ENVIRONMENTAL FLUID MECHANICS

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ENVIRONMENTAL FLUID MECHANICS Turbulent Jets http://thayer.dartmouth.

pdf Benoit Cushman-Roisin Thayer School of Engineering Dartmouth College

occurrence to see one fluid intruding into another.

convective thermals Some anthropogenic examples are: - wastewater

1 ENVIRONMENTAL FLUID MECHANICS Turbulent Jets Benoit Cushman-Roisin Thayer School of Engineering Dartmouth College One fluid intruding into another In environmental fluids, it is not a rare occurrence to see one fluid intruding into another. Some naturally occurring examples are: - plumes from volcanoes - convective thermals Some anthropogenic examples are: - wastewater discharges from pipes into rivers or lakes - plumes exiting from industrial smokestacks. In every case, a fluid with some momentum and/or buoyancy exits from a relatively narrow orifice and intrudes into a larger body of fluid with different characteristics, such as different speed, temperature or contamination level. 1

It is helpful to categorize the various types of intrusion according to - whether they inject momentum, buoyancy or both in the ambient fluid, and - whether or not they persist in time.

2 It is helpful to categorize the various types of intrusion according to - whether they inject momentum, buoyancy or both in the ambient fluid, and - whether or not they persist in time. Continuous injection Intermittent injection Momentum only Jet Puff Buoyancy only Plume Thermal Both momentum and buoyancy Buoyant jet or forced plume Buoyant puff These flows can be characterized as partly turbulent because they create situations where a definite structure exists while the turbulence level is much higher in the vicinity of the intrusion than in the surrounding fluid. Turbulent Jets Whenever a moving fluid enters a quiescent body of the same fluid, a velocity shear is created between the entering and ambient fluids, causing turbulence and mixing. In nature, the situation occurs where a river empties in a lake or estuary, or occasionally when a wind blows through an orographic gap. Grand River plume entering Lake Michigan at Grand Haven, Michigan USA. Perhaps the most clearly defined jets are those produced when a fluid is discharged in the environment through a relatively narrow conduit, such as an industrial discharge released through a pipe on the bank of a river, lake, or coastal ocean. Jet produced in the laboratory (Benoit Cushman-Roisin)

Turbulent Jet Theory Laboratory investigations of jets penetrating into a quiescent fluid of the same density consistently reveal that the envelope containing the turbulence caused by the jet adopts

Further, the opening angle is always the same, regardless of the nature of the fluid (air or water) and of other circumstances (such as diameter of outlet and discharge speed).

3 Turbulent Jet Theory Laboratory investigations of jets penetrating into a quiescent fluid of the same density consistently reveal that the envelope containing the turbulence caused by the jet adopts a nearly conical shape. In other words, the radius R of the jet is proportional to the distance x downstream from the discharge location. Further, the opening angle is always the same, regardless of the nature of the fluid (air or water) and of other circumstances (such as diameter of outlet and discharge speed). This universal angle is 11.8 o giving approximately 4 o from side to opposite side. It follows that the coefficient of proportionality between the jet radius R and the downstream distance x from the outlet is tan(11.8 o ). = 1/5. R=x/5 x 4 o Note that since the initial jet radius is not zero but the finite nozzle radius, equal to half the exit diameter d, the distance x must be counted not from the orifice but from a distance 5d/ into the conduit. This point of origin is called the virtual source. Observations suitably averaged over the many turbulent fluctuations reveal that the velocity in the jet obeys a law of similarity.

