CHAPTER 2

KINEMATICS OF FLUIDS

Kinematics of liquids is the section of aerodynamics, which studies types and forms of movement of liquids and gases, regardless the action of effecting forces.

The primary aim of kinematics is studying velocities of particles. Set of particles' velocities of fluid environment forms a field of velocities. Therefore, to have full representation about liquid or gas movement character it is necessary to obtain values of velocities in all points of researched area. Thus, in general, the velocity value will be a function of coordinates and time.

2.1. Methods of research of fluid movement

There are two methods of studying fluid movement: Lagrange method and Euler's method.

In Lagrange method studying fluid movement is conducted by supervision over fluid separate particles moving along their trajectories. Each particle is considered as a material point. As there is an infinite set of particles; they are characterized by coordinates at the initial time.

Let the coordinates of the given particle are: , , when (in the initial time). It means, that among all trajectories, the particle will possess the one, that passes through coordinates: , and (fig. 2.1). Particle's coordinates , , are functions of coordinates , , and time :

(2.1)

where , , are the parameters determining the trajectory of the particle, the point special values at the initial time .

The equations (2.1) are the family of trajectories (in the parametrical form), which fill the whole space occupied with fluid environment.

Equations (2.1) completely define kinematics of flow. Indeed, knowing these equations it is easy to

15

Fig. 2.1. Movement of liquid particle

determine velocities and accelerations of fluid particles in the arbitrary time moment by time differentiation of trajectory equations:

(2.2)

A concept about the particle trajectory corresponds to the Lagrange method. A trajectory of fluid particle is named a curve, which is drawn by a particle moving in space for some period of time. A great variety of particles’ trajectories may pass through each point of space. Trajectories can cross themselves and be very complex and intricate, therefore considering them is rather difficult problem.

But there is no necessity to know a particle trajectory for solution of practical problems of aerohydrodynamics.

For example, it is possible to determine pressure in each point of wing surface with the help of the energy equation, if a flow velocity is predefined in each point. Apparently, the practical task of fluid kinematics issues the problem to define the velocity for certain point of space, depending on personal features of particles passing through this point.

Euler’s method is used to regard stationary space filled with moving fluid, and to study velocity variation in time in the given point.

Thus, the Euler’s method represents particles’ velocities as function of time and coordinates , , of points of space, or in specifying the velocity field. Then

(2.3)

Let's assume, that fluid motion is non-separable, therefore we shall consider functions shown in equations (2.3) as simple, continuous and differentiable functions of coordinates , , and time . In that case to define a trajectory of fluid particles we shall (2.3) replace , , in

16

equations to , , correspondingly and integrate the system of

differential equations:

(2.4)

After integration we shall obtain:

(2.5)

These equations have three arbitrary constants , , , which shall be determined from the initial conditions.

Let's consider a fluid flow characterized by the given velocity vector field . Let there be some scalar field designated as in the space occupied with moving fluid, which varies with time. Let's define a rate of change of value with time, observing the given fluid particle, moving according to the law , , .

It is necessary to take a total derivative with respect to time of value to determine a rate of its change:

.

Since functions , , represent the coordinates of a moving

particle, than , , .

Finally for we shall obtain the following expression:

. (2.6)

For the vector field varying with respect to time we shall

receive

. (2.7)

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The first summand in formulas (2.6) and (2.7) characterizes variation of considered value with respect to time in the fixed point of space and therefore it can be named a local derivative. The other summands are associated with movement of a fluid particle and therefore they are called as a convective derivative. It characterizes the field heterogeneity of considered value at the given point of time.

Using expressions (2.6) and (2.7), it is possible to calculate a rate of change of any values of scalar and vector fields.

For example, we'll define a rate of change of moving particle density by means of formula (2.6):

. (2.8)

We shall get the fluid particle acceleration vector, which equals to the derivative from velocity vector of this particle with respect to time by (2.7):

(2.9)

Projecting on coordinate axes we can write down the following expressions:

;

; (2.10)

.

In general case of fluid motion, projections of velocity , , , pressure and density will be functions of coordinates , , and time

, that is;

;

; (2.11); .

If projections of a velocity , , , pressure and a density in the fixed point of space having coordinates , , will be functions of time

, than such fluid motion is unsteady.

18

If values , , , and in the fixed point of space do not vary with time, such fluid motion is called steady. It means, that an arbitrary fluid particle which comes to the given fixed point of space, will have the same values of , , , and in this point as former fluid particles in this point. In this case expressions (2.11) will be written as follows:

;

;

; (2.12);

For steady motion the field of considered values is stationary, and local derivatives are equal to zero:

; ; etc.

But total derivatives of these values differ from zero, as in general case the velocity, pressure and density change with transition from point to point. Only in the special case, when we consider the uniform rectilinear motion of an incompressible homogeneous fluid, we have

and .

2.2. Streamline. Fluid tube. Flow filament

The Euler's method feature is the concept of streamlines.

