Aircraft intake aerodynamics

Aircraft Intake Aerodynamics Revised 4/22/2014 13:37:00

Divas Gupta1 Arjun Agrawal2 Prabhat Kumar3 1Department of Mechanical Engineering, Indian Institute of Technology, New Delhi

Delhi,, INDIA 110016

e-mail: [email protected] 2Department of Mechanical Engineering, Indian Institute of Technology, New Delhi


e-mail: [email protected] 3Department of Mechanical Engineering, Indian Institute of Technology, New Delhi


e-mail: [email protected]

1. INTRODUCTION

In power systems design of an aircraft proper selection and design of a suitable intake system is very crucial for

smooth running on the engine at all times (take-off, cruise, landing). For proper combustion to happen in an engine

the flow in the combustion chamber should be almost stagnate and this calls for inclusion of a diffuser in front of the

combustion chamber. The main aim for an intake design is:

1. Make sure that air is available to engine at all times without any suffocation.

2. To minimize the losses in total pressure along the path from free stream to the combustion chamber as this

will tend to increase the amount of charge entering into the engine and will in return produce equivalent

amount of power.

3. Also minimize the drag acting on the intake system due to external flow around the engine.

Most of the time in this paper, we will be talking about the flow losses (total pressure loss) due to an intake system

which will be quantified by:

Pressure recovery factor, 1i

P

q

We will look into the cases of subsonic, transonic and supersonic intake flows.

2. SUBSONIC INTAKES

2.1 Types of Intakes

Two Categories based on location of engine on the air craft:

i. “Podded” installations

Boeing 707 Engine

Source: airliners.net

ii. “Integrated” installations

British Aerospace Harrier GR7

Source: flugzeuginfo.net

Three Categories based on Method of intake in subsonic flows:

i. Direct or fully ducted intakes

ii. Plenum chamber installations

iii. Propeller-turbine engines

mailto:[email protected]



2.2 Approximate theory of frictional losses p+1/2dp Following analysis done by Seddon (1952).

Assumptions: A+dA

- No flow separation, only frictional loss A,v,p dl v+dv - Inclination of boundary of streamtube to its axis is small. p+dp

- g is the local perimeter which may be or may not be the

complete perimeter of that station.

Fig 2.1 : streamtube with boundary viscosity.

F is therefore equal to q Cf g dl, where q=1/2 V2, and Cf is the local friction coefficient.

Applying the momentum theorem to the streamtube element of initial area A gives:

Av(v + dv – v) = pA + (p + ½ dp)dA – (p + dp)(A +dA) – F

Whence

Av dv = - Adp – F

Or

d(p + ½ v2) = -F/A

dP = - F/A,

where,

Total Pressure, P = p + ½ v2

P =

2 2

1 1

/ fg

F A qC dlA

Non dimensionally and using continuity equation (Av=1A1v1)

2 2

1 1 2

1 11 1

( )f fP q g A g

C dl C dlq q A A A

(2.2)

We have an approximate expression for loss of total pressure along a streamtube bounded wholly or partly by a

solid surface.

Wetted area, S : for an intake on the side of the fuselage, S is taken to be the surface area between entry to the engine

to the foremost point of the fuselage nose which comes in contact with the inlet flow to the engine. This definition is

somewhat imprecise since actual streamline patterns ahead of the intake are not usually known precisely.

Consider now the flow into an intake:

fuselage nose

A,v Ac,vc

Af,vf

0 lc

lf

v

l = 0 to l = lc : Approach

l = lc to l = lf : Duct v

0

Fig 2.2. Representation of internal flow with external wetted surface

Approach Duct

Pre-entry

Retardation

Taking q1 = qc , dynamic pressure at entry, and considering incompressible flow since subsonic, we can write:

2

0

( )

flc

f

c

P A gC dl

q A A

(2.3)

Approach Loss :

2

0

( )lc

a cf

c

P A gC dl

q A A

(2.4)

Duct Loss: 2( )

fld c

f

clc

P A gC dl

q A A

(2.5)

Approximation for approach loss: 3.( )a

a cF

c c

P A SC k

q A A

Where, CFa – overall friction coeeficient of approach

S = gdl

k – empirical factor taken to incorporate the difference from v of the assumed contant approach velocity.

We take, corrected position ratio, J = kS/Ac & hence,

3

.a

a cF

c

P AJC

q A

Approximation for duct loss: 2

. ( )

fld c

Fd Fd

clc

P A gC dl IC

q A A

Where

Duct Integral, 2( )

flc

lc

A gI dl

A A

& CFd – effective overall friction coefficient whose value can be defined for a conical diffuser from fig 2.3 and

following correlations:

f

2( )tan

2 ( )

c f f c

c f c

a g A g A

g g l l

& e

4R

cff

c

Ac V

g v

Reff is used to calculate Cf for flat plate used in the graph.

