Print version ISSN 0100-7386
J. Braz. Soc. Mech. Sci. vol.22 n.1 Rio de Janeiro 2000
Mila R. Avelino
Mechanical Engineering Program (PEM/COPPE/UFRJ), C.P. 68503, 21945-970, Rio de Janeiro, Brazil
Also: Department of Mechanical Engineering, Rio de Janeiro State University (UERJ), Rua São Francisco Xavier, 524, sala 5023, Bloco A, 20550-013, Rio de Janeiro, Brazil
A theory for the description of turbulent boundary layer flows over surfaces with a sudden change in roughness is considered. The theory resorts to the concept of displacement in origin to specify a wall function boundary condition for a k-e model. An approximate algebraic expression for the displacement in origin is obtained from the experimental data by using the chart method of Perry and Joubert(J.F.M., vol. 17, pp. 193-122, 1963). This expression is subsequently included in the near wall logarithmic velocity profile, which is then adopted as a boundary condition for a k-e modelling of the external flow. The results are compared with the lower atmospheric observations made by Bradley(Q. J. Roy. Meteo. Soc., vol. 94, pp. 361-379, 1968) as well as with velocity profiles extracted from a set of wind tunnel experiments carried out by Avelino et al.( 7th ENCIT, 1998). The measurements are found to be in good agreement with the theoretical computations. The skin-friction coefficient was calculated according to the chart method of Perry and Joubert(J.F.M., vol. 17, pp. 193-122, 1963) and to a balance of the integral momentum equation. In particular, the growth of the internal boundary layer thickness obtained from the numerical simulation is compared with predictions of the experimental data calculated by two methods, the "knee" point method and the "merge" point method.
Keywords: Turbulence, Boundary Layer, Surface Roughness, Atmospheric Flows, k-e model.
A complete understanding of the effects that a step change in surface roughness has on the properties of a turbulent boundary layer has been the object of several experimental and theoretical investigations in recent years, specially in micrometeorology. Most air flows of practical interest occur in situations where the roughness, the elevation and the temperature of the terrain is changing. Unfortunately, these conditions often occur at the same time, giving rise to complex flow configurations in which necessarily a large number of parameters to describe the roughness is required.
The present work is concerned with flows that develop over flat terrain with changing surface conditions. In particular, we will be looking at flows which present abrupt changes in surface conditions from one extensive uniform surface to another; the theory to be developed here is, therefore, expected to account for these effects. Here, we will use the k-e model to describe the properties of the boundary layer in the surface layer. The model will use a wall function to represent the velocity profile near to the wall so that a local analytical solution for the inner region will be used as a boundary condition for the outer solution. For this reason, this inner solution must take into account for local changes in the flow such as those provoked by the changes on the surface roughness. The local changes are then accounted for by logarithmic profiles that take as a characteristic length the displacement in origin, which has been previously experimentally studied by the present author in Avelino et. al.(1998). In that work, an algebraic expression was proposed for the specification of the displacement in origin, which is here recovered to propose a law of the wall expression with a stream-wise variable characteristic length. This expression is then applied to the wind tunnel data of Avelino et al.(1998) and to the atmospheric flow data of Bradley(1968) to validate the proposed procedure.
The present work is, therefore, twofold. It will show the reader how the k-e model stands for flows over step changes in surface roughness, in both wind tunnel and in natural flow conditions; it will also present some new experimental data specially obtained for the validation of the numerical simulation. Four geometrical configurations will be considered, uniformly smooth and rough surfaces, and surfaces with smooth to rough and rough to smooth changes.
In the following, we present a short review of the work recently published on the subject. Next, the experiments and the numerical simulations will be presented.
The understanding of the effects that changes on the surface nature have on the flow properties has rested mainly upon field experiments. A difficulty with most studies is the absence of reliable measurements of surface flux. One of the few field studies that has overcome this difficulty has been that of Bradley(1968), who made some simultaneous measurements of both velocity profiles and surface shear stresses in a neutral flow at several positions relative to a discontinuity separating surfaces made of grass, of tarmac and of spikes. Bradleys data are ideal for comparison with theoretical and numerical results. His measured velocity profiles were compared with the computation of Rao et al.(1974), whereas the surface shear stresses were compared with the theories of Peterson(1969) and of Panofsky and Townsend(1964).
