Water Science and Engineering, 9, (): 19-31 doi:1.388/j.issn.1674-37.9..3 http://kkb.hhu.edu.cn e-ail: wse@hhu.edu.cn Developent of point source ethod and its practical significance Bidya Sagar PANI* Civil Engineering Departent, The niversity of Hong Kong, Hong Kong, P. R. China Abstract: The advantages of Reichardt s hypothesis in dealing with single and ultiple circular jets in a stagnant environent are highlighted. The stages involved in the developent of the point source ethod, an offshoot of the new hypothesis, are presented. Previous results of experients on ultiple circular jets in a stagnant environent justify the ethod of superposition. As a prelude to discussion of ultiple jets in a co-flowing strea, results on the excess-velocity decay, the growth of the shear layer, and the dilutions for a single jet based on Reichardt s hypothesis are presented. The spreading hypothesis is generalized by introducing a link factor k l to account for the co-flowing strea. The distribution of excess-oentu flux uδu is shown to be Gaussian in nature. Based on the principle of superposition, the decay of the axiu excess velocity and the dilution are predicted for odd and even nubers of jets in an array. The predictions see to be in good agreeent with observed data. Key words: point source ethod; ethod of superposition; co-flowing jets; Reichardt s hypothesis; ultiple circular jets 1 Introduction Circular jets are frequently used in any engineering applications. Hence, the diffusion of circular jets has been studied extensively in the past and continues to draw attention. The early theoretical works of Tollien in 196 and Goetler in 194 were based on Prandtl s ixing length hypothesis and the eddy viscosity odel, respectively (Abraovich and Schindel 1963). The experiental easureents of circular jets provided by Trupel and Reichardt also played a significant role in validating the theoretical results (Abraovich and Schindel 1963). The theory of turbulent jets started with very siple cases of jets discharging into a stagnant abient fluid of the sae type. For the special case of a zero-pressure gradient, it can be shown that the oentu flux of the jet along the direction of flow is conserved. Based on nuerical integration of the velocity distributions proposed by Tollien and Goetler, the velocity decay relationship can be worked out fro the oentu conservation principle (Rajaratna 1976). The easured velocity distribution sees to follow the Gaussian distribution while satisfying the condition of self-siilarity. The differences between the theoretical distributions and the Gaussian distribution are sall for all practical purposes. The *Corresponding author (e-ail: bspani@civil.iitb.ac.in) Received Apr. 7, 9; accepted Jun. 6, 9
atheatical operations dealing with the Gaussian distribution are sipler than those with analytical distributions. It is therefore quite pragatic to assue the distribution is Gaussian while applying the oentu conservation equation. This procedure is coonly known as the integral technique. Interestingly, the final results of the integral ethod are not sensitive to the shape of the assued profile. Thus, the Gaussian distribution has played a central role in ost integral odels used for jets and plues. This paper presents the case of the stagnant abient in brief, followed by a ore general case of the co-flowing abient. Basic integral ethod Though the Gaussian distribution has been discussed in the literatures in different fors, the one that is used ost often is described by the following equation: where u is the velocity at the radial distance r, b g r bg u = e (1) u u is the value of u on the axis of the jet, and is the characteristic width representing the radial distance fro the axis at which u u = e. Making use of the continuity equation, the equation of otion for a zero-pressure gradient case can be integrated, resulting in the oentu conservation equation: π π ru dr = d () 4 where d is the diaeter of the nozzle, and The decay law for u is the jet velocity in the efflux section. and the law of dilution, as functions of distance x, can be derived fro any closure schee. The two popular schees are the spread relationship and the entrainent velocity. In the spread type of closure, the characteristic width is assued to grow linearly with distance x, i.e., bg = β gx. That the width b g should vary linearly with x can be established fro the requireent that the turbulent flow field ust be self-siilar. Even a siple diensional analysis will provide the required relationship. In the entrainent hypothesis of Morton and Taylor (Rajaratna 1976), the entrainent velocity v e is assued to be proportional to the difference between the centerline velocity and the outer abient velocity. For a stagnant abient, this gives v = u e α, where α is the entrainent coefficient. The link between β g and α can be easily established: βg = α. In any case, one has to depend on experiental results to find the value of either of the two coefficients. Coonly adopted values are β g =.114 and α =.57. Measureents show that the diffusion of scalar quantities like the tracer concentration and teperature results in a wider distribution profile than the velocity profile. To account for it, the concentration profile is often expressed as c λ = e (3) c r bg Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31
