Interaction of Round Turbulent Buoyant Jets Discharged Vertically from Sources Forming

Interaction of round turbulent buoyant jets discharged vertically from sources forming an equilateral triangle

ARISTEIDIS A. BLOUTSOS AND PANAYOTIS C. YANNOPOULOS Department of Civil Engineering University of Patras University Campus, 265 00 Patras Greece http://www.civil.upatras.gr

Abstract: - The present work deals with the application of the integral method for the prediction of the mean flow and mixing characteristics of a group of three buoyant jets discharged vertically upwards in a still ambient fluid from sources laying on the apexes of an equilateral triangle. The integral forms of the momentum and tracer equations were integrated on the reduced cross-sectional area of the one jet of the group with restricted entrainment periphery, utilizing the assumption used for single buoyant jets regarding the spreading coefficient function. The analysis led to analytical expressions concerning the non-dimensional axial velocity and concentration distributions along the jet axis. The present findings are discussed and compared with the single buoyant jet solutions reported in the literature. Findings should be useful for designing purposes, laboratory simulation studies and verification of numerical models.

Key-words: - Three buoyant jets; Rosette riser; Interacting buoyant jets; Buoyant jet merging; Mean axial velocity; Mean concentration; Integral model.

1 Introduction the slot buoyant jet. Unfortunately, this solution is Discharge from tunnel outfalls with rosette risers are inappropriate for rosette-shaped buoyant jets. used to dispose wastewater into large bodies of water. However, the JETLAG model combined with the Such outfalls consist of a submarine underground tunnel VISJET model visualizes virtual reality simulations of and a series of risers each resulting in a rosette-type 3-D rosette-shaped jet groups issuing from outfalls in arrangement of nozzles, forming a horizontal equilateral moving ocean waters [6]. Due to the merging polygon. They can achieve quite high dilutions to assumption employed, this model cannot be used to satisfy environmental requirements. Also, chimneys or simulate vertical rosette-shaped buoyant jets in calm cooling towers emitting smoke and other air pollutants ambient environments. Another option of application or heat into the atmosphere might be located in the of the integral method in predicting the merging apexes of equilateral polygons. characteristics of multiple buoyant jets in a moving The integral method aided by similarity assumptions ambient fluid is briefly described by the second author for the axial velocity and concentration profiles and [7]. utilizing a suitable entrainment or spreading closure The present work is based on integral forms of the hypothesis is the most popular technique applied in momentum and tracer equations for the mean flow [4, analysing these flows. The principles for a reasonably 8]. The Entrainment Restriction Approach (ERA), accurate application of the jet integral method and which does not distinguish the major zones of jet proposal of the CORJET integral model have been merging (zone prior to merging, transition zone during established [1]. The development of the JETLAG merging and zone well after complete merging), is (Lagrangian jet) integral model that adopts “top-hat” conveniently applied, as each buoyant jet of the system profiles and the projected area entrainment hypothesis is geometrically and hydraulically symmetrical [8]. The has been made [2, 3]. On the basis of the integral method can be applied in the entire range of the initial 1/ 2 method with Gaussian profiles and closure assumptions densimetric Froude number defined as F0 =V0 g 0D , for the spreading rates, vertical round turbulent buoyant where V0 is the jet exit velocity, jets have been examined and unique semi-empirical g0 =g0 a  0 / 0  g0 / 0 the apparent gravity expressions of mean axial velocities and mean acceleration, g the gravitational acceleration,  and concentrations have been proposed [4]. However, there 0 exists one paper that deals with a second order integral a the initial jet and ambient fluid densities. model for a round turbulent buoyant jet [5]. Unfortunately, there aren’t available experimental data The merging of buoyant jets from multi-port for actual model verification. The results are discussed, diffusers has been handled as an equivalent solution to and the practical use of the theoretical findings is x illustrated. t t 2 Theoretical analysis t (ζ) O΄ t 2.1 Basic equations and boundary conditions x΄ Flow configuration for three interacting vertical round t turbulent buoyant jets issuing from sources laying on t the apexes O, O', O'' of an equilateral triangle are shown K in Fig.1. The shaded jet area is considered to be isolated r θ0 from the entire field. The effect of interaction is mathematically introduced by the reduction of this φ0 cross-sectional area, which reflects the entrainment O O΄΄ y restriction due to the lesser contact of the jet to the ambient fluid [8]. The coordinate system (z vertical) and all other pertinent quantities are defined in the same figure. For the analytical description of the mean flow characteristics, the usual approximations are accepted, (ε) and for steady turbulent flow, the usual partial as K  [K (  K )]1/ 2  0.12 , with   0.24 and differential equations in the Cartesian coordinate system c w w Z F 1 z/D (O,xyz) may be used; w, u, v and w, u, v are the 0   dimensionless dynamic axial distance. mean and fluctuating velocity components Fig.1 Flow configuration showing jet area reduction correspondingly in z, y, x direction;  mean density; and entrainment two-sided restriction due to buoyant jet merging. c  (  )/(  0 ) mean concentration;  zy , zx the mean turbulent shear stresses [4, 8]. Taking into consideration the axial symmetry of the flow with Another assumption made is that an approximate axisymmetry may be considered with respect to the jet respect to the vertical planes ( ε ) and ( ζ ), the following axis [4]. Thus, the effect of using equation (1) would boundary conditions have to be satisfied: be limited in the computation of fluxes. The mass is - Forx=y=0, zero entrainment, zero shear stresses, zero assumed to be spread axisymmetrically and the turbulent diffusion of tracer mass and w = wm3 , c = cm3 concentration spreading coefficient Kc remains essentially constant in the whole range of variation of - For x→-, w = 0 , v = v , c = 0, τ zx = 0, cv = 0 e the Froude number, even in interacting buoyant jets [8]. The value of the tangent of the half-jet angle is - For y→-, w = 0, u = ue , c = 0, τ zy = 0, cu = 0 tan  0.18 or  10.2 , as for a single round - For y=t, i.e. on the vertical plane ( ε ), as well as on the buoyant jet [4]. vertical plane ( ζ ), zero entrainment, zero shear stresses and zero turbulent diffusion of tracer mass, where ue, ve are entrainment velocity components in the 2.3 Integration of equations y, x direction, correspondingly. The momentum and tracer partial differential equations The integral method in conjunction with the have been integrated on the reduced area (grey colour in similarity assumption has been used in the present work Fig.1) on which axisymmetry has been assumed, across to reduce the partial differential equations to a set of the main flow direction, with respect to x and y, ordinary differential equations. utilizing equation (1), prescribed boundary conditions and the following definitions of the specific fluxes (Fig.1 and 2): 2.2 Special considerations The transverse profiles of mean axial velocities and Momentum flux mean concentrations are a priori assumed to be at least    m = 2 w 2rdr4  w 2rdr  2  w 2rdr quasi-similar, according to the relation: 3   0  0 (2) 2 0 t r Φ   r   0 = exp  -    (1) Weight deficit flux Φ  K z  m3   Φ         = g2 crdr  4  crdr  2  crdr  where  stands for either w or c; the spreading 3 0    0  0  (3)  0 t r0  coefficients K are considered equal to those of a single round buoyant jet and are given for velocities as 2 Kw  0.12  0.02exp(0.05Z ) and for concentrations y΄ To integrate the set of the two ordinary differential equations, the spreading concept was utilized. Integration of the tracer equation (6) with respect to z, taking into consideration that  / g  D2V / 4 and O΄ 0 0 0 solving for cm3, yields:

-2 -1   z   w  c =  m3  (7) m3 2     2 Kw Kc I wc  D   V0  2 2 where   K w  Kc / K w  0.24 [4]. Replacing this

dr expression of cm3 in the momentum equation (5), then r integrating the resulting equation with respect to z, θ 0 θ between ze and z>ze (ze is the core length or the distance O Κ beyond which the flow is considered fully established), r0 x΄ and dividing both sides of the resulting equation by Buoyancy flux 3/ 2 2 2 3/ 2 , the following is obtained: m0  D V0 / 4       = g2 wcrdr  4  wcrdr  2  wcrdr  (4) 3/ 2 3/ 2 3 0    0  0      z / D 0 t r mN me 3 2 2 2  0    =   +  F I i dz / D (8)  m   m  4 0   0   0  ze / D where I  I I / 21/ 2 I 1 . Fig.2 The mathematical definition of integral fluxes in i c w wc 2 2 the common jet area of three buoyant jets in a Taking Iid(z/D) =Fi[(z/D) ] and considering that the triangle, in a polar coordinate system. half jet spacing t/D is adequate for merging to start beyond the jet core, thus for no merging entrainment is with initial values not restricted, and that Iw= Ic= Iwc=2, hence Ii=1, the 2 2 2 D 2 D D integral in equation (8) may be written as: m0 = V0 ,  0 = g0 , 0 = g0 V0 , z / D 4 4 4 2 2 2 I i dz / D  Fi z / D  Fi ze / D  (9) 1    z / D  r   2 3 e 0  arccos   , r  t  t   0 3 For a round buoyant jet, the following substitution can   also be made,  1 2  3/ 2  3  r  1  r    m  3 2 1 2  = arccos    1    e and 0   , M1 =   -  Fi ze / D  (10) 2  t  2  t   m  4 F 2    0  0 as shown in Fig.1 and 2. Thus, the following set of two where M1 can be evaluated by the simpler formula ordinary differential equations is obtained: 2 M1=13F0 [4]. Then, the momentum equation (8), Momentum solved for the velocity ratio wm3/V0, may be written in d m the following convenient form: 3 1/ 3 =   2 d z 3 w z / D1  3 2 1  z   m3 = M +  F    (11) d   V K I 1/ 2 1 4 F 2 i D  2 2 2  2 2 0 w w  0     K w z wm3 I w  = g0 Kc z cm3 I c (5)   d z  4  2 Equations (11) and (7) constitute the problem solution, Tracer i.e. expressions for the centreline axial velocity and 2 2 d  β3  d   Kw Kc 2  concentration distributions for one jet of the group of   =  z w c I wc  = 0 (6) 2    2 2 m3 m3  three interacting buoyant jets. The values of Fi[(z/D) ] d z  g0  d z  2 Kw  Kc  can be calculated numerically using equation (9) and where the jet core characteristics [4]. I w = I 21 I 31 , I c = I 22 I 32 , I wc = I 23 I 33 ,   t  2 2 = 2erf   erf( ) = e  d I 2 j   ,  , 3 Presentation and discussion of results  K j z   0 The centreline distributions of mean axial velocities and 2  2t 2 2 = mean concentrations can be determined using equations I 3 j = 0 exp d , 0 j ,   3K z 0 j j (11) and (7), which constitute analytical expressions defining uniquely these distributions along the axis of with j=1, 2, 3, and K  2 K / 2 , K K , 1 w 2 c one jet of the group of three buoyant jets. Computations 2 2 K3 Kw Kc / Kw  Kc . of these distributions have been made in the range 6

