Laminar boundary layers [Fluid Mechanics #13]
Summary
TLDRThis video delves into the concept of laminar boundary layers in fluid mechanics, focusing on external flows where fluid interacts with a solid surface. It explains how boundary layers form due to viscosity and the no-slip condition at the wall, leading to a velocity gradient near the surface. The video outlines the theoretical approach to analyzing these layers, including assumptions and boundary conditions. It also discusses the significance of boundary layer thickness and how it influences forces like drag and lift. Prandtl and Blasius' solutions for the velocity profile within these layers are highlighted, along with the calculation of wall shear stress and drag force, emphasizing the importance of understanding boundary layers in various applications such as aerodynamics and climate modeling.
Takeaways
- 🌊 The video discusses external flows, which are open on one or more sides and can extend infinitely in at least one direction, unlike enclosed flows.
- 🔍 The focus is on boundary layers, which are regions of velocity gradient near a surface due to viscosity and the no-slip condition at the wall.
- 🎯 The video aims to solve for the velocity and pressure fields in laminar boundary layers, with turbulent boundary layers to be covered in a subsequent video.
- 📚 External flows include boundary layers, jets, wakes, and shear layers, all of which require specific analysis in fluid mechanics.
- 🛠 Assumptions made for analysis include incompressible, steady, two-dimensional, and two-component flow, with no body forces.
- 📉 Observational assumptions are used to simplify the conservation equations, focusing on the gradual growth region of the boundary layer.
- 🧮 The conservation of mass and momentum equations are simplified using these assumptions, leading to a manageable set of equations for boundary layer analysis.
- 📉 Prandtl and Blasius solved the simplified equations, revealing the self-similar shape of the laminar boundary layer's velocity profile.
- 📏 Three methods to measure boundary layer thickness are discussed: disturbance thickness, displacement thickness, and momentum thickness.
- 💨 The video concludes with the calculation of wall shear stress and drag force due to the fluid on the surface, emphasizing the importance of understanding boundary layers in various applications.
Q & A
What is the primary focus of the video?
-The primary focus of the video is on laminar boundary layers, which are regions of lower velocity fluid that grow on an external surface in a flow field due to viscosity.
What are the key differences between enclosed flows and external flows?
-Enclosed flows have boundaries on all sides, while external flows can be open and extend effectively into infinity in at least one direction. External flows often exhibit growth or development in the streamwise direction, leading to more complicated flows.
What is a boundary layer in the context of fluid mechanics?
-A boundary layer is a thin layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant, resulting in a velocity gradient from the surface to the free-stream velocity.
Why is the no-slip condition important in boundary layer analysis?
-The no-slip condition is important because it states that the fluid velocity at the wall is equal to the wall's velocity, which is crucial for understanding the velocity profile near the surface and the associated shear stress.
What assumptions are typically made when analyzing laminar boundary layers?
-When analyzing laminar boundary layers, it is assumed that the fluid is incompressible, steady, two-dimensional, and two-component. Additionally, it is assumed that there are no body forces, and the flow develops in the x-direction.
How do Prandtl and Blasius contribute to the understanding of boundary layers?
-Prandtl and Blasius are credited with pioneering boundary layer research. They derived and solved the conservation equations for a laminar boundary layer, leading to the understanding of the self-similar shape of the velocity profile.
What are the three main strategies to measure the boundary layer height?
-The three main strategies to measure the boundary layer height are disturbance thickness (δ), displacement thickness (δ*), and momentum thickness (θ), each providing a different perspective on the boundary layer's impact on the flow.
How is the Reynolds number used in boundary layer analysis?
-The Reynolds number is used to characterize the flow and determine when a flow transitions to turbulence. For a smooth flat plate, the transition is estimated to occur around a Reynolds number of five hundred thousand.
What is the significance of the skin friction coefficient in boundary layer analysis?
-The skin friction coefficient (c_f) is a non-dimensional parameter that represents the wall shear stress relative to the flow inertia. It is important for understanding the drag force on a surface due to the fluid flow.
