Free surface profiles of near-critical instabilities in open channel flows: undular hydraulic jumps

Abstract

When the Froude number F of a free-surface flow ranges between 0.3 and 3, the flow is unstable and frequently characterised by free surface undulations, with the undular hydraulic jump being a seminal flow in hydro-environmental mechanics. The presence of the free surface undulations significantly affects the flow field, with major velocity and pressure field redistributions between successive crests and troughs. All the current theoretical models to simulate undular hydraulic jumps are limited to two-dimensional flow conditions, ignoring all the relevant three-dimensional flow effects, namely shock-wave drag, turbulent breaking and turbulent stresses. These aspects are critically accounted for in this review article, where a depth-averaged Boussinesq model which approximately accounted for 3D flow effects was presented, constituting the first attempt in this line. The model predictions were compared with experimental results from different sources for F₁ < 1.5, with F₁ the inflow Froude unnumber, resulting a reasonable agreement with observations. The curvature distribution parameter was found to controlling the wave length, and an approximate value was obtained based on ideal fluid flow computations. The new depth-averaged model did not include the effect of flow concentration in the centerline, observed physically, but this 3D feature is analysed at the first wave crest based on an improved treatment of flow curvature, highlighting the impact of the ratio q_CL/q on the velocity profile features with q_CL the centerline discharge. The main limitation of the new model, presented in this critical review, originated from approximating a complex 3D flow by a depth-averaged model. However, the predictions with the new approximate treatment of 3D effects produced better results than those previously reported.

Appendix: Development of 2D Boussinesq equations with boundary shear

The viscous velocity distribution for two-dimensional wavy free surface flow is [30]

$$ V = V_{s} \mu ^{N} {\text{exp}}\left( { - \varepsilon _{0} \frac{{1 - \mu ^{{K + 1}} }}{{K + 1}}} \right), $$

(37)

where V_s is the velocity at the free surface, μ = z/h, z the elevation, h the flow depth, K the curvature parameter, N the power-law exponent, h_x = dh/dx and h_xx = d²h/dx² and ε₀ = hh_xx/(1 + h_x²). For small arguments of the exponential function, Eq. (37) is approximated by

$$ V = V_{s} \mu ^{N} {\text{exp}}\left( { - \varepsilon _{0} \frac{{1 - \mu ^{{K + 1}} }}{{K + 1}}} \right) \approx V_{s} \mu ^{N} \left( {1 - \varepsilon _{0} \frac{{1 - \mu ^{{K + 1}} }}{{K + 1}}} \right). $$

(38)

Projection of this result into the (x, z) directions yields the velocity components (u, v) [30]

$$ u \approx \left( {1 + N} \right)\frac{q}{h}\mu ^{N} \left[ {1 - \frac{{\varepsilon _{0} }}{{K + 1}}\left( {\frac{{1 + N}}{{K + 2 + N}} - \mu ^{{K + 1}} } \right) - \frac{{\varepsilon _{1} }}{2}\left( {\frac{{N + 1}}{{N + 3}} + \mu ^{2} } \right)} \right], $$

(39)

$$ v \approx \left( {1 + N} \right)\frac{q}{h}\mu ^{{N + 1}} \varepsilon _{1}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \left[ {1 - \frac{{\varepsilon _{0} }}{{K + 1}}\left( {\frac{{1 + N}}{{K + 2 + N}} - \mu ^{{K + 1}} } \right) - \frac{{\varepsilon _{1} }}{2}\frac{{N + 1}}{{N + 3}}} \right], $$

(40)

and the pressure distribution

$$ \frac{p}{{\rho gh}} = 1 - \mu + \frac{{\varepsilon _{0} \left( {1 + N} \right)^{2} }}{{K + 2N + 1}}\frac{{q^{2} }}{{gh^{3} }}\left( {1 - \mu ^{{1 + 2N + K}} } \right), $$

(41)

where ε₁ = h_x²/(1 + h_x²). Now, the specific momentum S is given by

$$ S = \int\limits_{0}^{h} {\left( {\frac{{V^{2} }}{g} + \frac{p}{{\rho g}}} \right){\text{d}}z} . $$

(42)

From Eq. (38)

$$ V^{2} \approx \left( {1 + N} \right)^{2} \frac{{q^{2} }}{{h^{2} }}\mu ^{{2N}} \left[ {1 - \frac{{2\varepsilon _{0} }}{{K + 1}}\left( {\frac{{1 + N}}{{K + 2 + N}} - \mu ^{{K + 1}} } \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right]. $$

(43)

Integration of Eq. (43) yields

$$ \int\limits_{0}^{h} {\frac{{V^{2} }}{g}{\text{d}}z} = \left( {1 + N} \right)^{2} \frac{{q^{2} }}{{gh}}\left( {\frac{1}{{2N + 1}} - \frac{{2\varepsilon _{0} }}{{K + 1}}\left( {\frac{{1 + N}}{{K + 2 + N}}\frac{1}{{2N + 1}} - \frac{1}{{K + 2N + 2}}} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}\frac{1}{{2N + 1}}} \right) $$

(44)

