Boundary layer develops in the flow direction, δ = δ (x) τ

Transcription

1 58:68 Trblent Flos Handot: Bondar Laers Differences to Trblent Channel Flo Bondar laer develops in the flo direction, not knon a priori Oter part of the flo consists of interittent trblent/non-trblent otion Bt behavior of the flo in the inner laer. < is ver siilar to channel flo In the defect laer. >, the departres fro the log la are ore significant Definition Assptions: Fig. Sketch of a flat-plate bondar laer - Statistics independent of z: <W> - As the bondar laer continosl develops in the direction, statistics depend on both and. - Bernolli eqation for free strea: p const p Ths, accelerating flo corresponds to a negative or favorable pressre gradient.

2 Decelerating flo ields a positive or adverse pressre gradient Bondar laer thickness defined as the vale at hich,.99 This qantit depends on sall velocit differences. More reliable as to characterize the thickness of bondar laer are: Displaceent thickness Moent thickness * d θ d Relevant Renolds nbers: Re, Re, Re * *, Re θ θ Critical Renolds nber for zero-pressre gradient bondar laer: Flo is lainar fro to a location hich defines the start of transition hich corresponds to Re, crit bt this vale is also dependent on level of distrbances in the free strea. The bondar laer tpicall becoes fll trblent over soe distance ~3% of the distance fro the leading edge to the start of transition Mean Moent Eqations - Flo develops in the direction - Aial stress gradients are sall copared to cross-strea gradients The lateral oent eqation redces to: 6

3 p v Integrate fro the all to freestrea here the velocit flctations are eqal to zero: Integrate fro to p p all pressre The Mean Aial Moent Eqation is: p v p p p v and sing p v v 3 here the shear stress is defined as: v 4 v and as neglected as ere all the other contribtions fro streaise gradients of Renolds stresses bondar laer approiation in the original for of the aial oent eqation. In contrast to channel flo, convective ters are non-zero and cannot be deterined easil! At the all se 3 and the fact that the convective ters are zero, to obtain dp d

4 If the freestrea pressre gradient is zero and sing the definition of the shear stress 4 and the fact that <v> increases fro zero at the all proportional to 3 : hich shos that the ean streaise velocit profile varies linearl ith near. Integration of oent eqation leads to von Karan integral oent eqation Derivation for p so is not a fnction of se continit eqation and rerite 3 for zero pressre gradient as: Integrate fro to ith : d d d Add/sbstract d and integrate second and last ters in the free strea <> and the shear stress { d d Continit : ] ] ] Recall definition of θ ] θ d 443 θ

5 Mean elocit Profiles Fig. Mean velocit profiles in all nits eperients and silations La of the all still holds in the log-la region, bffer laer and the viscos sblaer. Qestion: What for does the la of the all take in the bffer laer 5< <3-5? an Driest Daping fnction for bffer laer Miing length hpothesis: - < v > So fro 4: t < > l < > t l

6 ith l l / For inner laer Soltion, < > /, /, / v 4 l l l l 5 To integrate 5 all hat e have to do is to specif l l Bt e kno that in the Log Laer: l k l 6 hich can be sed to deterine in the log-laer. If sae specification of the iing length old be sed in the viscos sb-laer: v t k ~ hereas it is ell knon that 3 <v>~ incorrect or dependence shold be 3 : So the specification l k shold be redced, or daped, near the all. an Driest daping fnction assres proper transition to viscos sblaer l ep ] ith A 6 7 A This is a prel epirical forla, bt it orks reasonabl ell and it is sed in an all odels, especiall in LES. For large, the daping fnction tends to nit and the log la is recovered.

7 Eqation 7 can be sed to integrate eqation 5 over both the viscos sblaer and the log laer to deterine and ths the ean velocit profile <>. elocit Defect La In the defect laer />., sa the ean velocit deviates fro the log la as can be seen fro Fig. 3. Fig. 3 Mean velocit profile in a trblent bondar laer Fro an etensive eaination of bondar-laer data, Coles 956 shoed that the ean velocit profile over the hole bondar laer is ell predicted b the s of to fnctions: f 443 La of the all 443 La of the ake

8 The ake fnction / is assed to be niversal sae for all bondar laers, and is defined to satisf the noralization conditions and The ake strength paraeter Π is flo dependent A convenient approiation for / is: π sin Approiate f b the log la ln B 8 For ln B 9 lnre B For given Re this eqation can be solved for / Skin friction coefficient is / c f elocit defect la sbtract 8 fro 9 ] ln

9 Edd iscosit in Defect Laer Edd viscosit definition t / Edd viscosit odel iing length t l Defect laer: the shear stress is less than and the velocit gradient than the vale / given b the log la. is larger This eans that the vale of t is less than the one given b the log-la forla t and, conseqentl, the iing length l is saller than in the defect later. This is confired b reslts fro DNS in Fig. 4 Fig. 4 Trblent viscosit and iing length variation in a trblent bondar laer. Ths l has to be adjsted. One siple a to do that is l in,.9