CC BY
2Magazine of Civil Engineering. 2024. 17(1). Article No. 12508
Magazine of Civil Engineering issn
2712-8172
journal homepage: http://engstroy.spbstu.ru/
Research article UDC 626.31
DOI: 10.34910/MCE.125.8
Determining parameters of high-velocity open water flow
O.A. Burtseva1 , S.I. Evtushenko2 B , V.N. Kokhanenko1
1 Platov South-Russian State Polytechnic University (NPI), Novocherkassk, Russian Federation
2 Moscow State University of Civil Engineering (National Research University), Moscow, Russian Federation
Keywords: mathematical model, 2D water flow, equations of motion, resistance forces, open-channel hydraulics, analytical solution, simple wave, characteristics method
Abstract. In the planar hydraulics, the problems with previously unknown boundaries are most difficult. The best adequacy in terms of flow parameters is provided by simplified analytical methods based on a potential flow model. A mathematical model of a stationary, potential, 2D planar high-velocity open water flow of an ideal fluid, freely spreading back from a non-pressure orifice is studied. The boundary problem of flow free spreading in plane were formulated. Studying the system of dimensionless equations of motion resulted in identification of the criteria influencing the process of flow spreading. A critical analysis was carried out and a description of various methods for solving the problem of free spreading of a high-velocity water flow was given. The problem in an analytical form was solved in the velocity hodograph plane. All flow parameters are determined in the physical plane. For the first time, the conjugating flow "simple wave" was applied. The proposed analytical method for solving the problem of flow free spreading is effective, unambiguous and has no singularities and discontinuities, particularly, at the outlet of a non-pressure pipe. The adequacy of the mathematical model was verified on a test example. The relative error of the flow parameters does not exceed 10 % compared to the experimental data.
Citation: Burtseva, O.A., Evtushenko, S.I., Kokhanenko, V.N. Determining parameters of high-velocity open water flow. Magazine of Civil Engineering. 2024. 17(1). Article no. 12508. DOI: 10.34910/MCE.125.8
1. Introduction
In modern hydraulic engineering construction, the hydraulic structures for passing water from elevated areas to lower ones are frequently used. These can be large hydro power plants, road drainage channels, spillways, junctions of various channels for changing flow parameters, small bridges that pass water flows during river floods or high water.
Due to inaccurate or irrational modeling in the construction of hydraulic structures, the operational reliability of a structure as a whole decreases; this causes collapses of fastenings of culverts under roads, small bridges under railways. Improper design of spillways leads to environmental disasters in Russia. There is a practical need for a reliable method for calculating the parameters of a water flow.
It was revealed that the methods of I.A. Sherenkov and G.A. Lilitsky, referred to as the most famous, accessible and described in the reference literature ones, do not always give results with sufficient accuracy for practical calculations when we compared the experimental and calculated contours of the border streamlines of flow free spreading downstream of rectangular cross-section pipes according to these methods.
Among the many tasks in the hydraulics of planar flows, the tasks with previously unknown flow boundaries are the most difficult. At the outlet of the flow from the pipe, the best adequacy in terms of flow
© Burtseva, O.A., Evtushenko, S.I., Kokhanenko, V.N., 2024. Published by Peter the Great St. Petersburg Polytechnic University.
parameters is represented by simplified analytical methods based on the model of the potential flow of the current. Further, the flow resistance forces increase and it is necessary to make a transition to numerical methods.
However, the fastening of the structure is carried out precisely at the culvert outlet for a flow, where an analytical solution may be enough. If the boundary value problem is immediately solved by numerical methods, then due to the characteristics of the problem (discontinuity of parameters in the pipe outlet area), the adequacy of the solution of the problem decreases. Therefore, first of all, it is necessary to use analytical methods as a basis for further use of numerical methods.
Therefore, there is a need for a simple analytical method for calculating the parameters of the water flow, which allows obtaining sufficient adequacy in terms of its parameters. Next, we consider a stationary, potential, high-velocity 2D planar open water flow of an ideal fluid freely spreading downstream of non-pressure culverts (hereinafter referred to as the water flow). Such a flow is characterized by local averaged velocities in depth and local depths at each point in the flow. The presence of velocities perpendicular to the planar flow plan distinguishes 3D flows from 2D planar flows. 2D water flows are studied and analyzed mathematically much easier than 3D spatial flows.