How does the drag force on a surface due to a boundary layer get calculated?
-The drag force on a surface due to a boundary layer is calculated by integrating the shear stress over the surface area. For laminar boundary layers, this is a function of the velocity, plate length, and the Reynolds number based on the plate length.
Outlines
🌊 Introduction to External Flows and Boundary Layers
This paragraph introduces the concept of external flows, contrasting them with enclosed flows by highlighting their open nature and potential to extend infinitely in at least one direction. The focus is on boundary layers, which are regions of velocity gradient near a surface due to viscosity and the no-slip condition at the wall. The aim is to understand laminar boundary layers, which are critical for analyzing fluid forces on external vehicles, aerodynamics, and atmospheric flows. The paragraph sets up the theoretical framework for studying boundary layers, including assumptions such as incompressibility, steady flow, two-dimensionality, and the absence of body forces. It also discusses the importance of the boundary layer's growth, velocity profile, and its impact on force exerted on a surface.
🔍 Assumptions and Conservation Equations in Boundary Layer Analysis
The second paragraph delves into the assumptions and boundary conditions necessary for solving the equations governing boundary layer flow. It acknowledges the need to move beyond the steady and 2D assumptions due to the development of the boundary layer in the streamwise direction. Key boundary conditions include the no-slip and no-penetration conditions at the wall and the behavior of the flow as it moves away from the wall towards infinity. The paragraph outlines the conservation of mass and momentum equations, simplifying them using the assumptions made. It introduces observational assumptions to further simplify the equations, focusing on the dominance of certain terms based on the growth characteristics of the boundary layer. The result is a set of equations that, while complex, can be solved numerically to understand the behavior of boundary layers.
📏 Quantifying Boundary Layer Thickness and Growth
This paragraph discusses the methods for quantifying the thickness and growth of boundary layers. It introduces three main strategies for measuring boundary layer height: disturbance thickness, displacement thickness, and momentum thickness. Each method is defined and explained in the context of its application and significance. The paragraph also touches on the historical context of boundary layer research, crediting Prandtl and Blasius for their pioneering work in solving the boundary layer equations and defining the self-similar shape of the laminar boundary layer. The importance of the Reynolds number in boundary layer analysis is highlighted, particularly its role in determining when a flow transitions to turbulence.
💨 Shear Stress, Drag Force, and Practical Implications of Boundary Layers
The fourth paragraph focuses on the practical implications of boundary layer analysis, particularly in terms of shear stress and drag force. It explains how the velocity field within the boundary layer contributes to wall shear and the resulting drag force. The concept of the skin friction coefficient is introduced as a non-dimensional measure of wall shear stress relative to flow inertia. The paragraph also discusses the general drag coefficient and how it relates to the force of drag, the surface area, and the flow inertia. The importance of understanding boundary layers in aerodynamics, climate modeling, and other industries is emphasized, highlighting the relevance of the equations and concepts discussed in the video.
🏁 Conclusion and Summary of Laminar Boundary Layers
In the final paragraph, the video concludes with a summary of the key points covered regarding laminar boundary layers. It reiterates the importance of understanding these layers in fluid mechanics and their impact on various fields such as aerodynamics and climate modeling. The paragraph also emphasizes the significance of the theoretical approach and the assumptions made in the analysis. It acknowledges the complexity of boundary layers and their role in phenomena like lift and drag on aircraft wings, as well as the onset of separation. The video ends with an encouragement for viewers to appreciate the intricacies of boundary layer analysis and its broader applications.
Mindmap
Keywords
💡Enclosed Flows
💡External Flows
💡Boundary Layer
💡No-Slip Condition
💡Free-Stream Velocity
💡Reynolds Number
💡Laminar Flow
💡Momentum Thickness
💡Skin Friction Coefficient
💡Prandtl and Blasius
Highlights
Introduction to external flows, which are open on one or more sides and can extend to infinity.
