Five Asymptotic Regimes of a Round Buoyant Jet

in Stratified Crossflow

 

GERHARD H. JIRKA

 

Institute for Hydromechanics, University of Karlsruhe

76128 Karlsruhe, Germany

 

 

ABSTRACT

The discharge in form of a round buoyant jet into a non-turbulent stratified crossflow may contain five asymptotic regimes in which self-similar flow conditions exist: (i) pure jet, (ii) pure plume, (iii) pure wake, (iv) advected line puff, and (v) advected line thermal. When formulating jet integral models for buoyant jet predictions it is a necessary condition that the model agrees with the five self-similar regimes. It is demonstrated that the buoyant jet model CORJET that forms part of the CORMIX expert system for mixing zone analyses indeed satisfies this requirement when compared to recent good quality data for these regimes.

 

Keywords: Turbulent jets, mixing, buoyant jets, plumes, pollutant discharges, water quality, air quality

 

INTRODUCTION

Despite some 50 years of past research the buoyant jet in a crossflow remains a topic of continuing investigation as attested by recent publications and specialty conferences (e.g. Davies and Neves, 1994). This interest is motivated in part by the important role that buoyant jets play as the initial mixing phase for pollutant discharges in air and water quality modeling. It is also stimulated by improved data sources and better experimental insight due to novel measurement techniques and last, but not least, by the boundless possibilities for formulating yet another buoyant jet model with a "new" closure hypothesis for turbulent entrainment.

The buoyant jet is a flow phenomenon with "free turbulence". It represents a gradually evolving flow along its axis and thus exhibits boundary layer characteristics with its possibilities for mathematical simplification including self-similarity techniques (Schlichting, 1968). However, because of the variety of forcing elements buoyant jet motions are, in general, not self-similar. They are self-similar only in five possible asymptotic regimes in which they have an invariant internal force balance and invariant turbulence and entrainment properties as will be examined in this paper. In between these regimes, the buoyant jet properties are variable and cannot be scaled uniquely by local jet parameters. Different assumptions can be made to describe these transition phases leading to different types of jet models.

Jet integral models in which the evolution of integrated flux parameters for fluid mass, fluid momentum and tracer mass is formulated along the jet axis are particularly simple, versatile and hence popular tools for discharge mixing prediction. A large number of such integral models can be found in the literature. Most are based on an Eulerian formulation describing the convective changes of the flux variables along the jet path, a few employ a Lagrangian formulation by which a jet element is advected in a time-dependent framework. We prefer here the Eulerian approach as more straightforward for a steady flow problem. Some models describe the evolution of the turbulent jet through a spreading hypothesis, others through an entrainment hypothesis. We prefer the latter as more directly linked to the physical growth process.

In any case, whatever its formulation and turbulence closure hypothesis, a jet integral model contains one or more coefficients that have to be specified from experimental data. From a pragmatic viewpoint, it must be stated that most integral models perform reasonably well in predicting jet trajectories, growth and dilution characteristics when compared to the bulk of experimental data (that has its own inaccuracies) even though some glaring model failures can be documented in a few instances. From a more rigorous scientific viewpoint, however, we must argue that a jet integral model must meet the test of a precise formulation for the asymptotic regimes and should furthermore contain some adequate transitions between these regimes. The model presented below is formulated according to these postulates.

 

JET FORCING FUNCTIONS AND ASYMPTOTIC REGIMES

The round buoyant jet (Fig. 1) is forced by an interplay of ambient and discharge conditions. The ambient conditions are given by a uniform non-turbulent crossflow with velocity u_a and a stable density distribution r_a(z) with a buoyancy parameter
e = -(g/r_a) (dr_a/dz) in case of a linear gradient.

The discharge occurs through a round port with diameter D, velocity U_o, density r_o and pollutant (tracer) mass concentration c_o. The vertical angle of the port is q_o, while the angle in the horizontal projection onto the xy-plane is s_o. The volume flux Q_o = U_o D² p/4 is dynamically unimportant except in the immediate vicinity and will be neglected herein. The dynamically important forcing functions are the (kinematic) momentum flux M_o = Q_o U_o, together with its orientation q_o and s_o, and the buoyancy flux J_o = Q_o g'_o, in which g'_o = g(r_a - r_o)/r_a is the initial buoyant acceleration.

Thus, the interplay given by u_a, e, M_o, q_o, s_o,J_o describes buoyant jet behavior. This can also be represented by appropriate length scales (Jirka and Doneker, 1991). For the following, it is useful to define alternatively the transverse momentum flux M_ot = M_o (1 - cos²q_o cos²s_o)^1/2 and the excess longitudinal momentum flux M_oe = Q_o (U_o cos q_o cos s_o - u_a) for the momentum forcings away from and along the x-direction, respectively.

