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The turbulent boundary layer in a compressible fluid. (English) Zbl 0125.18102

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fluid mechanics
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[1] F. W. Matting, D. Chapman, J. Nyholm, and A. Thomas, NASA Technical Report R-82 (1960).
[2] Stewartson, Proc. Roy. Soc. (London) A200 pp 84– (1949)
[3] There is always the possibility that the usual boundary-layer approximation for the stress tensor may fail under certain conditions. In addition, there is room for argument about the importance in high-speed turbulent flows of various correlation terms (involving fluctuations in density, temperature, or viscosity) which are neglected here in order to treat the mean value of a product such as {\(\rho\)}u or {\(\rho\)}T (but not necessarily {\(\rho\)}{\(\upsilon\)}) as a product of mean values.
[4] The author is aware that there exists no general affine transformation which is compatible with the experimental evidence and which allows the reduction of the turbulent boundary-layer equations to ordinary differential equations, as for laminar flow. The term similarity is used here in a broader sense not requiring uniform validity for the whole of the flow.
[5] An identical result is obtained if the flow at infinity is assumed to be a uniform stream, but the condition onȳseems to be unnecessary if the flow is bounded at infinity by a fluid at rest. A closer examination of this case,i.e., of the jet entering a stagnant fluid, is not undertaken here. Even for the other cases mentioned the transformation in question may not be the most general or the most useful one which can be found; first, because the sufficient conditions used to bound the right-hand side of Eq. (2.6) for largeȳhave not been shown to be necessary, and second, because these conditions have been applied not only for largeȳbut throughout the barred flow. Furthermore, it has not been proved that the two flows involved in the transformation must always have the same character, i.e., that if one flow is a wake, so is the other; if one flow is a boundary layer with pressure gradient and mass transfer, so is the other; and so on. On the other hand, the transformation considered here is more general than any that has been considered previously, and has several properties that will prove to be useful in any study of turbulent flow. The possibility of a more general transformation is therefore not investigated further.
[6] The transformation derived in this report has been designed specifically for turbulent flow, and strict attention has been paid from the beginning to the physical principle that it must be possible to observe experimentally any flow whose properties are supposed to be known. This principle, which makes it essential to keep in close touch with the physical variables and with the conservation laws connecting them, is most appropriate for problems that cannot be treated by analytical means. For problems that can be so treated, on the other hand, almost any transformation or change of variable is acceptable if it leads to equations that are mathematically more tractable than the equations of motion in their original form. The principle involved in such cases is a mathematical principle, however, not a physical one, and may be as much out of place in discussions of turbulence as is the physical principle in most discussions of laminar flow. An example of the use (and misuse) of the alternative mathematical principle is supplied by the transformation first proposed by Stewartson in Ref. 2 and later cited for turbulent flow by some writers. The transformation in question may be obtained by simply dropping the subscript in Eqs. (2.14) and (2.15) of Sec. II. For a perfect gas the quantity in parentheses in Eq. (2.14) can then be written as RT0dpp dx, where T0 is the local stagnation temperature. In this formulation the principle of physical existence for the barred flow seems to require T0 = constant if dpdx as before, whether the flow is laminar or turbulent. However, this requirement is entirely without force for the turbulent case, because the physical principle is being applied one stage too late in the argument. Even for laminar flow the requirement in question is largely irrelevant, and Stewartson’s transformation has been used (e.g., by Cohen and Reshotko7and by Coles8) to obtain physically meaningful similarity solutions for laminar flow with heat transfer and pressure gradient, in spite of the fact that the transformed equations do not correspond to any flow that can be observed experimentally.
[7] C. B. Cohen and E. Reshotko, NACA Technical Note TN 3325 (1955).
[8] D. Coles, inProceedings of the Heat Transfer and Fluid Mechanics Institute(Stanford University Press, Stanford, California, 1957), p. 119.
[9] Dorodnitsyn, Compt. Rend. (Doklady) URSS 34 pp 213– (1942)
[10] Dorodnitsyn, Prikl. Matem. Mekh. 6 pp 449– (1942)
[11] Cope, Phil. Trans. Roy. Soc. (London) A241 pp 1– (1948)
[12] Howarth, Proc. Roy. Soc. (London) A194 pp 16– (1948)
[13] Illingworth, Proc. Roy. Soc. (London) A199 pp 533– (1949)
[14] Although Fig. 4 is equivalent to Fig. 1 of Sec. I, the earlier figure emphasizes the dependence of skin friction on Mach number, whereas Fig. 4 emphasizes the dependence on Reynolds number. In a different sense Fig. 5 is equivalent to Fig. 2, since the coordinates in one figure are essentially reciprocals of the coordinates in the other. In this case, however, the earlier treatment of the data is simply a normalization of Cf\=Cf with respect to temperature ratio, whereas Fig. 5 represents an inquiry into the properties of the scaling function {\(\sigma\)}(x) of the transformation. In the construction of Figs. 4 and 5, the ratio TwT has been referred to a recovery factor of 0.885 in all cases where no experimental value is given in the original reference. Values of viscosity have been taken from standard NBS-NACA tables. The usual practice has been followed of assuming constant stagnation temperature T0 for the purpose of computing u(y) and {\(\rho\)}(y) and thus {\(\theta\)}(x) when only M(y) can be inferred from the measurements. The floating-element data have been corrected for gap effect by using the combined area of element and gap rather than the area of the element alone to relate force and shearing stress. Finally, the values of \=Cf for low-speed flow have been estimated by fitting measured mean-velocity profiles to the law of the wall.
