2. A brief history of models of two-layer hydraulic jumps

The principal objective is to obtain an expression for the speed of a hydraulic jump between two two-layer flows in terms of its amplitude. Certain conditions are assumed, namely (1) the jump is steady in a reference frame moving with it, (2) the top and bottom surfaces are horizontal through the jump, with no surface stress, and (3) the layers maintain their identity through the jump, with negligible exchange of fluid between them. The conventional picture of a two-layer hydraulic jump is given in figure 1, with two flowing layers with densities ${\it\rho}_{1}$ and ${\it\rho}_{2}$ . These densities may have any value, the only restriction being that ${\it\rho}_{2}<{\it\rho}_{1}$ . The flow has a rigid upper boundary with uniform total depth $D$ , and the jump may be regarded as an essentially turbulent structure (which dissipates energy) between a supercritical upstream flow state (in which waves cannot propagate upstream, away from the bore) with layer velocities $u_{1u},u_{2u}$ and thicknesses $d_{1u}$ , $d_{2u}$ , and a downstream state with velocities $u_{1d},u_{2d}$ and thicknesses $d_{1d},d_{2d}$ (which may be sub- or supercritical, as discussed in § 4). In principle the upstream flow may have any profile, but the upstream layer velocities are assumed to be the same in all cases considered here. Assuming that these flows are steady in the frame of the jump, we have

(2.1a,b ) $$\begin{eqnarray}d_{1d}+d_{2d}=D=d_{1u}+d_{2u},\quad d_{id}u_{id}=q_{i}=d_{iu}u_{iu},\quad i=1,2,\end{eqnarray}$$

where $q_{i}$ denotes the volume flux in the $i$ th layer. From assumption (2) we also have that the momentum flux $S$ must be uniform, so that

(2.2) $$\begin{eqnarray}S=\int _{0}^{D}p+{\it\rho}u^{2}\,\text{d}z=\text{constant}\end{eqnarray}$$

(where $p$ denotes pressure), which implies

(2.3) $$\begin{eqnarray}\displaystyle S & = & \displaystyle p_{su}D+{\textstyle \frac{1}{2}}{\rm\Delta}{\it\rho}gd_{1u}^{2}+{\it\rho}_{1}u_{1u}^{2}d_{1u}+{\it\rho}_{2}u_{2u}^{2}d_{2u}+{\textstyle \frac{1}{2}}{\it\rho}_{2}gD^{2}\nonumber\\ \displaystyle & = & \displaystyle p_{sd}D+{\textstyle \frac{1}{2}}{\rm\Delta}{\it\rho}gd_{1d}^{2}+{\it\rho}_{1}u_{1d}^{2}d_{1d}+{\it\rho}_{2}u_{2d}^{2}d_{2d}+{\textstyle \frac{1}{2}}{\it\rho}_{2}gD^{2}.\end{eqnarray}$$

This equation introduces an additional variable, $p_{su}-p_{sd}$ , which needs to be determined by other considerations. The first attempt at this was by Yih & Guha (Reference Yih and Guha1955), who made the following three additional assumptions about the dynamics within the jump: (i) interfacial stresses are negligible, (ii) the flow is effectively hydrostatic, and (iii) the surface pressure $p_{s}$ varies linearly with upper layer thickness $d_{2}$ . These assumptions lead directly to the relation (Yih & Guha Reference Yih and Guha1955; Baines Reference Baines1995)

(2.4) $$\begin{eqnarray}p_{su}-p_{sd}=2{\it\rho}_{2}(d_{2d}u_{2d}^{2}-d_{2u}u_{2u}^{2})/(d_{2d}+d_{2u}).\end{eqnarray}$$

Writing

(2.5a,b ) $$\begin{eqnarray}r_{u}=d_{1u}/D,\quad r_{d}=d_{1d}/D,\end{eqnarray}$$

and $g^{\prime }={\rm\Delta}{\it\rho}/{\it\rho}_{1}g=(({\it\rho}_{1}-{\it\rho}_{2})/{\it\rho}_{1})g$ , for the jump speed $c_{J}$ equations (2.1)–(2.5) give

(2.6) $$\begin{eqnarray}\frac{c_{J}^{2}}{g^{\prime }D}=\frac{(r_{u}+r_{d})/2}{\left[\displaystyle \frac{r_{u}}{r_{d}}+\displaystyle \frac{{\it\rho}_{2}}{{\it\rho}_{1}}\displaystyle \frac{(1-r_{u})(r_{u}+r_{d})}{(1-r_{d})(2-r_{u}-r_{d})}\right]}.\quad (\text{YG})\end{eqnarray}$$

