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 is 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.
Supersonic Flow of a Cone
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 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 derivative. 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 derivative, 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 oblique 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 is a Java program that performs the Taylor-Maccoll analysis. You can use this simulator to study the flow past a cone.
Please note: the simulation below is best viewed on a desktop computer. It may take a few minutes for the simulation to load.
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.