Nonlinear Differential Equations of Flow Motion Considering Resistance Forces

Abstract

For a stationary potential 2D planar open high-velocity water flow of the ideal liquid, we propose a closed system of nonlinear equations considering the resistance forces to the flow from the channel bottom. Tangential stresses on jet interfaces are ignored. The resistance force components are expressed in terms of velocity components. In this case, the flow equations can be solved through the method of characteristics, and the surface forces are reduced to equivalent volumetric forces. The system of non-linear equations is solved in the velocity hodograph plane; further, the transition to the physical plane takes place. Since the value of the hydrodynamic pressure decreases downstream of the flow, the friction forces to the flow in the first approximation can be considered by using the integral laws of resistance. At that, the form of the equations of motion in the plane of the velocity hodograph does not change. This fact is proved in the article. An example of calculating the water flow is provided. The kinecity, ordinates, and velocities of the flow along its extreme line are calculated without considering resistance forces. Validation of the model in the real flow is performed. Acceptable accuracy relative to experimental data is obtained.

1. Introduction

The research object is a stationary potential 2D planar open high-velocity (supercritical) water flow [1]. The subject of this research is a mathematical model of a freely spreading high-velocity flow, with the resistance forces from the channel taken into account. Such flows occur during the free spreading of water from open channels or free-flow pipes into a wide outlet channel; when water flows in the junctions of channels of different widths; in open spillways; in the back of hydro-power plants, thermal and nuclear power plants (inlet and outlet conduits, technical pipelines); back from culverts during river floods and weather anomalies.

1.1. Consideration of the Channel Resistance Forces in the Problem of Determining the Parameters of the Water Flow

When calculating high-velocity flows in long conduits, the effect of resistance forces should not be neglected, the presence of which causes a vortex flow structure and leads to a reduction in the total hydrodynamic pressure downstream. In addition, in many cases, the slope of the bottom is of great significance [1,2,3,4]. The predominant approach in calculating uniform currents is based on a 1D presentation of an open flow. Here, the average flow characteristics are considered: the average section velocity of the flow, the average tangential stress in terms of the length of the wetted perimeter and the depth (hydraulic radius). The basic calculated dependency in the case of uniform flow is the Shézy formula which connects the kinematic and dynamic flow characteristics. The main task of the hydraulic resistance theory is to determine the Shézy coefficient formula, or the coefficient of hydraulic friction associated with it [2,5,6,7].

A great number of empirical relationships have been suggested for determining the Shézy coefficient, the most common of which are formulas of exponential type (formulas by Pavlovsky and Manning) [8]. A conclusion about the influence of the section form of the open flow on the local distribution laws of averaged velocities and hydraulic resistance in smooth channels was offered in [2]. In hydrotechnical construction practice, we often encounter watercourses with variable roughness along the perimeter (hydrotechnical and meliorative canals with partial anchoring of the channel (pressure and non-pressure tunnels for water passage during construction and operational periods)). In the hydraulic calculation of considered watercourses according to the Shézy formula, the concept of the average roughness coefficient of the channel is used, which allows the replacement of a real channel by some simulated one with constant roughness [8,9].

The most important requirement to calculation formulas for Shézy coefficients still remains the full consideration of specific (local) flow conditions [10,11,12]. Therefore, the dependencies obtained by the statistical data processing method are unacceptable. The hydraulic resistances of the flows of spatial mode experiencing an essential braking effect of the banks are not sufficently investigated [13,14]. Though the determining influence of depth on the flow mode is established, the relation of flow resistance and depth has an ambiguous character [15]. Therefore, there appears a need to consider additional factors, including relative channel width to eliminate this uncertainty [16].

1.2. Relevance of Determining the Parameters of High-Speed Open Water Flows

The relevance of the research carried out is determined by the need to solve a number of problems related to the design and operation of hydraulic structures, which, in particular, include the issues of hydraulic calculation of uniform and non-uniform flows in rigid and erosive channels.

Objects of heat and nuclear power engineering (supply and discharge conduits, technical pipelines, etc.) need optimisation of design solutions on the basis of improvement of hydraulic calculation methods. Flood protection is becoming an important factor in modern life in all regions. The risk of flooding due to significant weather anomalies has increased.

In connection with the necessity to solve the most important problems of rational use, regulation, and protection of water resources of the country, the requirements to the accuracy of hydraulic calculations, with the purpose of optimisation of constructive decisions and increase in the operation reliability of designed hydraulic structures of various purposes are increasing.

All the aspects mentioned above confirm the topicality of the research works aimed at taking into account the drag forces in the mathematical model of a two-dimensional, exposed, potential, high-velocity (supercritical) flow and in establishing regularities of the processes taking place in nonpressure culverts.

The problem of hydraulic drag is one of the traditional problems of hydraulics, and a considerable amount of research is devoted to solving it. Nevertheless, some important aspects are still not sufficiently understood, even in the case of uniform flow rates. To a greater extent, this conclusion applies to nonuniform flows, flows in channels of complex cross-sectional shapes with variable roughness along the perimeter, etc.

1.3. Problem Definition

The aim of this work is to obtain a nonlinear system of equations of flow motion in partial derivatives conidering friction forces. In order to obtain analytical dependences of the flow parameters, we should convert the nonlinear system of motion equations to a more convenient form—a linear system of flow motion equations in partial derivatives. This equation system should be solved by the method of characteristics; calculated dependencies for flow parameters must be obtained and the adequacy of the resulting mathematical model must be justified.

2. Research Methods

2.1. Initial Equations of 2D Planar Water Flows

For ease of material presentation, we proceed from the system of 2D planar flows in the following form [17]:

$$\left\{\begin{array}{c}{U}_x\frac{\partial U_x}{\partial x}+U_y\frac{\partial U_y}{\partial y}=-g\frac{\partial }{\partial x}\left(h\right)-T_x;\hfill \\ U_x\frac{\partial U_y}{\partial x}+U_y\frac{\partial U_y}{\partial y}=-g\frac{\partial }{\partial y}\left(h\right)-T_y;\hfill \\ \frac{\partial }{\partial x}\left(h{U}_x\right)+\frac{\partial }{\partial y}\left(h{U}_y\right)=0,\hfill \end{array}\right.$$

(1)

where $U_x,U_y$ are the projections of the local velocity vector on the axes $Ox$, $Oy$ in planar flow; h is the local flow depth; $T_x,T_y$ are the components of resistance forces related to the unit mass of liquid. The flow is assumed to be stationary and vortex-free. Channel bottom is assumed to be horizontal.

The first two equations in the system (1) are directly dynamic equations of a flow movement, and the third one is the equation of flow continuity.

2.2. Derivation of the Flow Energy Equation

Let us prove the existence of the model in case of vortex-free flow considering the resistance forces. In the entire region of the flow the D. Bernoulli integral is valid for 2D planar flows [17]:

$$H=\frac{U^2}{2g}+h,$$

(2)

where h is the local flow depth; $U^2=U_x^2+U_y^2$ is the velocity local modulus of the particle liquid; g is the acceleration of gravity.

Then,

$$\left\{\begin{array}{c}\frac{\partial H}{\partial x}=\frac{\partial h}{\partial x}+\frac{1}{g}\left(U_x\frac{\partial U_x}{\partial x}+U_y\frac{\partial U_y}{\partial x}\right);\\ \frac{\partial H}{\partial y}=\frac{\partial h}{\partial y}+\frac{1}{g}\left(U_x\frac{\partial U_x}{\partial y}+U_y\frac{\partial U_y}{\partial y}\right).\end{array}\right.$$

(3)

We exclude in Equation (1) the terms containing the derivatives of h. After transformations, we obtain

$$\left\{\begin{array}{c}-g\frac{\partial H}{\partial x}-T_x=U_y\left(\frac{\partial U_x}{\partial y}-\frac{\partial U_y}{\partial x}\right);\\ -g\frac{\partial H}{\partial y}-T_y=U_x\left(\frac{\partial U_y}{\partial x}-\frac{\partial U_x}{\partial y}\right),\end{array}\right.$$

(4)

or, by entering the vortex vector modulus $\Omega =\frac{\partial U_y}{\partial x}-\frac{\partial U_x}{\partial y}$,

$$\left\{\begin{array}{c}g\frac{\partial H}{\partial x}=-T_x+U_y\Omega ;\\ g\frac{\partial H}{\partial y}=-T_y-U_x\Omega .\end{array}\right.$$

(5)

Multiplying the first of these equations by $U_x$, the second one by $U_y$, and summing up, we find

$$U_x\frac{\partial H}{\partial x}+U_y\frac{\partial H}{\partial y}=-\frac{1}{g}\left(T_x{U}_x+T_y{U}_y\right),$$

or, in a vector form,

$$\left(\overrightarrow{U}gradH\right)=-\frac{1}{g}\left(\overrightarrow{U}·\overrightarrow{T}\right),$$

where $\overrightarrow{U}=U_x\overrightarrow{i}+U_y\overrightarrow{j};\overrightarrow{T}=T_x\overrightarrow{i}+T_y\overrightarrow{j};\overrightarrow{i},\overrightarrow{j}$ are the unit orthogonal coordinate axes $Ox$, $Oy$.

We multiply the first of the Equation (5) by an arbitrary increment $dx$, and the second by $dy$, and add them together. As a result, we obtain

$$gdH=g\left(\frac{\partial H}{\partial x}dx+\frac{\partial H}{\partial y}dy\right)=-\left(T_{x}dx+T_{y}dy\right)+\Omega \left(U_{y}dx-U_{x}dy\right).$$

(6)

Introducing the current function, which follows from the flow continuity equation, we obtain

$$h{U}_x=\frac{\partial \psi }{\partial y};h{U}_y=-\frac{\partial \psi }{\partial x}.$$

Then, Equation (6) takes the form

$$gdH=-\left(T_{x}dx+T_{y}dy\right)-\frac{\Omega }{h}d\psi .$$

(7)

Along the stream-line $d\psi =0$ and from (7), we obtain the equation of the hydrodynamic pressure decrease:

$$dH=-\frac{1}{g}\left(T_{x}dx+T_{y}dy\right)=-\frac{1}{g}\left(\overrightarrow{T}·d\overrightarrow{S}\right),$$

where $\left(\overrightarrow{T}·d\overrightarrow{S}\right)$ is a directed segment of the stream-line with projections $dx$, $dy$.

By integration, we obtain the D. Bernoulli’s equation,

$$H_2-H_1=-\frac{1}{g}\underset{S_1}{\overset{S_2}{\int }}\left(\overrightarrow{T}·d\overrightarrow{S}\right),$$

which demonstrates the law of hydrodynamic pressure decrease along the stream-line.

On expressing $T_x$, $T_y$ through the velocity components and introducing the coefficient of hydraulic friction $\lambda =2g/C^2$, resistance factor C can be determined by the formula of N. N. Pavlovsky:

$$C=\frac{R^y}{n},$$

where y is a degree index depending on the value of the roughness coefficient and the hydraulic radius [3,8]:

$$y=2.5\sqrt{n}-0.13-0.75\sqrt{R}\left(\sqrt{n}-0.1\right),$$

where C is the Shezi’s ratio; R is the hydraulic radius; n is the roughness coefficient of the channel bottom [17].

With value $y=\frac{1}{6}$, Shezi’s formula is reduced to Manning’s formula agrees well with the practice of 2D planar flows. Since vectors $\overrightarrow{U}$ and $d\overrightarrow{S}$ are collinarnae, $\overrightarrow{U}·d\overrightarrow{S}=UdS$, and therefore

$$dH=-\frac{\lambda }{2}·F·dS,$$

(8)

where $F=\frac{U^2}{gh}$ is the Froude number [6].

Equation (8) is a hydrodynamic pressure decrease equation in a scalar form. As it follows from Equation (5), the flow can be potential at $\Omega =0$; then, the following model is valid:

$$\left\{\begin{array}{l}{U}_x=\frac{\partial \phi }{\partial x};U_y=\frac{\partial \phi }{\partial y};\\ \frac{h}{h_0}{U}_x=\frac{\partial \psi }{\partial y};-\frac{h}{h_0}{U}_y=\frac{\partial \psi }{\partial x};\\ \frac{U^2}{2g}+h=H;\frac{dH}{dS}=-\frac{\lambda }{2}F,\end{array}\right.$$

(9)

where the potential function $\phi =\phi \left(x,y\right)$ follows from the vortex-free conditions $\Omega =0$; $\psi =\psi \left(x,y\right)$ is the current function, it follows from the continuity equation; H is a hydrodynamic pressure; $dS$ is an element of the arc length along the stream-line. The hydrodynamic pressure decrease equation is valid along the stream-line.

2.3. Transition from the System of Equations of the Planar Water Flow $(x,y)$ to Equations of Flow Motion in the Plane of the Velocity Hodograph $(\tau ,\theta )$. The Property of the Mutual Position of Stream-Lines and Equipotentials

As it follows from System (9),

$$d\phi =U_{x}dx+U_{y}dy;d\psi =\frac{h}{h_0}\left(-U_{y}dx+U_{x}dy\right);$$

(10)

$grad\phi ·grad\psi =0$—stream-lines and equipotentials are mutually orthogonal. Along the stream-line $d\psi =0$ and

$$tg\theta =\frac{dy}{dx},$$

$\theta$ is an angle characterizing the direction of the velocity local vector to the flow symmetry axis $OX$.

Multiplying both parts of the second equation of System (10) by an imaginary unit and adding it to the first equation, we obtain the following equation:

$$d\phi +i\frac{h_0}{h}d\psi =\left(U_x-i{U}_y\right)d\left(x+iy\right),$$

(11)

where $i=\sqrt{-1}$ is an imaginary unit.

Equation (11) respresents a complex differential relationship between the plan of the flow and the velocity hodograph plane, in which $U,\theta$ are the independent variables, $\phi =\phi (U,\theta ),\psi (U,\theta )$ are the dependent variables.

Equation (11) can be rewritten in the form of [1,13,17]

$$dZ=\left(d\phi +i\frac{h_0}{h}d\psi \right)·\frac{e^{i\theta }}{V},$$

(12)

where $Z=x+iy$.

Then, from the expression for the hydrodynamic pressure (see Expression (2)), we find

$$h=H\left(1-\tau \right) \mathrm{and} U={\tau }^{1/2}\sqrt{2gH}.$$

(13)

Passing on in Equation (12) to variables $\tau$, $\theta$, we obtain

$$dZ=\left[d\phi +i\frac{h_0}{H\left(1-\tau \right)}d\psi \right]·\frac{e^{i\theta }}{{\tau }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2gH}}.$$

(14)

From (14), we obtain the following system:

$$\left\{\begin{array}{c}\frac{\partial Z}{\partial \tau }=\left[\frac{\partial \phi }{\partial \tau }+i\frac{h_0}{H\left(1-\tau \right)}\frac{\partial \psi }{\partial \tau }\right]·\frac{e^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}};\\ \frac{\partial Z}{\partial \theta }=\left[\frac{\partial \phi }{\partial \theta }+i\frac{h_0}{H\left(1-\tau \right)}\frac{\partial \psi }{\partial \theta }\right]·\frac{e^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}},\end{array}\right.$$

(15)

where $\phi =\phi \left(\tau ,\theta \right)$, $\psi =\psi \left(\tau ,\theta \right)$; $\tau ,\theta$—independent parameters: $\tau =\frac{U^2}{2gH}$—kinecity of the flow depending on the velocity U of the flow particle.

