A rapid method for the prediction of flow separation results from an approximate solution of the equations of motion; a single empirical factor is required. The equations are integrated by a modified ‘inner and outer solutions’ technique developed recently for laminar boundary layers, the criterion for separation being obtained as a simple formula applying directly to the separation position. At Reynolds numbers of the order of 106, the criterion is $C_p(xdC_p|dx)^{\frac {1}{2}} = 0 \cdot 39 (10^{-6}R)^{\frac {1}{10}},$ when d2p/dx2 [ges ] 0 and Cp [les ] 4/7; the coefficient 0·39 is replaced by 0·35 when d2p/dx2 < 0.The prediction of the pressure rise to separation is likely to be from 0 to 10% too low, which puts it second in accuracy to those methods, such as Maskell's (1951), which utilize the Ludweig-Tillmann skin friction law. However, the convenience of the method makes the present error acceptable for many applications, while a greater accuracy should be attainable from an improved allowance for the quantity d2p/dx2.The main derivation is for arbitrary pressure distributions, while an extension leads to the pressure distribution which just maintains zero skin friction throughout the region of pressure rise.The concept of a turbulent inner layer with zero wall stress is put forward, and it is deduced that in the neighbourhood of the wall the velocity is proportional to the square root of the distance from the wall.

A flow has been produced having effectively zero skin friction throughout its region of pressure rise, which extended for a distance of 3 ft. No fundamental difficulty was encountered in establishing the flow and it had, moreover, a good margin of stability. The dynamic head in the zero skin friction boundary layer was found to be linear at the wall (i.e. u ∞ y½), as predicted theoretically in the previous paper (Stratford 1959).The flow appears to achieve any specified pressure rise in the shortest possible distance and with probably the least possible dissipation of energy for a given initial boundary layer. Thus an aerofoil which could utilize it immediately after transition from laminar flow would be expected to have a very low drag. A design pressure distribution (besides having the usual safety margin against stall) should have a slightly more gradual start to the pressure rise than in the present experiment, as small errors close to the discontinuity can cause difficulty.

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SummaryA choked nozzle with an appropriate wall contour has adischarge coefficient, CD, so close to unity that a theoretical calculation of (I—CD) would allow the nozzle to be used as an absolute meter for air flow. The high discharge coefficient results basically from the fact that ∂(ρv)∂p=0 at M=1.Simplified calculations yield formulae for the boundary layer displacement thickness and for the flow reduction resulting from the variation in static pressure across the throat. The optimum profile for the wall at the throat of an absolute meter is suggested to be a circular arc of radius of curvature equal to about twice the throat diameter. For such a meter the theoretical discharge coefficient is found to be within ¼ per cent of 0·995 over a wide range of Reynolds numbers.The uncertainty in the discharge coefficient for a steady flow at Reynolds numbers of 106 and over appears to be less than ±0·15 per cent, both when the boundary layer is known to be entirely turbulent and when it is known to be entirely laminar. When the state of the boundary layer is not known the corresponding figure appears to be ±0·25 per cent. Experimental information might therefore be helpful on transition—under the appropriate conditions of flow unsteadiness and rig vibration. Available experimental results with known boundary layers tend to confirm the theoretical discharge coefficients down to a Reynolds number of 0·4x106.A pressure ratio of about 1·1/1 or less would probably be sufficient to establish fully supersonic flow if the nozzle were followed by a suitable diffuser.