Effect of different surface roughnesses on a turbulent boundary layer

Abstract

The classical treatment of rough wall turbulent boundary layers consists in determining the effect the roughness has on the mean velocity profile. This effect is usually described in terms of the roughness function delta U+. The general implication is that different roughness geometries with the same delta U+ will have similar turbulence characteristics, at least at a sufficient distance from the roughness elements. Measurements over two different surface geometries (a mesh roughness and spanwise circular rods regularly spaced in the streamwise direction) with nominally the same delta U+ indicate significant differences in the Reynolds stresses, especially those involving the wall-normal velocity fluctuation, over the outer region. The differences are such that the Reynolds stress anisotropy is smaller over the mesh roughness than the rod roughness. The Reynolds stress anisotropy is largest for a smooth wall. The small-scale anisotropy and interniittency exhibit much smaller differences when the Taylor microscale Reynolds number and the Kolmogorov-normalized mean shear are nominally the same. There is nonetheless evidence that the small-scale structure over the three-dimensional mesh roughness conforms more closely with isotropy than that over the rod-roughened and smooth walls.

Turbulence; boundary layer; roughness

Effect of Different Surface Roughnesses on a Turbulent Boundary Layer

R. A. Antonia

Department of Mechanical Engineering, University of Newcastle, N.S.W., 2308, Australia

P-A. Krogstad

Department of Mechanics, Thermo and Fluid Dynamics, Norwegian University of Science & Technology, N-7034, Trondheim, Norway

The classical treatment of rough wall turbulent boundary layers consists in determining the effect the roughness has on the mean velocity profile. This effect is usually described in terms of the roughness function DU^+. The general implication is that different roughness geometries with the same DU⁺ will have similar turbulence characteristics, at least at a sufficient distance from the roughness elements.

Measurements over two different surface geometries (a mesh roughness and spanwise circular rods regularly spaced in the streamwise direction) with nominally the same DU⁺ indicate significant differences in the Reynolds stresses, especially those involving the wall-normal velocity fluctuation, over the outer region. The differences are such that the Reynolds stress anisotropy is smaller over the mesh roughness than the rod roughness. The Reynolds stress anisotropy is largest for a smooth wall.

The small-scale anisotropy and interniittency exhibit much smaller differences when the Taylor microscale Reynolds number and the Kolmogorov-normalized mean shear are nominally the same. There is nonetheless evidence that the small-scale structure over the three-dimensional mesh roughness conforms more closely with isotropy than that over the rod-roughened and smooth walls.

Keywords:Turbulence, boundary layer, roughness.

Introduction

The technological importance of wall-bounded turbulent flows is well accepted. In many situations, turbulent boundary layers develop over surfaces that are hydrodynamically rough over at least some part of their length. The major impact of surface roughness is to perturb the wall layer in such a way as to lead, in general, to an increase in the wall shear stress. This has obvious implications to both the shipbuilding and aviation industries (e.g. Schlichting, 1968). It also adversely affects the overall performance of turbines, compressors and other bladed turbomachinery (e.g. Acharya et al., 1986).

The increase in the wall shear stress is almost invariably accompanied by an increase in the wall heat or mass transfer rate. This has major implications in engineering, e.g. in terms of improving the efficiency of heat exchangers or in meteorology, in the context of the atmospheric surface layer over vegetated surfaces. Pimenta et al. (1979) mentioned a number of applications, including nose-tips on re-entry vehicles and transpiration-cooled turbine blades where there is heat transfer to or from pervious rough walls.

Clauser (1954,1956) presented a method of analysing the effects of surface roughness on the mean velocity distribution; the scheme has proved to be robust and continues to be used. He argued that the inner portion for rough walls must have a logarithmic region with the same slope as for a smooth surface. According to this now "classical" scheme, the sole effect of the roughness is to shift the log-law intercept C as a function of the roughness Reynolds number k⁺ºUt k/n . With the exception of the "roughness sublayer", the inner mean velocity distribution on a rough wall is then described by

where DU⁺ represents the roughness-caused shift, as illustrated in Figure 1. (Note that the shift is generally downward although certain surfaces, e.g. longitudinal riblets, can, under certain conditions, produce a positive shift). The outer velocity, when expressed in defect form, viz.