N. Bernadsky and V. Makkaveev were the first to set the problem of planar hydraulics. They also developed an approximate solving method for calm (precritical) flows. The theory of 2D planar flows was further developed in the works of Russian researchers N. Meleshchenko, G. Sukhomel, I. Levi, S. Numerov,
F. Frankl, I. Sherenkov, B. Emtsev, A. Tursunov, N. Kartvelishvili, M. Mikhalev, V. Lyakhter, L. Vysotsky and foreign scientists A. Ippen, H. Rauz, D. Harleman, D. Dowson, R. Knapp, D. Liggett, T. Akatai, etc. The theory and methods for solving planar hydraulic problems are most fully described in monographs of
G. Sukhomel, I. Levi, B. Yemtsev [1], I. Sherenkov [2], A. Yesin [3] and V. Kokhanenko [4].
There are two main types of engineering problems in the 2D planar hydrodynamics [2]. The first type, or the direct problems, includes the configuration of the planar channel, i.e. the shape of its banks, in addition to the bottom surface (topography). To supplement these data with known values of velocities and depths in one of the boundary cross-sections, it is necessary to find the form of the free surface and velocity distribution within the selected section of the channel. The second type of problems is the inverse problems, where the law of estimating certain hydraulic parameters is given and other flow parameters are found as well as geometric characteristics of the bed (relief of bottom) forming the indicated flow. At the same time, it is necessary to take into account the flow's dynamic features and properties. The authors of this paper consider the second type of problems.
The mathematical basis of the problems mentioned above are systems of quasi-linear partial differential equations. The analytical solution is difficult to obtain in most cases due to the complexity of the motion. Therefore, a numerically analytical approach to solving such problems is considered promising.
Furthermore, the use of approximate methods of solution is recommended. The most famous approximate solution method is the method of characteristics developed by S. Chaplygin and borrowed from gas dynamics [4].
Sherenkov's method based on using a universal graph [2] constructed by means of the characteristics method is considered to be one of the most known approaches to determining parameters of a high-velocity flow. However, this method was not adequate enough for practical use. The discrepancy between the calculated and experimental values reached 50 % [2]. The characteristic method was further developed in works of V. Kokhanenko and his students.
In 1997, V. Kokhanenko [4] proposed the idea of an analytical method for determining flow parameters using the velocity hodograph plane, where such natural coordinates were proposed: t - flow kineticity (the square of the flow velocity coefficient) and 9 - angle of direction of the velocity vector in relation to symmetry axis OX of the flow. As the unknown functions we considered the potential function 9 = ^(t, 9) and the current function y = y (t, 9). The basic system of differential equations of motion
for the flow in the plane of the variables t, 9 has become linear, allowing a wide set of solutions to be defined.
E. Duvanskaya [5] calculated hydraulic parameters considering the friction forces and the slope of the bottom of 2D stationary precritical flows for solving problems at designing melioration networks and road structures.
This idea was further developed in the works of N. Kosichenko [6]. She set the problem about free flow spreading, solved it and indicated the limits of applicability of the results of solving of the planar problem. She proposed an analytical method for calculating the geometry of the flow spreading area and its parameters inside and at the boundary of the flow area.
N. Papchenko has developed new mathematical methods for modeling 2D planar flows. He determined a set of previously unknown analytical solutions for systems of 2D planar flows in the plane of the velocity hodograph. He founded solutions of the boundary problem of a free flow spreading both in the plane of velocity hodograph, and in the physical plane of the flow by both analytical and numerical methods [7].
The main contribution of D. Kelekhsaev to theory of planar flows was the definition of the inertial front length formula at the pipe outlet [8].
Substantial development of an analytical method for determining flow parameters was presented in the works of O. Burtseva and M. Aleksandrova [9-10]. They have formulated a boundary problem of free spreading of a planar flow. The system of equations of motion of water flow in dimensionless form was obtained in two ways. Criteria influencing the process of flow spreading were revealed and their universality was proved. The problem has been solved in the analytical form in the velocity hodograph plane. The transition to the physical plane made it possible to determine all parameters of plane water flow.
Improvement of the accuracy of the analytical method by splitting the flow scheme into four main sections: uniform flow, "simple wave", section limited by the characteristics of the 1st family and the radial flow section. Conjugation the sections and obtaining an adequate mathematical model for determining the parameters of a high-velocity open water flow.