Further assuming that $H=H\left(\phi ,\psi \right)$, we determine from (15) the second partial derivatives $\frac{{\partial }^{2}Z}{\partial \tau \partial \theta },\frac{{\partial }^{2}Z}{\partial \theta \partial \tau }$:

$$\begin{array}{c}\frac{{\partial }^{2}Z}{\partial \tau \partial \theta }=\left[\frac{{\partial }^2\phi }{\partial \tau \partial \theta }+i\frac{h_0}{H\left(1-\tau \right)}·\right.\frac{{\partial }^2\psi }{\partial \tau \partial \theta }-i\frac{h_0}{H^2\left(1-\tau \right)}·\frac{\partial \psi }{\partial \tau }\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \theta }+\right.\\ +\left.\left.\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \theta }\right)\right]·\frac{e^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}}+\left[\frac{\partial \phi }{\partial \tau }+i\frac{h_0}{H\left(1-\tau \right)}\frac{\partial \psi }{\partial \tau }\right]·\frac{i{e}^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}};\end{array}$$

(16)

$$\begin{array}{c}\frac{{\partial }^{2}Z}{\partial \theta \partial \tau }=\left[\frac{{\partial }^2\phi }{\partial \theta \partial \tau }+i\frac{h_0}{H\left(1-\tau \right)}·\right.\frac{{\partial }^2\psi }{\partial \theta \partial \tau }+i\frac{h_0}{H{\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\\ -i\frac{h_0}{H^2\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }+\right.\left.\left.\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \tau }\right)\right]·\frac{e^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}}+\\ +\left[\frac{\partial \phi }{\partial \theta }+i\frac{h_0}{H\left(1-\tau \right)}\frac{\partial \psi }{\partial \theta }\right]·e^{i\theta }\left[-\frac{1}{{2\tau }^{3/2}\sqrt{2gH}}\right.-\frac{1}{2}·\frac{1}{{\left(2gH\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\tau }^{1/2}}\times \\ \times 2g·\left.\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }+\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \tau }\right)\right].\end{array}$$

(17)

Due to the equality $\frac{{\partial }^2\phi }{\partial \theta \partial \tau }=\frac{{\partial }^2\phi }{\partial \tau \partial \theta }$, which is valid for continuous functions $\phi \left(\tau ,\theta \right)$ from (16), (17) there follows equation

$$\begin{array}{c}-i\frac{h_0}{H^2\left(1-\tau \right)}·\frac{\partial \psi }{\partial \tau }\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \theta }+\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \theta }\right)·\frac{e^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}}+\\ +\left[\frac{\partial \phi }{\partial \tau }+i\frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \tau }\right]·\frac{e^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}}=\left[i\frac{h_0}{H{\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\right.\\ -i\frac{h_0}{H^2\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }+\right.\left.\left.\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \tau }\right)\right]·\frac{e^{i\theta }}{{\tau }^{1/2}\sqrt{2gH}}+\\ +\left[\frac{\partial \phi }{\partial \theta }+i\frac{h_0}{H\left(1-\tau \right)}\frac{\partial \psi }{\partial \theta }\right]·e^{i\theta }\left[-\frac{1}{{2\tau }^{3/2}\sqrt{2gH}}\right.-\frac{1}{2}·\frac{1}{{\left(2gH\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\tau }^{1/2}}\times \\ \times 2g·\left.\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }+\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \tau }\right)\right].\end{array}$$

(18)

Simplifying (18), we obtain

$$\begin{array}{c}-i\frac{h_0}{H^2\left(1-\tau \right)}·\frac{\partial H}{\partial \phi }·\left(\frac{\partial \psi }{\partial \tau }·\frac{\partial \phi }{\partial \theta }-\frac{\partial \phi }{\partial \tau }·\frac{\partial \psi }{\partial \theta }\right)+i\left(\frac{\partial \phi }{\partial \tau }+i\frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \tau }\right)=\\ =\frac{i{h}_0}{H{\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\left(\frac{\partial \phi }{\partial \theta }+i\frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }\right)·\frac{1}{2\tau }+g·\left.\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }+\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \tau }\right)\right].\end{array}$$

(19)

In a particular case of $H=H_0=const$ from (19), there follows equation

$$i\left(\frac{\partial \phi }{\partial \tau }+i\frac{h_0}{H_0\left(1-\tau \right)}·\frac{\partial \psi }{\partial \tau }\right)=\frac{i{h}_0}{H_0{\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\left(\frac{\partial \phi }{\partial \theta }+i\frac{h_0}{H_0\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }\right)·\frac{1}{2\tau }.$$

(20)

Dividing the real and imaginary parts in (20), we obtain the following system:

$$\left\{\begin{array}{c}\hfill \frac{\partial \phi }{\partial \tau }=\frac{1}{2}·\frac{h_0}{H_0}·\frac{3\tau -1}{\tau {\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta };\hfill \\ \hfill \frac{\partial \phi }{\partial \theta }=2·\frac{h_0}{H_0}·\frac{\tau }{1-\tau }·\frac{\partial \psi }{\partial \tau },\hfill \end{array}\right.$$

(21)

where $h_0,H_0$—constant for the entire flow.

Thus, in general Equation (19) we derive a special case which coincides with the previously known one [17].

