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A simple graphical solution for rocket wake boundaries
Planet. Space Sci. 1973, Vol. 21, pp. 1625 to 1632. Fergatttott Pms.
Printed in
Northern Ireland
A SIMPLE GRAPHICAL SOLUTION ROCKET WAKE BOUNDARIES D. J. STIJRGEW Department of Electron Physics,The Universityof Bi~ingh~,
FOR
England
Ab~ct-asking a simple conical ‘aerodynamic’wake extending downstream from any point on the surface of a sounding rocket of typical configuration, equations are given to calculate whether a boom-mountedprobe is within the wake, for varying wake angles. The calculation is s~aightfo~~d for wake angles 0” and 90”. For the general case, as an alternative to soivingquartic equations,a simplegraphicalmethod of solutionis rovided, affording quick solutions in generalizedcoordinates applicable to a wide range oP rocket and boom configurations. INTRODUCTION In estimating possible effects of vehicle wakes on experimental probes carried on sounding rockets, a major difficulty is caused by the lack of a simple means of calculating whether the probe is within the wake at any given moment, as well as by limited knowledge of the character of such wakes. In a recent paper (Sturges, 1973) a way was suggested in which correlations between wake effects and rocket attitudes may be sought, based on simple assumptions about the plausible character of low-velocity ionosperic wakes. A simple method of determining wake boundaries in rocket-based coordinates, compatible with such a wake theory, is described here. The Mach cone, spreading out from direct contact with the rocket, is assumed to delineate the approximate location of the forward edge of the wake. This simplified view seems justifiable since knowledge of the exact nature of the wake at ionospheric altitudes is so limited (Sturges, 1973).t The method was applied first to a particular Skylark rocket, and the relevant dimensions are used here as an example, although the solution is readily applicable to other rockets of generally similar configuration. A sketch of the rocket and payload is given in Fig. 1 (not to scale}. The overall length of the 22 cm-radius rocket was 827 cm. The nose-cone was 136 cm long. At a location 0,217 cm back from the tip A of the cone, the 134 cm long booms OP were deployed. The three fins such as CDEF were 96 cm in radius, 45 cm long at their tips CD and 86 cm long at their bases EF, and set in 10 cm from the rear of the rocket. The rocket attitude is described by the angular coordinates cc and 8, also shown in Fig. 1; OLmeasures the angle between one boom OP and the plane through the velocity vector and the longitu~nal axis of the rocket; 8 is the angle between that axis and the velocity vector. In calculating whether or not a probe, mounted in a fixed position with respect to the rocket axes, lies within the Mach cone of the rocket at any given instant, the problem is essentially one of identifying for a given value of the wake half-angle o the critical combinations of values of the attitude coordinates 13and u at which the probe crosses the surface of the Mach cone. * Now at the General Electric Company, Lynn, Massachusetts,U.S.A. t If the existenceof a wake, and the wake angle, are clearly established by other evidence, the method is applicable without modification. Identification of the wake boundary as a Mach cone becomes unnecessary and even the existence of phenomena such as shock stand-off can be allowed for by effectively increasing the rocket body dimensions. 1625
1626
D. J. STURGES
Fm.1.
9O”wm.
ROCKETOEOMETRYFORCALCULATXONOFTHE
40 a0 60
80 -20
0
20
40
60
8"
Fiff. 2,
VALUESOPTHEA~ECOORDINATESe
AND~FORWHICfI A PROBE ATRADIUS WILLCROSSTHFi~0DWAKE,DUETOVARIOUSSECITONSOPTHEROCKET.
134 cm
For the case w = 90” this can readily be done by geometrical calculation, considering in turn the effect of the six points A-F shown in Fig. 1. Take, for example, point A, the tip of the nose cone. For a particular value of a, the probe P will cross the w = 90” plane when 134 cos a 6 = arctan (0 217 ’ This curve is plotted in Fig. 2 together with the curves resulting from similar relations for points B-E; the curve for F fahs below that for E, and is not shown. From the envelope of these curves is constructed the final o = 90’ curve. For the opposite extreme of w = O’, (i.e. the rocket casts a ‘shadow’), geometrical Consider, for example, the shadow cast into the calculations are also straightforward. plane of the probes by the forward section of the rocket, for positive values of 8. As shown in Fig. 3, the shadow may be considered in three sections: {a) the triangle cast by the cone, (b) the rectangle cast by the body and (c) the circle cast by the cross-section of the rocket at the plane conjoining the body and cone. (The entire shadow is, of course, the superposition of such circular shadows cast by adjoining cross-sections of the rocket, and this concept will be used in studying the more general case below.) Then, for the rectangular shadow of the rocket body, it follows that the critical value of a, for all 8 < 6, = arccot y/(1342 - 222)/812 is given by a = arc sin 221134 = 9” 27’.
(2)
A SIMPLE GRAPHICAL
SOLUTION
FOR ROCKET
WAKE
BOUNDARIES
1627
FIG. 3. GEOMETRY FOR CALCULATION OF THE SHADOW CAST BY THE FORWARD SECX’IONOF THE ROCKET.
-WJ
-60
-40
-20
0
20
40
60
8 FIG. 4. VALUES OF THE AI-ITTUDE COORDINATES 0 AND Q FOR WHICH A PROBE AT RADIUS 134 Cm WILL ENTSR THE SHADOW, DUB TO VARIOUS SBCITONS OF THE ROCKET.
This line may be found in Fig. 4, for 8 between 0 and + 31” 30’. Similar expressions are readily derived for the rest of the shadow, for both positive and negative 0, the latter shadow being composed of rectangular sections corresponding to (d) the body of the rocket and (e) the major section of the fins (which are treated as being equivalent to the annulus formed by their rotation about the axis of symmetry of the rocket, thus facilitating both the calculation and presentation of the wake data, with only a small loss in accuracy in what is already a hypothetical rather than exact wake representation); and curves (f), (g), and (h) corresponding respectively to the cross-sections through points C, D and E
in Fig. X+ Curve fi) sfrows the shadow due to the section CF of the leading edge of the fins. The resultant curves are knotted in ‘Fig*4; in most cases the section of the rocket body to chief they correspond is self-evidentS The envelope of these curves in turn leads to the cu = 0” curve for the entire rocket. For the general case of a wake of half-angle 0 < w < 90’, no such simple solution is practicable, due to the complicated nature of the equations describing the intersec~ou of the Mach cone with the probe plane (or, more simply, due to the d~~c~~ty of id~utif~ing exa&y which point on the surface of the rocket wiff give rise to the most ~~gui~~t wake).
&3.%
~BOMETRY FOR CALCXJLATXONOP THE WAKE XN THE PLANE OF THE PROBES DUli TO PCJlNT F, FOR A GENERAL WAKE-ANGLE 0.
Consider the wake due to a single point F, as &own in Fig. 5. F is at height k above the JKLMN plax~, WIG& is the plane of the probe. Angie LHF is 8, and angle HFN = o, Let
LL=yandIJ=x.
Then
Wf=hcot&-~
FC=h&-
OH==hsin~+ycos8
A SIMPLE GRAPHICAL
SOLUTION FOR ROCKET WAKE BOUNDARIES
1629
point F there corresponds a curve such as that described by Equation (3) but displaced by distances xl = r sin 6, Yl = r cos CT. To find the critical value of a, we need to construct a line OJ’ of length b equal to that of the probe boom, with J’ a point similar to J but on the envelope of this family of curves. In general there will be four such points from which we wish to select that giving the maximum value J’OQ, which corresponds to a. Rather than attempt to repeatedly solve quartic equations, a graphical method of solution to this problem has been devised, which has the additional advantage that it is
x/
x/
i FIG. 6. THE INTERSECTION OF THE WAKE OF A POXNT OBJECT, HEIGHT h AROVE A PLANE, WITH THAT PLANE, FOR VARIOUS WAKE-ANGLES W AND ATTXTUDE ANGLES 6 BETWFZN 0 AND 60”. (CURVES FOR w=15” IN BROKEN LINES.)
presented in normalized coordinates that can be directly applied to solving wake problems for rocket and boom confi~rations other than those presently under consideration. Thus we write Y = y(h) x2 = sinB+~cos8)8tanlO-(cosB-KsinSr
07; (
and
b = b(h),
normalizing all lengths in terms of the height h of the circular cross-section considered. The functions (sin 8 + y/h cos @2 and (cos 8 - y/h sin r9)% have been tabulated to 4 decimal place accuracy for - 1 G y/h < + 13, allowing curves of the form described in Equation (3) to be quickly plotted for any chosen value of o and 0. Three families of such curves for 0 GZ8 G 90” and o = W, 30” and 45”, are given in Figs. 6-8. Each curve represents the intersection of the wake of point F with the plane of the probes. For maximum clarity and utility, some curves have been plotted twice, with the larger values of y/h appearing in Fig. 6, and the smaller values in Figs. 7 and 8.
1630
D. J. STURGES
0.6. ~-15~ x/h 0.4.
FIG. 7.
THEINTERSECTIONOF THE
THAT
PLANE,
-08
FIG.
FOR WAKE-ANGLES
-0 4
8. THBINTERSB(;TION THAT
PLANE,
WAKE
OF
A POINT
W OF 15” AND
0
OF THE WAKE.
FOR WAKE-ANGLE
04
OF W =
OBJECT,
30”, AND
HEIGHT
h ABOVB
A PLANE,
VARIOUS
ATrITUDE
ANGLES
08
A POINT
2.0
1.2
OBJECT,
45”, AND
HEIGHT
VARIOUS
WITH 8.
h ABOVE
ATTITUDE
A PLANE,
ANGLES
WITH
6.
The critical value of a is then found by drawing a circle of radius r/h around the origin of any of these graphs; marking off a distance b/h on a rule or dividers, and moving one end of the rule around the circumference of the circle while keeping the marked distance set on one curve in the chosen family until the angle between the rule and the horizontal axis is maximised; and measuring and recording this angle as the critical angle a for that particular combination of o, 8, r, and h. This construction is exactly equivalent to that
A SIMPLE CEtAPHICAL SOLUTION FOR ROCKET WAKE Robots
I60
60
Fm. 9. VaLuEs OF THE A-ITITUD B CGGRDINATES 8 A?4R U FGR WHICH A PROBE AT RADIUS 134 cm W&L CROSS TRS 45” WAKE DUE TO CIRCLES OF VARIOUS RAJXUS AND ?BXGHT ABGVE THE PROBZ PLAN&
k4XXUrS AND HIZGHT RESPECTIVELY ARE SPECIFIED BY THE TWO NUMBERS GIVEN FOR EACH CURVE.
a*
134 Cm AITITUD E COORDINATES @AND II FOR WHICH A PRGBE AT RADIUS WILL CROSS TWII 45” WAKE DUB TO CIRCLES OF VARIOUS RADIUS AND IiHGHT ABGVE THE PROBE PLANE. &UXUS AND HEIGHT RESPECTIVELY ARB SPECIPISD BY THE TWO NUMBERS GIVEN FOR EACH CURVE. Fro. 10. VALUES0s ‘rm
1631
1632
D. J. STURGES
previously described for finding the line OJ’, but substitutes drawing a circle for the laborious task of plotting the envelope of the family of intersection curves. In order to use this method to obtain the desired wake curves in (0, CC)space, one more step is necessary. A choice must be made of appropriate values of r and h to ensure that the maximum wake is found. As an illustrative example the case for w = 45” has been very thoroughly evaluated, and the results are shown in Fig. 9 for the body of the Skylark, and in Fig. 10 for the fins and nose-cone; each curve is labelled with two numbers indicating the values used for r and h in cm, and these should be compared with the rocket dimensions previously given. The detail shown may be of interest in showing how the wake builds up as the vehicle size increases,* and also suggests why the analytical solution is difficult, but obviously in practice far fewer curves than this are needed to form a satisfactory envelope, and not all of each curve needs measuring or plotting, so that the technique is considerably quicker to use than it may here appear. REFERENCE STURGES,D. J. (1973). Planet. Space Sci. 21, 1029, 1049. l The curves for w = 45” in Fig. 4 (Sturges, 1973) were obtained from the present Figs. 9 and 10, and the various sections of the curves and the parts of the rocket body generating them may easily be identified. The curves for CLI= 15” and 30” in that Fig. 4 were obtained in a similar manner. Curves for wake angles between 45” and 90” were not plotted since in the ionosphere the meaningful existence of a shock-front as M approaches unity seems doubtful.
Printed in
Northern Ireland
A SIMPLE GRAPHICAL SOLUTION ROCKET WAKE BOUNDARIES D. J. STIJRGEW Department of Electron Physics,The Universityof Bi~ingh~,
FOR
England
Ab~ct-asking a simple conical ‘aerodynamic’wake extending downstream from any point on the surface of a sounding rocket of typical configuration, equations are given to calculate whether a boom-mountedprobe is within the wake, for varying wake angles. The calculation is s~aightfo~~d for wake angles 0” and 90”. For the general case, as an alternative to soivingquartic equations,a simplegraphicalmethod of solutionis rovided, affording quick solutions in generalizedcoordinates applicable to a wide range oP rocket and boom configurations. INTRODUCTION In estimating possible effects of vehicle wakes on experimental probes carried on sounding rockets, a major difficulty is caused by the lack of a simple means of calculating whether the probe is within the wake at any given moment, as well as by limited knowledge of the character of such wakes. In a recent paper (Sturges, 1973) a way was suggested in which correlations between wake effects and rocket attitudes may be sought, based on simple assumptions about the plausible character of low-velocity ionosperic wakes. A simple method of determining wake boundaries in rocket-based coordinates, compatible with such a wake theory, is described here. The Mach cone, spreading out from direct contact with the rocket, is assumed to delineate the approximate location of the forward edge of the wake. This simplified view seems justifiable since knowledge of the exact nature of the wake at ionospheric altitudes is so limited (Sturges, 1973).t The method was applied first to a particular Skylark rocket, and the relevant dimensions are used here as an example, although the solution is readily applicable to other rockets of generally similar configuration. A sketch of the rocket and payload is given in Fig. 1 (not to scale}. The overall length of the 22 cm-radius rocket was 827 cm. The nose-cone was 136 cm long. At a location 0,217 cm back from the tip A of the cone, the 134 cm long booms OP were deployed. The three fins such as CDEF were 96 cm in radius, 45 cm long at their tips CD and 86 cm long at their bases EF, and set in 10 cm from the rear of the rocket. The rocket attitude is described by the angular coordinates cc and 8, also shown in Fig. 1; OLmeasures the angle between one boom OP and the plane through the velocity vector and the longitu~nal axis of the rocket; 8 is the angle between that axis and the velocity vector. In calculating whether or not a probe, mounted in a fixed position with respect to the rocket axes, lies within the Mach cone of the rocket at any given instant, the problem is essentially one of identifying for a given value of the wake half-angle o the critical combinations of values of the attitude coordinates 13and u at which the probe crosses the surface of the Mach cone. * Now at the General Electric Company, Lynn, Massachusetts,U.S.A. t If the existenceof a wake, and the wake angle, are clearly established by other evidence, the method is applicable without modification. Identification of the wake boundary as a Mach cone becomes unnecessary and even the existence of phenomena such as shock stand-off can be allowed for by effectively increasing the rocket body dimensions. 1625
1626
D. J. STURGES
Fm.1.
9O”wm.
ROCKETOEOMETRYFORCALCULATXONOFTHE
40 a0 60
80 -20
0
20
40
60
8"
Fiff. 2,
VALUESOPTHEA~ECOORDINATESe
AND~FORWHICfI A PROBE ATRADIUS WILLCROSSTHFi~0DWAKE,DUETOVARIOUSSECITONSOPTHEROCKET.
134 cm
For the case w = 90” this can readily be done by geometrical calculation, considering in turn the effect of the six points A-F shown in Fig. 1. Take, for example, point A, the tip of the nose cone. For a particular value of a, the probe P will cross the w = 90” plane when 134 cos a 6 = arctan (0 217 ’ This curve is plotted in Fig. 2 together with the curves resulting from similar relations for points B-E; the curve for F fahs below that for E, and is not shown. From the envelope of these curves is constructed the final o = 90’ curve. For the opposite extreme of w = O’, (i.e. the rocket casts a ‘shadow’), geometrical Consider, for example, the shadow cast into the calculations are also straightforward. plane of the probes by the forward section of the rocket, for positive values of 8. As shown in Fig. 3, the shadow may be considered in three sections: {a) the triangle cast by the cone, (b) the rectangle cast by the body and (c) the circle cast by the cross-section of the rocket at the plane conjoining the body and cone. (The entire shadow is, of course, the superposition of such circular shadows cast by adjoining cross-sections of the rocket, and this concept will be used in studying the more general case below.) Then, for the rectangular shadow of the rocket body, it follows that the critical value of a, for all 8 < 6, = arccot y/(1342 - 222)/812 is given by a = arc sin 221134 = 9” 27’.