2.. Methods
2.1. Motion equations of a 2D planar water flow in the physical region of a flow
(OXY)
The initial physical assumptions for the 2D planar water flow model are [1]:
a) vertical components of local averaged velocities and accelerations are small;
b) velocity vectors of liquid particles located on the one vertical line are located in the same plane;
c) velocity distribution on any vertical is almost uniform.
In practice, in most open water flows the assumptions a), b), c) are satisfied by the high-velocity flows, the transverse dimensions of which are several times greater than the flow depth and there are no return currents [11-13].
Background of the paragraph c). The uneven distribution of open flow velocities along the vertical is conditioned by the retarding effect of the channel bottom. Herewith, a boundary layer develops which spreads its influence up to the free surface. However, the velocity distribution in the boundary layer has an asymptotic character and its conditional thickness is decreased with an increase in the flow kineticity. Therefore, in high-velocity flows (Froude number F > 1), the longitudinal component of the velocity at some point becomes almost equal to the surface one at a small distance from the bottom. Hence, there appears an almost uniform distribution of velocities along the vertical [18].
To obtain the dynamic equations of motion of a 2D open planar flow N.M. Bernadsky [1] suggested to proceed from L. Euler's equations, supplemented by components considering the flow resistance forces. For the flows satisfying the assumptions (a-c), the form of the motion equations for the average values of parameters in terms of depth and time coincides with the L. Euler's equations for ideal liquid supplemented by the flow resistance force per mass unit of the liquid. This model has been obtained artificially but it has the right to both theoretical and practical use.
The system of dynamic equations of motion of a non-stationary open 2D planar flow has the form
[1, 3]:
du du d / , \ _
^+uy +* dX(z* + *)+r* = 0;
(1)
duy duy dt \
" + S ~dT ( ** + h ) + Ty = a
dx dy dy
where ux, Uy are projections of the local velocity vector on the axes of a Cartesian coordinate system -OXY; z* is the mark of the channel bottom; h is flow depth; g is acceleration of gravity; Tx, Ty are the projections of resistance forces to the flow, referred to the unit mass of the liquid.
In the case of a flat horizontal channel bottom zg = 0, without taking into account the flow resistance forces Tx = Ty = 0, from (1) there follows the system of equations:
u„
dux
dx du,
■ + u
du*
y dy du
ux — + uy'
dx
+g %(h)=0; ay+ g dy( h )=0.
(2)
y
Adding to the system (2) the flow continuity equation, which has a specific form (3) for 2D open water planar flows [1]:
d(uxh)j(uyh)
dx
dy
= 0
and the flow potentiality equation
Q =
duv dux
= 0
dy dx
we obtain a closed system of equations relatively to variables ux, Uy, h.
2.2. Boundary value problem of flow free spreading in plane The system of equations describing the flow is as follows:
(3)
(4)
dux dx du„
du.
dh
+uy+g— = 0; dy dx
du.
- + u,
dh n
+g— = 0;
c)x y c)y " dy d(uxh) d(uyh)
(5)
dx
dy
= 0,
duv duy
dy dx
= 0.
One can go from ux, uy to the local velocity vector V with modulus V = ^u2x + u2 the angle 9 characterizing the direction of the velocity vector to the longitudinal axis of flow symmetry [4, 18]. The boundary conditions for a flow: 1) at the outlet of a non-pressure rectangular pipe:
x = 0; -2<y <b; h = h; v = V,; e = 0,
(6)
where b is the width of the culvert; h§, Vo are the depth of the flow and rate of its velocity at the outlet of the pipe into a diversion channel;
2) along the boundary streamline
y = f (x); yX = tga a = 0, (7)
a is wave angle;
3) at
x h ^ 0; V ^ K" (8)
where Vmax, 9max are maximum flow velocity and spreading angle.
2.3.
Research of the boundary value problem without its complete solution
Reducing the system of motion equations of 2D planar water flow and boundary conditions to dimensionless form.
Obtaining a system of motion equations of a water flow in a dimensionless form is necessary to identify dimensionless criteria that influence the process of flow spreading, and to answer the question, whether the type of the system and the boundary value problem are universal in dimensionless coordinates as a whole.
