Experimental study on turbulent boundary-layer flows with wall transpiration
by
Marco Ferro
October 2017 Technical Reports Royal Institute of Technology Department of Mechanics SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillst˚andav Kungliga Tekniska H¨ogskolan i Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen fredag den 24 November 2017 kl 10:15 i Kollegiesalen, Kungliga Tekniska H¨ogskolan, Brinellv¨agen 8, Stockholm.
TRITA-MEK 2017:13 ISSN 0348-467X ISRN KTH/MEK/TR-17/13-SE ISBN 978-91-7729-556-3
c Marco Ferro 2017 Universitetsservice US–AB, Stockholm 2017 Experimental study on turbulent boundary-layer flows with wall transpiration
Marco Ferro Linn´eFLOW Centre, KTH Mechanics, Royal Institute of Technology SE-100 44 Stockholm, Sweden
Abstract Wall transpiration, in the form of wall-normal suction or blowing through a permeable wall, is a relatively simple and effective technique to control the be- haviour of a boundary layer. For its potential applications for laminar-turbulent transition and separation delay (suction) or for turbulent drag reduction and thermal protection (blowing), wall transpiration has over the past decades been the topic of a significant amount of studies. However, as far as the turbulent regime is concerned, fundamental understanding of the phenomena occurring in the boundary layer in presence of wall transpiration is limited and consid- erable disagreements persist even on the description of basic quantities, such as the mean streamwise velocity, for the rather simplified case of flat-plate boundary-layer flows without pressure gradients. In order to provide new experimental data on suction and blowing boundary layers, an experimental apparatus was designed and brought into operation. The perforated region spans the whole 1.2 m of the test-section width and with its streamwise extent of 6.5 m is significantly longer than previous studies, allowing for a better investigation of the spatial development of the boundary layer. The quality of the experimental setup and measurement procedures was verified with extensive testing, including benchmarking against previous results on a canonical zero-pressure-gradient turbulent boundary layer (ZPG TBL) and on a laminar asymptotic suction boundary layer. The present experimental results on ZPG turbulent suction boundary layers show that it is possible to experimentally realize a turbulent asymptotic suction boundary layer (TASBL) where the boundary layer mean-velocity profile becomes independent of the streamwise location, so that the suction rate constitutes the only control parameter. TASBLs show a mean-velocity profile with a large logarithmic region and without the existence of a clear wake region. If outer scaling is adopted, using the free-stream velocity and the boundary layer thickness (δ99) as characteristic velocity and length scale respectively, the logarithmic region is described by a slope Ao = 0.064 and an intercept Bo = 0.994, independently from the suction rate (Γ). Relaminarization of an initially turbulent boundary layer is observed for Γ > 3.70 × 10−3. Wall suction is responsible for a strong damping of the velocity fluctuations, with a decrease of the near-wall peak of the velocity-variance profile ranging from 50% to 65% when compared to a canonical ZPG TBL at comparable Reτ . This decrease in the turbulent activity appears to be explained by an increased stability of the near-wall streaks.
iii Measurements on ZPG blowing boundary layers were conducted for blowing rates ranging between 0.1% and 0.37% of the free-stream velocity and cover the range of momentum thickness Reynolds number 10 000 / Reθ / 36 000. Wall-normal blowing strongly modifies the shape of the boundary-layer mean- velocity profile. As the blowing rate is increased, the clear logarithmic region characterizing the canonical ZPG TBLs gradually disappears. A good overlap among the mean velocity-defect profiles of the canonical ZPG TBLs and of the blowing boundary layers for all the Re number and blowing rates considered is obtained when normalization with the Zagarola-Smits velocity scale is adopted. Wall blowing enhances the intensity of the velocity fluctuations, especially in the outer region. At sufficiently high blowing rates and Reynolds number, the outer peak in the streamwise-velocity fluctuations surpasses in magnitude the near-wall peak, which eventually disappears.
Key words: Turbulent boundary layer, boundary-layer suction, boundary-layer blowing, wall-bounded turbulent flows, self-sustained turbulence.
iv Experimentell studie av turbulenta gr¨ansskikt med v¨aggenomstr¨omning
Marco Ferro Linn´eFLOW Centre, KTH Mekanik, Kungliga Tekniska H¨ogskolan SE-100 44 Stockholm, Sverige
Sammanfattning Genom att anv¨anda sig av genomstr¨ommande ytor, med sugning eller bl˚asning, kan man relativt enkelt och effektivt p˚averka ett gr¨ansskikts tillst˚and.Genom sin potential att p˚averka olika str¨omningsfysikaliska fenomen s˚asom att senarel¨agga b˚adeavl¨osning och omslaget fr˚anlamin¨ar till turbulent str¨omning (genom sugning) eller som att exempelvis minska luftmotst˚andet i turbulenta gr¨ansskikt och ge kyleffekt (genom bl˚asning),s˚ahar ett otaligt antal studier genomf¨orts p˚a omr˚adetde senaste decennierna. Trots detta s˚a ¨ar den grundl¨aggande f¨orst˚aelsen bristf¨allig f¨or de str¨omningsfenomen som intr¨affar i turbulenta gr¨ansskikt ¨over genomstr¨ommande ytor. Det r˚aderstora meningsskiljaktigheter om de mest element¨ara str¨omningskvantiteterna, s˚asommedelhastigheten, n¨ar sugning och bl˚asningtill¨ampas ¨aven i det mest f¨orenklade gr¨ansskiktsfallet n¨amligen det som utvecklar sig ¨over en plan platta utan tryckgradient. F¨or att ta fram nya experimentella data p˚agr¨ansskikt med sugning och bl˚asninggenom ytan s˚ahar vi designat en ny experimentell uppst¨allning samt tagit den i bruk. Den genomstr¨ommande ytan sp¨anner ¨over hela bredden av vindtunnelns m¨atstr¨acka (1.2 m) och ¨ar 6.5 m l˚angi str¨omningsriktningen och ¨ar d¨armed betydligt l¨angre ¨an vad som anv¨ants i tidigare studier. Detta g¨or det m¨ojligt att b¨attre utforska gr¨ansskiktet som utvecklas ¨over ytan i str¨omningsriktningen. Kvaliteten p˚aden experimentella uppst¨allningen och valda m¨atprocedurerna har verifierats genom omfattande tester, som ¨aven inkluderar benchmarking mot tidigare resultat p˚aturbulenta gr¨ansskikt utan tryckgradient eller bl˚asning/sugningoch p˚alamin¨ara asymptotiska sugningsgr¨ansskikt. De experimentella resultaten p˚aturbulenta gr¨ansskikt med sugning bekr¨aftar f¨or f¨orsta g˚angen att det ¨ar m¨ojligt att experimentellt s¨atta upp ett turbulent asymptotiskt sugningsgr¨ansskikt d¨ar gr¨ansskiktets medelhastighetsprofil blir oberoende av str¨omningsriktningen och d¨ar sugningshastigheten utg¨or den enda kontrollparametern. Det turbulenta asymptotiska sugningsgr¨ansskiktet visar sig ha en medelhastighetsprofil normalt mot ytan med en l˚anglogaritmisk region och utan f¨orekomsten av en yttre vakregion. Om man anv¨ander yttre skalning av medelhastigheten, med fristr¨omshastigheten och gr¨ansskiktstjockleken som karakt¨aristisk hastighet respektive l¨angdskala, s˚akan det logaritmiska omr˚adet beskrivas med en lutning p˚a Ao = 0.064 och ett korsande v¨arde med y-axeln p˚a Bo = 0.994, som ¨ar oberoende av sugningshastigheten. Om sugningshasigheten normaliserad med fristr¨omshastigheten ¨overskrider v¨ardet 3.70×10−3 s˚a˚aterg˚ar det ursprungligen turbulenta gr¨ansskiktet till att vara lamin¨art. Sugningen genom v¨aggen d¨ampar hastighetsfluktuationerna i gr¨ansskiktet med upp till
v 50 − 60% vid direkt j¨amf¨orelse av det inre toppv¨ardet i ett turbulent gr¨ansskikt utan sugning och vid j¨amf¨orbart Reynolds tal. Denna minskning av turbulent aktivitet verkar h¨arstamma fr˚anen ¨okad stabilitet av hastighetsstr˚aken n¨armast ytan. M¨atningar p˚aturbulenta gr¨ansskikt med bl˚asninghar genomf¨orts f¨or bl˚asningshastighetermellan 0.1 och 0.37% av fristr¨omshastigheten och t¨acker Reynoldstalomr˚adet(10−36)×103, med Reynolds tal baserat p˚ar¨orelsem¨angds- tjockleken. Vid bl˚asninggenom ytan f˚arman en stark modifiering av formen p˚a hastighetesf¨ordelningen genom gr¨ansskiktet. N¨ar bl˚asningshastigheten ¨okar s˚a kommer till slut den logaritmiska regionen av medelhastigheten, karakt¨aristisk f¨or turbulent gr¨ansskikt utan bl˚asning,att gradvis f¨orsvinna. God ¨overens- st¨ammelse av medelhastighetsprofiler mellan turbulenta gr¨ansskikt med och utan bl˚asningerh˚allsf¨or alla Reynoldstal och bl˚asningshastighetern¨ar profil- erna normaliseras med Zagarola-Smits hastighetsskala. Bl˚asning vid v¨aggen ¨okar intensiteten av hastighetsfluktuationerna, speciellt i den yttre regionen av gr¨ansskiktet. Vid riktigt h¨oga bl˚asningshastigheteroch Reynoldstal s˚akommer den yttre toppen av hastighetsfluktuationer i gr¨ansskiktet att ¨overskrida den inre toppen, som i sig gradvis f¨orsvinner.
Nyckelord: Turbulent gr¨ansskikt, gr¨ansskiktssugning, gr¨ansskiktsbl˚asning, v¨aggbundna turbulenta fl¨oden, sj¨alv-f¨ors¨orjande turbulens.
vi Other publications The following paper, although related, is not included in this thesis. Marco Ferro, Robert S. Downs III & Jens H. M. Fransson, 2015. Stagnation line adjustment in flat-plate experiments via test-section venting. AIAA Journal 53 (4), pp. 1112–1116.
Conferences Part of the work in this thesis has been presented at the following international conferences. The presenting author is underlined. Marco Ferro, Robert S. Downs III, Bengt E. G. Fallenius & Jens H. M. Fransson. On the development of turbulent boundary layer with wall suction. 68th Annual Meeting of the APS Division of Fluid Mechanics. Boston, 2015. Marco Ferro, Bengt E. G. Fallenius & Jens H. M. Fransson. On the turbulent boundary layer with wall suction. 7th iTi Conference in Turbulence. Bertinoro, 2016. DOI: 10.1007/978-3-319-57934-4 6. Marco Ferro, Bengt E. G. Fallenius & Jens H. M. Fransson. On the scaling of turbulent asymptotic suction boundary layers. 10th international symposium on Turbulence and Shear Flow Phenomena (TSFP10). Chicago, 2017.
vii
Contents
Abstract iii
Sammanfattning v
Introduction 1
Chapter 1. Basic concepts and nomenclature 3 1.1. Nomenclature 3 1.2. Definition of the problem 4 1.3. Turbulent boundary layers without transpiration 7
Chapter 2. Boundary-layer flows with wall transpiration 13 2.1. Laminar asymptotic suction boundary layers 13 2.2. Turbulent boundary layers with transpiration 14 2.2.1. The development of turbulent boundary layers with wall transpiration 14 2.2.2. The turbulent asymptotic suction boundary layer 15 2.2.3. Self-sustained turbulence in suction boundary layers 16 2.2.4. Mean-velocity profile 17 2.2.5. Reynolds stresses 27
Chapter 3. Experimental setup and measurement techniques 29 3.1. Wind tunnel 29 3.1.1. Test-section modifications 29 3.1.2. Traverse system 31 3.2. Perforated flat plate 33 3.2.1. Design and construction 33 3.2.2. Measurement station 36 3.3. Suction/blowing system 38 3.4. Instrumentation 39 3.4.1. Air properties 39
ix 3.4.2. Differential pressure measurements 39 3.5. Hot-wire anemometry 39 3.5.1. Introduction 39 3.5.2. Sensors characteristics 41 3.5.3. Sensors operation and calibration procedure 42 3.6. Transpiration velocity determination 43 3.7. Skin-friction measurement 45 3.7.1. Oil-film interferometry 46 3.7.2. Hot-film sensors 51 3.7.3. Miniaturized Preston tube 52
Chapter 4. Measurement procedure and data reduction 57 4.1. Preparation of an experiment 57 4.2. Heat transfer to the wall and outliers rejection 57 4.3. Estimation of friction velocity and absolute wall distance 59 4.3.1. Non-transpired turbulent boundary layers 60 4.3.2. Laminar/transitional suction boundary layers 60 4.3.3. Turbulent suction boundary layers 60 4.3.4. Turbulent blowing boundary layers 61 4.4. Intermittency estimation 62
Chapter 5. Results and discussion 65 5.1. Zero-pressure-gradient turbulent boundary layer 65 5.1.1. Assessment of the canonical state 65 5.1.2. Skin-friction coefficient 67 5.1.3. Statistical quantities 68 5.2. Zero-pressure-gradient suction boundary layers 75 5.2.1. Laminar ASBL 75 5.2.2. Self-sustained turbulence suction-rate threshold 76 5.2.3. Development of turbulent boundary layer with suction 79 5.2.4. Mean-velocity scaling for the turbulent asymptotic state 89 5.2.5. Profiles of streamwise velocity variance 100 5.2.6. Spectra 108 5.2.7. Higher order moments 109 5.3. Zero-pressure-gradient turbulent blowing boundary layers 114 5.3.1. Mean-velocity and velocity-variance profiles 115 5.3.2. Spectra and higher-order statistics 119
Concluding remarks 127
x Acknowledgements 131
Bibliography 133
xi
Introduction
This thesis deals with the study of the low subsonic (incompressible) flow regime of viscous fluids in the immediate vicinity of a wall. This region, called boundary layer by Prandtl (1904), is where the relative velocity of the fluid with respect to the surface transitions from a finite value to the zero value at the surface. This deceleration of the fluid is a consequence of the non-negligible action of the frictional forces, which impose the no-slip condition at the wall. The theory of boundary layers has evident engineering relevance because it explains and provides the tools necessary to predict the friction drag and phenomena such as the boundary-layer separation, responsible for the form drag (also denoted as the pressure drag) of an object in relative motion in a fluid. In addition, turbulent boundary layers in simplified geometries (such as circular pipes or flat plates) has become very important for the theoretical investigation on the nature of turbulence, providing well-defined standards against which various theories can be tested. In particular, this thesis is devoted to boundary layers spatially developing on a permeable surface, through which wall-transpiration (suction or blowing) is applied. Methods to modify and control the boundary-layer behavior have been sought from the earliest stage of boundary-layer studies and, in this respect, wall-normal suction and/or blowing immediately appeared as a relatively simple and very effective control technique. Already in Prandtl’s very first paper on boundary-layer theory, he showed the possibility of avoiding flow separation on one side of a circular cylinder with the application of a small amount of suction through a spanwise slit on the surface (see Prandtl 1904). Localized suction has been explored as a technique to postpone separation on wings and hence to increase the maximum lift coefficient (Schrenk 1935; Poppleton 1951). Furthermore, wall suction has a strong stabilizing effect on boundary layers, and has also been investigated as a technique to delay laminar-turbulent transition in order to accomplish drag reduction by the inherent lower friction drag of a laminar boundary layer in comparison with a turbulent boundary layer. Studies on flat-plate flows have, for instance, been performed by Ulrich (1947) and Kay (1948), while more recently Airbus carried out a series of tests where transition delay was sought applying suction through a micro-perforated surface on the leading edge of the A320-airliner vertical fin (Schmitt et al. 2001; Schrauf &
1 2 Introduction
Horstmann 2004). Distributed blowing has been investigated as a skin-friction drag reduction technique for turbulent boundary layers (see Kornilov 2015 for a review on the topic), while localized blowing, known as film cooling, is commonly adopted for the thermal protection of surfaces exposed to high-temperature flows such as the turbine blades of jet engines (see e.g. Goldstein 1971). Despite the practical interests of boundary layers with wall-normal mass transfer and the numerous investigations on the topic, fundamental understand- ing on the phenomena occurring in turbulent boundary layers in presence of wall transpiration is limited. Considerable disagreement persists in the literature even on the description of basic quantities, such as the the mean streamwise velocity, for the rather simplified case of flat-plate boundary-layer flow with uniform transpiration and no pressure gradient. The objective of this research is to expand the knowledge on this type of flows providing new experimental evidence and generating a database available to the research community. In order to meet this objective, a significant part of this research project was devoted to the design and construction of an experimental apparatus capable to generate well-defined transpired boundary layers, which now remains available for future investigations on this type of flows. Chapter 1
Basic concepts and nomenclature
In this thesis incompressible boundary layers spatially developing on a permeable flat plate are considered and in this chapter the main physical quantities of the problem are defined. A brief introduction to the common notation in wall- bounded turbulent flows is also given, together with a short summary on the non-transpired zero-pressure-gradient turbulent boundary layer, denoted ZPG TBL in the following. For a more thorough introduction the interested reader is referred to turbulence or boundary-layer textbooks (see e.g. Monin & Yaglom 1971; Pope 2000; Schlichting & Gersten 2017). The description of boundary layer flows in presence of wall-transpiration and a review of the previous studies on the topic will instead be given in Chapter 2.
1.1. Nomenclature
2 Cf : friction coefficient 2τw/ρU∞ (-); Rex0 : streamwise-coordinate Reynolds Cp: pressure coefficient number corrected for virtual 2 0 2(P − P∞)/ρU∞ (-); origin U∞x /ν (-); f: indicates both frequency (Hz) or Reδ∗ : displacement-thickness Reynolds ∗ a generic function; number U∞δ /ν (-); fcut: cut-off frequency of anemometer Reθ: momentum-thickness Reynolds low-pass filter (Hz); number U∞θ/ν (-); fmax: maximum resolved frequency, Reτ : friction Reynolds number defined as min(fsmp, fcut) (Hz); uτ δ99/ν (-); fsmp: sampling frequency (Hz); Suu: one-sided power-spectral- H12: boundary-layer shape factor density estimate of the δ∗/θ (-); streamwise-velocity fluctua- 2 2 −1 Lw: hot-wire sensor length (m); tions (m /s Hz ); ∗ ` : viscous length ν/uτ (m); T : temperature (K); P : mean pressure (Pa); t: time (s); R: specific gas constant of tsmp: sampling time (s); air (J kg−1 K−1) or electrical U: mean streamwise velocity (m/s); resistance (Ω); u0: streamwise-velocity fluctua- Re: representative Reynolds num- tions (m/s); p ber (-); uτ : friction velocity τw/ρ (m/s); Rex: streamwise-coordinate Reynolds V : mean wall-normal veloc- number U∞x/ν (-); ity (m/s);
3 4 1. Basic concepts and nomenclature
V0: spatially-averaged wall-normal η: wall-normal distance normalized velocity at the surface (m/s); with an outer length scale (-); v0: wall-normal-velocity fluctua- θ: boundary-layer momentum thick- tions (m/s); ness (m); W : mean spanwise velocity (m/s); κ: von K´arm´anconstant (-); 0 w : spanwise-velocity fluctua- λl: wavelength of the light (m); tions (m/s); λx: streamwise wavelength of the x: streamwise position (m); velocity fluctuations (m); x0: streamwise position corrected for µ: dynamic viscosity (Pa s); virtual origin (m); ν: kinematik viscosity (m2/s); y: wall-normal position (m); Π: wake parameter (-); z: spanwise position (m); ρ: density (kg/m3); τ: mean total shear stress (N/m2); 2 Greek Symbols: τw: mean wall shear stress (N/m ); 0 τw: wall shear stress fluctua- tions (N/m2); Γ: transpiration rate |V0|/U∞ (-); Γsst: maximum suction rate for self- Superscripts: sustained turbulence (-); : denotes time average; γ: intermittency of the velocity +: denotes normalization with vis- signal (-); cous scales; ∆: Rotta-Clauser length scale ∗ Subscripts: δ U∞/uτ (m); δ: generic boundary-layer thick- ∞: denotes the free-stream condi- ness (m); tions; δ99: 99% boundary-layer thick- s: denotes the conditions at the ness (m); suction/blowing start location; δ∗: boundary-layer displacement as: denotes the asymptotic condi- thickness (m); tion;
1.2. Definition of the problem Figure 1.1 provides a sketch of a turbulent boundary layer developing on a permeable flat plate. The origin of the coordinate system is the leading edge of the flat-plate, with x indicating the streamwise direction and y the wall normal direction. The ideal model to which we refer to extends infinitely in the spanwise and streamwise direction, with constant velocity U∞ in the free stream and a transpiration velocity V0 uniform in space (V0 > 0 indicates blowing while V0 < 0 indicates suction). For an experimental realization of this flow case, however, porous or perforated surfaces must be used to approximate the ideal fully permeable surface, hence in a portion of the surface the vertical velocity is zero and the uniformity of V0 in space cannot be guaranteed in a strict sense. In the case of experiments, as in this investigation, V0 represents the mean flow velocity in the wall normal direction defined as the ratio between the flow-rate through the surface and the total area of the surface. Moreover, when in the following the word uniform will be used in the framework of experimental studies, it will indicate a condition in which the local spatial average of V0 is constant in space, i.e. no intentional variation of V0 in space are present other 1.2. Definition of the problem 5 than the ones that unavoidably accompany the use of a porous or perforated surface. The transpiration rate Γ is defined as
Γ ≡ |V0|/U∞ . (1.1) Since it is a positive quantity, the context will clarify whether it refers to the suction or blowing rate. The flow is governed by the incompressible continuity equation and Navier-Sokes equation, representing the conservation of momentum. These equations can be specialized for 2D turbulent boundary layers by applying the Reynolds decomposition, the condition ∂/∂z = 0 and the boundary layer approximation obtaining ∂U ∂V + = 0 (1.2) ∂x ∂y ∂U ∂U 1 dP µ ∂2U ∂u0v0 ∂ U + V = − ∞ + − − u02 − v02 , (1.3) ∂x ∂y ρ dx ρ ∂y2 ∂y ∂x with the capital letters U and V indicating the time-averaged velocity component in the streamwise and wall-normal directions respectively, while u0 and v0 represent the fluctuations around the mean. P∞ indicates the pressure outside of the boundary layer, hence the term dP∞/dx = 0 in a zero-pressure-gradient (ZPG) flow. Finally µ is the dynamic viscosity of the fluid, while ρ is the density. The boundary conditions for the above equations are 0 0 U = u = v = 0 ,V = V0 for y = 0 (1.4) 0 0 U = U∞ , u = v = 0 for y → ∞ . (1.5) The second and third terms of the R.H.S. of eq. (1.3) are often expressed as the wall-normal variation of the total shear stress τ µ ∂2U ∂u0v0 1 ∂τ − = , (1.6) ρ ∂y2 ∂y ρ ∂y with ∂U τ = µ − ρu0v0 , (1.7) ∂y corresponding to the sum of the viscous shear stress, µ∂U/∂y, and the Reynolds shear stress, −ρu0v0. The last term of the R.H.S. of eq. (1.3) is of secondary importance and is often neglected, however it becomes significant if a region of separation is approached (Rotta 1962). In order to describe the problem, a measure of the boundary-layer thickness is needed. A turbulent boundary layer, contrary to the laminar case, has a definite edge separating the region where the flow is turbulent and the region where the flow is irrotational. The nature of turbulent flow makes this edge strongly irregular in space and unsteady in time, hence it is not a good choice for the statistical description of the flow. Several definitions of the boundary-layer thickness δ can (and will) be used. A natural choice is the 99% thickness δ99, defined as
δ99(x) = y : U(x, y) = 0.99U∞ . (1.8) 6 1. Basic concepts and nomenclature
U∞
y δ V x 0
Figure 1.1. Turbulent boundary layer developing on a per- meable flat plate with wall-normal transpiration (not to scale).
Since the determination of δ99 requires the measurements of small velocity differences and the use of interpolation between data points, integral measures of the boundary-layer thickness are sometimes preferred, such as the displacement thickness Z ∞ U(x, y) δ∗(x) = 1 − dy , (1.9) 0 U∞ and the momentum thickness Z ∞ U(x, y) U(x, y) θ(x) = 1 − dy . (1.10) 0 U∞ U∞
The shape factor H12 is defined as the ratio between the displacement and ∗ momentum thicknesses H12 = δ /θ and provides an indication of the “fullness” of the velocity profile. When calculating the displacement and momentum thicknesses from experimental data, is common practice to fix the upper limit of the integrations in eq. (1.9) and (1.10) to the boundary layer-edge instead of the total height of the measurement domain (see e.g. Titchener et al. 2015). Measurement uncertainty leads to a scatter around U∞ of the velocities measured outside of the boundary layer, which reflects in an error in the determination of the integral quantities if the data outside of the boundary layer are not excluded from the integration domains. In this work the upper limit of the integrals in eq. (1.9) and (1.10) was set to δ99.5, which was preferred to δ99 due to the particularly long tails of the mean-velocity profiles of suction boundary layers. Various Reynolds numbers are defined using different length scales, such as the streamwise coordinate or the integral boundary layer thicknesses introduced: ∗ U∞x U∞δ U∞θ Re = , Re ∗ = , Re = . (1.11) x ν δ ν θ ν Another important parameter in the description of the boundary layer is the mean (streamwise) wall shear stress
∂U(x, y) τw(x) = µ , (1.12) ∂y y=0 representing the shear force per unit area exchanged between the surface and the fluid. A natural normalization of the wall shear stress with the dynamic 1.3. Turbulent boundary layers without transpiration 7 pressure gives the skin-friction coefficient τ C = w . (1.13) f 1 2 2 ρU∞ Integrating the boundary-layer momentum equation eq. (1.3) from the wall to infinity, the von K´arm´anmomentum integral is derived, providing an ex- pression for the skin-friction coefficient. In presence of uniform streamwise wall-transpiration but in absence of pressure gradients one obtains Z ∞ Cf dθ V0 1 ∂ 02 02 = − − 2 u − v dy . (1.14) 2 dx U∞ U∞ 0 ∂x For turbulent boundary layers in absence of wall transpiration, the omission of the last term in eq. (1.14) appears justified, (see e.g. Johansson & Castillo 2002 and Schlatter et al. 2010). This result applies also to suction boundary layers, characterized by smaller intensity of velocity fluctuations, but should be extended with care to turbulent boundary layers with blowing, for which the intensity of velocity fluctuations is larger.