Subsequently, instead of assumptions (ii) and (iii) above, Chu & Baddour (Reference Chu and Baddour1977) and Wood & Simpson (Reference Wood and Simpson1984) made the assumption that all of the energy dissipation within the jump occurred in the expanding layer (the lower layer in figure 1). For Wood and Simpson this was suggested (to them) by their experiments, and it implies that the Bernoulli equation may be applied to the contracting layer. If this is the upper layer, it implies

(2.7) $$\begin{eqnarray}p_{su}-p_{sd}={\textstyle \frac{1}{2}}{\it\rho}_{2}(u_{2d}^{2}-u_{2u}^{2}),\end{eqnarray}$$

and if it is the lower layer, it is

(2.8) $$\begin{eqnarray}p_{su}-p_{sd}={\textstyle \frac{1}{2}}{\it\rho}_{1}(u_{1d}^{2}-u_{1u}^{2})+{\rm\Delta}{\it\rho}g(d_{1d}-d_{1u}).\end{eqnarray}$$

Using (2.7), with the configuration of figure 1 this gives

(2.9) $$\begin{eqnarray}\frac{c_{J}^{2}}{g^{\prime }D}=\frac{(r_{u}+r_{d})/2}{\left[\displaystyle \frac{r_{u}}{r_{d}}+\displaystyle \frac{{\it\rho}_{2}}{{\it\rho}_{1}}\displaystyle \frac{r_{u}+r_{d}-2r_{u}r_{d}}{2(1-r_{d})^{2}}\right]}.\quad (\text{WS})\end{eqnarray}$$

The resulting bore speeds between these two expressions are similar unless the amplitudes are close to the maximum possible.

More recently, Klemp, Rotunno & Skamarock (Reference Klemp, Rotunno and Skamarock1997) have made the opposite assumption to that of Wood and Simpson, namely that all of the dissipation occurs in the contracting layer, at least in the case of a Boussinesq fluid (where the densities are effectively equal except when multiplied by  $g$ ). This is approximately consistent with their own numerical simulations, and with the assumptions of Benjamin (Reference Benjamin1968) regarding gravity currents. In the context of figure 1, this implies using (2.8) instead of (2.6) or (2.7), giving, for the full density range,

(2.10) $$\begin{eqnarray}\frac{c_{J}^{2}}{g^{\prime }D}=\frac{[1-(r_{u}+r_{d})/2](1-r_{d})r_{d}^{2}}{(1-r_{d})[(r_{d}+r_{u})/2-r_{u}r_{d}]+\displaystyle \frac{{\it\rho}_{2}}{{\it\rho}_{1}}r_{d}^{2}(1-r_{u})}.\quad (\text{KRS})\end{eqnarray}$$

The models are only applicable in situations where hydraulic jumps exist: generally for $r_{u}<0.5$ and a limited range of $r_{d}>r_{u}$ . These three models give different answers, particularly for small $r_{u}$ . The YG and WS models give similar results but diverge at large amplitude, with YG the larger. The KRS model gives significantly lower jump speeds than both, but is believed to be more realistic, at least near the Boussinesq limit.

From the viewpoint of energy dissipation, there is something wrong with each of these models. An examination of the YG solution (2.6) shows that there is a net gain in energy flux in the upper (contracting) layer across the jump (Li & Cummins Reference Li and Cummins1998), which would seem unphysical. For the other two models, an absence of energy loss in either layer also cannot be correct in general terms. A numerical study by Borden, Meiburg & Constantinescu (Reference Borden, Meiburg and Constantinescu2012) of conditions in the Boussinesq limit showed that mixing at the interface could cause a gain of potential energy in the expanding (lower) layer, and they proposed a new model in this limit that involved the Reynolds and Schmidt numbers, which they denote as the BMC model.

3. Models based on vorticity generation

Subsequent to all of the above, Borden & Meiburg (Reference Borden and Meiburg2013) proposed a new method of solving the problem of the unknown surface pressure change across the jump. By making the Boussinesq approximation, they could write the two-dimensional vorticity equation for steady flow of incompressible fluids as

(3.1) $$\begin{eqnarray}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\omega}=-g^{\prime }\frac{\partial {\it\rho}^{\ast }}{\partial x}+{\it\nu}{\rm\nabla}^{2}{\it\omega},\end{eqnarray}$$

where ${\it\omega}$ is the vorticity in the $y$ -direction, ${\it\rho}^{\ast }=({\it\rho}(z)-{\it\rho}_{2})/({\it\rho}_{1}-{\it\rho}_{2})$ and the pressure term does not appear. By integrating this vorticity equation in a control volume of fluid that includes the jump, and then employing the Gauss divergence theorem, they were able to obtain an equation for the vorticity generation within the jump, which provides a new equation linking upstream and downstream conditions. This provides a new expression for the jump speed in the Boussinesq limit, which Borden and Meiburg term the vortex sheet (VS) model. It is shown here (and this is the central point of this paper) that this procedure may be extended to all density ratios, providing (in particular) a general relation for internal hydraulic jump speeds, as follows.