Focus on boundary layers, the layer of velocity gradient near a surface due to viscosity and the no slip condition at the wall.
Exploration of the importance of boundary layers in understanding fluid forces on external vehicles, aerodynamics, and atmospheric flows.
Assumptions made for analysis, including incompressible, steady, two-dimensional flow without body forces.
Explanation of the no slip boundary condition at the wall, where the fluid velocity must match the wall's velocity.
Discussion on the gradual growth of boundary layers and the importance of viscosity in these layers.
Introduction of the conservation of mass and momentum equations for boundary layer analysis.
Simplification of the momentum equations using observational assumptions about the growth characteristics of boundary layers.
Derivation of the final form of the conservation equations for a laminar boundary layer by Prandtl and Blasius.
Presentation of the self-similar shape of the laminar boundary layer and its velocity profile.
Quantification of boundary layer height using disturbance thickness, displacement thickness, and momentum thickness.
Calculation of the wall shear stress resulting from the boundary layer and its impact on drag force.
Introduction of the skin friction coefficient and its role in non-dimensionalizing the wall shear stress.
Explanation of how to calculate the total drag force on a surface due to the fluid's boundary layer.
Discussion on the significance of laminar boundary layers in aerodynamics, including their role in lift and drag on aircraft wings.
Highlight of the role of boundary layer analysis in climate modeling and predicting global flow patterns.
Conclusion summarizing the importance of understanding laminar boundary layers in fluid mechanics.
Transcripts
hi
and welcome to another video in fluid
mechanics
up until this point we have dealt
primarily with enclosed flows
these are flows that have boundaries on
all sides and include both
channels and pipes in analysis of these
flows we have come
to solve the velocity field u v and w
and also learned a bit about the
pressure field as well
we studied these flows in both laminar
and turbulent states
today we move on to the world of
external flows
unlike enclosed flows these flows can be
open and extend effectively into
infinity in at least one direction
specifically we'll be interested in
boundary layers the layer velocity
gradient near the surface due to
viscosity acting on the fluid
and the no slip condition at the wall
like with enclosed flows we will try and
solve for the velocity field and the
pressure field
today our focus will be on laminar
boundary layers and we'll save turbulent
boundary layers for the next video
let's jump right in external flows are
our next focus
these flows are open on one or more
sides and often exhibit growth or
development in the streamwise direction
leading to more complicated flows than
we saw with enclosed flows
these types of flows include boundary
layers a velocity gradient near a solid
surface
also jets a concentration of moving air
wakes which is a velocity deficit region
that's often
unsteady and downstream of a body moving
in a fluid
and shear layers where two flows with
different velocity come together
all these types of flows exist in the
world of fluid mechanics and warrant
specific analysis
today our aim is to better understand
the laminar boundary layer
consider a plate moving in a fluid or a
fluid passing by a solid plate
there is some free stream velocity u
infinity
near the wall we have what's called the
no slip boundary condition
physically this means velocity super
close to the wall
has to have the velocity of the wall
itself due to friction
so u at y equals zero is zero
now we know far from the wall the
velocity field does not feel the
pressure of the wall so the velocity
field remains u infinity which is the
free stream velocity
in our fluid there must be a continuous
region where the fluid changes from the
zero velocity
at the wall to the free stream velocity
far from the wall
this region is called the boundary layer
and it grows as flow
moves downstream boundary layers are
essentially molecular diffusion which is
viscosity
creeping out slowly further and further
from the wall
in these layers viscosity is deemed to
be important
due to the meaningful vertical velocity
gradients
these layers are generally quite small
relative to the surface moving through
the fluid
but despite their small size they are
absolutely critical in understanding how
fluid forcing works
on external vehicles and aerodynamics
heat transfer
and even atmospheric flows
our goal is to be able to describe how a
boundary layer grows downstream
what the thickness is what the velocity
profile looks like
and how that imparts a force on a
surface
as we did with all of our flows let's
see how far we can get with theory using
just the conservation equations
let's set up our flow with a schematic
of a boundary layer
we have a free stream velocity coming in
u infinity
the dashed white line is the line that
represents the upper bound of the
boundary layer
we can mark our u equals zero wall
condition
and that's it notice that here the flow
goes past the surface
remember this still applies to moving
vehicles we have just attached our
coordinate
system to the vehicle itself
to start analysis we're going to need
some assumptions
here we will use the some familiar
assumptions and some newer ones midway
through our analysis
first our fluid is incompressible so
density is a constant
second the fluid is steady which means
nothing changes in time in our flow
we also assume it's two dimensional and
two-component
so nothing substantial happens in the
