Boundary layer thickening

That the flow may be considered essentially as unidirectional (2f-direction) and that the effects of velocity components perpendicular to the surface within the boundary layer may be neglected (that is, uy Reynolds numbers where the boundary layer thickens rapidly. 

It is seen from equation 11.66 that the heat transfer coefficient theoretically has an infinite value at the leading edge, where the thickness of the thermal boundary layer is zero, and that it decreases progressively as the boundary layer thickens. Equation 11.66 gives the point value of the heat transfer coefficient at a distance x from the leading edge. The mean value between. v = 0 and x = x is given by ... 

Levich (L3) obtained an asymptotic solution to Eq. (3-39) for Pe oo, using the thin concentration boundary layer assumption discussed in Chapter 1. Curvature of the boundary layer and angular diffusion are neglected (i.e., the last term in Eq. (3-39) is deleted), so that the solution does not hold at the rear of the sphere where the boundary layer thickens and angular diffusion is significant. The asymptotic boundary layer formula, Eq. (1-59), reduces for a sphere to ... 

This transition has profound effects in all fluid dynamics, and certainly so in aerodynamics. The velocity profile in (he boundary layer becomes fuller neat the surface on account of Ihe higher average kinetic energy of the layer created by turbulent energy exchange from layer lo layer. The effective viscosity is therefore larger in turbulent than laminar flow, ihe turbulent boundary layer thickens more rapidly downstream, the skin friction increases. 

When a gas enters a smooth pipe from a large reservoir through a well-faired entry, a laminar boundary layer forms along the walls. The velocity profile in the main body of the How remains flat. The velocity boundary layer thickens with distance downstream from the entry until it eventually fills the pipe. If the Reynolds number based on pipe diameter is less than 2100, the pipe boundary layer remains laminar. The flow is said to be fully... 

The boundary layer can be either laminar or turbulent. At the forward stagnation point, it will always be laminar. However, as energy is removed in the form of heat, the boundary layer thickens, which, in turn, gives rise to an increase in the mean Reynolds Number for the boundary layer. Upon reaching a critical value of Reynolds Number, the boundary layer becomes turbulent. 

That the stream velocity does not change in the direction of flow. On this basis, from Bernoulli s theorem, the pressure then does not change (that is, dP/dx — 0). In practice, 3P/ dx may be positive or negative. If positive, a greater retardation of the fluid will result, and the boundary layer will thicken more rapidly. If dP/ dx is negative, the converse will be true. 

The rate of thickening of the boundary layer is then obtained by differentiating equation 11.17 ... 

It is of interest to compare the rates of thickening of the streamline and turbulent boundary layers at the transition point. Taking a typical value of Rexc — 105, then ... 

Thus the turbulent boundary layer is thickening at about four times the rate of the streamline boundary layer at the transition point. 

When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fulty developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist, as shown in Figure 11.8. 

Another important case is where the heat flux, as opposed to the temperature at the surface, is constant this may occur where the surface is electrically heated. Then, the temperature difference 9S — o will increase in the direction of flow (x-direction) as the value of the heat transfer coefficient decreases due to the thickening of the thermal boundary layer. The equation for the temperature profile in the boundary layer becomes ... 

The boundary layer thickness gradually increases until a critical point is reached at which there is a sudden thickening of the boundary layer this reflects the transition from a laminar boundary layer to a turbulent boundary layer. For both types, the flow outside the boundary layer is completely turbulent. In that part of the boundary layer near the leading edge of the plate the flow is laminar and consequently this is known as a... 

Fontaine et al. [81] concluded that the increase in crystallinity by further heating material, crystallized at 200 °C, to 215 °C involves a crystal (lamellae) thickening process which is probably due to crystal perfection at the boundary layers. Further annealing of this material at temperatures above 215 °C led to a melting temperature increase that was attributed to crystal perfection alone and not to crystal thickening. 

For the flow of a viscous fluid past the cylinder, the pressure decreases from A to B and from A to C so that the boundary layer is thin and the flow is similar to that obtained with a non-viscous fluid. From B to D and from C to D the pressure is rising and therefore the boundary layer rapidly thickens with the result that it tends to separate from the surface. If separation occurs, eddies are formed in the wake of the cylinder and energy is thereby dissipated and an additional force, known as form drag, is set up. In this way, on the forward surface of the cylinder, the pressure distribution is similar to that obtained with the ideal fluid of zero viscosity, although on the rear surface, the boundary layer is thickening rapidly and pressure variations are very different in the two cases. 

B 90° is due to the transition from laminar to turbulent flow. The later decrease ill Nu is again due to the thickening of the boundary layer. Nug reaches its second minimum at about 6 140°, which is the flow separation point in turbulent flow, and increases with 0 as a result of die intense mixing in the turbulent wake region. 

In addition to reduced friction factors (reduced momentum transfer), the heat transfer abilities of DR solutions are also greatly reduced. This may be caused by the thickened viscous boundary layers of DR flow and/or by reduced velocity fluctuations perpendicular to the flow.f Heat transfer reduction is defined as ... 

A multinational is a horizontal power system organizing a thin layer of human activities without precise geographical boundaries. The horizontal power (the multinational) generates confusion and loss of control by the vertical one (the state), and, with the layers thickening, will inevitably lock with it. The ostensible muck-raking against multinationals is a clear symptom of that. But what will be the outcome ... 

Near the leading edge of a flat plate immersed in a fluid of uniform velocity, the boundary layer is thin, and the flow in the boundary layer is entirely laminar. As the layer thickens, however, at distances farther from the leading edge, a point is reached where turbulence appears. The onset of turbulence is characterized by a sudden rapid increase in the thickness of the boundary layer, as shown in Fig. 3.7. 

Sensible heat and the latent heat of freeing are removed from the water at the liquid/solid interface. Under the prevailing static conditions heat will pass from the water to the cold sink by conduction. The resistance to this heat flow initially, will be a combination of the thermal resistances in the liquid and due to the cold solid. Immediately a layer of ice be ns to form on the cold surface a further resistance is added to the other thermal resistances. As further heat is extracted from the water, the ice layer thickens representing an advancing boundary between the solid ice and the liquid water, i.e. a transient condition. The transient condition, coupled with complex geometries and different forms of ice structure dependent in turn on the rate of cooling, constitute severe problems of mathematical analysis. 

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