ZPG TBLs - Results PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs

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2 ZPG TBLs - Results PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs

3 ZPG TBLs - Results PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs

4 Logarithmic composite profile Compared with past proposals ! " # $! % #.8 8 6! " #%$ & '.6.., The composite form can now be used to fit experimental data to determine, uτ, Π and δ PhD defense Preliminaries ZPG TBLs PG TBLs Pipes & Channels DNS Extra slides 6

5 Inner profile (Monkewitz, Chauhan, & Nagib, 7), Physics of fluids duinner dy P3 = b = b = b = P5 h h h3 h h5 = P3 P5, Ξ = y du dy 5 b y b y b b y b y κb b y 3. κ (κ =.38).. 3 h y h y = ( b ) h y h y h3 y 3 h y h5 y 5 =. = 6. 3 = =. 5 =. 6. Ξ3 Ξ5 ( ); Ξ3 ( ); Ξ5 ( ); KTH (M); NDF ( ) ; Carlier & Stanislas (5) ( ); T. Nickels et al., 7( ). Comparison of new fit Ξ3 Ξ5 ( ) with previous proposals in the literature. Other functional forms shown for κ =.38 and B =.7 are :, Musker (979);, Modified Musker from Chauhan, Monkewitz, & Nagib, 7;, Spalding (96);, Van Driest from White (6);, Gersten & Herwig from Schlichting & Gersten (). PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs 3 6 of 37

6 Outer profile (Monkewitz, Chauhan, & Nagib, 7), Physics of fluids η = y/ i h h i w E (η)w tanh( w η w8 η 8 ) = κ η U Uouter w w w w8 E (η) = Z η = = = = exp t dt = γ ln(η) η O(η ) for η t i exp η h O(η ) for η, η η NDF ( ); KTH (M);, exponential wake function of Chauhan, Nagib, & Monkewitz, 7;, quartic polynomial of Lewkowicz (98); , sine wake function of Coles (956);, wake function of T. B. Nickels (). γ = Euler-Mascheroni constant PhD defense - Kapil Chauhan Monkewitz, Chauhan, & Nagib, 7, Physics of fluids.6.. Preliminaries ZPG TBLs PG TBLs of 37

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8 FIG. : (Color) Comparison of U and shape factor, H with the logarithmic theory. (a) U versus Re δ. U for experiments obtained using the power law relation (B7) and shown in red symbols. Solid black symbols are direct oil-film measurements from KTH and NDF. (b) H versus Re δ. Shape factor for the theory obtained by integration of logarithmic composite profile. Blue, Prediction of logarithmic theory, and Cyan with low Re terms of Chauhan et al. 8. Data points shown for various experiments are represented as (listed alphabetically) :, Bell;, Bruns 7 ;, Carlier & Stanislas 8 ;, DeGraff & Eaton 9 ;, Erm ;, Hites ;, Karlsson ;, Knobloch & Fernholz 3 ;, Nagib et al., ;, Naguib 3 ;, Nickels et al. ;, Österlund6 ;, Purtell et al. 5 ;, Smith 7 ;, Smith & Walker 6 ;, Wark 7 ;, Wieghardt. Additional symbols in (b) represent :, data of Petrie et al. from Fernholz & Finley 8 ;, Priyadarshana & Klewicki 9 ;, Spalart 3 ; Data of Bell and Wieghardt obtained from Coles 3.

9 Channel forms & Pipefor Flows Other functional ZPG TBLs PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs

10 Channels & Pipes The composite profile Channels & Pipes : uv u τ du inner dy = U = U inner Π κ du dy = y Re τ s y κ y s Re τ s y κ y 3 W(η), η y/h d n U dy n = η= n is odd n is even W pipe = [ exp ] 3 (p 6p 3 7p )η 3 p η p 3 η 6 p η 7 ( exp [ (p 3p 3 p )/3 ] ln(η) ) Π where, p =.75, p 3 = 6.9 and p =.876. W channel = [ exp ] (3c 6c 3 7c )η c η 3 c 3 η 6 c η 7 ( exp [ (c c 3 5c )/ ] ln(η) ) Π where, c =., c 3 = 7. and c =.7. The composite profiles for Channels and Pipes fit accurately to both DNS and experimental data. Fit provides the local κ, B and Π. PhD defense Preliminaries ZPG TBLs PG TBLs Pipes & Channels DNS 57