F0=2.5 1 t/D=1.0 W m 3 t/D=3.0 V0

t/D=6.0

Single Plume (a)

F0=25 1

t/D=1.0

W m 3 V0

t/D=3.0

t/D=6.0

(b) Single Buoyant Jet

0.05 5 10 z 100 500  D

Fig.3 Variation of the predicted centreline mean-axial velocities for a buoyant jet of a group of three interacting buoyant jets with parameter the jet half-spacing t/D, compared to the corresponding variation of a single buoyant jet: (a) F0=2.5, (b) F0=25. 1

F0=2.5 F0=25

t/D=1.0

t/D=3.0 0.1 t/D=1.0

t/D=3.0

Cm 3  t/D=6.0 C0

0.01 t/D=6.0 Single Buoyant Jet

Single Plume

(a) (b)

0.001 5 10 100 500 5 10 100 500 z z   D D Fig.4 Variation of the predicted centreline mean concentrations for a buoyant jet of a group of three interacting buoyant jets with parameter the jet half-spacing t/D, compared to the corresponding variation of a single buoyant jet: (a) F0=2.5, (b) F0=25. before merging coincide with the single buoyant jet three buoyant jets discharged vertically in a still case, as it should, and that merging is delayed more and ambient fluid from sources located on the apexes of more as the jet spacing t/D becomes larger and larger. equilateral triangle. The behaviour of one buoyant jet of For large axial distances it is observed that all these the group is found similar to the single buoyant jet for cases tend to coincide with one curve being parallel to either short or large axial distances; for large distances the single buoyant jet curve. This indicates that the jet of the group seems to forget its origin behaviour obtained for the group of three buoyant jets is characteristics. Due to the lack of experimental similar to the behaviour of the single buoyant jet for evidence, the need of experiments is acknowledged for large axial distances. model verification. Unfortunately, investigation for experimental data available in the literature was unsuccessful. Therefore, experimental work on this field may be planed in the References: near future. [1] Jirka, G.H., Integral Model for Turbulent Buoyant Jets in Unbounded Stratified Flows - Part I: Single Round Jet, Environmental Fluid Mechanics, Vol.4, 4 Conclusion 2004, pp. 1-56. The application of the integral method, based on the [2] Frick, W.E., Non-empirical Closure of the Plume axial momentum and tracer conservation equations, a Equations, Atmos. Environ., Vol.18, No.4, 1984, closure assumption for spreading rates and a quasi- pp. 653-662. similarity assumption for the transverse profiles of [3] Lee, J.H.W. and Cheung, V., Generalized mean axial velocities and mean concentrations, Langrangian Model for Buoyant Jets in Current, succeeds to develop analytical solutions defining J. Environ. Engin. ASCE, Vol.116, 1990, pp. uniquely the distributions of mean axial velocities and 1085-1106. mean concentrations along the axis of one jet of the [4] Noutsopoulos, G. and Yannopoulos, P., The Round [7] Yannopoulos, P.C., Superposition Model for Vertical Turbulent Buoyant Jet, J. Hydr. Res., Multiple Plumes and Jets Predicting End Effects, Vol.25, No.4, 1987, pp. 481-502. J. Geophys. Res., Vol.101, No.D10, 1996, pp. [5] Wang, H. and Law, A. W-K., Second-order 15153-15167. Integral Model for a Round Turbulent Buoyant Jet, [8] Yannopoulos, P.C. and Noutsopoulos, G.C., J. Fluid Mech., Vol.459, 2002, pp. 397-428. Interaction of vertical round turbulent buoyant jets [6] Cheung, S.K.B., Leung, D.Y.L., Wang, W., Lee, - Part I: Entrainment Restriction Approach, J. J.H.W. and Cheung, V., VISJET – A Computer Hydr. Res., 2005 (submitted). Ocean Outfall Modeling System, IEEE, 2000, pp. 75-80.