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J O U R N A L O F T H E A E R O / S P A C E S C I E N C E S - J U L Y, 1 9 5 9 where h(x) is the boundary-layer thickness. The unknowns are s"u --s:, the parameter a and the function h(x); as a first approximation h was taken to be
h(x) = kV(x + a)(v/U); - a::;; x
namely, that of Eq. (3b) shifted a distance a. The value of k is taken as in Eq. (3c). The only parameter tc be determined 10 15 20 25 30 35 40 therefore, is a.
Tbe terms of a stream function, the Kavier-Stokes equations FIG. 1.
require that L(,f;) = v'v ,f; - (o/oy)[(o,J;/oy)(o ,f;/oxoy) - therefore the same solution as above, will, of course, result. The 2 2 (o,J;/ox)(o ,t,/ou")l (o/ox)[(o,J;/ox)(o ,J;/oxoy) - method is relatively insensitive to different chcices in the assumed
(o,J;/oy)(o',J;/o; )l = o (6)
function, as may be verified by performing the calculations on the This is dearly satisfied if basis of a different function h. For example, if we arbitrarily select L fih(x) h(x) = ax (4) (7}
f L[,f;]o,f; dydx = 0
-a 0 with a the required parameter, then the result is for any arbitrary virtual variation o,f;. Substitution of Eq. (5) a = (96v/7 U) L1 (4a) into expression (7) gives upon integration and simplification tbe following for Ra = a U / v: and, therefore, depends here (as it did not in the previous case) on
the length L. The results have meaning only for x < L, where
c1Ra'c2Ra + cs((c4 + c 5 RL + c6R.)/[(RdR 0 ) + 1]5 ) = 0
they are in good agreement, except near the leading edge, with where the ones of Eqs. (3b ), ( c); this can be seen by the comparison of the boundary-layer thicknesses, obtained by the two methods, RL = LU/v and C1 = 425/1872; C = 2399/1055; plotted in Fig. l. Of course, for better accuracy for large L, a c, = 153/400; c4 = 63/80; c, = 19/48; c, = 1;3 form for h more general than that cf Eq. ( 4)-i.e., with more The only admissible root for all cbcices of RL is that plotted in parameters, needs to be chosen. The drag coefficients correspond-
Fig. 2, and is virtually independent of the choice of RL for RL >
ing to the solutions of Eq. (3b), of Eq. ( 4a), and of Blasius are all 40. The boundary-layer height at x = 0 is given by hU/v = of the same form and are, respectively, 1.4/\/ R1,, 1.23/-V RL and k-V R ; away from the leading edge, the boundary layer over the l.33/-V RL where RL = UL/v.
plate very closely resembles that of the classical theory. As a In the leading-edge regime, the Prandtl equations are not valid consequence, the drag is almost unaffected by the introduction of and give rise to singularities in the velocity field; the use of the the nose region.
2• 3 complete Navier-Stokes equations can be expeeted to eliminate
this type of behavior. t We start by assuming that the boundary REFERENCES
layer extends a distance a (as yet unknown) ahead of the leading Carrier, G. F., and Lin, C. C., On the Nature of the Boundary Layer }\'ear edge along the axis of the plate (y = 0). Along y = 0, the velocity the Leading Edge of a Flat Plate, Quart. Appl. Math., Vol. 6, No. 1, pp. o:,- 68, April, 1948.
deereases continuously from the free-stream value U at x = -a
Schlichting, H., Boundary Layer Theory, McGraw-Hill Book Co., Inc., to zero at the leading edge x = 0. At tbe same time, the bound- New York, 1955.
ary layer grows from zero tbickness at x = -a in a manner to be Goldstein, S., Some Developments of Boundary Layer Theory in Hydrody- discussed. namics, Lecture Series No. 33, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1955.
