Rise and development of velocity profile in channel due to internal friction: The entropy of the flowing fluid increases, which is caused by the internal friction of the fluid, which causes part of its kinetic energy to be transformed into the internal energy of the fluid (referred to as the loss heat). Other effects of internal friction are pressure loss as the fluid flows through the channel and a lower flow velocity at the channel walls and a higher velocity at the channel center - the distribution of fluid velocity in the channel section under investigation is called the velocity profile. However, the velocity profile develops gradually. Figure 1 shows the gradual development of the fluid velocity profile in a pipe at the outlet of a vessel under the action of internal friction. The effect of internal friction starts at the inlet of the pipe, where the fluid friction against the walls of the channel occurs, this loss of kinetic energy of the fluid propagates away from the walls and thus the velocity profile gradually develops. Simultaneously, the flow velocity in the core of the flow increases to maintain flow continuity, as the velocity is conversely very low near the walls. The region affected by the existence of a wetted wall is called the flow boundary layer. In the case of closed channels, the boundary layers of opposite sides, as they continuously grow, merge after a certain length x_e.

E-region of fully developed boundary layer. V [m·s^-1] flow velocity at investigated location of channel; V_∞ [m·s^-1] flow velocity at inlet of channel section under investigation; x distance from inlet to pipe; x_e [m] [m] entrance length (not completed boundary layer development); δ [m] boundary layer thickness.

Frictionless ideal fluid flow: For the purpose of basic calculations of complex flow and comparison problems, we define an ideal fluid in which there is no internal friction and its heat capacity is constant. Ideal fluid flow models are closer to the actual flow the smaller the ability of the actual fluid to produce internal friction, see chapter Frictionless flow.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.4

Superfluidity of Helium: The closest to an ideal fluid is liquid Helium, which at temperatures below 2 K does not produce internal friction, this property is called superfluidity. Superfluidity also allows for the existence of opposing flows in the same channel without friction.

Difference between laminar and turbulent flow with respect to the flow trajectory: The development of the boundary layer and velocity profile is influenced by the type of flow. There are two types of flow according to the principle of interaction between the flow particles and the transfer of kinetic energy between them. These are laminar flow and turbulent flow. In laminar flow, the fluid forms parallel stream lines, and these lines slide over each other (the fluid forms tiny vortices within the lines). The fluid in neighbouring streamlines does not mix. In a turbulent flow, individual stream lines can no longer be identified and the motion of the elementary fluid particles is random–particles with higher energy move to areas with lower energy and vice versa.. Figure 2 shows the trajectories (streamline) of particles that are drifted by laminar flow and turbulent flow, see also the photographs in [Wilkens et al., 2009]. These particles are at the same time significantly more massive than the fluid molecules so that they cannot be affected by Brownian motion, but at the same time they are not significantly affected by gravitational acceleration. However, even in turbulent flow, lower velocities prevail near the walls and higher velocities in the core of the flow. Under what conditions laminar or turbulent flow can be expected is discussed in the chapter Turbulent flow.

(a) typical characteristics of laminar flow and its velocity profile; (b) typical characteristics of turbulent flow and its velocity profile.

Mean flow velocity

A large number of fluid flow parameters are calculated from the mean flow velocity, which can be related to the velocity profile, to the mass flow, to the fluid momentum, or to the kinetic energy of the fluid.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.5

Determining mean velocity from velocity profile: The mean flow velocity derived from the velocity profile corresponds to the average value of the velocity profile, see Formula 3c.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.6

Boundary layer thickness

The thickness of the boundary layer is analysed in terms of its effect on the mass flow (Displacement thickness), momentum (Momentum thickness) and energy of the flow (Energy thickness). These thicknesses are also used in solving Problem 1.

Calculation of boundary layer thickness from mass flow: The equivalent flow area through which the working fluid would flow at the maximum velocity and mass flow equal to the difference between the frictionless flow mass and the actual flow mass is called the displacement thickness, see Equation 4a.