[15] Schultz-Granow, Luftfahrtforsch. 17 pp 239– (1940)
[16] D. W. Smith and J. Walker, NACA Technical Note TN4231(1958).
[17] D. Coles, thesis, California Institute of Technology (1953).
[18] R. Hakkinen, thesis, California Institute of Technology (1954).
[19] R. H. Korkegi, thesis, California Institute of Technology (1954).
[20] W. H. Shutts, W. Hartwig, and J. Weiler, University of Texas Report DRL-364 (1955).
[21] C. J. Stalmach, Jr., University of Texas Report DRL-410 (1958).
[22] E. R. G. Eckert, Wright Air Development Center Technical Report 54–70 (1954).
[23] It is important to note that the end to be served is the completion of the transformation, not the completion of the equations of motion through the introduction of an explicit relationship connecting the shearing stresses {\(\tau\)} and {\(\tau\)\=} with the other dependent or independent variables, as in the case of laminar flow. On the contrary, the use of any such relationship is quite likely to prejudice the concepts of turbulent shearing stress and turbulent heat transfer in much the same way that these concepts are prejudiced by the mixing analogies of the older literature.
[24] The author has previously attempted an extension of the first interpretation to compressible flow,25using only part of the full transformation given here. This earlier paper, in which the quantity now called {\(\sigma\)}{\(\mu\)}w{\(\mu\)\=} was denoted by {\(\rho\)}{\(\tau\)}{\(\rho\)}w, includes a demonstration that this quantity (and hence {\(\sigma\)}) must be independent ofxif u({\(\tau\)}w{\(\rho\)}{\(\tau\)})12 = f is constant on mean streamlines of the compressible flow. Except for the one negative conclusion just quoted, the discussion of compressibility in Ref. 25 may therefore be suppressed.
[25] D. Coles,50 Jahre Grenzschichtforschung, edited by H. Görtler and W. Tollmien (Vieweg, Braunschweig, Germany, 1955), p. 153.
[26] Unfortunately this proposal is simply an invention, and like most inventions is not unique. For example, the sub-layer Reynolds number might equally well have been defined, say, as the definite integral offdzrather than as the productfz. The mean density {\(\rho\)}s would then be replaced by a mean mass flow, and the transition to a mean temperature and a mean viscosity would become more difficult. Even the latter transition is ambiguous unless it is assumed, as in the text, that mean thermodynamic variables for the sublayer are related by the same laws which apply for corresponding local variables. This assumption, however, requires that the mean temperature be treated as a derived rather than a fundamental quantity. In this respect, as well as in the application to the sublayer and the formation of the mean in terms of an integral determined by the structure of the incomplete equations of motion, the present definition of mean temperature differs from similar but more empirical definitions in the heat-transfer literature.
[27] For turbulent flow the validity of the Crocco energy integral Eq. (4.14)–or of any other energy integral in analytical form, for that matter–is a matter for conjecture. The most that can be argued from the linear behavior ofTandunear y = 0 is that the integral Eq. (4.14), if is it valid at all for turbulent flow with heat transfer, can only be valid if the laminar Prandtl number is unity. It follows that calculations based on this integral, which is to say calculations based on Eqs. (4.17) and (4.19), cannot be quantitatively correct for gases such as air, although they may be useful as a means of estimating the effects of compressibility and heat transfer on skin friction. With regard to Eq. (4.17), it should be remembered that the quantity Ts is actually defined, e.g., by Eq. (4.15) or more generally by Eq. (4.10), without reference to conditions in the free stream. On the other hand, Eq. (4.17) has the advantage of being independent of the form of the viscosity law. Furthermore, Eq. (4.17) is readily applied to various special cases of practical interest, including the case of low-speed heat transfer (M = 0 or T0 = T); the case of adiabatic flow (qw = 0 or Tw = T0); and the case of a very cold wall (Tw = 0, say). Finally, Eq. (4.17) expresses the sublayer mean temperature Ts as a linear combination of the three characteristic temperatures T, T0, and Tw and thus bears a strong resemblance to certain empirical formulas that are discussed at greater length in the Appendix.
[28] G. J. Nothwang, NACA Technical Note 3721 (1956).
[29] A. L. Kistler, Ballistic Research Lab. (Aberdeen) Report 1052 (1958).
[30] The accuracy of this approximation for both low-speed and high-speed flows with heat transfer has to be tested experimentally. It should be emphasized that the object here is to estimate the effect of Prandtl number on skin friction, not to determine the heat transfer. Neither the derivative (Ty)w, which is zero for adiabatic flow, nor the integral of {\(\rho\)}u(T0-T0) through the boundary layer, which is constant for adiabatic flow, can be correctly obtained from the empirical formula (4.27).
