Boundary-Layer Characteristics for Smooth and Rough Surfaces

Boundary-Layer Characteristics for Smooth and Rough Surfaces BY FRANCIS R. HAMA, t VISITOR After reviewing analyses of boundary-layer development on smooth plates with zero pressure gradient, wind-tunnel measurements of the velocity distribution along a plate covered by a series of artificial roughnesses of controlled characteristics are described. A systematic method of analyzing theeffect of roughness on the velocity profile and on the surface-resistance coefficient is established. The essential effect of the surface irregularities is shown to be universal for the boundary layer as well as for flow in pipes or channels, and the preparation and interpretation of charts for the mean surface-resistance coefficient is discussed. The pronounced effect on ship drag due to paint roughness reported by Todd is substantiated by the present investigation. A more convenient technique than boundary-layer surveys or towing-tank tests is suggested for further investigation of the roughness problem. SUMMARY Recent investigations on the turbulent boundary layer along a smooth flat plate with zero pressure gradient are first reviewed. Two universal laws for the velocity distribution (the velocity-defect law and the wall law) are the essential features underlying these analyses. The intermediate zone, in which the velocity distribution is logarithmic and both laws apply, plays a fundamental role in the analysis of the boundary layer for the smooth plate, and as well in the further study of the effect of surface roughness. In the course of presenting this material, 1 Research Engineer, Iowa Institute of Hydraulic Research, State University of Iowa, Iowa City, Iowa. Now, Assistant Research Professor, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md. Presented at the Annual Meeting, New York, N.Y., November 10-13, 1954, of TH~ SOCII~TY OF NAVAL ARCHITECT8 AND MARINB ]~N- GINBERS. several new formulas are introduced, and the entire picture of the interrelationship between velocity distribution and resistance is clearly drawn. In order to study the fundamental characteristics of the turbulent boundary layer along a rough surface, velocity-distribution surveys along a plate covered by a series of artificial roughnesses of controlled magnitude were conducted in a wind tunnel. The velocity-defect law is shown to be universal for either smooth or rough surfaces, the logarithmic part being shifted for rough surfaces by an amount depending upon the roughness parameter. This displacement is a direct indication of the roughness effect and is directly related to the increase of the surface-resistance coefficient produced by the roughness. A method of analyzing the effect of surface roughness on the boundary layer is thus firmly established. The roughness effect is further shown to be the same for boundary layers and for flow in pipes or 333 channels. Based upon this experimental confirmation of previous assumptions, the use of pipes or channels is suggested for further investigations of the roughness problem, as this is simpler and more exact than boundary-layer or towing-tank experiments. A general resistance formula is given and charts for portraying the surface-resistance coefficient for smooth and rough surfaces are discussed. Comparison also is made between the estimated roughness effect based on the present investigation and that evaluated from actual ship tests. The pronounced increase in the surface resistance produced by certain paints reported by Todd appears to be quite reasonable. INTRODUCTION The effect of surface roughness on boundarylayer resistance is of ever-increasing importance in the fields of marine, aeronautical, and hydraulic engineering. Whereas natural watercourses and airplane wings represent known extremes of rough-

554 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES hess, that of a ship is both variable and unknown. Only recently has a paper by Todd (1) 2 indicated the pronounced effect of certain paints upon the measured drag of full-scale vessels. According to evidence which he presents, the increase of surface resistance due to these paints sometimes exceeds 50 per cent of the surface resistance, or 25 per cent of the total drag, of clean ships. Information available for clarifying the effect of roughness on the boundary layer is, unfortunately, quite limited. As a matter of fact, most analyses have been based on the results of flow in pipes. Now there is no obvious reason why pipe flow and boundary-layer flow should be identical or even similar. First, a pressure gradient is essential for flow through a pipe but not along a plate. Second, pipe flow is confined and perforce uniform, while flow along a plate develops semi-freely and bears no such a priori guarantee of displaying similar velocity profiles at successive sections. Finally, the diameter and roughness size are the only geometrical dimensions of established flow in pipes, whereas at least three linear quantities are necessary to characterize the boundary layer. Evidently, it is urgently necessary to explore the effect of surface roughness on the boundary layer as systematically as already has been done for pipes. The purpose of the present investigation, therefore, is to establish the law of the effect of surface roughness on the mean velocity profile and on the surface-resistance coefficient of the turbulent boundary layer at zero pressure gradient for a wide range of roughness size. Because this is most easily accomplished with an artificial roughness of regular pattern, a secondary purpose is then the determination of the effect of a roughness distribution simulating actual prototype surfaces. Finally, it will be of practical interest to investigate the long-assumed similarity between roughness effects on the boundary layer and on flow in pipes or channels. In order to provide the answer to these and other roughness problems, a long-term project has been in progress at the Iowa Institute of Hydraulic Research since 1948. This report is a continuation of the boundary-layer surveys conducted by Baines (2) on a plate with sand roughness and by Moore (3) on plates with transverse-bar roughness of various sizes, both of which were summarized in Baines' subsequent review of boundary-layer literature (4). The present study first utilized plates roughened by geometrically similar wire screens covering a 28-fold range of scale, and comparison tests were then made with bars and screens in uniform channel flow. Finally, a series 2 Numbers in parentheses refer to the Bibliography at the end of the paper. of heterogeneous roughnesses was studied in the effort to simulate naturally roughened surfaces. This paper incorporates the experimental results into a unified analysis of boundary-layer resistance. NOMENCLATURE The following nomenclature is used in the paper: x--distance downstream from leading edge y--distance perpendicular to plate U--temporal mean velocity in x-direction at y Ul--temporal mean velocity in x-direction outside boundary layer 0--spatial mean velocity for pipe or channel flow A U--velocity decrement due to surface roughness k--roughness size (thickness of wire screen; root-mean-square deviation for simulated paints; height of square strips) k~--equivalent sand-grain size v--kinematic viscosity of fluid 3--nominal boundary-layer thickness 8*--displacement thickness 0--momentum thickness v*--shear velocity (v* = Ulx/dO/dx) cy--local surface-resistance coefficient at x(c I = 2dO/dx) C1--mean surface-resistance coefficient over x(cs = 2o/x) f--darcy-weisbach resistance coefficient for pipe and flume (f = 8Rg~h/O2L, g being the gravitational acceleration, R the hydraulic radius, i.e., flume depth or quarter diameter of pipe, and 3h change in piezometric head over distance L) Other symbols are defined in the text. REVIEW OF EXISTING KNOWLEDGE Boundary Layer Along Smooth Surfaces Although the present paper is primarily concerned with the effect of surface roughness, it is first essential to recapitulate what is known about the turbulent boundary layer developing along a smooth surface with zero pressure gradient, for this is fundamental to the more complicated problem of the turbulent boundary layer developing along a rough surface. A considerable amount of experimental information for the smooth case has been accumulated in the past several years by Schultz-Grunow (5), Hama (6), Ludwieg and Tillmann (7), Baines (2, 4), and Klebanoff and Diehl (8). Since the fundamental equations of motion cannot yet be solved for the boundary conditions in question, the information obtained in these experimental investigations is important. In addition, semi-

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 555 20 J J4" + mo-. " ppi0 U v,~ /0 =a e*.i fl 0..." o Laufer [4// ('dj * Ludw/eg & Tillmal~n [7] Oo I I / 2,3 ~/V '~ /og ~IG. 1 WALL LAW FOR THE VI~LOCITY DISTRIBUTION empirical analyses by Rotta (9, 10, 11), Hama (12, 14), Landweber (13), Clauser (15), and Coles (16, 17, 18) have helped to clarify the problem. The more' recent analyses of the turbulent boundary layer for smooth surfaces presume the existence of two universal laws, the wall law and the velocity-defect law. The wall law, which was first introduced by Prandtl (19) in connection with his mixing-length theory, states that the dimensionless velociw U/v* is a unique function of the dimensionless idistanee from the wall yv*/v in the immediate wall vicinity. Recently, Coles (17) claimed to have proved theoretically that the wall law must prevail if a universal relation between the velocity U and the distance y is to exist; this follows, nevertheless, from dimensional reasoning alone, once the pertinent variables are assumed. On the other hand, Schultz-Grunow (5) showed that a universal relation exists between the dimensionless velocity defect (U1 -- U)/v* and y/~ for the outer portion of the turbulent boundary layer, as was demonstrated previously for flow in pipes and channels by yon K~rm~n through his similarity concept (20). However, the nominal boundary-layer thickness ~ is not only arbitrarily defined but very difficult to determine aeeura.tely. Rotta (10, 11) found that the only possible well-defined thickness is ~*UJv* if the velocity defect (U1 -- U)/v* really has a universal relation to the dimensionless distance from the wall. The well-known logarithmic profile was obtained originally by the already obsolete mixing-length theory based on unreal assumptions. However, Millikan (21) derived the logarithmic law on the assumption that the velocity profile for pipe and channel flows in the region in which the wall law and the velocity-defect law overlap had to be commensurate with both if the two laws were to be universal. Clauser (15) showed the same to be true for the turbulent boundary layer, and propose d the formulas and U yv* v- ~ = 5.6 log--p + 4.9... [1] U1- U ( yv* ) v~ = -- 5.6 log~ + 0.6...[2] for this region from an examination of existing experimental data. Equation [1] expresses the wall law and Equation [2] the velocity-defect law; both apply to the same overlapping region. Because of the insufficiency of information on the turbulent boundary layer, arbitrariness still remains in the choice of the numerical factors involved, for different authors give different values. Nevertheless, the existence of an intermediate logarithmic portion of the velocity profile is well 'substantiated, and Clauser's' values in the foregoing formulas are adopted in the present paper as acceptable approximations. Deviations from the logarithmic profile, however, are definite for the inner part close to the surface and for the outer part near the essentially