Other atmospheric observations are those based on the bushel-basket experiments over the ice of Lake Mendota in the USA (Stearns(1964) and Lettau(1963)), the studies on the modification of the low level wind profiles based on the Riso Tower observations (Panofsky and Petersen(1972) and Petersen and Taylor(1973)), and a study of flows downwind of a wheat crop leading edge (Munro and Oke(1975)).
In more controlled conditions, in a laboratory environment, detailed measurements have been made about the turbulence over rough surface changes, involving both zero and adverse pressure gradient conditions (Antonia and Luxton(1971 and 1972), Antonia and Wood(1975), Schofield(1975) and Mulhearn(1976, 1978)). All these experiments have concentrated on the development of the internal boundary layer and its internal mean and turbulent structure. Wind tunnel observations can also be found in Krogstad et al.(1992), Krogstad and Antonia(1994) and Shafi and Antonia(1997).
The results obtained in the laboratory for boundary layers have frequently been extended to describe the properties of atmospheric boundary layers with good agreement. However, the inhomogeneity of the earths surface greatly complicates the application of results and theories established for uniform surfaces. To overcome this difficulty, most information comes from studies of the coastal boundary layer, where extensions of the small-scale approach to the mesoscale, specifically the internal boundary layer (IBL) at the coast, have been discussed by Venkatran(1986) and Shao et al.(1991).
The experiments of the last thirty years have been accompanied by a great deal of theoretical effort. Much of this work is discussed in a recent review by Garratt(1990). Here, we will mention a few.
The atmospheric boundary layer and the problem of surface heterogeneity can be considered on several scales, where different characteristics are attained. For a neutrally stable boundary layer, the flow is normally separated into two regions. On the smallest scale there exists an inner layer, where the effects are confined to the surface layer and the velocity profile is observed to have a logarithmic form; in this case of small-scale flow and of a neutral IBL responding to changes in surface roughness, analytical solutions were provided in studies by Elliott(1958), Taylor(1968, 1969), Panofsky and Townsend(1964), Plate and Hidy(1967) and Mulhearn(1977). For non-neutral flows, the IBL response to a change in surface roughness has been presented in an analytical solution (Townsend(1965, 1966)). Numerical approaches to the problem include those of Venkatran(1976, 1985), of Peterson(1969), of Shir(1972), of Rao et al.(1974) and of Beljaars et al.(1987).
For more stable or unstable flows, the velocity profile deviates from its logarithmic behaviour. For this kind of problems, where many characteristic scales can be found, asymptotic techniques can be evoked to show that in most situations the velocity profile assumes a logarithmic form for distances sufficiently close to the wall.
A good review text on turbulent boundary layers subjected to sudden perturbations is the article of Smits and Wood(1985); a review of the relevant work on the internal boundary layer is given by Garratt(1990). In the review of rough-wall turbulent boundary layers by Raupach et al.(1991) the effect of the roughness on the mean velocity is reported, considering wind tunnel experiments over rough surfaces as well as natural vegetated surfaces in the atmosphere.
The experiments reported in Avelino et al.(1998) were performed in the low-turbulence wind tunnel of the Laboratory of Turbulence Mechanics of the Mechanical Engineering Department of COPPE/UFRJ. The tunnel is an open circuit wind tunnel with a 0.17% turbulence intensity level in the main flow, and a working section 0.3m wide, 0.3m high, and 2m long. A 3:1 contraction section feeds the working section, which has an adjustable roof, to permit the control of the longitudinal pressure gradient.
To avoid any undesirable adverse pressure gradient, the first (or last) roughness element was always depressed below the smooth surface, its crest being aligned with the smooth surface. The rough surface configurations are described below. In Case I, two 1.0 m long aluminium sheets were used to provide a long uniform smooth surface. In Case II, a 1.0 m long smooth surface was placed downstream of an aluminium sheet consisting of transversally grooved surfaces with rectangular slats of dimensions 3mm high, 12mm wide, and pitch of 24mm. In Case III, the converse to Case II was realised; the upstream aluminium sheet was now smooth, followed by a rough sheet. Case IV was configured by two 1.0 m long aluminium rough sheets, resulting in a uniform rough surface. The present roughness elements characterise a roughness of the type K with W=3K, where W is the cavity width. This geometry is slightly different from those of Perry and Joubert(1963), Perry et al.(1969), Antonia and Luxton(1971), Antonia and Luxton(1972) and Bandyopadhyay(1987).