11 DNS of wall-bounded flows Near wall region Z y ν duτ uv du = U dy ZPG TBLs : uτ dy uτ dx ν P uv du PG TBLs : = 3 y uτ dy uτ x y uv du = Channels & Pipes : uτ dy H du 3 c y dy du ν P 3 3 y c y dy uτ x y du 3 c3 y dy Reτ Near the wall DNS of various flows show a cubic behavior for Reynolds shear stress, uv. 6 8 However, differences between DNS of ZPG TBLs by Spalart (988) and Skote () are observed for du /dy.! "$# % & ' ( )*,-. / $ 3 & ' ( )*, ' ( )*, ' ( )*,-. / $ 398;:< A #B C $ 6D6 E $"$> F " E We have also found differences at higher y for the gradient of total stress for these two DNS.. M. 6 6 PhD defense 8 GIHKJKL Preliminaries ZPG TBLs PG TBLs Pipes & Channels DNS 6

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14 Superpipe data Further evaluation Ψi is almost constant across y for κ =. but the magnitude of Ψi decreases with increasing Reynolds number. Ψi for κ =. is has a very good constant behavior for y > 6. Ψo is clearly constant across y/r for κ =., except the data with Reτ., Ψi = U.5 Ψo = (Uc U ) ln(y/r) Bo κ.5 6 ln(y ) B κ 3 < y <.5Reτ PhD defense Preliminaries ZPG TBLs PG TBLs Comment on consistency, and single κ for all Re versus composite profile approach yielding κ(re). Pipes & Channels DNS 65

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22 L U 9 U = = K = K K,,,,,, K.,, N Channels & Pipes The wake parameter, Π Π for pipe flows is higher than that for channel flows, although they have the same governing equation and boundary condition. Parallel flows do not show the development of Π at low Reynolds numbers "!#%$ &(' /3 "!#6587"9": )*-,/.,. "!#658;""< 9"7"=>58;"3 ADCFEGIH L J.K 3M// M/ :3M/NOJ 3QPR S3 / 7"3 M/NO / BADCFEGIH /! T. WVXA/CFEGIHJ L Y. VXA/CFEGIH J J,/, PR7"3 M//J M/\A/CFEGIH.Z. KZ[ KJ, WVXA/CFEGIHJ]%3^Q. ^ ^J3^ _5=//3Na`b FN TN/ TdcfegihSjkHml ^ ^J3^ _5=//3NaN NT/ dcfegihsjkhml,/..,/, ^ ^J3^ _5n TN/ TdcoegihpjkHml.,/, qsr>t6jkhvul.,/, qswxytijkhvz{ PhD defense Preliminaries ZPG TBLs PG TBLs Pipes & Channels DNS 59

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25 Superpipe data Further evaluation The 9 data points that are fitted to get κ =.65 are within ±σ (σ=standard deviation), while the points that fitted to get κ =.6 are only within.8σ. Fit to the intermediate Reynolds numbers is better. Value of κ =.6 is again in consistent agreement with those observed in previously. As the experimental Reynolds number increases, the accuracy required in obtaining wall shear stress and hence Uc needs to increase too. $ # "! $ # "! 8 *,.-/ 63A56E78:9;8=<>GF6/HE FI B BC D Uc = Ruτ ln B Bo κ ν *,.-/ 36578:9;8=<>?5 7 B Bo 6.5 Uc Ruτ ln κ ν PhD defense %'&)( 5 6 Preliminaries ZPG TBLs PG TBLs Pipes & Channels DNS 66

26 Other functional forms for ZPG TBLs PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs

27 Typical description of wall-bounded flows The two layer approach At least two length and velocity scales needed Inner : Viscous effects are dominant U U si = F inner (y/l si, Re, ) Outer : Inertial effects are dominant. Effects of viscosity are negligible U U U so = F outer (y/l so, Re, ) Matching : U = U si u τ [ ] lim F inner y/l si U so u τ [ ] lim F outer = F (Re) y/lso Also, scaling velocity for the Reynolds stresses, u i u j, (R si, R so ) are needed. Equilibrium : Self-similarity of mean flow & Self-preservation of turbulence U si L si U so L so Classical u τ ν/u τ u τ δ, George & Castillo (997) u τ ν/u τ U δ 99 Zagarola & Smits (998) u τ ν/u τ U δ /δ δ Other mean flow descriptions by T. B. Nickels (), Oberlack (), Barenblatt, Chorin, & Prostokishin () Completely self-consistent description of ZPG TBLs based on the classical (logarithmic) theory recently established by Monkewitz et al., 7 and Chauhan, Monkewitz, & Nagib, 7 PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs 7 of 37

28 Mean flow similarity George & Castillo (997) With original coefficients With modified coefficients PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs 3 9 of 37

29 Composite form (Monkewitz, Chauhan, & Nagib, 7), Physics of fluids = U H Z Z W dη IW W = 7.75 O H(Reδ ) = U =.5 W dη ln(reδ ),,...) Reδ Reδ κ ln(reδ ) C κ ln(reδ ) C IW W. ln(reθ ) ln(h) C κ κiw W κ IW W (IW W C) ln(reθ ) ln (Reθ ) κ IW W (κiw W IW W κiw W C κc ) ln3 (Reθ )... O Reθ.3 H(Reθ ) ln(h) κ IW W (IW W C) κiw W... O ln(reθ ) Reθ ln (Reθ ) κ =.38, C = 3.3, IW W = 7. (a) Shape factor H versus Reθ. KTH(M);NDF ( ); ( ) and H determined numerically from additive ( ) and multiplicative ( ) composite profiles. (b) ln(h)/κ minus the leading term IW W / ln(reθ ) versus Reθ.NDF oil-film ( ); KTH oil-film (N) experiments. PhD defense. 5, 3, 5,. 6, , 3, Preliminaries ZPG TBLs PG TBLs Pipes & Channels DNS Extra slides 5, 6, 5

30 Shape Factor, H = δ /θ H=, IW W (uτ /U ) 9 IW W = Z y (U U ) d 7. 8 Integral IW W asymptotes to a constant in ZPG TBLs. 7 The shape factor for various experiments show consistent behavior and decreases with Reynolds number The equation for H is approximated using classical theory to agree remarkably well with experimental data. 5 The asymptotic limit of shape factor is predicted as unity PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs 5 of 37

31 The Coles-Fernholz type relation and velocity defect in the outer part ,, 6, 8, U = ln(reδ ) C κ U = Finner (y ) lim F ln(y ) B inner = U = y uτ κ U U = = Fouter (y/ ) lim Fouter = U U = ln(y/ ) C η uτ κ Matching : U = lim Finner lim Fouter = U (Reδ ) Inner : Outer : W η y For ZPG TBLs, constant. δ Therefore, either of = U δ (Rotta-Clauser Delta) or δ (local boundary layer thickness) can be used in the outer part. κ =.38, PhD defense - Kapil Chauhan B =.7, C = 3.3 Preliminaries ZPG TBLs PG TBLs of 37

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33 FIG. 7: (Color) The dependence of δ 99 /δ on U : Red, integrated GC97 composite profile with original coefficients, Red , integrated GC97 composite profile with modified coefficients, Blue, integrated logarithmic composite profile, and Cyan logarithmic composite with low Re terms of Chauhan et al. 8. For explanation of symbols see caption in Fig..

34 Clauser ratio, /δ Ratio of over δ is an important parameter indicating the validity of Clauser theory. /δ reaches its asymptotic value near 3.5 and remains a constant consistent with classical theory Invalidates arguments of power law theory of George & Castillo (997) D h i Reδ = = exp Π κ(b C ) δ δ PhD defense - Kapil Chauhan Preliminaries ZPG TBLs PG TBLs 5 of 37