How the speed of the current changes on a river. Movement of fluid through pipes

In the previous paragraphs, the laws of equilibrium of liquids and gases were discussed. Now let's look at some phenomena associated with their movement.

The movement of fluid is called with the current, and a collection of particles of a moving fluid is a stream. When describing the movement of a liquid, the speeds at which liquid particles pass through are determined. this point space.

If at each point in space filled with a moving fluid, the speed does not change with time, then such motion is called steady, or stationary. In a stationary flow, any fluid particle passes through a given point in space with the same speed value. We will consider only the steady flow of an ideal incompressible fluid. Ideal called a liquid in which there are no frictional forces.

As is known, a stationary liquid in a vessel, according to Pascal’s law, transmits external pressure to all points of the liquid without change. But when a fluid flows without friction through a pipe of variable cross-section, the pressure at different places in the pipe is not the same. The pressure distribution in a pipe through which liquid flows can be assessed using an installation schematically shown in Figure 1. Vertical open pressure gauge tubes are soldered along the pipe. If the liquid in the pipe is under pressure, then in the pressure tube the liquid rises to a certain height, depending on the pressure at a given place in the pipe. Experience shows that in narrow areas of the pipe the height of the liquid column is less than in wide areas. This means that there is less pressure in these tight spots. What explains this?

Let us assume that an incompressible fluid flows through a horizontal pipe with a variable cross-section (Fig. 1). Let us mentally select several sections in the pipe, the areas of which we will denote by and . In a steady flow, equal volumes of liquid are transferred through any cross section of a pipe over equal periods of time.

Let be the velocity of the fluid through the section, and let be the velocity of the fluid through the section. Over time, the volumes of liquids flowing through these sections will be equal to:

Since the fluid is incompressible, then . Therefore, for an incompressible fluid. This relationship is called the continuity equation.

From this equation, i.e. fluid velocities in any two sections are inversely proportional to the cross-sectional areas. This means that liquid particles accelerate when moving from the wide part of the pipe to the narrow part. Consequently, a certain force acts on the liquid entering the narrower part of the pipe from the liquid still in the wide part of the pipe. Such a force can only arise due to the pressure difference in different parts of the liquid. Since the force is directed towards the narrow part of the pipe, the pressure in the wide section of the pipe should be greater than in the narrow section. Taking into account the continuity equation, we can conclude: during stationary fluid flow, the pressure is less in those places where the flow speed is higher, and, conversely, it is greater in those places where the flow speed is lower.

D. Bernoulli was the first to come to this conclusion, which is why this law is called Bernoulli's law.

Application of the law of conservation of energy to a flow of moving fluid allows us to obtain an equation expressing Bernoulli’s law (we present it without derivation)

- Bernoulli's equation for a horizontal tube.

Here and are the static pressures and the density of the liquid. Static pressure is equal to the ratio of the pressure force of one part of the liquid on another to the area of ​​contact when the speed of their relative motion is zero. This pressure would be measured by a pressure gauge moving with the flow. A stationary monometric tube with an opening facing the flow will measure pressure

The average depth velocity is the ratio of the hodograph area to the maximum river depth. The area of ​​the hodograph can be calculated either from the palette, or by calculating the area of ​​the living cross-section of the river (see task 2).

Task 2

Determine the open cross-sectional area of ​​the river using the data in Table 8:

Table 8

Cross-sectional depth of the river

Option I		Option II
	River depth, m	Distance from the permanent start of the target, m	River depth, m

The living cross-sectional area of ​​a river is calculated as the sum of a number of elementary geometric figures (Fig. 9).

The figures A 1 A 2 B 1 and A 5 B 4 A 6 are triangles, the area of ​​each of them is equal to half the product of the base and the height. The remaining figures are trapezoids. The area of ​​each trapezoid is equal to the product of half the sum of the bases and the height.

Rice. 9. Cross section of the river

Points A 1, A 2, A 3, etc., at which depth measurements were carried out, are called measuring points. The starting point from which measurements A 1 are made is called the permanent beginning of the alignment.

Task 3

Calculate the water flow in the river if it is known that the open cross-sectional area is 42.2 m2, the maximum water speed in the river is 0.5 m/s, and the average depth of the river is 4.5 m.

Calculation of the average river speed based on the maximum surface speed is carried out using the formula:

,

where, V av - average speed; V max - maximum speed, K - coefficient of transition of maximum speed to average. The coefficient K is presented in table. 9.

Table 9

Values ​​of the coefficient of transition from maximum speed to average

Task 4

Determine using the Chezy formula (
, Where WITH speed coefficient, R– hydraulic radius, i– the average slope of the river), the average speed of the river, if it is known that in a given section the bottom of the channel is composed of sandy material, there are islands and shoals. The average slope of the river is 0.000056, hydraulic radius is 1.8 m.

The speed coefficient C in the Chezy formula is determined by the Bazin formula
.

The roughness coefficient y is determined according to table 10.

When a liquid moves in a round pipe, the speed is zero at the pipe walls and maximum at the pipe axis. Assuming the flow to be laminar, we find the law for the change in speed with distance from the pipe axis.

Let us select an imaginary cylindrical volume of liquid of radius and length l (Fig. 77.1). In a stationary flow in a pipe of constant cross-section, the velocities of all fluid particles remain unchanged. Consequently, the sum of external forces applied to any volume of liquid is equal to zero. The bases of the cylindrical volume under consideration are subject to pressure forces, the sum of which is equal to This force acts in the direction of fluid movement. In addition, on lateral surface the cylinder is subject to a frictional force equal to (This refers to the value at a distance from the axis of the pipe). The stationarity condition has the form

The speed decreases with distance from the pipe axis. Therefore, it is negative and Taking this into account, we transform relation (77.1) as follows:

Dividing the variables, we get the equation:

Integration gives that

The integration constant must be chosen so that the speed becomes zero on the pipe walls, i.e. - the radius of the pipe).

