ENGINEERING RESEARCH INSTITtTEI ` er'aot UNIVERSITY OF MICHIGAN Report No. EMB-3 Copy No. THE APPLICATION OF THE THREE-DIMiE1TSIONAL MTETHOD OF CHARACTrERISTICS TO THE SOLUTION OF THE FLOW ABOUT AN AXIALLY SYMI4ETRPIC BODY CF REVOLUTION. Prepored by 7 A7YH. Dorran',e Approved by _ __ June 21, 1948

introductih on A previous report dealt with this method as applied to a nozzle solution (Ref. 1). The solution as discussed in Reference 1 did not trke' into account an occurrence of a shock wave such as appears at the nose of a missile moving at supersonic velocity. This report discusses the application of the three-dimensional method of characteristics to problems which have a shock wave present. The occurrence of a shock'wave necessitates the use of a shock polar in the velocity plane which serves to satisfy the demands of the Rankine-Hugonilot equations relating flow velocities,before and. after the shock wave and shock inclination. This report will cover the application of this method to finding the velocity field around an axially symmetric body of revolution moving at a supersonic velocity. Because known relations exist connecting the velocity at the surface of a missile with the pressure at the surface, this solution can be used to obtain the pressure distribution over the body. The solution will determine a Mach net made up of the two families of characteristic lines in the physical plane and the velocities associated with each intersection of the characteristic lines. The angle of the shock wave will also be determined as it decreases from the shock angle associated with the tip of the missile investigated to the free stream Mach angle at large distances from the tip. This procedure is a step-by-step solution in a meridian plane.

ENCINEERXNG RESEARCH INSTITUTE Page ~~1WB1~3 tftrNvERsrtY OF MICkIGANPa 2 -1 iO = local Xach angle - sin 1/N e =l1ocal velocity of sound o* v-uelocity of sound when XM 1' = ~ g > 1.4 for air M: Mach number s c u - component of velocity along the local Mach line = w cos8 v component of velocity across the local Mach line - w sin ro w = local free stream velocity Wmax maximwn velocity attainable by exhausting a adiabatically into a vacuum.

ENGINEERING RESEARCH INSTIITTE Page ~EM3-3 l~ ~ UNIVERSITY OF MICHIGAN _ Discussion This graphical-numerical solution mnakes use of a physical plane and a velocity plane. These two planes are connected by difference equations which will be discussed later. The method reauires a larie amount of arithmetical. calculation and careful measuring. It is recommended that a scale of at least I inch - 3 inches be used in the physical plane, and e scale of at least w 10 inches for the critical velocity of w* 1 be used in the velocity plane. If a larger size surface than the standard drafting table is available, it should be used in order that the scale be as large as possible. The large scale helps to eliminate inaccuracies that can occur when measuring the small quantities that arise in using the difference equations. Tables are used t~hroughout the method of keep the procedure as orderly -as possible and to prevent redundant calculations. The fundamental difference equations employed throughout this graphical solution arise from a mathematical manipulation of the hyperbolic supersonic differential equation for axially symmetrical flow. R. Sauer and others have treated the mathematical background thoroughly and methodically in several references. References 2 and 4 present such discussions in detail. The working equations derived through manipulation of the axially symmnetrical supersonic differential equation are given below. Tllese equations connect the physical plane and the velocity plane. do.v sin d?2 (la) dq = v sin2 CK d (lb) r o where, as illustrated in Figure 1, dp - increment of the component of stream velocity in the direction of the local Mach line at the angle- ( K - ) with x axis.,lq - increment of the component of stream velocity in the direction of the local Mach line at the angle ( " + OK ) with x axis. v - radial component of the stream -etocity or component perpendicular to x axis. - angle of stream velocity with the x axis.