(2)
A SIMPLE GRAPHICAL
SOLUTION
FOR ROCKET
WAKE
BOUNDARIES
1627
FIG. 3. GEOMETRY FOR CALCULATION OF THE SHADOW CAST BY THE FORWARD SECX’IONOF THE ROCKET.
-WJ
-60
-40
-20
0
20
40
60
8 FIG. 4. VALUES OF THE AI-ITTUDE COORDINATES 0 AND Q FOR WHICH A PROBE AT RADIUS 134 Cm WILL ENTSR THE SHADOW, DUB TO VARIOUS SBCITONS OF THE ROCKET.
This line may be found in Fig. 4, for 8 between 0 and + 31” 30’. Similar expressions are readily derived for the rest of the shadow, for both positive and negative 0, the latter shadow being composed of rectangular sections corresponding to (d) the body of the rocket and (e) the major section of the fins (which are treated as being equivalent to the annulus formed by their rotation about the axis of symmetry of the rocket, thus facilitating both the calculation and presentation of the wake data, with only a small loss in accuracy in what is already a hypothetical rather than exact wake representation); and curves (f), (g), and (h) corresponding respectively to the cross-sections through points C, D and E
in Fig. X+ Curve fi) sfrows the shadow due to the section CF of the leading edge of the fins. The resultant curves are knotted in ‘Fig*4; in most cases the section of the rocket body to chief they correspond is self-evidentS The envelope of these curves in turn leads to the cu = 0” curve for the entire rocket. For the general case of a wake of half-angle 0 < w < 90’, no such simple solution is practicable, due to the complicated nature of the equations describing the intersec~ou of the Mach cone with the probe plane (or, more simply, due to the d~~c~~ty of id~utif~ing exa&y which point on the surface of the rocket wiff give rise to the most ~~gui~~t wake).
&3.%
~BOMETRY FOR CALCXJLATXONOP THE WAKE XN THE PLANE OF THE PROBES DUli TO PCJlNT F, FOR A GENERAL WAKE-ANGLE 0.
Consider the wake due to a single point F, as &own in Fig. 5. F is at height k above the JKLMN plax~, WIG& is the plane of the probe. Angie LHF is 8, and angle HFN = o, Let
LL=yandIJ=x.
Then
Wf=hcot&-~
FC=h&-
OH==hsin~+ycos8
A SIMPLE GRAPHICAL
SOLUTION FOR ROCKET WAKE BOUNDARIES
1629
point F there corresponds a curve such as that described by Equation (3) but displaced by distances xl = r sin 6, Yl = r cos CT. To find the critical value of a, we need to construct a line OJ’ of length b equal to that of the probe boom, with J’ a point similar to J but on the envelope of this family of curves. In general there will be four such points from which we wish to select that giving the maximum value J’OQ, which corresponds to a. Rather than attempt to repeatedly solve quartic equations, a graphical method of solution to this problem has been devised, which has the additional advantage that it is
x/
x/
i FIG. 6. THE INTERSECTION OF THE WAKE OF A POXNT OBJECT, HEIGHT h AROVE A PLANE, WITH THAT PLANE, FOR VARIOUS WAKE-ANGLES W AND ATTXTUDE ANGLES 6 BETWFZN 0 AND 60”. (CURVES FOR w=15” IN BROKEN LINES.)
presented in normalized coordinates that can be directly applied to solving wake problems for rocket and boom confi~rations other than those presently under consideration. Thus we write Y = y(h) x2 = sinB+~cos8)8tanlO-(cosB-KsinSr
07; (
and
b = b(h),
normalizing all lengths in terms of the height h of the circular cross-section considered. The functions (sin 8 + y/h cos @2 and (cos 8 - y/h sin r9)% have been tabulated to 4 decimal place accuracy for - 1 G y/h < + 13, allowing curves of the form described in Equation (3) to be quickly plotted for any chosen value of o and 0. Three families of such curves for 0 GZ8 G 90” and o = W, 30” and 45”, are given in Figs. 6-8. Each curve represents the intersection of the wake of point F with the plane of the probes. For maximum clarity and utility, some curves have been plotted twice, with the larger values of y/h appearing in Fig. 6, and the smaller values in Figs. 7 and 8.
