CH6 OBLIQU SHOCK WAVES：第6节的斜激波

1、ch.6 oblique shock wavesch.6 oblique shock waves6.1 introduction6.1 introduction1. definition of oblique shock - a straight compression shock wave inclined at an angle to the upstream flow direction - in general, the oblique shocks produce a change in flow direction as indicated in fig. 6.1. (p119)2

2、. occurrence1 external flow - due to the presence of wedge in a supersonic flow - due to the presence of concave corner in a supersonic flow2 internal flow - in supersonic flow through an over-expanded nozzle6.1 introduction6.1 introduction3. distinction1 2, 3 dimensional shock 1) 2-dimensional shoc

3、k; due to the presence of wedge etc. 2) 3-dimensional shock; due to the presence of cone etc.2 attached, detached shock 1) attached oblique shock ; = straight line for a given 2)detached shock ; = curved shock where = deflection angle4. momentum consideration1 statement the oblique shock relations c

4、an be deduced from the normal shock relations by noting that the oblique shock can produce no momentum change parallel to the plane in which it lies.max1mmax2 proof 1) control volume (see p120 fig. 6.2) 2) because there is no momentum change parallel to the shock, must equal . 3) flow normal to an o

5、blique shock wave (see p120 fig. 6.3) ; all the properties of oblique shocks can be obtained by modification and manipulation of the normal shock relations provided that angle of the shock relative to the upstream flow is known.1l2l6.2 oblique shock wave relations6.2 oblique shock wave relations1./

6、basic assumptions * frictionless surface * steady 2-dimensional planar adiabatic flow * no external work, negligible effect of body forces2./ governing equations (see 1 p124-125) 1. control volume; (t p121 fig. 6.4, 1 p124 fig. 6.3) - unit area parallel to the oblique shock wave - ; shock wave angle

7、 ; deflection angle or turning angle or wedge angle (=change in flow direction induced by the shock wave) 2. continuity equation (6.1)2211nnoavs6.2 oblique shock wave relations6.2 oblique shock wave relationsanglewaveshock:angleturningorangledeflection:3. momentum equation 1 normal momentum equation

8、 (6.1) 2 tangential momentum equation 4. energy equation (eq. (6.3), (6.4)2112222211)(nnapapnnmfinoutn121 aa21122221nnpp2111112222)(lllnalnaollmfinoutt22222222221211222211lnhlnhvhvhconstht) 3 . 6(1212)(222222211121222211vpvpllnhnh; if eqs. (6.1), (6.2), and (6.4) are compared with the equations deri

9、ved for normal shock waves it will be seen that they are identical in all respects except that andreplace and respectively.5. rankine-hugoniot relations for oblique shock waves) 4 . 6()(122222221222221211122nlvnnnlnlpp1n2n1v2v) 5 . 6(111111,111111121212121212pppppp1111111111,111111211212121212121221

10、ppppppppppttppppnn6. relations between the changes across the shock wave and the upstream mach number 1 geometric relation 2 eqs. (6.1), (6.2), (6.4) becomes ; these are, of course, again identical to those used to study normal shocks, except that occurs in place of and occurs in place of . cos,sin1

11、111vlvn) 7 . 6 ()(sinsin2211vvsin1v)sin(2v2v)6 . 6()cos(,)sin(2222vlvn) 8 . 6 ()sin()sin(22121121vvpp) 9 . 6 ()sin()sin(1222211122vvpp1vhence, if in normal shock relations is replaced by and by the following relations for oblique shocks are obtained using equations given in ch.5.3 relations in terms

12、 of upstream mach number and wave angle , turning angles sin1m)sin(2m1m)10. 6 (112111sin2*2122112nmmpp1m2m12. 6sin12sin11sin2*221222122112mmmtt13. 611sin212sin)(sin*221221222mmm)11. 6 (tantansincossincos121sinsin2sin1sin1*2121212122122112nnmmvvnnmm7. limit values of 1) for normal shock ; - for obliq

13、ue shock ; 2) for normal shock ; - for oblique shock ; hence, for an oblique shock wave, can be greater than or less than 1.3)-the minimum value that can have is, therefore, i.e., the minimum shock wave angle is the mach angle. when the shock has this angle, eq. (6.10) shows that is equal to 1, i.e.

