Laminar flows with complications

The case of a compressible fluid is more complicated since it is the inventory and not the volume that scales with A. The case of laminar flow is the simplest and is one where scaling with geometric similarity can make sense. 

When running a CFD simulation, a decision must be made as to whether to use a laminar-flow or a turbulent-flow model. For many flow situations, the transition from laminar to turbulent flow with increasing flow rate is quite sharp, for example, at Re — 2100 for flow in an empty tube. For flow in a fixed bed, the situation is more complicated, with the laminar to turbulent transition taking place over a range of Re, which is dependent on the type of packing and on the position within the bed. 

Imagine two parallel plates of area A between which is sandwiched a liquid of viscosity r). If a force F parallel to the x direction is applied to one of these plates, it will move in the x direction as shown in Figure 4.1. Our concern is the description of the velocity of the fluid enclosed between the two plates. In order to do this, it is convenient to visualize the fluid as consisting of a set of layers stacked parallel to the boundary plates. At the boundaries, those layers in contact with the plates are assumed to possess the same velocities as the plates themselves that is, v = 0 at the lower plate and equals the velocity of the moving plate at that surface. This is the nonslip condition that we described in Chapter 2, Section 2.3. Intervening layers have intermediate velocities. This condition is known as laminar flow and is limited to low velocities. At higher velocities, turbulence sets in, but we do not worry about this complication. 

For quartz spheres, the maximum particle radius compatible with Eq. 23-10 is only 24 pm. In contrast, biogenic particles have a much smaller excess density, ps - pw, and complicated shapes (a small). Typical settling velocities for these particles are 0.1 md 1 for 1-pm particles and lOOmd-1 for 100-pm particles (Lerman, 1979). The 100-pm particles just meet the laminar flow limit (Re = 0.1). 

The analysis of Sec. 5-10 has shown how one might analytically attack the problem of heat transfer in fully developed laminar tube flow. The cases of undeveloped laminar flow, flow systems where the fluid properties vary widely with temperature, and turbulent-flow systems are considerably more complicated but are of very important practical interest in the design of heat exchangers and associated heat-transfer equipment. These more complicated problems may sometimes be solved analytically, but the solutions, when possible, are... 

Entrance effects for turbulent flow in tubes are more complicated than for laminar flow and cannot be expressed in terms of a simple function of the Graetz number. Kays [36] has computed the influence for several values of Re and Pr with the results summarized in Fig. 6-6. The ordinate is the ratio of the... 

For systems involving turbulent flows accurate modeling is much more complicated in comparison with those involving laminar flow and here it is considered very important to interpret and use the computational results with great care, especially when one deals with turbulent flows in complex geometries. Whenever... 

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word hydrodynamic is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition. 

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