Let's consider an arbitrary point in space filled with fluid in arbitrary

moment of time (fig. 2.2). Let is the

vector of velocity at point . Let us put aside a small line segment from the

point in the direction of vector

and mark point . is the vector of

velocity in the point . Having put aside a small line segment from point along

the vector we'll receive point at

the end of the line segment. Continuing Fig. 2.2. Drawing of streamline

19

such drawing further we shall receive in the aggregate some polyline . Increasing number of line segments and considering length of

each segment as infinitesimal we shall receive a smooth curve instead of a polyline, which will be one of streamlines.

The streamline is a totality of fluid particles, which velocity vectors are tangent to it at given point of time.

The fluid particle draws a trajectory during its movement. Streamlines do not coincide with trajectories during the unsteady movement and they are identical, when the movement is steady. It is necessary to note, that it is possible to draw only one streamline through each point of the space filled with moving fluid at the fixed moment of time.

The trajectory of a particle fixes changing position of the same particle eventually, and the streamline specifies a velocity direction of various particles at the same moment of time. Trajectories can intersect. Streamlines neither intersect themselves, nor any another, as the velocity vector would have two various directions in the cross point at given time, that is physically impossible. Exception is only so-called flow special points, where the value of velocity equals zero or is theoretically infinite.

The set of streamlines gives a picture of flow at present time that is an instant photograph of flow velocity directions. A number of such pictures for various moments of time represent the geometrical flow image corresponding to the Euler’s method.

Hence, it has been established, that velocity vectors are tangent to the streamline at each moment of time. Therefore if we should take any point on the curve, and an elementary line segment located close to it with projections , , (fig. 2.3) on corresponding axes, then the velocity vector and the direction of this line segment in the given point would coincide, or in other words they would be parallel.

Let's find the differential equation of a streamline. It follows from parallelism condition of streamline and vector of velocity in the given point

. (2.13)

Fig. 2.3. Streamline

20

Let's represent the vector product as determinant

,

deploying the first line of determinant we shall obtain

,

and from this equation,

,

.

The obtained expressions can be written as follows

. (2.14)

The system (2.14) is called as differential equations of streamlines.

Thus, the problem of streamline determination by the given field of velocities is reduced to integrating the system of differential equations.

Let's introduce the concepts of the fluid tube. For this purpose we shall draw some closed contour within fluid (fig. 2.4), which is not a streamline, and draw streamlines through each point of the contour. A set of these streamlines forms a surface, which is named fluid tube. Fluid flowing inside the fluid tube is called as flow filament.

It is necessary to note, that fluid tube formed by streamlines in stationary motion, does not change with respect to time and is similar to impermeable tube in which the fluid flows as in a tunnel with solid walls, which limit its contents. The fluid does not flow out from the fluid tube through lateral surface and is not added to it as in all points of flow filament actual velocity is directed along the streamline.

Within elementary flow filament velocities in all points of the same cross section can be assumed as equal to each other and to local velocities. Elementary flow filament is a visual kinematic representation, which

Fig. 2.4. Fluid tube

21

significantly facilitates studying fluid motion, and it is put in the basis of so-called jet model of fluid flow.

According to this model, entered in aerohydrodynamics even in a period of its scientific formation, the space filled with moving fluid is considered as a set of many elementary flow filaments. The set of elementary flow filaments, which flow through the area large enough, forms a fluid flow. Today the flow filament model is one of the basic fluid models.

2.3. Continuity equation

The continuity equation expresses a law of mass conservation as applied to a moving fluid, which approves an invariance mass of fluid volume

with respect to time ( ).

Let's assume that moving fluid completely fills entire space or its fixed area, i.e. that voids or gaps are absent. This requirement is called as continuity condition.

Let's consider some fixed closed surface of an arbitrary shape limiting volume through which a compressible fluid flows (fig. 2.5). Let's determine the fluid mass flowing through the given surface per unit of time.

Let's consider the fluid mass flowing out from the volume as positive and the fluid mass that flowing in as negative. The fluid mass equals to product shall flow through the surface element per unit time, where is the projection of velocity vector to perpendicular drawn to surface element .

All the fluid mass flowing through surface per unit time can be determined by the following integral

. (2.15)

Fig. 2.5. Explanation of continuity equation

22

On the other hand, this fluid mass can be obtained as change of mass in volume per unit time:

, (2.16)

where represents change of density per unit time.

Hence,

. (2.17)

Applying the Ostrogradsky-Gauss formula to the integral in the left part

of equation (2.17) ,

we receive

or

. (2.18)

Provided that the written equation (2.18) is valid for any arbitrary volume , the integrand should be equal to zero

, (2.19)

which is the continuity equation in the differential form for unsteady motion of compressible fluid.

In the detailed form the continuity equation (2.19) looks like:

. (2.20)

The equation (2.20) can also be written in the other form. Taking into account, that

;

;

23

,

we shall obtain:

,

or

, (2.21)

The vectorial form of the last expression looks like

. (2.22)

For different types of flows the continuity equation (2.19) becomes as follows:

for steady-state motion of compressible fluid ( , )

(2.23)or

( ) ( ) ( )V

x

V

y

V

zx y z 0 ; (2.24)

for steady-state motion of incompressible fluid ( )(2.25)

or

V

x

V

y

V

zx y z 0 . (2.26)

Taking into account the equation (2.26) it is possible to approve, that requirement of conservation of mass demands the volumetric deformation rate of incompressible fluid to be equal to zero when fluid moves.