where c is the tracer concentration at the radial distance r and is the centerline concentration. Though the spread coefficient λ varies fro one experient to another, a value of 1. is ost often found in the literatures. Following the integral technique and aking use of the experiental coefficients, the decay of the axiu velocity concentration where C c for circular jets can be obtained as follows: u c C x = 6. d x = 5.6 d 1 1 u c and the centerline (4) (5) is the tracer concentration in the efflux section. As the dilution S along the centerline is the reciprocal of the quantity of the left hand side in Eq. (5), we can write S.19 x = (6) d For a Gaussian distribution, with a value of λ = 1., it can be proved that the average dilution is nearly 1.7 ties the centerline dilution given in Eq. (6). Thus, we can write the average dilution as (Lee and Chu 3) S.3 x = (7) d The jet loses its eory regarding the geoetric properties of the outlet a short distance away fro the efflux section. It is only the oentu flux that affects the characteristics of the diffusing jet. Hence, soe researchers prefer to use the specific oentu flux, = π π M d, in place of the diaeter d. Furtherore, the volue flux, Q = d, is 4 4 adopted at the outlet. Thus, fro Eq. (4) and Eq. (7), one can derive expressions for and the average dilution in the following alternate fors: 3 Reichardt s hypothesis u 7. M = (8) S x.9x M = (9) Q It was propounded by Reichardt (1943) that an inductive ethod, i.e., adapting a theore to the experiental facts, ight be ore suitable for analyzing turbulent flows. Noting the siilarity between the distribution of easured oentu curves and the curves of the theral conduction theory, a new hypothesis was proposed. The hypothesis is that the distribution of the turbulent oentu of the principal otion is capable of being represented by a partial differential equation, akin to that of theral conductivity. Therefore, the lateral transport of oentu is proportional to the transverse gradient of the horizontal coponent u Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31 1
of oentu: V = Λ y where Λ is a length scale that is a linear function of the distance x, and and V are the instantaneous velocities in the x and y directions, respectively. sing Reichardt s hypothesis, the equation of otion for a circular jet can be recast as 1 Λ + x r r r = Eq. (11) is a linear differential equation describing the variation of. Let us assue that the dynaic pressure distribution satisfies the self-siilarity condition and can be described as follows: = f ( ξ), (1) (11) r ξ = (1) b where b is the length scale in the oentu distribution function, and is the axiu instantaneous velocity on the axis of the jet. Eq. (1) shows that the siilarity function satisfies the conditions that if ξ =, then f ( ξ ) = 1; and if ξ, then f ( ξ ) =. With the help of Eq. (1), the governing Eq. (11) can be solved and the final solution is (Abraovich and Schindel 1963) ξ The square of the instantaneous velocity can be expressed as the fluctuating coponent of. = e (13) = u + u', in which u ' is If the contribution of the fluctuating coponent is neglected, we can replace with the ean velocity coponent u. Thus, one can write the distribution of the square of the ean velocity of the jet as u u 1 r b = e (14) Fro conservation of the forward oentu flux, given in Eq. (), the decay of the axiu dynaic pressure occurring on the axis of the jet is deterined: u A = (15) π b In Eq. (15), the area of the outlet A is retained in place of the square of the diaeter as it will serve an iportant purpose later. Coparing Eqs. (1) and (14), one can infer that both give the sae Gaussian distribution for uu, provided the length scale in the oentu distribution is b = b g. Since b, or b g, will anyway have to be found through experients, it really does not atter which of the two is used in practice. Pria facie, Reichardt s new theory offers no advantage as such in Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31