(d is the boundary layer thickness) is identical for both smooth and rough walls. Although (1) and (2) have received widespread experimental verification, a few remarks are necessary, especially with respect to DU⁺. The significance of this quantity cannot be overstated since, as Hama (1954) showed,

for the same magnitude of the Reynolds number U_1d ^*/n (d^* is the displacement thickness). The magnitude of DU⁺ is not uniquely determined by k, even for the so-called k-type roughness, for which

The additive constant C₁ depends, inter alia, on the roughness density, e.g. Dvorak (1969) and Raupach et al. (1991). The latter authors showed that (3) is very well supported by both laboratory and atmospheric surface layer data over an impressive range of k⁺ A distinction has been made between k-type roughness and d-type roughness for which

However, experimental support for (4) is lacking, at least for a boundary layer. While there is mild support for a correlation based on Î⁺ (Î is the error in surface origin), e.g. Perry et al. (1969), Wood and Antonia (1975), Osaka et al. (1984), Bandyopadhyay (1987), such a proposal is only tenable over a limited range of x (Raupach et al., 1991). Nonetheless, a surface comprising two-dimensional transverse bars with narrow cavities, which is generally associated with a "d-type" roughness, is of interest since it appears to satisfy the conditions for exact self-preservation, as set out by Rotta (1962), i.e. both Ut /U₁and dd/dx are independent of x. Evidence in support of this was provided by Tani (1986,1987), Osaka et al. (1982), Osaka and Mochizuki (1988) and Djenidi and Antonia (1998). While this evidence is hard to refute, cogent physical reasons as to why this surface is closer to equilibrium than either a smooth wall or other rough walls have not been formulated; some work is being done in this direction (Djenidi and Antonia, 1998), especially in the context of self-sustaining energy production mechanisms. Although we subscribe to Raupach et al.'s assessment that there are difficulties with the division of roughness into k and (especially) d-type classes, we see no reason why Clauser's proposal, encaptured by (1), (2) and (3), shoud not continue to be useful. We emphasise however that the scheme only addresses the mean velocity distribution. According to the Reynolds number similarity hypothesis (Townsend, 1976) or the wall similarity hypothesis (Perry and Abell, 1977), turbulent motions outside the roughness sublayer are independent of the wall roughness at sufficiently large Reynolds number. This is of course consistent with the universality of f (y/d ) in (2). The validity of this hypothesis has recently been challenged by Krogstad et al. (1992), also Krogstad and Antonia (1994). The experimental evidence presented in the last two papers indicates that the outer layer distributions of the wall-normal turbulence intensity and the Reynolds shear stress are markedly different between a mesh roughness and a smooth wall. There was also evidence of major differences in the characteristics of the large scale motion between the two surfaces.

One implication of these results is that the communication between the wall and the outer region is more important than has hitherto been thought. Another, possibly more serious, inference is that there may be a fundamental difference in the momentum transport process contrary to what the equality of the log-law slope, Eq. (1), between smooth and rough surfaces may imply. As a consequence, classical mixing length calculations, albeit allowing for a shift in origin, are unlikely to explain the difference, in the Reynolds shear stress - á uvñ (angular brackets denote time averaging), in the outer region of the layer.

This paper continues to examine possible differences between smooth and rough. walls. A particular strategy we adopt is to consider different types of roughness geometries but with the important requirement that DU⁺ is kept constant. We assess the influence of different surface conditions on the larger-scale (shear-stress carrying), as well as the smaller scale motions. Although the generally accepted wisdom is that the latter are less likely to be affected by the nature of the surface than the former, especially if the Taylor microscale Reynolds number Rl is kept constant, the possibility that a reduced anisotropy of the large-scale motion (e.g. Krogstad and Antonia, 1994: Shafi and Antonia, 1995) could impact on the anisotropy of the small-scale motion should not be dismissed.

Following a brief description (Section 2) of the surfaces we consider, we address mainly the anisotropy of both the large scale (Sections 3 and 4) and the small scale motion (Section 5). We do not consider here the effect of the roughness on the heat transfer characteristics of the boundary layer nor do we treat the implications that the present observations may have on calculation methods. These topics merit to be addressed separately, at a future date.

Experimental Conditions

Two rough surfaces are considered. One is essentialy three-dimensional, consisting of a woven stainless steel mesh screen. The other is basically two-dimensional, consisting of circular rods placed in a spanwise direction at regular streamwise intervals. Dimensions for these two geometries are given in Table 1, where details on experimental conditions are also included. Both the screen and the rods were glued, in separate experiments, on to the aluminium wall of the wind tunnel working section (5.4 m long, inlet area º 0.9 m x 0.15 m). The ceiling of the working section was adjusted to set the pressure gradient to zero. The mesh screen covered a streamwise distance of 3.5m; for the rods, the distance was 3.2m. Further details for the mesh screen and rod roughness experiments can be found in Krogstad et al. (1992) and Krogstad and Antonia (1998) respectively. For reference, measurements were also made on a smooth wall, though in a different wind tunnel; only a few relevant experimental details are shown in Table 1. Whenever possible, the results obtained by Spalart (1988) for a smooth wall boundary layer direct numerical simulation are shown; only results at the highest value of R_q =1410) have been used.

U was measured both with a Pitot tube (0.81 mm outer dia.) and single and X-hot wires (Pt-10% Rh). Most of the wires used has a diameter (d_w) 2.5 m m and a length l_w of 0.5 mm; the measured frequency response was 12.5 kHz at U=7 m/s. For the small-scale experiments, smaller diameter wires (d_w = 1.2mm and l_w= 0.22 mm) were used; the frequency response was about 23 kHz at U = 7 m/s. The X-wires had an included angle of 110° .

In-house constant temperature circuits were used to operate the hot wires. The output signals from the circuits were filtered (cut off frequency f_c) amplified to optimise the input range of the 12 bit (16 channel sample and hold) A/D converter and sampled (sampling frequency f_s). For most measurements, fc5 kHz, fs 10 kHz and the record duration was about 30s. For the small-scale measurements significantly larger values of f_c and f_s were used, with f_c set close to the Kolmogorov frequency f_K (e.g. Table 2); record durations up to 180s were used.

Mean Velocity and Velocity Fluctuation Moments

Mean velocity distributions are shown in Figure 2 for the three surfaces, with wall variable normalization. Both the present smooth wall results at R_q =12570 and the DNS smooth wall results of Spalart (1988) for R_q = 1410 are shown. These two distributions overlap, in agreement with the law of the wall. The two rough wall distributions also overlap in the inner region indicating that nominally the same value of DU⁺ (see Table 1) was indeed for each of the two rough surfaces and the experimental conditions chosen. As originally anticipated by Clauser [1954,1956, see Eq. (2)] and subsequently verified by many investigators, the velocity defect (U₁⁺-U⁺) [Figure 3], is the same for rough and smooth walls.

Reynolds stresses are plotted in Figures 4 (á(u⁺)²ñ), ⁵ (á(v⁺)²ñ) and ⁶ (-áu⁺ v⁺ñ) in terms of y /d . If we focus our attention primarily on the outer layer, several salient observations can be made

1. There is much closer agreement between the different surfaces for á (u⁺)²ñthan either á(v⁺)²ñor - á u⁺ v⁺ñ. Even in the case of á (u⁺)²ñ(Figure 4), there are some differences between the three surfaces, á (u⁺)²ñtending to be larger over the rod roughness. The differences are believed to be genuine, since they fall outside the range of experimental uncertainty.

2. The most pronounced differences seem to occur on á (v⁺)²ñ(^{Figure 5}) and -á u⁺v⁺ñ(^{Figure 6}), implying that the wall-normal motion is the most affected by the type of surface. Note that á (v⁺)²ñand - á u⁺v⁺ñare also largest for the rod roughness, implying a much stronger momentum transport for this particular surface. Relative to the smooth surface, there is clearly much more activity associated with the wall-normal velocity fluctuation over the rough surface.

3. Although DU⁺ is the same for the two roughnesses, the Reynolds stress distributions are different. This alone considerably limits the generality of DU⁺ as a descriptor of the effect that different surface conditions have on the momentum transport.

Outer layer differences for the turbulent kinetic energy k⁺º 1/2(á (u⁺)^2ñ +á (v⁺)^2ñ +á(w⁺)²ñ)

[Figure 7] between the rough surfaces are less pronounced than those for á (v⁺)²ñor -á u⁺v⁺ñ, reflecting the dominant contribution from á (u⁺)²ñto á k⁺ñ. Both rough wall distributions lie significantly above the smooth wall distribution.