2.4. Flow consumption equation

To solve many practical aerohydrodynamics problems it is necessary to have the continuity equation in the form, which would settle the connection between velocity and the flow filament cross-sectional area. Such equation is the flow consumption equation, representing the law of conservation of mass as well as the continuity equation.

24

The flow consumption is the quantity of fluid running through actual flow cross-section or flow filament section per unit time. The flow consumption can be measured in units of volume, weight or mass. Accordingly three types of the flow consumption are distinguished: volumetric, weight and mass.

For elementary flow filament (fig. 2.6,а) the velocity distribution is accepted to be considered as uniform then the volumetric flow consumption is ; the weight flow consumption ; and the mass flow consumption .

For the finite size flow, where the velocity within the actual cross-section is variable (Fig. 2.6,b), expressions for three types of flow consumption will look like:

;

;

.

As always, considering such cases the concept of mean velocity shall be entered:

. (2.27)

а

b

Fig. 2.6. а – elementary flow filament; b – finite size flow

25

Let's receive the equation of flow consumption. The fluid mass running through arbitrary surface and limiting volume (fig. 2.7) can be determined by the integral (2.15). For steady-state motion we also have the

mass , i.e. mass flowing in volume is equal to the mass

flowing out. Let's apply the obtained result to elementary flow filament.

Starting from the said above . Fluid mass , because on lateral surface of the fluid tube (from property of

streamline). Having written the values for and , we shall receive

,

and it follows that. (2.28)

Fig. 2.7. Explanation of development of flow consumption equation for elementary flow filament

Expression (2.28) is called the flow consumption equation for elementary flow filament of the compressible fluid. For incompressible fluid (

) so. (2.29)

For the finite size channel (fig. 2.8), it is possible to write:

, (as ), then the flow consumption through cross-sections and will be equal to

.

Using concept of mean velocity (2.27), we shall receive

. (2.30)

26

Expression (2.30) represents the flow consumption equation for moving compressible fluid within the finite size channel.

For incompressible fluid the flow consumption equation is

. (2.31)

Fig. 2.8. Explanation of the development of flow consumption equation for the finite size channel

The flow consumption equation is widely used to solve the aerohydrodynamics problems; it gives simple dependence between the velocity and flow filament sectional area. The equation (2.29) shows, that for incompressible fluid when fluid tube becomes narrower (thickening streamlines) velocity increases, and with expanding fluid tube (streamlines divergence) velocity decreases.

In the event of compressible fluid volume-flow persistence may not have place. Under certain conditions fluid flow consumption through two cross-sections of the flow filament may differ due to density change between them, but thus the law of mass conservation is not been broken.

According to flow consumption equations the fluid tubes should be closed or end on fluid boundaries as at we get ; the infinite velocity does not exist in nature.

2.5. Motion of fluid particle. Angular velocity

In a kinematics of a solid body it is proved, that it general case motion of a solid body can be separated into its constituent parts: translation and rotation around some axis called instantaneous rotation axis.

27

Motion of fluid particle is much more complex. When moving fluid particle can change one’s shape – it can distort, in this case resulting motion of fluid particle will comprise three motion types: translation, rotation and deformation.

Rotary movement of fluid particle is called vortex motion. It is accepted to designate angular velocity of rotation as . This value is vectorial, so it is possible to separate it to three components ( , , ) parallel the corresponding coordinate axes.

Apparently, absolute value of full angular velocity can be determined in the known way:

.

Let's express angular velocity , , by means of linear

velocities , , . For this purpose we shall consider motion of the elementary fluid parallelepiped with edges , and (fig. 2.9) during rather small period of time .

Fig. 2.9. Motion of the elementary fluid parallelepiped

28

Let's consider at first the face of parallelepiped parallel to coordinate plane (fig. 2.10). Let the point of this edge at some instant time

moment has the traveling velocity equal to along the axis, and

along the axis - equal to ; then the point distanced from point on

along the axis will have velocity

directed along the axis , equal to:

.

At the same time the point distanced from the point on value equal to along axis has the velocity

.

Due to velocity difference in infinitesimal period the point will move along the axis the following distance relatively to point

,

and for the same period the point due to difference of velocitys will move along the axis with respect to the point on distance

.

For the same time line segment will rotate on infinitesimal angle

, (2.32)

and line segment will rotate, accordingly, on angle

Fig. 2.10. The fluid parallelepiped edge movement in the general case

29

. (2.33)

If parallelepiped was not deformated and if it would rotate about the edge , line segments and would rotate in the same direction on the same angle (fig. 2.11). On the contrary, if they would not rotate and the element would be distorted only so line segments and would rotate on the same angle , either towards each other, or in the opposite directions (fig. 2.12).

Fig. 2.11. Rotation of the fluid parallelepiped edge

Fig. 2.12. Deformation of the fluid parallelepiped edge

Generally it is possible to consider, that sides and turn around edge on angle due to rotation of the element and additionally turn on opposed angles due to parallelepiped-to-lozenge deformation and finally take the positions and . In that case we shall have using fig. 2.13

, ,

and

30

.

