New Approach for the Calculation of Critical Depth in a U-Shaped Channel

The U-Shaped channel is a high performance structure widely used in practice. Currently, the problem of the critical depth in such profile does not have a direct solution. Current methods of calculation are based on complex mathematical procedures or optimal fitting methods, often generating unacceptable errors in practice, knowing that the calculation of the critical depth requires a high accuracy. The complexity of the problem stems from the fact that the flow governing equations are complicated due to the shape of the profile. In this study, the form of the flow equation is simple through the intake of the properties of the triangle. Furthermore, even if the equation is implicit, its resolution is possible by applying the fixed-point method with an initial value judiciously chosen. The process converges after the seventh step of calculating only and leads to an almost exact solution. A calculation example is presented that highlights the simplicity of the calculation procedure.


INTRODUCTION
Critical depth plays a major role in determining the subcritical or supercritical nature of the flow and in the classification of varied flow [1][2][3]. In this regard, several studies have been devoted to the calculation of critical depth in various geometric profiles of channels [4][5][6][7][8][9][10][11][12][13], offering solutions with varying degrees of accuracy. Several methods were used to calculate the critical depth such that numerical method, graphical method, explicit regression-based equations, fitting curves and Newton-Raphson iterative method. Regarding the U-channel, a relatively recent study has been devoted to the calculation of the critical depth using fitting curves [14]. The relative error caused by this method is about 0.7%, which can be an important relative error for a certain number of practical cases. The calculation of critical depth requires a much smaller relative error. The methods of calculating the critical depth in the U-Shaped channel are complex due to the geometrical shape of the channel. The U-Shaped channel may be considered as a triangular channel with a rounded bottom. Using the properties of a triangle, one obtains equations of simple form such that the critical water area and the critical top width that come into consideration in the criterion for critical flow. The form of the equation of critical flow is handy, contrary to that obtained in previous studies.
This equation is implied towards the nondimensional critical depth. However, its form is so simple that one can apply standard methods of resolution, such as the fixed-point method. This is the method which is used in the present study, with a suitable initial value. The iterative process is not constraining since almost exact solution is obtained after the seventh step of calculation. A practical example is provided showing both the high accuracy and the simplicity of the calculation.  The critical top width T c can be written as:

CRITICAL FLOW EQUATION
Where m = cotg.
The critical water area is expressed as: Where: The well known criterion for critical flow states that: Eq. (5) can be rewritten as: Let us assume the relative conductivity * Q as: Assume also the following aspect ratio: / y r   (8) Thus, Eq. (6) is reduced to: Adopt the following change in variables: Eq. (9) can be then simply written as: Or else: Eq. (12) is implicit towards the variable z. To solve Eq. (12) we suggest a numerical method which consists in approaching successively the solution. The calculation process is iterative and operates on Eq. (12) after selecting a first value of z. Assume that the first value of z is o 1 z   .
As a result, the next values of z are obtained such that: The calculation process stops when z i and z i+1 are sufficiently close. It is obvious that the speed of convergence of the described iterative process depends strongly on the value of z o initially selected. With o 1 z   , intensive calculations showed that almost exact value of z is obtained, in the worst case, at the end of the seventh step of calculating only. The suggested procedure of calculation is not therefore constraining.
We tested 1120 numerical examples which showed that the proposed iterative method converges. We did not test all cases that may arise in practice in studying the convergence of the recommended method. Like any iterative method, the advocated method may not converge.
Once the final value of z is determined, the aspect ratio η is worked out from Eq. (10) as: The non-dimensional critical depth can be expressed as: Where:

PRACTICAL EXAMPLE
Compute the critical depth y c in the U-Shaped channel shown in Fig. 1 for the following data: (For the sake of calculation, the counts will not be rounded off) 1. According to Eq. (7), the relative conductivity * Q is: As we can see, the first derivative is less than unity, which confirms that the method converges.
We can even calculate the number of iterations needed to solve the problem. For this, consider an absolute error and an interval for z. Choose a relative error such that The iterative process converges after the seventh step of calculating.
Inserting the obtained values of * Q and 1  in Eq. (12) and adopting the described iterative process for 0 1 z   , the final value of z is such that: 6 7 5.260588152 z z z    3. According to Eq. (15), the aspect ratio  is as: