# A Nonlinear System of Differential Equations in Supercritical Flow Spread Problem and Its Solution Technique

## Abstract

**:**

## 1. Introduction

- the development of up-to-date application software packages making it easier to use mathematical methods when solving practical tasks in hydraulics of open water flows;
- new approaches in finding the solution of hydraulic structure calculation problems;
- the introduction of new normative indices, requiring detailing of practical HS calculations at large Froude numbers.

## 2. Research Methods

#### 2.1. System of Equations of Motion for Two-Dimensional in Plan Supercriticals Potential, Stationary, Streams Open Water

#### 2.2. On the Boundary Problem of the Free Supercritical Flow behind an Unpressurised Culvert When It Spreads into a Wide Discharge Channel

#### 2.3. Description of the Method for Solving the Problem in the Velocity Hodograph Plane

#### 2.3.1. Solving the Problem in the Uniform Flow Area (Section I)

- Since at ${M}_{0}$ the flow velocity and the parameter ${\tau}_{0}$, then the lines angle ${\alpha}_{0\phantom{\rule{0.166667em}{0ex}}}$ is also determined at this point:$${\alpha}_{0}=arcsin\left(\sqrt{\frac{1-{\tau}_{0}}{2{\tau}_{0}}}\right),$$
- Knowing the length of the inertial front [33,34,35,36,37], the geometry of the uniform flow section can be determined.$${X}_{D}^{I}=trunc\left[\frac{\sqrt{{F}_{0}-1}}{sin{\theta}_{max}\left({F}_{0}+2\right)}{h}_{0}\right]+1\phantom{\rule{0.277778em}{0ex}}cm,$$${X}_{D}^{I}$—front length of the inertial section along the flow symmetry axis $OX$.The geometry of section I is determined by the parameters ${X}_{D}^{I}$, ${\alpha}_{0}=arcsin\sqrt{\frac{1-{\tau}_{0}}{2{\tau}_{0}}}$, and pipe width b.Since the flow in area I is uniform, then $\tau ={\tau}_{0}$, $\theta =0$, $V={V}_{0}$, $h={h}_{0}$.
- Let us determine parameter values $\tau $, $\theta $ on the characteristic of the 1st family. From the characteristic equation of the 1st family [20,38], the angle $\theta $ can be determined at the known $\tau \in \left[{\tau}_{0},1\right]$$$\theta \left(\tau \right)=\sqrt{3}\xb7arctg\sqrt{\frac{3\tau -1}{3\left(1-\tau \right)}}-arctg\left(\sqrt{\frac{3\tau -1}{1-\tau}}\right)+{C}_{1},$$$${C}_{1}=arctg\sqrt{\frac{3{\tau}_{0}-1}{1-{\tau}_{0}}}-\sqrt{3}\xb7arctg\left(\sqrt{\frac{3{\tau}_{0}-1}{3\left(1-{\tau}_{0}\right)}}\right).$$Setting the spacing $\Delta \tau =\frac{1-{\tau}_{0}}{N}$, we get:$${\tau}_{i}={\tau}_{0}+i\Delta \tau $$The angle ${\theta}_{\mathrm{max}}$ is also determined from the system of Equation (12).
- Let us determine the flow coefficients at the intersection points i of the current line with the characteristic of the 1st family. From the equation equitable along the current line$$\frac{sin\theta}{{\tau}^{1/2}}=Ksin{\theta}_{max}$$$${K}_{i}=\frac{sin{\theta}_{i}}{{\tau}^{1/2}sin{\theta}_{max}}.$$

#### 2.3.2. Problem Solving in the General Flow Area (Section III)

- Let us define the parameters $\tau $, $\theta $ in the flow area of section III. The base flow is given by the equations in the velocity hodograph plane (Figure 5):$$\psi =A\frac{sin\theta}{{\tau}^{1/2}};\phantom{\rule{1.em}{0ex}}\phi =A\frac{{h}_{0}}{{H}_{0}}\xb7\frac{cos\theta}{{\tau}^{1/2}\left(1-\tau \right)}.$$This section is bounded by the 1st family characteristic and the flow symmetry axis $OX$. The characteristic runs through a point ${M}_{0}$ with parameters $\tau ={\tau}_{0};\theta =0.$ This is the main flow characteristic that runs through the entire flow. It has the form [20]:$$\theta =\sqrt{3}\xb7arctg\sqrt{\frac{3\tau -1}{3\left(1-\tau \right)}}-arctg\left(\sqrt{\frac{3\tau -1}{1-\tau}}\right)+{C}_{1},$$$${C}_{1}=arctg\sqrt{\frac{3{\tau}_{0}-1}{1-{\tau}_{0}}}-\sqrt{3}\xb7arctg\left(\sqrt{\frac{3{\tau}_{0}-1}{3\left(1-{\tau}_{0}\right)}}\right).$$
- Setting the parameters ${\theta}_{i}$, ${\tau}_{i}$ at the points of intersection of the characteristics of the 1st family and the corresponding current line, it is possible to determine the parameters of the intersection points i of the current line and j of the equipotentiality from the system$$\left\{\begin{array}{c}\hfill \frac{sin{\theta}_{ij}}{{\tau}_{ij}^{1/2}}={K}_{i}sin{\theta}_{max};\hfill \\ \hfill \frac{cos{\theta}_{ij}}{{\tau}_{ij}^{1/2}\left(1-{\tau}_{ij}\right)}=\frac{cos{\theta}_{i}}{{\tau}_{i}^{1/2}\left(1-{\tau}_{i}\right)}.\hfill \end{array}\right.$$Herewith$${\tau}_{0}\le {\tau}_{ij}\le 1;\phantom{\rule{1.em}{0ex}}\theta \le {\theta}_{ij}\le {\theta}_{max}.$$
- Coordinates of the points ${x}_{ij}$, ${y}_{ij}$ in region III of the flow is determined from the differential relation (5).If moving along the corresponding current line, Equation (5) by virtue of the condition $d\phi =0$ is recomposed after separating the variables in the form of:$$\left\{\begin{array}{c}\hfill dx=\frac{d\phi cos\theta}{{\tau}^{1/2}\sqrt{2g{H}_{0}}};\hfill \\ \hfill dy=\frac{d\phi sin\theta}{{\tau}^{1/2}\sqrt{2g{H}_{0}}}.\hfill \end{array}\right.$$
- Determination of parameters at points ${C}_{1}$, ${L}_{1}$. Let us draw the equipotential through the point C. Then the equation of the equipotential passing through the point C:$$\frac{cos{\theta}_{C}}{{\tau}_{C}^{1/2}\left(1-{\tau}_{C}\right)}=\frac{cos{\theta}_{{C}_{1}}}{{\tau}_{{C}_{1}}^{1/2}\left(1-{\tau}_{{C}_{1}}\right)}.$$At point ${C}_{1}$ we assume ${\tau}_{{C}_{1}}={\tau}^{*}.$ Consequently,$$cos{\theta}_{{C}_{1}}=\frac{cos{\theta}_{C}\xb7{\tau}_{{C}_{1}}^{1/2}\left(1-{\tau}_{{C}_{1}}\right)}{{\tau}_{C}^{1/2}\left(1-{\tau}_{C}\right)}.$$$${\theta}_{{C}_{1}}=arccos{\theta}_{{C}_{1}}.$$Alternatively,$$\begin{array}{c}sin{\theta}_{{C}_{1}}=\sqrt{1-{cos}^{2}{\theta}_{{C}_{1}}};\\ {\theta}_{{C}_{1}}=arcsin\sqrt{1-{cos}^{2}{\theta}_{{C}_{1}}}.\end{array}$$Similarly, we determine the parameters at the point ${L}_{1}$.
- Determining the coordinates of the points ${C}_{1}$, ${L}_{1}$. Parameters at a point ${C}_{1}$ are ${\tau}_{{C}_{1}},\phantom{\rule{0.166667em}{0ex}}{\theta}_{{C}_{1}}$.The equation of the current line passing through the point ${C}_{1}$:$$\frac{sin{\theta}_{{C}_{1}}}{{\tau}_{C}^{1/2}}=Ksin{\theta}_{max}.$$

#### 2.3.3. Determination of Flow Parameters in the Simple Wave Region (Section II)

- We determine $tg\left(\theta +\alpha \right)$ as an angular coefficient of slope of the tangent to the 1st family characteristic $tg\left(\theta +\alpha \right)={f}^{*}\left(\tau \right)$ and a section of its uniformity. Since $tg\left(\theta +\alpha \right)$ is a monotonically increasing function of the argument $\tau $, then we determine the monotonicity areas of the function:$$f\left(\tau \right)=\theta +\alpha =\theta \left(\tau \right)+\alpha \left(\tau \right),$$$$\begin{array}{c}\theta \left(\tau \right)=\sqrt{3}\xb7arctg\sqrt{\frac{3\tau -1}{3\left(1-\tau \right)}}-arctg\left(\sqrt{\frac{3\tau -1}{1-\tau}}\right)+{C}_{1};\\ \alpha \left(\tau \right)=arcsin\sqrt{\frac{1-\tau}{2\tau}}.\end{array}$$To do this, we solve the equation$${f}_{\tau}^{\prime}={\theta}^{\prime}\left(\tau \right)+{\alpha}^{\prime}\left(\tau \right)=0.$$Root of the Equation (14) $\tau ={\tau}^{*}$ defines areas of uniformity of the function $f\left(\tau \right)$, and consequently the functions ${f}^{*}\left(\tau \right)$:$$\left[{\tau}_{0},{\tau}^{*}\right],\phantom{\rule{1.em}{0ex}}\left[{\tau}^{*},1\right].$$At the site $\left[{\tau}_{0},{\tau}^{*}\right]$ the function ${f}^{*}\left(\tau \right)$ monotonically decreases, and in the area $\left[{\tau}^{*},1\right]$ monotonically increases.
- Similarly, to section II with simple waves, we connect the Froude line points ${M}^{*}$ and A of equal numbers $A{M}^{*}$, the line on which is $\tau ={\tau}^{*}$, and the angle $\theta $ is determined from the solution to the problem.Equal Froude number lines convey perturbations in the presence of discontinuities in the flow parameters.
- From the equation of the extreme current line determine the angle$${\theta}_{A}^{*}=\mathrm{arcsin}\left({\tau}_{*}^{1/2}sin{\theta}_{max}\right).$$As $tg\theta $ increases ultimately along the extreme current line $ACL{A}_{n}$, the points A and ${M}^{*}$ can be connected by Froude’s equal number perturbation waves.Perturbations by equal Froude number lines are more generic disturbances than a simple wave. In a simple wave$$\theta =constA,\phantom{\rule{1.em}{0ex}}\tau =constA.$$In a wave of Froude equal numbers line $\tau =const.$Point ${M}^{*}$ should necessarily be connected to point A, as the minimum possible value at point A must be $\tau ={\tau}^{*}$ and further increase downstream.
- Further conducting an equipotential ${M}_{0}C$, let us determine the flow parameters at point C by solving the system:$$\left\{\begin{array}{c}\hfill \frac{cos{\theta}_{C}}{{\tau}_{C}^{1/2}\left(1-{\tau}_{C}\right)}=\frac{1}{{\tau}_{0}^{1/2}\left(1-{\tau}_{0}\right)};\hfill \\ \hfill \frac{sin{\theta}_{C}}{{\tau}_{C}^{1/2}}=sin{\theta}_{max}.\hfill \end{array}\right.$$Similarly, we determine the flow parameters at the point L:$$\frac{sin{\theta}_{L}}{{\tau}_{L}^{1/2}}=sin{\theta}_{max},$$$$\theta \left({\tau}^{**}\right)=\left({\tau}^{**}\right).$$
- Choice of steps in sections: Step selection $\Delta {\tau}_{1}$ on a characteristic ${M}_{0}{M}_{n}$ between the points ${M}_{0}$ and ${M}^{*}$:$$\Delta {\tau}_{1}=\frac{{\tau}^{*}-{\tau}_{0}}{{N}_{1}}.$$Then,$${\tau}_{i}={\tau}_{0}+\Delta {\tau}_{1}\xb7i;\phantom{\rule{1.em}{0ex}}i=0,1,2,\dots ,{N}_{1}.$$Selecting the sampling step $\Delta {\tau}_{2}$ between the points ${M}^{*}$ and ${M}^{**}$:$$\Delta {\tau}_{2}=\frac{{\tau}^{**}-{\tau}^{*}}{{N}_{2}}.$$Then,$${\tau}_{j}={\tau}^{*}+\Delta {\tau}_{2}\xb7j;\phantom{\rule{1.em}{0ex}}j=0,1,2,\dots ,{N}_{2}.$$Selecting the sampling step $\Delta {\tau}_{3}$ between the points ${M}^{**}$ and ${M}_{n}\left(\tau \right)$:$$\Delta {\tau}_{3}=\frac{1-{\tau}^{**}}{{N}_{3}}.$$Then,$${\tau}_{k}={\tau}^{**}+\Delta {\tau}_{3}\xb7k;\phantom{\rule{1.em}{0ex}}k=0,1,2,\dots ,{N}_{3}.$$
- The right lines ${A}_{0}{M}_{0}$, ${A}_{1}{M}_{1}$ in a simple wave are determined from the condition that the characteristic of the 2nd family passes through the point ${A}_{i}$ and has an angular coefficient $tg\left({\theta}_{i}-{\alpha}_{i}\right)$ [5]. Extreme current line points ${A}_{i}$ are determined by the distance ${\rho}_{i}$ on the corresponding line ${M}_{i}{A}_{i}$:$${\rho}_{i}=\frac{b\left(1-{K}_{i}\right)\sqrt{{\tau}_{0}}\left(1-{\tau}_{0}\right)}{2\sqrt{{\tau}_{i}}\left(1-{\tau}_{i}\right)sin{\alpha}_{i}},$$

#### 2.3.4. Improvement of the Proposed Algorithm

## 3. The Discussion of the Results

- initial flow velocity ${V}_{0}$ [cm/s];
- initial depth of the flow relative to the bottom ${h}_{0}$ [cm];
- gravity acceleration $g=981$ [cm/s${}^{2}$];
- pipe width b [cm].

#### 3.1. Solving the Problem in the Uniform Flow Area (Section I)

- Froude number ${F}_{0}=2.4$;
- initial flow velocity ${V}_{0}=\frac{Q}{{h}_{0}b}=147.654$ cm/s;
- hydrodynamic head ${H}_{0}=20.382$ cm;
- initial flow kinetics (block 1, item 4) ${\tau}_{0}=0.545$;
- wave angle at the point where the flow exits the pipe ${\alpha}_{0}=0.702\phantom{\rule{0.166667em}{0ex}}p$ or ${\alpha}_{0}={40}^{\circ}23$;
- angle of the velocity vector of the liquid flow to the OX axis at infinity ${\theta}_{\mathrm{max}}=0.981p$ or ${\theta}_{\mathrm{max}}={56}^{\circ}23$;
- length of inertial front ${X}_{D}^{I}=3$ cm;
- distance from the end of the inertial section to the point along the flow symmetry axis ${M}_{0}\phantom{\rule{1.em}{0ex}}A{M}_{0}=9.457$ cm;
- the length of the straight-line segment of the 2nd family characteristic between the points ${A}_{0}$ and ${M}_{0}$${A}_{0}{M}_{0}=12.387$ cm.

#### 3.2. Problem Solution in the General Flow Area (Section III) and in the Simple Wave Area (Section II)

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic of a potential supercritical flow behind an unpressurised pipe into a wide, plain horizontal channel.

**Figure 7.**Plots of extreme current line and some equipotentials (solid) obtained from numerical calculations and extreme current line obtained from in situ experiments (dashed).

**Table 1.**Kinetics values ${\tau}_{i}$, flow abscissa ${X}_{i0}$, its velocity ${V}_{i0}$ and depth ${h}_{i0}\phantom{\rule{0.166667em}{0ex}}$ at the symmetry axis points.

