Qualitative Effects of Viscosity

It is a common feature of most flows of engineering interest that the viscosity of the fluid is extremely small. The Reynolds number, Re = TJJ,/v, which gives an overall measure of the ratio of inertia forces to viscous forces, is in typical aeronautical applications of the order 10® or more. For large ships, Reynolds numbers of the order 109 are common. Viscosity can then only produce significant forces in regions of extremely high shear, i. e., in extremely thin shear layers where there is a substantial variation of velocity across the streamlines. The thickness of the laminar boundary layer on a flat plate of length l is approximately ~ o///Re. For Re = 108, 8/1 ~ 0.005, which is so thin that it cannot even be illustrated in a figure without expanding the scale normal to the plate. A turbulent boundary layer has a considerably greater thickness, reflecting its higher drag and therefore larger momentum loss; an approximate formula given in Schlichting (1960), p. 38, is 8/1 ~ 0.37/Re1/s. For

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Re = 10® this gives 6/1 = 0.023, which is still rather small. Viscosity is only important in a very small portion of the turbulent boundary layer next to the surface, in the “viscous sublayer. ”

The transition from a laminar to a turbulent boundary layer is a very complicated process that depends on so many factors that precise figures for transition Reynolds numbers cannot be given. For a flat plate in a very quiet free stream (i. e., one having a rms turbulent velocity fluctua­tion intensity of 0.1% or less) transition occurs at approximately a distance from the leading edge corresponding to Re ~ З X 10®. With a turbulent intensity of only 0.3% in the oncoming free stream the transition Reynolds number decreases to about 1.5 X 10®. These distances are far beyond that for which the boundary layer first goes unstable. Stability calculations show that on a flat plate this occurs at Re ~ 10s. The complicated series of events between the point where instability first sets in until transition occurs has only recently been clarified (see Klebanoff, Tidstrom, and Sargent, 1962). Transition is strongly influenced by the pressure gradient in the flow; a negative (“favorable”) pressure gradient tends to delay it and a positive (“adverse ”) one tends to make it occur sooner. As a practical rule of thumb one can state that the laminar boundary layer can only be maintained up to the point of minimum pressure on the airfoil. On a so-called laminar-flow airfoil one therefore places this point as far back on the airfoil as possible in order to try to achieve as large a laminar region as possible. Laminar-flow airfoils work successfully for moderately high Reynolds numbers (<107) and low lift coefficients but require ex­tremely smooth surfaces in order to avoid premature transition. At very high Reynolds numbers transition starts occurring in the region of favor­able pressure gradient.

Both laminar and turbulent boundary layers will separate if they have to go through extensive regions of adverse pressure gradients. Separation will always occur for a subsonic flow at sharp corners, because there the pressure gradient would become infinite in the absence of a boundary layer. Typical examples of unseparated and separated flows are shown in Fig. 4-1. In the separated flow there will always be a turbulent wake

behind the body. In principle one could have instead a region of fairly quiescent flow in the wake, separated from the outer flow by a thin laminar shear layer attached to the laminar boundary layer on a body. However, a free shear layer, lacking the restraining effect of a wall, will be highly unstable and will therefore turn turbulent almost immediately. Because of the momentum loss due to the turbulent mixing in the wake the drag of the body will be quite large. On a thin airfoil at a small angle of attack the boundary layer will separate at the sharp trailing edge but there will be a very small wake so that a good model for the flow is the attached flow with the Kutta condition for the inviscid outer flow determining the circulation.

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