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A similarity theory for the turbulent plane wall jet without external stream
- WILLIAM K. GEORGE, HANS ABRAHAMSSON, JAN ERIKSSON, ROLF I. KARLSSON, LENNART LÖFDAHL, MARTIN WOSNIK
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- Journal:
- Journal of Fluid Mechanics / Volume 425 / 25 December 2000
- Published online by Cambridge University Press:
- 01 December 2000, pp. 367-411
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A new theory for the turbulent plane wall jet without external stream is proposed based on a similarity analysis of the governing equations. The asymptotic invariance principle (AIP) is used to require that properly scaled profiles reduce to similarity solutions of the inner and outer equations separately in the limit of infinite Reynolds number. Application to the inner equations shows that the appropriate velocity scale is the friction velocity, u∗, and the length scale is v/u∗. For finite Reynolds numbers, the profiles retain a dependence on the length-scale ratio, y+1/2 = u∗y1/2/v, where y1/2 is the distance from the wall at which the mean velocity has dropped to 1/2 its maximum value. In the limit as y+1/2 → ∞, the familiar law of the wall is obtained. Application of the AIP to the outer equations shows the appropriate velocity scale to be Um, the velocity maximum, and the length scale y1/2; but again the profiles retain a dependence on y+1/2 for finite values of it. The Reynolds shear stress in the outer layer scales with u2*, while the normal stresses scale with U2m. Also Um ∼ yn1/2 where n < −1/2 and must be determined from the data. The theory cannot rule out the possibility that the outer flow may retain a dependence on the source conditions, even asymptotically.
The fact that both these profiles describe the entire wall jet for finite values of y+1/2, but reduce to inner and outer profiles in the limit, is used to determine their functional forms in the ‘overlap’ region which both retain. The result from near asymptotics is that the velocity profiles in the overlap region must be power laws, but with parameters which depend on Reynolds number y+1/2 and are only asymptotically constant. The theoretical friction law is also a power law depending on the velocity parameters. As a consequence, the asymptotic plane wall jet cannot grow linearly, although the difference from linear growth is small.
It is hypothesized that the inner part of the wall jet and the inner part of the zero-pressure-gradient boundary layer are the same. It follows immediately that all of the wall jet and boundary layer parameters should be the same, except for two in the outer flow which can differ only by a constant scale factor. The theory is shown to be in excellent agreement with the experimental data which show that source conditions may determine uniquely the asymptotic state achieved. Surprisingly, only a single parameter, B1 = (Umv/Mo)/ (y+1/2Mo/v2)n = constant where n ≈ −0.528, appears to be required to determine the entire flow for a given source.
A theory for turbulent pipe and channel flows
- MARTIN WOSNIK, LUCIANO CASTILLO, WILLIAM K. GEORGE
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- Journal:
- Journal of Fluid Mechanics / Volume 421 / 25 October 2000
- Published online by Cambridge University Press:
- 02 November 2000, pp. 115-145
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A theory for fully developed turbulent pipe and channel flows is proposed which extends the classical analysis to include the effects of finite Reynolds number. The proper scaling for these flows at finite Reynolds number is developed from dimensional and physical considerations using the Reynolds-averaged Navier–Stokes equations. In the limit of infinite Reynolds number, these reduce to the familiar law of the wall and velocity deficit law respectively.
The fact that both scaled profiles describe the entire flow for finite values of Reynolds number but reduce to inner and outer profiles is used to determine their functional forms in the ‘overlap’ region which both retain in the limit. This overlap region corresponds to the constant, Reynolds shear stress region (30 < y+ < 0.1R+ approximately, where R+ = u*R/v). The profiles in this overlap region are logarithmic, but in the variable y + a where a is an offset. Unlike the classical theory, the additive parameters, Bi, Bo, and log coefficient, 1/κ, depend on R+. They are asymptotically constant, however, and are linked by a constraint equation. The corresponding friction law is also logarithmic and entirely determined by the velocity profile parameters, or vice versa.
It is also argued that there exists a mesolayer near the bottom of the overlap region approximately bounded by 30 < y+ < 300 where there is not the necessary scale separation between the energy and dissipation ranges for inertially dominated turbulence. As a consequence, the Reynolds stress and mean flow retain a Reynolds number dependence, even though the terms explicitly containing the viscosity are negligible in the single-point Reynolds-averaged equations. A simple turbulence model shows that the offset parameter a accounts for the mesolayer, and because of it a logarithmic behaviour in y applies only beyond y+ > 300, well outside where it has commonly been sought.
The experimental data from the superpipe experiment and DNS of channel flow are carefully examined and shown to be in excellent agreement with the new theory over the entire range 1.8 × 102 < R+ < 5.3 × 105. The Reynolds number dependence of all the parameters and the friction law can be determined from the single empirical function, H = A/(ln R+)α for α > 0, just as for boundary layers. The Reynolds number dependence of the parameters diminishes very slowly with increasing Reynolds number, and the asymptotic behaviour is reached only when R+ [Gt ] 105.