out-of-plane direction
remember the steady and the 2d2c
assumptions are very common for laminar
flows
at this point we almost instinctively
write fully developed
but for boundary layers and most
external flows this is not the case
these flows develop in the x direction
and we can no longer
ignore x gradients in the velocity field
so regrettably we have to leave this
assumption behind because we can't use
it
last let's assume there are no body
forces for now
after assumptions which help us simplify
our equations
we will need boundary conditions which
we need to solve the equations
at the wall we have the no slip and no
penetration conditions
meaning the streamwise and vertical
velocities are zero
where they meet the surface these are
specifically wall conditions
however now that our fluid is unbound in
the positive y direction
we have some new conditions we can
utilize as it goes off into infinity
we take the limits on the flow behavior
as y approaches infinity or gets so far
from the wall
it no longer feels the wall as y goes to
infinity the streamline's velocity
becomes the free stream velocity u
infinity
as a result the free stream is an
idealized location where there is no
vertical velocity
so we set v to be zero
lastly we also just say there is no
pressure gradient in the free stream
direction as
y goes to infinity as the free stream is
just a constant
x velocity this condition on the
pressure gradient will be critical in
being able to deal with the pressure
terms later on
as always we make our start with the
conservation of mass for an
incompressible fluid
we can only remove the term dwdz due to
the assumption that the flow is two
dimensional
and that's as far as we can get we can't
say anything about the v
velocity as we would for the enclosed
flows because we have two gradients here
to keep track of
so we move right on to our conservation
of momentum in the x direction to see
what that tells us
let's write them out in their entirety
the material derivative representing
flow acceleration is on the left
then that is balanced by the forces of
pressure viscosity
and a body force on the right to
simplify
let's apply our assumptions we're going
to be able to remove four terms here the
du dt term because the flow is steady
both the derivatives in the z direction
because the flow is two dimensional
and the body force term and what we're
left with is a rather intimidating
balance of momentum terms
pressure and viscosity terms let's call
this equation one
if we did the same process on the
conservation momentum in the y direction
and applied our assumptions we would
find a similar result here
these two equations are really hard to
work with in the current state
in order to move forward we need to be
able to do something else to make them
more simple
so we're going to make some
observational assumptions that are based
on
observations we've made for this type of
flow in the past
let's consider the growth
characteristics of the boundary layer
and maybe we can say something about the
sizes of some terms compared to others
in the beginning of the boundary layer
development growth is rapid
however not long into development the
growth gets dramatically more gradual
the bulk of the flow is gradual growth
so let's claim that
in our analysis we're considering only
this gradual growth region
this leads us to some more possible
assumptions
because growth is slow in the x
direction we claim that the velocity
field
changes in the wall normal direction y
are much larger than changes in the
streamwise
direction x remember vertically the
boundary layer is small
so things have to change rapidly to go
from zero velocity to the free stream
velocity in the y direction
this means that if we ever see these two
terms together in an equation
for example d u d y versus d u d x
we assume one dominates the other the d
u d y
term and we remove the smallest term d u
d
x keep in mind here you can only compare
like terms meaning only u terms compared
to u terms
and v terms compare to v terms for
example
we can't compare d u d x and d v d
y next we're going to assume that the
streamlines velocity
is much larger than other velocity
components which is typically a fine
claim for these flows since the majority
of the flow is in the streamwise
direction
soon we will be comparing terms and
deciding which to get rid of based on
size
so let's sort the terms into whether
they are considered to be big
small or tiny u and y derivatives of the
u velocity are the biggest
next up we have v and y derivatives on v
as well as the streamwise derivatives on
u these are small
but not too small last we have the tiny
terms
streamwise gradients of v
we can organize these like this because
of generally how we've observed boundary
layers to behave
now we can write both conservation of
momentum equations we have
equations 1 and 2 and label each term
with their size
in equation one we have some terms that
are big times as small
and some that are just small and some
that are just
big let's keep the big and the big time
small
terms we remove only the smallest term
in the equation
in equation two we have some tiny terms
now
we keep everything that is small or a
big
times a tiny term
we can remove the smallest term which is
just tiny
notice this is the same term we removed
above
now if we compare equation 1 to equation
2 term by term
we notice that all terms in equation one
are much bigger than their partner term
in equation two
and we don't know anything about the
pressure terms
however if we know all the terms in
equation one
are much greater than equation two we
also know that the pressure gradient in
1 is bigger than the pressure gradient
in 2.
