Equilibrium Similarity and Turbulence at Finite Reynolds Number

Transcription

1 Equilibrium Similarity and Turbulence at Finite Reynolds Number William K. George Professor of Turbulence Chalmers university of Technology Gothenburg, Sweden

2 The classical theory tells us (and most modern texts as well) all jets are alike asymptotically only the initial momentum matters. large scales: energy containing eddies small scales: energy dissipation (mm or less) High Re large scale separation Even the Reynolds number has been believed to be nearly irrelevant.

3 But jets really don t all look alike.

4 And Reynolds number certainly plays a role especially if it is too small. Low Reynolds # 300 High Reynolds #

5 Equilibrium similarity provides a way to understand this... Consider the fully-developed axisymmetric jet U U x + V U r = r r < uv r >, U x + r r rv = 0

6 SINGLE POINT EQUILIBRIUM SIMILARITY Change variables to: U = U s etc., where: ( x) f ( η), η = r / ( x < ) uv >= Momentum equation for jet transforms to: U s du dx s f U s All explicit x-depen-dence in square brackets. d + dx = U s du dx R s s f Rs ( x) ( x) g( η) ' η 0 f ( ηg)' η ( η ) η dη

7 EQUILIBRIUM SIMILARITY HYPOTHESIS All of the terms in square brackets of the equation must evolve with downstream distance in exactly the same way (unless they are identically zero). There are no further assumptions. If no solution consistent with boundary conditions, none will be found.

8 Copyright William K. George, 003 If we divide by second term, momentum equation for jet reduces to: η η η η η η )' ( )/ ( ) ( ) ( ) ( ' 0 g dx x d x U x R d f f f s s = η η η η η η )' ( ) ( ) ( ' 0 g x R d f f dx du U dx d U f dx du U s s s s s s = + The single bracketed term remaining can at most depend on the upstream conditions.

9 f Rs ( x) ( x) d ( x) / ( η )' η η g f ' f ( η ) η dη = 0 U s dx We can redefine the Reynolds stress profile function g to absorb the constant; i.e., η f f ' f ( η ) η dη = 0 ( η g~ ) Clearly the solutions are dependent only on, since there are no parameters left. η ' / η

10 Mean momentum equation: η f f ' f ( η ) η dη = 0 ( η g~ ) Generally not possible to do this for Reynolds stress equations. So, profiles of turbulence quantities will be different for different upstream conditions. ' / η

11 =b ( x x ), o Half-width: / Centerline velocity: U c ( x x ) = a o Copyright William K. George, 003 Thus mean velocity and properly scaled Reynolds shear stress profiles should be universal. BUT the growth rate d / dxand centerline velocity decay can reflect source conditions, i.e., U c Constants a and b depend on source conditions (c.f., Grinstein (00), Boersma et al. 998, Slessor et al. (998), Nobis et al. 00, Cater & Soria 00, Westerweel et al. 00).

12 Even in late 80 s, it was clear the two-point equations must play a major role. Dissipation scaling determined (or reflected) the spreading rate and its dependence on source Reynolds number: ε 3 c U / vs ε νu / / Differences obscured by fact that turbulent jet evolves at constant Re, no matter the source Reynolds number i.e., c / Re = U / ν = const c / Consequence of momentum conservation, M=M o.

13 This opens interesting new opportunities: Complete similarity analysis of the two-point Reynolds stress equations possible since axisymmetric jet evolves at constant local Reynolds number, U c /ν (Ewing 995). Streamwise growth is removed by stretching the coordinate system logarithmically. Similarity variables are: ξ = ln ( x x )/ D, η = o r / ( x), θ

14 Copyright William K. George, 003 The two-point equations are quite complicated, and even resulting constraints are not trivial... ( ) 3, 3 3, 3, 3,,,,,,,,,, 3,,, 3,, ) ( ) ( ) ( ) ( ) ( ) ( i j i j i j i j j i j i j i j i i s j i s j i s j i s j j i j i j i j i i i j i j i j s j i j i s Q Q Q Q Q x Q x x Q x Q x U Q dx d x U Q x U Q x U Q x T x T x T x T x x x x x U Q x Q x U ν ν ν ν ν ν ν ν + Π Π Π PLUS an additional set for the other point,. ' x

15 Two-point velocity correlation tensor become < u ( x, r, θ, t) u ( x', r', θ ', t) > i where = j U c ( x) U c d ( x') dx / d ' dx / ' Q i,j = Q i,j (ξ -ξ, η,η, θ -θ), ij Q i, j ξ = ln[(x-x 0 )/D] and η = r/ / Scaled turbulence moments are homogeneous in ξ -ξ and θ -θ.

16 TWO-POINT CORRELATIONS Copyright William K. George, 003 x/d N LDA/HW Fronapfel et al. 003 APS

17 Two-point correlations of streamwise velocity ξ ' ξ = ln[( x x o ) /( x' x o )], ξ ln[( x x o ) / D] Re D =30,000 PIV: Eljack et al. 003 APS Re D =,000 data of Westerweel et a LDA/HW Fronapfel et al. 003 APS

18 IMPLICATIONS POD application to inhomogenous flows of infinite extent (like streamwise direction of jet) is problematical since solutions depend on the domain chosen. BUT since the streamwise direction of the transformed jet is homogeneous in similarity variables... solution to POD integral in the streamwise direction is Fourier modes in. ξ = ln x / D

19 Copyright William K. George, 003 Jet flow field (PIV data of Westerweel et al. 00) Physical coordinates Similarity coordinates

20 Reconstruction using POD modes -0

21 Reconstruction using radial POD modes - and all streamwise wavenumbers.

22 Reconstruction of instantaneous field using POD only radial mode and all streamwise wavenumbers

23 Now for a more difficult problem...: all wakes don t look alike either.

24 In fact, upstream (or initial) conditions are remembered far downstream! Wakes behind four different axisymmetric generators... suggesting strongly that far wakes retain dependence on initial conditions. (Cannon 99).

25 Cannon porous disk 99 Johansson disk 00 Gourlay et al. DNS 00 These high Reynolds number axisymmetric wake profiles are nearly identical -as both the equilibrium similarity and classical theories predict.

26 But only equilibrium similarity can explain this. From P. B. V. Johansson et al. 003 Phys. Fluids

27 Clearly structure does matter... These pictures were telling us something after all.

28 But there remains a problem for the axisymmetric wake... The local Reynolds number for the high Re solution decreases with downstream distance, i.e., Re = ( U U ) / ν ( x x cl o Thus our infinite Re equations slowly become invalid if they ever were valid in the first place. ) /3

29 This animation gives a clue. Note how the vortex cores thicken as wake evolves downstream. DNS of axisymmetric wake: wake generator moves from right to left. Gourlay et al. (00)

30 u' max U o High local Re, far wake, ~ x /3 regime Equilibrium similarity of Reynolds stress equations demands that u ' /U o be constant. x x x x 0 3 x 0 4 max Low local Re, far wake ~ x / regime 0 0 x x x x 0 5 x/θ Johansson et al 003 Phys. Flds

31 F /<u >l c U o * ν R = 00 R = 400 R = 600 R = 5300 High local Re, far wake, ~ x /3 regime Copyright William K. George, 003 Local Reynolds number diminishes with distance. Velocity Spectra of Uberoi and Freymuth 970-5/3 slope Deviations from high Re solution occur quite early not nea turbulence Re of unity, but several hundred! x x x x 0 3 x 0 4 Low local Re, far wake ~ x / regime x x x x x/θ kl c Johansson et al 003 Phys. Flds. NOT laminarization, but low Re turbulence.