4:
Defining equations for characteristic thicknesses of boundary layers

(a) displacement thickness; (b) momentum thickness; (c) energy thickness; (d) definition of boundary of affected area in case of profile wrapping. A* [m²] flow area of displacement thickness; A** [m²] flow area of momentum thickness; A*** [m²] flow area of energy thickness; V_max [m·s^-1] maximum flow velocity; V_∞ [m·s^-1] attack velocity (velocity in front of profile). The equations are derived in App. 6.

Calculation of boundary layer thickness from fluid momentum: The equivalent flow area through which the working fluid would flow at a maximum velocity with momentum equal to the difference between the frictionless fluid momentum and the actual fluid momentum is called the momentum thickness, see Equation 4b.

Calculation of boundary layer thickness from fluid kinetic energy: The equivalent flow area through which a working fluid would flow at a maximum velocity of the same kinetic energy as the difference between the kinetic energy of the fluid in frictionless flow and the kinetic energy of the actual fluid is called the energy thickness, see Formula 4c.

Definition of boundary layer thickness in the case of wrapped bodies: The characteristic thicknesses of the boundary layer in the vicinity of the isolated profiles and bodies determine the attack velocity, while the boundary of the affected area to which the flow is determined is at a distance where the flow velocity is already very close to the attac velocity (attac velocity reaching 99% of the velocity V_A), see Figure 4d.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.7

Application of characteristic boundary layer thicknesses: The given definitions of characteristic thicknesses are used when comparing different types of channels with each other in terms of velocity, momentum and energy losses, because there are applications where, for example, the smallest possible momentum loss is important and in others, energy loss, etc.

Viscosity

The influence of internal friction on the velocity profile in laminar flow can be qualified by a quantity called dynamic viscosity (abbreviated as viscosity see Viscosity definition). The viscosity values of the fluids under investigation are used to calculate flow parameters including pressure loss.

Viscosity definition: Dynamic viscosity is the ratio between the tangential stress and the velocity tensor, see definitional Equation 5. This definition was introduced by Issac Newton based on a simple experiment with internal fluid friction, which is described in Appendix 7.

F [N] frictional force acting on element; η [Pa·s] dynamic viscosity of fluid; τ [Pa] shear stress between streamlines caused by frictional force (friction between streamlines); ν [m²·s^-1] kinematic viscosity; S [m²] frictional area between investigated streamlines.

Newtonian vs. non-Newtonian fluids: Fluids for which the above definition of viscosity can be applied are called Newtonian fluids and, conversely, fluids in which the viscosity changes with velocity are called non-Newtonian fluids (fluids containing larger clusters of molecules such as colloidal solutions, suspensions, emulsions, gels, etc.).

Newton's viscosity law: The definition of viscosity written by Formula 5 is based on the very simple case of plane flow. If we are investigating spatial flow, where the velocity profile changes in multiple directions, we must assume a tensor of tangential stresses in the fluid from internal fluid friction (see Problem 2). The relationship between the individual shear stresses and viscosity in spatial flow is called Newton's law of viscosity, which is given, for example, in [Bird et al., 1965] for different coordinate systems.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.8

Viscosity values are function of temperature and pressure: The dynamic viscosity of fluids is measured using viscometers, of which there are several types. The results of the measurements are entered into thermodynamic tables which are used in calculations. However, viscosity varies with temperature and pressure, complicating data collection. Dynamic viscosity of liquids decreases with temperature and increases with pressure, though the pressure effect is negligible except at extremely high pressures (megapascals). The dynamic viscosity of gases increases with temperature and is independent of pressure, except at extremely low or high pressures. For these reasons, dynamic viscosities of fluids for engineering purposes are given only as a function of temperature, see Tables 6, 7 for water and steam and Tables 9, 10 for dry and moist air. In the absence of data, the approximate viscosity can be calculated as the product of the reduced and critical viscosities. The reduced viscosity is a function of pressure and temperature, see charts in [Bird et al., 1965]. The critical viscosity is the viscosity of the substance at the critical point.

t	0	10	20	30	40	50	60	70	80
η	1770,2	1303,9	1001,9	797,3	652,6	546,8	466,5	404,2	354,7
ν	1769,7	1303,7	1003,3	800,46	657,46	553,2	474,28	413,22	364,84
t	90	100	110	120	130	140	150	160	170
η	314,7	281,8	254,7	232,05	212,9	196,54	182,46	170,24	159,55
ν	325,87	293,92	267,84	246,05	227,74	212,22	198,97	187,6	177,78