this is important if we say dp dx
is much much bigger than dpdy then we
know that p
is not a function of y and that it is
approximately only a possible function
of x
so p is a constant in y
and if we pair this with our boundary
condition on the pressure gradient which
says as
y goes to infinity the pressure gradient
goes to zero
this means the pressure gradient is zero
everywhere because it's a constant in
y essentially
through this deduction we can ignore the
streamwise pressure gradient
this brings us to our final form of the
conservation equations for a laminar
boundary layer
they're still rather complicated but
they're totally solvable numerically
back in 1908 these were originally
derived and solved by prandtl and his
student blasius
a famous pair of researchers who
pioneered boundary layer research
their solutions led to the following
discretized form of the boundary layer
shape
here's a table showing their results the
left column is effectively the
y-coordinate
the profile is self-similar in x meaning
it doesn't change
shape in x it just grows this is why we
see the x variable come into the y
column
because the x variable is acting to
stretch the vertical coordinate
next to this is the velocity normalized
by the free stream
so it goes to one far from the wall
if you plotted this data it would
represent the shape of a laminar
boundary layer
[Music]
these equations and a ton of analysis
and boundary layers
require the reynolds number for boundary
layers the reynolds number is the
product of density
the free stream velocity which acts as
our velocity scale
and the x location of interest which is
sometimes our length scale
all divided by the viscosity
the reynolds number tell us when flows
transition to turbulence
for a smooth flat plate flows this is
estimated to be around five hundred
thousand
now that we know the boundary layer
shape let's determine how we can discuss
its size we measure the boundary layer
height with three main strategies
depending on our needs
first the most simple strategy is to
define a physical distance above the
surface and mark its height
this is called the disturbance thickness
delta and is the y
location where the velocity in the
profile is equal to 99
of the free stream velocity sometimes
people use 95 percent instead of 99 but
regardless it's the same procedure
people use this because it's simple and
easy to find and measure
the second strategy is called the
displacement thickness
consider the boundary layer the velocity
deficit is the region in purple
where the flow velocity is less than the
free stream velocity
this space has an area which we're going
to call area a
we can equate this area a to a rectangle
that has
one length as the free stream velocity
which is known and
some height which is the displacement
thickness which we're trying to find
physically interpret this as the height
from the ground
that the surface would have to extend in
order to push the fluid back into being
uniform
it's a bit harder to calculate and
requires the integral of the velocity
profile
people use this in aircraft design where
they consider this delta star as part of
the surface and ignore viscosity above
this distance
lastly we have the most complex way to
define the height the momentum thickness
similar to the displacement thickness we
consider the area beneath
velocity profile but instead we're
considering the momentum deficit so u
squared
and not the velocity deficit
this means if we found a rectangle with
an area a
equal to the momentum deficit the length
of the rectangle is u infinity squared
which is known
and the height of the rectangle is our
theta the momentum thickness
the calculation is similar and also
needs an integral
this is generally used when considering
the surface drag
because the momentum deficit is more
heavily connected to drag force than
just
velocity
for laminar flows we already have the
solution numerically and we can exactly
define the behavior of these three
thicknesses as they vary in the
streamwise direction
and they're all functions of the local
reynolds number
as you can see the boundary layer grows
downstream non-linearly
because x appears in the square root of
the reynolds number
so now we have a good handle on the
boundary layer size shape
and growth in aerodynamics and in
industry
we're generally interested in the force
