Question

In high speed aerodynamics, give an example of a strong oblique shock solution.

Answer 1

In the high speed aerodynamic gives an example of strong oblique solution -:

s 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 theentropy 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 wavesare generated in the flow. Shock waves are very small regions in the gas where the gas propertieschange by a large amount. Across a shock wave, the static pressure, temperature, and gas densityincreases almost instantaneously. The changes in the flow properties are irreversible and the entropyof 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 as shown on the slide. Because total pressure changes across the shock, we can not use the usual (incompressible) form of Bernoulli's equation across the shock. The Mach number and speed of the flow also decrease across a shock wave.

If the shock wave is perpendicular to the flow direction, it is called a normal shock. There are equations which describe the change in the flow variables. The equations are derived from the conservation of mass, momentum, and energy. Depending on the shape of the object and the speed of the flow, the shock wave may be inclined to the flow direction. When a shock wave is inclined to the flow direction it is called an oblique shock. On this slide we have listed the equations which describe the change in flow variables for flow across an oblique shock. The equations presented here were derived by considering the conservation of mass, momentum, and energy for a compressible gas while ignoring viscous effects. The equations have been further specialized for a two-dimensional flow without heat addition. The equations only apply for those combinations of free stream Mach number and deflection angle for which an oblique shock occurs. If the deflection is too high, or the Mach too low, a normal shock occurs. For the Mach number change across an oblique shock there are two possible solutions; one supersonic and one subsonic. In nature, the supersonic ("weak shock") solution occurs most often. However, under some conditions the "strong shock", subsonic solution is possible.

Oblique shocks are generated by the nose and by the leading edge of the wing and tail of a supersonic aircraft. Oblique shocks are also generated at the trailing edges of the aircraft as the flow is brought back to free stream conditions. Oblique shocks also occur downstream of a nozzle if the expanded pressure is different from free stream conditions. In high speed inlets, oblique shocks are used to compress the air going into the engine. The air pressure is increased without using any rotating machinery.

On the slide, a supersonic flow at Mach number Mapproaches a shock wave which is inclined at angle s. The flow is deflected through the shock by an amount specified as the deflection angle - a. The deflection angle is determined by resolving the incoming flow velocity into components parallel and perpendicular to the shock wave. The component parallel to the shock is assumed to remain constant across the shock, the component perpendicular is assumed to decrease by the normal shock relations. Combining the components downstream of the shock determines the delflection angle. Then:

cot(a) = tan(s) * [{((gam+1) * M^2)/(2 * M^2 * sin^2(s) - 1)} - 1]

where tan is the trigonometric tangent function, cotis the co-tangent function:

cot(a) = tan(90 degrees - a) .