The mean velocity profile in the smooth wall turbulent boundary layer : 1) viscous sublayer

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Peregrine West
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1 The mean velocity profile in the smooth wall turbulent boundary layer : 1) viscous sublayer u = τ 0 y μ τ = μ du dy the velocity varies linearly, as a Couette flow (moving upper wall). Thus, the shear stress is constant: τ 0

2 scaling near wall turbulence The smooth wall TBL We can define a velocity scale u* = τ ρ [m/s] characteristic of near wall turbulence u* = shear velocity or friction velocity we can rewrite the linear profile in the viscous sublayer as where υ u is a length scale (very small, remember υ =O( ) m 2 /s, while u* is a fraction (~10-20%) of the undisturbed velocity U 0 u u = yu υ δ boundary layer height we already have 2 velocity scales: 1) u* 2) U 0 How many length scale? 1) υ u 2) δ

3 viscous sublayer continued How thick is the viscous sublayer? it depends on the flow... as u* and υ define the viscous length scale, we can quantify the extension of the viscous sublayer in terms of multiples of the viscous scale (viscous wall units) δ υ = 5 υ u Note that as u* δ υ : the viscous sublayer becomes thinner Note: roughness protrusion (fixed physical scale) may emerge from the viscous sublayer and change th near wall structure of the flow δ υ

4 The mean velocity profile in the smooth wall turbulent boundary layer : 2) the logarithmic region here is another velocity scale: the standard deviation or r.m.s. velocity velocity scale of the energy containing eddies The mixing length theory: fluid particles with a certain momentum are displaced throughout the boundary layer by vertical velocity fluctuation. This generate the so called Reynolds stresses τ = ρu v

5 If we know the stress, we can obtain by integration the velocity profile τ = ρu v mixing length assumption (Prandtl: u = l du dy ) What does it mean? A displaced fluid parcel (towards a faster moving fluid) will induce a negative velocity u ~ v such that τ = ρu v = ρl 2 du/dy 2 l represent the scale of the eddy responsible for such fluctuation very important: we also assume that the size of the eddies l varies with the height l=ky: very reasonable, farther from the wall eddies are larger

6 we thus have τ = ρk 2 y 2 du/dy 2 with u* = τ ρ integrating we obtain : u u = 1 k ln yu υ +C Logarithmic law of the wall where u* depends on the flow and the surface k is the von Karman constant(?)= (k=0.41 is a good number) C is the smooth wall constant(?) of integration (C=5.5 is a good number) note that for a rough wall boundary layer = 1 ln u k where y 0 is the aerodynamic roughness length: it is a measure of aerodynamic roughness, not geometrical (surface) roughness u y y0 relating with y 0 is complicate

7 The mean velocity profile: where is it valid? from about 60 viscous wall units to about 15% of he boundary layer height it makes sense that the extension of the log layer has to be determined by both inner scaling and outer scaling

8 What is turbulence? turbulence is a state of fluid motion where the velocity field is : highly 3D, varying in space and time, hardly predictable, varying over a wide range of scales non Gaussian, anisotropic but somehow statistically organized coherent structures +

9 Coherent structures in wall turbulence Smallest scale of the flow: kolmogorov scale (in the near atmosphere about 1mm) Largest scale of the flow: several times the boundary layer height (in the atmosphere may go up to O(1-10 Km ) There are 6-7 orders of magnitude! However IF, we understand how turbulent structures behave and IF these structures truly play a major role (statistically) on momentum, scalar and energy fluxes, mixing, etc.... Then we could propose low dimensional models, smart closures, control systems Short term goal: understand and control near wall processes (relevant for drag, lift, particle resuspension, near surface processes) Long term goal: shift turbulent closure to larger scales, in order to solve large domain accurately (atmosphere, rivers, oceans)

10 Flow visualization and sketches

11 Kline 1967 (near wall streaks) (log and outer layer) F L O W

12 flow (towards the screen) (1981) flow

13 Acarlar and Smith, 1987, downstream of a fixed hemisphere flow downstream of a low momentum fluid ejection Laminar flow upstream

14 Kline 1967 Flow visualization (hydrogen bubbles, flow markers) Robinson 1991

For a vortex line of infinite length, the induced velocity at a point is given by: V = 2 πγ /d where Γ is the strength of the vortex d

15 Hairpin vortex detection: track of a strongly 3D structure on the 2D streamwise - wall normal laser sheet: Adrian et al 2000 The Biot-Savart law is used to calculate the velocity induced by vortex lines. For a vortex line of infinite length, the induced velocity at a point is given by: V = 2 πγ /d where Γ is the strength of the vortex d is the shortest distance from a point P to the vortex line For a arch-like vortex line, there is a combined induction towards its center (ejection of low momentum fluid u v Q2 event Q2 event vortex Shear layer

16 A brief summary... Single hairpin vortices can explain the observed features of low and high speed streaks, bursting phenomena and lift up of structures (viscous & buffer layer) What is still missing so far is the outer layer, Structures were observed to form bulges with ramp-like features.