2.4. Simplification of Equation (19)

Distinguishing the real and imaginary parts in Equation (19), we obtain the following system:

$$\left\{\begin{array}{c}\frac{\partial \phi }{\partial \tau }-\frac{h_0}{H^2\left(1-\tau \right)}\frac{\partial H}{\partial \phi }·\left(\frac{\partial \psi }{\partial \tau }·\frac{\partial \phi }{\partial \theta }-\frac{\partial \phi }{\partial \tau }·\frac{\partial \psi }{\partial \theta }\right)=\hfill \\ =\frac{h_0}{H{\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }\left[\frac{1}{2\tau }+\frac{1}{2H}·\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }+\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \tau }\right)\right];\hfill \\ \frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \tau }=\frac{\partial \phi }{\partial \theta }·\left[\frac{1}{2\tau }+\frac{1}{2H}·\left(\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }+\frac{\partial H}{\partial \psi }·\frac{\partial \psi }{\partial \tau }\right)\right].\hfill \end{array}\right.$$

(22)

This system differs from System (21) by the inclusion of the hydrodynamic pressure H and derivatives $\frac{\partial H}{\partial \phi },\frac{\partial H}{\partial \psi }$, in the particular case, without resistance force $H=const$; therefore,

$$\frac{\partial H}{\partial \phi }=\frac{\partial H}{\partial \psi }=0.$$

Since the hydrodynamic pressure varies only along the stream-line, then $\frac{\partial H}{\partial \psi }=0$, and we rewrite System (22) as

$$\left\{\begin{array}{c}\frac{\partial \phi }{\partial \tau }-\frac{h_0}{H^2\left(1-\tau \right)}·\frac{\partial H}{\partial \phi }·\left(\frac{\partial \psi }{\partial \tau }·\frac{\partial \phi }{\partial \theta }-\frac{\partial \phi }{\partial \tau }·\frac{\partial \psi }{\partial \theta }\right)=\hfill \\ =\frac{h_0}{H{\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }\left[\frac{1}{2\tau }+\frac{1}{2H}·\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }\right];\hfill \\ \frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \tau }=\frac{\partial \phi }{\partial \theta }·\left[\frac{1}{2\tau }+\frac{1}{2H}·\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }\right].\hfill \end{array}\right.$$

(23)

One can make sure that the equality is valid for the entire flow area:

$$\frac{\partial \psi }{\partial \tau }·\frac{\partial \phi }{\partial \theta }-\frac{\partial \phi }{\partial \tau }·\frac{\partial \psi }{\partial \theta }=0,$$

(24)

and System (23) is transformed into

$$\left\{\begin{array}{c}\hfill \frac{\partial \phi }{\partial \tau }=\frac{h_0}{H{\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }\left[\frac{1}{2\tau }+\frac{1}{2H}·\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }\right];\hfill \\ \hfill \frac{\partial \psi }{\partial \tau }=\frac{H\left(1-\tau \right)}{h_0}·\frac{\partial \phi }{\partial \theta }·\left[\frac{1}{2\tau }+\frac{1}{2H}·\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }\right]\hfill \end{array}\right.$$

or

$$\left\{\begin{array}{c}\hfill \frac{\partial \phi }{\partial \tau }=\frac{h_0}{2H}·\frac{3\tau -1}{\tau {\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta }-\frac{h_0}{H\left(1-\tau \right)}·\frac{\partial \psi }{\partial \theta }·\frac{1}{2H}·\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau };\hfill \\ \hfill \frac{\partial \psi }{\partial \tau }=\frac{H\left(1-\tau \right)}{h_0}·\frac{1}{2}·\frac{\partial \phi }{\partial \theta }+\frac{H\left(1-\tau \right)}{h_0}·\frac{\partial \phi }{\partial \theta }·\frac{1}{2H}·\frac{\partial H}{\partial \phi }·\frac{\partial \phi }{\partial \tau }.\hfill \end{array}\right.$$

(25)

Given Equality (24),

$$\frac{\partial \phi }{\partial \tau }·\frac{\partial \psi }{\partial \theta }=\frac{\partial \phi }{\partial \theta }·\frac{\partial \psi }{\partial \tau },$$

and the D. Bernoulli integral (2), System (25) is finally transformed into

$$\left\{\begin{array}{c}\hfill \frac{\partial \phi }{\partial \tau }=\frac{h_0}{2H\left(\phi \right)}·\frac{3\tau -1}{\tau {\left(1-\tau \right)}^2}·\frac{\partial \psi }{\partial \theta };\hfill \\ \hfill \frac{\partial \psi }{\partial \theta }=\frac{2h_0}{H\left(\phi \right)}·\frac{\tau }{1-\tau }·\frac{\partial \psi }{\partial \tau },\hfill \end{array}\right.$$

(26)

since the partial derivative $\frac{\partial H\left(\phi \right)}{\partial \theta }=0$ along the fixed equipotential.

Thus, the system of Equation (26) in appearance is identical to System (21) with the only difference that the hydrodynamic pressure does not remain constant throughout the flow area and varies, remaining constant along the equipotentiality.

2.5. Method for Determining the Hydrodynamic Pressure along Equipotentiality $H\left(\phi \right)$

The flow is supposed to have a longitudinal axis of symmetry. Since [3,18]

$$d\phi =dS·V,$$

(27)

Equation (8) should be rewritten in the form of

$$\frac{dH\left(\phi \right)}{d\phi }=-\frac{\lambda }{2}·\frac{F}{U}.$$

(28)

Since

$$\lambda =\frac{K}{h^{1/3}};F=\frac{2\tau }{1-\tau };U={\tau }^{1/2}\sqrt{2gH};h=H\left(1-\tau \right),$$

Equation (28) transforms into

$$\frac{dH\left(\phi \right)}{d\phi }=-\frac{K·\tau }{\left(1-\tau \right)}·\frac{1}{H^{1/3}{\left(1-\tau \right)}^{1/3}{\tau }^{1/2}\sqrt{2gH}}=-\frac{K·{\tau }^{1/2}}{{\left(1-\tau \right)}^{4/3}{H}^{1/3}\sqrt{2gH}},$$

(29)

where K is a constant coefficient.

