Liu Shihe and Qu Bo
College of Water Consevancy and Hydro-eletric Power,
Wuhan
University, Hubei, 430072, China
Abstract:
Aerated jet, such as the jet flow behind the flip bucket of the overflow dam,
widely exists in hydraulic engineering. Until now model test and prototype
observation are the two main methods to study this aerated jet for a special
hydraulic project. In this paper a 3-D mathematical model for the aerated jet
was firstly suggested, and the trajectory as well as the energy dissipation rate
of the aerated jet were also discussed. By comparison with the results of model
test and prototype observation data it seems that our suggested model has high
predictive power, which is very useful in the study of energy dissipation and
jet flow atomization problems.
Keywords: hydraulics, jet
Jet flow widely exists in nature and practical engineering, thence the study of this kind of flow is very important in both theory and practice, and much work has been done for the characteristics, numerical models as well as control of the jet flow [1]. Yet until now attention has been paid much to the jet emits into the environment which has the same physical properties as the jet itself, and the work for the jet emits into the ‘foreign’ environment is relatively less though the latter widely exists. For example, the jet flow emits from the flip bucket of the overflow dam into the atmosphere in hydraulic engineering, this kind of jet usually has high enough velocity as the surrounding air often entrains into the jet, thus forms the so-called aerated jet. Until now much work has been done for this aerated jet by model test and prototype observation for special hydraulic projects, and the work by theoretical analysis as well as numerical simulation is quite less [2,3,4]. It is well known that the aerated jet is fundamental to the determination of the atomization source resulted from the aerated jet impinging with the water downstream of the overflow dam, as well as to gain an insight into the energy dissipation mechanism. In Ref. [2] a mathematical model for the plane aerated jet had been suggested. Based on this original work a 3-D mathematical model was suggested in this paper considering the effect of flow resistance and air entrainment, and the trajectory as well as the energy dissipation rate of the aerated jet were also discussed.
The aerated jet could be classified into partly aerated jet, fully aerated jet and completely aerated jet based on its aerated extent. Using Hw to represent the length scale of the jet core (the region where no bubbles entrain in the sense of time averaging), for partly aerated jet Hw≠0, for fully aerated jet Hw =0, yet for completely aerated jet the air concentration is so high that the jet almost breaks into pieces, and the jet flow might be regarded as the fluid flow driven by the moving water droplets[5]. In this paper the results for the partly and fully aerated jet are briefly introduced owing to the limited space of the paper. We choose the natural coordinate system along the jet axis to describe the aerated jet. In this coordinate system the longitudinal coordinate x is in coincidence with the jet axis, and y and z are used to represent the lateral and vertical coordinates respectively. Simplifications and assumptions used in this model are as follows:
(1) the velocity and air concentration in the aerated jet is usually so high that its cross section is not rectangular any more, yet for convenience and application in hydraulic engineering, the complicated cross section of the aerated jet is simplified as a rectangle with vertical thickness 2H and lateral width 2B, and the simplified rectangle is of the same area as the aerated jet itself;
(2) in the vertical direction the partly aerated jet could be divided into jet core region, bubble entraining region and mixing region, each with the distance between the jet axis Hw, Hm and H, for detail see Ref.[2];
(3) the initial Froude number of the aerated jet is so large that for the curvature radius R of the aerated jet we have H/R<<1, thence the curvature effect can be omitted under the first order approximation[2];
(4) for fully aerated jet the mean velocity and mean water concentration obey self-similar distribution, using U and Um to represent the mean velocity in the longitudinal direction and its maximum, and using β and βm to represent the mean water concentration and its maximum, based on Ref.[3,4] the following formula were got
where δ is a special length scale which represents the difference between the locations of the maximum mean velocity in the longitudinal direction and the mean water concentration.
(5) for partly aerated jet Eq. (1) is still useful, yet the mean water concentration should be represented as
(6) the mass flow rate of the entraining air in unit area at the outer edge of the aerated jet is where α1 is the entraining coefficient, andρa is the density of the surrounding air;
(7) the drag force along the jet axis at the outer edge of the aerated jet in unit area is
where Cf represents the drag coefficient, andρw is the density of the water.
Under the simplifications and assumptions above, based on the principle of mass conservation, momentum conservation and the geometric conditions at the jet axis, the governing equations for the aerated jet can be simplified as a set of partial differential equations. The corresponding characteristics of the aerated jet can be got by numerical integration for the giving initial conditions. To verify the mathematical model, comparison with the experimental results in Ref. [3,6,7] and the prototype observation results in Ref. [8] were made. In Fig.1 the variation of the predicted jet axis and the mean velocity with the longitudinal coordinate x were given, following by the experimental results in Ref. [3] and the predicted results by the projectile formula.
(6a)
(6b)
Fig. 1 Variation of jet axis and mean velocity
— present model for U ● experimental data for U;
┈ projectile formula for U —— present model for Y
┉ projectile formula for Y ▲ experimental data for Y.
As can be seen from Fig.1 that the predicted results by our model are in coincidence with the experimental ones, yet the results predicted by the projectile formula are not since aeration and drag force of the surrounding air were not considered. In Ref. [3] velocity coefficient of the aerated jet was also studied and analyzed, based on experimental results the following formula was got
(7)
The predicted results of
is given in Fig.2, following by the
experimental results of Ref.[3]. As can be seen from this figure that the
predicted results also coincide with the experimental ones quite well.
