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OBLIQUE SHOCK WAVE
The process of oblique shock isadiabatic and irreversible i.estagnation temperature remainsconstant across the oblique shockwhile stagnation pressure deceasesacross the shock. The oblique shock
relations can be deduced fromnormal shock relations by noting thatthe oblique can produce no
momentum change parallel to the
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OBLIQUE SHOCK WAVE
Consider control volume shown below
L1 N1 v2 v1 L2
N2
ince there is no change in momentum
!arallel to the wave L1" L
2. Normal
component
of the velocity makes the normal #ach No
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OBLIQUE SHOCK WAVE
which e$ects the properties acrossthe shock.
Conservation of mass gives
%1N1" %2N2 &1'
#omentum (quation
!1) !2" %2N22) %1N
21 &2'
The *ow through the control volumemust be adiabatic.
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OBLIQUE SHOCK WAVE
The energy equation
+2,-,)1 !1-%1 / 021" +2,-,)1 !2-%2 / 022 &'
ut 02" L2/N2
o because L1" L2 &' can be rearranged as
+2,-,)1+!2-%2 ) !1-%1 " L21/N21) L22/N22
"N21) N22 &3'
Normal component of #ach no is consideredfor in coming #ach No of normal shock andthen
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OBLIQUE SHOCK WAVE
then all the relations developed fornormal shock are applicable tooblique shock.
!2-!1 " +&,/1- ,)1' %2 - %1 4 1-+&,/1- ,)
1')%2 -%1
%2 - %1" +&,/1- ,)1' !2-!1 /1-+&,/1- ,)1' / !2-!1
N1- N2"+&,/1- ,)1' !2-!1 / 1-+&,/1- ,)
1' / !2-!1
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OBLIQUE SHOCK WAVE
The changes across the shock wave and the upstream #achNo. Now since N1"01in7 and N2"02in&7 4 8' &9' as
shown below
7 02 8
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OBLIQUE SHOCK WAVE
(quation &1' can be written as
%101in7" %202in&7)8' &:'
(q&2' can be written as !1 ) !2 " %1 &01in&7'
2 ) %12+02in&7)8'2
&;'
(q&' can be written as2,-,)1+!2-%2)!1-%1" &01in&7'
2 )
+02in&7)8'2 &
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OBLIQUE SHOCK WAVE
=f in the normal shock relations #1 is replaced
by #1 in7 and #2 = by #2 in &7)8' the
following relations for oblique shocks are
obtained !2-!1 " 2, # 21 in27 4 &,)1'-,/1 &1>'
%2-%1" &,/1' # 21 in27 -+2/ &,)1' #21in27
&11'
T2-T1"+2 /&,)1' # 21in27+2, #21in274&,)1'
&12
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OBLIQUE SHOCK WAVE
=t should be noted that for oblique shock
#2in &748'E1
Fence for oblique shock wave 6 #2 can be greater
than or less than 1.5s tan 7"N1-L1 tan &7 48' "N2-L2 &13'
ince L1 "L2 and from continuity N1-N2 " %1- %2
(q&13' can be used to give
tan &7 48'- tan 7 " %1- %2
" +2/& ,)1' # 21 in27 -&,/1' # 21 in27 "G say
&1H'
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OBLIQUE SHOCK WAVE
ince
tan &7)8' "+ tan7) tan8-+1/tan7 tan8
=t follows that
tan &7)8'-tan7 " +1)&tan8-tan7'-+1/tan7tan8
ubstituting this into eq&1H' then gives
1) &tan8-tan7' " G / G tan7 tan8hich becomes on rearrangement
tan8 " +tan7&1)G'-+Gtan27 /1
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OBLIQUE SHOCK WAVE
These two limits being 6 of course6 anormal shock and inJnitely weak#ach wave. Kblique shock lies
between a normal shock and a #achwave . =n both of these limitingcases6 there is no turning of the *ow.
etween these two limits 8 reaches amaDimum.
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OBLIQUE SHOCK WAVE
The relation between 86 #1 and 7 as
given by eq&1H' is usually presentedgraphically and resembles thatshown below
#21 #aDimum turningangle
7 9> #2 M1 #
2"1
>
> 8
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OBLIQUE SHOCK WAVE
=t can be seen from the Jgure that6 there is amaDimum angle through which a gas can beturned at a given #1. The value of this
maDimum turning angle for a given #1 canbe obtained by di$erentiating eq &19'withrespect to 8 for a fiDed #1 and setting d8-d7
equal to Iero. This leads to the following
eDpression for the maDimum turning anglein27maD" &,/1-3,' 4 1- ,# 21.?1) +&,/1'&1/&,)
1' #21 -2 / &,/1' #31-19>.H@
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OBLIQUE SHOCK WAVE
where 7maD is the shock angle that eDists when 8
has its maDimum value for given #1. Knce 7maD
is found using this equation 6 eq &19' can beused to find the value of 8maD .
