A Discussion of the Application of the Prandtl-Glauert Method to Subsonic Compressible Flow over a Slender Body of Revolution

The Prandtl-Glauert method for subsonic potential flow of a compressible fluid has generally been believed to lead to an increase in the pressures over a slender body of revolution by a factor 1/([sqrt](1-M[sub]1^2)) (where M[sub]1 is Mach number in undisturbed flow) as compared with the pressures in incompressible flow. Recent German work on this problem has indicated, however, that the factor 1/([sqrt](1-M[sub]1^2)) is not applicable in this case. In the present discussion a more careful application of the Prandtl-Glauert method to three-dimensional flow gives the following results: The Prandtl-Glauert method does not lead to a universal velocity or pressure correction formula that is independent of the shape of the body. The factor 1/([sqrt](1-M[sub]1^2)) is applicable only to the case of two-dimensional flow. The increase with Mach number of the pressures over a slender body of revolution is much less rapid than for a two-dimensional airfoil. An approximate formula from which the increase can be estimated is derived theoretically. The increase with Mach number of the maximum axial interference velocity on a slender body of revolution in a closed wind tunnel is given approximately by the factor 1/((1-M[sub]1^2)^-3/2), rather than by the factor 1/([sqrt](1-M[sub]1^2)) previously obtained by Goldstein and Young and by Tsien and Lees

The Stability of the Laminar Boundary Layer

The present papcr is a continuation of a theoretical investigation of the stability of the laminar boundary layer in a compressible fluid. An approximate estimate for the minimum critical Reynolds number Re[sub]cr[sub-sub]min, or stability limit, is obtained in terms of the distribution of the kinematic viscosity and the product of the mean density [rho][super][bar]* and mean vorticity [formula] across the boundary layer. With the help of this estimate for Re[sub]cr[sub-sub]min it is shown that withdrawing heat from the fluid through the solid surface increases RRe[sub]cr[sub-sub]min and stabilizes the flow, as compared with the flow over an insulated surface at the same Mach number. Conduction of heat to the fluid through the solid surface has exactly the opposite effect. The value of Re[sub]cr[sub-sub]min for the insulated surface decreases as the Mach number increases for the case of a uniform free-stream velocity. These general conclusions are supplemented by detailed calculations of the curves of wave number (inverse wave length) against Reynolds number for the neutral disturbances for 10 representative cases of insulated and noninsulated surfaces. So far as laminar stability is concerned, an important difference exists between the case of a subsonic and supersonic free-stream velocity outside the boundary layer. The neutral boundary-layer disturbances that are significant for laminar stability die out exponentially with distance from the solid surface; therefore, the phase velocity c* of these disturbances is subsonic relative to the free-stream velocity [symbol] or [symbol], [symbol] where is the local sonic velocity. When [symbol]<1, (where M[sub]0 is free-stream Mach number), it follows that [inequalities] and any laminar boundary-1ayer flow is ultimately unstable at sufficiently high Reynolds numbers because of the destabilizing action of viscosity near the solid surface, as explained by Prandtl for the incompressible fluid. When M[sub]0 >1, however, [inequalities]. If the quantity [forumla] is large enough negatively, the rate at which energy passes from the disturbance to the mean flow, which is proportional to [formula], can always be large enough to counterbalance the rate at which energy passes from the mean flow to the disturbance because of the destabilizing action of viscosity near the solid surface. In that case only damped disturbances exist and the laminar boundary layer is completely stable at all Reynolds numbers. This condition occurs when the rate at which heat is withdrawn from the fluid through the solid surface reaches or exceeds a critical value that depends only on the Mach number and the properties of the gas. Calculations show that for M[sub]0 > 3 (approx.) the laminar boundary-layer flow for thermal equilibrium -- where the heat conduction through the solid surface balances the heat radiated from the surface -- is completely stable at all Reynolds numbers under free-flight conditions if the free-stream velocity is uniform. The results of the analysis of the stability of the laminar boundary layer must be applied with care to discussions of transition; however, withdrawing heat from the fluid through the solid surface, for example, not only increases Re[sub]cr[sub-sub]min but also decreases the initial rate of amplification of the self-excited disturbances, which is roughly proportional fo 1/[sqrt]Re[sub]cr[sub-sub]min. Thus, the effect of the thermal conditions at the solid sufice on the transition Reynolds number Re[sub]tt, is similar to the effect on Re[sub]cr[sub-sub]min. A comparison between this conclusion and experimental investigations of the effect of surface heating on transition at low speeds shows that the results of the present paper give the proper direction of this effect. The extension of the results of the stability analysis to laminar boundary-layer gas flows with a pressure gradient in the direction of the free stream is discussed