the case of a single circular jet in a stagnant abient. For a considerable tie, no great attention was given to Reichardt s theory. Hinze (1975) did ention the advantages of the new theory in the case of ultiple jets as the superposition ethod could be adopted. Siultaneously, soe of the liitations of the proposed theory were presented. The ain objection is that the oentu transport relationship does not satisfy Newton s relativity principle, which requires that the forces in a echanical syste ust be independent of the addition of a constant translation velocity. 3.1 Multiple circular jets in a stagnant abient fluid Knystautas (1964) investigated the ean flow field downstrea of an array of circular jets arranged in a line. It was clear that Reichardt s hypothesis was suitable for predicting not only the decay of axiu velocity but also the transverse variation of the strea velocity. Experiental results supported the efficacy of the new hypothesis in superposition. Many others (Narain 1976; Deissie and Maxwell 198; Pani and Dash 1983) followed a siilar approach and adopted the principle of superposition. Soe even extended the new hypothesis to the case of three-diensional (rectangular) outlets by introducing two different length scales, Λ y and Λ z, in the y and z directions, respectively (Pani and Dash 1983). All the investigations were conducted for the fully developed zone of single and ultiple outlets, and the presence of the potential core and the characteristic decay region were never in doubt. The work initiated by Sforza et al. (1966) on three-diensional jets fro a different perspective was contrasted with the solutions of the new hypothesis. Iportantly, in all the aforeentioned work, the abient fluid was stagnant and the results pertained to the fully developed flow region. Theoretically, it was difficult to erge the developing and the developed regions. 3. Point source ethod Revisiting soe existing data and their own experiental observations, Pani and Paraeswaran (1994) ade an iportant detour and introduced the concept of a point source. Though a siilar ter was used by Narain (1976), it had a different connotation. If we shrink the size of the nozzle to da, the expression in Eq. (15) takes the following for: u da = (16) π b Eq. (16) is designated the point source equation. The difference between Eqs. (15) and (16) is very subtle, yet the advantages of Eq. (16) are substantial while it derives theoretical expressions based on the principle of superposition. This will be deonstrated in the following discussion. By considering an aggloeration of a large nuber of point sources at the three-diensional outlet sized BB L, the variation of the axiu dynaic pressure along the axis of the jet can be expressed as (Pani and Dugad ) Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31 3
u B L = erf erf cx cx In Eq. (17), cx represents the length scale b used in Eq. (14) and u is the axiu value of the ean velocity u on the x-axis of the three-diensional jet. With the square root of the right hand side of the above equation, one can evaluate the ratio u /. For a circular jet, u = u. It is iportant to reeber that, in deriving Eq. (17), no distinction is ade between the developing and the developed zones, and only a single coefficient of expansion of the jet is used for all the directions. The value of the coefficient of expansion is fro that of a circular jet, and independent of the shape and size of the three-diensional outlet. Since we are aking use of a point source, it is unnecessary to have any special provision for a virtual origin. In the past, the location of the virtual origin played an iportant role in explaining the disparity between the results of various investigators. The single jet result, given in Eq. (16), acts as the building block for developing an expression for the decay of the axiu velocity of ultiple jets. The distribution of the dynaic pressure and the ean velocities easured in the laboratory agree with the predictions (Pani and Praeswaran 1994). By adopting a siilar approach and extending the new hypothesis, expressions for scalar quantities like the excess teperature and tracer concentration can be worked out. For exaple, the transverse distribution of the heat flux and its changes along the axis of a circular jet are as follows: 1 r b t uδt = e uδt uδt A = ΔT πb t where b t = λb, ΔT is the excess teperature at distance r fro the axis, Δ T is the excess teperature on the axis, and ΔT is the excess teperature at the efflux section. Measured teperature and velocity distributions (Pani and Paraeswaran 1994) can be found to be in agreeent with those obtained by Eqs. (18) and (19). 4 Co-flowing jets In any applications, especially when effluent is discharged into water bodies, the abient fluid is also in otion. If the jet is directed along the direction of the strea flow, it is called a co-flowing jet. Although single port discharge can be used in the field, ultiport diffusers see to be ore coon because of better ixing with the abient fluid. Before we discuss ultiport diffusers, it will be useful to understand the basic features of a single co-flowing jet. 4.1 Single jet forulation The ean flow characteristics of a single co-flowing jet have been studied coprehensively (Chu et al. 1999; Wang 1996; Davidson 1989). In these investigations, the (17) (18) (19) 4 Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31
excess velocity in the x direction, Δu = u, is assued to follow a Gaussian distribution. a The assuption has been substantiated by experiental observations. The concentration profile also follows the Gaussian distribution, although the tracer spreads at a faster rate than the oentu. We shall consider the case when the initial jet velocity larger than the co-flowing abient velocity shown in Fig. 1. a is significantly. The definition sketch of a co-flowing jet is Fig. 1 Sketch of co-flowing jet In the near field, the jet will behave ore like a pure jet. Hence, for an axially syetric jet, the axiu value of the excess velocity Δu is inversely proportional to distance x. With increasing distance, the axiu excess velocity Δ u becoes sall copared to the co-flowing abient velocity a. This zone is known as the far field and the jet is designated a weak jet. The weak jet is of considerable iportance in carrying out an environental ipact assessent when effluent is discharged into rivers. 3 Theoretically, Δu x for a weak jet. However, unlike the strong jet case where 13 bg x, for a weak jet the characteristic width bg x. As far as the interediate zone is concerned, no theoretical relationship exists. pon integration, the equation of otion will lead to the excess-oentu flux conservation equation: d πru( u a ) r= dx d () In ters of the conditions prevailing at the outlet, we can rewrite Eq. () as d M e = Me = πru( u a ) dr = ( a ) (1) 8 where M e is the specific oentu flux at any downstrea location and M e is the specific oentu flux at the nozzle. A very useful length scale, the oentu length scale θ, is introduced, which satisfies the following relationship: θ a = M e () The oentu length scale depends only on the velocity ratio, β =a. The Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31 5
characteristic width of the jet bg and the observed decay of the excess velocity Δu for different values of a look very different when plotted against the distance x. However, all of the collapse into unified curves when x is noralized by θ. Hence, it is assued that b θ = f ( x θ ), and Δ u = f ( x θ ). These variations expressed in the functional for g 1 a have been predicted nuerically fro the equation of otion and a spreading hypothesis in the Lagrangian frae of reference (Chu et al. 1999). 4. Single jet solutions The general spreading hypothesis (Chu 1994; Wright 1994) assues that the change in the characteristic width of the jet in a Lagrangian frae of reference is proportional to the relative velocity of the jet eleent and that of its surroundings. When odified according to the Eulerian reference, the sae relationship can be written as dbg Δu = cg (3) dx a +Δu In Eq. (3), c g represents the growth rate of the jet in a stagnant abient. An expression siilar to Eq. (3) was proposed by Abraovich and Schindel (1963). As Δu decays with distance x, the growth rate of the jet changes along with the flow. Because the coherent doinant eddies are responsible for the transport of ass and oentu, it is felt that the average excess velocity Δ u can better represent the physics of the proble. The top-hat profile, shown in Fig. 1, is based on the average excess velocity Δ and a radius equal to B. It can be shown that Δ = Δu and B = b g. The excess-oentu conservation equation and the spreading hypothesis were expressed in ters of the top-hat profile paraeters Δ and B and solved as an initial value proble (Chu et al. 1999). The solutions are shown in Figs. and 3 and can be seen to be in agreeent with the experiental data procured with non-intrusive easureent techniques. Fig. Decay of axiu excess velocity Fig. 3 Growth of characteristic width in a co-flowing strea However, these solutions cannot be extended directly to the case of ultiple jets in a co-flowing strea. There is a need to reforulate the single jet case when it is used for superposition. In the past, the excess velocity and the centerline dilution of ultiple jets were predicted by siply adding the individual velocity and concentration contributions of N jets. 6 Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31
