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Bernoulli
Daniel Bernoulli was a Swiss Math Mathematician and physicist in the early 1700s who is best
remembered for his mathematic applications to fluid mechanics. One of his well-known contributions is
the Bernoulli Principle which describes the mathematics of the mechanism underlying the operation of
two important technologies of the 20the century: the carburetor and the airplane wing.
As a child, Bernoulli was pressured, by his father, to pursue a degree in business as there were no
substantial benefits to being a mathematician. After a certain amount of time, Bernoulli shifted his
career to the field of medicine where he earned a PhD in botany and anatomy. Bernoulli was friends
with Leonhard Euler and became a professor of mathematics in 1724.
Bernoulli’s contributions to the world include 1/2 ρ μ ^2+P=constant where p is the density of a fluid, u
signifies the velocity of said fluid, and P signifies pressure. This is exploited by the wing of an aeroplane
which is designed to create an area above its surface where the air velocity increases. The pressure in
this area is lower than that under the wing, so the wing is pushed upwards by the relatively higher
pressure under the wing.
B.L. Viscous/Inviscid Interaction
In physics and fluid mechanics, a boundary layer is the layer of fluid in the immediate vicinity of
a bounding surface where the effects of viscosity are significant. In the Earth's atmosphere,
the atmospheric boundary layer is the air layer near the ground affected by diurnal heat, moisture or
momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the flow
close to the wing, where viscous forces distort the surrounding non-viscous flow
Inviscid flow is the flow of an ideal fluid that is assumed to have no viscosity.
Viscosity is the state of being thick, sticky, and semifluid in consistency, due to internal friction.
The difference in velocity between adjacent layers of the fluid is known as a velocity gradient and is
given by v/x, where v is the velocity difference and x is the distance between the layers.
The boundary layer is not a static phenomenon. It is dynamic. The thickness of boundary layer (the
height from the solid surface where we first encounter 99% of free stream speed) continuously
increases.
The velocity of fluid on the wing of an airplane relative to the airplane is 0 as a result of the boundary
layer whereas the velocity of the fluid increase until it reaches the free stream velocity the further
away it gets from the wing and ultimately the boundary layer.
Drag force on the fluid due to shear stress, or skin friction, at the wall increases from zero at the start of
the body to a maximum in the fully developed flow region where it remains constant. Like the height of
the boundary layer, the drag force gradually increases as one moves across the object.
Shear Stress is the force causing two components or layers to slide upon each other in opposite
directions parallel to the plane of contact. In this instance, sheer stress is between the moving body
and the boundary layer.
As a fluid passes over a greater length more fluid is slowed by friction between the fluid layers close
to the boundary. Hence the thickness of the slower layer increases.
As the boundary layer thickness becomes greater, so the velocity gradient become smaller and the shear
stress decreases until it is no longer enough to drag the slow fluid near the surface along. If this viscous
force was the only action then the fluid would come to a rest.
It, of course, does not come to rest but the second mechanism comes into play. Up to this point the flow
has been laminar and Newton's law of viscosity has applied. This part of the boundary layer is known as
the laminar boundary layer
The viscous shear stresses have held the fluid particles in a constant motion within layers. They become
small as the boundary layer increases in thickness and the velocity gradient gets smaller. Eventually they
are no longer able to hold the flow in layers and the fluid starts to rotate.
This causes the fluid motion to rapidly becomes turbulent. Fluid from the fast moving region moves to
the slower zone transferring momentum and thus maintaining the fluid by the wall in motion.
Conversely, slow moving fluid moves to the faster moving region slowing it down. The net effect is an
increase in momentum in the boundary layer. We call the part of the boundary layer the turbulent
boundary layer.
Laminar flow has high viscous forces whereas Turbulent flow has high inertial forces
The Reynold's Number of a system determines whether or not a flow is laminar or turbulent by showing
the ratio of momentum forces to viscous forces. Laminar flow usually occurs at low Reynolds' Numbers
where the viscous forces are dominant. Consequently, turbulent flow occurs when Reynolds' Numbers
are seemingly high as a result of greater momentum forces.
v = mean velocity of object relative to fluid
L=traveled length of fluid
myu = dynamic viscosity of the fluid
v= kinetic viscosity (v=myu/rho)
rho=density of fluid
Laminar flow: Re < 2000
Transitional flow: 2000 < Re < 4000
Turbulent flow: Re > 4000
At points very close to the boundary the velocity gradients become very large and the velocity
gradients become very large with the viscous shear forces again becoming large enough to maintain
the fluid in laminar motion. This region is known as the laminar sub-layer. This layer occurs within the
turbulent zone and is next to the wall and very thin - a few hundredths of a mm.
In turbulent flow if the height of the roughness of a pipe is greater than the thickness of the laminar sublayer then this increases the amount of turbulence and energy losses in the flow. If the height of
roughness is less than the thickness of the laminar sub-layer the pipe is said to be smooth and it has
little effect on the boundary layer.
Boundary layer separation is the detachment of a boundary layer from the surface into a broader wake.
Boundary layer separation occurs when the portion of the boundary layer closest to the wall or leading
edge reverses in flow direction. The separation point is defined as the point between the forward and
backward flow, where the shear stress is zero.
One phenomenon that greatly affects the flight of planes occurs when the flow separation of the
boundary layer is precedent almost immediately when the fluid comes in contact with the body in
motion.
The dramatic altercation in flow yields a detrimental change in pattern pressure and results in stalling.
There are three conventional ways to allay the stalling effects on an airplane foil. One is to arrange
the engine intakes so that they draw air from the rear to keep the boundary layer close. Two is to
have holes on the greater pressure side of the wing so that the pressure on top decreases so the
adverse pressure gradient reduces boundary layer separation. Three is installing a flap at the end of
the wing to increase fluid velocity again reducing boundary layer separation.
https://en.wikipedia.org/wiki/Inviscid_flow
https://en.wikipedia.org/wiki/Boundary_layer
https://en.wikipedia.org/wiki/Daniel_Bernoulli
https://www.grc.nasa.gov/www/k-12/airplane/boundlay.html
http://wwwmdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/introvisc/node7.html
http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section4/boundary_layer.htm
https://en.wikipedia.org/wiki/Reynolds_number