ENGINEERING RESEARCH INSTIT'TE 'Payge ',E.B-3. UNIVERSITY OF MICHIGAN. 4 O - local Mach angle - sln-l 1 r - the radial distance of a point in the physical field from the x axis. do- length of side of Mach quadrangle in the Mach line direction at the angle ( j 1( ) with x axis. d - length of side of Mach quadrangle in the Mach line direction at the angle ( 6-O( ) with x axis. x axis - axis of symmetry. All of these terms appear in Figure 1. Presume that the velocities of points 1 and 2 -are known and that the velocity at point 3 is desired. By making use of equations (1) which connect the physical and velocity planes the velocity at unknown point 3 can be determined. That is, through measuring v, r, dj o(, and d, enough quantities in equation (1) are known todetermine dpp and dq3 which locate the velocity of point 3 in 13 23 the velocity planes Before proceeding with a detailed discussion of the various techniques employed in this solution, a few recommendations will be made. 1. Use a graph paper with lines spaced 20 to the inch if a scale for W* = 1 of 10 inches is used. 2. Use P drafting machine with the scales marked in tenths of an inch and finer divisions. This makes angular measurements and fractional measurements simpler. 3. Use tables throughout to keep the numerous measurements and calculations in ordor and prevent needless repetition of measurements. Figure 2 illustrates the scheme of such a table. 4. Use the "adiabatic ellipse" throughout to measure sin2(K in the velocity plane. This eliminates a tremendous amount of calculation which would occur if these quantities were to be determined individually. The ellipse should be drawn on transparent paper. The adiabatic ellipse is discussed in simple detail in Reference 4, pages 55-56. The equation for the ellipse arises from the energy equation in the form:

ENGINEERING RESE.ARC'H IN'TI I t- '.r. Page "UNIVERSITY OF MICHIGAN 5 U2 A_2 v2 w2!!. ~~~~~+ o2 _!(2) 2 z- 1 2 This can be reduced to the equation of the ellipse: 2 V2..... =1 (3) w~ max c*2 3Because the velocities in the velocity plane are represented in terms of c*, the velocity of sound when M = 1, plot the semi-major axis as = 2.45, and the semi-minor as c- = 1 to be consistent. This ellipse can be utilized in the velocity plane to determine t anrd sin2 OK for any particular velocity vector as shown in the sketch below: Vel c t ane When the ellipse is adjusted to a Velocity vector as shown above, the local Mach angle ~ and the corresoonding sin2~CO can be read on the scale as the angle between the velocity vector and the major axes of the ellipse. Sin' 0( apoears in the difference equations (1).

cen~~X -' UNIV'ERSITY OF NICHI(;AN\ I?1 tuhe solution1 use ib,CT of the:hock polar in ithe 'V>>ocity plane. This nolar deenrts iup.on t he free stream M!ach nmSber M4, an is constructed in the velocity plane uslnig the -eonetric l construction of a stroohoid described in Reference X' P. 107-110, expresslng all velocities in terms of c*. This diagrai will appear in the velocity plane as shown.by the sketch belo, v e:!~o~ciy:ane mr C*C*. I To make use of this method, the velocities 'of at least two points behind the shock wave must be knonm. Taylor and Maccoll in Reference 7 have presented the solution for an axially symmetric cone. Use is made of this valuable solution to give the velocities at certain points behind the shock by approximating the tip of the missile to be investigated with a cone. This can be done without sacrificing an undesirable amount of accuracy in the region of the tip. Reference 5 presents this data in excellent detail for a large range of Mach numbers. After using the information from the above sources to determine two or tore initial velocities behind the shock, the solution employing the fundamental working equations (1) proceeds. During the solution, three distinct cases occur which differ in detail in their handling. These are: 1. The case where the velocity to be determined lies in the free stream behind the shook wave. 2. The case where the velocity to be determined lies on the boundary or surface. 3. The ease where the velocity to be determined lies on the shock wave.

— f 4 o t '~~~~~~~.1U''lX 0' lll(\FHIG. AN- 1 Eac. of these different cases merits a discussion on the te:hnique of handling them when they arie. Cobisider Case 1. Assume the velocities are knovm at points 1 and 2 and the velocity at point 3 is to be determined as is shown in Figure 1. Here use is made of the table in Figure 2. Fill in columns (I) through (6) for points 1 and 2 by measuring the appropriate quantities in the physical and velocity plane as showrn in Figures 1 and 2. Next, rouglhly estimate the position of the end Doint of velocity vector 3 in the velocity plane and thus establish the approxinate values of columns (1 ), (3) (4 ), and (5) for point 3. Now7 take the mean value of ( $t- I and ( 6- ' ) for points 1 rand 3 and points 2 and 3 and put in colunns (9) and (10) for row 17 and:, i.e. (+~ = tD 2 () — i 77 2 Construct the first aprDroxlmation to the CMach quadran le by, drar.r? ng lirne ror 2 at anrle ( -) and I7 ine from 1 at angle (g-Z). The intersection of these two lines locates the first approximation to noint 3 in the ohvsical plane and ennbles the value of r3 fo point 3 in column (6) to be measured. as well as the values d and d. That is, d _ is the distance between polnts 2 and 3 and d7_ is the distance between points 1 and 3. These valuee are put in proper columns for 17 and 7 as shown in Figure 2. Now all columns (1) through (6) are filled in for rows s1 and Z'j by tar-inr the mean of the values for points 1 and 3 and 2 and n. Then all of the quantities are cetermined for equations (1). T'a t is, for - and 2 columns (1) and (12): V sin d Now, dp__ is layed off slong a line from point 1 in the velocity plane in direction ( ^ - )__. A Ierpendicu la'r is 13

