[mm-m?— -►.m.leaoois RM No. L9C02 NACA RESEARCH MEMORANDUM AN APPROXIMATE METHOD FOR ESTIMATING THE INCOMPRESSIBLE LAMINAR BOUNDARY-LAYER CHARACTERISTICS ON A FLAT PLATE IN SLIPPING FLOW By Coleman duP. Donaldson Langley Aeronautical Laboratory Langley Air Force Base, Va. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 1 20 MARSTON SCIENCE LIBRARY RO. BOX 117011 GAINESYtU.E. FL 32611-7011 USA NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON May 2, 1949 /jr '^vf^ 3^/^ NACA EM No. L9C02 NATIONAL ADVISORY COMMITTEE FOE AERONAUTICS RESEAECH MEMORANDUM AN APPEOXIMATE METHOD FOR ESTIMATING THE INCOMPEESSIBLE LAMINAE BOUNDARY-LAYER CHARACTERISTICS ON A FLAT PLATE IN SLIPPING FLOW By Coleman duP. DonaldBon SUMMARY An approximate method is presented for the estimation of the properties of the incompressihle iBmlinav "boundary layer on a flat plate in the slip— flow region iising Earman's momentum, method. At eq^uivalent stations on the same "body at the same Eeynolds numher, the total thicJoiess and the skin friction of a slipping "boundary layer are less than that of the normal hoijndary layer at the same Reynolds n-umber. However^ the difference "between the slip and normal "boundary layer is amn 1 1 until slip velocities at the wall are encountered which are large in comparison with the free— stream velocity; that is, u-jf ^ 0.3Uq. An Important effect of slip is that on the displacement thickness of the "boundary layer. The following criterion is presented for determining the importance MX .ip phenomena: If — > de3crl"blng viscous phenomena, MX of slip phenomena: If — > 0.0^4-, slip becomes an important factor in INTRODUCTION Recent interest in very high altitude flight has led to an interest in and considerable speculation as to the nature of gas flows when the mean free path of the gas molecules is of the order of magnitude of the boundary— layer thickness on a body and also for which the mean free path of the molecules is of the order of magnitude of the length of the body itself. Tsien (reference l) has described these two types of flow, the former being called the slip— flow regime and the latter the free- molecule— flow regime . MCA RM No. L9C02 There has "been considerahle work done on shear or drag forces in the slip— flow region (see references 2, 3, and k) , hut most of this work has heen done on hounded flows such as the flow hetween two concentric rotating cylinders or the flow through long tubes. It is the purpose of this paper to investigate the general nature of the slip flow on a flat plate when the mean free path is of the order of ^ hut less than, the "boundary— layer thickness. The analysis is only an approximation, hut the properties of a laminar boundary layer in the slip— flow regime and the magnitude of the drag reduction due to slip are evaluated. Insofar as a simple method of evaluation is useful, such an approximate analysis may he Justified. SYMBOLS C-Q drag coefficient k ratio (s/X) I length of flat plate L length of order of mean free path m mass of molecule u horizontal velocity V vertical velocity X horizontal coordinate y vertical coordinate 5 boundary— layer thickness \ mean free path H viscosity V kinematic viscosity p density T shear stress NACA EM No. L9C02 Subscripts: n normal flow o free-stream values s slip f low w wall values ANALYSIS Karman's momeiitum theory for the "boundary layer on a flat plate may be expressed (see reference 5) by the formula d \ = j^ I p^(^o - ^)^y (1) This formula may be used if the normal boundary— layer assumptions are valid; that is^ that the boundary— layer thickness is small compared with the distance to the leading edge of the plate, that the flow in the boundary layer is almost parallel to the surface, and that the major viscous terms are of the same order of magnitude as the inertia terms. It will be seen as the analysis progresses that these conditions are met, and so a velocity profile for the laminar boundary layer consistent with the boundary conditions in slip flow will be assumed and used to solve eq^uation (l) for the rate of growth of the boundary layer and the value of the surface friction at a given station. The velocity at the wall in a slip flow from elementary kinetic considerations (see reference k, pp. 291—299) ™a-y ^Q taken as % = 4^) (2) ^ '^ w so that if a boundary— layer profile of the form Ji- = A +Bf + Cfff (3) u„ 5 \S/ MCA EM No. L9C02 is assumed, the following "boundary conditions may "be taken y = u = y = 5 u = - = = W The application of these "boundary conditions results in Uq 2X, + & 2\ + 6 5(2X + 6) jzi (5) The friction at the wall is w -.('^ 2|au, Vdy/w 2\ + 8 (6) "Upon su"bstituting eq^uations (5) and (6) in equation (l) and assuming the flow to "be incQmpressihle, the following result is obtained "by carrying out the integration 2nu o ^^ " 2X, + 5 dx 2 2 2 3 3 15 (2A. + &)2 (7) Equation (7) is differentiated to o'btain g^^^o 2\ + b = PUo ^X5 + 2^2 3 5 3 ^6^ , A^3 15 (2\ + 6)2 (2;^ + 5)3 d5 dx (8) NACA RM No. L9C02 This eq^uation is ncfw integrated and yields X = 51 - Ax% 16 \- 1 s3 8. 10 15 15 (2X- + S) + -^X log 22l_L^ 15 2X. + 6 15 2X (9) This equation relates the distance from the leading edge of a flat plate and the "boundary— layer thickness for a given mean free path. Equation (9) may he simplified by the introduction of the plate length I and the ratio I- wherehy equation (9) "becomes X T ao _8_ 16 1 15 "*■ 15 2 + k 1 k3 8 , 2 + kA ,,-^ or T - «n(T)'f "=' (11) Equation (ll) gives the position on a flat plate at which the "boundary layer is k times thicker than the mean free path. The value of f (k) is plotted against k in figure 1. From equation (2) the velocity at the wall at any point is given "by 2u. "w = 2 + k (12) and from equation (6) the friction stress at the wall divided "by twice the dynamic pressure is 'w piio'^ EnVV2 + k (13) 6 NACA RM No. L9C02 The displacement thickness of the "bouadary layer is found to "be (li^) It is readily seen that if the mean free path X is placed eq^ual to zero in equation (5) the ho-undary— layer profile 'beGomea and the boundary— layer thickness is foiuid, "by putting X = in equation (9)j "to "be 6 = 5-^8 /? which is in general agreement with the Blasius solution. EXAMPLE A specific example is now worked out to illustrate for a particular case the difference "between a slip and a normal flow. Thus a Reynolds num"ber of 100 was assumed and the ratio of plate length to mean free path was chosen as 25- This might correspond to a 1— foot— chord plate traveling at a Mach num'ber around 2.9 a-"t an altitude of 250,000 feet, since (see appendix A) M = ^ ^ (15) A lower velocity might have been chosen for a 1— foot body at 250,000 feet, so that the flow would be incompressible, but the resulting lower Reynolds number would have given a larger boimdary layer and the slip— flow region would have been confined to a somewhat smaller region near the leading edge so that it would have been more difficult to demonstrate the results of the NACA RM No. L9C02 slip. Indeedj this fact indicates the desiralDility of extending the analysis to include the effects of compressihillty. Figure 2 shows the solution for the thickness of the "boundary layer along the plate in terms of the dimensionless ratios b/X and x/Z . The velocity profiles are also plotted at their proper positions. It is seen that there is a slip velocity at the wall over the entire plate. Figure 3 shows this solution compared with the normal boundary— layer solution at the same Reynolds number. It is seen that the slipping houndary layer is thinner at eq^ulvalent stations than the normal "boundary layer (taken in this example to "be the solution when X, = O) . Figure k shows a comparison of the displacement thiclmesses in termB of the ratio B*/l for the two cases. It can "be seen that there is a large effect on the displacement thickness due to slip, as might be expected. Finally, figure 5 shows a comparison of the local skin frictions in the two cases. It may "be seen that as the thickness of the "boundary layer "becomes large with respect to the mean free path, the slip skin friction approaches the normal value. But at the leading edge where, since there is no "boundary layer, the flow must "be a free-molecule flow, the result of this analysis is compared with the free-molecule stress coefficient given "by (see appendix B): M 2.91 It is seen that the present method yields a stress at the leading edge that is just twice the value derived from free molecule considerations. If these skin frictions are integrated over the surface of the plate, the drag coefficient for the slip flow on one surface is found to be Cd3 = 0.1312 while for the normal flow at the same Reynolds number it is found to be Cfl = 0.1460 °-n The figure shows that the effect of slip has little effect on the skin friction over most of the plate. 8 MCA EM No. L9C02 DISCUSSION Strictly speaking, the eq^uation for the velocity at the vail M ^" ^ Hit I w holds only for the houndary layer when the mean free path is conalderahly less than the boimdary— layer thickness. The error is principally that, in deriving the equation for the velocity at the vail, the momentuni "brought in to the vail hy molecules at an average distance L from the wall ia (see reference h, p. 1^4-0) m -^K^. It may be seen from the shape of the velocity profile in figure 6 that this assumption becomes feasible when the boundary— layer thickness is approximately twice the length L or approximately twice the mean free path. It iSj therefore, obvious that this type of analysis is only applicable to boundary— layer regions when the mean free path is about one— half or less than one— half the boundary— layer thickness. From the analysis it is seen that the thicknesses and rate of growth of the slip boundary layer are of the same order of magnitude as those of a normal boundary layer at the same Reynolds number, and hence the boundary— layer assumptions