On the other hand, we decompose $H\left(\phi \right)$ to a Taylor series

$$H\left(\phi \right)=H{\left(\phi \right)}_0+{\left(\frac{dH}{d\phi }\right)}_0·d\phi +\dots .$$

(30)

Keeping in (30) the summands of the first degree relative to $d\varphi$, we obtain

$$H\left(\phi \right)=H{\left(\phi \right)}_0+{\left(\frac{dH}{d\phi }\right)}_0·d\phi .$$

(31)

Considering Equations (29)–(31), we obtain

$$H\left(\phi \right)=H_0-\frac{K}{H_0^{1/3}\sqrt{2g{H}_0}}·f\left(\tau \right)d\phi ,$$

where

$$f\left(\tau \right)=\frac{{\tau }^{1/2}}{{\left(1-\tau \right)}^{4/3}}.$$

Averaging function $f\left(\tau \right)$ along the equipotential, we obtain

$$f\left({\tau }_{mean}\right)=\frac{\underset{{\tau }_{axis}}{\overset{{\tau }^{*}}{\int }}f\left(\tau \right)d\tau }{{\tau }^{*}-{\tau }_{axis}},$$

where ${\tau }_{axis}$ is the value $\tau$ on the flow symmetry axis, which is corresponding to equation

$${\phi }_0=\phi \left({\tau }_{axis},0\right);$$

${\tau }^{*},\theta$* correspond to equation

$${\phi }_0=\phi \left({\tau }^{*},{\theta }^{*}\right).$$

${\tau }^{*}$ is the value $\tau$ on the extreme stream-line; $\theta$* is the value $\theta$ on the extreme stream-line.

And finally, we obtain

$$H\left(\phi \right)=H_0-K_0·d\phi ,$$

(32)

where $K_0=\frac{K·f\left({\tau }_{mean}\right)}{H_0^{1/3}\sqrt{2g{H}_0}}.$

3. Results and Discussion

3.1. Description of the Experimental Facility and Flow Parameters

The experiment and the experimental facility are described in detail in [17]. The experimental facility had a form of a hydraulic horizontal trough, 250 cm wide and 830 cm long. Water from the tank with constant pressure was supplied to the receiving tank through a calibrated triangular spillway. The studies were performed on models of steel culverts of rectangular cross-section. The water current from the pipe was free-flowing, the flooding from the downstream pool was not modeled. The outgoing flow was measured using a triangular spillway which was installed at the outlet of the constant pressure tank. The flow spreading was studied at relative expansions of the downstream pool, $\beta =B/b_{pipe}=3÷7$, where B is the width of the flow.

Let us consider a model of a stationary, potential, 2D planar, open, high-velocity water flow of the ideal liquid, with free spreading back from the culvert. Data on spreading coordinates, depth and flow velocity are given in

Table 1. Experimental data on the coordinates of the water flow, its depth and velocity.

$X_E$ (cm)	4	24	44	64	71
$Y_E$ (cm)	11	38	59	76	80
V (cm/c)	151.928	186.461	191.243	192.714	191.49

and borrowed from [17].

The flow has the following parameters:

• initial flow velocity $U_0=147.654$ (cm/s);
• initial flow depth relative to the bottom $h_0=9.27$ (cm);
• acceleration of gravity $g=981$ (cm/s${}^2$);
• pipe width $b=16$ (cm);
• flow outgo at the outlet of the culvert $Q=2.19$ (cm${}^3$/s);
• relative flow expansion $\beta =5$;
• Froude number $F_0=2.397$.

Next, we find:

• hydrodynamic pressure (see Formula (2)) $H_0=20.382$ cm;
• initial flow kinecity according to the formula ${\tau }_0=\frac{U_0^2}{2g{H}_0}=0.545$;
• the length of the inertial front $X_D^I=3$ cm [19].

3.2. Calculation of Flow Parameters on the Symmetry Axis and along the Extreme Stream-Line, with Resistance Forces Not Considered

One of the solutions for System (26) is the flow functions and the potential function in the following form [20]:

$$\psi =\frac{A}{{\tau }^{\frac{1}{2}}}sin\theta ;\phi =A\frac{h_0}{H_0}\frac{cos\theta }{{\tau }^{\frac{1}{2}}(1-\tau )}.$$

(33)

Considering that $d\psi =0$ along the stream-line and $\theta =0$ on the flow symmetry axis, and (12), (33), we obtain the following ordinary differential equation associating $dx$, $d\tau$:

$$dx=\frac{{Ah}_0}{{2H}_0\sqrt{2g{H}_0}}\frac{(3\tau -1)}{{\tau }^2{(1-\tau )}^2}d\tau ,$$

(34)

where $A=\frac{U_{0}b}{2sin{\theta }_{max}}$ is denoted as a constant for the entire flow; $\tau$ is the flow kinecity parameter.

