oblique shock wave lecture

8/18/2019 Oblique Shock Wave lecture

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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


2/27

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 &


8/27

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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10/27

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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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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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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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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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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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.


25/27


SHOCK WAVES

=nteracting oblique shock

weak

re*ected

shock

sliplines


26/27

=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.

oblique shock wave lecture

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