1.3. Turbulent boundary layers without transpiration It can be shown that for ZPG TBL it exists a layer for which the shear stress τ is approximately constant in the wall-normal direction. This observation is in close analogy with the near-wall region of pressure-driven internal flows (pipe flow or channel flow) for which
τ(y) = τw (1 − y/δ) , (1.15)
(δ here indicates the pipe radius or the channel half-width) and hence τ(y) ≈ τw as long as y/δ 1. In the layer of approximately constant shear stress, the boundary layer thickness δ is not important in the description of the flow, leaving exclusively the quantities y, U, τw, µ and ρ. Dimensional analysis suggests that two non-dimensional parameters can fully describe the problem. Introducing the friction velocity as rτ u = w , (1.16) τ ρ it is possible to write U yuτ = fw . (1.17) uτ ν ∗ The lengthscale ` = ν/uτ is called viscous length scale and together with the friction velocity it defines the viscous units, sometimes referred to as inner or wall units. Normalization by the viscous units is commonly indicated with the superscript “+” such that eq. (1.17) can be written as + + U = fw(y ) . (1.18) The above equation is commonly indicated as law of the wall and was originally formulated by Prandtl (1925). Very close to the wall, the Reynolds shear stress is small compared to the viscous shear stress. This region is called viscous 8 1. Basic concepts and nomenclature sublayer and a Taylor series expansion of the mean velocity profile gives for ZPG flows (Monin & Yaglom 1971) U + = y+ + O(y+ 4) , (1.19)
+ which is valid in the region y / 5. In the outer part of the boundary layer, instead, the outer length scale given by the boundary layer thickness δ becomes important in the description of the flow. With the assumption that the velocity distribution depends only on the local conditions and not on the streamwise evolution (i.e. the streamwise coordinate enters the problem just through the local wall shear stress τw(x) and the local boundary-layer thickness δ(x)), we can write (Rotta 1962) U∞ − U y U∞ = Φ1 , . (1.20) uτ δ uτ
Empirical evidence suggests that the role of the parameter U∞/uτ = f(Re) in eq. (1.20) is small in the whole outer part of the boundary layer and can be neglected at “high enough” Reynolds number, obtaining the classical form of the velocity-defect law U∞ − U y = Φ1 , (1.21) uτ δ in complete analogy with what proposed by von K´arm´an(1930) for pipe flow. The above expression provides a good description of the flow down to the vicinity ∗ of the wall as long as δ ` . Choosing now δ99 as boundary-layer thickness, the ratio δ u δ 99 = τ 99 = Re , (1.22) `∗ ν τ is another possible definition of a Reynolds number describing the flow and is known as the friction Reynolds number or the K´arm´annumber. In the classical literature on turbulent boundary layers (e.g. Clauser 1956; Townsend 1961, 1976; Tennekes & Lumley 1972), turbulent boundary layer flows obeying eq. (1.21), i.e. without Reynolds-number dependency in the outer part of the boundary layer, are called equilibrium or self-preserving boundary layers. Since the equilibrium conditions are expected to be maintained for Reynolds number approaching infinity, observations at high but finite and practically realizable Reynolds number can be used to infer the asymptotic behaviour of the boundary layer. As already discussed above, defining a representative length scale for the outer part of the boundary layer is problematic. Rotta (1950) and Clauser (1956) derived an integral length scale from the similarity description in eq. (1.21). The displacement thickness eq. (1.9) can be written as Z δ ∗ uτ + + δ = U∞ − U dy (1.23) U∞ 0 Z 1 uτ = δ Φ1 d(y/δ) , (1.24) U∞ 0 1.3. Turbulent boundary layers without transpiration 9 which for an equilibrium layer at high Reynolds number (i.e. neglecting the deviation of the inner layer in eq. 1.21) becomes u δ∗ = δ τ K, (1.25) U∞ where K is the integral of Φ1 from 0 to 1. The Rotta-Clauser length scale is defined as δ∗U ∆ = ∞ , (1.26) uτ and provides an integral length scale for the similarity description of the outer flow. The Rotta-Clauser length scale is related to the boundary-layer thickness as δ = ∆/K, and hence the velocity-defect law eq. (1.21) can be rewritten as
U∞ − U = Φ2(η) , (1.27) uτ where η = y/∆. As already argued by Millikan (1938) for sufficiently high Reynolds number there should be an overlap region between the inner and outer layer, where y δ and y `∗ simultaneously. By matching the derivatives of eq. (1.18) and eq. (1.27) we obtain
+ y ∂U + dfw(y ) dΦ(η) = y + = −η = const. (1.28) uτ ∂y dy dη From the above equation a logarithmic velocity profile in the overlap region is immediately derived, which can be expressed as 1 U + = ln y+ + B, (1.29) κ or as
U∞ − U 1 = − ln η + B1 . (1.30) uτ κ The logarithmic behavior of the velocity profile in the boundary layer was originally derived by von K´arm´an(1930) making use of Prandtl’s mixing-length model, hence the constant κ is known as von K´arm´anconstant. As reviewed thoroughly in the book by Monin & Yaglom (1971), the logarithmic behaviour of the mean-velocity profile can also be obtained by different arguments than the one presented above, i.e. either by dimensional arguments (Landau & Lifshitz 1987) or by the invariance of the dynamic equations of an ideal fluid to similarity transformations. A logarithmic behaviour of the mean velocity profile was also derived by analytical methods by Fife et al. (2009) and Klewicki et al. (2009) for plane Couette flow and pressure-driven internal channel flow respectively. An important consequence of the log law is that as long as B and B1 are taken to be independent of the Reynolds number, a logarithmic behaviour of the skin-friction coefficient with the Reynolds number is obtained. Combining 10 1. Basic concepts and nomenclature eq. (1.29) and (1.30) we can write 1 U + = ln y+ − ln η + B + B (1.31) ∞ κ 1 1 U + = − ln ∆+ + C∗ , (1.32) ∞ κ ∗ where C = B + B1. Recalling eq. (1.12) and eq. (1.26), we get 2 ∗ 2τw uτ + δ U∞ ∗ Cf = 2 = 2 and ∆ = ∗ = Reδ . ρU∞ U∞ uτ ` Hence, we can rewrite eq. (1.32) as r 2 1 ∗ = ln Reδ∗ + C , (1.33) Cf κ or −2 1 ∗ C = 2 ln Re ∗ + C . (1.34) f κ δ Since inaccuracies in the wall-position determination provoke a larger uncertainty on the displacement thickness in comparison to the momentum thickness (see e.g. Titchener et al. 2015), a slightly different form of eq. (1.34) is sometimes preferred: 1 −2 C = 2 ln Re + C . (1.35) f κ θ Recent experiments (Osterlund¨ 1999; Nagib et al. 2007; Marusic et al. 2013) indicate that eq. (1.34) or eq. (1.35) can be used to describe the Reynolds number behaviour of the directly measured skin-friction coefficient for the whole Reynolds-number range explored by the measurements. For a turbulent boundary layer the logarithmic law is valid in a limited portion of the boundary layer, with the lower and upper bounds being a question of debate in the turbulence community (see Orl¨u¨ et al. 2010 for an overview). The upper-bound limit ranges between y = 0.1δ to y = 0.2δ, while the estimates of the lower bound varies more significantly between y+ = 30 (Tennekes & Lumley 1972; Pope 2000) to y+ = 200 Nagib et al. (2007) or even y+ > 600 proposed by Zagarola & Smits (1998a√) for pipe flow. Recently Marusic et al. + (2013) adopted the expression y > 3 Reτ for the lower bound, on the base of the results by Klewicki√et al. (2009) which indicates that viscous forces can + be neglected for y ' 2.6 Reτ . Since neither the law of the wall, the velocity defect law or the log-law are able to provide an appropriate description of the mean velocity profile in the whole boundary layer, Coles (1956) proposed the use of a composite profile Π y U + = f (y+) + W , (1.36) w κ δ with Π and W known as wake parameter and wake function respectively. 1.3. Turbulent boundary layers without transpiration 11
A final remark should be made on the experimental realization of turbulent boundary layers. While the local approach is justified by dimensional arguments for the ideal turbulent boundary layer of Figure 1.1, it is well-known from experiments that significant history effects, originating from the presence of tripping devices and of a physical leading edge with its related pressure gradient, can be responsible for an alteration of the behaviour of the boundary layer, especially in its outer part. History effects results in significant discrepancies between different experimental or numerical data sets even when the local parameters are matched (Chauhan & Nagib 2008; Schlatter & Orl¨u2010,¨ 2012; Marusic et al. 2015). In presence of history effects, hence, the Reynolds number and the normalized distance from the wall are not the only parameters of the problem and the flow cannot be considered fully developed or canonical. The large amount of experiments on ZPG TBL has however allowed the derivation of practical criteria to assess whether a specific boundary layer can be consid- ered fully developed or not and hence correctly represents the canonical flow (Chauhan et al. 2009; Alfredsson & Orl¨u2010;¨ Sanmiguel Vila et al. 2017).
Chapter 2
Boundary-layer flows with wall transpiration
In this chapter a description of boundary-layer flows with uniform wall transpi- ration is provided, together with a review of previous works on the topic. After a short description of the rather special case of the laminar asymptotic suction boundary layer, the focus will be on turbulent boundary layers.
2.1. Laminar asymptotic suction boundary layers The laminar regime of suction boundary layers is one of the few cases for which an analytical solution of the Navier-Stokes equation can be derived. The application of uniform suction at the wall can lead to a state for which the momentum loss due to wall friction is exactly compensated by the entrainment of high-momentum fluid due to the suction, hence the boundary layer thickness remains constant in the streamwise direction. Applying the condition ∂/∂x = 0 and V (y = 0) = V0 < 0 on the two-dimensional and steady continuity and Navier-Stokes equations we obtain ∂U ∂2U V = ν , (2.1) 0 ∂y ∂y2 from which, together with the boundary conditions
U(y = 0) = 0 ,U(y = ∞) = U∞ , (2.2) the mean velocity profile for an asymptotic suction boundary layer (ASBL) is readily obtained U = 1 − eyV0/ν . (2.3) U∞ Originally derived by Griffith & Meredith (1936), the exponential profile of eq. (2.3) was experimentally verified by Kay (1948) and later by Fransson & Alfredsson (2003) over a streamwise distance of more than 400δ99. Integrating eq. (2.1) from the wall to infinity, the wall shear stress can be obtained, with
τw = −ρU∞V0 , (2.4) which is valid independently of the flow regime, i.e. both for the ASBL and for a possible turbulent asymptotic suction boundary layer. An exact measure of the boundary-layer displacement and momentum thicknesses can be derived from the expression of the mean velocity profile
13 14 2. Boundary-layer flows with wall transpiration
(eq. 2.3): Z ∞ U ν δ∗ = 1 − dy = − , (2.5) 0 U∞ V0 and Z ∞ U U 1 ν θ = 1 − dy = − , (2.6) 0 U∞ U∞ 2 V0 from which follows δ∗ H = = 2 . (2.7) 12 θ In the literature, it is common to characterize the laminar asymptotic suction boundary layer with its displacement thickness Reynolds number, which will be indicated in the following as ReASBL. A simple relation between ReASBL and the suction rate can be derived ∗ U∞δ U∞ 1 ReASBL = = − = . (2.8) ν V0 Γ
ReASBL is sometimes used also for the characterization of turbulent asymptotic suction boundary layers (see e.g. Schlatter & Orl¨u2011,¨ Bobke et al. 2016 and Khapko et al. 2016). This use will here be avoided, since, eq. (2.8) is defined with a length scale derived for the laminar regime, hence not representative of the boundary-layer thickness of a turbulent asymptotic suction boundary layer.
2.2. Turbulent boundary layers with transpiration 2.2.1. The development of turbulent boundary layers with wall transpiration For a canonical developing turbulent boundary layer, dimensional analysis suggests that the problem can be fully described by three non-dimensional parameters (e.g. U/U∞, y/δ, Re; see Rotta 1962), while if wall transpiration is applied, an additional parameter (V0/U∞) has to be considered. For turbulent suction boundary layers, though, exactly as for its laminar counterpart, it is possible to hypothesize that a streamwise-invariant state is reached, for which the momentum loss at the wall is compensated by the entrainment of high- momentum fluid due to the suction. For this state, known as the turbulent asymptotic suction boundary layer (TASBL) one of the physical variables of the problem, namely x, disappears, and a link between two of the non-dimensional parameters is established, i.e. Re = f(V0/U∞). The TASBL appears to be considerably more difficult to obtain experimentally than its laminar counterpart. It has been known from the earliest experiments on suction boundary layers (Dutton 1958; Black & Sarnecki 1958; Tennekes 1965) that at high-enough suction rate an initially turbulent boundary layer would relaminarize, hence the asymptotic state obtained for x → ∞ would in that case be the laminar ASBL (see §5.2.2). Even in the range of suction rates for which turbulence is self-sustained, the existence of an asymptotic state for any suction rate Γ has been questioned (see §2.2.2). However, if a turbulent asymptotic state is proven 2.2. Turbulent boundary layers with transpiration 15 to exist for any suction rate below the relaminarization threshold, it means that the asymptotic state is the only “fully developed” state for a certain suction rate, and no self-similarity is expected between non-asymptotic and asymptotic boundary layer at the same suction rate. While suction decreases, and eventually eliminates, the boundary-layer growth, wall-normal blowing significantly increases it, contributing also to a decrease of the wall shear stress. The limiting behavior as x → ∞ (or Rex → ∞) for the blowing turbulent boundary layer is to my knowledge unclear. Boundary layer separation occurs in the case of strong wall-normal uniform blowing: Glazkov et al. (1972) (based on experimantal results) proposed that the separation occurred for a blowing rate V0/U∞ > 0.02, while Coles (1971) estimated the value of V0/U∞ > 0.035 from an analogy between a separated blowing boundary layer and a plane mixing layer between a uniform stream and a fluid at rest. McLean & Mellor (1972) reported that weak uniform blowing (V0/U∞ < 0.003) hastened the approach to separation in a strong adverse- pressure-gradient boundary layer. It is unclear, though, whether any value of uniform blowing rate will eventually lead to separation of a turbulent boundary- layer at a certain downstream distance from the leading edge, as expected for laminar boundary layer with blowing according to the analytical analysis by Catherall et al. (1965). Understanding the asymptotic behaviour of boundary layers with wall blowing is rather important if we want to extend to this flow case the concept of Reynolds-number similarity mutuated from canonical ZPG TBLs. Regarding the experimental realization of turbulent boundary layer with blowing, it should be kept in mind that another source of history effect is often present in addition to those commonly present in any turbulent boundary layer experiment (see §1.3). As a matter of fact, wall-transpiration is usually applied downstream of a certain impermeable streamwise-development length, hence the achievement of the fully developed state should depend also on the distance from the location where blowing starts to be applied. At the current state, the amount of data available is however not sufficient to define analogous criteria identifying fully-developed blowing boundary layers and care should hence be taken in generalizing the experimentally-observed behaviour.
2.2.2. The turbulent asymptotic suction boundary layer Already in the first experimental investigation on suction boundary layers by Kay (1948), mainly devoted to the laminar regime, some turbulent velocity profiles were measured and it was conjectured that “an asymptotic turbulent suction profile may be closely approached at sufficient values of suction velocity”. This conclusion was, however, drawn from a very limited set of experimental conditions and measurement locations, as was later noted by Dutton (1958), who undertook an experimental study exclusively dedicated to the turbulent regime of suction boundary layers. Dutton concluded that a spatially invariant turbulent boundary layer can be established just for a specific suction rate, its value dependent on the nature of the porous surface: for a lower value of suction 16 2. Boundary-layer flows with wall transpiration rate the boundary layer was found to grow continuously, while for larger values the boundary layer thickness continually decreased, slowly approaching the laminar asymptotic suction boundary layer. Black & Sarnecki (1958) proposed instead that for every suction rate there is an asymptotic value of the momentum thickness Reynolds number Reθ = f(Γ): this state is reached rapidly when the asymptotic momentum thickness is close to the one at the beginning of the suction, otherwise a large development length is required to reach the asymptotic condition. The slow approach to the asymptotic state was also reported by Tennekes (1965, 1964), who furthermore suggested that a minimum + suction rate is necessary for obtaining the asymptotic state (−V0 & 0.04). More recently, a numerical study by Bobke et al. (2016) numerically obtained two TASBLs through LES simulations and raised doubts on the possibility of obtaining an asymptotic suction boundary layer in a practically realizable experiment due to the very long streamwise suction length required, claiming that a “truly TASBL is practically impossible to realise in a wind tunnel”. It should be noticed, however, that the initial condition chosen for the simulations was the laminar ASBL while the common approach in the experimental studies is to start the suction downstream of an initial impermeable entry length where a turbulent boundary layer has been allowed to grow. Even in this case the evolution towards the asymptotic state appears to be slow, nevertheless it can be hastened choosing a boundary layer thickness at the beginning of the suction close to the asymptotic one.
2.2.3. Self-sustained turbulence in suction boundary layers As already reported above, it is known since the earliest studies on suction boundary layers that an initially turbulent boundary layer would relaminarize for large enough suction rate. However, there are considerable differences in the reported values for the threshold suction rate Γsst below which a self- sustained turbulent state is observed. While Dutton (1958) and Tennekes (1964) 1 suggested Γsst ≈ 0.01, Watts (1972) proposed the lower value of Γsst = 0.0036, which was closely confirmed in recent numerical simulations by Khapko et al. (2016), who reported Γsst = 0.00370. The present experimental investigations confirms the results by Watts (1972) and Khapko et al. (2016) (see §5.2.2). Figure 2.1 shows a summary of the reported state (laminar/relaminarizing or turbulent) in function of the suction rate for some previous works on the topic2. We notice that all the boundary-layers reported as turbulent by Dutton (1958), 8 out of 10 of the ones in Black & Sarnecki (1958) and 7 out of 12 of the ones in
1It should be observed, however, that in Tennekes (1964) two measurement cases with Γ ≥ 0.00543 were already considered by the author to be in a “early state of reversal to laminar flow”. 2The different terminology and procedures used by the different investigators make a strict comparison difficult: Favre et al. (1966) instead of “relaminarization” used the concept of “progressive destruction of the boundary layer”, while Black & Sarnecki (1958), even if aware of the possibility of a relaminarization, did not discuss the phenomena in the data analysis, applying the proposed turbulent mean-velocity scaling to all of the experimental results. 2.2. Turbulent boundary layers with transpiration 17
Current Exp. Khapko et al. (2016) [DNS] Bobke et al. (2016) [LES] Yoshioka & Alfredsson (2006) [Exp.] Watts (1972) [Exp.] Favre et al. (1966) [Exp.] Tennekes (1964) [Exp.] Black & Sarnecki (1958) [Exp.] Dutton (1958) [Exp.] Kay (1948) [Exp.]
0 5 10 15 20 Γ × 10−3
Figure 2.1. Suction boundary layer reported as turbulent (filled symbols) or relaminarizing/laminar (empty symbols) in the current and in some previous works on suction boundary layers. Gray filled area:Γ > Γsst according to Khapko et al. (2016) and the present study.
Tennekes (1964), were obtained with Γ > Γsst. It is thus possible to speculate that those boundary layers were undergoing relaminarization, also considering that the above investigators were using Pitot tubes as measurement devices, making the fluctuating velocity component inaccessible and the traces of a relaminarization process hard to recognize. This possibility should be kept in mind in the critical review of the proposed mean velocity scaling for suction boundary layers.
2.2.4. Mean-velocity profile As all other turbulent boundary layers, also the boundary layer with transpira- tion has a two-layers structure. In the viscous sublayer the molecular momentum transfer, hence the viscous shear stress, is dominant, while in the largest part of the boundary layer the turbulent momentum transfer, hence the Reynolds stresses, is prevalent.
The viscous sublayer Close to the wall ∂U ∂U U V , (2.9) ∂x ∂y and, in presence of wall transpiration
V ≈ V0 . (2.10) 18 2. Boundary-layer flows with wall transpiration
The streamwise Reynolds equation for boundary-layer approximation eq. (1.3) reduces thus to (Rubesin 1954) ∂U 1 ∂τ V = , (2.11) 0 ∂y ρ ∂y where τ is the total shear stress defined in eq. (1.7). Equation (2.11) can be integrated from the wall to an arbitrary wall-normal position where eq. (2.9) continues to hold, obtaining τ − τ V U = w . (2.12) 0 ρ In viscous units eq. (2.12) can be rewritten as τ u2 + V U = . (2.13) τ 0 ρ It should be noted that while eq. (2.13) is only approximately valid for a generic boundary layer with wall transpiration, it is exact for the whole boundary-layer in the case of a TASBL, since it can be derived from the full Reynolds equation with the assumption ∂/∂x = 0. In the viscous sublayer, the viscous stress dominates over the Reynolds stress and eq. (1.7) is simplified to ∂U τ = µ . (2.14) ∂y Eq. (2.13) can then be rewritten as + + + ∂U 1 ∂ + + 1 + V0 U = + = + + (1 + V0 U ) . (2.15) ∂y V0 ∂y Making use of the no-slip boundary condition, eq. (2.15) becomes (Rubesin 1954; Mickley & Davis 1957; Black & Sarnecki 1958)
1 + + + y V0 U = + (e − 1) , (2.16) V0 describing the velocity profile in the viscous sublayer for a transpired boundary layer.
The turbulent near-wall region - Logarithmic or bi-logarithmic form? In the literature on turbulent boundary layer flows with wall transpiration two main categories of scaling laws for the mean-velocity profile can be distinguished. In a number of works a dependency of the streamwise velocity with the logarithm of the wall-normal distance is suggested for the near-wall turbulent region, analogously to the non-transpired turbulent boundary layers. In other works the streamwise velocity is proposed to be described by the series of logarithmic functions a ln2 y + b ln y + c. Due to the presence of a quadratic logarithmic term expressions of this family are sometimes referred to as bi-logarithmic laws. These two results originated from four different approaches to the problem: 2.2. Turbulent boundary layers with transpiration 19
– the use of a closure hypothesis for the Reynolds stresses such as the momentum transfer (Rubesin 1954; Clarke et al. 1955; Mickley & Davis 1957; Black & Sarnecki 1958; Stevenson 1963a; Simpson 1970) or the vorticity transfer (Kay 1948), – asymptotic matching of expressions valid in the inner and outer region of the boundary layers and derived from dimensional arguments and characteristic scales (Tennekes 1964, 1965; Andersen et al. 1972), – analytical methods based on matched asymptotics expansions (Vig- dorovich 2004; Vigdorovich & Oberlack 2008; Vigdorovich 2016), – empirical induction (Dutton 1958; Schlatter & Orl¨u2011;¨ Bobke et al. 2016). In the following paragraphs a review of the proposed mean-velocity scalings will be given, while a summary is presented in Table 2.1. For TASBLs, using Taylor’s vorticity-transfer theory and a mixing length defined being proportional to the wall-normal distance L = κy, Kay (1948) obtained U 1 V0 y = 1 − 2 ln , (2.17) U∞ κ U∞ δ in which a logarithmic dependency of the streamwise velocity to the wall-normal distance is observed. It should be noted that since this analysis is restricted to asymptotic suction cases, the proposed scaling extends until the boundary layer edge. Rubesin (1954) was the first to apply Prandtl’s momentum-transfer theory to the (compressible) boundary layer with blowing, deriving an integral expression for the near-wall turbulent region. For incompressible flow using L = κy as mixing-length, Prandtl’s momentum transfer theory gives τ ∂U 2 = κy , (2.18) ρ ∂y which can be used in eq. (2.13) to obtain ∂U 2 u2 + V U = κy . (2.19) τ 0 ∂y Eq. (2.19) can be rewritten as 2 2 2 κ ∂(uτ + V0U) uτ + V0U = . (2.20) V0 ∂ ln y The solution of this differential equation is V 2 u2 + V U = 0 (ln y + C )2 , (2.21) τ 0 4κ2 1 where C1 is an integration constant. One possible way to express eq. (2.21) in viscous scaling is (Stevenson 1963a) q 2 + + 1 + 2 + 1 + U V0 − 1 = ln y + C2 − + , (2.22) V0 κ V0 20 2. Boundary-layer flows with wall transpiration
∗ where C2 = (C1 + ln ` )/κ. Equation (2.22) reduces to the canonical logarithmic law of the wall (eq. 1.29) as long as
2 + C2 → B + + for V0 → 0 , (2.23) V0 where B is the log-law intercept for the no-transpiration case. Stevenson (1963a) has however reported that the dependency on the transpiration rate of the + + term C2 − 2/V0 is weak and he chose C2 − 2/V0 = B for the description of all his experimental results on blowing boundary layers. An expression similar to eq. (2.22) has been derived by many other authors (Clarke et al. 1955; Mickley & Davis 1957; Black & Sarnecki 1958; Stevenson 1963a; Simpson 1970), and has recently been used by Kornilov (2015) to describe his experimental data on turbulent boundary-layers with blowing. Rotta (1970) followed the same procedure, including a damping function following van Driest (1956) in order to account for the viscous stresses near the wall. The difference between the expressions proposed by the different authors is just in the values and in the way of expressing the integration constant: a summary on the topic can be found in Stevenson (1963a). As pointed out already by Rubesin (1954) and Clarke et al. (1955), both the mixing-length parameter κ and the integration constant should in general be regarded as functions of the suction or blowing rate. Nevertheless, it seems that all the supporters of the bilogarithmic law assumed the value of κ to be constant or just weakly depending on the transpiration rate, fixing it to the value for the turbulent boundary layer without mass transpiration. Mickley & Davis (1957), though, specified that “at values of V0/U∞ above 0.005 the value of κ increases with increasing values of V0/U∞”. The LHS of eq. (2.22) is sometimes referred to as the pseudo-velocity: if the mixing length parameter κ is independent of the transpiration parameter, a semilogarithmic plot of the pseudo-velocity against the wall-normal distance for the inner turbulent region of boundary layers with mass transfer should result in a series of parallel lines. The bilogarithmic law has also been derived through an analytical approach by Vigdorovich (2004), Vigdorovich & Oberlack (2008) and Vigdorovich (2016). The application of momentum transfer theory to boundary-layer flows with mass transfer and the resulting bilogarithmic law appears to be the predominant view for the first decade of research on the topic. Doubts about the application of the mixing-length model to boundary-layer flows with mass transfer were raised in Tennekes (1964) and Tennekes & Lumley (1972), stating in the latter that “mixing-length models are incapable of describing turbulent flows containing more than one characteristic velocity with any degree of consistency”. Mickley & Smith (1963) were the first to propose an alternative scaling, extending Coles (1956) decomposition of the canonical turbulent boundary-layer eq.(1.36) to boundary layers with wall transpiration and suggesting an empirical velocity- defect law of the form
U∞ − U 1 y Π(x) y ∗ = − ln + W , (2.24) uτ κ δ κ δ 2.2. Turbulent boundary layers with transpiration 21 where a dependency on the first power of the logarithm is evident. In eq. (2.24) ∗ uτ is a characteristic shear velocity based on the maximum shear stress. Con- sidering a boundary-layer flow without pressure gradient, the maximum shear stress does not coincide with the wall shear stress just in presence of blowing, while for the suction case eq. (2.24) would revert to the common velocity defect law for flows on non permeable surfaces, as long as κ is taken as constant. Tennekes (1964, 1965), Coles (1971) and Andersen et al. (1972) also suggested a dependency of the streamwise velocity to the first power of the logarithm of the wall normal distance. Their rationale is that in presence of mass transfer it is possible to adopt the same type of argument used by Millikan (1938) to derive the log-law for turbulent boundary-layer flow on impermeable surfaces. The boundary layer can be divided in a wall region which can be described with a law of the wall U y = f , (2.25) u0 `0 and an outer region where the velocity profile has the form of a defect law U − U y ∞ = g . (2.26) u0 δ
The two regions share the same velocity scale u0, which can be related to the characteristic stress level close to the wall. If there is an overlap region where both descriptions are valid, then the velocity profile must have the logarithmic shape U 1 y = ln + B2 , (2.27) u0 κ y0 or, equivalently,
U∞ − U 1 y = − ln + B3 . (2.28) u0 κ δ For the case of a turbulent boundary layer flow without wall-normal mass transfer (see §1.3), r τw ∗ ν u0 = uτ = and `0 = ` = . (2.29) ρ uτ In presence of mass transfer, instead, since the viscous sublayer is described by eq. (2.16), an attractive choice of velocity and length scale is (Tennekes 1965) 2 τw/ρ uτ ν u0 = = and `0 = , (2.30) V0 V0 V0 so that eq. (2.16) can be written in the form of eq. (2.25) as U = ey/`0 − 1 , (2.31) u0 independently of the suction ratio. If this choice of velocity scale proves to be correct also for the outer part of the boundary layer, so that the velocity profile is correctly represented by eq. (2.26), then a logarithmic profile is expected to hold 22 2. Boundary-layer flows with wall transpiration in the overlap region between inner and outer region. Tennekes (1965) tested this hypothesis on velocity profiles that he identified as TASBLs, concluding that indeed the scaling for the streamwise velocity profile applied in eq. (2.26) is appropriate for this kind of flow. However, it should be noticed that for an asymptotic suction boundary layer 2 uτ u0 = = −U∞ , (2.32) V0 hence the conclusion by Tennekes simply means that the streamwise velocity profiles of TASBLs scales in the outer layer when normalized with the free- stream velocity. Nevertheless, since the inner and outer region show different length scales but a common velocity scale, Millikan (1938) argument is valid and a semilogarithmic velocity profile with a slope independent of the suction ratio is expected to hold for TASBLs. Using the normalization parameters in eq. (2.30), Tennekes (1965) proposed the semi-empirical expression UV0 V0y + − 2 = 0.06 ln − − 11 V0 + C , (2.33) uτ ν where C is a function of the surface roughness. Equation (2.33) fits Tennekes’ + experimental data just in the range 0.04 < −V0 < 0.1, which led him to the + tentative conclusion that no asymptotic state is possible for −V0 < 0.04, while + for −V0 > 0.1 he found that a relaminarization process occurs. Furthermore Tennekes also noticed that the normalization in eq. (2.30) makes u0 and `0 diverge for V0 → 0 and is hence unlikely to hold for very small suction rates. The Taylor’s series expansion of eq. (2.16) around y = 0 1 1 U + = y+ + V +y+2 + V +2y+3 + ... (2.34) 2 0 6 0 illustrates that suction and blowing appears as second order term when the viscous scaling is used, suggesting that there is no advantage in using the normalization parameters in eq. (2.30) instead of the classical viscous scales of + + eq. (2.29), as long as V0 y << 1 at the edge of the viscous sublayer (Tennekes 1965). For small suction rates and for the blowing cases, both Tennekes (1965) and Andersen et al. (1972) proposed a logarithmic scaling of the type U 1 y ∝ ln , (2.35) uc κ `c + with κ having the value obtained for non-transpired case, uc = uτ (1 + αV0 ) 2 and `c = νuc/uτ (Tennekes 1965) or `c = ν/uc (Andersen et al. 1972). The form of eq. (2.35), might lead the reader to the wrong opinion that eq. (2.35) has been derived by similarity argument in a similar fashion than the log-law for boundary layer without mass transfer. However, eq. (2.35) is a purely empirical expression with the choice of velocity scale uc made by the authors with the specific purpose of obtaining a constant slope of the logarithmic region. Moreover, the choice of length scale in Tennekes (1965) and Andersen et al. 2.2. Turbulent boundary layers with transpiration 23
(1972) do not have any specific role in the description of the flow and were defined by analogy with the length scale for a boundary layer with or without mass transfer respectively. As a result, it is not possible to express the viscous sublayer as U/uc = f(y/`c). Formulations of the type of eq. (2.35) are hence equivalent to the empirical log-law with modified coefficient of the type U + = A ln y+ + B, (2.36) used by Dutton (1958) to fit his experimental data. Watts (1972) and Bobke et al. (2016) also favours this empirical logarithmic scaling, with the coefficients A and B being functions of the suction rate.