[31] Figure 7 shows the same data as Fig. 3, except that the ordinate in Fig. 3 is {\(\mu\)}w{\(\mu\)}s rather than TwTs(1+). Here and elsewhere in this section the sublayer parameters have been computed from the definitions, e.g., Eqs. (4.5), (4.13), or (4.16), using the particular function f(z) tabulated in Ref. 25, but withfreduced by 2%.
[32] The study referred to can be found in Appendix A of Rand Report R-403-PR (1962). Except for the one appendix, this Rand Report is essentially the same as the present paper and has the same title.
[33] Preston, J. Fluid Mech. 3 pp 373– (1958)
[34] R. A. Dutton, dissertation, Cambridge University (1955).
[35] The termwind-tunnel conditionsmeans that the stream stagnation temperature for air has been taken to be 550 {\(\deg\)}R for M<4.7, and the stream static temperature to be 100 {\(\deg\)}R for M>4.7. The increase in Cf\=Cf with increasing Mach number, when \=Cf is large, corresponds qualitatively to the change in sign of the factor multiplying M2 in the denominator of Eq. (A18) of the Appendix. For the special conditions represented by the latter formula it then follows that TsT is less than unity whenever \=Cf exceeds the value 22= 0.0066. Since this condition cannot be met in an adiabatic flow in whichTdecreases monotonically from Tw at the wall to T in the free stream, a definite upper bound in the neighborhood of \=Cf = 0.0066 is seen to be inherent in the formalism of the substructure concept itself.
[36] The correction for gap effect (the inclusion of the gap area as part of the element area) has not been made for Dhawan’s data,37as it amounts to a decrease in \=Cf of some 15%. Figure 9 also confirms a conclusion which the author reached and reported at the time of his own experiments; in Ref. 17, this conclusion is that the fence tripping device sometimes had the peculiar effect of inhibiting rather than stimulating transition.
[37] S. Dhawan, thesis, California Institute of Technology (1951).
[38] The real origins of this concept are to be found in early work on heat transfer in enclosed channels. For such problems the idea of a bulk or mixing-cup temperature arises quite naturally in any heat-balance calculation. This review, however, is restricted to the concept of mean temperature as it has developed in the context of the boundary-layer equations for compressible flow.
[39] T. von Kármán, inProceedings of the Fifth Volta Congress(Reale Accademia d’Italia, Rome, 1936).
[40] Rubesin, Trans. ASME 71 pp 383– (1949)
[41] G. B. W. Young, and E. Janssen, Rand Corporation Report P-214 (1951).
[42] E. R. G. Eckert, Wright Air Development Center Technical Report 59–624 (1960).
[43] M. Tucker, NACA Technical Note 2337 (1951).
[44] S. C. Sommer and B. Short, NACA Technical Note 3391 (1955).
[45] N. Rott, Ramo-Wooldridge Corporation GMRD Report GM-TR-211 (1957).
[46] O. R. Burggraf, Lockheed Aircraft Corporation Report LSMD-895080 (1961).
[47] R. J. Monaghan, inProceedings of the Fifth International Aeronautical Conference, edited by R. J. Turino (Institute of Aeronautical Science, New York, 1955), p. 277.
[48] Prandtl, Physik. Z. 11 pp 1072– (1910)
[49] G. I. Taylor, Report and Memoranda 272 (1916).
[50] Prandtl, Physik. Z. 29 pp 487– (1928) · JFM 54.0916.06
[51] C. duPont Donaldson, NACA Technical Note 2692 (1952).
[52] In the opinion of the author Donaldson’s paper deserves a prominent place in any review of the analytical literature of the compressible turbulent boundary layer. The reason is that this paper has stood almost alone, pending the first applications of coordinate transformations to turbulent flow, in recognizing the increase in the relative thickness of the sublayer at large Mach numbers as the dominant effect of compressibility.
[53] Tables of the ratio {\(\theta\)}{\(\delta\)}, which depends on M, on TwT, and on the exponentnin the power-law profile, have been published by several authors; e.g., Tucker (Ref. 43), and Persh and Lee [J. Persh and R. Lee, Navord Report 4282 (1956)], for the more general density-velocity relationship given by the Crocco energy integral.
[54] D. A. Spence, Royal Aircraft Establishment, Farnborough, Report Aero 2631 (1959).
[55] Van Le, J. Aeron. Sci. 20 pp 583– (1953)
[56] Mager, J. Aeron. Sci. 25 pp 305– (1958)
[57] Culick, J. Aeron. Sci. 25 pp 259– (1958)
[58] In the present analysis the equations corresponding to Eqs. (A15) through (A17) are Eqs. (2.17), (2.20), and (4.24); the solution proposed for the problem of a reference temperature is the sublayer hypothesis; and thexdependence of the reference properties is assigned to the parameter {\(\sigma\)}(x) = {\(\mu\)\=}{\(\mu\)}s(x) rather than to the parameter {\(\mu\)\=} alone.
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