One of the most intriguing engineering feats humankind have achieved is the ability to go into
outer space. Watching a rocket launch event – even the televised versions are always interesting. The
hot plumes of air ejected out of the launch rocket’s nozzle is a type of external flow.
In most external flow examples, a solid object is immersed in and is moving through
a stationary fluid. Under certain scenarios, such as air flow over stationary high-rise buildings,
we have a fluid which is flowing around a solid. There is, however, another class of external flows
which is not constrained by solid wall boundaries. Such flows are called free shear flows or shear
layer flows. There are typically three types of free shear flows – Jets, Wakes, and Mixing Layers.
In this lesson, we will focus our attention on jets. In engineering,
we want to know the spreading rate of a jet. This requires knowing the width of a jet
and its velocity scales as a function of distance from the jet outlet. It is extremely critical in
engineering applications for designing rocket nozzles for providing the necessary thrust to
take a trip into outer space. Some of these jets are planar, like in the case of Aerospike Engine,
while others are round. Even though planar jets have limited engineering applicability,
these develop the necessary fundamentals to analyze round jets. Applications involving jet
engine noise where acoustics measurements are required use these jet width calculations to
identify optimal locations to place microphones to capture these acoustic waves.
In the case of an automobile exhaust, these calculations
are necessary to meet emission standards. ---------------------------------------------
Jets are flows exiting a confined source into the ambient environment. This planar
jet is exiting from an opening whose dimension is 2bo. As there are no walls,
the jet continues to expand at constant pressure. Similar to the boundary layer on a flat plate,
the jet forms a 2D shear layer. Just downstream of the opening, there is a potential core, where
the fluid can be treated as inviscid. In this region, the fluid velocity is relatively constant
and equals the exit velocity of the jet. Outside this inviscid zone, we have a developing region
where the fluid velocity decreases from the exit velocity V0 to the ambient air
velocity at the outer boundary of the jet. This velocity decrease is because of fluid viscosity.
After a certain length, the velocity of the jet becomes smooth and its shape does not change
anymore. Here, velocity is said to have achieved a ‘self-similar’ profile and the jet region
is often referred to as a fully developed self-similar jet. In this region,
the dominant velocities along the x direction are characterized by large Reynolds numbers.
For all practical purposes, the boundary layer approximations to obtain the flat
plate solution can be reused to simplify the governing equations in the self-similar region.
Following these approximations, a simplified set of equations
are obtained in both 2D planar and axisymmetric forms. As the jet spreads at constant pressure,
the momentum flux at every cross-section in this region is always a constant. This can
be evaluated from the constant inlet velocity profile to obtain the following relationship.
The obtained value of momentum flux is applicable in both laminar and turbulent flow regimes.
------------------------------------------------ Depending on the exit velocity, jets can show
laminar or turbulent characteristics. In this lesson, we are going to learn about
both these flow regimes. I’m going to be the laminar guy and analyze the laminar jet solution,
and … I’m the turbulent guy and will talk to you about turbulent jets.
Laminar K: Similar to the Blasius approach in laminar boundary layers,
we use the similarity solution to solve for the velocity profile of a laminar plane jet.
Herman Schlichting proposed the following similarity variables
for the stream function. The corresponding velocity components are given by u and v.
These variables help us reduce the second-order partial differential equation in two variables
to a third-order ordinary differential equation in just one variable. We obtain
an analytical solution using the following boundary conditions. The constant a is evaluated
from the integral of momentum flux, M. Turbulent K shows enthusiasm right before
his turn. Turbulent K:
To understand the behavior of turbulent jets, we use the mean jet velocities instead of actual.
The boundary layer continuity and momentum equations are slightly modified. We have
a shear term in the momentum equation which is related to the turbulent eddy viscosity
of the fluid. For turbulent jets,
the centerline velocity and jet width are functions of jet momentum, density, and distance.
As there are no walls, these variables are not influenced by the molecular
viscosity of the fluid. Dimensional analysis of the momentum equation
gives us the following results for plane and axisymmetric jets.
Unlike laminar jets, these constants are unique and independent of the flow Reynolds number
and are obtained experimentally. The simplest model for eddy viscosity
was proposed by Prandlt where the viscosity varies as a function of square root of x.
Gortler in 1942 proposed similarity variables to compute the jet width for turbulent jets.
Similar to the laminar jet analysis, a single ordinary differential equation is solved
using the same boundary conditions. Laminar K shows enthusiasm right before
his turn. Laminar K:
On solving the laminar equations, the solution gives us the velocity distribution as well as
the maximum centerline velocity. When the obtained velocity profiles are plotted
at various axial lengths, we obtain the self-similar profile of the laminar jet.
This jet continues to spread as it travels away from the issued location.
The centerline velocity of a laminar jet decreases at the rate of one-third power of distance from
the jet exit. The jet width is defined as twice the distance where the local jet velocity becomes
1 percent of its maximum velocity. The jet width increases at the rate of two-thirds power
based on the location of the jet from its exit. A plane jet can quickly transition to turbulence
at a very small Reynolds number of 30. Turbulent K shows enthusiasm
right before his turn. Turbulent K: Since the velocity of a turbulent jet
drops asymptotically, the jet width is not very well-defined. Therefore, we define half-velocity
point along the y direction where the jet-velocity becomes half of its maximum value at the jet axis.
Using this, the jet growth rate is estimated as tan 13 degrees and is independent
of Reynolds Number. In a different approach,
we assume a velocity profile shaped like a Gaussian curve and come up with a different set
of similarity variables. Using the jet’s integral equations, we estimate the centerline velocity.
In this case, C1 is obtained from experiments. For example, Albertson in
1950 computed the value of C1 to characterize the velocity distribution in turbulent jets.
The obtained velocity profiles at various axial locations are plotted. In the
case of turbulent jets, its centerline velocity decrease as it travels farther.
This decay occurs as square root of x.
Laminar K shows enthusiasm right before his turn.
From an engineering perspective, turbulent round jets are more widely used compared to laminar.
Because of this, we are going to say good-bye to the laminar guy and talk
about turbulent round jets for the rest of the lesson. (laminar guy puts up a sad face)
Round jets are more resilient to instabilities compared to plane
jets and transition to turbulence at a much larger Reynolds number of 2000.
A different set of similarity variables are used to solve for round turbulent jets.
Without going into the similarity analysis again, we will discuss the obtained results here.
As the jet diameter increases linearly along x, the centerline velocity decreases. The eddy
viscosity for a round turbulent jet is a constant and is obtained using experimental curve fitting.
Based on experiments, the following jet velocity correlation was obtained.
The self-similar velocity profiles at various axial locations are plotted as a function of
y. The maximum velocity decays along the axial length as the jet flows downstream.
Because of this, the jet continues to slow down
until the jet velocity drops to the quiescent velocity of the fluid.
These correlations are used to this day to obtain first order estimates during the
design phase on any engineering application. With the availability of better numerical techniques
and today’s computational resources, we are able to obtain a more accurate
solutions to these jets using the modern-day computational fluid dynamics.
Having said that, engineering designers use these correlations obtained from both similarity
approach as well as experiments to verify and validate the insights obtained from CFD models.
Once that is done, the CFD code is now ready to explore different complex engineering scenarios.