6:
Viscosity of water at pressure of 101 325 Pa

t [°C] temperature; η [μPa·s]; ν [nm²·s^-1]. Values from 100 °C and above are for saturated water, i.e. at higher pressures corresponding to saturated liquids.

t	0	10	20	30	40	50	60	70	80
η	9,24	9,461	9,7272	10,01	10,307	10,616	10,935	11,26	11,592
ν	1778	1005,8	561,81	329,12	201,15	127,68	83,837	56,747	39,474
t	90	100	110	120	130	140	150	160	170
η	11,929	12,269	12,612	12,956	13,301	13,647	13,992	14,337	14,681
ν	28,141	20,511	15,251	11,547	8,8853	7,9770	5,4912	4,3983	3,5615

7:
Viscosity of saturated steam

t [°C]; η [μPa·s]; ν [nm²·s^-1].

Zde článek jak tlak může snižovat viskozitu tuhých látek a jak je to důležité pro tektoniku zemských desek.https://t.co/wRSp0SHXYN
— Jiří Škorpík (@jiri_skorpik) July 2, 2024

Equations for calculating viscosity of mixture: In engineering practice, mixtures of gases or liquids are common, with viscosity depending on the molar concentrations of their components (see Equation 8 and Problem 3).

Equations for calculating viscosity of mixture

η_i [Pa·s] dynamic viscosity of individual mixture component; δ_i [1] mole fraction of individual mixture component. The equation is valid for cases where the individual viscosities are independent of the partial pressures of the individual components.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.9

t	-20	0	10	20	40	60	80	100	150
η	16,28	17,08	17,75	18,24	19,04	20,10	20,99	21,77	23,83
ν	11,93	13,70	14,70	15,70	17,60	19,60	21,70	23,78	29,50
t	200	300	400	500	600	700	800	900	1000
η	25,89	29,70	33	36,20	39,10	41,70	44,40	46,60	49,30
ν	35,82	48,20	63	79,30	96,80	115	135	155	178

9:
Dry air viscosity at 0,1 MPa

t [°C]; η [μPa·s]; ν [μm²·s^-1].

t	10	20	40	60	80	100
ϕ	η	η	η	η	η	η
0,2	17,73	18,20	18,91	19,75	20,15	20,12
0,4	17,71	18,16	18,79	19,43	19,45	18,96
0,6	17,69	18,12	18,67	19,13	18,86	18,10
0,8	17,67	18,09	18,56	18,85	18,35	17,43
1	17,65	18,05	18,45	18,59	17,91	16,90
	ν	ν	ν	ν	ν	ν
0,2	14,67	15,63	17,35	18,86	19,77	19,66
0,4	14,63	15,56	17,11	18,17	18,16	16,75
0,6	14,60	15,49	16,87	17,53	16,80	14,60
0,8	14,57	15,43	16,64	16,93	15,62	12,93
1	14,54	15,36	16,42	16,38	14,60	11,61

10:
Viscosities of moist air at 0,1 MPa

t [°C]; η [μPa·s]; ν [μm²·s^-1]; ϕ [1] relative humidity

Entrance length

Calculation of entrance length and Reynolds number: The velocity profile along the length under investigation may not be constant, especially if it is the entrance section of the channel where the boundary layer has yet to develop, see Figure 1. The entrance length x_e of the channel at which the boundary layer develops is a function of the channel shape and the ratio between the dynamic pressure and the shear stress in the fluid, which we refer to as the Reynolds number Re, see Equation 11.

Calculation of entrance channel length and Reynolds number

11:

C_h [m] entrance length coefficient; L [m] characteristic dimension; Re [1] Reynolds number (Re is added to formula for x_e when boundary layer is fully developed) - formula for Reynolds number is derived in Appendix 8.

Entrance length coefficient for pipe: The values of the entrance length coefficient for a circular pipe are in the range C_h≈0,025...0,065 - the value 0,065 was derived by the French physicist and mathematician Joseph Boussinesq (1842-1929), the value 0,025 by the German physicist Ludwig Schiller (1882-1961). It can be said that higher values correspond to shorter and lower to longer x_e [Bauer et al., 1950, p. 143]. C_h factors for channels of non-circular flow area are given in [Latif, 2006, p. 208] and a selection in Table 12.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.10

	C_h		C_h
t=h	0,09	h=4·t	0,075
h=2·t	0,085	h·t^-1≈∞	0,011

12:
Entrance length coefficients of rectangular channels

C_h [m]; h [m] longer side of rectangle; t [m] shorter side of rectangle.

Calculation of characteristic dimension (Equivalent diameter) of non-circular channels: The characteristic dimension in Formulas 11 takes into account the dimension of the flow channel or wrapped body. It is the dimension to which any measurements are made. The characteristic dimension of closed channels is most often defined by Formula 13 - in the case of a circular flow area it is the diameter, therefore the characteristic dimension is also called the equivalent diameter. However, there are also a number of atypical cases where a different definition of the characteristic dimension is used. In general, the characteristic dimension of a body is the dimension that has the greatest influence on the flow (for example, in the case of blade profiles, it is the length of the chord).

13:

A [m²] flow area; u [m] wetted circumference of channel (perimeter of channel flow area in contact with flowing fluid).

Laminar flow

The fundamental parameters of laminar flow can be determined using the Navier-Stokes equation. The Euler equation of hydrodynamics is also a special form of the Navier-Stokes equation for the case of insignificant viscosity effects. In addition, the Navier-Stokes equation can be used to derive equations for pressure loss or loss heat for the case of channels of simple shapes, for example, Poiseuille law for pressure loss in a circular pipe, the relationship between mean velocities determined from mass flow and kinetic energy of the fluid, etc.

General form of Navier-Stokes equation for incompressible fluid: The amount of loss heat increases in the direction of flow, hence, and using the definition of viscosity, the equation of laminar fluid motion, also known as the Navier-Stokes equation, can be derived, see Equation 14. The Irish mathematician George Gabriel Stokes (1819-1903) is added in the title to honor him because he experimented further with the equation and described its possibilities in depth, although there were more scientists who developed it.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.11

14:

g [m·s^-2] gravitational acceleration; grad L_q [J·kg^-1·m^-1] gradient of loss heat ( amount of loss heat released in 1 kg of fluid when displaced 1 m in given direction); p [Pa] pressure; s^→ [m] unit direction vector; V·∇)V [J·kg^-1·m^-1] change (gradient) of kinetic energy in flow direction. The equation is derived for the case of steady laminar flow of a viscous fluid at constant density in Appendix 9; for the general case of non-steady flow with variable density, the Navier-Stokes equation is derived in [Bird et al., 1965], where it is referred to as the equation of motion.

Meaning of loss heat in the Navier-Stokes equation: The loss heat L_q is the cause of the increasing entropy of the fluid. The increase in the loss heat can occur not only during friction, but also during swirling between individual streamlines. In gases, part of the loss heat respectively internal thermal energy can be transformed back into pressure, kinetic or potential energy and work. This is due to the fact that the specific volume of the gas increases when the temperature increases. For this energy, the concept of re-usable heat is used in turbomachinery theory [Škorpík, 2024]. The loss heat equation also implies that a gas at very low density or pressure can have very high internal friction.

Euler equation of hydrodynamics as simplified Navier-Stokes equation for frictionless flow: The Euler equation of hydrodynamics (Equation 15) is an equation derived from the force equilibrium of a fluid element without considering the effect of frictional forces on this element. It can be easily obtained from Equation 14 for the case that the left-hand side equals 0. From Equation 15, it is well seen that the scalar multiple of the velocity vector and the velocity divergence (V·∇)V is the fluid acceleration, which can be broken down into the form of Equation 15(left) as the sum of the velocity kinetic energy gradient and the velocity vector products - that is, laminar flow is not potential flow because in the case of potential flow, the acceleration is only equal to the kinetic energy gradient.

15:

The derivation of the Euler equation of hydrodynamics for vortex flow and the relation to potential flow are shown in Appendix 10.