From this condition

Substituting the value of C into (77.2) leads to the formula

The speed value on the pipe axis is equal to

Taking this into account, formula (77.3) can be given the form

Thus, in laminar flow, the speed changes with distance from the pipe axis according to a parabolic law (Fig. 77.2).

In turbulent flow, the speed at each point changes in a random manner. At constant external conditions The average (over time) speed at each point of the pipe section turns out to be constant. The average velocity profile for turbulent flow is shown in Fig. 77.3. Near the walls of the pipe, the speed changes much more strongly than with laminar flow, but in the rest of the section the speed changes less.

Assuming the flow is laminar, we calculate the fluid flow Q, i.e., the volume of fluid flowing through the cross section of the pipe per unit time. Let's divide the cross-section of the pipe into rings of width (Fig. 77.4). A volume of liquid will pass through the radius ring in a second equal to the product of the area of ​​the ring and the flow velocity at points located at a distance from the axis of the pipe.

Taking into account formula (77.5), we obtain:

To obtain the flux Q, you need to integrate expression (77.6) over the range from zero to R: i 9

Pipe cross-sectional area). From formula (77.7) it follows that in laminar flow the average (over the cross section) value of the velocity is equal to half the value of the velocity at. pipe axis.

Substituting into (77.7) the value (77.4) for

We obtain the formula for the flow

This formula is called Poiseuille's formula. According to (77.8), the fluid flow is proportional to the pressure drop per unit length of the pipe, proportional to the fourth power of the pipe radius and inversely proportional to the viscosity coefficient of the liquid. Recall that Poiseuille's formula is applicable only for laminar flow.

Relation (77.8) is used to determine the viscosity of liquids. By passing a liquid through a capillary of known radius and measuring the pressure drop and flow Q, one can find

Average flow velocities vary along the length of the river due to the variability of the cross-sectional dimensions of the channel. In a particular transverse section, the average velocity is found by averaging local velocities measured at individual points of the flow along the depth and width of the river. In turn, local velocities at different points of the flow differ significantly from each other. They are usually larger at the surface than at the bottom, and at the banks, on the contrary, they are smaller than in the middle part of the river.

This distribution is strongly influenced by the shape of the cross-section of the channel and the conditions of water movement in the area.

The presence of vegetation or other additional roughness at the river bottom leads to a decrease in bottom water flow velocities. The formation of ice cover on the free surface in winter creates additional resistance to water movement. As a result, the surface current velocities decrease, and the maximum velocities move into the flow thickness. This leads to the fact that the average speeds in the cross section of the river in winter also decrease compared to in summer time, other things being equal.

To analyze the distribution of local flow velocities along the living cross section, in practice they are measured at individual points along the flow depth on a whole series high-speed verticals, outlined along the width of the river. In Fig. Figure 4.4 shows the cross-sectional profile of the river bed with measured flow velocities on the verticals. IN in this example current velocities were measured in 5 points along the flow depth. The river profile shows isotachs – lines of equal velocities in the cross section of the channel.

The top part of the construction shows diagram distribution of average flow velocities on verticals along the width of the river, and the dotted line is the value of the average current speed over the open section.

Based on measurements of water flow velocities at individual points along the depth of the flow, it can be constructed diagram their vertical distribution. An example of such a construction is shown in Fig. 4.5. The vertical axis on this graph shows the distances from the free surface of the water to the speed measurement points on a scale, and the horizontal axis shows the values ​​of these speeds. The average vertical speed is usually at a distance 0.4h, counting from the bottom of the river.

In each specific case, the distribution of flow velocities vertically and across the width of the channel depends on the conditions of water movement in the area. Typically, the maximum surface flow velocities and the highest average current velocities on verticals are observed in the area of ​​maximum depths in the living section of the channel. On riffles, the diagram of average current speeds is aligned with the width of the river compared to the reaches of the valleys. The greatest uneven distribution of velocities across the width of the river is observed in areas where the channel turns. In this case, the maximum flow velocities are concentrated near the concave - pressed bank of the river. In Fig. Figure 4.6 shows diagrams of the distribution of vertical average flow velocities in the riffle section of the river.

Rice. 4.6. Distribution of average current speeds

on a riffle section of the river

An analysis of the distribution of flow speeds across the width of the river shows that at the core of the flow, in the deepest part of the channel, the actual flow speeds of water are always greater than the average for the live cross section.

Therefore, when performing technical and economic calculations, the concept is introduced operating current speed, the value of which can be found from the following relationship:

, (4.8)

Where: Vav – average flow velocity along the living cross section in the river section under consideration, m/s;

D.V.– the difference between the flow speed on the axis of the navigation channel and the average speed along the open section in a given river section, m/s.

The average current speed can be determined using the Chezy formula or based on field measurements. Current speeds in the river are measured with special instruments - hydrometric meters(Fig. 4.7) or by launching floats. Determine the value of a quantity D.V. direct measurements along an extended section of the river seem to be very difficult.

Rice. 4.7. Hydrometric turntable:

1 – blades; 2 – body; 3 – tail section;

4 – rod; 5 – electrical terminals

In practice, the operating speed for a separate section of the river is determined by measuring the speed of the vessel relative to the shore when following the current Vin and against the tide Vвв according to the formula

. (4.9)

For approximate calculations it is often taken

Knowing the operating speed of the current, you can find the speed of the vessel relative to the shore:

when moving downstream

, (4.11)

when moving upstream

, (4.12)

Where: Vс – speed of the vessel in calm water (in the absence of current), m/s.