Integration of Equation (34), with consideration of the initial conditions $x=X_D^I$, $\tau ={\tau }_0$, allows the obtention of a dependence in the following form:

$$x=X_D^I+\frac{{Ah}_0}{{2H}_0\sqrt{2g{H}_0}}\left[\frac{1+\tau }{\tau (1-\tau )}-ln\frac{1-\tau }{\tau }-\frac{1+{\tau }_0}{{\tau }_0\left(1-{\tau }_0\right)}+ln\frac{1-{\tau }_0}{{\tau }_0}\right].$$

(35)

Thus, knowing the abscissa of the flow, one can obtain its kinecity ${\tau }_{axis}$ on the axis of symmetry.

The system of equations

$$\left\{\begin{array}{c}A\frac{sin\theta }{{\tau }^{1/2}}=sin{\theta }_{max};\hfill \\ \hfill \frac{cos\theta }{{\tau }^{1/2}(1-\tau )}=\frac{1}{{\tau }_{axis}^{1/2}(1-{\tau }_{axis})}\hfill \end{array}\right.$$

(36)

is to be solved in the plane of the velocity hodograph along the extreme stream-line to determine parameters $\theta$, $\tau$, where ${\tau }_{axis}$ is the flow kinecity on the axis of symmetry, ${\theta }_{max}$ is the maximum spreading angle. System (36) is to be solved as a cubic equation [17,20].

We suppose that the flow abscissa step is equal to 4 sm and determine the kineticity ${\tau }_{axis}$ of the flow at these points on the axis of symmetry using Equation (35). Knowing ${\tau }_{axis}$ from System (36), we find the kinecity parameter $\tau$ and the angle of inclination $\theta$ of the local velocity vector to the flow symmetry axis. Using Formula (13), we determine the value of the velocity and the depth along the extreme stream-line.

Table 2. Value kinetics ${\tau }_i$, abscissa $X_i$ and ordinates $Y_i$ of water flow, its velocity $U_i$ at points along the extreme stream-line of the flow with resistance forces not considered, as well as their integral values: $Y_{i,in}$, $U_{i,in}$.

No.	${\mathit{X}}_{\mathit{i}}$	${\mathit{\tau }}_{\mathit{i}}$	${\mathit{\tau }}_{\mathit{I},\mathit{i}\mathit{n}}$	${\mathit{Y}}_{\mathit{i}}$	${\mathit{Y}}_{\mathit{i},\mathit{i}\mathit{n}}$	${\mathit{U}}_{\mathit{i}}$	${\mathit{U}}_{\mathit{i},\mathit{i}\mathit{n}}$
1	0	0.545	0.545	8	8	147.654	147.654
2	4	0.767	0.623	9.029	9.029	175.145	157.847
3	8	0.866	0.776	13.637	13.637	186.071	176.206
4	12	0.908	0.843	18.687	18.687	190.54	183.655
5	16	0.931	0.881	23.973	23.973	192.908	187.653
6	20	0.945	0.904	29.402	29.402	194.354	190.121
7	24	0.954	0.92	34.926	34.926	195.322	191.576
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
17	64	0.983	0.97	92.257	92.257	198.304	194.809
18	68	0.984	0.972	98.085	98.085	198.406	197.176
19	72	0.985	0.974	103.922	103.922	198.497	194.907
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮

represents model values of kinecity ${\tau }_i$, abscissas $X_i$ and ordinates $Y_i$, its velocity $V_i$ along the extreme stream-line, with resistance forces not considered, as well as their integral values.

3.3. Consideration of Resistance Forces in the Calculation of Water Flow Parameters

To increase the adequacy of the model to the real process, we use the method of replacing the planar flow with a one-dimensional one considering the flow resistance forces [2,20].

The hydraulic friction coefficient for steel pipes is 0.02, which corresponds to the Shezy coefficient $C=65$ and the roughness coefficient $n=0.014\dots 0.017$.

Let us consider the Bernoulli integral taking into account the flow resistance forces

$$dH=-\frac{\lambda \left(x\right)}{2}·F\left(x\right)·dx,$$

(37)

where $\lambda$ is the coefficient of hydraulic friction to the flow determined by the Manning formula:

$$\lambda \left(x\right)=\frac{8g{n}^2}{{h\left(x\right)}^{1/3}};$$

$F\left(x\right)=\frac{U{\left(x\right)}^2}{gh\left(x\right)}$ is the Froude number; H is the total hydrodynamic pressure (see Expression (2)); x is the longitudinal coordinate of the considered flow point; h is the average depth in the cross-section of the flow; g is the acceleration gravity; U is the average velocity in the cross-section of the flow; n is the roughness coefficient of the channel bottom.

Flow continuity Equation (38) is

$$BUh=B_0{U}_0{h}_0,$$

(38)

where B is the width of the flow stream; $B_0,U_0,h_0$ are the values of parameters $B,U,h$ at the culvert outlet of the flow.

The system of Equations (37) and (38) is a system with three unknowns. Using the trapezoid rule, Equation (37) is transformed into the following form [17]:

$$\frac{U_0^2}{2g}+h_0-\frac{{U\left(x\right)}^2}{2g}-h\left(x\right)=\frac{1}{4}x\left[{\lambda }_0{F}_0+\lambda \left(x\right)F\left(x\right)\right].$$

(39)

At numerical integration, function $f\left(x\right)=\lambda \left(x\right)F\left(x\right)=\frac{8n^2{U\left(x\right)}^2}{{h\left(x\right)}^{4/3}}$ is assumed to be monotonic, which is true for the section of increasing velocities and decreasing depths. That is, the algorithm admits correcting the parameters of the flow when it expands up to $\beta =$ 3 ÷ 5 in the region far from the expansion of the flow close to the limit expansion.