This had no theoretical basis. Nevertheless, the observed excess velocity for 11 and 6 jets in the array were reported to agree with the predictions (Wang 1996). Even the observed dilutions for 15 jets were found to agree with the theoretical predictions (Wang ). Reichardt s hypothesis helps develop a sound basis for superposition of single jet results. Certain changes are needed, as deonstrated below. 5 Reichardt s hypothesis applied to a co-flowing jet Let us extend Reichardt s hypothesis to the case of the axisyetric co-flowing jet retaining the original for eant for strong jets. Furtherore, let us assue that the excess-oentu flux satisfies the self-siilarity condition and can be described by a Gaussian distribution: ξ uδ u = e (4) uδu where ξ is the non-diensional distance, ξ = r/b; b is the radial distance fro the axis at which uδu is.66 ties u Δ u ; and uδu = ( a + Δu)Δu, where a has a constant value. Experients have shown that the distribution of Δu is Gaussian and is accepted by all researchers. However, it cannot be deterined whether the distribution of the product uδu is Gaussian or not. It is difficult to find rigorous proof to this effect. Eq. (4) is an iportant assuption based on which the characteristics of co-flowing jets can be analyzed, since the governing equation of otion is linear as regards uδu. In order to test the validity of Eq. (4), Δu Δ u values were generated based on an assued Gaussian distribution. A velocity ratio a was assigned and the excess-velocity values were odified to include u and u. In the final stage, the products uδu and uδu were recorded in a tabular for and their ratios re-plotted. sing the length scale b fro the new plot, the distribution of uδu ( uδ u) was obtained as a function of ξ. Plots of the coputation results and the experiental data are shown in Fig. 4. Both fit very well the Gaussian distribution proposed in Eq. (4). Fig. 4 Transverse distribution of uδu/(u Δu ) Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31 7
Though this process is not atheatically rigorous, it provides convincing proof of the validity of Eq. (4). The other steps are identical to those of the stagnant abient case that led to Eqs. (14) and (15). Based on Eq. (4), Eq. (5) is derived for a source of eleent area da: uδu da = Δ πb By superposing uδu of each eleent area (da = rdrdφ), which spreads over the cross-section of the circular outlet, it can be shown that e ξ u Δ u 1 1 exp d = Δ 4 cx For given values of and a, Eqs. (4) and (6), in conjunction with Eq. (7) below, can be used to predict uδu for a single jet at a distance r fro its axis. When applying Reichardt s hypothesis, there is a ajor constraint to the large eddy spreading hypothesis. In the case of a stagnant abient fluid, the relationship between the Gaussian velocity scale bg and the Reichardt s oentu scale b is siple: b = bg for all values of x. This is not true for a co-flowing jet. The spreading hypothesis applicable to b needs to be odified as follows: db kc l g Δu = (7) dx Δu + a where k is a link factor and c, with a value of.114, is the spread rate in the velocity l g profile in stagnant abient fluid. The link factor can be proved to be (Pani et al. 8) k = l Δu a Δu a a + + 4 1 The coefficient k l changes along the jet trajectory as Δ u a becoes saller with increasing distance x. Eq. (8) shows that, in the case of a stagnant abient fluid and a strong jet, k l.5. However, for a weak jet, k l.77 as Δu a. In the transition zone, the link factor has values within the two aforeentioned liits. This variation is shown in Fig. 5. (5) (6) (8) Fig. 5 Variation of link factor with distance x 8 Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31
6 Results for co-flowing ultiple jets For the sake of siplicity, we consider N jets and let the axis pass through the nozzle at the iddle. Accounting for contributions of all the jets at a point on the x-axis, we can superpose the values of uδu and write ( u Δ u ) = uδu N For an odd nuber of jets, n = ( N 1) ; for an even nuber of jets, n = N. By siultaneously solving Eqs. (7) and (9), the variation of the jet width b and the velocity ratio Δu/ a were obtained using a 4th order Runge-Kutta ethod. Along siilar lines, the product uc calculated for each individual nozzle was superposed and, fro the consolidated su, the dilution of the tracer was obtained for λ = 1.. The details of the coputations were reported elsewhere (Pani et al. 8). n n The results on the decay of the axiu excess velocity in the case of an odd nuber of jets are shown in Fig. 6, where s is the port spacing, and d is the port diaeter. The effect of N on the variation of the width of the jet was very sall. The predicted centerline dilutions for a range of velocity ratios, β, agree with the observed values as shown in Fig. 7. (9) Fig. 6 Decay of axiu excess velocity Fig. 7 Dilution downstrea of co-flowing for ultiple jets ultiple jets (N=15, s/d=34) The siple point source ethod and the ethod of superposition have been adopted to solve any probles of practical interest, including wall jets on a sooth boundary, ultiple three-diensional wall jets, ultiple three-diensional free jets, energy loss in a junction-box, energy loss in a settling chaber, single and twin jets, finding the length of the potential core, velocity distribution in a two-diensional jet boundary layer, and the proxiity effect of the channel boundary. 7 Conclusions Application of Reichardt s inductive hypothesis linearizes the governing equation of otion of jets in ters of u. The analytical solutions of u in a stagnant abient can be superposed if there is ore than one source. The point source concept is an offshoot of Reichardt s hypothesis and the oentu conservation principle. Because of the iniature Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31 9