17 Numerical Simulation (Zhou, Adrian et al. 1996, 1999) isovorticity surface Self sustaining mechanism (see also Waleffe 1990) and vortex alignment Limitation : low Re with initial perturbation

18 Experimental evidence of hairpin packets in smooth wall turbulence (Adrian, Meinhart, Tomkins JFM, 2000)

19 Instantaneous flow fields: U-Uc (convection velocity) Vortex marker: swirling strength Q 4 Q 4 Ramp packet Q 2

20 Detection of zones of uniform momentum associated to the streamwise alignment of hairpin vortex: mutual induction of Q 2 event

21 Okubo-Weiss parameter Q S S S 2 n s S u x w x 2 n where : S 2 S w z u z 2 z 2 s Vortex identification u From the local velocity gradient tensor u x w x u z w z See also Chong & Perry, 1990 Jeong and Hussain 1995 Swirling Strength analysis Imaginary eigenvalues c cr i ci We select the region where 0 ci

22 Numerical simulation Adrian, PoF 2008 multimedia appendix Flow visualization, flow

24 Statistical Signature 1)Relevance 2)Physical mechanisms 3)Connection with quadrant analysis (Lu & Willmart, 1973, Wallace 1972, Nezu & Nakagava 1977) 4)Vortex identification in 2D and 3D 5)Zones of uniform momentum 6)Consistency with observed resuspension events (strong correlation between c w and u w events)

25 Besides instantaneous realizations Is it possible to obtain some quantitative information about turbulent structures? See also Proper Orthogonal Decomposition (Holmes & Lumley ) 2 point correlation 2 point correlation tensor R ρ ij * r, y, y' u x, y u x r, y' correlation coefficien t (normalized) ij x R σ ij i rx y, σ y j i, y' y' j x for i, j u, v Linear stochastic estimate Estimate of the flow field Statistically conditioned To the realization of a known event : 1) II quadrant (u < 0, v > 0) 2) IV quadrant (u > 0, v < 0) 3) Vortex identified by the swirling strength : complex part of the eigenvalue of the local velocity gradient tensor.

26 Comparison A B center (reduction of the streamwise lengthscale: lost of coherence within the structures of the packets) (see also Krogstad e Antonia 1994 rough wall) A B

27 Two point correlation streamwise velocity fluctuation Comparison A B center A B

28 Linear Stochastic Estimate: Question: What is the average flow field statistically conditioned to the realization of a vortex with a spanwise axe (identified as the signature of the hairpin vortex On the laser sheet)? The best (linear) estimate is given by u j con x' λ x x ( x, y) λ L λ x x u j x' x λ x x u x' λ x, y' u x r, y j j Adrian, Moin & Moser, 1987 Adrian 1988, Christensen 2000 Note: Information about conditioned probabilistic variables are obtained from unconditioned statistical moments λ λ x λ x

29 Linear stochastic Estimate : E known event assumed at a fixed y See Christensen 2000 Flow field obtained from a statistical analysis (conditioned to the realization of a E event) E

30 Kim and Adrian 1999

31 Spanwise alignment of hairpin structures leading to long coherent regions of uniform momentum Kim & Adrian 1999

32 VLSM Contribution : turbulent kinetic energy and Reynolds stresses Guala et al. 06

33 Net force exerted by Reynold stress in the mean momentum equation Pipe : Guala et al, 06 Pipe flow Turbulent Boundary layers channel flow channel: Turb. B. layer: Balakumar, (2007)

34 A brief summary... Large scale motion participate significantly to the Reynolds stress, thus contribute not only to TKE but also to TKE production. In terms of momentum balance, close to the wall, VLSM push the flow forward, while smaller scales slow down the flow. Such features are observed for turbulent pipe, channel and boundary layers flows

35 Atmospheric Surface Layer Reτ=O(10 6 ) Marusic & Hutchins 2008

36 Hutchins & Marusic 2007 High Re Large scale influence on the near Wall turbulence intensity: Amplitude modulation Low Re High Re Low Re Note that in different research field some type of very large scale motions are addressed with different names e.g. streamwise rolls (atmospheric science) or secondary current (river hydraulics)

37 VLSM : A visual inspection PIPE FLOW ATMOSPHERIC SURFACE LAYER (ASL) Lekakis 88, Guala et al. 06 Metzger et al. 07; Guala, Metzger, McKeon 08

Summary: dominant structural populations in turbulent boundary layers (above the near wall streaks / roughness sublayer, i.e. where vortex structure organization really matters) Adrian et al.