A comment with regard to the smooth wall DNS distribution shown in this and previous figures seems appropriate. The agreement between the DNS distribution at R_q = 1410 and the present smoth wall measurements at R_q =12570 is generally quite good in the outer region, implying that the moderately low R_q DNS results of Spalart can serve as a reliable smooth wall reference against which the effect of the rough wall can be assessed.

Velocity triple products are expected to be a more sensitive indicator of the effect of surface condition than second-order moments. Andreopoulos and Bradshaw (1981) noted that triple products were spectacularly altered for a distance up to 10 roughness heights above a surface covered with floor-sanding paper. Bandyopadhyay and Watson (1988) reported that instantaneous motions involved in the shear stress flux near the wall in smooth and transversely grooved surfaces are opposite in sign to those over a three-dimensional roughness. Distributions of á(u⁺)² v⁺ñand á(v⁺)³ñare shown in Figures 8a and 8b respectively. There are major differences both for á(u⁺)² v⁺ñand á(v⁺)³ñbetween the two types of roughness in the region y/d 0.5. In particular, both triple products change sign above the rod-roughened wall. For the mesh roughness, only á (u⁺)² v⁺ñchanges sign but negative values á(u⁺)² v⁺ñoccur only very near the surface (y/d 0.03). Contrary to what happens over the mesh-screen roughness (or indeed the sandgrain roughness investigated by Andreopoulos and Bradshaw, 1981), there is an important transport of u² and v² towards the wall over the rods. Although distributions of á(w⁺)² v⁺ñand á u⁺(v⁺)²ñare not shown here, the transport of turbulent energy and momentum flux towards the wall over the two-dimensional roughness contrasts with a transport away from the wall for the three-dimensional roughness. The y-derivatives of áu⁺(v⁺)²ñand á(v⁺)³ñcontribute to the term which represents the turbulent energy diffusion associated with v. Although not shown here, the term is negative over the rods - thus representing a gain of turbulent energy - but positive over the mesh screen.

The skewness (S_a º áa³ñ / áa²ñ ^3/2) and flatness factor (F_a º áa⁴ñ / áa²ñ²) of u and v are shown in Figures 9 and 10 respectively. While S_u changes sign, irrespectively of the type of surface, near the wall, S_v, changes sign only over the rods. In contrast to S_v, F_v is, like F_u, practically unaffected by the surface up to y/d > 0.6. There are nevertheless differences between the three surfaces in both F_u, and F_v, as the edge of the turbulent/non-turbulent interface is approached.

Anisotropy of Reynolds Stresses

Ratios such as áv²ñ / áu²áv²ñ / á w²ñ, áv²ñ / áw²ñor -áuvñ / áv²ñprovide a rough indication of the departure of the Reynolds stresses from isotropy. Figure 11a shows that, while there may not be large differences in á v^{2ñ / á} u^2ñ between the two rough surfaces, the magnitude of this ratio is significantly smaller for a smooth wall (the DNS data of Spalart, 1988, is used here and in subsequent figures). The inference is that the anisotropy - at least for the Reynolds normal stresses - is reduced over a rough wall. Note that áv²ñ /áu²ñis somewhat larger, in the region y/d <0.4, for the mesh than for the rod roughness. This closer tendency towards isotropy for the mesh roughness is better illustrated by the ratio áv²ñ /áw²ñin Figure 11b over a significant fraction of the layer. The ratio -á uvñ / áv²áv²ñ / áw²ñ, Figure 11c, is also smaller for the mesh than the rods. The smooth wall values of -á uvñ / á v²ñare largest, reflecting the greater anisotropy for this surface.

A better measure of the anisotropy of the Reynolds stresses is provided by the second (II) and third (III) invariants of the Reynolds stress anisotropy tensor bit defined by

where á u_iu_iñ º á q²ñ is the mean turbulent energy and d _ij is the Kronecker delta tensor. The states that characterise b_ij can be identified on a plot of -II (º b_ijb_ji) vs III (1/3b_ijb_jkb_ki), as originally proposed by Lumley and Newman (1976). The limiting values of the invariants delineate an anisotropy invariant map or partially curvilinear triangle as shown in Figure 12. The upper linear boundary of the triangle characterises 2-component turbulence, such as might be expected in the vicinity of a smooth wall; the DNS boundary layer data of Spalart (1988) shown in the figure and the DNS channel flow data of Kim et al. (1987), shown in Antonia et al. (1991), confirm this behaviour.