The outer region Following Coles (1956), Black & Sarnecki (1958) proposed a description of the outer region of a turbulent boundary layer with transpiration by summing an inner velocity component Ui coinciding with the bilogarithmic law of the wall and a wake component Uw negligible in the inner part of the boundary layer. Differently from Coles (1956) they did not use uτ as the single normalization parameter for the wake function, suggesting instead the use of the local shear velocity of the bilogarithmic law y∂Ui/∂y. Mickley & Smith (1963) proposed the use of a velocity defect law (eq. 2.24) in which a Coles-type wake function is evident. Stevenson (1963b), instead, proposed an extension of the bilogarithmic law to the wake region as a velocity defect law in the form: q q 2 + + + + + 1 + V0 U∞ − 1 + V0 U = g(y/δ) , (2.37) V0 which was adopted also by Simpson (1970). Tennekes (1965), instead, proposed a velocity defect law for turbulent asymptotic states in the form + + + V0 (U∞ − U ) = g(y/δ) . (2.38) Generalizing eq. (2.38) for non-asymptotic suction or blowing boundary layers with pressure gradient, he proposed the tentative expression
+ + + V0 (U∞ − U ) = g(y/δ, Λ, Π)e , (2.39) where Λ is a transpiration parameter and Πe is a pressure-gradient parameter. More recently, Cal & Castillo (2005) extended to boundary layers with transpiration and pressure gradient the use of the empirical scaling proposed by Zagarola & Smits (1998b) for ZPG TBLs, concluding that “the dependencies on the upstream conditions, pressure gradient, and the blowing parameter are nearly removed from the mean deficit profiles when normalized by the Zagarola-Smits ∗ scaling U∞(δ /δ)”, even though differences are observed between the blowing and the suction cases. Kornilov & Boiko (2012) tested the Zagarola-Smits scaling on their experimental data on ZPG boundary layer with blowing, obtaining a good overlap among the measured mean-velocity profiles. 24 2. Boundary-layer flows with wall transpiration ) ) ) . ) + + + + 0 0 0 0 (Γ) V V V (Γ) V f (surface) ( ( ( 419 f 23 ( (Γ) . . f f f f f f 0 = = = const ======0 = + ≈ a + a κ A A B κ λ κ U y B κ 2 + y ln + + a y y + 0 V 2 ln ) 2 + + 1 κ 1 κ y 4 y ln = + (ln + 0 + + 0 V + y 0 V κ 1 V 2 2 ln + 1 κ a κ λ 4 + U δ y λ + + ln 1 + ) + 1 τ 0 y B ∞ κ q V U − U y/δ ln + 2 ( − 2 = 1 λ B + g κ + ) 0 y i + x V − + ln 0 ∂y 1 + ∂U A V A y U = 1 = = Π( = = = + ∗ i ∞ w 1 + ∗ + i i + U U U U U U U U q with + 0 2 V bilogarithmic layer wake region w U + i U = law of the wall law of the wall turbulent layer U Proposed formulations for the mean-velocity profile of turbulent boundary layers with suction (1955) B or blowing. The acronymssuction; in AS, the asymptotic column suction “Case” state; indicate B, the blowing. range of validity indicated by the authors: S, Table 2.1. et al. InvestigatorKay (1948)Mickley & Case Davis (1957) B AS Clarke Dutton (1958)Black & Sarnecki (1958) Mean S, Velocity B Formulation AS Parameters 2.2. Turbulent boundary layers with transpiration 25 ) + 0 V ) + = 11 x ∞ P + 0 d d U 3(1+9 + a . V 8 + y ∗ ( 419 44 w . 0 41 (surface) . . δ . τ f V f = = 2 = 5 = = = 0 = 0 + c τ = 0 a u u e C κ B Π = Λ = B U κ κ C ) θ + + ) Re + + a 0 , y y V Π y/δ 11 B ( g ln − + δ y/δ, y 1 κ = e ( Π) + , g + y 0 = Λ V = ) W + ln + ) U 1 y κ x + 0 κ + y/δ, y/δ + 0 − B ( ( V = Π( V g g U + a + + 0 + U 1 V 1 + ) = ) = δ y 06 ln c 2 τ . − + + q 1 + ln νu yu U U 1 + − q = 0 1 κ + 1 − − q + ∞ − + − + + + − ∞ ∞ U U + U = 0 U U + = ln + + 0 ∞ ( ( + V 0 0 V c + + U V U + V 0 0 U u + − V V U 0 ∗ τ − q V u 1 + ∞ q 1 + U + 0 2 1 + q 1 V . q + 0 0 2 V < + 0 2 + + 0 V 0 04 2 . V V 0 − < < + 0 04 V . logarithmic region 0 logarithmic region − generalrium state equilib- asymptotic state law of the wall velocity defect law law of the wall law of the wall velocity defect law velocity defect law velocity defect law (blowing) S, B ) b , a InvestigatorStevenson CaseMickley & Smith (1963) S, B Tennekes (1964, 1965) S, B Mean Velocity FormulationSimpson (1970) S, B Parameters (1963 26 2. Boundary-layer flows with wall transpiration 1)] − + 0 V 75 . 0014 . 10 0 − / e Γ 390Γ) + 4 0 × . − V 5 − . 7 π 2 ( . 5 . 4(1 . ∞ 6 + 12 05 157 0 . . . 0 5 + 4 + (Γ) (Γ) 41 41 . / . . . . f f 0 = 1 + 7 = 2 = 2 → = 5 = 1 = = = 7 = 13 = 0 arctan [2 = 0 = 0 0 1 c τ + u κ u A A κ A κ B B α > B y B B 2 /A + U ) + 0 α V − ) 1+ + q y + (( y O − e + ) − ) + 1 2 ) U ) ) + + 0 0 + 1 V V y/δ U ( (( − g + 0 O c τ V u u + 2(1 + ) + + + = 0 0 (1 + V 1 ,V 1 +2 + 14 + B y y − 2 ( B − ) κ f 0 + + 0 ) B V B B = + 0 c + + 1 + 4 + + + ν U ,V + yu s + + U y + y y y ( ln ln ln ln (1 + 1 + f 1 κ 1 A A κ q = = = = = + + + 0 c + + 2 U df U dy u V U U law of the wall law of the wall law of the wall (1972) S, B (2016) AS et al. et al. InvestigatorRotta (1970) Case S, B Andersen Mean Velocity FormulationWatts (1972)Bobke S Parameters Vigdorovich (2016) S 2.2. Turbulent boundary layers with transpiration 27
2.2.5. Reynolds stresses Dutton (1958) calculated the Reynolds shear stress u0v0 profile in an turbulent asymptotic suction boundary layer from the measured streamwise velocity gradient through eq. (2.13) with τ ≈ −ρu0v0, reporting a strong decrease of the peak value of u0v0+ when suction was applied. He then used the calculated Reynolds shear stress and the measured velocity gradient to obtain an estimate for the turbulence production and the viscous dissipation term of the mean-flow energy equation, concluding that in presence of suction the larger velocity gradient at the wall enhances the viscous dissipation, decreasing the relative amount of mean-flow energy transferred to the turbulent motion. Similar results were also obtained by Rotta (1970), who also included in the analysis boundary layers with blowing, reporting a large increase of the near-wall turbulence production term in presence of blowing. To my knowledge, the first study reporting direct measurements of the Reynolds stresses in boundary layers with suction is by Favre et al. (1961). Profiles of u02, v02, and u0v0 were reported, concluding that in presence of suction the Reynolds stresses are damped in the whole boundary layer compared to the no-transpiration case. Favre et al. (1966) concluded the same behaviour also for the spanwise component w02. Similar results were obtained by Andersen et al. (1972), even if complicated by pressure gradient effects. The various components of the Reynolds stress tensor are affected differently by wall-suction, with the near-wall anisotropy increasing with the suction rate. This increase in anisotropy is explained by the increased organization of the near-wall flow showing a “more orderly behavior of low-speed and high-speed streaks and a greater longitudinal coherence of the low-speed streaks” (Antonia et al. 1994). Fulachier et al. (1977) reported X-wire anemometry results showing that the streamwise velocity variance u02 was damped the most by the suction, while w02 was affected the least, in agreement with Antonia et al. (1988) but in contrast with El´ena(1975) and Fulachier et al. (1982) (as reported by Antonia et al. 1988), who conjectured that w02 should be more damped by the suction than u02. Finally Antonia et al. (1994), analyzing the DNS simulation results by Mariani et al. (1993), concluded that the component of velocity fluctuation most affected by suction was v02, followed respectively by w02 and u02. These differences between numerical and experimental results were explained by Antonia et al. (1994) with the difficulties in obtaining reliable measurements of the near-wall fluctuations with hot-wire anemometry through X- or V-probes. The Reynolds stresses are magnified by wall-normal blowing, as observed in the experimental data by Andersen et al. (1972) and Kornilov (2015) and in the LES by Kametani et al. (2015). In general blowing increases the magnitude of the Reynolds stresses particularly in the outer part of the boundary layer: in the range of blowing rate and Reθ explored by Andersen et al. (1972) a secondary peak in the u02 profiles emerges in the outer part of the boundary layer, which in some case is larger in magnitude than the near-wall peak. For the blowing 28 2. Boundary-layer flows with wall transpiration rate reported in Kornilov (2015) a single peak in the u02 profiles located in the outer part of the boundary layer is observed. Since with increasing blowing the wall shear stress decreases (and hence the viscous length-scale increases), these observations cannot be explained by spatial filtering of the hot-wire probe.
The Reynolds shear-stress distribution in a TASBL In the special case of the turbulent asymptotic suction boundary layers, it is possible to derive a relation between the Reynolds shear-stress and the mean velocity profile. As already noted above, for an asymptotic boundary layer eq. (2.12) is exact. Dividing both sides of eq. (2.12) with the asymptotic wall shear stress τw = −ρU∞V0 we get τ U = 1 − . (2.40) τw U∞ For a turbulent boundary layer τ ≈ −ρu0v0 everywhere but in the near-wall region, hence for a large portion of the boundary layer u0v0 U − 2 ≈ 1 − , (2.41) uτ U∞ relating the inner-scaled Reynolds shear stress to the outer-scaled mean velocity profile. Chapter 3
Experimental setup and measurement techniques
This chapter presents a description of the experimental setup built for this study together with a summary on the measurement techniques employed. The main component of the apparatus is a flat plate with a permeable top surface installed in the Minimum Turbulence Level wind tunnel of the Fluid Physics Laboratory at the Department of Mechanics of KTH - Royal Institute of Technology. A suction/blowing system providing the necessary air flow through the permeable surface and two automated traverse systems complete the apparatus. Each of these parts will be described in detail in the following sections and the main design choices will be motivated. Thermal anemometry and oil-film interferometry will briefly be introduced and, finally, an account on the measure of the wall shear stress on permeable surfaces with a miniaturized Preston tube will be given.
3.1. Wind tunnel The Minimum Turbulence Level (MTL) wind tunnel is a closed loop wind tunnel with a 7 m long test section having a cross-sectional area of 1.2 × 0.8 m2. The maximum streamwise turbulence intensity for an empty test section in the speed range from 10 m/s to 60 m/s is less than 0.04% and a cooling system maintains the temperature of the flow constant with a maximum variation around the mean in space and time of ±0.07K. The adjustable shape of the ceiling and floor of the test section allows the regulation of the pressure gradient. The interested reader can find more details on the wind-tunnel design and characteristics in Johansson (1992) and Lindgren & Johansson (2002).
3.1.1. Test-section modifications Considerable modifications to the wind-tunnel test section were required to allow the desired installation of the present experimental apparatus. In pres- ence of wall suction/blowing over a large area of a wind-tunnel model, the significant ejection/injection of mass flow from the test section results in an acceleration/deceleration of the flow along the streamwise direction. In order to compensate for this effect, the ceiling, originally made of 30 mm thick wood
29 30 3. Experimental setup and measurement techniques Filled ) Oil-film (b) ) Landing f c
650 (a) (c) ) leading-edge bleed slot; ( b (d) ) Wall-mounted traverse system; ( e ). e, f ) Impermeable leading-edge section; ( a ) Ceiling-height adjustment station; ( d ); ( y − (e) x Drawing of the experimental setup mounted in the MTL wind-tunnel test section. (f) : Perforated surfaces; ( Figure 3.1. areas traverse system ( measurement station. Note:hot-wire measurements The at Landing the traverse downstream station system ( was unmounted when performing oil-film or 3.1. Wind tunnel 31 panels, was exchanged with a series of perforated steel sheets with hole diame- ter 2 mm and a hole spacing giving an open area of 29.6%. The idea behind adopting this largely perforated ceiling, was to impose constant static pressure along the whole test section, thus obtaining a zero-pressure-gradient boundary layer on the test surface. However, when the wind tunnel was run with the new ceiling, very large velocity fluctuations were observed, originating from the flow being alternately discharged through the ceiling or through the wind-tunnel diffuser. The solution of the problem was found by covering a large extent of the perforated area of the ceiling with adhesive plastic foil, leaving just 100 mm long open slots every 1 m. Apart from the ejection/injection effect, the flow accelerates inside the test section due to the growth of the boundary layers on all the walls. This effect is typically taken care of by adjusting the shape of the ceiling such that the total cross-sectional area of the test section increases with the downstream distance. Preliminary experiments showed however that the open area of the ceiling together with the largest allowed regulation of the shape of the ceiling was not sufficient to obtain a ZPG region extending over the whole streamwise length of the test plate. The problem was solved installing a 1.2 m long wall liner inside the wind-tunnel contraction section with the function to decrease the inlet cross-sectional height of the test section to 0.7 m from the original 0.8 m and, hence, allowing a larger expansion of the cross-sectional area. With this configuration a large ZPG region could be obtained for any experimental condition examined through a joint regulation of the ceiling shape and of a bleed-slot opening beneath the plate leading edge (see §3.2.1 for more details). The pressure gradient was checked either with a hot-wire traverse or with pressure-taps readings before each measurement and a regulation of the ceiling shape and/or of the bleed-slot opening was performed whenever needed. With this procedure the variation of U∞ was typically limited to less than ±0.5% on the whole plate model, with somewhat larger variation limited to the first meter from the leading edge. The test-section modifications described above and the installation of the plate model increased the free-stream disturbance level compared to the original empty test section. The streamwise turbulence intensity in the free-stream increases along the streamwise direction, reaching a maximum level of 0.2% at the most downstream measurement location (6.06 m downstream of the leading edge).
3.1.2. Traverse system The MTL wind tunnel can be equipped with a fully automatic 5-axis traverse system, able to accurately position a probe mounted on a sting penetrating into the test section from a slot in the middle of the ceiling. At the flow velocity considered in this investigation (up to 45 m/s), flow-induced vibrations on the sting would cause erroneous data and frequent hot-wire probe breakage in close proximity to the wall. In order to reduce probe vibrations, a new traverse system was designed such that the traverse arm could be supported by the plate, 32 3. Experimental setup and measurement techniques
I. y (a) (g) (f) x
(b)
(h) (c)
(d) (e)
II. y x 2 9 7 9 3
204 345
Figure 3.2. Drawing of the Landing traverse system. I. Lifted Secondary stage; II. Landed secondary stage. (a) vertical sting traversable in the x − y direction from the main wind-tunnel traverse; (b) rotating shaft; (c) spring: (d) rubber feet; (e) wheel; (f ) DC servomotor; (g) stop switch; (h) hot-wire probe. instead of simply hanging from the ceiling. Figure 3.2 reports a drawing of this new traverse system, which will in the following be referred to as the Landing traverse system. It is made of two main components, a wing-shaped vertical sting (a), which is attached to the main traverse chassis on the top of the test-section, and a secondary traverse stage pivoting around a small shaft (b). 3.2. Perforated flat plate 33
The vertical wing can be moved in the streamwise and wall-normal direction by the main wind-tunnel traverse system. A motion routine starts with raising the vertical sting until the secondary stage is lifted from the surface, as indicated in Figure 3.2I. The vertical wing is then moved in the desired streamwise x location and lowered toward the surface, with the wheel (e) facilitating the motion of the secondary stage on the surface. The vertical motion is interrupted by the electrical switch (g), regulated so that it activates when the rubber feet (d) touches the surface. The secondary stage is now pressed in position on the plate by a spring (c) as depicted in Figure 3.2II. Finally the desired position of the probe (h) is adjusted rotating a leadscrew with the DC servomotor (f ). The servomotor is equipped with a rotary encoder ensuring a relative accuracy of the vertical displacement of the probe of ±1 µm. The vertical range of the secondary traverse stage, and hence of the probe, is 180 mm. The extent of the upstream region influenced by the traverse system was checked moving the traverse system progressively closer to a fixed Prandtl tube. The length of the horizontal sting supporting the probe was then chosen such that the deviation in velocity measured by a Prandtl tube was less than 0.5% from the undisturbed value.
3.2. Perforated flat plate 3.2.1. Design and construction A drawing of the flat plate installed inside the wind-tunnel test section is shown in Figure 3.1. The flat plate is 6.6 m long and spans the whole 1.2 m width of the wind-tunnel test section. It starts with a 122 mm long impermeable elliptical leading edge followed by 8 individual plate elements, each one extending 812 mm in the streamwise direction. Finally, a 1.2 m long linear diffuser (not shown in Fig. 3.1) extends inside the wind-tunnel diffuser section from the plate trailing edge, expanding the flow to the full local cross-sectional area. The plate is installed in the test section such that the test surface constitutes the wind-tunnel bottom surface. Such arrangement was preferred to the more common installation of the flat plate in the mid-height of the test-section for two reasons: first, the extra blockage originating from the suction/blowing tubing was deemed problematic, moreover since this configuration maximizes the distance between test-section floor and ceiling, it guarantees a larger region of free stream reducing blockage effects. In order to remove the boundary layer developed in the wind-tunnel contraction and to allow the development of a fresh boundary layer with a well-defined origin on the test plate, the flow beneath the plate leading edge is vented through a bleed slot with an adjustable opening (see Fig. 3.3). Thirteen pressure taps allow the measurement of the pressure distribution on the leading-edge top and bottom surface. The variation of the leading-edge pressure coefficient as function of the bleed-slot opening is illustrated in Figure 3.4b, showing that quite a large regulation is possible with the permissible adjustment of the bleed-slot opening. Figure 3.4a show instead the variation of the leading-edge pressure distribution 34 3. Experimental setup and measurement techniques
122 345 26 23
(a) 280 (b) = 25 ÷ 50 o
h 285
20 3
Ø 120 80 40 10 0 0.5 26
122
Figure 3.3. Top: Detail view of the leading-edge bleed slot. The right edge of the drawing corresponds to the start of the test-section. A hinged surface (a) allows the adjustment of the bleed-slot opening (ho) through the regulation of two turnbuckles (b). Bottom: Magnified view of the elliptical leading edge with its pressure taps. at fixed bleed-slot opening but varying free-stream velocity, showing that a regulation of the bleed-slot opening is needed in order to maintain a fixed leading-edge pressure distribution at different experimental conditions. The possibility of using an active venting system for the regulation of the leading- edge pressure coefficient was explored during the preliminary design of the current experimental apparatus and is described in Ferro et al. (2015). Each individual plate element is a sandwich construction: a 0.9 mm thick perforated titanium sheet is glued on a frame of square, L- and T-beams bolted on a 6 mm thick bottom plate (see Fig. 3.5 and 3.6). The T-beams elongate in the spanwise direction, with a streamwise spacing of 57.5 mm, chosen to limit the maximum plate deflection at the maximum suction level to less than 5 µm (calculated with classical beam theory). The webs of the T-beams are perforated to ensure a uniform pressure in the plate inner chamber. The width of the webs of the L- and T-beams was chosen to be 2 mm to minimize the 3.2. Perforated flat plate 35
−1 (a) h = 30 mm o −0.5 p
C 0
U U 0.5 ∞ = 15.0 m/s ∞ = 30.0 m/s U U ∞ = 20.0 m/s ∞ = 40.0 m/s 1 0 20 40 60 80 100 120 x (mm)
−1 (b) U ∞ = 20 m/s −0.5 p
C 0 h = 25 mm h = 40 mm o o h = 30 mm h = 45 mm 0.5 o o h = 35 mm h = 50 mm o o 1 0 20 40 60 80 100 120 x (mm)
Figure 3.4. Leading-edge pressure coefficient. (a): fixed bleed-slot opening (ho) and varying free-stream velocity; (b): fixed free-stream velocity and varying bleed-slot opening. streamwise direction where suction/blowing is interrupted, while providing sufficient surface for the glue to hold. In each plate an access port for a 10 mm diameter cylindrical plug is provided. Originally conceived for hot-film wall- shear-stress measurements, the plugs were instead mainly used to host pressure taps for measuring the streamwise pressure gradient. Inside each plate four pressure taps measure the static pressure in the inner chamber: in order to check the pressure uniformity, two of these pressure taps are located close to the centre of the plate, while the other two are located 25 mm from the edges of the plate in two opposite corners. The titanium sheets (provided by CAV Advanced Techonologies in Consett, UK) are laser drilled with 64 µm diameter holes with a centre-to-centre spacing of 0.75 mm in both spanwise and streamwise direction, giving a total open area of 0.56%. The holes are not aligned in the streamwise direction, but rather a random pattern with fixed centre-to-centre distance and fixed row spacing was chosen, in order not to introduce any preferential spanwise scale in the flow. Since smoothness at the joint is important, the titanium sheets were designed 2 mm shorter than the frame, so that a gap originates between two adjacent plates on assembly. This gap is then filled with boxing 36 3. Experimental setup and measurement techniques
(a) 1197 (f)
810
0,75 (b)
Ø 0,064
812
(c)
(d)
To/From Fan
(e)
Figure 3.5. Exploded view of one plate element. The perfo- rated titanium sheet (a) is supported by a hollow frame (b) mounted on the bottom plate (c). Below, three suction/blowing channels (e) from which air is driven to/from the fan. d) sur- face measurement access plug (pressure tap, hot-film probe or Preston tube); f ) magnified photography of the laser-drilled titanium sheet. wax and polished smooth. Since laser drilling may introduce unwanted and uncontrolled curvature to the titanium sheet, a controlled large curvature along the spanwise direction (R ≈ 3.6 m) was imposed on the sheet after laser-drilling, so that they would appear concave if lied on top of the supporting frame. The desired flatness is then achieved fastening the sheets to the frame.
3.2.2. Measurement station One of the plate element (see Fig. 3.6) is different from the others and it hosts a wall-mounted traverse system, a glass insert for oil-film interferometry 3.2. Perforated flat plate 37
57,5
26
40 Ø 130 The most downstream plate element with the wall-mounted traverse system, the oil-film
150 58 Figure 3.6. interferometry measurement station and the surface measurement access plug. 38 3. Experimental setup and measurement techniques
Figure 3.7. Photography depicting the bottom side of the plate, with the suction/blowing channel and the flexible hoses. skin-friction measurements and the 10 mm diameter plug also present in the other plate elements. In order to avoid leakage when suction/blowing is applied, the oil-film-interferometry plug, the access plug and the traverse system are mounted into sleeves protruding in the inner chamber of the plate. This plate was mounted in the most downstream position, so that the wall-mounted traverse allows the measurement of boundary-layer profiles in a location unaccessible to the wind-tunnel traverse system. The traverse has a range of 500 mm and a relative accuracy of ±1 µm. The positioning is obtained with a DC servomotor controlled with a rotary optical encoder. Since the traverse is fixed to the plate and the traverse mechanism is not exposed to the flow, probe vibrations are kept at a minimum.
3.3. Suction/blowing system The suction/blowing system is the system of hoses, valves and a fan which allows to generate and regulate the airflow through the perforated plate. For each plate section eighteen flexible hoses with a diameter of 25 mm depart from the side of the three suction channels on the bottom side of the plate (see Fig. 3.7). These flexible hoses are connected to 8 manifolds (one for each plate) from which rigid steel pipes 125 mm in diameter drive the flow to a wooden suction chamber (see Fig. 3.8). Eight regulation valves allow the adjustment of the volume flow in each one of the steel pipe, so that a uniform transpiration velocity can be ensured even in presence of differences in the permeability of each titanium sheet. The chamber is connected to a 7.5 kW AC centrifugal fan through a single pipe 200 mm in diameter equipped with a flowmeter measuring 3.5. Hot-wire anemometry 39 the total volume flow driven by the system. The regulation of the volume flow is obtained adjusting the fan rotation speed with a variable-frequency drive.
3.4. Instrumentation 3.4.1. Air properties
The atmospheric pressure Patm was measured with a Druck PTX 520 absolute pressure transmitter (accuracy of ±180 Pa) connected to a Furness FCO510 micromanometer. The air temperature T was obtained measuring with a Fluke- 45 multimeter the resistance of Pt-100 sensor positioned in proximity of the hot-wire probe. The estimated accuracy on the temperature is ±0.15 K. Air density is obtained from the ideal gas law
Patm = ρRT , (3.1) where the specific gas constant is R = 287 J kg−1 K−1. The dynamic air viscosity was obtained from Sutherland’s formula µ T 3/2 T + S = 0 , (3.2) µ0 T0 T + S −5 with constants S = 111 K, T0 = 273 K and µ0 = 1.716 × 10 (constants from White 1991).
3.4.2. Differential pressure measurements Dynamic pressure The dynamic pressure in the free stream was measured with a Prandtl tube and monitored during the experiments. The Prandtl tube was also used as the reference for the calibration of the hot-wire probes. The differential pressure between the total and static port was measured with a Furness FCO510 pressure transducer with an accuracy of ±0.25% of the reading for differential pressures from 20 Pa to 2000 Pa and of ±0.05 Pa for differential pressures smaller than 20 Pa.
Static pressure taps The pressure taps on the surface of the plate, the leading-edge pressure taps and the pressure ports inside the plate chamber were monitored with a 16 inputs Scanivalve DSA 3217 with a nominal accuracy of ±5 Pa. Given the limited amount of pressure inputs available on the pressure transducer, not all the pressure taps can be measured simultaneously but a choice was made according to the specific measurement requirements.
3.5. Hot-wire anemometry 3.5.1. Introduction Due to the small length scale and time scale encountered in the investigated flow cases, with the viscous length scale as small as l∗ ≈ 6 µm for some turbulent 40 3. Experimental setup and measurement techniques
(d)
(c)
(b)
(a)
Figure 3.8. Photography of a portion of the suction/blowing system. The wind-tunnel test section is on the left of the photo. (a) manifold; (b) regulation valve; (c) suction/blowing chamber; (d) centrifugal fan. 3.5. Hot-wire anemometry 41 suction boundary layers, the most suitable measurement technique is hot-wire anemometry. Hot-wire (and hot-film) anemometry relies on the dependence of the heat transfer from a heated surface on the fluid motion of the surrounding fluid. In practice, Joule heating is used to heat an electrically conducting element which is simultaneously cooled by a fluid stream. An electrical circuit measures the changes in electrical resistance of the sensor, which are related to its temperature variation and in turn to the fluid velocity. The way the electrical circuit is designed differentiates the Constant Current Anemometry (CCA), the Constant Voltage Anemometry (CVA) and the Constant Temperature Anemometry (CTA), which was used for this investigation. If the heated sensor has the shape of a wire suspended between two conducting support (prongs) it is referred to as hot-wire, if it has the shape of thin film deposited on a insulating substrate it is called hot-film. Hot-wire are the preferred choice for measurements in gasses, while hot-film are used in liquids or for wall-shear-stress measurements. A common choice for the material of hot-wire and hot-film probes is tungsten, or platinum and its alloys. The diameter of a hot-wire is usually in the order of 1 µm to 5 µm and its length ranges from 0.25 mm to 2 mm, with the length over diameter ratio lw/dw ' 200 in order to limit the magnitude of the conductive heat transfer to the prongs (Ligrani & Bradshaw 1987), which degrades the time-response of the sensor. Measurements of multiple velocity components and of flow vorticity are also possible if multiple wires are used. The small physical dimension of the probe makes the frequency response of typical hot-wires much faster than other measurement techniques, ranging in the order of tens of kilohertz, a characteristics that make hot-wire anemometry well-suited for measurements in turbulent or unsteady flows. However spatial resolution, due to the size of the sensing element, can become a limitation for hot-wire measurements (as for any other measurement technique) in high Reynolds number turbulent flows. For this reason efforts to build nanoscale sensing element have been initiated in the last decade, leading to the production of the NanoScale Thermal Anemometry Probe (NSTAP) (see e.g. Fan et al. 2015). Since hot-wire anemometry has been a commonly used measurement technique for more than a century, the literature on the topic is rich and the interested reader is referred to the textbooks by Perry (1982), Lomas (1985), Bruun (1995) and Tropea et al. (2007, §5.2).
