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ROUGHNESS EFFECTS O N
BOUNDARY-LAYER TRANSITION
FOR BLUNT-LEADING-EDGE
PLATES AT MACH 6
by Paul F, Holloway and E . Leon Morrisette
Langley Research Center
Lczngley S t d o n , €€ampton, Va, N A T I O N A L A E R O N A U T I C S AND SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. A U G U S T 1 9 6 6 NASA TN D-3517 J ROUGHNESS EFFECTS ON BOUNDARY-LAYER TRANSITION FOR BLUNT-LEADING-EDGE PLATES AT MACH 6 By Paul F. Holloway and E. Leon Morrisette Langley Research Center Langley Station, Hampton, Va.
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t ROUGHNESS EFFECTS ON BOUNDARY-LAYER TRANSITION FOR BLUNT-LEADING-EDGE PLATES AT MACH 6 By Paul F. Holloway and E. Leon Morrisette Langley Research Center SUMMARY An investigation has been conducted to determine the effects of controlled roughness (spheres) on boundary-layer transition for unswept, blunted plates at a free-stream Mach number of 6. The location of boundary-layer transition w a s determined by heating-rate distributions downstream of the roughness element on the center line of the plates.
Experimental data are presented for leading-edge bluntnesses of 0.125 and 0.375 inch (0.318 and 0.953 cm). Tests were made for an angle of attack of 0 ' and for a test unit Reynolds number per foot (per 30.5 cm) between 1.2 x lo6 and 9.2 X lo6.
Blunting the leading edge of a plate has been found to affect the roughness height required to t r i p the boundary layer by changing the distance that transition must be moved (i.e., the natural transition location). The definition of an effective roughness height for which the end of transition is an arbitrarily chosen constant distance downstream of the roughness location has been utilized in the analysis of the experimental data. With this definition of effective roughness height, it has been shown that blunting the leading edge of a plate reduces the required effective roughness Reynolds number. However, the required ratio of effective roughness height to boundary -layer thickness at the roughness location is essentially constant for both sharp- and blunt-leading-edge plates. The required value of effective roughness height has been shown to decrease with increasing unit Reynolds number for blunt-leading-edge plates. This parameter was found to be essentially constant with varying unit Reynolds number for sharp-leading-edge plates.
Correlation of data f o r both sharp- and blunt-leading-edge plates by the method of Potter and Whitfield was successful. An evaluation of the application of this correlation technique has been discussed.
INTRODUCTION The determination of the location of boundary-layer transition in the high super- sonic and hypersonic Mach number range is complicated by the effects of possible surface discontinuities. These discontinuities may result from fabricational processes (for example, rivet heads), from buckling of the skin material, or from ablation of protective heat shields. One technique of studying the results of surface discontinuities on transi- Investigations of tion location is that of determining the effects of surface roughness.
effects of surface roughness on boundary-layer transition may be found in the litera- the The studies in references 2 to 21 were, however, primarily ture. (See refs. l to 21.)
concerned with relatively sharp leading-edge models at Mach numbers below 5. Refer- 1 is an investigation of the effects of controlled three-dimensional surface rough- ence ness on boundary-layer transition and heat transfer on a relatively sharp leading-edge (leading-edge thickness < 0.004 inch (0.010 cm)) flat-plate model at Mach numbers 4.8 and 6 . 0 .
Because the previous data available in the literature a r e primarily concerned with sharp-leading-edge models, the purpose of the present report is to extend the work of reference 1 to include the effects of surface roughness on boundary-layer transition on unswept flat plates with blunt leading edges. Current design trends of winged reentry vehicles indicate that the leading edges of the wings must be blunted significantly because of heating considerations. This investigation has two functions: First, to increase knowledge of the effects of surface roughness on boundary-layer transition for models with significant degrees of leading-edge bluntness; and second, to serve as a guide for future high-speed experiments in which it is desired to t r i p the boundary layer with the minimum size roughness necessary to move the turbulent flow near the roughness elements.
The correlation techniques available in the literature vary considerably as to the definition and determination of the most important parameter in assessing the effective- ness of surface roughness as a boundary-layer trip. Several correlation techniques a r e applied o r discussed for both the present data obtained with blunt -leading-edge models and for the data o f reference 1 obtained with sharp-leading-edge models at a free-stream Mach number of 6.
A comparison of the experimental heat-transfer data with theoretical predictions has been made for laminar flow over blunt-leading-edge plates without roughness. Also, the turbulent heat-transfer data for the plates with roughness have been compared with the theoretical predictions by assuming the virtual origin of turbulent flow t o be located at the roughness elements.
The experimental investigation w a s conducted in a variable-density Mach 6 . 2 blow- down jet at the Langley Research Center for a range of free-stream unit Reynolds number per foot (per 30.5 cm) of approximately 1 . 2 x 106 to 9 . 2 x 106. The two models tested were unswept flat plates with two degrees of leading-edge bluntness. The models were instrumented w i t h thermocouples so that the transition location could be determined from the local -heating -rate results .
SYMBOLS Measurements for this investigation were taken in the U.S. Customary System of Factors Units. Equivalent values a r e indicated herein in the International System (SI).
relating the two systems are given in reference 22.
b thickness of cylindrical leading edge indicating amount of bluntness specific heat cP specific heat of wall material CW d diameter of roughness elements h heat-transfer coefficient k vertical height of roughness above plate M Mach number Stanton number NSt pressure P experimental heating rate Reynolds number based on fluid conditions at top of roughness elements and Rk pkukk
height of roughness, -
pk correlation parameter (see eq. (5) or ref. 10)
R;z
unit Reynolds number per foot (per 30.5 cm) at outer edge of boundary layer, RO
pouo -
IJO local free-stream Reynolds number based on distance from roughness
R O , x *
P O U O X *
location, IJO
unit free-stream Reynolds number per foot (per 30.5 cm), poouo3 -
CLOO lateral spacing of roughness S T temperature time t velocity component of flow parallel to surface of plate U X longitudinal distance from leading edge distance from leading edge to roughness location xk distance from leading edge to end of transition for model with roughness
xt
distance from leading edge to end of natural transition 3 9 0 X * distance from roughness location ratio of specific heats Y 6 calculated undisturbed boundary-layer thickness at roughness location based on velocity E (see eq. (6)) correlation parameter recovery factor local w a l l thickness viscosity correlation parameter (see eq. (6)) density P w exponent in viscosity -temperature relation b Subscripts: aw adiabatic wall cr critical eff effective k conditions a t top of roughness elements 0 local conditions at outer edge of boundary layer laminar plateau P r recovery V distance from virtual origin W wall distance from leading edge to roughness location xk distance from leading edge to end of natural transition xt Q) free-stream conditions APPARATUS, TEST METHODS, AND DATA REDUCTION Wind Tunnel The test program was conducted in a variable-density Mach 6.2 blowdown jet at the Langley Research Center. The tunnel is of the intermittent type exhausting to a 40,000-cubic-foot (1130-m3) sphere which can be pumped to pressures as low as 1 milli- meter o f mercury absolute. The tunnel h a s a rectangular test section of 12 inches (30.5 cm) in width and 14 inches (35.6 cm) in height. The model was tested in a position approximately at the center line of the tunnel. Tests were run with tunnel stagnation pressures of approximately 65, 165, 265, 365, 515, and 615 pounds per square inch absolute (448, 1137, 1827, 2516, 3550, and 4240 kN/m2) with an approximate stagnation- temperature range of 860' to 1020' R (480° to 565O K). The resulting free-stream unit Reynolds numbers per foot (per 30.5 cm) a r e approximately 1.2 x 106, 2.7 x lo6, 4.1 x lo6, 5.6 X lo6, 7.7 X lo6, and 9.2 X lo6, respectively. A more detailed description of the tunnel is given in reference 23.