As a basis, we take the system of equations (5). Let us consider various variants for bringing it to a dimensionless form.
Variant I. We introduce the following dimensionless coordinates and dimensionless flow parameters
- x _ y - h - ux - uy x =-; y =—; h = —; Vx =—; Vy , b b h/ x V0 y V0
where x, y are coordinates of the flow.
After transition in the system (5) to the dimensionless quantities x, y, h, Vx, Vy, we obtain the following system:
f
F
F
Vt •
dx
+ Vr •■
dy
_ V - V
3 = 0;
j dx
\
Vv •■
- + Vr •-
dx dy j
d(Vx • h) a(Vy • h)
=0,
dy
(9)
dx
dy
= 0;
V dVy
where F0 =
VL
gk)
dy dx , is Froude criterion of a flow at the outlet from of a pipe.
Analyzing the system (9), we see that the first and the second equations are not reduced to a universal form as they contain the dimensionless parameter Fo. The third and the fourth equations have a universal form. Therefore, without solving the boundary value problem, we can say that its solution will depend on the Froude criterion Fo of a flow at the outlet of a pipe and is not reduced to a universal form1.
In case of the boundary value problem is universal, it can be solved one time. For the different boundary conditions, we can use only recalculation of the parameters describing the problem in dimensionless parameters.
Variant II. Further we show that from I.A. Sherenkov's graph [2] in coordinates
- y — x
y=b x=bFo'
for the system (5) that he used, and for the boundary conditions (6), there follows a transformation:
1 A universal form is a form of the boundary value problem in the dimensionless coordinates, which does not depend on any dimensionless criteria.
h = -; V,
V x
V
Vq Kx
Vy
Vy
Vq Ky
The system of equations (5) is not reduced to the universal dimensionless form in this case:
the first three equations at Kx = ■
1 1
Ky = — are reduced to a universal form, except the fourth
F0
one.
- dV- -V-•—x + V-
Vx dx Vy
- V _ Vx —- + Vy
x dx y
dVx dh n x + — = 0;
dy dVy
dx
y +ddh = 0;
dy dy
d(Vx • h) s(v- • h)
dx
dVx
dy
= 0;
1 dvy = q.
dy Fq dx
The boundary values (6-8) cannot be reduced to universal dimensionless form at the outlet of the pipe (6). That means that the problem of free spreading of the 2D flow cannot be reduced to the dimensionless form. Therefore, the universal graphic can be used only with the Froude criterion values close to 1 (see Fig. 1).
/
3 A
\
2
bjF,
0.5
1,5
1.5
3.5
Figure 1. Graphs of boundary streamlines: 1 - according to the method of G.A. Lilitsky;
2 - according to the method of I.A. Sherenkov; 3 - according to the experimental data
at F0 = 2.184.
2.3.1. Methods for solving problem of free spreading of a water flow
I.A. Sherenkov's method. He developed a method based on the use of universal graphic with the additional D. Bernoulli equation for the total hydrodynamic pressure. The graphic was given in dimensionless coordinates, and at the first approximation, it allowed the design organizations to obtain the entire spectrum of flow parameters. However, the adequacy of this solution was rather low and it made the researchers of 2D planar water flows search for new ways to solve the problem. The development of numerical solution of the problem based on integral equation system has not been brought to possibility of practical use in hydraulic construction.
G.A. Lilitsky's method is not universal, it is based on the experimental data processing.
The most promising method of solving problems may be the analytical method using the velocity hodograph plane [4]. However, it has its own drawbacks.
2.4. Description of the analytical methods using the velocity hodograph plane
According to S.A. Chaplygin's method of [10], the system of equations (5) is converted to the variables t and 9 by using a complex differential connection between the physical plane Y(XY) and the
velocity hodograph plane T(t, 9) :
d ( x + iy ) = —
1
1/2
V2iH
d 9 + i
h>
H o (1 -T)
d y
je
(10)
where t is flow kineticity; 9 is angle characterizing the direction of the velocity vector to the longitudinal axis of symmetry of the flow; i = •vTI is the imaginary unit. We obtain a system of equations [1, 4]:
Ö9 _ h0 5T =
3t- 1
2ho t(1 -t)
2 ho
dy
2 "de;
dy
(11)
de
H 0 1 - t st
where 9 = 9(1,9) is the potential function; y = y(x,9) is the current function; the hydro-dynamic pressure is
^2 o_
2g
Vit V Hn = — + h = — + h = H ; 2g
the flow kineticity parameter is
t =
V2
2 gH 0'
the local depth and velocity (average over the live section) is:
h = H0 (1 -T); V = T1l2y/2gH^.