336 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES /5 o~ /0 ~-U 5 o4 o 6mooth Rough I -2 V NX.\ -/ 0 FIG. 2 VELOCITY-DEFECT LAW TABLE 1 TURBULENT BOUNDARY LAYER ALONG SMOOTH PLATE Governing formula for velocity Subdivision Range of y profile Outer part.. 0.15 =< y/a = 1 l~q. [41 with [5] Logarithmic - ~yv*/a*u, < 0"0457 " Eqs. [1] and [2] part... Viscous zone (32.5 < yv*/v Y 0 <= yv*/v 6 32.5 Eq. [3] with a = 0.ii undisturbed flow. The universal velocity distribution in the immediate wall vicinity, where the viscosity cannot be neglected and which may hence be called the viscous zone, was given by the present author (6, 12) semi-empirically; that is 2{) -- w/1 --4-z 4 ( 1 )} =~ z~ +F ~,~.[3] in which ~o -- w/2 a(u/v*), z ~ x/2 a(yv*/u), cos ~b ~ (1 -- z2)/(1 4- z2), and F is the incomplete elliptic integral of the first kind. If the value of a is chosen as 0.11, Equation [3] smoothly connects with the logarithmic law of Equation [1] at yv*/v = 32.5. Equation [3] represents the universal velocity distribution in the viscous zone, including the laminar sublayer and transition region, for smooth surfaces. Equations [1] and [3] are shown in Fig. 1. A different method of repre- senting the velocity distribution in the transition region, it may be noted, was given by Hudimoto (22), Rotta (9), and Deissler (23). On the other hand, the outer part, which also differs from the logarithmic law, occupies most of the layer, and its velocity distribution can be given by the empirical formula (14) in which v~-- = 9.6 1 -- U1 a = 0.30 a* ~... [5] Equation [4] connects smoothly with the logarithmic velocity-defect law [2] at y/a = 0.15, or yv*/ a*u~ -- 0.045, contrary to Ross' 3/2-power law (24), which intersects the logarithmic law and which requires a somewhat ambiguous concept of a blending region. Equations [2] and [4] are shown in Fig. 2. Thus, the overlapping region, in which both logarithmic Formulas [1] and [2] hold, has as its outer limit yv*/6*u1 = 0.045 and as its inner limit yv*/v = 32.5. The complete picture of the velocity profile for the turbulent boundary layer along a smooth plate is seen from Table 1. The entire profile evidently consists of three parts. As dearly shown by Landweber (13), however, the logarithmic part

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 337 0 do jfl ZO J o llama }6j7 * 3chultz-Grunow Z~5~ 7 Ludwieg & 7711mann [7~-. W/egmrdt (rro. H) % 8 4 ~e l 5 6 FIO. 3 I~.ESISTANCE FORMULA DERIVED FROM VELOCITY LAWS IN COMPARISON WITH ]~XPERIMBNTAL RESULTS can no longer exist if the value of the Reynolds number UI~*/u is less than 725, for at this value the inner and outer limits of the logarithmic part coincide. This represents the lower limit of present knowledge, since the existence of the logarithmic part is the backbone of the recent analyses, and there is no information or method of analysis for Reynolds numbers below this limit. Since the velocity-defect law is universal, if the thickness of the viscous part is considered negligible the form parameter H can be found (4) as ~* - (1 -- G = - H= o. uu. ~2/,............. [6} in which the new universal form parameter is a function of the pressure gradient alone and approximately equal to 6.1 for the ease of zero pressure gradient (15). From Equations [1] and [2] and the definition equations for v* and c f, the local surface-resistance formula for smooth surfaces ~J~ = 5.6 log UI**+ 4.3... [8] ~[cr l.s is immediately obtained (15). By virtue of the relation for the form parameter of Equation [6], the foregoing formula can be transformed into ~c~+5.61og(1--g~) =5.61og U10p +4.3 The simple logarithmic formula ~cs Y... [9] is found to be an excellent approximation to Relation [9] for the Reynolds-number range of 102 < UiO/v < 106. The approximation is so close that they can hardly be distinguished in Fig. 3, in which both Relations [9] and [10] are plotted in comparison with experimental data. Squire and Young (25), by a combination of the K&rm~n asymptotic formula with the Prandfl- Sehliehting law, had already obtained a relationship of the same type as [10]. It may be of interest to mention here that all of the "reliable" formulas that exist for the local surface-resistance coefficient of the turbulent boundary layer along a smooth plate with zero pressure gradient can be reduced to the form ~J~ = A log--u10 +B... [11] v

338 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 40 3O % 20 J - - $choenherr [27].... 8chul/z-Grunow - - - co/es [SJ /0 2 Fla. 4 3 4 5 /og e COMPARISON OF LOCAL-SURFACE-REsISTANCB :FORMULAS 6 TABLI~ 2 NUMERICAL VALUES FOR A AND B Author Original formula A ~g U~* Clau~er (15)... ~ = 5.6 log --v + 4.3 5.45 Squire and Young (25) 4 2 = 5.89 log Uj _[_ 3.58,c~ p 5. 89 Prandtl and Schlichting (26, 37) Cs 0.455 -(log -UlxJ - = 5. 7 5 Schoenherr and Granville (27, 28) cs = 0.310 [ln2(2-uv~o) +21n (2 Ul~)J-15.85 Landweber (13)... Numerical 5.85 Schultz-Grunow, (5, 7). c, 0.334(log?0) - - -L838......... = 6. 3 0 Coles (16, 18)... Numerical 5.60 B 5.55 3.58 4.15 4.15 4.10 2.40 5.80 with extremely good approximation for the same Reynolds-number range as for Equation [10]. Numerical values of A and B for each formula are tabulated in Table 2. This at once provides a common basis for comparison. One can see from Fig. 4, in which some of the logarithmic formulas jl~st introduced are plotted, that the discrepancy between them over a wide Reynolds-number range is rather marked. This may have resulted from inaccurate determination of cf and O, insufficient pressure control, three-dimensionality of the flow, or the actual inadequacy of the distance x used in the original formulas as a primary variable. In fact, this distance has only a vague meaning for the turbulent boundary layer except in practmal applications, since the boundary layer is usually made turbulent artificially and its origin hence does not coincide with the leading edge. However, it should be noted that the scatter of experimental points is greater than the discrepancy between the formulas More refined measurement, especially the direct determination of boundary shear, to the end of establishing the "final" surface-resistance formula is urgently needed. Integration of Equation [11] with respect to 0 provides an asymptotic relation for the mean surface-resistance coefficient for large Reynolds numbers which is approximately of the form

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 559 do J / -zo 17 J I J /05 6 Fro. 5,, ~/A" /og 7-8 Fq. d3)'3ckoenherr [2# ---- q. 04).'//aI~/~pprox) -'-- Eq. ~5): Yughe3 [31] comparison OF Mt~AN-SURFACt~-R]?,SISTANCt~ FORMULAS I 9 /O 1 A U:O _.]_ b [12] = ~ log ~... A special case of Equation [12] is the K~rm~n- Sehoenherr law (27) 1 = 4.13 log g: O _[_ 1.24 = 4.13 log(c,... [13] which is widely accepted by praetical engineers. However, since the direct calculation of CI as a function of UlX/U is not possible, one has to rely on a chart or table for this purpose (29, 30). The author has found that the simple formula 1 _ 3.46 log U:x _ 5.60... [14] is an excellent approximation to the Schoenherr law [13], the'discrepancy being less than =t=1.8 per cent for the Reynolds-number range of 105 < U:x/u < 10:0 Beeause of its simplicity, this may find a versatile utility among practical engineers. It should be added that Hughes (31) recently obtained a formula of similar structure from his experimental results = 3.89 log U:x _ 7.90... [15] v/us ~ However, superposition of Formulas [13], [14], and [15] as in Fig. 5 shows that Hughes' formula is markedly different from Schoenherr's. Effects of Surface Roughness In analyzing the effect of surface roughness on flow in pipes, the ratio of the roughness dimension to the thickness of the laminar sublayer has long been accepted as the governing factor. Thus, if the roughness elements are so small that the laminar sublayer enclosing them is stable against the perturbation, the roughness will have no dragincreasing effect. This is called the effectively smooth case. On the other hand, if the size of the roughness is so large as to disrupt the laminar sublayer completely, the surface resistance will then be independent of the viscosity. This is called the case of fully developed roughness action. Between these two extremes there exists an intermediate region in which only a fraction of the roughness elements disturbs the laminar sublayer. Consequently, the resistance law in this intermediate region depends upon both the roughness magnitude and the thickness of the laminar sublayer. Since the quantity kv*/~ is proportional to the ratio between the roughness size k and the thickness of 11.6 ~/v* of the laminar sublayer, it must be the primary parameter for the analysis of geometrically similar roughnesses. Nikuradse (32), for example, conducted experiments on pipes roughened by uniform sands of various sizes and proved the consistency of this concept for a wide range of the roughness parameter kv*/u. On the

340 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 0 U v-~ I0.~,s t11"" / 00 / 2 J yg ~ /og -V--- FIG. 6 TYPICAL VELOCITY PROFILE FOR ROUGIt SURFACB other hand, Colebrook and White (33) demonstrated that the nature of the effect of surface roughness in the intermediate region depends as well on the geometrical characteristics of the roughness pattern; i.e., the spacing between sand grains and the composition of grain sizes. In order to systematize this difficult situation, Morris (34) recently proposed a new concept, in which the spacing of the roughness elements was introduced as a primary dimension instead of the usual roughness height k. Since it is quite certain that at least one additional dimension should be introduced, a mere replacement can be no better than the original parameter. For most commercial materials, such as wrought iron, cast iron, galvanized metal, etc., Colebrook (35), and later Rouse (36), showed that the roughness effect appears to be universal in the intermediate region. Prandtl and Sehlichting (37), in preparing their resistance chart for rough surfaces, proceeded on the basis of the Nikuradse data for pipes, because of lack of further information as to either the boundary-layer characteristics or the effect of roughness upon them. This required the tacit assumption that the roughness effect would be universal and independent of outside flow con- ditions. Even today available information to clarify the roughness effect on the boundary layer remains quite limited, including only the exploratory papers by Tillmann (38) and Wieghardt (39), and the more quantitative ones by Baines (2, 4), and Moore (33). Moore is so far the only investigator to have carried out boundary-layer experi- merits using controlled roughness sizes. Baines and Moore attempted to analyze their experimental results following the Prandfl-Schlichting method; they were not entirely successful, because use of the distance from the leading edge as in the Prandtl-Sehlichting analysis cannot be expected to yield a systematic result. The most significant conclusion in Moore's paper, which was based on tests with two-dimensional square-strip roughness, seems to be that the velocity-defect law is universal either for smooth or rough surfaces, provided that the origin of y is located at the proper point between the top and bottom of the roughness protuberances. Encouraged by this conclusion, Clauser (15) presented a method of analyzing the roughness effect which is the most consistent and useful to date. By virtue of Moore's conclusion, the velocitydefect distributions [2] and [4] hold either for smooth or rough surfaces. Hence, the relation for the form parameter [6] also holds. On the other hand, the velocity profile U/v* versus yv*/j, undergoes a parallel downward shift AU/v* for rough eases, as shown in Fig. 6, so that the velocity profile in the logarithmic part can be expressed generally as follows: U * AU [16] v~ = 5.6 log yw~, --t- 4.9 -- v- ~-... This downward shift of the logarithmic profile due to surface roughness had already been mentioned by Nikuradse (32) for the case of pipe flow. Instead of A U/v* itself, he plotted, essentially

I BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 341 40 d do RO.t t" /O " J / I / J / /.. (re;l-" I 1< ooo ".. 3 4 5 /ogv f / / / k" ~ j/,/ 1~" / 6 FIG. 7 SIGNIFICANCE OF THE QUANTITY A U/V* x v* 5.6 log 5.0 log kv* + 4.9 -- Lx U u v*... [17] as a function of kv*/v. It may be conjectured that x and AU/v* are functions of kv*/v for boundary layers, too. These quantities thus become alternative measures of the roughness effect. The author prefers A U/v* to X, because it is more directly related to the resistance increase, as will be shown, and can be evaluated more easily from the experimental results. From Equations [2] and [16] and the definition equations for v* and c~, the local surface-resistance formula for rough surfaces ~2 = 5.6 log UI~*u + 4.3 -- A Uv. ""[18] is immediately obtained. Subtraction of Equation [18] from Equation [8], for the same magnitude of U16*/u, provides which directly indicates an important meaning of the quantity AU/v*. The significance of Equation [18] or [19] is shown in Fig. 7. Values of AU/v* calculated by Clauser for Moore's experi- ments displayed good correlation with kv*/v, as shown in Fig. S. Since Nikuradse's x tends toward a constant value for large values of kv*/u, AU/v* has the corresponding asymptotic form A U kv* + const... [20] v* - 5.6 log in the fully developed roughness region, which is evidently the case for Moore's experiment. Consequently, the logarithmic velocity distribution in that region becomes U - 5.6 log y + const... [21] g* R Furthermore, substitution of Equation [20] into [18] gives ~2_ 5.61og~ ~ ~* @ = 5.6 log ~ + const... [22] which states that the local surface-resistance coefficient c~ is independent of the Reynolds number in the region of fully developed roughness action for a constant value of the ratio ~*/k, instead of the ratio x/k in the Prandtl-Schlichting resistance chart. Or, by means of Equation [6], Equation [22] also can be written as

(Moore) 342 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 30 2O AU Boundary Layer o Flume ~and) J J /0 f f f f f f O of f f f f J J / Z d A.v ~' /o9' V 4 FtG. 8 ROUGHNESS EFFECT A U/v* FOR TRANSVERSE BARS 0 5.6 log ~ + const... [23] which means that c I is independent of the Reynolds number for a constant value of ratio O/k as well. The primary aim of the present investigation is to verify the law of fully developed roughness action through use of roughness elements which are three-dimensional in form and which can be reproduced at various scales without losing geometrical similarity. This not only would confirm the general principle introduced by Clauser on the basis of Moore's two-dimensional rougbness tests but also would yield a dependable method for further analyses of the roughness problem. A secondary aim is to establish the long-conjectured universality of the roughness effect regardless of outside flow conditions by means of comparable experiments in uniform flow; the relatively vast amount of experimental data already obtained in rough pipes would then supplement the relatively limited information now available for rough plates. As previously described, the nature of the roughness effect in the intermediate region depends to a considerable degree on the roughness pattern; hence, the final purpose of the present investigation is to explore the effect of roughness distributions simulating actual prototype surfaces. The eventual goal of the long-term project conducted at Iowa is a detailed understanding of the roughness effect, to the end of being able to predict the resistance for any combination of flow and roughness characteristics. EXPERIMENTAL EQUIPMENT AND PROCEDURE Wind Tunnel I Experiments to accomplish the foregoing ends were performed in the same wind tunnel as that previously used by Baines and Moore. The tunnel was of the closed-circuit type, with a basic working section 5 ft wide, 5 ft high, and 22 ft long. The flow was produced by a propeller driven by a constant-speed, 65-hp electric motor through a Waterbury hydraulic transmission, which permitted both accurate control and continuous variation of the wind velocity. In order to regulate the longitudinal pressure distribution within the working section of the tunnel, adjustable ceiling and floor sections with a well-faired approach were added. The usable portion of the test section was thereby decreased to 3.7 ft in height and 12 ft in length, the maximum velocity being increased to 120 fps. Screens installed in the 12-ft approach section reduced the relative turbulence level in the test section to 0.15 per cent.

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 343 TABLE 3 SCREEN DIMENSIONS Number of meshes Mesh length, Wire diameter. Roughness Series per inch M, in. d, in. height k = 2d, ft d/m I 1 1 O. 207 O. 0345 O. 207 II 3 ~ O. 072 O. 0120 O. 216 III 9 ~,~ O. 023 O. 00383 O. 207 IV 28 ~8 O. 0075 O. 00125 O. 210 Instruments Velocities were monitored by a standard Prandtl-type Pitot-static tube located at the test section or by the differential pressure between two small piezometer holes before and after the contraction. Within the boundary layer, the velocity was determined from the difference in pressure between a stagnation tube and a static tube. The outside diameters of the two hypodermic needles used interchangeably as stagnation tube were 0.024 and 0.029 in. and that of the static tube was 0.095 in. Each stagnation tube, slightly S-shaped to permit boundary contact, was mounted on traversing carriage so that it could be moved longitudinallyand transversely in addition to the vertical traversing for the boundary-layer survey. The static tube was fixed to the carriage approximately 3~ in. above the plate surface. The minimum reading of the vertical traversing mechanism was 0.005 in. For all the pressure measurements, multitube manometers with 30 deg inclination were first used, ethyl alcohol being the liquid. The minimum reading of the manometers was 0.01 in. In the later stage of the investigations for the intermediate region, an improved type of Wahlen gage was used, kerosene and methanol being the liquids. The minimum reading of the manometer was then 0.001 in. of methanol. Surface Roughnesses After the preceding, more or less exploratory investigations by Baines and Moore, wire screen was chosen as the most effective type of surface roughness for the verification of the effect of scale upon the resistance law. The geometrical dimensions of the screens tested are given in Table 3. They are three-dimensional and geometrically closely similar to each other, and the ratio between the maximum to minimum size is 28. Through use of these artificial wire-screen roughnesses, a systematic survey of the effect of surface roughness on the turbulent boundary layer could be made over a sufficient range of the roughness parameter to provide a thorough substantiation of the analysis. All screens were 9 ft long and 34 in. wide. The 1-, 3-, and 9-mesh screens were tied down on 3 X 10-ft panels of 1/~-in. hardboard by stove wire, brass wire, and nylon thread, respectively. The ties were placed carefully in the shadow of wooves, so that they would not disturb the flow. The finest or 28-mesh screen was fastened on a 3 X 12- ft sheet of l~-in, plate, glass by double-coated Scotch talse. Each wire screen was attached to the panel or plate as flat as possible, but a slight waviness due to initial deformation could not be avoided. In the first stage of the experiment, the surface roughness started right at the leading edge; the hardboard panels had a circular leading edge, while an elliptic leading edge was fitted to the glass plate. In order to explore the effectiveness of the surface roughness in the intermediate region, an artificially thickened boundary layer was required, since a sufficiently fine yet geometrically similar screen was not commercially available. (Even if it h~td been available, undesirable initial waviness would have become far more serious than the primary roughness itself.) A moderate increase in boundary-layer thickness was obtained by extending the plate 10 ft ahead of the test plate and covering the first 6 ft with corrugated paper. Since the boundary layer requires a certain lengt h to adjust itself to a new boundary condition,'i.e., a particular surface roughness, the degree to which the layer should be thickened was restricted by the length of the test plate. In order to confirm the universality of the roughness effect independent of outside flow conditions, comparative experiments were carried out in a tilting flume by W. Rand and T. Sarpkaya of the Institute staff. The flume had glass side walls and a smooth steel bottom, with a working section 30 ft long, 2 ft wide, and 1 ft deep. The roughnesses tested were l~-in, square bars with 1-in. spacing and 28-mesh wire screen, which were identical to two of the roughnesses studied in the wind tunnel These were installed over the entire bottom of the flume. The measurements consisted of the depth, slope, and rate of discharge for various states of uniform flow. Although the bar and screen types of artificial roughness were the most convenient that could be devised for verifying the law of fully developed roughness action, they are seldom, if ever, en-

544 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES Z.5 0 H 2,0 O0 0 e Smooth i/schultz-grunow 3mooth ['Hc/ma [6]) Smooth ).., {Hama) o ~'ougnj S /,5 o 1,0 IO 15 2o Z6-3o FIG. 9 FORM PARAMETER H AS A FUNCTION OF LOCAL SURFACE-RESISTANCE COEFFICIENT countered in practice. Moreover, the roughness effect in the intermediate region can be expected to vary with the roughness pattern. In order to determine the nature of this effect, it was evident that more practical types of roughness, which are ordinarily irregular in both the normal and the longitudinal direction, would have to be tested. In naval circles particular importance is attached to the roughness effect of the hot- and coldplastic, zinc-chromate, and anticorrosive paints which are commonly applied to ship hulls (1). For wind-tunnel tests, however, the roughness size of such paints is too small to permit their investigation over the entire intermediate range. Therefore, test plates with artificial heterogeneous roughnesses, rather than the actual paints themselves, were prepared "oy carefully air-blasting fine glass beads of controlled size-frequency distribution onto a tacky varnished surface with the desired areal frequency and then applying several additional coats of sprayed varnish. Unlike the Nikuradse roughness, the resulting elements were neither uniform in size nor tightly clustered on the plate. In fact, the probability-density distribution of the roughness pattern so obtained was closely Gaussian, as was found to be the case for the available records of the natural paints (1, 40). In order to secure complete geometrical similarity, further statistical analyses of the roughness pattern of the natural paints were also attempted, but the available records were so limited that the convergence was unsatisfactory. Nevertheless, the artificial roughnesses thus prepared were fairly similar in geometry to the natural paints so far as could be judged from oscillograph records. The root-mean-square deviations from their mean elevation were 0.007 in. and 0.004 in., respectively. A forerunning plate of 2-ft length with a 1-in-thick circular leading edge was installed ahead of the test plate, the first 6 in. being covered by corru- gated paper. Two additional rough plates were also prepared in a similar manner, the one having a skewed probability-density distribution and the other a more pronounced areal dispersion of the glass beads. Experimental Procedures After installing the test plate, the movable boundaries of the tunnel were adjusted to produce a constant pressure along the working section. This was verified by means of a longitudinal traverse of the free stream with the static-pressure tube along the center line, the pressure distribution being found to be uniform within the limits of experimental error. The pressure distribution across the boundary layer normal to the plate was also measured, and the results were sufficiently

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 345 30 ZO-- Boundary Layer /'Harna) o Flume #arpha#o)../ z~u i/. ~" /6 f / J # y 0 0 /og Z,6./~' 3 4 FIG. 10 ROUGHNESS EFFECT AU/v* FOR WIRE SCREENS uniform to justify the assumption of constancy. The velocity profile at several sections off the center line was also measured, and the twodimensionality of the flow was confirmed within 1 per cent over a central zone at least 22 in. wide. For the wire-screen roughnesses, the velocity profiles were first measured at several positions relative to an individual mesh. A difference was found only in the immediate vicinity of the screen. Thereafter, all the velocity-profile measurements were made at fixed geometrical positions, nominally 2, 3, 4, 5, 6, and 7 ft from the beginning of each test plate. The corresponding absolute distance from the leading edge Was not essential to the fundamental analysis presented herein. In the flume experiments, the central third of the 30-ft length was used for the determination of the depth and the slope of the fully developed flow, a point gage reading to 0.001 ft being the indicator. The flow rate was measured by weir and orifice. The depth was never greater than 3 in., and the two-dimensionality of the flow was checked by velocity-profile measurements. ANALYSIS OF EXPERIMENTAL RESULTS From the velocity profiles measured in the wind tunnel, the displacement thickness and the momentum thickness were calculated. OI'he local surface-resistance coefficient and the shear velocity were then evaluated by graphical differentiation of the momentum-thickness development with respect to distance along the plate. In other words ( do [24] c~ = 2 \~] = 2~... It was at once confirmed that the velocity-defect distribution is universal for both smooth and rough surfaces, as shown in Fig. 2, in which some of the typical results are plotted. An immediate consequence of the universal velocity-defect law is the fact that the Relation [6] between the form parameter H and the local surface-resistance co- efficient c I is also universal. Fig. 9 confirms this conclusion for a very wide range of H. From the resulting values of c I and U16*/v for the wire screens, values of A U/v* were determined by means of Equation [18] and plotted in Fig. 10. This plot is convincing evidence that the quantity AU/v* is a function of the roughness parameter kv*/u alone for boundary layers as well as pipes. This would seem, at first glance, a quite natural conclusion. However, it involves more than a simple adaptation of the information already established for pipes. In the boundary layer the parameter kv*/v varies as the boundary layer develops with x, whereas it maintains a constant value in pipe or channel flow. In other words, Fig. 10 indicates that under conditions of fully de-

346 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES veloped roughness action A U/v* depends solely on the parameter kv*/u even when the latter is changing from section to section. Under the existence of the universal law of the wall, the roughness effect may be expected to be independent of the outside flow conditions, since it depends only on the parameter kv*/u. This expectation is confirmed by the flume data obtained by Rand and Sarpkaya with roughnesses identical to those tested in the boundary layer by Moore and the present author. Similar to the Relation [19] for the boundary layer, the relation for uniform flow has the form = - - [25]... 73"... in which the difference should be taken for the same value of v*r/~,. The values of AU/v* for the flume data were calculated by means of Equation [25] and the surface-resistance formula for the smooth flume v*r ~f 5.77 log + 3.05... [26] v and are plotted also in Figs. 8 and 10, in which excellent agreement is seen to exist between the two flow conditions for either roughness, at least in the zone of fully developed roughness action. This is truly an encouraging fact, as it indicates the universality of the quantity 5U/v* for the two otherwise dissimilar conditions. It thus seems to be safe to extend the universality to other boundary conditions as well; e.g., flow in pipes. As a matter of fact, such universality had long been conjectured but had never before been tested experimentally, because no comparative experiments on the effect of surface roughness in both uniform and nonuniform flow had ever been made. Since experiments in pipes or two-dimensional channels are much easier to make and have a higher accuracy than those in the boundary layer, the former have a certain advantage over the latter in determining the roughness effect AU/v*, which is sufficient as far as practical engineering problems are concerned. For the boundary layer, elaborate profile surveys and inaccurate graphical differentiations of the momentum-thickness distribution are required, while only the piezometrie gradient or surface slope, the rate of flow, and the cross-sectional dimensions are necessary measure- ments for the other. Furthermore, towing-tank tests with long planks or pontoons not only provide mean characteristics including ambiguous leadingedge conditions, but they are also inevitably ac- companied by three-dimensional effects of an indeterminate nature; if one uses a longer plate to minimize the former, the latter effect becomes worse,, and vice versa. Consequently, the author would recommend experiments in pipes or channels for further roughness investigations. The sole dependence of the quantity AU/v* on the roughness parameter kv*./u and its universality independent of outside flow conditions have now been well established for a series of geometrically similar roughnesses for conditions of fully developed roughness action. The nature of the roughness effect in the intermediate region, however, is different for different kinds of roughness pattern, as reported by Colebrook and White (33). On the other hand, Colebrook (35) and Rouse (36) have shown that most of the natural roughnesses appear to have an almost universal trend in the intermediate region. If it is assumed that the universality of the roughness effect applies to the intermediate zone as well, the experimental data for rough pipes can be used for the boundary layer or vice versa. For example, the Colebrook formula (35, 36) for the so-called natural roughnesses, such as wrought iron, cast iron, galvanized metal, and asphalt paints, can be expressed in the present notation as v- ~- = 5.66 log -t- 3.30 -- 2.92... [27] for which the surface-resistance formula for smooth pipes v*r ~ = 5.66 log q- 3.71... [28] lp is assumed. Equation [27] is shown in Fig. 11. The next important step is to investigate the effect of natural surface roughnesses, and in particular to determine what kind of roughness distribution produces the Colebrook type of inter- mediate region. The experimental results for the artificial random roughnesses with Gaussian probability-density distribution were analyzed in a similar manner to those for the wire-screen roughnesses. The quantity AU/v* is shown in Fig. 11 based upon the equivalent sand-grain size, the ratio of the equivalent sand-grain size to the rootmean-square value being 5 for these roughnesses. From the available records (1, 40) the author estimated the root-mean-square sizes of zincchromate and hot-plastic paints as approximately 0.0008 in. and 0.002 to 0.004 in., respectively. The corresponding equivalent sand-grain sizes of these two paints are 0.004 in. and 0.01 to 0.02 in., respectively. On the other hand, Todd (1) estimated th~ equivalent sand-grain sizes of these paints from actual ship tests as 0.0048 in. and

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 347 20 AU 7 /o 0 -/ FIG. 11.] Colebrook [,.t5] \ -- Nikurad~e [J2] ~ D.I Y /4v ~' /og 7 f J 2 3 ROUGHNESS EFFECT A U/v* FOR VARIOUS SURFACE CONDITIONS 0.0072 to 0.0175 in., respectively, which values are fairly close to those estimated from the present investigation. Although the geometrical similarity between the artificial random roughnesses and the actual paints is not fully guaranteed, and the equivalent sand-grain size is significant only in the region of fully developed roughness action, it seems safe to conclude that the remarkable increase in the surface resistance due to paints reported by Todd (1) is quite within reason. However, as seen from Fig. 11, the effect of the artificial random roughness in the intermediate region is quite different from that of the so-called natural roughnesses represented by the Colebrook formula and even from that of Nikuradse's sandgrain roughness. In order to find what kind of roughness pattern produces the Colebrook type of transition function, two other roughnesses were also tested, one having a skew probability-density distribution and the other being dispersedly spaced grains. Both results had characteristics similar to those of the previous Gaussian roughnesses, and resolution of the problem as to just what produces the Colebrook type of function hence remains the primary task of the next phase of the Institute project. The ultimate goal of boundary-layer analysis, of course, is the.preparation of a resistance diagram for practical use. As Equation [19] and Fig. 7 show, the local surface-resistance law for rough surface is given by parallel lines shifted downward by an amount AU/v* from that for the smooth surface in a plot of x/2/c s versus UI~*/v, kv*/u being the significant parameter. From this family of curves of constant kv*/v, the corresponding family for constant values of ~*/k can be drawn by graphical interpolation. In a similar manner, the family of curves for constant Ulk/v also can be obtained. A similar chart can be drawn with respect to the momentum thickness 8 instead of the displacement thickness ~*, as shown in Fig. 12. No less significant than the former, this chart is considerably handier, particularly for '~determining the mean surface-resistance coefficient from the boundarylayer development over a given longitudinal distance. ~, The significance'0f each group of curves in Fig. 12 is the following : :(a) Any curve for which the roughness Reynolds number Ulks/v is constant represents that according to which the boundary layer along a particular surface of constant roughness actually develops. (b) Constancy of the quantity k~v*/v indicates that the degree of roughness effect is the same for all points on the curve no matter what the Reynolds number is; for example, one of these curves, for which ksv*/v = 60, marks the border between the intermediate region and the region of fully developed roughness action. (c) The group of curves for which O/k~ is constant, though comparable in form to those for pipe flow, merely indicate a fixed geometric ratio between the momentum thickness and the roughness magnitude; however, it is these curves which become independent of the Reynolds number in the region of fully developed roughness, and they are hence of particular use in evaluating the roughness magnitude from actual tests. For a plate uniformly covered by the same roughness, the ratio O/ks necessarily increases

348 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES Z J C-"" FIG. 12 SCHEMATXC LOCAL-StrRFAc~-REsmTANCB CHART FOR SMOOTH AND ROUGH StrRFAeEs from the front to the rear (i.e., along a line of constant Ulks/v), since the boundary-layer thickness increases with increasing distance x. Correspondingly, the parameter k~v*/v decreases from the front to the rear as the local surface-resistance coefficient decreases. Therefore, a boundary layer which begins as one with fully developed roughness action near the leading edge gradually passes into the intermediate region and, finally, if the surface is long enough or the roughness size is small enough, into the effectively smooth region. For flow in a rough pipe, on the contrary, since the diameter-to-roughness ratio is fixed, the resistance coefficient, would follow a line on which R/ks, rather than O/ks, is constant. Consequently, as the pipe Reynolds number increases, the effectively smooth region comes first, followed by the intermediate region, and then by the region of fully developed roughness action. Based upon this chart for the local characteristics, a general diagram of the mean surface-resistance coefficient as a function of the Reynolds number can be drawn as shown in Fig. 13. This is essentially similar to the Prandtl-Schlichting chart and to that normally used in naval research. Such a diagram differs from that on which it is based in two ways: (a) The coefficient refers to the total resistance over the distance x rather than to its local intensity at a given value of x. (b) The distance x thereby becomes the significant length used in the several key parameters. As a particular result of the latter difference, at least one of these parameters takes on added significance. The lines in Fig. 13 along which Ulk~/v is constant again indicate the development of the boundary layer from section to section on a given surface at constant velocity, progressing from fully developed roughness action (if the surface is sufficiently rough) through the intermediate region to effective smoothness (if the: surface is sufficiently long). Any line of constant :ksv*/v, of which the line marking the limit of the intermediate region is typical, likewise indicates a locus of. points of similar roughness action. But lines of constant x/ks now take on added significance, for they represent the variation in the resistance of a given surface as the velocity is changed; in this instance the progression is from effective smoothness (if the roughness is not too great) to fully developed roughness action (if the velocity becomes sufficiently high). The author firmly believes that a clear understanding of each kind of diagram and of each type of function that it portrays is essential to the proper interpretation of laboratory and field data. In this regard, the fact should be emphasized strongly that the forms of the curves may be expected to vary in the intermediate region with the characteristics of the roughness distribution. The probable upper limit of such variation is indicated in Fig. 13, by the Colebrook transition curves shown as broken lines. Only if it is eventually found that all natural surfaces of practical interest yield the same transition function will it be possible to prepare a universal resistance diagram. CONCLUSION Analysis of experimental investigations conducted on the boundary layer along a plate covered by a series of artificial roughnesses of controlled

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 349... ~.. " ---~ ~-----" --.-~, @ " - - - f~.-2~_ ~. "7--- - - ~ -.-~--- - - - Co/ebrook x :~_~. ~ ~ /.~/X /og FIG. 13 SCHEMATIC MEAN-SURFACE-RESISTANCE CHART FOR SMOOTH AND ROUGH SURFACES geometric characteristics has led to the following conclusions: 1 Two universal laws, the wall law and the velocity-defect law, and an intermediate region common to the two in which the velocity profile is logarithmic, are essential features of the turbulent boundary layer. The velocity-defect law is uni- versal for either smooth or rough surfaces, whereas the logarithmic velocity-distribution curve is shifted by an amount &U/v* depending on the magnitude of the roughness effect. 2 For geometrically similar roughnesses, the quantity &U/v* is a function of the roughness parameter kv*/u alone and is directly related to the increase in surface resistance which the roughhess produces. Knowledge of this function is thus sufficient for determining the resistance coefficient. 3 The long-conjectured universality of the roughness effect regardless of outside flow conditions has now been substantiated--at least for the case of fully developed roughness action at zero pressure gradient. Hence, the relatively vast amount of experimental data already obtained for rough pipes should be useful in' extending the relatively limited information available for rough plates. Moreover, boundary-layer and towingtank tests might conveniently be supplemented-- or even replaced--by simpler and more accurate experiments in pipes or channels. 4 The nature of the effect of the roughness pattern--particularly that of natural roughnesses--still requires systematic investigation. However, the pronounced increase in surface re- sistance produced by certain paints as reported by Todd appears to be within reason. ACKNOWLEDGMENTS This work is a part of a long-term project conducted by the Iowa Institute of Hydraulic Research under the general direction of Dr. Hunter Rouse and sponsored by the Office of Naval Research through contract NSonr-500. The author is greatly indebted to Dr. Rouse for his continuous advice and encouragement. The author also acknowledges the assistance of Mr. M. F. Andrews in the conduct of the boundary-layer experiments, of Mr. S.-C. Ling in the preparation of the rough plates, and of Drs. W. Rand and T. Sarpkaya in the conduct of the flume experiments, as well as the critical suggestions provided by Dr. F. H. Clauser, of the Johns Hopkins University, and by Dr. L. Landweber, of the David Taylor "Model Basin. BIBLIOGRAPHY 1 "Skin Friction Resistance and the Effect of Surface Roughness," by F. H. Todd, Trans. SNAME, vol. 59, 1951, p. 315. 2 "An Exploratory Investigation of Boundary-Layer Development on Smooth and Rough Surfaces," by W. D. Baines, Ph.D. dissertation, State University of Iowa, 1950. 3 "An Experimental Investigation of the Boundary-Layer Development along a Rough Surface," by W. F. Moore, Ph.D. dissertation, State University of Iowa, 1951..

350 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 4 "A Literature Survey of Boundary-Layer Development on Smooth and Rough Surfaces at Zero Pressure Gradient," by W. D. Baines, Project Rep., Iowa Inst. Hydraulic Res., State University of Iowa, 1951. 5 "Neues Reibungswiderstandsgesetz ffir glatte Platten," by F. Schultz-Grunow, Luftfahrt- Forschung (NACA TM 986), vol. 17, 1940, p. 239. 6 "Turbulent Boundary Layer along a Flat Plate," by R. Hama, Parts I and II (in Japanese), Rept. Inst. Sci. Techn., University of Tokyo, vol. 1, 1947, pp. 13 and 49. 7 "Untersuchungen fiber die Wandschubspannung in turbulenten Reibungsschichten," by H. Ludwieg and W. Tillmann, Ingenieur-Archiv (NACA TM 1285), vol. 17, 1949, p. 288. 8 "Some Features of Artificially Thickened Fully Developed Turbulent Boundary Layers with Zero Pressure Gradient," by P. S. Klebanoff and Z. W. Diehl, NACA TM 2475, 1951 (superseded bytr 1110). 9 "Das in Wandn~ihe gfiltige Geschwindigkeitsgesetz turbulenter Str6mungen," by J. Rotta, Ingenieur-Archiv, vol. 18, 1950, p. 277. 10 "l)ber die Theorie der turbulenten Grenzschichten," by J. Rotta, Mitteilungen Max- Planck-Inst. ffir Str6mungsforschung, G6ttingen, No. 1, 1950 (NACA TM 1344). 11 "Beitrag zur Berechnung der turbulenten Grenzschichten," by J. Rotta, Ingenieur-Archiv, vol. 19, 1951, p. 31. 12 "On the Velocity Distribution in the Laminar Sublayer and Transition Region in Turbulent Shear Flows," by F. R. Hama, Journal of the Aeronautical Sciences, vol. 20, 1953, p. 648. 13 "Der Reibungswiderstand der l~ngsangestr6mten ebenen Platte," by L. Landweber, Jahrbuch Schiffbautech. Ges., vol. 46, 1952, p. 137. "The Frictional Resistance of Flat Plates in Zero Pressure Gradient," by L. Landweber, Trans..SNAME, vol. 61, 1953, p. 5. 14 Discussion to Ref. 13, by F. R. Hama, 1953. 15 "Turbulent Boundary Layers in Adverse Pressure Gradients," by F. H. Clauser, Journal of the Aeronautical Sciences, vol. 21, 1954, p. 91. 16 "The Problem of the Turbulent Boundary Layer," by D. Coles (in English), Zeitschrift fi~r angewandte Mathematik und Physik, vol. 5, 1954, p. 181. 17 "The Law of the Wall in Turbulent Shear Flow," by D. Coles, GALCIT, 1954. 18 "Measurements in the Boundary Layer on a Smooth Flat Plate in Supersonic Flow--I. The Problem of the Turbulent Boundary Layer," by D. Coles, JPL, CIT. Rep. No. 20-69, 1953. 19 "The Mechanics of Viscous Fluids," by L. Prandtl, Article G, vol. 3; "Aerodynamic The- ' ory," edited by W. F. Durand, 1934. :20 "l~ echanische "A_hnlichkeit und Turbulenz," by Th. yon Kgmn(m, Nachr. Ges. Wiss. G6ttingen, Math.-Phys. Klasse, 1930, p. 58, (NACA TM 611). 21 "A Critical Discussion of Turbulent. Flows in Channels and Circular Tubes," by C. B. Millikan, Proc. 5th I.nternational Congress of Applied Mechanics, 1938, p. 386. 22 "On the Turbulent Boundary Layers," by B. Hudimoto (in Japanese), Trans. Japan Society of Mechanical Engineers, vol. 7, part III, 1941, p 1. 23 "Analytical and Experimental Investigation of Adiabatic Turbulent Flow in Smooth Tubes," by R. G. Deissler, NACA TN 2138, 1950. 24 "A Study of Incompressible Turbulent Boundary L)dyers," by D. Ross, Ph.D. dissertation, Harvard University, TM NR 062-139-1, Pennsylvania State College, 1953. 25 "The Calculation of the Profile Drag of Aerofoils," by H. B. Squire and A. D. Young, ARC R & M 1838, 1937. 26 "Grenzschicht-Theorie," by H. Schlichting, Karlsruhe, 1951. 27 "Resistance of Flat Surfaces Moving Through a Fluid," by K. E. Schoenherr, Trans. SNAME, vol. 40, 1932, p. 279. 28 "A Method for the Calculation of the Turbulent Boundary Layer in a Pressure Gradient," by P. S. Granville, David Taylor Model Basin Report 752, 1951. 29 "Recommended Definition of Turbulent Friction in Incompressible Fluids," by F. W. S. Locke, Jr., NAVAER DR Report 1415, 1952. 30 "A Reanalysis of the Original Test Data for the Taylor Standard Series," by M. Gertler, David Taylor Model Basin Report 806, 1954. 31 "Friction and Form Resistance in Turbulent Flow, and a Proposed Formulation for Use in Model and Ship Correlation," by G. Hughes, Institution of Naval Architects, London, preprint, 1954. 32 "Str6mungsgesetze in rauhen Rohren," by J. Nikuradse, VDI Forschungsheft, no. 361, 1933, (NACA TM 1292). 33 "Experiments with Fluid Friction in Roughened Pipes," by C. F. Colebrook and C. M. White, Proc. Royal Society of London, Series A, vol. 161, 1937, p. 367. 34 "A New Concept of Flow in Rough Conduits," by H. N. Morris, American Society of Civil Engineers, vol. 80, 1954, separate 390. 35 "Turbulent Flow in Pipes, with Particular

