ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR GROWTH OF TEE TURBULENT REGION AT THE LEADING EDGE OF RECTANGUIAR OBSTACLES IN SHOCK WAVE DIFFRACTION REPORT 51 - 2 ROBERT N. HOLLYER, JR. RUSSELL E. DUFF Supervised by OTTO LAPORTE January 18, 1951

THIS REPORT PUBLISHED FOR,~ C.& adocped 49-~a Sfteict % 6^4T~wee

TABLE OF SYMBOIS = Po _- P1 S a = local velocity of sound. t = time measured in microseconds, where t = 0 when primary shock shock wave is at the leading edge of the model. xD = distance from leading edge of model to leading edge of boundary layer. M = Mach number behind primary shock. *P = pressure ahead of primary shock P1 = pressure behind primary shock in the undistrubed flow. S P 1 *T = time for primary shock to travel 500 millimeters. -*T = measured delay time of photograph. T' = delay time to place the primary shock at the leading edge of the model. U = flow velocity behind primary shock wave. *X = length of turbulent or vortex region. *Y = height of turbulent or vortex region. *Z = distance from leading edge of the model to the center of the vortex. * Measured quantities.

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN GROWTH OF TBE TRBULETf REGION AT TEE IEAD2NG EDGE OF ECTANGILIAR OBSTACLES IN SEOCK WAVE DIFFRACTION I. INT ODUCTION This report presents the results of an investigation of the growth of the vortex or turbulent region at the leading edge of a rectangular block following the passage of a shock wave over the block. The primary purpose of the study is to determine the dependence of the growth upon the various parameters of the problem, namely, model height, shock strength, and flow velocity. The length of the block is assumed to be infinite. A representative sequence of schlieren photographs of the phenomenon under investigation is included as Fig. 10. II. EXPEREIMENTAL PROCEDURE The data were obtained in the University of Michigan 2-inch by 7inch rectangular shock tube. The models used were 8-inch by 2-inch rectangular steel blocks of various heights placed on the floor of the tube (see Fig. 1). Both schlieren and shadow photography were used. Table I contains the data for the three values of shock velocity used.

ENGINEERING RESEARCH INSTITUTE Pa 2 UNIVERSITY OF MICHIGAN liI Model h A 4" B 2" C 1/2" 17"8 h Fig; 1 TABLE I Gas V Po/PI P1/Po U M a Tcorrect Air 0.4062 0.6916 1.446 0.0936 0.257 0.364 1231 N2 0.4062 0.7005 1.427 0.0908 0.248 0.366 1231 Air 0.4482 0.5557 1.800 0.1524 0.403 0.378 1115 N2 0.4482 0.5628 1.777 0.1494 0.395 0.378 1115 Air 0.5682 0.3332 3.001 0.2992 0.774 0.386 880 N2 0.5682 0.3388 2.951 0.2963 0.717 0.413 880 (All velocities are in mm per Asec.)

ENGINEERING RESEARCH INSTITUTE Page 3 UNIVERSITY OF MICHIGAN Data for both air and nitrogen are tabulated. Nitrogen was employed for those cases in which a greater density of gas seemed desirable. It was necessary to use nitrogen instead of air to accomplish this because hydrogen was used in the compression chamber of the shock tube for these higher density shots. In some cases, a variation of density was used to alter the kinematic viscosity to determine the dependence of the phenomenon upon this parameter. In other cases, the density was increased to enhance the optical effects of the disturbance. This made it possible to extend the data to larger values of t than would have been possible with the use of air alone. The parameters chosen as characteristic of the region under study are the length X and the height Y of the turbulent region, as illustrated in the following diagram. - X'->l