${\mathit{\tau}}_{\mathit{i}}$ | 0.545 | 0.596 | 0.646 | 0.697 | 0.747 | 0.798 | 0.848 | 0.899 | 0.949 | 1 |

${X}_{i0}$ | 6.299 | 7.269 | 8.547 | 10.259 | 12.642 | 16.169 | 21.942 | 33.244 | 66.336 | 3233 |

${V}_{i0}$ | 147.654 | 154.345 | 160.759 | 166.926 | 172.873 | 178.622 | 184.192 | 189.599 | 194.855 | 199.963 |

${h}_{i0}$ | 9.274 | 8.234 | 7.215 | 6.176 | 5.157 | 4.117 | 3.098 | 2.059 | 1.039 | 0 |

Point No. | Abscissa on the Symmetry Axis at ${\mathit{X}}_{0\mathit{i}}$ | Parameter of Kinetics, ${\mathit{\tau}}_{{\mathit{C}}_{\mathit{i}}}$ | Angle of Inclination of the Flow Velocity Vector to the Axis of Symmetry | Flow Ordinates on the Extreme Line | Experimental Data at Some Points | Relative Algorithm Error, % |
---|---|---|---|---|---|---|

1 | 0 | 0.545 | 0.661 | 8 | 8 | 0 |

2 | 4 | 0.767 | 0.815 | 10.442 | 11 | 5.073 |

3 | 8 | 0.866 | 0.884 | 13.637 | ||

4 | 12 | 0.908 | 0.914 | 18.687 | ||

5 | 16 | 0.931 | 0.93 | 23.973 | ||

6 | 20 | 0.945 | 0.94 | 29.402 | ||

7 | 24 | 0.954 | 0.947 | 34.926 | 38 | 8.090 |

8 | 28 | 0.961 | 0.952 | 40.519 | ||

9 | 32 | 0.966 | 0.956 | 46.162 | ||

10 | 36 | 0.97 | 0.959 | 51.846 | ||

11 | 40 | 0.973 | 0.961 | 57.56 | ||

12 | 44 | 0.975 | 0.963 | 63.3 | 59 | 7.288 |

13 | 48 | 0.978 | 0.965 | 69.06 | ||

14 | 52 | 0.979 | 0.966 | 74.84 | 73 | 2.458 |

**Table 3.**Kinetics, slope angle of the fluid velocity vector to the OX axis and flow coefficients at the points on the characteristic ${M}_{0}{M}_{n}$ (area III).

Step No. | Kinetics | Angle of Inclination of Velocity Vector | Fluid Flow Coefficient |
---|---|---|---|

1 | 0.5452 | 0.145 | 0.212 |

2 | 0.5756 | 0.157 | 0.229 |

3 | 0.6059 | 0.17 | 0.246 |

4 | 0.6363 | 0.184 | 0.263 |

5 | 0.6667 | 0.197 | 0.28 |

6 | 0.6765 | 0.211 | 0.297 |

7 | 0.6862 | 0.224 | 0.314 |

⋮ | ⋮ | ⋮ | ⋮ |

31 | 0.9208 | 0.697 | 0.788 |

32 | 0.9306 | 0.734 | 0.819 |

33 | 0.9404 | 0.778 | 0.853 |

34 | 0.9501 | 0.834 | 0.896 |

35 | 0.9599 | 0.937 | 0.97 |

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Evtushenko, S.
A Nonlinear System of Differential Equations in Supercritical Flow Spread Problem and Its Solution Technique. *Axioms* **2023**, *12*, 11.
https://doi.org/10.3390/axioms12010011

**AMA Style**

Evtushenko S.
A Nonlinear System of Differential Equations in Supercritical Flow Spread Problem and Its Solution Technique. *Axioms*. 2023; 12(1):11.
https://doi.org/10.3390/axioms12010011

**Chicago/Turabian Style**

Evtushenko, Sergej.
2023. "A Nonlinear System of Differential Equations in Supercritical Flow Spread Problem and Its Solution Technique" *Axioms* 12, no. 1: 11.
https://doi.org/10.3390/axioms12010011