3.5.2. Sensors characteristics In this investigation in-house built boundary-layer type single-wire probes were used and an account on the manufacturing process can be found in Ferro (2012). The wires were made out of platinum, which is commercially available in the form of Wollastone wire (a platinum wire clad in silver). For probes with wire-length Lw > 0.5 mm, wires with diameter dw = 2.54 µm were used. In this case the Wollastone wire was immersed in nitric acid to fully remove the silver coating, then the platinum core was directly soldered on the prongs. Probes with Lw < 0.5 mm required instead the use of a thinner wire with diameter dw = 1.27 µm: in this case the un-etched Wollastone wire was soldered on the 42 3. Experimental setup and measurement techniques
(a) (b) 1 mm
H2NO3 (6% m/m) 6 V
Figure 3.9. Photography of the Wollastone wire electroetch- ing (a) and of the finished hot-wire probe (b). prongs, spaced of approximately 1 mm, and the platinum core was exposed for the desired length by electroetching with a small jet (dj = 0.15 mm) of diluted nitric acid (see Westphal et al. 1988), as shown in Figure 3.9.
3.5.3. Sensors operation and calibration procedure The hot-wire probes were operated in constant temperature mode with a Dantec StreamLine 90N10 frame in conjunction with a 90C10 CTA module. The resistance overheat OH was set to 70% or 80%, with R − R OH = h c , (3.3) Rc where Rh indicates the electrical resistance of the sensor at operating tempera- ture and Rc the resistance at flow temperature. For probes with dw = 2.54 µm, the square-wave test gave a frequency response between 35 kHz to 85 kHz at flow velocities from 0 m/s to 45 m/s. In these cases an analogue low-pass fil- ter with 30 kHz cut-off frequency was applied on the signal. For probes with dw = 1.27 µm the square-wave test gave a frequency response between 85 kHz to 150 kHz at flow velocities from 0 m/s to 40 m/s and the low-pass filter cut-off frequency was set to 100 kHz. The hot-wire signal was amplified and finally acquired with a 16bit National Instruments PCI-6259 acquisition card. The probes were calibrated in-situ in the free stream against a Prandtl tube connected to a Furness FCO510 pressure transducer. The number of calibration points varied between 12 and 20, increasing for larger velocity ranges, with the lowest velocity point having a speed U ≈ 1.4 m/s. Below this speed the steady operation of the tunnel and the accuracy in the differential pressure determination cannot be guaranteed in the present setup. At lower velocity the tunnel cannot be maintained at constant temperature, and the measurement 3.6. Transpiration velocity determination 43
40 pre−calibration 35 post−calibration calibration law 30
25
20 (m/s) U 15
10
5
0 0.45 0.5 0.55 0.6 0.65 E (V)
Figure 3.10. Typical pre- and post-calibration of one of a hot-wire probes used for the experiments. uncertainty on the differential pressure measured at the Prandtl tube port becomes larger. A fourth order polynomial fit through the calibration points (including the hot-wire signal at zero velocity, E0) was used as calibration law. The flow temperature was kept the same between calibration and mea- surements, hence no temperature correction of the measured data is required. The calibration procedure was repeated many times in one day in order to check for calibration drift. While for probes with wire diameter dw = 2.54 µm the calibration law appeared to be very stable in time, it was observed that probes with wires with diameter dw = 1.27 µm were more prone to show drift problems, as already reported by Hites (1997) and Discetti & Ianiro (2017). When these probes were in use, a full calibration was performed before and after each boundary-layer profile measurement and the hot-wire signal at a reference location in the free stream was acquired several times during the experiment to check and quantify the possible drift. The calibration law used for these experiments has been obtained from interpolated calibration points, average of the pre- and post-calibration laws (see Fig. 3.10). The largest variation in the measured free-stream velocity between the start and the end of the measurement has been 1.4%, even though for most of the cases the variation was limited to less than 1%.
3.6. Transpiration velocity determination Two different methods were used to determine the transpiration velocity. The total volume flow through the eight plates was measured with a flow meter during all the experiments and could be related to the mean transpiration velocity on the whole plate. Moreover the differential pressure across the titanium sheets was monitored and, in combination with previously determined 44 3. Experimental setup and measurement techniques
0.2 (a) 7 (b)
0.15 6 3 ρ s) 5 / 10 P/ × (m
0.1 | ∆ | 0
0 4 V V √ | | 3 0.05 T = 295.1 K P = 100.9 kPa 2 atm 0 1 0 200 400 600 800 1000 20 40 60 80 100 120 ∆P (Pa) ∆ dh√ρ P µ
Figure 3.11. Suction velocity V0 vs. the differential pressure across the titanium sheet ∆P for all the eight plate element expressed in physical units (a) and normalized form (b). The solid line are third-order polynomial fit thorugh the measured points. plate permeability, it can be related to the transpiration velocity through each plate element. The flow meter used was a Lindab FMU-200 nozzle-type flow meter, specifically calibrated for this experiment with extended uncertainty of ±1.5%. The permeability of each assembled plate element for the cases of suction and blowing was determined applying suction and blowing to one plate at a time while measuring the total flow rate and the pressure drop (∆P ) across the sheet. The uniformity of the pressure in the chamber of the plate could be checked comparing the pressure measured at four different locations (see §3.2.1) and the variation was found to be less than ±1%. The volume flow rate was measured with a Meriam 50MC2-6 laminar flow element (accuracy ±1.5%) and could directly be related to the suction/blowing velocity (V0). The results for the suction case are reported in Figure√ 3.11 for all the eight plates. The use of the two normalized parameters dh ρ∆P /µ (where dh is the p diameter of one hole) and |V0|/ ∆P/ρ allows to compensate for the variation of ambient condition between the permeability measurement and the actual experiments. A third order polynomial fit through the measured points is used to calculate the vertical velocity from the measured ∆P , ρ and µ. Plate-to-plate variation can be observed from Figure 3.11 and concerns on the uniformity of the permeability of each plate were raised. Due to the impossibility of measuring the local permeability on the whole 7.5 m2 of perforated surface, permeability measurements were performed on four circular samples of diameter 60 mm obtained from a 150 mm × 150 mm test piece manufactured in the same 3.7. Skin-friction measurement 45
0.14
0.12
0.1
0.08 | (m/s) 0 0.06 V | 0.04
0.02
0 0 100 200 300 400 500 ∆P (Pa)
Figure 3.12. Suction velocity V0 vs. the differential pressure across the titanium sheet ∆P for four samples obtained from a single test piece. way as the eight larger sheets used in the plate elements. The results are shown in Figure 3.12 and the uniform behaviour of the different samples can be observed. As a conclusion, the plate-to-plate variation in permeability observed in Figure 3.11 is likely due to a systematic variation of the mean hole size among the different sheets, which were not laser drilled in a single batch but one by one. Since the permeability of each plate is known, a uniform suction velocity on the whole perforated surface can be obtained regulating the differential pressure across each of the eight sheets with the regulation valves of the suction/blowing system (see §3.3). The valves need to be regulated for each transpiration velocity, and, of course, whenever a portion of a plate is covered to partially reduce the suction area. With this procedure a deviation of the suction velocity smaller than ±2.5% could be obtained between the different plate elements. During the experimental campaign an increasing deviation between the suction velocity measured from the differential pressure across the sheets and the one measured with the flow meter was observed. Cleaning the surface with acetone strongly reduced the discrepancies, hence it was concluded that the cause for the deviation had been the deposition of dust and dirt on the plate surface. For this reason the suction velocity used for the data analysis was obtained from the flow-meter measurements, which measure directly the volume flow rate, while the differential pressure measurements have been used exclusively to check and regulate the uniformity of the suction velocity between the different plate elements. 3.7. Skin-friction measurement In boundary-layer studies the wall shear stress is a quantity of great importance. As a measure of the forces exchanged between the flow and an object, it captures the results of all the physical phenomena occurring in the boundary layer. For 46 3. Experimental setup and measurement techniques
α U
oil
Figure 3.13. Schematics of the oil-film interferometry working principle. The incident light must be monochromatic. this reason, in addition to the obvious practical relevance for engineering applications, it has a fundamental role in the theoretical description of turbulent boundary layers, often as a scaling parameter in the form of uτ . For turbulent boundary layers on impermeable surfaces, oil-film interferometry has in the last decade become one of the reference techniques to measure the wall shear stress. However, since it relies on the observation of the motion of a thin film of oil on a surface, it cannot be used on permeable surfaces. As a result, in this investigation oil-film interferometry has been employed exclusively for the case of canonical turbulent boundary layer. Measuring shear stress on permeable surfaces present significant challenges: efforts to use hot-films or miniaturized Preston tube for shear-stress measurement on permeable surfaces were initiated, but proved to be unsuccessful. Few attempts of measuring wall shear stress over permeable surfaces with floating elements with a porous surface can be found in the literature (Dershin et al. 1967; Depooter et al. 1977). However, the technological difficulties in realizing such a balance (especially for the necessity of an air-tight but mechanically isolated air supply) and the problematics related to the errors originating from misalignements and gap-sizes discouraged the author from the use of this technique. The following section will report a brief summary on the theory of oil-film interferometry together with details on the specific experimental arrangement. Next, the attempts to use a hot-film probe and a miniaturized Preston tube for the determination of shear stress on porous surfaces are documented.
3.7.1. Oil-film interferometry Oil-film interferometry is a technique which allows the direct measurement of the mean wall shear stress through the observation of the motion of a very thin layer of oil (< 10 µm) deposited on a surface and stretched by the action of the flow. The local thickness of the oil film is visualized and made measurable by the use of Fizeau interferometry. This technique was first introduced by Tanner & Blows (1976) and further developed by Monson (1983) and Zilliac (1996) among others. Its evolution has been strictly related to the technological 3.7. Skin-friction measurement 47 improvement in imaging techniques: first photographic films were used to capture the interferograms, then single point techniques using photodetectors were developed, and finally, with the availability of affordable CCD cameras, digital processing of the full oil-film interferogram became possible. A review on the technique and its historical developments can be found in Naughton & Sheplak (2002). The motion of a thin oil sheet stretched by an external flow is governed by the equation (Squire 1961) " 3 2 # ∂h 1 ∂ h ∂P h ∂Uair = − µair + ∂t µoil ∂x 3 ∂x 2 ∂y y=h " 3 2 # 1 ∂ h ∂P h ∂Wair − µair , (3.4) µoil ∂z 3 ∂z 2 ∂y y=h where h = h(x, z, t) is the local thickness of the oil. The underlying assumptions are that the oil is two-dimensional, incompressible and characterized by a very small Reynolds number (creeping flow) and that the external air flow can be described by the boundary-layer equations. Moreover, gravity forces and surface tension are neglected. The no-slip boundary condition was imposed at the wall and the coupling of the oil and air flowfield was obtained imposing continuity of the velocity and of the shear stress at the oil-air interface. Since the viscous- scaled oil thickness is small (h = O(`∗)) we can make the additional hypothesis that the distortion of the surface caused by the oil is negligible, hence
∂Uair µair = τw , (3.5) ∂y y=h 1 where τw is the local shear stress in absence of the the oil film . Considering a bi-dimensional external flow with sufficiently small pressure gradient2 eq. (3.4) can be rewritten as ∂h 1 ∂(h2τ ) = − w , (3.6) ∂t 2µoil ∂x which relates the evolution in time of the oil thickness to the skin friction of the undisturbed flow. Even if this analysis has been obtained for laminar flow, the results holds also for turbulent flow, with τw now representing the time-averaged wall shear stress (Zilliac 1996; Fernholz et al. 1996). It appears from eq. (3.6) that if we could measure the thickness of the oil-sheet we could obtain a measurement of the skin-friction. If we choose a transparent oil in combination with a reflective wall, Fizeau interferometry can serve to the purpose. Illuminating the oil with a monochromatic light, part of the light is reflected at the external surface of the oil (oil-air interface), while another portion is transmitted into the oil drop and reflected by the wall (see Fig. 3.13). At the oil-air interface constructive or destructive interference occurs
1This assumption is discussed in greater detail by Segalini et al. (2015). 2Zilliac (1996) reported that “the pressure gradient terms are at least two order of magnitude smaller than the shear stress terms for most flows of aerodynamic interest”. 48 3. Experimental setup and measurement techniques
Figure 3.14. Typical sequence of interferograms (detail): flow direction is top to bottom. Each frame corresponds to a physical space of about 5 × 5 mm2. Time interval between frames is ∆t = 160 s, µoil = 0.105 Pa · s, τw = 0.58 Pa. between light rays coming from the different paths: an interference pattern originates, where dark and light fringes alternates depending on the local oil thickness (see Fig. 3.14). The difference of the film thickness between two successive dark (or light) fringes is
λl ∆h = q , (3.7) 2 2 2 noil − sin α where λl is the wavelength of the light, noil is the refractive index of the oil and α is the viewing angle of the observer measured from the wall normal direction. The oil-thickness at the kth dark fringe is hence given by
hk = h0 + k∆h , (3.8) with h0 as the height at the zeroth dark fringe, dependent on the wall material (Fernholz et al. 1996). Fernholz et al. (1996) lists five different methods through which a single or a time-series of interferograms can be used to obtain the skin-friction τw. The one used in this investigation is obtained from eq. (3.6) for the case with τ w constant in space on top of the oil-film. With this assumption eq. (3.6) can be 3.7. Skin-friction measurement 49 rewritten in the form of an advection equation ∂h τ h ∂h + w = 0 , (3.9) ∂t µoil ∂x which tells us that the kth dark fringe moves with speed
τwhk uk = . (3.10) µoil Combining eq. (3.10), (3.8), (3.7) the shear stress can finally be expressed as q 2 n2 − sin2 α h0 oil τw k + = µoiluk . (3.11) ∆h λl From a time-series of interferograms, it is possible to follow the movement of many fringes, hence many fringe speeds uk can be determined. For each fringe, eq. (3.11) can be written, hence an overdetermined system of linear equations can be built with unknowns τw and h0/∆h. Solving this system the mean wall shear stress can be determined without any a priori knowledge of h0/∆h.
Oil-film interferometry setup and data processing The oil-film interferometry measurements were conducted on a glass plug inserted in the plate element equipped with the wall-mounted traverse (see §3.2.2). The plug is cylindrical with diameter 48 mm and is made of an inner glass cylinder mounted inside an outer aluminum ring equipped with a pressure-tap. The inner cylinder is made of N-BK7 borosilicate glass 40 mm in diameter and 4 mm thick. In order to measure the temperature of the surface during a run, a thermocouple is pressed in contact to the bottom surface of the glass. The plug can be accurately aligned with the surface of the plate with three set screws. A mistake in the design phase led to a streamwise misalignment between the location of the hot-wire probe and the oil-film interferometry plug. However, 6 for the range of Rex considered (Rex ' 5 × 10 ), the misalignment results in a variation of τw < 0.2%, smaller than the estimated uncertainty on the skin-friction determination via oil-film interferometry.
The light source is a sodium-vapour lamp (λl = 589 nm) and the interfer- ograms were recorded with a Nikon D7100 camera placed on the roof of the wind-tunnel with a 200 mm focal-length objective, resulting in a resolution of 70 px/mm. The view-angle α of the camera was measured with a digital angle gauge with resolution ±0.1◦. The photos were taken at interval ranging from 4 s to 15 s depending on the shear-stress level. The oil employed was silicone oil DOW CORNING 200 fluid, with a nominal kinematic viscosity of 100 mm2/s. The oil-viscosity variation with temperature has been measured with a Ubbelo- hde viscometer (accuracy ±0.1%) immersed in a temperature-controlled heated bath and the results are shown in Figure 3.15. The viscometer provides a measurement for the kinematic viscosity, from which the dynamic viscosity can 50 3. Experimental setup and measurement techniques
0.115
0.11
0.105 s) ⋅ 0.1 (Pa oil µ 0.095
0.09
0.085 290 292 294 296 298 300 302 304 T (K)
Figure 3.15. Variation of the oil kinematic viscosity with temperature. Solid line: Fit of eq. (3.13) through the measured points. be calculated ρoil(T0) µoil(T ) = ρoil(T )νoil(T ) = νoil(T ) , (3.12) 1 + C1(T − T0) where the values of the constants were provided by the manufacturer (ρ0 = 3 −3 964 kg/m , T0 = 298.15 K, C1 = 0.96 × 10 ). To calculate the viscosity at temperatures other than the calibration points, the relation
C3/T µoil = C2 e (3.13) was used3, with the value of the constants obtained from a fit through the −3 measured points (C2 = 218.4 × 10 Pa·s ; C3 = 1819 K). The oil refraction index has been obtained from the data provided by the manufacturer. Before starting a measurement, the wind tunnel was run for a long time (more than one hour) so that all the components reached a steady temperature. Since the lower surface of the test plate is not immersed in the flow, but is facing the laboratory, the temperature in the test section was set to be close to room temperature (∆T < 1 ◦C), in order to limit the heat transfer at the plug. The flow was then stopped and one or several drops of oil were quickly positioned on the glass plug with the help of a needle. The wind tunnel was turned on again and the acquisition of the images was started as soon as the air and the plug reached a steady temperature. The maximum temperature variation observed on the plug during a run was 0.1 K. The quality of the oil flow can be severely decreased by the presence of dust particles, and care must be taken to work in a clean environment and with clean tools. Cleaning the
3The relation is commonly known as Andreade equation, even if it was originally proposed by de Guzm´an(1913) (see Viswanath et al. 2007). 3.7. Skin-friction measurement 51
20
16
12 (mm) x 8
4
0 200 400 600 800 1000 t (s)
Figure 3.16. Typical x − t diagram obtained from a time series of interferograms. Red dashed lines: user-identified fringe centre. glass plug and the needle before each run with a lens tissue wet in acetone helped to reduce the number of runs largely affected by dust deposition. From each series of interferograms several x−t diagrams can be constructed: one line of pixels parallel to the flow direction is selected and extracted from all the interferograms. An x − t diagram is then obtained assembling all the extracted lines in a single figure as shown in Figure 3.16. The x − t diagrams were then analyzed with a semi-automatic computer program, originally written by Osterlund¨ (1999) and further modified by Ruedi et al. (2003) and by the present author. The user is required to manually locate the fringe centre (peak of the greyscale intensity), an example is shown with the red dashed line in Figure 3.16. To improve accuracy, each measurement was repeated between 3 to 5 times and from each of the runs, showing a large area not contaminated by dust particle 3 different x − t diagrams were obtained and analyzed. The reproducibility of the measured τw proved to be better than ±2%, in close agreement with the accuracy estimate provided in Nagib et al. (2004) (±1.5%).
3.7.2. Hot-film sensors During this investigation, an attempt to measure the wall shear stress in presence of suction and blowing with hot-film sensors was conducted. Tao Systems Senflex SF0303 hot-films were glued on eight 10 mm diameter cylindrical plugs and aligned flush to the plate with the help of two set screws. Particular care was 52 3. Experimental setup and measurement techniques
Figure 3.17. Photography of the hot film sensor. taken to position the sensing element as close as possible to the leading edge of the plug, to minimize the length of transpiration interruption upstream of the sensor, which resulted to be about 3 mm (see Fig. 3.17). The cables were driven through a small cavity at the downstream edge of the plug, which was sealed on the bottom side with tape. The hot-films were operated with Dantec 90C10 CTA modules at resistance overheat ratio of 20%. To validate the method, one of the hot-film probes was calibrated against the shear stress measured with oil film interferometry for a non-transpired turbulent boundary layer. Suction was then applied on the surface, so that a laminar ASBL was obtained at the measurement location. In such a way a boundary layer with known velocity profile could be generated and another calibration law could be obtained from the shear stress obtained from the analytical solution of the ASBL. As apparent from Figure 3.18, the two calibration laws deviate significantly between each other, proving that the hot-film probes cannot be used to measure shear stress for transpired boundary-layer. The probable cause for the discrepancy is the transpiration interruption upstream and downstream of the sensing element. The technical difficulties in reducing the transpiration interruption length were considered to be insurmountable in the framework of this project and the use of hot-film sensors was abandoned.
3.7.3. Miniaturized Preston tube In order to measure the shear-stress in presence of wall-transpiration, the possibility to use a miniaturize Preston tube was explored. The method was initially developed by Preston (1954) and relies on the measurement of stagnation pressure with a tube placed on the wall: the difference between the stagnation pressure and the surface static pressure can be related to the shear stress with a law of the type 2 ∆P d τw = f 2 . (3.14) τw ρν 3.7. Skin-friction measurement 53
3 laminar calib. (ASBL) turbulent calib. (OFI) 2.5 +97%
2 ) 2 1.5 (N/m w τ 1
0.5
0 2.26 2.28 2.3 2.32 2.34 2.36 2.38 E (V)
Figure 3.18. Comparison between the hot-film calibration law obtained in a non-transpired TBL and the one obtained for a laminar ASBL.
Preston tube method does not provide a direct measurement of the skin-friction, since the measured quantity is the mean velocity in the tube in disguise. However, the method proved to work as long as the tube is fully immersed in the region where the law of the wall is valid. Preston tubes are typically calibrated in a fully-developed pipe flow, where a direct measurement of the skin friction can be obtained from the pressure-drop, with the Preston-tube diameter d+ lying mainly in the logarithmic region of the turbulent boundary layer (see Patel 1965; Head & Vasanta Ram 1971; Bechert 1996). In presence of wall transpiration, however, the law of the wall for non-suction turbulent boundary layer does not represent accurately the mean velocity profile outside the viscous sublayer, as depicted in Figure 3.19 where DNS results in the inner region of suction and impermeable turbulent boundary layer are compared. Consequently, as long as the Preston tube lies outside of the viscous sublayer, a different calibration law would apply for transpired or impermeable boundary layers, in analogy to what occurs in pressure gradient boundary layers (Patel 1965; Hirt & Thomann 1986). Simpson & Whitten (1968) proposed a calibration for Preston tube measurements in presence of wall transpiration, basing their conclusions on the Simpson’s law of the wall (Simpson 1967): as a consequence, this approach is biased and cannot be used to validate a mean-velocity profile scaling. A different Preston tube correction for wall transpiration was later proposed by Baker et al. (1971), a modified version of which was later recommended by Depooter et al. (1978), on the basis of shear-stress data measured with an ingeniously built porous floating element. If the tube opening resides fully in the viscous sublayer, a single calibration curve is expected to apply in all cases, independently of transpiration rate or 54 3. Experimental setup and measurement techniques
20 TASBL ZPG TBL
15
+ 10 U
5
0 100 101 102 y+
Figure 3.19. Comparison between the mean velocity profile of a ZPG TBL (DNS data by Schlatter & Orl¨u2010,¨ for Reθ = 4060) and a TASBL (DNS data by Khapko et al. 2016 for Γ = 3.571 × 10−3). pressure gradients. For this reason an attempt to build a miniaturized Preston tube always lying entirely in the viscous sublayer was explored. The main advantage of the use of a Preston tube for our purpose is that the tube can be placed directly on the transpired region, without the need to interrupt the suction/blowing for a certain distance. Due to the large shear stress level encountered in suction boundary layer, however, the size limitation on the tube diameter are stringent. In order to obtain an inner-scale diameter d+ < 3 in an −3 asymptotic suction boundary layer with U∞ = 25 m/s and Γ = 3 × 10 in air ν = 1.55 × 10−5, the diameter of the Preston tube must be ν ν d < 3l∗ = 3 = 3 √ = 34 µm . (3.15) −1 uτ U∞ Γ Pipettes used for in-vitro fertilization have diameters ranging from 6 µm to 40 µm and are easily available, representing an appealing choice. Such a Preston tube could be calibrated in a conventional ZPG TBL, and later used in transpired boundary layer providing reliable results. As a verification an ASBL at moderate speed could be used, for which an analytical expression for the shear-stress is available. To test this approach a Wallace ICSI WBB-30Z-30 polar biopsy pipette with 32 µm external diameter, and 1 µm wall thickness was used (see Fig. 3.20 and Fig. 3.21 for a sketch of the geometry). The pipette was mounted in a support that could be inserted in the 10 mm plate access plug at the plate equipped with the oil-film interferometry plug. Accurate positioning of the pipette on the wall was obtained traversing the support in the plate-normal direction with a micrometer screw while observing the position of the pipette tip with a microscope. The differential pressure between the stagnation and 3.7. Skin-friction measurement 55
Figure 3.20. Photography of the miniaturized Preston tube. In the inset a magnified view of the tip of the pipette is shown.
30°
Ø 30 μm 56 - 60 mm Ø 32 μm
Figure 3.21. Sketch of the Wallace ICSI WBB-30Z30 polar biopsy pipette. the static pressure measured at the wall (in a different spanwise position) was measured with a Furness FCO560 pressure calibrator with range 200 Pa. Due to the small tube opening, concerns on the frequency response of the pressure measurement raised. To estimate the time response of the system, the wind-tunnel was run for long times (hours) and then rapidly halted while the responses of the Preston tube and of a Prandtl tube in the free stream were monitored. The results are shown in Figure 3.22. It can be noticed that two hours after the flow was stopped, the stagnation pressure did not still reach the zero value. The time response of the miniaturized Preston tube appeared to 56 3. Experimental setup and measurement techniques
20
(Pa) 15 P ∆ 10
5 Preston 0 0 0.5 1 1.5 2
30
20 (m/s) ∞
U 10
0 0 0.5 1 1.5 2 t (hours)
Figure 3.22. Time response of the miniaturized Preston tube. be insufficient for any practically interesting laboratory application. The cause of the slow response is probably the very small diameter and comparably long extent of the opening. In fact, if the tip of the pipette was cut such that the external diameter became ≈ 100 µm, the time response considerably improved but the size limitation of eq. (3.15) could not be met. Due to the insufficient time response of the miniaturized Preston tube, the method was abandoned. Chapter 4
Measurement procedure and data reduction
4.1. Preparation of an experiment A typical experiment consisted in the acquisition of velocity time series for several boundary-layer profiles. As a first operation the extent of the wall transpiration was regulated either by disconnecting the upstream plate elements from the suction/injection system or by covering a portion of the surface with standard household aluminium foil, which was taped to the plate at its four edges. The regulation of the transpiration region with aluminium foil could only be performed for the suction cases, since in presence of wall injection the foil would detach from the surface. The desired suction/injection rate was obtained by regulating the fan rotation speed, while the desired uniformity was achieved by adjusting eight valves on the suction chamber (one for each plate element). For all the measurements performed employing the supported traverse system, the zero-pressure-gradient condition was checked with streamwise traverses of the hot wire in the free stream. Readings from the pressure taps at the plate surface were instead used when the wall-mounted traverse was in use and the supported traverse unmounted. A rather tedious iterative regulation of the ceiling shape and/or of the bleed-slot opening allowed to establish a zero-pressure-gradient free stream velocity distribution for all the experimental conditions for 1.5 m / x / 6.5 m, as shown in Figure 4.1. Once the experimental conditions were set, all the measured quantities were logged and the hot-wire scans of the velocity field were performed by a fully automatic computer program. A full recalibration of the hot-wire probe after the acquisition of a minimum of two and a maximum of five boundary-layer profiles was also automatically performed.
4.2. Heat transfer to the wall and outliers rejection It is well known that hot-wire measurement are affected by the proximity of a solid wall, since additional heat transfer occurs between the heated plume emanating from the wire and the colder surface. If the conventional hot- wire free-stream calibration is used, the additional heat transfer manifests as an unphysical increase of the measured velocity with decreasing wall-normal distance. In order to identify the data points affected by wall-proximity effects the method proposed by Orl¨u¨ et al. (2010) is followed. It relies on the observation
57 58 4. Measurement procedure and data reduction
1.03 a) 1.02
i 1.01 ∞ U h 1 / ∞ U 0.99
0.98
0.97
1.03 V =0 b) 0 1.02 V <0 0 V >0 0
i 1.01 ∞ U h 1 / ∞ U 0.99
0.98
0.97 0 1 2 3 4 5 6 x(m)
Figure 4.1. Variation of the free-stream velocity with the streamwise coordinate. hU∞i indicates the streamwise aver- aged value of the free-stream velocity. a) Hot-wire velocity- measurement in the free stream for all the experiments per- formed with the supported traverse system. b) Pressure- taps reading for all the experiments performed with the wall- mounted traverse system at x = 6.06 m. Dashed lines: hU∞i ± 0.5%.
p that the local turbulence intensity u02/U in the viscous sublayer increases monotonically reaching its peak value at the wall (see Fig. 4.2). A decrease of the measured local turbulence intensity in proximity of the wall can hence be explained by an additional heat transfer to the wall, responsible to an apparent p increase of U and/or a decrease of u02. In the current experiments all the data points closer to the wall than the location of the measured peak of local turbulence intensity were considered to be outliers and rejected (see Fig. 4.4). 4.3. Estimation of friction velocity and absolute wall distance 59
0.6 TASBL V /U = −2.50 × 10−3 [1] 0 ∞ TASBL V /U = −3.00 × 10−3 [1] 0.5 0 ∞ TASBL V /U = −3.45 × 10−3 [2] 0 ∞ 0.4 TASBL V /U = −3.57 × 10−3 [2] 0 ∞
/U −3 [2] 2 V U × ′ TASBL / = −3.70 10 0.3 0 ∞ u [3] p ZPG TBL V /U = 0, Re =3626 0 ∞ θ 0.2 Blowing TBL V /U = +1.00 × 10−3, 0 ∞ Re = 2395 [4] 0.1 θ
0 0 10 20 30 y+
Figure 4.2. Near-wall local turbulence-intensity distribution for TASBLs, a non-transpired boundary layer and a blowing boundary layer. [1]: LES by Bobke et al. (2016); [2]: DNS by Khapko et al. (2016); [3]: DNS by Schlatter & Orl¨u(2010);¨ [4]: LES by Kametani et al. (2015).