(12)
(13)
The system of equations (11) is already a linear system admitting to obtain analytical solutions. The authors in [4, 9, 18] chose from these solutions the following solution:
.sin e .ho
y = A•-j/^; 9 = A
cos 9
Ho t12 (1 -T):
(14)
where the constant A is determined according to the theory described in [4, 5]:
A = ■
Vob
2sin em
here b is the width of the culvert.
Therefore, by solving the problem in the plane of the velocity hodograph and using (11), we can define the solution over the entire spectrum of flow parameters in the flow plane.
The main necessary requirements to perform boundary conditions of the problem of free flow spreading at t ^ 1 are as follows:
h ^o; V ^VmaX =V2gH~ h = Ho (1 -t) .
(15)
There are two appropriate types of flows to fulfill conditions (15): radial spreading of the flow and flow of the type (14).
The additional condition is flow spreading with a free surface curved from the bottom towards the atmosphere along the channel (the axis of symmetry of the flow). This condition follows from experimental studies of flow spreading. The flow of type (14) also corresponds to this condition but the radial spreading of the flow does not. This is why the authors have chosen the scheme (Fig. 2) and the common type flow (14).
However, solution (14) satisfies the boundary conditions on infinity at t but it does not satisfy
the boundary conditions at the outlet of the pipe at 9 = 0, t0 =
V2
2 gH 0
An,Mn
Mo
0
x
AqMq, are characteristics of the 2nd family;
Mq, M, • • • Mn are characteristic of the first family; OX -flow symmetry axis
Figure 2. Scheme of combining a uniform flow and the flow (14): I - steady-state flow in the pipe;
II - simple wave; III - general flow.
The authors tried to eliminate this contradiction by searching for other types of solutions and by switching from a straight edge of the outlet pipe to a curvilinear shape as well. The authors obtained a sufficient adequacy to the real process in terms of the form of the border streamline (but not in all parameters) [5-7]. A detailed study of the theory of 2D planar water flows provided the following conclusion [10, 19]: a uniform flow in a pipe can be combined with a flow of the general type to which the flow belongs (14), only by a simple wave. This is why when using the scheme of free spreading of the flow (region 2, see Fig. 2) it satisfies the boundary conditions at the outlet of the pipe.
2.5. Determination of flow parameters in various areas
Let us consider the region of flow spreading in the plane of the velocity hodograph, see Fig. 3. Here are the following indications:
Mq (xq,0) is the point corresponding to the region I of a uniform flow current (see Fig. 2) with
parameters Tq, 9 = Q. On the velocity hodograph plane the point Mq corresponds to the entire region I of the uniform flow current.
MqMu is the characteristic of the first family, the conjugation line of the flow of the general form (14) III and the simple wave flow (see Fig. 2).
n Ty
Figure 3. Flow spreading region in the velocity hodograph plane.
1. Determining the boundaries of the uniform flow area.
Point M0 in the sector I (see Fig. 2) belongs to the characteristic of the 1st family with parameters t0, 6 = 0. The wave angle at this point is
a0 = arcsin ^
1 -T
0
2to
AM0 =
b
2tgao
m K A0__
j b/2 L ' 0 1 x I : \ j ao II KMo
i b/2 \ L XD i .X .................1.
K x a'o
Figure 4. Boundaries of the of uniform flow area.
Since M0AO and M0Aq are straight lines, the pentagon is K'OKAqMqAO in Fig. 4 is symmetrical
about the OX axis. Its upper part is a rectangular trapezoid. Thus, with known distance xD, the geometry and kinematics of sector l of the flow spreading are determined.
To determine the parameter XD we use the formula given in [8]
X d = trunc
VFO^ï
sin ömax (F0 + 2)
ho
+1,
(16)
where 1 is the depth measuring unit of Hq. In calculations this is 1 cm, in natural experiments it is 1 m.
