Turbulent Separation Criteria
Turbulent separation criteria are the most useful since most pressure
recovery is done using turbulent boundary layers. There are many criteria
that are used.
Minimum Cp
The simplest criterion is that used to estimate when the flow will separate
from the leading edge of an airfoil. This rule of thumb states that there
is a minimum value of Cp that can be tolerated. Numbers such as -10 to -13
are sometimes used, but this is a very crude rule and applies only to cases
of leading edge separation.
Loftin's criterion
A related, but somewhat more sophisticated method is attributed to Loftin
and states that the maximum value of Cp', the canonical
pressure coefficient, after the start of recovery is +0.88. This is
not a conservative estimate, however, and cannot be relied on for a wide
range of airfoils.
Shape Factor
Perhaps the most reliable criterion is that based on the computed boundary
layer quantities. It has been shown that separation is very likely when
the value of the shape factor, H exceeds 2.2 to 2.4.
Stratford's Criterion
In 1959 Stratford devised a rather simple criterion for the separation of
turbulent boundary layers. Similar to his laminar separation criterion,
this rule states that separation will occur when:
Where the constant S is 0.35 when d2p/dx2 < 0 (concave recovery)
and 0.39 when d2p/dx2 > 0 (convex recovery)
The Reynold's number in the Stratford formula is based on the local
effective
length of the boundary layer, x', and the maximum velocity, Um.
The formula is based on a great deal of empirical data and is only valid
for Cp' < 4/7, but it is very useful in the design of airfoil sections.
Stratford's method usually is conservative, predicting separation just a
bit before the methods based on explicit computation of the shape factor.
Comparison with the corresponding laminar flow formula shows how turbulent
boundary layers are very much more resistant to separation. Note also that
the expression for x is different for turbulent boundary layers. More detail
on the definition of effective boundary layer length is presented at the
end of this section.
Stratford's criterion may be used to compute the shape of the pressure distribution
that is everywhere on the edge of separation. This is a useful distribution
for many reasons. Most importantly, it permits the most rapid possible recovery
from a given minimum pressure. This, or something approaching it, would
be used in the design of sections with maximum extent of laminar flow or
sections with maximum lift or maximum thickness. This will be discussed
in a subsequent section, but here we show how the particular Cp distribution
is derived.
We start by taking Stratford's criterion as a differential equation describing
the Cp variation and integrate the expression for the resulting Cp. This
is not as straightforward as it appears since the formula is only valid
for Cp' < 4/7.
Stratford effectively assumed a constant value of the boundary layer shape
factor (e.g. H = 2.0) over this section and derived:
The values of a and b are chosen so that the slope and value of Cp' match
at Cp' = 4/7.
Upper surface pressures with Stratford recovery to Cp = 0.20 at trailing
edge.
Laminar Rooftop, Re = 5 x 106
Upper surface pressures with Stratford recovery to Cp = 0.20 at trailing
edge.
Turbulent Rooftop, Re = 5 x 106
Liebeck airfoils with Stratford pressure recoveries designed for maximum
lift.