38 Summary: dominant structural populations in turbulent boundary layers (above the near wall streaks / roughness sublayer, i.e. where vortex structure organization really matters) Adrian et al. 2000, 2007 hairpin vortex (exact shape? transitional?) Ramp like structures (widely accepted) Schlatter & Orlu 2010 Hommema 2001

39 Very large scale motion Kim & Adrian 99 Balakumar 07 Guala et al. 06, 10, 11 Ret = Ret = 10 6 Ret = 5884 Ret = 520 Superstructures Ret = 10 6 Hutchins et al 2007, 2013 Monty et al 2009 Mathis et al Marusic, et al Ret = 10 6

The scientific and engineering relevance of large scale structures is due to their dominant

cal process e.g. surface hoar formation and related avalanche risk (Stoessel et al.

WE, 2015-16 How far can we represent the atmospheric surface layer in wind tunnel study?

40 The scientific and engineering relevance of large scale structures is due to their dominant contribution to the Reynolds stresses and to TKE production: micrometeorological process e.g. surface hoar formation and related avalanche risk (Stoessel et al. WRR, 2010) and wind turbine siting optimization and turbine lifetime e.g. Howard et al. WE, How far can we represent the atmospheric surface layer in wind tunnel study? Among the many relevant issues: 1) Inner Outer Scale separation In general, scaling of structural types 2) Thermal stability effects

At moderate-low Reynolds number they scrape the boundary layer height (Adrian 2000, Christensen 2001, Adrian 2007, Wu 2009)

41 1) About scaling and structural types 1) how far from the wall do we expect hairpin and hairpin packets to extend? Adrian et al, ) what is the correct scaling? At moderate-low Reynolds number they scrape the boundary layer height (Adrian 2000, Christensen 2001, Adrian 2007, Wu 2009) At high Reynolds we do not know for sure, mixed evidence Morris 2007, Hommema 2001, Marusic 2007 Ramp structures YES single hairpins??? do we expect ramp like coherent structures to extend up to =50m in the ASL?

42 Z [m] Comparison between the Atmospheric Surface Layer (ASL) the flat plate turbulent boundary layer (TBL) y/ = 1.2 Lehew et al.2011, 2013, Re = 4 * 10 2 T U MAX / = 30 y/ = 0.06 Re = 5 * 10 5 T U MAX / = 10 Guala et al.2010, 2011

43 Experimental methods, Statistics, Spectra ASL: 29 simultaneous hotwire probes (1 velocity component, coarse vertical resolution, time resolved) Symbols Metzger and Klewicki, Guala Metzger McKeon 2009 ASL: Production - dissipation TBL: Production - dissipation

44 ASL Two point correlation of the streamwise velocity fluctuation: the signature of ramp like structures. y/ = y/ = 0.01 TBL y/ = 0.07 y/ = 0.24

45 However, recent evidence may provide a different view (Hutchins et al. 2013) Z ref Condition at 2.14m, corresponding to z + ~ 10 4 From conditional average: evidence of dominant roll mode It is legitimate to ask if these structures are attached or non- attached (in the sense of Towsend)

CS are responsible for most of the Reynolds

A non-ordinary example on the relevance of

46 CS are responsible for most of the Reynolds stresses, thus contributing to near surface processes, such as momentum heat and vapor fluxes. A non-ordinary example on the relevance of coherent structures in the Atmospheric Surface Layer. Stoessel et al Images of deployed sonic anemometer

47 Coherent structures vs vortices Some Questions 1) What are the relevant scales for CS (inner, outer)? 2) How VLSM relate to hairpin packets (is it Reynolds number dependent)? 3) why near wall peak can be affected by outer layer structures? 4) How roughness in general can perturb CS self organization, how about complex terrain? 5) How CS grow in size? 6) Is there a hope to reproduce CS in a non-navier-stokes environment? 7) Can we really define a coherent structure? 8) Can we describe coherent structures evolution in unambiguous quantitative (not handwavy) terms? 8) can we go beyond geometrical characteristics (exp) and vorticity contour (num)? More questions: 1) How spanwise mean vorticity relates to streamwise fluctuating vorticity? 2) Do CS both scale with Kolmogorov (core) and the integral lengthscale? 3) Are CS more or less stable as compared to worms (vortex filaments) in isotropic 3D turbulence? What is the effect of a non zero mean strain ( and perhaps also mean vorticity)?

48 Soria 94 Chong & Perry 90 Chacin & Cantwell 2000 (Turb. Boundary Layer) A different view: the small scales of turbulence Luthi 2005 PTV isotropic turbulence

DAY 19: Boundary Layer

DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence