- Email: [email protected]
Transverse curvature effects in turbulent boundary layer
Progress in Aerospace Sciences 35 (1999) 661}672
Transverse curvature e!ects in turbulent boundary layer J. Piquet *, V.C. Patel Laboratory of Fluid Mechanics, Ecole Centrale de Nantes, UMR 6598 CNRS, 1 Rue de la Noe, B.P. 92101, 44321 Nantes Cedex, France IIHR, University of Iowa, USA Received 30 April 1999
Contents 1. Introduction . . . . 2. Mean velocity data 3. Flow reH gimes . . . 4. Flow structures . . 5. Turbulence data . 6. Conclusions . . . . References . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
661 662 665 667 668 671 671
Abstract The e!ect of transverse surface curvature on the turbulent boundary layer is reviewed by recourse to experiments on axial #ow along a circular cylinder. Three #ow regimes are identi"ed depending on values of the two controlling parameters, namely, the Reynolds number and the ratio of the boundary layer thickness to cylinder radius. The boundary layer #ow resembles a wake when both parameters are large. As expected, the e!ect of curvature is small when the Reynolds number is large and the boundary layer is thin. When the boundary layer is thick and the Reynolds number is small, which is typical of laboratory investigations, the e!ect of transverse curvature is felt throughout the boundary layer with evidence for relaminarization at the low Reynolds numbers. This review describes the experimental evidence and points out gaps that remain. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Transverse curvature e!ects in turbulent boundary layers are associated with curvature of the wall in planes orthogonal to the mean direction of the #ow. The simplest, canonical situation in external #ow is that of a long cylinder, of radius a, immersed in a uniform #ow of velocity ; . Then the no-slip condition implies the pres ence of a boundary layer along the cylinder. This boundary layer is subject to a negligible axial pressure gradient so that transverse curvature e!ects are isolated from other e!ects. Experimental data available in this case are rather limited [1}5], because it is di$cult to align the axis
* Tel.: #33-2-40-37-16-33; fax: #33-2-40-37-25-23. E-mail address: [email protected] (J. Piquet)
of the cylinder with the #ow direction and to keep it straight against elastic deformation. There is however an intrinsic interest in the subject for the description of boundary layer #ows past bodies including towed submerged cables, vehicles, missiles, glass or polymer "bers during fabrication. Also, a knowledge of transverse curvature e!ects is signi"cant in estimating the surface resistance of ship models and of other three-dimensional elongated aerodynamic and hydrodynamic shapes which share a transverse radius of curvature comparable in size with the boundary layer thickness. The data pertaining to the axial #ow along a circular cylinder are reviewed here to study the e!ects of transverse curvature on the boundary layer development. Transverse curvature e!ects that arise in internal #ows, such as circular pipes, are rather limited and are largely excluded from this review.
0376-0421/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 6 - 0 4 2 1 ( 9 9 ) 0 0 0 0 7 - X
662
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Nomenclature x, r a ;, < k, o q u , v , w C D ;,; O C d, h dH
axial and radial coordinates radius of the cylinder axial and radial mean velocity components Dynamic viscosity, density of the #uid total stress (see Eq. (1c)) axial, radial and azimuthal velocity #uctuations skin friction factor (see Eq. (2a)) friction velocity, far-"eld mean velocity boundary layer thickness, momentum thickness displacement thickness
B , B
Inner and outer intercept of the logarithmic law i Von Karman constant in the logarithmic law l mixing length (see Eq. (4)) l eddy viscosity coe$cient 2 k turbulent kinetic energy (see Eq. (17e)) e turbulent dissipation (see Eq. (17e)) S skewness factor F #atness factor w (index) refers to the wall boundary #(superscript) refers to near-wall conditions
The momentum integral obtained from the above set of equations is ; dh C , O " , ; dx where the momentum thickness is de"ned by
; r ?>B ; 1! dr, (2b) ; ; a ? C is the skin friction coe$cient, ; "(q /o) is the O friction velocity, d is the boundary layer thickness (de"ned as the distance from the wall where ;"0.995; ) which, like h, increases slowly with x. The continuity equation yields similarly h"
Fig. 1. Schematic sketch of the problem.
2. Mean velocity data
The boundary layer approximation for the mean motion of an incompressible #ow along the exterior surface of a circular cylinder can be written as
d < dd* , " 1# dx a ; where
R R (r;)# (r<),0, Rx Rr
d*"
;
R; R; 1 R #< " (rq), Rx Rr or Rr
(1a)
(1b)
where x and r are the axial and radial coordinates (the cylinder surface is r"a), as indicated in Fig. 1, ; and < are the axial and radial mean velocity components, respectively, q is the total stress (molecular#turbulent): q"k
R; !ou v Rr
(2a)
(1c)
and o and k are the density and viscosity of the #uid.
?>B
; r 1! dr ; a ? is the displacement thickness which also increases slowly with x, so that the entrainment velocity, < , at the edge of the boundary layer is positive. The momentum thickness allows de"nition of the small parameter e"O(h/d), which is a function of x and represents a momentum defect much less than unity for a><1, where the index # denotes the wall scaling using l and ; . This parameter is shown by Afzal and O Narasimha [6] to be "xed by the ratio ; /; . These O authors established the limits of validity of the logarithmic law and indicated that the inner intercept, B , in this law, B ";>!i\ ln y>
(3a)
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
663
tively, we use the classical form l>"iy>, we "nd [16]
(1#y>/a>) !1 1 #B , ;>" ln 4a> ? i (1#y>/a>) #1
Fig. 2. X, WY, Willmarth and Yang [4]; O, RK, Rao and Keshavan [16]; R, Richmond [17] (from [6]).
is a function of 1/a>, while the outer intercept, B, in the velocity-defect layer,
; !; y #i\ ln (3b) B" d ; O is a decreasing function of d/a. This latter result is con"rmed by Fig. 2. Thus, if a><1, transverse curvature mainly shifts the log law. We may now try to estimate the level of the shift in terms of a. A plausible estimation of the velocity pro"le is obtained by Patel [7] from mixing length theory which states that, in the wall region
!u v "l
R; . Ry
W>
2q> dy
. 1#(1#4(l>)q>
(7c)
If a continuous mixing length distribution from the sublayer to the log region is prescribed, Eq. (7c) may be replaced by a velocity pro"le connecting (7a) and (7b) by quadrature [8]. Denli and Landweber [10] propose
y> l>"ia> ln 1# (tanh[j(y>)]. a>
(8)
(5)
1 1 ;>" E (ln(1#y>/a>)# ln(2a>)!c#B (a>) i G i
(6)
Eq. (6) implies that for su$ciently small values of a>, the stress falls monotonically to very low levels from its value at the wall, a situation similar to that occuring in a twodimensional accelerated boundary layer. This analogy will be exploited further. It then remains to "x l. If we integrate Eq. (5) while neglecting the turbulent shear stress with respect to the viscous stress (l"0), we obtain a logarithmic velocity distribution of the form ;>"a> ln(1#y>/a>),
1 ;>" ln[a> ln(1#y>/a>)]#B . i
(4)
A velocity pro"le for the wall region is obtained if distributions of q(y) and l(y) are given or assumed. If we neglect the acceleration terms in Eq. (1b), we "nd a non-dimensional stress distribution throughout the wall region under the form rq"aq "const. or a> q>" . a>#y>
where B is a function of a# reducing to B (+5.45) as ? a>PR. The departure of the ln argument in [2] of Eq. (7b) from a> ln(1#y>/a>) is rather small. Patel [8] indicates less than 1% for y>"a> and about 8% for y>"6a>. Rao's form [9] (7c) yields a result very close to (7b), namely
Using again Eq. (6), the resulting expression (5) for ;> may be solved numerically. For y>'y>, where y> is the value of y> at which the viscous stress becomes negligible in comparison with the Reynolds stress, the velocity is a function of the logarithmic integral E (ln z): G
Substitution of Eq. (4) into Eq. (1c) gives ;>"
(7b)
(7a)
a necessary byproduct of the constant shearing force per unit length, 2prkR;/Rr, upon the cylinder. If alterna-
1 + ln a# i
>> y> 1# ln 1# a> a>
#B (a>),
(9)
where c is the Euler constant 0.5772156, while [11] B (a>)"5.45!41.76/a>. In the approximate form (9), the square root term is obtained from the dominant term in the in"nite series of E . When neglected, the expression G becomes similar to (7c). The argument [2] for the logarithmic term in Eq. (9) is very close to y> when y>)a> [10], justifying the use of the Preston tube technique with the usual calibration curve for the measurement of the wall shear stress. Denli and Landweber [10] have shown that it was possible to retain acceleration e!ects in Eq. (1b), resulting in a more complete law of the wall, ;>"f (y>, a>). Then the key assumption is that the dependence of ;> on x through a> can be neglected as compared with the dependence on y>. The total shear stress can be obtained as a modi"ed form of Eq. (6): a> q>" [1!c(y>, a>, p )], a>#y>
(10a)
664
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
be sustained. This implies that Eq. (7a) is valid. Close enough to the wall, this law degenerates into the linear sublayer pro"le. To "nd a signi"cant departure from the linear sublayer pro"le would require a>"O(1). Another related in#uence of transverse curvature lies in the collapse of the velocity defect, as shown in Fig. 3b, where ; is evaluated at y"0.99d. It is clear that the very full velocity proxle is a direct result of the cylindrical geometry of the #ow since there is no acceleration of the free stream. It is thus plausible to admit a Clauser constant-eddy-viscosity estimate in the wake-like part of the boundary layer. For d/a'20, the velocity distribution exhibits similarity based on outer variables, suggesting that the boundary layer is controlled by the outer #ow. Denli and Landweber [10] have formulated a velocitydefect law starting from the linearized (wake-type) mean#ow equation. Substituting into it: x , ; !;"Re\ ; ; (m) f (g) with m" ? Re a ? r ;a Re " , g" ? l d#a
(11)
and using an eddy viscosity closure, q"!ol (m)R;/Rr, 2 they obtain for f Fig. 3. (a) In#uence of transverse curvature on the velocity pro"les. **, (bold) #at plate, law of the wall. } } }, log law (7b) for a>"33.4. 䉭, d/a"4.7; 䊐, d/a"16; *, d/a"37.5. Smallest value of a>"33.4 (from [3]). (b) Velocity-defect pro"les. **, (bold) #at plate, Coles law. £, d/a"1.8; 䉭, d/a"4.7; 䊐, d/a"16; *, d/a"37.5 (from [3]).