4.3. Estimation of friction velocity and absolute wall distance
A good estimate of the friction velocity uτ and of the absolute wall position is essential in the description of the flow, especially when viscous scaling is used as normalization. In some cases, namely non-transpired boundary layer measured at the most downstream measurement station, a direct measurement of the wall shear stress with oil film interferometry was available, but for all the other remaining cases the wall shear stress needed to be estimated indirectly. A direct measurement of the wall position with sufficient accuracy was unavailable in most of the cases: the rotary encoder of the traverse system used can provide an accurate measurement of the relative displacement of the hot-wire probe but not of the absolute position of the probe in respect to the wall. For measurements at the most downstream measurement station (x = 6.06 m, see §3.2.2), a measurement of the wall position was obtained observing the wire and the surface with a microscope mounting a micrometer focus gauge. The estimated uncertainty is ±20 µm, which is insufficient for most of the non-transpired turbulent boundary layers (`∗ = 11 µm to 34 µm) and for all the turbulent suction boundary layer (`∗ ≈ 7.5 µm) measured at that location. In the following paragraphs the procedure followed to determine uτ and the absolute wall position is given for each type of experiments reported in the results section. 60 4. Measurement procedure and data reduction
4.3.1. Non-transpired turbulent boundary layers For cases measured at the most downstream measurement station, a direct measurement of uτ was available through OFI (see §5.1.2 for details). The measured values of U + could then be used to find the absolute wall normal position applying a wall-normal shift to the viscous scaled velocity profile fitting it in a least-square sense to DNS data in the inner region. This procedure + + was applied just to data-points with U < 10, corresponding to y / 13, and the simulation data employed were the ones by Schlatter & Orl¨u(2010)¨ at Reθ = 4060. The result of the procedure is illustrated in Figure 4.4. For cases measured at streamwise positions different from x = 6.06 m, no direct measurement of the wall shear stress was available: in these cases a least-square fit extended up to y+ ≈ 35 to the aforementioned DNS data was performed to determine both the y-shift and the friction velocity.
4.3.2. Laminar/transitional suction boundary layers For the laminar ASBL profiles (see Fig. 5.12), the absolute wall position was obtained applying a y-shift determined with a fit to the ASBL solution to the measured y. The absolute wall positions determined for each profile agree between each other with a maximum deviation of ±5 µm around their mean value, which in turn deviates less than 20 µm from the microscope observations of the wire distance from the wall. Since the fitting procedure acts as a rigid shift of the profile, it does not alter the shape of the profile. The agreement observed in Figure 5.12 between the experimental profiles and the ASBL solution should therefore not be consider an artifact of the fitting procedure. For the transitional profiles shown in Figure 5.13, the fitting procedure to the ASBL solution could not be applied: the wall position was hence fixed to the mean value of the one determined for the laminar ASBL profiles, measured in the same set of experiments.
4.3.3. Turbulent suction boundary layers For turbulent suction boundary layers the wall shear stress was calculated using the von-K´arm´anmomentum integral equation modified for mass-transfer. In absence of pressure gradients and neglecting the streamwise variation of the Reynolds normal stresses difference u02 − v02 the following relation holds u 2 C dθ V τ = f = − 0 . (4.1) U∞ 2 dx U∞
Once uτ is determined for each profile, the absolute wall position is determined with a fit to simulation data of TASBL in the near-wall region (U + < 10.5, + corresponding to y / 18). The law of the wall for suction boundary layers is in general a function both of the absolute wall-normal position y+ and + of the suction velocity V0 . In the range of suction rates considered (Γ = 2.5 × 10−3 − 3.70 × 10−3), however, no significant difference is observed up to y+ = 20 (see Fig. 4.3). 4.3. Estimation of friction velocity and absolute wall distance 61
12 Γ = 3.70 × 10−3 [1] −3 [2] 10 Γ = 3.00 × 10 Γ = 2.50 × 10−3 [2] 8
+ 6 U
4
2
0 0 5 10 15 20 y+
Figure 4.3. DNS and LES of TASBL in the suction-rate range Γ = 2.50 × 10−3 − 3.70 × 10−3. [1]: DNS by Khapko et al. (2016); [2]: LES by Bobke et al. (2016).
The determination of uτ and the absolute wall position follows an iterative procedure. With an initial estimate of the wall position it is possible to calculate the momentum thickness for each of the measured profiles, from which is possible to calculate the term dθ/dx from a fit of the measured momentum thicknesses b to an exponential law of the type Reθ = aRex (see e.g. Fig. 5.15). Since for the reported profiles the first term of the R.H.S. of eq. (4.1) is at least one order of magnitude smaller than the second term, Cf has the same uncertainty as the suction rate. Once the value of uτ is calculated for each profile from eq. (4.1), the wall position is determined with the fit to the LES data by Bobke et al. (2016) for Γ = 3.00 × 10−3. Since the y-shift applied on the data determines a small variation of θ, the procedure is iterated until the variation in the estimated wall position is less than `∗/2. Figure 4.4 reports the results of this procedure for some of the measured turbulent suction boundary layers.
4.3.4. Turbulent blowing boundary layers Measurements of turbulent blowing boundary layers were performed at a single streamwise location, hence the calculation of the skin friction coefficient by means of eq. (4.1) was not possible. In the literature it is possible to find empirical models describing the variation of the skin-friction coefficient with the Reynolds number in presence of blowing (Simpson et al. 1969; Andersen et al. 1972; Depooter et al. 1977), but since the Reynolds number range of the current experiments is not covered by the Reynolds number range for which the aforementioned empirical correlations were derived, their use was avoided. For the turbulent blowing boundary layers, in absence of an estimate of uτ , no fitting of the velocity profile in the near-wall region could be performed, and 62 4. Measurement procedure and data reduction
0.4 (a)
0.3 /U 2 ′ 0.2 u p 0.1
0 0 5 10 15 20 25 y+
15 (b)
10 + U 5
0 0 5 10 15 20 25 y+
Figure 4.4. (a): Near-wall local turbulence intensity; (b): Near-wall inner-scaled mean velocity. Black Symbols: measured canonical ZPG TBL cases at x = 6.06 m (see Tab. 5.1); Colored symbols: some of the measured turbulent suction boundary layers (data in Tab. 5.3 for x = 4.8 m); Empty Symbols: data considered affected by additional heat transfer to the wall and rejected. Black solid line: DNS of a canonical ZPG TBL at Reθ = 4060 by Schlatter & Orl¨u(2010);¨ Red solid line: LES of a TASBL for Γ = 3.00 × 10−3 by Bobke et al. (2016); Black dashed line: upper velocity threshold for the fit to the near-wall DNS data for canonical ZPG TBLs; Red dashed line: upper velocity threshold for the fit to the LES near-wall data for turbulent suction boundary layers. the absolute wall-position was determined by microscopy, with an accuracy of ±20 µm. 4.4. Intermittency estimation In order to properly distinguish between laminar, transitional and turbulent boundary layers, an estimate of the intermittency of the velocity signal, indicated in the following by γ, is required. The intermittency measures the fraction of 4.4. Intermittency estimation 63 time in which the signal is in a turbulent state, taking the value γ = 0 when the signal is fully laminar and γ = 1 when the signal is fully turbulent. In order to estimate the velocity-signal intermittency, the method proposed by Fransson et al. (2005) was used. It is an objective and user-independent method which requires as only parameter a cut-off frequency for a high-pass filter operation on the velocity signal. An accurate description of the procedure can be found in Fransson et al. (2005). In this work, the intermittency estimation procedure was applied to two different cases: firstly it was used to determine the smallest suction rate for which no turbulent spots are observed in an initially laminar boundary layer, secondly it was used to determine at which suction rate an initially turbulent boundary layer starts to show traces of relaminarization. In the first set of experiments, no tripping tape is used and suction is applied immediately downstream of the leading edge: in this configuration the boundary-layer profile evolves toward the ASBL solution, with a characteristic length scale δ99,ASBL = ln(0.01)ν/V0. The cut-off frequency was set to be U∞/(4δ99) (close to the value U∞/(5δ99) used by Fransson et al. 2005). In the second set of experiments the boundary-layer is tripped at the leading-edge and the suction location is varied at different downstream location. The characteristic boundary-layer length scale can be approximated with δ99,TASBL, which for the range of suction rate considered corresponds to roughly 10 to 20 times δ99,ASBL. The cut-off frequency was in this case chosen to be fcut = U∞/(4δ99,TASBL) approximated by U∞/(40δ99,ASBL). To verify that these choices of cut-off frequency are appropriate to obtain a good estimate of the intermittency of the velocity signal, the calculated value γcalc are compared with the intermittency obtained from a visual inspection of the velocity signal γvis. Figure 4.5 shows two different velocity signals, the top one (a) was measured for an initially laminar boundary layer with traces of turbulent spots, while the bottom one (b) was measured for an initially turbulent boundary layer undergoing relaminarization. The gray-shaded areas are the portion of the signal identified as turbulent by visual inspection. For case (a) γvisual = 0.22 and γcalc = 0.20, while for case (b) γvisual = 0.68 and γcalc = 0.79, demonstrating that the choices of fcut are appropriate. 64 4. Measurement procedure and data reduction
10 (a) 8
6 (m/s) U
4
2 0 1 2 3 4 5 6 7 t (s)
(b) 25
20 (m/s) U 15
10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t (s)
Figure 4.5. Velocity-signal for two transitional boundary layers. (a): initially laminar boundary layer (γvisual = 0.22 and γcalc = 0.20). (b): initially turbulent boundary layer (γvisual = 0.68 and γcalc = 0.79). Gray-filled areas: portion of the signal identified as turbulent by visual inspection. Chapter 5
Results and discussion
5.1. Zero-pressure-gradient turbulent boundary layer This section reports results obtained for zero-pressure-gradient turbulent bound- ary layers in absence of wall-transpiration: the purpose is not to offer additional data or new insight on zero-pressure-gradient turbulent boundary layers, a topic which has been the subject of a large number of dedicated studies, but rather to test the capability of the current setup to reproduce this well-known flow case. Benchmarking the measured non-transpired TBL against the canonical ZPG TBL is a first proof of the quality of the setup and of the experimental procedures. If successful, it is possible to conclude, for instance, that the threshold on the variation of local free-stream velocity used to define the ex- perimental approximation of a zero pressure gradient is sufficiently low, that the history effects from the leading-edge pressure-distribution and from the tripping devices are small enough and finally that the perforated surface can indeed be considered hydraulically smooth. Hot-wire measurements were con- ducted for different free-stream velocities at different streamwise locations (0.55 m < x < 5.15 m) with the landing traverse system (see §3.1.2). Additional measurements were also performed at the measurement station on the most downstream plate element (x = 6.06 m) using the wall-mounted traverse. For the measurement at x = 6.06 m the skin friction, could be directly measured with oil-film interferometry, allowing a more careful analysis of the velocity profiles.
5.1.1. Assessment of the canonical state A turbulent boundary layer is here defined to be canonical when it is completely described by the governing parameters of the flow, which for ZPG TBLs are the properly normalized wall-normal distance and the local Reynolds number. In other words with this definition the canonical state is achieved when the real flow case studied can be considered representative of the ideal flow case that we intended to study, despite all the experimental imperfections such as wall roughness, three dimensionality, free-stream turbulence and history effects originating from the leading-edge pressure gradient and from turbulence- triggering devices. To assess the quality of the measured turbulent boundary layer, the procedure delineated by Chauhan et al. (2009) was followed. There,
65 66 5. Results and discussion
30 Composite profile Exp. Reθ = 14740
25
20 +
U 15
10
5
0 0 1 2 3 4 10 10 10 10 10 y+
Figure 5.1. Comparison of one of the ZPG TBL profile at x = 6.06 m with the composite profile by Chauhan et al. (2009). The fitted parameters were uτ , δ, Π. a large experimental data-set of ZPG TBL was analyzed on the basis of an analytic formulation of the mean-velocity profile as a composite profile of the type (Coles 1956) 2Π y U + = U + (y+) + W . (5.1) composite inner κ δ It was concluded that a boundary layer can be considered canonical1 if its shape factor H12 and wake parameter Π differ less than a certain threshold from the shape factor H12,num and the wake parameter Πnum of the composite profile at the same Reynolds number. The shape factor for all the measured non-transpired cases is shown in Figure 5.2, and compared with the one obtained from the composite velocity profile. The proposed threshold ±0.008 for the maximum deviation permissible for the profile to be considered in an equilibrium state is also reported with dashed lines. Figure 5.3 reports instead the experimentally determined wake parameters, together with Πnum and the proposed threshold Πnum ± 0.05. The wake parameters were determined from a fit of all the points with y+ > 50 and y < δ99 to the composite profile in Chauhan et al. (2009). It can be noticed that the majority of the measured profiles appear to respect the criteria proposed for the wake parameter, with more frequent deviations for Reθ / 7500. The
1In Chauhan et al. (2009) a different terminology is used, indicating with equilibrium what here we denote with canonical. 5.1. Zero-pressure-gradient turbulent boundary layer 67
1.5
1.45
1.4 12 H 1.35
1.3
1.25 5 10 15 20 Re × 10−3 θ
Figure 5.2. Shape factor H12 against the momentum thick- ness Reynolds number Reθ for all the measured non-transpired cases. Filled symbols: profiles at x = 6.06 m; Solid line: H12,num obtained from the integration of the composite profile proposed in Chauhan et al. (2009); Dashed line: H12,num±0.008; Dotted line: H12,num ± 1% .
shape-factor criteria appears to be more stringent than the wake-parameter criteria, with a larger number of profiles outside of the proposed bounds, again mainly for Reθ / 7500. These deviations at lower Reynolds number can be an indication of over or under tripping or of history effects originating from the leading-edge pressure gradient. For large enough Reynolds number, however, the measured profiles can be considered to be equilibrium ZPG TBL profiles, proving the quality of the present apparatus and of the experimental procedures. As additional proof, the next sections will report the measured skin-friction coefficient and the velocity profiles at the most downstream measurement location.
5.1.2. Skin-friction coefficient Skin-friction measurements with oil-film interferometry were performed at the most downstream measurement station (xOFI = 6.13 m) for different free-stream velocities. Each measurement was repeated from three to five times, and several x − t diagrams (hence several value of τw) were obtained from each run (see §3.7.1). The mean value of the measured skin friction τw, was used to obtain 68 5. Results and discussion
0.8
0.7
0.6
0.5
0.4 Π
0.3
0.2
0.1
0 5 10 15 20 Re × 10−3 θ
Figure 5.3. Wake parameter Π against the momentum thick- ness Reynolds number Reθ for all the measured non-transpired cases. Filled symbols: profiles at x = 6.06 m; Solid line:Πnum for the composite profile proposed in Chauhan et al. (2009); Dashed line:Πnum ± 0.05 . the skin-friction coefficient τ C = w , (5.2) f 1 2 2 ρU∞ with U∞ measured with a Prandtl tube located in the free stream above the measurement location. The results are reported in Figure 5.4 together with a power-law fit through the data and the 1/7th power law with the modified coefficient proposed in Nagib et al. (2007). The power-law fit was then used to calculate the skin-friction coefficient, and hence the friction velocity uτ , for a series of velocity profiles measured with the wall-mounted traverse at the location x = 6.06 m. The skin-friction coefficients obtained with this procedure are plotted against the momentum-thickness Reynolds number in Figure 5.5 and compared with the Coles-Fernholz skin friction law eq. (1.35) with parameters κ = 0.384 and C = 4.127 as proposed by Nagib et al. (2007). Good agreement between the OFI measurements and the Coles-Fernholz skin-friction law is found, with larger deviation (even though limited to less than 2%) for the two smallest Reynolds numbers considered.
5.1.3. Statistical quantities Seven boundary-layer profiles were measured at the most downstream measure- ment location x = 6.06 m for different free-stream velocities, with the main 5.1. Zero-pressure-gradient turbulent boundary layer 69
2.7 C = 0.022 Re−0.138 f x C = 0.024 Re−1/7 2.6 f x
2.5 3
10 2.4 ×
f C
2.3
2.2
2.1 4 6 8 10 12 14 16 18 20 Re × 10−6 x
Figure 5.4. Skin friction coefficient measured with oil-film interferometry. The error bars show a ±2% variation in Cf . Solid line: power-law fit through the measured data; Dashed line: 1/7th law with coefficient proposed by Nagib et al. (2007).
experimental parameters listed in Table 5.1. For these profiles uτ was estimated from the oil-film interferometry measurements. The mean-velocity profile in inner scaling is shown in Figure 5.6, together with the linear and logarithmic law of the wall U + = y+ (5.3) and 1 U + = ln y+ + B (5.4) κ with constants κ = 0.384 and B = 4.173 as proposed by Nagib et al. (2007). Good overlap of the data in the inner region is observed for all the profiles considered, as expected from the classical turbulent-boundary-layer theory. Figure 5.7, shows the mean velocity defect scaled with the friction velocity + + ∗ U∞ − U in outer scaling, with the Rotta-Clauser length scale ∆ = δ U∞/uτ as the outer length scale. The good collapse of the data when looking at the velocity defect in outer scaling gives additional confidence that the equilibrium state was attained for the boundary-layer profiles considered. The dashed line in Figure 5.7 represents the log law expressed in the velocity-defect formulation 1 U + − U + = − ln η + B , (5.5) ∞ κ 1 70 5. Results and discussion
2.7
2.6
2.5 3
10 2.4 ×
f C
2.3
2.2
2.1 8 10 12 14 16 18 20 22 24 Re × 10−3 θ
Figure 5.5. Skin friction coefficient vs. Reθ for all the profiles measured at the measurement station x = 6.06 m. The error bars show a ±2% variation in Cf . Dashed line: Coles-Fernholz skin-friction law eq. (1.35) with coefficient κ = 0.384 and C = 4.127 (Nagib et al. 2007).
with η = y/∆, κ = 0.384 and B1 = −0.87 (Monkewitz et al. 2007). The streamwise-velocity-variance profiles in inner scaling are shown in Figure 5.8. Spacial filtering effects due to the finite size of the hot-wire probe is + apparent for viscous-scaled wire length Lw > 10 as an attenuation of the peak value of the measured velocity variance. The spatial-filtering correction method proposed by Smits et al. (2011) was applied on the data and the results are shown in Figure 5.8 with solid lines. The intensity of the near-wall peak of the inner-scaled velocity variance for corrected and uncorrected data is shown in Figure 5.10 together with the DNS data by Schlatter & Orl¨u(2010).¨ The peak of the velocity variance in the range of Reynolds number considered, when spatial-filtering effects are corrected for, is larger than the one obtained at the lower Reynolds number covered by the results of the simulations, in + 02 agreement with the view that u peak grows with Reynolds number (see e.g. Metzger & Klewicki 2001, Marusic & Kunkel 2003 and Hutchins et al. 2009). + However, all the corrected magnitudes of the peak of u02 for the Reynolds- number range explored by the current experiments fall in the quite narrow band + 02 8.54 < u peak < 8.75. The expression for the Reynolds number variation of + 02 u peak proposed by Hutchins et al. (2009), Marusic et al. (2010), Monkewitz 5.1. Zero-pressure-gradient turbulent boundary layer 71
30
25
20 +
U 15
10
5
0 100 101 102 103 104 + y
Figure 5.6. Inner-scaled mean-velocity profiles for x = 6.06 m. Dashed-dotted line: linear-law eq. (5.3); Dashed line: log-law eq. (5.4) with constants κ = 0.384 and B = 4.173 (Nagib et al. 2007). Symbols as in Tab. 5.1.
25
20
15 + −U + ∞ U 10
5
0 10−3 10−2 10−1 100 y/∆
Figure 5.7. Outer-scaled velocity-defect profiles for x = 6.06 m. Dashed line: log-law eq. (5.5) with constants κ = 0.384 and B1 = −0.87 (Monkewitz et al. 2007). Symbols as in Tab. 5.1. 72 5. Results and discussion
Table 5.1. Experimental parameters for the ZPG TBL profiles measured at x = 6.06 m.
Case
U∞ (m/s) 12.5 15.0 20.0 25.0 29.9 34.9 39.8 uτ (m/s) 0.450 0.531 0.696 0.855 1.011 1.167 1.321 3 Cf × 10 (-) 2.58 2.52 2.42 2.34 2.29 2.24 2.20 `∗ (µm) 33.5 28.3 21.6 17.5 14.8 12.9 11.4 θ (mm) 9.90 9.63 9.29 8.84 8.36 8.22 8.05 δ∗ (mm) 13.42 13.00 12.37 11.68 10.99 10.78 10.50 δ99 (mm) 83.74 84.99 81.87 78.29 77.99 77.88 76.42 Reθ (-) 8230 9580 12360 14740 16650 19090 21330 Reτ (-) 2500 3000 3790 4470 5250 6050 6710 H12 (-) 1.36 1.35 1.33 1.32 1.31 1.31 1.30 Π (-) 0.48 0.47 0.44 0.42 0.40 0.39 0.39 Lw (mm) 0.280 0.280 0.280 0.280 0.280 0.280 0.280 + Lw (-) 8.4 9.9 13.0 16.0 18.9 21.7 24.6 + 1/fmax (-) 0.17 0.23 0.40 0.61 0.85 1.13 1.45 tsmpU∞ (-) 13500 15900 22000 25500 30700 31300 36500 δ99
& Nagib (2015), together with the constant value reported by Vallikivi et al. (2015) are shown in Figure 5.10 for a comparison with the present data. 5.1. Zero-pressure-gradient turbulent boundary layer 73
9
8
7
6
5 + 2 ′
u 4
3
2
1
0 101 102 103 104 + y
Figure 5.8. Symbols: measured inner-scaled streamwise velocity-variance profiles for x = 6.06 m, symbols as in Tab. 5.1; Lines: data corrected for spatial-filtering effects with the method by Smits et al. (2011).
0.012
0.01
0.008 2 ∞ 0.006 /U 2 ′ u
0.004
0.002
0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y/∆
Figure 5.9. Outer-scaled streamise-velocity variance profiles for x = 6.06 m. Symbols as in Tab. 5.1. 74 5. Results and discussion
10 + ′ 2 [ 1] (a) u p e ak = 4.84 + 0.47 lnReτ + 9.5 ′ 2 [ 2] u p e ak = 4.8 + 0.38 lnReτ + 9 ′ 2 [ 2] u p e ak = 2.65 + 0.69 lnReτ + ′ 2 [ 3]
+ p e ak u p e ak = 8.4 2 8.5 ′ u 8
7.5
7 103 104 Re τ
11 + ′ 2 [ 3] (b) u p e ak = 8.4 10 + ′ 2 − + [ 4] u p e ak = 22 340/U∞ 9 + p e ak
2 8 ′ u 7
6
5 20 22 24 26 28 30 + U∞
Figure 5.10. Variation of the near-wall peak of the streamwise-velocity variance with the Reynolds number Reτ + (a) and with U∞ (b). Filled symbols: current experiments (x = 6.06 m), data corrected with the method by Smits et al. (2011); Open symbols: current experiments, measured data; Black crosses: DNS data by Schlatter & Orl¨u(2010);¨ Lines: logarithmic trends proposed by [1] Hutchins et al. (2009) (ex- + trapolated to Lw = 0) and [2] Marusic et al. (2010), con- stant value proposed by [3] Vallikivi et al. (2015) for the range 2600 < Reτ < 8300, expression proposed by [4] Monkewitz & Nagib (2015). 5.2. Zero-pressure-gradient suction boundary layers 75
5.2. Zero-pressure-gradient suction boundary layers In this section the results obtained for boundary layer with wall-normal suction are reported. Initially, laminar boundary layers are considered, with the inten- tion to verify that the apparatus is able to reproduce the laminar ASBL. Later, the range of suction rate for which turbulence can be maintained is investigated and finally turbulent suction boundary layer are described, with the main focus on turbulent asymptotic suction boundary layers.
5.2.1. Laminar ASBL A series of suction boundary-layer profiles were measured at the downstream measurement location x = 6.06 m for a free-stream velocity U∞ ≈ 10 m/s, with the suction applied immediately downstream of the leading-edge section (xs = 0.18 m) where no tripping tape was applied. In this configuration the Falkner-Skan boundary layer developing on the leading-edge section is expected to evolve towards the ASBL velocity profile, which is obtained at a certain downstream distance on the plate. Figure 5.11 shows the measured shape factor H12 corresponding to different suction rates Γ, together with the intermittency value at y ≈ δ∗ calculated from the velocity-signal (see §4.4). The displacement thickness Reynolds number of the ASBL solution ReASBL (eq. 2.8) is also reported alongside the suction rate in the following figures, since it is the most commonly used parameter in the description of the flow. We observe that for all the profiles characterized by fully laminar velocity (γ = 0), the measured shape factor coincide with the theoretical value H12 = 2. For this subset of measurements the full velocity profile is illustrated in Figure 5.12 and compared to the analytical ASBL solution. Excellent agreement is observed, proving that an ASBL was indeed obtained at the measurement location for all the suction −3 rates Γ ≥ 3.38 × 10 (ReASBL ≤ 296), providing a second proof of the quality of the experimental apparatus and procedures. Figure 5.13 depicts the three boundary-layers profiles for which an intermittent velocity was observed. For −3 the profile at Γ = 2.92 × 10 (ReASBL = 343) the intermittency is still low (γ = 4% at y = δ∗) and the mean-velocity profile is still in good agreement with the analytical ASBL solution. For increasing values of intermittency, the mean-velocity profile departs from the ASBL solution, showing larger normalized boundary-layer thickness and lower shape factor, related to the occurrence of turbulent mixing in the boundary layer. From the present data ona may conclude that in this particular setup intermittency in the velocity signal appears at a suction rate as high as Γ = −3 2.92 × 10 (ReASBL = 343). Care should be exercised in the generalization of these results. Since the critical Reynolds number according to linear stability is two order of magnitude higher than the values of ReASBL for which turbulent profiles were observed2, the transition of the ASBL is subcritical. The Reynolds
2 Bussman & M¨untz (1942) reported ReASBL,crit = 70 000 (see Schlichting & Gersten 2017, §15.2.4c), Hocking (1975) reported ReASBL,crit = 47 000 and Fransson & Alfredsson (2003) reported ReASBL,crit = 54 382. 76 5. Results and discussion
Re ASBL 500 400 300 200 100 2.2 (a) 2
12 1.8 H
1.6
1.4 2 3 4 5 6 7 8 9 10 Γ × 103
Re 500 400 300 ASBL 200 100 1 (b) 0.8 )
* 0.6 δ
≈
y 0.4 ( γ 0.2
0 2 3 4 5 6 7 8 9 10 Γ × 103
Figure 5.11. Boundary-layer shape factor H12 (a) and velocity-signal intermittency γ (b) at x = 6.06 m for U∞ ≈ 10 m/s and xs = 0.18 m. In (a), Solid line: theoretical value of shape factor for an ASBL (H12 = 2); Dashed lines: H12 = 2 ± 1%. number at which transition is observed depends on the perturbation level to which the boundary layer is exposed (surface roughness, free-stream turbulence intensity, leading-edge pressure gradients, smoothness at the plate joints etc.) and the extent of the streamwise distance along which the disturbances are allowed to grow. It is hence likely that a fully laminar ASBL exists for Γ < −3 2.92 × 10 (ReASBL > 343) under different conditions, as reported by Fransson (2010), where ASBL profiles at Re as high as ReASBL = 600 where obtained in a different experimental setup.
5.2.2. Self-sustained turbulence suction-rate threshold Before discussing the turbulent state of suction boundary layer, it is crucial to define the range of suction rate for which a turbulent state is self sustained. It is known since the earliest studies on suction boundary layers that an initially turbulent boundary layer would relaminarize for large enough suction rates. 5.2. Zero-pressure-gradient suction boundary layers 77
10 Γ = 9.41 × 10−3 9 Γ = 8.40 × 10−3 −3 8 Γ = 7.51 × 10 Γ = 6.62 × 10−3 7 Γ = 4.78 × 10−3 6 Γ = 3.84 × 10−3
ν −3 /
0 Γ = 3.61 × 10 5 −3 y V Γ = 3.38 × 10 − 4
3
2
1
0 0 0.2 0.4 0.6 0.8 1 / U U∞
Figure 5.12. Velocity profiles for fully laminar boundary layers at x = 6.06 m. Solid line: ASBL analytical velocity profile.
