Unsteady Wave Motion Shock Tube Problem - Shock Reflection

Size: px

Start display at page:

Download "Unsteady Wave Motion Shock Tube Problem - Shock Reflection"

Bertina Alexander
5 years ago
Views:

1 Unsteady Wave Motion Shock Tube Problem - Shock Reflection Niklas Andersson Division of Fluid Dynamics Department of Applied Mechanics Chalmers University of Tecnology 8 februari 09 Shock Reflection When the incident shock wave reaches the wall, a shock propagating in the opposite direction is generated with a shock strength such that the velocity of the induced flow behind the incident shock is reduced to zero. The flow can not go through the wall and thus the velocity must be zero in the vicinity of the wall. The properties of the incident shock wave are directly related to the pressure ratio over the shock wave. Therefore, it would be convenient to have a relation between the reflected shock wave and incident shock wave. t contact surface reflected shock W r expansion wave 3 W u p 4 incident shock x left wall diaphragm location right wall Figur : shock reflection at solid wall located at x4.0)

2 The Incident Shock Wave The pressure ratio over the incident shock in Fig. can be obtained as p + γ M p γ + s ) ) where M s is the wave Mach number, which is calculated as M s W a ) In Eqn., W is the speed with which the incident shock wave travels into region and a is the speed of sound in region see Fig. ). Solving Eqn. for M s, we get M s ) γ + p + 3) γ p Anderson derives the relations for calculation of the ratio T /T T T p p γ + γ + p + γ + γ p p p 4) From Eqn. 4 it is easy to get the corresponding relation for ρ /ρ ρ ρ + γ + p γ p γ + γ + p ) p Anderson also shows how to obtain the induced velocity, u p, behind the incident shock wave, i.e. the velocity in region see Fig. ). u p W ρ ) M s a ρ ) ρ ρ 6)

3 The Reflected Shock Wave The pressure ratio over the reflected shock can be obtained from Eqn. by analogy p + γ M p γ + r ) 7) where M r is the Mach number of the reflected shock wave defined as M r W r + u p a 8) where W r is the speed of the reflected shock wave and a is the speed of sound in region see Fig. ). Solving Eqn. 7 for M r gives M r ) γ + p + 9) γ p The ratios T /T and ρ /ρ can be obtained from Eqns. 4 and by analogy T T p p γ + γ + p + γ + γ p p p 0) ρ ρ p p + γ + γ γ + γ + p ) p The velocity in region which is the same as the induced flow velocity behind the incident shock wave can be obtained as ) ρ u p W r M r a ρ ) ρ ρ ) 3

4 Reflected Shock Relation With the relations for the incident shock wave and reflected shock wave defined, we now have the tools to derive a relation between the incident and reflected shock waves. The induced flow velocity u p calculated using the relation obtained for the incident shock wave must of course be the same as when calculated using reflected wave properties, i.e. the result of Eqn. 6 is identical to that of Eqn. M r a ρ ) M s a ρ ) ρ ρ 3) rewriting gives M r ρ ) M s ρ ) a 4) ρ ρ a Assuming calorically perfect gas gives a γrt and thus M r ρ ) M s ρ ) T ) ρ ρ T Let s first look at the term on the left hand side of Eqn. M r ρ ) ρ Using the ρ /ρ and p /p from Eqns. and 7 and simplifying gives M r ρ ) ) ) M r ρ γ + M r 6) Using the same approach on the corresponding term for the incident shock wave on the right hand side of Eqn. gives M s ρ ) ) ) M s ρ γ + M s 7) 4

5 Now, inserting 6 and 7 in Eqn. gives ) ) M r γ + M r γ + ) M s M s ) T T 8) Simplifying and inverting gives ) ) Mr T Mr 9) T The rightmost term in Eqn. 9 T /T ) needs to be rewritten. Inserting in 4 and expanding all terms gives T γ + ) + γ + )γ )M s + 4γ ) + γγ ) ) T γ + ) γ + ) + γ + )γ )M s + 4γM s ) γ + ) M s + γ ) γ + ) M s )γ γ + ) + γ + )γ )M s + 4γ ) γ ) )) γ + ) + γ ) γ + ) M s ) γ + ) Finally we end up with the following relation T γ ) + T γ + ) M s ) γ + ) 0) The temperature ratio over the incident shock wave is now totally defined by the incident Mach number M s and the ratio of specific heats γ. With 0 in 9 we get the sought relation between the reflected and incident Mach numbers. ) ) Mr γ ) Mr + γ + ) M s ) γ + ) ) It should be noted that Eqn. is valid for calorically perfect gases only.

AA210A Fundamentals of Compressible Flow. Chapter 13 - Unsteady Waves in Compressible Flow

AA210A Fundamentals of Compressible Flow Chapter 13 - Unsteady Waves in Compressible Flow The Shock Tube - Wave Diagram 13.1 Equations for irrotational, homentropic, unsteady flow ρ t + x k ρ U i t (