Application of high-level Green–Naghdi theory to sill-controlled flows

Abstract

Sill-controlled flows involving the acceleration across a crest or control section with significant flow curvature are characterized by the Euler equations of an inviscid fluid. The problem is of both theoretical and practical interest in fluid mechanics research, given the role of such flows in water discharge measurements structures, underwater oceanic currents, river flows and shallow bar-built estuary inlets, among others. While three-dimensional computations are feasible, vertically averaged solutions are sought to gain efficiency. The vertically averaged representation of sill overflows based on the Serre–Green–Naghdi (SGN) theory is limited to very shallow flows, and a good averaged approach for large overflows is currently not available. High-level Green–Naghdi (GN) theory forms a hierarchy of theories of increasing accuracy based on expanding the kinematic field and vertically-averaging in a weighted-residual sense. These theories have been successfully applied to ocean research but so far, they have not been applied to flow in open channels and structures. In this work, the high-level Green–Naghdi theories, of which SGN equations are the lowest level possible (GN level I theory), are formulated and newly applied to the sill-controlled flow problem. High-level GN theory is compared with detailed experiments from the literature and new ones conducted, and with a new fully non-linear vertically resolved potential flow solver which uses a x–ψ mapping. It was found that the GN expansions are convergent in the sill problem, in contrast to former perturbation solutions, which are asymptotic. The GN level V theory was found to be in good agreement with experiments for the sill-controlled flow problem, which excellently reproduced the free surface, bottom pressure, and vertical distributions of velocity and pressure. The GN level V theory was found to be applicable for relatively wide range of overflows up to E/R = 3, where E is the minimum specific energy and R the sill crest radius of curvature. From a practical viewpoint, large improvements were observed when using GN level II theory, instead of GN level I (e.g. the SGN theory), producing good results up to E/R = 1.75, which is the minimum level thus recommended in practice.

Appendices

Appendix A: Matrix formulation for GN II equations

In this section we provide the symbolic output for the modeling approach GN II. Green–Naghdi theory of level II starts assuming that u = u₀ + u₁ϕ₁. The expansions for (u, w, p) are as follows:

Note u is a polynomial of degree 1 in σ, w is a polynomial of degree 2 in σ and p is a polynomial of degree 4 in σ. Using these expansions for (u, w, p), the Green–Naghdi evolution equations found are:

$$\left( {\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\ {A_{21} } & {A_{22} } \\ \end{array} } \right){\mathbf{f}} + \left( {\begin{array}{*{20}c} {B_{11} } & {B_{12} } \\ {B_{21} } & {B_{22} } \\ \end{array} } \right)\frac{{\partial {\mathbf{f}}}}{\partial x} + \left( {\begin{array}{*{20}c} {C_{11} } & {C_{12} } \\ {C_{21} } & {C_{22} } \\ \end{array} } \right)\frac{{\partial^{2} {\mathbf{f}}}}{{\partial x^{2} }} = \left( \begin{gathered} g_{1} \hfill \\ g_{2} \hfill \\ \end{gathered} \right)$$

$${\mathbf{f}} = \frac{{\partial {\mathbf{u}}}}{\partial t},\quad {\mathbf{u}} = \left( \begin{gathered} u_{0} \hfill \\ u_{1} \hfill \\ \end{gathered} \right),$$

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where the coefficients are stated below:

Equation 1 (j = 1)

Equation 2 (j = 2)

Equation 1 (j = 1)

Equation 2 (j = 2)

Appendix B: Discrete free-surface Bernoulli equation

The determination of the free surface position with the discrete Bernoulli equation is a complex non-linear problem. To our knowledge, the difficulties are commented in some works [22, 56, 79], but these are nor explored neither explained. Here, we provide an outlook to the problem and some practical advice.

The discrete free surface Bernoulli equation at a position i of the mesh is using the x-ψ mapping

$$H\left( i \right) = z\left( {i,M} \right) + \frac{1}{2g}\left[ {\frac{2\Delta \psi }{{3z\left( {i,M} \right) - 4z\left( {i,M - 1} \right) + z\left( {i,M - 2} \right)}}} \right]^{2} \left[ {1 + \left( {\frac{{z\left( {i + 1,M} \right) - z\left( {i - 1,M} \right)}}{2\Delta x}} \right)^{2} } \right]$$

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For a given value of H(i), this is a cubic equation with three possible roots for z(i, M), the coordinate of the free surface at that node. It was found that the three possible roots z(i, M) for a given H(i) value are real, and of a magnitude very similar. This makes exceedingly difficult to determine with a root finding algorithm like the Newton–Raphson method which one is the physically correct root. Further analysis was undertaken here to get a better understanding.

We have considered the flow over the Gaussian hump test in Fig. 7. Using the final converged two-dimensional solution, the function H(i) is plotted versus z(i, M) in Fig. 

Fig. 15

Discrete free surface Bernoulli equation diagram at two sections for the gaussian flow test of Fig. 7 indicating the actual 2D flow solution and the three possible roots

15 at the crest section (x = 0 m) and upstream of the sill crest (x =  − 0.5 m), in the zone of uncertainty according to Cheng et al. [22]. At both sections, the actual flow solution is plotted. We have also determined the three roots for the target energy head, which are for x = 0 m:

(S1, S2, S3) = (0.349512991389396, 0.302113856953000, 0.294138840891281).

For the section at x =  − 0.5 m the three roots are:

(S1, S2, S3) = (0.344987598039410, 0.343003371274220, 0.326689053566463).

The roots are also marked in Fig. 15.

At the sill crest two of the roots are found to be remarkably close to each other (S2 and S3), one of them being unphysical and corresponding to u_s < 0 (solution S3). The condition of zero surface velocity (u_s = 0) is given analytically by z(i, M) =  − [− 4z(i, M − 1) + z(i, M − 2)]/3, which is plotted as a dotted line in Fig. 15. It is in general not possible to decide with the Newton–Raphson iteration scheme which of the roots is the physically correct one, given that all are real and very close to each other at the weir crest. Upstream of the crest the situation is even more complex, given that the two roots with u_s > 0 (S1 and S2) are almost identical from a numerical standpoint and close to the point of minimum H(i). Therefore, a small variation in H(i) during the iteration of the surface node there will produce a very large variation in z(i, M) given that the changes dz(i, M)/dH(i) are maximal in this zone of the curve. Note that the curves presented in Fig. 15 are a representation of the discrete free surface Bernoulli equation and are unrelated to the minimum specific diagram of hydrostatic flows. A minimum in H(i) is unrelated to critical flow.

The above considerations suggest the necessity of using a good initial solution for all the free surface nodes in the 2D model, e.g., that given by GN V, and during the iterations, a damping factor of the order of 0.1 is to be applied to the correction vector ∆z_s, ensuring that the physically correct root is approached in the zone of uncertainty upstream of the sill crest. The corrections are not allowed to drop below z(i, M) =  − [− 4z(i, M − 1) + z(i, M − 2)]/3, which is therefore a lower bound.