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A hydrodynamic profile is a contour with an elongated shape with respect to the direction of stream, rounded at the front edgecalled leading edgeand having a peak at the back edge, called trailing edge.
In what follows we shall stress on some of the elements, which characterise the profile.
a) The chord of the profile is defined as the straight line which joins the trailing edge A, with the point B, in which the circle
Fig.9.1
with the centre in A is tangent to the leading edge; the length of the chord will be noted by c (fig.9.1).
b) The thickness of the profile is measured on the normal to the chord and is noted by e. This thickness varies along the chord and reaches a maximum in a section which is called section of maximum thickness, situated at the distance _{} to the leading edge.
c) Relative thickness, _{}, and maximum relative thickness, _{}, are defined by the relations:
_{} (9.1)
d) The framework of a profile, or the line of mean curvature, is the curve that joins the mean thickness points. The shape of the framework is an important geometric parameter and is linked to the curvature motion of the profile.
From this point of view, profiles can be with simple curvature (fig.9.1) or with double curvature (9.2).
e) The arrow of the profile, f, is the maximum distance, measured on the normal to the chord, between the framework and the chord of the profile.
f) The extrados and intrados of the profile represent the upper and lower part of the profile, respectively.
By the geometric shape of the trailing edge, which plays an important part in the theory of profiles, we may distinguish among three categories of profiles:
Fig.9.2
Jukovski profiles, profiles with a sharp edge, for which the tangents to the trailing edge at extrados and intrados superpose (fig.9.3 a)
KarmanTrefftz profiles, or profiles with a dihedral tip, for which the
tangents to the extrados and the intrados make an angle _{} in the
trailing edge (fig.9.3 b),
Carafoli profiles, or profiles with the rounded tip, for which the trailing
edge ends in a rounded contour, with a small curvature radius.
(fig.9.3c).
It is generally studied the plane potential motion around the hydrodynamic profile, considered as the intersection of the complex plane of motion with a cylindrical object (called wing), normal on this plane and having an infinite length (called span).
In reality, wings have a finite span and, from a geometrical point of view, they are characterised by the section of the wing, which, generally, alters
Fig.9.3 a, b, c
the length of the wing and the shape of the wing in plane.
By the shape of the wing in plane, there are: rectangular wings (fig.9.4), trapezoidal wings (9.4 b), elliptical (9.4 c), and triangular wings (9.4 d).
Fig.9.4 a, b, c, d
An important parameter of the wing is the relative elongation defined by the relation:
_{} (9.2)
where l and S represent the span and the surface of the wing, respectively.
In the particular case of rectangular wing, the length of the chord is constant _{} and relation (9.2) becomes:
_{}
since:
_{}
We can classify wings by their elongation _{}; into:
wings of infinite span, when _{};
wings of finite span, when _{}.
KuttaJukovski’s relation (5.62) can be applied to any solid body in relative displacement with respect to a fluid.
It indicates that whenever there is a circulation _{}around a body, there arises a lift force _{}, whose value is determined, under the same circumstances of environment (_{}), by the intensity of circulation.
To get a higher circulation around bodies, we can act in two ways:
for geometrical symmetric bodies: they are asymmetrically placed with respect to _{} direction or a rotational motion is induced (an infinitely long cylinder, sphereMagnus effect).
for asymmetrical bodies: study of shapes more proper to circulation.
On the basis of many theoretical and experimental studies, we have come to designing wings with a high lift, called hydrodynamic profiles.
Fig.9.5
In fig.9.5, the arising of circulation around the hydrodynamic profile, alters the spectre of lines of rectilinear stream, of velocity _{} as follows: on the extrados the sense of circulations coincides with that of motion and is seen as a supplement of velocity _{}, and on the intrados velocity is decreased with _{}.
According to Bernoulli’s law, the velocities asymmetry brings about the static pressures asymmetry (high pressure on the intrados, low pressure on the extrados) as well as the arising of lift force.
Applying Bernoulli’s relation between a point at _{} and a point on the profile, we get:
_{} (9.3)
The pressure coefficient is defined by the relation:
_{} (9.4)
In fig. 9.6 it is shown the distribution of pressure and of the pressure coefficient on a hydrodynamic profile at a certain angle of incidence, _{}.
Fig.9.6
The alteration of the incidence angle leads to the shift in the pressures distribution.
The forces which act upon hydrodynamic or aerodynamic profiles: lift, shape resistance, friction force or the force due to the detachment of the limit layer give a resultant _{} which decomposes by the direction of velocity in infinite and by a direction which is perpendicular on it (fig.9.7). Component _{} is called resistance at advancement, and component _{}, lift force.
They are usually written in the form:
_{} (9.5)
where _{} is called the coefficient of resistance at advancement, and _{} the lift coefficient (_{} for profiles of constant chord).