Models The models tested were flat plates constructed from stainless steel with two degrees of leading-edge bluntness (see fig. 1). For convenience, the leading-edge pieces are referred to herein as leading edge A and leading edge B. Leading edge A w a s 0.375 inch (0.953 cm) in diameter and leading edge B, which consisted of a 14.5O wedge The model that tapered to a near hemicylinder, w a s 0.125 inch (0.318 cm) in diameter.
7.5 inches (19.05 cm) wide and 10.4 inches (26.42 cm) long. A s is shown assembly was in figure l(a), the roughness elements (which were mounted on interchangeable roughness strips) were alined equidistantly from the leading edge (Xk = 2.87 inches (7.29 cm)). The spheres were glued into small spherical indentations in the roughness strips. The spacing (s), height above the plate (k), and diameter (d) of the spheres a r e also given in figure 1.
The instrumentation w a s located along the center line of the rear plate. A rough- ness element of each roughness strip was located on the center line so that the instru- mentation lay directly in the wake of a roughness element. Two models of the instru- mented plate were constructed - one being instrumented with 0.050-inch (0.127-cm)
pressure orifices and the other with 30-gage iron-constantan thermocouples. The under -
surface of the plate instrumented with thermocouples was slotted along the center line to a width of 0.6 inch (1.52 cm) and a surface skin thickness of approximately 0.020 inch (0.051 cm).
Test Methods and Data Reduction Pressure tests.- P r e s s u r e distributions along the center line of the models were obtained for the smooth plate, for use in the reduction of the heat-transfer data. The local static pressures on the plates were measured by connecting the orifices to pressure transducers. The changes in the electrical signals f r o m the transducers were recorded on a digital-readout recorder. The range of the transducers was 0 to 1 pound per square inch absolute (0 to 6.9 kN/m2). All pressure tests were run on the same support system as was used for the heat-transfer tests.
The pressure data are presented in figure 2. During the pressure tests, mechanical failure of the tunnel prevented testing of leading edge B above R , = 4.1 X 106 .
However, t e s t s in the Langley 20-inch hypersonic tunnel (Mach 6) have indicated that variation of ' Reynolds numbers R, of 4.1 X lo6 to 8.5 X lo6 has negligible effects on the pressure distribution for leading edge B. Also, comparisons of the data for R, = 4.1 X lo6 and R , =. 7.7 x lo6 for plates with the more blunt leading edge (leading edge A) have indi- cated very little variation in the pressure distribution. Therefore, the data obtained for leading edge B at R, = 4.1 X lo6 were utilized i n the reduction of heat-transfer results at higher Reynolds numbers. For the same reason, the data obtained for leading edge A at R, =. 7.7 X lo6 were used in the reduction of heat-transfer data at R, = 9.2 X lo6.
Heat-transfer tests.- The aerodynamic heating was determined by the transient calorimetry technique by which the rate of heat storage in the model skin is measured.
The models, originally at room temperature or slightly cooler, were suddenly exposed to the established hypersonic airflow by quick injection from a sheltered position beyond the tunnel wall. Injection was accomplished in less than 0.25 second and the model remained in the tunnel for a maximum of 4 seconds.
The electrical outputs from the thermocouples were recorded on a high-speed digital readout recorder. The reading from each thermocouple was recorded at 0.025- second intervals, converted to a binary digital system, and recorded on magnetic tape.
The temperature-time data were fitted to a second-degree curve by the method of least squares, and the time derivative of temperature was computed on a card-programed computer.
The tunnel-stagnation-temperature range w a s 860' to 1020° R (480° to 565' K) and the wall temperature of the plate w a s approximately 550° R (305' K ) . Because of the short time required for the injection of the model, the plates were considered to have been subjected to a step function in the applied heat-transfer coefficient. The thin-skin equation used to calculate the local surface heating rate (neglecting conduction) w a s The local heat-transfer coefficient w a s then calculated by the relation ; I h =
Tr - Tw
where Tr is the calculated recovery temperature defined as (3) Tw is the measured wall temperature, and Mo is the local Mach number outside the boundary layer calculated from the measured pressure distribution assuming a normal-shock pressure loss. This method w a s considered adequate since the measured heat-transfer coefficient is rather insensitive to small e r r o r s in Mo. Roughness w a s considered to have a negligible effect on the pressure distribution of the instrumented plate. Therefore, in the calculation of Mo for equation (3), the smooth-plate distribu- tion was used for all tests. The recovery temperature w a s calculated by assuming a recovery factor o f 0.830 for a laminar boundary layer and 0.883 for a turbulent boundary layer. The Stanton number, based on free-stream conditions ahead of the model, w a s calculated by the use of the equation h NSt = (4) PaJU00CP, 00
The experimental heat-transfer parameters 4, h, and NSt presented in this
report were determined by calculating the slope of the temperature-time curve approxi- mately 0.20 second after the model was in position in the tunnel. The nearly isothermal conditions of the tests kept the lateral conduction to a minimum.
Determination of transition.- The method used herein to determine the location of I boundary-layer transition from laminar to turbulent flow is the same as that previously described in reference 1. That is, the location of the beginning and the ending of transi- tion has been determined by noting a change in the heat-transfer parameters with longi- tudinal distance as illustrated in figure 3. The local heating rate decreases for laminar flow until transition begins, which causes the heating rates to increase rapidly. When transition ends (beginning of fully developed turbulent flow), the heating rate peaks and begins to decrease with increasing distance from the leading edge. The transition loca- tion xt as used in this report refers to the end of transition (see fig. 3).