Time factors in slowing down the rate of growth of demand for primary energy in the United States

The purpose of this report is to identify the time scales involved in slowing down the rate of growth of primary energy consumption in the U.S., as one component of an overall energy/environment strategy designed to limit the required volume of energy imports from overseas. Two important energy-consuming sectors of the economy are chosen as illustrative examples: (1) the "automobile" as a total system (25%); (2) space heating, air conditioning and water heating in the residential sector (22%). Efficient, light-weight vehicles are introduced into the automobile population by allocating an increasing percentage of new car production to such vehicles year by year until some fixed percentage is attained. Parametric calculations show that significant reductions in the annual rate of energy consumption by automobiles can be achieved if (a) the fuel consumption of efficient vehicles is 60% or less of "standard" vehicles; (b) the increment in percentage of new car production devoted to efficient vehicles is not less than 8% per year; (c) the efficient vehicles are "frozen" at not less than 80% or more of all new car production at the end of an eight to ten year period. In the residential sector the "turnover" rate is comparatively low, and the calculated reduction in annual energy growth rate produced by energy-conserving measures is modest, as expected, unless a "retrofit" rate of older living units of at least 2% per year can be attained. These two components of an energy-conserving policy taken together would bring the growth rate in U. S. primary energy demand down from its present rate of 4.2% per year to about 2.8% per year by 1985. Reductions in the annual growth rate of the remaining 50% of U.S. primary energy consumption that seem quite feasible would bring the overall growth rate down to about 2.5% per year by 1985. If reductions in growth rate of this magnitude could in fact be achieved, energy imports would peak in the mid-1980s at a level no higher than about 60% above the present (1973) volume of imports. Incentives and disincentives designed to bring about this slowdown in the rate of U. S. energy consumption are discussed briefly

Investigation of the Stability of the Laminar Boundary Layer in a Compressible Fluid

In the present report the stability of two-dimensional laminar flows of a gas is investigated by the method of small perturbations. The chief emphasis is placed on the case of the laminar boundary layer. Part I of the present report deals with the general mathematical theory. The general equations governing one normal mode of the small velocity and temperature disturbances are derived and studied in great detail. It is found that for Reynolds numbers of the order of those encountered in most aerodymnic problems, the temperature disturbances have only a negligible effect on those particular velocity solutions which depend primarily on the viscosity coefficient ("viscous solutions"). Indeed, the latter are actually of the same form in the compressible fluid as in the incompressible fluid, at least to the first approximation. Because of this fact, the mathematical analysis is greatly simplified. The final equation determining the characteristic values of the stability problem depends on the "inviscid solutions" and the function of Tietjens in a manner very similar to the case of the incompressible fluid. The second viscosity coefficient and the coefficient of heat conductivity do not enter the problem; only the ordinary coefficient of viscosity near the solid surface is involved. Part II deals wlth the limiting case of infinite Reynolds numbers. The study of energy relations is very much emphasized. It is shown that the disturbance will gain energy from the main flow if the gradient of the product of mean density and mean vorticity near the solid surface has a sign opposite to that near the outer edge of the boundary layer. A general stability criterion has been obtained in terms of the gradient of the product of density and vorticity, analogous to the Rayleigh-Tollmien criterion for the case of an incompressible fluid. If this gradient vanishes for some value of the velocity ratio of the main flow exceeding 1 - 1/M (where M is the free stream Mach number), then neutral and self-excited "subsonic" disturbances exist in the inviscid fluid. (The subsonic disturbances die out rapidly with distance from the solid surface.) The conditions for the existence of other types of disturbance have not yet been established to this extent of exactness. A formula has been worked out to give the amplitude ratio of incoming and reflected sound waves. It is found in the present investigation that when the solid boundary is heated, the boundary layer flow is destabilized through the change in the distribution of the product of density and vorticity, but stabilized through the increase of kinematic viscosity near the solid boundary. When the solid boundary is cooled, the situation is just the reverse. The actual extent to which these two effects counteract each other can only be settled by actual computation or some approximate estimstes of the minimum critical Reylolds number. This question will be investigated in a subsequent report. Part III deals with the stability of laminar flows in a perfect gas with the effect of viscosity included. The method for the numerical computation of the stability limit is outlined; detailed numerical calculations will be carried out in a subsequent report