1630
D. J. STURGES
0.6. ~-15~ x/h 0.4.
FIG. 7.
THEINTERSECTIONOF THE
THAT
PLANE,
-08
FIG.
FOR WAKE-ANGLES
-0 4
8. THBINTERSB(;TION THAT
PLANE,
WAKE
OF
A POINT
W OF 15” AND
0
OF THE WAKE.
FOR WAKE-ANGLE
04
OF W =
OBJECT,
30”, AND
HEIGHT
h ABOVB
A PLANE,
VARIOUS
ATrITUDE
ANGLES
08
A POINT
2.0
1.2
OBJECT,
45”, AND
HEIGHT
VARIOUS
WITH 8.
h ABOVE
ATTITUDE
A PLANE,
ANGLES
WITH
6.
The critical value of a is then found by drawing a circle of radius r/h around the origin of any of these graphs; marking off a distance b/h on a rule or dividers, and moving one end of the rule around the circumference of the circle while keeping the marked distance set on one curve in the chosen family until the angle between the rule and the horizontal axis is maximised; and measuring and recording this angle as the critical angle a for that particular combination of o, 8, r, and h. This construction is exactly equivalent to that
A SIMPLE CEtAPHICAL SOLUTION FOR ROCKET WAKE Robots
I60
60
Fm. 9. VaLuEs OF THE A-ITITUD B CGGRDINATES 8 A?4R U FGR WHICH A PROBE AT RADIUS 134 cm W&L CROSS TRS 45” WAKE DUE TO CIRCLES OF VARIOUS RAJXUS AND ?BXGHT ABGVE THE PROBZ PLAN&
k4XXUrS AND HIZGHT RESPECTIVELY ARE SPECIFIED BY THE TWO NUMBERS GIVEN FOR EACH CURVE.
a*
134 Cm AITITUD E COORDINATES @AND II FOR WHICH A PRGBE AT RADIUS WILL CROSS TWII 45” WAKE DUB TO CIRCLES OF VARIOUS RADIUS AND IiHGHT ABGVE THE PROBE PLANE. &UXUS AND HEIGHT RESPECTIVELY ARB SPECIPISD BY THE TWO NUMBERS GIVEN FOR EACH CURVE. Fro. 10. VALUES0s ‘rm
1631
1632
D. J. STURGES
previously described for finding the line OJ’, but substitutes drawing a circle for the laborious task of plotting the envelope of the family of intersection curves. In order to use this method to obtain the desired wake curves in (0, CC)space, one more step is necessary. A choice must be made of appropriate values of r and h to ensure that the maximum wake is found. As an illustrative example the case for w = 45” has been very thoroughly evaluated, and the results are shown in Fig. 9 for the body of the Skylark, and in Fig. 10 for the fins and nose-cone; each curve is labelled with two numbers indicating the values used for r and h in cm, and these should be compared with the rocket dimensions previously given. The detail shown may be of interest in showing how the wake builds up as the vehicle size increases,* and also suggests why the analytical solution is difficult, but obviously in practice far fewer curves than this are needed to form a satisfactory envelope, and not all of each curve needs measuring or plotting, so that the technique is considerably quicker to use than it may here appear. REFERENCE STURGES,D. J. (1973). Planet. Space Sci. 21, 1029, 1049. l The curves for w = 45” in Fig. 4 (Sturges, 1973) were obtained from the present Figs. 9 and 10, and the various sections of the curves and the parts of the rocket body generating them may easily be identified. The curves for CLI= 15” and 30” in that Fig. 4 were obtained in a similar manner. Curves for wake angles between 45” and 90” were not plotted since in the ionosphere the meaningful existence of a shock-front as M approaches unity seems doubtful.