The obtained values ​​of vessel speeds are used in practice when planning cargo delivery times and drawing up dispatch schedules.

SEE MORE:

When constructing many engineering structures on rivers, it is necessary to know the amount of water flowing in a particular place per second, or, as they say, water flow. This is necessary to determine the length of bridges, dams, as well as for irrigation and water supply.

Water flow is usually measured in cubic meters per second. Water flow during high water is very different from flow during low water, that is, at low summer levels. Table 7 shows the flow rates for some rivers as an example.

If we mentally cut the river across the flow, we get the so-called “living cross-section” of the river. The distribution of flow velocity across the living cross-section of the river is very uneven. The speed of the flow is influenced by the depth of the channel, its shape, and the obstacles that the river encounters along its path, for example, a bridge support, an island, etc.

Usually the speed is lower near the banks, but in the middle, in the deeper part of the river, the speed is much higher than in the shallow part. In the upper part of the flow, the speeds are greater, and the closer to the bottom, the less. On a flat section of the river, the highest speed is usually somewhat below the surface of the water, but sometimes the highest speed is also observed on the surface.

If the current encounters an obstacle, for example, a bridge support or an island, then the highest speeds can move closer to the bottom of the river. On oxbow lakes during high water, velocities near the bottom drop to zero.

Figure 14 shows the distribution of current velocities along the live cross-section of the Volga near Saratov during high water. The speed on the surface in the left arm is 1.3 per second, and in the right arm 1.7 per second. Over the island, which is covered with water during high water, the speeds drop to 0.5 per second. At the river bottom, velocities drop to 0.4. In summer, the highest speed in this section in the main channel was no more than 0.4 per second.

Along the river, speeds can also vary greatly depending on the contours of the live section. For example, fourteen kilometers below Saratov, near Uvek, where the channel has no islands and is constrained by dams, during high water the surface speed reached 3 per second, while at Saratov the speed was up to 1.8 per second.

In deep places on the river, which are called reaches, the living cross-section is larger. In shallow places or riffles, the living cross-section is much smaller. Since in a short section along the length of the river the water flows are equal, and the cross-sections on the reach are larger than on the rift, the flow speeds will be different: in a deep place the water flows quietly, but on the rift it flows much faster.

The speed of the current also depends on the slope of the flow, bottom roughness and depth. The greater the slope, the smoother the bed and the more regular its outline, the higher the flow speed. Approximate speed values ​​on rivers are shown in Table 8.

The table shows "average speed". This speed is determined by dividing the water flow by the cross-sectional area of ​​the river. The highest surface speed is usually one and a half times greater, and the bottom speed is one and a half times less than the average speed.

The science of hydrometry deals with measuring the speed and flow of river water.

The speed of water flow can be measured in a very simple way.

To do this, you need to measure a certain distance along the shore, at least in steps, set marks and throw a float or just a piece of wood into the water slightly above the top mark. The time it takes the float to travel from one mark to another is measured by a clock with a second hand. By dividing the distance between the marks by the time the float floated from one mark to the next, we obtain the surface velocity of the flow at that location.

During surveys, the passage of the floats is detected with a special goniometer tool.

The most accurate way to measure speed is with hydrometric meters (Fig. 15). These turntables on a metal rod (at depths up to 4) or on a cable (at any depth) are lowered from specially equipped vessels into the water to different depths. As soon as the turntable makes a certain number of revolutions, the electrical wires in it close, current flows through the turntable, and a short bell is produced at the top. The time interval between individual calls corresponds to a certain flow speed. By lowering the turntable lower and lower, you can measure velocities throughout the entire depth of the river at a given vertical.

The water flow on the river is calculated as follows. At each of 10–20 verticals located across the flow at the same distance from each other, the average flow speed is determined, which is then multiplied by the cross-sectional area of ​​the river between the verticals. The individual private costs obtained in this way between the verticals are added up. The sum gives the total flow of the river, expressed in cubic meters per second.

In conclusion, we will provide some information about fording rivers.

Wading can be done, depending on the speed, at different depths. As a rule, at a speed of 1.5 you can wade at a depth of 1, on horseback at a depth of 1.2, and by car at a depth of 0.5. At speed 2, you can wade at a depth of 0.6, cross the river on horseback at a depth of 1, and by car at a depth of 0.3. If the water is still, greatest depth for wading is determined only by the height of the person and the design of the machine.

There are several ways to measure river speed. This can be done by solving mathematical problems, when there is some data, and this can be done by applying practical actions.

River flow speed

The speed of the current depends directly on the slope of the riverbed. The slope of the channel is the ratio of the difference in heights of two sections, points to the length of the section. The greater the slope, the greater the speed of the river flow.

You can find out what the speed of a river's current is by sailing a boat upstream and then downstream. The speed of the boat with the current is V1, the speed of the boat against the current is V2. To calculate the river flow speed you need (V1 - V2): 2.

To measure the speed of water flow, a special lag device is used, a pinwheel, consisting of a blade, body, tail section, and rotor.

There's another one the simplest way how to find the speed of a river.

You can measure 10 meters upstream, in steps. Your height will be more accurate. Then make a mark on the bank with a stone or branch and throw a sliver of wood into the river above the mark. After the sliver reaches the mark on the shore, you need to start counting the seconds. Then divide the measured distance of 10 meters by the number of seconds over this distance. For example, a sliver traveled 10 meters in 8.5 seconds. The river flow speed will be 1.18 meters per second.