The element angular velocity with respect to edge can be determined using the following relation:

,

That is resulting in the following form with the help of equations (2.32) and (2.33):

.

Considering motion of edges of similar parallelepiped, which faces are parallel to planes and we shall receive expressions for all components of angular velocity:

(2.34)

With the help of these formulas it is easy to determine components of angular velocity by elementary differentiation if components of linear velocity are known as functions of coordinates.

Fig. 2.13. Explanation how to define angles of face rotation

31

Angular velocity of rotation can be written in the vectorial form as

, (2.35)

where is a vortex of velocity.

It should be note that both translational and rotational motions and deformation of fluid element occur simultaneously, irrespective of each other.

The particular case when fluid particles do not revolve around the instantaneous axis is named as vortex-free or potential (the meaning of the latter will be explained in subsection 2.7).

2.6. Fluid vortex-type flow

In aerohydrodynamics the significant place is allocated to the theory of vortex-type flow – flow where vortex of velocity is not equal to zero.

As it is known, movement of a fluid particle can be divided into three components: translation, rotation and deformation movement. Rotation of a fluid around some instant axis is defined by the vortex of velocity. The vortex of velocity represents a vector of the double instant angular velocity

. (2.36)

If the considered part of fluid is totally whirling, it is possible to speak about the vortex field: the vector representing angular velocity of fluid particle located at this place at present time can be drawn for each point in space.

2.6.1. Vortex line. Vortex tube. Vortex core

As streamlines give the concept about field of velocities, vortex lines give analogous concept of vortex field.

Fig. 2.14. Vortex line

32

Vortex line is such line within flow where the vector of angular velocity or vector of vortex of velocity is directed along tangent to this line in its

each point (fig. 2.14). Vortex lines can change their form and position in space with time. Vortex lines, similarly to streamlines, cannot intersect in the flow: only one vortex line may be drawn through each point of vortex flow.

It follows from definition of vortex line:

, (2.37)

where is a radius-vector, determining position of points located on vortex line with respect to some center.

The differential equations of vortex lines can be written down as follows by analogy to the differential equations of streamlines

. (2.38)

After substituting the expressions for components of angular velocity into these equations (2.34) we shall receive the system of independent differential equations, integration of which gives equations of vortex lines in the final form.

Generally vortex lines and streamlines do not coincide and can intersect. It is necessary to note, that the streamline can be ideally drawn in any fluid flow, and the vortex line can not always and everywhere be drawn. For example in case of potential flow the angular-velocity vector is equal to zero in all points and vortex lines in such flow do not exist.

At constant flow vortex lines do not vary with time, similarly to streamlines. If vortex lines and streamlines coincide, vectors of linear and angular velocities coincide. Such fluid flow is called helical flow. The equation of helical lines obtained from a requirement of vector parallelism of linear and angular velocity vectors has the following form:

.

If we should draw vortex lines through each point of some line ,

which is not being a vortex line; their combination would form a vortex surface. If the line is a closed loop the vortex surface turns into a vortex tube (fig. 2.15). The vortex tube together with rotating fluid enclosed within forms

Fig. 2.15. Vortex tube

33

a vortex core. Thin vortex core is sometimes called a vortex trunk (vortex line).

2.6.2. Vortex core strength

Let's consider a thin vortex line (fig. 2.16) and split it by plane , perpendicular to filament axis.

Strength or vortex line consumption is determined by velocity vortex flow through the area of filament section normal to vortex vector

or by doubled flow of vortex velocity:

. (2.39)

Let's now split the vortex line by plane , making the arbitrary angle with the plane of normal section . Then the area of inclined section will be concerned with the area of normal section by the equation:

.

Let be the normal to the area , and is the angular rate component normal to this area.

Taking into account two latest equations we shall write down the expression (2.39) in more general form:

, (2.40)

where is the unit vector normal to surface .

Strength of vortex line or its consumption can serve as a standard measure of fluid vorticity happening within the vortex filament. Consumption of a vortex core is:

Fig. 2.16. Explanation of development of equation of vortex filament

consumption

34

. (2.41)

2.6.3. First Helmholtz vortex theorem

Values of angular-velocity vector and sectional area may vary along the given vortex core, however vortex core consumption along its whole length remains constant. This is a content of the first Helmholtz theorem.

This theorem is only of kinematic type and is valid for any continuum provided that the field of velocities is a continuous function of coordinates. We shall prove this theorem using Zhukovsky method. Starting from equations (2.34), we shall write partial derivatives with respect to coordinates , and for angular velocity

Having summed them we obtain the equation:

(2.42)

or. (2.43)

Last equation is similar to the continuity equation (2.25) if we should assume that incompressible fluid moves within the vortex tube with vector . Thus, the equation (2.42) is the continuity equation for vector .

By analogy with the flow consumption equation (2.29) for vortex filament for which angular rate may be considered as constant value over the section, it is possible to write down

. (2.44)

For all cross-sections of vortex line strength is a constant value, therefore at reduction of cross-sectional area angular rate will increase and on the contrary. If at angular rate that is physically

35

impossible. Thus, the vortex line can not be needle point at its end in a fluid, it only can lean against its solid boundaries, or on a free surface or to swing in a ring.

