We will analyze realistic internal flows across all fluid regimes from laminar to turbulent.
The main difference between internal flows and external flows is that internal flows
are confined by solid walls that surround the fluid.
The presence of the walls affect the behavior of the fluid, especially its velocity is affected
by the viscous effects.
Internal flows can be found in numerous applications, from typical industrial applications such
as the gas flowing through pipelines, water been pumped and ventilation systems to maintain
air quality in buildings, to biomedical applications to convey medicines into the body or study
the bloodstream through our circulatory system.
So let's analyze general characteristics of internal flow.
First we need solid walls that will contain the fluid.
In this case the fluid will enter from the left with a certain velocity, and, through
the interaction with the walls, a boundary layer will start to form and grow.
Hence, we can identify a region where the boundary layer and viscous effects are relevant,
and also a region, highlighted here, that is inviscid.
As the boundary layers grow the inviscid region shrinks and the profile of velocity changes,
until the boundary layer on the opposite sides of the wall eventually meet each other.
The velocity profile changes its shapes as the fluid propagates through the channel up
to the end of the inviscid region.
Past that point the flow velocity profile does not change anymore and we call it a fully
developed velocity profile.
In this way we can define two regions: a hydrodynamic entrance region where along a section of the
channel we have both inviscid and viscous flow, and then the fully developed region
where the viscous effects are relevant in the entire channel section.
Now, internal flows can be either laminar or turbulent, however most of common internal
flows are turbulent.
In the late 19th century, Osborne Reynolds performed an experiment injecting dye into
a water stream to demonstrate the classification of these flows.
At low speed, and low Reynolds numbers, the flow is laminar.
Indeed, if we inject a dye into the stream it would not mix with the surrounding water.
As the flow becomes transitional will see that some mixing start occurring and the dye
streak starts fluctuating.
As the flow becomes turbulent the dye diffuses and spreads in the water due to the strong
turbulent mixing of the flow.
This experiment give us an idea that internal flows can be laminar, transitional or turbulent.
The transition from laminar to turbulent occurs at a Reynolds number of 2300 for internal
flows.
However, this number gives just an order of magnitude for transition since small perturbances
in the flow and the surface conditions of the pipe can delay or anticipate the transition
to turbulent flow.
For internal flows the Reynolds number is normally based on the hydraulic diameter of
the pipe or channel.
The hydraulic diameter is defined as the ratio of four times the cross-sectional area divided
by the wetted perimeter.
In the next lessons when we refer to the Reynolds number it will be based on the hydraulic diameter
as characteristic length.
Internal flow geometries can be simple or complex.
We can have simple geometries like basic piping components with straight pipes and bends that
we are commonly used in our houses.
We can also have extremely complex internal flows geometries, like the piping system used
in jet engines, where a maze of pipes and connections transport different fluids to
let the engine operate.
Internal flows comprehend any case where a fluid is transported from a point to another.
Even the water that is pumped from the municipal water source to your home.
The flow that travels through channels or pipes encounters is subjected to viscous effects
and turbulence that consume its energy.
The further it travels, the larger is the loss of energy and more power is required
to pump the fluid and maintain a given flow rate.
The importance of analyzing internal flows is to estimate the amount loss that an internal
passage will develop, in order to design the pumping systems accordingly and sustain the
correct flow rate.
This will be the focus of the next set of lessons.