Elements of the water regime and methods of observing them

(according to L.K. Davydov)

Under the influence of a number of reasons, which will be discussed below, water flows in rivers, the position of the level surface, its slopes and flow speeds change. The cumulative change in water flow rates, levels, slopes and flow velocities over time is called the water regime, and changes in flow rates, levels, slopes and speeds individually are called elements of the water regime.

Water flow (Q) is the amount of water that flows through a given living section of a river per unit time. The flow rate is expressed in m3/s. Water level (H) is the height of the water surface (in centimeters), measured from some constant comparison plane.

Observations of levels and methods for their processing

Observations of level fluctuations are carried out at water measuring posts (Fig. 73) and consist of measuring the height of the water surface above a certain constant plane, taken as the initial, or zero. Such a plane is usually taken to be a plane passing through a mark slightly below the lowest water level. The absolute or relative elevation of this plane is called the zero of the graph, in excess of which all levels are given.

Rice. 73. Pile water-measuring station (a) and water level reading using a portable rail (b).

Measurements are made using a water measuring rod with an accuracy of 1 cm. There are two types of rods - permanent and portable. Permanent slats are attached to bridge abutments or to a pile driven into the bottom of the riverbed near the bank. With flat banks and large amplitudes of level fluctuations, observations are carried out using a portable staff. To do this, a number of piles located in the alignment are driven into the river bed and floodplain.

The marks of the pile heads are connected by leveling with a water-measuring station benchmark installed on the shore, the absolute or relative mark of which is known. The water level is measured using a portable rod mounted on the pile head. Knowing the elevation of the head of each pile, it is possible to express all measured levels in excesses above the zero surface, or the zero of the graph. Observations at water measuring posts are usually carried out 2 times a day - at 8 and 20 o'clock. During periods when levels change rapidly, additional observations are made at 1, 2, 3 or 6 hours throughout the day. For continuous recording of levels throughout the day, level recorders are used, a description of which can be found in a hydrometry textbook (V.D. Bykov and A.V. Vasiliev). There you can also get acquainted with the automatic regime recording (water level and temperature) hydrological post. Go to automated system observations speeds up the acquisition of hydrological information and increases the efficiency of its use.

Based on all measurements, average levels for each day are calculated and tables of daily average levels for the year are compiled. In addition, these tables contain the average levels for each month and year and select the highest and lowest levels for each month and year.

The average, highest and lowest levels are called characteristic levels. Level observation data are published in the USSR in special publications—hydrological yearbooks. In the pre-revolutionary period, these data were published in “Information on water levels on the inland waterways of Russia based on observations at water measuring posts.”

Based on daily observations of levels, graphs of their fluctuations are constructed, giving a clear idea of ​​the level regime for given year.

Methods for measuring river flow velocities

River flow velocities are usually measured using either floats or gauges. In some cases, the average speed for the entire living section is calculated using the Chezy formula. The simplest and most commonly used floats are made of wood. Floats are thrown into the water on small rivers from the shore, on large rivers - from a boat. Using a stopwatch, the time t for the passage of the float between two adjacent targets, the distance l between which is known, is determined. The surface speed of the current is equal to the speed of the float

More accurately, current velocities are measured using a hydrometer. It allows you to determine the average flow velocity at any point in the flow. There are different types of turntables. In the USSR, the modernized hydrometric turntables of Zhestovsky and Burtsev GR-21M, GR-55, GR-11 are currently recommended for use.

When measuring speeds, a turntable on a rod or cable is lowered into the water to various depths so that its blades are directed against the current. The blades begin to rotate, and the faster the flow speed increases. After a certain number of revolutions of the turntable axis (usually 20), using a special device, light or beep. The number of revolutions per second is determined by the time interval between two signals.

The turntables are calibrated in special laboratories or at the factories where they are manufactured, i.e., a relationship is established between the number of rotations of the turntable blade per second (n rev/s) and the flow speed (v m/s). From this dependence, knowing n, we can determine v. Measurements with a pinwheel are made on several verticals, at several points on each of them.

Methods for determining water flow

Water flow in a given open section can be determined by the formula

Where v is the average speed for the entire living section; w is the area of ​​this section. The latter is determined as a result of measurements of the depths of the river bed along a transverse section.

Using the above formula, the flow rate is calculated only if the speed is determined using the Chezy formula. When measuring velocities with floats or a turntable on individual verticals, the flow rate is determined differently. Let the average speeds for each vertical be known as a result of measurements. Then the scheme for calculating water consumption is reduced to the following. Water flow can be represented as the volume of a water body - a flow model (Fig. 76 a), limited by the plane of the living section, the horizontal surface of the water and the curved surface v = f (H, B), showing the change in speed along the depth and width of the flow. This volume, and therefore the flow rate, is expressed by the formula

Since the law of change v = f(H,B) is mathematically unknown, the flow rate is calculated approximately.

Rice. 76 Scheme for calculating water consumption. a — flow model, b — partial flow.

The flow model can be divided by vertical planes perpendicular to the open section area into elementary volumes (Fig. 76 b). The total flow rate is calculated as the sum of partial flow rates AQ, each of which passes through a part of the open section area wi, contained between two speed verticals or between the edge and the vertical closest to it.

Thus, the total flow rate Q is equal to

where K is a variable parameter that depends on the nature of the coast and varies from 0.7 to 0.9. In the presence of dead space K = 0.5.

The average speed for the entire living section at a known water flow rate Q is calculated by the formula vcр =Q/w.

Other methods are also used to measure water flow, for example mountain rivers The ion flood method is used.

Detailed information on determining and calculating water flow rates is presented in the hydrometry course. There is a certain relationship between water flow rates and levels, Q - f(H), known in hydrology as the water flow curve. A similar empirical curve is presented in Fig. 77 a.