4 Similarity in velocity profile: All cross-sections appear identical, except for a stretching factor, and the velocity profile across the jet exhibits a nearly Gaussian shape (bell curve). Therefore, we can write: u( x, r) u r exp u 5r exp x 4

5 Now, we need to determine u (x). When a jet enters a fluid at rest, the sole source of momentum is that of the jet itself, and the absence of external accelerating or decelerating forces implies that the momentum flux in the jet's cross-section remains constant downstream. Since this flux is the momentum per unit volume, u (where is the fluid density and u the velocity), times the velocity u itself cumulated over the jet's cross-section, the statement that momentum is constant downstream is: u rdr U 4 d in which U is the exit velocity and d the nozzle diameter. With u( x, r) we obtain: and u u 5r exp x 5d U x u u u rdr R 1 5d U x Question from an environmental perspective: How much dilution and mixing is taking place? To answer this question, we first determine the amount of ambient fluid that the jet entrains. The volumetric flowrate Q in the jet is Q( x) u rdr u 5 which obviously grows with distance x along the jet. x du x 1 The entrainment rate E can be defined as: dq E Ud dx 1 a constant Alternatively, we can define an entrainment velocity v that feeds the jet from the side: increment in flow rate piece of lateral area dq v da v Rdx Ud u v.1 u 4x 5

6 The distribution of a contaminant s concentration can be described as: For concentration, Gaussian profiles, just like velocity c( x, r) c 5r exp x with c 5d c x in which c is the upstream concentration at the discharge nozzle. Engineering question: If c 1 is the imum allowable value by law, where does the jet become legal? Answer: 5d Set c c equal to the threshold value c 1 x and solve for distance x c x 5 d c 1 Note how this answer depends on the diameter d of the nozzle from which the jet exits. The smaller the diameter, the shorter the distance to the legal concentration. Side consideration: The smaller the diameter, the more pressure needs to be exerted to get the fluid out through the nozzle (more momentum to impart to the fluid by virtue of the Bernoulli principle); so, there is a price to pay. 6

7 Puffs A puff is a finite injection of momentum in an otherwise undisturbed environment, an interrupted jet so to speak. A puff does not need to be described in all its details but may be represented by only three variables: -its volume V, - its average radius R, and - its bulk velocity u. The radius and volume can be related to each other by: V mr in which the constant m of proportionality is not 4/ = 4.19, as it would be for a sphere, but assumes a lower value (m ) because puffs tend to be a bit flatter. To describe the behavior of a puff, we need to establish its mass and momentum budgets. 1) Momentum budget (the easier of the two!): d ( V u) dt V u constant R u R u R u u R ) Mass budget: d ( V ) entrainment dt d ( mr ) (surface area)(perpendicular velocity) R u dt a R u Some algebra: dr Ru mr ar dt R dr aru R dt m 1 4 aru R t 4 m with t = being a virtual origin, preceding the time when the puff actually begun with radius R and velocity u. Now, solve for radius R and velocity u as functions of time: 4au t R R mr mr u u 4 au t Extract position x from the velocity u: dx dt 1/ 4 / 4 / 4 t mr 1/ 4 u x u dt 4u t 4au which grows over time as t 1/4, just like R. 7

Form the ratio of position over radius: x m R a which combines both parameters (m and a) of the model Laboratory evidence confirms the similarity (x R) but is somewhat uncertain about the coefficient

8 Form the ratio of position over radius: x m R a which combines both parameters (m and a) of the model Laboratory evidence confirms the similarity (x R) but is somewhat uncertain about the coefficient of proportionality: x n R with n 4 but varying between 1.7 and 6.. The difficulty associated with the determination of n is due to the fact that puffs easily turn into vortex rings. Adopting x/r = 4, we deduce the value of the entrainment coefficient: a.5 (uncertainty: 1.4 < a < 5.) 8

Jets and Plumes. Agenda. Definition of jets, plumes, and others. Engineering applications of Jets/Plumes. Basic properties of Jets/Plumes

Jets and Plumes. Agenda. Definition of jets, plumes, and others. Engineering applications of Jets/Plumes. Basic properties of Jets/Plumes Dr. (IfH) Environmental Fluid Mechanics II Stratified Flow and Buoyant Mixing Jets and Plumes Dong-Guan Seol INSTITUTE FOR HYDROMECHANICS National Research Center of the Helmholtz Association www.kit.edu