1 −3 25 Γ = 2.92 × 10 −3 Γ = 2.47 × 10 −3 Γ = 1.94 × 10 0.8 20
0.6
ν 15 / 0 γ y V − 0.4 10
5 0.2
0 0 0.2 0.4 0.6 0.8 1 U/U ∞
Figure 5.13. Velocity profiles for transitional boundary layers at x = 6.06 m. Solid line: ASBL analytical velocity profile. 78 5. Results and discussion
1
0.8 (a)
0.6
γ (b) 0.4
0.2
0
3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 Γ × 10−3 Re = 0.34 × 106; ∆x/δ ≈ 950 x,s s Re = 0.52 × 106; ∆x/δ ≈ 1030 x,s s Re = 1.99 × 106; ∆x/δ ≈ 300 x,s s Re = 0.34 × 106; ∆x/δ ≈ 680 x,s s Re = 1.67 × 106; ∆x/δ ≈ 200 x,s s Re = 0.52 × 106; ∆x/δ ≈ 740 x,s s Re = 2.02 × 106; ∆x/δ ≈ 210 x,s s
16 (a) (b) 14 12 20 10 (m/s) (m/s)
U 8 U 15 6 4 10 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 t (s) t (s)
Figure 5.14. Top: Intermittency factor γ of the near-wall velocity signal vs. the suction rate Γ for different suction start locations Rexs and streamwise evolution lengths ∆x/δs. Black −3 solid line:Γsst = 3.70 × 10 (Khapko et al. 2016); Black −3 dashed line:Γsst = 3.70 × 10 ± 4%; Red dashed-dotted line: −3 Γsst = 3.6 × 10 (Watts 1972). Bottom: Time series of the velocity signal for the two sample cases indicated with (a) and (b). 5.2. Zero-pressure-gradient suction boundary layers 79
However, there are considerable differences in the literature regarding the values of the threshold suction rate Γsst (see §2.2.3). While the determination of Γsst is interesting per se, its value is also important to avoid the inclusion of undesired data in the analysis of turbulent suction boundary layers, since relaminarizing profiles can carry misleading information in the study of the scaling of turbulent suction boundary layers.
To obtain a measure of Γsst, experiments were conducted in which suction was applied starting from the normalized streamwise location Rex,s downstream of an impermeable entry length on which a turbulent boundary layer developed (boundary-layer tripping is applied on the leading-edge section). At a down- stream distance ∆x from the commencement of suction, the velocity signal is + measured in the inner region of the boundary layer (9 / y / 15) and the intermittency is calculated to determine whether the signal is fully turbulent, relaminarizing or fully laminar. The measurement is repeated for different values of Γ and Rex,s and the results are shown in Figure 5.2.2. We observe that for all the initial conditions and evolution length considered the measured self-substained turbulence suction-rate threshold fall in a ±4% bound from the −3 value Γsst = 3.70 × 10 reported by Khapko et al. (2016).
5.2.3. Development of turbulent boundary layer with suction Turbulent suction boundary layers are expected to evolve towards an asymp- totic condition, for which the boundary layer becomes independent from the streamwise coordinate. Earlier works suggested that to experimentally obtain an asymptotic turbulent state is difficult (or even impossible, Bobke et al. 2016), mainly because the evolution towards the asymptotic state is slow, i.e. occurring over a streamwise distance many times larger than the initial boundary layer thickness. The evolution to the asymptotic state can however be hastened if the boundary layer thickness at the location of the suction start is chosen to be close to the asymptotic one (Dutton 1958; Black & Sarnecki 1958; Tennekes 1964). In order to test whether an asymptotic state could be obtained in the current setup, a series of experiments where conducted in which the suction rate was kept constant while the streamwise Reynolds number of the suction- start location Rex,s was gradually varied with a regulation of the free-stream speed and of the physical suction-start location. The latter regulation was obtained either disconnecting the upstream plate elements from the suction system or, when finer adjustment was needed, covering a portion of the surface with standard households aluminium foil. The results for different suction rates are shown in Figures 5.15 to 5.17, while in Table 5.2 the main experimental 0 parameters are listed. In Figures 5.15 to 5.17 x = x − xvo represent the streamwise coordinate corrected for the virtual origin xvo calculated from the downstream development of the canonical ZPG TBL cases. For all the suction rates considered here, it was possible to experimentally realize a boundary layer with approximately constant boundary-layer thickness, moreover for 4 out of 5 80 5. Results and discussion
Table 5.2. Experimental parameters for the measurement cases in Fig. 5.15 and 5.16.
3 −6 Case U∞ −V0 Γ × 10 xs Rex,s × 10 (m/s) (m/s) (-) (m) (-)
25.1 0.082 3.26 0.94 1.56 25.0 0.082 3.27 0.59 0.97 15.0 0.049 3.27 0.59 0.58 15.1 0.049 3.26 0.19 0.19 35.1 0.115 3.27 0.30 0.70 24.9 0.081 3.26 0.19 0.31
35.1 0.107 3.05 0.94 2.20 35.1 0.109 3.10 0.59 1.38 35.9 0.111 3.09 0.36 0.86
25.0 0.071 2.83 0.19 0.32 25.0 0.070 2.80 0.94 1.57 35.0 0.099 2.83 0.94 2.19 30.0 0.084 2.80 0.94 1.88 35.1 0.099 2.83 0.59 1.38
35.1 0.093 2.65 1.23 2.85 35.0 0.093 2.65 1.75 4.07 37.6 0.099 2.65 0.94 2.27
35.0 0.089 2.54 1.75 4.09 45.1 0.116 2.58 0.94 2.82 39.0 0.100 2.56 0.94 2.40
cases the same boundary-layer momentum thickness Reynolds number could be −3 obtained for different Rex,s (case and for Γ ≈ 3.27 × 10 , case and for Γ ≈ 3.07 × 10−3, case and for Γ ≈ 2.65 × 10−3, case and for Γ ≈ 2.56 × 10−3), suggesting that the turbulent asymptotic state was indeed −3 reached. For Γ ≈ 2.82 × 10 no exact overlap of Reθ is observed for different initial conditions, however the downstream evolution of one measurement case ( ) appear to be bounded between a case showing a slow decrease ( ) and a case showing a slow increase ( ) of the momentum thickness along the streamwise coordinate, thus suggesting that case ( ) can be representative of the asymptotic state for this suction rate. In Figure 5.18, the velocity mean and variance are compared for the boundary-layer profiles measured at the most downstream measurement lo- cation (x = 4.80 m) for the subset of cases listed above. An excellent collapse in 5.2. Zero-pressure-gradient suction boundary layers 81 the mean velocity profiles between the cases with matching suction rate is ob- served. The variance profiles for all the suction rates excluding Γ = 2.82 × 10−3 also show excellent collapse in the outer region of the boundary layer, while the observable deviation in the inner region can be explained by hot-wire spatial filtering effects. For Γ = 2.82 × 10−3 the velocity variance profile show small but observable differences in the outer region of the boundary layer, with the case ( ) having variance values between the ones of case ( ) and of case ( ). It is concluded that all the cases reported in Figure 5.18 excluding ( ) and ( ) can be considered turbulent asymptotic states. Figure 5.19 and 5.20 respectively show the mean and variance velocity profiles at the three most downstream measurement locations for some of the cases identified as asymptotic. In the streamwise-coordinate interval considered, corresponding to a streamwise dis- tance ∆x exceeding 20 times the boundary layer thickness δ99, the variation of momentum thickness is less than ±1.5% for all the cases considered. The varia- tion considering the full mean-velocity profiles is minimal and good overlap in the outer part of the velocity variance profile is also observed. For completeness the full evolution from a canonical ZPG TBL to the TASBL is illustrated in Figure 5.21 for case ( ). As expected the inner region adapt to the suction in a short downstream distance, while a longer distance is required for the outer part of the boundary layer to reach the asymptotic condition. This is particularly evident in the velocity variance profiles. Concluding, for all the suction rates considered, it was possible to obtain a turbulent asymptotic state towards the downstream end of the flat plate: this was assessed observing that Reθ reached a constant value and that the mean velocity and the outer part of the velocity variance profiles became invariant along the streamwise direction. For four out of the five suction rates considered, the asymptotic Reθ and the asymptotic mean and variance velocity profile could be obtained with two different initial conditions at the suction start, additional proof that the asymptotic state was indeed reached in a strict manner. For one suction rate (Γ ≈ 2.82 × 10−3) the asymptotic value of Reθ and the velocity-variance profile could not be exactly reproduced with different initial conditions, however Reθ and the velocity-variance profile appear to be bounded from two measurement cases with respectively slightly higher and lower streamwise coordinate Reynolds number at the suction start Rex,s . Table 5.3 summarizes the experimental conditions for which the asymptotic state could be obtained and the boundary-layer parameters at the most downstream measurement station (x = 4.80 m). In Figure 5.22 the change in momentum- thickness Reynolds number and shape factor with the suction rate for the asymptotic states in Table 5.3 are plotted and compared with the simulations results by Bobke et al. (2016) and Khapko et al. (2016). 82 5. Results and discussion
Γ ≈ 3.27 × 10−3 8000
6000
θ 4000 Re
2000
0
1.4 12
H 1.2
1 0 2 4 6 8 10 12 Re × 10−6 x’ Γ ≈ 3.07 × 10−3 8000
6000
θ 4000 Re
2000
0
1.4 12
H 1.2
1 0 2 4 6 8 10 12 Re × 10−6 x’
Figure 5.15. Momentum-thickness Reynolds number Reθ and shape factor H12 evolution for different initial condition at the suction-start location. Dashed lines: Reθ = f(Rex0 ) (Nagib b et al. 2007). Solid lines: power-law fit Reθ = aRex0 . 5.2. Zero-pressure-gradient suction boundary layers 83
Γ ≈ 2.82 × 10−3 8000
6000
θ 4000 Re
2000
0
1.4 12
H 1.2
1 0 2 4 6 8 10 12 Re × 10−6 x’ Γ ≈ 2.65 × 10−3 8000
6000
θ 4000 Re
2000
0
1.4 12
H 1.2
1 0 2 4 6 8 10 12 14 Re × 10−6 x’
Figure 5.16. Momentum-thickness Reynolds number Reθ and shape factor H12 evolution for different initial condition at the suction-start location. Dashed lines: Reθ = f(Rex0 ) (Nagib b et al. 2007). Solid lines: power-law fit Reθ = aRex0 . 84 5. Results and discussion
Γ ≈ 2.56 × 10−3 8000
6000
θ 4000 Re
2000
0
1.4 12
H 1.2
1 0 2 4 6 8 10 12 14 Re × 10−6 x’
Figure 5.17. Momentum-thickness Reynolds number Reθ and shape factor H12 evolution for different initial condition at the suction-start location. Dashed lines: Reθ = f(Rex0 ) (Nagib b et al. 2007). Solid lines: power-law fit Reθ = aRex0 . 5.2. Zero-pressure-gradient suction boundary layers 85
Γ ≈ 3.27 × 10−3 Γ ≈ 3.07 × 10−3 25 25
20 20 + 2 ′ + + u ≈ 52 ≈ 78 15 Lw 15 Lw × + ≈ 80 + ≈ 79 Lw Lw
; 10 10 + U 5 5
0 0
Γ ≈ 2.82 × 10−3 Γ ≈ 2.65 × 10−3 25 25
20 20 + 2
′ + ≈ 75 + u Lw ≈ 72 15 15 Lw × + ≈ 76 Lw + ≈ 75 Lw ; 10 + 10
+ ≈ 64 Lw U 5 5
0 0 100 101 102 103 104 Γ ≈ 2.56 × 10−3 + 25 y
20 + 2 ′ + u ≈ 79 15 Lw × + ≈ 92 Lw
; 10 + U 5
0 100 101 102 103 104 + y
Figure 5.18. Inner-scaled velocity mean and variance profiles at x = 4.80 m. Dashed lines: Viscous sublayer. Colors and symbols as in Tab. 5.2. 86 5. Results and discussion
Γ = 3.27 × 10−3 Γ = 3.10 × 10−3 20 20 ∆ = 57 δ ∆ = 37 δ x 99 x 99 15 15
+ 10 10 U 6 6 Re = 9.60 × 10 Re = 9.67 × 10 x x 6 6 5 Re = 10.42 × 10 5 Re = 10.50 × 10 x x 6 6 Re = 11.25 × 10 Re = 11.33 × 10 x x 0 0
Γ = 2.80 × 10−3 Γ = 2.65 × 10−3 20 20 ∆ = 24 δ ∆ = 20 δ x 99 x 99 15 15
+ 10 10 U 6 6 Re = 8.27 × 10 Re = 9.58 × 10 x x 6 6 5 Re = 8.98 × 10 5 Re = 10.40 × 10 x x 6 6 Re = 9.69 × 10 Re = 11.22 × 10 x x 0 0 100 101 102 103 104 Γ = 2.58 × 10−3 + 20 y ∆ = 24 δ x 99 15
+ 10 U 6 Re = 12.42 × 10 x 6 5 Re = 13.48 × 10 x 6 Re = 14.55 × 10 x 0 100 101 102 103 104 y+
Figure 5.19. Inner-scaled mean-velocity profiles for some asymptotic cases at the three most downstream measurement locations. ∆x represents the streamwise distance between the most upstream and the most downstream boundary-layer profile shown in each graph. Colors as in Fig. 5.18. Dashed lines: Viscous sublayer. 5.2. Zero-pressure-gradient suction boundary layers 87
Γ = 3.27 × 10−3 Γ = 3.10 × 10−3 6 6 Re = 9.60 × 10 Re = 9.67 × 10 2.5 x 2.5 x 6 6 Re = 10.42 × 10 Re = 10.50 × 10 x x 2 6 2 6 Re = 11.25 × 10 Re = 11.33 × 10 x x 1.5 1.5 + 2 ′ u 1 1
0.5 0.5
0 0
Γ = 2.80 × 10−3 Γ = 2.65 × 10−3 6 6 Re = 8.27 × 10 Re = 9.58 × 10 2.5 x 2.5 x 6 6 Re = 8.98 × 10 Re = 10.40 × 10 x x 2 6 2 6 Re = 9.69 × 10 Re = 11.22 × 10 x x 1.5 1.5 + 2 ′ u 1 1
0.5 0.5
0 0 101 102 103 104 −3 Γ = 2.58 × 10 y+ 6 Re = 12.42 × 10 2.5 x 6 Re = 13.48 × 10 x 2 6 Re = 14.55 × 10 x 1.5 + 2 ′ u 1
0.5
0 101 102 103 104 y+
Figure 5.20. Inner-scaled velocity-variance profiles for some asymptotic cases at the three most downstream measurement locations. Colors as in Fig. 5.18. 88 5. Results and discussion
6 Re = 0.95 × 10 30 x 6 Re = 1.54 × 10 x 6 Re = 2.71 × 10 25 x 6 Re = 3.88 × 10 x = 5.05 × 106 + 20 Re 2
′ x
u 6 Re = 6.22 × 10 × x 15 = 7.39 × 106 ; 5 Re
+ x 6 U Re = 7.98 × 10 x 10
5
0 100 101 102 103 104 + y
Figure 5.21. Evolution of the velocity mean and variance 6 profiles from a ZPG TBL (Rex = 1.35 × 10 ) to a TASBL for −3 6 Γ = 3.10 × 10 and Rex,s = 1.38 × 10 . 5.2. Zero-pressure-gradient suction boundary layers 89
8000
6000 , as
θ 4000 Re
2000
0 2.4 2.6 2.8 3 3.2 3.4 3.6 Γ × 103
1.4
1.3
1.2 12, as H
1.1
1 2.4 2.6 2.8 3 3.2 3.4 3.6 Γ × 103
Figure 5.22. Momentum thickness Reynolds number Reθ and shape factor H12 variation with the suction rate. Filled symbols: asymptotic cases in Tab. 5.3; Open Blue Squares: LES data by Bobke et al. (2016); Open Red Diamonds: DNS data by Khapko et al. (2016).
5.2.4. Mean-velocity scaling for the turbulent asymptotic state Once a series of TASBL profiles is identified, the problem of the appropriate mean-velocity scaling can be addressed. Figure 5.23 show the viscous-scaled mean-velocity profile for some of the measured TASBLs. The profiles appear to be characterized by a large logarithmic region and by the absence of a clear wake region. The disappearance of the wake region was already reported in previous studies and appears to be such a fundamental characteristics of TASBLs that the presence of a wake region can be considered a symptom that the boundary layer has still not reached its asymptotic state (Black & Sarnecki 1958; Simpson 1970; Bobke et al. 2016). These observations suggest that the logarithmic law U + = A(Γ) ln y+ + B(Γ) (5.6) is a valid choice as an empirical description of the mean-velocity profile. However, a fairly large database of TASBLs at different suction rates is necessary to determine the functions A = f1(Γ) and B = f2(Γ), while the amount of 90 5. Results and discussion
Table 5.3. Experimental parameters for all the measurement cases for which a TASBL was obtained and boundary-layer parameters for the profile at x = 4.80 m.
Case
U∞ (m/s) 25.0 35.1 35.1 35.9 30.0 35.1 37.6 39.0 45.1 −V0 (m/s) 0.082 0.115 0.109 0.111 0.084 0.093 0.099 0.100 0.116 Γ × 103 (-) 3.27 3.27 3.10 3.09 2.80 2.65 2.65 2.56 2.58 xs (m) 0.59 0.30 0.59 0.36 0.94 1.23 0.94 0.94 0.94 −6 Rex,s × 10 (-) 0.97 0.70 1.38 0.86 1.88 2.85 2.27 2.40 2.82
uτ (m/s) 1.43 2.01 1.95 2.00 1.58 1.81 1.94 2.00 2.30 3 Cf × 10 (-) 6.48 6.51 6.17 6.19 5.61 5.30 5.29 5.20 5.19 `∗ (µm) 10.6 7.5 7.7 7.6 9.4 8.4 8.0 7.6 6.5 θ (mm) 1.16 0.77 1.16 1.16 1.86 2.22 2.07 2.18 1.86 δ∗ (mm) 1.39 0.94 1.38 1.38 2.20 2.60 2.43 2.57 2.19 δ99 (mm) 20.47 12.40 19.24 18.86 29.35 35.70 33.61 36.00 29.31 Reθ (-) 1920 1800 2730 2740 3740 5170 5030 5600 5620 80 m . Reτ (-) 1930 1660 2510 2480 3120 4270 4220 4720 4510
= 4 H12 (-) 1.20 1.21 1.18 1.19 1.18 1.17 1.17 1.18 1.18
x Lw (mm) 0.55 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 + Lw (-) 52 80 78 79 64 72 75 79 92 + 1/fmax (-) 4.5 8.9 8.5 8.8 5.6 7.2 8.1 13.1 11.8 tsmpU∞ (-) 73500 125000 72900 66700 50900 44200 39400 38100 53900 δ99 experimental or numerical data available is indeed limited. In the following another approach is attempted: observing the mean-velocity profiles for the same boundary layers plotted in outer scaling (see Fig. 5.24), a good overlap in the inner region can be noticed between all the TASBLs considered, independently from the suction rate. It follows that, at least in the range of suction rate considered, the TABLs profile can be described with the logarithmic law
U/U∞ = Ao ln η + Bo (5.7) with the slope Ao and the intercept Bo constant for any suction rate. Regarding ∗ the choice of the outer length-scale, three different choices (δ99, δ and θ) are compared in Figure 5.24. No substantial difference between them can be observed. To further verify the mean-velocity scaling proposed in eq. (5.7), the indi- cator function d(U/U ) Ξ = y ∞ (5.8) dy was calculated for all the TASBLs listed in Table 5.2. Since the data were sam- pled nonequidistantly (namely with a logarithmic spacing) along the wall-normal coordinate, the derivative was calculated with a weighted central-difference scheme of the type j=i+2 dU X = wjUj (5.9) dy y=yi j=i−2 5.2. Zero-pressure-gradient suction boundary layers 91
20
18
16
14
12
+ 10 U 8 Γ=3.27 × 10−3 6 Γ=3.10 × 10−3 4 Γ=2.80 × 10−3 Γ × −3 2 =2.65 10 Γ=2.58 × 10−3 0 100 101 102 103 104 y+
Figure 5.23. Viscous-scaled mean-velocity profiles of some of the measured TASBLs at x = 4.80 m. Dashed line: Viscous sublayer for Γ = 2.80 × 10−3. Symbols as in Tab. 5.3.
with the weights wj calculated from the values of yi=j−2 , ... , j+2 following the procedure proposed by Fornberg (1998) in order to maximize the accuracy at y = yi. The results are illustrated in Figure 5.25. A clear plateau of the + indicator function is observed for y ' 150 and y/δ99 / 0.5 (corresponding to y ≈ δ95), indicating that the mean-velocity profile show indeed a logarithmic behaviour along the wall-normal coordinate. The extent of this logarithmic region is particularly large, extending for more than 40% of the boundary-layer thickness already for the lowest Reynolds (largest suction rate) considered (Reτ = 1760). The slope Ao of the logarithmic region has been calculated for all the TASBLs profiles in Table 5.2 as the mean value of the indicator function for + y > 150 and y/δ99 < 0.5. The results are shown in Figure 5.27: no clear trend of the value of the slope with the suction ratio can be distinguished and for all the TASBLs profile considered Ao = 0.064 ± 5%. With this choice for Ao, the intercept Bo of the log-law can be calculated for each profile as the mean value + for y > 150 and y/δ99 < 0.5 of
Ψ = U/U∞ − Ao ln η . (5.10)
For the choice of outer scale η = y/δ99 and Ao = 0.064, Figure 5.26 illustrates the function Ψ in inner- and outer-scaled wall-normal coordinate, while Figure 5.28 reports the calculated value for Bo. The averaged value of the intercept Bo 92 5. Results and discussion
1
/δ 99 0.9 =y ) * η θ /δ /(4 =y η=y η
0.8 ∞
U/U 0.7
=3.27 10−3 0.6 Γ × Γ=3.10 × 10−3 Γ=2.80 × 10−3 0.5 Γ=2.65 × 10−3 Γ=2.58 × 10−3 0.4 10−2 10−1 100 101 η
Figure 5.24. Outer-scaled mean-velocity profiles of some of the measured TASBLs at x = 4.80 m, for three different choices of the outer length scale. The multiplicative coefficients of the length scales were chosen solely for illustration purposes. Dashed line: eq. (5.7) with Ao = 0.064 and Bo = 0.994. Sym- bols as in Tab. 5.3. between all the profiles was found to be Bo = 0.994, 0.826 and 0.815 for ∗ η = δ99, δ and θ respectively. From this analysis is also possible to derive an expression for A(Γ) and B(Γ) of eq. (5.6). We can rewrite eq. (5.7) as + + U = U∞(Ao ln η + Bo) , (5.11) hence, choosing η = δ99 + + + U = U∞(Ao ln y − Ao ln Reτ + Bo,δ99 ) . (5.12) √ + Since for a TASBL U∞ = 1/ Γ, comparing eq. (5.6) and eq. (5.12) A B − A ln Re (Γ) A = √o B = o,δ99 √o τ . (5.13) Γ Γ
Non-asymptotic turbulent suction boundary layers and mean-velocity scaling Figure 5.29 and 5.30 illustrates, respectively in inner and outer scaling, two non-asymptotic states at Γ ≈ 3.27 × 10−3 together with the asymptotic state at the same suction rate. In the two non asymptotic cases the presence of a small but distinguishable wake region, compromises the validity of the proposed 5.2. Zero-pressure-gradient suction boundary layers 93
0.25
0.2
0.15 Ξ 0.1
0.05
0 102 103 104 y+
0.25
0.2
0.15 Ξ 0.1
0.05
0 10−2 10−1 100 /δ y 99
Figure 5.25. Indicator function Ξ vs. the inner-scaled (top) and outer-scaled (bottom) wall-normal coordinate for all the TASBLs in Tab. 5.2 at x = 4.8 m. Red dashed line: Ξ = 0.064; Gray dashed-dotted line: limits of the logarithmic region + y = 150 and y/δ99 = 0.5. mean-velocity scaling. In particular, if suction is applied too early upstream, the wake region appears as an overshoot above the log law, a behavior similar to the effect of insufficient box size in the simulations by Bobke et al. (2016), while if suction is applied too late downstream, the departure from the log law takes the form of an undershoot. These deviations are probably linked to not fully developed outer structures in the case of Reθ,s < Reθ,as (Bobke et al. 2016) or to an excess of low-wavenumber turbulent energy in the case of Reθ,s > Reθ,as (Coles 1971), hypotheses which find a confirmation in the behaviour of the most outer part of the velocity-variance profiles shown in Figure 5.29. From Figures 5.29 and 5.30, we observe that the inner and outer part of the boundary layer evolve over different downstream distances, with the near-wall region adapting to the application of the suction in a shorter downstream distance than the outer part. History effects originating from the condition upstream of the suction-start location persists for large downstream distance, 94 5. Results and discussion
1.15
1.1
1.05
1 Ψ
0.95
0.9
0.85 102 103 104 y+
1.1
1.05
1 Ψ
0.95
0.9
0.85 10−2 10−1 100 /δ y 99
Figure 5.26. Ψ function for Ao = 0.064 vs. the inner-scaled (top) and outer-scaled (bottom) wall-normal coordinate for all the TASBls in Tab. 5.2 at x = 4.8 m. Red dashed line: Ψ = 0.0994; Gray dashed-dotted line: limits of the logarithmic + region y = 150 and y/δ99 = 0.5.
with the evolution toward the asymptotic state depending on whether Reθ,s > Reθ,as or Reθ,s < Reθ,as. Non-asymptotic boundary layers cannot hence be considered equilibrium layers and a scaling of turbulent suction boundary layers can only be sought for turbulent asymptotic suction boundary layers.
Comparison with other experiments or simulations The proposed mean-velocity scaling for TASBL is compared with previous numerical and experimental results in Figure 5.31. The asymptotic profiles obtained numerically by Khapko et al. (2016) and Bobke et al. (2016) appear to show outer-scaling similarity for all the suction rates considered, excluding the case Γ = 3.70×10−3 representing their maximum Γ for self-sustained turbulence. Good agreement on the slope of the logarithmic region is found with the profile measured by Kay (1948) at the suction rate for which he reported that a constant boundary layer thickness was achieved. However the boundary-layer thickness 5.2. Zero-pressure-gradient suction boundary layers 95
0.075
0.07
o 0.065 A
0.06
0.055 2.5 2.75 3 3.25 3.5 Γ × 103
Figure 5.27. Slope of the logarithmic region of the individual TASBLs mean-velocity profiles. Solid line: Ao = 0.064; Dashed line: Ao = 0.064 ± 5%;. Symbols as in Tab. 5.3.
1.04
1.02 99 δ
o, 1 B
0.98
0.96 2.5 2.75 3 3.25 3.5 Γ × 103
Figure 5.28. Intercept of the logarithmic law of the individual TASBLs mean-velocity profiles for Ao = 0.064. Solid line:
Bo,δ99 = 0.994; Dashed line: Bo,δ99 = 0.994 ± 1%;. Symbols as in Tab. 5.3.
Reτ (observable by extent of y/δ99 in the logarithmic plot) appears small if compared with the simulation data or current experiments. The profile from Tennekes (1964) deviates considerably from the one measured in the current experiments. This profile represents however a case where the boundary-layer momentum thickness was still weakly growing and hence the asymptotic regime was not fully established (see Fig. 5.30 for a comparison). Figure 5.32 show the indicator function Ξ (eq. 5.8) calculated with the same procedure applied on the current experimental data for the LES by Bobke et al. (2016) and the DNS by Khapko et al. (2016). A plateau of Ξ can be observed for 96 5. Results and discussion
18
16
14
12 + 2 ′
u 10 ×
; 5 8 + U 6
4
2
0 100 101 102 103 104 y+
Figure 5.29. Viscous-scaled velocity mean and variance for two non-asymptotic states (symbols) compared with the asymp- totic state at the same suction rate (solid line). Dashed line: Viscous sublayer. Symbols as in Tab. 5.2.
1
0.9
0.8 ∞
U/U 0.7
0.6
0.5
0.4 10−2 10−1 100 /δ y 99
Figure 5.30. Outer-scaled mean velocity for two non- asymptotic states (symbols) compared with the asymptotic state at the same suction rate (solid line). Dashed line: Log- law as in eq. (5.7) with A = 0.064 and B = 0.994. Symbols as in Tab. 5.2. 5.2. Zero-pressure-gradient suction boundary layers 97
1
0.9
0.8 ∞ U / Γ=2.50 × 10−3 [1] U 0.7 Γ=3.00 × 10−3 [1] Γ=3.45 × 10−3 [2] 0.6 Γ=3.57 × 10−3 [2] Γ=3.70 × 10−3 [2] 0.5 Γ=3.12 × 10−3 [3] Γ=3.32 × 10−3 [4] 0.4 10−3 10−2 10−1 100 y/δ 99
Figure 5.31. Comparison between the proposed mean-velocity scaling and other experimental and numerical data. Black dashed line: log-law as in eq. (5.7) with Ao = 0.064 and Bo = 0.994; [1]: LES simulations by Bobke et al. (2016); [2]: DNS simulations by Khapko et al. (2016); [3] Experiments by Tennekes (1964) (run 2-312; x = 878 mm); [4]: Experiments by Kay (1948). all cases, even though for the highest suction case (γ = 3.70×10−3) the extension + is limited. In Figure 5.33 the value of Ao = hΞi(y > 150∧y/δ99 < 0.5) obtained for the current experiments are compared with the one obtained from the cited −3 simulations. For the three data points with Γ < 3.5 × 10 , Ao is approximately constant with Ao = 0.0614 ± 0.5%, a value 4% lower than Ao = 0.064 obtained from the current experiments.
