Chapter 6 Momentum Transfer in an External Laminar Boundary Layer

Transcription

1 6. Similarit Soltions Chapter 6 Momentm Transfer in an Eternal Laminar Bondar Laer Consider a laminar incompressible bondar laer with constant properties. Assme the flow is stead and two-dimensional aligned with a flat plate (Fig ). Figre Laminar bondar laer on a flat plate The - component of the conseration of momentm; ~ ~ ~ ~ p ~ ~ ~ (5.a) From an Order of Magnitde Analsis (OMA), the nd iscos term is determined to be >> than the st iscos term, and therefore the st term is neglected. p ~ Since = constant, one can show b Bernolli's eq.: simplifing eq(5.a) ~ ~ Recall the continit eq; Bondar conditions: = ~ ~ ~ ~ ~ (6.) (6.)

2 Simplif eq (6.) b appling the following similarit transformation. Facilitate the soltion b introdcing the stream fnction, Define: and Since these definitions directl satisf the continit eq (6.), now sbstitte these definitions into the momentm eq (6.), 3 3 (6.3) Bondar = Althogh the bondar laer eqations are mch simpler to sole, eq(6.3) is still a non-linear P.D.E. Blasis, a stdent of Prandtl, was able to maniplate these eqations and obtain a series soltion. The general strateg Blasis emploed follows:. Introdce two new spatial coordinates. = (, ) and = (, ) (6.4). Epress and its deriaties in terms of and (6.5) 3. Now sbstitte the following general soltion into eq 6.3 and separate ariables. (, ) = G () F () (6.6)

3 Based on eperience the following coordinate transformation was defined, and (6.7) the deriaties become, 3 / (6.8a) and (6.8b) In a similar manner repeat this process for the higher order deriaties. Then sbstitte back into eq (6.3) and simplif, 3 3 (6.9) Based on step 3 in Blasis s strateg, separate ot the and dependence. GF' F' F' G' G'FF" F' ' ' 3

4 Let G ; which is onl a fnction of, the eqation redces to; FF + F = (6.) where F is onl a fnction of The bondar conditions (6.) or in terms of F sing eq (6.6), the first b.c becomes: F' and a second b.c. = (6.a) = (6.b) and finall a third b.c. F = GF ' So can be rewritten as, F' (6.c) F GF, F (6.3) 4

5 and F' (6.4a) F' F (6.4b) These reslts show is a f() onl, whereas depends on both and. sing a series epansion Blasis (in 98) fond the soltion to eq (6.). The following table presents a newer, more accrate nmerical soltion; Soltion of the Bondar-Laer Eqations for Flow oer a Flat Plate with Zero Pressre Gradient F' F F F = / F From this table it is obsered that = = 5. This ale b definition has been taken as the edge of the bondar laer. From eq.(6.7) or and if = 5, then 5 (6.5) 5

6 6 Inspection of the following figre proides insight into F and F ariations (i.e., the elocit profile and shear stress distribtion, respectiel). sing the chain rle the shear stress at the wall can be ealated, F ' ' and therefore, wall wall F ' ' F ' ' from the aboe table, F = ;.33 (6.6) F F =/ o

7 .33 Non-dimensionalizing to the local skin friction coefficient (c f ), or c f c f.664 (6.7a&b) c.664 f Re So the resltant Drag/nit width on one side of a flat plate of length L follows; Drag nit width L d Drag nit width L.33 d Drag nit width.664 L/ L The corresponding dimensionless drag coefficient, is also referred to as the skin friction coefficient (Cf); C f Drag L C f.664 L / Re L L 7

8 .38 Cf Re (6.8) L For completeness sole for the displacement thickness (d) and the momentm thickness (): d d and d (6.9a&b) In terms of the new ariables and sing the fact that the flow is incompressible; d F' d and F' F' d Recall is reached at approimatel = 5 5 d F' d and = d d F 5 5 F 5 (6.a). 7 d And 5 F' F' d and = 8