where the c-term accounts for inertia terms: c(y>, a>, p )"p
W>
yf (y) f (y) f (y)# dy a>
(10b)
1 + p (y>), (10c) 3 where a Taylor series expansion of c gives the "rst order correction (10c). This correction is rather weak in the wall region because a typical value of p is about 6;10\: at y>"30, the error introduced by the approximation (6) is about 5.4%. Fig. 3a presents the evolution of velocity pro"les in wall coordinates and shows how the law of the wall is a!ected by transverse curvature. Since the skin friction factor decreases signi"cantly as the cylinder diameter is reduced, the scaling of both the velocity and the distance from the wall is highly dependent on the friction velocity. Data of Fig. 3a and of [1], as well as DNS [7] indicate that, as the diameter of the cylinder is decreased, the extent of the region where the transverse curvature in#uences the velocity pro"le increases. For a small enough radius, i.e. a> small, the erosion of the logarithmic layer is made possible by the fact that turbulence e!ects cannot
1 ! gf
R dt/dm gf# 2# t dR/dm
g f dg E v (m) a#d ; m "2 2 , R" , t" . v R dR/dm a (2; Re O ? (12)
Self-similarity then reduces Eq. (12) to an ordinary di!erential equation in g if R dt/dm 2# "2b or t"t R@\ t dR/dm and m v (m) "c. 2 2 v R dR/dm
(13)
Eq. (12) is then solved with [10] l (m)/l"DmK\. Once 2 the evolutions of R and t are expressed in terms of b and m, it remains to solve the ordinary di!erential equation. Analytical solutions may be found in terms of the hypergeometric con#uent (Kummer) function [12]. Retaining the function which decreases exponentially as gPR, it is found that f (g)"e\X;(b, l, z) with z"g/(2c), while (1') b'0 (otherwise d would not increase with d). Denli and Landweber [10] "t all undetermined constants (only) with experimental data [23]: b"0.35, c"1/2, m"7/13. The resulting velocity-defect
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
law does not match the logarithmic law of the wall because small velocity gradients and radial velocity have been assumed, as g becomes small. This appears to be of no serious consequence, the agreement with data remaining good where both expressions tend to fail simultaneously, about y/(d#a)"0.125. The fuller velocity pro"les (Fig. 3b) are indicative of a more quiescent outer #ow when d/a is high, as compared with plane boundary layer #ow. Willmarth [13] has explained cyclic occurrence of bursts from the massaging action of the large eddies passing over the sublayer which produce, during random periods of time, an unstable in#exional velocity pro"le in localized regions near the wall. Bursts occur in these regions of locally unstable in#exional pro"les. On a cylinder, the structure of turbulence is at a uniformly reduced scale [4]. Hence, there is no reduction in the time scale of the massaging action of the large eddies on the sublayer relative to the characteristic time scale in the sublayer. The reduction of the size of turbulent eddies relative to those of a #at plate is determined by two e!ects. The "rst e!ect is the &fullness' of the pro"le which causes eddies at a given convection speed to be of reduced size, because a given mean speed is attained at a smaller distance from the wall. The second e!ect is the limited lateral extent of the axisymmetric boundary layer which results in lateral shearing action by the freestream on the larger eddies when they extend or move laterally. As d/a increases, the large-scale surface pressure #uctuations are correlated over a greater fraction of the cylinder circumference; the outer perimeter of the large eddies being very large compared to the inner perimeter. Hence for large d/a, the structure and position of large eddies do not depend on the presence of the wall, in contrast to the yat plate case where the large eddies are completely bounded on one side by the wall.
3. Flow reH gimes There are two length parameters in the transversecurvature problem, namely d/a, the ratio of the boundary layer thickness to the radius of the cylinder and a>. Fig. 4 presents available experiments in the plane of these parameters. While it would be of interest to determine the separate e!ects of these two parameters at a given Reynolds number, this is not in general possible because the wall friction coe$cient and the normalized free stream velocity ; /; have a very low sensitivity to O changes in the Reynolds number. This implies [2] a roughly linear variation of a>"Re ; /; with Re , ? O ? well correlated by ; /; "0.11 Re\ . Of course, this O ? correlation is limited to values of a> such that the #ow is fully turbulent. Using an argument [14] borrowed from accelerated boundary layers for which relaminarization occurs when the non-dimensional stress gradient, K ("lRq/o ;Ry) approaches approximately !0.009, O O
665
Fig. 4. Curvature parameter d/a and a> in axial turbulent #ow along a cylinder. 䉭, Luxton et al. [2]; #, Lueptow and Haritonidis [21] and Lueptow et al. [1]; ;, Willmarth et al. [3] and Willmarth and Yang [4]; 䊐, Rao and Keshavan [16]; *, Afzal and Narasimha [6]; **, Richmond (17); *, Neves et al. [7] (from [7]).
relaminarization should occur for a>(28, with transition for 28(a>(106. The threshold value of 28 is obtained if we compute the stress gradient at the edge of the blending region and set it equal to K . The criterion for O relaminarization suggests that some of the measurements indicated in Fig. 4 may not be in fully turbulent yow. Also, Lueptow and Haritonidis [15] have shown that C was an increasing function of transverse curvature and thus was higher than that for a plate boundary layer at the same Reynolds number, R . C was also a slightly F decreasing function of the Reynolds number based on x. This means that the cylindrical wall is more e!ective in converting mean #ow energy into turbulent energy than a planar wall. This di!erence may perhaps be attributed to the spanwise vorticity which is intensi"ed by the sweep, while being altered by the transverse curvature. The transverse curvature may then enhance turbulence generation during the burst so that an increase in friction results. Another reason could be that the large-scale outer structures may pass closer from the cylinder wall than on a #at-plate boundary layer. There are two major technical di$culties in determining the wall shear stress. First, it is necessary to ensure axial symmetry; if not, the determination of ; from the O momentum integral relation (2) would be inaccurate. This is probably the case in experimental data [16]. Second, there is the question of the status of the law of the wall. If axial symmetry is not guaranteed, experimental data in the bu!er zone may be "tted to the #at-plate law of the wall if there is only a gradual departure from it. This is the Clauser method which gives a good approximation to the actual wall shear stress (but may overestimate friction velocities, as in [3] since they "t their experimental data with Coles' law of the wall for a planar boundary layer). Otherwise, a law of the wall accounting for transverse curvature e!ects is required. Three #ow reH gimes may be distinguished: (i) The re& gime where d/a and a> are large. This case applies to a long slender cylinder such that the boundary
666
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
layer is almost all a wake-like #ow. Unfortunately, data with large d/a were restricted to low values of a> such that transitional e!ects are suspected (see low Re -data of Richmond [17] and Rao and Keshavan ? [16] where 4.1(d/a)11.6, Luxton et al. [2] where 9)a> )47 and 26)d/a)42, although Willmarth et al. [3] where 2(d/a(42, a>+33 show no evidence of relaminarization). This reH gime has been studied by Lueptow et al. [1] who have shown the validity of the logarithmic near-wall behavior (7a) for y>(10, approximately. The sublayer however does not depend on transverse curvature. The e!ect of curvature is primarily felt in the outer layer, through a probable e!ect on the viscous superlayer. Transverse curvature acts to lower turbulence intensities away from the wall (y>'20) although u is sometimes reported to be 10% higher than in the #at-plate case (see e.g. [18]). Large-scale structures sweep past the cylinder and they have a strong e!ect on the #ow. (ii) The re& gime where a> is large and d/a is small, or of order one, corresponds to the case of a large cylinder with a small e!ect of transverse curvature. In this case the logarithmic law is still valid [6], with asymptotic behaviors for i and the intercept (namely, B "5#236/a>) which are constant if d/a"O(1), so that the boundary layer may be regarded as planar when d/a)1. Most available experimental data fall on the boundary of this reH gime however (like those of Yasuhara [19] where d/a+0.6; Willmarth and Yang [4] where 1.8)d/a)16; Afzal and Singh [11] where 0.6)d/a)2; Wietrzak and Lueptow [20] where d/a"5.7 and a>"30), as do the DNS simulations of Neves et al. [7] at d/a"5 and a>"43 and d/a"11, a>"21. The calculations [7] have been performed at exceedingly low Reynolds numbers (Re "674 and 311 for d/a"5 and 11, respectively). ? Moreover, they consider the #ow between concentric cylinders, with a zero-shear stress condition applied at the outer cylinder located at r"d. Thus a mean streamwise pressure gradient is necessary to drive the #ow (the mass #ux through any crosssection is constant) and the #ow is treated as homogeneous in the axial direction with the following mean total stress:
a y q"o; 1# O a#y 2a#d
y d
1!
(14)
instead of q"o;(1!y/d) in the plane-channel case. O Comparison of this expression with Eq. (14) suggests the following velocity scale:
a y ;H"; 1# O a#y 2a#d
.
(15)
However, such a scaling should be considered valid only in the near-wall region: the calculations performed by Neves et al. using this parallel-#ow assumption cannot properly describe the outer region of the boundary layer and of its outer boundary. This is because the axial
Fig. 5. Schematic sketch of the boundary layer pro"le.
velocity gradient generates a mean radial velocity component in any boundary layer, an e!ect that can be neglected only within the wall region. (iii) The re& gime where d/a large and a> small is similar to that of an axisymmetric wake with an inner layer (Fig. 5). In this case, there are extreme di$culties in measurements. (1) An accurate alignment of the cylinder in the tunnel is critical, especially for long cylinders, because even a 13-misalignment of the cylinder to the mean #ow produces a spectacular loss of axial symmetry in the measurements [16], with an accuracy of at most $10}15% in shear stress. (2) This case requires such a long cylinder compared with its diameter that #ow non uniformities or free stream disturbances become important. (3) The total shear force on the #uid at the cylinder surface becomes vanishingly small and di!usion does not dominate or control the #ow: a large eddy passing from one side of the cylinder to the opposite side would not `seea the cylinder. (4) The initial development of the boundary layer is likely to be in#uenced by the shape of the nose wake but the boundary layer should be rather independent of it for x/a*5000, downstream of the nose. (5) Other problems of major concern include structural isolation of the cylinder, possible aeroelastic interaction with the #ow, and cylinder sag. Concerning the #ow in the wall region, there is no consensus on the e!ect of transverse curvature on the logarithmic law. From Fig. 3, it is evident that the entire velocity pro"le is directly or indirectly a!ected by curvature. Eq. (7b) derived from simple mixing length arguments is known to adequately describe the departures from standard logarithmic law, Eq. (3a) with due consideration for the accuracy of the wall shear stress. Some authors have, on the other hand, retained Eq. (3a) and allowed the constants to vary from their #at-plate values. For example, Lueptow et al. [1] increase i to unity for d/a'30 and let B take values as high as 11}13. When analyzed by overlap arguments, the e!ect of curvature on the outer #ow is a!ected by what model is adopted for the inner layer. Since the inner, near-wall log behavior yields: ;>+y>!y>/2a>, curvature e!ects become signi"cant if a> is su$ciently small, i.e. less than 50. The skin friction is increased with
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 6. Contours of axial streamwise vorticity #uctuations, normalized by ; and l. d/a"11, a>"21. Bold contours denote O negative vorticity (from [7]).
respect to the #at-plate case, in spite of strong measurement uncertainties. This implies that the cylindrical wall is more e!ective that the planar wall in converting mean #ow energy into turbulence energy.
4. Flow structures An instantaneous view of longitudinal vorticity contours obtained from DNS [7] is presented in Fig. 6. The #ow in the outer region has large structures wrapping around the cylinder alternating with regions of quiescent #ow. The regions of signi"cant vorticity are seen to penetrate to about 10 radii into the quiescent #ow region. The corresponding transverse velocity correlation is far lower than in a #at-plate boundary layer, as is the shear stress correlation, indicating a tendency towards stabilization. Also the ratio between the transverse length scale and the cylinder radius is large. To investigate how the shear stress is in#uenced by low transverse curvature e!ects, the power spectral density
( f ) of the wall shear stress #uctuations was considered by Wietrzak and Lueptow [20]. Root mean square values of the shear stress #uctuations, q , then result from
q "
f ( f ) d(ln f ).
(16)
In this form, the energy between any two frequencies is proportional to the area under the curve bounded by these frequencies. Fig. 7 shows the normalized psd using inner variables. The most prominent feature of the spectrum is the higher energy content of planar bounded #ows and pipe #ows at lower frequencies compared to that of a boundary layer on a cylinder. In the latter case [21], the frequency band of the streamwise velocity spectra where the maximum occurs is centered at
667
Fig. 7. Normalized power spectral density. Bold, Wietrzak and Lueptow [20]; cylindrical boundary layer d/a"5.7; R "3050 F (1); M, Madavan et al. [22] planar boundary layer R "10 630 F (1.19); H, Mitchell and Hanratty [23] pipe, Re "22 900 (1.06); S, Sreenivasan and Antonia [24] channel #ow Re "11 780 " (0.92); K, Keith and Bennett [25] planar boundary layer R "8200 (0.88). Italic numbers give the area under each curve F (from [20]).
fl/;+10\ or fd/; +0.4, suggesting that the strucR tures responsible for most of the shear stress #uctuations are the large-scale ones. The reduction of energy at low frequencies in the wall shear-stress spectrum is similar to that for the wall pressure spectrum [4], so that the convection velocity for the pressure producing eddies is independent of transverse curvature, whereas the velocity pro"le is fuller in a cylindrical boundary layer, implying a faster motion of eddies. To maintain the same convection velocity, the pressure-producing eddies must be smaller in a cylindrical boundary layer. A similar e!ect could reduce the #uctuations of wall shear stress at lower frequencies. A VITA technique was used by Wietrzak and Lueptow [20] to detect positive events, i.e. events such that Ru /Rt'0. The conditional average of positive events (ensemble average of events exceeding the threshold level) indicated no signi"cant di!erence with channel #ow but the averaging time ¹ at which the maximum number of positive events was detected in q was 10% larger in the duct-#ow case than in the cylindrical boundary layer. Hence, turbulent events detected at the wall of a cylindrical boundary layer occur more frequently (and are of a shorter time scale, as suggested by higher frequencies in Fig. 7) than in planar wall-bounded #ows. However, the burst-sweep cycle is not substantially altered by transverse curvature, in agreement with results [21]. A second signi"cant e!ect of transverse curvature is the appearance of a double hump in the cross correlations of u and q when probes are o!set in the transverse direction by one-half streak spacing. This fact could be the footprint of an arrowhead structure or of a structure with an inclined edge (with respect to the cylinder axis) possibly initiated during the burst-sweep cycle, as
668
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 8. (Left) Conceptual model of large-scale boundary layer structure based on correlations between the #uctuating wall pressure and streamwise velocities in frame of reference with mean velocity removed. (Right) Iso-correlation contours in the (x, y)-plane (from [18]).
a probable consequence of the #uid washing over the cylinder. Snarski and Lueptow [18] measured cross correlations between the wall pressure and streamwise velocity for zero time delay, i.e.
;
R R R< R< v #< v "!2 u v #v Rx Rr Rx Rr
!2o\v Rp /Rr!! o (x, y),p (t, 0, 0)u (t, x, y)/p u . NS Isocontours, which then represent a &snapshot' of the averaged eddy structure, show a series of negative contours (!0.07 to !0.01) at an angle of approximately 183 to the wall and a band of positive correlation that extends out from the wall past the edge of the boundary at an angle of approximately 453. While positive correlations are often associated with low-speed #uid (u (0), Snarski and Lueptow [18] seem to favor the association of positive correlation with high-speed #uid. Then the isocontours are consistent with the conceptual view (in a frame moving with the convection velocity, +0.83; ) of a large inclined vortex (about 1.5d average height) that would rotate in the direction of the mean shear with a trailing face at an angle of 453 to the wall and a leading face at an angle about 183 to the wall. Lines at these two angles would then correspond to the locii of maximum positive and negative u (Fig. 8).
5. Turbulence data
#l
;
;
R 1 R !2o\u Rp /Rx! u # (ru v ) Rx r Rr
#l
1 R R u # (ru ) !2e . VV Rx r Rr
(17a)
!2e . PP
(17b)
R 1 R w u # (rw v ) Rx r Rr
R 1 R R w # r w Rx r Rr Rr
#l
!2e . FF
(17c)
R R R< R< R; u v #< u v "! u #u v # Rx Rr Rx Rr Rx #v
#l
R R R; R; u #< u "!2 u #u v Rx Rr Rx Rr
R 1 R R v # r v Rx r Rr Rr
!
;
R 1 R v w v u # (rv )!2 Rx r Rr r
R R < w #< w "!2 w !2o\w Rp /(rRh) Rx r Rr
! Before investigating experimental data, we "rst list the Reynolds-stress equations:
R; !o\(u Rp /Rr#v Rp /Rx) Rr
R 1 R u w u v # (ru v )! Rx r Rr r
R R 1 R u v # (ru v Rx Rr r Rr
!e . PV
(17d)
Besides left-hand-side convective terms, we "nd on the right-hand side a "rst bracket gathering production terms. The following terms are the velocity}pressure-gradient correlation and turbulent transport terms. Finally, we "nd the molecular diwusion terms and the last contribution which is due to dissipative terms. If we take the half sum
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
669
of Eqs. (17a)}(17c), we obtain the equation for the turbulent kinetic energy, k,q/2: ;
Rk Rk #< Rx Rr
"! u
R; R; R< R< < #u v # #v #w Rx Rx Rr Rr r
!o\[u Rp /Rx#v Rp /Rr#w Rp /(rRh)]
1 R 1 R ! (qu )# (rqv ) 2 Rx r Rr #l
Rk 1 R Rk r # !e, Rx r Rr Rr
(17e)
where q"u #v #w . The right-hand side of Eq. (17e) involves the production ("rst bracket), the velocity}pressure-gradient correlation and turbulent transport. The two last brackets involve viscous di!usion and the rate of dissipation, e, with e/l"(Ru /Rx)#(Rv /Rx)#(Rw /Rx)#(Ru /Rr) 1 #(Rv /Rr)#(Rw /Rr)# [Ru /Rh)#(Rv /Rh) r 1 #(Rw /Rh)]# (v #w ). r It is interesting to see that there is no explicit (inviscid) inyuence of transverse curvature on production terms except in the transverse normal stress equation (17c). Thus transverse curvature involves a local curvature ewect, through the additional e!ects of R;/Rx and R
Fig. 9. Pressure-strain term in the equation for u ; comparison with plane-channel results of [26] (from [7]).
lower than in a #at-plate boundary layer. The absence of a u plateau, related to the absence of a log law in ;, is associated with a signi"cant decrease of the level of u over most of the boundary layer [2]. In view of Eq. (17a), this is because the reduced velocity defect lowers R;/dr,!u v and axial and shear-stress production terms. This phenomenon does not involve the near-wall region. There, the maximum of u occurs at a constant distance from the wall, y>"13}16, and is about 3.2; , O a high maximum with respect to the #at-plate case [21]. This increase of the u maximum becomes more marked as d/a increases [2]. In contrast, the maximum v is about ; , as for a plane #ow. O For reasons which are not clear, these experimental results are at variance from the DNS data [7] which "nd a weak reduction in the azimuthal and radial turbulent normal stresses in comparison with that for the longitudinal normal stress, and a related signi"cant increase of K "2u /(v #w ). The attenuation of the normal and azimuthal velocity #uctuations is strongest for y>(30 and increases with curvature. This may be because mean shear stress a!ects mainly the axial component, while the pressure strain term in the budget of the axial streamwise intensity decreases signi"cantly, as indicated in Fig. 9. Hence, while transverse curvature inhibits pressurestrain transfer to radial and azimuthal normal stresses, it does not seem to diminish the ezciency of production, nor its location, but it distributes it over a larger volume: the large-scale structures sweeping the cylinder are much larger than the scale of the cylinder. Because the #ow is less constrained by the wall in a boundary layer on a cylinder (with a signi"cant w component) than in a #at-plate boundary layer, turbulent transport of #uid (from one side of the cylinder to the other) is very di!erent from that on a #at plate. The #uid is transported from one side of the cylinder to the opposite side by the large-scale, cross#ow structures. The distributions of the skewness factor for u are mostly negative, indicating
670
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 10. (a) (left) Pro"le of Reynolds stress. Bold, Planar case. Shaded, axially symmetric case, for 4.59)d/a)8.53 (from [1]). (b) (right) Correlation coe$cient. Bold, plane channel [15] (from [7]).
a domination of negative u with respect to the mean, together with a high degree of similarity for small y/h. The overall shape of the Reynolds stress pro"le is quite di!erent from that for a #at plate. The decrease of the Reynolds shear stress is very important in the outer layer (Fig. 10a), mainly because the velocity defect is severely reduced, as already mentioned. In fact, the pro"les resemble those found in a strongly accelerated boundary layer (favorable pressure gradient) on a plane wall. The maximum of u v is a function of d/a; it moves towards the wall with increasing curvature, although the maximum of production of k is not a!ected and remains at about y>"12 [7]. With u reduced and v quite una!ected, the correlation coe$cient, R , remains between 0.4 and 0.5 except near the wall and the outer edge of the boundary layer, its shape is strongly altered for d/a"11 (Fig. 10b), in contrast with d/a"0 or 5 [7]. As a result, the eddy viscosity coe$cient !u v /[d; *;/*r] is reduced R to values of about 0.27 [27]. Although the mechanism of turbulence production is similar to that occurring in the planar case, some details are di!erent and should explain the increase of friction. First, the transverse curvature enhances the perturbation vorticity before the burst. Second, the interaction between the outer #ow and burst events near the wall is
stronger in the case of a cylindrical wall than in the planar boundary layer. Such results are again at variance from DNS [7] where the partition of Reynolds stresses is very similar to that of the plane channel in transversely curved #ows. Also their Reynolds stress is dominated by second quadrant events (u (0, v '0) in the outer layer and by fourth-quadrant events (u '0, v (0) in the inner layer. The crossover between the dominance of second- and fourth-quadrant occurs at y>"12, as in the plane channel. Also the fractional contributions to the Reynolds shear stress are similar to that in the plane channel as is the bursting frequency at the same Reynolds number. Moreover, the mean streak spanwise spacing, estimated from the two-point correlation of u is about 100 viscous units, the plane channel value, and is roughly proportional to r [21]. Finally, the axial integral scale increases with increasing curvature for y>(10 and near-wall streamwise vortices become longer, where the mean shear is largest [7]. The intermittency function is found by Lueptow and Haritonidis [21] to be Reynolds-number independent and well approximated by an error function c(y)" 0.5 erfc[(g!g )/p(2] where g is the average position of the interface and p its standard deviation. The value of g is found at d instead of 0.8d in the planar case, the best "t with data yields p"0.18g . The value of c is nearly one out to y"0.7d compared to y"0.4d in the planar case. Such a larger intermittency level may be attributed to more energetic velocity #uctuations in the outermost part of the axisymmetric boundary layer as compared with a planar one. This might imply that turbulent eddies have a longer life and distribute their energy over a larger region before being dissipated. Also, the cylinder does not constrain the motion of eddies like a planar wall does; thus these eddies can move across the axisymmetric boundary layer `"lling it outa with turbulent eddies. Hence, transport mechanisms are rather di!erent because of this signi"cant cross#ow motion of large structures. Third-order moments appear to be strongly in#uenced by transverse curvature. The skewness factors S and S , S T associated respectively with the streamwise and wall normal velocity #uctuations, are presented in Fig. 10. Away from the wall, S is positive and increases slightly with T increasing curvature [2]. Near the wall, S decreases T sharply with increasing curvature, indicating low levels of intermittency, while it is positive away from the wall. S is S positive near the wall and not signi"cantly in#uenced by curvature. Hence cylindrical boundary layer measurements are more negatively skewed than those of a yat plate throughout the entire boundary layer, as a consequence of spots of low-speed inner #uid which is stripped away from the cylinder surface by large-scale cross#ows. Large negative streamwise velocity (fourth quadrant) #uctuations occur for y>'12 (Fig. 10a). F decreases also S sharply in the near-wall region, but it is not a!ected by curvature for 5(y>(30 (Fig. 10b), while F increases T
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 11. (a) Skewness pro"les S and S (from [7]). (b) (right) S T Flatness pro"les F and F . Italics, values of d/a; bold, plane S T channel [15] (from [7]).
with increasing curvature away from the wall (y>'30) (Fig. 11). Fluctuating vorticity intensities normalized with the mean wall shear stress (they do not collapse when scaled with ;H) decrease with increasing curvature in the nearwall region, with the same trends as for a plane channel; the axial rms vorticity #uctuations, normalized with l/;, exhibits a near-wall maximum (close to the edge of O the sublayer) and a near-wall maximum (about y>"20). Such extrema are only slightly shifted outwards when the curvature is increased, while the maximum of the radial rms vorticity #uctuation, at about y>"15 for a plane channel, is shifted inwards. This indicates that the mean position of streamwise vortex cores is not signi"cantly a!ected by transverse curvature, even though the strength of the vorticity is reduced (but such results might be simply a consequence of the rather weak level of curvature in the low-Reynolds number simulations). The radial vorticity is not in#uenced by curvature in the sublayer region, while the other vorticity components decrease.
6. Conclusions The prototypical problem of the thick axisymmetric boundary layer in axial #ow along a long circular cylin-
671
der is reviewed. Three #ow regimes were identi"ed on the basis of the two parameters that describe this #ow: small d/a and large a>; large d/a and small a>; and large d/a and a>. The "rst regime is the thin boundary layer at typical laboratory Reynolds numbers; the second is the thick boundary layer at low Reynolds numbers; and the third is the thick boundary layer at high Reynolds numbers. Experiments in the "rst regime reveal little e!ect of transverse curvature. Strong e!ects of transverse curvature are found in the second regime but often in combination with low Reynolds number (transitional) e!ects. There is little data in the third regime of thick boundary layers at large Reynolds numbers, which is of interest in applications such as towed cables. This #ow resembles that in an axisymmetric wake with a di!erent inner boundary condition, namely, no-slip on cylinder. This case is also or more fundamental interest insofar as it enables the study of interaction between large unrestricted outer eddies and the smaller wall-bounded eddies in general turbulent #ow. Experimental studies in special facilities along with large-eddy simulations are needed to fully explore this #ow reH gime. Available experiments reveal that transverse curvature e!ects increase skin friction and stabilize the boundary layer #ow. In the outer region, the mean #ow structure shows a signi"cant reduction of the velocity defect so that the wake component of the mean velocity is greatly reduced, as are the wall-scaled turbulent shear stress, and normal stresses to a lesser degree. Velocity distributions in the inner region show characteristic departures from the standard logarithmic law. For small values of a>, stabilization mechanisms are strong enough and relaminarization of the boundary layer is possible. However, "rm conclusions are hampered by measurement uncertainties. Available (low-Reynolds number) direct numerical simulations are rather limited and indicate only moderate di!erences in turbulence structure parameters such as bursting frequency and mean streak spacing, at least for low transverse curvature. References [1] Lueptow RM, Leehey P, Stellinger. The thick turbulent boundary layer on a cylinder: Mean and #uctuating velocities. Phys. Fluids 1985;28:3495}505. [2] Luxton RE, Bull MK, Rajagopalan S. The thick turbulent boundary layer on a long "ne cylinder in axial #ow. Aeronaut J 1984;88:186}99. [3] Willmarth WW, Winkel RE, Sharma LK, Bogar TJ. Axially symmetric turbulent boundary layers on cylinders: Mean velocity pro"les and wall pressure #uctuations. J Fluid Mech 1976;76:35}64. [4] Willmarth WW, Yang CS. Wall-pressure #uctuations beneath turbulent boundary layers on a #at plate and a cylinder. J Fluid Mech 1970;41:47}80. [5] Yu YS. E!ect of transverse curvature on turbulent boundary layer characteristics. J Ship Res 1958;2:33}51.
672
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
[6] Afzal N, Narasimha R. Axisymmetric turbulent boundary layer along a circular cylinder at constant pressure. J Fluid Mech 1976;74:113}28. [7] Neves JC, Moin P, Moser RD. E!ects of convex transverse curvature on wall-bounded turbulence. Part 1. The velocity and vorticity. J Fluid Mech 1994;272:349}81. [8] Patel VC. A uni"ed view of the law of the wall using mixing-length theory. Aeronaut Quart 1973;24:55}70. [9] Rao GNV. The law of the wall in a thick axisymmetric turbulent boundary layer. Trans ASME J Basic Eng Ser D 1967;89:237}8. [10] Denli N, Landweber L. Thick axisymmetric boundary layer on a circular cylinder. J Hydronaut 1979;13:93}104. [11] Afzal N, Singh KP. Measurements in an axisymmetric turbulent boundary layer along a circular cylinder. Aeronaut Quart 1976;27:217}28. [12] Abramowicz M, Stegun IA. Handbook of Mathematical Functions. New York: Dover Publ. Inc., 1972. [13] Willmarth WW. Structure of turbulent boundary layers. Adv Appl Mech 1975;15:159}254. [14] Neves JC, Moin P. E!ects of convex transverse curvature on wall-bounded turbulence. Part 2. The pressure #uctuations. J Fluid Mech 1994;272:383}406. [15] Kim J, Moin P, Moser RD. Turbulence statistics in fullydeveloped channel #ow at low Reynolds number. J Fluid Mech 1987;177:133}66. [16] Rao GNV, Keshavan NR. Axisymmetric turbulent boundary layers in zero pressures gradients. Trans ASME J Appl Mech 1972;39E:25}32. [17] Richmond, RL. Experimental investigation of thick, axially symmetric layers on cylinders at subsonic and
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
hypersonic speeds. Hypersonic Research Proj. Memo. 39, CA Calif. Inst. Tech, Pasadena, 1957. Snarski SR, Lueptow RM. Wall pressure and coherent structures in a turbulent boundary layer on a cylinder in axial #ow. J Fluid Mech 1995;286:137}71. Yasuhara M. Experiments on axisymmetric boundary layers along a cylinder in incompressible #ow. Trans Jpn Soc Aerosp Sci 1959;2:33. Wietrzak A, Lueptow RM. Wall shear stress and velocity in a turbulent axisymmetric boundary layer. J Fluid Mech 1994;259:191}218. Lueptow RM, Haritonidis JH. The structure of the turbulent boundary layer on a cylinder in axial #ow. Phys Fluids 1987;30:2993}3005. Madavan NK, Deutsch S, Merkle CL. Measurements of local skin friction in a microbubble-modi"ed turbulent boundary layer. J Fluid Mech 1985;156:237}56. Mitchell JE, Hanratty TJ. A study of turbulence at a wall using an electrochemical wall shear-stress meter. J Fluid Mech 1966;26:199}221. Sreenivasan KR, Antonia RA. Properties of wall shear stress #uctuations in a turbulent duct #ow. Trans ASME J Appl Mech 1977;44:389}95. Keith WL, Bennett JC. Low frequency spectra of the wall shear stress and wall pressure in a turbulent boundary layer. AIAA J 1991;29:523}30. Mansour NN, Kim J, Moin P. Reynolds stress and dissipation-rate budgets in a turbulent channel #ow. J Fluid Mech 1988;192:363}77. Lueptow RM, Leehey P. The eddy viscosity in a turbulent boundary layer on a cylinder. Phys Fluids 1986;29:4232}3.
Transverse curvature e!ects in turbulent boundary layer J. Piquet *, V.C. Patel Laboratory of Fluid Mechanics, Ecole Centrale de Nantes, UMR 6598 CNRS, 1 Rue de la Noe, B.P. 92101, 44321 Nantes Cedex, France IIHR, University of Iowa, USA Received 30 April 1999
Contents 1. Introduction . . . . 2. Mean velocity data 3. Flow reH gimes . . . 4. Flow structures . . 5. Turbulence data . 6. Conclusions . . . . References . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
661 662 665 667 668 671 671
Abstract The e!ect of transverse surface curvature on the turbulent boundary layer is reviewed by recourse to experiments on axial #ow along a circular cylinder. Three #ow regimes are identi"ed depending on values of the two controlling parameters, namely, the Reynolds number and the ratio of the boundary layer thickness to cylinder radius. The boundary layer #ow resembles a wake when both parameters are large. As expected, the e!ect of curvature is small when the Reynolds number is large and the boundary layer is thin. When the boundary layer is thick and the Reynolds number is small, which is typical of laboratory investigations, the e!ect of transverse curvature is felt throughout the boundary layer with evidence for relaminarization at the low Reynolds numbers. This review describes the experimental evidence and points out gaps that remain. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Transverse curvature e!ects in turbulent boundary layers are associated with curvature of the wall in planes orthogonal to the mean direction of the #ow. The simplest, canonical situation in external #ow is that of a long cylinder, of radius a, immersed in a uniform #ow of velocity ; . Then the no-slip condition implies the pres ence of a boundary layer along the cylinder. This boundary layer is subject to a negligible axial pressure gradient so that transverse curvature e!ects are isolated from other e!ects. Experimental data available in this case are rather limited [1}5], because it is di$cult to align the axis
* Tel.: #33-2-40-37-16-33; fax: #33-2-40-37-25-23. E-mail address: [email protected] (J. Piquet)
of the cylinder with the #ow direction and to keep it straight against elastic deformation. There is however an intrinsic interest in the subject for the description of boundary layer #ows past bodies including towed submerged cables, vehicles, missiles, glass or polymer "bers during fabrication. Also, a knowledge of transverse curvature e!ects is signi"cant in estimating the surface resistance of ship models and of other three-dimensional elongated aerodynamic and hydrodynamic shapes which share a transverse radius of curvature comparable in size with the boundary layer thickness. The data pertaining to the axial #ow along a circular cylinder are reviewed here to study the e!ects of transverse curvature on the boundary layer development. Transverse curvature e!ects that arise in internal #ows, such as circular pipes, are rather limited and are largely excluded from this review.
0376-0421/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 6 - 0 4 2 1 ( 9 9 ) 0 0 0 0 7 - X
662
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Nomenclature x, r a ;, < k, o q u , v , w C D ;,; O C d, h dH
axial and radial coordinates radius of the cylinder axial and radial mean velocity components Dynamic viscosity, density of the #uid total stress (see Eq. (1c)) axial, radial and azimuthal velocity #uctuations skin friction factor (see Eq. (2a)) friction velocity, far-"eld mean velocity boundary layer thickness, momentum thickness displacement thickness
B , B
Inner and outer intercept of the logarithmic law i Von Karman constant in the logarithmic law l mixing length (see Eq. (4)) l eddy viscosity coe$cient 2 k turbulent kinetic energy (see Eq. (17e)) e turbulent dissipation (see Eq. (17e)) S skewness factor F #atness factor w (index) refers to the wall boundary #(superscript) refers to near-wall conditions
The momentum integral obtained from the above set of equations is ; dh C , O " , ; dx where the momentum thickness is de"ned by
; r ?>B ; 1! dr, (2b) ; ; a ? C is the skin friction coe$cient, ; "(q /o) is the O friction velocity, d is the boundary layer thickness (de"ned as the distance from the wall where ;"0.995; ) which, like h, increases slowly with x. The continuity equation yields similarly h"
Fig. 1. Schematic sketch of the problem.
2. Mean velocity data
The boundary layer approximation for the mean motion of an incompressible #ow along the exterior surface of a circular cylinder can be written as
d < dd* , " 1# dx a ; where
R R (r;)# (r<),0, Rx Rr
d*"
;
R; R; 1 R #< " (rq), Rx Rr or Rr
(1a)
(1b)
where x and r are the axial and radial coordinates (the cylinder surface is r"a), as indicated in Fig. 1, ; and < are the axial and radial mean velocity components, respectively, q is the total stress (molecular#turbulent): q"k
R; !ou v Rr
(2a)
(1c)
and o and k are the density and viscosity of the #uid.
?>B
; r 1! dr ; a ? is the displacement thickness which also increases slowly with x, so that the entrainment velocity, < , at the edge of the boundary layer is positive. The momentum thickness allows de"nition of the small parameter e"O(h/d), which is a function of x and represents a momentum defect much less than unity for a><1, where the index # denotes the wall scaling using l and ; . This parameter is shown by Afzal and O Narasimha [6] to be "xed by the ratio ; /; . These O authors established the limits of validity of the logarithmic law and indicated that the inner intercept, B , in this law, B ";>!i\ ln y>
(3a)
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
663
tively, we use the classical form l>"iy>, we "nd [16]
(1#y>/a>) !1 1 #B , ;>" ln 4a> ? i (1#y>/a>) #1
Fig. 2. X, WY, Willmarth and Yang [4]; O, RK, Rao and Keshavan [16]; R, Richmond [17] (from [6]).
is a function of 1/a>, while the outer intercept, B, in the velocity-defect layer,
; !; y #i\ ln (3b) B" d ; O is a decreasing function of d/a. This latter result is con"rmed by Fig. 2. Thus, if a><1, transverse curvature mainly shifts the log law. We may now try to estimate the level of the shift in terms of a. A plausible estimation of the velocity pro"le is obtained by Patel [7] from mixing length theory which states that, in the wall region
!u v "l
R; . Ry
W>
2q> dy
. 1#(1#4(l>)q>
(7c)
If a continuous mixing length distribution from the sublayer to the log region is prescribed, Eq. (7c) may be replaced by a velocity pro"le connecting (7a) and (7b) by quadrature [8]. Denli and Landweber [10] propose
y> l>"ia> ln 1# (tanh[j(y>)]. a>
(8)
(5)
1 1 ;>" E (ln(1#y>/a>)# ln(2a>)!c#B (a>) i G i
(6)
Eq. (6) implies that for su$ciently small values of a>, the stress falls monotonically to very low levels from its value at the wall, a situation similar to that occuring in a twodimensional accelerated boundary layer. This analogy will be exploited further. It then remains to "x l. If we integrate Eq. (5) while neglecting the turbulent shear stress with respect to the viscous stress (l"0), we obtain a logarithmic velocity distribution of the form ;>"a> ln(1#y>/a>),
1 ;>" ln[a> ln(1#y>/a>)]#B . i
(4)
A velocity pro"le for the wall region is obtained if distributions of q(y) and l(y) are given or assumed. If we neglect the acceleration terms in Eq. (1b), we "nd a non-dimensional stress distribution throughout the wall region under the form rq"aq "const. or a> q>" . a>#y>
where B is a function of a# reducing to B (+5.45) as ? a>PR. The departure of the ln argument in [2] of Eq. (7b) from a> ln(1#y>/a>) is rather small. Patel [8] indicates less than 1% for y>"a> and about 8% for y>"6a>. Rao's form [9] (7c) yields a result very close to (7b), namely
Using again Eq. (6), the resulting expression (5) for ;> may be solved numerically. For y>'y>, where y> is the value of y> at which the viscous stress becomes negligible in comparison with the Reynolds stress, the velocity is a function of the logarithmic integral E (ln z): G
Substitution of Eq. (4) into Eq. (1c) gives ;>"
(7b)
(7a)
a necessary byproduct of the constant shearing force per unit length, 2prkR;/Rr, upon the cylinder. If alterna-
1 + ln a# i
>> y> 1# ln 1# a> a>
#B (a>),
(9)
where c is the Euler constant 0.5772156, while [11] B (a>)"5.45!41.76/a>. In the approximate form (9), the square root term is obtained from the dominant term in the in"nite series of E . When neglected, the expression G becomes similar to (7c). The argument [2] for the logarithmic term in Eq. (9) is very close to y> when y>)a> [10], justifying the use of the Preston tube technique with the usual calibration curve for the measurement of the wall shear stress. Denli and Landweber [10] have shown that it was possible to retain acceleration e!ects in Eq. (1b), resulting in a more complete law of the wall, ;>"f (y>, a>). Then the key assumption is that the dependence of ;> on x through a> can be neglected as compared with the dependence on y>. The total shear stress can be obtained as a modi"ed form of Eq. (6): a> q>" [1!c(y>, a>, p )], a>#y>
(10a)
664
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
be sustained. This implies that Eq. (7a) is valid. Close enough to the wall, this law degenerates into the linear sublayer pro"le. To "nd a signi"cant departure from the linear sublayer pro"le would require a>"O(1). Another related in#uence of transverse curvature lies in the collapse of the velocity defect, as shown in Fig. 3b, where ; is evaluated at y"0.99d. It is clear that the very full velocity proxle is a direct result of the cylindrical geometry of the #ow since there is no acceleration of the free stream. It is thus plausible to admit a Clauser constant-eddy-viscosity estimate in the wake-like part of the boundary layer. For d/a'20, the velocity distribution exhibits similarity based on outer variables, suggesting that the boundary layer is controlled by the outer #ow. Denli and Landweber [10] have formulated a velocitydefect law starting from the linearized (wake-type) mean#ow equation. Substituting into it: x , ; !;"Re\ ; ; (m) f (g) with m" ? Re a ? r ;a Re " , g" ? l d#a
(11)
and using an eddy viscosity closure, q"!ol (m)R;/Rr, 2 they obtain for f Fig. 3. (a) In#uence of transverse curvature on the velocity pro"les. **, (bold) #at plate, law of the wall. } } }, log law (7b) for a>"33.4. 䉭, d/a"4.7; 䊐, d/a"16; *, d/a"37.5. Smallest value of a>"33.4 (from [3]). (b) Velocity-defect pro"les. **, (bold) #at plate, Coles law. £, d/a"1.8; 䉭, d/a"4.7; 䊐, d/a"16; *, d/a"37.5 (from [3]).
where the c-term accounts for inertia terms: c(y>, a>, p )"p
W>
yf (y) f (y) f (y)# dy a>
(10b)
1 + p (y>), (10c) 3 where a Taylor series expansion of c gives the "rst order correction (10c). This correction is rather weak in the wall region because a typical value of p is about 6;10\: at y>"30, the error introduced by the approximation (6) is about 5.4%. Fig. 3a presents the evolution of velocity pro"les in wall coordinates and shows how the law of the wall is a!ected by transverse curvature. Since the skin friction factor decreases signi"cantly as the cylinder diameter is reduced, the scaling of both the velocity and the distance from the wall is highly dependent on the friction velocity. Data of Fig. 3a and of [1], as well as DNS [7] indicate that, as the diameter of the cylinder is decreased, the extent of the region where the transverse curvature in#uences the velocity pro"le increases. For a small enough radius, i.e. a> small, the erosion of the logarithmic layer is made possible by the fact that turbulence e!ects cannot
1 ! gf
R dt/dm gf# 2# t dR/dm
g f dg E v (m) a#d ; m "2 2 , R" , t" . v R dR/dm a (2; Re O ? (12)
Self-similarity then reduces Eq. (12) to an ordinary di!erential equation in g if R dt/dm 2# "2b or t"t R@\ t dR/dm and m v (m) "c. 2 2 v R dR/dm
(13)
Eq. (12) is then solved with [10] l (m)/l"DmK\. Once 2 the evolutions of R and t are expressed in terms of b and m, it remains to solve the ordinary di!erential equation. Analytical solutions may be found in terms of the hypergeometric con#uent (Kummer) function [12]. Retaining the function which decreases exponentially as gPR, it is found that f (g)"e\X;(b, l, z) with z"g/(2c), while (1') b'0 (otherwise d would not increase with d). Denli and Landweber [10] "t all undetermined constants (only) with experimental data [23]: b"0.35, c"1/2, m"7/13. The resulting velocity-defect
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
law does not match the logarithmic law of the wall because small velocity gradients and radial velocity have been assumed, as g becomes small. This appears to be of no serious consequence, the agreement with data remaining good where both expressions tend to fail simultaneously, about y/(d#a)"0.125. The fuller velocity pro"les (Fig. 3b) are indicative of a more quiescent outer #ow when d/a is high, as compared with plane boundary layer #ow. Willmarth [13] has explained cyclic occurrence of bursts from the massaging action of the large eddies passing over the sublayer which produce, during random periods of time, an unstable in#exional velocity pro"le in localized regions near the wall. Bursts occur in these regions of locally unstable in#exional pro"les. On a cylinder, the structure of turbulence is at a uniformly reduced scale [4]. Hence, there is no reduction in the time scale of the massaging action of the large eddies on the sublayer relative to the characteristic time scale in the sublayer. The reduction of the size of turbulent eddies relative to those of a #at plate is determined by two e!ects. The "rst e!ect is the &fullness' of the pro"le which causes eddies at a given convection speed to be of reduced size, because a given mean speed is attained at a smaller distance from the wall. The second e!ect is the limited lateral extent of the axisymmetric boundary layer which results in lateral shearing action by the freestream on the larger eddies when they extend or move laterally. As d/a increases, the large-scale surface pressure #uctuations are correlated over a greater fraction of the cylinder circumference; the outer perimeter of the large eddies being very large compared to the inner perimeter. Hence for large d/a, the structure and position of large eddies do not depend on the presence of the wall, in contrast to the yat plate case where the large eddies are completely bounded on one side by the wall.
3. Flow reH gimes There are two length parameters in the transversecurvature problem, namely d/a, the ratio of the boundary layer thickness to the radius of the cylinder and a>. Fig. 4 presents available experiments in the plane of these parameters. While it would be of interest to determine the separate e!ects of these two parameters at a given Reynolds number, this is not in general possible because the wall friction coe$cient and the normalized free stream velocity ; /; have a very low sensitivity to O changes in the Reynolds number. This implies [2] a roughly linear variation of a>"Re ; /; with Re , ? O ? well correlated by ; /; "0.11 Re\ . Of course, this O ? correlation is limited to values of a> such that the #ow is fully turbulent. Using an argument [14] borrowed from accelerated boundary layers for which relaminarization occurs when the non-dimensional stress gradient, K ("lRq/o ;Ry) approaches approximately !0.009, O O
665
Fig. 4. Curvature parameter d/a and a> in axial turbulent #ow along a cylinder. 䉭, Luxton et al. [2]; #, Lueptow and Haritonidis [21] and Lueptow et al. [1]; ;, Willmarth et al. [3] and Willmarth and Yang [4]; 䊐, Rao and Keshavan [16]; *, Afzal and Narasimha [6]; **, Richmond (17); *, Neves et al. [7] (from [7]).
relaminarization should occur for a>(28, with transition for 28(a>(106. The threshold value of 28 is obtained if we compute the stress gradient at the edge of the blending region and set it equal to K . The criterion for O relaminarization suggests that some of the measurements indicated in Fig. 4 may not be in fully turbulent yow. Also, Lueptow and Haritonidis [15] have shown that C was an increasing function of transverse curvature and thus was higher than that for a plate boundary layer at the same Reynolds number, R . C was also a slightly F decreasing function of the Reynolds number based on x. This means that the cylindrical wall is more e!ective in converting mean #ow energy into turbulent energy than a planar wall. This di!erence may perhaps be attributed to the spanwise vorticity which is intensi"ed by the sweep, while being altered by the transverse curvature. The transverse curvature may then enhance turbulence generation during the burst so that an increase in friction results. Another reason could be that the large-scale outer structures may pass closer from the cylinder wall than on a #at-plate boundary layer. There are two major technical di$culties in determining the wall shear stress. First, it is necessary to ensure axial symmetry; if not, the determination of ; from the O momentum integral relation (2) would be inaccurate. This is probably the case in experimental data [16]. Second, there is the question of the status of the law of the wall. If axial symmetry is not guaranteed, experimental data in the bu!er zone may be "tted to the #at-plate law of the wall if there is only a gradual departure from it. This is the Clauser method which gives a good approximation to the actual wall shear stress (but may overestimate friction velocities, as in [3] since they "t their experimental data with Coles' law of the wall for a planar boundary layer). Otherwise, a law of the wall accounting for transverse curvature e!ects is required. Three #ow reH gimes may be distinguished: (i) The re& gime where d/a and a> are large. This case applies to a long slender cylinder such that the boundary
666
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
layer is almost all a wake-like #ow. Unfortunately, data with large d/a were restricted to low values of a> such that transitional e!ects are suspected (see low Re -data of Richmond [17] and Rao and Keshavan ? [16] where 4.1(d/a)11.6, Luxton et al. [2] where 9)a> )47 and 26)d/a)42, although Willmarth et al. [3] where 2(d/a(42, a>+33 show no evidence of relaminarization). This reH gime has been studied by Lueptow et al. [1] who have shown the validity of the logarithmic near-wall behavior (7a) for y>(10, approximately. The sublayer however does not depend on transverse curvature. The e!ect of curvature is primarily felt in the outer layer, through a probable e!ect on the viscous superlayer. Transverse curvature acts to lower turbulence intensities away from the wall (y>'20) although u is sometimes reported to be 10% higher than in the #at-plate case (see e.g. [18]). Large-scale structures sweep past the cylinder and they have a strong e!ect on the #ow. (ii) The re& gime where a> is large and d/a is small, or of order one, corresponds to the case of a large cylinder with a small e!ect of transverse curvature. In this case the logarithmic law is still valid [6], with asymptotic behaviors for i and the intercept (namely, B "5#236/a>) which are constant if d/a"O(1), so that the boundary layer may be regarded as planar when d/a)1. Most available experimental data fall on the boundary of this reH gime however (like those of Yasuhara [19] where d/a+0.6; Willmarth and Yang [4] where 1.8)d/a)16; Afzal and Singh [11] where 0.6)d/a)2; Wietrzak and Lueptow [20] where d/a"5.7 and a>"30), as do the DNS simulations of Neves et al. [7] at d/a"5 and a>"43 and d/a"11, a>"21. The calculations [7] have been performed at exceedingly low Reynolds numbers (Re "674 and 311 for d/a"5 and 11, respectively). ? Moreover, they consider the #ow between concentric cylinders, with a zero-shear stress condition applied at the outer cylinder located at r"d. Thus a mean streamwise pressure gradient is necessary to drive the #ow (the mass #ux through any crosssection is constant) and the #ow is treated as homogeneous in the axial direction with the following mean total stress:
a y q"o; 1# O a#y 2a#d
y d
1!
(14)
instead of q"o;(1!y/d) in the plane-channel case. O Comparison of this expression with Eq. (14) suggests the following velocity scale:
a y ;H"; 1# O a#y 2a#d
.
(15)
However, such a scaling should be considered valid only in the near-wall region: the calculations performed by Neves et al. using this parallel-#ow assumption cannot properly describe the outer region of the boundary layer and of its outer boundary. This is because the axial
Fig. 5. Schematic sketch of the boundary layer pro"le.
velocity gradient generates a mean radial velocity component in any boundary layer, an e!ect that can be neglected only within the wall region. (iii) The re& gime where d/a large and a> small is similar to that of an axisymmetric wake with an inner layer (Fig. 5). In this case, there are extreme di$culties in measurements. (1) An accurate alignment of the cylinder in the tunnel is critical, especially for long cylinders, because even a 13-misalignment of the cylinder to the mean #ow produces a spectacular loss of axial symmetry in the measurements [16], with an accuracy of at most $10}15% in shear stress. (2) This case requires such a long cylinder compared with its diameter that #ow non uniformities or free stream disturbances become important. (3) The total shear force on the #uid at the cylinder surface becomes vanishingly small and di!usion does not dominate or control the #ow: a large eddy passing from one side of the cylinder to the opposite side would not `seea the cylinder. (4) The initial development of the boundary layer is likely to be in#uenced by the shape of the nose wake but the boundary layer should be rather independent of it for x/a*5000, downstream of the nose. (5) Other problems of major concern include structural isolation of the cylinder, possible aeroelastic interaction with the #ow, and cylinder sag. Concerning the #ow in the wall region, there is no consensus on the e!ect of transverse curvature on the logarithmic law. From Fig. 3, it is evident that the entire velocity pro"le is directly or indirectly a!ected by curvature. Eq. (7b) derived from simple mixing length arguments is known to adequately describe the departures from standard logarithmic law, Eq. (3a) with due consideration for the accuracy of the wall shear stress. Some authors have, on the other hand, retained Eq. (3a) and allowed the constants to vary from their #at-plate values. For example, Lueptow et al. [1] increase i to unity for d/a'30 and let B take values as high as 11}13. When analyzed by overlap arguments, the e!ect of curvature on the outer #ow is a!ected by what model is adopted for the inner layer. Since the inner, near-wall log behavior yields: ;>+y>!y>/2a>, curvature e!ects become signi"cant if a> is su$ciently small, i.e. less than 50. The skin friction is increased with
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 6. Contours of axial streamwise vorticity #uctuations, normalized by ; and l. d/a"11, a>"21. Bold contours denote O negative vorticity (from [7]).
respect to the #at-plate case, in spite of strong measurement uncertainties. This implies that the cylindrical wall is more e!ective that the planar wall in converting mean #ow energy into turbulence energy.
4. Flow structures An instantaneous view of longitudinal vorticity contours obtained from DNS [7] is presented in Fig. 6. The #ow in the outer region has large structures wrapping around the cylinder alternating with regions of quiescent #ow. The regions of signi"cant vorticity are seen to penetrate to about 10 radii into the quiescent #ow region. The corresponding transverse velocity correlation is far lower than in a #at-plate boundary layer, as is the shear stress correlation, indicating a tendency towards stabilization. Also the ratio between the transverse length scale and the cylinder radius is large. To investigate how the shear stress is in#uenced by low transverse curvature e!ects, the power spectral density
( f ) of the wall shear stress #uctuations was considered by Wietrzak and Lueptow [20]. Root mean square values of the shear stress #uctuations, q , then result from
q "
f ( f ) d(ln f ).
(16)
In this form, the energy between any two frequencies is proportional to the area under the curve bounded by these frequencies. Fig. 7 shows the normalized psd using inner variables. The most prominent feature of the spectrum is the higher energy content of planar bounded #ows and pipe #ows at lower frequencies compared to that of a boundary layer on a cylinder. In the latter case [21], the frequency band of the streamwise velocity spectra where the maximum occurs is centered at
667
Fig. 7. Normalized power spectral density. Bold, Wietrzak and Lueptow [20]; cylindrical boundary layer d/a"5.7; R "3050 F (1); M, Madavan et al. [22] planar boundary layer R "10 630 F (1.19); H, Mitchell and Hanratty [23] pipe, Re "22 900 (1.06); S, Sreenivasan and Antonia [24] channel #ow Re "11 780 " (0.92); K, Keith and Bennett [25] planar boundary layer R "8200 (0.88). Italic numbers give the area under each curve F (from [20]).
fl/;+10\ or fd/; +0.4, suggesting that the strucR tures responsible for most of the shear stress #uctuations are the large-scale ones. The reduction of energy at low frequencies in the wall shear-stress spectrum is similar to that for the wall pressure spectrum [4], so that the convection velocity for the pressure producing eddies is independent of transverse curvature, whereas the velocity pro"le is fuller in a cylindrical boundary layer, implying a faster motion of eddies. To maintain the same convection velocity, the pressure-producing eddies must be smaller in a cylindrical boundary layer. A similar e!ect could reduce the #uctuations of wall shear stress at lower frequencies. A VITA technique was used by Wietrzak and Lueptow [20] to detect positive events, i.e. events such that Ru /Rt'0. The conditional average of positive events (ensemble average of events exceeding the threshold level) indicated no signi"cant di!erence with channel #ow but the averaging time ¹ at which the maximum number of positive events was detected in q was 10% larger in the duct-#ow case than in the cylindrical boundary layer. Hence, turbulent events detected at the wall of a cylindrical boundary layer occur more frequently (and are of a shorter time scale, as suggested by higher frequencies in Fig. 7) than in planar wall-bounded #ows. However, the burst-sweep cycle is not substantially altered by transverse curvature, in agreement with results [21]. A second signi"cant e!ect of transverse curvature is the appearance of a double hump in the cross correlations of u and q when probes are o!set in the transverse direction by one-half streak spacing. This fact could be the footprint of an arrowhead structure or of a structure with an inclined edge (with respect to the cylinder axis) possibly initiated during the burst-sweep cycle, as
668
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 8. (Left) Conceptual model of large-scale boundary layer structure based on correlations between the #uctuating wall pressure and streamwise velocities in frame of reference with mean velocity removed. (Right) Iso-correlation contours in the (x, y)-plane (from [18]).
a probable consequence of the #uid washing over the cylinder. Snarski and Lueptow [18] measured cross correlations between the wall pressure and streamwise velocity for zero time delay, i.e.
;
R R R< R< v #< v "!2 u v #v Rx Rr Rx Rr
!2o\v Rp /Rr!! o (x, y),p (t, 0, 0)u (t, x, y)/p u . NS Isocontours, which then represent a &snapshot' of the averaged eddy structure, show a series of negative contours (!0.07 to !0.01) at an angle of approximately 183 to the wall and a band of positive correlation that extends out from the wall past the edge of the boundary at an angle of approximately 453. While positive correlations are often associated with low-speed #uid (u (0), Snarski and Lueptow [18] seem to favor the association of positive correlation with high-speed #uid. Then the isocontours are consistent with the conceptual view (in a frame moving with the convection velocity, +0.83; ) of a large inclined vortex (about 1.5d average height) that would rotate in the direction of the mean shear with a trailing face at an angle of 453 to the wall and a leading face at an angle about 183 to the wall. Lines at these two angles would then correspond to the locii of maximum positive and negative u (Fig. 8).
5. Turbulence data
#l
;
;
R 1 R !2o\u Rp /Rx! u # (ru v ) Rx r Rr
#l
1 R R u # (ru ) !2e . VV Rx r Rr
(17a)
!2e . PP
(17b)
R 1 R w u # (rw v ) Rx r Rr
R 1 R R w # r w Rx r Rr Rr
#l
!2e . FF
(17c)
R R R< R< R; u v #< u v "! u #u v # Rx Rr Rx Rr Rx #v
#l
R R R; R; u #< u "!2 u #u v Rx Rr Rx Rr
R 1 R R v # r v Rx r Rr Rr
!
;
R 1 R v w v u # (rv )!2 Rx r Rr r
R R < w #< w "!2 w !2o\w Rp /(rRh) Rx r Rr
! Before investigating experimental data, we "rst list the Reynolds-stress equations:
R; !o\(u Rp /Rr#v Rp /Rx) Rr
R 1 R u w u v # (ru v )! Rx r Rr r
R R 1 R u v # (ru v Rx Rr r Rr
!e . PV
(17d)
Besides left-hand-side convective terms, we "nd on the right-hand side a "rst bracket gathering production terms. The following terms are the velocity}pressure-gradient correlation and turbulent transport terms. Finally, we "nd the molecular diwusion terms and the last contribution which is due to dissipative terms. If we take the half sum
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
669
of Eqs. (17a)}(17c), we obtain the equation for the turbulent kinetic energy, k,q/2: ;
Rk Rk #< Rx Rr
"! u
R; R; R< R< < #u v # #v #w Rx Rx Rr Rr r
!o\[u Rp /Rx#v Rp /Rr#w Rp /(rRh)]
1 R 1 R ! (qu )# (rqv ) 2 Rx r Rr #l
Rk 1 R Rk r # !e, Rx r Rr Rr
(17e)
where q"u #v #w . The right-hand side of Eq. (17e) involves the production ("rst bracket), the velocity}pressure-gradient correlation and turbulent transport. The two last brackets involve viscous di!usion and the rate of dissipation, e, with e/l"(Ru /Rx)#(Rv /Rx)#(Rw /Rx)#(Ru /Rr) 1 #(Rv /Rr)#(Rw /Rr)# [Ru /Rh)#(Rv /Rh) r 1 #(Rw /Rh)]# (v #w ). r It is interesting to see that there is no explicit (inviscid) inyuence of transverse curvature on production terms except in the transverse normal stress equation (17c). Thus transverse curvature involves a local curvature ewect, through the additional e!ects of R;/Rx and R
Fig. 9. Pressure-strain term in the equation for u ; comparison with plane-channel results of [26] (from [7]).
lower than in a #at-plate boundary layer. The absence of a u plateau, related to the absence of a log law in ;, is associated with a signi"cant decrease of the level of u over most of the boundary layer [2]. In view of Eq. (17a), this is because the reduced velocity defect lowers R;/dr,!u v and axial and shear-stress production terms. This phenomenon does not involve the near-wall region. There, the maximum of u occurs at a constant distance from the wall, y>"13}16, and is about 3.2; , O a high maximum with respect to the #at-plate case [21]. This increase of the u maximum becomes more marked as d/a increases [2]. In contrast, the maximum v is about ; , as for a plane #ow. O For reasons which are not clear, these experimental results are at variance from the DNS data [7] which "nd a weak reduction in the azimuthal and radial turbulent normal stresses in comparison with that for the longitudinal normal stress, and a related signi"cant increase of K "2u /(v #w ). The attenuation of the normal and azimuthal velocity #uctuations is strongest for y>(30 and increases with curvature. This may be because mean shear stress a!ects mainly the axial component, while the pressure strain term in the budget of the axial streamwise intensity decreases signi"cantly, as indicated in Fig. 9. Hence, while transverse curvature inhibits pressurestrain transfer to radial and azimuthal normal stresses, it does not seem to diminish the ezciency of production, nor its location, but it distributes it over a larger volume: the large-scale structures sweeping the cylinder are much larger than the scale of the cylinder. Because the #ow is less constrained by the wall in a boundary layer on a cylinder (with a signi"cant w component) than in a #at-plate boundary layer, turbulent transport of #uid (from one side of the cylinder to the other) is very di!erent from that on a #at plate. The #uid is transported from one side of the cylinder to the opposite side by the large-scale, cross#ow structures. The distributions of the skewness factor for u are mostly negative, indicating
670
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 10. (a) (left) Pro"le of Reynolds stress. Bold, Planar case. Shaded, axially symmetric case, for 4.59)d/a)8.53 (from [1]). (b) (right) Correlation coe$cient. Bold, plane channel [15] (from [7]).