REVIEW O F THE LITERATURE is a great deal of variance in the literature as to the choice of parameters There for the best correlation of the effect of a given roughness condition on boundary-layer transition. Therefore, a review of the often-used techniques from the literature that a r e employed in the evaluation of roughness effects is presented to lay the proper background for analysis of the present results.
The concept of a critical roughness Reynolds number appears to have been originally ~ based on experimental results presented by Schiller in reference 3. The results indicated that roughness had no effect on the nature of the flow within the boundary layer until the Reynolds number of the element (based on the characteristic height k) reached a definite critical value at which vortices appeared. An increase in roughness Reynolds number slightly above this critical value, by either increasing the roughness height or by increasing the local unit Reynolds number, should then cause transition to occur at the roughness element itself. A review of published data on the effect of roughness on transition from laminar to turbulent flow has been presented in reference 4. The methods of Braslow, Knox, and Horton presented in references 5 t o 7 for determining the distributed roughness have defined the critical roughness Reynolds number as the value at which turbulent "spots" are initiated behind the roughness and at which a small increase in roughness Reynolds number above this value is required to move the fully developed turbulent boundary layer substantially up to the roughness particles. The experimental values of critical roughness Reynolds number based on this definition were found to vary from 250 t o 600 for subsonic and supersonic Mach numbers up to 2. This range of values results in a variation of the parameter of the square root of the critical roughness Reynolds number as first used by Schiller of about 16 to 25, which is an essen- tially invariant magnitude with Mach number change up to M = 2.
Other investigations - for example, the work of Fage, with various-shaped two-
dimensional t r i p s (ref. 8) - have indicated that the concept of an almost instantaneous shift of the transition location from its undisturbed position to the roughness position is generally erroneous. In the analysis presented in reference 4, Dryden made use of the ratio of roughness height to the displacement thickness of the boundary layer at the roughness location. Reference 9 reported that although this parameter does serve t o correlate roughness effects of two-dimensional trips, it is not sufficient to correlate the roughness effects of a single row o f spheres (three-dimensional trip).
Smith and Clutter (refs. 13 and 14) have suggested that the roughness Reynolds number based on roughness height and actual disturbed conditions at the top of the rough- ness element is a useful correlation parameter in evaluating roughness effects. Thus, if the Mach number is greater than 1 at the top of the element, the roughness Reynolds number should be based on conditions behind a shock.
More recently, Potter and Whitfield (refs. 10 to 12) have demonstrated that both the critical roughness Reynolds number as defined in references 5 to 7 and the ratio of roughness height to boundary -layer thickness are inadequate parameters for the correla- tion of the effects of three-dimensional roughness on boundary-layer transition at high supersonic and hypersonic Mach numbers. Additional evidence of this conclusion was presented in reference 1.
A review of previous studies of the effect of surface roughness led Potter and Whitfield (ref. 10) to present a semiempirical correlation parameter which may be defined by where the subscript p r e f e r s t o the laminar separation plateau value in the region of the roughness element, the subscript k r e f e r s to the conditions of height k in the undisturbed boundary layer at station xk, and the subscript o refers to the edge of the w represents the exponent in the viscosity-temperature boundary layer. The symbol relation. Experimental results were then correlated (ref. 10) by considering the varia- tion of R;( as a function of where E is a constant which essentially represents t..e roughness Reynolds number required to move transition forward to the roughness location (Le., where Xk = 3).
Using the correlation parameter Rk, Potter and Whitfield were able to correlate two-
dimensional and three-dimensional roughness effects for both subsonic and supersonic flow.
The analysis presented in reference 16 by Van Driest and Blumer indicates that after a certain roughness height has been reached, increasing the Reynolds number of the flow will result in a rapid forward movement of the transition position until the transition reaches the region of the roughness. At this point, a "knee" in the curve occurs and a further increase in the Reynolds number will cause the transition to slowly approach the roughness location asymptotically. By the definition of an effective roughness as that required to move transition to the region of the "knee" in the xt curve, height the experimental results were correlated successfully in references 16 and 17.
The application of the correlation technique of Potter and Whitfield and that of Van Driest t o the present data will be discussed in more detail in the section of this "Results and Discussion."
paper entitled RESULTS AND DISCUSSION Effects of Roughness on Boundary-Layer Transition The predominant variable in producing transition by three -dimensional spherical surface roughness is the height of the roughness elements relative to the local boundary- layer conditions. A discussion of the effects of increasing roughness height on transition location relative to the effect of free-stream disturbances is given in reference 1. The work of Van Driest and McCauley in reference 17 indicated that lateral spacing of a single row of spheres has little effect on boundary-layer transition provided the spheres a r e not so close that they act as a two-dimensional trip. Therefore, the effect of sphere spacing has not been considered in this program. Another variable is the free-stream turbulence level which is strongly influenced by the turbulence input from the tunnel walls. It has not been attempted in this report to account for the free-stream turbulence effects on the experimental results. However, all data for the present investigation were obtained in one tunnel under similar test conditions. Also, much of the information from reference 1, which is compared to the present data, was obtained from the same tunnel at similar test conditions.
Figures 4 and 5 present the heating-rate distributions and Stanton number varia- tions for various size roughness elements as a function of distance from the leading edge of the model for leading edges A and B, respectively. The Stanton numbers plotted in fig- u r e s 4 and 5 were calculated from the experimental data by assuming a laminar recovery factor for the smooth plate. In the reduction of the data for the plate with roughness to Stanton number form, a turbulent recovery factor w a s assumed. Also shown in the fig- u r e s are the theoretical laminar and turbulent Stanton number distributions t o serve as a guide in determining the effectiveness of a particular roughness trip. The theoretical Stanton number distributions are based on free-stream conditions, that is, and were calculated by the Monaghan reference temperature method as reviewed in refer- ence 1. The local Mach number w a s assumed to be 3.16 (obtained by taking a normal- shock loss in pressure and assuming that the flow had expanded t o the conditions of pw/p, = 1). The virtual origin for the turbulent-flow case was assumed to be located at the roughness position as was done in reference 1. The actual unit free-stream Reynolds number, the ratio of roughness height to boundary-layer thickness as determined by the calculated Monaghan velocity profile of reference 24, and the local unit Reynolds number as calculated by the previously mentioned assumptions in obtaining the theoretical distri- butions are tabulated in these figures for each test.