Comparison with bilogarithmic law Figure 5.34 depicts the profiles of pseudo-velocity q 2 + + Up = + V0 U + 1 − 1 , (5.14) V0 as defined by Stevenson (1963a). If a bilogarithmic law is assumed for the mean-velocity profiles of turbulent asymptotic suction boundary layers, the + pseudo velocity profiles would exhibit an extended region where Up ∝ ln y . Moreover, Stevenson (1963a) proposed that a log-law 1 U = ln y+ + B, (5.15) p κ 98 5. Results and discussion
0.25 Γ=2.50 × 10−3 [1] 0.2 Γ=3.00 × 10−3 [1] Γ=3.45 × 10−3 [2] 0.15 Γ=3.57 × 10−3 [2] Ξ −3 [2] 0.1 Γ=3.70 × 10
0.05
0 102 103 104 y+
0.25
0.2
0.15 Ξ 0.1
0.05
0 10−2 10−1 100 y/δ 99
Figure 5.32. Indicator function Ξ vs. the inner-scaled (top) and outer-scaled (bottom) wall-normal coordinate for all the simulations data by Bobke et al. (2016) [1] and Khapko et al. (2016) [2]. Gray dashed-dotted line: limits of the logarithmic + region y = 150 and y/δ99 = 0.5. with κ and B equal to the no-transpiration case (proposing the values κ = 0.419 and B = 5.8) represents the velocity profile independently of the suction or blowing velocity. In Figure 5.34, eq. 5.15 is shown for two different choices for the constants, the one proposed by Stevenson (1963a) and the one adopted by Nagib et al. (2007) for the description of canonical ZPG TBLs. It is evident that with these two choices for the value of the constants, the bilogarithmic law in eq. (5.15) does not describe the experimental data on asymptotic suction boundary layers. Moreover, even though there is a region in which the profile of Up appears linear in a semi-logarithmic plot, the extent of the logarithmic region of the pseudo-velocity profiles region is considerably smaller than the one observed for the inner- or outer-scaled mean-velocity profiles (see Fig. 5.23 and 5.24). In conclusion, a logarithmic law for the mean velocity profile of asymptotic suction boundary layer is able to describe a larger portion of the boundary-layer than a bilogarithmic law. 5.2. Zero-pressure-gradient suction boundary layers 99
0.075
0.07
o 0.065 A
0.06
0.055 2.25 2.5 2.75 3 3.25 3.5 3.75 Γ × 103
Figure 5.33. Slope of the logarithmic region of TASBLs mean- velocity profiles. Filled symbols: current experiments as in Tab. 5.3; Open squares: DNS data by Khapko et al. (2016); Open circles: LES data by Bobke et al. (2016); Solid line: Ao = 0.064; Dashed line: Ao = 0.064 ± 5%
40
35 2 1 30 − + 1
+ 25 U + 0 V
p 20 1 + 0 2 V Γ=3.27 × 10−3
= 15 −3
p Γ=3.10 × 10 U Γ=2.80 × 10−3 10 Γ=2.65 × 10−3 Γ=2.58 × 10−3 5 101 102 103 104 + y
Figure 5.34. Profile of pseudo velocity Up as defined by Stevenson (1963a) for the same asymptotic profiles of Fig. 5.23 and 5.30. Dashed line: log-law eq. (5.15) with κ = 0.419 and B = 5.8 as proposed in Stevenson (1963a); Dashed-dotted line: log-law with κ = 0.384 and B = 4.17. Symbols as in Tab. 5.3. 100 5. Results and discussion
5.2.5. Profiles of streamwise velocity variance In Figure 5.35 the profiles of streamwise-velocity variance for the turbulent asymptotic cases measured at x = 4.8 m are plotted and compared with one non-transpired ZPG TBL. The ZPG TBL profile chosen for the comparison has a Reynolds number Reτ = 4470, comparable to the two TASBLs with the lowest suction rates (Reτ = 4220 and 4720). It is evident that wall-normal suction strongly damps the intensity of the velocity fluctuations in the whole boundary layer. The shape of the velocity-variance profile is also significantly altered: an inner-peak is still observable close to the wall but the “shoulder” observable in the overlap region of canonical ZPG TBLs (which becomes more pronounced at larger Reynolds numbers) disappears, replaced by a monotonic decrease from the inner peak to the boundary-layer edge. In the profiles considered, however, the large wall shear stress leads to + insufficient spatial resolution of the hot-wire probe, with Lw = 65 − 80. The temporal resolution of the measurement is also insufficient for temporally fully- + resolved measurement, with values of 1/fmax, where fmax is the largest resolved + frequency, exceeding the criterion 1/fmax < 3 proposed by Hutchins et al. (2009). In order to overcome these limitations, a series of velocity profiles was measured with the wall-mounted traverse system at x = 6.06 m using hot-wire probes with smaller wire-length and wire-diameter, increasing hence both the spatial and temporal resolution. For each value of Γ considered, the measurement has been repeated with three different hot-wire probes, in order to quantify and correct for the spatial filtering effects. Moreover the experimental conditions + were chosen such that matching values of Lw could be obtained for different values of Γ in order to allow for direct comparison between profiles at different suction rates. The experimental parameters for this set of experiments are listed in Table 5.4. Since just one velocity profile at a fixed streamwise location has been obtained, the term dθ/dx of the von-K´arm´anmomentum-integral specialized for boundary-layer with wall transpiration
C dθ V f = − 0 (5.16) 2 dx U∞ remains unknown. For all the suction cases reported in §5.2.3, however, in the region x > 4.0 m, dθ/dx < 0.02 × |V0/U∞|. Since the measurement cases considered in this section are similar for suction rate and suction start-location to those in §5.2.3, the additional systematic error on the wall shear stress introduced by neglecting the momentum-thickness derivative can hence be estimated to be less than 2% in Cf or less than 1% in uτ . To assess whether the velocity profile measured are representative of asymptotic states, in Figure 5.36 the mean-velocity profiles are compared with the log law proposed in §5.2.4 (eq. 5.7). Good agreement with the proposed scaling is observed for all the profiles, with somewhat larger deviation for the cases with Γ ≈ 2.56 (yellow symbols), suggesting that the asymptotic state is just approximated but not fully obtained for this suction rate. 5.2. Zero-pressure-gradient suction boundary layers 101
2.5
8
6 2 4
1.5 2 Γ=3.27 × 10−3; L+ = 80 + 2 w 0 ′ 1 2 3 4 u Γ=3.10 × 10−3; L+ = 78 10 10 10 10 1 w Γ=2.80 × 10−3; L+ = 64 w Γ=2.65 × 10−3; L+ = 75 0.5 w Γ=2.56 × 10−3; L+ = 79 w Γ = 0; Re = 5250; L+ = 19 0 τ w 101 102 103 104 y+
Figure 5.35. Symbols: Inner-scaled velocity-variance profiles at x = 4.8 m for TASBLs at various suction rates. Symbols as in Tab. 5.3. In the inset a comparison with one of the canonical ZPG TBL case reported in Fig. 5.8 with Reτ = 5250.
1
0.9
0.8 ∞ U / 0.7 U
0.6
0.5
0.4 10−3 10−2 10−1 100 101 /δ y 99
Figure 5.36. Outer-scaled mean-velocity profile measured at x = 6.06 m. Dashed line: log law as in eq. (5.7) with Ao = 0.064 and Bo = 0.994. Symbols as in Tab. 5.4: the same color is used for cases with matching suction rate. 102 5. Results and discussion 06 m. . = 6 x Experimental parameters for the suction cases at Table 5.4. m) 7.3 7.3 7.2 7.4 7.3 7.4 7.5 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 (-) 248100 231400 237100 158200 150100 155400 75900 70800 70900 62000 62200 59300 59900 59300 60900 (-)(-) 3.74(-) 3.72 0.43 3.70 0.43 7.47 3.44 0.43 7.44 3.43 0.70 7.40 3.48 0.71 6.88(-) 2.90 0.69 6.87(-) 2.92 2.31 6.97 2.90 78 2.28 9.5 5.81 2.72 2.29 5.84 56 2.71 3.6 3.13 5.81 2.75 3.14 32 3.6 5.45 2.55 3.13 5.42 2.57 77 9.3 3.22 5.49 2.57 3.20 5.10 3.5 55 3.23 5.13 3.5 31 5.14 8.9 76 3.3 53 3.3 30 8.7 75 3.3 54 3.2 30 8.6 3.2 75 3.2 53 30 (-)(-)(-) 1170 1130 1210 1.23 1220 1220 1.23 1190 1890 1.22 1800 1970 1900 1.21 1880 1820 4340 1.20 3940 4480 1.20 4150 4490 1.17 4180 5660 1.17 4860 5780 4860 1.17 5710 5010 6560 1.17 5140 6590 1.17 5130 6240 1.17 5040 1.17 1.17 1.18 µ (m) 0.19 0.19 0.19 0.30 0.30 0.30 0.94 0.94 0.94 1.24 1.24 1.25 1.24 1.24 1.25 ( (mm) 0.52 0.54 0.54 0.81 0.85 0.82 1.76 1.84 1.83 2.24 2.28 2.27 2.52 2.55 2.41 (mm)(mm) 0.64 8.23 0.66 8.83(mm) 0.66 8.62 0.57 0.98 13.25 0.41 1.02 13.97 0.23 13.51 0.99 29.63 0.57 2.06 31.63 31.71 2.15 0.41 36.84 2.14 36.80 0.23 2.62 38.14 0.57 39.19 2.66 0.41 39.25 2.65 38.38 0.23 2.96 0.57 2.98 0.41 2.83 0.23 0.57 0.41 0.23 (m/s)(m/s) 34.0 0.127 34.1 0.127(m/s) 0.126 34.1 0.120 2.08 34.9 0.120 2.08 35.0 0.122 2.07 0.109 35.0 0.109 2.05 37.5 0.109 0.104 37.3 2.05 0.103 37.5 0.104 2.07 0.100 38.0 2.02 0.100 38.1 0.100 2.02 37.7 2.02 39.1 1.99 38.8 1.99 38.9 1.98 1.98 1.97 1.97 6 − 10 3 3 × 10 ∞ s 10 U + max 0 99 × x, θ τ δ × 12 V f ∞ w + w τ s /f ∗ 99 smp ∗ t U x u ` θ δ 1 − Γ Re C δ Re Re H L L Case 5.2. Zero-pressure-gradient suction boundary layers 103
Figure 5.37 illustrates the dependency of the measured inner-scaled velocity variance profile to the viscous-scaled wire length for all the cases at x = 6.06 m. The correction method proposed by Segalini et al. (2011) has been applied on the measured data in order to compensate for filtering effects and obtain an estimate of the actual velocity-variance profile. The correction scheme relies on the dependence of the spatially-averaged streamwise turbulence intensity on the wire length and on the spanwise correlation coefficient (Dryden et al. 1937), which in turn can be related to the local transverse Taylor microscale (Frenkiel 1949; Segalini et al. 2011): if the same flow field is measured with two or more hot-wire length, an estimate of the actual velocity variance can be obtained together with an estimate of the Taylor microscale. The results of the correction method are reported in Figure 5.37 for all the suction rates investigated. Even though other correction methods for hot-wire spatial filtering exist (Monkewitz et al. 2010; Smits et al. 2011), they were developed and calibrated for wall-bounded flow in absence of wall transpiration and cannot hence be applied on turbulent suction boundary layers. Figure 5.38 reports, for all the suction rates considered, the inner-scaled velocity variance profiles corrected from spatial-filtering effects, together with + the measured profile at matched Lw ≈ 31 for the suction cases. Figure 5.38 also depicts a comparison with DNS and experimental corrected and uncorrected velocity-variance profiles for canonical ZPG-TBLs. The velocity variance is strongly damped by the suction if compared to a canonical ZPG TBL, as already noted in Figure 5.35. For the suction rates investigated the reduction of the (corrected) near-wall peak ranges from about 50% to 65% in respect to a canonical ZPG TBL with approximately the same Reτ . The magnitude of the near-wall peak of the inner-scaled velocity variance profile is clearly dependent on the suction rate, with larger peak value for lower suction rates. The variation of the velocity-variance peak with the suction rate is reported in Figure 5.40 in inner and outer scaling and compared with numerical simulations results by Bobke et al. (2016) and Khapko et al. (2016). A simple linear fit through the + 02 experimental data (u peak = −1170 Γ + 7.18) describes reasonably well (max. deviation < 2.5%) the inner-scaled velocity-variance peak from the current experiment and from the LES by Bobke et al. (2016), but not from the DNS by Khapko et al. (2016). In the range of suction rates considered, the velocity variance peak from the experiments and the LES (but not from the DNS) can 02 2 also be described with the same accuracy by the relation u peak/U∞ = 0.0108. Since for an asymptotic state
+ 02 02 02 u u 1 u = = 2 , (5.17) −U∞V0 U∞ Γ these two simple empirical relations are in contradiction with each other and will diverge for decreasing suction ratio. From eq. (5.17) it is also apparent that + 02 2 02 if u peak/U∞ is taken as constant, u peak → ∞ if Γ → 0, which is unphysical. 104 5. Results and discussion
4.5 4 Γ = 3.72 × 10−3 Γ = 3.45 × 10−3
3.5 + + L = 32 L = 31 3 w w + + L = 56 L = 55 2.5 w w + 2 ′ + = 78 + = 77 u 2 Lw Lw 1.5 1 0.5 0 4.5 4 Γ = 2.91 × 10−3 Γ = 2.73 × 10−3
3.5 + + L = 30 L = 30 3 w w + + L = 53 L = 54 2.5 w w + 2 ′ + = 76 + = 75 u 2 Lw Lw 1.5 1 0.5 0 101 102 103 104 4.5 + y 4 Γ = 2.56 × 10−3
3.5 + L = 30 3 w + L = 53 2.5 w + 2 ′ + = 75 u 2 Lw 1.5 1 0.5 0 101 102 103 104 + y
Figure 5.37. Inner-scaled velocity-variance profiles at x = 6.06 m for different suction rates. Symbols: measured data for + various hot-wire length Lw ; Colored lines: data corrected from spatial-filtering effects with the method by Segalini et al. (2011) using all three measured profiles (dashed lines) and using just data measured with the smallest and the largest probe (solid lines). 5.2. Zero-pressure-gradient suction boundary layers 105
9 Γ × 103 = 2.57, = 5040 Reτ 8 Γ × 103 = 2.75, = 5010 Reτ 7 Γ × 103 = 2.90, = 4180 Reτ 6 Γ × 103 = 3.48, = 1820 Reτ 5 Γ × 103 = 3.70, Re = 1190
+ τ 2 ′
u Γ = 0, = 5250 4 Reτ Γ = 0, = 1145 3 Reτ
2
1
0 101 102 103 104 y+
Figure 5.38. Inner-scaled velocity-variance profiles at x = 6.06 m for different suction rates compared with no- + transpiration cases. Circles: TASBLs (measured data, Lw ≈ 31); Solid lines: TASBLs, corrected data (method: Segalini et al. 2011); Triangles: canonical ZPG TBL at Reτ = 5250 + (measured data, Lw ≈ 19); Dashed-dotted line: canonical ZPG TBL at Reτ = 5250, corrected data (method: Smits et al. 2011); Dashed line: canonical ZPG TBL at Reτ = 1145, DNS by Schlatter & Orl¨u(2010)¨
From Figure 5.38 we observe that the streamwise velocity-variance profiles decrease monotonically from the inner-peak location to the zero value at the boundary-layer edge. For non-transpired canonical boundary layers (canonical ZPG TBL, turbulent pipe flow and turbulent channel flow) a shoulder in the outer part of the inner-scaled velocity-variance profiles is present and it becomes more evident with increasing Reynolds number, finally taking the shape of a plateau for high enough Reynolds numbers3. It has been shown (Marusic et al. 2010; Ng et al. 2011) that this increase with Reynolds-number of the magnitude of the broadband velocity-variance profiles in the outer region can be attributed to the large and very-large scale motions. It can be speculated, based on the present data, that even small values of wall suction such as those applied in this experiment are very effective in reducing the strength of the larger scales of a turbulent boundary layer. This tentative conclusion finds support in the analysis of the frequency spectra of the velocity signal (see §5.2.6).
3Whether or not an outer peak of the streamwise-velocity variance appears at high enough Reynolds number is a matter of debate, see e.g. Hutchins et al. (2009), Vallikivi et al. (2015) and Monkewitz & Nagib (2015). 106 5. Results and discussion
0.01 Γ × 103 = 2.57, = 5040 Reτ Γ × 103 = 2.75, = 5010 Reτ 0.008 Γ × 103 = 2.90, = 4180 Reτ Γ × 103 = 3.48, = 1820 Reτ 0.006 3 2 ∞ Γ × 10 = 3.70, = 1190 Reτ /U 2 ′ Γ = 0, Re = 5250
u τ 0.004 Γ = 0, = 1145 Reτ
0.002
0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 y/δ 99
Figure 5.39. Outer-scaled velocity-variance profiles at x = 6.06 m for different suction rates compared with no- + transpiration cases. Circles: TASBLs (measured data, Lw ≈ + 31); Triangles: canonical ZPG TBL, measured data (Lw ≈ 19); Dashed line: canonical ZPG TBL, DNS by Schlatter & Orl¨u¨ (2010)
The outer-scaled velocity-variance profiles, reported in Figure 5.39, show a good overlap for a large portion of the boundary-layer thickness (y/δ99 > 0.2) for all the profiles excluding the case with Γ = 2.57. For this particular case, however, the full achievement of the asymptotic state was considered doubtful from the analysis of the mean velocity profile. The observed similarity between outer-scaled velocity variance profiles at different suction rates is however in contradiction with the outer-scaled similarity of the mean-velocity profile proposed in §5.2.4. Recalling eq. (2.41), describing the Reynolds shear-stress distribution in the outer region of a TASBL, a similarity between the outer- scaled streamwise mean velocity profile of TASBL at different suction rates + implies a similarity between the viscous scaled Reynolds stress −u0v0 (and vice versa), giving
U 0 0+ = f1(η) ⇐⇒ −u v = 1 − f1(η) . (5.18) U∞ For consistency with the proposed similarity of the mean-velocity profile in outer-scaled variables (see §5.2.4) and considering that the various components of the Reynolds-stress tensor should share the same scaling if similarity is observed, + it is expected that u02 = f(η). In Figure 5.41 the inner- and outer-scaled velocity-variance profiles are plotted vs. the outer-scaled wall-normal distance for the current experiments and the simulations data by Bobke et al. (2016) 5.2. Zero-pressure-gradient suction boundary layers 107
5
4.5
4 p e ak
2 3.5 + 2 ′ u
1 3
2.5
2 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 Γ × 103
0.012
0.011 p e ak 2 2 ∞ 0.01 /U 2 ′ u
1 0.009
0.008 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 Γ × 103
Figure 5.40. Maximum of the velocity variance in inner (top) and outer (bottom) scaling. Open circles: current experiments, + measured data for Lw ≈ 31; Filled circles: current experiments, corrected data (method by Segalini et al. 2011); Blu squares: LES simulations by Bobke et al. (2016); Red diamonds: DNS simulations by Khapko et al. (2016); Black dashed-dotted line: linear fit through the experimental data; Red dashed-dotted 02 2 line: u /U∞ = 0.0108; Vertical dashed line: self-sustained turbulence threshold (Khapko et al. 2016). and Khapko et al. (2016). The experimental data show larger scatter if inner scaling is adopted, in apparent contradiction with the observed behaviour of the mean-velocity profiles. However, the opposite is observed for the simulation + 02 data, for which good scaling of u occurs in the region (y/δ99 > 0.4) for all the suction rates excluding the lowest (Γ = 2.50 × 10−3). The reason of this discrepancy between experiments and simulations is at the present state unclear, even though it can be hypothesized that, analogously to what happen in pipe flow (Doherty et al. 2007), the achievement of a fully developed (asymptotic) turbulent state for the velocity-variance profile requires a longer streamwise distance than for the mean-velocity profile. 108 5. Results and discussion
x 10−3 2 0.8 Γ = 2.57 × 10−3 Γ = 2.50 × 10−3 [1] Γ = 2.75 × 10−3 0.7 Γ = 3.00 × 10−3 [1] −3 −3 [2] 1.5 Γ = 2.90 × 10 0.6 Γ = 3.45 × 10 Γ = 3.48 × 10−3 Γ = 3.57 × 10−3 [2] −3 0.5 −3 [2]
2 Γ × Γ ×
∞ = 3.70 10 = 3.70 10 + 2 ′
/U 1 0.4 2 u ′ u 0.3
0.5 0.2
0.1
0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 y/δ y/δ 99 99
Figure 5.41. Outer- (left) and inner- (right) scaled velocity variance profiles for the current experiments (symbols) and available numerical simulations ([1]: Bobke et al. (2016); [2]: Khapko et al. (2016)).
Given the small amount of experimental or numerical data on TASBLs, firm conclusion on the scaling of the velocity variance in TASBL should wait for data covering a larger range of suction rates (and hence Reτ ) from more independent sources. If a Reynolds-number similarity of the Reynolds-stresses tensor is confirmed for this flow, it would constitute a solid ground to explain the observed outer scaling of the mean-velocity profile, as apparent from the expression 5.18, derived form the Navier-Stokes equations for a 2D TASBL with the only assumption of negligible viscous stress.
5.2.6. Spectra Figure 5.42 depicts the inner-scaled premultiplied power-spectral-density (P.S.D.) + map for all the suction cases for x = 6.06 m obtained with matching Lw ≈ 31. The P.S.D. were estimated from the streamwise-velocity time series using Welch’s method (Welch 1967). The wavelengths were inferred from the time-series of velocity using the Taylor’s “frozen turbulence” hypothesis (Taylor 1938) and the local mean velocity as the convective velocity of the waves. The applicability of Taylor’s hypothesis to wall-bounded flows has been recently questioned: Del Alamo´ & Jim´enez(2009) showed with DNS of turbulent channel flow that the hypothesis holds for the small eddies (except near the wall), but is violated by eddies with long wavelength, which are advected at velocity close to the bulk velocity. For suction boundary layers, as can be observed from Figure 5.42, the energetic contribution of the low frequency (large wavelength) components is 5.2. Zero-pressure-gradient suction boundary layers 109 small compared to the high frequency ones, providing a justification for the application of Taylor’s hypothesis. As conjectured from the analysis of the velocity variance profiles, suction is very effective in reducing the strength of the large scale structures present in turbulent boundary layers, so that TASBLs (at least at the suction rates considered) appear to be fundamentally dominated by the near-wall cycle. This is particularly evident if the power-spectra maps of the two lowest suction cases are compared with a canonical ZPG TBL at comparable Reτ (Fig. 5.43), for which a clear contribution of the large energy components is observed in a large portion of the boundary-layer. + The streamwise wavelength λx,p related to the peak of the premultiplied power spectral density increases with the suction rates, as clearly illustrated + + in Figure 5.44. For the non-transpired case the peak in f Puu occurs at + λx,p ≈ 1000 (in agreement with Jim´enez et al. 2004; Hutchins & Marusic 2007 + among others) and it increases with the suction rate reaching λx,p ≈ 2700 for Γ = 3.70 × 10−3. The peak in the premultiplied power spectral density corresponds to the signature of near-wall motion of the high- and low-speed + streaks observed firstly by Kline et al. 1967. The increase of the λx,p with the suction rate is hence in qualitative agreement with the measurements and flow visualization of a (localized) suction boundary layer by Antonia et al. (1988), who reported that “low-speed streaks tend to oscillate less in a spanwise direction while their streamwise persistence is increased [compared to the non transpired case]”. Since the instability of the near-wall streaks plays a major role in the production of turbulence (see Kim et al. 1971; Jim´enez& Pinelli 1999, among others), the increased stability of the near-wall streaks was related by Antonia et al. (1988) to the decrease of the Reynolds stresses in presence of wall-normal suction.
5.2.7. Higher order moments To conclude this description of turbulent suction boundary layers, the profiles of skewness and kurtosis of the streamwise velocity are reported for the suction cases at x = 6.06 m and compared with canonical ZPG TBLs at different Reτ in Figure 5.45 and 5.46 respectively. In the inner region of the boundary layer, the skewness of the velocity becomes more negative with increasing suction rates, meaning that in this region the ejections of low momentum fluid from the wall are more frequent than sweeps of high momentum fluid from the outer part of the boundary layer. This observation is in accordance with the view of turbulent suction boundary layers (at least in the range of Reτ considered) as fundamentally dominated by the near wall cycle deducted from the analysis of the spectral maps. Similarly, the increase of the near-wall minimum value of the skewness profile with increasing Reτ for the canonical ZPG-TBL cases can be related to the more frequent sweeps of high-momentum fluids toward the near-wall region caused by the larger relevance for increasing Reynolds number of the outer large-scale motions. For the suction cases the near-wall 110 5. Results and discussion
Figure 5.42. Premultiplied power-spectral-density maps in inner scaling for the suction cases measured at x = 6.06 m with + matching Lw ≈ 31. 5.2. Zero-pressure-gradient suction boundary layers 111
Figure 5.43. Premultiplied power-spectral-density maps in inner scaling for the ZPG TBL case measured at x = 6.06 m + with Reτ = 5250 and Lw ≈ 19.
2 Γ = 3.70 × 10−3 + 1.8 Γ = 3.48 × 10−3 y ≈ 15 −3 1.6 Γ = 2.90 × 10 Γ = 2.75 × 10−3 1.4 Γ = 2.57 × 10−3 1.2 Γ = 0 + uu
S 1
+
f 0.8
0.6
0.4
0.2
0 101 102 103 104 105 λ+ x
Figure 5.44. Premultiplied power-spectral-density maps in inner scaling for y+ ≈ 15 and x = 6.06 m. Colored lines: + suction cases with Lw ≈ 31; Black line: canonical ZPG TBL + with Reτ = 5250 and Lw ≈ 19. 112 5. Results and discussion
1 Γ × 103 = 2.57, = 5040 Reτ 0.5 Γ × 103 = 2.75, = 5010 Reτ Γ × 103 = 2.90, = 4180 0 Reτ Γ × 103 = 3.48, Re = 1820 −0.5 τ Γ × 103 = 3.70, = 1190 Reτ u’ −1 +
Sk Γ = 0; Re = 5250; L = 19 τ w −1.5 Γ = 0; Re = 2500; L+ = 8 τ w Γ = 0, = 1145; DNS −2 Reτ
−2.5
−3 101 102 103 104 y+
Figure 5.45. Velocity-skewness profiles at x = 6.06 m for the suction cases and for canonical ZPG TBLs. Circles: TASBLs + (Lw ≈ 31); Black symbols: canonical ZPG TBL, measured data; Dashed line: canonical ZPG TBL, DNS by Schlatter & Orl¨u¨ (2010) minimum of the skewness profile becomes more negative with increasing suction rate, while its location moves outwards, with values ranging from y+ ≈ 40 for Γ = 2.57 × 10−3 to y+ ≈ 60 for Γ = 3.70 × 10−3. The velocity-kurtosis profiles of the suction cases show a near-wall peak increasing in magnitude with the suction rate. Its wall-normal location increases with the suction rate, with values ranging from y+ ≈ 70 for Γ = 2.57 × 10−3 to y+ ≈ 85 for Γ = 3.70×10−3. The distinctive minimum and distinctive maximum observed respectively for the skewness and flatness profile in correspondence of the boundary-layer edge are indicators of the highly intermittent behaviour of the flow in this region. As a final observation, the smoothness of the skewness and kurtosis profile serves as a proof that the sampling time chosen for the measurement was long enough to obtain well-converged statistics. 5.2. Zero-pressure-gradient suction boundary layers 113
30 Γ × 103 = 2.57, = 5040 Reτ Γ × 103 = 2.75, = 5010 20 Reτ Γ × 103 = 2.90, = 4180 Reτ Γ × 103 = 3.48, = 1820 Reτ 10 Γ × 103 = 3.70, = 1190 Reτ u’ + Ku Γ = 0; Re = 5250; L = 19 τ w Γ = 0; Re = 2500; L+ = 8 τ w Γ = 0, = 1145; DNS Reτ
2 101 102 103 104 y+
Figure 5.46. Velocity-kurtosis profiles at x = 6.06 m for the suction cases and for canonical ZPG TBLs. Circles: TASBLs + (Lw ≈ 31); Black symbols: canonical ZPG TBL, measured data; Dashed line: canonical ZPG TBL, DNS by Schlatter & Orl¨u¨ (2010); Solid line: normal-distribution kurtosis Kuu0 = 3. 114 5. Results and discussion
5.3. Zero-pressure-gradient turbulent blowing boundary layers A series of blowing boundary-layer profiles was measured at the most downstream measurement location (x = 6.06 m). For all the measurement cases uniform blow- ing was applied in the whole region downstream of the first plate element (xs = 0.94 m). The blowing rates considered were Γ ≈ (1.00; 1.46; 1.95; 2.95; 3.74)×10−3 and for each blowing rate boundary layers with different Re number were ob- tained regulating the free-stream velocity in the range from 10 m/s to 40 m/s. The experimental parameters for the measurements of blowing boundary layers are listed in Table 5.5. To assess the magnitude of the spatial filtering of the hot-wire probe, selected measurements were repeated with a different wire length. Unfortunately, given the difficulties in measuring the shear-stress in a boundary layer with transpiration (see §3.7.2 and §3.7.3), the friction velocity could not be estimated for these experiments, hence all the results will be presented exclusively in outer scaling. No boundary-layer separation could be observed by means of tufts visualization for the whole range of blowing rates and Reynolds numbers considered.