Hladkost Navier-Stokesových rovnic byla zpochybněna https://t.co/bKE39DKZze
K tomu bych dodal zamyšlení, že E. rovnici hydrodynamiky bych za podobnou s rovnicí NS z pohledu fyzikální reality moc nepovažoval. ERH vychází ze silové rovnováhy proudění, NS z energetické rovnováhy.
— Jiří Škorpík (@jiri_skorpik) January 14, 2023

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.12

Solution of Navier-Stokes equation for laminar flow in pipe is referred to as Poiseuille law: Deriving equations for the pressure loss and fluid velocity for laminar flow in channels of simple shapes is not difficult using the Navier-Stokes equation, see Problem 4 for flow between two plates and Equation 16 for a pipe. The equations for laminar flow through a circular flow area were first derived by the German engineer Gotthilf Hagen (1797-1884) and the French physicist Jean Poiseuille (1797-1869), hence they are sometimes referred to as Poiseuille law. The validity of this equation (except for very short sections) was confirmed by the German-Georgian engineer Johann Nikuradse (1894-1979).

16:

L_p [Pa] pressure loss on investigated length of pipe; l [m] length of pipe; r_e [m] inner radius of pipe; Q [m³·s^-1] volumetric flow; r [m] distance of investigated radius from centre (axis) of pipe; V [m·s^-1] axial velocity component (in direction of pipe axis). The relation is derived in Appendix 11 for the case of steady flow of incompressible fluid in a circular pipe, neglecting the effect of potential energy from the Navier-Stokes equation.

Equations for the average velocity of laminar flow: The derivation of the equations for the mean velocity of laminar flow according to their defining Equations 3 is easy for simple channel shapes, see Equation 16 and Equation 17 for the mean velocities in the pipe and laminar flow between two plates.

17:

(a) equation of mean velocity for laminar flow between two plates; (b) equation of mean flow velocity for laminar fluid through pipe. The equations were derived for a constant fluid density ρ=const. The derivation of the equations is shown in Appendix 12.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.13

Turbulent flow

Between the streamlines of laminar flow, a pair of forces acts on the fluid elements (see Figure 5), this pair of forces drives the elements into rotation. This means that a series of tiny vortices are formed between the streamlines, which dissipate their energy by friction, respectively their kinetic energy is constant, but at higher velocities the energy in the vortices gradually increases. Eventually, the vortices may gain such energy that they begin to disrupt the boundaries of the streamlines, causing the flow to mix and share energy. Turbulent flow occurs, which usually has a very different velocity profile from that of laminar flow. The speed at which this occurs is called the critical velocity of laminar flow, which can be calculated from the critical Reynolds number for a given case. At this velocity, the inertial forces of the particles dominate over the frictional force.

Turbulent velocity profile in the pipe: The turbulent velocity profile is characterized by the fact that it does not show such a significant difference between the velocity at the core and at the margins of the flow as the laminar velocity profile, see Figure 18. This is due to the migration of fluid particles and therefore kinetic energy throughout the flow area (see Figure 2). The shape of the turbulent flow velocity profile can be determined by the equations given, for example, in [Bird et al., 1965].

18:

1-velocity profile of laminar flow; 2-velocity profile of turbulent flow. V_max [m·s^-1] maximum velocity in turbulent profile. Data for velocity ratios [Maštovský, 1964, p. 78], [Mikula et al., 1974, p. 57].

Entrance length of turbulent flow: The critical flow velocity does not guarantee the existence of turbulence in the entire fluid volume under investigation. Turbulent flow develops from laminar flow as the inertial forces in the boundary layer increase, see Figure 20. For example, fully developed turbulent flow in a pipe is found only in the region of the pipe 10 to 60 pipe diameters away from the mouth [Jícha, 2001, p. 66].

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.14

20:
Development of turbulence during plate wrapping

LBL-laminar boundary layer; TBL-turbulent boundary layer. δ [m] local thickness of boundary layer; x [m] distance from edge; x_crit [m] start of transition from laminar to turbulent boundary layer (formula according to [Latif, 2006, p. 296]).

