Turbulent Jets

Turbulence in Fluids

Benoit Cushman-RoisinThayer School of Engineering

Dartmouth College

Some naturally occurring examples are:- plumes from volcanoes- convective thermals

One fluid intruding into another

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.

In environmental fluids, it is not a rare occurrence to see one fluid intruding into another.

Some anthropogenic examples are:- wastewater discharges

from pipes into rivers or lakes - plumes exiting

from industrial smokestacks.

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 momentumand buoyancy

Buoyant jetor 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.

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.

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 Michiganat Grand Haven, Michigan USA.

Jet produced in the laboratory (Benoit Cushman-Roisin)

Turbulent Jets: Laboratory Observations

The turbulent jet can de described by only two input variables:

The diameter d of the orifice, and

The velocity UJ of the exiting fluid.

When the viscosity of the fluid is included, the three variables can be gathered into a single dimensionless number, the Reynolds number:

dU JRe

Source of figures and equations:Pope, 2000, Chapter 5

Forward (axial) jet velocity as a function of radial distance

),( rxuU

An important piece of this profile is the velocity along the centerline of the jet:

),0()(0 rxuxU

Similarity in velocity profile:

All velocity profiles appear identical in shape, except for a stretching factor.

If we non-dimensionalize the velocity with the centerline velocity U0 (hence =1 at the center) and the radial distance with the distance r1/2 at which the velocity has fallen off by 50%, all profiles collapse on a single curve.

We can say that the jet is self-similar.

velocitycenterlinetyjet veloci

)(0

xU

U J

The linear relationship tells us that the centerline velocity U0(x) varies inversely proportionality with distance x along the jet:

dxxBUxU J /)(

)(0

0 with B = 5.8 (page 101)

Fig 5.4

???

Furthermore…

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 11.8o giving approximately 24o 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(11.8o) ≈ 0.2 = 1/5.

24ox

R=x/5

We write: )()(constant 02/12/1 xxSxrS

dxdr

with S = 0.094 (page 101)

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/2 into the conduit. This point of origin is called the virtual source.

Axial velocity scaled by centerline velocity

0

22

0

)1(1)(

)()(),(

xxr

af

fxUrxu

Radial velocity scaled by centerline velocity, obtained from

Note: At its maximum, the radial velocity is inward and ~0.025 U0 whereas the radial velocity is ~0.75 U0 there.

Thus, the radial velocity is no more than about 3% of the axial velocity.

0)(1

rvr

rxu

(Schlichting’s solution, page 119)

Reynolds stress values:

Reynolds stress tensor is anisotropic:

______

____________

____________

''000''''0''''

wwvvvuvuuu components

If we define :______

* ''uuuu

This argues for looking for an eddy viscosity that is independent of the radial direction (but only ok for inner half of the jet width…).

In the axial (x-) direction, one can use scaling to argue that the eddy viscosity should be x-independent as well.

Slightly better: make the eddy viscosity be the product of a friction velocity (u*) and a length scale ℓ, with ℓ given to the right.

Follow with Pope’s book, Section 5.2, pages 111 to 122.