Five regimes with self-similar turbulent flow characteristics can be defined as special cases of these forcing functions:

(i) Pure jet: u_a = 0, e = 0, J_o = 0; source of M_o only. A jet proceeds in the direction given by q_o and s_o.

(ii) Pure plume: u_a = 0, e = 0, s_o = 90°; source of J _o (in equilibrium with M_o so that a constant internal Froude number exists; Holley and Jirka, 1986). A plume rises vertically.

(iii) Pure wake: e = 0, J_o = 0, q_o = 0, s_o = 0, M_ot = 0; source of M_oe only (whereby M_oe << M_o; i.e. a weak momentum excess only). A wake motion gradually develops in the ambient flow.

(iv) Advected line puff: e = 0, J_o = 0, M_oe = 0; source of M_ot only. The transverse momentum injected into the flow sets up a cylindrical line puff that propagates transversely into the flow while being advected.

(v) Advected line thermal: e = 0, M_oe = 0, M_ot = 0 (ie. s_o = 0, q_o = 0); source of J_o only. The buoyancy flux causes a vertically rising cylindrical thermal that is advected.

Of these regimes, the first three are free turbulent flows dominated by transverse shear (shear normal to jet axis), and the last two by azimutal shear (parallel to jet axis). In the latter case an internal double-vortex structure is generated within the jet.

No self-similar regime is possible in the presence of density stratification, e ¹ 0. In that case, axial pressure forces influence and finally destroy the boundary-layer evolution of the flow and lead to strong horizontal spreading, the so-called collapse motion during the terminal layer phase of a buoyant jet in stratified surroundings. The flow is no longer jet-like. This makes futile past attempts to extend jet model predictions into the terminal layer phase.

In actual discharge situations one or more of the five regimes can occur as asymptotic regimes. Regime (i) is often the initial regime (whenever U_o >> u_a) and regime (v) is usually - but not always - the concluding regime.

 

INTEGRAL MODEL FORMULATION

The jet integral model CORJET forms part of the CORMIX expert system (Jirka et al., 1996) for initial mixing analyses in water bodies and is a generalisation of an earlier model by Jirka and Fong (1981).

The model assumes Gaussian profiles for jet velocity u, buoyancy g and concentration c (Fig. 1)

(1)

in which the subscript c denotes centerline values, b is the jet width and l > 1 describes a dispersion ratio for the scalar quantities. Integration across the jet leads to four flux qualities at any jet position s, the volume flux Q, the momentum flux M, the buoyancy flux J and the tracer mass flux Q_c. These are summarized in Table 1 as well as the conservation equations for the change along the jet trajectory s in which q and s are the local vertical and horizontal angles, respectively.

There are three fluid forces acting on the jet element, an entrainment force Q_a in the x-direction, a buoyant force pb²g'_c in the z-direction, and a drag force F_D normal to the jet.

The turbulent interaction of the buoyant jet with its surroundings is represented by two expressions, the entrainment rate E in the continuity equation and the drag force F_D in the momentum equations. The entrainment is parameterized as

(2

in which the first part represents the three transverse shear entrainment components (with u_c as the velocity scale) and the second part the azimutal shear components (with u_a). F_l² = u_c²/(g'b) is a local densimetric Froude number. The drag force is parameterized as

(3)

in analogy to a flow around a cylindrical object with radius . C_D represents a drag coefficient. Over the years there has been considerable controversy over whether a turbulent drag effect is consistent with the ambient flow around an entraining jet. Recent evidence (Coelho and Hunt, 1989; Fric and Roshko, 1994), however, shows vortex structures in the lee of the jet consistent with a rotational shear effect represented by Eq. 3.

 

VALIDATION WITH DATA FOR ASYMPTOTIC REGIMES

The CORJET model contains the following coefficients describing the various turbulence related mechanisms:

l = 1.2, C_D = 1.3, a₁ = 0.055, a₂ = 0.6, a₃ = 0.055, a₄ = 0.5

For the jet, plume and wake regimes (i) to (iii), the entrainment velocity u_e is proportional to u_c.

(i) Pure jet: u_e = a_jet u_c in which a_jet = a₁ = 0.055.

(ii) Pure plume: u_e = a_plume u_c. With an asymptotic Froude number F_lp = 4.66 and sinq = 1 this leads to a_plume = 0.083.

Data comparisons for these much studied and routine reference cases (Fischer et al., 1979; Holley and Jirka, 1986) will be omitted here.

(iii) Pure wake: u_e = a_wake u_c . In the pure co-flowing wake u_c << u_a so that a_wake = a₁ + a₃ = 0.11 from Eq. 2. The round wake has the following evolutionary laws:
b ~ x^1/3 and u_c ~ x^-2/3 (Schlichting, 1986). High quality experimental data on the jet/wake transition have not been available until the recent study of Nickels and Perry (1996) in which special precautions were taken to eliminate the negative wake effect of the discharge nozzle. Fig. 2 shows their data for width growth b and excess velocity u_c/u_a in which lengths have been normalized by L_me = M_oe^1/2/u_a. The agreement is satisfactory.

