WHAT IS A BOUNDARY LAYER

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WHAT IS A BOUNDARY LAYER?
A boundary layer is a layer of flowing fluid near
a boundary where viscosity keeps the flow
velocity close to that of the boundary. If the
boundary is not moving, then the normal and
tangential flow velocities at the boundary
vanish, and the normal and tangential flow
velocities near the boundary are small compared
to those of the invisicid (potential) flow far
from the boundary. Consider the potential flow
around the body below. Constant rectilinear
potential flow prevails far upstream and
downstream of the body. The potential flow is
forced to accelerate around the body. The
potential flow field satisfies the condition of
zero normal velocity at the boundary, but not
zero tangential velocity at the boundary.
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POTENTIAL FLOW AROUND THE BODY
The potential flow is forced to accelerate around
the body. Thus u(?u/?s) gt 0 along a streamline
near the body from near the upstream stagnation
point to section A A. The velocity profile
along the line A A reflects this.
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BUT POTENTIAL FLOW IS NOT THE WHOLE STORY
If the body is not moving, there must be some
region (perhaps very thin) where the flow
velocity drops to zero. This region is called a
boundary layer. We will find that boundary layer
thickness depends on an appropriately defined
Reynolds number of the flow, and as this number
gets larger the boundary layer gets
thinner. Here we are interested in the case of
large Reynolds number (but not so large that the
flow becomes turbulent).
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BOUNDARY-ATTACHED COORDINATES AND BOUNDARY LAYER
THICKNESS
Let x denotes a boundary-attached streamwise
coordinate, and y denote a boundary-attached
normal coordinate, as noted below. Furthermore,
let ??(x) be defined as some measure of boundary
layer thickness, i.e the thickness of the zone of
retarded flow near the boundary (to be defined in
more detail later). Then the boundary layer can
be illustrated as follows (red line)
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2D NAVIER-STOKES AND CONTINUITY EQUATIONS
Since the x-y coordinate system is
boundary-attached, it defines a curvilinear
rather than cartesian system. If the curvature is
not too great, however, the governing equations
for steady 2D flow around the body can be
approximated as if in a cartesian coordinate
system
Note that only the dynamic pressure pd is used
here. The total pressure p has been decomposed
as p ph pd,, and the static pressure ph has
been cancelled out against the gravitational
terms.
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SCALING
The scale for flow velocity in the x direction is
the free-stream velocity U far from the body.
The length scale in the x direction is the body
length L. Boundary layer thickness (which might
vary in x) is simply denoted as ?.
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SCALING OF THE CONTINUITY EQUATION WITHIN THE
BOUNDARY LAYER
The continuity equation is The term ?u/?x
scales as Now consider continuity within the
boundary layer. If the length scale in the
direction normal to the body is denoted as ?
(boundary layer thickness) and the normal
velocity scale is denoted as V, it follows from
the estimate ?v/?y V/? and the
relation that It furthermore follows that
if ?/L ltlt 1 then V/U ltlt 1.
Note the sign is not important in order of
magnitude estimates
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SCALING OF THE STREAMWISE EQUATION OF MOMENTUM
BALANCE WITHIN THE BOUNDARY LAYER
The streamwise equation of momentum balance is
Recalling from the Bernoulli equation that pd
can be scaled as ??U2 and from the previous slide
that V (?/L) U, the terms scale as
Note ?2u/?x2 scales as U/L2, and not U2/L2.
Remember why?
Now multiply the scale estimates by L/U2 to get
the dimensionless scale estimates for each term
relative to U2/L
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SCALING OF THE STREAMWISE EQUATION OF MOMENTUM
BALANCE contd.
From the previous slide, our estimates of the
terms in the streamwise momentum balance equation
in the boundary layer are where denotes
the Reynolds number of the flow. The case of
interest to us here is that of large Reynolds
number, so that 1/Re ltlt 1.
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SCALING OF THE STREAMWISE EQUATION OF MOMENTUM
BALANCE contd.
The viscous terms in the Navier-Stokes equations
are in general small compared to the other terms
when the Reynolds number is large. But this
cannot be true everywhere there must be at least
some small region where viscosity is powerful
enough to bring the tangential velocity to zero
at the boundary. The small zone where viscosity
remains important even at large Reynolds number
denotes the boundary layer.
Thus this term can be neglected (Re ltlt, 1),
But this term must be retained regardless of how
small Re is, or there are no viscous effects
anywhere.
Hence the estimate
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WHAT DOES THIS ESTIMATE SAY?
The scale relation, or estimate says that the
boundary layer ? gets ever thinner compared to
the length of the body L as the Reynolds number
Re (UL)/? gets large, but never goes to 0 as
long as Re is finite! Outside of the boundary
layer viscosity can be completely neglected in
the streamwise momentum equation, and within
it the streamwise momentum equation approximates
as
This term is needed to bring u to 0 at the
boundary!
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SCALING OF THE NORMAL EQUATION OF MOMENTUM
BALANCE WITHIN THE BOUNDARY LAYER
The normal equation of momentum balance is
Recalling from Bernoulli that pd can be
scaled as ??U2, the terms scale as Further
recalling from Slide 7 that V (?/L)U and
multiplying all terms by (?/U2), the following
dimensionless scale estimates are obtained
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SCALING OF THE NORMAL EQUATION OF MOMENTUM
BALANCE contd.
From the previous slide But from Slide 14,
(?/L) (Re)-1/2, so that the estimates
become or thus in the limiting
approximation as Re becomes large,
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WHAT DOES THIS ESTIMATE OF PRESSURE SAY?
At sufficiently large Reynolds number the
equation of momentum balance in the normal
direction reduces to the approximation This
says that the pressure pd(x,y) can be
approximated as constant in y in the boundary
layer.
Now let ppd(x,y) denote the dynamic pressure
obtained from the potential flow (inviscid)
solution. It then follows that pd(x) within the
boundary layer can be obtained from ppd(x,0)
Thus the solution of any boundary layer problem
also requires the inviscid (potential flow)
solution.
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THE BOUNDARY LAYER APPROXIMATIONS OF THE
NAVIER-STOKES EQUATIONS
The boundary layer equations are thus
where ppdp(x) denotes the dynamic pressure
obtained from the potential flow solution
extrapolated to the surface of the body. The
boundary conditions are
so that both the tangential and normal velocities
vanish on the boundary. In addition, u must
approach the potential flow value as y becomes
large.