In general, the steady-state form of the vorticity equation (in the frame of the jump) has the form

(3.2) $$\begin{eqnarray}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\omega}=\frac{1}{{\it\rho}^{2}}\boldsymbol{{\rm\nabla}}{\it\rho}\times \boldsymbol{{\rm\nabla}}p=-\boldsymbol{{\rm\nabla}}\times \left(\frac{1}{{\it\rho}}\boldsymbol{{\rm\nabla}}p\right).\end{eqnarray}$$

We assume that the flow is on a sufficiently large scale so that the viscous term may be neglected. Since the fluids are incompressible so that $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0$ , if (3.2) is integrated over the region of the jump (as represented in figure 1) one obtains for the left-hand side

(3.3) $$\begin{eqnarray}\int _{A}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\omega}\,\text{d}A=\int _{A}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\omega}\boldsymbol{u})\,\text{d}x\,\text{d}z=\int {\it\omega}\boldsymbol{u}\boldsymbol{\cdot }\text{d}\boldsymbol{S},\end{eqnarray}$$

where area $A$ is the region containing the jump as depicted in figure 1, and $\boldsymbol{S}$ denotes the boundary surfaces of $A$ upstream and downstream. The last term of (3.3) denotes the flux of vorticity across surface $\boldsymbol{S}$ . For jumps moving into stationary fluid, or fluid in uniform motion, there is no upstream vorticity. On the downstream side the vorticity is concentrated in the vortex sheet between the two fluids, and the flux of it is given by the vortex sheet strength times the mean velocity of the sheet, which is (Saffman Reference Saffman1992; Borden & Meiburg Reference Borden and Meiburg2013)

(3.4) $$\begin{eqnarray}\int {\it\omega}\boldsymbol{u}\boldsymbol{\cdot }\text{d}\boldsymbol{S}=-(u_{2d}-u_{1d})(u_{2d}+u_{1d})/2=-\frac{1}{2}(u_{2d}^{2}-u_{1d}^{2}).\end{eqnarray}$$

The area integral of the right-hand side of (3.2) may be reduced by Stokes’s theorem to a line integral around the boundary of area $A$ enclosing the jump

(3.5) $$\begin{eqnarray}-\int _{A}\boldsymbol{{\rm\nabla}}\times \left(\frac{1}{{\it\rho}}\boldsymbol{{\rm\nabla}}p\right)\text{d}A=-\int \left(\frac{1}{{\it\rho}}\boldsymbol{{\rm\nabla}}p\right)\boldsymbol{\cdot }\text{d}\,\boldsymbol{l},\end{eqnarray}$$

in the anticlockwise sense. Since the density is uniform in each layer, the integral reduces to

(3.6) $$\begin{eqnarray}-\int \left(\frac{1}{{\it\rho}}\boldsymbol{{\rm\nabla}}p\right)\boldsymbol{\cdot }\text{d}\,\boldsymbol{l}=\left(\frac{1}{{\it\rho}_{2}}-\frac{1}{{\it\rho}_{1}}\right)(p_{iu}-p_{id}),\end{eqnarray}$$

where $p_{iu}$ and $p_{id}$ denote the upstream and downstream pressures at the interface respectively (see figure 1).

The upstream pressure at the interface is equal to $p_{su}+g{\it\rho}_{2}(D-d_{1u})$ and the corresponding downstream pressure is $p_{sd}+g{\it\rho}_{2}(D-d_{1d})$ . Putting these terms into (3.6) and equating it with (3.4) gives

(3.7) $$\begin{eqnarray}\frac{1}{2}(u_{2d}^{2}-u_{1d}^{2})=\left(\frac{1}{{\it\rho}_{2}}-\frac{1}{{\it\rho}_{1}}\right)[p_{su}-p_{sd}+{\it\rho}_{2}g(d_{1d}-d_{1u})]=\frac{{\rm\Delta}{\it\rho}}{{\it\rho}_{1}}\frac{p_{su}-p_{sd}}{{\it\rho}_{2}}+g^{\prime }(d_{1d}-d_{1u}).\end{eqnarray}$$

This is the same expression as that derived by Borden and Meiburg, except that it has the additional term containing the difference in surface pressure, which would be discarded in taking the Boussinesq approximation. For previous models the difference in surface pressure was an unknown quantity, but (3.7) provides a value for it that may be combined with the value obtained from the conservation of momentum flux  $S$ in (2.4). Eliminating $p_{su}-p_{sd}$ from these two equations gives the expression for the jump speed $c_{J}$ (which in present notation is equal to $u_{1u}$ and $u_{2u}$ ) in terms of its amplitude, which takes the form

