Circular cylinder boundary layer

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

There are some cases where this approach fails. One such case is that in which significant regions of separated flow exist. In this case, although the boundary layer equations are adequate to describe the flow upstream of the separation point, the presence of the separated region alters the effective body shape for the outer inviscid flow and the velocity outside the boundary layer will be different from that given by the inviscid flow solution over the solid surface involved. For example, consider flow over a circular cylinder as shown in Fig. 2.16. Potential theory gives the velocity, ui, on the surface of the cylinder as ... 

Solutions based on the use of the boundary layer approximations to the full governing equations have been discussed in the above sections. These boundary layer approximations qfe, however, often not applicable. For example, the boundary layer equations do not apply if there are significant areas of reversed flow or if the Reynolds number is low. Even with two-dimensional flow over a circular cylinder, the bound-... 

FIGURE 716 Laminar boundary layer separation witli a turbulent wake flow over a circular cylinder at Re = 7.000. 

Problem 9-12. Thermal Boundary Layer for a Circular Cylinder at Hr < 1 and... 

To illustrate this latter point, we first derive equations for the inner boundary-layer region for the specific problem of streaming flow past a circular cylinder, starting from the equations of motion expressed in a cylindrical coordinate system. These are... 

First, however, it is important to recognize that the form of equations (10 28), (10-30), and (10 32) is independent of the geometry of the body (i.e., independent of the cross-sectional shape for any 2D body). Although we started our analysis with the specific problem of flow past a circular cylinder, and thus with the equations of motion in cylindrical coordinates, the equations for the leading-order approximation in the inner (boundary-layer) region reduce to a local, Cartesian form with Y being normal to the body surface and x... 

The cause of large drag in the case of a body like a circular cylinder is the asymmetry in the velocity and pressure distributions at the cylinder surface that results from separation. All bodies in laminar streaming flow at large Reynolds number are subjected to viscous stresses that boundary-layer analysis shows must be... 

The potential-flow solution for streaming motion past a circular cylinder was obtained earlier and given in terms ofthe streamfunctionin(10-17). To calculate the pressure gradient in the boundary layer, we first determine the tangential velocity function, ue, as defined in (10-37) ... 

The boundary-layer problem for the specific case of a circular cylinder is (10-40), (10 41), (10-43), and (10-47), with ue and 3p/dx given by (10-122) and (10-123). The first point to note is that a similarity solution does not exist for this problem. Furthermore, in view of the qualitative similarity of the pressure distributions for cylinders of arbitrary shape, it is obvious that similarity solutions do not exist for any problems of this general class. The Blasius series solution developed here is nothing more than a power-series approximation of the boundary-layer solution about x = 0. 

At the beginning of this section, we started out to analyze the boundary-layer solution for the specific case of a circular cylinder. Thus it is of interest to apply (10-148) and (10-149) to that problem by using the un coefficients given by (10-133). The result is... 

Figure 10-9. The dimensionless shear stress as a function of position on the surface of a circular cylinder as calculated with the approximate Blasius series solution. Note that x is measured in radians from the front-stagnation point. The predicted point of boundary-layer separation corresponds to the second zero of du/dY 0. and is predicted to occur just beyond the minimum pressure point atx = jt/2.

Experimental observations of the flow past a circular cylinder show that separation does indeed occur, with a separation point at 0S — 110 . It should be noted, however, that steady recirculating wakes can be achieved, even with artificial stabilization,24 only up to Re 200, and it is not clear that the separation angle has yet achieved an asymptotic (Re —> oo) value at this large, but finite, Reynolds number. In any case, we should not expect the separation point to be predicted too accurately because it is based on the pressure distribution for an unseparated potential flow, and this becomes increasingly inaccurate as the separation point is approached. The important fact is that the boundary-layer analysis does provide a method to predict whether separation should be expected for a body of specified shape. This is a major accomplishment, as has already been pointed out. 

The problem of start-up flow for a circular cylinder has received a great deal of attention over the years because of its role in understanding the inception and development of boundary-layer separation. An insightful paper with a comprehensive reference list of both analytical and numerical studies is S. I. Cowley, Computer extension and analytic continuation of Blasius expansion for impulsive flow past a circular cylinder, J. Fluid Mech. 135, 389-405 (1983). 

