DISTRIBUTION OF MERIDIONAL FLUID VELOCITIES IN THE INTERBLADE CHANNELS OF A FLUID COUPLING

Yu. I. Kapustin

(Presented by the scientific seminar of the Department of Applied Mechanics)

Existing methods of hydraulic calculation of the impellers of fluid couplings assume that the work performed in the circulation circle is equivalent to the work of the mean stream filament. Therefore, the design and calculation of the impellers of a fluid coupling are carried out according to the mean stream filament. The location of this filament is determined from the assumption of constancy of the meridional velocity both in sections of lines orthogonal to the streamlines and along the streamline.

Investigations of the direction of the velocity of the circulating flow, carried out by the author, have shown that the mean meridional velocity in a section orthogonal to the streamline does not remain constant along the streamline. At the same time, the coefficient of variation of the mean meridional velocity may be taken as constant for almost all slip regimes. Measurements of the meridional velocities of the flow, carried out by S. N. Kozlov [1], showed that in sections of lines orthogonal to the streamlines there is also no constancy of velocity. Analyzing the diagram of the distribution of meridional velocities in these sections, the following can be established:

a) the diagram of the distribution of velocities over the meridional section is almost identical for all slip regimes;

b) the radius of the position of the maximum velocities of the fluid flow at the inlet to and outlet from the turbine wheel does not depend on slip.

This makes it possible to approximate the diagram of the distribution of fluid velocities over the meridional section by an empirical dependence. The approximation may be performed as follows.

As has been pointed out by many authors [2], [3], the flow in a fluid coupling is turbulent in character. In the case of turbulent flow, the tangential stresses in the fluid depend on the circumstances of the motion, which are different for different distances from the wall of the channel [4]. As a result, the analytical calculation of the velocity diagram can be solved only approximately, by using various assumptions and hypotheses. Similar problems have been solved by many authors (L. Prandtl, T. Karman, A. N. Patrashëv, and others).

L. Prandtl obtained a logarithmic law for the distribution of velocities over the live cross section of a flow. These formulas, which give good agreement with experimental data for the main part of the core of the flow, sometimes do not satisfy the boundary conditions. In practice, approximate formulas expressing the law of velocity distribution are more convenient.

flow in the form of power functions. T. Kármán proposed writing this law in the form

\[ u = u_{\max}\left(1-\frac{r}{r_0}\right)^{\frac{1}{m}}, \tag{1} \]

where \(u_{\max}\) is the maximum flow velocity in the section,
\(r_0\) is the radius of the live section,
\(r\) is the radius of an elementary stream filament,
\(m\) is a coefficient.

This formula was proposed for the distribution of velocities in turbulent flow in pipes of circular cross section, in which the maximum flow velocity is located at the center of the section. The flow in a fluid coupling has a somewhat different velocity distribution, if only the inlet section of the flow into the turbine wheel is considered. In the vortex center, which may be compared with the center of a pipe of circular cross section, the flow velocity is zero. In order that relation (1) satisfy this boundary condition, we write it in the following form:

\[ w = aw_{\mathrm{avg}}\left(1-\frac{y}{L}\right)^{\frac{1}{m}}\cdot\frac{y}{L}, \tag{2} \]

where \(y\) is the coordinate of the position of an elementary stream filament at the inlet to the turbine wheel,
\(L\) is the length of the inlet section of the flow in the radial direction (Fig. 1),

Fig. 1

\(w_{\mathrm{avg}}\) is the mean meridional velocity at the inlet to the turbine wheel,
\(a\) and \(m\) are dimensionless coefficients.

We shall find the coefficients \(a\) and \(m\) in the following way. As the experiments of S. N. Kozlov showed, the position of the maximum meridional velocity at the inlet to the turbine wheel does not depend on slip, and it is located at a distance equal to approximately \(0.7L\) from the center of the vortex zone (Fig. 2). Consequently, at \(y = 0.7L\) we have

\[ \frac{dw}{dy}=0. \]

Proceeding from this condition, we find the value of the coefficient \(m\). Taking the first derivative with respect to \(y\) of expression (2), setting it equal to zero and carrying out the obvious simplifications, we obtain

\[ m=\frac{L}{L-y}. \]

For the coupling tested by S. N. Kozlov, the coefficient is equal to

\[ \frac{1}{m}=\frac{L-0.7L}{L}=0.3. \]

Fig. 2

Fig. 3

The coefficient \(a\) is found from the condition

\[ 2\pi \int_{0}^{L} w_{\mathrm{av}}(R_{\mathrm{c}}+y)\,dy = 2\pi \int_{0}^{L} a w_{\mathrm{av}} \left(1-\frac{y}{L}\right)^{\frac{1}{m}} \cdot \frac{y}{L}(R_{\mathrm{c}}+y)\,dy. \]

Carrying out the simplifications and solving with respect to \(a\), we obtain

\[ a= \frac{ \displaystyle \int_{0}^{L}(R_{\mathrm{c}}+y)\,dy }{ \displaystyle \int_{0}^{L} \left(1-\frac{y}{L}\right)^{\frac{1}{m}} \cdot \frac{y}{L}(R_{\mathrm{c}}+y)\,dy }. \]