(3.8) $$\begin{eqnarray}\frac{c_{J}^{2}}{g^{\prime }D}=\frac{r_{d}^{2}(1-r_{d})^{2}\left[1+\left(\displaystyle \frac{{\it\rho}_{1}}{{\it\rho}_{2}}-1\right)\displaystyle \frac{r_{d}+r_{u}}{2}\right]}{\left[\displaystyle \frac{r_{d}+r_{u}}{2}-r_{u}r_{d}+\left(\displaystyle \frac{{\it\rho}_{1}}{{\it\rho}_{2}}-1\right)r_{u}r_{d}(1-r_{d})^{2}-\left(1-\displaystyle \frac{{\it\rho}_{2}}{{\it\rho}_{1}}\right)r_{d}^{2}(1-r_{u})(1-r_{d})\right]}.\end{eqnarray}$$

This model may be termed the ‘full vortex sheet (FVS)’ model. If ${\it\rho}_{2}={\it\rho}_{1}$ in this expression, giving the Boussinesq limit, it reduces to (2.9) of Borden and Meiburg for their ‘vortex sheet (VS)’ model. It also satisfies other required limits, as follows.

(3.9) $$\begin{eqnarray}\text{If}~r_{d}\rightarrow r_{u}:\quad c_{J}^{2}\rightarrow c_{0}^{2}\equiv \frac{{\rm\Delta}{\it\rho}g}{\displaystyle \frac{{\it\rho}_{1}}{d_{1u}}+\displaystyle \frac{{\it\rho}_{2}}{d_{2u}}},\end{eqnarray}$$

where $c_{0}$ is the linear long-wave speed.

(3.10) $$\begin{eqnarray}\text{If}~{\it\rho}_{2}\rightarrow 0:\quad c_{J}^{2}\rightarrow \frac{gd_{1d}}{2}\left(1+\frac{d_{1d}}{d_{1u}}\right),\end{eqnarray}$$

which is Rayleigh’s relation for single-layer jumps (see Baines Reference Baines1995, § 2.3.1).

(3.11) $$\begin{eqnarray}\text{If}~D\rightarrow \infty :\quad c_{J}^{2}\rightarrow \frac{2g^{\prime }d_{1d}^{2}}{d_{1u}+d_{1d}}.\end{eqnarray}$$

None of the previous models (YG, WS and KRS) listed in § 2 conforms with all of these limits.

For some purposes it useful to compare the jump speed with the linear wave speed $c_{0}$ in (3.9), and (3.8) then becomes

(3.12) $$\begin{eqnarray}\frac{c_{J}^{2}}{c_{0}^{2}}=\frac{r_{d}^{2}(1-r_{d})^{2}\left[1-r_{u}\left(1-\displaystyle \frac{{\it\rho}_{2}}{{\it\rho}_{1}}\right)\right]\left[1+\left(\displaystyle \frac{{\it\rho}_{1}}{{\it\rho}_{2}}-1\right)\displaystyle \frac{r_{d}+r_{u}}{2}\right]}{r_{u}(1-r_{u})\!\left[\displaystyle \frac{r_{d}+r_{u}}{2}-r_{u}r_{d}+\!\left(\displaystyle \frac{{\it\rho}_{1}}{{\it\rho}_{2}}-1\!\right)\!r_{u}r_{d}(1-r_{d})^{2}-\!\left(\!1-\displaystyle \frac{{\it\rho}_{2}}{{\it\rho}_{1}}\right)\!r_{d}^{2}(1-r_{u})(1-r_{d})\right]}.\end{eqnarray}$$

Some representative results for wave speeds from this model up to the point of maximum speed are shown in figure 2 for a range of density ratios, for $r_{u}=0.1,0.2$ and 0.35. For ${\it\rho}_{2}/{\it\rho}_{1}=0.99$ the results are very similar to those from the Boussinesq model of Borden and Meiburg and the KRS model. For ${\it\rho}_{2}/{\it\rho}_{1}=0.79$ the results may be compared with the observations of Baines (Reference Baines1984) with water and kerosene. Comparisons are shown in figure 3 for $r=r_{u}=0.035$ , 0.1 and 0.2, which give a reasonable fit to the data, significantly better than those with the model of Yih and Guha, with which the comparisons were presented in 1984.

The upper limit of $r_{u}$ values for which upward jumps may occur is 0.5 for the Boussinesq limit, which is also observed numerically for ${\it\rho}_{2}/{\it\rho}_{1}=0.99$ . However, for smaller values of ${\it\rho}_{2}/{\it\rho}_{1}$ this limit increases, and it reaches 0.75 for ${\it\rho}_{2}/{\it\rho}_{1}=0.1$ . As the upper layer density vanishes, jumps (of modest sizes) are possible for $r_{u}$ values approaching unity. Downward hydraulic drops are possible for $r_{u}$ values ranging from 0.5 to 1 in the Boussinesq limit (and ${\it\rho}_{2}/{\it\rho}_{1}=0.99$ ), but as ${\it\rho}_{2}/{\it\rho}_{1}$ decreases the range shortens, to 0.54:1 for ${\it\rho}_{2}/{\it\rho}_{1}=0.79$ , 0.765:1 for ${\it\rho}_{2}/{\it\rho}_{1}=0.1$ , and it vanishes as ${\it\rho}_{2}/{\it\rho}_{1}\rightarrow 0$ .