2.6.4. Velocity circulation

Both theoretical and applied aerodynamics widely use a concept of velocity circulation designated as .

Let's draw an arbitrary closed contour within moving fluid flow and take any element belonging to this contour (fig. 2.17). We shall mark the velocity in the middle of element (point ) as , and the angle between vector and tangent to the contour as

. Let's consider the product

,

this is called an elementary velocity circulation.

Let’s take the curvilinear integral along the arc :

. (2.45)

This expression is termed as a velocity circulation along arc . Circulation is usually calculated along the whole closed loop

. (2.46)

We shall consider the direction of integration as positive if the area, enclosed by contour remains to the left during integration.

Expression (2.46) for circulation can be written in another form. The velocity vector projection on the direction of tangent to the contour in its any point will be equal to the sum of projections of vector components on the same direction, i.e.

,

Fig. 2.17. To a velocity circulation determination

36

where is tangent direction.

For such substitution elementary circulation will be written down as

.

Taking into account, that

, , ,

we shall receive the following expressions:

, (2.47)

. (2.48)

Integrals (2.46) and (2.48) have no defined physical meaning. However, as we shall see further, the velocity circulation is very important value in aerohydrodynamics. It is related to the vortex consumption and strength; as N.E. Zhukovsky had proved, the value of wing lift directly depends on the velocity circulation value.

2.6.5. Relation of elementary circulation with vortex strength. Stock’s theorem

Let’s draw in the fluid flow infinitely small closed contour in the form of right-angled triangle which legs are parallel to coordinate axes and and determine velocity circulation over this contour. We shall be passing around the triangle contour in the counter-clockwise direction, having taken it as a positive direction of tangent to the contour. We shall suppose, that velocities , and

at the infinitely small sides of triangle vary by the linear law and applied to the middle of the sides.

Elementary circulation by the contour of triangle equals to:

.

It is seen from fig. 2.18.

Fig. 2.18. Explanation how to develop velocity circulation of the contour of

elementary triangle

37

and ,

therefore

.

Taking into account that

, , ,

where is the triangle area we obtain:

. (2.49)

But the angular rte is normal to area , therefore

, (2.50)

where is a strength of vortex filaments crossing area .

Comparing equations (2.49) and (2.50), we can finally write down

. (2.51)

So, velocity circulation calculated along simple closed contour in the form of right-angled triangle, is equal to the strength of vortex filaments

, enclosed by the triangle contour.

Obtained result can be easily generalized for an arbitrary infinitely small triangle (Fig. 2.19,a). Such triangle always can be split on two right-angled triangles and by dotted line , and we can apply equation (2.51) to each of them, i.e.

, ,

or

, (2.52)

where is a velocity circulation over the contour of given triangle ; is strength of vortex filament enclosed by the contour of given triangle.

Equation (2.52) follows from fig. 2.19,a, but when summing circulations and over contours of right-angled triangles and , the circulation

along the is taken twice, but with the opposite signs, so it is cancelled. The circulation over sides of the given triangle is not cancelled and equals to

in sum.

So, as it follows from the latter equation, for any infinitely small triangle

38

dГ dI ,

i.e. equation (2.51) is true.

Moreover, it is true for any infinitely small tetragon (fig. 2.19,b), as it can always be divided into two triangles by dotted line , for each triangle we can apply equation (2.52). When summing circulations over both triangles like in the former case, we can see that the circulation over internal dotted line will be taken twice, but with the opposite sings and while summation it will be cancelled. But the circulation over the sides of the given tetragon will not be cancelled and it will be equal to strength of vortexes that intersect area of tetragon. Hence, for infinitely small tetragon we obtain equation (2.51).

Let's consider the general case. Examine arbitrary closed contour in fluid flow with continuous surface located within the fluid flow and leans against it (fig.2.19,c). On surface let's draw a grid formed by two families of intersecting lines, which divide the surface into a number of areas, each of them will be bounded by its own closed contour .

Fig. 2.19. Proving Stock’s theorem.

If we should sum circulations over all internal contours, we could see that the circulation will be taken twice over every line drawn on the surface

, but with opposite signs and under summation these circulations for all internal lines will be mutually cancelled. But circulations over sections of given contour would not be cancelled. As these circulations for all sections of contour are taken in the same direction as shown on fig. 2.19,c, the sum of circulations over all internal contours will be equal to the circulation of velocity over given contour, i.e.

,

where is a velocity circulation over any internal closed contour.

39

For the extreme case, when every internal area becomes infinitely small and each bounding contour is a triangle or tetragon, which are infinitely small, equation (2.51) for any closed contour will have the final form

. (2.53)

In the same manner one can show that this relation is true for non-planar contour and part of surface , but only if the contour is simply and part of surface does not go beyond boundaries of the fluid. So finally Stock’s theorem can be formulated in the following way: the velocity circulation over any arbitrary contour equals to the vortex vector flow through the surface leaning against this contour and not going beyond the fluid bounds, or equals to the sum of strength of vortex filaments intersecting the surface.