For both cases of advected line puff and thermal the entrainment velocity, u_e is proportional to the propagation velocity .

(iv) Advected line puff: u_e = a_puff u_p in which a_puff = a₄ = 0,5. Consistent with the basic work of Richards (see Scorer, 1978). Data on advected line puffs can be found in the recent study of Chu (1996) with which the CORJET predictions are compared (Fig. 3) for both trajectory (normalized by length scale L_mt = M_ot^1/2/u_a) and centerline dilution.

(v) Advected line thermal: u_e = a_thermal u_p in which a_thermal = 0.5 approximately equal to the puff value, again based on Richard's work suggesting roughly similar entrainment mechanisms for these two flows. Fig. 4 shows the comparison of CORJET with the data of Fai (1991). Here lengths are normalized by L_b = J_o/u_a³. Both Fig. 3 and 4 show satisfactory agreement even though the data scatter is considerable in these flows with large turbulent elements. Furthermore, the determination of centerline dilutions is somewhat tenuous due to the internal vortex structure leading to lower concentrations - and hence higher dilutions - along the centerline.

 

In summary, it can be concluded that the CORJET integral model formulation meets the consistency test of predicting the five truly self-similar regimes for the buoyant jet in a crossflow. Beyond that necessary requirement the entrainment hypothesis, Eq. 2, clearly contains somewhat arbitrary formulations for the transition ranges. These cannot be tested in detail with the attendant data material. Here the plausibility test is simply given by the large number of data comparisons the model has been subjected to in the past. These comparisons indicate overall satisfactory performance for a wide range of flow conditions.

The CORJET model can be downloaded from the U.S. Environmental Protection Agency (http://www.gky.com/_downloads/cormix.htm).

 

 

Table 1: Jet integral model CORJET: Definition of flux quantities and conservation equations

 

REFERENCES

Chu, C.K., 1996, Mixing of Turbulent Advected Line Puffs, Ph.D. Thesis, University of Hong Kong

Coehlo, S.L.V., and Hunt, J.C.R., 1989, The dynamics of the near field of strong jets in crossflows, J. Fluid Mech. 200, 95-120

Davies, P.A., and Navies, M.J.V., Eds., 1994, Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plumes, Kluwer Academic, Dordrecht/Bos-ton/London

Fai, W.C., 1991, Advected Line Thermals and Puffs, Ph.D. Thesis, University of Hong Kong

Fischer, H.B., et al., 1979, Mixing in Inland and Coastal Waters, Academic Press.

Fric, T.F., and Roshko, A., 1994, Vortical structure in the wake of a transverse jet, J. Fluid Mech., 279, 1-47

Holley, E.R., and Jirka. G.H., 1985, Mixing and solute transport in rivers, Technical Report E-85, U.S. Army Engr. Waterways Experiment Station, Vicksburg, Miss.

Jirka, G.H., and Doneker, R.L., 1991, Hydrodynamic Classification of Submerged Single Port Discharges", J. Hydraulic Engineering, 117, No. 9, 1095-1112

Jirka, G.H., Doneker, R.L., and Hinton, S.W., 1996, User's Manual for CORMIX: A Hydrodynamic Mixing Zone Model and Decision Support System for Pollutant Discharges into Surface Waters, Tech. Rep., U.S. Environmental Protection Agency, Tech. Rep., Environmental Research Lab, Athens, Georgia

Jirka, G.H., and Fong, H.L.M., 1981, Vortex dynamics and bifurcation of buoyant jets in cross flow, J. Eng. Mech. Div., ASCE, 107, EM3, 479-499

Nickels, T.B., and Perry, A.E., 1996, An Experimental and Theoretical Study of the Turbulent Coflowing Jet, J. Fluid Mech., 309, 157-182

Schlichting, H., 1968, Boundary layer theory, 6^th ed., McGraw-Hill

Scorer, R.S., 1978, Environmental Aerodynamics, Ellis Horwood

 

 

Fig. 1: Round buoyant jet in stratified crossflow.

 

Fig. 2: Transition from pure jet to pure wake regime. a) Decay of excess velocity u_c/u_a, b) growth of width b. Comparison of experiments by Nickels and Perry (1996) and CORJET predictions.

 

 

Fig. 3: Advected line puff regime. a) Centerline trajectory z, b) centerline dilution S_c. Comparison of experiments by Chu (1996) and CORJET prediction.

 

 

Fig. 4: Advected line thermal regime. a) Centerline trajectory z, b) centerline dilution S_c. Comparison of experiments by Fai (1991) and CORJET prediction