KUNDU Fluid Mechanics 2E Episode 17 potx

which M = 1 at the exit station. The flow is then said to be choked, and no more masdtime can flow through that duct. This is analogous to flow in a convergent duct. Imagine pouring liquid through a funnel from one container into another. There is a maximum volumetric flow rate that can be passed by the funnel, and beyond that flow rate, the funnel overllows. The same thing happens here. If f or q is too large, such that no (steady-state) solution is possible, there is an external adjustment that reduces the mass flow rate to that for which the exit speed is just sonic. Both for MI .e 1 and MI > 1 he limiting curves for .f and q indicating choked flow intersect M2 = 1 at right angles. Qualitatively, the effect is the same as choking by area contraction. 9. Mach Cbne So Iar in this chapter we have considered one-dirncnsional flows in which the flow properlies varied only in the direction of flow. Tn this sixtion we begin our study of wave motions in more than one dimension. Consider a point source emitting infinites- imal pressure disturbances in a still fluid in which the spccd of sound is c. If the point disturbance is stationary, then the wavefronts are concentric spheres. This is shown in Figure 16.17a, whcre the wavefronts at intervals of At are shown. Now supposc that the source propagates to the left at speed U .e c. Figurc 16.17b shows four locations oI the source, that is, 1 through 4, at equal intervals of time At, with point 4 being the present location of the source. At point 1, the sourcc cmitted a wave that har spherically expanded to a radius of 3c At in an interval of dmc 3 At. During lhis time the source has moved to location 4, at a distance of 3U Af from point 1. The figure also shows the localions of thc wavefronts emitted while the SOUKC was at points 2 and 3. It is clear that the wavefronts do not intersect because U .e c. As in thc casc of the stationary source, the waveIronts propagate everywhere in the flow ficld, upstream and downstream. It thereforc follows that u body mowing al a subsonic speed influences the entireflowjeld; information propagates upstream as well as downstrcam of the body. Now consider a case where the disturbance moves supmonkally at U > c (Figure 16.17~). Tn this case the spherically cxpanding wavefronts cannot catch up with the faster moving disturbance and form a conical tangent surface called theMach cone. In plane two-dimensional flow, the tangent surFace is in thc form of a wedge, and the tangent lines are called Mach fines. An examination of thc figure shows that the half-anglc of thc Mach cone (or wedge), called the Mach angle p, is given by sinp = (c At)/(U At), so that . smp= I M !* (1 6.38) The Mach cone becomcs wider as M decreases and becomes a plane front (that is, p = 9W) when M = 1. Thc point source considered hcrc could be part oI a solid body, which sends out pressurc wavcs as it moves bough thc fluid. Moreover, Figurc 16.17~ applies equally 9. .Wadi Ciww 695 Mach cone (C) Figure 16.17 Wavefronts emined by a point source in a still fluid when the source speed U is: (a) V = 0; (b) U -z c; and (c) U =- c. if the point source is stationary and thc fluid is approaching at a supersonic speed CJ. Tt is clcar that in a supersonic flow an observer outside the Mach cone would not “hcar” a signal emitted by a point disturbance, hence this region is called the zone Qfsilence. In contrast, the region inside the Mach conc is called the zone ojacfion, within which the effects of the disturbance are felt. This explains why the sound of a supersonic airplane does not reach an observer until the Mach conc anives, aJer the plane has passed overhead. At every point in a planar supersonic flow thcre are two Mach lines, oriented at fl.~ to the local direction of flow. Information propagates along these lines, which are the churucferisrics of the governing diffcrcntial equation. It can be shown that the nature of the governing differential equation is hyperbolic in a supcrsonic Row and elliptic in a subsonic flow. 10. Oblique Shock Waui! In Section 6 we examined the case of a normal shock wave, orientcd pcrpcndicular to the direction of flow, in which the velocity changes from supersonic to subsonic values. Howcver, a shock wave can also be oricntcd obliquely to the flow (Figure 16.18a), the velocity changing from VI to V2. The flow can be analyzed by considering a normal shock across which the normal velocity varies from u I to up and superposing a vclocity u parallel to it (Figure 16.18b). By considering conservation of momentum in a direction tangential to the shock, we may show that v is unchanged across a shock (Exercise 12). The magnitude and direction of the velocities on the two sides or the shock are VI = ,/- oricntcd at r7 = tan-'(uI/v), V2 = JUZ + v2 The nod Mach numbers are orientcd at r7 - 6 = tan-'(u2/v). Mnl = uI/q = it41 sin m > 1, M,,2 = UZ/C~ = M2 S~(O - 8) < 1. Because u2 u1, them is a suddcn change of direction of flow across the shock; in fact the flow turns towurtl the shock by an amount S. The angle u is called the shock angle or wuve mgle and S is called the deflection angle. Supcrposition of the tangential velocity v does not affect thc static properties, which are therefore the same as those for anormal shock. The expressions for the ratios p2/p1, P~/PI, T~/TI, and (S2 - Sl)/C, are therefore those given by Eqs. (16.31), (16.33)-(16.35), if MI is replaced by the normal component of he upstrcam Mach number MI sin u . For example, P2 2Y PI Y+l - = 1 + -(M,2sin2u - 1)$ (16.39) Figure 16.18 (a) Oblique shock wavc in which 8 = deflection anglc and u = shock angle; and (h) uniil- yxis by considering a normal shock and superposing a vclocity u parallel to Lhc shock. Thc normal shock table, Table 16.2, is therefore also applicable lo obliquc shock waves if we use MI sin CT in place or MI. The relation between the upstream and downstream Mach numbcrs can be found from Eq. (16.32) by rcplacing MI by MI sin o and Mz by M2 sin (a - 6). This gives (y - I)M: sin2 a + 2 2y~;sin'o + 1 - y' 2 M: sin (a - 6) = (16.41) An imporlant relation is that between the deflection angle S and the shock angle for a givec MI, given in Eq. (16.40). Using the trigonometric identity for tan (a - S), this becomes ( 16.42) X plot of this relation is given in Figure 16.19. The curves represent S vs a for constant MI. The value of M2 varies along the curves, and the locus of points corresponding to M2 = I is indicated. It is apparent that there is a maximum deflection angle S,, M: sin2rJ - 1 tanS=2cota M?(y +cos2a) +2' 0" 10" 2oo 30" 40" 50" 60" 70" 80" 90" Wave angle (r Figure 16.19 Plot of obliquc shock solution. Thc stmng shock branch is indicated by dashed lines, and the heavy dotlcd linc indicaks the maximum deflection anglc for oblique shock solutions to bc possible; for example, S,, = 23 ' for MI = 2. For a given MI, S becomes zero at cr = n/2 comsponding to a normal shock, and at IT = 1-1 = sin-'(I/M~) comsponding to thc Mach angle. For a fixed MI and 6 .c 8-, thcrc arc two possiblc solutions: a weak shock corresponding to a smaller IT, and a strong hock comsponding to a largcr 6. Tt is clear that the flow downstream of a strong shock is always subsonic; in contrast, the flow downstrcim of a weak shock is generally supersonic, except in a small range in which S is slightly smaller than Sm. Generation of Oblique Shock Waves Consider the supersonic flow past a wedge of half-angle S, or thc flow over a wall that turns inward by an angle S (Figure 16.20). If MI and 6 arc givcn, then 0 can be obtained from Figure 16.19, and M,,z (and therefore M2 = M,,2/sin(a - 6)) can be obtained from the shock table, Tablc 16.2. An attached shock wave, corresponding to the weak solution, forms at he nose of the wedge, such that the flow is parallcl to the wedge after turning through an anglc 6. The shock angle CT decrcascs to thc Mach angle 1.11 = sin-'(] /MI) as thc dcflection S tends to zero. It is intcrcsting that the comer velocity in a supersonic flow is finitc. In contrast, the corner velocity in a subsonic (or incompressible) flow is either zcro or infinite, depending on whcthcr the wall shape is concave or convex. Moreover, thc strcamlines in Figure 16.20 arc siraight, and computation of the field is easy. By conlrast, the streamlines in a subsonic flow are curved, and thc computation of the flow field is not casy. The basic reaqon for this is that, in a supersonic flow, the disturbances do not propagate upstream of Mach lines or shock waves emanating from the disturbances, hcnce the flow field can bc constructed step by step, proceeding downstwm. In contrast, thc disturbances propagate both upstream and downstream in a subsonic flow, so that all features in the cntire flow field are related to each othcr. As 6 is incrcascd beyond S,,,, attached oblique shocks are not possible, and a detached curved shock stands in front of the body (Figure 16.21). The central strcamline goes through a normal shock and generates a subsonic flow in €on1 of the wedge. The strong shock solution of Figure 16.1.9 therefore holds ncar the nose ol the body. Farher out, the shock angle decreases: and the weak shock solution applies. If the wedge angle is not too largc, then the curved dctached shock in Figure 16.21 Fiyrc 16.20 Ohliquc shocks in supersonic flow. weak shock strong shock I I I Figure 16.21 Dclachcd shock. becomes ar, oblique attached shock as the Mach number is increased. In the case of a blunt-noscd body, however, the shock at the leading edge is always dctached, although it moves closer to 1he body as thc Mach number is increased. We see that shock waves maj7 exist in supersonic flows and their location and orientation adjust to satisfy boundary conditions. In external flows, such as those just described, the boundary condition is that streamlines at a solid surface musl be tangent to that surface. In duct flows the boundary condition locating the shock is usually the downstream pressure. The Weak Shock Limit A simple and useful expression can be derived for the pressure change across a weak shock by considering thc limiting casc of a small dcflcction angle 6. We first nced to simplify Eq. (1 6.42) by noting hat as S +. 0, the shock angle a tends to the Mach anglc 1~1 = sin-'(I/Ml). sin2 Q - 1 + 0, (as 0 + .u and S +. 0). Then from Eqs. (16.39) and (16.42) Also from Eq. (16.39) we note that (p? - p,)/pl + 0 as (16.44) The interesting point is that dation (1 6.44) is also applicable to a weak expansion wavc and not just a wcak comprcssion wave. By this we mean that thc prcssure inmase due Lo a small deflection of thc wall toward the flow is the samc as the pressure decrease due to a small dcflwtion of the wall wuy from the flow. This is because the entropy change across a shock goes to zero much fastcr than the rate at which the pressure dimerence across thc wavc dccwases as our study of nod shock waves has shown. Very weak “shock waves” arc thcmfore approximately isentropic or reversible. Relationships for a weak shock wave can thcrcfore be applied to a weak expansion wave, except for some sign changes. In Scction 12, Eq. (16.44) will be applied in estimating the lift and drag of a thin airfoil in supersonic flow. 11. Fkpansion and Cornpmtwion W iSupi?rsonic Flow Consider the supersonic flow over a gradually curved wall (Figure 16.22). The wave- fronts are now Mach lines, inclined at an angle of ,Y = sin-’ (1 /M) to the local direction of flow. The flow orientation and Mach numbcr arc constant on each Mach line. Tn the case of compression, the Mach numbcr dccrcases along the flow, so that the Mach angle increases. The Mach lines therefore coalcscc and form an oblique shock. In the case of gradual cxpansion, the ‘Mach number increases along the flow and the Mach lines diverge. Tf thc wall has a sharp deflection away from the approaching stream, then thc pattern of Figure 16.22b takes the form of Figurc 16.23. The flow expands through a “fan” of Mach lines centered at the corner, callcd thc Prandtl-Meyer expansion fun. The Mach number incrcases through the fan, with M2 > MI. The first Mach linc is inclined at an anglc of 1.11 to the local flow direction, while the last Mach linc is inclined at an anglc of ,~2 to the local flow direction. Thc pressure falls gradually along a streamline through thc fan. (Along the wall, however, thc pressure remains constant along the upstream wall, falls discontinuously at the comer, and thcn remains constant along the downstmam wall.) Figure 16.23 should be compared with Figure 16.20, in which the wall turns inward and generates a shock wavc. By contrast, the expansion in Figure 16.23 is gradual and iscntropic. . . . :.::, 1 :, ,:: ,:, .’. Figure 16.22 Gradual cornprcsrion and expansion in supcrronic flow: (a) gradual compression. resulting in shock formation; and (h) gradual cxpansion. F'igurc 16.23 Thc PrandU-Mcycr expansion h. The flow through a Prandll-LMeyer €an is calculated as follows. From Figure 16.18b, conservation of momentum tangential to the shock shows that Ihc tangential velocity is unchanged, or VI cos CT = V2 cos(a - S) = V~(COS cr cos S + sin cr sin S). We are concerned here with very small dcflcctions, 6 + 0 so cr + p. Hcrc, cos S % 1, sin6 = S, VI 2 ~2(1 +~tana), so (~2 - vI)/vI Regarding this as appropriate for infinitesimal change in V for an infinitesi- mal deflection, we can write this as dS = -dVm/V (first quadrant deflec- tion). Because V = Mc, dV/ V = dM/M + dc/c. With c = for a perfect gas, dcjc = dT/2T. Using Eq. (16.20) for adiabatic flow of a perfect gas, dT/T Stan0 = -s/ = -(v - l)MdM/[,I + ((v - 1)/2)M2]. Then d@=T dM d6 = - M 1 + ((v - 1)/2)M2' Intcgdting 6 horn 0 (radians) and M from 1 gives S + v(M) = const., where is called thc Prandtl-Meyer function. Thc sign of dm originatcs from the idcntification or tan fi = tan IL = 1 /dm lor a first quadrant dcflcc- tion (uppcr half-plane). For a fourth quadrant deflection (lower half-plane), tan ,u = - 1 /dm. For example, in Figurc 16.22 we would writc 61 + v1 (MI) = 62 + k(Mz), whcrc, for cxample, SI,&, and MI are given. Then v2Wz) = 61 - 62 + vl(MI). In pancl (a), 61 - 82 < 0, so y < VI and MZ < MI. In panel (b), 61 - 81 > 0, so y > VI andMz >MI. 12. Thin Airfoil Y%eory in Siqcrsonic Jlow Simplc cxprcssions can be derived for the lift and drag coefficients of an airfoil in supersonic flow if the thickness and angle of attack are small. The disturbances caused by a thin airfoil are small, and the total flow can be built up by superposition of small disturbances emanating from points on the body. Such a lincarizcd theory of lift and drag was developed by Ackerct. Because all flow inclinations are small, we can use the relation (I 6.44) to calculate the pressure changes due to a change in flow direction. We can write this relation as (16.46) where pm and MJc refer to the properties of the he stream, and p is the prcssure at a point where the flow is inclined at an angle S to the liec-stream direction. The sign of S dctermines the sign of (p - pea). To see how the lift and drag of a thin body in a supersonic stream can be estimated, consider a flat plate inclined at a small angle (r to a stream (Figure 16.24). At the leading cdgc thcrc is a weak expansion fan on the top surfacc and a weak obliquc shock on the bottom surface. The streamlines ahead of these waves are straight. Thc streamlines above the plate turn through an angle (r by expanding through a centered Ian, downstream of which they become parallel to the plate with a pressurc p~ < pw. The uppcr streamlines then turn sharply across a shock cmanathg from hc trailing edge, becoming parallel to the free stream once again. Opposite features occur for the streamlines below the plate. The flow first undergoes compression across a shock coming from the leading edge, which results in a pressurc p3 > pW. It is, however, not important to distinguish between shocks and expansion waves in Figurc 16.24, because thc linearized theory trcats them the samc way, except for the sign of thc prcssure changes hey produce. The pressures above and below the platc can be found from Eq. (16.46), giving P3-POo - YMkU Pw -Jm. The pressurc difference across the plate is Lhcrefore p2 Figure 16.24 Jnclined flat plate in a supersonic stream. Thc uppcr ptlncl sbows tbc flow pattern and the lowcr pancl shows the pressure distribution. If h is the chord length, then the lift and drag forces per unit span are (16.47) 'The lift coefficient is defincd L - L c- '. = (1/2)p,U&h - (1/2)yp,M&b' where wc have used the relation pU2 = ypM2. Using Eq. (16.47), the lift and drag coefficients for a Rat lifting surface arc (16.48) [...]... douhlet/dipole 157-159 forces on two-dimensional body, 166 -170 i m p , mclhod or, 143 ,170 -171 numcrical zolulion orplanc, 176 181 ovcr elliplic cylinder, 173 -174 past circularcyliodcr wih circulation, 163-166 past circular cylinder withoui circulation, 160-163 past half-body, 159-160 relevance of, 148-150 sources and sinks, 156 uniqueness of, 175 -176 unsteady 113-1 14 veltrily potential and Laplace equation,... kg m-' s-' m2/s m2/s 1.787E - 3 1.307'6 - 3 1.0 02E - 3 0.7998 - 3 0.653E - 3 0.548B - 3 1.78E - 6 1.307E-6 I W E - 6 0.8 02E - 6 0.658E - 6 0.5558 - 6 0 10 20 30 40 50 lo00 loa0 997 995 9 2 988 - 0 a -4 M.9E - 4 2.IE - 4 3.0E - 4 3.8E - 4 4.33 - 4 _ R C P - J@-'K-' 1.33E - 7 1.38E - 7 1. 42E - 7 1.468-7 1. 52E- 7 1.58E - 7 V/K 4 217 4192 4182 4178 4178 4180 13.4 9.5 7.1 5.5 4.3 3.5 Latent heat... Taylor, 7 Much 1886-27 June 1975.” Journal o Fluid f Mechanics 173 1-14 Oswatitsch, K and K Wicghardt (1987) “Ludwig Prandtl and his I(iri~-Wilhelm-Tnstitute:’Annual Review < $Fluid Mechanics 1 1-25 Y Von Karman, T (1954) Aedjnaniics New Y r :McGmw-Hill ok Index Ackeret, Jacob, 663,702 Acoustic waves, 665 Adiabatic dcnsity gradient, 541,557 Adiabatic process, 17 Adiabatic temperalum gradient, 19,541,557... applied mechanics at Gottingen from 1904 to 1953; the quiet university town of Gotthgen becamc an international center of aerodynamic research In 1904,h d t l conceived the idea of a boundary layer, which adjoinsthe surface of a body moving through a fluid, and is perhaps the greatest single discovery in the history of fluid mechanics He showed that frictional effects in a slightly viscous fluid are... Hupniot, P e r Henry, 681 ire 723 Index Hydraulic jmnp, 227-229 Hydroskilics, I 1 Hydrostatic waves, 2 2 1 Hypcrwnic flow, 664 Images, method of, 143 ,170 -17 1 Incomprcssihleaerodynamics See Acrodynanics Tncompressihle fluids, 81,96 Incompressible viscous fluid flow, 393 convection-dominatedproblems 394-396 Glowinski scheme, 403-404 incompressibilitycondition, 396 MAC scheme 396400 ITLXC~ clcmcnt, 404-406... 234-237 in slralificd fluid, 245-253 in s~atificd fluid with rotation 598-608 W K B solution 60-603 Internal Kosshy radius or dcrormtllion,5 W Intrinsic frequency, 198,607 Inversion, atmospheric, 1 9 Inviscid stability of parallel f o s 471475 lw, htational fow,59 application of complex variables, 1 2 1 54 5m n body ofrevolution, IX7-IX9 ud axisymmetric, 1x1-187 conformal mapping, 171 -173 douhlet/dipole... atmosphcrc, 2 1 Compression wavcs, 194 Computational fluid dynamics (CF'D) advantiigcsof, 379-380 conclusions, 424427 defined, 378 cxamples of, 406424 finitc dill-cmnce method, 38S38.5 finite elcmcnt method, 385-393 incornprcssiihlc viscous fluid tlow, 393-4(K, sources of error, 379 Concentric cylinders laminar flow hetween, 279-282 Conformal mapping 171 -173 application t airfoil, 638-642 o 720 Bl&~ Conscrvdon... connected region, 175 Singuluiiies, 153 Singular pxturbalion, 36371,477 Skan,S.W., 329 Skin frictioc cocficicnl, 328-329 Sloping ccnvwtion, 622 Solenoidal vector, 38 Solid-body rotation, 65-66, 127 Solids, 3-4 Soliujns, 231-232 Sonic condilionr, 672 Sonic properties, comprcssiblc flow, 671-675 Sound speed of, 15, 17, 665-667 waves, 665-667 Source-sink axisymmetric, I86 near a wall, 17& 171 plane, 156 Spatial... pages); Volume 1 contains “Meteorology, Oceanography, and Turbulent How” (45 papers, 5 15 pages); Volume Ill contains “Aerodynamics and the Mechanics of Projectiles and Explosions” (58 papers, 559 pages); and Volume 1V contains “Miscellaneous Papers on Mechanics of Fluids” (49 papers, 579 pages) Pcrhaps G 1 Taylor is best known for his work on turbulence When asked, however, what gave him maximum snrisfncrion,... mechanics Later, Prandtl became Foppl’s son-in-law, following the good German academic tradition in those days T 1901, he became professor of n mechanics at the University of Hanover, where he continued his earlier efforts to provide a sound theontical basis for fluid mechanics Thc famous mathematician Felix Klein, who stressed the use of mathematics in engineering education, became interested in Prandtl . - . 1.33E - 7 1.38E - 7 1. 42E - 7 1.468-7 1. 52E- 7 1.58E - 7 CP R J@-'K-' V/K 4 217 13.4 4192 9.5 4182 7.1 4178 5.5 4178 4.3 4180 3.5 Latent heat of vaporization. moves bough thc fluid. Moreover, Figurc 16 .17~ applies equally 9. .Wadi Ciww 695 Mach cone (C) Figure 16 .17 Wavefronts emined by a point source in a still fluid when the. . . 1.787E - 3 1.307'6 - 3 1.0 02E - 3 0.7998 - 3 0.653E - 3 0.548B - 3 v m2/s 1.78E - 6 1.307E-6 I .WE - 6 0.8 02E - 6 0.658E - 6 0.5558 - 6 . _. K