Steady Flow Loading on a Solid Obstacle in a Laboratory Flume
2.2 One-dimensional model
In this section it is argued that the force on a square cylinder can be adequately modelled through a distributed drag force and a blockage ratio. As demonstrated in Section 1.2, there are a number of ways that the equations of motion can be simplified through various approximations and assumptions. A starting point is chosen as the 2D non-linear shallow water equations described in Section 1.2.2 (1.10 - 1.12), which are appropriate for modelling long-period waves in shallow water where pressure distributions are close to hydrostatic. We can re-write (1.10 - 1.12) in the following form, ignoring advection, substituting η = h + zb and including a
distributed drag term; where the system is expressed in a flux conserving form. On the right hand side, the contribution due to bottom shear stresses τb = (τbx, τby) and a depth-averaged resistive force per unit mass, fD = (fDx, fDy) are included. A roughness model is applicable when the water depth is an order of magnitude larger than the height of the roughness elements, which in turn may also require the introduction of a roughness height of the flow above. These roughness models are only applicable when the roughness changes smoothly and so rapid changes due to localised groups of buildings may require additional closures.
In order to test rapidly the importance of various assumptions, it is helpful to sim-plify the problem further. Where blockage ratios are small, the transverse velocities are many times smaller than the inline velocities, so it makes sense to focus on the important behaviour. Hence, the general model can be described in one-dimension by averaging the flow in the y- direction. Across the flume a local blocking frac-tion is defined to be φ = Rw/2
−w/2Sbdy. When φ = 0, there is no blocking; usually φ = b/w, meaning the void fraction available for flow and mass to occupy is 1− φ.
The flow model is modified by multiplying all terms by φ to provide a reduction to the available conveyance in the region of the block in the following way;
∂
Bed shear stresses are modelled through a Manning roughness parameterisation as (1.13). The flow equations (2.7) and (2.8) can be thought of as average quantities
Front face
Figure 2.2: Average flow conditions and reduction of cross-sectional area due to blocking.
(a) View of channel cross-section at the upstream face of the block from upstream to downstream (b) View of channel cross-section at the downstream face of the block from downstream to upstream.
per unit width, so φ can be applied to locally vary the average channel conveyance by multiplying all the terms by φ as shown in Figure 2.2. By the product rule,
∂(1−φ)h
∂t = (1 − φ)∂h∂t + h∂(1−φ)∂t . The (1 − φ) term here is invariant in time, so
∂(1−φ)
∂t = 0. Similarly as we are discussing a block of constant b with its front-face perpendicular to the flow, across the domain φ is zero in the regions without a block, and b/w in the regions with a block, so in these regions ∂(1−φ)∂x = 0. It is therefore possible to simplify (2.7) and (2.8) by moving the (1− φ) terms outside the partial differentials. This gives a 1D flow model with blocking and drag as;
(1− φ)∂h
The form of these equations are very similar to those used to described bubbly two-phase flow, using the so-called two fluid model (Drew and Passman, 2006). Note that here φ is defined to be constant and uniform with its gradients zero. It is possible to explore the link between change in momentum flux and drag force for steady flows by integrating (2.10) over the whole domain to give
For steady flows, a drag coefficient can be identified:
CD = FD
(1/2)ρbh1u21. (2.12)
The drag is related to the change in the momentum flux and hydrostatic force between the upstream and downstream:
The closure used by Qi et al. (2014), was formed from a combination of form drag and hydrostatic force, (1.28). It was found that even for choked flows where large differences in water depth exist between the front and rear of a block, the dominant
contribution comes from the hydrodynamic drag component for lower values of φ.
As such the hydrostatic component is firstly ignored. For low F r it is expected that;
ρ(1− φ)wfDh1L = 1
2ρCDbh1|u1|u1 (2.17) so that
fD = CD 2L
φ
(1− φ)|u1|u1. (2.18)
The form of this closure bears a similarity with (1.17). It is expected that it should provide reasonable results for subcritical flows, and because for low φ the hydrostatic component is small it is likely to be sufficient for choked flow also. The next section describes a set of experiments carried out to verify this behaviour and parameterise it for a different φ = 0.25. The tests of Qi et al. (2014) were conducted on a tall cylinder where the building height, Hb > h at all times. The experiments described in this these allow for flow conditions to occur where Hb < h, so that it is possible to observe over-topping.
For higher blockage ratios (b/w & 0.4), transverse velocities are more significant and a two- or three-dimensional approach would therefore be more appropriate. A-two dimensional approach forms part of future work and is outside the scope of this thesis. Additional current work within the group at UCL is also examining higher blockage ratios while looking at three-dimensional effects and over-topping (Bahmanpour et al., 2015). As such higher blockage ratios are outside the scope of this thesis and do not feature in any of the experiments conducted and described as part of this work.