Formula (16) was derived from the compilation of the structural formula and the determination of the coefficients by the regression analysis method. To derive this formula we used information from 70 experiments, some of which was published in [4].
The flow parameters Vq, Hq, Tq in the closed region I are constant and are defined by formulas (12) and (13).
2. Determining flow parameters in terms of the MQMn characteristic. The angle of inclination of the first family characteristicsin the plane of the velocity hodograph has the following form [4, 18]:
e = V3 •arctg^
3t-1
- arctg
3t-1
A
J 1 -t,
C,
'3 (1 -T)
where the constant C' is found from the condition of the characteristic's passing through the point Mq :
(17)
C ' = arctg i3^—1 - V3 • arctg Vi -T
3t0 -1
V3V1 -
'0 ,
The angle 9 takes the maximum value at the boundary condition
h = H0(1 -x)^0; V = t1^2gH0 ^2gH0; e^e
and is equal to
emax = C ' + (V3-l)-2.
Note that at t ^ 1 according to the scheme (Fig. 2), the boundary streamline (free boundary) tends to the characteristic MQMn and the points An and Mn approach each other, since the wave angle tends to zero:
1-T
an = arcsin-n ^ 0.
n i 2Tn
The parameters t, 9 on the characteristic of the 1st family are also parameters of the "simple wave" flow. In a simple wave the characteristics of the second family are straight lines [1].
Given the parameter Tm e[to,1], we define the corresponding angle 9m by formula (17) and calculate the depths and velocities by formulas (13).
2. Determining the coordinates of the points Mi along the characteristic of the first family, see Fig. 5.
The specific volumetric flow rate coefficient of an arbitrary streamline is equal to:
Q = Kb, 0 < K < 1, 2
where K is the flow rate coefficient.
To determine the kineticity Tm along the characteristic of the 1st family we substitute function (17) into the streamline equation (14) and solve it [18]:
sin 9m
TM
= K sin 9maX. (18)
We find Tm and then 9m .
To find the coordinates of points Mt, we use the relation equation (10) between the physical plane Y(XY) and the plane of the velocity hodograph r(r, 9). Separating the imaginary and real parts in (10) and considering that d y = 0 along the streamline, we obtain
^ d 9 a
(19)
T' 1/2gH 0
d y
r- V2gH0
From the equation (18) we have
sin 9m = K • TM sin 9max. (20)
Then, by differentiating both parts of the expression (20) and omitting the index "m", we obtain:
dy = ,1/2 ^ ^ sin 9. ■1/2,
1 dT
cos 9d 9 = — K •—¡^ sin 9max (21)
2 Vt
or
1 ^ dt sin 9max
d 9 = — K -max. (22)
2 Vt cos 9
To determine d<p we use the expression for the potential function in equations (14)
9 = A-^-,^. (23)
^ H0 t (1 -T) ( )
Differentiating 9 = 9 (t,9) , we obtain
d 9 =
Aho
H
- sin 0d0 +cos 0(3t-1) dT
7-1/2
(1 -T)
(24)
(25)
2t32 (1 -t)2
Considering expressions (20)-(24) we can transform the system (19) to:
[ dx = f1 (t) d t; \dy = f2 (t)dT
where f (t), f (t) are the functions of the flow kinetics parameter defined after transformations.
Integrating the system (25) at the initial values we obtain the values of the coordinates of the selected flow point:
X = XM„ + T fl(T) dt; Y = YM„ + T f2 (t) dT.
Thus, we find the coordinates of the point M that is located of the intersection of the streamline with the flow rate coefficient K and the characteristics of the first family in the planar flow.
3. Determining flow parameters in the flow region III (see Fig. 2).
We have to solve the system of equations in the plane of the velocity hodograph to determine the parameters 9, t at the point of intersection of an arbitrary streamline and an arbitrary equipotential:
. sin 0 .
A~\h~ = K Sin 0max;
T '
cos 0
cos 0,
(26)
T12 (1 -T) 72 (1 -Tm )
V2,
where K is given; 9m, Tm are parameters at the point of intersection of the characteristics of the first
family with the streamline determined by the flow rate coefficient K. The solution of the system (26) is reduced to the solution of the cubic equation [4].
4. Determining the parameters on the rectilinear characteristics of the second family, in a simple wave II and determining the coordinates of the points A1, A2,..., An (Fig. 2).