A simple flow field whicb satisfies these conditions is given by Kuo, Y. H., On the Flow of an Incom.pressible Viscous Fluid Past a Flat tbe stream function Plate at Moderate Reynolds Numbers, J. Math. & Phys., Vol. 32. No. 2-3, pp.
83-101, July-October, 1953.
2 2
U/ [y (x + a) /2ha ] +
2 2
y - [y(x + a) /a ]); -a ::;; x ::;; 0
,f;
0::::; y::;; h
()::::; X < L
U/2[(y/h) J 0 ::::; y ::;; h (5) Experimental Verification of Boundary-Layer Corrections in Hypersonic Nozzles t This problem was considered by Kuo. His solution, obtained by a series approach combined with Lighthill's method for improving higher approxi- Donald L. Baradell mations, shows the great complexity of the problem. K uo's results are in Aeronautical Research Engineer, Langley Research Center, general agreement with the ones described here.
NASA, Langley Field, Va.
February 2, 1959 24,-----y--...,------~-~--~- -~--- R,r~- HE PROBLEM of accurately predicting boundary-layer growth
T
for supersonic nozzle design assumes increased importance as the design Mach Number of the nozzles is pushed into the by- personic regime. Several methods of calculating this growth bave been advanced which vary in ease of computation and accuracy of results. Results of the calibrations performed on two hyper- sonic helium nozzles recently put into operation in the 11-in.
Hypersonic Tunnel Section at the NASA Langley Research Cen- 9.22 ter indicate that the method employed for boundary-layer calcu- lations in these nozzles is adequate. The flow in the test region of botb nozzles has been found to be of good quality and the Mach Numbers obtained agree very satisfactorily with tbe de- sired design Mach Number.
These nozzles were designed by the method of characteristics to produce uniform parallel flow at Macb Numbers of 10 and 18, FIG. 2.
R E A D E R S' F O R U M the computations of the characteristic net being performed on an 0 Rake vertical Pt • 170 peia D Rake horizontal IBM 704 calculator. Both nozzles are axisymmetric with a 10.5- pt ,s" 190 paia .6 Rake vertical Pt "' 390 psi& in. diameter at the center of the test region. The flow angles at ,(12
j::°"o
,0
·~
the inflection points of the nozzles were specified as 10° for the DC 0.5 M ,o □ < 'o □~ □ l i l ~ 6 11.0 Mach-10 nozzle and 12° for the Mach-18 nozzle. The method of
computing turbulent boundary-layer displacement effects devised I I
2 (a) Mach 10 nozzle by Persh and Lee was used to obtain the boundary-layer displace-
I I
ment thickness in these nozzles after being adapted for high- speed machine computation. This method, which is based on a finite difference solution of the von Karman momentum equation, 0 Rake vertical pt • 990 psia □ Rake horizontal Pt =- 990 paia is applicable to either two-dimensional or axisymmetric nozzles 6 Rake vertical Pt .,,.l.590 psis and includes the effects of heat transfer. The two problems were
,_
717.0 - -
-
set up so that after the ordinates of the flow field were obtained, .Q ::Jl.8.0 M A L t □~ □~ A these results could be used directly as input for the boundary-layer 19.0
-
!& calculations. In order to obtain the desired test section dimen- .002 sions, several iterations were necessary.
As shown in Fig. l(a), the Mach-10 nozzle was designed to (b) Mech 18 non.le operate at a stagnation pressure (p,) of 100 psi. Because of pres- .000 -3 ~ -1 0 +l +2 sure ratio difficulties the nozzle has not been calibrated at this +3 Distance :from nozzle axis of symmetry (in) pressure, but an extrapolation of the results obtained at higher Figure ]. -Results of a survey with an impact pressure rake in pressures indicates that a Mach Number of 10 would be obtained the center of the test region in two hypersonic helium nozzles.