The roof of the wind tunnel was carefully adjusted to assure a constant pressure. The mean velocity at the center-line of the working section was close to 5.5m/s in all cases, and the measurements were made at distances 0.2m, 0.4m, 0.6m, 0.8m, 1m, 1.05m, 1.1m, 1.15m, 1.2m, 1.3m, 1.4m, 1.6m and 1.8m from the beginning of the working section.
The mean velocity profiles were obtained using a Pitot static probe and a Mensor Pressure Gauge. The mean velocity was also measured with a boundary layer hot-wire operated by a linealized constant temperature anemometer.
Mean velocity profiles and turbulence intensity levels were obtained using a DANTEC hot-wire system series 56N. The boundary layer probe was of the type 55P15. The mean velocity data had a precision of 0.6%.
A detailed description of the experimental set up can de found in Avelino et al.(1998).
Two Types of Roughness
Nikuradse(1933) investigated the effects of sand-roughened surfaces on the mean velocity profile in pipes, and found that, at high Reynolds number, the near wall flow becomes independent of viscosity, and is a function of the roughness scale, K, of the pipe diameter, D, as well as of Reynolds number, R. From dimensional arguments and comparison with Prandtls law of the wall, Nikuradse described the velocity profile in the near wall region as
where ut is the friction velocity, k is the von Kármán constant(=0.41), and B is a function of the surface roughness.
Equation (1) was written in an alternative form by Clauser(1954), who cast it as
Hama(1954) showed that
which immediately shows that equations (1) and (2) are just the same but written in a different form.
Flows that follow the behaviour set by equations (1) to (3) are said to occur over surfaces of the "K" type. Flows, on the other hand, which are apparently insensitive to the characteristic scale K, but depend on other global scale of the flow are termed flows over surfaces of the "D" type. In the latter case, the roughness is geometrically characterised by a surface with a series of closely spaced grooves within which the flow generates stable vortical configurations. To describe the part of the velocity profile that deviates from the logarithmic law in the defect region, we consider that, in the flow region above the rough elements, the mean motion is independent of the characteristic scales associated with the near wall flow. Thus, equation (2) may be re-written as
In principle, there is no physical reason why the functions appearing in equations (2) and (4) should have the same form. In fact, the distinct length scales used in the representation of the "K" and "D" type rough wall flows may suggest that a single framework for the description of both types of roughness cannot be devised. However, Moore(1951) showed that a similarity law can be written in a universal form provided the origin for measuring the velocity profile is set some distance below the crest of the roughness elements.
Writing the above expressions in a more general form, valid for both types of roughness, we find
and Ci, i = K, D; is a constant characteristic of the roughness.
Having worked out an expression for the representation of the velocity profile in the near wall region (expression (5)), let us now argue about its domain of validity and how it relates to the classical asymptotic structure of the turbulent boundary layer.
From an asymptotic point of view, one the important factors in the determination of the flow structure is the correct assessment of the order of magnitude of the fluctuating quantities. For the velocity field, a classical result is that, for flow over a smooth surface, both the longitudinal and the transversal velocity fluctuation components scale with the friction velocity, ut . The direct implication of this result is that the fully turbulent region is limited by the scales (u2t / U2¥)L and n/ut.
All these arguments can easily be formalised through application of the single limit concept of Kaplun(1967). Indeed, an application of the theory of Kaplun (Silva Freire and Hirata(1990)) to the equations of motion, shows that the flow structure consists of two distinct regions determined by specific regions of validity obtained through passage of the single limiting process. The domains defined by the limits quoted above are just the overlap domains of the inner and the outer regions.
For flows over rough surfaces, we have seen that the lower bound of the overlap regions must change, being now a function of the surface geometry. Indeed, in this situation, the viscosity becomes irrelevant for the determination of the inner wall scale because the stress is transmitted by pressure forces in the wakes formed by the crests of the roughness elements. We have also seen that the characteristic length scale for the near wall region must be the displacement in origin. It is also clear that, in either case, roughness of the type "K" or roughness of the type "D", the roughness elements penetrate well into the fully turbulent region so that the new origin for the velocity profile will always be located in the overlap fully turbulent region. Therefore, concerning the k-e model, it appears that an adequate description of the flow can be given provided the wall boundary condition is written according with equation (5).
The Displacement in Origin
The determination of the displacement in origin, e, is crucial for the evaluation of the properties of the flow over a rough surface, including all local and global parameters such as, e.g., the skin-friction coefficient and the momentum thickness. All graphical methods for its determination, however, assume the existence of a logarithmic region, which may not occur near to a step change in roughness.