Fig.9.7
Force _{} can also decompose by the direction of chord (component _{}) and by a direction perpendicular on the chord (component _{}).
These components may also be expressed with the help of coefficients:
_{}  the coefficient of tangent force and _{}  the coefficient of normal force.
For a certain angle _{} is the distance between the leading edge and the pressure centre (the application point of hydrodynamic force).
The relation expresses the moment of the force R with respect to the leading edge:
_{} (9.7)
Also, moment M can be expressed by an analytic form similar to that used for the components of hydrodynamic force:
_{} (9.8)
Using (9.5), (9.7), and (9.8), we get:
_{} (9.9)
In the case of small incidence angles:
_{} (9.10)
The usage of coefficients _{}, _{} and _{} is often met in actual practice. Their variation is studied in different conditions and given in the form of tables and graphics of great importance for the calculus and design of systems, which deal with profiles.
Coefficients _{}, _{} and _{} depend on the following main elements:
the shape of the profile;
the span of the profile (finite or infinite, finite of small span or great span);
the type of the flow (Reynolds’ number);
rugosity of surfaces;
the angle of incidence.
For each shape of profile, at certain different relative elongation, _{}, (see paragraph 9.1), in the case of certain flow velocities (numbers Re variable), there are diagrams experimentally established _{}.
Fig.9.8
In fig. 9.8 there are plotted the diagrams of coefficients for resistance at advancement and for lift force for a NACA 6412 profile, of relative elongation 3, at a number Re = 85,000.
Another type of diagram often used is the polar profile, namely the function _{} at different slanting angles (fig.9.9). The polar allows us to define two characteristics of the profile:
the floating or gliding coefficient:
_{} (9.11)
aerodynamic accuracy:
_{} (9.12)
Fig.9.9
For wings of great span, considered infinite _{}, the motion around the profile is plane. Circulation _{} may be replaced by a whirl.
In reality, at the tips of the wing, because of the difference in pressure, there arises a motion of fluid from intrados to extrados (9.10). The greater the weight of this motion, the smaller the wing span is.
Fig. 9.10
As a consequence, circulation _{} is no longer constant; at the tips there is a minimum. (fig.9.11).
This leads to an alteration of hydrodynamic parameters, through the arising of the socalled induced resistance.
In fig.9.12 the scheme of hydrodynamic forces for the wing of finite span is plotted.
Due to the arising of an induced velocity _{}, created by the free whirl, perpendicular on the velocity in infinite _{}, the resultant velocity becomes:
_{} (9.13)
Fig.9.12
As a consequence there will appear an induced incidence angle _{}, which thus decreases the incidence angle _{}.
The alteration of direction and value of velocity bring about the corresponding alteration of lift, which, as we have already shown, is perpendicular on the direction of stream velocity.
If _{} is the lift of the infinite profile and _{} is the lift under the circumstances of an induced velocity (perpendicular on the direction of velocity _{}), then:
_{} (9.14)
In the conditions of very small values of _{}, we may assume that _{}, namely lift does not alter.
Component _{} acting on the direction Ox is called induced resistance and may be written in the form:
_{} (9.15)
The total resistance of the wing of infinite span is the sum between the resistance of wing of infinite span _{} and the induced resistance _{}.
Several profiles that are in the stream of fluid are in reciprocal influence, behaving in a different manner within the assembly, rather than solitary. Networks of profiles are often met in practice in the hydraulic or pneumatic units, propellers, etc.
To study the behaviour of profiles in network, let us consider a system made up of several identical profiles, of span l and control contour ABCD (fig.9.13). The pitch of the network is t.
Fig.9.13
Velocities v in points 1 and 2 have the components _{} and _{}, according to the system of axes shown in the figure. Assuming that the density of fluid doesn’t alter in a significant way when passing through the network, _{}, then _{}.
Indeed, applying the equation of continuity:
_{} (9.16)
it results _{}.
We have denoted by m the massic flow. Applying the theorem of impulse, we get component _{} of the lift force in the network:
_{} (9.17)
The circulation of velocity on the control contour will be:
_{} (9.18)
The integrals on the segments of contour BC and AD cancel reciprocally. There only remains:
_{} (9.19)
Therefore:
_{} (9.20)
Replacing (9.20) into (9.17), we get:
_{} (9.21)
The axial component _{} is due to the difference of pressure:
_{} (9.22)
Applying Bernoulli’s equation between the points 1 and 2, we get:
_{} (9.23)
or else:
_{} (9.24)
Replacing (9.24) into (9.22). we get:
_{} (9.25)
The resultant force will be:
_{} (9.26)
In relation (9.26) we have denoted by _{} the coefficient of the network and by _{} the mean velocity in the network (fig.9.14).
_{} (9.24)
Fig.9.14
The lift force is perpendicular on _{}. Coefficient _{} is different from the hydroaerodynamic coefficient corresponding to a separate profile.
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