KUNDU Fluid Mechanics 2 Episode 5 potx

1 54 lirututional How which implies (6.18) It is easy to show that taking Sz parallel to the y-axis leads to an identical result. The dcrivativc dw ldz is therefore a complex quantity whose real and imaginary parts give Cartcsian components of the local velocity; dw/dz is therefore called the complex vebciry. Ifthc local velocity vector has a magnitude y and an angle a! with the x-axis, then (6.19) It may be considered rcmarkable that any twice differentiable function w(z), z = x + iy is an identical solution to Laplace's equation in the plane (x, y). A general function of the two variables (x, y) may be written as f (z, z*) where z* = x - iy is the complex conjugate of z. It is the very special case when f (z, z*) = w(z) alone that we consider here. As Laplace's equation is linear, solutions may be superposed. That is, the sums of clemental solutions are also solutions. Thus, as we shall see, flows over specific shapes may be solved in this way. 4. Flow a1 a Wall Angle Consider the complex potential w = Az" (n 2 i), (6.20) where A is a real constant. If r and 8 represent the polar coordinaks in the z-plane, then w = A(re'@)" = Ar"(cosn8 + i sinno), giving qi = Ar" cos n8 = Ar" sin ne. (6.21) For a given n, lines of constant II. can be plotted. Equalion (6.21) shows that II. = 0 for all values of r on lincs 8 = 0 and 8 = n/n. As any streamline, including the $ = 0 line, can be regarded as a rigid boundary in the z-plane, it is apparent that Eq. (6.20) is the complcx potential for flow between two plane boundaries of included angle a! = n/n. Figure 6.4 shows the flow patterns for various values of n. Flow within a certain sector of the z-plane only is shown; that within other scctors can bc found by symmetry. It is clear hat thc walls form an angle larger than 180" for n e 1 and an angle smaller than 180" lor n > 1. The complex velocity in terms of a! = n/n is which shows that at thc origin dwldz = 0 for a! e K, and diiildz = eo for a! > n. Thus, he comer is a stagnation paint forfiw in a wall angle smaller than 180"; w = Az’~ w=A9 w =AS \ 1 - w =A.P w = Az’n Figure 6.4 Irrotational flow at a wall anglc. Equipotcntial lincr arc hhcd. Fm 6.5 Stagnation flow itpresented by UI = AzZ. in contrust, it is a point of inJinile velocily for wull angles larger than 180“. In both cases the origin is a singular point. Thc pattcm for n = 1/2 corresponds to flow around a semi-infinite platc. Whcn la = 2, Ihe pattern represcnts flow in a region bounded by perpcndicular walls. By including the field within the second quadrant of the z-planc, ir is clear that n = 2 also represcnts thc flow impinging against a flat wall (Figure 6.5). Tbe streamlincs and equipotential lines are all rectangular hyperbolas. This is called a stagqnafionJluw bccause it represents llow in thc ncighborhood of the slagnation point of a blunt body. Real flows ncar a sharp change in wall slopc arc somewhat different than those shown in Figurc 6.4. For n 1 the irrotational flow velocity is infinitc at the origin, implying that thc boundary streamline (+ = 0) accelerates before rcaching this point and dccclcrslles alter it. Bernoulli’s cquation implies that thc pressure force down- stream of the corner is “adverse” or against the flow. It will be shown in Chapter 10 that an adverse pressure gradient causes separation of flow and generation of station- ary eddies. A real flow in a corner with an included angle larger than 180” would therefore separate at the comer (see the right panel of Figure 6.2). 5. Sources and Sinh Consider the complex potential i9 w = -hz= m -ln(re m ). 21s 2a The real and imaginary parts are from which the velocity components arc found as m UT = - 2ar Ug = 0. (6.22) (6.23) This clearly represents a radial flow from a two-dimensional line source at the origin, with a volume flow rate per unit depth of m (Figure 6.6). The flow represents a line sink if m is negative. For a source situated at z = a, the complex potential is m 2x w = -ln(z-a). (6.25) I ’\ Figure 6.6 Plane SOUKC. 7. Ihubkt 157 'I Figure 6.7 Plane irrotational vortcx. 