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# Model application to waves on sheared currents

One of the advantages of the Lagrangian formulation is that it offers a simple treatment of flows with vorticity and is, therefore, suitable for modelling waves on sheared currents. A sheared current can be defined by specifying the vorticity that depends solely on the vertical Lagrangian coordinate c. For our choice of the Lagrangian labels the parallel current can be specified as x = a + V(c)t; z = c, where E(c) = 'F(-ao) is the current profile. Substitution to (3) gives

Therefore, the waves on a sheared current with an undisturbed profile V(20) are described by Equations (1,3) with the free surface boundary condition (4), the bottom condition (5) and the vorticity distribution given by (16). As before, the numerical implementation is based on a form of the governing equations given by (7) and the free surface boundary condition with the dispersive correction (14) with additional terms on the right-hand side. For wave-current calculations we use the following additional terms

The first term of (17) is the modified dissipation term and the second term is the time varying surface pressure gradient that is used for wave generation. The breaking model is not implemented in the wave-current version of the solver.

The numerical wave-current flume is created by specifying inlet and outlet boundary conditions, distribution of the surface dissipation к (a) and the surface pressure gradient Px(x,t). The NWT design should provide free current inflow and outflow to and from the computational domain, wave generation on the current, and absorption of waves incident to the domain boundaries to eliminate reflections. The dissipation coefficient is set to zero in the working section of the flume. It gradually increases to a large value near the input and output boundaries to ensure a stable horizontal free surface that remains at the initial position 2 = 0. This provides parallel input and output flows and serves a double purpose. Firstly, the reflections from the boundaries are significantly reduced. Secondly, the inlet and outlet boundary conditions can be specified as the undisturbed velocity profile at the inlet and as a parallel flow at the outlet

The waves are generated by creating an area in front of one of the wave absorption zones where the pressure distribution of a prescribed shape is defined. The time varying amplitude of the pressure disturbance is used as the control input for this pneumatic wave generator. It. should be noted that the generated waves propagate in both directions, but the waves propagating backwards are damped by the first absorption zone. Figure 11 illustrates the setup of the Lagrangian numerical wave-current flume.

An additional difficulty with the numerical realisation of the Lagrangian formulation on sheared currents is the continuous deformation of the original physical domain. The accuracy of the calculations for highly deformed meshes decreases considerably. If the deformation is too strong, this can lead to calculations breakdown. To avoid these difficulties, we carry out shear deformation of the Lagrangian domain to compensate for the deformation of the physical domain. The deformation takes place after several time steps and brings the boundaries of the physical domain back to the vertical lines. After that, we re-label the fluid particles with new values of Lagrangian coordinates in order to preserve the rectangular shape of the Lagrangian computational domain with the vertical and horizontal lines of the computational grid. The procedure is illustrated in Figure 12.

We use the numerical wave-current flume to reproduce the results of an experimental study of focused wave groups over sheared currents. The experimental flume is 1.2 m wide and the distance between two piston wavemakers is about 16 m. The depth for all tests is h = 0.5 m. A recirculation system with three parallel pumps and vertical inlets 13 m apart is

Figure 11: Schematic representation of the Lagrangian numerical wave-current flume. The upper graph demonstrates the shapes of distributions for the surface pressure P and the dissipation coefficient k from Equation (17). Wave and current directions and wave generation and absorption zones are indicated.

Figure 12: Diagram of the procedure of deformation and re-labelling of the Lagrangian mesh.

used to create a current. A paddle on the right end of the flume is used as a wave generator and the opposite paddle as an absorber. Trapezoidal wire mesh blocks are installed above the inlet and outlet to condition the flow and create a desired current profile. The surface elevation at selected points along the flume is measured by resistance wave probes and a PIV system is used to measure flow kinematics. An iterative procedure (Buldakov et al., 2017) is used to focus the wave group at a prescribed time and place. We use the same coordinate system as previously with the origin on the water surface at the centre of the flume, the ж-axes directed towards the wave generator and the 2-axis directed upwards. The wave probe at position x = 4.7 m is used to match the linearised amplitude spectrum with the target spectrum, and the wave probe at x = 0 for focussing the phase of the generated wave group. A broadband Gaussian spectrum with peak frequency fp = 0.6 Hz is used as the target spectrum. Wave groups having different linearised focus amplitudes A on opposing and following sheared currents with different surface velocities Vo are generated in the experimental study. We use a moderately steep wave with A = 7 cm propagating on currents with Vo ~ 0.2 m/s as a test case for comparison with numerical results. More details of the experimental setup and methodology can be found in Stagonas et al. (2018a).

Since the experimental and numerical wave flumes have different wave generators and the flow conditioner can not be modelled adequately, direct replication of the experiment in the numerical flume is not possible. We, therefore, apply in the numerical flume the same iterative wave generation procedure as in the experimental flume using the linearised experimental spectrum as a target. This makes it possible to generate the wave which reproduces the linearised experimental wave with the accuracy of the iterative procedure. This also ensures that the higher order bound wave components are also modelled with the corresponding accuracy. However, the higher order spurious components generated by the experimental and numerical wavemakers are different. This is one of the main sources of difference between experimental and numerical results. The current profiles applied in the Lagrangian model are obtained from PIV measurements of the current velocity, as shown in Figure 13. PIV data only cover the upper part of the water depth (z > —0.3 m). The shape of the lower part of the profiles is reconstructed from the ADV measurements available for currents with a slightly higher discharge.

The comparison of the numerical results with the experiment is shown in Figures 14 and 15. Both the surface elevation (Figure 14) and the combined wave and current velocity profiles (Figure 15) demonstrate good agreement. The contribution of spurious free components to the difference between the measured and calculated surface elevation is clearly visible in Figure 14. For the opposing current, one can also observe the effect of the dispersive error on the phase difference of the results at x = 4.7 m. Because the wave is focused at the centre of the flume both in the experiment and in the calculations, the dispersive error increases with the increasing distance from the focus position x = 0. This error is higher for the opposing current due to the longer effective path of the wave travelling against the current.

Figure 13: Horizontal velocity profiles for following (left) and opposing (right) currents. PIV measurements (grey dots) and profiles used as the input to the Lagrangian model (solid).

Figure 14: Time history of surface elevation for a wave group at x = 4.7 m (left) and x = 0 (right) over a following (top) and opposing (bottom) currents. Experiment (dashed) and Lagrangian solver (solid).

Figure 15: Horizontal velocity profiles under the crest of a focussed wave group over a sheared current. PIV measurements (grey dots) and Lagrangian calculations (solid), (a) - following current; (b) - no current; (c) - opposing current.

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