3D Free Surface Model of Laboratory Channel with Rectangular Broad-Crested Weir

 

MD. AKHTARUZZAMAN SARKER¹ and DAVID G. RHODES²

 

¹PhD Student, ²Lecturer

Cranfield University, Engineering Systems Department

RMCS Shrivenham, Swindon SN6 8LA, UK

Tel: 441793 785643, Fax: 441793 783192, Email: sarkerma@rmcs.cranfield.ac.uk

 

 

ABSTRACT

Measurements of the free surface profile in a laboratory channel housing a broad-crested weir were carried out at the maximum flow rate of the facility. The experimental arrangement was then numerically modelled by CFD commercial software (FLUENT), using the standard k-e turbulence model and wall function, in conjunction with the volume of fluid free surface model. Predicted results agreed closely with experiment near the critical section on the weir and in the super-critical flow downstream of the weir. However, the CFD under-predicted the upstream water level.

 

Keywords: CFD, k-e model, volume of fluid method, free surface flow, hydraulic structures.

 

INTRODUCTION

Numerical models applied to river engineering frequently represent extensive river reaches with comparatively little attention being given to flow details at local features. Often the effects of such features are subsumed in an overall resistance coefficient or some other expedient, which is usually quite satisfactory for the purpose at hand. However, when local flow structures (eg. rapidly varying free surface flow profiles, 3-dimensional velocity distributions and bed shear stress distributions on 3-dimensional surfaces) are required, numerical models of this kind are unsuitable.

 

With the development of refined flow modelling techniques in CFD, the facility to model local features in a river became commercially available and there was a consequent increase in river engineering applications. However, such applications in many cases treated the free surface as a boundary condition, the geometry of which had to be defined by the user in advance of the numerical solution procedure. Generally these boundary conditions were plane surfaces, at a prescribed height and with various associated properties, eg. the rigid lid approximation and the plane of symmetry were two of the simplest plane surface boundary conditions in common use, and in the research community attempts were also made to model the damping or anisotropy of local turbulence intensity in the vicinity of the free surface.

 

However, if the free surface profile is not physically plane, as in a gradually varied or rapidly varied flow, this approach is no longer appropriate, and indeed it is not possible to know the precise free surface profile in advance of the calculation of the general flow structure, the two being inextricably linked. Therefore in recent years a number of CFD commercial software packages have incorporated one or more models for calculating the free surface profile, while simultaneously solving the water internal flow structure by the usual methods. The two processes are interactive: the turbulence model, influential in developing the internal flow structure of the water phase, consequently affects the development of the free surface profile through, among other processes, the resistance and energy losses that it predicts; the free surface profile in effect defines the flow domain of the water phase for the turbulence model.

 

This paper describes the first stages of a feasibility study into the application of commercial CFD to the calculation of local features in a river. Physical measurements of the 3-dimensional free surface profile were carried out in a laboratory channel housing a rectangular broad-crested weir. The experimental data were compared with the predictions of the commercial software FLUENT, using the standard k-e turbulence model with a wall function, and the volume of fluid method for free surface prediction.

 

EXPERIMENTAL PROCEDURE

 

 

Figure 1: General arrangement of laboratory flume with broad-crested weir.

 

The general arrangement of the recirculating flume is given in Figure 1. It consisted of a 105 mm wide ´ 3340 mm long open channel with a full-width rectangular broad-crested weir 400 mm long ´ 100 mm high located as shown. The bed, which was hydraulically smooth, was set horizontal and the flow rate adjusted to the pump capacity of 4.684 l s^-1. This was measured by a turbine flow meter, the latter being checked against a timed known volume. The channel exit control weir was removed for the purposes of this study, with the result that downstream of the broad-crested weir the flow was entirely supercritical, discharging into an exit sump over a sharp brink. Before measurements were taken, the flume was run for an hour to warm up and thereafter a near steady water temperature was held at about 18 ^°C.

 

A pointer gauge and vernier scale were used to measure the free surface local height above bed level. A longitudinal free surface profile was obtained on the centreline of the flume, readings being taken at intervals varying from 50 mm down to 5 mm, at the highest resolution required in the region of rapidly varied flow downstream of the weir. With the origin of the longitudinal x-coordinate at the centre of the weir, transverse free surface profiles at transverse z-coordinate intervals of 5 mm or less were taken at x = -300, -200, 0, 200, 440 and 800 mm, the first and the two last coordinates being in the regions of gradually varied flow upstream and downstream of the weir respectively, and the rest in the rapidly varying flow around the weir.

 

COMPUTATIONAL FLUID DYNAMICS

Using the commercial software FLUENT, the standard k-e turbulence model was selected; this is a linear eddy viscosity model in which k, the turbulence kinetic energy per unit volume, and e, the turbulence kinetic energy dissipation rate per unit volume, are solved by transport equations in order to determine the turbulence velocity and length scales respectively. The eddy viscosity thus calculated is used to derive the turbulent stresses in the Reynolds-averaged momentum transport equations and with the continuity equation thereby develop a closed system. The k and e transport equations are closed by using a number of semi-empirical procedures in which standard empirical constants are employed (Rodi 1984). By this means the high Reynolds number fully turbulent flow remote from the wall is solved, while the viscosity affected near-wall region is bridged by a function which for velocity is based on the law of the wall for a smooth surface. The wall function assumes local equilibrium for k and e.

 

The volume of fluid method (Hirt and Nichols 1981) was selected in order to predict the free surface. In this method the previously mentioned transport equations are solved as a single set of equations for the air and water phases, together with a transport equation for F, the volume fraction of the water phase. This is equal to 1 or 0 for water filled and air filled cells respectively, and for cells spanning the free surface it takes an intermediate value. The properties appearing in the transport equations, eg r and n, are the volume fraction averaged values.