32 u' max U o * θ There is a second similarity state... to which the flow evolves and stays forever x x x x 0 5 a) b) x/θ U o * ν U o 500 U x x x x x x x x 0 5 x x x x 0 5 x/θ U -U U o x/θ c) d) x/θ r/ * x/θ =.96 x 0 5 x/θ =.4 x 0 5 x/θ =.9 x 0 5 x/θ = 3.48 x 0 5 x/θ = 4.0 x 0 5 x/θ = 4.79 x 0 5 x/θ = 5.55 x 0 5 x/θ = 6.40 x 0 5 x/θ = 7.35 x 0 5 x/θ = 8.4 x 0 5 Eddy viscosity Curve, Johansson

33 Other examples of evolving flows The outer part (90% plus) of turbulent boundary layer flows do this in reverse. As the Reynolds number increases downstream, they evolve toward the infinite Re solution (which is the only equilibrium similarity solution). This can be quite frustrating to those who want simple scaling laws for finite Re experiments or DNS.

34 Even so, sometimes the asymptotic equilibrium similarity solutions can work quite well especially if driven by a strong shear, imposed strain rate or pressure gradient. 0 4 Max velocity vs half-width Mean velocity U m ν/m AVKL pulsed HW 0,000 AJL HW 5,000 AJL HW 0,000 KEP LDA 0,000 WKH HW 0,000 WKH HW 9,000 eqn. (5.), using n= 0.58, B = y / M 0 /ν Plane walljet, George et al. 000 JFM

35 Result of using near-asympototics to match outer equilibrium similarity solution to inner is that normal stresses scale differently than shear. U d / dx u * Reynolds shear stress Streamwise normal Plane walljet data Abrahamsson et al. 997

36 Equilibrium similarity of pressure gradient boundary layer implies / Λ U Λ ρd / dx = const Λ( f ' op + ( Λ ) f + ' op f 0, op y f Λ 0 dp dx Castillo and George 00 AIAAJ ) + ( Λ ) yf op ( yˆ) dyˆ = ~ r ' op ' op log 0 (U oo /U ooi ) log 0 (U oo /U ooi ) log 0 (/θ i ) Copyright William K. George, 003 Bradshaw: Mild APG Clauser: Mild APG Bradshaw: Mod APG Clauser: Mod APG Ludwieg & Tillmann: Very Strong APG (event. separation) Fit: 0.(/θ)+0.7 Fit: 0.(/θ) +0. Fit: 0.(/θ) Adverse pressure gradient Herring & Norbury: Mild FPG 0.0 Ludwieg & Tillmann: Mod FPG Herring & Norbury: Strong FPG Kline et al.: Mild FPG Kline et al.: Strong FPG Fit:.95(/θ).98 Fit:.95(/θ) log 0 (/θ i ) Favorable pressure gradients

37 The velocity profiles collapse pretty well too in equilibrium similarity variables especially with the Smits-Zagarola scaling to remove the Reynolds number and upstream dependence. Λ( f ' op + ( Λ ) f + ' op f op 0 y f ) + ( Λ ) yf op ( yˆ) dyˆ = ~ r ' op ' op [( U ] U )/ U ( * / ) Castillo and George 00 AIAAJ Ludwieg & Tillman: Mod FPG: R θ = Herring & Norbury: Mild FPG: R θ = Favorable pressure gradients y/ Clauser: Mild APG Newman: Very Strong APG (sep.) Clauser: Mod APG Ludwieg and Tillman: Very Strong APG (sep.) Bradshaw: Mild APG Bradshaw: Mod APG Adverse pressure gradients y/

38 ... and even for zero pressure gradient. yf ' o f ' o 0 y f = ~ r o ' o ( yˆ) dyˆ [( U ] U )/ U ( * / ) Wosnik 000, Castillo et al Osterlund (999): ZPG y/ 95 R θ =668, x=5.5m, U oo =0.36 m/ s R θ =804, x=5.5m, U oo =3.06 m/ s R θ =9556, x=5.5m, U oo =5.78 m/ s R θ =308, x=5.5m, U oo =.3 m/ s R θ =564, x=5.5m, U oo =6.89 m/ s R θ =783, x=5.5m, U oo =3.40 m/ s R θ =056, x=5.5m, U oo =37.88 m/ s R θ =3309, x=5.5m, U oo =43.57 m/ s R θ =5767, x=5.5m, U oo =49.0 m/ s R θ =730, x=5.5m, U oo =54.04 m/ s R θ =5688, x=4.5m, U oo =0.38 m/ s R θ =6930, x=4.5m, U oo =3.0 m/ s R θ =805, x=4.5m, U oo =5.8 m/ s R θ =0386, x=4.5m, U oo =.0 m/ s R θ =886, x=4.5m, U oo =6.90 m/ s R θ =497, x=4.5m, U oo =3.36 m/ s R θ =70, x=4.5m, U oo =37.80 m/ s R θ =935, x=4.5m, U oo =43.5 m/ s