Figure 12

A further measure of the anisotropy of the Reynolds stresses is provided by the parameter F(º 1+27III+9II) which is proportional to the product of the three eigen values of áu_iu_jñ/áu_iu_iñ. F is bounded between 0 and 1. It is zero along the linear boundary of the triangle, which describes a 2-component state of turbulence. It is 1 when both II and III are zero, i.e. for isotropic turbulence. Figure 13 clearly shows that, almost everywhere in the layer, F is largest for the mesh roughness and smallest for the smooth wall.

Small-Scale Anisotropy and Intermittency

Ideally, the anisotropy of the small scale-motion should be quantified by evaluating the anisotropy invariants of the energy dissipation rate tensor, in similar manner to the way the Reynolds stress invariants were obtained in the previous section. Anisotropy invariants of e _ij require all velocity derivatives to be known. This is not yet feasible experimentally (it is possible from direct numerical simulations, e.g. Kim and Antonia, 1993, but these have yet to be performed for surface geometries of comparable complexity to those considered here). We therefore limit ourselves here to considering relatively simple measures of small-scale anisotropy, as provided for example by the ratios á(d v)²ñ^c /(d v)²ñ^m and f_v^c(k₁) /f_v^m(k₁), where the superscripts c and m denoted calculated (using isotropy) and measured values. As noted in the introduction, the small-scale anisotropy is expected to depend on a number of factors, such as R_l , the mean shear, the proximity to; the wall or the presence of a turbulent/non-turbulent interface. Here we consider data at y/d 0.2, a location where the influence of the last two factors should not be significant' Experimental conditions were chosen so that the magnitude of RN is about the same for the three surfaces considered. Also, the magnitude of S*º S(n/áeñ)^1/2 where Sº¶áUñ/¶y is the mean shear, is nominally the same for the three surfaces. The experimental conditions are summarised in Table 2.

Distributions of á(d u^*)²ñ^m, á (d v^*)²ñ^m and á(d v^*)²ñ^c are shown in Figure 14 for the rod roughness. The values of á(d v^*)²ñ ^c were calculated using the isotropic relation

As expected, the ratio á(d v^*)²ñ ^c /á(d v^*)²ñ ^m, also shown in the figure for the other two surfaces, is close to 1 at small r* and increases as r* increases. As r*® ¥ (or, to a reasonable approximation r* t L*, where L is the integral length scale), á(d u^*)²ñ ^® 2á(u^*)²ñ and á(d v^*)²ñ =2á(v^*)²ñ . Since á(d u^*)²ñ^m was used as input in Eq. (5), á(d v^*)²ñ^c = / á(d u^*)²ñ^m when r* is large enough for ¶ á (d u^*)²ñ^m /¶ r^* to be zero. The different levels of the ratio á(d v)²ñ^{c /} á(d v)²ñ^m in Figure 14 reflect the different values of the ratio á v²ñ^m/ áu²ñ^m for the three different surfaces (Table 2).

Distributions of (f _u^*)^m(k₁^*), (f _v^*)^m(k₁^*) and (f _v^*)^c(k₁^*) are shown in Figure 15 as a function of k₁^*, where k₁, is the one-dimensional wavenumber. (f _v^*)^c(k₁^*) was evaluated from the isotropic relation

with f u^m(k₁)^m(k₁) as input. The departures of f _v^c(k₁)/f _v^m(k₁)^c(k₁)/f _v^m(k₁) from 1 at high wavenurnbers of course reflects that of á (d v)²ñ^{c / á} (d v)²ñ^m from 1 at small separations (Figure 14).

It is difficult to select unambiguously, from ratios in Figures 14 and 15, a surface for which the anisotropy is smallest. One could conclude that all three surfaces satisfy isotropy equally well to a rough approximation. Alternately, it may be argued that isotropy is somewhat better satisfied, especially in terms of the extent of the range - in either r* or k₁* - for the mesh roughness. A more detailed examination of the small-scale structure for this roughness supports this argument, in particular, transverse vorticity spectra were found (Antonia and Shafi, 1998) to satisfy isotropy at least as well, as the boundary layer measurements (Ong and Wallace, 1995) in the NASA-Ames wind tunnel at R_l 1400. Speculatively, the small-scale structure over three-dimensional surfaces such as the mesh screen may satisfy isotropy more closely than that over two-dimensional roughnesses such as the rods. Although more work is needed, for example, vorticity measurements have yet to be made over the rods or, for that matter, over the present smooth wall, the experimental conditions (in particular the magnitudes of h and f_k) in Table 2 clearly indicate that small-scale statistics should be measured more accurately over rough surfaces than smooth walls.