Fig 2.3 – Effective friction coefficient in duct deduced from

experiments of “Squire”. 1

a

Total Loss: 3

Fd Fa

c

PIC JC

q

(2.6)

=Ac/A which is inverse flow ratio

( 1)

ps pm

f MC C

J

2.3 Examination of variation

We compare the results for Wing leading-edge intakes for various angles of sweep. Sweepback angle,

ICFd JCFa k

0 0.14 .010 .80

30 0.11 .021 .65

40 .08 .033 .72

52 .05 .054 .80

variation agrees with the experimental results for >1 and for some values below =1 except for large

sweep back angles. This is due to the inclusion of additional loss of flow separation. This additional loss

reaches a maximum in ground running conditions, =0, when the engine is being run at zero forward speed.

Duct integral, I, decreases with increasing sweepback due to inclination of the transverse plane or effective

entry plane which is seen when the duct first becomes fully closed and taken from the rearmost point of

the lip.

As sweepback is decreased, a progressively larger proportion of the pre-entry retardation occurs ahead of

the sweptback surfaces, which is taken into account by decreasing the value of k approx. linearly with

sweptback angle. The value of k shown in the table is compound of the variation of k due to body and

swept surface.

Normally the design point for high speed level flight, determined by the sizing of the entry, lies in the range

of values between 1.4 – 2.0. In this range the theoretical variation complies with experimental results.

Fig 2.4 – theoretical variation loss on leading edge intakes at various angles of sweep.

The above is the results obtained from theoretically calculating the Total pressure loss from equation 2.6. The Matlab

code for calculation is attached in the appendix 2.1

2.4 Pressure recovery characteristics

a. A comparison with measured loss coefficient with that calculated from equation 2.6 shows reveals the

extent of lip separation at low speeds and that of pre-entry separation at high speeds.

b. In terms of practical design, the entry area Ac is an exploitable variable so we cannot use qc for non-

dimensionalyzing the total pressure. It is usual to express the loss in terms of free-stream dynamic head

and make use of recovery factor ƞσi.

2

2

c FdFd Fa Fa

c

P A A ICIC JC JC

q A A

So that

1i

P

q

21 1i

FdFa

P ICJC

q

c. If there were no approach surface, then ƞσi would tend to value 1.0 as is increased to . This will be the

case of a pitot tube registering free stream total pressure.

2.5 Plenum chambers

The analysis of the effect of installing an engine downstream of a sudden enlargement in duct cross-sectional area.

Due to practical space restrictions:

i. It is not possible to diffuse efficiently to the full cross sectional area of plenum chamber so sudden

enlargement is done.

ii. Velocities within the chamber itself are by no means negligible.

plenum Fig 2.5 plenum chamber installation

chamber

Intake duct

c d f

Pressure loss at a sudden enlargement with initially uniform flow - 2 2

1 2 1 1d d d d

d f f f f

P A A A A

q A A A A

Pressure loss in terms of dynamic head at entry – 22

1c d

c d f

P A A

q A A

(2.7)

It can be seen that a modest reduction in effective area Ad can result in large increase in theoretical loss at the

enlargement.

Although plenum chambers intakes are unlikely to come up in future jets but a general knowledge of the effects of

sudden enlargement can be helpful in designing other ducted systems.

2.6 Propeller turbines

Because of their potentially good fuel economy, propeller turbine engines are always in interest of designers.

General intake arrangement is that of an annular intake located directly behind the propeller.

There is a large additional loss as compared to the direct intakes, some 15% of free stream dynamic head at typical

model Reynolds number. Wind tunnel testing shows that the additional loss is attributed to the presence of propeller

blade roots ahead of the intake. The flow over the roots is complicated in character due to large thickness/chord ratio

and the action of centrifugal forces on the boundary layer. An approx. formula for blade root loss:

0.62

bP Nt

q r

where, N – number of blades, t – profile thickness, r – radius of rotation of the section

In annular intake flow when pre-entry flow separation occurs it does so in patches rather than uniformly round the

circumference. So there is a limiting value of above which circumferentially uniform flow cannot exist.

lim 1 1.8A

LR ; where, A – annular entry area, L – length of projecting spinner and the hub, R is its max. radius.

A breakdown of uniformity in the intake has a potentially detrimental effect both on total pressure recovery and also

on engine compatibility.

3. TRANSONIC EFFECTS IN PRE-ENTRY FLOW

3.1 Initial Expectations

M<1 M>1 M<1

a) Subsonic free stream b) supersonic free stream

Fig.3.1 Flow pattern in intake with forward external surface

Total losses in a transonic flow are approach and duct losses as discussed earlier corrected for compressibility effects

in approach loss, together with loss due to shock.