if the nozzle were run at design pressure. The results of the cali- bration of the Mach-18 nozzle, as shown in Fig. l(b), indicate Static pressures obtained along the nozzle wall during the cali- that at the design pressure of 1,000 psi the average Mach Num- bration runs agree well with the design wall pressures and sub- ber in the center of the test region is 17.8, which differs from de- stantiate the results obtained with the impact pressure rake. The sign Mach Number by only 1 per cent.
values of Mach Number, M, in the figures were obtained from the In order to obtain a larger testing area, the length of the Mach- impact pressure ratios and are true values in all cases considered 18 nozzle was limited by a method similar to that presented by here, except those where the boundary layer influences the Kenny and Yu. The agreement between the design longitu- measured impact pressure_ Accordingly, in Fig. 3(b), the Mach dinal Mach Number distribution for the Mach-18 nozzle and Number scale applies only to a region within about 2 in. of the values obtained experimentally is shown in Fig. 2. The Mach nozzle axis of symmetry.
Number gradient along the nozzle axis in the test region is only 0.05 per in. in the Mach-18 nozzle and 0.01 per in. in the Mach-10 REFERENCES nozzle.
1 Beckwith, Ivan E., Ridyard, Herbert W., and Cromer, Nancy, The Aero- The pressure distribution in the test region of both nozzles was dynamic Design of High Mach Number Nozzles Utilizing Axisymmet,ic Flow determined by a survey with an impact pressure rake. As shown with Application to a Nozzle of Square Test Section, NACA TN 2711, June, in Fig. 3, the distribution in both nozzles is good, and a moderate 1952.
change in stagnation pressure effects only a overall shift in the 2 Persh, Jerome, and Lee, Roland, A Method for Calculating Turbulent Boundary Layer Development in Supei-sonic and Hypersonic Nozzles Including level of the distribution.
the Effects of Heat Transfer, Naval Ordnance Laboratory Rep. No. 4200.
Kenny, James T., and Yu, Ying-Nien 1 On the Length of Hypersonic Nozzles, Journal of the Aero/Space Sciences, Readers' Forum, Vol. 25, No.
0 Design value 11, p. 724, November, 1958.
□ Experimental values
-
_.-cl "'
A
M 10 Remarks on "The Solution of the Laminar Boundary-Layer Equations" ·(a) Mach 10 nozzle 400 600 800 1000 0 200 M. Z. v. Krzywoblocki . ··--.
Professor, University of Illinois, Urbana, Ill.
February 13, 1959 1-'-E ~ M 18 c- EPSTEIN DEVELOPED a new iteration procedure for solving the so-called Prandtl boundary-layer equation. I would like to (b) Mach 18 nozzle add a few remarks on this subject and on the subject of the boundary-layer equations in general.
1000 1200 By means of a not one-to-one transformation of coordinates, Figure 1. - Comparison Of Mach numbers obtained by Prandtl and Blasius passed from a partial differential equation to impact pressure surveys with design Mech numbers an ordinary one. All the procedures referring to a solution of the for two hypersonic nozzles.
I I I Prandtl-Blasius equation (Epstein's as well) refer to an ordinary I
{:L1 Along axis
Experimental values . □ Average in flow core
differential equation. Due to the existence of complex boundary 0 Edge of boundary layer I I conditions and the lack of a one-to-one transformation, it is not ,·
I ·· I
possible-using the available methods of the theory of functions- Design M on axis to prove anything back in the domain of the partial differential ~ I I -
- -
M 18 equation. This can be done by means of some algebraic methods
-·
"'
2 3 -. (Michal, Morgan, et al.) but only for an equation without any
-
17 llr" I boundary conditions. Thus, an exact solution in the domain of ~gn Mon wall an ordinary differential equation does not need to be necessarily ' ' ' -6 -Ji -2 0 +2 +4 an exact solution of the original partial differential equation. As Distance from ¢_ of test section on nozzle axis (in~ a matter of fact, absolutely nothing can be said on that at the Figure 2. - Longitudinal Mach number distribution present time. There are some voices here and there, that the in the Mach 18 helium nozzle.