Conclusion following from the Stock’s theorem is: if we should draw a closed contour on the surface of vortex tube, enclosing vortex tube the circulation over such closed contour will be equal to vortex tube strength; if we should draw a contour on its surface that does not enclose a vortex tube, the circulation over this contour will be equal to zero.

This theorem is kinematic one and can be applied to any continuous system if the movement of this system is continuous.

2.6.6. The generalization of circulation theorem

When proving this theorem some certain assumptions were made, such as:

– velocity of flow and the first derivatives of velocity components with respect to coordinates are continuous functions;

– velocities vary under a linear law along every side of infinitely small right-angled triangle;

– surface leaning against the contour is continuous and is entirely located within fluid.

Accepted assumptions do not always take place, let’s determine how we can use this theorem for such cases.

Let’s show, that requirement of continuity of velocity derivatives is not essential. Assume, that on some line belonging to the surface (fig. 2.20), the first derivative of velocity components undergo rupture, in the way that when transferring from one side of this line to another side they suddenly change their value, but keep continuously along this line.

40

Then we always can draw a grid of lines on the surface , so that one of lines could coincide with line .

In that case the line will be a boundary of a number of infinite small internal areas, but derivatives are continuous along the line itself and theorem proving is truly expressed by the formula (2.53).

We would come to another result when assuming that on some line velocities undergo rupture when transferring from one side of the line to another and is surface has ruptures (fig. 2.21). For example, surface is leaning against the given contour from its face and has a cut inside, bounded by contour . In that case formula (2.53) can not be applied directly. But it can be used in the following way. Let’s join outer contour with internal contour by means of two lines and which are very close to each other (fig. 2.21). As a result we shall obtain the combined closed contour leaning against continuous surface , for which we can use formula (2.53)

,

Fig. 2.20.

Fig. 2.21.

41

where is a velocity circulation along the compound contour .

As it follows from fig. 2.21,

.

Bringing together lines and , we shall obtain in the limit:

,

since circulations and are taken along the same line but in different directions.

Further, for considered limit case we shall also obtain

, ,

where is a velocity circulation over the contour ; is a circulation over contour . For these substitutions we shall obtain for circulation

,

and

. (2.54)

The theorem of velocity circulation is generalized in this way for a case of ruptured surface leaning against outer contour and internal contour .

Let’s consider a case when line is located on the surface , on

which velocities undergo rupture when transferring from one side of it to another (fig. 2.22).

In this case we shall encircle the line with closed contour and designate a part of surface as , enclosed within contour . So we can apply formula (2.54) to the surface equals to difference between i.e.

Рис. 2.22.

42

.

Now we shall pull together the contour to the line in such a way that in the limit its sides would coincide with opposite sides of line . Then the surface will be equal to zero and we shall come to formula (2.54), where mean velocity circulation on velocity rupture line , calculated over its two sides, shaping together one closed contour .

As velocity circulation is determined only by velocities tangent to contour, so circulation could differ from zero only if there is a rupture of tangent velocities on the line . If tangent velocities are continuous, circulation will be equal to zero and we again return to formula (2.53).

If surface has more than one cuts bounded by internal contours with circulation , then generalized formula will be written down similar to formula (2.54) as follows:

. (2.55)

Contours may include rupture lines of tangent velocities.

2.7. Foundations of potential flow theory

2.7.1. Concept of potential flow

Fluid flow where vortex velocity equals to zero is called irrotational or potential fluid motion, i.e. flow free of local rotations of fluid particles.

It should be noted that in real fluids formation of vortex motions is observed permanently. The reason is a number of facts, including existence of fluid internal friction. Despite this fact the potential motion scheme gives us picture very close to real in many cases, important for solving practical problems.

So, assuming that flow is vortex-free, let’s consider main features of potential motion.

Basing on definition of fluid potential flow ( ) we shall obtain for ratios (2.34)

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(2.56)

In vector form this condition will have the following form:

. (2.57)

It follows from equations (2.56):

(2.58)

Let’s consider the differential trinomial . As it is known, equations (2.58) are required and sufficient conditions that this differential trinomial would be a total differential of function , which is continuous function including its partial derivatives up to the second-order, i.e.

.

The function is called velocity potential and has great importance in aerohydrodynamics. Let’s disclose its total differential

.

Comparing coefficients near , , (increments of arguments), we shall receive

44

(2.59)

i.e. velocity projection onto coordinate axis is equal to partial derivative of velocity potential with respect to corresponding coordinate. This important potential property is kept for arbitrary direction also. Let’s consider in fluid point located on arbitrary curve (fig. 2.23). Let velocity in the point is equal to . Draw the tangent to the curve in the point . As we consider potential flow, there is velocity potential so we can write

.

Since

, , ,

so finally we shall obtain

, (2.60)

i.e. velocity projection onto arbitrary direction is equal to derivative of velocity potential along this direction. In particular case in polar coordinates on plane we shall have

(2.61)

where , are projections of velocity vector of point onto direction of polar radius-vector and onto direction that is perpendicular to polar radius-vector (fig. 2.24).

