froude numbers 2.0 and 4.0 by solving Reynolds averaged Navier-Stokes equations [16]. Turbulence was modeled through the k -ε closure equations. Galerkin finite element method with three-noded triangular elements is used for spatial discretization. A detailed study of the internal and external characteristics of hydraulic jump is done and compared with experimental values where possible. The FLOW 3D was used to simulate the hydraulic jump in the convergence stilling basin [17]. The software was applied to numerically solve the Navier-Stokes equations for solution domains, namely the shout, the stilling basin and the downstream of dam, and to estimate the turbulence flow, the standard k-ε and RNG models was used. These models are based on the volumeof-fluid method, and they are capable of simulating the hydraulic jump. The calculated results such as the pressure, the velocities, the flow rate, the surface height air entranced, the kinetics energy, the kinetics energy dissipated, and the Froude number were compared with the scale model data where available. The physical model and CFD model results showed good correlations. The primary goal of this present study is to numerically simulate hydraulic jump in a sloping channel using the MacCormack method [18] to solve the St. Venant equation which is used as the governing equations. A source code is written in MATLAB (matrix laboratory), which is a proprietary product of MathWorks, to do the numerical computations.

Governing equations
Unsteady flow phenomenon such as hydraulic jump in rectangular channel are often modeled as one dimensional flow which can be described by a set of quasi-linear, hyperbolic partial differential equations, called the Saint-Venant equations. The derivation of the equations involves the following assumptions: a) The pressure distribution is hydrostatic. b) The channel bottom slope is small enough such that the depth measured normal to the channel bed is approximately equal to the depth measured vertically. c) The velocity distribution is uniform over the entire channel cross-section. d) The channel is prismatic. e) Friction losses for a given flow velocity during unsteady flow is same as that during steady flow. The one dimensional St. Venant equation may be written as follows: Continuity: where; x is the distance along the channel bottom (considered positive in the downstream direction), t is the time, u is the flow velocity in the x-direction, h is the flow depth, g is the acceleration due to gravity, S 0 is the channel bottom slope, and S f is the slope of the energy grade line. Manning equation

Numerical scheme
A hydraulic jump can be simulated by solving the above governing equations with appropriate boundary conditions. The initial conditions are specified and the iterations are continued until steady state is reached. If shock capturing numerical technique is used, then the jump forms as a part of the steady state solution. MacCormack's scheme [18] is a two level predictor corrector scheme. In the predictor part, forward finite difference is used for the spatial derivative terms and backward finite difference approximation is used in the corrector part. The finite difference expressions are: Predictor step: where i is the node in x direction, k is the node in time direction, asterisks refers to the predicted values of the variable; and ∆x and ∆t are the spatial grid size and time step size respectively. Based on the above finite difference approximation (1) and (2) Corrector step: Two double asterisks denote the values of the variable after the corrector step. Based on the above finite difference approximations the variables after the corrector step can be written as: where; The channel is divided into n equal reaches. Thus if the upstream end is numbered section 1, then the downstream end will be n+1. The flow velocity and flow depth is specified at time t=0 as the initial condition. The flow is assumed to be supercritical initially in the entire channel. The initial steady state flow depth is determined by integrating the gradually varied flow equation; starting with the specified flow depth and velocity at section 1: The MacCormack's scheme is used to compute the variables at the interior nodes. At the boundaries the flow conditions are specified. The flow depth and velocity are specified at the upstream boundary and they are same as the initial conditions. At the downstream boundary, a constant flow depth is specified and the flow velocity is calculated from the characteristic form of (1) and (2) [19] shows that higher order terms are introduced which are not present in the governing partial differential equations. These terms represent the truncation errors which affect the behavior of the scheme and contribute to the scheme what is known as diffusion and dispersion. The solution has dissipative errors if the leading term in the truncation error contains an even derivative and dispersive errors if the leading term has odd derivatives [20]. For C n less than that required by the Courant-Friedrich-Lewy (CFL) limit, the dispersive errors result in introducing numerical oscillations in the solution. Therefore it becomes necessary to add artificial viscosity to smooth these oscillations. Several procedures have been reported for this purpose. A procedure given in [21] used herein, has the advantage of smoothing regions where the solution has large gradients while leaving relatively smooth areas undisturbed; i.e., high-frequency oscillations are smoothed. A parameter Ξ is first computed from a normalized form of the gradients of one variable.
For the studies reported herein, the depth of flow h was selected for determining the parameter: In which k is used to regulate the amount of dissipation. The computed variables are then modified as