An initial estimate of e can be made based on the physics of the problem. For rough surfaces of the K-type, the type of surface studied here, the ratio e/K ® 1.0 according to the relation (Bandyopadhyay, 1987)
The asymptotic value of e was observed by Bandyopadhyay to be reached at a distance of about 1000K downstream of the point of surface change.
Here, the values of e were calculated according to the method of Perry and Joubert(1963). Systematically adding an arbitrary displacement in origin to the original profiles, the least square method could be applied to the near wall points to search for the best straight line fit. As mentioned by other authors, this method is extremely sensitive, as small departures in the true value of e will give large differences in the calculated values of Cf.
The estimated values of e are shown in Figures 1 and 2 compared with equation (6). In opposition to the results of Bandyopadhyay(1987), we have found here different values for the exponent in the power law. For the uniformly rough case we found m = 1.04, whereas for the smooth-to-rough case we found m = 0.81. In the present experiments, the asymptotic value of e was reached at about x = 400.
The result is that in the numerical computations the following expressions will be used to represent e:
Case uniformly rough.
Case smooth to rough.
The Numerical Scheme
The present theory, based on equations (5) to (8), was numerically implemented through the computer code CAST (Computer Aided Simulation of Turbulence, Peric and Scheuerer(1989)). This program has the same structure of other existing fluid flow prediction schemes such as TEAM and TEACH. It is thus a conservative finite-volume method in primitive variables. Differences from those codes arise in the co-located variable arrangement, the discretization scheme, the solution algorithms for the linear equation systems resulting form the discretization, and in the pressure coupling which is adopted to the co-located variable storage.
For turbulent flow, the code solves the Reynolds averaged Navier-Stokes equations in connection with the k-e differential turbulence model of Launder and Spalding(1974). The five empirical constants appearing in the code take on the standard values. Since CAST uses the wall function method for specification of the boundary conditions at the wall, an extension of the program to our case of interest was a relatively straightforward affair. Changes were basically made in the momentum balance at the adjacent to the wall control volumes. Here, we will spare the reader the main implementation details.
In all flow simulations, the major modification in the code relied entirely on the manner in which the boundary condition was implemented. The concept of displacement in origin was incorporated to the original code, with a carefully chosen expression for its description (equations (7) and (8)).
Mean Velocity Profiles
The computed mean velocity profiles are shown in Figures 3 to 4 in dimensional co-ordinates and in a logarithmic form. As expected, both the angular and the linear coefficients of the straight lines are observed to decrease as the flow progresses. In Figure 3, the linear regions cease to exist after the change in surface roughness at x = 0. The most interesting feature of this figure, indeed, is the large distortion in the velocity profile at x = 0. At this point, no logarithmic behaviour of the velocity profile can be noted. In fact, at the would be logarithmic region a strong "kink" in velocity can be seen. In the rough-to-smooth case (not shown here) the level of the velocity curves were observed to decrease until x = 0; at that point, the velocity started to recover to its undisturbed conditions, raising the values of the linear coefficients to the values of the reference uniformly smooth surface curve. Figure 4 displays the flow behaviour for the uniformly rough surface case.
The next figure, Figure 5, shows the calculated values of Cf .
In principle, Clausers chart method can be used to evaluate Cf for flows over a smooth surface. In fact, if the classical formulation for the law of the wall is assumed to hold and if the von Karman constant, k , is really considered constant and equal to 0.41, then the wall shear stress can be estimated directly from the slopes of the straight lines defined in logarithmic graphs. With the values of Cf, the value of the additive "constant", A, in the law of the wall can then be determined. The resulting As are not constant for some flow conditions but vary with x.
For the flows over rough surfaces, on the other hand, the task of evaluating Cf is much more complex for two parameters, the displacement in origin and the roughness function are previously unknown (Perry and Joubert(1963), Perry, Schofield and Jouber(1969)). If the flow is in a near state of energy equilibrium condition, the chart method of Clauser can be extended to calculate Cf (Perry and Joubert, 1963). The difficulties are many. The most serious one is that the value of Cf is confirmed only by the slope of the logarithmic line and not by its position. In some of our experiments, however, the flow in the vicinity of the point of change in surface roughness is not in equilibrium condition. Thus, any method which presumes the existence of a logarithmic region and searches for values of Cf by distorting the measured velocity profile into a logarithmic curve must be seen with caution.