It was based on measured water flows in the river during the ice-free period. The points corresponding to winter water flows lie to the left of the summer curve, since the flows measured during freeze-up Qwinter (at the same level height) are less than summer QL. The decrease in flow rates is a consequence of an increase in the roughness of the riverbed due to ice formations and a decrease in the open cross-sectional area. The relationship between Qwin and Ql, expressed by the transition coefficient

It does not remain constant and changes over time with changes in the intensity of ice formations, ice thickness and the roughness of its lower surface. The course of changes in Kzim=f(T) from the beginning of freezing to opening is shown in Fig. 77 b.

The flow curve makes it possible to determine the daily flow of river water based on known levels observed at water metering stations. For the ice-free period, using the curve Q = f(H) does not cause any difficulties. Daily costs during freeze-up or other ice formations can be determined using the same curve Q = f(H) and the chronological graph Kzim = f/(T), from which Kzim values ​​are taken for the desired date:

QZIM = Kzim Ql

There are other ways to determine winter costs, for example, using the “winter” flow curve, if it can be constructed.

In a number of cases, the unambiguity of the water flow curve is also violated during the ice-free period. This is most often observed when the channel is unstable (alluvium, erosion), as well as when variable backwater occurs caused by a discrepancy between the levels of a given river and its influx, the operation of hydraulic structures, overgrowing of the channel with aquatic vegetation and other phenomena. In each of these cases, one or another method for determining daily water flows is chosen, as set out in the hydrometry course.

Based on daily water consumption data, you can calculate the average consumption for a decade, month, or year. The average, highest and lowest expenses for a given year or for a number of years are called characteristic expenses. Based on daily flow data, a calendar (chronological) graph of water flow fluctuations, called a hydrograph, is constructed (Fig. 78).

Rice. 78. Hydrograph.

River flow mechanism

(according to L.K. Davydov)

Movement is laminar and turbulent

In nature, there are two modes of fluid movement, including water: laminar and turbulent. Laminar movement is parallel to the jet. With a constant flow of water, the velocities at each point of the flow do not change over time, either in magnitude or in direction. In open flows, the speed from the bottom, where it is zero, smoothly increases to its greatest value at the surface. Movement depends on the viscosity of the fluid, and the resistance to movement is proportional to the speed to the first power. Mixing in a flow is of the nature of molecular diffusion. The laminar regime is characteristic of underground flows flowing in fine-grained soils.

In river flows the movement is turbulent. Characteristic feature The turbulent regime is the pulsation of speed, i.e. its change over time at each point in magnitude and direction. These velocity fluctuations at each point occur around stable average values, which are usually used by hydrologists. The highest velocities are observed on the surface of the flow. Towards the bottom they decrease relatively slowly and in the immediate vicinity of the bottom they still have quite large values. Thus, in a river flow, the speed at the bottom is practically non-zero. Theoretical studies of turbulent flow note the presence of a very thin boundary layer at the bottom, in which the velocity sharply decreases to zero.

Turbulent motion is practically independent of the viscosity of the fluid. Resistance to movement in turbulent flows is proportional to the square of the speed.

It has been experimentally established that the transition from laminar to turbulent mode and back occurs at certain relationships between the velocity vav and the depth Hav of the flow. This relationship is expressed by the dimensionless Reynolds number

denominator (ν) - coefficient kinematic viscosity.

For open channels, the critical Reynolds numbers at which the motion mode changes vary approximately within the range of 300-1200. If we take Re = 360 and the coefficient of kinematic viscosity = 0.011, then at a depth of 10 cm the critical speed (the speed at which laminar motion becomes turbulent) is 0.40 cm/s; at a depth of 100 cm it decreases to 0.04 cm/s. Low values ​​of critical speed explain the turbulent nature of water movement in river flows.

According to modern concepts (A.V. Karaushev and others), inside a turbulent flow in different directions and with different relative speeds elementary volumes of water (structural elements) of different sizes move. Thus, along with the general movement of the flow, one can notice the movement of individual masses of water, leading, as it were, an independent existence for a short time. This obviously explains the appearance on the surface of a turbulent flow of small funnels - whirlpools, quickly appearing and just as quickly disappearing, as if dissolving in total mass water. This also explains not only the pulsation of speeds in the flow, but also the pulsations of turbidity, temperature, and concentration of dissolved salts.

The turbulent nature of water movement in rivers causes mixing of the water mass. The intensity of mixing increases with increasing flow speed. The phenomenon of mixing is of great hydrological importance. It helps to equalize the temperature, concentration of suspended and dissolved particles along the live cross-section of the flow.

Rice. 65. Examples of a curved water surface flow. a - screaming support, b - decline curve (according to A.V. Karaushev).

Movement of water in rivers

Water in rivers moves under the influence of gravity F'. This force can be decomposed into two components: parallel to the bottom Fx and normal to the bottom F’y (see Fig. 68). The force F' is balanced by the reaction force from the bottom. The force F'х, depending on the slope, causes the movement of water in the stream. This force, acting constantly, should cause acceleration of movement. This does not happen, since it is balanced by the resistance force that arises in the flow as a result of internal friction between water particles and friction of the moving mass of water against the bottom and banks. Changes in slope, bottom roughness, narrowing and widening of the channel cause a change in the ratio driving force and resistance forces, which leads to changes in flow velocities along the length of the river and in the living section.

The following types of water movement in streams are distinguished: 1) uniform, 2) uneven, 3) unsteady. With uniform movement of the flow velocity, open cross-section, and water flow rate are constant along the length of the flow and do not change over time. This kind of movement can be observed in channels with a prismatic cross-section.