Modified frictional loss equations:

c c

Fd Fa

c

P AIC JC

q A

(3.1)

Or alternatively

2c

Fd Fa

c c

P A AIC JC

q A A

(3.2)

Shock losses:

1

1 ( 1)

22 1

1 2 21 1

1*

1 21 1 1 ( 1)

1 1

P M

PM M

(3.3)

Ps, shock loss is given by the following equation:

2

1

1sP P

P P

(3.4)

From here onwards it is convenient to express pressure loss in terms of total absolute pressure P, so the frictional

loss equation needs to be converted in terms of P by the following relation:

12

2

2 11

2

PM

q M

(3.5)

Shock Front

Total intake loss for this case would be

d a sP P P P (3.6)

3.2 Experiments of Davis et al. (1948)

A comparison between the measured pressure recoveries and those estimated according to equation 3.6 were made.

The very large discrepancy, the measured losses being twice and three times the calculated ones, show that the present

model is incomplete and a major source of total pressure loss exists in addition to those already postulated. Such a

loss can occur only as a result of large scale turbulence associated with flow separation.

3.3 The real nature of pre-entry flow

For subsonic free stream mach number, reducing the flow ratio, Ac/A, imposes a pre-entry pressure gradient, the

boundary layer at entry thickens to a point distorts and then, beyond a certain decrease in flow ratio, boundary layer

separates. Further reduction of flow ratio results in forward movement of separation point in the pre-entry field.

For supersonic free stream mach number a similar characteristics are observed but with two important differences,

a) at flow ratio 1.0 a normal shock wave is sitting at the entry plane, b) the critical flow ratio for boundary layer

separation is now much higher which means that separation will occur much easily.

3.4 Pressure Coefficient at separation

Gadd (1953) obtained an expression for the pressure coefficient at separation based on the assumption that the

separation pressure is that pressure which is just sufficient to bring to rest without friction , the fluid at the knee of the

normal turbulent velocity profile.

Gadd’s formula:

12

22

11

2 2 11

1 .642

pm

MC

MM

(3.7)

(Cpm)incompressible=.36 putting, M=0 and =infinity

This expression may be taken for applying at boundary layer passing through a sudden jump, or infinite pressure

gradient. But in a finite pressure gradient, some energy is entrained into the boundary layer from the main stream in

the distance over which the gradient acts. As the boundary layer distorts, the friction at the surface falls towards zero

and hence an energy surplus is available for increasing the pressure rise.

So,

Cps=Cpm + pressure recovery due to energy fed to BL, Cp

After the derivation we get the following expression for separation pressure coefficient,

( 1)

ps pm

f MC C

J

Where, = gc/ga; J – corrected position ratio as used earlier

4. LIP SEPARATION AND TRANSONIC THROAT FLOW

Pressure loss or recovery characteristics combine the effects of two flow parameters – the mean intake throat

Mach number Mt (which controls compressibility effects) and the position of the dividing (stagnation) streamline at

the cowl lip for a given ratio A∞/Ac (which controls whether or not there is lip separation). The following two graphs

(Fig. 4.1) illustrate how the pressure recovery characteristic is related to Mt and M∞.

4.1 Subsonic diffuser

Aircraft intakes often have ducts with varying shape of cross-section, usually in the shape of an S-bend. Also, in

cases where not much length is available, wall curvature may have to be at a large rate that causes local flow

separation. The resulting additional losses (i.e. in addition to the calculated skin friction losses) can be determined

by fitting the duct with a bellmouth entry and performing a suction test or from computers using Navier-Stokes

equations.

Fig. 4.2 shows the increase in pressure loss with increase of mean throat Mach number, increase of duct length

and changing cross-sectional shape and curving of the duct. When Mt exceeds 0.6 or 0.7, Mach numbers can locally

be transonic at the walls in the region near the throat. In case of curved ducts, there can be some flow separation and

then reattachment. Thus, the loss increases significantly beyond a mean throat Mach number of around 0.6.

In S-bend ducts, the first bend is generally the major source of total pressure loss. Total pressure loss for a given

Mt, length of duct and final bend radius can be considerably reduced by decreasing the amount of turning in the first

bend.

Figure 4.1 Interrelationship between pressure recovery characteristics and Mt and M∞ for axisymmetric pitot intakes having (a) sharp and (b) elliptic lip.1

4.2 Intake plus subsonic diffuser

Instead of performing a bellmouth test, one could obtain a similar result from measurements in an intake plus duct

if A∞/Ac < 1. This condition being satisfied would mean the stagnation streamline being on the lip’s underside,

which would in turn mean the loss of total pressure being only due to skin friction, which for a pitot intake is

typically a function of only Mt for M∞ between 0.6 and 2. Mt can be found from

*

*

t

ct

c t

PA

A PA

AAA

A A

(4.1)

where Pt/P∞ = 1 for subsonic speeds, and it is the ratio of total pressures across a normal shock (NS) for supersonic

speeds. Fig. 4.3 shows a typical plot of total pressure loss, which is defined as

1 at subsonic speeds

and at supersonic speeds

NS

f

f

PP

P P

PP

P P

(4.2)

versus mean throat Mach number. The comparison with a bellmouth test result shows good agreement for values

of Mt below approximately 0.6.

Figure 4.2 Basic duct loss. Increasing duct length (from lower curve to middle curve) increases

skin friction loss. Loss further increases due to varying cross-section shape and duct curving (upper

curve). Also, for Mt > 0.6 or 0.7, compressibility greatly increases the loss.1

As M∞ goes below 0.6 and flow rate increases, the capture streamtube would start growing larger than Ac and

stagnation would occur outside the cowl. There would be flow separation from the cowl’s inner surface, causing

total pressure loss to rise with Mt much more rapidly than with bellmouth intake because of turbulent mixing.

Calculated peak Mach number and lip suction force both correlate with Ac/ A∞. Calculated peak Mach numbers

also correlate with the minimum radius of curvature that the cowl lip has, especially at the static condition. But the

most significant factor that affects the loss is the contraction ratio CR=At/Ac.

This loss, which is mainly controlled by the lip contraction ratio, is termed lip loss,

f

lip SF

PP P

P P P

(4.3)

i.e. it is the difference between the total loss and the component of loss due to skin friction.

4.3 Factors influencing lip loss

As mentioned in Section 4.2, contraction ratio is the single most important factor affecting lip loss. However, the

initial shape of the subsonic diffuser and the lip shape also have a role to play.

Lip losses are lower if circular arc lips are used rather than elliptic lips. The eccentricity (or the ratio a/b) of the

ellipse also has a small effect. Fig. 4.4 shows the variation of the loss with these factors.

Figure 4.3 Comparison of duct skin friction loss from free-stream pitot intake tests and measurements with

a static bellmouth entry.1

4.4 Total pressure loss in case of separated flow at entry

When there is flow separation at the lip, total pressure loss can generally not be calculated for an arbitrary cowl

shape. However, it is useful to assume the intake entry to be a sharp-lipped cowl that is cylindrical. When A∞/Ac > 1,

the situation can be analyzed using momentum equations.

As shown in Fig. 4.5, the momentum theorem is first applied to the external flow. The cowl is taken as a thin-

walled cylinder extending indefinitely to the right. ABCDEA is the control surface, which includes the pre-entry

streamtube AB, cowl surface BC up to a point C where flow velocity has regained free stream level, perpendicular

planes AE and DC and a large outer cylinder ED which has velocity of the free stream level. AE has an area of A0

and DC A0 + A∞ - Ac.

From the continuity equation, inflow over ED = (A∞ - Ac)ρ ∞V∞. Then, from the momentum theorem,

0 0

2 2 2( ) ( ) ( )c cABC

A A p pA V V A VdA A A (4.4)

which yields ( ) 0ABC

p p dA (4.5)

Since ( ) 0BC

p p dA , we have ( ) 0AB

p p dA . (4.6)

Now applying the momentum equation to the internal flow, we consider a control volume bounded by the control

surface ABCDEA, as shown in Fig. 4.6. Station 1 (points C and D) is enough down the duct to have uniform flow.

A∞ Ac

A B C

D E

Figure 4.4 Effect of lip shape.1

Figure 4.5 Application of momentum theorem to

external flow.

Momentum flux at station 1 = Momentum flux at ∞ + Lip suction force (F) + ( ) 0AB

p p dA (4.7)

F=0 since the lip is infinitely thin. The pressure integral was proven above to be zero. So

( 1)/2( 1)

11

1/22

1 1 1

/

1 /

t tP

P M t t M M

1 1 1 12 ( ) 2q A p p A q A (4.8)

which can be rewritten as 1 1

1 1 1 1 1

2 2. . .

q p p P q P A

P P P P P P A

(4.9)

Thus, 1 1

1 1

1 1

2.

2

q A p

P P A P

q pP

P P

(4.10)

Continuity yields 1 1P A P A

(4.11)

so that 1 1

1 1

. .A P A A

A P A A

(4.12)

and 1

*

1 1 1

*

1 1 1

2 2. .

p

P P

q p q A AP

P P P A A

(4.13)

In terms of Mach numbers,

( 1)/2( 1)

11

1/22

1 1 1

/

1 /

t tP

P M t t M M

(4.14)

where t = (1 + (γ-1)M2/2).

Taking γ as 1.4 and four different values for M1, Fig. 4.7 shows how the lip loss for a sharp lip varies with M∞, and

also with M1.

This graph has been plotted using MATLAB code in Appendix 4.1.

A∞ Ac

A B

C

D E

Figure 4.6 Application of momentum theorem to isentropic

internal flow from station ∞ (AE) to station 1 (CD).

As expected, the lip loss is higher for higher Mach numbers inside the intake (at a cross-section after uniform flow

has been reached) and lower for higher free-stream Mach numbers.

5. EXTERNAL & INTERNAL SUPERSONIC COMPRESSION

5.1 Pitot Intake Let us consider an aerodynamic duct with c as entry point and f being point of maximum cross-section and e being

exit of duct. Area of duct increases from c to f and at exit, a tapered plug translate and control the exit area. M∞ , P∞

denotes the pressure at entry of duct.

c f e

Let us analyze flow through this duct by varying exit area starting from zero.