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 35l Reference to the Transition Region between the Smooth and Rough Pipe Laws," by C. F. Colebrook, Journal of the Institution of Civil Engineers, London, vol. 11, 1939, p. 133. 36 "Evaluation of Boundary Roughness," by H. Rouse, Proc. 2nd Hydraulics Conference, State University of Iowa, 1943, p. 105. 37 "Das Widerstandgesetz rauher Platten," by L. Prandtl and H. Schlichting, Werft, Reederei, T[afen, vol. 15, 1934, p. 1. 38 "Untersuchungen fiber Besonderheiteh bei turbulenten Reibungschichten an Platten," by W. Tillmann, Kaiser-Wilhelm-Institut ffir Str6mungsforschung, UM 6627, 1945. 39 "Zur Reibungswiderstand rauher Platten," by K. Wieghardt, Kaiser-Wilhelm-Institut ffir Str6mungsforschung, UM 6612, 1944. 40 "Note on Studies of the Resistance of Hydraulically-Rough Surfaces," by P. Eisenberg, David Taylor Model Basin Report 726, NS715-086, 1950, p. 1. 41 '!The Structure of. Turbulence in Fully Developed Pipe Flow," by J. Laufer, NACA TN 2954, 1953. Discussion DR. DONALD ROSS,*.Visitor: The Society is indeed fortunate in receiving this excellent paper on the effects of roughness on turbulent boundary layers. This paper, together with Dr. Landweber's paper (13), gives a comprehensive picture of the relationship of boundary-layer characteristics to turbulent skin friction for smooth and rough plates. It should be clear now, even to the most confirmed skeptic, that these boundarylayer investigations are at last putting a firm scientific foundation under our empirical skinfriction formulas. The writer (24) has approached equilibrium boundary-layer problems in a different manner than Rotta, Landweber, and Clauser, as presented by the author. However, the results are essentially the same, and we need not dwell on these small differences at this time. They are mentioned in the discussion of Dr. Landweber's paper (13). The writer agrees with the author that his selection of the second power in Equation [4] is best for flat-plate boundary layers. The writer (24) chose the 3/2-power formula because of the desire of fitting a much larger selection of profiles including those near separation, for which the second power gives a miserable fit. The major point of the paper is the demonstration of the universality of the effects of surface roughness. Having shown this experimentally, the author suggests the use of experiments in pipes and two-dinlensiona.1 channels to determine roughness effects on plates. The writer enthusiastically endorses this point of view and submits the following physical analysis to support it: The universal wall law for the velocity distribution near a smooth surface (Fig. 1 of the paper) is a consequence of the existence of the thin laminar sublayer in which viscous forces predominate. The logarithmic profile (Equation The Bell Telephone Laboratories, Ine., Whippany, N.J. [1 ] of the paper) extrapolates to zero velocity at a distance from the wall which is a fixed fraction 3, of the laminar sublayer thickness, 6t. From Equation [1], and the expression for the thickness of the laminar sublayer, we have u /a.r* V* 5.6 log --v + 4.9 = 5.6 log_ (11.6 ~) -p 4.9 = 0... [29] whence ~ ~ 0.010 for all smooth surfaces. In the case of a fully rough surface, the laminar sublayer plays no part, if it exists at all. Instead, the roughness elements govern the innermost flow, and the logarithmic velocity distribution extrapolates to zero velocity at a distance, ak, that is, a function of the roughness height and the type of roughness. For the fully rough surface, Equation [16] applies and extrapolation yields. U akv* A U v- ~ = 5.61Og--v + 4.9 v* 0...[30] whence we may derive an expression for the downward shift of the logarithmic profile AU_ (4.9+5.61oga)+5.61ogkV* [31] 17. - p ". Here ~ is not a universal constant, but is a function of the type of geometric roughness and the definition of the roughness height. Thus, this equation fits the Iowa experimental data if we use ~ = 0.117 for the transverse bars and a = 0.22 for the wire screens. The difference between Figs. 8 and 10 of the paper is explained in terms of the different values of the coefficient a characteristic of the different types of geometrical roughness. For the natural roughnesses, a 0.04. The important result of this derivation is the

352 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES confirnlation of the author's major result; namely, the sole dependence of the velocity shift on the roughness Reynolds number, kv*/v, even when the latter is changing from section to section. As a can only be a function of the geometry of the roughness, it follows that Equation [:31] of this discussion is equally valid for plates, channels and pipes. It also should apply equally well to flows involving sizable pressure gradients. The problem of fully rough, boundary-layer friction is reduced to the experimental evaluation of the coefficient a for the various types of roughness. One word of caution concerning the relation of pipe and flat-plate boundary layers. In cor- relating these two flows it is important to use the correct cylindrical co-ordinate definitions of the displacement and momentum thicknesses 8" = ~ lfo ( 1 -- ~ ") (R -- y) dy... [321 0 = ~ 1 -- u u (R-- y) dy.[33] The writer has shown in a recent paper* that when these definitions are used Fig. 3 and Equation [10] of the text are equally applicable to Nikuradse's smooth-pipe friction data. As the shape parameter H is also a universal function of the proper momentmn-thickness Reynolds number, it follows that the author's Fig. 9, showing H as a function of cf, is universal for pipe and plate flows. In conclusion, the author is to be complimented for confirming the unity of all types of boundarylayer flows and for recognizing the practical importance of this unity to the naval architect. DR. S. F. HOERNER, l],[ember: This paper is extremely interesting. Following are some comments: (a) Transition to Fully Developed Roughness Flow. I. Nikuradse (NACA Teeh. Memo 1292) and Hama found a dip in the function of resistance coefficient against Reynolds number, evidently on uniformly rough surfaces. Other investigators, such as Moody (Trans. ASME, 1944), Colebrook (Proe. Royal Society, London, Series A, 1937) and Schultz-Grunow (Yearbook STG, 1938) did not find the dip in their experiments on evidently nonuniform, but mostly "natural" roughnesses. It is suggested that a possibly small number of single and larger grains makes the dip disappear. The author's attempts with respect to nonuniform a "A New Analysis of Nikuradse's Experiments on Turbulent Flow in Smooth Pipes," by Donald Ross, Proceedings of the Third Midwestern Conference on Fluid Mechanics, University of Minnesota, 1953, pp. 651-667. grain sizes were possibly not bold enough to produce the "natural" characteristics. Evidently certain coatings and other physical surfaces do not have a Gaussian distribution. Rather, they may consist of "wavy" but more or less smooth areas combined with a number of higher, but single protuberances. (b) Roughness Concentration. ]~nowledge ota roughness has been greatly advanced by the sandgrain technique. Unfortunately, however, there is hardly any physical or natural surface condition which is truly equal or similar to sand roughness. One conclusion from sand experiments has been the expectation that from then o11 every rough surface should have a constant terminal drag coefficient. As early as 1924, it has been demonstrated (by Hopf and Fromm in Zeitschr. Angew. Math. Mech., 1923, pp. 329 and 339) that certain types of roughness do not show any constant coefficients. Recently this discusser has suggested (p.'497 of the Journal of the American Society of Naval Engineers, May, 1954) correlation among the available material on the basis of the roughnessgrain density or concentration. It is found that a constant drag coefficient applies only to concentrations higher than some 60 or 70 per cent of the surface considered. For smaller concentrations, the slope of Cf(R) is between that of the smoothturbulent function and zero (as for "fully" rough condition). It is believed that such a mechanisnl is the only realistic one, suitable for predicting the frictional resistance of ships at full scale. It is hoped that the author may have the opportunity to continue his research in the direction of roughness having low concentration. DR. DONALD COLES, 5 Visitor: The author is to be congratulated on a thorough demonstration that turbulent boundary-layer flow over rough surfaces can be treated by methods which have been successful in other phenomenological studies of the effects of roughness. There is considerable point in experimental investigations of the drag and mean-velocity distribution associated with regular two-dimensional roughness, particularly a sinusoidal wall contour. As is pointed out in the paper, there is yet no completely satisfactory way of representing, or for that matter even of measuring, the statistical properties of random roughness in terms for example of the moments of the distribution function for an arbitrary nearly plane surface y(x, z). If the drag of a rough surface is ever to be predicted independently of measurements off similar surfaces under similar conditions, more attention should 6 Guggenheim Aeronautical Laboratory, California Institute of Technology, Pasadena, Cal.