Shortening entrance length using vortex generators: The length of the section over which the flow begins to turbulise also depends on the geometry of the entrance, where the streamlines may interfere with the entrance edges, and also on the surface roughness. This is the principle behind vortex generators, which are designed to shorten the entrance length, for example for the purpose of mixing the streams, or for the purpose of evenly distributing the kinetic energy and momentum of the stream as one of the measures to reduce the sensitivity to flow separation from diffuser walls, etc.

Critical Reynolds number values in pipes: The critical velocity at which turbulence begins to form can be determined from the critical Reynolds number Re_C. The value of the critical Reynolds number for the pipe obtained by experiments is Re_C=2320. However, it can be higher in some cases, so the range Re=2320 to Re=5000 to 6000 is referred to as the transition region. This means that at these values there is some probability for both laminar and turbulent flow. From Re=6000 onwards (the so-called upper critical Reynolds number) the flow is always turbulent. It should be noted that in practice these values will be lower, as the values given here are from measurements in laboratories on perfectly seated pipes without vibration.

Probability of exceeding the critical Reynolds number: Figure 19 is a nomogram for the calculation of the Reynolds number with the transition region in the pipe flow marked. The nomogram shows, among other things, that laminar flow normally occurs only at very high kinematic viscosities and low velocities - most likely to be encountered in small diameter ducts - otherwise the Reynolds numbers are significantly greater than the critical Reynolds number.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.15

19:
Nomogram for reading off Reynolds numbers

V‾ [m·s^-1]; L [mm]; ν [m²·s^-1]; Re [1]. a-range of kinematic viscosities of water between 0 °C and 100 °C; b-range of kinematic viscosities of dry air between 0 °C and 100 °C. Re_C [1] range of critical Reynolds numbers for pipe.

Disappearance of turbulence

Laminar boundary layer formation: The turbulent flow can revert to laminar flow if the Reynolds number drops below the critical Reynolds number. For example, if we embed a plate in a turbulent flow, a laminar boundary layer will form on both sides of the plate, see Figure 20. Another example is the change in pipe diameters, as indicated in Figure 21. In this case, a turbulent flow is sucked through an embedded pipe at the mouth of which a laminar boundary layer is formed which, if the Reynolds number is low enough in this channel, can merge to form a laminar profile throughout the flow area. The same effect of laminar layer formation can be observed in blade passage flow, even if the inlet flow is turbulent. The inserts in the flow in which a laminar layer is to be formed or maintained is called a laminar flow element.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.16

21:
Example of transition of turbulent flow into laminar flow

1-fully developed turbulent profile; 2-areas of laminar boundary layer formation.

Frictionless flow

Models of ideal fluid frictionless flow, especially analytical ones, are used in the first iteration of fluid machinery design. Based on the results of these models, a CAD computer model can be constructed, which can be further optimized using numerical models. Frictionless flow is based on the assumption of non-vortex motion. This assumption is based on observation and on the forms of energy transformation in the fluid. Non-vortex fluid motion satisfies the conditions of a potential vector field, and therefore, to describe frictionless flow, we use a mathematical model of potential flow, i.e., its quantities such as velocity and velocity vector, density, pressure, etc. are potential quantities dependent only on coordinates, which is a property we expect from an analytical description. This gives rise to many mathematically unique properties. The potential flow model is easy to solve and, under suitably chosen conditions, gives results close to real flow with friction.

Construction of potential flow streamlines: The function that describes streamlines is called the stream function, and velocity vectors are tangents to this function. The velocity vectors of potential flow are also gradients of some potential quantity u, which we call the velocity potential, see Figure 22. The gradients in the potential vector field are perpendicular to the potential surfaces, so the stream function must be perpendicular to the function describing the curve u=const. Both functions can be derived using mathematical rules for potential vector fields, see Figure 22. For example, to construct streamlines, it is sufficient to know the curve of constant velocity potentials and vice versa, see also the application in deriving velocity in a irrotational vortex below. The curve u=const. can also be obtained graphically using the normals of the flow surfaces, because the boundary streamlines copy these surfaces, etc.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.17

22:
Equation of potential flow in plane

u=f₁(x, y) [m²·s^-1] velocity potential; ψ=f₂(x, y) stream function. The derivation of equations for potential flow in a plane is shown in Appendix 13.