In an attempt t o analyze the present data by the method of reference 16, the distance between the transition location and the roughness location (3 - xk) w a s plotted as a func- tion of the free-stream unit Reynolds number per foot (per 30.5 cm) in figure 6(a). Shown
in figure 6(b) is the variation of 5 - xk with roughness height k. A s expected, an
increase in R, o r k resulted in a decrease in the value of 3 - xk (Le., a forward
movement of transition). However, the data are insufficient to indicate a definite "knee" in the variation of R , for most roughness heights as the roughness loca-
3 - xk with
tion is approached. From figure 6(a), it can be seen that the location of the "knee" (or
the effective roughness height as defined in ref. 16) in the 3 - X k variation with
Reynolds number was determined only for the largest size roughness (k = 0.0091 foot (0.2774 cm)) for both leading edge A and B and for the second largest roughness (k = 0.0067 foot (0.2042 cm)) for leading edge A.
(The variation of 3 - Xk with R , has been faired for these three cases in fig. 6(a).) For the remaining combinations of roughness height and leading edge, the Reynolds numbers necessary to obtain the knee in the curve were beyond the maximum capability of the tunnel. The first thermocouple was located at a distance of 0.57 inch (1.448 cm) from the roughness location. More instru- mentation i n this region would be desirable f o r accurate determination of the effective as defined in reference 16. In addition to the geometry differences roughness height between the model used in the investigation of reference 16 and the model used in the pres- ent investigation, that is, a sharp cone as compared with a blunt-leading-edge flat plate, a further difference exists which may have significant influence on correlation attempts by the method of reference 16. In reference 16, Van Driest and Blumer greatly reduced the influence of free-stream disturbances as a variable by testing far upstream of the natural transition location. The ratio of roughness height t o boundary-layer thickness (k/b) tabulated i n figures 4 and 5 for the present results shows that, generally, the rough- ness extends outside the boundary layer. Thus, the roughness not only affects the boundary layer but also has a very definite effect on local turbulence level in the inviscid -f low region.
Analysis o f the results of Van Driest and Blumer suggests that transition cannot occur at the roughness location but instead approaches this position asymptotically.
Because of this result and the difficulties of applying other definitions of critical rough- ness Reynolds number to the present data for blunt-leading-edge models, the present authors have chosen to define an effective roughness height as the one for which
9 - Xk = 0.10 foot (3.05 cm). This definition will allow an analysis of the data that is
believed to be adequate for most applications. It is doubtful that roughness-height requirements necessary to move the end of transition closer than 0.10 foot (3.05 cm) from the roughness location a r e desirable from any but a purely theoretical viewpoint.
It should be noted that the utilization of certain dimensionless parameters such as ' - x k or R - w a s not practical in the consideration of the present experi- 0 9 % RO,xk
%,o - Xk
mental results. In particular, the quantity 3 xt - Xk w a s an
in the parameter
9 0 3.0 - xk
unknown that could not be determined experimentally with the models available.
.
Based on the definition 5 - Xk = 0.10 foot (3.05 cm), the effective roughness
heights may now be determined from figures 4 and 5. For example, from figure 5(a) for leading edge B and for Ro = 1.10 X lo6, the value k = 0.0054 foot (0.1646 cm) would be slightly less than effective whereas the value k = 0.0067 foot (0.2042 cm) would be
slightly greater than effective. The reader should be cautioned that the value of 3 is
determined by fairing the heating-rate curves in the manner shown schematically in fig- ure 3. The technique of fairing has been consistent, but a slightly different technique might yield different quantitative values of The differences, however, would be 5.
small and would not affect the qualitative conclusions of this paper.
Figure 6 indicates that the effective roughness height (that is, the height for which
5 - xk 5 0.10 foot (3.05 cm)) is generally greater for leading edge B than for leading
edge A for a given free-stream Reynolds number. Unpublished data obtained in an inves- tigation conducted in the Langley 20-inch hypersonic tunnel (Mach 6) to determine the bluntness effects on natural transition for a 16-inch plate at an angle of attack of 8' (com- pression) have shown that an increase in bluntness from a near sharp leading-edge con- dition causes a delay in transition. However, above a certain bluntness, increasing the bluntness further causes the transition Reynolds number based on free-stream conditions xt). This result has led to the speculation that to decrease (Le., a forward movement of xt - Xk to 0.10 foot (3.05 cm) for leading the larger roughness heights required to move edge B, as compared with that required for leading edge A, may be primarily a result of
a larger value of 5 for the l e s s blunt leading edge at a given set of free-stream
9 0 conditions.
Effective Roughness Correlations A s was mentioned previously, both the roughness Reynolds number and the ratio of roughness height to boundary-layer thickness at the roughness location are often used as correlation parameters for the effects of roughness on boundary-layer transition. For the present investigation and for the results of reference 1, the roughness height is gen- erally greater than the boundary-layer thickness at the roughness location. Therefore,
f o r a given local Mach number, %,eff and (k/6)eff may be considered t o vary pri-
marily as follows: In figure 7, t h se parameters are plotted as functions of the local unit Reynolds number for both sharp and blunt leading edges. (Note that the t e r m "sharp leading edge" is used in connection with the data of reference 1 for which b < 0.004 inch (0.010 cm).)
Analysis of this figure indicates several differences between the results for the models
with blunt and sharp leading edges. The trend of %,eff for blunt-leading-edge models
is less sensitive to local Reynolds number change than is the variation of %,eff f o r be expected because of the large differ- sharp-leading-edge models. This trend might f o r between the two types of leading edge. That is, the value of x
ences in 3
7 0 t, 0 the blunt-leading-edge plate is several times greater than the value for the sharp- leading-edge plate. This difference has been established in an experimental investiga- tion (unpublished) of leading edges A and B on a 16-inch plate at a Mach number of 6.0.
The results showed that transition did not occur on the plate even a t the highest Reynolds number. Based on the 3 locations for the sharp-leading-edge plate of reference 1, 9 0
therefore, the values of 3 for the blunt-leading-edge plates must be at least on the
9 0 order of three times greater than those values for the sharp-leading-edge plate.
The ratios of the value of xt,O - xk for the sharp-leading-edge plates to that for
the blunt-leading-edge plates are necessarily larger than the ratios of the value of xt,O for the sharp-leading-edge plate t o that for the blunt-leading-edge plate, since the xk locations were generally equal in the two tests. Therefore, for the Reynolds number range under consideration, it seems reasonable that the roughness height would be the more predominant factor in determining Rk,eff for the blunt-leading-edge plate. How- ever, as is shown in figure 7(b), for increasing Ro, the value of (k/6)eff decreases for the blunt-leading-edge plate.