14、, the shock wave is a mach wave.-the maximum value that can have is, of course, , the wave then being a normal shock wave. hence (6.15)8. relation between and (see p125 fig. 6.6),21mm11m12m2msin11 m12/pp9090/ 1sin11m,1m11nm12nm1 formula (6.19) 2 meaning of eq. (6.19) (see p124 fig. 6.5, fig. 6.6)1)

15、the turning angle , is zero when and also when is equal to 1, i.e. , when : normal shock and : mach wave; thus an oblique shock lies between a normal shock and a mach wave. in both of these two limiting cases, there is no turning of the flow. between these two limits reaches a maximum.2) the normal

16、shock limit and mach wave limit on the oblique shock at a given value of are given by the intercepts of the curves with the vertical axis at (see p125 fig. 6.8)2cos21sincot2tan21221mm)90.,.(coteiosin1mo90)/1 (sin11m1mo3 ; value of (= maximum turning angle) for a given 1) derivation ; p1242) variatio

17、n of maximum turning angle with upstream mach number for (see p126 fig. 6.8)4 remarks-for flow over bodies involving greater angles than this, a detached shock occurs. (see p126 fig. 6.10)-it should also be noted that as increases, increases so that if a body involving a given turning angle,accelera

18、tes from a low to a high mach number, the shock can be detached at the low mach numbers and become attached at the higher mach numbers.9. strong and weak (=non-strong) shocks 1 two possible solutions for a value if is less than , there are two possible solutions, 1maxmmax1m4.11mmaxmaxi.e., two possi

19、ble values for , for a given and .(see p127 fig. 6.11)2 classification 1) strong shock ; larger : dotted line in fig. 6.6 2) weak shock ; smaller 3 experimental results- experimentally, it is found that for a given and in external flows the shock angle, , is usually that corresponding to the weak or

20、 non strong shock solutions. - under some circumstance, the conditions downstream of the shock may cause the strong shock solution to exist in part of the flow. in the event of no other information being available, the non-strong shock solution should be used.4 physical meaning of 1) meaning (physic

21、al interpretation)1m1,2m1,2m1m1nmedisturbancpressureofvelocitynpropagatioedisturbancpressurefiniteofvelocitynpropagatiomnmalinfinitesi12) remarks * if ; shock wave = mach wave * greater - greater discontinuity * intensity of shock 5 general relation of 1) for both case : 2) * strong shock : * weak s

22、hock :6 occurrence of weak shock and strong shock 1) whether weak or strong shock = f (boundary condition)1nm21, mm1,121nnmmlarge,1,121mmsmallmm,1,1211nm),(1nmf 2) weak shock * typically occurs in external aerodynamic flows * of the two choice for , it is an experimental fact that the one correspond

23、ing to the weak shock usually occurs.3) strong shock * the strong shock wave occurs if the downstream pressure is sufficiently high. the high downstream pressure may occur in flows in wind tunnels, in engine inlets, or in other ducts.10. characteristics of the oblique shock wave 1 reason for the def

24、lection of stream direction * velocity component * so is deflected from the direction of , i.e., fluid stream is deflected toward the oblique shock wave.2121211,1llandmmnnnn2v1v2 distinction between mach wave and shock wave by normal velocity component 1) mach wave (= shock wave of zero intensity) ;

25、 2) shock wave ;3 deflection angle 1) formula 2) application * applicable to conical shock as well as plane shock * valid only for ; 3) case of a) (mach angle) ; mach wave b) ; normal shock wave421nn 21nn 1) 1sin(2) 1(tancot22121mmmaxo)/1 (sin11m)2/(2/maxmax1m1) 2) if ; the basic relation previously

26、 presented are not applicable.5 in 2, 3 dimensional shock wave1)2-dimensional shock (= plane shock) = angle of wedge = angle of concave corner2) 3- dimensional shock angle of cone; in this case streamlines after the conical shock must be curvedin order that the 3-dimensional continuity eq. be satisf