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 355 be paid to the simplest kinds of regular roughness. In this connection, the writer would like to take exception to a statement in the section "Review of Existing Knowledge" of the paper. What the writer claims to have proved in reference (17) is the following: For two-dimensional incompressible mean flow past a smooth plane surface, the necessary and sufficient conditions for a universally valid wall law are: (a) Both velocity components vanish at the wall; (b) the friction is Newtonian at the surface; (c) the ratio U/v* is constant on streamlines of the mean flow. The edge of the sublayer is thus a mean streamline. This result provides a means for tentative generalization of the wall law to other geometries or to compressible flow, a means which is free of the ambiguities arising in the application of diniensional analysis to systems with one or more ad- ditional parameters. In particular, it is possible in principle to determine explicitly the way in which certain kinds of regular roughness should affect the law of the wall in the transition region between completely smooth and completely rough surface conditions. PROF. L. TROOST, Member: The merit of this paper is the establishment of a systematic method of analysis of roughness effect on the velocity profile of the boundary layer and, therewith, on the wall-friction coefficient. It is experimentally shown that this effect is universal as well for the flat plate as for pipe and channel flow. On the strength of this confirmation, the author concludes that boundary-layer and towing tank tests might be supplemented or even replaced by simpler and more accurate experiments in pipes or channels. This recommendation can be supported heartily in view of the fact that a group of well-known German scientists has done exactly this sort of thing, and for the very reasons, in the years from 1935 to the beginning of the second World War. In Professor Prandtl's laboratory in G6ttingen, a flow channel was built under his direction by H. Schlichting, in order to define the flow resistance of various kinds of practical ship roughness. It had a rectangular section of 6.7 X 1.57 in. Over the measuring length of about 10 ft, one of the smooth walls with a width of 6.7 in. could be replaced by a rough plate. By letting the water flow with various speeds and by measuring the pressure decrease past the channel wall and the velocity distribution in the channel center, the specific resistance of the surface under test could be determined for various and quite high Reynolds numbers. With this installation, Schlichting and Schulz- Grunow have measured a great number of systematic roughness configurations and also roughnesses of less density than the Nikuradse sand roughness ("moderate roughness"). A description of the channel and a report on the first experiments has been given by H. Schlichting in two German papers which have been translated into English and published in the United States (Transactions ASME, 1936). The reports of Schulz-Grunow's measurements on "moderate roughness" configurations, and on actual ship's hull roughness, also have been published (Schifffahrtstechnische Forschungshefte, October, 1936, and the Transactions of the Sehiffbautechnische Gesellschaft, 1938).,Since these references do not appear in the author's bibliography, it might be of advantage to take notice of these additional data. DR. F. H. TODD, dl~rember: The problem of the skin-friction resistance of smooth and rough surfaces is of special importance to naval architects for two principal reasons: (a) The variation of the resistance of smooth surfaces with absolute size of surface and with speed, i.e., with Reynolds number, forms the basis for the correlation of model resistance experiments on different sized models and for the prediction of ship resistance from that of the model. (b) Having predicted the smooth ship resistance by this means, the naval architect is then faced with the.question of making a proper allowance for the roughness of the actual ship hull. The first of these problems is one which is very much alive today. The International Towing Tank Conference (ITTC) since 1948 has asked all members, in their published work, to use either the Froude friction coefficients or the Schoenherr formulation. Both of these methods make the assumption that the viscous drag of a curved ship surface is the same as that of a flat surface of the same length and area. This was the basis of Froude's original justification for the use of models in this work, and has stood the test of time so well that it is still the basis of all model basin predictions today, after some 80 odd years. However, it has been realized for many years that while it may be sufficiently accurate for engineering estimates it is basically not correct, and that some of the remaining model resistance, called residuary by Froude, is really viscous in nature, and should be scaled by Reynolds law rather than Froude's. Many proposals to accomplish this have been made in th~ past, and the investigator today is faced with an embarrassing number of friction lines and methods. The Skin Friction Committee of the ITTC at the recent meeting in Scandinavia was instructed to consider all the relevant data and prepare pro-

554 BOUNDARY-LAYER.CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES posals for a satisfactory engineering solution by the next conference in 1957, and we are already struggling with this problem. As chairman of the committee, the writer therefore read this paper with great interest, especially to see what light it might throw on these two basic problems. So far as the first is concerned, the extrapolation from model to ship Reynolds numbers, it does not give any new data, but suggests a new approximate formula, Equation [14], to fit the Schoenherr line reasonably well, and which expresses C I explicitly rather than implicitly as does the Schoenherr formula. The values of C I and R derived from the latter have been published so frequently and at such small increments of R that the writer does not believe there is any great need today for calculating Ci from any formula. As regards the second problem, the effects of roughness on resistance, if the writer understands conclusions 1 and 2 aright, the effect of roughness is to reduce the nondimensional velocity U/v* at the same Reynolds number yv*/v, from its value for a smooth surface, by a constant amount 2XU/v*. This decrement zxu/~* is found to be a function of the roughness parameter, kv*/v, and when the form of the function is known, the author states that the resistance coefficient for the rough surface may be determined. This would seem to offer a means of correlating roughness of a specific character with the measured increase of resistance and eventually to the prediction of the latter for new surfaces of known character. The Taylor Model Basin has a number of paint surfaces under test at the present time on a 20-ft friction plane, and also work on roughness is being carried out for the Basin under contract. It will be interesting to apply these methods to the results to see if by this means we can come to a better understanding of the Navy's paint problem. In Fig. 13 of the paper the lines showing Cf for constant values of the parameter x/k~ presumably apply to a ship of given length x, and given roughness k~, over a range of speed. These curves for the higher ranges of Reynolds number give a constant value of Cf, which is not in accord with the ship data or the friction-plane data as presently available at TMB, where the increment of Cf above the smooth line appears to be nearly constant rather than the total Cf. It would be interesting to have the author's views on this point. The author has made reference to some results which the writer presented to the Society in 1951, and he is glad to know that the author has found the very large increases in resistance as measured on actual ship trials to be quite compatible with his work on the effects of roughness, and we look forward to reading of his further work in this very important field. MR. MARSHALL P. TULIN, 6 Visitor: The author has provided an excellent scientific review of some aspects of turbulent boundary-layer flow along smooth and rough surfaces. It should be noticed that in this paper an important fact about smooth-plate flows has again been implicitly emphasized; that is, that the Schoenherr friction line may be derived on the basis of modern boundary-layer theory combined with experimental results describing the shape of the boundary-layer velocity distribution, and need not, therefore, be justified solely on the basis of flat-plate towing tests. It is to be hoped, as does the author, that more careful and cleverer experimentation will resolve the differences involved in the comparison of the variously derived friction curves of Fig. 4, but the possibility cannot be overlooked, and there is even some reason to believe, that such experimental studies will only lead to an understanding of how usually ignored influences such as the outer stream turbulence level and previous flow history have slight but perceptible influence on the boundary-layer and resulting frictional resist- ance. At any rate, reflection on the other complexities involved in boundary-layer flow on actual ship forms should convince us that such irksome questions as those inherent in the set of curves of Fig. 4 are not, by themselves, the major obstacles in the way of improved resistance prediction methods. One of those "real-ship" complexities mentioned is the influence of the ship hull roughness on the frictional resistance, this being the motivation for the major part of the author's very fine experi- mental research. People who must make resistance predictions may well ask what the meaning of the roughness curves like those represented in Fig. 13 is for their problems, and it is probable that the boundary-layer scientist should reply: "These curves, although otherwise important, are not directly useful for the ship roughness problem." This follows frohl the fact that the way in which the roughness influences the boundary layer and finally the friction curve depends very much on the ratio of the lateral spacing of the roughness to its mean height, and only for a limited although much studied range of values of this ratio does the roughness produce the effects illustrated in Fig. 13. For spacing ratios on the high side of this range, roughness curves above and more parallel (than in Fig. t3) to the smooth 6 Mechanical Branch, Office of Naval Research, Washington, D.C.

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 355 friction line, and thus much more resembling those found in ship trials are obtained. Incidentally, the work of Morris, part of which is mentioned as reference (34) in the paper and together with which should be mentioned Morris' original thesis at the University of Minnesota (1950), although empirical and speculative in nature is a comprehensive study and quite revealing with regard to the variety of ways in which different roughness distributions manifest themselves as increased resistance. For the future it seems a pressing need that from a variety of ship roughness records, spectra or suitably defined scales be determined electronically and then analyzed with regard to the lateral spacing of the roughness. A start toward this end has been made at the David Taylor Model Basin. It is especially useful that the author's results put a sounder foundation under suggestions to carry out boundary-layer roughness research in pipes and channels--for such facilities have been and probably will be used to advantage in the future. For those who wish especially to study real ship roughness in any kind of facility, a word of warning is her6 recorded. There may be very large differences between the actual roughness of a ship hull as painted in a shipyard dry'dock and the roughness of a sample plate prepared in a laboratory, even though the paints are identical, and carefully specified methods of application are utilized in both cases. The influence of the unpainted condition of the ship's plates, of the presence of sand in dry docks as the residue of sandblasting operations, and of temperature and gravity effects, all may contribute toward creating large differences between actual and synthesized roughnesses. Finally, it should be remarked that workers in the field of naval architecture research must feel grateful to the author for turning his attentions and talents toward problems of importance in the field of naval architecture. We all hope that his interest in such problems will continue, and that we shall hear from him again. I)R. F. H. TODD, ~lember: I should like to add just one note in extension of something that Mr. Tulin said in his discussion. He remarked upon the effects of various other matters on roughness, such as the way the paint was applied and I think this can be well illustrated by the fact that on the naval ships for which we have full-scale trial data the increase in resistance due to using, say, hot plastic paint, varies quite largely. We have ships in which the increase in resistance when they are painted with hot plastic--i am talking of the frictional resistance--has been increased by any- thing from 15 to,50 per cent over that with zinc chromalt and this variation we believe is due to the way the paint is applied, the conditions of temperature at the shipyard, and so on, much in the way that Mr. Tulin mentioned. I just wanted to give these rough figures to show how great that variation can be. MR. F. EVERETT REED, Member: Within recent months I noticed an article on boundary-layer research entitled "Turbulent Processes as Observed in Boundary Layer and Pipe," written by G. B. Schubauer of the National Bureau of Standards. 7 Because of its contribution to at least my understanding of boundary-layer phenomena. I would have liked to see it included as a reference. The interesting point made by this article is that the boundary layer is not a smoothly widening surface of transition between the flow influenced by the surface and that beyond, but rather a very irregular and turbulent boundary. In the neighborhood of the mean position of this edge the flow is intermittently turbulent and nonturbulent as the irregular boundary moves past at approximately free-stream velocities. When the nature of this turbulent area. is taken into account Mr. Schubauer shows that the structure of the boundary layer is similar to that of the turbulent flow in a pipe. This bears out the recommendations of the author of the present paper that pipe and channel tests be used to investigate roughness phenomena. At the same time I would like to raise the question as to whether Mr. Schubauer's mathematical considerations of turbulence generation and dissipation might not offer an effective approach to an analytical consideration of rough- Hess. The problems of frictional resistance and the effect of roughness upon friction are important economically because of the multitude of their applications. I am pleased to see that the naval architects of this country are still studying the problems and that scientists like Mr. Hama are contributing to the expansion of knowledge in the field. DR. HAMA" I am very much pleased to have received so many thoughtful discussions of my paper, particularly since in general they provide valuable supplementary information or corroboration of my own conclusions. In spite of Dr. Ross's hearty support of some of my results~, I fear that I cannot follow the logic involved in hi.s "physical analyses." First, his 7 Journal of Applied Physics, wfl. 25, Feb. H)54, op, 188-19.