After manipulation, Eq. (44) is rewritten as

$$ \int\limits_{0}^{h} {\frac{{V^{2} }}{g}{\text{d}}z} = \beta \frac{{q^{2} }}{{gh}}\left( {1 - \frac{{2\varepsilon _{0} }}{{K + 1}}\left( {\frac{{1 + N}}{{K + 2 + N}} - \frac{{2N + 1}}{{K + 2N + 2}}} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right), $$

(45)

where β is the Boussinesq momentum velocity correction coefficient for the power-law velocity profile,

$$ \beta = \frac{{(1 + N)^{2} }}{{2N + 1}}. $$

(46)

Using Eq. (41) one obtains

$$ \int\limits_{0}^{h} {\frac{p}{{\rho g}}{\text{d}}z} = \frac{{h^{2} }}{2} + \frac{{\varepsilon _{0} \left( {1 + N} \right)^{2} }}{{K + 2N + 1}}\frac{{q^{2} }}{{gh}}\left( {1 - \frac{1}{{2 + 2N + K}}} \right) = \frac{{h^{2} }}{2} + \beta \varepsilon _{0} \frac{{2N + 1}}{{K + 2N + 2}}\frac{{q^{2} }}{{gh}}. $$

(47)

Inserting Eqs. (45) and (47) in Eq. (42) yields for the specific momentum S

$$ \begin{aligned} S & = \beta \frac{{q^{2} }}{{gh}}\left( {1 - \frac{{2\varepsilon _{0} }}{{K + 1}}\left( {\frac{{1 + N}}{{K + 2 + N}} - \frac{{2N + 1}}{{K + 2 + 2N}}} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right) + \frac{{h^{2} }}{2} + \beta \varepsilon _{0} \frac{{2N + 1}}{{K + 2N + 2}}\frac{{q^{2} }}{{gh}} \\ & = \frac{{h^{2} }}{2} + \beta \frac{{q^{2} }}{{gh}}\left( {1 + \varepsilon _{0} \left( {\frac{{2N + 1}}{{K + 2 + 2N}} - \frac{{2\left( {N + 1} \right)}}{{\left( {K + 1} \right)\left( {K + 2 + N} \right)}} + \frac{{2\left( {1 + 2N} \right)}}{{\left( {K + 1} \right)\left( {K + 2 + 2N} \right)}}} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right) \\ & = \frac{{h^{2} }}{2} + \beta \frac{{q^{2} }}{{gh}}\left( {1 + \varepsilon _{0} \left( {\frac{2}{{K + 2 + N}} + \frac{{2NK^{2} - K^{2} + 2N^{2} K + 5NK - 3K + 2N^{2} + 3N - 2}}{{\left( {K + 1} \right)\left( {K + N + 2} \right)\left( {K + 2 + 2N} \right)}}} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right) \\ & = \frac{{h^{2} }}{2} + \beta \frac{{q^{2} }}{{gh}}\left( {1 + \varepsilon _{0} \left( {\frac{2}{{K + 2 + N}} - \frac{{1 - 2N}}{{K + 2 + 2N}}} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right). \\ \end{aligned} $$

(48)

The depth-averaged specific energy head E is

$$ E = \frac{1}{h}\int\limits_{0}^{h} {\left( {\frac{{V^{2} }}{{2g}} + \frac{p}{{\rho g}} + z} \right){\text{d}}z} . $$

(49)

Using Eqs. (45) and (47), Eq. (49) yields

$$ \begin{aligned} E & = h + \beta \frac{{q^{2} }}{{2gh^{2} }}\left( {1 - \frac{{2\varepsilon _{0} }}{{K + 1}}\left( {\frac{{1 + N}}{{K + 2 + N}} - \frac{{2N + 1}}{{K + 2N + 2}}} \right) + 2\varepsilon _{0} \frac{{2N + 1}}{{K + 2N + 2}} - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right) \\ & = h + \beta \frac{{q^{2} }}{{2gh^{2} }}\left( {1 + \frac{{2\varepsilon _{0} }}{{K + 2N + 2}}\left( {\frac{{2N + 1}}{{K + 1}} - \frac{{\left( {N + 1} \right)\left( {K + 2N + 2} \right)}}{{\left( {K + N + 2} \right)\left( {K + 1} \right)}} + 2N + 1} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right) \\ & = h + \beta \frac{{q^{2} }}{{2gh^{2} }}\left( {1 + \frac{{2\varepsilon _{0} }}{{K + 2N + 2}}\left( {\frac{N}{{K + N + 2}} + 2N + 1} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right) \\ & = h + \beta \frac{{q^{2} }}{{2gh^{2} }}\left( {1 + \frac{{2\varepsilon _{0} }}{{K + 2N + 2}}\left( {1 + N\left( {2 + \frac{1}{{K + N + 2}}} \right)} \right) - \varepsilon _{1} \frac{{N + 1}}{{N + 3}}} \right). \\ \end{aligned} $$

(50)

Equations (48) and (50) are the extended momentum and energy equations for viscous wavy free surface flow. Some typos in the original paper by Montes and Chanson [30] makes the detailed derivation presented here useful.