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Fig. 2.23. Development of main property of velocity potential

Fig. 2.24. Velocity components in polar coordinate system.

Availability of velocity potential simplifies solving many problems in aerodynamics, since in this case three unknown values , and can be expressed using partial derivatives of one unknown function – velocity potential according to equations (2.59).

Surface defined by equation

(2.62)

is called equipotential surface. It follows from definition that velocity potential has constant value on such surface. Line drawn on equipotential surface will be called an equipotential line respectively.

Due to condition (2.62) derivative will be equal to zero in

any direction tangent to equipotential surface direction.

It follows that velocity vectors are perpendicular to equipotential surface. Streamlines are also perpendicular to it as velocity vectors are tangent to streamlines.

There is close association between velocity potential and circulation in vortex-free flow. For example, velocity circulation over any closed contour connecting two arbitrary points and can be expressed in a form of curvilinear integral

.

Substituting , and with their expressions taken from formulas (2.59) in the latter equation we shall obtain

46

. (2.63)

It means that velocity circulation in potential flow over some contour does not depend on contour shape and is equal to difference between velocity potential in the initial point and velocity potential in the final point of contour. If contour is closed velocity potential is simple coordinate function and circulation becomes equal to zero.

It should be noted that velocity potential could be both single- and multiple-valued coordinate function.

2.7.2. Multiple-valued velocity potential

Such parameters as pressure, density, and velocity components in steady motion must be single-valued coordinate functions by their physical implication. Velocity potential is an auxiliary function and it can be both single-valued and multiple-valued coordinate function. Let’s determine in what cases velocity potential will be single-valued and in what cases it will be multiple-valued.

Potential flow domain is called simply connected if any closed curve drawn within it can be pulled together into a point by means of continuous deformation without rupture and do not spread over boundaries of potential domain.

Otherwise domain is called multiply connected. For example potential flow, which contains a vortex filament with strength is two connected domain (fig. 2.25).

In this case contour enclosing vortex filament cannot be pulled together in a point without rupture and without spreading over bounds of potential domain.

47

Let’s show that velocities potential is a single-valued only in simply connected domain.

Let’s consider a segment of contour between points and (see fig. 2.25) and determine velocity circulation over this segment passing from point to in direction shown by arrows. We shall denote velocity potential value as in point and as in point . Then as it follows from formula (2.63)

.

Pulling together points and in direction shown by arrow until they become complete coincident we shall obtain the following equation on the base of circulation theorem

.

Due to this substitution we shall obtain value of velocity potential in point from the later equation.

.

We can see that if vortex filament is present of in flow velocity potential has two different values and in the same point (point coincides with point ), differing one from another on value .

Let’s assume obtained value is the initial value of velocity potential in point and make full round over contour directing to point . Then according to the later equation new velocity potential value in point will be equal to

.

Fig. 2.25. Multiply connected domain

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If we continue this process further we shall come to conclusion that velocity potential in the same point of multiply connected domain can have a number of different values:

, (2.64)

where is any integer.

If flow is free of vortexes ( ), then potential flow will be simply connected. In this case and velocity potential will be single-valued coordinate function.

We should come to the same result if any bodies in spite of one or several vortex filaments would be placed in potential flow, for example a wing creating vortex circulation (not equal to zero) over its closed contour. In this case potential flow is not simply connected and velocity potential will be multiple-valued.

Any multiply connected domain can be artificially transformed into simply connected by drawing additional boundary surfaces in flow that eliminate foreign inclusions (vortexes, bodies) from considered domain.

2.7.3. Continuity equation for potential fluid motion in Cartesian coordinate system.

It was proved in subsection 2.3 that continuity equation in general form can be written in the following way

10

d

d t

V

x

V

y

V

zx y z .

Substituting values of velocity components according to formulas (2.59), we shall have

. (2.65)

For incompressible fluid i.e. continuity equation will have the following form

. (2.66)

Obtained equation is called Laplace's equation, function satisfying this equation is called harmonic function. So, for potential flow of

49

incompressible fluid velocity potential will be harmonic function of coordinates , , .

Laplace's equation (2.66) is a linear differential equation expressed in second order partial derivatives. Now methods to solve this equation are well known. Own velocity potential corresponds to each specific potential fluid flow. As there is infinite number of fluid flows so equation (2.66) has infinite number of solutions. Boundary conditions are introduced into practice to obtain the solution of Laplace's equation that corresponds to body of given shape and desired condition on external boundaries of fluid flow.

Supposing, that solid body which surface is given by function is streamed by fluid flow with velocity parallel to axis on infinity ( ) and equals to . In this case it is necessary that the following boundary condition would work: when velocities have the following values

, .

For inseparable streaming we have the second boundary condition: normal velocity component equals to zero on body surface.

For plane potential motion continuity equation for incompressible fluid (Laplace's equation) will take the following form

. (2.67)

In conclusion it should be noted that only harmonic functions could define potential flow of incompressible fluid.