This trend could indicate an increased importance of R, relative to roughness height in determining effective roughness criterion. From fig- (k/6)eff f o r both sharp- and blunt-leading-edge models (at ure 7(b), the values of M , = 6) lie within the range of 1.5 to 3.0. It can be clearly noted from figure 7 that both the required value of Rk,eff and of (k/6)eff are slightly greater for leading edge B than for leading edge A.
F $ correlation.- Study of the literature has indicated that the most applicable correlation technique for the present results and those of reference 1 on the effects of roughness on boundary-layer transition is that given by Potter and Whiffield in refer- ence 10. However, application of this correlation to the present data and to the sharp- leading-edge data of reference 1 has demonstrated several points that are of interest.
The data a r e presented in figure 8(a) in t e r m s of the parameter F $ as suggested for _ _
correlation i n reference 10. (&e eq. (5).) The ratio Mp - p~ - in equation (5) was taken
Mk as having a value of 1.0 as w a s suggested in reference 10. The actual value varies with and is 1.15 f 0.05. Both the sharp- and the blunt-leading-edge data correlate on Ro reasonably smooth curves, but neither agree with the correlation curve from reference 10.
Differences exist between the data of the present investigation and those upon which the ' correlation curve was based. First, the model w a s not at adiabatic conditions as in reference 10. However, the correlation equations take into account the difference in Tw/Taw conditions; therefore, these differences should not greatly affect the correla-
tion. Second, the relatively large value of 3 for the blunt-leading-edge model, as
9 0 compared with the sharp-leading-edge model of reference 1, should not necessarily cause
large e r r o r s since the % effect is included in the correlation. (Values of xt,o for
9 0 the blunt -leading-edge model could not be obtained experimentally. For the correlation of data in figure 8, the value of xt,o was obtained by assuming a constant local transi- tion Reynolds number of 3 x 106. Consideration of several values of xt,O indicated that this parameter affected the correlation of data by slightly varying the shape of the curve but that it had no effect on the indicated value of E . Therefore, a constant transi- tion Reynolds number was considered to be sufficient for the present analysis.) Finally, the local Mach number of 6 for the sharp-leading-edge data is beyond the range Mo = 1.9 to 5.0 for which E w a s defined in reference 10. However, since E was defined as constant for this particular range, it does.not seem reasonable t o expect the value of E t o change drastically at a Mach number of 6.
The following sketch shows the trends of a typical variation in transition location with increasing unit Reynolds number for a smooth plate and a plate with roughness (see ref. 16):
t
X e Region A L R e g i o n B
-----------
-E- xk Ro- E used in the correlation Potter and Whitfield (ref. 10) indicated that the exact value of is not critical. Although they recognized that E may be somewhat in e r r o r , they stated that such an e r r o r would affect results in correlation only in the region of this sketch
where 9 = xk (region B). Therefore, in the region (region A) where the characteristic
"knee" is used to define effective roughness size by the method of Van Driest and Blumer (ref. 16), e r r o r s in E would be insignificant in correlating the data according to reference 10.
The present results indicate that an error in E can affect results in an area other E of 3000 as than of region B (see sketch). Also, the assumption of a constant value of was done in reference 10 can lead to an erroneous concept. In figure 8(a), the variation
of R ; is presented for both positive and negative values of 5 for E = 3000. This
figure shows conclusively that the use of E = 3000 results in negative values of the
correlation parameter 5 , which would indicate transition prior to the roughness ele-
ments. The values of 3 - Xk as a function of 5 are also shown in figure 8(b). The
magnitude of these values of 3 - Xk for which 5 < 0 would indicate that the data for
which 5 is negative are either in region A o r t o the left of region A. The reader is
reminded that the characteristic "knee" in the variation of 5 - xk with Ro w a s not
obtained for most of the present data (see fig. 6). Also, from figure 6, it can be seen that the %nee" was not reached for the data from reference 1 (shown as diamond flagged symbols in fig. 8). Therefore, the assumption of E = 3000 would rule out of considera- tion a substantial amount of data which is definitely not in region B of the preceding 5 is negative were eliminated. (It is interesting to note, sketch if the data for which
however, that i f the data for which 5 < 0 were ignored in figure 8(a), the data for
leading edges A and B would indicate a value of E approximately equal t o 3000.)
The near asymptotic approach of 3 to the xk location in region B of the pre-
ceding sketch as demonstrated by Van Driest and Blumer (ref. 16) would indicate that
% might be expected to increase to very large values as 3 approaches xk. The
concept of an instantaneous beginning of turbulent flow at the roughness location is prob- ably fictitious since transition to turbulent flow should be expected to occur over some finite distance. Therefore, E should be determined by fairing the results to zero for data prior to the occurrence of region B. (Realization of this fact led the present authors to define effective roughness Reynolds number based on an arbitrary distance of
3 - Xk = 0.10 foot (3.05 cm).)
From figure 8(a), fairing of the present data and those from reference 1 suggests the use of E = 5500. The data a r e seen to correlate very well in figure 9 based on E = 5500. The e r r o r s which can result from the selection of E a r e relatively insignifi- cant when a correlation of data is desired. However, in the practical application of the correlation method, that is, the estimation of the minimum height of roughness necessary to move transition approximately to the roughness location, it is suggested that the value of l $ be determined without consideration of a value of E . In applying the correlation of Potter and Whitfield to roughness data, the suggested value of E can be used to reduce the data (for example, E = 3000 for three-dimensional roughness). From the results of the first correlation, a new value of E may be determined which will lead t o a more satisfactory final correlation result.
Critical roughness height.- It has been stated previously that a definition of
Xt - Xk = 0.10 foot (3.05 cm) is considered adequate for determining effective roughness
heights f o r most applications. In figure 10, the effective roughness heights determined from the present data and reference 1 a r e plotted as a function of free-stream Reynolds
I 16
number. If the critical roughness height is defined as that roughness for which the end of transition is essentially at the roughness element, then the variation of with kcr free-stream Reynolds number can be calculated from the Potter and Whitfield correla- tion results. The curves representing the variation of kcr with R, a r e also shown in figure 10 for Mo = 3.16 and Mo = 6.0 (E = 5500). From the figure it can be seen that kcr is much larger than the measured keff at the lower free-stream Reynolds numbers. For example, when R, =: 2.5 X lo6, the value of kcr is approximately three times larger than the value of keff for a sharp leading edge. This would lead t o a value for kcr/d of the order of 6 to 9. The value of kCr, of course, would be expected to be significantly larger than that of keff because of the difference in defini- tion of the two parameters.