~rom th e';:'' '......'"'....i. ' I. 1 iq is lyed 2? ilorm point 2 in the direction ( + K) and a perpendiculsr:..ected from the end point. The intersection of the perpendicuL.Trs locates velocity point 3. If the velocity point 3 corresponds closely with the first estimated location, no further calculation is needed. However, this occurs rarely and further calculations are necessary. Using the values of this velocity point 3 in Columns (1) through (5), the process described above is repeated. If the results of this recalculation yields a point 3 closely approximating the location of the previously calculated point 3, then no further calculations are needed as point 3 is determined within the accuracy of this method. All points in the flow field behind the shock wave not on either the boundary or shock wave are determined in this manner. Consider case 2 where the point to be determined lies on tlie boundary.,. Referring to Figure 3, assume that the velocity has been determined at point 5 and- that the velocity at some noint 6 on the surface is to be determined. Columns (1) through (6) ancd (9) and (10) in Figure 2 for known point 5 can be filled in. Draw a line from point 5 at angle ( - )5 to determine a first apnroximation to the location of point 6 on the surface, (61). Measure the angle from the axis of Symmetry made by the tangent to the surface at point 6. Because the flow is tangent to the surface this angle is a first approximation to the desired angle for the velocity plane. Next take the mean of 6 this ~ with $ and find ( 5 - ( ) and lay off another line 6 5 in the physical plane from 5 at this angle to give a closer approximation to point- 6 on the surface (i.e. 6 in Figure 3). 2 Again measure the angle of the tangent and record it in the table as $6 for point 6. Lay this off in the velocity plane and reke an estimate of the location of velocity point 6 somewhere along it. Measure the values for columns (1), (3) (4), and (5) from the velocity plane and the value for column t6) in the physical plane and record for point 6. Take the means of each of these values and record in row 3I columns (I) though (6). Measure df in the physical plane and record in column (7). Use columns (2) and (3) row 3 to determine columns (9) and (10) row vie All quantities are now known to determine dq using equation (lb) i.e. ciq _ sin20 d3 (lb) 3 r 1

ENGINEERING RESEARCH INSTITUTE Page I B-,' I IllIlI I UNIVERSITY OF MICHIGAN This quantity is layed off in the velocity plane from known velocity point 5 in the direction of (, + 5- ). A perpendicular is erected fromethe end point of dq_ as shown in Figure 3 and its intersection with the 6 line locates point 6. 6 If 'this point 6 coincides with point 6 estimated, no further calculations are necessary. If not, use the new values for point 6 columns (1) through (5) to construct a new lihe -) in the physical plasne to locate a new noint 6 (i.e. point 6 in 3 Figure 3). Measure a new,6 here and repeat the process determining dq exactly as done for the first approximation making 56 use of the values for point 6 resulting for the first approximation. Generally this second approximation will yield a location of point 6 in the velocity plane very close to the previously located velocity point 6 indicating that no further calculations are necessary. All points on the missile surface are found using this method. It is of interest to note here that 5 will not necessarily always be of larger magnitude than 6 as is shown in Figure 3. The last case to be considered in detail is the case where the velocity to be found is on the shock wave. This is the case that makes use of the shock polar in the velocity plane. Assume that the velocities for ooints 8 and 9 in Figure 4 are known. Record the values of columns (1) through (6) in Figure 2 for points 8 and 9. We know that any point that aopears on the sho'k wave in the physical plane must be represented on the shock polar in the velocity plane. Since the shock wave bends in the clockwise sense, the angle 6 as measured in the physical plane is decreasing. This determines that the point 10 must be on the shock polar below point 8 as 9 lO will be less than 9 8 as shown in Figure 4. The first step in case three is to estimate the position of point 10 on the shock polar. Measure and record columns (1) through (5) in Figure 2. Measure e0 and l10 in, the velocity plane as shown in Flture 4 and take the mean to get 8 as shown in Figure 4. This is the first approxinmtion to the shock line segment i. Fill in the columns (2) through (5) for row 91I by averaging the appropriate values for points 9 and 10 columns (2) through (5). Obtain columns (10) and (11) row 4l by making use of columns (2) and (3). Drawr a line from known point 9 in the physical plane at the angle ( ~ ) _. Where 910 this line intersects the shock line 810 locates the first approxi

ENGINEERINGt REISELARCH ISLrITI rr; [ Pagtv 10 E,,- -, "...........i..UNIVERSITY OF MICHIGAN 10 mation to the nosition of point 10 in the phy sical plane. Mea sure r and record it in column (6) row 10 and use it to determine 10 r_9. IMeasure d'( anrd record it in row 910. Now calculate 910 910 dp i.e. 910 V dp = 920 sif2 0., d7 (1e) 910 Lay this value off in the velocity plane along a line fron knowin point 9 in the direction of ( - ). Erect a perpen910 diculcar at the end of dp_. Where the perpendicular intersects 910 the shock polar as shown in Figure 4 locates the new position of velocity point 10. If this point closely approximates the original estimated position of point 10 nc further calculation is needed. However, this rarely occurs and the procedure is repeated making use of the new velocity point 10 throughout instead of the velocity point 10 originally estinated. One more calculation is usually sufficient to converge the velocity noint to the solution within the accuracy of the method. Thti completes the discussion on the three varlations of solution encountered in dealing rwith configurations of the axially symrmetric tyoe. This ster by sten solution yields results in the physicrl plane similar to Figure 5. The corresponding velocities associated with each Mach line or characteristic line intersection is recorder In the tables used in the method such as. FigTure 2 column (1).J The velocities of noints on the surface can be used to deter-ine the pressure coefficient at each noint. All. points will rrovide data for a il.ot of the pressure coefficient in a meridian oLne. Such a nlot is shown in Figure 6 for an -c~ivrl nose at a Miach number slirfhtly less than 3. This solution is time-consuming and requires careful attention to details. The results obtained, however, are very nccurate and closely apnroximate exper'ir,ental results.

[NGEE;IN-F:RI9tN( J'EARVIT0'If IfNSTiR"IiTl ~r J:!,...'.NIVEISIT~ OF MJCHIGAN Conclul on This report p1resents the essential details of the mechainics of sapplication of the three-dimensional method of characteristics in determining the velocity distribution over an axially symmetric body immersed in a supersonic strearm. The report treats in detail the technique employed in determining; an unknown velocity behind a shock wave for three distinctly different ccases: 1. A noint in the free stream behind the shock wave, 2. A point located on the surface of the body. 3. A point located on the shock wave. This method is time-consuming and laborious in application, but may well.:erit use in certain specific cases. The purpose of this paner is to present an explanation of the procedure of the method including certain refinemrents which help to keep the labor involved to a miniinumt It is anticipated that this method w.ll be employed to some extent to determine the pressure distribution over missile configurations prior to and in conjunction with wind tunnel tests.

EN GINEE:RING RESE.ARCH ixsT''I'rt'rJE | Pa-i E14> -3 i UNIVERSITY OF MICHIGAN Fe ferences: L. EMB-1 "The Practical Application Of the ThreeDimensional Method of Characteristics for Axially Symmetric Flow to the Conical Nozzle." W. H. Dorrance and H. C. Tinney. 2. JPL-R-4-34 "Characteristic Methods for Quasilinear Hyperbolic Differential Equations and an Application to Axially Symmetric Supersonic Flow Past an Ogive." H. K. Forster. 3. APL-CM-393-N "A Numerical-Graphical Method of Cha3racteristics in Axially Symmetric Isentropic Flow Problems." L. L. Cronvich. 4. "Theoretical Gas Dynamics.W R. $auer. Edwards Bros. MIT-TR-1 "3upersonic Flow of Air Around Cones." 6. "Aerodynamics of a Compressible Fluid." Liepmann and Puckett. Wiley Bros. 7. "The Air Pressure on a Cone Moving at High Speeds." G. I. Taylor and J. W. Maccoll. The Royal Society, 1932.

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