Cylinder boundary layer

Fig. 4. Boundary layer development around a circular cylinder where A represents the point of separation.

Cylindrical Boundary Layer Laminar boundary layers on cylindrical surfaces, with flow parallel to the cylinder axis, are described by Glauert and LighthiU Proc. R. Soc. [London], 230A, 188-203 [1955]), Jaffe and Okamura (Z. Angety. Math. Phys., 19, 564—574 [1968]) and Stewartson ((J. Appl Math., 13, 113-122 [1955]). For a turbulent boundaiy layer, the total drag may be estimated as... 

For a turbulent boundary layer, the total drag may be roughly estimated using Eqs. (6-184) and (6-185) for finite cylinders. Measured forces by Kwon and Prevorsek ]. Eng. Jnd., 101, 73-79 [1979]) are greater than predicted this way. 

Fig. 9.4 A turbulent boundary layer forms behind a cylinder for TZ 10. ...

When velocity gradients are small, for example, near the boundary layer separation point and at the rear of a cylinder in separated flow, Eq. (33) is inaccurate. The separation point was determined with an accuracy of 1 degree by using twin strip electrodes of 125 /im length, separated by... 

Weder s experiments were carried out with opposing body forces, and large current oscillations were found as long as the negative thermal densification was smaller than the diffusional densification. [Note that the Grashof numbers in Eq. (41) are based on absolute magnitudes of the density differences.] Local mass-transfer rates oscillated by 50%, and total currents by 4%. When the thermal densification dominated, the stagnation point moved to the other side of the cylinder, while the boundary layer, which separates in purely diffusional free convection, remained attached. 

As seen in Fig. 11-2, the drag coefficient for the sphere exhibits a sudden drop from 0.45 to about 0.15 (almost 70%) at a Reynolds number of about 2.5 x 105. For the cylinder, the drop is from about 1.1 to about 0.35. This drop is a consequence of the transition of the boundary layer from laminar to turbulent flow and can be explained as follows. 

Figure 1 shows the geometry. Strictly speaking, equation 2 applies only to single cylinders in slow cross flow, for point particles with negligible deposition and with the thickness of boundary layer much less than the radius of the cylinder. 

The temperature of a liquid metal stream discharged from the delivery tube prior to primary breakup can be calculated by integrating the energy equation in time. The cooling rate can be estimated from a cylinder cooling relation for the liquid jet-ligament breakup mechanism (with free-fall atomizers), or from a laminar flat plate boundary layer relation for the liquid film-sheet breakup mechanism (with close-coupled atomizers). 

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. 

As Re increases further and vortices are shed, the local rate of mass transfer aft of separation should oscillate. Although no measurements have been made for spheres, mass transfer oscillations at the shedding frequency have been observed for cylinders (B9, D6, SI2). At higher Re the forward portion of the sphere approaches boundary layer flow while aft of separation the flow is complex as discussed above. Figure 5.17 shows experimental values of the local Nusselt number Nuj c for heat transfer to air at high Re. The vertical lines on each curve indicate the values of the separation angle. It is clear that the transfer rate at the rear of the sphere increases more rapidly than that at the front and that even at very high Re the minimum Nuj. occurs aft of separation. Also shown in Fig. 5.17 is the thin concentration boundary layer... 

The boundary layer equations for an axisymmetric body, Eqs. (1-55), (10-17), and (10-18) have been solved approximately for arbitrary Sc (L4). For Sc oo the mean value of Sh can be computed from Eq. (10-20). Solutions have also been obtained for Sc oo for some shapes without axial symmetry, e.g., inclined cylinders (S34). Data for nonspherical shapes are shown in Fig. 10.3 for large Rayleigh number. The characteristic length in Sh and Ra is analogous to that used in Chapters 4 and 6 ... 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

Beginning with the innovative work of Tsuji and Yamaoka [409,411], various counter-flow diffusion flames have been used experimentally both to determine extinction limits and flame structure [409]. In the Tsuji burner (see Fig. 17.5) fuel issues from a porous cylinder into an oncoming air stream. Along the stagnation streamline the flow may be modeled as a one-dimensional boundary-value problem with the strain rate specified as a parameter [104], In this formulation complex chemistry and transport is easily incorporated into the model. The chemistry largely takes place within a thin flame zone around the location of the stoichiometric mixture, within the boundary layer that forms around the cylinder. 

Knaff and Schlunder [9] studied the evaporation of naphthalene and caffeine from a cylindrical surface (a sintered metallic rod impregnated with the solute) to high-pressure carbon dioxide flowing over an annular space around the rod. They studied the diffusion flux within the bar and in the boundary layer. The mass-transfer coefficient owing to forced convection from cylinder to the gas flowing in the annular duct was correlated, using the standard correlation due to Stephan [7]. For caffeine, it does not require a free-convection correction, as the Reynolds dependence is that expected by a transfer by forced convection. This is... 

Movement of the Huid may be generated by means external to the heat transfer process, us by fans, blowers, or pumps. It may also be created by density differences connected with the heat transfer process itself. The first mode is culled timet cniireeiirtn the second one natural or free t ttttveclion. Convection heal transfer may also be classified as heat transfer in iltni /fnn. or in interna flow (over cylinders, spheres, air foils, and similar objects). In ilie case of external flow, the heal transfer process is essentially concentrated in a thin fluid layer surrounding the object (boundary layer . 

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