2 max 2 max 1 *h h h h Nh hh wall* / / ' /( ) = + (96)

*<sup>h</sup> Fr* (97)

*<sup>h</sup> Fr* (98)

*<sup>h</sup> Fr* (99)

=[(1+8*Fr*<sup>1</sup>

2


number, suggesting the empirical equation (94), for *Fr*1 between 1.98 and 8.5.

the standard deviation (multiple given by the factor *N*), that is:

Using equations (95) and (96), no bed shear force, that is *h*#

the predicted value of 0.27 to the observed value of 0.32, we have:

<sup>2</sup> <sup>2</sup> <sup>1</sup>

<sup>2</sup> <sup>2</sup> <sup>1</sup>

<sup>2</sup> <sup>2</sup> <sup>1</sup>

each curve may have its own *β*.

106 Hydrodynamics - Concepts and Experiments

**3.5. Depth fluctuations and total depths (wall heights)**

*3.5.1. Predictions based on empirical formulation*

**Figure 21.** a) Oscillations of the free surface due to a wave in a hydraulic jump for *Fr*1=3.5, b) Relative depth attained by the wave shown in Figure 21a iv.

Equations (97, 98, 99) use 1, 2 and 3 standard errors (68.2%, 95.5% and 99.6% of the observed events, respectively, see Figure 13). Figure 22 presents the curves of the three equations, showing that the additional height observed in Figure 21b is better represented by equation (99), for 99.6% of the cases. Equation (97) furnishes values closer to the suggestion of [46] (*hwall*/ *h*2=1.25).

**Figure 22.** Equations (97, 98, 99), for the maximum depth as a function of the Froude number.

#### *3.5.2. Predictions based on theoretical reasoning*

Equations (85, 86) and (91) are used, with *N*=0.8, *M*=3.3 and *α*=0.82. *h*' and *hwall* are given by:

$$\frac{h^\*}{h\_2 - h\_1} = 0.18 \sqrt{\left(1 - e^{-2.9x/L} + \frac{1}{h^\* - 1}\right) \left(e^{-2.9x/L} + \frac{0.0207h^\*}{h^\* - 1}\right)}\tag{100}$$

$$\frac{h\_{\text{wall}}}{h\_{\text{2}}} = 1 - \left(\frac{h^{\ddagger} - 1}{h^{\ddagger}}\right) e^{-2.9x/L} + 0.594 \sqrt{\left[1 - \left(\frac{h^{\ddagger} - 1}{h^{\ddagger}}\right) e^{-2.9x/L}\right] \left[\left(\frac{h^{\ddagger} - 1}{h^{\ddagger}}\right) e^{-2.9x/L} + 0.0207\right]}\tag{101}$$

Predictions of equation (101) were compared with depth data obtained by [17]. The comparison is shown in Figure 23. The normalized mean depth of the data increases continuously with *x*/ *Lr*, but when added to three times the standard error, the points show a very slight maximum. For the predictive calculations, it was assumed *H*1=0 (a limiting value), but any value 0≤*H*1≤*h*<sup>1</sup> is possible. The value *M*=3.3 multiplied by the rms function of the RSW method is close to the "three standard error" used for *hwall*/*h*2. For *Fr*1=2.36 the maximum calculated value is *hwall*/ *h*2=1.11, lower than the suggested value of [46], that is, *hwall*/*h*2=1.25. Figure 24 shows the data measured by [17], where the vertical lines are the measured points.

**Figure 23.** Measured surface profile (lower "o"), and 3 standard errors added to it (upper "o"). Data of [17]. Calculated profile (equation 91) and 3.3 times the rms value of the RWS method added to it (equation 101).

Predictions similar to those presented in Figure 22 need only the maximum value of the function along *x*/*Lr*. Indicating this value as "*J*(max)", equations (101) and (61) furnish, for no bed shear force:

$$\frac{h\_{\text{wall}}}{h\_{\text{2}}} = \frac{2}{\sqrt{1 + 8Fr\_{\text{1}}^{2}} - 1} + \left(\frac{\sqrt{1 + 8Fr\_{\text{1}}^{2}} - 3}{\sqrt{1 + 8Fr\_{\text{1}}^{2}} - 1}\right) J\left(\max\_{1}\right), \qquad \text{adjusted} \quad J(\max\_{1}) = 1.515 \tag{102}$$

**Figure 24.** Measurements of [17] used in Figure 23.

*3.5.2. Predictions based on theoretical reasoning*

108 Hydrodynamics - Concepts and Experiments

2 1

2

bed shear force:

'

Equations (85, 86) and (91) are used, with *N*=0.8, *M*=3.3 and *α*=0.82. *h*' and *hwall* are given by:

2.9 / 2.9 /

2.9 / 2.9 / 2.9 /

*h h h h* (101)

1 11 <sup>1</sup> 0.594 1 0.0207 - - - æ ö - -- <sup>é</sup> æö æö ù é <sup>ù</sup> =- + - ç ÷ <sup>ê</sup> ç÷ ç÷ ú ê <sup>+</sup> <sup>ú</sup> è ø <sup>ë</sup> èø èø û ë <sup>û</sup>

Predictions of equation (101) were compared with depth data obtained by [17]. The comparison is shown in Figure 23. The normalized mean depth of the data increases continuously with *x*/ *Lr*, but when added to three times the standard error, the points show a very slight maximum. For the predictive calculations, it was assumed *H*1=0 (a limiting value), but any value 0≤*H*1≤*h*<sup>1</sup> is possible. The value *M*=3.3 multiplied by the rms function of the RSW method is close to the "three standard error" used for *hwall*/*h*2. For *Fr*1=2.36 the maximum calculated value is *hwall*/ *h*2=1.11, lower than the suggested value of [46], that is, *hwall*/*h*2=1.25. Figure 24 shows the data

**Figure 23.** Measured surface profile (lower "o"), and 3 standard errors added to it (upper "o"). Data of [17]. Calculated

Predictions similar to those presented in Figure 22 need only the maximum value of the function along *x*/*Lr*. Indicating this value as "*J*(max)", equations (101) and (61) furnish, for no

profile (equation 91) and 3.3 times the rms value of the RWS method added to it (equation 101).


<sup>1</sup> 0.0207 0.18 1