356 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES Equation [29] seems to give only a definition of ~. It cannot contribute, either mathematically or by physical reasoning, to the experimentally confirmed universality of Equation [1] of the paper. Without the experimental evidence, how can it be claimed that ~ is a universal constant? Secondly, Equation [16] and hence the discusser's Equation [30] apply equally well to the intermediate region and to the region of fully developed roughness action. As the experimental evidence shows, AU/v* and consequently ~ are functions of the geometry of surface roughnesses as well as of the roughness parameter kv*/v in general. It is quite true that &U/v* has a logarithmic relation to kv*/v and, therefore, ~ is a function of the geometry of surface roughnesses alone in the region of fully developed roughness action. I think, however, that no one can claim a priori that ~ is a function of the geometry of surface roughness alone without experimental confirmation. Here also it seems to me that only a definition of an intercept coefficient ~ has been introduced. There appears to' be no physical analysis or derivation of significant results. As the velocity-defect distributions are different for boundary layers and for fully developed pipe or channel flows, the relation between the form parameter H and the local surface-resistance coefficient c I should be different, even if only slightly, for the two cases. Although I have never used the new definitions of fi* and O, it might be possible to show that the relation between the newly defined H and U~O/v is universal regardless of outside flow conditions. If this were true, however, the universal relation between H and c I should no longer be expected, since the surfaceresistance fornmlas with respect to UiO/v for the two cases are not identical. Therefore, there is no apparent reason why the relation for H should be universal. Nevertheless, the experimental accuracy of H is ordinarily poor and the universality of Hregardless of the outside flow conditions nmy be approximately true. As clearly described in the text, our artificial random roughnesses are neither closely packed nor made of uniform grains, one of them being fairly dispersedly spaced grains. In spite of the difference in geometry, all the results showed similar characteristics, having a dip in the intermediate region. Dr. Hoerner is quite right when he says that natural roughnesses may consist of wavy but more or less smooth areas combined with a number of higher, but single protuberances. Our roughnesses might be still too densely packed and different from natural roughnesses. I would repeat that resolution of the problem as to just what produces the Colebrook type of function remains the primary task of the next phase of the project. Concerning Dr. Hoerner's second comment, however, I am not quite sure that the effect of roughnesses of smaller concentrations is the decreasing tendency of surface resistance in the fully rough region; possibly the test was made in a much extended intermediate region. The investigations of the simplest kinds of regular roughness suggested by Dr. Coles are of great interest and are exactly the type of experiments I would like to undertake if I were in a position to conduct further investigations of this problem. I firmly believe, however, that the wire screens which were wisely chosen by Dr. Rouse of the Iowa Institute were suited for this phase of the investigations, that of exploring the general principle of the roughness effect on boundary layers, since they were conveniently available for a wide range of roughness sizes. I regret that I did not refer fully in the paper to Dr. Coles' logical discussion of the conditions for validity of a universal wall law. The significance of his contribution is that it may furnish a procedure, more powerful than that of dimensional analysis, for extending the boundary-layer laws to more general three-dimensional flows. ProfessSr Troost's comments emphasize the contributions to the roughness problem by Prandtl, Sehlichting, and Schultz-Grunow. The fundamental and brilliant work of the G6ttingen school in the field of boundary layers is well known. Since it is fully described in text books such as Goldstein's "Modern Developments in Fluid Dynamics" and Sehlichting's "Grenzschicht-Theorie," it seeflled unnecessary to dis- cuss it more fully in my paper. I must disagree, however, with Professor Troost's statement that these "German scientists have exactly done this sort of thing." When a method based on certain assumptions has been used for many years, there is a tendency to forget these assumptions, and perhaps even a reluctance to investigate their validity. One of my goals was to justify experimentally their adaptation of the laws for the effect of roughness in pipe and channel flows to fiat-plate boundary layers. I believe that this motivation has been adequately discussed in the sections "Introduction" and "Effects of Surface Roughness" of my paper. I am particularly grateful for the discussions made by Mr. Tulin and by Dr. Todd, emphasizing the necessity of this sort of fundamental investigation to more practical engineering problems. As mentioned in the text, careful experiments on smooth flat plates at very high P, eynolds numbers are of the utmost importance. The analysis of turbulent boundary layers depends on the uni-

BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES 357 versal law of the wall and the universal velocitydefect law. While the wall law has a foundation of at least dimensional and physical considerations, the velocity-defect law is supported merely by experimental results in a limited Reynoldsnumber range. Therefore, further investigations at higher Reynolds numbers would provide a more definite basis for application of boundary-layer analysis to practical problems--particularly to the prediction of ship resistance, for which Reynolds numbers are much higher than those for the existing boundary-layer tests. Once the smoothplate resistance is established, it does not necessarily require high Reynolds numbers to investigate the roughness effect even on real ships, because this can be systematized in terms of the roughness parameter kv*/u and can be investigated with moderate Reynolds numbers. I would agree with Morris' paper mentioned by Mr. Tulin on one point; namely, that the roughness effect depends on the spacing as well as the height of roughness elements. In this regard, the paper is of interest. However, I cannot agree with Morris' proposal of a new roughness parameter based upon spacing to replace the ordinary one based'upon height, since it actually gives poorer agreement than he claims, as was clearly pointed out in the discussion of his paper by Dr. Rand. Moreover, it may be remarked that Morris' proposal is in contradiction to Schlichting's experimental results. On the other hand, the spacing parameter shofild be a useful secondary variable in co-ordinating the effect of various kinds of roughnesses. The effect of geometrically similar surface ronghnesses has been shown to be a function of the roughness parameter kv*/p alone. We are in no position, however, to predict the effect of an arbitrarily given roughness. To the end of being able to predict the resistance of any roughness, more detailed descriptions of roughness characteristics--such as, perhaps, the spectrum suggested by Mr. Tulin--are surely required for future investigations. This, I believe, is directly related to Dr. Coles' suggestion and to a private communication from Professor Liepmann of C.I.T. The present paper was able to provide a basis for further studies of the effect of surface roughness on boundary layers and to verify that the effect is identical for different outside flow conditions, so that the relatively vast amount of experimental data already obtained in rough pipes could then supplement the relatively limited information now available for rough plates. However, since the present investigation must be considered as fundamental or exploratory, many questions must be answered in order to make the entire problem clear and directly useful for practical engineers. Some of them are clearly pointed out by Dr. Todd. The most critical question concerning Fig. 13 is the following: The increment of CI due to surface roughness above the smooth line appears to be nearly constant; i.e., the Cz curves for ship tests are almost always parallel to the smooth line, quite contrary to Fig. 13. I can think of several explanations of this result : (a) Boundary layers of real ships are very thick and the effect of roughness is almost always in the intermediate region between practically smooth and fully developed roughness action. Even for high-speed vessels, therefore, constancy of the total surface-resistance coefficient CI should not be expected. Furthermore, surface roughnesses for real ships might not belong to any of the kinds of roughnesses presented in the text. The nature of the effect of such roughnesses might be quite different from those shown in Fig. 11 or 13 in the intermediate region, and it might provide fairly parallel lines as found for ship tests, particularly at high Reynolds numbers. (b) The present and most of the existing investigations are for two-dimensional flows, whereas boundary layers of real ships are threedimensional in nature. The effect of threedimensionality or of finite width is only slightly explored by Hughes and, quite recently, by Townsend. This certainly would affect the smooth line and consequently the roughness chart, and is a very important and interesting problem for further study. (c) Instead of another possible explanation, I would rather pose a question for naval architects, since I am merely a boundary-layer man and I do not know much about ship testing. Are there no uncertainties in deducing the surface resistance from full-scale tests? In particular, is there no ambiguity in evaluating the so-called residual resistance? It occurs to me that some of the unknown or unexplained aspects of drag are arbitrarily classed as surface resistance." In any event, I am firmly convinced that the method of systematizing the roughness effect presented in the paper is equally valid for any series of geometrically similar roughnesses and, hence, that it offers a means of predicting the increase of surface resistance for any particular flow conditions. The study of the mechanism of turbulence in shear flows such as that referred to by Mr. Reed is certainly of great importance to clarify the entire problem. Particularly, the exploration of flow phenomena in detail near the roughness ele-

358 BOUNDARY-LAYER CHARACTERISTICS FOR SMOOTH AND ROUGH SURFACES ments should be an essential part of further investigations. However, I have intentionally limited the scope of the present investigation to the engineering approach, as the present status of turbulence research is, unfortunately, still far from being able to provide any practical information for engineering purposes. CHAIRMAN DAVmS0N: On behalf of th e Society I should like to express the thanks of the Society for your very interesting paper. You claim to be a boundary-layer man and have no idea of the practical consequences. I am sure, as matters stand today, all of those concerned with the practical side very much welcome the interest and cooperation of those in boundary-layer work to learn a little more about it. I, myself, ha~:e felt that is the only way we are going to get much further with this problem, although it may be a rather long process. V.