KUNDU Fluid Mechanics 2 Episode 9 doc

334 HOumhy Idpm und lielated Tw’a over a flat plate where U(dU/dx) = 0. Using definition (10.17) for 8, Eq. (10.43) reduces to (10.44) Assume a cubic profile U Y Y2 Y3 - = U+ h- + C- +d U 6 62 63 The four conditions that we can satisfy with this profile me chosen to be a% ay2 u = 0, -=U aty=O, au a)? u = LI, -=0 aty=6. The condition that a2u/ay2 = 0 at the wall is a requirement in a boundary layer over a flat plate, for which an application of the equation of motion (10.8) gives v(a2u/i3y2)() = U(dU/dx) = 0. Dctcnnination of the four constants reduces the assumed profile to -=-(-) ( u 3y 1y3 ). u 26 2s The term on the left- and right-hand sides of the momentum equation (10.44) are then [(U - u)udy = -U26, 39 280 3 uv P Substitution into the momentum integral equation gives 39U2d6 3uv - - 280 dx 2 6 . Integrating in x and using the condition S = 0 at x = 0, we obtain 6 = 4.64JqT, which is remarkably close to thc cxact solution (10.35). The friction factor is (3/2)U v/6 0.646 to - =- CJ = (1/2)PU2 - (1/2)U2 &’ which is also very closc to the exact solution of Eq. (10.37). pohlhausen found that a fourthdegme polynomial was necessary to exhibit sen- sitivity of the velocity profile to the pressure gradient; Adding mother term below w. (10.44). e(y/814 requires m additional boundary condition, azu/ay2 = o at y = 8. Wilh the assumption of a form for thc velocity profile, Eq. (10.43) may be reduced to an equation with one unknown, 8(x) with V(x), or tfie pressure gradient -4. This equation was solved approximately by Pohlhausen in 1921. This is described in Yih (1977, pp. 357-360). Subsequent improvements by Holstein and Bohlen (1940) are recounted in Schlichting (1979, pp. 357-360) and Rosenhead (1988, pp. 293-297). Sherman (1990, pp. 322-329) mlated the approximate solution due to Thwaites. 7. LgkGtOfh?Wx? CdmL So far we have consided the boundary layer on a flat plate, for which the pssm gradienl of the external stream is m. Now suppose that the surface of the body is curved (Figure 10.1 1 ). Upstream of the highest point the streamlines ofthe outer flow converge, resulting in au increase of the free siream velocity V(x) and a consequent fall of pressure with x. Downstream of the highest point the streamlines diverge, resultinginadecreawof U(x) andariseinpressure.Iuthissection weshallinvestigate theefFectofsuchapressuregradientcmtheshapeoftheboundarylayerprofileu(x, y). Thc boundary layer equation is aU aU 1 ap a2u u- + v- = + v- ax ay pax ay2’ where the pressure gradient is found from the external velocity field as dp/dx = -pU(dU/dx), wilhx taken along the surface of the body. At the wall, theboundary layer equation becomes In an accelerating stream dp/dx < 0, and therefore < 0 (accelerating). ($) Wall ( 10.45) As the velocity profile has to blend in smoothly with the external profilc, the slope au/ay slightly below the edge of the boundary layer decreases withy from apositive valuc to zero; therefore, a2u/i)y2 slightly below thc boundary layer edge is negative. Equation (10.45) then shows that a2u/ay3 has the same sign at both the wall and the boundary layer edge, and presumably throughout the boundary layer. In contrast, for a decelerating external stream, the curvature of the velocity profile at thc wall is > 0 (decelerating). ($) wall (10.46) so that the curvature changes sign somewhere within the boundary layer. In other words, the boundary layer profile in a decelerating flow has a point of inflection where a2u/ay2 = 0. In the limiting case ol a flat plate, the point of inflection is at the wall. Thc shape of the velocity profiles in Figure 10.1 1 suggests that a decelerating pressure gradient tends to increase the thickncss of the boundary layer. This can also be seen from the continuity equation u(y) = - IY dy. 0 ilx Comparcd to a flat plate, a decelerating external sham causes a larger -au/ax within the boundary laycr bccause the deceleration of the outm flow adds to the viscous deceleration within the boundary layer. It follows from the foregoing equation that the u-field, directed away rTom the surface, is larger for a dccelerating flow. The boundary layer hercforc thickens not only by viscous diffusion but also by advcction away from the surface, resulting in a rapid increase in thc boundary layer thickncss with x. If p falls along thc dircction of tlow, dp/dx < 0 and we say thal thc pressure gradient is “favorable.” If, on the other hand, the pressure rises along the direction of flow, dp/dx > 0 and wc say that the pressure gradient is “adverse” or “uphill.” The rapid growth of the boundary layer thickness in a decelcrating stream, and thc associated large v-field, causes the imporlant phenomenon of separation, in which the exbrnal stream ccascs to flow nearly parallcl to the boundary surfacc. This is discussed in the next section. 8. Separation We have sccn in the last section that the boundary layer in a decelerating stream has a point of inflection and grows rapidly. The existencc of the point of inflcction implics a slowing down of the region next to the wall, a conscquence of the uphill pressure gradient. Under a strong enough adverse pressure gradient, the flow ncxt Figme lOJ2 Stmmlincs and vclaciiy pfiles near a -on piru S. Poi d inoection is indicated by 1. The dashed linerepmenm u = 0. to the wall mses direction, resulting in a region of backward flow (Figure 1.0.22). The zevezsed flow meets the forward flow at some point S at which the fluid near the dice is transported out into the mainstream. We say hat the flow sepamtes h the wall. The separation point S is defined as the boundary between the forward flow and backward flow of the fluid near the wall, where the stcess vanishes: It is apparent hm the figure that one streamline intersects the wall at a definite angle at the point of separation. At lower Reynolds numbem the ~wersed flow downstream of the paint of sep don forms part of a large steady vortex behind the surface (see Figure 10.15 in Section 9 for the range 4 < Re < 40). At higher Reynolds numbers, when the flow has boundary layer characteristics, the flow downsheam of separation is unsteady and How strong an adverse presm gradient the boundary layer can withstand with- out undergoing sepamtion depends on the geometry of lhe flow, and whether the boundary layer is laminar M turbulent. A steep pressure gdient., such as that behind a blunt body, invariably leads to a quick separation. In contrast., the boundary layer on the trailing surface of a thin body can overcome the weak pressure gradients involved. Therefore,toavoidseparationandlargedcag,thetrailingsectionofasubmergedbody should be gmdudly reduced in size, giving it a so-called stnamlined shape. Evidence indicates Ihat the point of separation is insensitive to the Rcynolds number as long as the boundary layer is laminar. However, a rmnsirion fo furbuknce &Zap hunahy rclyer sepamtbn; that is, a turbulent boundary layer is more capable of withstanding an adverse p%sm gradient. This is because the velocity profile in a turbulent boundary layer is "fuller" (Figure 10.13) and has more energy. Fa example, the laminar boundary layer over a circular cylinder separates at 82" from frequently chaotic. Figure 10.13 Coinparison of laminar and turbulcnt vclocity pmfiles in a boundary layer. . Figure 10.14 Separation or flow in B highly divergent chsmncl. the forward stagnation point, whercas a turbulent layer ovcr the same body separates at 125" (shown later in Figure 10.15). Experiments show that the pressure rcmains fairly uniform downstrcarn of separation and has a lower value than thc pressures on the forward face of the body. The resulting drag due to pressure forccs is calledfimn drag, as it depends crucially on the shape of the body. For a blunt body the form drag is larger than the skin €riction drag because of the occurrence of separation. (For a streamlined body, skin friction is generally larger than the form drag.) As long as the separation point is located at the same place on the body, he drag coefficient of a blunt body is nearly constant at high Reynolds numbers. However, the drag coefficient drops suddcnly when the boundary layer undergoes transition to turbulence (see Figure 10.20 in Section 9). This is because thc separation point thcn moves downstream, and thc wake becomes narrower. Separation takes place not only in external flows, but also in internal flows such as thal in a highly divergent channel (Figure 10.14). Upstream of the throat the prcssure gradient is favorable and the flow adheres to the wall. Downstream of the hat a large enough adverse pressure gradient can cause separation. The boundary layer equations are valid only as Iar downstream as the point of separalion. Bcyond it the boundary layer becoma so thick that the basic underly- ing assumption bccome invalid. Moreover, the parabolic character of the boundary layer equations qujnx that a numerical integration is possible only in the dkc- tion of advection (along which information is propagated), which is rcpstrecun within the wed flow region. A farward (downstream) integration of the boundary layer equation. therefore breaks down after the separation point. Last, we can no longer apply potential thcory to find the pressure distribution in the separated region, as the effective boundary or thc irrotational flow is no longer the solid surface but some unknown shape cncompassing part of the body plus the separated regia In gcncral, analytical soluticms of viscous flows can be found (possibly in terms of perturbation series) only in two limiting cases, namely Re << 1 and Re >> 1. Tn the Re << 1 limit the inertia forax are negligible over most of the flow field; the Stokes-Oseen solutions discusscd in the pdng chapter are of this type. In the witc limit of Re >> 1 , the viscous forces are neagible everywhere except close io thc surfacc, and a solution may be attempted by matching an irrotational outcr flow with a boundary layer near the surface. In the intexmediate range of Reynolds numbers, finding aualytical solutions becomes almost an impossible task, and one has to depend on experimentation and numerical solutions. Some of these experimental flow patterns will be described in thi section, taking the flow over a circular cylinder as an example. Instead of discussing only the intermediate Reynolds number range, we shall describe the experimental data for the entire range of small to very high Reynolds numbers. Low Reynolds Numbers ZRt us start with a consideration of the creeping flow around a circular cylinder, charactcrizcd by Rc < 1. (Hen: we shall define Re = U,d/u, based on he upstream velocity and the cylinder diamctcr.) Vorlicity is gcnmed close to the surface because of the neslip boundary conditioL In the Stokes approximation this vorticity is sim- ply diffuscd, not advccted, which results in a lore and dt symmetry. The Oseen approximation partially takes into account the advection of vorticity, and resulk in an asymmetric velocity distributionfurihm the body (which was ShowninFigure 9.17). Thc vorticity distribution is qualitatively analogous to the dye distribution caud by a sow of colored fluid at the position of the body. The color diffuscs symmelrisally in very slow flows, but at higher flow speeds hc dye source is mn6ned behind a parabolic boundary with thc dyc source at the focus. Aq Re is increased beyond l., the Oseen approximation breaks down, and the vor- ticity iu inueasingly coujined behind the cylinder becawc of advection. For Re > 4, two small auacbed or “standing” eddies appcar behind the cylinder. The wake is com- pletely laminar and the vortices act like ‘Wuidynamic rollers” over which the main stream flows (Figure 10.1 5). The eddies gct longer as Re is increased. 4eRec40 Re<4 80 <Re e 200 laminar boundary layer turbulcnt boundary layer nt Re<3x 10s Re>3xl@ Figure 10.15 Some regimes or flow over a circular cylindcr. von Karman Vortex Street A very interesting sequcnce of events begins to develop when the Reynolds number is incrcased beyond 40, at which point the wake behind the cylinder becomes unstable. Pholographs show that the wake develops a slow oscillation in which the velocity is periodic in time and downstrcam distance, with the amplitudc of the oscillation increasing downstrcam. The oscillating wake rolls up into two staggered rows of vortices with opposite scnse of rotation (Figure 10.16). von Karman investigated the phenomenon as a problem of supcrposition olirrotational vortices; he concluded that a nonstaggered row of vortices is unstable, and a staggered row is stable only if the ratio of lateral distance between the vorlices to their longitudinal distance is 0.28. Because of thc similarity of the wake with footprints in a street, the staggered row of vortices behind a blunt body is called a von Kurmara vorrex street. The vortices move downstream at a speed smaller than the upstream velocity U,. This mcans that the vortex pattern slowly follows thc cylinder iC it is pulled Lhrough a stationary fluid. In the range 40 < Re < 80, the vortex street does no1 interact wilh thc pair of attached vortices. As Re is increased beyond 80 the vortex street €oms closer to he cylinder, and the attached eddics (whose downstream length has now grown to be about twice thc diameter of thc cylinder) themselves begin to oscillate. Finally the attached eddies periodically break off alternatcly from the two sides of the cylinder. Figure 10.16 von Karman vortex street downstream of a circular cylinder at Re = 55. Flow visualized by condensedmilk.S.’IBneda, Jour:Phys.Soc., Jlrpanu): 1714-1721,1%5,andreprintedwiththepermission of The Physical society of Ja~#m and Dr. !Watosh ‘Taneda I Figme 10.17 Spiral blades used for breaking up the spanwise coherence of vortex shedding fmm a cywcalrod. While an eddy on one side is shed, that on the other side forms, resulting in an unsteady flow near the cylinder. As vortices of opposite circulations are shed off alternately from the two sides, the circulation around the cylinder changes sign, resulting in an oscillating “lift” or lateral force. If the frequency of vortex shedding is close to the natural frequency of some mode of vibration of the cylinder body, then an appreciable lateral vibration has been observed to result. Engineering structures such as suspension bridges and oil drilling platforms are designed so as to break up a coherent shedding of vortices from cylindrical structures. This is done by including spiral blades protruding out of the cylinder surface, which break up the spanwise coherence of vortex shedding, forcing the vortices to detach at different times along the length of these structures (Figure 10.17). The passage of regular vortices causes velocity measurements in the wake to have a dominant periodicity. The frequency n is expressed as a nondimensional parameter known as the Strouhal number, defined as Experiments show that for a circular cylinder the value of S remains close to 0.2 1. for a large range of Reynolds numbcrs. For small values of cylinder diameter and moderate values of U,, the rcsulting frequencies of the vortex shedding and oscillating lift lie in the acoustic range. For example, at U, = 10m/s and a wire diameter of 2mm, the frequency corresponding to a Strouhal number of 0.21 is n = 1050 cyclcs per second. The “singing” of telephone and transmission lincs has been attributed to this phenomenon. Wcn and Lin (2001) conducted very careful experiments that purported to be strictly two-dimcnsional by using both horizontal and vertical soap film water tun- nels. They give a revicw of the recent literaturc on both the computational and exper- imental aspects of this problem. The asymptote cited here of S = 0.21 is for a flow including three-dimensional instabilities. Their experiments are in agreemcnt with two-dimensional computations and the data are asymptotic to S = 0.2417. Below Re = 200, the vortjces in the wake im laminar and continue to be so for very large distances downstnam. Above 200, thc vortex street becomcs unstable and irregular, and the flow within the vortices themselves becomes chaotic. However, the flow in the wake continues to have a strong frequency component corresponding to a Strouhal number d S = 0.21. Above a very high Reynolds number, say 5000, thc periodicity in the wake becomcs imperceptible, and the wake may bc described as completely turbulent. Striking examples of vortex streets have also been obscrved in the atmosphere. Figure 1.0.18 shows a satellite photograph of the wakc bchind several isolated moun- tain peaks, through which the wind is blowing toward thc southeast. Thc mountains picrce through the cloud Icvel, and the flow pattern becomes visible by thc cloud pattern. The wakes behind at least two mountain peaks display the characteristics ofa von Karman vortex street. Thc strong density stratification in this flow has prcvented a vertical motion, giving the flow the two-dimensional character necessary for the formation of vortex streets. High Reynolds Numbers At high Rcynolds numbers thc frictional elTects upstream of scparation are confined near the surface of the cylinder, and the boundary layer approximation becomes valid a. far downstream as thc point of scpamtion. For Re c 3 x 16, the boundary layer remains laminar, although the wake may be completely turbulent. Thc laminar boundary layer separates at % 82” from thc forward stagnation point (Figure 10.15). The pressure in the wake downstream or the point of separation is nearly constant and lower than Lhc upstream pressure (Figure 10.19). As Lhc drag in this range is primarily due to the asymmetry in thc pressure distribution caused by scparation, and as the point or separation remains fairly stationary in this range, the drag coeflicient also stays constant at CD 21 1.2 (Figure 10.20). Importanl changcs take place bcyond the critical Reynolds number or Re, - 3 x lo-’ (circular cylindcr). In the range 3 x l.05 -= Re < 3 x lo6, the laminar boundary layer hecomcs unstable and undergoes transition to turbulcnce. We have seen in thc preceding scction that ofHowpaataChdarQ+k%r 343 9. LksqMwn Figore 10.18 A von Kannan vortex street downstream of mountain peaks in a strongly stratified atmo- sphexe. There are several mountain peaks along the linear, light-colored feature Nnning diagonally in the upper lefi-hand corner of the photograph. North is upward, and the wind is blowing toward the southeast. R E. Thomson and J. E R. mer, Monfhly Wenther Review 105: 873-884,1977, and reprinted with the permission of the American Meteorlogical Society. because of its greater energy, a turbulent boundary layer, is able to overcome a larger adverse pressure gradient. In the case of a circular cylinder the turbulent boundary layer separates at 125" from the forward stagnation point, resulting in a thinner wake and a pressure distribution more similar to that of potential flow. Figure 10.19 com- pares the pressure distributions around the cylinder for two values of Re, one with a laminar and the other with a turbulent boundary layer. It is apparent that the pressures with the wake are higher when the boundary layer is turbulent, resulting in a sudden drop in the drag coefficient from 1.2 to 0.33 at the point of transition. For values of Re > 3 x lo6, the separation point slowly moves upstream as the Reynolds number is increased, resulting in an increase of the drag coefficient (Figure 10.20). It should be noted that the critical Reynolds number at which the boundary layer undergoes transition is strongly affected by two factors, namely the intensity [...]... m/s, Re = 0.85 x 1 6 R Mehta, Ann Rev Fluid Mech 17 151-1 89. 198 5 Photograph reproduced w t permissionfrom theAnnua1 Review of Fluid Mechanics, Vol 17 @ 198 5 Annual Reviews ih ww AnnualReviews.org in cricket literature, in contrast to an “inswinger” for which the seam is oriented in the opposite direction so as to generate an upward force in Figure 10 .22 .) Figure 10 .23 , photograph of a cricket ball in... Eq (10. 49) into the momentum constraint (10 4 ) giving 7, + M = pa2b'-Ix21n n L oc: ff2dr,J indcpcndcnt oFn, = which can be true only if 2m - n = 0 The exponents are therefore m=f, n = 2 3 The valuc of n shows that the jet width increases as x2/3 The factors u and b in Eq (10. 49) can now be chosen so that r] and f are dimensionless These constants can depend only on the external parameter M and fluid. .. approximation is improved as E (say) tends to its limit The value of an asymptotic expansion becomes clear if we comparc thc convcrgcnt series for a Bessel function Jo(x), given by J()(X)= 1 x2 x4 2' 22 42 - - + -+ X6 22 426 2 (10.64) ~ with the first term of its asymptotic expansion (1 0.65) The convergent scrics (1 0.64) is useful when x is small, but more than cight lcrms are needed for three-place accuracy... (1 0.56) and (1 0.57) yields f'" + ff" + 2jl2 = 0: f (0) = 0, f'(0) = 0, f ' 9 )= 0 (0 (1 0. 59) The introduction of foe as a normalization parameter (so that in dimcnsionlcss form f(cc) = 1) indicates that the trivial solulion of Eq (10. 59) is to bc excluded One integration of Eq (10. 59) and an evaluation or the constant of integration yields f f " - $.f '2 f 2f' = 0 Here and in the following, f is madc... Eq (10 .90 ) the boundary condition equations (10.85) and (10.86) give uo(~)+~o(0)+s[ul(o)+li1(0)1+ =o, uo(l)+ 0 S[Ul(1) 01 * * = 1 + +- + (10 .94 ) (10 .95 ) Eyuating like powers of 6, we obtain thc following conditions uo(0) + fiO(0)= 0, uo(1) = 1: Ul(0) + il(0) = 0, u1(1) = 0 (1 0 .96 ) (1 0 .97 ) We can now solvc Eq (1 0 .93 ) along with the first condition in Eq (1 0 .97 ), obtaining uo(p) = 1 (1 0 .98 ) Next,... the following sets: Order E': ( 10. 79) uo(0) = 0, U()(l) = 1 Order E ' : d2u, duo dY2 u,(O) = 0, dy ' (1 0.80) Ul(1) = 0 The solution of thc zero-order problem (1 0. 79) is uo = y (10.81) Substituting this into the first-order problem (10.80), we obtain the solution Y u1 = -(I -y) 2 The complete solution up to order E is then U(Y) = Y + $Y(l E - Y)l + O(E9 (10. 82) In this expansionthe second term... form 01ajet of fluid ejected from an orifice, a wakc 12 7bo-Dimensional Jets Figure 10 .25 Smoke photograph of flow around a spinning baseball Flow is from left to right, flow speed is 21 m/s, and the ball is spinning counterclockwise at 15rev/s [Photograph by E N M Brown, University of Notre Dame.] Photograph reproduced with permission, from the Annual Review of Ftuid Mechanics, Vol 17 @ 198 5 by Annual... ball oricnted as in Figure 10 .22 is called an “outswinger” - Re lo5 d=7.2m m = O M 6 kg I!Xgurc 10 .22 The s i g of a cricket ball The seam is oriented in such a r a y that the lateral force on the wn hall is downward in UIC l i p 347 348 Boundary h p r s and Related 7bpieR Figve 1 3 Smoke photograph of flow over a cricketball Flow is from left to right Seam angle is 40” 02 flow speed is 17 m/s, Re =... O(1)) and d + 0, we obtain the behavior within the boundary layer Multiplying Eq (10 .91 ) by 6 and taking the limit as 6 + 0, with q = 0(1), wc obtain (10. 92 ) which governs the first term of the boundary layer correction Next, the limit of Eq (10 .91 ) as 6 + 0, with y = 0(1),gives duo - = 0, dY (I 0 .93 ) 16 ,In f-de - 3 69 o$a %gular hJurbaliorr Avdrl?m - which governs the first term of thc outer solution... scparates at 2 85’, whereas that on thc turbulent side separates at 120 ‘ Compared to region B, thc surface pressure near rcgion A is therefore closer to that given by the potcntial flow theory (which predicts a suction pressure of (Pmin - p x ) / ( i p U & ) = - 1 .25 ; see Eq (6. 79) ) In other words, thc prcssurcs are lower on side A, resulting in a downward force on the ball (Notc that Figurc 10 .22 is a view . recounted in Schlichting ( 197 9, pp. 357-360) and Rosenhead ( 198 8, pp. 29 3 - 29 7). Sherman ( 199 0, pp. 322 - 3 29 ) mlated the approximate solution due to Thwaites. 7. LgkGtOfh?Wx?. Pohlhausen in 1 92 1 . This is described in Yih ( 197 7, pp. 357-360). Subsequent improvements by Holstein and Bohlen ( 194 0) are recounted in Schlichting ( 197 9, pp. 357-360). to -=-(-) ( u 3y 1y3 ). u 26 2s The term on the left- and right-hand sides of the momentum equation (10.44) are then [(U - u)udy = -U26, 39 28 0 3 uv P Substitution into