Case 1:- As the exit area is zero, hence duct would act as a solid body. Supersonic flow reaching the duct will

undergo a shock ahead of duct and a region of subsonic flow would form between shock and the nose of the body,

bounded by sonic line, after which flow would again be supersonic. Shock is normal only at the center with subsonic

mach number Mw ∞, elsewhere it is oblique.

Fig 5.1 Area Profile of Duct

M<1 M∞>1

Sonic Lines

M>1

Shock Fig 5.2 Flow for Case 1

Figure 4.7 Calculated lip loss for sharp lip with different values of M1 and M∞.

Case 2:- As the exit area start to increase from zero, a narrow streamline start to develop though the duct and

therefore `allowing the shock to stand closer. Assuming frictionless flow in duct, total pressure be Pw ∞ behind the

normal shock. Flow first decelerates from Mw ∞ inside duct and reaccelerate in the exit portion. The flow will exit at

Mach 1 or we can say that exit is chocked and total pressure throughout the flow is Pw ∞.

Case 3:- As Ae is increased further with its value being less than Ac, shock continuously approaches duct entry and

a point reaches when the ratio Ac/Ae becomes equal to A/A* and the corresponding mach number at c, lets call Mc,

which is now equal to Mw ∞. Hence, flow is subsonic throughout the nozzle and is choked at exit. The flow ratio A∞/Ac

at this point become equal to 1.

Case 4:- On increasing Ae further, more flow will try to go out of duct, hence this depression suck the shock inside

duct. As shock travels along the diffuser, corresponding mach number increases and hence total pressure loss

increases. Flow ratio will remain unaltered. Since mass flow rate cannot be further increased, hence equilibrium would

be achieved, when total pressure downstream of shock has fallen enough to compensate for the increase in area. Mass

flow rate remains constant and hence

3 4w e w eP A P A (5.1)

Me=1 M∞>1

Mw ∞

Shock

Pw ∞

Fig 5.3 Flow for Case 2

Me=1 M∞>1

Shock

Pw ∞

Mc=Mw ∞


Shock

Pw

Mc<Mw ∞

M> M∞


No solution exist for flow ratio greater than 1. If shock were to be out in front of duct with A∞ greater than Ac, then

flow must have to converge after shock and pressure would have to decay from shock to entry, hence it is not possible,

because any small disturbance would try to move shock toward entry. Next, if the shock is inside of duct, the flow

information could not travel upstream of shock, hence no solution for flow ratio greater than 1 exist.

As we can see from graph, if intake inside pitot occurs at Stage 3, we can have maximum pressure recovery.

Intermediate stages between Stage 1 and Stage 3 would introduce more drag produced by pitot tube as compared to

Stage 3, even if these intermediate stages create maximum pressure recovery same as Stage 3.

P2/P1

To design a pitot tube, we can reduce its drag, but will suffer due to loss in pressure recovery. Losses also occur in

real due to friction in duct. Suppose we have defined pressure recovery ratio, then we can find intake mach number

M1 using matlab code in appendix 3.1

22

1

1

21 ( 1)

1

pM

p

(5.2)

1/22

12 2

1

1 ( 1) / 2M

( 1) / 2

M

M

(5.3)

/( 1)2

2 2 2

2

1 1 1

1 ( 1) / 2

1 ( 1) / 2

P p M

P p M

(5.4)

Stage 2

Ae /Ac A*/A for Mw ∞

Stage 3 Stage 4 1

1

Fig 5.6 Graph of A∞/Ac vs Ae/Ac

1

Stage 2

(Subcritical) 1

Stage 3

(Critical)

Stage 4

(Supercritical)

Fig 5.7 Graph of η∞ vs Ae/Ac

Fig 5.8 Graph of M2 vs M1 and P2/P1 vs M1 for normal

shock

We can analyze from graph (Matlab code in appendix 5.1), increasing M1 by small amount initially from 1 result

in very low pressure losses, but at higher M1 pressure recovery decay very fast. If we allow 4% losses in system, then

intake mach number should be 1.4.

Pressure Recovery 0.96 0.94 0.90 0.85 0.80

M1 1.39 1.46 1.58 1.71 1.82

If we want to design our intake for higher mach number, then we should introduce oblique shocks before normal

shock, which makes entry of air inside duct.

5.2 Two-Shock Intakes

Before flow reaches duct, it undergoes one shock with interaction of flow with wedge type surface. Maximum flow

ratio is achieved, when the boundary of free streamtube A∞ arrives at lip without any disturbance, which means that

c

A

A

=1 (5.5)

This condition is known as full flow, but for two shock

intake maximum flow condition has to be defined using

position of lip relative to wedge. Let β be angle of oblique

shock produced due to wedge and βD be angle subtended by

lip at the apex of wedge.

The pressure recovery of the two shock intake at maximum

pressure recovery is product of separate total pressure ratios

across oblique and normal shocks. The figure 5.10 shows the

graph of maximum pressure recovery of two shock intake.

