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˚and av 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
Marco Ferro 2017c
Universitetsservice US–AB, Stockholm 2017
Experimental study on turbulent boundary-layer flows with wall transpiration
Marco Ferro
Linn´e FLOW 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´e FLOW 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˚a som att senarel¨agga b˚ade avl¨osning och omslaget fr˚an lamin¨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˚a har ett otaligt antal studier genomf¨orts p˚a omr˚adet de 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˚ader stora meningsskiljaktigheter om de mest element¨ara str¨omningskvantiteterna, s˚asom medelhastigheten, n¨ar sugning och bl˚asning till¨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˚a gr¨ansskikt med sugning och bl˚asning genom ytan s˚a har 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˚ang i 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˚a den experimentella uppst¨allningen och valda m¨atprocedurerna har verifierats genom omfattande tester, som ¨aven inkluderar benchmarking mot tidigare resultat p˚a turbulenta gr¨ansskikt utan tryckgradient eller bl˚asning/sugning och p˚a lamin¨ara asymptotiska sugningsgr¨ansskikt.
De experimentella resultaten p˚a turbulenta 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˚ang logaritmisk 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˚a kan 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 × 10−3s˚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˚an en ¨okad stabilitet av hastighetsstr˚aken n¨armast ytan.
M¨atningar p˚a turbulenta gr¨ansskikt med bl˚asning har genomf¨orts f¨or bl˚asningshastigheter mellan 0.1 och 0.37% av fristr¨omshastigheten och t¨acker Reynoldstalomr˚adet (10 − 36) × 103, med Reynolds tal baserat p˚a r¨orelsem¨angds- tjockleken. Vid bl˚asning genom ytan f˚ar man 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˚asning erh˚alls f¨or alla Reynoldstal och bl˚asningshastigheter n¨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˚asningshastigheter och Reynoldstal s˚a kommer 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
Cf: friction coefficient 2τw/ρU∞2 (-);
Cp: pressure coefficient 2(P − P∞)/ρU∞2 (-);
f : indicates both frequency (Hz) or a generic function;
fcut: cut-off frequency of anemometer low-pass filter (Hz);
fmax: maximum resolved frequency, defined as min(fsmp, fcut) (Hz);
fsmp: sampling frequency (Hz);
H12: boundary-layer shape factor δ∗/θ (-);
Lw: hot-wire sensor length (m);
`∗: viscous length ν/uτ (m);
P : mean pressure (Pa);
R: specific gas constant of air (J kg−1K−1) or electrical resistance (Ω);
Re: representative Reynolds num- ber (-);
Rex: streamwise-coordinate Reynolds number U∞x/ν (-);
Rex0: streamwise-coordinate Reynolds number corrected for virtual origin U∞x0/ν (-);
Reδ∗: displacement-thickness Reynolds number U∞δ∗/ν (-);
Reθ: momentum-thickness Reynolds number U∞θ/ν (-);
Reτ: friction Reynolds number uτδ99/ν (-);
Suu: one-sided power-spectral- density estimate of the streamwise-velocity fluctua- tions (m2/s2 Hz−1);
T : temperature (K);
t: time (s);
tsmp: sampling time (s);
U : mean streamwise velocity (m/s);
u0: streamwise-velocity fluctua- tions (m/s);
uτ: friction velocitypτw/ρ (m/s);
V : mean wall-normal veloc- ity (m/s);
3
4 1. Basic concepts and nomenclature
V0: spatially-averaged wall-normal velocity at the surface (m/s);
v0: wall-normal-velocity fluctua- tions (m/s);
W : mean spanwise velocity (m/s);
w0: spanwise-velocity fluctua- tions (m/s);
x: streamwise position (m);
x0: streamwise position corrected for virtual origin (m);
y: wall-normal position (m);
z: spanwise position (m);
Greek Symbols:
Γ: transpiration rate |V0|/U∞ (-);
Γsst: maximum suction rate for self- sustained turbulence (-);
γ: intermittency of the velocity signal (-);
∆: Rotta-Clauser length scale δ∗U∞/uτ (m);
δ: generic boundary-layer thick- ness (m);
δ99: 99% boundary-layer thick- ness (m);
δ∗: boundary-layer displacement thickness (m);
η: wall-normal distance normalized with an outer length scale (-);
θ: boundary-layer momentum thick- ness (m);
κ: von K´arm´an constant (-);
λl: wavelength of the light (m);
λx: streamwise wavelength of the velocity fluctuations (m);
µ: dynamic viscosity (Pa s);
ν: kinematik viscosity (m2/s);
Π: wake parameter (-);
ρ: density (kg/m3);
τ : mean total shear stress (N/m2);
τw: mean wall shear stress (N/m2);
τw0: wall shear stress fluctua- tions (N/m2);
Superscripts:
: denotes time average;
+: denotes normalization with vis- cous scales;
Subscripts:
∞: denotes the free-stream condi- tions;
s: denotes the conditions at the suction/blowing start location;
as: denotes the asymptotic condi- 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 V0uniform 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, V0represents 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 V0in 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
∂x +∂V
∂y = 0 (1.2)
U∂U
∂x + V∂U
∂y = −1 ρ
dP∞ dx +µ
ρ
∂2U
∂y2 −∂u0v0
∂y − ∂
∂x
u02− v02
, (1.3) 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
U = u0 = v0 = 0 , V = V0 for y = 0 (1.4) U = U∞, u0= v0= 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 ρ
∂2U
∂y2 −∂u0v0
∂y = 1 ρ
∂τ
∂y, (1.6)
with
τ = µ∂U
∂y − ρu0v0, (1.7)
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
y δ x
U∞
V0
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
δ∗(x) = Z ∞
0
1 − U (x, y) U∞
dy , (1.9)
and the momentum thickness θ(x) =
Z ∞ 0
U (x, y) U∞
1 − U (x, y) U∞
dy . (1.10)
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:
Rex=U∞x
ν , Reδ∗=U∞δ∗
ν , Reθ= U∞θ
ν . (1.11)
Another important parameter in the description of the boundary layer is the mean (streamwise) wall shear stress
τw(x) = µ ∂U (x, y)
∂y y=0
, (1.12)
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
Cf = τw
1
2ρU∞2 . (1.13)
Integrating the boundary-layer momentum equation eq. (1.3) from the wall to infinity, the von K´arm´an momentum 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
Cf
2 = dθ dx− V0
U∞ − 1 U∞2
Z ∞ 0
∂
∂x
u02− v02
dy . (1.14)
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
uτ=r τw
ρ , (1.16)
it is possible to write
U uτ
= fwyuτ ν
. (1.17)
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
uτ = Φ1 y δ,U∞
uτ
. (1.20)
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 uτ = Φ1
y δ
, (1.21)
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
δ99
`∗ =uτδ99
ν = Reτ, (1.22)
is another possible definition of a Reynolds number describing the flow and is known as the friction Reynolds number or the K´arm´an number.
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
δ∗= uτ
U∞ Z δ
0
U∞+ − U+dy (1.23)
= δ uτ
U∞ Z 1
0
Φ1d(y/δ) , (1.24)
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
δ∗= δ uτ
U∞K , (1.25)
where K is the integral of Φ1 from 0 to 1. The Rotta-Clauser length scale is defined as
∆ = δ∗U∞ uτ
, (1.26)
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
uτ = Φ2(η) , (1.27)
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 uτ
∂U
∂y = y+dfw(y+)
dy+ = −ηdΦ(η)
dη = const. (1.28)
From the above equation a logarithmic velocity profile in the overlap region is immediately derived, which can be expressed as
U+= 1
κln y++ B , (1.29)
or as
U∞− U uτ
= −1
κln η + B1. (1.30)
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´an constant. 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 U∞+ = 1
κ ln y+− ln η + B + B1 (1.31) U∞+ = −1
κln ∆++ C∗, (1.32)
where C∗= B + B1. Recalling eq. (1.12) and eq. (1.26), we get Cf= 2τw
ρU∞2 = 2 uτ U∞
2
and ∆+= δ∗U∞
uτ`∗ = Reδ∗. Hence, we can rewrite eq. (1.32) as
r 2 Cf = 1
κln Reδ∗+ C∗, (1.33)
or
Cf = 2 1
κln Reδ∗+ C∗
−2
. (1.34)
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:
Cf= 2 1
κln Reθ+ C
−2
. (1.35)
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
U+= fw(y+) +Π κWy
δ
, (1.36)
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¨u 2010, 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¨u 2010; 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
V0
∂U
∂y = ν∂2U
∂y2 , (2.1)
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
U∞ = 1 − eyV0/ν. (2.3)
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):
δ∗= Z ∞
0
1 − U
U∞dy = −ν V0
, (2.5)
and
θ = Z ∞
0
U U∞
1 − U
U∞
dy = −1 2
ν
V0, (2.6)
from which follows
H12=δ∗
θ = 2 . (2.7)
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
ReASBL=U∞δ∗
ν = −U∞ V0
= 1
Γ. (2.8)
ReASBL is sometimes used also for the characterization of turbulent asymptotic suction boundary layers (see e.g. Schlatter & ¨Orl¨u 2011, 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) suggested1 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
Γ × 10−3
0 5 10 15 20
Kay (1948) [Exp.]