2.7.4. Continuity equation for potential motion of incompressible fluid in polar coordinate system

Let’s consider plane potential flow of incompressible fluid. Pick out an infinite small area bounded by two infinitely close polar radiuses and two infinitely close circles with radiuses and (fig. 2.26). Let’s try to calculate masses of fluid inflowing through edges and and flowing out through edges and . We shall consider mass of fluid flowing out as positive and mass of fluid flowing in as negative.

Fluid mass that flows in through edge in a unit time is

,

through edge is

.

Fluid mass that flows out through edge in a unit time is

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,

through edge is

.

Immutability of mass in time is required for continuity

condition of incompressible fluid

Making summation of fluid flowed in and flowed out

we shall have

.

Substituting velocities and with equations (2.61), we shall obtain

.

Making differentiation and dividing through by finally we shall have

. (2.68)

Equation (2.68) is the Laplace's equation in polar coordinate system.

Fig. 2.26. Determination of continuity equation in polar coordinate system

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2.7.5. Stream (Flow) function

In aerodynamics the so called stream function has great importance in analysis of streams. Let’s clear up its meaning for potential plane-parallel steady motion of incompressible fluid.

Plane-parallel flow is called flow where fluid particles moving parallel to fixed plane, and at the same time gas-dynamic variables have equal values in corresponding points of all planes that are parallel to it.

Differential equation of streamlines (2.14) has following form for plane-parallel fluid motion

or

. (2.69)

Substituting in this equation values and expressed in function of coordinates and making integration we can obtain equation that connects coordinates and with arbitrary constant. To each value of arbitrary constant will be correspond definite line of fluid flow. Differential binomial in the left part of equation (2.69) is total differential of function

.

Continuity equation (2.26) for considering case can be written in following way

. (2.70)

As it follows from ratio velocities and can be expressed by function

(2.71)

In fact substituting (2.71) in (2.70) we obtain identity

.

Then substituting values and from (2.71) to (2.69) we shall have

52

,

from which by integration we shall find equation of streamlines

, (2.72)

where is an arbitrary constant.

Equation (2.72) is an equation of streamlines family. Giving different values to constant we shall obtain different flow lines that belong to this family. Function is called stream (flow) function.

Comparing of formulas (2.59) and (2.71) leads to important ratios

(2.73)

that are called Coushy-Rimans’s or D’Alamber-Euler’s condition.

Following ration between functions and can be established by multiplying from formulas (2.73):

As it known from mathematics this ratio is condition of perpendicular curves and (fig. 2.27). So, families of flow lines and equipotential lines are mutually orthogonal in potential plane steady fluid flow. Flow function for potential fluid flow and as velocity potential , corresponds to Laplace's equation. Actually using condition of potentiality (2.56)

and ratios (2.71) we obtain Laplace's equation

Fig. 2.27. Family of equipotential lines and streamlines

53

. (2.74)

If consider flow function as velocity potential so, according to Coushy-Riman’s condition (2.73) velocity potential of initial fluid flow will become flow function i.e. equipotential lines of initial flow will become flow lines in new flow. Velocity vectors of particles in new flow will turn on angle

with respect to velocity vectors of particles of initial flow.

Thus, functions and can be permuted. Two fluid flows described by these functions are called conjugated.

It should be noted that flow function exists under plane-parallel fluid flow independently of if fluid flow is vortex or potential, as only continuity of fluid flow was supposed under flow function determination.

Let’s clear up physical meaning of flow function . Consider cylindrical surface with single altitude leaning against contour that connects points and in space filled by plane potential fluid flow (fig. 2.28). Volumetric fluid consumption through surface element which has contour element in its base, is equal to

where

, ,

minus sign before had been taken because while moving along

segment from point to point .

Fig. 2.28. Determination of physical meaning of streamlines

54

Substituting velocities and by their expressions according to (2.71) we shall obtain

,

from which fluid consumption through contour we shall determine in following form

, (2.75)

i.e. volumetric fluid consumption of incompressible fluid flowing through contour between two streamlines does not depend on contour form end is equal to difference of flow lines values in endpoints.

From it follows if points and coincide fluid consumption is equal to zero ( )for single valued flow function . If curve is flow line section, fluid consumption through it will be also equal to zero (as along flow line ). If under full bypassing over closed contour flow function does not come to its initial value it means that flow function is multiple valued and fluid consumption will be differ from zero. It can take place if advance or retract of fluid is carried out inside of fluid, i.e. there are so called sources or drains inside contour. From formula (2.75) and fig. 2.28 follows that fluid volume flowing between two arbitrary flow lines in a unit time is constant value, numerically equal to constants value difference that correspond to these flow lines.

2.6.7. Complex potential

As it was shown, the basic functions characterizing the properties of potential plane motion, (i.e. stream function and velocity potential ), are interconnected by the equations

,

.

In the theory of complex variable function these equations show the condition that the complex combination of these 2functions of 2 actual variables, i.e. , is a complex variable function . Let’s put this function through :

. (2.76)

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The function is an analytical function of a variable and is called complex potential or flow characteristic function.

Let’s consider the derivative from complex potential with respect to complex variable . The expression for derivative will take the form

. (2.77)

By substituting the derivatives in the expression (2.76) by the formulae (5.73), we’ll obtain:

. (2.78)

This expression is called complex velocity.

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