For S. N. Kozlov’s coupling \(a=2.86\), and the distribution of meridional velocities at the inlet to the turbine wheel can be approximated by the following relation:

\[ w=2.86\,w_{\mathrm{av}}\left(1-\frac{y}{L}\right)^{0.3}\cdot\frac{y}{L}, \tag{3} \]

in this case the maximum flow velocity is equal to

\[ w_{\max}=1.39w_{\mathrm{av}}. \]

Figure 3 gives diagrams of the distribution of meridional velocities for various slips at the inlet to the turbine wheel, obtained from relation (3) (solid lines) and from the data of S. N. Kozlov (dashed lines).

Knowing the dependence of the distribution of flow velocities at the inlet to the turbine wheel, it is not difficult to obtain the distribution of meridional velocities in the outlet section of the turbine wheel.

Introduce the following notation (Fig. 1):

\(y_1\) — the coordinate of an elementary stream at the outlet from the turbine wheel,

\(k'\) — a coefficient accounting for the change in the area of the inlet flow relative to the area of the outlet flow, bounded by the elementary stream and the vortex center in the turbine wheel.

The flow area at the inlet to the turbine wheel can be represented in the form

\[ F_1=2\pi R_{\mathrm{c}}\cdot y+\pi y^2-y\delta n, \tag{4} \]

at the outlet

\[ F_2=2\pi R_{\mathrm{c}}y_1-y_1^2-y_1\delta n. \tag{5} \]

Here \(n\) is the number of blades in the turbine wheel,
\(\delta\) is the blade thickness.

Proceeding from the continuity equation, for the flow bounded by the elementary stream and the vortex zone, one may write:

\[ F_2=k'F_1. \tag{6} \]

The coefficient \(k'\), when \(y\) is variable, will also vary because the change in the meridional velocity in an elementary stream depends on its difference at the inlet and outlet in the turbine wheel. To simplify the calculations, let us assume that the coefficient \(k'\) varies with \(y\) according to a linear law, which we write in the following form:

\[ k'=1+\frac{y}{L}(k-1), \tag{7} \]

where \(k\) is the coefficient characterizing the total change in the inlet flow area relative to the outlet area, or, in other words, the coefficient of change in the mean meridional velocity.

Substituting relations (4), (5), and (7) into (6), we obtain

\[ 2\left(R_{\mathrm{c}}-\frac{n\delta}{2\pi}\right)y_1-y_1^2 = \left(1+\frac{y}{L}k-\frac{y}{L}\right) \left(2R_{\mathrm{c}}y+y^2-\frac{\delta ny}{\pi}\right) \]

or

\[ y_1^2-2\left(R_{\mathrm{c}}-\frac{n\delta}{2\pi}\right)y_1 + \left(1+\frac{y}{L}k-\frac{y}{L}\right) \left(2\cdot R_{\mathrm{c}}y+y^2-\frac{\delta ny}{\pi}\right). \]

Determining \(y_1\) from this equation and carrying out the obvious transformations, we obtain

\[ y_1=R_{\mathrm{c}}\left[ 1-\frac{n\delta}{2\pi R_{\mathrm{c}}} - \sqrt{ \left(1-\frac{n\delta}{2\pi R_{\mathrm{c}}}\right)^2 - \left(1+\frac{y}{L}k-\frac{y}{L}\right) \times \left(2\frac{y}{R_{\mathrm{c}}}+\frac{y^2}{R_{\mathrm{c}}^2}-\frac{\delta ny}{\pi R_{\mathrm{c}}^2}\right) } \right]. \]

Given the inlet radius of an elementary stream tube, one can find its outlet radius and determine the magnitude of the meridional velocity at the outlet of the turbine wheel from relation (2), replacing in it

\[ w_{\mathrm{av}} \quad \text{by} \quad \frac{w_{\mathrm{av}}}{\kappa}. \]

Fig. 4

Figure 4 shows the calculated (solid lines) and experimental (dashed lines) curves of the distribution of the meridional velocities of the fluid flow at the outlet from the turbine wheel.

As can be seen from Figs. 3 and 4, the calculated curves have only small deviations from the experimental curves of the distribution of the meridional flow velocities at the inlet to the turbine wheel for all slip regimes, and somewhat larger deviations for the outlet flow, especially in the region of small slips.

Thus, the proposed relations reflect fairly well the law of distribution of meridional velocities over the section of the fluid coupling and therefore can be used to correct the position of the equivalent stream tube and in other calculations.

REFERENCES

1. S. N. Kozlov. Measurement of the flow between the impellers of hydrodynamic couplings. Proceedings of VIGM, issue XXXIII, Moscow, 1963.
2. M. N. Etingof. Theory and Calculation of Turbo-Couplings. Oborongiz, 1947.
3. D. Ya. Aleksanpolskii. Hydrodynamic Transmissions. Mashgiz, 1963.
4. R. R. Chugaev. Hydraulics. Gosenergoizdat, 1963.