Due to the uncertainties of the chart method, at least one alternative estimate of Cf had to be provided; this method was based on the application of the momentum integral equation
where q is the momentum thickness and H = d1/ q. This equation was used considering the normal stress difference gradients negligible.
Figures 6 and 7 were prepared with the numerical and the experimental data in outer scales and in logarithmic form. The agreement provided by the computations was very reasonable showing that the k-e model responses well for the law of the wall formulation of expression (6).
The Internal Layer
In addition to the velocity profiles, we want to show how the numerical predictions for the wind-tunnel data can be used to trace some analogy with atmospheric data. To this end, we will compare de present results with the atmospheric data of Bradley(1968).
In literature, several methods have been proposed to determine di. Here, two methods will be used (Antonia and Luxton, 1971). In the first method, di is inferred from the position of merging between two consecutive mean velocity profiles. The resulting points closely coincide with the merging of the turbulence intensity profiles yelling a physically realistic procedure. In the second method, the velocity profiles are plotted against y1/2. Under these co-ordinates, two distinct linear regions appear with different slope coefficients. The intersection of the two straight lines defines the edge of the internal layer.
Considering the diffusive character of the growth of the internal layer and a logarithmic expression for the mean velocity profile, Panofsky and Dutton(1984) derived a logarithmic expression for the growth of di. The resulting numerical values of di are shown in Figures 7 and 8 compared with logarithmic and power-law expressions and the data of Bradley(1968). The physical evidence is that for the rough-to-smooth case the growth rate is much slower than that observed for the smooth-to-rough case. For the rough-to-smooth surface, estimates from the "knee" point method and from the "merge" point method furnished respectively n = 0.41 and 0.43. For the smooth-to-rough case, we found n = 0.77 and 0.87. In this case, to apply Panofsky and Dutton equation we replaced z2 by e.
Overall the agreement shown by the computations was very good.
A comparison of the present numerical computation with the data of Avelino et al.(1998) and with the data of Bradley(1986) shows that, apparently, the k-e model can be used to provide good predictions of wind-tunnel and atmospheric flow data over changing surfaces. Overall, the present data are consistent with the data of other authors; the values of Cf , of e and of di are of the order of the data of Perry and Joubert(1963), of Perry et al.(1969) and of Antonia and Luxton(1971, 1972). Currently, the k-e model is under further scrutiny by the present author in order to demonstrate its capability of predicting flows over rough surfaces.
The author is grateful to Prof. A. P. Silva Freire for the useful discussions undertaken during the course of the present work. Prof. P. P. M. Menut was very helpful in helping with the experiments.
Antonia, R. A. and .Luxton, R. E.; The Response of a Turbulent Boundary Layer to a Step Change in Surface Roughness. Part 1: Smooth to Rough, J. Fluid. Mech., vol. 48, pp. 721-761, 1971. [ Links ]
Antonia, R. A. and .Luxton, R. E.; The Response of a Turbulent Boundary Layer to a Step Change in Surface Roughness. Part 2: Rough to Smooth, J. Fluid. Mech., vol. 53, pp. 737-757, 1972. [ Links ]
Antonia, R. A. and .Wood, D. H.; Measurements in a Turbulent Boundary Layer over a d-type Surface Roughness, J. Appl. Mech., pp. 591-596, 1975. [ Links ]
Avelino, M.R., Menut, P. P. M. and Silva Freire, A. P.; Experimental Characterisation of a Turbulent Boundary Layer Subjected to a Step Change in Surface Roughness, Proceedings of the 7th Brazilian National Meeting on Thermal Sciences, November, 1998. [ Links ]