a domination of negative u with respect to the mean, together with a high degree of similarity for small y/h. The overall shape of the Reynolds stress pro"le is quite di!erent from that for a #at plate. The decrease of the Reynolds shear stress is very important in the outer layer (Fig. 10a), mainly because the velocity defect is severely reduced, as already mentioned. In fact, the pro"les resemble those found in a strongly accelerated boundary layer (favorable pressure gradient) on a plane wall. The maximum of u v is a function of d/a; it moves towards the wall with increasing curvature, although the maximum of production of k is not a!ected and remains at about y>"12 [7]. With u reduced and v quite una!ected, the correlation coe$cient, R , remains between 0.4 and 0.5 except near the wall and the outer edge of the boundary layer, its shape is strongly altered for d/a"11 (Fig. 10b), in contrast with d/a"0 or 5 [7]. As a result, the eddy viscosity coe$cient !u v /[d; *;/*r] is reduced R to values of about 0.27 [27]. Although the mechanism of turbulence production is similar to that occurring in the planar case, some details are di!erent and should explain the increase of friction. First, the transverse curvature enhances the perturbation vorticity before the burst. Second, the interaction between the outer #ow and burst events near the wall is
stronger in the case of a cylindrical wall than in the planar boundary layer. Such results are again at variance from DNS [7] where the partition of Reynolds stresses is very similar to that of the plane channel in transversely curved #ows. Also their Reynolds stress is dominated by second quadrant events (u (0, v '0) in the outer layer and by fourth-quadrant events (u '0, v (0) in the inner layer. The crossover between the dominance of second- and fourth-quadrant occurs at y>"12, as in the plane channel. Also the fractional contributions to the Reynolds shear stress are similar to that in the plane channel as is the bursting frequency at the same Reynolds number. Moreover, the mean streak spanwise spacing, estimated from the two-point correlation of u is about 100 viscous units, the plane channel value, and is roughly proportional to r [21]. Finally, the axial integral scale increases with increasing curvature for y>(10 and near-wall streamwise vortices become longer, where the mean shear is largest [7]. The intermittency function is found by Lueptow and Haritonidis [21] to be Reynolds-number independent and well approximated by an error function c(y)" 0.5 erfc[(g!g )/p(2] where g is the average position of the interface and p its standard deviation. The value of g is found at d instead of 0.8d in the planar case, the best "t with data yields p"0.18g . The value of c is nearly one out to y"0.7d compared to y"0.4d in the planar case. Such a larger intermittency level may be attributed to more energetic velocity #uctuations in the outermost part of the axisymmetric boundary layer as compared with a planar one. This might imply that turbulent eddies have a longer life and distribute their energy over a larger region before being dissipated. Also, the cylinder does not constrain the motion of eddies like a planar wall does; thus these eddies can move across the axisymmetric boundary layer `"lling it outa with turbulent eddies. Hence, transport mechanisms are rather di!erent because of this signi"cant cross#ow motion of large structures. Third-order moments appear to be strongly in#uenced by transverse curvature. The skewness factors S and S , S T associated respectively with the streamwise and wall normal velocity #uctuations, are presented in Fig. 10. Away from the wall, S is positive and increases slightly with T increasing curvature [2]. Near the wall, S decreases T sharply with increasing curvature, indicating low levels of intermittency, while it is positive away from the wall. S is S positive near the wall and not signi"cantly in#uenced by curvature. Hence cylindrical boundary layer measurements are more negatively skewed than those of a yat plate throughout the entire boundary layer, as a consequence of spots of low-speed inner #uid which is stripped away from the cylinder surface by large-scale cross#ows. Large negative streamwise velocity (fourth quadrant) #uctuations occur for y>'12 (Fig. 10a). F decreases also S sharply in the near-wall region, but it is not a!ected by curvature for 5(y>(30 (Fig. 10b), while F increases T
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
Fig. 11. (a) Skewness pro"les S and S (from [7]). (b) (right) S T Flatness pro"les F and F . Italics, values of d/a; bold, plane S T channel [15] (from [7]).
with increasing curvature away from the wall (y>'30) (Fig. 11). Fluctuating vorticity intensities normalized with the mean wall shear stress (they do not collapse when scaled with ;H) decrease with increasing curvature in the nearwall region, with the same trends as for a plane channel; the axial rms vorticity #uctuations, normalized with l/;, exhibits a near-wall maximum (close to the edge of O the sublayer) and a near-wall maximum (about y>"20). Such extrema are only slightly shifted outwards when the curvature is increased, while the maximum of the radial rms vorticity #uctuation, at about y>"15 for a plane channel, is shifted inwards. This indicates that the mean position of streamwise vortex cores is not signi"cantly a!ected by transverse curvature, even though the strength of the vorticity is reduced (but such results might be simply a consequence of the rather weak level of curvature in the low-Reynolds number simulations). The radial vorticity is not in#uenced by curvature in the sublayer region, while the other vorticity components decrease.
6. Conclusions The prototypical problem of the thick axisymmetric boundary layer in axial #ow along a long circular cylin-
671
der is reviewed. Three #ow regimes were identi"ed on the basis of the two parameters that describe this #ow: small d/a and large a>; large d/a and small a>; and large d/a and a>. The "rst regime is the thin boundary layer at typical laboratory Reynolds numbers; the second is the thick boundary layer at low Reynolds numbers; and the third is the thick boundary layer at high Reynolds numbers. Experiments in the "rst regime reveal little e!ect of transverse curvature. Strong e!ects of transverse curvature are found in the second regime but often in combination with low Reynolds number (transitional) e!ects. There is little data in the third regime of thick boundary layers at large Reynolds numbers, which is of interest in applications such as towed cables. This #ow resembles that in an axisymmetric wake with a di!erent inner boundary condition, namely, no-slip on cylinder. This case is also or more fundamental interest insofar as it enables the study of interaction between large unrestricted outer eddies and the smaller wall-bounded eddies in general turbulent #ow. Experimental studies in special facilities along with large-eddy simulations are needed to fully explore this #ow reH gime. Available experiments reveal that transverse curvature e!ects increase skin friction and stabilize the boundary layer #ow. In the outer region, the mean #ow structure shows a signi"cant reduction of the velocity defect so that the wake component of the mean velocity is greatly reduced, as are the wall-scaled turbulent shear stress, and normal stresses to a lesser degree. Velocity distributions in the inner region show characteristic departures from the standard logarithmic law. For small values of a>, stabilization mechanisms are strong enough and relaminarization of the boundary layer is possible. However, "rm conclusions are hampered by measurement uncertainties. Available (low-Reynolds number) direct numerical simulations are rather limited and indicate only moderate di!erences in turbulence structure parameters such as bursting frequency and mean streak spacing, at least for low transverse curvature. References [1] Lueptow RM, Leehey P, Stellinger. The thick turbulent boundary layer on a cylinder: Mean and #uctuating velocities. Phys. Fluids 1985;28:3495}505. [2] Luxton RE, Bull MK, Rajagopalan S. The thick turbulent boundary layer on a long "ne cylinder in axial #ow. Aeronaut J 1984;88:186}99. [3] Willmarth WW, Winkel RE, Sharma LK, Bogar TJ. Axially symmetric turbulent boundary layers on cylinders: Mean velocity pro"les and wall pressure #uctuations. J Fluid Mech 1976;76:35}64. [4] Willmarth WW, Yang CS. Wall-pressure #uctuations beneath turbulent boundary layers on a #at plate and a cylinder. J Fluid Mech 1970;41:47}80. [5] Yu YS. E!ect of transverse curvature on turbulent boundary layer characteristics. J Ship Res 1958;2:33}51.
672
J. Piquet, V.C. Patel / Progress in Aerospace Sciences 35 (1999) 661}672
[6] Afzal N, Narasimha R. Axisymmetric turbulent boundary layer along a circular cylinder at constant pressure. J Fluid Mech 1976;74:113}28. [7] Neves JC, Moin P, Moser RD. E!ects of convex transverse curvature on wall-bounded turbulence. Part 1. The velocity and vorticity. J Fluid Mech 1994;272:349}81. [8] Patel VC. A uni"ed view of the law of the wall using mixing-length theory. Aeronaut Quart 1973;24:55}70. [9] Rao GNV. The law of the wall in a thick axisymmetric turbulent boundary layer. Trans ASME J Basic Eng Ser D 1967;89:237}8. [10] Denli N, Landweber L. Thick axisymmetric boundary layer on a circular cylinder. J Hydronaut 1979;13:93}104. [11] Afzal N, Singh KP. Measurements in an axisymmetric turbulent boundary layer along a circular cylinder. Aeronaut Quart 1976;27:217}28. [12] Abramowicz M, Stegun IA. Handbook of Mathematical Functions. New York: Dover Publ. Inc., 1972. [13] Willmarth WW. Structure of turbulent boundary layers. Adv Appl Mech 1975;15:159}254. [14] Neves JC, Moin P. E!ects of convex transverse curvature on wall-bounded turbulence. Part 2. The pressure #uctuations. J Fluid Mech 1994;272:383}406. [15] Kim J, Moin P, Moser RD. Turbulence statistics in fullydeveloped channel #ow at low Reynolds number. J Fluid Mech 1987;177:133}66. [16] Rao GNV, Keshavan NR. Axisymmetric turbulent boundary layers in zero pressures gradients. Trans ASME J Appl Mech 1972;39E:25}32. [17] Richmond, RL. Experimental investigation of thick, axially symmetric layers on cylinders at subsonic and
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
hypersonic speeds. Hypersonic Research Proj. Memo. 39, CA Calif. Inst. Tech, Pasadena, 1957. Snarski SR, Lueptow RM. Wall pressure and coherent structures in a turbulent boundary layer on a cylinder in axial #ow. J Fluid Mech 1995;286:137}71. Yasuhara M. Experiments on axisymmetric boundary layers along a cylinder in incompressible #ow. Trans Jpn Soc Aerosp Sci 1959;2:33. Wietrzak A, Lueptow RM. Wall shear stress and velocity in a turbulent axisymmetric boundary layer. J Fluid Mech 1994;259:191}218. Lueptow RM, Haritonidis JH. The structure of the turbulent boundary layer on a cylinder in axial #ow. Phys Fluids 1987;30:2993}3005. Madavan NK, Deutsch S, Merkle CL. Measurements of local skin friction in a microbubble-modi"ed turbulent boundary layer. J Fluid Mech 1985;156:237}56. Mitchell JE, Hanratty TJ. A study of turbulence at a wall using an electrochemical wall shear-stress meter. J Fluid Mech 1966;26:199}221. Sreenivasan KR, Antonia RA. Properties of wall shear stress #uctuations in a turbulent duct #ow. Trans ASME J Appl Mech 1977;44:389}95. Keith WL, Bennett JC. Low frequency spectra of the wall shear stress and wall pressure in a turbulent boundary layer. AIAA J 1991;29:523}30. Mansour NN, Kim J, Moin P. Reynolds stress and dissipation-rate budgets in a turbulent channel #ow. J Fluid Mech 1988;192:363}77. Lueptow RM, Leehey P. The eddy viscosity in a turbulent boundary layer on a cylinder. Phys Fluids 1986;29:4232}3.