9 F ' F ' d (6.b) Note: i) d is obsered to be /3 ii) The relationship for drag proides an estimate which is er close to that obsered in the laborator for ReL > 4 6. Similarit Soltions for the Laminar Incompressible Bondar Laer when =c m The following famil of two-dimensional Potential Flows are addressed sing similarit transformations. Soltions to these problems hae shown that the elocit along the srface aries as = c m, where, / d m (6.) / d 9

10 Soltions to these flows are commonl referred to as wedge or Falkner-Skan soltions to the bondar laer problem. The case of constant freestream elocit (corresponding to = ) was preiosl soled b the Blasis approach. Also inclded in this special class of problems is the famil of D stagnation point flows; =. Starting with a rearranged bondar laer momentm eq dp (5.4a) d sing Bernolli's eq, the pressre gradient term can be epressed as, p cons tant dp d d d Copled with the epression for elocit, dp d c m c m m Sbstitte back into eq (5.4a) m (6.) sing the same procedre as Blasis, sbstitte,eq (6.6) into eq (6.) and transform (,) to (, ), F and where F = F () / (6.3a & b)

11 Combining eqs (6.3a & b) and differentiating: = F ( ) Eq (6.) becomes, F' '' m FF' ' m F' the etra term (6.4) sing the same bondar conditions as the flat plate problem; F () = F () = and F () = The soltion is then obtained b soling (eq 6.4) nmericall. The following table presents ales of m,, and F"() Soltions to the laminar constant-propert bondar laer with an impermeable wall and = C m m F ()..33 Stagnation Flat Plate Separation Note: - Soltions for m > are niqe. - Two families of soltions eist for m < : a) Decelerating mainstream to the point of incipient separation. b) Flows in a laminar bondar laer after separation. A local skin friction coefficient can be estimated: / m / c f F' '() Re (6.5)

12 6.3 Similarit Soltions for the Laminar Incompressible Bondar Laer when V Consider the case when the elocit normal to the srface is nonzero, i.e.,v Sch applications inclde flow oer poros srfaces (i.e., blowing in film cooling applications, sction for bondar laer separation control, and condensation or eaporation on a srface). The momentm eqation (eq 5.4a) is still the appropriate eqation to sole; howeer, the bondar conditions are now changed. Starting with eq 6. (bondar laer momentm eq) and 6.3a & b, where; (recall if the Re is large we can eliminate the st iscos term) and elocities, F and m c Sole for V b combining these relations; V F m m / m c c F' m Bt: = and therefore at =, so the initial elocit in the -direction becomes, m m / V F c (6.6) Rearranging sing = c m and m V F (6.7) /

13 F V Re (6.8) m sing F() = constant as a bondar condition to eq(6.4) prodces a famil of similarit soltions for V. Howeer, for F() eqal to a nonzero constant, V mst ar along the srface in the following fashion, V is a constant for stagnation point flow (m = ) m V / (6.9) For the case when m =, = constant and V ½ gies similarit in the elocit profiles. A new parameter freqentl sed in blowing or sction applications is V Re (6.3) Eq 6.4 has been soled nmericall for arios ales of this parameter and has been listed in the following table, Soltions to the laminar constant-propert bondar laer with blowing or sction at the wall and = const Re F () Flow Condition (Shear Stress) Sction Blowing (Separation) 3

14 4 6.4 Integral Method Approach to Bondar Laer Flows Starting with the Conseration of Momentm eq(5.4a) modified for the iscos term, d dp (5.4a) Replacing the pressre gradient term sing Bernolli's eq: d dp Integrate w.r.t. "" d d d (6.3) Separatel integrate, recall integrating b parts: d a b ab d b a with = d d d d d Sbstitte back into eq(6.3) d d d after some maniplation the bondar laer momentm integral becomes,

15 5 d d d d Recall eq(6.9a&b), displacement and momentm thickness d d and d Sbstitting; d d d (6.3) Note: For a trblent flow inclde the iscos term with the Renolds stress, i.e., ' ' Since is a constant for a flat plate eq 6.3 becomes, the rate of increase in the is eqal to the normalized wall shear stress. f c (6.33) Eample : Assme a bondar laer elocit profile is gien b, Determine the wall shear and (Remember from the similarit soltion, Cf =.664 for a flat plate)