size of the point source, it is unnecessary to consider the location of the virtual origin. Without aking any distinction, a single spread coefficient suffices to predict the velocity decay relationships in the developing and developed zone of a jet. Thus, the traditional practice of having a potential core in the analysis is not necessary. The new ethod can be used to predict the velocity field of three-diensional jets. The dynaic pressure field of ultiple jets can be predicted based on the principle of superposition. In the case of a stagnant abient fluid, the proble is siple as the spread coefficient is constant throughout the path of the jet and the shear layer grows linearly with distance x. For co-flowing jets, the local flow conditions govern the spread rate and the shear layer growth with x is nonlinear. For a co-flowing circular jet, the original spreading hypothesis, based on large eddy otion, is odified with the introduction of a link factor. The decay of the excess velocity and the dilution of tracer concentration have been predicted for odd and even nubers of jets in an array. Agreeent was found between the predicted and the observed values. The ethod of superposition is versatile and has been applied to a host of probles of varied nature. References Abraovich, G. N., and Schindel, L. 1963. The Theory of Turbulent Jets. Cabridge: MIT Press. Chu, P. C. K., Lee, J. H. W., and Chu, V. H. 1999. Spreading of turbulent round jet in coflow. Journal of Hydraulic Engineering, 15(), 193-4. [doi:1.161/(asce)733-949(1999)15:(193)] Chu, V. H. 1994. Lagrangian scalings of jets and plues with doinant eddies. Davies, P. A. and Valente Neves, M. J., eds., Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plues, 45-7. Dordrecht: Kluwer Acadeic Publishers. Davidson, M. J. 1989. The Behaviour of Single and Multiple Horizontally Discharged Buoyant Flows in a Non-turbulent Coflowing Abient Fluid. Ph. D. Dissertation. Christchurch: niversity of Canterbury. Deissie, M. and Maxwell, W. H. C. 198. Three diensional slot jets. Journal of Hydraulic Division, 18(), 47-51. Hinze, J. O. 1975. Turbulence (nd Edition). New York: McGraw-Hill. Knystautas, R. 1964. The turbulent jet fro a series of holes in line. Aeronautical Quarterly, 15, 1-8. Lee, J. H. W., and Chu, V. H. 3. Turbulent Jets and Plues: A Lagrangian Approach. Dordrecht: Kluwer Acadeic Publishers. Narain, J. P. 1976. Moentu flux developent fro three diensional free jets. Journal of Fluids Engineering, 96(), 56-6. Pani, B. S., and Dash, R. N. 1983. Three-diensional single and ultiple free jets. Journal of Hydraulic Engineering, 19(), 54-69. [doi:1.161/(asce)733-949(1983)19:(54))] Pani, B. S., and Paraeswaran, P. V. 1994. Moentu and heat flux characteristics of three diensional jets based on point-source concept. Journal of Hydraulic Research, 3(1), 53-66. Pani, B. S., and Dugad, S. B.. Turbulent jets: Application of point source concept. Prasad, R., and Vedula, S., eds., Research Perspectives in Hydraulics and Water Resources Engineering, 1-37. Singapore: World Scientific Publishing Copany. Pani, B. S., Lee, J. H. W., and Lai, A. 8. Mixing of coflowing jets in rivers. Advances in Water Resources and Hydraulic Engineering, Proceedings of the 16th IAHR-APD Congress and 3rd Syposiu of IAHR-ISHS, 651-655. Beijing: Tsinghua niversity Press. 3 Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31
Rajaratna, N. 1976. Turbulent Jets. Asterda: Elsevier Scientific Publishing Copany. Reichardt, H. 1943. On a new theory of free turbulence. Journal of Royal Aeronautical Society, 47, 167-176. Sforza, P. M., Steiger, M. H., and Trentacoste, N. 1966. Studies on three-diensional viscous jets. AIAA Journal, 4(5), 8-86. [doi:1.514/3.3549] Wang, H. J. 1996. An Experiental Study of Single and Multiple Jets in a Coflowing Environent. M. Phil. Dissertation. Hong Kong: Hong Kong niversity of Science and Technology. Wang, H. J.. Jet Interaction in a Still or Co-flowing Environent. Ph. D. Dissertation. Hong Kong: Hong Kong niversity of Science and Technology. Wright, S. J. 1994. The effect of abient turbulence on jet ixing. Davies, P. A., and Valente Neves, M. J., eds., Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plues, 13-7. Dordrecht: Kluwer Acadeic Publishers. Bidya Sagar PANI Water Science and Engineering, Jun. 9, Vol., No., 19-31 31