The trend of keff for the blunt-leading-edge plates follows approximately that given by the Potter and Whitfield correlation method, but the values a r e lower as expected. (See fig. 1 0 . ) For the sharp-leading-edge plates, however, keff is essen- tially constant over the complete Reynolds number range. This result is thought to be primarily caused by the 3 location for the sharp-leading-edge plates. That is, is very close to xk for the sharp-leading-edge test conditions and Ro is the 3 9 0 predominant parameter in the determination of keff.
The differences in magnitude between add justification to the keff and kcr utilization of an arbitrary definition of effective roughness height, such as that for which xt - xk = 0.10 foot (3.05 cm), if minimum flow distortions a r e t o be obtained.
CONCLUSIONS An investigation has been conducted to determine the effects of controlled surface roughness on boundary-layer transition determined by heating -rate distributions f o r unswept, blunted plates at a free-stream Mach number of 6. The location of boundary- layer transition was determined by heating-rate distributions downstream of the rough- ness element on the center line of the plates. Data are presented for a free-stream Reynolds number per foot (per 30.5 cm) between approximately 1 . 2 X lo6 and 9.2 X lo6 and for a nominal angle of attack of Oo. Analysis of the experimental results is based on a definition of effective roughness height as being that for which the distance between the end of transition and the roughness location is an arbitrarily chosen constant. These results and a comparison with theory and previous results from the literature have led t o the following conclusions: 1 . The required value of effective roughness height decreases with increasing unit Reynolds number for blunt-leading-edge plates, but is essentially constant with varying unit Reynolds number for sharp-leading-edge plates.
-
2. The effective roughness height as defined in this report is considerably smaller , than that required to reach conditions for which the end of transition occurs approxi- mately at the roughness element based on a previously derived correlation, particularly at the lower test Reynolds numbers.
3. For a constant free-stream Mach number of 6, blunting the leading edge had only a small effect on the required values of the ratio of effective roughness height to boundary -layer thickness at the roughness location. For both sharp- and blunt-leading- edge plates, the values of this ratio were within the range of 1.5 to 3 . 0 .
4. Blunting the leading edge of the plates resulted in a considerably smaller value of effective roughness Reynolds number than was previously found for a sharp-leading- edge plate at similar test conditions.
5. It has been shown that the Potter and Whitfield method will correlate roughness- induced-transition data for both sharp- and blunt-leading-edge plates at a free -stream Mach number of 6.0.
Langley Research Center, National Aeronautics and Space Administration, I Langley Station, Hampton, Va., February 28, 1966.
REFERENCES 1. Holloway, Paul F.; and Sterrett, James R.: Effect of Controlled Surface Roughness on Boundary-Layer Transition and Heat Transfer at Mach Numbers of 4.8 and 6.0.
NASA T N D-2054, 1964.
2. Sterrett, James R.; and Holloway, Paul F.: Effects of Controlled Roughness on Boundary-Layer Transition at a Mach Number of 6.0. AIAA J. (Tech. Notes Comments), vol. 1, no. 8, Aug. 1963, pp. 1951-1953.
StrCimung in Rohren. Handb. Experimentalphys., Bd. 4, 4. Teil, Akad.
3. Schiller, L.: Verlagsgesellschaft m.b.H. (Leipzig), 1932, pp. 189-192.
4. Dryden, Hugh L.: Review of Published Data on the Effect of Roughness on Transition From Laminar to Turbulent Flow. J. Aeron. Sci., vol. 20, no. 7, July 1953, pp. 477-482.
5. Braslow, Albert L.: Review of the Effect of Distributed Surface Roughness on Boundary-Layer Transition. AGARD Rept. 254, Apr. 1960.
6. Braslow, Albert L.; and Knox, Eugene C.: Simplified Method f o r Determination of Critical Height of Distributed Roughness Particles for Boundary-Layer Transition at Mach Numbers From 0 to 5. NACA TN 4363, 1958.
7. Braslow, Albert L.; Knox, Eugene C.; and Horton, Elmer A.: Effect of Distributed Three-Dimensional Roughness and Surface Cooling on Boundary-Layer Transition and Lateral Spread of Turbulence at Supersonic Speeds. NASA T N D-53, 1959.
(Supersedes NACA RM L58A17.)
8. Fage, A.: The Smallest Size of a Spanwise Surface Corrugation Which Affects Boundary-Layer Transition on an Aerofoil. R. & M. 2120, Brit. A.R.C., 1943.
9. Klebanoff, P. S.; Schubauer, G. B.; and Tidstrom, K. D.: Measurements of the Effect
of Two-Dimensional and Three -Dimensional Roughness Elements on Boundary -
Layer Transition. J. Aeron. Sci. (Readers' Forum), vol. 22, no. 11, Nov. 1955.
pp. 803-804.
10. Potter, J. Leith; and Whitfield, Jack D.: Effects of Unit Reynolds Number, Nose Bluntness, and Roughness on Boundary Layer Transition. AEDC-TR-60-5, U.S. Air Force, Mar. 1960.
11. Potter, J. Leith; and Whitfield, Jack D.: The Relation Between Wall Temperature and the Effect of Roughness on Boundary-Layer Transition. J. Aerospace Sci.
(Readers' Forum), vol. 28, no. 8, Aug. 1961, pp. 663-664.
12. Potter, J. Leith; and Whitfield, Jack D.: Effects of Slight Nose Bluntness and Rough- J. Fluid Mech., vol. 12, ness on Boundary-Layer Transition in Supersonic Flows.
pt. 4, Apr. 1962, pp. 501-535.
13. Smith, A. M. 0.; and Clutter, D. W.: The Smallest Height of Roughness Capable of Affecting Boundary-Layer Transition in Low-Speed Flow. Rept. No. ES 26803 (Contract No. NOa(s) 54-773c), Douglas Aircraft Co., Inc., Aug. 31, 1957.
14. Smith, A. M. 0.; and Clutter, Darwin W.: The Smallest Height of Roughness Capable of Affecting Boundary-Layer Transition. J. Aero/Space Sci., vol. 26, no. 4, Apr.
1959, pp. 229-245, 256.
15. Brinich, Paul F . : Boundary-Layer Transition at Mach 3.12 With and Without Single Roughness Elements. NACA T N 3267, 1954.
16. Van Driest, E. R.; and Blumer, C. B.: Effect of Roughness on Transition in Super- sonic Flow. MD 60-329 (AFOSR TN 60-1164), North Am. Aviation, Inc., Mar. 1960.
