889:
443:
25:
884:{\displaystyle {\begin{aligned}{\frac {T_{2}}{T_{1}}}&=\left({\frac {1+{\frac {\gamma -1}{2}}M_{1}^{2}}{1+{\frac {\gamma -1}{2}}M_{2}^{2}}}\right)\\{\frac {p_{2}}{p_{1}}}&=\left({\frac {1+{\frac {\gamma -1}{2}}M_{1}^{2}}{1+{\frac {\gamma -1}{2}}M_{2}^{2}}}\right)^{\frac {\gamma }{\gamma -1}}\\{\frac {\rho _{2}}{\rho _{1}}}&=\left({\frac {1+{\frac {\gamma -1}{2}}M_{1}^{2}}{1+{\frac {\gamma -1}{2}}M_{2}^{2}}}\right)^{\frac {1}{\gamma -1}}.\end{aligned}}}
3198:). By the time the object reaches point B, the pressure disturbances from point A have travelled a distance c·t and are now at circumference of the circle (with centre at point A). There are infinite such circles with their centre on the line AB, each representing the location of the disturbances due to the motion of the object. The lines propagating outwards from point B and tangent to all these circles are known as Mach lines.
1331:
1817:
2709:
90:
2168:
3011:
1083:
2310:
2100:
If the flow turns enough so that it becomes parallel to the wall, we do not need to worry about pressure boundary condition. However, as the flow turns, its static pressure decreases (as described earlier). If there is not enough pressure to start with, the flow won't be able to complete the turn and
1757:
1326:{\displaystyle {\begin{aligned}\nu (M)&=\int {\frac {\sqrt {M^{2}-1}}{1+{\frac {\gamma -1}{2}}M^{2}}}{\frac {\,dM}{M}}\\&={\sqrt {\frac {\gamma +1}{\gamma -1}}}\arctan {\sqrt {{\frac {\gamma -1}{\gamma +1}}\left(M^{2}-1\right)}}-\arctan {\sqrt {M^{2}-1}}.\\\end{aligned}}}
2906:. Since this is not possible, it means that it is impossible to turn a flow through a single shock wave. The argument may be further extended to show that such an expansion process can occur only if we consider a turn through infinite number of expansion waves in the limit
3040:) as it moves from point A to B (distance u·t), the disturbances originating from point A travel a distance c·t. The corresponding angle is known as a Mach angle and the lines enclosing the disturbed region are known as Mach lines (in 2-D case) or Mach cone (in 3-D).
292:
with respect to final flow direction. Since the flow turns in small angles and the changes across each expansion wave are small, the whole process is isentropic. This simplifies the calculations of the flow properties significantly. Since the flow is isentropic, the
2143:
of the velocity (normal component being zero). Beyond the slipstream the flow is stagnant (which automatically satisfies the velocity boundary condition at the wall). In case of real flow, a shear layer is observed instead of a slipstream, because of the additional
2008:
3151:
2704:{\displaystyle {\begin{aligned}{\frac {\Delta s}{R}}&=\ln \left\\&\approx {\frac {\gamma +1}{12\gamma ^{2}}}\left({\frac {p_{2}-p_{1}}{p_{1}}}\right)^{3}\\&\approx {\frac {\gamma +1}{12\gamma ^{2}}}\left^{3}\end{aligned}}}
1577:
2080:
448:
290:
229:
3167:
To understand the concept better, consider the case sketched in the figure. We know that when an object moves in a flow, it causes pressure disturbances (which travel at the speed of sound, also known as
93:
When a supersonic flow encounters a convex corner, it forms an expansion fan, which consists of an infinite number of expansion waves centred at the corner. The figure shows one such ideal expansion fan.
2315:
2175:
Impossibility of expanding a flow through a single "shock" wave: Consider the scenario shown in the adjacent figure. As a supersonic flow turns, the normal component of the velocity increases (
1088:
2085:
One can also look at it as follows. A flow has to turn so that it can satisfy the boundary conditions. In an ideal flow, there are two kinds of boundary condition that the flow has to satisfy,
1034:
116:
Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single "shock" wave because this would violate the
2933:
2302:
1845:
1918:
3076:). They are a pair of bounding lines which separate the region of disturbed flow from the undisturbed part of the flow. These lines occur in pairs and are oriented at an angle
1926:
2875:
2213:
2904:
1510:
1473:
1436:
1370:
2253:
3257:
1569:
1067:
384:
54:
3228:
3074:
2752:
1869:
966:
431:
3459:
3196:
3038:
1890:
2772:
2126:
1806:
1537:
1399:
946:
919:
411:
357:
326:
3082:
2832:
2812:
2792:
2732:
1779:
2096:
Pressure boundary condition, which states that there cannot be a discontinuity in the static pressure inside the flow (since there are no shocks in the flow).
1752:{\displaystyle v_{r}={\sqrt {2(h_{0}-h)-c^{2}}},\quad v_{\phi }=c,\quad {\text{where}}\quad \phi =-\int {\frac {d(\rho c)}{\rho {\sqrt {2(h_{0}-h)-c^{2}}}}},}
2101:
will not be parallel to the wall. This shows up as the maximum angle through which a flow can turn. The lower the Mach number is to start with (i.e. small
113:, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point.
2132:
3269:
Landau, L. D., & Lifshitz, E. M. (2013). Fluid
Mechanics: Landau and Lifshitz: Course of Theoretical Physics, Volume 6 (Vol. 6). Elsevier.
3433:
2019:
3449:
2834:
is the component of flow velocity normal to the "shock". The suffix "1" and "2" refer to the initial and final conditions respectively.
234:
173:
3410:
3360:
3337:
76:
3379:
3484:
3387:
117:
974:
37:
2171:
An expansion process through a single "shock" is impossible, because it will violate the second law of thermodynamics.
1070:
47:
41:
33:
2975:
58:
3489:
2909:
2139:(shown as the dashed line in the figure). Across this line there is a jump in the temperature, density and
3230:). In case of subsonic flows, the disturbances will travel faster than the source and the argument of the
2258:
2013:
This places a limit on how much a supersonic flow can turn through, with the maximum turn angle given by,
1823:
329:
2003:{\displaystyle \nu _{\text{max}}={\frac {\pi }{2}}\left({\sqrt {\frac {\gamma +1}{\gamma -1}}}-1\right).}
1895:
1539:) and the other flow properties. The velocity field in the expansion fan, expressed in polar coordinates
3479:
3306:
2140:
2840:
2178:
2880:
1478:
1441:
1404:
434:
298:
3425:
1339:
3352:
3329:
2936:
2218:
3233:
3172:). The figure shows an object moving from point A to B along the line AB at supersonic speeds (
1542:
1042:
362:
3429:
3406:
3383:
3356:
3333:
2955:
2952:Über zweidimensionale Bewegungsvorgänge in einem Gas, das mit Überschallgeschwindigkeit strömt
2145:
3207:
3053:
2737:
1854:
951:
416:
386:) remain constant. The final static properties are a function of the final flow Mach number (
3175:
3146:{\displaystyle \mu =\arcsin \left({\frac {c}{u}}\right)=\arcsin \left({\frac {1}{M}}\right)}
3017:
2992:
2090:
1874:
294:
144:
2973:
Prandtl, L. (1907). "Neue
Untersuchungen über die strömende Bewegung der Gase und Dämpfe".
2757:
2104:
1784:
1515:
1377:
924:
897:
389:
335:
304:
147:
properties (e.g. the total pressure and total temperature) remain constant across the fan.
128:
3372:
2817:
2797:
2777:
2717:
1764:
155:
1816:
3473:
3291:
151:
2089:
Velocity boundary condition, which dictates that the component of the flow velocity
105:, is a centered expansion process that occurs when a supersonic flow turns around a
3281:
2996:
3463:
3402:
3301:
1074:
132:
124:
102:
3311:
3296:
2167:
140:
106:
3286:
3169:
2959:
167:
123:
Across the expansion fan, the flow accelerates (velocity increases) and the
110:
2093:
to the wall be zero. It is also known as no-penetration boundary condition.
89:
2954:(Doctoral dissertation) (in German). Georg-August Universität, Göttingen.
231:
with respect to the flow direction, and the last Mach line is at an angle
3010:
166:
The expansion fan consists of an infinite number of expansion waves or
136:
1077:= 1) must turn to reach a particular Mach number (M). Mathematically,
413:) and can be related to the initial flow conditions as follows, where
2075:{\displaystyle \theta _{\text{max}}=\nu _{\text{max}}-\nu (M_{1}).\,}
3422:
The
Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1
2135:
which separates the final flow direction and the wall is known as a
3204:
These concepts have a physical meaning only for supersonic flows (
3160:). In case of 3-D flow field, these lines form a surface known as
2128:), the greater the maximum angle through which the flow can turn.
1073:. This function determines the angle through which a sonic flow (
3453:
3050:
are a concept usually encountered in 2-D supersonic flows (i.e.
285:{\displaystyle \mu _{2}=\arcsin \left({\frac {1}{M_{2}}}\right)}
224:{\displaystyle \mu _{1}=\arcsin \left({\frac {1}{M_{1}}}\right)}
18:
3156:
with respect to the direction of motion (also known as the
154:
on his thesis dissertation in 1908, along with his advisor
158:, who had already discussed the problem a year before.
3236:
3210:
3178:
3085:
3056:
3020:
2912:
2883:
2843:
2820:
2800:
2780:
2760:
2740:
2720:
2313:
2261:
2221:
2181:
2107:
2022:
1929:
1898:
1877:
1857:
1826:
1787:
1767:
1580:
1545:
1518:
1481:
1444:
1407:
1380:
1342:
1086:
1045:
977:
954:
927:
900:
446:
419:
392:
365:
338:
307:
237:
176:
2215:), while the tangential component remains constant (
3371:
3251:
3222:
3190:
3145:
3068:
3032:
2927:
2898:
2869:
2826:
2806:
2786:
2766:
2746:
2726:
2703:
2296:
2247:
2207:
2120:
2074:
2002:
1912:
1884:
1863:
1839:
1800:
1773:
1751:
1563:
1531:
1504:
1467:
1430:
1393:
1364:
1325:
1061:
1028:
960:
940:
913:
883:
425:
405:
378:
351:
320:
284:
223:
109:corner. The fan consists of an infinite number of
3164:, with Mach angle as the half angle of the cone.
1029:{\displaystyle \theta =\nu (M_{2})-\nu (M_{1})\,}
97:A supersonic expansion fan, technically known as
2162:
2160:
46:but its sources remain unclear because it lacks
3349:Mathematical theory of compressible fluid flow
3370:Courant, Richard; Friedrichs, K. O. (1999) .
8:
2255:). The corresponding change is the entropy (
3014:For an object moving at supersonic speeds (
1847:) through which a supersonic flow can turn.
2935:. Accordingly, an expansion process is an
2754:is the ratio of specific heat capacities,
3235:
3209:
3177:
3129:
3102:
3084:
3055:
3019:
2911:
2882:
2861:
2848:
2842:
2819:
2799:
2779:
2759:
2739:
2719:
2691:
2673:
2663:
2657:
2638:
2627:
2622:
2612:
2605:
2590:
2569:
2553:
2541:
2530:
2517:
2510:
2496:
2475:
2440:
2436:
2424:
2414:
2408:
2384:
2372:
2362:
2356:
2318:
2314:
2312:
2288:
2275:
2260:
2239:
2226:
2220:
2199:
2186:
2180:
2112:
2106:
2071:
2059:
2040:
2027:
2021:
1958:
1943:
1934:
1928:
1909:
1903:
1897:
1881:
1876:
1856:
1831:
1825:
1792:
1786:
1766:
1735:
1713:
1701:
1678:
1660:
1644:
1628:
1606:
1594:
1585:
1579:
1544:
1523:
1517:
1501:
1492:
1480:
1464:
1455:
1443:
1427:
1418:
1406:
1385:
1379:
1361:
1341:
1302:
1296:
1268:
1233:
1231:
1197:
1174:
1171:
1162:
1140:
1120:
1113:
1087:
1085:
1058:
1044:
1025:
1016:
994:
976:
953:
932:
926:
921:) is related to the initial Mach number (
905:
899:
855:
842:
837:
815:
801:
796:
774:
765:
745:
735:
729:
706:
693:
688:
666:
652:
647:
625:
616:
596:
586:
580:
563:
558:
536:
522:
517:
495:
486:
467:
457:
451:
447:
445:
418:
397:
391:
370:
364:
343:
337:
312:
306:
270:
261:
242:
236:
209:
200:
181:
175:
77:Learn how and when to remove this message
3009:
2166:
1815:
88:
3460:Prandtl- Meyer expansion fan calculator
3397:Anderson, John D. Jr. (January 2001) .
3324:Liepmann, Hans W.; Roshko, A. (2001) .
2156:
1820:There is a limit on the maximum angle (
2989:Ludwig Prandtl Gesammelte Abhandlungen
1512:one can obtain the final Mach number (
2928:{\displaystyle \Delta s\rightarrow 0}
1808:is the stagnation specific enthalpy.
1374:Thus, given the initial Mach number (
170:. The first Mach line is at an angle
7:
3403:McGraw-Hill Science/Engineering/Math
2297:{\displaystyle \Delta s=s_{2}-s_{1}}
1840:{\displaystyle \theta _{\text{max}}}
3259:function will be greater than one.
1913:{\displaystyle \nu _{\text{max}}\,}
2913:
2884:
2321:
2262:
1858:
14:
3044:Mach lines (cone) and Mach angle:
1851:As Mach number varies from 1 to
894:The Mach number after the turn (
23:
3380:Springer Science+Business Media
3374:Supersonic flow and shock waves
2734:is the universal gas constant,
2304:) can be expressed as follows,
1665:
1659:
1639:
139:decrease. Since the process is
3246:
3243:
2919:
2870:{\displaystyle w_{2}>w_{1}}
2208:{\displaystyle w_{2}>w_{1}}
2065:
2052:
1725:
1706:
1693:
1684:
1618:
1599:
1558:
1546:
1498:
1485:
1461:
1448:
1438:and using the turn angle find
1424:
1411:
1352:
1346:
1100:
1094:
1055:
1049:
1022:
1009:
1000:
987:
1:
2899:{\displaystyle \Delta s<0}
1781:is the specific enthalpy and
1505:{\displaystyle \nu (M_{2})\,}
1468:{\displaystyle \nu (M_{2})\,}
1431:{\displaystyle \nu (M_{1})\,}
3399:Fundamentals of Aerodynamics
3347:Von Mises, Richard (2004) .
2997:10.1007/978-3-662-11836-8_78
2987:Riegels, F. W., ed. (1961).
1365:{\displaystyle \nu (1)=0.\,}
150:The theory was described by
118:second law of thermodynamics
16:Phenomenon in fluid dynamics
3420:Shapiro, Ascher H. (1953).
2248:{\displaystyle v_{2}=v_{1}}
99:Prandtl–Meyer expansion fan
3506:
2146:no-slip boundary condition
437:of the gas (1.4 for air):
359:) and stagnation density (
3252:{\displaystyle \arcsin()}
2976:Physikalische Zeitschrift
1564:{\displaystyle (r,\phi )}
1062:{\displaystyle \nu (M)\,}
379:{\displaystyle \rho _{0}}
2794:is the static pressure,
32:This article includes a
3326:Elements of Gasdynamics
3223:{\displaystyle u\geq c}
3069:{\displaystyle M\geq 1}
2877:, this would mean that
2774:is the static density,
2747:{\displaystyle \gamma }
1892:takes values from 0 to
1864:{\displaystyle \infty }
961:{\displaystyle \theta }
426:{\displaystyle \gamma }
61:more precise citations.
3485:Conservation equations
3253:
3224:
3192:
3191:{\displaystyle u>c}
3147:
3070:
3041:
3034:
3033:{\displaystyle u>c}
2929:
2900:
2871:
2828:
2808:
2788:
2768:
2748:
2728:
2705:
2298:
2249:
2209:
2172:
2122:
2076:
2004:
1914:
1886:
1885:{\displaystyle \nu \,}
1865:
1848:
1841:
1802:
1775:
1753:
1565:
1533:
1506:
1469:
1432:
1395:
1366:
1327:
1071:Prandtl–Meyer function
1063:
1030:
962:
948:) and the turn angle (
942:
915:
885:
427:
407:
380:
353:
330:stagnation temperature
322:
286:
225:
94:
3307:Schlieren photography
3302:Shadowgraph technique
3254:
3225:
3193:
3148:
3071:
3035:
3013:
2930:
2901:
2872:
2829:
2809:
2789:
2769:
2767:{\displaystyle \rho }
2749:
2729:
2706:
2299:
2250:
2210:
2170:
2123:
2121:{\displaystyle M_{1}}
2077:
2005:
1915:
1887:
1866:
1842:
1819:
1803:
1801:{\displaystyle h_{0}}
1776:
1754:
1566:
1534:
1532:{\displaystyle M_{2}}
1507:
1470:
1433:
1401:), one can calculate
1396:
1394:{\displaystyle M_{1}}
1367:
1328:
1064:
1031:
963:
943:
941:{\displaystyle M_{1}}
916:
914:{\displaystyle M_{2}}
886:
428:
408:
406:{\displaystyle M_{2}}
381:
354:
352:{\displaystyle T_{0}}
323:
321:{\displaystyle p_{0}}
287:
226:
127:increases, while the
92:
3234:
3208:
3176:
3083:
3054:
3018:
2991:. Berlin: Springer.
2910:
2881:
2841:
2818:
2814:is the entropy, and
2798:
2778:
2758:
2738:
2718:
2311:
2259:
2219:
2179:
2141:tangential component
2105:
2020:
1927:
1896:
1875:
1855:
1824:
1785:
1765:
1578:
1543:
1516:
1479:
1475:. From the value of
1442:
1405:
1378:
1340:
1084:
1043:
975:
952:
925:
898:
444:
417:
390:
363:
336:
305:
235:
174:
101:, a two-dimensional
2632:
847:
806:
698:
657:
568:
527:
435:heat capacity ratio
299:stagnation pressure
3353:Dover Publications
3330:Dover Publications
3249:
3220:
3188:
3143:
3066:
3042:
3030:
2950:Meyer, T. (1908).
2937:isentropic process
2925:
2896:
2867:
2824:
2804:
2784:
2764:
2744:
2724:
2701:
2699:
2618:
2294:
2245:
2205:
2173:
2118:
2072:
2000:
1910:
1882:
1861:
1849:
1837:
1812:Maximum turn angle
1798:
1771:
1749:
1561:
1529:
1502:
1465:
1428:
1391:
1362:
1323:
1321:
1059:
1026:
958:
938:
911:
881:
879:
833:
792:
684:
643:
554:
513:
423:
403:
376:
349:
318:
282:
221:
95:
34:list of references
3435:978-0-471-06691-0
3137:
3110:
2827:{\displaystyle w}
2807:{\displaystyle s}
2787:{\displaystyle p}
2727:{\displaystyle R}
2679:
2644:
2597:
2547:
2503:
2456:
2430:
2400:
2378:
2331:
2043:
2030:
1984:
1983:
1951:
1937:
1906:
1834:
1774:{\displaystyle h}
1744:
1741:
1663:
1634:
1314:
1285:
1257:
1223:
1222:
1185:
1169:
1156:
1132:
871:
849:
831:
790:
751:
722:
700:
682:
641:
602:
570:
552:
511:
473:
276:
215:
87:
86:
79:
3497:
3439:
3416:
3401:(3rd ed.).
3393:
3377:
3366:
3343:
3270:
3267:
3261:
3258:
3256:
3255:
3250:
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3142:
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3072:
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3000:
2984:
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2941:
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2932:
2931:
2926:
2905:
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2902:
2897:
2876:
2874:
2873:
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2866:
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2830:
2825:
2813:
2811:
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2805:
2793:
2791:
2790:
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2773:
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2770:
2765:
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2750:
2745:
2733:
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2725:
2710:
2708:
2707:
2702:
2700:
2696:
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2690:
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2685:
2681:
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2678:
2677:
2668:
2667:
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2645:
2643:
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2633:
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2617:
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2606:
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2595:
2594:
2581:
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2557:
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2548:
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2545:
2536:
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2534:
2522:
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2511:
2504:
2502:
2501:
2500:
2487:
2476:
2468:
2464:
2460:
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2455:
2441:
2435:
2431:
2429:
2428:
2419:
2418:
2409:
2402:
2401:
2399:
2385:
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2379:
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2376:
2367:
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2357:
2332:
2327:
2319:
2303:
2301:
2300:
2295:
2293:
2292:
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57:this article by
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162:Flow properties
129:static pressure
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38:related reading
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3490:Fluid dynamics
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156:Ludwig Prandtl
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42:external links
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3292:Oblique shock
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2985:Reprinted in
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2979:(in German).
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152:Theodor Meyer
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67:November 2020
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3480:Aerodynamics
3426:Ronald Press
3421:
3398:
3373:
3348:
3325:
3282:Gas dynamics
3265:
3201:
3200:
3166:
3161:
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2136:
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2099:
2084:
2012:
1850:
1761:
1373:
1335:
1038:
893:
165:
149:
122:
115:
98:
96:
73:
64:
53:Please help
45:
3464:Java applet
133:temperature
125:Mach number
103:simple wave
59:introducing
3474:Categories
3389:0387902325
3318:References
3312:Sonic boom
3297:Shock wave
3170:Mach waves
3158:Mach angle
3048:Mach lines
2137:slipstream
2133:streamline
295:stagnation
168:Mach lines
145:stagnation
141:isentropic
111:Mach waves
3287:Mach wave
3241:
3215:≥
3162:Mach cone
3123:
3096:
3087:μ
3061:≥
2920:→
2914:Δ
2885:Δ
2762:ρ
2742:γ
2655:−
2610:ρ
2588:γ
2573:γ
2567:≈
2524:−
2494:γ
2479:γ
2473:≈
2450:−
2447:γ
2443:γ
2438:−
2422:ρ
2412:ρ
2394:−
2391:γ
2344:
2322:Δ
2282:−
2263:Δ
2050:ν
2047:−
2038:ν
2025:θ
1987:−
1977:−
1974:γ
1963:γ
1946:π
1932:ν
1901:ν
1879:ν
1859:∞
1829:θ
1729:−
1720:−
1699:ρ
1688:ρ
1676:∫
1673:−
1667:ϕ
1646:ϕ
1622:−
1613:−
1556:ϕ
1483:ν
1446:ν
1409:ν
1344:ν
1309:−
1294:
1288:−
1275:−
1248:γ
1240:−
1237:γ
1229:
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1213:γ
1202:γ
1147:−
1144:γ
1127:−
1111:∫
1092:ν
1047:ν
1007:ν
1004:−
985:ν
979:θ
956:θ
865:−
862:γ
822:−
819:γ
781:−
778:γ
743:ρ
733:ρ
716:−
713:γ
709:γ
673:−
670:γ
632:−
629:γ
543:−
540:γ
502:−
499:γ
421:γ
368:ρ
255:
240:μ
194:
179:μ
3276:See also
2983:: 23–30.
2960:77709738
1920:, where
2714:where,
1069:is the
1039:where,
433:is the
137:density
55:improve
3432:
3409:
3386:
3359:
3336:
3238:arcsin
3120:arcsin
3093:arcsin
2958:
2837:Since
2091:normal
1291:arctan
1226:arctan
968:) by,
252:arcsin
191:arcsin
143:, the
107:convex
3202:Note:
2152:Notes
1662:where
40:, or
3454:NASA
3430:ISBN
3407:ISBN
3384:ISBN
3357:ISBN
3334:ISBN
3183:>
3025:>
2956:OCLC
2891:<
2855:>
2193:>
2131:The
135:and
2993:doi
2042:max
2029:max
1936:max
1905:max
1833:max
328:),
120:.
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