necessary for eq^uation (l) must be eq\;ia.lly valid for the slip boundary layer. In general, the effect of slip is to decrease the drag and the boundary— layer thickness from what would be calculated for a nonnal boundary layer at the same Reynolds number. The greatest effect of slip is upon the displacement thickness. The growt-h of the boundary layer and the skin friction at the wall may be very close to the normal values even in the presence of a considerable slip velocity at the wall; that is, uv = O.3U0. It should be noted that further boundary conditions may be imposed by assuming higher powers of y/& in the eq^uation of the velocity profile. The most obvious condition neglected by this analysis is NACA RM Wo. L9C02 This condition leads to reBults in somewhat "better agreement with the Blasius results for the case of X, = where u^ = so that "but leads to difficulties in the analysis when the mean free path is apprecia'ble and there exists a slip velocity at the wall. From the example it may "be seen that if the "boundary layer at the end of the plate has a thicloaess less than a'bout 20 times the mean free path, it might "be expected that the effects of slip would "be important . From this fact it is possi"ble to construct, with the aid of eq^uations (11) and (15), an approximate criterion for the Importance of slip phenomena, "Upon su"bstituting equation (I5) into equation (11) and putting — = 1.0 for the trailing edge, there results I 1~ l.i*^f(k)iE If k is to "be less than 20 at the trailing edge, then f (k)i]Tg must "be leas than l6.5j or roughly ^>-^ (16) 1 25 From this approximate criterion for the importance of slip— flow effects, it may "be seen that slip phenomena will "be more Important at high Mach numbers and the present analysis should be extended to include the effects of compressihility . This criterion agrees well with the slip— flow regime as defined "by Tsien. Further, it may "be seen that the two fundamental varia"bles most useful to descri'be gas flows at low densities are Mach number and the ratio of mean free path to body length. 10 NACA EM No. L9C02 CONCLUSIONS 1. An approximate method of estimating the slip— flow "boundary layer on a flat plate has heen presented. 2. At eq^uivalent stations the total thickness and the skin friction of a slipping hoimdary layer are less than that of the norm.«?l houndary layer at the same Reynolds number. 3. The difference "between the slip and normal "boundary layer is small until slip velocities at the wall are encoimtered which are large in comparison with the free— stream velocity, that is, Uy = O.3U0. k. An important effect of slip is that on the displacement thickness of the "boundary layer. I 5. The following criterion is presented for determining the importance ip phencBEena: If — > descrihing viscous phenomena, of slip phenomena: If — > G.Qii, slip "becomes an important factor In Langley Aeronautical La"boratory National Advisory Committee for Aeronautics Langley Air Force Base, Va. NACA RM No. L9C02 11 APPENDIX A DERIVATION OF EQUATION (I5) Reynolds number is defined as R = P^2 _ u2 pk and J from the kinetic theory of gases, |i = O.U99pc\ so that practically R^ = 2-i 1 ^ c X, Since the mean molecular velocity c for air is 1.462 times the velocity of sound a, there results Rn = 1.37Mf A* or M = — 1.37 12 NACA EM No, L9C02 APPENDIX B DERIVATION OF EREE-MOLECULE FRICTION STRESS COEFFICIENT The momentum which strikes a unit area in a unit time of a flat plate in a free-molecule flov is l/i^-pcuQ. If all the molecules are reflected with zero velocity from the surface of the plate, the shear stress at the surface is ■"w = j;^°^o The stress coefficient is therefore \ _ i J_ _ 1 ^-^^^ _ 0.366 puo2 ~ i| Uq " 1+ M ' M 41 NACA RM No. L9C02 I3 EHPERENCES 1. Tslen, Hsue-Shen: SuperaerodynamlcSj Mechanics of Rarefied Gases. Jour. Aero. Sci., vol. 13^ no. 12, Dec. 1946, pp. 653-66^^. 2. Maxwell, James Clerk: Scientific Papers, vol. II, Cambridge Univ. Press 1890, p. 705. 3. Milllkan, R. A.: Coefficients of Slip in Gases and the Law of Reflection of Molecules from the Surfaces of Solids and Lic[uids. Phys. Rev., vol. 21, no. 3, 2d ser., March 1923, pp. 217-238. k. Eennard, Earle H.: Kinetic Theory of Gases. McGraw-Hill Book Co., Inc., 1938, pp. lijO and 291-299- 5. Prandtl, L.: The Mechanics of Yiscous Fluids. Theorem of Momentum and Karman's Approximate Theory. Vol. Ill of Aerodynamic Theory, div. G, sec. I7, W. F. Durand, ed., Julius Springer (Berlin), 1935. pp. 103-105. ll^ NACA EM No. L9C02 10 m / / / '' / / / / / / / / / / / / / ^^---^NACA,.,-^ 1000 800 600 400 200 100 80 60 40 20 10 10 20 40 60 80 100 Figure 1.- Function of k from equation (11) plotted against k. NACA EM No. L9C02 15 ■p a) ft O OJ -1^ o o H •H H m (D H a) g (D I, CVI o H CO vO _S CVJ 16 NACA RM No . L9C02 \ z/ \ H^ II H P. H CO / o \ II \ ^l'- \ ^ \ 7^ ^ i ■ 3 5 \\ ^ L ^o o Id o +3 OQ i/g 'q^Tiex e^B^d oq. 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