 

The numerical scheme for solving the transport equations is based upon the finite volume method using a general curvilinear grid which is non-staggered. In the present work the default power-law interpolation scheme (Patankar 1980) was chosen to treat the convection-diffusion problem (this scheme is first or second order accurate depending upon the cell Peclet number). An exception was that at the interface between the two phases, the advection flux through a cell face was calculated by a special "donor-acceptor" scheme (Hirt and Nichols 1981). The equations were solved using the default SIMPLE pressure-correction algorithm (Patankar 1980).

 

With the foreknowledge of the experimental results, the solution domain depth of 250 mm was chosen to give an air filled domain approximately 50 mm deep above the free surface of the sub-critical flow upstream of the broad-crested weir. The full length (3340 mm) of the channel bed was modelled which, with the additional length of the water entry (380 mm) and exit (100 mm) boundaries, gave a solution domain 3820 mm long. The half channel cross-section, modelled with a plane of symmetry, was 52.5 mm wide. The upper edge of the air filled domain was modelled as a zero gauge pressure boundary condition as was the water filled channel exit boundary condition, consisting of a 100 mm extension in the plane of the channel floor. The entry boundary condition, a 380 mm upstream extension in the plane of the channel floor was a fixed velocity boundary condition, the latter chosen to give the flow rate of 4.684 l s^-1 in the physical model. Flow entry and exit through the channel floor were thus similar to the physical model, though somewhat idealised in that the curvature of the inlet passage and the expanded widths of the inlet and outlet were not replicated in the numerical model. A 384 ´ 52 ´ 9 (x ´ y ´ z) non-uniform rectangular grid was used with refinement in the near wall regions. In order to develop the free surface profile, the flow was treated as unsteady and time-stepped at 0.01 s intervals until a steady state was achieved after 30 s. The solution was actually run on for a further 20 s. The user supplied initial condition was, upstream of the broad-crested weir, water filled to weir level and air filled elsewhere.

 

RESULTS AND DISCUSSION

Although here we focus on the steady state solution, it is interesting to note that in the physical and numerical models air temporarily trapped beneath the nappe of the broad-crested weir was gradually entrained and removed.

 

Figure 2: Longitudinal free surface profile on centre-line of channel.

 

Figure 2 shows the longitudinal free surface profile in which the region around critical depth on the broad-crested weir and the supercritical flow profile downstream of it were reproduced quite well by the CFD. In the latter case a discernible standing wave-like profile in the physical model was reproduced by the CFD in the immediate vicinity of the weir though moving out of phase further downstream. Upstream of the weir, the gradually varied flow water level was underpredicted by about 10.5 mm, the CFD value approximating that given by assuming zero energy loss. The CFD had underestimated the energy loss in the separated flow region immediately upstream of the weir, a flow feature that is not modelled very well by the standard k-e turbulence model/wall function combination.

 

The transverse profiles in Figures 3-6 in average terms reflect the longitudinal profile in Figure 2. The discrepancy is greatest at the furthest upstream position (x = -300 mm), diminishing in the rapidly varying flow region upstream of the weir (x = -200 mm) and reaching negligible proportions at the centre of the weir and in the supercritical flow downstream of the weir.

 

Figure 3 Transverse free surface profile upstream of weir (x = -300 mm).

 

 

Figure 4 Transverse free surface profiles downstream of weir (x = -200 mm) and at weir centre (x = 0 mm).

 

 

Figure 5 Transverse free surface profiles downstream of weir (x = 200, 440 mm)

 

The detailed transverse distributions were near invariant with z except in the supercritical flow region. At x = 440 mm, the depth of flow was nearly constant over the middle 60 mm width of channel (30 mm each side of the centreline) but increased quite sharply with proximity to the side wall. This trend observed experimentally was reproduced quite well in the CFD. At x = 800 mm, further downstream, the trend was less clear though a slight waviness could be discerned in the experimental results, which were fitted approximately by the numerical solution.

 

Figure 6 Transverse free surface profile downstream of weir (x = 800 mm)

 

SUMMARY AND CONCLUSIONS

Measurements of free surface profiles have been carried out in an open channel of rectangular cross-section housing a rectangular broad-crested weir, and the experimental arrangement has been numerically modelled using commercial software. The numerical predictions reproduced the physical details very well at the centre of the weir near the critical point and in the supercritical flow conditions downstream of it. However, water depth in the upstream subcritical flow was underpredicted probably reflecting the shortcomings of the standard k-e turbulence model/wall function combination when modelling flow separation and the associated energy losses.

 

The failure to accurately predict the upstream water depth is a significant and disappointing outcome of the present work, in that the calculation of the stage-discharge relationship is a fundamental requirement in the engineering of hydraulic structures. It seems likely that improvements in this area will be achieved by employing a turbulence model that more faithfully reproduces the region of separated flow upstream of the weir, and that work is currently in progress.

 

ACKNOWLEDGEMENTS

The work reported here was supported by a Cranfield University internally funded research studentship. The first author is on study leave from the Bangladesh Institute of Technology (BIT), Chittagong where he is a lecturer in civil engineering.

 

REFERENCES

Hirt, C.W. and Nichols, B.D. (1981). Volume of fluid (VOF) method for the dynamics of free boundaries. J. Computational Physics, 39, 201-225.

 

Patankar, S.V. (1980). Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, New York.

 

Rodi, W. (1984). Turbulence Models and their Applications in Hydraulics-A State of the Art Review. 2nd Edn., IAHR, The Netherlands.