In the past few years much attention has been paid to determine the jump length of aerated jet owing to the consideration of the safety of hydraulic structures, and many methods have been suggested, the simplest of which is the free projectile formula (see Eq.(6)). As has been discussed above, since aeration and air resistance were not considered there exist some difference between the experimental jump length and the theoretical ones calculated by Eq.(6), sometimes the latter is even 30% larger than the experimental ones. In this paper analysis has been carried out based on the mathematical model of the aerated jet suggested above.
Fig.2 Variation of velocity coefficient
—
present model for
; experimental data for Q=421 l/s
▲experimental data for Q=355 l/s; ●experimental data for Q=221 l/s
┉ Eq. (7).
The governing equations for the mean velocity of the aerated jet can be got by the simplification of the 3D mathematical model, which are
where u0,β0 and H0 are initial mean velocity, mean water concentration and half width of the aerated jet. The trajectory of the aerated jet can be got from Eq.(8) by perturbation method using Cf /β0 as the perturbation parameter, which is
Where K is a coefficient which reflects the aeration and air resistance effect and represents
as
From Eq.(6) the theoretical jump length L1 can be got for the difference in height between the bottom of the overflow dam bucket and the downstream water level to beΔs , which is
The jump length L suggested in this paper considering
aeration and air resistance is
Where
(13)
is the initial Froude number, and ξis a non-dimensional parameter which represents as
(14)
It can be concluded from Eq.(12) that the jump length L considering aeration and air resistance would be less than the theoretical jump length for positiveθ0, yet slightly larger than the theoretical ones for negativeθ0.
Zhexi hydropower station is located in Hunan province in China. Prototype observation for the aerated jet jumped from the bucket was carried out in 1964. The jump length calculated by Eq.(12) is 105.21m, yet the observed length is105m, which is in agreement with each other.
The flow resistance in pipe flow might be reduced duo to the addition of micro-bubbles, yet for aerated jet the entrainment of air would increase the flow resistance and change the turbulent structures in it. In this paper E0 and E are used to represent the total energy of the aerated jet at the bucket and downstream water level respectively, which are
Where y0 and yT are used to represent the vertical distances between the jet axis at the bucket and the downstream water level and the datum plane respectively. The definition of the energy dissipation rate of the aerated jet in the open air is
Thence
Of courseηalso depends strongly on the entrainment coefficientα1 and drag coefficient Cf, and these two coefficients have a close relation with the energy dissipation device used.
The energy dissipation rate of the aerated jet in the open air was experimentally investigated in Ref. [3]. The experimental conditions areθ0=25°, u0=3.44m/s. The experimental results and the numerical results of this paper are given in Fig.3, as can be seen from this figure that the two results are in good agreement with each other.
By further numerical simulation it seems that the energy dissipation rate would monotonously increase with the increase of the Froude number or initial water concentration, and if the other parameters are unchanged, there exist a certain initial jump angleθ0 under these conditions the energy dissipation rate in the open air becomes the largest.
Fig. 3 Variation of energy dissipation rate
— present model; ╳ experimental data of Ref. [3].
(1) In this paper a mathematical model for the aerated jet was suggested, comparison of the numerical results with experimental as well as prototype observation results was also made, these results seems in good agreement with each other.
(2) The jump length of the aerated jet can be calculated by using Eq.(12) of this paper.
(3) The energy dissipation rate of the aerated jet in the open air can be calculated by using Eq.(18).
(4) The energy dissipation rate would monotonously increase with the increase of the Froude number or initial water concentration, and if the other parameters are unchanged, there exist a certain initial jump angleθ0 under these conditions the energy dissipation rate in the open air becomes the largest.
References
[1] Xie Xiangchun, 1975, Theory and Computation of Turbulent Jet Flow, Publication of Science (in Chinese).
[2] Liu Shihe, Liang Zaichao, 1995, On the Properties of Plane Atomized Jet, J. Hydrodynamics, Ser.A, Vol.10, No.3,pp.274-280.
[3] Liu Xuanlie, Zhang Wenzhou, 1988: The Investigation of Dynamic Characteristics of Jet Flow in Open Air, Journal of Hydroelectric Engineering, No.2, pp.46-54.
[4] Wu Chigong, Yang Yongsen, 1994: a Study on Water Concentration Distribution in Cross-sections along the Free Jet, Journal of Hydraulic Engineering, No.7, pp.1-11.
[5] Liu Shihe, Zhu Genling, 1994: Fluid Motion Driven by Moving Particles, Journal of Hydrodynamics, Ser.A, Vol.9, No.5, pp.574-580.
[6] Jiang Xinhe, Zhang Ren, 1984: Preliminary Study on the Spreading Characteristics of the Aerated Jet, Journal of Hydraulic Engineering, No.7, pp.49-53.
[7] Liu Xuanlie, Liu Jun, 1989: Experimental Study on the Diffusion and Aeration of Three-dimensional Jet, Journal of Hydraulic Engineering, No.11, pp.10-17.
[8] Central and Southern Institute of Survey and Design, 1993, General Report of the Prototype Observation Data for the Hydraulic Problems in the Ski Jump Spillway of Dongjiang Hydropower Station.