Parameters Tm , 9m change along the characteristic of the 1st family, but not in simple waves -
section II (from the characteristic of the 1st family up to the flow boundary). In other words, the boundaries of the flow and characteristics of the 1st family have the same look in the velocity hodograph plane [1, 4].
Since the characteristics of the 2nd family are straight lines with constant parameters Tm , 9m , they will have the same values of the parameters ta = Tm, 9a = 9m in the points Aq, A1,..., An of their intersection with the flow boundary. As soon as the value of the kinetic parameter ta is determined it is possible to find the velocity and depth of the flow at the point At using the formulas (13).
To construct the boundary streamline, we have parameters Ta, 9a, and the direction of the segments AtAt+1 should be taken along the corresponding streamline passing through these points on the characteristic, with the wave angle taken into account [4].
ai = arcsin
1 — t,
"A
2t
(27)
i j
0
To determine the coordinates of the points of the boundary streamline, we write a system of equations in which the first equation determines a pencil of lines passing through the point Mi, and the
second equation is the equation of a circle centered at the point Mt.
The coordinates of the boundary streamline point are the solution of this system of equations, i.e. intersection point of a pencil of lines and a circle.
Omitting the index "i+1", we have
\Ya-Ym) = tg(0-a)(Xa -Xm);
2 2 2 (28)
(XA - XM ) + (YA - YM ) =pAM • Obviously, the following conditions must be met:
Y a > Y\4 > — ; Y A > Y. ; X A > X . ; X . > X . .
A M 2 4+1 A- ' A A' A+1 A
To determine the coordinates of points lying on the boundary streamline we solve the system of equations (28). As a result, we obtain:
YA = YM + k \XA - XM |;
Pam (29)
X A = X\4 —
Vi+k2
where k = |tg (9-a)|.
Let us consider a model of a non-stationary potential 2D planar open high-velocity water flow of an ideal fluid with free spreading back of a culvert, as a test example. The validation of the model is performed on the basis of real experimental data published in [4]. Data on spreading coordinates, depth and flow velocity are given in Table 1.
Table 1. Experimental data on the coordinates of the water flow, its depth and velocity.
Xe (cm) 9 24 44 64 71
Ye (cm) 9 38 59 76 80
V (cm/s) 151.928 186.461 191.243 192.714 194.49
The flow at the outlet of the pipe has the following initial parameters:
- initial flow velocity (at the outlet of a pipe) Vo = 147.654 (cm/s);
- initial flow depth relative to the bottom ho = 9.27 (cm);
- acceleration of gravity g = 981 (cm/s2);
- pipe width b = 16 (cm);
- flow rate at the outlet of a culvert Q = 2.19 -104 (cm3/s);
- relative flow expansion P = 5.
3. Results and Discussion
3.1. Parameters in the area of uniform flow and on the characteristic of
the 1st family
According to the above presented algorithm, we find:
- Froude criterion Fo = 2.397 (see formula (16));
- hydrodynamic head Ho = 20.382 (cm);
- initial flow kinetics T0 = 0.545;
- length of the inertial front XID = 3 (cm) [8];
- wave angle at the outlet of the pipe is aQ = 0.702 radian or a0 = 40°23;
- angle of inclination of the fluid flow velocity vector to the OX axis at infinity 9max = 0.981 radian or emax = 56°23;
- distance along the flow symmetry axis from the end of the inertial section to the
M0 : AM0 = 9.457 (cm);
- straight line length of a segment of the characteristic of the second family between the points Aq and M0 • AqM0 = 12.387 (cm).
We divide the width of the flow pipe b/ 2 into 40 parts with step length AY = 0.2. We find the flow parameters at the points of intersection of a particular streamline with the characteristic of the first family, see equation (18). Table 2 shows the calculated values of the flow coefficient K, kineticity Tm, the angle
of inclination of the velocity vector eM (radian), abscissa Xm (cm) and ordinate Ym (cm) at the points on the characteristic of the 1st family.
Table 2. Values of the flow parameters on the characteristic of the 1st family.
