Flow around a Cylinder

To illustrate the idea of potential flow and how to use it to calculate forces, let us calculate the pressure distribution on the surface of a cylinder which is immersed in a flow perpendicular to it. If this is a very long cylinder, then there will be negligible change in the flow in the direction of the cylinder s axis, and so the flow will be practically two-dimensional. To find the flow field, we must make a judicious combination of a steady flow, a source, and a sink. Consider first a source and a sink with equal flow rates located some distance A apart on the X axis See Fig. 10.16. The flow between them is given by 

This limiting case of a source and a sink at zero distance apart is called a doublet. If we now combine this doublet flow with a uniform flow given by 0 = Dx, we find 

At this point it is convenient to switch to polar coordinates, so Eq. 10.61 

This flow is sketched in Fig. 10.17, in which one of the streamlines is a circle. From Eq. 10.63, this is the circle for which = 0 that is, r = (C/DY Thus, this is an ideal-fluid flow which has a circular streamline. In ideal-flow theory, we can substitute a solid body for any streamline without affecting the flow outside that streamline so the flow for r (CIDY is the same as the ideal-fluid flow which would exist outside a circular cylinder oriented perpendicular to the flow. 

Now we can japply Bernoulli s equation to find the pressure at any point on the surface of the cylinder. Let us assume that far to the right in Fig. 10.17 the flow is undisturbed by the cylinder, so that it is moving from right to left at a uniform velocity Vq with a uniform pressure Pq and that the changes in 

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Flow around a cylinder with axis normal to the flow provides a good example of the many unusual features of such flows. These are illustrated in Figure 3. 

Figure 2.47 Micro mixer based on the excitation of an electro-osmotic flow around a cylinder by an oscillatory electric field (top). The bottom of the figure shows particle traces on both sides of the liquid/liquid interface with no electric field (above) and with the electric field switched on (below), as described in [145].

A. Lamuraand G. Gompper, Numerical study of the flow around a cylinder using multi-particle collision dynamics, Eur. Phys. J. 9, All (2002). 

FIGURE 2 Particle motion in aerosol flow around obstacles (dashed line), (a) Flow around a cylinder of radius a (b) flow around a flat plate inclined at an angle to the aerosol flow. 

Flow type Flow over a flat plate U() Flow in a pipe Flow around a cylinder... 

Krahn [76] explained how the rotation of the sphere would cause the transition from laminar to turbulent boundary layers at different rotational velocities at the two sides of a sphere. The direction of the asymmetrical wake was explained based on the separation points for laminar and turbulent boundary layers. Krahn studied the flow around a cylinder. For a non-rotating cylinder the laminar boundary layer separates at 82° from the forward stagnation point, while the turbulent boundary layer separates at about 130°. Due to the rotation the laminar separation point will move further back, while the turbulent separation point will move forward. For some value of v qaa/v between 0 and 1 the laminar and turbulent separation points will be at equal distance from the stagnation point. The pressure on the turbulent side will be smaller than on the laminar side causing a negative Magnus force. 

Referring to the nondimensional equation of convective diffusion (3.3), it is of interest to examine the conditions under which the diffusion term, on the one hand, or convection, on the other, is the controlling mode of transport. The Peclet number, dUfD, for flow around a cylinder of diameter r/ is a measure of the relative importance of (he two term.s. For Pe 1, transport by llte flow can be neglected, and the deposition rate can be determined approximately by solving the equation of diffusion in a non flowing fluid with appropriate boundary conditions (Carslaw and Jaeger, 1959 Crank, 1975). 

Since the flow conditions strongly influence the heat transfer (recall Nu Re112 for laminar flows and Nu Re0- for turbulent flows over a flat plate not involving liquid metals), let us recall from fluid mechanics the results of experimental observations on flow around a cylinder, which are sketched in Kg. 6.7. We learn from this figure that the flow around a cylinder may assume different forms, depending on the Reynolds number. The separation, beginning with the second sketch from the top, is associated... 

Figure 10.17 shows the perfect-fluid solution for flow around a cylinder. The flow splits at the upstream face, flows smoothly around the cylinder, and rejoins at the downstream face. Equation 10.64 shows that at B — and 6 = 180° the flow velocity is zero these are the points where the flow divides, hence the flow has no velocity here. Such zero-velocity points are commonly called stagnation points. 

Realize that dvo/dr)r=R and Kdy/dt)re]r=R are nonzero for potential flow around a cylinder and that the first-order term in the polynomial expansion for V(, does not vanish, but this first-order term is small relative to the leading zeroth-order term. Now the locally flat description of the equation of continuity allows one to calculate the radial velocity component. For example, integration from the nondeformable solid-liquid interface at y = 0, where Vy = 0, to any position y within the thin mass transfer boundary layer produces the following result ... 

A convenient method of relating to the phenomenon of wind excitation is to equate it to fluid flow around a cylinder. In fact this is the exact case of early discoveries related to submarine periscopes vibrating wildly at certain speeds. At low flow rates, the flow around the cylinder is laminar. As the stream velocity increases, two symmetrical eddies are formed on either side of the cylinder. At higher velocities vortices begin to break off from tlie main stream, resulting in an imbalance in forces exerted from the split stream. The discharging vortex imparts a fluctuating force that can cause movement in the vessel perpendicular to the direction of the stream. 

For the flow around a cylinder, a platinum wire ( ( > = 50 (jm) constituted the cylinder and acted as a microelectrode. The wire was placed along a diameter in a tube and was covered by a varnish except for a small portion around the tube axis. 

Tanner, R. I., Stokes paradox for power law flow around a cylinder, J. Non-Newtonian Fluid Mech. 50 217-224 (1993). 

There exist many theoretical predictions of the permeability of fibrous porous media in the literatures [7] - [10], From early works on the theoretical predictions of the permeabihty of fibrous porous medium we can emphasize the works of John Happel (1959) [7] and Hasimoto (1959) [8]. John Happel [7] foimd the theoretical prediction of the permeability of fibrous porous media by solving the Stokes equation for a fluid flow in fibrous porous medium. The flow around a cylinder investigated in his work (see Fig. 2). His theoretical prediction of the permeabihty of fibrous porous medium ... 

Bayraktar, E., Mierka, O., Turek, S. (2012). Benchmark compulations of 3D laminar flow around a cylinder with CFX, OpenFOAM and FeatFlow. International Journal of Computer Science and Engineering, 7, 253-266. doi 10.1504/ijcse.2012.048245. 

Figure 18 The flow around a cylinder—left the incident flow being laminar right in a free-stream vorticity. Pictures taken as early as 1931 by Ahiborn, reprinted from Torobin and Gauvin (1960b) with permission from Wiley.

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