the fluid applies on the moving surface
boundary layers have a wall shear and
the wall shear results in a shear stress
which makes a drag force
recall newton's law of viscosity if we
calculate it
at the wall we have the shear stress at
the wall
numerically we can do this derivative to
our flow field and come up with another
estimated expression for laminar
boundary layers
interestingly you might notice that the
further downstream you go
the less the fluid pulls on the surface
this is because as the boundary layer
grows the velocity gradient near the
wall
gets smaller
this is popularly represented in
non-dimensional form
where a form of the flow inertia one
half rho u squared
is used to normalize this stress this is
the skin friction coefficient
c sub f and it defines the behavior for
the wall skin frictions
relative to the flow inertia much like
reynolds number this is similar to
viscosity versus inertia
it's important to keep in mind the skin
friction coefficient is different from a
drag coefficient the skin friction
coefficient is just used as to
non-dimensionalize
the wall shear you cannot get drag from
it directly
however we can turn this shear stress
due to friction
into a drag force
it's important to point out that because
our shear stress is now a function of x
we can no longer use the shortcut that
total wall forcing is just the shear
stress times the wall area
we have to move to the general
definition where the force is the
integral of tau
over an area let's assume flow is
uniform in the z
direction and give the plate a width w
by assuming it's uniform w can come out
of the integral
we integrate from 0 to l which is the
plate length
over dx the final expression tells us
the wall force is a function of the
velocity
plate length and the reynolds number
based on the length
it's important to note that for boundary
layers when there is a subscript on the
reynolds number it's typically referring
to the length scale you use in the
reynolds number calculation
sometimes you'll see re sub l or re sub
x
or even r e sub theta which is a
reynolds number based on the momentum
thickness
this drag force can also be
non-dimensionalized using the flow
inertia
the general drag coefficient is the
force of drag divided by the area of the
surface times one half density times
velocity squared specifically for
laminar boundary layers this transforms
into the following function of reynolds
number based on plate length
and that's it for the equations these
equations are valuable to keep in your
tool belt when
analyzing boundary layers because even
when flow is turbulent we need to use
these to calculate the laminar portion
of the plate as we'll see in the next
video
in practice you will can certainly come
across boundary layers in some way shape
or form
in aerodynamics they are the direct
cause of lift and drag on aircraft wings
the growth of the boundary layer leads
to a phenomena called separation
which can dramatically change the forces
a body feels due to a fluid
last boundary layer analysis is the root
of climate modeling
trying to predict flow patterns on the
planet
so understanding what a boundary layer
is and how to work with it
is critical and that's it
let's review today we introduce laminar
boundary layers
a specific external flow that is a
region of lower velocity fluid that
grows on an
external surface in a flow field due to
viscosity
we used our typical theoretical approach
to understanding a flow with some unique
assumptions and boundary
conditions in the end of our analysis
using some creative assumptions and
comparing the size of different terms in
our equations
we were able to arrive at a solvable set
of equations that define the
velocity prandtl and blasius solve these
equations to get the self-similar shape
of the laminar boundary layer known as
the velocity profile
the boundary layer height was quantified
using three separate methods the
disturbance thickness the displacement
thickness
and the momentum thickness we then use
the velocity field to calculate the
stress that acts on the surface
and the resulting drag force on the
plate due to the fluid
while laminar boundary layers are
typically overshadowed by turbulent
boundary layers
they are still a critical aspect of
fluid mechanics and analysis
i hope you enjoyed the video and thanks
for watching
you
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