A measure of small-scale intermittency is provided by the departure of the IR exponents z _u(p) where

from the corresponding Kolmogorov (1941) values. According to Kolmogorov (1941),

when r lies within the IR. This departure, which is usually described as the "anomalous scaling", increases as p increases (e.g. Anselmet et al., 1984) but is detectable even for small values of p (e.g. Stolovitzky and Sreenivasan, 1993).

The magnitude of z _u(p) was estimated using the extended self-similarity (ESS) method of Benzi et al.(1993); the scaling range was however restricted to that over which á |du|³ñ r^*-1 was approximately constant (this range is loosely identified here with the IR). For consistency, the same range was used to determine z _v (p), the IR exponent of á |dv|^pñ , viz.

It is worth emphasising that ESS only yields relative, rather than absolute, estimates of z _u(p) and z _v(p) . The distributions of z _u(p) over the three different surfaces at y/d > 0.2 (Figure 16) are nearly the same, implying that the intermittency affects each flow in similar fashion. For comparison, the prediction of the lognormal model (Kolmogorov, 1962), viz.

where m (>0.2) is the intermittency factor, has been included in Figure 16. While it is in reasonable agreement with the experimental estimates of z _u(p), it does not differentiate between z _u(p) and z _v(p). The experimental magnitudes of z _v(p) are significantly smaller than those of z _u(p), possibly suggesting that transverse velocity fluctuations are more intermittent than longitudinal velocity fluctuations. Another possibility may be that v is more sensitive to anisotropy than u. In this context, we note that z _v(p) is generally bigger for the mesh roughness than the other two surfaces, in support of the earlier suggestion that the small-scale turbulence over three-dimensional surfaces satisfies isotropy more closely than that over either a smooth wall or a two-dimensional surface roughness.

As the Reynolds number (R_q or R_l ) increases, the anisotropy is expected to decrease, thus bringing z _v(p) into closer alignment with z _u(p). This tendency is illustrated in Figure 17 for the rod-roughness. Note that z _u(p) is hardly affected by the increase in R_l . It should be mentioned that the increase in z _v(p) with R_l in Figure 17 and the difference between the three z _v(p) distributions in Figure 16 are statistically significant, allowing for experimental uncertainty in estimating z _v(p).

Conclusions

The Clauser roughness function DU+ is a useful descriptor of the effect that the surface roughness has on the mean velocity distribution in the inner region of a boundary layer. Also, the mean velocity distribution in the outer region is, to a good approximation, unaffected by the roughness. However, there is as yet no adequate scheme which describes the effect the roughness has on the Reynolds stresses in the outer region of the boundary layer. In particular, we have noted that for two different roughness geometries for which DU+ is approximately the same, the outer layer distributions of the Reynolds stresses, especially those involving the wall-normal velocity distribution, are discernibly different. They also differ with respect to the smooth wall. The Reynolds stress differences are such that the Reynolds stress anisotropy is smallest over the mesh screen and largest for the smooth wall.

We have also shown that when R_l and a Kolmogorov-normalized value of the mean shear are kept constant, the differences in the small scale anisotropy are only small. Nonetheless, there is evidence indicating that, for the three-dimensional roughness, the small scales are more closely isotropic than for either the two-dimensional roughness or the smooth wall. Consistently, the difference between IR power-law exponents of longitudinal and transverse velocity structure functions appears to be smallest for the mesh screen roughness. A useful feature of rough walls is that a particular value of R_l can be attained at a particular value of y/d for a significant smaller free stream velocity relative to a smooth wall. There are consequently less severe experimental constraints in rough wall layers in the context of adequately resolving the small-scale motion.

Acknowledgements

RAA acknowledges the continued support from the Australian Research Council. We would like to acknowledge the contribution from Mr. R. Smalley to the rod roughness experiment and many useful discussions with Dr L. Djenidi in the context of a turbulent boundary layer over the so-called "d-type" surface.

Article presented at the 1st Brazilian School on Transition and Turbulence, Rio de Janeiro, September 21-25, 1998.

Technical Editor: Atila P. Silva Freire.

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Publication Dates

• Publication in this collection
  18 Jan 2001
• Date of issue
  2000