Table 5.5. Experimental parameters for all the blowing tur- bulent boundary layers measured at x = 6.06 m
3 −6 ∗ tsmpU∞ Case U∞ V0 Γ × 10 xs Rex,s × 10 θ δ δ99 Reθ H12 Lw δ99 (m/s) (m/s) (-) (m) (-) (mm) (mm) (mm) (-) (-) (mm) (-) 39.9 0.039 0.98 0.94 2.40 12.05 16.53 97.73 30820 1.37 0.57 32700 39.9 0.039 0.98 0.94 2.41 11.73 16.11 94.88 30130 1.37 0.28 33700 30.0 0.029 0.97 0.94 1.82 12.41 17.14 98.87 24130 1.38 0.28 27300 20.0 0.019 0.97 0.94 1.21 12.56 17.62 101.34 16280 1.40 0.28 19700 15.1 0.014 0.95 0.94 0.91 12.98 18.37 102.25 12550 1.42 0.28 17700 40.0 0.060 1.49 0.94 2.41 14.07 20.03 112.15 36220 1.42 0.57 28600 39.8 0.059 1.49 0.94 2.41 14.13 19.98 110.41 36390 1.41 0.28 28800 30.0 0.044 1.47 0.94 1.83 14.19 20.29 112.91 27720 1.43 0.28 23900 19.9 0.029 1.47 0.94 1.20 14.32 20.51 106.52 18270 1.43 0.28 18600 15.0 0.022 1.46 0.94 0.91 15.02 21.67 112.50 14550 1.44 0.57 13300 10.0 0.014 1.42 0.94 0.62 15.12 22.07 110.38 9940 1.46 0.28 10900 30.1 0.059 1.97 0.94 1.83 17.02 24.99 125.41 33230 1.47 0.28 21600 30.0 0.060 2.00 0.94 1.80 17.12 25.25 125.48 32960 1.48 0.57 21500 20.0 0.039 1.97 0.94 1.20 16.65 24.58 123.12 21320 1.48 0.28 16200 15.0 0.029 1.95 0.94 0.93 16.88 25.22 124.67 16740 1.49 0.57 10800 10.1 0.019 1.91 0.94 0.62 16.83 25.33 123.26 11140 1.51 0.28 9800 20.0 0.060 3.00 0.94 1.20 21.97 34.47 150.46 28140 1.57 0.28 13300 20.0 0.059 2.96 0.94 1.22 22.09 34.58 151.68 28710 1.57 0.57 11900 15.1 0.044 2.92 0.94 0.93 21.65 34.21 152.00 21480 1.58 0.57 10400 10.0 0.029 2.93 0.94 0.61 21.43 33.91 149.56 13870 1.58 0.57 8000 16.0 0.060 3.72 0.94 0.96 25.33 41.59 175.68 26050 1.64 0.28 10900 5.3. Zero-pressure-gradient turbulent blowing boundary layers 115
5.3.1. Mean-velocity and velocity-variance profiles Figures from 5.47 to 5.51 illustrate the measured mean-velocity profiles for different blowing rates and Reθ. A comparison with the lower Reynolds-number experiments by Andersen et al. (1972) is presented whenever data at matching blowing rate were available. For all the blowing rates considered excluding the lowest (Γ ≈ 1.00 × 10−3), overlap of the outer-scaled mean velocity is observed in a large portion of the boundary layer for large enough momentum-thickness Reynolds numbers. The range of Reθ among which the the outer-scaled mean- velocity profiles overlap is increasingly larger with increasing blowing rate. For the largest blowing rate considered, Γ ≈ 3.74 × 10−3, a good overlap of the outer-scaled mean-velocity profiles is noticed for 98% of the boundary layer- thickness between Reθ = 6670 and Reθ = 26050 (see Fig. 5.51), while for Γ ≈ 2.95 × 10−3 the outer-scaled mean-velocity profile are indistinguishable for the outer 99% of the boundary-layer thickness in the Reynolds number range from Reθ = 13870 and Reθ = 28710 (see Fig. 5.50). In absence of an + estimate for the friction velocity, the variation of U∞ with the Reynolds number is unknown, hence it is not possible to conclude whether scaling of the mean velocity with U∞ is to be preferred over the scaling with uτ . For the same reason, the applicability of the bi-logarithmic law of the wall (see §2.2.4) cannot be tested on the experimental database. Mean-velocity profiles at different blowing rates are compared in Figure 5.52 from which a clear dependency on the blowing rates of the outer-scaled mean-velocity profile is apparent. With increasing blowing rates, moreover, an increasing curvature of the semi-logarithmic plot of the mean velocity profiles becomes evident, suggesting that for this type of flow a logarithmic law cannot properly describe the mean-velocity distribution.
In Figure 5.53 the velocity-defect profiles U∞ − U normalized with the ∗ Zagarola-Smits velocity scale U∞δ /δ99 are plotted vs. the outer-scaled wall- normal distance for all the blowing cases measured and compared with one non-transpired ZPG TBL case and with the TASBL cases of Table 5.3. We observe a good overlap between all the blowing boundary layer profiles and the non-transpired ZPG TBL case, in agreement with Cal & Castillo (2005) and Kornilov & Boiko (2012), but not between blowing and suction profiles, as already noted by Cal & Castillo (2005). Even if for graphical clarity Figure 5.53 reports just one canonical ZPG TBL case, the above conclusions do not change if all the measured ZPG TBL cases measured are considered. Finally, Figure 5.54 shows the variation of the shape factor with the momentum-thickness Reynolds number for all the measured blowing boundary layers, illustrating that the shape factor at fixed Reθ increases with the blowing rate. Wall-normal blowing generally increases the magnitude of turbulent fluctu- ations in the whole boundary layer, as observed from Figure 5.55 showing the local turbulence intensity profiles for boundary layers at different blowing rates. The shape of the velocity-variance profiles is also altered by the application of wall-blowing, as evident from Figure 5.56, illustrating the velocity-variance profiles for blowing boundary layers at different blowing rates and Reynolds 116 5. Results and discussion
1 Reθ = 30130
0.9 Reθ = 24130 Re = 16280 0.8 θ Reθ = 12550 0.7 [*] Reθ = 4000 0.6 ∞ U / 0.5 U 0.4 0.3 0.2 0.1 Γ ≈ 1.00 × 10−3 0 10−3 10−2 10−1 100 y/δ 99
Figure 5.47. Outer-scaled mean-velocity profiles for blowing boundary layers with Γ ≈ 0.97 × 10−3. Filled Symbols: current investigation, x = 6.06 m (see Tab. 5.5); (*) Open Symbols: data from Andersen et al. (1972): case 100571-1 (x = 90 in; Γ = 1.04 × 10−3).
= 36390 1 Reθ = 27720 0.9 Reθ Re = 18270 0.8 θ Re = 14550 0.7 θ = 9940 Reθ 0.6 ∞ U / 0.5 U 0.4 0.3 0.2 0.1 Γ ≈ 1.46 × 10−3 0 10−3 10−2 10−1 100 /δ y 99
Figure 5.48. Outer-scaled mean-velocity profiles for blowing boundary layers with Γ ≈ 1.47 × 10−3, current investigation, x = 6.06 m (see Tab. 5.5). 5.3. Zero-pressure-gradient turbulent blowing boundary layers 117
1 Reθ = 33230
0.9 Reθ = 21320 Re = 16740 0.8 θ Reθ = 11140 0.7 [*] Reθ = 4740 0.6 ∞ U / 0.5 U 0.4 0.3 0.2 0.1 Γ ≈ 1.95 × 10−3 0 10−3 10−2 10−1 100 y/δ 99
Figure 5.49. Outer-scaled mean-velocity profiles for blowing boundary layers with Γ ≈ 1.97 × 10−3. Filled Symbols: current investigation, x = 6.06 m (see Tab. 5.5); (*) Open Symbols: data from Andersen et al. (1972): case 090171-2 (x = 90 in; Γ = 2.00 × 10−3).
= 28710 1 Reθ = 21480 0.9 Reθ Re = 13870 0.8 θ 0.7 0.6 ∞ U / 0.5 U 0.4 0.3 0.2 0.1 Γ ≈ 2.95 × 10−3 0 10−3 10−2 10−1 100 /δ y 99
Figure 5.50. Outer-scaled mean-velocity profiles for blowing boundary layers with Γ ≈ 2.95 × 10−3, current investigation, x = 6.06 m (see Tab. 5.5). 118 5. Results and discussion
1 Reθ = 26050 [*] 0.9 Reθ = 6670 0.8 0.7 0.6 ∞ U / 0.5 U 0.4 0.3 0.2 0.1 Γ ≈ 3.74 × 10−3 0 10−3 10−2 10−1 100 y/δ 99
Figure 5.51. Outer-scaled mean-velocity profiles for blowing boundary layers with Γ ≈ 3.74 × 10−3. Filled Symbols: current investigation, x = 6.06 m (see Tab. 5.5); (*) Open Symbols: data from Andersen et al. (1972): case 090871-3 (x = 90 in; Γ = 3.76 × 10−3). numbers: for increasing Reynolds number and fixed blowing rate the magnitude of the inner peak decreases, while an outer peak emerges. Already for the largest Reynolds number measured for Γ ≈ 1.46 × 10−3, the outer-peak magnitude is larger than the magnitude of the inner-peak, and at the largest Reynolds −3 numbers considered (Γ ' 2.95 × 10 ) the inner-peak completely disappears. These observations cannot be considered an artefact caused by the spatial filter- ing of the hot-wire probe, as evident from Figure 5.57 reporting measurements obtained with different wire-length at the largest Reynolds number considered −3 −3 for the blowing rates Γ 6 3.00 × 10 . For Γ ≈ 2.98 × 10 the observed overlap of the velocity variance profiles measured with different wire-length suggests that already for the larger wire no significant spatial filtering occurred. For smaller blowing rates some spatial filtering effect was observed, hence the measured velocity variance profiles were corrected for spatial resolution effects with the method by Segalini et al. (2011). From Figure 5.57 can be concluded that cases ( ), ( ) and ( ) can be considered free from spatial-filtering effects. This conclusion can be extended for case ( ), characterized by the same wire-length, larger blowing rate but smaller free-stream velocity than ( ), resulting in smaller wall shear-stress and larger `∗. Reconsidering Figure 5.55, we observe that the near-wall turbulence intensity increases significantly with blowing. The near-wall turbulence intensity can 5.3. Zero-pressure-gradient turbulent blowing boundary layers 119
1 Γ = 0.98 × 10−3 −3 0.9 Γ = 1.49 × 10 Γ = 1.97 × 10−3 0.8 Γ = 3.00 × 10−3 0.7 Γ = 3.72 × 10−3 Γ = 0 0.6 ∞ U / 0.5 U 0.4
0.3
0.2
0.1
0 10−3 10−2 10−1 100 /δ y 99
Figure 5.52. Colored symbols: mean-velocity profiles at x = 6.06 m for different blowing rates at the highest Reθ measured (see Tab. 5.5). Black triangles: canonical ZPG TBL at Reθ = 21330. be related to the relative level of the wall-shear-stress fluctuations as (see e.g. Alfredsson et al. 1988) q 02 p τw u02 = lim . (5.19) τw y→0 U For the largest blowing rate measured (Γ = 3.72 × 10−3, in which no spatial p filtering is expected to occur), the near-wall turbulence intensity is u02/U ≈ p 0.6, about 50% larger than the value of u02/U ≈ 0.4 usually reported for canonical boundary-layers (Alfredsson et al. 1988; Osterlund¨ 1999; Alfredsson & Orl¨u2010).¨ For suction boundary layers the opposite is observed, with the near-wall turbulence intensity strongly damped by the suction and reaching p values of u02/U ≈ 0.2 for suction rate Γ = 3.70×10−3 (data from Khapko et al. 2016, see Fig. 4.2). In Figure 5.55 the observed value of the local turbulence p intensity at the wall for the canonical ZPG TBL case is u02/U ≈ 0.3, instead of the expected value of 0.4, due to hot-wire spatial-filtering effects.
5.3.2. Spectra and higher-order statistics Figure 5.58 reports the premultiplied power-spectral-density map for the largest Reynolds number measured at each blowing rate. The results are shown in the frequency domain instead of the wavelength domain, because, as clearly seen in 120 5. Results and discussion
6 2 ) ∗
5 δ
∞ 1.5 U ( ) / ∗ δ 99 δ
∞ 4
) 1 U ( U / − 99 δ ∞ ) 3 0.5 U U ( −
∞ 2 0 U
( 0.2 0.4 0.6 0.8 1 /δ y 99 1
0 0 0.2 0.4 0.6 0.8 1 1.2 /δ y 99
Figure 5.53. Mean-velocity-deficit profiles for blowing, suc- tion and no-transpiration cases normalized according to Za- garola & Smits (1998a). Filled colored symbols: all the blowing boundary layers in Tab. 5.5. Black triangles: canonical ZPG TBL at Reθ = 21330. Open colored symbols: all the TASBL cases in Tab. 5.3. Inset: detail view of the outer region.
Figure 5.58 low-frequency components in the outer part of the boundary layer (representative of the large-scale motions) becomes predominant with increasing blowing rate. This observation rises doubts about the applicability of Taylor’s hypothesis (see §5.2.6) for boundary layers with blowing. With increasing blowing rate, the high-frequency near-wall peak of the premultiplied P.S.D., trace of the near-wall cycle, disappears, in agreement with the disappearance of the inner peak of the streamwise-velocity-variance profiles. The outer peak of the premultiplied P.S.D. starts to develop at wall normal location y/δ99 ≈ 0.05 and normalized frequency F ≈ 0.15, spreading with increasing blowing rates towards the outer portion of the boundary layers covering a wider frequency range. This observations are the opposite behaviour of what observed for suction boundary layers, where the turbulent activity remained fundamentally limited to the near-wall region. The skewness and kurtosis profiles for the largest Reynolds number measured at each blowing rate are reported and compared with canonical ZPG TBLs in Figure 5.59 and Figure 5.60 respectively. The near-wall minimum of the skewness profile observed for the canonical ZPG TBL cases disappears for the blowing boundary layers, replaced by a monothonic increase of the velocity- skewness profile. Positive values of skewness in the whole inner region are 5.3. Zero-pressure-gradient turbulent blowing boundary layers 121
1.7 Γ = 0 Γ ≈ 1.00 × 10−3 −3 1.6 Γ ≈ 1.46 × 10 Γ ≈ 1.95 × 10−3 Γ ≈ 2.95 × 10−3 1.5 Γ ≈ 3.72 × 10−3 12 H 1.4
1.3
1.2 5 10 15 20 25 30 35 40 Re × 10−3 θ
Figure 5.54. Shape factor H12 variation with the momentum- thickness Reynolds number Reθ for the blowing boundary layer in Tab. 5.5 (symbols) compared with the shape factor of the com- posite profile for canonical ZPG TBLs proposed by Chauhan et al. (2009) (solid line). observed for the blowing boundary layers, indicating the prevalence of sweeping motion of high-momentum fluid from the outer layer towards the near-wall region, commonly associated with large-scale motions. The streamwise-velocity-kurtosis profiles of blowing boundary layers and canonical ZPG TBLs show a good overlap in a large portion of the boundary layer independently of the blowing rate if plotted vs. the outer-scaled wall normal distance. In the near-wall region, instead, differences among cases at different blowing rates can be observed, with increasing values of the near-wall kurtosis for larger blowing rates. For wall normal distances 0.03 / y/δ99 / 0.3 the kurtosis values is approximately constant, with a value of Kuu0 ≈ 2.7, close to Kuu0 ≈ 2.8 reported by Fernholz & Finley (1996) for the log law region of canonical ZPG TBL. In this respect, considerable difference in the velocity kurtosis profiles of suction boundary layers (see Fig. 5.46) can be observed: at −3 large suction rate (Γ ' 3.5 × 10 ), in fact, the value of Kuu0 remains larger than the Gaussian value Ku = 3 in the region y+ > 40. 122 5. Results and discussion
Γ = 0.98 × 10−3 0.6 Γ = 1.49 × 10−3 Γ = 1.97 × 10−3 0.5 Γ = 3.00 × 10−3 Γ = 3.72 × 10−3 0.4 Γ = 0 /U 2 ′
u 0.3 p
0.2
0.1
0 10−3 10−2 10−1 100 /δ y 99
Figure 5.55. Colored symbols: local turbulence-intensity pro- files at x = 6.06 m for different blowing rates at the highest Reθ measured (see Tab. 5.5). Black triangles: canonical ZPG + TBL at Reθ = 21330 (Lw ≈ 25). Data are not corrected for spatial filtering. 5.3. Zero-pressure-gradient turbulent blowing boundary layers 123
0.012 Γ ≈ 1.00 × 10−3 = 0.28 mm Γ ≈ 1.46 × 10−3 = 0.28 mm Lw Lw 0.01
0.008 2 ∞ /U
2 0.006 ′ Re = 30130 Re = 36390
u θ θ = 24130 = 27720 0.004 Reθ Reθ Re = 16280 Re = 18270 0.002 θ θ = 12550 = 9940 Reθ Reθ 0
0.012 Γ ≈ 1.95 × 10−3 = 0.28 mm Γ ≈ 2.95 × 10−3 = 0.57 mm Lw Lw 0.01
0.008 2 ∞ /U
2 0.006 ′ u 0.004 = 33230 = 28710 Reθ Reθ Re = 21320 Re = 21480 0.002 θ θ = 11140 = 13870 Reθ Reθ 0 10−3 10−2 10−1 100 /δ 0.012 y 99 Γ = 3.72 × 10−3 = 0.28 mm Lw 0.01
0.008 2 ∞ /U
2 0.006 ′ u 0.004
0.002 = 26050 Reθ 0 10−3 10−2 10−1 100 /δ y 99
Figure 5.56. Outer-scaled velocity-variance profiles for blow- ing boundary layers measured at x = 6.06 m (see Tab. 5.5). 124 5. Results and discussion
1.2 Γ ≈ 0.98 × 10−3 Γ ≈ 1.49 × 10−3 ≈ 30500 ≈ 36300 1 Reθ Reθ 100
× 0.8 2 ∞ /U 2
′ 0.6 u ;
∞ 0.4
U/U = 0.57 mm = 0.57 mm 0.2 Lw Lw = 0.28 mm = 0.28 mm Lw Lw 0 1.2 Γ ≈ 1.98 × 10−3 Γ ≈ 2.98 × 10−3 ≈ 33100 ≈ 28400 1 Reθ Reθ 100
× 0.8 2 ∞ /U 2
′ 0.6 u ;
∞ 0.4
U/U = 0.57 mm = 0.57 mm 0.2 Lw Lw = 0.28 mm = 0.28 mm Lw Lw 0 10−3 10−2 10−1 100 10−3 10−2 10−1 100 /δ /δ y 99 y 99
Figure 5.57. Spatial-resolution effects on the velocity variance profile at the highest Reθ measured for each blowing rate. Filled Symbols: current investigation, x = 6.06 m (see Tab. 5.5); Solid line: velocity-variance profile corrected from spatial resolution effects with the method by Segalini et al. (2011). 5.3. Zero-pressure-gradient turbulent blowing boundary layers 125
Figure 5.58. Premultiplied power-spectral-density maps in 2 outer scaling (F = fδ99/U∞; Puu,n = Puu/U∞) for the blowing boundary layers at the highest Reτ measured for each blowing rate. 126 5. Results and discussion
1.5
1
0.5
0
−0.5 u’ Γ = 0.98 × 10−3 Sk −1 Γ = 1.49 × 10−3 Γ × −3 −1.5 = 1.97 10 Γ = 3.00 × 10−3 −2 Γ = 3.72 × 10−3 Γ = 0; = 5250 Reτ −2.5 Γ = 0; = 2500 Reτ −3 10−3 10−2 10−1 100 y/δ 99
Figure 5.59. Velocity-skewness profiles for the blowing bound- ary layers shown in Fig. 5.52. Symbols as in Tab. 5.5.
30 Γ = 0.98 × 10−3 Γ = 1.49 × 10−3 20 Γ = 1.97 × 10−3 Γ = 3.00 × 10−3 Γ = 3.72 × 10−3 Γ = 0; = 5250 10 Reτ Γ
u’ = 0; = 2500 Reτ Ku
2 10−3 10−2 10−1 100 y/δ 99
Figure 5.60. Velocity-kurtosis profiles for the blowing bound- ary layers shown in Fig. 5.52. Symbols as in Tab. 5.5. Solid line: kurtosis of a normal distribution Kuu0 = 3. Dashed line: Kuu0 = 2.7 Concluding remarks
Experimental apparatus An experimental apparatus for the study of boundary layers with wall suction and blowing was designed, built and tested. In order to better study the development of the boundary layer, the extent of the perforated area of the present apparatus was designed to be the largest possible in the MTL wind tunnel, obtaining a perforated area of 6.5 m × 1.2 m, longer than any previous study on the topic. For suction boundary layers this allowed to experimentally realize the turbulent asymptotic conditions, while for blowing boundary layers it allowed to provide measurements at Reynolds numbers larger than previous studies. The capability of this apparatus to accurately reproduce both the mean velocity profile of a laminar ASBL and some previous results on canonical ZPG TBLs, regarding both the skin-friction law and the distribution of the mean-velocity profile, established the quality of the experimental setup and of the measurement procedures.
Turbulent suction boundary layers Turbulent suction boundary layers are found to exist only for suction rates Γ < 3.7 × 10−3, in accordance with Watts (1972) and Khapko et al. (2016): at larger suction rates the boundary layer undergoes a process of relaminarization. It was proven that it is possible to experimentally realize a turbulent asymptotic state, provided that the boundary layer thickness at the streamwise location where suction is started is close to the asymptotic one: in the current experiments TASBLs were obtained for suction rates 2.55 × 10−3 < Γ < 3.7 × 10−3. The mean-velocity profiles of TASBLs are characterized by the disappear- ance of a clear wake region and by a logarithmic behaviour for a particularly large portion of the boundary layer (covering 40% of the boundary-layer thick- ness already at Reτ = 1760). A good overlap of the mean velocity profile in outer scaling (U/U∞ vs. η) independently from the suction rate is observed and a log law with slope Ao = 0.064 and intercept Bo = 0.994 (if η = y/δ99) describes accurately the outer-scaled mean-velocity profiles. A possible explanation of the observed mean-velocity scaling is proposed: if the shear-stress scales with the wall shear stress independently of the Reynolds number (or equivalently, of the suction rate), as observed in other parallel turbulent shear-flows (pipe
127 128 Concluding remarks
flow and channel flow), the mean velocity scaling with the free-stream velocity follows from eq. (2.40), derived from the Navier-Stokes equation for a TASBL. The application of suction leads to a strong damping of the velocity fluc- tuations, with a large decrease of the magnitude of the near-wall peak of the streamwise-velocity variance, characterized by values from 50% to 65% lower than canonical ZPG TBLs at comparable Reτ . The damping of the velocity fluctuations by suction appears to be primarily due to the increased stability of the near-wall streaks (in agreement with Antonia et al. 1988), as can be + concluded from the increase with suction of the streamwise wavelength λx,p related to the peak of the premultiplied power spectral density in the near-wall region. The analysis of the power-spectral-density maps and the disappearence of the shoulder in the streamwise-velocity variance profile suggests that suction is very effective in reducing the strength of the (outer) large-scale structures of the boundary layer. TASBLs appear, hence, to be fundamentally dominated by the near-wall cycle. This conclusion is supported by the negative values + taken by the skewness of the velocity fluctuations for the whole region y ' 10, indication of a flow dominated by the ejections of low-momentum fluid from the near-wall region. In order to confirm the proposed mean-velocity scaling and to verify the generality of the above conclusions, data on TASBLs for lower suction rates (hence larger Reynolds number) would be strongly beneficial. However, the larger Reθ,as expected when the suction rate is lowered, represents a considerable challenge. From an experimental perspective lowering Γ presents difficulties related to the size of the experimental facility needed. To reach larger asymptotic Reynolds numbers, in fact, it is required that a turbulent boundary layer is allowed to grow for a long downstream distance before suction is applied, so that the boundary-layer thickness at the suction-start location is comparable with the asymptotic one. Downstream of this location, a suction region must be provided, which extends multiple times the (larger) boundary-layer thickness. From the numerical perspective, instead, the limiting factor would be the large Reτ encountered when Γ is lowered (Bobke et al. 2016).
Turbulent blowing boundary layers The application of wall-normal blowing with small blowing rates, in the range of 0.1% to 0.37% of the free-stream velocity, significantly modifies the behaviour of the boundary layer. The shape factor increases with the blowing rate, indicating less full velocity profiles. An increasing curvature of the mean-velocity profile in a semi-logarithmic plot is observed with increasing blowing rate, suggesting that a logarithmic law cannot be used to effectively describe the mean-velocity profile of blowing boundary layers. The mean velocity-defect profiles of all the examined blowing boundary layers appear to overlap between each other and with canonical ZPG TBLs when normalized with the empirical Zagarola-Smits velocity scale, in agreement with Kornilov & Boiko (2012). Turbulent blowing boundary layers 129
Blowing enhances the velocity fluctuations, with an increased local turbu- lence intensity throughout the boundary layer. The largest increase in the energy of the turbulent fluctuations is, however, located in the outer layer, as evident from premultiplied power-spectra-density maps. At sufficiently large blowing rates and Reynolds number, the outer peak in the power spectra becomes predominant over the near-wall peak, which eventually disappears. Similarly, for all the blowing rates considered an outer peak of the streamwise-velocity variance appears at high enough Reynolds number, eventually becoming larger than the near-wall peak, which disappears completely for the largest Reynolds numbers and blowing rates investigated. The lack of an estimate for the friction velocity constituted a considerable limitation for the analysis of the results on blowing boundary layers, and additional experiments where the wall shear stress is measured would be useful. The determination of the wall shear stress over a permeable surface presents, however, considerable difficulties. Wall balances have been successfully used for this purposes in the past (see Depooter et al. 1977) and their use should be reconsidered. Another possibility is the adoption of a measurement technique (e.g. laser Doppler velocimetry) which allows the measurement of velocities in the viscous sublayer, from which the wall shear stress can be calculated. Some other aspects of blowing boundary layers deserve further investigation. In particular, it remains unclear under which condition a blowing boundary layer would remain attached, i.e. whether a minimum blowing rate is necessary to produce separation or if any uniform blowing rate would eventually lead to boundary-layer separation at a certain downstream position. Answering this question would be beneficial for the treatment of the scaling laws of the statistical quantities, because it would clarify whether an equilibrium asymptotic behaviour (here signifying Re number independence) is to be expected or not.
Acknowledgements
This work has been financially supported by the European Research Council (ERC) and by KTH, which are gratefully acknowledged. Furthermore, the travel stipends by the Petersohns minne foundation and the AForsk˚ foundation were greatly appreciated.
I would like to thank my supervisor Prof. Jens Fransson for giving me the opportunity of undertaking the doctoral studies under his guidance. The trust and the freedom I have received allowed me to follow my own pace, while having the right support when needed. My co-supervisor Dr. Bengt Fallenius has provided invaluable assistance in the set-up and tuning of the experimental apparatus. During the years I have learned to strongly appreciate his calm and friendly attitude, especially during the most nerve-wracking moments of the experimental campaigns. Dr. Robert Downs, Docent Ramis Orl¨uand¨ Dr. Antonio Segalini have always been the most available in sharing their broad knowledge and experience in the field of fluid mechanics and for this reason I want to thank them profoundly. Docent Ramis Orl¨uhas¨ also given me useful comments while reviewing this thesis and other manuscripts. Lic. Alexandra Bobke and Dr. Taras Khapko are thanked for sharing their numerical database on TASBLs. Our present and former skilled technicians Joakim Karlstr¨om,Jonas Vikstr¨om and Rune Lindfors are thanked for providing me well-crafted components to- gether with design suggestions and technical assistance. Renzo and Sohrab, being my officemates for more than three years, deserve a special mention for welcoming me on board of the research group, sharing their knowledge with me and for all the laughs and discussions we had in our room. Moreover, Renzo (here acting as Dr. Trip) volunteered for reading the manuscript of this thesis and his comments were much appreciated. The pleasant working environment of the Mechanics Department have contributed enormously to this project. I wish to thank all the present and former colleagues for that and for all the fun times during and outside working hours. Lastly, I want to thank my parents, my brother and my girlfriend Anya for their love and constant support.
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