If the shock wave tries to move to right with velocity u 1 relative to the upstream and the gas motion upstream with velocity u 1 to the left  the shock wave is stationary for observers fixed in the laboratory If the gas motion upstream is turned off. i.e We are watching a normal shock wave propagate with velocity W (crelative to the laboratory) into a quiescent gas  Induced velocity u p behind the moving shock Chapter 7 Unsteady Wave Motion 7.1 Introduction

7.2 Moving Normal Shock Waves change Coordinate system An important application of unsteady wave motion is a shock tube u=0

Shock Mach number Hugoniot equation(identically the same as eq(3.72) for a stationary shock) As expected,it is a pure thermodynamic relation which do not care abort the coordinate system

For a calorically perfect gas,,, Note : for a moving shock wave it becomes convenient to think of P 2 /P 1 as a major parameter governing change across the wave (instead of M s )

If P 2 /P 1 So r=1.4, u p /a as P 2 /P 1 M 2 can be supersonic M 2 supersonic or subsonic?

Also for a moving nomal shock So Also is not constant 7.3 Reflected Shock Wave Unsteady,

Note : a general characteristic of reflectted shock, W R <W So : in x-t diagram the reflected shock path is more steeply Inclined than the incident shock path Coordinates transform

Velocity jump Formula

Note : The local wave velocity w the local velocity of a fluid element of the gas, u Propagated by molecular collisions, which is a phenomenon superimposed on top of the mass motion of the gas 7.4 Physical Picture of Wave Propagation In general, ∴ the shape of the pulse continuously deforms as it propagates along the x axis

7.5 Elements of Acoustic Theory continuitymomentum Note : for a gas in equilibrium, any themodynamic state variable is uniquely by any two other state variable. perturbations ( in general, are not necessary small) Non-linear but exact eqn for 1-D isentropic flow

Now consider acoustic waves => & are very small perturbations => Momention eqn becomes Acoustic equations => Linear. Approximate eqs for small perturbations. Not exact More and more accurate as the perturbation become smaller and smaller => 1-D form of the classic wave equ Linearized Small Perturbation Theory

Let => If Note that & are not independent Let => similarly F, G, f, g, are arbitrary functions of their argument

The other way to derive the above equation : Summary : + : right – running waves – : left – running waves Note : 1. (+) => particles move in the positive x direction (–) => particles move in the negative x direction 2. In acoustic terminology, that part of a sound wave where >0 => condensation => in the same direction as the wave motion rarefaction => in the opppsite direction as the fashion

7.6 FINITE WAVES – Δρ and Δu are not small In contrast to the linearized sound wave, different parts of the finite wave propagate at different velocities relative to the laboratory. Consider a fluid element located at x 2 which is moving to the right with velocity u 2  Wave speed relative to the laboratory. Physically, the propagation of a local part of the finite wave is the local speed of sound superimposed on top of the local gas motion. Point moving to the left The wave shape will distort In fact, of u 1 > a 1 → W 1 moves to the left

The compression wave will continually steepen until it coalesces into a shock wave, whereas the distortion of the wave form is illustrated in Fig. 7.9

Governing equation for a finite wave : Continuity : For 1-D flow Momention : For 1-D flow

Consider a specific path so that Similarly The methed of characteristics – along specific paths, the P.D.E reduces to O.D.E C + characteristic C - characteristic

= (along C + characteristic) = (along C - characteristic) For a clalorically perfect gas isentropic Riemann Invaruants

(along a C + charcteristic) (along a C - charcteristic) 7.7 Incident and Reflected Expansion Waves

Prove theat the C - characteristics are straight lines In the constant – property region 4, and is a constant C + characteristics have the same slope & J + is the same everywhere in region 4 Is the same at all points → Straight line Also p,, T are constant along the given straight – line C - characteristic Note : 1. Such a wave is defined as a simple wave – a wave propagating into a constant – property region. Also, it is a centered wave – originetes at a given point. 2. C + cheracteristics can be curved.

3.For a simple centered expansion wave, the solution can be obtained is a closed analytical form. is constant through the expansion wave. constant through the wave Consider the C - characteristics for

4. In non – simple region, a numerical procedure is needed. The characteristic lines and the compatibility conditions are pieced together point by point. Non – simple region obtained from simple wave solution (for point 1, 2, 3, 4) The slopes of straight lines 3-6 & 5-6 are for line 3-6 for line 5-6

High Pressure Driver section Low Pressure Driver section Diaphragm pressure ratioDetermines uniquely the strengths of the incident Shock and expansion waves. 4 upup w 321 Contact surface 7.8 Shock Tube Relations

are implicit function of

The incident shock streugth will be made stronger as is made smaller We want as small as possible The driver gas should be a low – molecular – weight gas at high T The driver gas should be a high – molecular – weight gas at low T

7.9 Finite Compression Wave After the breaking of the diaphragm, the incident shock is not formed instantly. Rather, in the immediate region downstream of the diaphragm location, a series of finite compression waves are first formed because the diaphragm breaking process is a complex three – dimensional picture requir a finite amount of time. These compression wave quickly coalesce into incident shock wave.