Aor *ow over bodies involving greater angles thanthis a detached shock occurs. 5 detached shockis curved in general.
=t should be noted that if 8 is less than 8maD 6 thereare two possible solutions i.e two possiblevalues for 76 for given #1 and 8 as shown in the
Jgure.
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OBLIQUE SHOCK WAVE
The solution giving the larger 7 istermed the strong shock solution. #2
is always less than 1 in strongsolution.
(Dperimentally6 it is found that for agiven #1 and 8 in eDternal flows the
shock angle 7 is usually thatcorresponding to the weakO or nonstrong shock solution.
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REFLECTION OF OBLIQUE
SHOCK WAVES
• REFLECTION FROM PLANE WALL.
#2 !2 71 8
#1 !1
8 72
71 72)
8 # !
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REFLECTION OF OBLIQUE
SHOCK WAVES
5n oblique shock is assumed to begenerated from a body that turns the *owthrough an angle 8 as shown in the figure.
The entire *ow on passing through thiswave is then turned PdownwardsQthrough an angle 8. Fowever6 the flowadRacent to the lower *at wall must be
parallel to the wall. This only possible if aPre*ectedQ wave is generated as shownthat turns the flow back PupQ through 8.
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REFLECTION OF OBLIQUE
SHOCK WAVES
ince the *ow downstream of the re*ectedwave must again be parallel to the wallsboth waves must produce the same
change in *ow direction. Thus in order todetermine the properties of this re*ectedwaves6 the following procedure is used.
1. Aor given #1 and 8 determine #2 and !2-!1
2. Aor this value of #2 and value of 8
determine # and !-!2
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REFLECTION OF OBLIQUE
SHOCK WAVES
. The overall pressure ratio !-!1 is
then found from !-!1 " &!-!2 '&!2-!1 '
3. The angle that the re*ected wavemakes with the wall is 72/8 and
since 72/was found step 26 this angle
can be determined.
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INTERACTION OF OBLIQUE
SHOCK WAVES
• 5n oblique shock always decreases the#ach No. i6.e #2 #1
• Considering only non)strong solution6 the
shock angle 76 for given turning angle 8increases with decreasing #ach.No.
• The oblique shock waves generated ateach step in the concave wall will tend toconverg and coalesce into a single obliquewave which is stronger than any of initialwaves.
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INTERACTION OF OBLIQUE
SHOCK WAVES
Now6 the pressure and *ow direction must bethe same for all streamlines downstream oflast wave. ut two or more weaker waves
can not produce the same changes as asingle stronger wave and for this reasons there*ected shocks must be generated. Thesewaves are weaker than the initial waves.
hile these re*ected wave equaliIed thepressure and *ow direction6 but they can notequaliIe the velocity6 density6 and entropy.
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INTERACTION OF OBLIQUE
SHOCK WAVES
Aor this reason 6 the slip)lines eDistacross which there is Rump in theseproperties. =n theory6 these lines are
planes of discontinuity but in realitythey grow into thin regions overwhich the changes in properties
occur. The Jgure on the neDt slidshows series of oblique shock waves6re*ected wave and sliplines.
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INTERACTION OF OBLIQUE
SHOCK WAVES
=nteracting oblique shock
weak
re*ected
shock
sliplines
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=NT(S(CT=KN KA KL=U( FKCV50(
hen oblique shock waves of di$ering strengthgenerated by di$erent surfaces interact asshown in Jgure below6 the *ows in regions 3and H must be parallel to each other.
2 3
1 slipstream
H
Therefore6 conservation of momentum applied in a directionnormal to the *ows in these two regions indicates that the
pressures in regions 3 and H must be same.
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=NT(S(CT=KN KA KL=U( FKCV50(
The initial waves separating regions 1 and 2and regions 1 and are determined by the#ach number in region 1 and the turning
angles6 W and X. The properties of the Transmitted waves are then determinedfrom the condition that the pressure and*ow directions in regions 3 and H must be
same. The density6 velocity6 and entropywill then be di$erent in these regions andthe slipsteam must 6therefore eDist.