With uneven movement, the slope, speed, and open section do not change in a given section over time, but change along the length of the flow. This type of movement is observed in rivers during low water periods with stable water flows in them, as well as under conditions of backwater formed by a dam.

Unsteady motion is one in which all the hydraulic elements of the flow (slopes, velocities, open cross-sectional area) in the area under consideration change both in time and in length. Unsteady movement is typical for rivers during floods and floods.

With uniform movement, the slope of the flow surface I is equal to the bottom slope i and the water surface is parallel to the leveled bottom surface. Uneven movement can be slow or accelerated. As the flow slows down the river, the curve of the free water surface takes the form of a backwater curve. The surface slope becomes less than the bottom slope (I< i), и глубина возрастает в направлении течения. При ускоряющемся течении кривая свободной поверхности потока называется кривой спада; глубина убывает вдоль потока, скорость и уклон возрастают (I >i) (Fig. 65).

Rice. 68. Scheme for deriving the Chezy equation (according to A.V. Karaushev).

Water flow velocities and their distribution over the open section

Flow velocities in rivers are not the same at different points of the flow: they vary both along the depth and width of the living section. On each individual vertical lowest speeds are observed near the bottom, which is due to the influence of the roughness of the riverbed. From the bottom to the surface, the increase in velocity at first occurs quickly and then slows down, and the maximum in open flows is reached at the surface or at a distance of 0.2H from the surface. Curves of vertical velocity changes are called hodographs or velocity diagrams (Fig. 66). The vertical distribution of velocities is greatly influenced by unevenness in the bottom topography, ice cover, wind and aquatic vegetation. If there are irregularities at the bottom (hills, boulders), the velocities in the flow in front of the obstacle sharply decrease towards the bottom. Velocities in the bottom layer decrease with the development of aquatic vegetation, which significantly increases the roughness of the channel bottom. In winter, under the ice, especially in the presence of slush, under the influence of additional friction on the rough lower surface of the ice, the speeds are low. The maximum speed shifts to the middle of the depth and is sometimes located closer to the bottom. Wind blowing in the direction of the current increases speed at the surface. With the opposite relationship between the direction of the wind and the current, the velocities at the surface decrease, and the position of the maximum shifts to a greater depth compared to its position in calm weather.

Along the width of the flow, both surface and average velocities on the verticals change quite smoothly, basically repeating the distribution of depths in the live section: near the coast the speed is lower, in the center of the flow it is highest. A line connecting points on the river surface to at the highest speeds, is called a rod. Knowing the position of the rod has great value when using rivers for water transport and timber rafting. A visual representation of the distribution of velocities in a live section can be obtained by constructing isotahs - lines connecting points with the same velocities in a live section (Fig. 67). The region of maximum velocities is usually located at some depth from the surface. The line connecting the points of individual live sections along the length of the flow with the highest velocities is called the dynamic axis of the flow.

Rice. 66. Velocity diagrams. a - open channel, b - in front of an obstacle, c - ice cover, d - sludge accumulation.

The average vertical velocity is calculated by dividing the area of ​​the velocity diagram by the vertical depth or in the presence of measured velocities at characteristic points in depth (VPOV, V0.2, V0.6, V0.8, VDON) using one of the empirical formulas, for example

Average speed in the live section. Chezy formula

To calculate the average flow velocity in the absence of direct measurements, the Chezy formula is widely used. She has next view:

where Hav is the average depth.

The value of coefficient C is not a constant value. It depends on the depth and roughness of the riverbed. There are several empirical formulas for determining C. Here are two of them:

Maning's formula

formula of N. N. Pavlovsky
where n is the roughness coefficient, found according to special tables of M.F. Sribny. The variable indicator in Pavlovsky's formula is determined by dependence.

From Chezy's formula it can be seen that the flow velocity increases with increasing hydraulic radius or average depth. This happens because with increasing depth, the influence of bottom roughness on the velocity value at individual vertical points weakens and thereby the area on the velocity diagram occupied by low velocities decreases. An increase in the hydraulic radius also leads to an increase in the coefficient C. From the Chezy formula it follows that the flow velocity increases with increasing slope, but this increase is less pronounced during turbulent motion than during laminar motion.

Flow speed of mountain and lowland rivers

The flow of lowland rivers is much calmer than that of mountain rivers. The water surface of lowland rivers is relatively flat. Obstacles flow around calmly, the curve of the backwater that appears in front of the obstacle smoothly mates with the water surface of the upstream area.

Mountain rivers are characterized by extreme unevenness of the water surface (foamy ridges, reverse faults, dips). Reverse faults occur in front of an obstacle (a pile of boulders at the bottom of the channel) or with a sharp decrease in the bottom slope. The upsurge of water in hydraulics is called a hydraulic (water) jump. It can be considered as a single wave that appears on the water surface in front of an obstacle. The speed of propagation of a single wave on the surface, as is known, is c = , where g is the acceleration of gravity, H is the depth.

If the average flow velocity vav of the flow turns out to be equal to the wave propagation speed or exceeds it, then the wave formed at the obstacle cannot propagate upstream and stops near the place of its initiation. A stopped wave of displacement is formed.

Let vav = c. Substituting the value from the previous formula into this equality, we obtain vav = , or

The left side of this equation is known as the Froude number (Fr). This number allows us to estimate the conditions for the existence of a stormy or calm flow regime: at Fr< 1 — спокойный режим, при Fr >1 - stormy mode.

Thus, the following relationships exist between the nature of the flow, depth, speed, and, consequently, slope: with an increase in slope and speed and a decrease in depth at a given flow rate, the flow becomes more turbulent; with a decrease in slope and speed and an increase in depth at a given flow rate, the flow becomes calmer.