The graph was drawn using Matlab code referenced in

Appendix 5.3.

2 2

1

2

1

sin 1tan( ) 2cot

( cos 2 ) 2

M

M

(5.6)

1/22 2

12 2 2

1

1 ( 1) sin / 2M sin( )

sin ( 1) / 2

M

M

(5.7)

2 22

1

1

21 ( sin 1)

1

pM

p

(5.8)

/( 1)2

2 2 2

2

1 1 1

1 ( 1) / 2

1 ( 1) / 2

P p M

P p M

(5.9)

1/22

23 2

2

1 ( 1)M

( 1) / 2

M

M

(5.10)

23

2

2

21 ( 1)

1

pM

p

(5.11)

/( 1)2

3 3 3

2

2 2 2

1 ( 1) / 2

1 ( 1) / 2

P p M

P p M

(5.12)

Fig 5.9 Table of pressure recovery and M1 for normal shock

δ

β βD

Fig 5.9 Profile of two intake shock showing δ, β and βD

SolidWorks Drawing Tool

2

2

s sP PP

P P P

(5.13)

As we can observe from graph, 96% pressure recovery can be done at M∞=1.7 nearly with two shock intake, while

for same pressure recovery we need M∞=1.4 as discussed in Section 5.1.

Three condition of pressure recovery can be analyzed depending on the value of βD and β.

a) β >>βD. In this case, all streamline captured within are passed through system of two shock (Fig 5.11) .

Hence, on increasing (refer section 5.1) Ae, A∞/Ac increases and pressure recovery remains follows trend

as shown earlier in Fig 5.7.( also curve (a) of Fig 5.13).

b) β >βD, but a very little difference. As we move away from critical value, after certain flow ratio, pressure

recovery start to diminish and part of intake shock passes through single strong shock (Fig 5.12) and hence

a smaller pressure recovery is turned out. Thus curve (b) follows curve (a) trend in Fig 5.13 near critical

region.

c) β <βD. In this case, intersection of oblique shock line and intake profile occur inside the duct, hence flow

ratio is reduced (curve(c) of Fig 5.13) from the critical value toward the pitot intake level.

5.3 Multi Shock Intake

Ps P∞

∞

δ

Fig 5.10 Graph of Ps/P∞ vs Wedge angle (δ) for various M∞ of two shock intake

M∞

1.6

1.7

1.9

2.0

2.2

2.3

2.5

2.7

2.8

3.0

(a)

(b) (c)

1

1

(Critical)

Fig 5.13 Graph of η∞ vs Ae/Ac

Pitot intake level

Fig 5.11 Flow for Case (a)

Fig 5.12 Flow for Case (b) & Case(c)

Next stage to observe is three shock system, 2 oblique shock and 1 normal shock. Double wedge and double cone

type system. Pressure recovery of such a system with respect to two deflection angle (δ1 & δ2) has been tabled in Fig

5.15.

If we have n shock(n-1 oblique and 1 normal) in our inatake, then net pressure recovery can be written as product

of pressure recovery across each shock, that is

32

1 1 2 1

.......s s n s

n n

P P P P PP

P P P P P P

(5.14)

Oswatitsch showed that for such a system to have maximum pressure recovery, when all oblique shock are of equal

strength, hence

M1sinβ1= M2sinβ2= M3sinβ3=………..= Mn-1sinβn-1 (5.15)

Connors worked out for axisymmetric intakes, developed the graphical method for optimum shock system. Design

criterion can be decided with graph in Fig 3.16.

Fig 5.15 Shock Pressure Recovery of double wedge intake1

Fig 5.16 Shock Pressure Recovery with varying M∞ and no of shock1

Fig 5.14 Double wedge intake

δ1

δ2 β1 β2

5.4 Isentropic Compression

Let us first derive the Pradtl-Mayer relation for expansion fan and then we can relate it with isentropic compression.

Tangential velocity remain constant, hence

Vt,upstream=Vt,downstream (5.16)

cos( ) ( )cos( )v v dv d (5.17)

sin( )

cos( )

dvd

v

(5.18)

Also, 1

sin( )M

(5.19)

2

1

1

dvd

v M

(5.20)

Now, let us relate v and M using relation v=Ma

dv dM da

v M a (5.21)

Using a RT (5.22)

We get 2

dv dM dT

v M T (5.23)

Since, we are assuming isentropic flow, hence stagnation temperature will not change and relation between T0 and

M is given by

2

0

1(1 )

2T T M

(5.24)

Differentiating the given equation we get

2

0

20

1(1 )

201

(1 )2

d MdT dT

T TT M

(5.25)

Now substituting the value from equation (5.25) into equation (5.23)

2

2

1( )

21

(1 )2

Mdv dM dM

v M MM

(5.26)

Substituting (5.26) into (5.20),

2

2

1

1(1 )