17. Van Driest, E. R.; and McCauley, W. D.: The Effect of Controlled Three-Dimensional at Supersonic Speeds. J. Aero/Space Roughness on Boundary -Layer Transition Sci., vol. 27, no. 4, Apr. 1960, pp. 261-271, 303.
18. Jones, Robert A,: An Experimental Study at a Mach Number of 3 of the Effect of Turbulence Level and Sandpaper-Type Roughness on Transition on a Flat Plate.
NASA MEMO 2-9-59L, 1959.
19. Bidwell, Jerold M.: Roughness Effect on Boundary-Layer Transition With a Cold Wall. R-60-6, The Martin Co., Apr. 1960.
20. Howard, Paul W.; and Czarnecki, K. R.: Effect of Fabrication-Type Surface Rough- ness on Transition on Ogive-Cylinder Models at Mach Numbers of 1.61 and 2.01.
NASA TN D-1933, 1963.
21. Nagamatsu, H. T.; Graber, B. C.; and Sheer, R. E., Jr.: Roughness, Bluntness, and Angle of Attack Effects on Hypersonic Boundary Layer Transition. Rept. No.
No. 64-RL-(3829 C), Gen. Elec. Res. Lab., Nov. 1964. (Available from DDC as AD 615601.)
22. Mechtly, E. A.: The International System of Units - Physical Constants and Conver- sion Factors. NASA SP-7012, 1964.
23. Jones, Robert A.; and Gallagher, J a m e s J.: Heat-Transfer and P r e s s u r e Distributions on a 60° Swept Delta Wing With Dihedral at a Mach Number of 6 and Angles of Attack 0 ' to 5 2 ' . NASA TM X-544, 1961.
From 24. Monaghan, R. J . : An Approximate Solution of the Compressible Laminar Boundary Layer on a Flat Plate. R. & M. No. 2760, Brit. A.R.C., 1953.
Of instrrrmentation roughness strip (a) Flat-plate assembly.
Characteristics of spheres k I d I S I ' k I I .1646 .2042 cm) 2.65" (6.731 .440" (1.118 cm) 2.65" (6.731 cm)
I - -
.lW ( .478 cm) - r k .440" (1.118.cm) I Leading edge B, b = 0.125" (0.318 cm) Leading edge A , b = 0.375" (0.953 cm) ( b ) Leading-edge d e t a i l s .
Figure 1.- Sketch of model.
2 1 E E P, p, . 2 . 3 . 4 .5 .6 .7 . 2 . 3 . 4 .5 .6 .7 x , ft x, ft
I I - - I
L - I - - _ I -1
8 12 16 20 8 12 16 20 X , cm x , cm
0 5
. 2 .3 .4 .5 .6 2 . 3 . 4 .5 .6 .I x , ft x , ft
I-- _ I - - \ - J
L 1 .- ._ 1 - _I
12 16 20 8 1 2 16 20 x , cm x, cm C Leading edge A U Leading edge B n . 2 . 3 . 4 .5 .6 .7 x , ft
L - . L.--L-- 1
8 12 16 20 x , cm Figure 2.- Pressure distributions on flat plate with two degrees of leading-edge bluntness over a range of Reynolds numbers.
.
Beginning of fully t u r b u l e n t flow (end of t r a n s i t i o n ) or NSt X X t Figure 3.- Determination of transition location from heating-rate distribution.
k k/6 RO it I cm 9.04 x lo6 1.10 x 106 8 . 9 1.08 9.3h 1.11 9.01 9.08 9.04 8.74 - W b u l e n t theory, %," = %,x* - _ _ Iaminar theory N LI , s .d; ,2 Roughness
.4 1 location
I I I , .1 . 3 . 4 .5 .6 . 7 .2 x, ft U i L I I . -_I 8 10 12 14 16 18 20 x, cm (a) % 1.10 x 106.
Figure 4.- Heating-rate distribution on flat plate for various size roughness (spheres ) . Leading edge A.
t I !
P 2 N f t : .
5 1 s .8 .d .6 .4 x, f t (b) R, = 0.92 x lo6.
Figure 4 . - Continued .
k I 6 Rm Ro
ft 1 , crn
0 5.41 x lo6 0.65 x 106 .5O 5.54 .66 ,0018 ,0549 .89 6.12 .72
.0050 I .0914
1.25 5.58 .66 ,0044 ,1541 .w54 ,1646 1.53 5.70 .67 8 1 . 9 5.68 .67
,0067 1 ,2042
5.44 .66 2.55 ,0091 .2774 G Wrbulent theory, %," = %,x* 1 , - ~ -Laminar theory .8 .6 .4 .2 I I I I I I I I I I I 8 10 12 14 1 6 18 20 x. cm ( c ) R , = 0.67 x lo6, Figure 4. - Continued.
k b %.
Ro 5.95 x 106 0.48 x lo6 4.23 .45 .50 .71 4.13 .49 1.05 4.11 .48 1.26 4.01 1.56 4.09 2.15 4.19 ~ Turbulent theory, Ro,, = Ro,x* _ - - Laminar theory P 2 N c -.
2 1 m" .8 .d .6 .4 .2 I I 1 1 .1 l ' J l
1 I 1 1
. 1 .5 .6 .7 .2 . 5 .4 x, f t j////J_/ 5 10 12 14 16 18 20 XI ( d ) R , = 0.49 x lo6.
Figure 4. - Continued.
%.
Crn _- 2.66 x lo6 n 0.51 x 106 2.80 .54 .054 .54 2.75 .53 .56 2.66 ,1646 .80 . 52 2.66 .2042 1.20 . 5 1 2.76 1.69 .33 .2774 - Turbulent theory, %," = %,x" - - - Laminar theory ' 4
.02 'I
1- .2
I I I I I
.01 - .1
20 x 10-4 I I I I I - Roughness - .2 location I I I I .1 x, ft R, = 0.33 x lo6.
( e ) Figure 4. - Continued.
::~rRou@nes; I , I 1
location .1 . e .J . 4 .5 .6 .7 x, ft i I I I I I 1 8 10 12 14 16 18 80 XI (f) Ro = 0.15 x lo6.
Figure 4. - Concluded.
0 0 1.10 x 106 0 0 9.17 X lo6 ,0549 .70 9.93 .0914 1.13 9.08
;;; 1.08 ' I T 80 60
3 4 1 1.71 9.79 ,1646 2.00 8.93 ,2042 2.48 8.99 9.02 1.08 ,2774 3.37 Ln Turbulent theory, Ro," = Ro,x* I Laminar theory 1 I - .
