TT. We [4] 4.1 NORMAL SHOCK WAVES 441.1 Introduction When there is a relative motion between a body and fluid, the disturbance is created. Ifthe disturbance is of an infinitely small amplitude, that disturbance is transmitted through the fluid with the speed of sound. If the disturbance is finite amplitude shock waves are created, ‘Wave phenomenon is classified as follows Wave phenomenon Weal waves shoel waves Weak Weak Normal Obtique compression expansion shock shock wave wave wave wave Mach Waves ‘A mach waves are weak waves, Mach wave has been described in the previous Chapter (Chapter 1) Expansion Wave ‘A.wave which is at a lower pressure than the fluid in to which its, moving is called an expansion wave (or) rarefaction wave. Compression Wave ‘A wave whieh is at a higher pressure than the fluid in to whi moving is called compression wave. itis42 ‘Shock Wave ‘A shock wave is nothing but steep finite pressure wave. The shock wave may be deseribed as compression wave front in a supersonic flow field across which there is abrupt change in flow properties, The flow process through the shock wave is highly ieversible and cannot be approximated as being isentropic, t Steep wave | non Steep waves x (space) Fig 4.1 Development shock wave [Time-space diagram] Normal shock When the shock wave is at right angle to the flow, itis called normal shock. Oblique shock low, itis called When the shock wave is inclined at an angle to the oblique shock. 4.41.2. Prandtl ~ Meyer Relation andtl-Meyer relation which is the bass of other equation fr shock saves. It gives the relationship between the gas velocities before and afer the normal shock and the critical velocity of sound. Normal shock waves 4.3 Shock wave I ' area Ms | % goes renee —-" | | ly ; Before shock | | After shock ee Se cere ARR Fig? We know tat Stagnation enthalpy equation 2 @ guy fae Fol yet y= {From Chapter 1 - Equation no.1.25] ‘Applying this equailon to the flow before shock wave and after shock wave. Before shock wave = de yx (Ht) wdo-y = a (tye(42) \We know that from momentum equation (-BYAH mE) 2 a ESD Px-Py > | my mcs Norma Shock Waves 4.5 = Substitate = value in Equation (4.3) Gas equation, py= mRT For unt mass prRT st Hee Beer ei IRE pee icy bi2 id Pe Y > YP ag? 7 7 Ps (43)4.6 Gas Dynamics and Jet Propulsion ica = Substite ,2 and 2,2 values in equation no (4:5) 2 yp Bf X= 1f6,-0) - ES exty ‘Multiply by Normal Shock Waves 4.7 Multiplying by 2 > (4a + Neg =27 16.5) > (lat =27 [XG ]- Der ey > WA)at? =27 Ep 6~ Tepe Fey 2 = (Na =1e6,+e6 > Ota = gore) art = 04D a) ===) ----48) This is another useful form of Prandtl-Meyer relation.4.124 Gas Dynamics and Jet Propulsion 4.2 OBLIQUE SHOCK WAVES 4.2.1 Introduction When the shock wave is inclined at an angle to flow, it is called ‘oblique shock. It is also referred to as a two dimensional plane shock wave. 4.2.2 FlowThrough Oblique Shock Waves Consider a stationary wedge in a flow system as shown in fig.4.7. “The gas undergoes a change of direction due to this concave corner. A shock will appear atthe wedge as shown and this will make an angle o with the original flow direction. This angle is called the wave angle (@) andi will be greater than the deflection angle (6). {A stronger oblique shock wave is nearer to a normal shock wave nd has a larger wave angle (i.e,, nearer to 90°). ‘A.weak oblique shock wave has smaller wave angle and closer to the Mach waves. Oblique shocks usually occur when a supersonic flow is turned in to itself. The apposite ofthis, ie., when a supersonic flow is turned away from itself, expansion fan is formed. Narmat Shock Waves 4.125 The following assumptions are used for oblique shock flow 1, Flow is steady, adiabatic and frictionless. 2. The gas is perfect with constant specific heats. 3. Absence of work transfer across the boundaries. 4, Absence of body forees. 423 Formulae Used 1, Mach number at entry (on) Upstream Mach number of Oblique shock where M, ~ Upstream Mach number of normal shock o- Wave angle. 2. Mach number at exit , (or) Downstream Mach number of Oblique shock: ™, [Mer mew where M,— Downstream Mach number of normal shock 8 ~ Deflection angle (or) Wedge angle. 3. Deflection angle M,?sin? 6-1 tan 8=2coto > 2+7M,?+M,2(1-2sin"o) poet ie4.118 Gas Dynamics ard Jet Propulsion ‘Solution {Refer section 4.1.3] 3. Derive the static pressure ratio across the shock. By (Anna Univ -Dec'03, MU ~ Oct'97 & Oct'96) Solution [Refersection 4.1.4) 4. Derive the temperature ratio across the shock. Mi Ban Bay [MU -Apr96 & Apr Sotuion {Refer section 4.1.5] 5. Derive Rankine-Hugoniot equation, IMRU ~ Apr'96, MSU — Nov'95 & Nov'96, MU Oct'95 & Oct'99} Solution [Refer section 4.1.7] 6, Derive stagnation pressure ratto across the shock wave. Poy Normal Shock Waves 4.119 ‘Solution [Refer section 4.1.8) 7. Dertve change in entropy across the shock, Terao] 2 Solution [Refer section 4.1.9] 4 & Show that strength of shock wave is proportional to iz -1 | a Px Solution {Refersection4.1.10] 4.1.16 TWO MARKS QUESTIONS AND ANSWERS 1. What is meant by shock waiie? ‘A shock wave is nothing but a steep finite pressure wave, The shock Wave may be described as a compression wave front in a supersonic flow field across which there is abrupt change in flow properties, 2. What is normal shock? (Madras. Uni, Apr'2003] ‘When the shock wave is at right angle to the flow, it is called normal shock. 3. What is oblique shock? [Anna Univ ~ May’04 & Madras Univ, Apr'2003) When the shock wave is inclined at an angle to the flow, itis called oblique shock4.120 Gas Dyntmies-and Jet Propulsion 4, What is Prandtl —-Meyer relation? [Madras Univ; Apr'96) Prandtl-Meyer relation which is the basis of other equation for shock ‘waves, It gives the relationship between the gas velocities before and after the normal shock and the critical velocity of sound. M,*M,2= 1 Xen at? 5. Define strength of shock wave. [Anna Unix, Dec'2003, May'2004 & Madras Univ Apr'03] It is defined as the ratio of difference in downstream and upstream shock pressures (p,-p,) to upstream shock pressure (p,). Itis denoted bys. Py=Px By How the Mach number before and after the occurrence of « normal shock are related? [MK Univ Apr'96] ‘Mach number after the normal, shock an enstia 2 ya ri 7 What are applications of moving shock wave? It is used in EMS Univ Apr'96] 1. Jet engines 2. Shock tubes 3. Supersoni¢ wind tunnel 4. Practical admission turbines, Normal Shock Waves 4.121 8, Write the equation for efficiency of « diffuser. IMS Univ Apr'96} 11 Sam 9. Shock waves cannot develop in subsonic flow? Why? {MS Univ Nov'95} In subsonic flow, the velocity of fluid is less than the velocity of sound. Due to this reason, déeeleration is fot possible in subsonic flow. So shockwaves cannot develop in subsonic flow. Ty ‘across the normal shock. 10, Give the expression for = is {[Oct'96 MU and Nov'96 MKU] Me Me yey 2-1) Seve II. Define compression and rarefaction shocks? Is the latter possible. [ MKUniv Nov'96 de Bharathiyar Univ - Apr'97] ‘A shock wave which is at a higher pressure than the fluid in to which it is moving Is called compression shock wave. ‘A shock wave which is ata lower pressure than the fuid into which it is moving is called an expansion shock wave or rarefaction shock wave. It is not possible. 12, State the necessary conditions for a normal shock to occur in ‘compressible flov. [Bharathiyar Univ - Nov'96]4122 Gas Dynamics and Jet Propulsion 1. The compression wave is to be at right angle to the compressible flow. 2. Flow should be supersonic. 13, Write down the Rankine-Hugoniot equation. [Madras Untv - Oct’97} 14, Is the flow through a normal shock an equilibrium one. [MEUntv - Nov'97} No. Since the fluid properties like pressure, temperature and density are changed during normal shock. 15. Write down the static pressure ratio expression for a normal shock. [Bharathiyar Univ - Nov'97} 16, Give the difference berween Normal and Oblique shocks. [MS Univ - Nov'97, Oct'96 & MU-2000) Normal shock Oblique shock Shock wave is inclined at an angle to the flow 1. Shock wave is right angle to the flow 2. One dimensions! flow ‘Two dimensional flow 17, Thestagnation pressure anormal shock. ‘and statte pressure: across (0cr'95 MU} Ans : Decreases, Increases Normal Shock Waves 4.123 18, What are properties changes across a normal shock? (Apr96 MUT 1. Stagnation pressure decreases. 2, Stagnation temperature remains constant. 3, Static temperature and static pressure increases. 19. Calculate the strength of shock. wave when normal shock appears at M=2, [Apr'99 MU] Py Pr Strength of shock & = Refer Normal shocks table for M,=2 and y= 1.4 2. 450 [From gas tables page no.52] = ga4s-1 5-35 20. Show the normal shock in hs diagram with the help of Rayleigh ine and Fano line. [MU Apr'99 & Oc'99} Fano line h ~7 Shocks