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R& M No, 2 1 6

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AERONAUTICAL RESEARCH COUNCIL

REPORTS AND MEMORANDA

The Flow in an Axially-Symmetric

Supersonic Jet from a NearlySonic

Orifice into a VacuumBy

LONDON : HER MAJESTY'S STATIONERY OFFICE

I952

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The Flow in an Axially-Symmetric Supersonic Jetfrom aNearly-Sonic Orifice into a Vacuum

By

P. L. OWEN, B.A. (Cantab.)

and

C. K. THORNHILL, M.A. (Oxon.)

COMMUNICATED BY THE PRINCIPAL DIRECTOR OF SCIENTIFIC RESEARCH (AIR)

MINISTRY OF SUPPLY

Reports and Memoranda No. z 6 z 6

September, 1948

Introduction and Summary.--The numerical method of characteristics is used to calculate theflow in a steady supersonic jet of air issuing from a slightly supersonic circular orifice into avacuum. The calculations are entirely numerical, and no recourse is made to graphical methods.

The characteristic equations for steady supersonic flow with rotational symmetry are firstderived, and then special consideration is given to the flow near the axis of symmetry, wherethe normal step-by,step numerical process breaks down.

In the calculation, the Mach angle in the plane of the orifice is taken as 85 deg to obviate

the difficulties of a sonic orifice at which the initial characteristics would be perpendicular tothe flow, and the potential equation parabolic. The results should be practically the same asfor a sonic orifice. An alternative method of dealing with a sonic boundary-plane would havebeen the use of analytical solutions for the initial flow in this region (cf. Ref. 3).

The results of the calculations are presented in diagrams. The solution is a universal solutionin so far that it applies to any similar jet, flowing into any external pressure, in that regionbounded by the orifice and the first wave front which registers the existence of an externalpressure outside the jet.

This fact allows the calculated pressure distribution along the axis of symmetry to be compared

with experimental measurements in air jets with finite pressure ratios, and good agreement isobtained.

1. The Characteristic E quations for Steady Supersonic- f low with Rotational Symmetry.-Usingcylindrical co-ordinates (z, r, 'ip), with r = 0 as the axis of rotational symmetry, the motion is

independent of ' . Let asu, asv be the components of fluid velocity in the z and r directionsrespectively, where as is a constant speed. Then, for irrotational adiabatic flow, a velocitypotential 4 may be defined so that,

a^u=P- a

z

a^v=Q=ay

* A.R.E. Report 30/48. Received 14th October, 1948.

.. .. .. .. (1.01)

1(22303)

The speed of sound, A as, is given by,

(A a5)2 = ^ p = ? ? . .. (1.02)

The equations of motion integrate to give Bernoulli's relation,

2 as2(p2 _+ Q2) + A 2as2/ ( Y __.. 1) = constant

(Y + 1) as2 (103)2(Y_1)

chosen to be the local speed of sound when the fluid is flowing with sonic velocity, i. e.,if a s isif P2 -4- Q2 = 1 when A = 1.Hence, 2 2 2 • .

p2 2-A1A ^ .. ( 1.04 )

where ) is defined by,

2 Y -1--

Y± 1 . .. .. .. .. .. .. • .. .. (1.05)

The equation of continuity reduces to the potential equation,

(p2 -- A 2) R ± 2PQS ± (Q2 - A 2)T = A 2Q/i,, • u • . .. . , (1.06)

where .R = a2^bi az2, S = a 2c / ar az, T = a2^^ a y2 . .

The characteristic ions of this potential equation are real if the flow is supersonic, anddirectco-incident if the flow is sonic. In the supersonic or hyperbolic case, the Mach angle µ maybe defined by

A2.. .. .. .. .. ..V2=u2+V2= -

sin 2 µ

and the diff erential equation of the characteristic curves may then be written,

dy--=tan(0 +µ) ..dz

where 0 is the inclination of the flow to the z-axis, or

tan 0 = v/u. .. .. .. .. .. .. .. .

The differential relations which hold along the characteristic curves (1.08) are respectively,

d V sin 0 sine µ dy. ( 1.10)µ .^ tan • de - -- =0 ... .. ..

V + sin (0 ± µ) cos µ y

But, from (1.04) and (1.07)

V2(2 cost ± sine ) = 1 .. .. / .. .. .. .. .. (1.11)

and therefore

dV (1 - 2) sin µ. cos µ. dµ _TT ( ^2 cos2 + sin

2 - 0 ..

^ µ µ).. ( 1 . 1 2)

2

Substituting this result, the differential relations (1.10) may be written

sin o µ drv/2 __ a ^- o +

jrsin

µ-- = constant .. .. .. .. (1.13)

where v is an angle defined by,

A tan v = tan ,u. .. .. .. .. .. .. .. ( 1 . 14)

The characteristic relations (1.13) are used in the calculations, but, as is done in the followingSection, the limit of the integrand must be considered as the axis is approached and both r andsin 6 tend to zero.

In terms of the state of the fluid when flowing at sonic speed, namely pressure ^S, density PS,speed of sound as, the speed of sound a at any point is given by,

a;as=A=sinv; .. .. .. .. .. .. .. .. (1.15)

the resultant fluid velocity aSV is given by

sin vV= .. .. .. .. .. .. .. .. .. (1.16)

sin µ

with components,

1sin v cos 8 sin v sin o( 1 . 17u= v= . : .. .. .. .. ..

sin 1u

the pressure p is given by,

p/ps = (sin v)1 + 1/22

and the density p is given by,

PPS (sin µ) -1 + 1^^ .