Figure 2. Hydraulic jump speeds $c_{J}$ for the FVS model, for ${\it\rho}_{2}/{\it\rho}_{1}=0.1$ , 0.5, 0.79 and 0.99 as a function of amplitude $r_{d}=d_{1d}/D$ for initial layer thickness $r_{u}=d_{1u}/D$ (given by the starting value of $r_{d}$ ).  $c_{J}$ has been scaled with the linear wave speed  $c_{0}$ .

Figure 3. FVS model jump speeds for $r_{u}=0.035$ , 0.1 and 0.2 compared with experimental observations from Baines (Reference Baines1984) made with water and kerosene ( ${\it\rho}_{2}/{\it\rho}_{1}=0.79$ ). The results obtained with the Yih–Guha (YG) model are also shown, and the FVS model is clearly a better fit.

4. Jump amplitudes, speeds and energy losses

The results shown in figure 2 have been terminated at the point of maximum speed of the hydraulic jumps. This is not the limit of the formal solutions for such jumps, as the solutions continue for larger amplitude, but at decreasing jump speed. These solutions are theoretically possible, but have lesser significance for two reasons, as given below. Examples are shown in figure 4 for $r_{u}=0.035$ and 0.2, and in figure 5 for $r_{u}=0.965$ , 0.9, 0.8 and 0.65, where the jumps are downward (sometimes known as ‘hydraulic drops’). In these figures the extensions at larger amplitude and lower speed beyond the point of maximum speed are shown as dashed. Figure 6 shows the losses in the energy flux in each layer for $r_{u}=0.035$ , 0.2, 0.8 and 0.965. Here one notes that the energy flux loss in each layer is everywhere positive, that both rise to a maximum at a common point, and then decrease to zero, again terminating at a common point. In each case, the point of the double maximum coincides with the maximum in the jump speed, and the speed curves in figures 4 and 5 are terminated at the point of zero loss of energy flux.

Figure 4. Hydraulic jump speeds $c_{J}$ scaled with $c_{0}$ as in figure 2 but for the full range to the point of zero energy loss, for $r_{u}=0.035$ and 0.2. Amplitudes greater than those for maximum speed are shown dashed.

Figure 5. Hydraulic jump (hydraulic drop) speeds $c_{J}$ scaled with $c_{0}$ as in figure 4 for $r_{u}=0.65$ , 0.8, 0.9 and 0.965. Amplitudes greater than those for maximum speed are shown dashed.

Figure 6. Energy flux losses for each layer within the hydraulic jump for (a)  $r_{u}=0.035$ , (b)  $r_{u}=0.2$ , (c)  $r_{u}=0.965$ and (d)  $r_{u}=0.8$ , for ${\it\rho}_{2}/{\it\rho}_{1}=0.1$ , 0.2, 0.5, 0.79 and 0.99. The solid lines denote the upper layer, the dashed lines the lower layer. Note that the upper layer is contracting in (a,b), but the lower layer is contracting in (c,d), which show results for hydraulic drops.

The termination point represents a transition through the jump to a two-layer flow state with the same energy fluxes in each layer (and zero loss through the jump), which is described as a flow state that is ‘conjugate’ to the original upstream state (Benjamin Reference Benjamin1966; Lamb & Wan Reference Lamb and Wan1998). Solutions of (3.8) and (3.12) for larger amplitudes are possible but are essentially unphysical, since they would require a gain in energy flux within the jump. Hence the conjugate flow state represents the maximum jump amplitude.

The criterion for maximum speed of the hydraulic jump coincides (or very nearly coincides) with another property, namely that where the flow downstream of the jump becomes critical, in the frame of the jump. Figure 7 shows a representative example of this for $r_{u}=0.1$ , where the upper curves show the jump speeds, and the lower dashed curves the speeds of leftward-propagating waves (relative to the jump) on the downstream side. The zeros for these downstream wave speeds occur at $r_{d}$ values very slightly smaller $({\sim}0.1\,\%)$ than those for maximum jump speed. Similar diagrams can be derived for other $r_{u}$ values. This means that, for jump amplitudes smaller than that for maximum speed, disturbances downstream of the jump are able to propagate up to the jump, and potentially affect it by increasing or decreasing its amplitude. For larger jump amplitudes with lower jump speeds, the downstream flow is supercritical (relative to the jump), and disturbances from downstream cannot propagate to the jump and influence it. This means, in particular, that jumps with amplitudes larger than jumps with maximum speed cannot be created by forcing from downstream, whereas smaller-amplitude jumps can. This applies to upstream jumps that are forced by flow over topography either by a slow increase in the flow or by an obstacle of slowly increasing height. This implies that it may be difficult to realise these larger-amplitude jumps.