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Page 4 For the lower values of t, the boundary of the region is fairly well defined, but as. t increases, this boundary becomes less distinct. In all cases the measurement was made to last observable disturbance of the smooth flow. The probable error of the measurements for the largest values of t should not be considered to be less than + 3 mm. The errors are most probably in the negative direction, i.e., a low value of the variable is most probably reported. In the case of the two lower values (S = 1.44, 1.8) the region under consideration begins as a fairly well-defined vortex which eventually disintegrates. While this vortex is visible, the motion of its center can be tracedo This has been done where possible, and the distance from the leading edge of the block to the center of the vortex is included in the data as Z. Shadowgraph pictures of the two lower values of shock strengths (none were taken at S = 3) show the baodary layer behind the disturbed region. The distance between the leading edge of the model and the beginning of this boundary layer is shown in the data as D. Because this point can be located with accuracy only for lower values of t and because of the umcertainty concerning the interpretation of the data obtained, it was felt that it would be unwise to consider it seriously in this report. Except where otherwise indicated, the value of t associated with each photograph is obtained from the electronic control equipment, which simultaneously records on microsecond scalers the time, T, for the shock to travel 500 millimeters and the delay time, T*, of the photograph. From pictures containing the primary shock, the delay time, T', needed to place the shock wave at the leading edge of the model for a given shock velocity can be computed. For later photographs at this same shock velocity, the time, t, can then be obtained from the equation: t = T* - T' Since it is not always possible to reproduce shock velocities exactly it is necessary to correct the value of To, and thus t, to compensate for the

ENGINEERING RESEARCH INSTITUTE Page 5 UNIVERSITY OF MICHIGAN variations in the measured shock velocity. This is done by determining the value of T* which would have been needed to place the primary shock in its actual position at the time the photograph was taken if the correct value of the shock velocity had been obtained. This corrected value of T* is simply T*(corrected) = T*(measured) T{lrected T (measured) Table II shows the values of T (corrected) used in this report and the maximum deviations from this value in addition to the percentage error introduced in the flow parameters for this maximum deviation. TABLE II P1/Po T Maximum % Error in % Error in % Error in Corrected Deviation V U S 1.44 12351 sec 4 hsec 0.3 2.0 0.8 1.8 1115 6 0.5 2.4 1.3 3.0 880 5 o.6 1.5 1.6 A consideration of the errors inherent in the integer time-measuring system used and the experimental techniques employed, gives a maximum error in t of no more than + 5 microseconds. Since the errors due to all causes in the value of t are small by comparison to the errors in the measurements of the dimensions of the phenomenon, no corrections for shock velocity variations have been applied to the measured values of X, Y, Z, and D, These procedures have been used in handling data presented in the past and, unless otherwise indicated, will be used in the future for reports prepared for the Armed Forces Special Weapons Project.

ENGINEERING RESEARCH INSTITUTE Page 6 I UNIVERSITY OF MICHIGAN IIT. DISCUTSSION OF THE DATA In the early stages of this investigation it was noted that no significant difference existed between the data for the 2-inch block and the 4-inch block. Therefore, the major portion of the data has been collected for the 1/2-inch and 4-inch blocks. In addition, the fact that this phenomenon involves the viscosity of the gas, made it seem wise to vary the gas densities, and therefore the Reynolds number, in order to check any dependence of the data upon this parameter. The data obtained are shown in Tablets IIIVII. Following standard procedure, the measured quantities, X and Y, were plotted against various parameters of the problem in an attempt to determine empirically the dependence upon such parameters. By far the most successful of these attempts was to plot X against the product of t and the flow velocity, U, behind the primary shock. These curves for the 4-inch, 2-inch, and. 1/2-inch block are included as Figs. 2, 5, and 4. A composite curve of all values of X for the three different blocks is shown in Fig. 5. The curve of Y versus Ut for the 4-inch block is shown in Fig. 6. Figs. 7 and 8 are curves of Y versus Ut for the 2-inch and 1/2-inch block, respectively, while the composite curve of Y versus Ut for all models is shown in Fig. 9. The values of U used for these curves are those given in Table I. No corrections have been applied for variations of the actual flow velocity from the ideal values listed in that table. The following features of these curves are of particular importance. (a) The curves of Figs. 2, 35, 4, 6, 7, and 8 indicate that for a given block the size of the turbulent region is a function of the product of the flow velocity, U, and the time, t, and not of U or t separately. This

ENGINEERING RESEARCH INSTITUTE Page UNIVERSITY OF MICHIGAN 7 means that we have obtained the empirical result that: X = x(Ut), Y = Y(Ut). (b) The curves of X versus Ut for a given model are straight lines within the experimental error of the points. This is a rather surprising result, first, because it indicates that the change from vortex flow to apparently uniform turbulence does not effect the growth of the disturbed region, and secondly, it indicates an extremely weak dependence of the growth of X upon the presence of the upper wall of the shock tube. The approximate points at which the conversion from vortex flow to turbulence takes place can be obtained by noting those values of Ut in the data to which no value of the parameter, Z, has been assigned. (c) The slope of the curve X versus Ut is almost identical for all blocks (see Fig. 5). The plotted points indicate that the curve for the 1/2inch block is slightly lower than those for the 2- and 4-inch block. It is difficult to make any quantitative statement, however, because of several experimental factors. First, since the turbulence in the region under discussion is much less pronounced in the case of the 1/2-inch block, the measurement errors here would tend to lower the curve below those of the higher blocks. Secondly, the effect of the upper wall of the shock tube is considerably less for this model. The first reflected shock arrives at the top of the model at a value of Ut about twice the value for the corresponding flow using the 4inch block. Furthermore, since this shock is approximately cylindrical, it will be weaker than that for the 4-inch case. (d) The curve Y versus Ut (Fig. 6) is a straight line for the 1/2inch block, while it appears to be approaching some fixed value for the two higher models. We cannot quantitatively explain this result, but again the

ENGINEERING RESEARCH INSTITUTE P UNIVERSITY OF MICHIGAN age 8 effect of the top wall of the tube seems to be the most promising qualitative explanation. The reflection from the top of the tube interferes with the turbulent region at about 40, 60, and 100 mm Ut for shock strengths S = 1.44, 1.8, and 3.0, respectively, for the 4-inch block, and at about twice these values for the 1/2-inch block. It seems logical to expect the effect of the top wall to produce greater changes in vertical velocity components for the large blocks than for the small block. (e) Contrary to the situation we found for the X versus Ut curves that the slopes are not equal for all block in the case of Y versus Ut. This result is not unexpected, since the effect of the variable height of the models is naturally stronger for vertical measurement. There are not sufficient data, nor is the experimental accuracy good enough, to arrive at any reliable dependence of the slope on block height. The slopes for the 2- and 4-inch block seem to be almost equal, while that for the 1/2-inch block is approximately one half of these. In addition, the slope for S = 3.0 appears to be a bit lower than that for the other values of S. This latter observation is complicated by the fact that the region above the turbulence is clear supersonic, since a cluster of shock waves can be observed in all photographs (see Fig. 11). The presence of these extra disturbances makes the measurement of Y extremely difficult. Because of this and because of the small number of points plotted, nothing reliable can be deduced from this observation. Conclusions Within the range of the variables studied in this investigation, the following conclusions seem to be valid: (a) The curve of X versus Ut is a straight line whose slope is the same for all block heights, shock strengths, and gas densities.

ENGINEERING RESEARCH INSTITUTE Pae UNIVERSITY OF MICHIGAN (b) The curve of Y versus Ut is a straight line whose slope is independent of shock strength and gas density but is some undetermined functio of block height. More data must be obtained if this dependence is to be empirically determined. (c) There is no observable effect upon the growth as the flow in the region under question changes from a vortex flow to a turbulent flow.

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16 t = 205 microseconds t = 389 t = 609 t = 716 t = 1034 t = 1428 Fig. 10 S = 1.8, 4-inch block

17 Fig. 11 S = 3.0, t - 230, 4-inch block Fig. 12 Shadowgraph, S = 1.44, t - 388, 4-inch block

18 TABLE III S = 1.44 Schlieren Model t PO Gas Film X Y Z Ut (sec) (mm of Hg) No. (mm) (mm) (mm) (mm) A** 77* 475 N2 160 4 2 2 7 A 245 475 N2 161 11 6 6 22 A 444 475 N2 162 20 11 12 40 A 647 475 N2 163 28 15 18 59 A 860 475 N2 164 35 18 20 78 A 1050 475 N2 165 47 23 -- 95 A 1283 475 N2 166 58 23 -- 116 C** 71* 475 N2 191 3 2 2 6 C 278 475 N2 192 11 5 -- 25 C 476 475 N2 193 17 7 -- 43 C 680 475 N2 194 23 9 -- 62 C 902 760 Air 243 34 13 -- 84 C 1011 760 Air 240 33 15 -- 95 C 1063 760 Air 244 36 13 - 99 C 1359 760 Air 241 48 15 - 127 C 1694 760 Air 242 69 20 - 159 C 1482 760 Air 245 57 19 -- 139 C 1694 760 Air 246 67 - - 159 * For these photographs t was computed from measurements of the position of the primary shock. ** Model A = 4-inch block Model B = 2-inch block Model C = 1/2-inch block

20 TABLE V S = 1.8 Schlieren Model t P Gas Film X Y Z Ut (sec) (mm of Hg) No. (mm) (mm) (mm) (mm) A 171 760 2 182 13 6 8 26 A 177 760 N2 181 13 7 8 27 A 179 200 N2 178 13 6 7 27 A 205 200 Air 135 15 7 10 31 A 389 200 Air 132 25 14 17 59 A 593 760 N2 183 38 19 22 89 A 605 350 N2 177 40 20 22 90 A 609 200 Air 136 37 17 25 93 A 645 760 N2 184 42 23 27 96 A 702 350 N2 176 47 23 -- 105 A 716 200 Air 139 43 24 30 109 A 717 760 N2 185 49 25 -- 107 A 764 200 N2 179 58 24 -- 114 A 820 200 Air 137 57 24 -- 125 A 1028 760 N2 186 71 27 - 154 A 1034 200 Air 138 69 25 -- 158 A 1428 350 N2 140 94 26 - 213 B 401 200 Air 106 28 14 17 61 B 592 200 Air 109 41 20 27 90 B 611 200 Air 105 42 19 29 93 B 794 200 Air 108 53 22 -- 121 C 191* 200 Air 126 12 6 7 29 C 389 200 Air 125 24 9 14 59 C 599 200 Air 124 37 12 24 91 C 805 200 Air 123 49 15 -- 123 C 1000 200 Air 127 55 19 -- 152

21 TABLE VI S = 1.8 Shadowgraph Model t P0 Gas Plate X Y D Ut (sec) (mm of Hg) No. (mm) (mm) (mm) (mm) A 27* 200 Air 1013 2.5 -- 4 A 136* 200 Air 1015 10.5 4.5 20 A 291 200 Air 1017 20 10 16 44 A 421 200 Air 1014 30 15 21 64 A 524 -200 Air 1018A 36 17 26 80 A 693 200 Air 1018B - - 35 106 B 107* 200 Air 1036A 7.5 3.5 7 16 B 114* 200 Air 1039A 8.5 3 6 17 B 272 200 Air 1036B 19 9 14 41 B 438 200 Air 1037A 30 15 22 67 B 605 200 Air 1037B 39 -- 30 92 B 625 200 Air 1039B -- -- 31 95 C 65* 200 Air 1043A 5 2.5 -- 10 C 163* 200 Air 1043B 12 5 11 25 C 263 200 Air 1044A 20 7 18 40 C 363 200 Air 1044B -- 10 23 55 C 561 200 Air 1045A - 32 85 C 762 200 Air 1045B -45 116

22 TABLE VII S = 3.0 Schlieren Model t PO Gas Film X Y Z Ut (sec) (mm of Hg) No. (mm) (mm) (mm) (mm) A 40* 165 N2 169 6 -- 2 12 A 124* 106 N2 175 17 5 -- 37 A 128 165 N 174 17 5 -- 38 A 230 165 N2 170 51 10 -- 68 A 425 165 N2 171 56 19 -- 126 A 626 165 N2 172 82 24 185 A 824 165 N2 173 114 31 -- 244 C 18* 165 N 197 3 -- 1.5 5 C 117* 165 N2 198 17 5 -- 35 C 315 165 N 199 37 10 -- 93 C 522 165 N2 200 64 15 -- 155 C 710 165 N2 201 107 20 -- 210 C 921 165 N2 202 >109 20 -- 273