.. .. .. .. .. (1.18)

.. .. .. .. .. .. .. .. (1.19)

2. Flow near the Axis of Symmetry.-Using the suffix 0 to denote values on the axis ofsymmetry , r = 0, it follows at once that

o0 = 0 .. .. .. .. .. .. (2.01)

.. .. .. .. .. .. (2.02)

d Q^and therefore So = .dz

(2.03)

Thus, from (1.15),

A U = sin vo .. .. .. .. .. .. .. .. .. (2.04)

and f rom (1. 1,75,

sin vo

sin µ o.. (2.05)

The potential equation (1.06) then gives,

Sln2 vo2 __ sin vo R 0 -- sinsin ,u o

Qvo TO == sin vo Lt -r -> 0 7

.. .. (2.06)

whence, neglecting the trivial case sin vo = 0,

2 Q (2 .07 )- T0 Lt {} . . . . . . . . . . . . .r -. O I'

But QLt = Lt

S dz + T dr- -^

r-- 0 Y r O di

=T0

and therefore, from (2.07)

Lt

r -^ 0

= - R 0 cote .. .. .. .. .. (2.08)

Hence, the limit of the integrand in the characteristic relations (1.13), as the axis,approached, is given by,

sin 8 sin ,uLt {Y^.o lr sin (8 + i^)

sin 1t 0 Q 1Lt I -

sin v0 r -^ 0` y

sin 2= ± 2 sin v

• cot R0o

cost oLt

J sin_V cos 8--

2 sin vo sin ^o r 0 az sin ,u-

cost ^Ct0 d sin v0

2 sin v o sin µ o dz sin µ o

v = 0 , is

and, after some reduction, this may be expressed in the very useful form,

sin 8 sin ,u

LeoI

sin (e J ,u)cot /2 o d

(vo/A - o) . . . . . . . . . (2.09)2 dz

The relation (2.09) thus enables the integral term in the characteristic relations (1.13) to beevaluated accurately as r->0 along a characteristic, by making use of the gradient of (v/. - µ}along the axis of symmetry.

3. The Calculation.-In the calculation, the Mach angle in the plane of the circular orifice,was taken as 85 deg, and the external pressure zero. The familiar ` patching ' method ofcharacteristics was used, in which the known solution at two points P1i P2r leads to the solutionat a point P3 such that P1P3 and P2P3 are small arcs of characteristic curves. In the calculationthe small arcs P1P3i P2P3 were taken to be arcs of circles, i.e., the inclinations of the . chordsPip3i P2P3 were taken to be the means of the characteristic inclinations at P1, P3, and P2, P3respectively. Since the characteristic relations (1.13) involve the space co-ordinates and do not

4

integrate exactly, an iterative process is necessary. For convenience in the calculation, a tablegiving values of the angles ,u and v at equal small intervals of the function (v/2 - ) was used.This was prepared with the aid of National Accounting Machines. The integral term in therelations (1.13 was evaluated numerically by the trapezoidal rule.

The calculation of the points and conditions along the axis of symmetry was accomplishedby means of the relation (2.09) . The iterative process, in this case, was found to be much moredifficult, and it was harder to maintain accuracy in these arcs.

The results of the calculation, so far as it was taken, are shown in the diagrams. Fig. 1 showsthe pattern of the characteristic network; Fig. 2 shows the distribution of velocity and Machnumber along the axis of symmetry. It is worth remarking that the maximum possible valueof the non-dimensional velocity V is 1/2 = -/6 = 2.45, for y = 1.40.

Fig. 3 shows the calculated axial distribution of pressure, and, for comparison, some experi-mental measurements of the pressure along the axes of supersonic jets with sonic orifices andfinite pressure ratios are also plotted in this figure. This comparison is possible since thesolution calculated here for an infinite pressure ratio is, in some sense, a universal solution, inso far as it applies to that area, of any similar jet with finite pressure ratio, which is boundedby the orifice and the first wave front from the orifice boundary which registers the existenceof an external pressure outside the jet. The experimental measurements are of two kinds;Pitot measurements taken along the axis of a j et flowing through a 6-mm orifice from a tank ata pressure of 7 kg/sq cm, by Hartmann and Lazarus (Ref. 1) ; and interf erometric measurementsmade by Ladenburg and others with a jet flowing through a 10-mm orifice from a tank at apressure of 5 atmospheres (Ref. 2). The agreement is good, particularly when it is rememberedthat, for convenience in starting the calculation, a Mach angle of 85 deg was taken at the orifice.

REFERENCES

No. Author Title, etc.

1 J. Hartman and F. Lazarus .. The Air Jet with a Velocity Exceeding that of Sound. Phil. Mag.,Vol. 31, p. 35, 1941.

2 R. Ladenburg, C. C. Van Voorhis Interferometric Study of Supersonic Phenomena: A Supersonic Air Jetand J. Winckler at 60 lbf in.2 Excess Tank Pressure. NAVORD Report 69-46, Navy

Department, Bureau of Ordnance, Washington, U.S.A.

or

J. Winckler .. .. .. .. The Mach Interferometer Applied to the Study of an Axially SymmetricSupersonic Air Jet. Review of Scientific Instruments, Vol. 19, No. 5,p. 307, May, 1948.

3 R. Hill and D. C. Pack .. .. An Investigation, by the Method of Characteristics, of the LateralExpansion of the Gases behind a Detonating Slab of Explosive. Proc.Roy. Soc. A, Vol.191, p. 524, 1947.

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