27、ied.6 corresponding to (= maximum flow deflection angle for a given )oommf11max:)(4 . 158.451maxwithmformaxmaxmax1m2141212121max216121111411sinmmmmexample 6.1?,4,30,80, 22111pctkpapm4 .33:1.506gfigpfrom10. 1)4 .33(sin2sin11mmnkpapppmforcappendixpfromn6 .99)245. 1 (80245. 1/10. 1:4892121tableshocknor

28、malfromm:2)sin(91177. 022mmn)0 . 44 .33sin(2 m0 . 1857. 12m6.3 reflection of oblique shock waves6.3 reflection of oblique shock waves1. reflection of an oblique shock wave from a plane wall (see p129 fig. 6.12)2. wall pressure distribution near point of oblique shock wave reflection in ideal case (s

29、ee p131 fig. 6.13)3.wall pressure distribution near point of oblique shock wave reflection in real case (see p131 fig. 6.14) -boundary layer separation during shock wave-boundary layer interaction (p132 fig. 6.15)4. mach reflection (p132-133 fig. 6.16, 6.17, 6.18 )5. noreflection of wave (p135 fig.

30、6.19)(=neutralization, cancellation or absorption of an oblique shock wave)6.3 reflection of oblique shock waves6.3 reflection of oblique shock wavesexample 6.2?,4,20,60, 5 . 233111vpctkpapm6 .264, 5 . 2:1.5061mforgfigpfrom12. 16 .26sin5 . 2sin11mmn12. 1:4891nmfortableshocknormalpfrom0775. 1/,2961.

31、1/,897. 012122ttppmn334. 2)46 .26sin(/897. 0)sin(222mmmbutn5 .284,334. 21.22mforgfigfrom113. 15 .28sin334. 2sin222mmn113. 14892nmforptableshocknormalfrom0732. 1/,2787. 1/,9019. 023233ttppmn175. 2)45 .28sin(/9019. 0)sin(333mmmbutnkpappppppalso4 .99)60)(2961. 1)(2787. 1 (112233ktttttt6 .292)253)(0775.

32、 1)(0732. 1 (112233:where2787. 1)245. 12968. 1 (20/13245. 1/23pp0732. 1)06494. 107763. 1 (20/1306494. 1/23tt9019. 0)89656. 091177. 0(20/1391177. 032nmmsmrta/343)6 .292)(287)(4 . 1 (33smamv/746)343(176. 2333example 6.3reflectionmachwithoutmmax1, 5 . 212:caseconsider8 .334, 5 . 2:1.5061mwithgfigpfrom3

33、91. 18 .33sin5 . 2sin11mmnwithrelationshocknormalpfrom489391. 111nmm7436. 02nm)sin(22 mmn002. 2)128 .33sin(/7436. 02m:002. 21.mforgfigfrom01.23max01.111201.23maxexample 6.4?,4, 5 . 2,20,60111xmctkpap6 .264, 5 . 21.1mwithgfigfrommxxh99. 16 .26tan/6.4 interaction of oblique shock waves6.4 interaction

34、of oblique shock waves1. intersection of multiple left-running oblique shock waves (p136 fig. 6.20)2. intersection of oblique shock waves on a curved wall (p137 fig. 6.21)3. intersection of right- and left-running oblique shock waves (p138 fig. 6.23)6.4 interaction of oblique shock waves6.4 interact

35、ion of oblique shock wavesexample 6.5?,4342pp1 region 23 .224, 3:1mwithchartshockobliquefrom140. 13 .22sin3sin11mmn35. 1/,882. 014. 1:using1221ppmmfortableshocknormalnn799. 2)43 .22sin(/882. 0)sin(/22nmm3region26 .213, 3:1mwithchartshockobliquefrom104. 16 .21sin3sin11mmn255. 1/,908. 0104. 1:using1231ppmmfortableshocknormalnn848. 2) 36 .21sin(/882. 0)sin(/33nmm4region3-turning angle produced by the oblique shock wave between region2 and 42 ; -turning angle produced by the