Further, if jumps with magnitudes greater than that for maximum jump speed (and dissipation) are formed, they are probably unstable. This is because, if the amplitude of the leading part of such a jump is decreased slightly, it will travel faster, and the larger-amplitude component will lag behind. Conversely, an increase in the jump amplitude will cause it to travel more slowly, and again the highest part will lag behind. For these two reasons – difficulty of generation and inherent instability – the practical relevance of the jumps with speeds greater than the maximum is doubtful, although, like the conjugate states, there may be special situations where they may be formed and sustained. Dam-break flows are one situation in which the large-amplitude jumps may be formed, but they would then be subject to the instability that would lead to a jump of maximum speed followed by a wave field. Numerical results by White & Helfrich (Reference White and Helfrich2014) of jumps formed (by dam-break) in Boussinesq near-two-layer continuous stratification show maximum jump amplitude coinciding with maximum jump speed and zero energy loss for $r_{u}=0.1$ and 0.2, but not for 0.4. There is scope for further work to resolve these details.

Figure 7. Hydraulic jump speeds $c_{J}$ scaled with $c_{0}$ as in figures 2 and 4 for $r_{u}=0.1$ , together with (leftward-propagating) linear wave speeds (in the frame of the jump) on the downstream side. The flow is subcritical (wave speeds are negative, towards the jump) for jump heights less than the value for maximum speed, and supercritical (wave speeds are positive) for jump heights greater than this value.

In figure 6, (a,b) are for hydraulic jumps and (c,d) are for hydraulic drops. The energy flux losses (notionally due to turbulent dissipation) show that, if the densities of the two layers are comparable, most of the energy flux is lost in the contracting layer. As the relative density of the upper layer decreases, the energy flux loss there decreases in proportion, as may be seen in figure 6(b). In (c,d), the contracting lower layer dominates the energy flux loss regardless of the density difference. As ${\it\rho}_{2}/{\it\rho}_{1}\rightarrow 1$ in (a,b) the energy loss of the lower layer becomes small but not zero: the dissipation rates shown for ${\it\rho}_{2}/{\it\rho}_{1}=0.99$ are very close to the limiting values ( ${\it\rho}_{2}/{\it\rho}_{1}=0.999$ gives the same result). This provides some justification for the KRS approximation and Borden and Meiburg’s VS Boussinesq model when ${\it\rho}_{2}/{\it\rho}_{1}$ is close to unity, but the results are not precise. For larger $r_{u}$ , relatively more dissipation occurs in the lower layer: for $r_{u}=0.2$ , the changeover occurs near ${\it\rho}_{2}/{\it\rho}_{1}=0.25$ .

5. Profiles with a diffuse or mixing interface

In their study of Boussinesq internal bores, Borden and Meiburg extended their model to include a mixing layer between the two fluids by assuming a linear variation in both the density and velocity in place of a sharp interface. This may also be done here for the general case, but of course it only applies if the fluids are miscible. There are several options. One may regard the mixing as a process occurring within the hydraulic jump with an upstream two-layer flow as described above and a downstream flow with a mixed layer. Alternatively, one may also have a mixed layer in the upstream flow, whether or not the velocities in the two layers are the same. The analysis below covers both of these cases.

Figure 8. Definition sketch (similar to figure 1) for a hydraulic jump with a linear transition region between the upper and lower layers, both upstream and downstream. Velocities are given in the frame of the jump.

We consider the general case with jump structure as shown in figure 8, with a linear transition in both density and velocity between the upper and lower layers, both upstream and downstream. The notation is the same as in figure 1, except for the additional layer and the fact that the velocities and thicknesses of the upper and lower layers are denoted by dashes, to distinguish them from the case of figure 1. The central layer has the thickness ${\it\delta}_{u}$ upstream and ${\it\delta}_{d}$ downstream, and if $z^{\ast }$ is the vertical coordinate within the central layer the velocity and density there (in a frame of reference moving with the jump) have the form

(5.1a,b ) $$\begin{eqnarray}u(z^{\ast })=u_{1u}^{\prime }+\frac{z^{\ast }}{{\it\delta}_{u}}(u_{2u}^{\prime }-u_{1u}^{\prime }),\quad {\it\rho}(z^{\ast })={\it\rho}_{1}-\frac{z^{\ast }}{{\it\delta}_{u}}({\it\rho}_{1}-{\it\rho}_{2}),\quad 0<z^{\ast }<{\it\delta}_{u},\end{eqnarray}$$