2 - 20 e r - 10 . .
- 8 B- I I I '-11 x, cm ( a ) R, f s 1.10 x 106.
Figure 5.- Heating-rate d i s t r i b u t i o n on f l a t p l a t e f o r various s i z e roughness (spheres). Leading edge B.
- m b u l e n t theory, RO,“ = R o , ~ *
Laminar t h e o r y 1
8 2 4, 20 N I F c ??
.
2 1 - 10 ..
f .8 - 8 3N .6 7 6 . 4 - 1 4
::ILT; iocstion , , , 1
J.e .3 . 4 .5 .6 .7 x, ft I I I I I -L_J 18 20 10 12 1 4 1 6 8 XI (b) R, 0.94 x lo6.
Figure 5.- Continued.
Ft f cm U6 5.59 x 106 0.66 x lo6 5.82 .a 5.84 .a 6.27 .73 .66 5.53 .66 5.67 .66 5.65 - Turbulent theory, R = R _ - - Laminar theory O?" O J X N ei . .
* m c 4 .2 .3 .4 .5 .6 .i x, ft
u- L - L 1 - _ I _ - - I
6 I U 12 14 16 18 20 X . cm ( c ) R, = 0.68 x lo6.
Figure 5.- Continued.
k/6 Rm % 0 4.28 x lo6 0.50 x lo6 .44 " 5 9 .52 .72 4.24 .5O 1.07 4.38 .52 1.26 4 . 1 1 .48 1.56 4.15 .48 D .oOgl) .2774 .51 2.20 4.46 __ Turbulent theory, Ro," = Ro,,4 _ - - Laminar theary ,2 Roughness ,2 Roughness
: : .1 .2 location . 5 .4 .5 . 6 .7 : : .1 .2 location . 5 .4 .5 . 6 .7
X ) ft
u - 20
8 10 1 2 1 4 16 18 X I cm (d) Ro = 0.50 x 106.
Figure 5.- Continued.
.02 I
I ~ .1
. 01 ( e ) R , = 0.33 x 106.
Figure 5.- Continued.
.15 I 7 1 0 .u - 8 . 6 - 6 .15 .4 - 4 ~ Turbulent theory, R0," = B o , , % Laminar theory .d . I s .&
1- .2
.02 t
I I I I I .01 I - .1 20 x,1~-4 I I I I I - p 1 0 - - - . 8 - - - - .6
-
- - - . 4 - -
- Roughness -
,2 location I I I I .1 x, f t , 1 I I 1 I I 1 8 14 16 8 1 0 12 X. cm (f) R, 0.15 X lo6.
Figure 5 . - Concluded.
3 5 X t - R- (a) Variation of % - xk with R , for constant values of k.
.5 . 3 Xt - Xk , ft .2 .1 I I I J 0 .1 .2 . 3 k, (b) Variation of xt - Xk with k for constant values of R , .
Figure 6.- Variation of transition location with R , and k. M, = 6; Xk = 2.870 inches (7.290 cm). (Open symbols denote leading edge A; flagged symbols denote leading edge B; closed symbols denote sharp- leading-edge data of ref. 1.)
.1 . 2 . 4 . 6 .8 1 2 4 6 8 1 0 x 1 0 ’ % ( a ) Variation of Rk,eff with %.
.1 . 2 . 4 .6 .8 1 2 4 6 8 1 0 x 1 0 Ro ( b ) Variation of (k/6)eff with Ro.
Figure 7.- Variation of e f f e c t i v e roughness Reynolds number and (k/6)eff with l o c a l u n i t Reynolds number. M, = 6 . (Open symbols i n d i c a t e rough- ness height less than t h e e f f e c t i v e value; closed symbols indicate roughness height g r e a t e r than t h e e f f e c t i v e value. j Reference b E * Xk M m M, cm in. cm in.
3000 Present investigation .9525 0 6.0 3.16 2.870 7.290 .375 .3175 3000 Present investigation 0 6.0 3.16 2.870 7.290 .125 <.O& .0102 3000 Reference 1 2.000 5.dO 0 6.0 6.0 6.0 6.0 2.870 7.290 <.O& .0102 3000 Reference 1 3000 Reference 10 8Ooo R11 loo0
- -8 -.6 -.4 -.2 0 .2 .4 .8
(a) Variation of R i with correlation parameter.
.4 *5 Lr=
n l I "
- .8 -.6 -.4 -.2 0 .2 .4 .6 .8
(b) Variation of transition location with correlation parameter.
Figure 8. - Correlation of roughness-induced-transition data by method of reference 10. E = 3000.
b E R e f ere nce xk M - M, i n .
crn i n . c m 0 6.0 3.16 2.870 7.290 -375 .9525 5500 P r e s e n t i n v e s t i g a t i o n 3.16 7.290 0 6.0 2.870 -125 .3175 5500 P r e s e n t i n v e s t i g a t i o n
0 6.0 6.0 2.000 5.080 <.O& .0102 5500 Reference 1
6.0 7.290 <.O& 5500 Reference 1
0 6.0 2.870 . O l e
3000 Reference 10 I
0 0
A d ,.
3000 <?
I "
I 4
3.0 - 5 . 0
~0
Hollow cylinder . 6 .8 1 . 0 Figure 9.- Correlation of roughness-induced-transition data by method of reference 10. E = 5500.
0 k e f f , leading edge A 0 keff. leading edge B 0 keff, b 4 <0.004 i n . (0.0102 an), \ = 2.000 i n . (5.080 cm), M = 6 . 0 , ref. 1
a # keff. b = c0.004 i n . (0.0102 cm). xk = 2.870 i n . (7.290 cm), M = 6 . 0 , ref. 1
- kc,, leading edges A and B , M , = 3.16, E = 5500, xt = \, r e f . 10
---- 0.004 i n . (0.0102 cm), Mo = 6 . 0 , E = 5500, x = xk, r e f . 10 kcr, b = lo Lo-2: - - a .O .6 Sharp leading edge \ .4 \ \ effl cm c r ,
.9 0 " a * l
.8 .7 .2 .6 .5 .4 .1 . 3 .2 .1 Figure 10.- Effective and c r i t i c a l roughness s i z e s as a function of f r e e - stream Reynolds number.
% = 6 . (Open symbols i n d i c a t e roughness s i z e s l i g h t l y l e s s than t h e e f f e c t i v e value; closed symbols i n d i c a t e roughness s i z e s l i g h t l y g r e a t e r than t h e e f f e c t i v e value. )