The position of the separation point depends on the Reynolds number. At low Reynolds numbers, we have a flow without separation (see Figure 1.7). In a cross flow around a circular cylinder, the separation occurs if the Reynolds number (the cylinder diameter is taken as the characteristic length) is greater than 5 [486]. For 5 < Re < 40, a separation region with steady-state symmetric adjacent vortices is formed (there is no boundary layer yet). 

In what follows we consider the laminar boundary layer in a cross flow past a circular cylinder. This problem is also of practical importance, since tube elements are widely used in industrial equipment. 

In the range 103 < Re < 105 (here the radius a of the circular cylinder is taken as the characteristic length, Re = aUi/u), the laminar boundary layer approximation is valid, and the separation point do moves from 109° to 85° [427], In this case, retaining only the first three terms in the expansion (1.8.13) provides fairly good accuracy. 

In the book [117], some data are given on the hydrodynamic characteristics of bodies of various shapes these data mainly pertain to the region of precrisis self-similarity. The influence of roughness of the cylinder surface and the turbulence level of the incoming flow on the drag coefficient is discussed in [522]. In [211], the relationship between hydrodynamic flow characteristics in turbulent boundary layers and the longitudinal pressure gradient is studied. Analysis of the transition to turbulence in the wake of circular cylinders is presented in [333]. 

The solution of the corresponding mass exchange problem for a circular cylinder and an arbitrary shear flow was obtained in [353] in the diffusion boundary layer approximation. It was shown that an increase in the absolute value of the angular velocity Cl of the shear flow results in a small decrease in the intensity of mass and heat transfer between the cylinder and the ambient... 

In the mass exchange problem for a circular cylinder freely suspended in linear shear flow, no diffusion boundary layer is formed as Pe - oo near the surface of the cylinder. The concentration distribution is sought in the form of a regular asymptotic expansion (4.8.12) in negative powers of the Peclet number. The mean Sherwood number remains finite as Pe - oo. This is due to the fact that mass and heat transfer to the cylinder is blocked by the region of closed circulation. As a result, mass and heat transfer to the surface is mainly determined by molecular diffusion in the direction orthogonal to the streamlines. In this case, the concentration is constant on each streamline (but is different on different streamlines). 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

A particularly interesting phenomenon connected with transition in the boundary layer occurs with blunt bodies, e.g., spheres or circular cylinders. In the region of adverse pressure gradient (i.e., dP/dx > 0 in Fig. 1.9) the boundary layer separates from the surface. At this location the shear stress goes to zero, and beyond this point there is a reversal of flow in the vicinity of the wall, as shown in Fig. 1.9. In this... 

The thin-layer approximation fails because natural convective boundary layers are not thin. From the interferometric fringes in Fig. 4.2ft (which are essentially isotherms), the thermal boundary layer around a circular cylinder is seen to be nearly 30 percent of the cylinder diameter. For such thick boundary layers, curvature effects are important. Despite this failure, thin-layer solutions provide an important foundation for the development of correlation equations, as explained in the section on heat transfer correlation method. 

Darcy-like flows obtain except in a thin boundary layer near solid walls, where Darcy s equation is unable to satisfy the no-slip boundary condition [cf. Brinkman s treatment (B35) of flow through a porous medium bounded externally by a circular cylinder with solid walls]. 

In the range of Reynolds number Re = 103 to 107 (based on cyhnder diameter and free stream velocity), the flow aronnd a solid circular cylinder is periodic and transitional in character. The range of interest of the present work is located in a sub-critical flow regime (103 < Re < 105, corresponding to air velocities of - 0.1-10 m/s around a typically sized 0.1m diameter limb), in which, dne to the vortex shedding at the cylinder surface, the flow is highly unstable. The boundary layer remains fidly laminar up to the separation point and transition to turbulence... 

A closer analysis of this problem would reveal more complex situations, such as a fluid flowing around a solid body. In that case the streamlines may take off behind the body at the limit of zero viscosity of the fluid. However, all fluids exhibit some viscosity and no such phenomenon can be observed. Experiments show that vorticity is generally generated in a thin boundary layer, close to a solid surface. It is propagated from the wall by both viscous diffusion and convection. The vortices are transported with the fluid they are observable for some time after their appearance. If the experiment is made with a circular cylinder moving at a constant velocity, the eddies appear in the wake of the body and their regular distribution constitutes the famous, as well as beautiful, Karman vortex street . 

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