Dutton (1958) [Exp.]
Black & Sarnecki (1958) [Exp.]
Tennekes (1964) [Exp.]
Favre et al. (1966) [Exp.]
Watts (1972) [Exp.]
Yoshioka & Alfredsson (2006) [Exp.]
Bobke et al. (2016) [LES]
Khapko et al. (2016) [DNS]
Current Exp.
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
∂x V∂U
∂y , (2.9)
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)
V0
∂U
∂y = 1 ρ
∂τ
∂y , (2.11)
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
V0U = τ − τw
ρ . (2.12)
In viscous units eq. (2.12) can be rewritten as u2τ+ V0U = τ
ρ. (2.13)
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
τ = µ∂U
∂y . (2.14)
Eq. (2.13) can then be rewritten as 1 + V0+U+=∂U+
∂y+ = 1 V0+
∂
∂y+(1 + V0+U+) . (2.15) Making use of the no-slip boundary condition, eq. (2.15) becomes (Rubesin 1954; Mickley & Davis 1957; Black & Sarnecki 1958)
U+= 1
V0+(ey+V0+− 1) , (2.16) 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 ln2y + 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¨u 2011; 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
U∞ = 1 − 1 κ2
V0
U∞lny
δ, (2.17)
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 ρ =
κy∂U
∂y
2
, (2.18)
which can be used in eq. (2.13) to obtain u2τ+ V0U =
κy∂U
∂y
2
. (2.19)
Eq. (2.19) can be rewritten as
u2τ+ V0U = κ V0
∂(u2τ+ V0U )
∂ ln y
2
. (2.20)
The solution of this differential equation is u2τ+ V0U = V02
4κ2(ln y + C1)2, (2.21) where C1 is an integration constant. One possible way to express eq. (2.21) in viscous scaling is (Stevenson 1963a)
2 V0+
q
1 + U+V0+− 1
= 1
κln y++ C2− 2
V0+, (2.22)
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
C2→ B + 2
V0+ for V0+→ 0 , (2.23)
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 u∗τ = −1
κlny
δ +Π(x) κ Wy
δ
, (2.24)
2.2. Turbulent boundary layers with transpiration 21 where a dependency on the first power of the logarithm is evident. In eq. (2.24) u∗τ 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 u0
= f y
`0
, (2.25)
and an outer region where the velocity profile has the form of a defect law U∞− U
u0
= gy δ
. (2.26)
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 u0
= 1 κln y
y0
+ B2, (2.27)
or, equivalently,
U∞− U u0
= −1 κlny
δ + B3. (2.28)
For the case of a turbulent boundary layer flow without wall-normal mass transfer (see §1.3),
u0= uτ =r τw
ρ and `0= `∗= ν
uτ. (2.29)
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)
u0= τw/ρ V0
=u2τ V0
and `0= ν V0
, (2.30)
so that eq. (2.16) can be written in the form of eq. (2.25) as U
u0
= ey/`0− 1 , (2.31)
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