2

M dMd

MM

(5.27)

dM is change in mach number associated with dʋ change in turn angle. δ=ʋ2- ʋ1

which gives

Fig 5.17 Flow over wedge Schematics v

vt

vn vt

vn+dvn

v+dv

dʋ

µ

µ+dʋ

2

1

2

2 12

( 1)

(

1)1

2

M

M

M dM

MM

(5.28)

2

1

2 2

2 1

1 1arctan( ( 1) arctan( 1) |

1 1

M

MM M

(5.29)

Now, if we try to analyze the above derived equation, we would find that the equation can be equally valid for

compression (wedge problem), but instead all expansion would converge to form one oblique shock. This method is

also known as reversed Prandtl Mayer Expansion. With the help of above equation, we can find the maximum

deflection angle by which flow can be turned for maximum pressure recovery, i.e. when M2=1, hence

2 21 1arctan( ( 1) arctan(

1

1 1)M M

(5.30)

5.5 Flow Starting Problem

Let us consider the duct problem attempted in Section 5.1, with difference that duct initially contract and then

increase in area. We have already know with zero exit area, shock wave stand infront of duct and with smaller opening,

shock move towards entry with subsonic flow throughout the duct. There are now two cases, either before shock reach

to the entry, Mt becomes unity or shock reaches entry before shock in the throat.

(a) Shock reaches entry before with Mt<1.This attachment condition implies that A∞/Ac=1. If we further increase

the exit area, shock moves inside duct, but we know that shock can be made only stable inside duct diverging

area (Fig 5.19). An intake in this case is said to started in supersonic sense.

(b) Mt=1 before the shock reaches entry plane (Fig 5.20). If the duct throat area chokes, then further exit area

information cannot cross the throat, hence a second shock (Fig 5.21) will develop in system without having any

change in first shock. Finally, flow will be supersonic from forward detached shock to throat and intake in this

condition is said to be unstarted.

Mt<1

Fig 5.18 Shock reaches entry plane with Mt<1

Mt>1

Fig 5.19 Shock inside duct and flow started

Mt=1

Fig 5.20 Shock forward of entry plane with

Mt=1

Mt=1

Fig 5.21 Forward detached shock and second shock

with flow unstarted

5.6 Limiting Contraction Ratio

If we combine case (a) and case (b) to find limiting value of contraction ratio (At/Ac), then the forward shock would

have to reach entry plane at same time, when flow at throat reaches Mt=1. If ratio is less than limiting value, than case

(b) would be applied and flow would not be started. Mw ∞ being the subsonic mach number after shock.

( 1)/2( 1)

w 2

w

* 1 ( ) M ( )

( 1) M 2wMt

c

AA

AA

(5.31)

Substituting the value of Mw ∞ from normal shock equation (5.3), we get

1/2 1/( 1)

lim 2 2

1 2 2 1( )

1 ( 1) 1 ( 1)

t

c

A

A M M

(5.32)

The graph Fig 5.22 of limiting contraction ratio has been drawn with the help of equation (5.32) and of A*/A from

equation (5.31) with matlab code stated in Appendix 5.5. It seems that χ tends to 0.6 as M∞ tends to ∞. Also the

idealized full internal sonic area relation has been plotted in same graph.

Consider Fig 5.19, consider normal shock has been swallowed up and intake has been started, if we reduce the exit

area, then back pressure will increase causing the shock to move forward toward the throat and hence decreasing its

mach number, hence maximum pressure recovery can be done at throat will stable shock.

APPENDICES

Appendix 2.1 function loss() clear all close all clc u=0:0.01:3; ICFd=[0.14 0.11 0.08 0.05]; JCFa=[0.010 0.021 0.033 0.054]; axes('FontSize', 18) hold all

A*/A

------------- ᵡ

Fig 5.22 Limiting Contraction Ratio and A*/A vs M∞

for i=1:4 ploss(:, i) = ICFd(i) + JCFa(i).*u.^3; plot(u, ploss(:, i), '--') end xlabel('\mu', 'FontSize', 18) ylabel('\DeltaP/qc', 'FontSize', 18) title('Loss vs \mu', 'FontSize', 20) legend('0 degree', '30 degrees', '40 degrees', '52 degrees') end

Appendix 4.1 function lip() clear all close all clc g = 1.4; M1 = 0:0.001:0.7; Minf = [0.115 0.166 0.237 0.330]; t1 = 1+(g-1)*M1.^2/2; tinf = 1+(g-1)*Minf.^2/2; l1 = 1-(t1(91:701)/tinf(1)).^((g+1)/(2*g-2))./((1+g*M1(91:701).^2)

.*(tinf(1)./t1(91:701)).^0.5-g*M1(91:701)*Minf(1)); l2 = 1-(t1(181:701)/tinf(2)).^((g+1)/(2*g-2))./((1+g*M1(181:701).^2)



