4 Exact laminar boundary layer solutions

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Lilian Gray
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1 4 Eact laminar bondary layer soltions 4.1 Bondary layer on a flat plate (Blasis 1908 In Sec. 3, we derived the bondary layer eqations for 2D incompressible flow of constant viscosity past a weakly crved or flat srface. We now solve them for the case of a flat plate of length L, set at zero angle of incidence to a niform stream of velocity. (Recall Fig. 5. Looking back at the non-dimensional form of the BL eqations 50, 51 and 56, we see that the general form of the BL soltion is, v, fnctions(,y ; e(. (79 We assme that the plate is thin enogh not distrb the niform eterior flow, so that We then have In dimensional form, v Re1/2, e constant, i.e., e 1. (80, v, fnctions(,y. (81 fnctions( L L, y L Re1/2 in which Re L. (82 This can be simplified even frther sing the concept of self-similarity, which we motivate as follows. Looking again at the BL eqations, we see that the flow profile at any 1 is only affected by advection from pstream < 1, and by viscos diffsion in the y direction. The flow at 1 therefore only knows abot the previos history, < 1, and not abot the ftre, > 1. Mathematically, this is de to the bondary layer eqations being parabolic. In direct conseqence, at any the profile cannot know how far the plate etends into the ftre, L. We mst therefore replace L by another length scale. The only candidate for this is itself, so we now have in which, v Re1/2, fnctions( y Re1/2 Re and δ( Re 1/2 fnctions( y, (83 δ(. (84 The flow profile ths varies with the horizontal position only via a rescaling of y by the length δ( 1/2 : it is self-similar, Fig. 8. We identify δ( as the length scale that sets the local bondary layer thickness. Yo might recognise this as a diffsive process: information diffses a distance δ t 1/2 away from the plate de to viscos effects in time t, dring which it has been advected along the plate a distance t. Given that /, etc., can depend only on the single scaling variable yre 1/2 / y(/ 1/2, it mst be possible to recast the momentm eqation as an ordinary differential eqation with y(/ 1/2 as the only independent variable, rather than a partial differential eqation in and y. (By sing a stream fnction, we will not need to consider the continity eqation eplicitly. We do this now by changing variables to η ( 1/2 y, and ξ, (

2 y y δ( δ( 1 (,y δ( 1/2 O(Re single crve for all 1 Figre 8: Self similarity of the bondary layer soltion. and epressing the scaled stream fnction as a fnction f of η 2 f(η. (86 The factors of 2 are prely for convenience. Writing ξ is also a convenience to avoid confsion between old (, y and new (ξ, η variables. In any case, the final form of the momentm eqation will depend only on η and not on ξ, for the reasons jst given. In terms of the new variables (ξ,η, the partial derivatives and ( ( y ξ η ( ( y ξ η ( ( ξ + y η ξ ( ( ξ + y η ξ ( η y ξ η 2ξ η (87 ( ( η 1/2 y 2ξ η. (88 (Check these as an eercise. The velocity components, given in terms of the stream fnction by Eqns. 53, then become: ( 1/2 y 2ξ η {(2ξ1/2 f(η} df dη, (89 and v ( ξ η { f {(2ξ 1/2 f(η} (2 1/2 2ξ η 2ξ 1/2 η } df ξ1/2. (90 2ξ dη The convective operator is then + v y ( f ξ η 2ξ η ( f ξ f 2ξ η ( 1/2 ( 1/2 (f ηf 2ξ 2ξ η. (91 Inserting these into the momentm eqation 70 with e constant, we get ( f ξ f 2ξ ( 1/2 {f } η 2ξ η 20 { ( } 1/2 2ξ η {f }. (92

3 Becase f is a fnction of η only, we can ignore the term in /ξ 0 on the LHS. Dividing the remaining terms across by the constant 2 /2ξ, and tidying p, we obtain ff + f 0. (93 (Check this as an eercise. This is the final simplified form of the momentm eqation, Eqn. 73, epressed in terms of the scaled stream fnction f. The term in f comes from the viscos terms on the RHS of the original eqation; ff from advection. We recall that the BCs for the bondary layer eqations are v 0 at the solid srface y 0. e ( at the eterior edge y. For the flat plate e. We transcribe these into BCs for Eqn. 93 by recalling that η y/ and sing Eqn. 89 ( f and Eqn. 90 (v f ηf, to get: f 0, f 0 at the solid srface η 0. f 1 at the eterior edge η. There are ths three BCs, corresponding to the order of the differential eqation. Eqations sch as 93 generally can t be solved analytically. They mst be solved nmerically. The sal method is to start at η 0 and march along the η ais, nmerically generating sccessive vales of f. If all three BCs were specified at η 0 (say if we knew f(0, f (0 and f (0 this wold be simple: We wold only need to march once, becase the entire soltion f(η wold be specified by these three conditions. However we only know two BCs at η 0: f(0 0 and f (0 0. The other is at η. To resolve this apparent impasse, we gess the third bondary vale f (0 g (for gess, and generate (by marching the correspondingly specified f g (η. Of corse this will not, in general, satisfy the third (actal BC, f ( 0. We mst therefore systematically iterate g ntil the f g (η it generates does obey f ( 0. This is known as the shooting method for solving differential eqations, Fig. 9. η gess n gess 2 gess 1 1 f Figre 9: The shooting method for solving differential eqations. In the case of the flat plate (bt not for any other BL that we shall enconter, this process can actally be simplified considerably by a special property of the differential eqation 93 called the affine transformation. We pt aη X, f(η af(x, (94 21

4 where a is an arbitrary constant, so that df dη a df dx dx dη a2 F, d 2 f dη 2 d dx (a2 F dx dη a3 F, d 3 f dη 3 a4 F. (95 (Dashes now denote differentiation with respect to X. Eqn. 93 then becomes F + FF 0, (96 which is the same as the original Eqn. 93, bt now with BCs F(0 F (0 0, F ( 1/a 2. (97 The point is that we can solve this for F(X sing any vale of a that we choose, then back-ot the soltion f( by reversing the transformation 94. Pt another way, we can choose any F (0 (let s set F (0 1 to be definite, find the corresponding F g1 (X by the sal marching method, see what F g1 ( it has, calclate the corresponding a 1/ F g1 (, then reverse the transformation to find f(η. sing this trick, we only need to do the nmerical marching process once: we have removed the need to iterate! Having obtained the nmerical soltion, what do we want to etract from it? The most important thing is sally a vale for the srface shear stress, i.e. or ( ( 1/2 ( f (Π y y0 µ µ µ y y0 2 η η0 c f Re 1/2 2(Π y y0 ρ 2 ( 1/2 2 ( ( 1/2 f (0 (98 2 1/2 f (0 2f (0, (99 2 Notice that the dependence has been absorbed into the definition of Re. From the affine transformation we know that f (0 a 3 F (0 a 3 (F g1 ( 3/2. (100 sing the nmerically calclated vale for F g1 (, we get f ( , c f Re 1/ (101 Another important qantity is the thickness of the bondary layer. How do we define this? Looking back at the definition of the scaling variable η in Eqn. 85, we see that the characteristic length scale on which the profile attains the free stream is 2/. (This eqals the δ( of Eqn. 84 to within an nimportant prefactor. It therefore sets the typical scale of the BL s thickness. Strictly, however, the profile only trly attains the free stream in the limit η (where f 1. This gives y too, which is clearly not a practical measre. One way to resolve this is to arge that for all practical prposes the free stream is reached once f to 4 figre accracy. This trns ot to be when η η δ 5, giving a length ( 2 1/2 δ η δ or δ Re1/2 2η δ. (102 22

5 Strictly, thogh, for f to trly attain 1, η δ is infinite. The qantity jst defined is obviosly arbitrary to the etent that η δ depends on the vale of smallness imposed on f 1. Two measres that avoid this difficlty are the displacement and momentm thicknesses defined by Eqns. 77 and 78. In the case of the flat plate, e, so ( δ 1 ( dy, θ 1 dy. ( As an eercise, show sing the sal relations η y /2 and f that δ Re1/2 2 lim (η f , (104 η θ Re1/2 2f ( (105 (The final nmbers have to be etracted from the nmerical soltion and are given for interest only. Ths, both δ and θ show the same behavior as δ with respect to, and, bt have finite vales. The main point to note is the diffsive scaling of thickness, as discssed above. 23

UNIT V BOUNDARY LAYER INTRODUCTION

UNIT V BOUNDARY LAYER INTRODUCTION The variation of velocity from zero to free-stream velocity in the direction normal to the bondary takes place in a narrow region in the vicinity of solid bondary. This