Finite-Amplitude Instability of the Compressible Laminar Wake. Strongly Amplified Disturbances

The interaction between mean flow and finite‐amplitude disturbances in certain experimentally observed unstable, compressible laminar wakes is considered theoretically without explicitly assuming small amplification rates. Boundary‐layer form of the two‐dimensional mean‐flow momentum, kinetic energy and thermal energy equations and the time‐averaged kinetic energy equation of spatially growing disturbances are recast into their respective von Kármán integral form which show the over‐all physical coupling. The Reynolds shear stresses couple the mean flow and disturbance kinetic energies through the conversion mechanism familiar in low‐speed flows. Both the mean flow and disturbance kinetic energies are coupled to the mean‐flow thermal energy through their respective viscous dissipation. The work done by the disturbance pressure gradients gives rise to an additional coupling between the disturbance kinetic energy and the mean‐flow thermal energy. The compressibility transformation suggested by work on turbulent shear flows is not applicable to this problem because of the accompanying ad hoc assumptions about the disturbance behavior. The disturbances of a discrete frequency which corresponds to the most unstable fundamental component, are first evaluated locally. Subsequent mean‐flow and disturbance profile‐shape assumptions are made in terms of a mean‐flow‐density Howarth variable. The compressibility transformation, which cannot convert this problem into a form identical to the low‐speed problem of Ko, Kubota, and Lees because of the compressible disturbance quantities, nevertheless, yields a much simplified description of the mean flow

Inviscid Hypersonic Flow Over Blunt-Nosed Slender Bodies

At hypersonic speeds the drag/area of a blunt nose is much larger than the drag/area of a slender afterbody, and the energy contained in the flow field in a plane at right angles to the flight direction is nearly constant over a downstream distance many times greater than the characteristic nose dimension. The transverse flow field exhibits certain similarity properties directly analogous to the flow similarity behind an intense blast wave found by G. I. Taylor and S. C. Lin. Conditions for constant energy show that the shape of the bow shock wave R(x) not too close to the nose is given by R/d = K_1 (γ)(d/c)^(1/2) for a body of revolution, and by R/d = K_0(γ) (x/d)^(2/3) for a planar body, where d is nose diameter, or leading-edge thickness. A comparison with the experiments of Hammitt, Vas, and Bogdonoff on a flat plate with a blunt leading-edge at M_∞ = 13 in helium shows that the shock wave shape is predicted very accurately by this analysis. The predicted surface pressure distribution is somewhat less satisfactory. Energy considerations combined with a detailed study of the equations of motion show that flow similarity is also possible for a class of bodies of the form r_b ~ x^m, provided that m' ≤ m ≤ 1, where m' = 3/4 for a planar body and m' = (3/2(γ+1))/(3γ + 2) for a body of revolution. When m < m' the shock shape is not similar to the body shape, and except for the constant energy flows the entire flow field some distance from the nose must depend to some extent on the details of the nose geometry. Be again again utilizing energy and drag considerations one finds that at hypersonic speeds the inviscid surface pressures generated by a blunt nose are larger than the pressures produced by boundary layer growth on a flat surface over a distance from the nose of order ℓ, where ℓ/d ≃ 1/15 ((Re_d)/M_∞^2))^3 (Here Re_d is free-stream Reynolds number based on leading-edge thickness.) Thus at M_∞ = 15 the viscous interaction effects should be important for Re_d 3000 the inviscid pressure field is dominant and determines the boundary layer development, skin friction and heat transfer over the forward portion of the body. These rough estimates are in qualitative agreement with the experimental results of References 7 and 9