6. lrmlalionnl Yorim The complcx potential iT 2n In Z. = (6.26) represents a line vortex of counterclockwisc circulation r. Its mal and imaginary parts arc - - (6.27)' 1' 1. #=-O ~= inr, 2;r 2x from which the velocity components arc found to be I' u, =o ug = 2n r The flow pattern is shown in Figure 6.7. (6.28) 7. lloubbl A doublet or dipole is obtained by allowing a sourcc and a sink of equal strcngth to approach each othcr in such a way hat their slrengths incrcase as thc separation distance gocs to zero, and that he product lends to afinite limit. lhc complex potential 'Thc argument of transccndcntal functions such as thc logwithm must always he dimcnsionlcss. Thus a consttint must bc ddcd Lo @ in Fi. (6.27) to put Ihc logarithm in proper form. This is clonc cxplicitlp when we arc solving a problcm as in Section 10 in what follows. Figure 6.8 plwc doublet. for a source-sink pair on the x-axis, with the source at x = E and the sink at x = E, is in rn 2Yr 2K w = -h(z + E) - -In (z -E) = Defining the limit of mE/x as E + 0 to be p, the preceding equation becomes e I (6.29) w=-=P P -iB zr whose real and imaginary parts are The expression for @ in the prcceding can be rearranged in Ihc form (6.30) x2+(Y+&)’=($) 2 - The streamlines, reprcscntcd by * = const., arc thcrcforc circlcs whose centers lie on thc y-axis and are tangent Lo the x-axis a1 the origin (Figure 6.8). Dircction of flow at the origin is along the negative x-axis (pointing outward from the source of the limiting source-sink pair), which is called the axis of the doublet. It is easy to show that (Excrcisc 1) thc doublct flow Eq. (6.29) can bc cquivalently defined by superposing a clockwise vortex of strength -r on thc y-axis at y = E, and a counterclockwisc vortex of strcngth r at y = E. The complex potentials for concentrated source, vortex, and doublet are all sin- gular at the origin. It will be shown in the following sections that several interesting flow patterns can be obtained by superposing a uniform flow on thcsc conccntrated singularities. 8. Fk,w past a HuJJ-Body An internsting flow rcsulls lorn superposition of a source and a uniform stream. The complex potcntial for a uniform flow of strength U is u; = Ue, which follows from integrating the relation dw/dz = u - iv. The complex potential for a source at the origin of strcngth in, immersed in a uniform flow, is m. 2n u)=UZ+-hz, (6.3 1) whosc imaginary part is in 27c = Ur sin8 + -0. (6.32) From Eqs. (6.12) and (6.13) it is clear that there must be a stagnation point lo the left ol the source (S in Figure 6.9), wherc thc uniform stream cancels the velocity of flow hm the source. Tf thc polar coordinate or the stagnation point is (a, IC), then cancellation of velocity rcquircs m u =o, 2na giving m 2XU' a=- (This result can also be found by finding dw/dz and setting it to zcro.) The value of the smamfunction at the stagnation point is therefore in 112 m 21c 21C 2 $s = Ur sin 8 + -8 = Ua sin ?r + -1c = - . The equation ol the streamlinc passing through the stagnation point is obtaincd by setting $ = $s = m/2, giving m m 2n L IJr sin8 + -8 = T. (6.33) A plot of this smamline is shown in Figure 6.9. It is a semi-infinite body with a smooth nosc, generally callcd a hay-body. Thc stagnation stredine divides thc field mr I ____ __ -e - - __ Figure 6.9 Jrroiational tlow past a iwwdimensional halr-body. The boundary streamline is givcn by + = m/2. into a region cxternal to the body and a region internal to it. The internal flow consists entircly of fluid emanating from the source, and the external region contains the originally uniform flow. The half-body resembles several practical shapcs, such as the front part of a bridge pier or an airroil; the upper hall of the flow rcsembles thc Row over a cliff or a side contraction in a wide channcl. The half-width or the body is found to be m(x - 6) h =rsinQ = 2Ycu ’ where Eq. (6.33) has been used. The half-width tends to h,,, = m/2U as H + 0 (Figure 6.9). (This result can also be obtained by noting that mass flux from the source is contained entirely within thc half-body, rcquiring the balance m = (2hmax)U at a large downstream distancc where K = U.) Thc pressure distribution can be found from Bernoulli’s equation p + 4pq2 = px + ipU2. A convenient way of represcnting pressure is through the nondimensional excess pressurc (called P~ESSKIZ coeflcient) A plot of C, on the surface of the half-body is given in Figure 6.10, which shows that there is pressure excess near the nose of the body and a pressure deficit beyond it. Tt is easy to show by integrating p over the surface that the net pressure force is zero (Exercise 2). 9. Flow pas1 a Cimular Cflinder wil/zout Cimulation The combination of a uniform stream and a doublet with its axis directed against the stream gives the irrotational flow over a circular cylinder, for the doublet strength I1 Figure 6.10 indicald by Prcssurc distribulion in irrotational flow ovcr a half-body. Prcssun: cxccss near Ihc nosc is and prcssun: dcficit elsewhcrc is indicated by 8. chosen below. Thc complex potcntial [or this combination is u:=uz+-=u e+- 1 e ( 3 where u = m. The real and imaginary parts or w give ~lr = u (r - :)sinH. (6.34) (6.35) It is sccn hat $ = 0 at r = u for all values of H, showing that the streamlinc $ = 0 represents a circular cylindcr of radius N. The streamlinc pattern is shown in Figurc 6.1 1. Flow inside the cuclc has no influcnce on that outsidc the circle. Vclocity components are from which thc flow sped on the surfacc of the cylinder is found as 41,- = l~el, ~ = 2U sink): (6.36) where what is meant is the positivc value of sin 0. This shows that thcre are stagnation points on the surfxc, whose polar coordinates are (a, 0) and (a, x). The flow reaches a maximum vclocity of 2 U at he top and bottom or the cylindcr. Pressurc distribution on the surface of thc cylinder is given by Surface distribution of prcssure is shown by thc continuous line in Figure 6.12. Thc symmetry of the distribution shows that therc is no net pressure drag. In fact, a general X Figure 6.11 Irrotational flow past a circular cyhder without circulation. 0 90” 180“ Dee from forward stagnation pint Figure 6.12 Comparison of irrohtional and observed prcssure disuibutions ovcr a circular cylinder. The observcd disiribution changes with the Rcynolds numbcr Re; a lypical behavior at high Re is indicated hy thc dashed line. result of irrotational flow theory is that a steadily moving body experiences no drag. This result is at variancc with observations and is sometimes known as d’ Alembert’s pcrrdox. The existence of tangential stress, or “skin friction,” is not the only reason for the discrepancy. For blunt bodies, the major part of the drag comes from separation of the flow from sides and the resulting generation of eddies. The surface pressure in the wake is smaller than that predicted by irrotational flow theory (Figure 6.12), resulting in a pressure drag. These facts will be discussed in further detail in Chapter 10. The flow due to a cylinder moving steadily through a fluid appears unsteady to an observer at rest with respect to the fluid a1 infinity. This flow can be obtained by - +”+ 8 c +- u = Figure 6.13 Decomposition of irmtational flow pattcm duc to a moving cylindcr. supcrposing a uniform strcam along the negative x direction to the flow shown in Figurc 6.1 1. The resulting instantaneous flow pattcm is simply that of a doublet, as is clear from thc dccornposition shown in Figure 6.13. 10. Flow pad n Cimiilar C3indcr wilh CXmulalion It was seen in thc last section that there is no net form on a circular cylindcr in steady irrotational flow without circulation. It will now bc shown that a lateral force, akin to a lift .force on an airfoil, rcsults when circulation is introduccd into the flow. Tf a clockwise line vortex of circulation -r is added to the irrotational flow around a circular cylinder, the complex potential becomes ui = U z+ - + -ln(z/u)! ( :) 1: whose imaginary part is (6.37) (6.38) where we have added to 111 the term -(ir/2x) lna so that the argumcnl of the logd- rithm is dimcnsionless, as it must be always. Figurc 6.14 shows thc resulting streamline pattern for \w-ious valucs of r. The close sl.reamline spacing and higher velocity on top of thc cylinder is due to the addition of velocity fields of the clockwise vortcx and the uni€orm stream. In contrast, the smallcr velocities at the bottom of the cylinder are a result of the vortex field countcracling the uniform stream. Bernoulli’s cquation consequently implics a higher pressurc below thc cylinder and an upward ‘‘lift” lorce. Thc tangential vclocity component at any point in the flow is At the surface of the cylinder, velocity is entirely tangential and is givcn by (6.39) r ug Ira = -2U sin8 - -, 2rra [...]... sign, so that thc complcx potential is m rn m w = -In (z -a) -In(z a) - -h a 2 , 2r 5 2n 2n m m =- 1n(x~-y*-u~+i2xy) 11na~ (6.48) 25 r 2? r Wc know that the logarithm of any complex quantity C = I . ha2, 25 r 2n 2n m m 25 r 2? r =- 1n(x~-y*-u~+i2xy) 11na~. (6.48) Wc know that the logarithm of any complex quantity C = I< I exp (iQ) can be written as In 5 = In 15. far from the body. The constant terms U2, po/p, -U2 /2 integrate to zero around the closcd path. Thc quadratic terms u’u’: (uR + vR) /2 5 I/r2 as r + oc and thc perimeter of the. found from the Bernoulli equation P + P 92/ 2 = poc + pu2 /2. Using Eq. (6.39), the surface pressure is found to be p,,=poo+~p -2~ sine 2Yra )'I . (6.41) The symmetry of Row