Bandyopadhyay, P. R.; Rough-Wall Turbulent Boundary Layers in the Transition Regime, J. Fluid Mech., vol. 180, pp. 231-266,1987. [ Links ]
Bradley, E. F.; A Micrometeorological Study of Velocity Profiles and Surface Drag in the Region Modified by a Change in Surface Roughness, Quart. J. Roy. Meteor. Soc., vol. 94, pp. 361-379. [ Links ]
Clauser, F. H.; Turbulent Boundary layers in Adverse Pressure Gradients, J. Aero. Sci., vol. 21, pp. 91, 1954. [ Links ]
Elliott, W. P.; The Growth of the Atmospheric Internal Boundary Layer, Trans. Amer. Geophys. Un., vol. 39, pp. 1048-1054, 1958. [ Links ]
Hama, F. R.; Boundary Layer Characteristics for Smooth and Rough Surfaces, Trans. Soc. Nav. Arch. Mar. Engrs., vol. 62, pp. 333, 1954. [ Links ]
Kaplun, S.; Fluid Mechanics and Singular Perturbations, Academic Press, 1967. [ Links ]
Krogstad, P.-A., Antonia, R. A. and Browne, L. W. B., Comparison Between Rough- and Smooth-Wall Turbulent Boundary Layers, J. Fluid Mech., 245, pp. 599-617. 1992. [ Links ]
Krogstad, P.-A., Antonia, R. A., Structure of Turbulent Boundary Layers on Smooth and Rough Walls, J. Fluid Mech., vol. 277, pp. 1-21,1994 [ Links ]
Moore, W. L., An Experimental Investigation of the Boundary Layer Development Along a Rough Surface, Ph. D. Thesis, State University of Iowa, 1951. [ Links ]
Mulhearn, P. J.; A Wind-Tunnel Boundary-Layer Study of the Effects of a Surface Roughness Change: Rough to Smooth., Boundary Layer Meteorol., vol. 15, pp. 3-30, 1978. [ Links ]
Nikuradse, J.; Stromungsgesetze in Rauhen Rohren., V. D. I. Forshungsheft No 361, 1933. [ Links ]
Panofsky, H. A and Dutton, J. A. Atmospheric Turbulence, Wiley-Interscience, N.Y., 1984. [ Links ]
Peric, M. and Scheuerer, G.; CAST - A Finite Volume Method For Predicting Two-Dimensional Flow and Heat Transfer Phenomena, GRS- Technische Notiz Srr-89-01, 1989. [ Links ]
Perry, A. E. and Joubert, P. N.; Rough-Wall Boundary layers in Adverse Pressure Gradients. J. Fluid Mechanics, vol. 17, pp. 193-211, 1963. [ Links ]
Perry, A. E. and Joubert, P. N., Rough-wall Turbulent Boundary Layers, J. Fluid Mech., vol. 37, pp. 383-413, 1969. [ Links ]
Rao, K. S., Wyngaard, J. C. and Coté, O. R.; The Structure of Two-Dimensional Internal Boundary Layer over a Sudden Change of Surface Roughness, J. Atmos. Sci., vol. 31, pp. 738-746, 1974. [ Links ]
Schofield, W. H.; Measurements in Adverse-Pressure-Gradient Turbulent Boundary Layers with a Step Change in Surface Roughness. J. Fluid Mech., vol. 70, part 3, pp. 573-593, 1975. [ Links ]
Shao, Y., Hacker, J. M. and Schwerdtfeger, P.; The Structure of Turbulence in a Coastal Atmospheric Boundary Layer. Quart. J. Roy. Meteorol. Soc., vol. 117, pp. 1299-1324., 1991. [ Links ]
Silva Freire, A. P. and Hirata, M. H.; Approximate Solutions to Singular Perturbation Problems: the Intermediate Variable Technique, J. Math. Analysis and Applications, vol. 145, pp. 241-253, 1990. [ Links ]
Smits, A. J. and Wood, D. H. The Response of Turbulent Boundary Layers to Sudden Perturbations, Ann. Rev. Fluid Mech., vol. 17, pp. 321-58, 1985. [ Links ]
Townsend, A. A.; The Flow in a Turbulent Boundary Layer after a Change in Surface Roughness, J. Fluid Mech., vol. 26, part 2, pp. 255-266, 1966. [ Links ]
Townsend, A. A.; The Response of a Turbulent Boundary Layer to Abrupt Changes in Surface Conditions, J. Fluid Mech., vol. 22, part 4, pp. 799-822, 1965. [ Links ]
Venkatram, A. A.; Model of Internal Boundary-Layer Development. Boundary Layer Meteorol., vol. 11, pp. 419-437, 1977. [ Links ]
Venkatran, A.; An Examination of Methods to Estimate the Height of the Coastal Internal Boundary Layer. Boundary Layer Meteorol., vol. 36, pp. 149-156, 1986. [ Links ]
Walmsley, J. L. Internal Boundary-Layer Height Formulae - A Comparison with Atmospheric Data, Boundary Layer Meteorol. , vol. 47, pp. 251-262, 1989. [ Links ]
Article presented at the 1st Brazilian School on Transition and Turbulence, Rio de Janeiro, September 21-25, 1998.
Technical Editor: Atila P. Silva Freire.