39 Obviously which terms come into play and where is very important.? Best example of this are the strained wake DNS of Rogers (003 JFM). Ten different time-dependent equilibrium similarity solutions were identified, depending on the orientation and magnitude of the applied strain rate. Equilibrium similarity solutions appear to behave as very powerful attractors.

40 Some problems remain for isotropic decaying turbulence:... Spectral energy equation E = T ν k E t E(k,t) is three dimensional energy spectrum function (averaged over spherical shells of radius k). T(k,t) is the non-linear spectral transfer.

41 Copyright William K. George, 003 George 99 Phys. Fluids (G9) seeks single length scale similarity solutions of type: E k T t E ν = ) ( ) ( ) ( ), ( ) ( ) ( ), ( t kl k k G t T t k T k F t E t k E s s = = =

42 Spectral Similarity Equation L ν E E& s F + LL& ν temporal decay Equilibrium similarity hypothesis: All of the terms in square brackets of the equation must evolve with time in exactly the same way (unless they are identically zero). There are no further assumptions. L T kf' = G + νes non-linear spectral transfer [] k F viscous dissipation

43 Theory implies spectra collapse (for fixed initial conditions) when plotted as: F= E/u λ de Bruyn Kops/Riley DNS data (5 cubed) 0.9 < t < 0.8 George and Wang 00 kλ

44 de Bruyn Kops/Riley DNS data (5 cubed) 0.9 < t < 0.8 dissipation spectra enstropy production spectra Wang and George 00 JFM

45 G9 equilibrium similarity theory deduces that the non-linear transfer is related to spectrum by: G where = 5 [ ( F + kf') 0F] + n u t initial conditions n k F and n is determined by the There are no adjustable parameters since initial conditions determine the decay exponent, n.

46 Non-linear transfer spectrum at single time Wray DNS data G = λt / νu R λ = 50, n =.5 George and Wang 00 5 G = [ ( F + kf') 0F] + n kλ k F

47 T lam/nu/u vs klam 4 L / λ = 3.4 k Kol λ =. lam T/ nu u t = 0.87 (data) t = 0.87 (G 9) t = 0.87 (K 4) -6-8 k lam George and Wang 00 DeBruyn Kops/Riley 999 n =.7

48 Experiments of Helland, van Atta and Stegun 977 G = λt / νu R = 37, n= λ G = 5 [ ( F + kf') 0F] + n k F George and Wang 00 kλ

49 Copyright William K. George, 003 DNS spectral plots from Antonia and Orlandi 003 Spectral transfer - velocity Spectral transfer - scalar

50 In spite of this... still disagreements about whether the derivative skewness behaves as the theory dictates. And there is still discussion about the behavior of the overall scaling, integral scales, etc. Does this reflect a problem with the simulations and experiments? Or is there something missing in the theory? (E.g., like there was for the axisymmetric wake.)

51 Implications for Turbulence Models Integration of equilibrium similarity spectral equations for decaying homogeneous turbulence deduces directly that: dk dt = ε and C = ( n + ) / ε d ε dt n where. = C BUT the theory also deduces that n is determined by the initial conditions! ε ε k

52 Conclusions from Equilibrium Similarity The k-epsilon model is EXACT, at least for homogeneous turbulence. C ε BUT the coefficient depends on the initial conditions, in fact probably on the initial turbulence energy spectrum. VERY BAD news indeed, since there is no way to put this information into a single point turbulence model (structurebased models??).

53 Summary and Conclusions Initial and upstream conditions affect (and even dominate) the asymptotic state of many turbulent flows. Equilibrium similarity theory accounts for (and expects) this behavior. Two-point manifestations may provide the clues and tools to understand how and why.