(5.2a,b ) $$\begin{eqnarray}u(z^{\ast })=u_{1d}^{\prime }+\frac{z^{\ast }}{{\it\delta}_{d}}(u_{2d}^{\prime }-u_{1d}^{\prime }),\quad {\it\rho}(z^{\ast })={\it\rho}_{1}-\frac{z^{\ast }}{{\it\delta}_{d}}({\it\rho}_{1}-{\it\rho}_{2}),\quad 0<z^{\ast }<{\it\delta}_{d},\end{eqnarray}$$

where the first equation applies upstream and the second downstream. We also have

(5.3a,b ) $$\begin{eqnarray}d_{1u}^{\prime }+d_{2u}^{\prime }+{\it\delta}_{u}=D,\quad d_{1d}^{\prime }+d_{2d}^{\prime }+{\it\delta}_{d}=D.\end{eqnarray}$$

As before, our objective is to obtain expressions for the flow downstream of the jump in terms of the flow properties upstream. Since the fluids are incompressible, volume conservation gives the equation

(5.4) $$\begin{eqnarray}u_{1u}^{\prime }d_{1u}^{\prime }+u_{2u}^{\prime }d_{2u}^{\prime }+(u_{1u}^{\prime }+u_{2u}^{\prime })\frac{{\it\delta}_{u}}{2}=u_{1d}^{\prime }d_{1d}^{\prime }+u_{2d}^{\prime }d_{2d}^{\prime }+(u_{1d}^{\prime }+u_{2d}^{\prime })\frac{{\it\delta}_{d}}{2},\end{eqnarray}$$

and similarly mass conservation gives

(5.5) $$\begin{eqnarray}\displaystyle & & \displaystyle u_{1u}^{\prime }d_{1u}^{\prime }+\frac{{\it\rho}_{2}}{{\it\rho}_{1}}u_{1u}^{\prime }d_{1u}^{\prime }+\frac{{\it\delta}_{u}}{6}\left[u_{1u}^{\prime }\left(2+\frac{{\it\rho}_{2}}{{\it\rho}_{1}}\right)+u_{2u}^{\prime }\left(1+2\frac{{\it\rho}_{2}}{{\it\rho}_{1}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \quad =u_{1d}^{\prime }d_{1d}^{\prime }+\frac{{\it\rho}_{2}}{{\it\rho}_{1}}u_{1d}^{\prime }d_{1d}^{\prime }+\frac{{\it\delta}_{d}}{6}\left[u_{1d}^{\prime }\left(2+\frac{{\it\rho}_{2}}{{\it\rho}_{1}}\right)+u_{2d}^{\prime }\left(1+2\frac{{\it\rho}_{2}}{{\it\rho}_{1}}\right)\right].\end{eqnarray}$$

The remaining equation is the conservation of momentum flux (2.2); inserting the hydrostatic pressure as before, one obtains for the downstream side

(5.6) $$\begin{eqnarray}\displaystyle S & = & \displaystyle \frac{1}{2}{\it\rho}_{2}gd^{2}+p_{sd}+\frac{1}{6}g({\it\rho}_{1}-{\it\rho}_{2})[(D-d_{2d}^{\prime })^{2}+d_{1d}^{\prime }(D-d_{2d}^{\prime })+d_{2d}^{\prime 2}]+{\it\rho}_{1}d_{1d}^{\prime }u_{1d}^{\prime 2}\nonumber\\ \displaystyle & & \displaystyle +\,{\it\rho}_{2}d_{2d}^{\prime }u_{2d}^{\prime 2}+\frac{{\it\delta}_{d}}{12}[u_{1d}^{\prime 2}(3{\it\rho}_{1}+{\it\rho}_{2})+u_{2d}^{\prime 2}({\it\rho}_{1}+3{\it\rho}_{2})+2u_{1d}^{\prime }u_{2d}^{\prime }({\it\rho}_{1}+{\it\rho}_{2})],\end{eqnarray}$$

which is equal to the corresponding expression (with subscript ‘ $u$ ’ replacing subscript ‘ $d$ ’) on the upstream side.

As before, we need an expression for the difference in surface pressure $p_{su}-p_{sd}$ , and this is again obtained from the vorticity balance which, in the same manner as in § 3, yields

(5.7) $$\begin{eqnarray}\frac{{\rm\Delta}{\it\rho}}{{\it\rho}_{1}}\frac{p_{su}-p_{sd}}{{\it\rho}_{2}}=\frac{1}{2}(u_{2d}^{\prime 2}-u_{1d}^{\prime 2})-\frac{1}{2}(u_{2u}^{\prime 2}-u_{1u}^{\prime 2})-\frac{{\rm\Delta}{\it\rho}g}{{\it\rho}_{1}}(d_{1d}^{\prime }-d_{2d}^{\prime }-d_{1u}^{\prime }+d_{2u}^{\prime }).\end{eqnarray}$$

Substituting this into (5.5) gives the fourth equation for the downstream flow. If ${\it\delta}_{d}$ and ${\it\delta}_{u}$ have prescribed values, (5.2)–(5.6) may be solved numerically to determine the jump speed and downstream conditions in terms of those upstream, as in § 3.