Mountain rivers are characterized, as a rule, by rapid flows, while lowland rivers have a calm flow regime. A turbulent flow regime can also occur in rapids areas of lowland rivers. The transition to a rough flow sharply increases the turbulence of the flow.

Transverse circulations

One of the features of water movement in rivers is the non-parallel flow of currents. It clearly manifests itself on curves and is observed on straight sections of rivers. Along with the general movement of the flow parallel to the shores, there are generally internal currents in the flow, directed at different angles to the axis of movement of the flow and producing movements water masses in a direction transverse to the flow. The Russian researcher N.S. Lelyavsky drew attention to this at the end of the last century. He explained the structure of internal flows as follows. At the rod, due to the high speeds on the water surface, jets are drawn in from the side, as a result, a slight increase in level is created in the center of the flow. As a result, in a plane perpendicular to the direction of the flow, two circulation flows are formed along closed contours, diverging at the bottom (Fig. 69 a). In combination with forward motion, these transverse circulation currents take the form of helical movements. Lelyavsky called the surface current directed toward the core a faulty one, and the bottom diverging one a fan-shaped one.

In curved sections of the channel, jets of water, meeting a concave bank, are thrown away from it. The masses of water carried by these reflected jets, which have lower velocities, superimposed on the masses of water carried by the following jets impinging on them, increase the level of the water surface near the concave shore. As a result, a skew of the water surface occurs, and jets of water located near the concave shore descend along its slope and are directed in the bottom layers to the opposite convex shore. A circulation current occurs in curved sections of rivers (Fig. 69 b).

Rice. 69. Circulation currents on a straight (a) and on a curved (b) section of the channel (according to N. S. Lelyavsky). 1 - plan of surface and bottom jets, 2 - circulation currents in the vertical plane, 3 - helical currents.

The features of internal flows were studied by A.I. Losievsky in laboratory conditions. He established the dependence of the shape of circulation currents on the ratio of depth and width of the flow and identified four types of internal flows (Fig. 70).

Types I and II are represented by two symmetrical circulations. Type I is characterized by jet convergence at the surface and divergence at the bottom. This case is typical for watercourses with a wide and shallow channel, when the influence of the banks on the flow is insignificant. In the second case, bottom jets are directed from the shores to the middle. This type of circulation is typical for deep flows with high speeds. Type III with one-way circulation is observed in triangular-shaped channels. Type IV - intermediate - can occur during the transition from type I to type II. In this case, the jets in the middle of the flow can be converging or diverging, respectively, near the coast - diverging or converging. Further development ideas about circulation currents were obtained in the works of M. A. Velikanov, V. M. Makkaveev, A. V. Karaushev and others. Theoretical studies of the occurrence of these flows are presented in special courses on hydraulics and dynamics of channel flows. The appearance of transverse currents at channel bends is explained by the centrifugal force of inertia developing here and the associated transverse slope of the water surface. The centrifugal force of inertia arising on curves is not the same at different depths.

Rice. 70. Scheme of internal flows (according to A.I. Losievsky). 1 - surface jet, 2 - bottom jet.

Rice. 71. Scheme of the addition of forces causing circulation. a - vertical change in centrifugal force P1, b - excess pressure, c - resulting diagram of centrifugal and force acting on the vertical overpressure, g - transverse circulation.
At the surface it is greater, at the bottom it is less due to the decrease in longitudinal velocity with depth (Fig. 71 a).

Depending on the direction of the bend, the deflecting Coriolis force either strengthens or weakens the transverse flows at the bend. The same force excites transverse flows in straight sections.

At low levels at the curve, circulation currents are almost not expressed. As levels increase, speed and centrifugal force increase, circulation currents become distinct. The speed of transverse currents is usually small - tens of times less than the longitudinal component of the speed. The described character of circulation currents is observed before the water reaches the floodplain. From the moment water enters the floodplain, two flows are created in the river - an upper one, in the valley direction, and a lower one, in the main channel. The interaction of these flows is complex and still poorly understood.

Modern literature on the dynamics of channel flows (K.V. Grishanin, 1969) provides, apparently, a more rigorous explanation for the emergence of transverse circulations in a river flow. The origin of such circulations is associated with the mechanism of transmission of Coriolis acceleration to elementary volumes of water in the flow through a pressure gradient caused by the transverse slope (and constant vertically) and the difference in tangential stresses caused on the edges of elementary volumes of water by differences in vertical flow velocities.

A role similar to Coriolis acceleration is played by centripetal acceleration at a channel turn.

In addition to transverse circulation, vortex movements with a vertical axis of rotation are observed in the flow (Fig. 72).

Rice. 72. Diagram of vortices with vertical axes (according to K.V. Grishanin).

Some of them are mobile and unstable, others are stationary and have large transverse dimensions. More often they arise at the confluence of flows, behind steep coastal ledges, when flowing around some underwater obstacles, etc. The conditions for the formation of stationary vortices have not yet been studied. Grishanin suggests that the formation of a stable localized vortex is facilitated by the significant depth of the flow and the existence of an upward flow of water. These eddies in the flow, known as eddies, resemble air vortices- tornadoes.

Transverse circulations and eddy movements play a large role in the transport of sediment and the formation of river channels.

Hydraulic resistance.

When fluid flows through pipes, it has to expend energy to overcome the forces of external and internal friction. In straight sections of pipes, these resistance forces act along the entire length of the flow and the total energy loss to overcome them is directly proportional to the length of the pipe. Such resistances are called linear. Their magnitude (pressure loss) depends on the density and viscosity of the liquid, as well as on the diameter of the pipe (the smaller the diameter, the greater the resistance), flow speed (increasing speed increases losses) and the cleanliness of the inner surface of the pipe (the greater the roughness of the walls, the greater the resistance ).

