As an object moves through a gas, the gas molecules are deflected
around the object. If the speed of the object is much less than the
speed of sound
of the gas, the density of the gas remains constant and the flow of
gas can be described by conserving
momentum, and
energy.
As the
speed of the object approaches the speed of sound, we
must consider
compressibility effects
on the gas. The density of the gas varies locally as the gas is
compressed by the object.
For compressible flows with little or small
flow turning, the flow process is reversible and the
entropy
is constant.
The change in flow properties are then given by the
isentropic relations
(isentropic means "constant entropy").
But when an object moves faster than the speed of sound,
and there is an abrupt decrease in the flow area,
shock waves are generated in the flow.
Shock waves are very small regions in the gas where the
gas properties
change by a large amount.
Across a shock wave, the static
pressure,
temperature,
and gas
density
increases almost instantaneously.
The changes in the flow
properties are irreversible and the
entropy
of the entire system increases.
Because a shock wave does no work, and there is no heat addition, the
total
enthalpy
and the total temperature are constant.
But because the flow is non-isentropic, the
total pressure downstream of the shock is always less than the total pressure
upstream of the shock. There is a loss of total pressure associated with
a shock wave.
On this web page, we show a method for determining the supersonic flow past a cone. The
method was first developed by G.I. Taylor and J.W. Maccoll in 1933. The derivation of the
differential equation shown on the slide is fairly complicated.
The method assumes that the supersonic flow
along a cone is simplified because of symmetry considerations. We define a
polar coordinate system
through the point of the cone, with r being the radial coordinate along rays that
meet at the point, theta being the angle that the ray makes with a reference line
through the axis of the cone, and phi being the angle measured around the axis.
All changes with phi are equal to zero because the flow is axisymmetric.
d (variable) / d phi = 0
where d represents a partial derivitive. On the figure we show a two dimensional
cut at any representative value of phi. The analysis can be further simplified by
assuming that the flow variables do not change with r; flow values are constant
along a ray and therefore only vary with theta
d (variable) / d r = 0
By taking the conservation of
mass.
momentum, and
energy equations, written in polar coordinates, and applying
the simplifications given above, Taylor and Maccoll were able to derive the differential
equations given on the slide which express the change in flow variables with theta from
behind the oblique shock to the surface of the cone.
[(gamma -1) /2][1 - Vr^2 - (d Vr / d theta)^2][2Vr + cot(theta)d Vr /
d theta + d^2 Vr / d theta^2] -
d Vr / d theta * [Vr * d Vr / d theta + d Vr /
d theta * d^2 Vr / d theta^2] = 0
Vtheta = d Vr / d theta
where gamma is the ratio of specific heats, Vr is the radial component of the velocity,
and Vtheta is the angular velocity component
perpendicular to the rays. d Vr / d theta is an ordinary derivitive, since the radial velocity
varies only with theta. There is no known closed solution to this ordinary differential
equation, so the solution must be obtained by numerical analysis.
The Taylor-Maccoll analysis proceeds as follows. With a known free stream mach number M and known
cone angle c, assume a value for the obique shock angle s. The
oblique shock
relations provides the values of change in flow variables
across the shock, and the deflection a of the flow through the shock. Determine the
velocity components Vr and Vtheta immediately downstream of the shock. The differential
equation can be evaluated by selecting some incremental change in theta and integrating using
a fourth order Runge-Kutta scheme from the shock until the value of Vtheta is equal to zero.
Vtheta = 0 is the normal velocity condition at the surface of the cone. Knowing the value of
theta = theta surface for which Vtheta = 0 and the initial guess of the shock angle s,
gives a solution of the Taylor-Maccoll differential equation.
A cone of angle theta surface generates a shock of
angle s. In general, theta surface will not be equal to the desired cone angle c.
Modify the assumed shock angle and repeat the solution of the differential equation to
obtain a new pair of shock angle s and theta surface. Repeat this procedure until
theta surface = cone angle c. Then the resulting shock angle s is the
desired result for cone angle c. In determining the shock angle, the variation of flow
velocity and Mach number from the shock to the cone surface is also determined. The
isentropic flow
relations can then be used to determine the value of the flow variables from the shock to the
cone surface.
Here's a Java program that performs the Taylor-Maccoll analysis.
You can use this simulator to study the flow past a cone.
Due to IT
security concerns, many users are currently experiencing problems running NASA Glenn
educational applets. There are
security settings that you can adjust that may correct
this problem.
Input to the program can be made
using the sliders, or input boxes at the upper right. To
change the value of an input variable, simply move the slider. Or
click on the input box, select and replace the old value, and
hit Enter to send the new value to the program.
Output from the program is displayed
in output boxes at the lower right. The flow variables are constant along rays from
the leading edge of the cone. To display the rays. use the drop menu at the lower part of
the output panel next to the word "Ray Plot".
To select a given ray and display the value of the flow variables along the ray, use the
drop menu at the upper part of the output panel next to the word "Ray".
Variables are presented as ratios
to free stream values. The graphic at the left shows the cone (in red)
and the shock wave generated by the cone as a line. The line is colored
blue for an oblique shock and magenta when the shock is a normal shock. The black
lines show the streamlines of the flow past the cone. Notice that downstream
(to the right) of the shock wave, the lines are curved as the velocity components
vary along the rays. Downstream, the streamlines are closer together than upstream
which indicates an increase in the density of the flow.
There is more complete
shock simulation program
that is also avaliable at this web site. It solves for both flow past a
wedge
and flow past a cone.
If you are an experienced user of the simulator, you can use a
sleek version
of the program which loads faster on your computer and does not include the
user's manual. The
ShockModeler
program models the intersection and reflection of multiple oblique shock waves
generated by multiple wedges.
You can also download your own copy of the program to run off-line by clicking on this button:
