# #

(1 ) <sup>1</sup> ln ( 1)(1 ) ( 1)

. This implies that *φ*∞=(*h*#

*hh h h e <sup>h</sup>* (82)

= -

f

*MMM* or *VV u* = = *<sup>M</sup>* (77)

*s VL s L s* (78)

 w

*IJs h h e* (81)

& *DT K mh*

(76)

1

, where *β* is a constant

, so that the solution is

*IJs e* (80)


b

b

 *M M* = *h h Zh h h* f

<sup>111</sup> <sup>1</sup> *m hV h V* & = =

2

¶ ¶ ¶ -

¶¶ ¶ & *D K mh <sup>T</sup>*

1 2 2 1 2

= + = = ¶¶ ¶

f f(1 ) - = - ¥

followed assuming that at *s*=1 (length of the roller) we have *h*/*h*1=*βh*#


*IJs*

*h h <sup>e</sup> IJ*


*IJs*

 f

= +

water density. The integral mass conservation equation furnishes:

 r

1

f

[30] proposed *K´=K"h-2*. Equation (79) is then obtained:

ff

 w

2

¶¶ ¶

w

and defining *IJ*=(*ω*2-1)/*ω*1, is given by:

Equations (76) and (80) produce:

its use here). For *s* → ∞ we have *h*/*h*1=*h*#

1

2 1

given now by:

f

r r

94 Hydrodynamics - Concepts and Experiments

From equations (74) to (77) it follows:

r

The obtained mean profiles of the surface are useful to obtain the profile of the maximum fluctuations of the surface, which may be used in the design of the lateral walls that confine the hydraulic jump. In this sense, the method of Random Square Waves (RSW) for the analysis of random records is used here. Details of this method, applied to interfacial mass transfer, are found in [31, 32]. A more practical explanation is perhaps found in [33], where the authors present the rms function of random records, that is, the "equivalent" to the standard error function (RSW are bimodal records, but used to obtain approximate solutions for general random data).

Equations (70) and (82) express the normalized mean depth *h* along the hydraulic jump. The RSW approximation considers the maximum (*H*2) and minimum (*H*1) values of *h*, that is, it involves the local maxima of the fluctuations. Following the RSW method, the depth profile may be expressed as:

$$\frac{h - H\_1}{H\_2 - H\_1} = n\tag{83}$$

The normalized RSW rms function (a "measure" of the standard error) of the depth fluctuations is:

$$\frac{h^\*}{H\_2 - H\_1} = \sqrt{n\left(1 - n\right)}\left(1 - \alpha\right) \tag{84}$$

*h*' is the rms value, and *α* is called "reduction function", usually dependent on *x*. From its definition it is known that 0≤*α*≤1 (see, for example, [33]). [31] and [32] present several analyses using a constant *α* value, a procedure also followed here as a first approximation. Undulating mean surface profiles, however, need a more detailed analysis.

Equations (70) and (82) to (84) allows obtaining equations for the height attained by fluctua‐ tions like waves and drops along the hydraulic jump, using procedures similar to those of normal distributions of random data. For normal distribution of data, 68.2%, 95.5% or 99.6% of the observed events may be computed by using respectively, 1, 2 or 3 standard errors in the prediction (for example). Figure 11 illustrates the comments, where *µ* represents the mean value and *σ* represents the standard error. The RSW method uses modified bimodal records (thus, the mentioned percentages may not apply), but the general idea of summing the mean depth and multiples of the rms value is still valid.

In order to relate *n* from equations (83) and (84) to *Π* from equations (70) and (82) it was considered that *H*<sup>1</sup> is zero (no fluctuation crosses the bottom), *H*<sup>2</sup> is given by *H*2=*h*2+*Nh*2'(mean depth summed to *N* times the rms value), and *hwall*=*h*+*Mh*'. Algebraic operations lead to:

$$\frac{h^\*}{h\_2 - h\_1} = (1 - \alpha) \sqrt{\left(\Pi + \frac{1}{h^\* - 1}\right) \left(1 - \Pi + \frac{N^2 \left(1 - \alpha\right)^2 h^\*}{h^\* - 1}\right)}\tag{85}$$

$$\frac{h\_{\text{wall}}}{h\_2} = \frac{1}{h^\pi} + \left(1 - \frac{1}{h^\pi}\right) \left[\Pi + M\left(1 - \alpha\right)\sqrt{\left(\Pi + \frac{1}{h^\pi}\right)\left(1 - \Pi + \frac{N^2\left(1 - \alpha\right)^2 h^\pi}{h^\pi - 1}\right)}\right] \tag{86}$$

*hwall* is used here to represent the mean height attained by the fluctuations. *N* and *M* are constants multiplied by the rms value ("replacing" the standard error) which originally expresses the percentage of cases assumed in the solution. In the present study (1-*α*) is taken as an adjustment coefficient. When analyzing the maximum value of *h*2, the term between braces in equation (86) is substituted by *J*(max).

**Figure 11.** Normal curve showing the percentage of observed cases as the area limited by the curve.

#### **3. Data analyses**

#### **3.1. Length of the roller**

The data of [18] were first used to test equations (55-58). Different normalizations lead to different adjusted coefficients, because of the different distributions of errors. It was observed here that equation (57) shows the best adjustment to measured data, furnishing the coefficients shown in equation (87), and the adjustment of Figure 12. The obtained correlation coefficient was 0.996. *L* is used without index (*r* or *j*) because both lengths (roller and jet) are used in the following analyses.

Details of Hydraulic Jumps for Design Criteria of Hydraulic Structures http://dx.doi.org/10.5772/58963 97

$$\frac{L}{h\_{\rm l}} = 3.62 \frac{Fr\_{\rm l}}{\sqrt{h^\*}} - 0.157 \left(Fr\_{\rm l} \right)^3 \sqrt{h^\*} \tag{87}$$

**Figure 12.** Adjustment of the data of [18] to equation (87).

( ) ( )

ì ü æö æ ö ï ï æ ö - = + - P+ - P+ -P+ í ý ç ÷ ç÷ ç ÷

*hwall* is used here to represent the mean height attained by the fluctuations. *N* and *M* are constants multiplied by the rms value ("replacing" the standard error) which originally expresses the percentage of cases assumed in the solution. In the present study (1-*α*) is taken as an adjustment coefficient. When analyzing the maximum value of *h*2, the term between

èø è ø - - ï ï è ø î þ

*hh h <sup>h</sup> <sup>h</sup>* (86)

æ öæ ö - = - P+ -P+ ç ÷ ç ÷ - -- è øè ø *h N h*