No. step K TMi XMi YMi
0 0 0.545 0 0 0
1 0.25 0.559 0.016 12.633 4.162 • 10-4
2 0.05 0.574 0.031 12.899 1.77 -10-3
3 0.075 0.588 0.048 13.185 4.234*10-3
28 0.7 0.926 0.594 39.34 6.273
29 0.725 0.936 0.623 43.235 8.757
33 0.825 0.972 0.742 73.489 47.873
34 0.85 0.979 0.774 89.655 85.193
Next, we constructed a grid in the general flow region and the parameters are determined according to the equations (26). We do not present them here.
3.2. Determining the coordinates of points along the boundary streamline Ai Since the parameters Tm, eM do not change in simple waves, we assume that
A = tm > e A = e M.
We calculate the wave angle a( radian) (see expression (27)) for segments of the straight lines
AMi. From the system of equations (29) we find the radii of the circles p( cm) with centers at the points
Mt and define the coordinates of the points Ai on the boundary streamline.
The adequacy of the model was evaluated in terms of the error of the abscissa, ordinate, and velocity compared to the experimental data. The calculation results are summarized in Table 3.
Table 3. Values of the flow parameters along the boundary streamline.
No. step ai Pi XAi Xe SX, % YAi Ye SY, % VAl Ve sv , %
0 0.702 12.387 0 0 8 8 0 147.654
1 0.678 12.989 9.545 9 9.457 9 0.455 149.561 151.928 1.558
2 0.656 13.303 10.643 10.324 151.45
3 0.634 13.64 11.75 11.458 153.321
28 0.201 61.37 23.337 24 2.761 35.771 38 6.233 192.435 186.461 3.204
29 0.185 70.962 25.767 5.671 58.644 0.607 193.504 191.243 3.067
33 0.121 158.398 46.495 44 8.654 73.13 59 3.924 197.109 192.714 2.649
34 0.105 212.027 58.462 64 9.855 89.332 76 10.446 197.82 194.49 2.034
Figure 5. The graphs of the boundary streamlines obtained by the presented method (solid line) and experimental data (line labeled with circles), as well as the characteristic curves of the 1st family (line labeled with squares).
Fig. 5 shows the curves of the extreme streamline obtained by the presented algorithm (solid line) and experimental data (labeled with circles), as well as the characteristic curve of the 1st family (labeled with square markers).
Detailed development of programs was published by the authors in [22, 23].
This work develops the methods for the analytical study of problems of technical fluid and gas mechanics [1, 18-21], using a technique for solving nonlinear problems similar to those proposed in [2427].
4. Conclusions
1. The article formulates a mathematical model of a 2D high-velocity planar flow with justification and consideration of some physical assumptions. The boundary problem of free spreading of the flow in terms of the flow has been set. A system of equations for the motion of a water flow in a dimensionless form has been obtained in two ways. The criteria influencing the process of flow spreading were revealed, their versatility was proved. The conclusion is drawn that the known methods based on Sherenkov's universal graph do not provide sufficient adequacy for the solution to the free spreading water flow problem.
2. The theory in the article agrees with the main points of the theory of conjugation of 2D potential flows. The proposed method for solving the problem includes both previously known and new points. The novelty is the division of the flow into three main sections: a uniform flow, a general flow and a conjugating flow of a simple wave. The use of simple waves to determine the coordinates of the extreme line of the water flow is also new.
3. Using Chaplygin's method made it possible to obtain an analytical method for solving the problem of flow free spreading in the plane of the velocity hodograph. The method is effective, unambiguous and has no singularities and discontinuities at the outlet of a pipe.
4. A test case was calculated. The parameters of the flow and its coordinates on the characteristic of the 1st family and the boundary streamline at the points coinciding with the natural experiment are given (Table 2 and Table 3). Relative errors are calculated. A more complete calculation is presented in the form of a graph in Fig. 5. The adequacy of the obtained flow model to the natural experiment was proved. The calculation error is less than 10 %, which is quite acceptable.
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Information about authors:
Olga Burtseva, PhD in Technical Sciences ORCID: https://orcid.ora/0000-0003-4288-8709 E-mail: [email protected]
Sergej Evtushenko, Doctor of Technical Sciences ORCID: https://orcid.ora/0000-0003-3708-380X E-mail: [email protected]
Viktor Kokhanenko, Doctor of Technical Sciences ORCID: https://orcid.ora/0000-0002-9962-3966 E-mail: [email protected]
Received 18.05.2021. Approved after reviewina 18.12.2023. Accepted 10.01.2024.