In addition to friction in straight sections, additional resistance is encountered in pipelines in the form of flow turns, changes in cross-section, taps, branches, etc. In these cases, the structure of the flow is disrupted and its energy is spent on restructuring, turbulence, and impacts. Such resistances are called local. Linear and local resistances are two types of so-called hydraulic resistances, the determination of which forms the basis for the calculation of any hydraulic systems.

Fluid flow regimes. In practice, two characteristic fluid flow regimes are observed: laminar and turbulent.

In laminar mode, elementary streams of flow flow in parallel without mixing. If a stream of colored liquid is introduced into such a flow, it will continue its flow in the form of a thin thread among the flow of uncolored liquid, without being washed out. This flow regime is possible at very low flow rates. With an increase in speed above a certain limit, the flow becomes turbulent, vortex-like, in which the liquid within the cross-section of the pipeline is intensively mixed. With a gradual increase in speed, the colored stream in the stream first begins to oscillate about its axis, then breaks appear in it due to mixing with other streams, and then, as a result, the entire stream receives a uniform color.

The presence of one or another flow regime depends on the value of the ratio of the kinetic energy of the flow 1 1

(■п-гпi2=ч-рУу2) to the work of internal friction forces (/7 = р„5^/) - see. (2.9).

This is a dimensionless ratio

^-pVv21 (p,5^/) can be simplified keeping in mind that V is proportional to V. The quantities 1 and A/r also have the same dimension, and they can be reduced, and the ratio of the volume V to the cross section 5 is linear size y.

Then the ratio of kinetic energy to the work of internal friction forces, up to constant factors, can be characterized by a dimensionless complex:

which is called the Reynolds number (or criterion) in honor of the English physicist Osborne Reynolds, who at the end of the last century experimentally observed the presence of two flow regimes.

Small values ​​of Reynolds numbers indicate the predominance of the work of internal friction forces in the fluid flow and correspond to laminar flow. Large values Ye correspond to the predominance of kinetic energy and a turbulent flow regime. The boundary for the beginning of the transition from one mode to another - the critical Reynolds number - is 1?cr = 2300 for round pipes (the diameter of the pipe is taken as a characteristic size).

In technology, including diesel locomotive technology, in hydraulic (including air and gas) systems, turbulent flow of liquids usually occurs. Laminar flow occurs only in viscous liquids (for example, oil) at low flow rates and in thin channels (flat radiator tubes).

Calculation of hydraulic resistance. Linear pressure losses are determined using the Darcy-Weisbach formula:

where X (“lambda”) is the linear resistance coefficient, depending on the Reynolds number. For a laminar flow in a round pipe R, = 64/E (depends on the speed), for turbulent flows the value of k depends little on the speed and is mainly determined by the roughness of the pipe walls.

Local pressure losses are also considered proportional to the square of the speed and are defined as follows:

where £ (“zeta”) is the coefficient of local resistance, depending on the type of resistance (rotation, expansion, etc.) and on its geometric characteristics.

Local resistance coefficients are established experimentally; their values ​​are given in reference books.

The concept of calculating hydraulic systems. When calculating any hydraulic system, one of two problems is usually solved: determining the required pressure difference (pressure) to pass a given fluid flow rate or determining the fluid flow rate in the system at a given pressure difference.

In any case, the total pressure loss in the AN system must be determined, which is equal to the sum of the resistances of all sections of the system, i.e., the sum of the linear resistances of all straight sections of pipelines and local resistances of other elements of the system:

If the average flow velocity is the same in all sections of the pipeline, equation (2.33) is simplified:

Typically, there are sections in the system in which the flow velocities differ from each other. In this case, it is convenient to reduce equation (2.33) to another form, taking into account that the fluid flow rate is constant for all elements of the system (without branches). Substituting the values ​​u = C)/5 into condition (2.33), we obtain

hydraulic characteristic, or overall resistance coefficient of the system.

It must be borne in mind that pipeline calculation is not a solution to a problem with one definite answer. Its results depend on the choice of diameters of pipeline sections or velocities in them. Indeed, it is possible to take low speed values ​​into account and obtain small pressure losses. But then, at a given flow rate, the pipeline cross-sections (diameters) must be large, and the system will be bulky and heavy. Having accepted high speeds flow in the pipes, we will reduce their transverse dimensions, but at the same time the pressure losses and energy costs for operating the system will increase significantly (proportionally to the square of the speed). Therefore, in calculations, some average, “optimal” values ​​of fluid flow rates are usually specified. For water systems, the optimal speed is about 1 m/s, for low-pressure air systems - 8-12 m/s.

Water hammer is a phenomenon that occurs in a fluid flow when the speed of its flow changes rapidly (for example, when a valve in a pipeline suddenly closes or a pump stops). In this case kinetic energy The flow instantly turns into potential energy and the pressure of the flow in front of the valve increases sharply. Region high blood pressure then propagates from the valve towards the flow that has not yet been completely inhibited at a speed close to the speed of sound a in this medium.

A sharp increase in pressure leads, if not to destruction, then to elastic deformation of pipeline elements, which reduces the impact force, but increases fluctuations in fluid pressure in the pipe. The magnitude of the pressure jump during a complete stop of the fluid flow, which had a speed v, is determined by the formula of the outstanding Russian scientist - Professor N. E. Zhukovsky, obtained by him in 1898: Dr = paa, where p is the density of the liquid.

In order to prevent shock phenomena in large hydraulic systems (for example, water supply networks), shut-off devices are designed so that their closure occurs gradually.