Boundary layer unstable

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

Figure8.15. Freeandforcedconvectionregimes (o)freeconvectiveplume,where(6)convective plume deflected by wind (c) unstable boundary layer and (d) stable or neutral boundary layer, where Ts < (Adams et al., 1990).

The unstable boundary layer is a flow regime that exists at wind speeds greater than approximately 1.5 m/s when is positive. This is a commonly occurring regime... 

Figure E8.6.1. Computations of gas film coefficient for the unstable boundary layer of Example 8.6 and for a neutral boundary layer that is otherwise similar.

Figure 3.6. Schlieren photographs showing the changes in thickness of the diffusion boundary layer and the behavior of buoyancy-driven convection shown in relation to bulk supersaturation [1], [2]. The figure shows the (111) faceofaBa(N03)2 crystal from an aqueous solution. In region I, only the thickness of the diffusion boundary layer increases in region II, we see unstable lateral convection (HA) and intermittently rising plumes (IIB) and in region III we see steady buoyancy-driven convection.

It seems possible, therefore, that in this initial region the upper part of the film, outside the growing boundary layer, is in potential-like flow, and that once the boundary layer reaches the free surface, its vorticity is sufficient to trigger the wave disturbances, which can then propagate or not, depending on whether the flow is unstable or stable (jVFr > 1 or N < 1). 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

The Orr- Sommerfeld equation can be solved either as a temporal or as a spatial instability problem. For disturbance field created as a consequence of a localized excitation inside boundary layers, the temporal growth of the disturbance field is not realistic. It has been observed phenomenologically that for attached flows, instability is usually of a convective t3rpe and obtaining solution by spatial analysis is the appropriate one. In chapter 4, we will note that even for such a problem there can be spatio-temporally growing wave-fronts that dominate in attached boundary layers that are noted to be spatially stable. Such a problem is not evident for flows those are spatially unstable. The monograph by Betchov Criminale (1967) specifically talks about temporal growth of disturbances in shear layers and the readers are referred there for detailed expositions. 

Figure 4.2 Neutral curve for the Blasius boundary layer identifying stable and unstable regions. The marked points are further investigated.

Systematic differences appear to exist in sea-salt aerosol concentrations related to locations having significantly different sea surface temperatures. The predominant effect of sea surface temperature may pertain to the probability of formation of stable or unstable conditions in the marine boundary layer. 

The static stability of the air stream usually changes as it moves into and out of the urban area, typically becoming less and more stable, respectively. However it should not be assumed that the boundary layer profiles over the urban area and downwind are identical to the equilibrium states found in neutral, stable and unstable boundary layers over flat terrain. In fact as the flow adjusts characteristic distortions of the air flow profiles occur on these scales, such as blocked flow, unsteady slope flows, gravity currents and boundary layer jets especially near hills, coasts and urban/rural boundaries. These distorted profiles (which are ignored in most mesoscale atmospheric models) significantly affect dispersion (e.g. Hogstrom and Smedman, [274] Owinoh et al., [477]). 

Deardorff, J.W. (1972) Numerical investigations of neutral and unstable planetary boundary layers, J. Atmos. Sci. 18, 495-527. 

Deardorff JW (1972) Numerical Investigation of Neutral and Unstable Planetary Boundary Layers. Journal of the Atmospheric Science 29 91-115... 

Reed M, Johansen 0, Brandvik PJ, Daling P, Lewis A, Fiocco R, Mackay D, and Prentki R (1999) Oil spill modeling toward the close of the 20th century Overview of the state of art. Spill Science Technology Bulletin 5 3-16 Thomson TW, Liu WT, and Weissman DE (1983) Synthetic radar observation of ocean roughness from rolls in unstable marine boundary layer. Geophys Res Lett 10 1172-1175... 

---