Fluid acceleration: The fluid acceleration in potential flow is equal to the gradient of its kinetic energy, which can be proven using mathematical formulas derived for potential vector fields (see the article Engineering Mathematics [Škorpík, 2023]).

Circulation of velocity indicates lift of body: The circulation of the potential flow velocity along a closed curve must be zero. If there is a body or singularity inside the curve, then the circulation is non-zero, with the resulting number providing specific information about these entities inside the curve. If it is a body inside the potential flow (Figure 23a), respectively the curve along which we carry out the circulation, then the result of the circulation is directly proportional to the lift of this body.

23:
Examples of potential flow velocity circulation

(a) velocity circulation around wrapped body; (b) irrotational vortex (potential vortex). L [N·m^-1] lift of body profile; ρ_∞ [kg·m^-3] density in front of body; a₁ [m²·s^-1] constant; V_θ [m·s^-1] tangetial component of velocity; K, L-curves along which velocity circulation is carried out. More information on potential vortices and the derivation of the equation for tangetial velocity is shown in Appendix 14.

Irrotational vortex: Potential flow along circular streamlines creates a so-called irrotational vortex (Figure 23b) – a typical model of frictionless flow around an axis in turbines and other turbomachines. The result of circulation along a circle L around the center of the irrotational vortex is equal to the product of the velocity and the circumference of the circular path on which the circulation is carried out.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.18

Using irrotational vortex effect in vortex tubes: The irrotational vortex is very similar to vortices that occur naturally in water and air, such as tornadoes. Potential vortices are characterized by very high flow velocities near the center of the vortex, and therefore, according to the law of conservation of energy, the pressure and internal thermal energy must decrease towards the center, and vice versa. This effect is used in so-called vortex tubes (Figure 24) to separate cold and hot gas – this gas is then extracted depending on whether the application requires heating or cooling. In the case of liquids, a device that uses the energy distribution in the vortex is called a vortex pump, where the liquid is pumped through the center of the vortex tube.

24:
Schéma vírové trubice

a-tryska; b-plášť trubice; c-vnitřní kanál; d-štěrbina u obvodu trubice pro odvod horkého stlačeného plynu; C-výtok studeného plynu; H-výtok horkého plynu. 1-tangenciální vtok plynu do trubice; 2-odběr studeného plynu. ∂p/∂r [Pa·m^-1] gradient tlaku.

Paradox rychlosti laminárního proudění: An interesting paradox can be observed in the equations of laminar flow. This is a flow where some quantities are potential, such as the velocity of fluid in a pipe (Equation 16) or between two plates (Problem 4), but others are not, such as the vector of this velocity, which does not create a potential vector field—it is not the gradient of a potential function, see Problem 5.

Problems

Problem 1:

Calculate the characteristic boundary layer thicknesses for the flow between two plates if the velocity profile were parabolic. Choose the maximum flow velocity, channel width, channel height and fluid density. The solution to the problem is shown in Appendix 1.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.19

Parabolický rychlostní profil mezi dvěma deskami

t [m] distance of plates; δ [m] characteristic thickness of boundary layer.

Problem 2:

Determine the stress tensor in a fluid at laminar flow between two plates if the pressure p is at the point under investigation. The solution to the problem is shown in Appendix 2.

Problem 3:

Determine the viscosity of a mixture of nitrogen N₂ and oxygen O₂ under standard conditions. The mole fraction of nitrogen for this mixture is 0,785. The solution to the problem is shown in Appendix 3.

Problem 4:

Determine the equations for loss heat, pressure loss and velocity for the case of steady fully developed laminar flow of an incompressible fluid between two plates. The solution to the problem is shown in Appendix 4.

Probelm 5:

Determine the velocity gradient of laminar flow in a pipe and derive the equation for velocity increment. Make the velocity vector of laminar flow circulate along a closed curve and decide whether the velocity vector has potential. The solution to the problem is shown in Appendix 5.

INTERNAL FLUID FRICTION AND BOUNDARY LAYER DEVELOPMENT

7.20

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