It remains to determine appropriate values of the intermediate layer thickness, ${\it\delta}_{d}$ and ${\it\delta}_{u}$ . The value of ${\it\delta}_{u}$ will be dependent on processes upstream of the jump, and we assume it to be zero here. If we restrict consideration to the case where the upstream flow has uniform speed with no mixing layer (i.e.  ${\it\delta}_{u}=0$ , with $u_{2u}=u_{1u}$ ), for miscible fluids the thickness ${\it\delta}_{d}$ will be determined by mixing processes within the jump. In a general sense, these may be related to shear instability within the jump, which is governed by the Richardson number $R_{i}$ , defined by $R_{i}=N^{2}/(\text{d}u/\text{d}z)^{2}$ , where the buoyancy frequency $N$ is given by $N^{2}=-(g/{\it\rho})\text{d}{\it\rho}/\text{d}z$ . On the downstream side of the jump, this gives

(5.8) $$\begin{eqnarray}R_{i}(z^{\ast })=\frac{{\it\rho}_{1}-{\it\rho}_{2}}{(u_{2d}^{\prime }-u_{1d}^{\prime })^{2}\left({\it\rho}_{1}-\displaystyle \frac{z^{\ast }}{{\it\delta}_{d}}({\it\rho}_{1}-{\it\rho}_{2})\right)},\end{eqnarray}$$

which has maximum and minimum values at the top and bottom (respectively) of the central layer, with values

(5.9a,b ) $$\begin{eqnarray}R_{i\mathit{max}}=\frac{{\rm\Delta}{\it\rho}g{\it\delta}_{d}}{{\it\rho}_{2}(u_{2d}^{\prime }-u_{1d}^{\prime })^{2}},\quad R_{i\mathit{min}}=\frac{{\rm\Delta}{\it\rho}g{\it\delta}_{d}}{{\it\rho}_{1}(u_{2d}^{\prime }-u_{1d}^{\prime })^{2}}.\end{eqnarray}$$

If $R_{i}>1/4$ in a density-stratified shear flow, internal waves may propagate through it, but if $R_{i}<1/4$ they cannot. The instability of this shear layer depends on the mutual interaction of waves propagating on the vorticity interfaces at the top and bottom of the layer, and wave propagation in the fluid between them disrupts this process. Hence, if $R_{i\mathit{min}}>1/4$ the flow will be stable, but if $R_{i\mathit{max}}<1/4$ it will be unstable. Here we base the value of ${\it\delta}_{d}$ on the second criterion, and obtain

(5.10) $$\begin{eqnarray}{\it\Delta}_{d}={\it\delta}_{d}/D=\frac{{\it\rho}_{2}(u_{2d}^{\prime }-u_{1d}^{\prime })^{2}}{{\rm\Delta}{\it\rho}gD}.\end{eqnarray}$$

(This choice is, to some extent, arbitrary, and may underestimate ${\it\delta}_{d}$ , but the scaling is robust.) To compute these values numerically for increasing values of $r_{d}^{\prime }=d_{1d}^{\prime }/D$ , one takes ${\it\delta}_{d}=0$ for the initial increment and uses the values from the previous increment to give the next value of ${\it\delta}_{d}$ ; the error involved is negligible if the increments are sufficiently small.

Figure 9. Comparisons of jump speeds between flows without a mixed transition layer, as previously (solid curves), and flows with a downstream transition layer (dashed curves) with thickness derived via (5.10), for some representative cases. The abscissa is $r_{d}$ for the solid curves, $r_{d}^{\prime }+{\it\Delta}_{d}/2$ for the dashed curves.

Figure 10. The thickness ${\it\Delta}_{d}={\it\delta}_{d}/D$ obtained from (5.10) for the cases with jump speeds as given in figure 9.

Some representative results from such calculations are shown in figures 9 and 10. Figure 9 shows comparisons between jump speeds for $r_{u}=0.1$ and 0.2 between cases with no (partially) mixed layer (solid lines) and flows with a downstream mixed layer (dashed curves) as defined in (5.10). Here the jump speeds with mixing are less than those without, except when ${\it\rho}_{2}/{\it\rho}_{1}$ is small. Thicknesses of the mixed layer for the same parameter values are shown in figure 10. It is seen that, even with the conservative estimate of (5.10), the transition layer thickness becomes quite substantial with increasing jump amplitude.