KUNDU Fluid Mechanics 2 Episode 8 pot

3. Difliion of a Virh Sheet Consider the case in which the initial velocity field is in the form of a vortex shcct with u = U €or y > 0 and u = -U for y < 0. We want to investigate how the vortex sheet decays by viscous dflusion. The governing equation is au a2u - = v- at i)y' subject to U(Y, 0) = U sgn(y), u(x. t) = u, u(-x, t) = -u, where sgn(y) is the "sign function," defined a,. 1 €01 positive y and -1 for negative Y. As in thc previous section, the parameter U can be eliminated €om the governing set by regarding u/U as the dependent variable. Then u/U must bc a function of (y, t, v), and a dimcnsional analysis reveals that there must exist a similarity solution in the form The detailed arguments for the existence of a solution in this form are given in the preceding section. Substitution dhc similarity form into the governing set transforms it into the ordinary differential equation F" = -2qF'. F(+oo) = I, F(-m) = - 1 ~ whose solution is w?) = emo. The velocity distribution is therefore (9.38) A plot of the velocity distribution is shown in Figure 9.11. If we define the width of he transition layer as the distance between the points where u = f0.9SU, then the corresponding value of r,~ is f 1.38 and consequently the width of the transition layer is 5.52,';i. It is clear that the flow is essentially identical to that duc to the impulsive start of a flat plate discussed in the preceding section. In fact, each half of Figure 9.1 1 is idcntical to Figure 9.10 (within an additive constant of fl). In both problems 290 hfflinar I;yuIL. Figure 9.11 Viscous decay of a vortex shect. Thc right panel shows thc nondimcnsional solution and thc left panel indicatcs hc vorticity distribution at two tirncs. the initial delta-function-like vorticity is diffused away. Tn the presenl problem the magnitude of vorticity at any time is (9.39) This is a Gaussian distribution, whose width increases with timc as a, while the maximum value decreascs as I/&. The total amount of vorticity is which is independcnt of time, and equals the y-integral of the initial (delta-function-like) vorticity. 9. Decay of a Line hrkx In Section 6 it was shown that when a solid cylinder of radius R is rotated at angu- lar specd s2 in a viscous fluid, the resulhg motion is irrotational with a velocity distribution US = !2R2/r. The velocity distribution can be writkn as r Uo = -9 wherc r = 2n SZ R2 is thc circulation along any path surrounding the cylinder. Suppose the radius of the cylinder goes to zero while its angular velocity correspondingly 2x1- inmeases in such a way that the product r = 2irQR' is unchanged. In the limit we obtain a line vortex of circulation r, which has an infinite velocity discontinuity at thc origin. Now suppose that the limiting (infinitely thin and fast) cylinder suddenly stops rotating at r = 0, thereby reducing the velocity at the origin to zero impulsively. Then the fluid would gradually slow down from the initial distribution because of viscous diffusion from the region near the origin. The flow can therefore be regarded as that of the viscous decay of a line vortex, for which all the vorticity is initially concentrated at the origin. The problem is the circular analog o€ the decay of a plane vortex sheet discussed in the preceding section. Employing cylindrical coordinates, the governing equation is subject to ug(r, 0) = r/27rr, (9.41) (9.42) (9.43) We expect similarity solutions here because there are no natural scales for Y and t introduced from the boundary conditions. Conditions (9.41) and (9.43) show that the dependence of the solution on the parameter r/21rr can be eliminated by defining a nondimensional velocity (9.44) which must have a dependence of the form u' = f(r, t, u). As thc lcft-hand side of the preceding equation is nondimensional, the right-hand side must be a nondimensional function of r, t, and u. A dimensional analysis quickly shows that the only nondimensional group formed from thcsc is r/Jvb. Therefore, the problem must have a similarity solution d the form u' = F(q), (9.45) (Notc that we could have defined q = r/2& a$ in the previous problems, but the algebra is slightly simpler if we define it as inEq. (9.45).) Substitution of thc similarity solution (9.45) into the governing set (9.40X9.43) givcs F" + F' = 0, subject to F(0) = 1, P(0) = 0. 292 Imminar Flow r Rprc! 9.12 Viscous dccrry of a line vortcx showing the iangcnlial velocity at diJTcrent times. The solution is The dimcnsional Velocity distribution is therefore F = 1 - e-q. (9.46) A sketch of the velocity diskibution for various values of f is given in Figurc 9.12. Near the center (r << 2fi) the flow has the form of a rigid-body rotation, whilc in the outcr region (r >> 2fi) the motion has the form of an irrotational vortex. The foregoing discussion applies to the &cay of a line vortex. Consider now the case where a line vortcx is suddenly introduced into a fluid at rest. This can be visualized as the impulsive start of an infinitely thin and fast cylindcr. It is easy to show that the velocity distribution is (Exercise 5) (9.47) which should be compared to Eq. (9.46). The analogous problem in heat conduction is the sudden introduction of an infinitely thin and hot cylinder (containing a finite amount of heat) into a liquid having a different tcmperature. 10. Flow llue to an Oscillahg Plate The unsteady parallel flows discussed in the three preceding sections had similarity solutions, because there were no natural scales in space and time. We now discuss IO. Jhu Ilue to an 0.wilhtitig !'hie an unsteady parallel flow that does not have a similarity solution bccause of the existence ora natural time scale. Consider an idmite flat plate that executes sinusoidal oscillations parallel to itself. (This is sometimes called Stokes' secondproblem.) Only the steady periodic solution a~let- the slarting transients have died will be considcred, thus there are no initial conditions to satisfy. The governing equation is 293 subject to u(0, t) = u cos wt, u(00: r) = bounded. (9.48) (9.49) (9.50) In the stcady statc, thc flow variables must have a periodicity equal to the periodicity of the boundary motion. Consequently, we use a separable solution of the form = pr .f (Y), (9.51) where what is meant is the real part of the right-hand side. (Such a complex form of represcntation is discussed in Chapter 7, Section 15.) Here, f (y) is complex, thus u(y, t) is allowed to have a phase difference with the wall velocity U cos wl. Substitution of Eq. (9.51) into the governing equation (9.48) gives (9.52) This is an equation with constant coefficients and must have exponential solu- tions. Substilution of a solution of the form f = exp(ky) gives k = m = &(i + l)-, where the two square roots of i have been used. Consequently, the solution of Eq. (9.52) is (9.53) The condition (9.50), which requires that the solutionmustremain boundcd a1 y = 30, needs B = 0. The solution (9.51) then becomes = A eiw~ ,-(l+i)y,hP (9.54) The surface boundary condition (9.49) now givcs A = U. Taking the real part of Eq. (9.54), we finally obtain the velocity distribution for the problem: u = Ue-J-cos wt - y ( E). (9.55) The cosine term in Eq. (9.55) represents a signal propagating in the direction of y, while the exponcntial term represents a dccay in y. The flow thercfore resem- bles a damped wave (Figure 9.13). However, this is a dfision problcm and nor a 294 imminar I-liiw U -1 0 1 Figore 9.13 Velocity dishbution in laminar flow near an osdllating plalc. The distributions at wf = 0, x/2, n, and 3n/2 are shown. Thc dillilsive distmcc is of order d = 4m. wave-propagation problem because there are no rcstoring forces involved here. The apparent propagation is merely a result of the oscillating boundary condition. For y = 4m, ihc amplitude of u is U exp(-4/&) = O.O6U, which means that the influence of the wall is confined within a distance of order s ‘c 4-, (9.56) which decreases with frequency. Note that the solution (9.55) cannot be mpresented by a single curve in krms of the nondimensional variables. This is expected because the frequency of the bound- ary motion introduces a natural time scale l/o into the problem, thereby violating the requiremcnts of self-similarity. There are two parameters in the governing set (9.48)-(9.50), namely, U and w. The parameter U can be eliminated by regarding u/U as the dependent variable. Thus the solution must have a form U - = .f(Y, t, 0: VI. (9.57) As there are fivc variables and two dimensions involved, it follows that there must be three dimensionless variables. A dimensional analysis of Eq. (9.57) gives u/U, of, and ym as the three nondimensional variables as in Eq. (9.55). Self-similar solu- tions exist only when there is an absence of such naturally occurring scalcs requiring a reduction in the dimcnsionality of the space. An interesting point is that the oscillating plate has a constant diffusion dis- tance 6 = 4m that is in contrast to the casc of the impulsively started platc U in which the diffusion distance increases with time. This can be understood from the govcming cquation (9.48). In thc problcm of sudden accelcration of a plate, i12u/i)y2 is positive for all y (see Figure 9.10), which results in a positive au/at everywhere. The monotonic acceleration signifies that momentum is constantly diffused outward, which results in an ever-increasing width of flow. In contrast, in thc casc of an oscillating plate, a2u/i3y2 (and therefore au/ar) constantly changes sign in y and t . Therefore, momentum cannot diffuse outward monotonically, which results in a constant width of flow. The analogous problem in heat conduction is that of a semi-infinite solid, the surhce of which is subjected to a periodic fluctuation of temperature. The resulting solution, analogous to Eq. (9.59, has been used to estimate the effective “eddy” diffusiviry in thc upper layer of the ocean from measurements of the phase difference (that is, he time lag between maxima) between the temperature fluctuations at two depths, generated by the diurnal cyclc of solar heating. 11. Hifih and 1,ow Reynolds :I:Wnber 1~’Lowx Many physical problems can be describcd by ihe behavior of a system when a certain parameter is either very small or very large. Consider the problem of steady flow around an object dcscribed by pu vu = -vp + pv2u. (9.58) First, assume that the viscosity is small. Then the dominant balance in thc flow is between the pressure and inertia forces, showing that pressure changcs are of order pU2. Consequently, we nondimensionalize the governing cquation (9.58) by scaling u by the frcc-strcam velocity U, pressure by pU2, and distance by a representative lcngth L of the body. Substituting the nondimensiond variables (denoted by primcs) the equation of motion (9.58) becomes 1 Re uf Vu’ = -Vp’ + -V2U’, (9.59) (9.60) where Re = ULv is thc Reynolds number. For high Reynolds number flows, Eq. (9.60) is solved by treating 1/Re as a small parameter. As a hst approxima- lion, we may set 1/Re to zero everywhere in thc flow, thus reducing Eq. (9.60) lo the inviscid Euler equation. However, this omission of viscous terms cannot be valid near the body because thc inviscid flow cannot satisfy the no-slip condition at the body surface. Viscous forces do become important near the body becausc of the high shcar in a layer near the body surfacc. The scaling (9.59), which assumes that veloc- ity gradients are proportional to U/L, is invalid in thc boundary layer near the solid surface. We say that there is a region of nonunifornib): near the body at which point a perturbation expansion in terms of the small parameter 1 /Re becomes singulur. The proper scaling in the boundury luyer and the procedure of solving high Reynolds number Rows will be discussed in Chapter 10. 296 Laminar Flow Now consider flows in the opposite limit of very low Rcynolds numbers, that is, Re + 0. It is clear that low Reynolds number flows will have ncgligible inertia forces and therefore the viscous and pressure forces should be in approximate balancc. For the governing equations to display this fact, we should have a small parameter multiplying the inertia forces in this case. This can be accomplished if thc variables are nondimensionalized properly to take into account the low Reynolds number nature of the flow. Obviously, the scaling (9.59), which leads to Eq. (9.60), is inappropriatc in this case. For if Q. (9.60) were multiplied by Re, then the small parameter Re would appear in front of not only the incrtia force term but also the pressure €mc term, and the governing equation would reduce to 0 = pVzu as Re + 0, which is not thc balance for low Reynolds number flows. Thc source of the inadequacy of the nondimemionalization (9.59) for low Reynolds number flows is that thc pressure is not of order pU2 in this case. As we noted in Chapter 8, for these extcrnal flows, pressure is a passive variable and it must be normalized by the dominant efFcct(s), which here are viscous forces. The purpose of scaling is to obtain nondimensional variables that are of order one, so that pressure should be scaled by pUz only in high Reynolds number flows in which the pressure forccs are of the order of the inertia forces. In contrast, in a low Reynolds numbcr flow the pressure forces are of the order of the viscous forces. For Vp to balance pVzu in Eq. (9.58), the pressure changes must have a magnitudc of the ordcr p - LpPu - pU/L. Thus the proper nondimensionalization for low Reynolds number flows is (9.61) The variations of the nondimensional variables u‘ and p’ in the flow ficld are now of ordcr one. The pressure scaling also shows that p is proportional to p in a low Reynolds number flow. A highly viscous oil is used in the bearing of a rotating shaft because the high pressure developed in the oil film of thc bearing “lifts” the shaft and prevents metal-to-metal contact. Substitution of Eq. (9.61) into (9.58) gives the nondimensional equation Re uf . Vu’ = -Vp’ + v2u’. (9.62) In the limit Re + 0, Eq. (9.62) becomes the linear equation vp = pvh: (9.63) where the variables have been converted back to thcir dimensional hm. Flows at Re << 1 are called creeping motions. They can bc due to small velocity, large viscosity, or (most coinmonly) the small sizc of the body. Examplcs of such flows are the motion of a thin film of oil in the bearing of a shaft, settling of sediment particles near the ocean bottom, and the fall of moisture drops in the atmosphere. In thc next section, we shall examine the creeping flow around a sphere. Sumrmri-y: The purpose of scaling is to generate nondimensional variables that are of order onc in the flow field (except in singular regions or boundary layers). The proper scales depend on the nature of theJlav ad are obtained by equating the terms thut are most important in the flow field. For a high Reynolds number flow, thz dominant terms are the inertia and pressure forces. This suggests the scaling (9.59). resulting in the nondimensional equation (9.60) in which the small parameter multiplies the subdominant term (except in boundary layers). In contrast, the dominant terms for a low Reynolds number flow are the pressure and viscous forces. This suggests the scaling (9.611, resulting in the nondimensional equation (9.62) in which the small parameter multiplies the subdominant term. 12. &?c?ping Flow murid a Sphere A solution for the creeping flow around a sphere wa, iirst given by Stokes in 185 1. Consider the low Reynolds number flow around a sphere of radius a placed in a uni- form stream CJ (Figure 9.14). Thc problem is axisymmetric, that is, the flow patterns are idcntical in all planes parallel to U and passing through the center of the sphere. Since Re + 0, as a first approximation we may ncglect the inertia forces altogether and solve the equation We can form a vorticity equation by taking the curl of the preceding equation, obtain- ing Here, we have used the fact hat thc curl of a gradient is zero, and that the order of thc operators curl and V2 can be interchanged. (The reader may verify this using indicia1 notation.) The only component of vorticity in this axisymmetric problem is q,, the component perpendicular to (p = const. planes in Figure 9.14, and is given by vp = pv2u. 0 = v20. In axis,mmetric flows we can dehe a streamfunction I,$; these are given in Sec- tion 6.1 8. in spherical coordinates, it is defined as u = -V(p x V$, so 1 a$ r sine ar ua = 1 ' - r2sine ae u = In terms of the streamfunction, the vorticity becomes - -1 [ 1 a2$ up- r sine ar* The governing equation is Combining the last two equations, we obtain v2w, = 0. 2 [$+-+-)I sin0 a 1 a $=O. r2 ae sin0 a0 298 Larninurlylrw -re 9.14 Creeping flow ovee a sphcrc. The uppmpanel shows lhc blur slrars componena at the sllrke. nlclowcr~ shows h~distrihtirn in rmanial (p = amst.) plane. The boundary conditions on the preceding equation ~IE Ma, e) = 0 [u, =o atsurface], (9.65) *(oo, e) = ;ur2 sin2 e [uniform at 001. (9.67) The last condition follows from the fact that the stream function for a uniform flow is (1/2)Ur2 sin2 8 in spherical coordinates (see Eiq. (6.74)). ag/ar(a, e) = 0 [ue=O atsllrfacel, (9.66) The upsheam condition (9.67) suggests a separable solution of the form @ = f(r) sinz e. Substitution of this into the governing equation (9.64) gives whose solution is D r f = Ar4+ Biz + Cr + The upstream boundary condition (9.67) requks that A = 0 and B = U/2. The surface boundary condition then gives C = -3 Ua/4 and D = Ua3/4. The solution (9.68) [...]... thereforc [-p cos 8 + or,.cos H - ore sin O],.,, , (9.71) which can be understood from Figure 9.14 TIic viscous stress componcnls are E:] ilu, = 2pu cos#: ; [ - - 2 ar or,.= 2p- gro = 1.1 [ r ; a (9. 72) I,; 1 aUr (7) + ug 3puu3 = -sin 8 , 2r4 so that Eq (9.7 1 ) becomes 3pu -cos2f9 2a +o+ 3pu 3 -sin 28 = - 2a w 2a Thc drag Iorce is obtaincd by multiplying this by the surface area 4na2 ofthc sphere, which... r m Snhion 32 1 5 Iloundq l q w on a Fht Phur: R l a s h S ohtion 323 Similarity S o l u t i o n 4 l ~ r n ~ v e hrcdiirc 324 Matching with Extcmal bivuri 327 Transvme Velocity 327 Iknindq L y : r ' I ' h i k 327 Skirt hktion 3 28 Fdknw-Slan SoliLou of thc 1 a r r h w t3ouncitu-y I.ayer Etpnnns 329 Bmakdown of h i n r u Solution 330 6 L Y ~ IKivman Mvmcnhn lnkgml 3 32 7 cl o Pre#~iun?... ( R2) 2 ' 3 08 Imakrfb wherex isjust aparameter Therefore, t e sfrcamkx correspondingtothisvelocity h potentialace identical to the potentialBOW s h r e m of E (6.35) This allows for q the ccmstmction of an apparam to visualize such potential flows by dyc injection between two closely spaced glass PI- The velocity diyiributim of h i s f o is lw 1 , u + Obutthcreisaslipvelocityue + -2YinO(l -x2) /2 ~... sphere d a circular cylinder." J FluidMeclr 2: 23 7 -26 2 Semntalsehlichhg, H (1979) & q d Luyr IlrC0r)iNew Y & McGraw-Hill Chapter 10 Boundarv Layers and Related Topic's 1 l n h c h e h n 3 12 2 Uawuhq-lqw Apptruhdbn .3 13 3 IX&?fll MeaSuN??ofl3l>un* I A ~'I'hicknms 3 1 8 T T h u =0.99UTh:kncss 3 18 l pxcnicnt 'lbickriws 3 19 h l Morncnt~im 'I-biclams 320 4 R o w I q e r on a I.ht I'ht ltc isih... IP2 2 * ( p l 0; Re) = zsin 0 Rc’ 1 + QI(p, Rc 0) +o Substituting in Eq.(9.77) and taking the limit Re + 0 yields (9. 82 ) where the opcrator The solution to Eq (9. 82 ) is round to be 2 where the constant of integration C is determined by matching in the overlap region between the inner and outer regions: I < r < 1/Re, Re < p < 1 < < < < The matching gives C2 = 314 and CI = -3116 Using this in E (9 .81 )... i I-pzar rap a2 &=-+-ar2 1 -p2 r2 a2 ap2’ We have seenthat the right-hand side o 4.(9.76) M(9.77) becomesof the same order f a the left-hand side when Re r/u 1 M r/u $ l/Re W will define the ”inner e ~gim” r/u . [r; (7) + ;,I = sin 8, - - - ar (9. 72) a ug 1 aUr 3puu3 2r4 so that Eq. (9.7 1 ) becomes 3pu -cos2f9 +o+ - 3pu sin 28 = 3w 2a 2a 2a Thc drag Iorce is obtaincd. *(oo, e) = ;ur2 sin2 e [uniform at 001. (9.67) The last condition follows from the fact that the stream function for a uniform flow is (1 /2) Ur2 sin2 8 in spherical. with those fourelments, p = me, and the operators p a ia a2 1 -p2 a2 L= + , &=-+ I-pzar rap ar2 r2 ap2’ We have seen that the right-hand side of 4. (9.76) M (9.77)