The Stability of the Laminar Boundary Layer
 Creators
 Lees, Lester
Abstract
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[subsub]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[subsub]min it is shown that withdrawing heat from the fluid through the solid surface increases RRe[sub]cr[subsub]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[subsub]min for the insulated surface decreases as the Mach number increases for the case of a uniform freestream 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 freestream velocity outside the boundary layer. The neutral boundarylayer 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 freestream velocity [symbol] or [symbol], [symbol] where is the local sonic velocity. When [symbol]<1, (where M[sub]0 is freestream Mach number), it follows that [inequalities] and any laminar boundary1ayer 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 boundarylayer 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 freeflight conditions if the freestream 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[subsub]min but also decreases the initial rate of amplification of the selfexcited disturbances, which is roughly proportional fo 1/[sqrt]Re[sub]cr[subsub]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[subsub]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 boundarylayer gas flows with a pressure gradient in the direction of the free stream is discussed.
Additional Information
NACA Report 876; NACA TN 1360Attached Files
Published  LEEnacarpt876.pdf
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 Eprint ID
 459
 Resolver ID
 CaltechAUTHORS:LEEnacarpt876
 Created

20050621Created from EPrint's datestamp field
 Updated

20191002Created from EPrint's last_modified field
 Other Numbering System Name
 NACA Technical Report
 Other Numbering System Identifier
 876