.*(tinf(4)./t1(301:701)).^0.5-g*M1(301:701)*Minf(4)); axes('FontSize', 18) plot(M1(91:701), l1, M1(181:701), l2, M1(241:701), l3, M1(301:701), l4) xlabel('M_1', 'FontSize', 18) ylabel('\DeltaP/P_i_n_f_i_n_i_t_y', 'FontSize', 18) title('Loss vs M_1', 'FontSize', 20) legend('M_i_n_f_i_n_i_t_y=0.115', 'M_i_n_f_i_n_i_t_y=0.166',

'M_i_n_f_i_n_i_t_y=0.237', 'M_i_n_f_i_n_i_t_y=0.330') end

Appendix 5.1 function M1_net=normalshock(Pr_net) %Pr_net=pressure recovery for which system need to be designed M1=linspace(1,4,100); r=1.4; pr=zeros(1,100);%static pressure ratio pr=p2/p1 Pr=zeros(1,100);%stagnation pressure ratio Pr=P2/P1 M2=zeros(1,100); for i=1:100 pr(i)=1+(2*r/(r+1)*(M1(i)^2-1)); M2(i)=sqrt((1+(r-1)/2*M1(i)^2)/(r*M1(i)^2-(r-1)/2)); Pr(i)=((1+(r-1)/2*M2(i)^2)^(r/(r-1)))*pr(i)/((1+(r-1)/2*M1(i)^2)^(r/(r-

1))); end M1_net=1.2; %initial guess M1_net=fsolve(@(M1)mach1(M1,Pr_net),M1_net); plot([1 4],[Pr_net Pr_net],'--cyan',[M1_net M1_net],[0 1],'--r') hold on plot(M1,M2,M1,Pr) xlabel('M1')

ylabel('M2 & P2/P1') grid end

Appendix 5.2 function fun=mach1(M1,Pr) r=1.4; fun=Pr-((1+(r-1)/2*(sqrt((1+(r-1)/2*M1^2)/(r*M1^2-(r-1)/2)))^2)^(r/(r-

1)))*(1+(2*r/(r+1)*(M1^2-1)))/((1+(r-1)/2*M1^2)^(r/(r-1))); end

Appendix 5.3 function Pr=obliqueshock() n1=100; th1=linspace(.1,60,n1); n2=10; M1=linspace(1.6,3,n2); th2(1:n1)=th1(1:n1)*3.14/180; r=1.4; b1=zeros(n2,n1); Pr=zeros(n2,n1); for i=1:n2 m1=M1(i); for j=1:n1 th=th2(j); b1(i,j)=fsolve(@(b)betashock(b,m1,th),th); b=b1(i,j); m2=sqrt((1+(m1^2*(sin(b)^2)*(r-1)/2))/(sin(b-

th)^2)/(r*m1^2*(sin(b)^2)-(r-1)/2)); pr1=1+(2*r/(r+1)*((m1*sin(b))^2-1)); %p2/p1 Pr1=((1+(r-1)/2*m2^2)^(r/(r-1)))*pr1/((1+(r-1)/2*m1^2)^(r/(r-1)));

%P2/P1 m3=sqrt((1+(r-1)/2*m2^2)/(r*m2^2-(r-1)/2)); pr2=1+(2*r/(r+1)*(m2^2-1)); %p3/p2 Pr2=((1+(r-1)/2*m3^2)^(r/(r-1)))*pr2/((1+(r-1)/2*m2^2)^(r/(r-1)));

%P3/P2 Pr(i,j)=Pr1*Pr2; %P3/P1 end for i=1:n2 plot(th1(1:1.6*M1(i)*10),Pr(i,1:1.6*M1(i)*10)) hold on end plot([0 30],[.96 .96],'--r') xlabel('Wedge Angle') ylabel('Ps/P1') grid

Appendix 5.4 function fun=betashock(b,m1,th) r=1.4; fun=tan(th)-2*cot(b)*(m1^2*(sin(b))^2-1)/(m1^2*(r+cos(2*b))+2); end

Appendix 5.5 function isentropic() n2=100;

M1=linspace(1,5,n2); M11=linspace(1.5,5,n2); r=1.4; v1=zeros(1,n2); v2=zeros(1,n2); for i=1:n2 m1=M1(i); v1(i)=sqrt((r+1)/(r-1))*atan(sqrt((r+1)/(r-1)*(m1^2-1)))-atan(sqrt(m1^2-

1)); m2=1.4; v2(i)=-sqrt((r+1)/(r-1))*atan(sqrt((r+1)/(r-1)*(m1^2-1)))+atan(sqrt(m1^2-

1))+sqrt((r+1)/(r-1))*atan(sqrt((r+1)/(r-1)*(m2^2-1)))-atan(sqrt(m2^2-1)); end v1(1,:)=180/3.14*v1(1,:); v2(1,:)=180/3.14*v2(1,:); plot(M1,v1,M11,v2) xlabel('Mach Number(M1)') ylabel('Turn Angle') grid

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Aircraft intake aerodynamics

Technology

p v2 p

engine source

intake design

new delhi delhi

p dp g

location of engine

method of intake

loss of total pressure