As a result, we obtain a system of equations for two unknowns $F\left(x\right)$ and $h\left(x\right)$:

$$\frac{F_0{h}_0}{2}+h_0-\frac{F\left(x\right)h\left(x\right)}{2}-h\left(x\right)=2n^{2}gx\left(\frac{F_0}{h_0^{1/3}}+\frac{F\left(x\right)}{h^{1/3}\left(x\right)}\right);$$

(40)

$$B\left(x\right)=\frac{b_0{U}_0{h}_0}{h\left(x\right)\sqrt{F\left(x\right)gh\left(x\right)}}.$$

System (40) is solved under the condition of minimizing the functional

$${\left(U_i-U\left(x\right)\right)}^2+{\left(B_i-B\left(x\right)\right)}^2+{\left(h_i-h\left(x\right)\right)}^2\to min.$$

Table 3. Model values of ordinates and velocities water flow at the controlled points without considering resistance forces $Y_i$, $V_i$ and considering resistance forces $Y_{R,i}$, $U_{R,i}$. Values of errors in calculating these parameters relative to experimental data $\delta Y$, $\delta U$.

${\mathit{X}}_{\mathit{E},\mathit{i}}$	${\mathit{Y}}_{\mathit{E},\mathit{i}}$	${\mathit{Y}}_{\mathit{i}}$	${\mathit{Y}}_{\mathit{R},\mathit{i}}$	$\mathit{\delta }\mathit{Y}$, %	${\mathit{U}}_{\mathit{E},\mathit{i}}$	${\mathit{U}}_{\mathit{i}}$	${\mathit{U}}_{\mathit{R},\mathit{i}}$	$\mathit{\delta }\mathit{U}$, %
4	9	9.029	9.028	0.316	151.928	175.145	157.84	3.891
24	38	34.926	34.94	8.052	186.461	195.322	191.576	2.743
44	59	63.3	63.305	7.297	191.243	197.505	194.678	1.796
64	76	92.257	92.259	21.393	192.714	198.304	194.809	1.087
71	80	103.922	103.909	29.886	194.49	198.497	194.907	0.21

represents the model values of ordinates $Y_{R,i}$ and velocities $U_{R,i}$ of the water flow at the controlled points, with resistance forces taken into account, as well as errors in calculating these parameters relative to experimental data. The following programs are used in the calculations [21,22,23].

Analyzing the data in Table 3, the following conclusions can be drawn:

• Consideration of resistance forces has little effect on the flow ordinates. However, the error in calculating the flow velocity decreases when resistance forces are taken into account.
• There is a significant error in calculating the flow ordinates at point $X=64$ and $X=72$. This can be explained by the transition of the flow to the area of limit expansion. The rest of the values are within the acceptable error.
• The results obtained have a significantly smaller error in comparison with the results obtained in [9,10,12].

4. Conclusions

A system of closed Equations (26) and (32) is proposed, which allows, in the first approximation, the consideration of the resistance forces to the flow. Their consideration is to use the integral laws of resistance. Friction forces are taken into account only at the bottom of the flow, and tangential stresses at the jet interfaces are ignored. Therefore, the resistance force components are expressed in terms of the velocity components. In this case, the flow equations can be solved by the method of characteristics. The expressions for the components of the resistance forces are adopted similarly to those already used by S. N. Numerov and F. I. Frankl. This way of representing the resistance forces reduces the surface forces to equivalent volume ones, as it is assumed in [2,3]. Methods for solving System (26) at a constant pressure $H_0$ are provided in [17,19,20].

It is proved that at the first approximation, it is possible to consider the flow resistance forces, assuming that the value of its hydrodynamic pressure decreases downstream of the flow, and the form of the equations of motion in the plane of the velocity hodograph does not change. When the roughness coefficient is equal to zero, from the resulting System (26), there follows a system of equations in the plane of the velocity hodograph without considering the flow resistance forces.

In the model for considering the flow resistance forces, the hydrodynamic pressure is assumed to be unchangeable along the equipotential, but it changes only along the stream-line downstream of the flow. It is proposed to divide the flow into separate successive parts downstream. At each segment of the flow, the hydrodynamic pressure is assumed to be constant. As a result, a piecewise continuous solution of the problem is obtained, as a whole. This assumption agrees well theoretically with the previously known results on the flow of 2D water flows [2,5,13,18].

An example of calculating the water flow is given. The experimental facility and characteristics of the studied water flow are described (Table 1). The kineticity, ordinates, and velocities of the flow along its extreme line are calculated without consideration of resistance forces, as well as their integral values (Table 2).

An algorithm for considering resistance forces in calculating the parameters of the water flow is given. Corrected values of ordinates and flow velocities are obtained. Experimental data errors are calculated. The admissible accuracy is obtained in the region far from the expansion of the flow close to the limit (Table 3).

This paper represents the experimental data obtained by the Department of Hydraulic Engineering and Melioration of the Don State Agrarian University. This work is the development of analytical methods for solving problems of technical mechanics of liquids and gases [1,17,19], using a technique for solving nonlinear problems similar to those proposed in [24,25,26,27,28,29,30].