THC Science

Template:Noncompliant

A shock is called an oblique shock when the flow through the shock is not normal to the shock. Such shocks are formed when flow is 'turned into itself'. A typical example of when this happens is when flow is turned by a wedge and an oblique shock forms at the tip of the wedge. The conservation of mass, momentum and energy across a shock are the basic relations that describe the properties of an oblique shock are called the Rankine-Hugoniot equation . Let $p,u,\rho$ and $e$ denote pressure, velocity, density and internal energy per unit mass. Let the subscript 't' denote the tangential velocity component and 'n' denote the normal velocity component and let '1' denote unshocked quantities and let '2' denote shocked quantities. Then

\rho _{1}u_{1n}=\rho _{2}u_{2n}\,

p_{1}+\rho _{1}u_{1n}^{2}=p_{2}+\rho _{2}u_{2n}^{2}

e_{1}+{\frac {p_{1}}{\rho _{1}}}+{\frac {1}{2}}u_{1n}^{2}=e_{2}+{\frac {p_{2}}{\rho _{2}}}+{\frac {1}{2}}u_{2n}^{2}

u_{1t}=u_{2t}

Note that these are general equations that applies to any Equation of state. In the extreme case the disturbance appears in front of the flow and the created shock is referred as normal shock. The oblique shock appears as a shock in angle to flow direction. This shock is common in many situations which include space vehicles (airplane, etc) and many internal flow. The first to discover this shock was Ernst Mach in his original picture showing a bullet in a supersonic flow that cause the oblique shock.

The oblique shock analyzed in the same way as the normal shock when flow is "broken" into two components, one in the flow direction and one perpendicular to the flow. The component perpendicular to the flow doesn't go any change and the component in the flow direction undergoes shock. The mathematical treatment for this component is the same as for normal shock. However, because two dimensional nature of the problem, as oppose to normal shock, two parameters have to be provided in order to solve the problem. The first who analytically investigate this physical situation was Ludwig Prandtl with his student Theodor Meyer 190x. They also discovered the isentropic flow (flow without shock) region. However, they were not able to express the limit of their equations. Due to the mathematical complications, the solution to these equations declared unattainable by NASA in the famous NACA 1135 report page 10. Nevertheless, scientists attempted to solve these equations with various techniques. For example, some mathematicians suggested to solve a more complicated equations with viscosity and find the solution when the viscosity is approaching zero. Other suggested numerical approaches. In last ten years or so partial and limited solution was suggested by Emmual. Recently, Bar-Meir found the complete analytical solution.

File:ObliqueD.png

Fig. 1: View of the flow direction for oblique shock

Bar-Meir's solution suggests that, contrary to popular believe, oblique shock cannot occur at zero inclination. The other interesting part of Bar-Meir's solution is the explanation for the detached shock. The solution also explains the transition to Prandtl-Meyer function. The governing equation basically states that the results of flow component in the new flow direction gone through shock with the other component are parallel to the inclination see the Figure 1. The governing equation is $x^{3}+a_{1}x^{2}+a_{2}x+a_{3}=0$ where

   $x=\sin ^{2}\theta$ 
and

$a_{1}=-{{M_{1}}^{2}+2 \over {M_{1}}^{2}}-k\sin ^{2}\delta$ , and $a_{2}=-{2{M_{1}}^{2}+1 \over {M_{1}}^{4}}+\left[{(k+1)^{2} \over 4}+{k-1 \over {M_{1}}^{2}}\right]\sin ^{2}\delta$ , and

 $a_{3}=-{\cos ^{2}\delta  \over {M_{1}}^{4}}$

File:LimitedTheta.jpg

Fig. 2: shock angle as a function of upstream Mach number

The solution of this equation is three pair sets of angle that satisfies the above conditions. The mathematical derivations of the solution can be found in Bar-Meir's book in the external link below. Every pair symmetrical angles. The first pair is thermodynamically unstable because of increase of entropy. Yet, there isn't evidence that for unsteady state conditions this angle cannot be achieved as it is commonly believed. The two other pairs have been shown to be stable. The question which of these pair will actually occur depends on the boundary conditions.

The second angle referred to as the week angle/solution in which upstream and down stream are remain in supersonic flow (mostly) for ideal gas model. The third angle is referred to as the strong solution in which flow transformed in to subsonic flow. When the change angle flow direction is above critical value it has been shown that the mathematical solution [[[Bar-Meir]], 2005] requires a root of negative nubmer. Physically, this means that no solution is possible and the shock is dittached from the physical body which was observed even by Mach.

External links

Fundamentals of compressible flow mechanics - Genick Bar-Meir, Ph.D.
Shockwave calculator (Java applet).

THC Science

Bringing Science to the Cannabis Conversation!

Cannabis Ruderalis

See also

External links

Leave a Reply