The stability of the laminar boundary layer in a compressible fluid

Report is a continuation of a theoretical investigation of the stability of the laminar boundary layer in a compressible fluid. An approximate estimate for the minimum critical Reynolds number, or stability limit, is obtained in terms of the distribution of the kinematic viscosity and the product of the mean density and mean vorticity across the boundary layer. The extension of the results of the stability analysis to laminar boundary-layer gas flows with a pressure gradient in the direction of the free stream is discussed

State Power Plant Siting: a Sketch of the Main Features of a Possible Approach

Work on various phases of power plant technology and siting has been underway within the Environmental Quality Laboratory (EQL) at the California Institute of Technology for some time. Of particular relevance to this memorandum, a good deal of effort has been devoted to institutional aspects of the siting process. Our purpose in what follows is to draw from our past work -- and from the discussions and work of others -- a sketch of the major outlines of one possible approach to power plant siting for the state. We hope in doing so to give our present views about the issues and how they might rationally be resolved, not so much to convince as to inform, stimulate fruitful ideas, and help provide the basis for constructive debate. We ourselves are not necessarily wedded to any of the discussion that follows; we find our own minds changing from time to time as we study the problem further or confront sound suggestions from others. Part I of this memorandum briefly outlines the major features of what we see as a fruitful approach to the siting problem. Sections A through E of Part I describe some elements of the approach; Section F sketches the actual siting decision process we suggest, and in doing so shows how the elements play into the process. Section G comments briefly on a suggested role for judicial review. In Part II we attempt to reduce our ideas to a fairly precise outline for a state siting statute, and to deal with certain matters of detail not covered in Part I. Section A of Part II introduces the statutory outline by summarizing each of its provisions; Section B sets forth the outline itself. The Appendix to this memorandum depicts our suggested approach in time-line fashion; it should be helpful in reading and understanding the proposal

Kinetic Theory Description of Plane, Compressible Couette Flow

By utilizing the two-stream Maxwellian in Maxwell's integral equations of transfer we are able to find a closed-form solution of the problem of compressible plane Couette flow over the whole range of gas density from free molecule flow to atmospheric. The ratio of shear stress to the product of ordinary viscosity and velocity gradient, which is unity for a Newtonian fluid, here depends also on the gas density, the plate temperatures and the plate spacing. For example, this ratio decreases rapidly with increasing plate Mach number when the plate temperatures are fixed. On the other hand, at a fixed Mach number based on the temperature of one plate, this ratio approaches unity as the temperature of the other plate increases. Similar remarks can be made for the ratio of heat flux to the product of ordinary heat conduction coefficient and temperature gradient. The effect of gas density on the skin friction and heat transfer coefficients is described in terms of a single rarefaction parameter, which amounts to evaluating gas properties at a certain "kinetic temperature" defined in terms of plate Mach number and plate temperature ratio. One interesting result is the effect of plate temperature on velocity "slip". In the Navier-Stokes regime most of the gas follows the hot plate, because the gas viscosity is larger there. As the gas density decreases the situation is reversed, because the velocity slip is larger at the hot plate than at the cold plate. In the limiting case of a highly rarefied gas most of the gas follows the cold plate