Experimental details
The experiments were carried out in a rectangular perspex flume 36.5 cm width, 45 cm height and 5 m long ( Figure 1). The discharge was measured by a digital (magnetic type) flow meter. The flume has a tail gate to control the water depth. By adjusting the tail water depth the jump position was varied. The flow depths at the upstream end and in the section of the flume with metal walls were measured at equally spaced intervals by a point gauge having an accuracy of 1 mm. There were continuous undulations in the water surface downstream of the jump. The maximum and minimum levels of these undulations at a location were marked and an average of these levels was considered the depth at that location. Twenty laboratory experiments were carried out with the Froude number upstream of the jump ranging from 2.17-7.00 for three different slopes 0.0435, 0.02174 and 0.  the sides are made up of glass. First, the initial steady-state depth and velocity at every computational node were computed by assuming the flow to be supercritical throughout the flume. Then, the unsteady computations were started by increasing the downstream depth to the value measured during the experiment (see Table 1). The computations were continued until they converged to the final steady state for the specified end conditions. One very important parameter in the simulation of a hydraulic jump is the size of the spatial grid size, ∆x. Its value was varied from 0.1 m to 0.4 m. Values greater than 0.4 m and in some cases 0.3 m could not be used since they resulted in the jump forming at less than three computational nodes. For Fr = 7, and slope 0.0435 the jump was computed to form between 1.60 m and 2.10 m from the upstream end of the flume, with the average distance of four different values of ∆x being 1.95 m. In other words, the jump location may be predicted in a satisfactory manner for typical engineering applications by using a reasonable value of ∆x. For the range of Froude numbers tested, these values varied from 0.014 to 0.016 depending upon the flow depth since the bottom of flume is made up of metal and the sides are made up of glass. First, the initial steady-state depth and velocity at every computational node were computed by assuming the flow to be supercritical throughout the flume. Then, the unsteady computations were started by increasing the downstream depth to the value measured during the experiment (Refer Table 1). The computations were continued until they converged to the final steady state for the specified end conditions. One very important parameter in the simulation of a hydraulic jump is the size of the

Comparative analysis between experimental and numerical results
Once the numerical solution converged to a steady state -the depths at the corresponding grid points is obtained, which gives the flow profile alongwith the jump.    Equations describing these flows were derived in [22] and [23] assuming that the vertical velocity varies from zero at the channel bottom to its maximum value at the free surface. These equations referred to as the Boussinesq equations. But the Boussinesq terms have little effect in predicting the jump location [24]. Numerical models gave the same result for the St. Venant equation and Boussinesq equations when fourth order accurate numerical scheme is used. Thus the accuracy was more dependent on the order of the numerical scheme. MacCormack scheme being second order accurate in space and time could not predict the jump location precisely for higher Froude numbers but gave good results for low Froude numbers as is evident from the comparison of the results. The comparison between the computed and the numerical results shows that as the Froude number increases the jump forms downstream of the location obtained experimentally. For the slope 0.0435 for Froude numbers more  figure 4, the deviation of the jump location obtained numerically from the measured location is within ±25%.

General equation using regression analysis
Sometimes it may be useful to form an empirical relationship between the parameters determining the jump location. The results obtained from the empirical equation are then compared with the numerical results. It is assumed that the dependent variable (L), which is the distance from the beginning of the flume to the location of hydraulic jump is a function of the following independent variables: density of flow (ρ), the upstream water depth (h u ), the tail water depth (h t ) at the end of the channel, the upstream velocity (u), the acceleration of gravity (g), bed slope (S 0 ). The general function relationship between the above variables can be written as: Using the dimensional analysis, the π terms obtained are, 1 These π terms may be arranged in the following non dimensional form: The general form of equations relating a dependent π-term with a number of independent π terms using regression analysis in this work is in the form of the product of powers of relevant π terms, i.e., 3 (20) Finally, the equation can be rewritten in matrices form as follows: where, N is the number of observations. Then, the values of the parameters C, a 2 , a 3 , ………, a m are obtained and can be replaced back into equation 19. If the term (L/h u ) is taken as a dependent term, the equation form will be as follows:     