A common external flow configuration involves the circular cylinder or tube in crossflow, where the flow is normal to the axis of the cylinder. If an inviscid fluid is considered, the velocity distribution over the cylinder is given by
where U_{∞} is the velocity far upstream the cylinder, r_{o} the cylinder radius, r the radial coordinate and θ the angle measured from the forward stagnation point. At the tube surface one finds
and thus for θ = π/2 the velocity on the tube surface is twice the freestream velocity.
By applying the Bernoulli Equation for an inviscid fluid, the pressure coefficient C_{p} is found to be
For an inviscid fluid, the pressure coefficient is distributed symmetrically and an integration of the pressure distribution results in zero drag and lift forces. This is an example of the d'Alembert paradox for inviscid flow past immersed bodied. In Figure 1, the inviscid flow past a tube is shown.
For viscous fluids the flow pattern is much more complicated and the balance between inertia forces and viscous forces is important. The relative importance is expressed by the Reynolds number Re_{D} defined as
where D is the tube diameter and ν the kinematic viscosity of the fluid.
As the fluid approaches the front side of the tube, the fluid pressure rises from the freestream value to the stagnation point value. The high pressure forces the fluid to move along the tube surface and boundary layers develop on both sides. The pressure force is counteracted by viscous forces and the fluid cannot follow the tube surface to the rear side but separates from both sides of the tube and form two shear layers. The innermost part of the shear layers are in contact with the tube surface and moves slower than the outermost part. As a result, the shear layers roll up. The flow pattern is dependent on the Reynolds number and in Figure 2 a principal description of the various occurring flow phenomena is provided.
Figure 2. Regimes of fluid flow across a smooth tube. From Blevins, R. D. (1990), Flow Induced Vibration, 2nd Edn., Van Nostrand Reinhold Co.
At Reynolds numbers below 1, no separation occurs. The shape of the streamlines is different from those in an inviscid fluid. The viscous forces cause the streamlines to move further apart on the downstream side than on the upstream side of the tube. In the Reynolds number range of 5 ≤ Re_{D} ≤ 45, the flow separates from the rear side of the tube and a symmetric pair of vortices is formed in the near wake. The streamwise length of the vortices increases linearly with Reynolds number as shown in Figure 3.
Figure 3. Streamwise length of vortices. From Taneda S. (1956) J Phys. Soc. Japan 11,302 with permission.
As the Reynolds number is further increased the wake becomes unstable and Vortex Shedding is initiated. At first, one of the two vortices breaks away and then the second is shed because of the nonsymmetric pressure in the wake. The intermittently shed vortices form a laminar periodic wake of staggered vortices of opposite sign. This phenomenon is often called the Karman vortex street. von Karman showed analytically and confirmed experimentally that the pattern of vortices in a vortex street follows a mathematical relationship, namely,
where h and 1 are explained in Figure 4.
In the Reynolds number range 150 < Re_{D} < 300, periodic irregular disturbances are found in the wake. The flow is transitional and gradually becomes turbulent as the Reynolds number is increased.
The Reynolds number range 300 < Re_{D} < 1.5·10^{5} is called subcritical (the upper limit is sometimes given as 2·10^{5}). The laminar boundary layer separates at about 80 degrees downstream of the front stagnation point and the vortex shedding is strong and periodic.
With a further increase of Re_{D}, the flow enters the critical regime. The laminar boundary layer separates on the front side of the tube, forms a separation bubble and later reattaches on the tube surface. Reattachment is followed by a turbulent boundary layer and the separation point is moved to the rear side, to about 140 degrees downstream the front stagnation point. As an effect the drag coefficient is decreased sharply.
For Re_{D} ≥ 6·10^{5}, the laminar to turbulent transition occurs in a nonseparated boundary layer, and the transition point is shifted upstream.
The Re_{D}-range 2·10^{5} < Re_{D} < 3.5·10^{6} is in some references named the transitional range. Three-dimensional effects disrupt the regular shedding process and the spectrum of shedding frequencies is broadened.
In the supercritical Re_{D}-range, Re_{D} > 3.5·10^{6}, a regular vortex shedding is re-established with a turbulent boundary layer on the tube surface. (Some references name the regime Re_{D} > 6·10^{5} the supercritical regime.)
The various flow phenomena are reflected in the pressure distribution on the tube surface. Figure 5 provides a few distributions of the pressure coefficient C_{p} and the changes in the distributions are explained by the flow mechanisms described previously.
Figure 5. Pressure distribution around the circumference of a tube or circular cylinder. From Batchelor, G. K. (1970). An Introduction to Fluid Mechanics, Cambridge University Press, with permission.
Particularly for Re_{D} = 6.7·10^{5}, the separation of the laminar boundary layer and the reattachment is believed to be reflected in the behavior between θ ≈ 100−110 degrees downstream the front stagnation point.
The nonsymmetrical pressure distribution result in a net force on the tube, and the existence of this force is the main cause of the pressure drop across the tube.
The total drag is generated by the friction forces and pressure forces acting on the tube. At very low Reynolds number, the drag is mainly due to friction. With an increase of Re_{D} the contribution of the inertia forces begin to grow so that at high Reynolds numbers the skin friction constitutes just a few per cent of the total drag. A dimensionless expression of the total drag is the drag coefficient defined by
where L is the length of the tube.
Figure 6 shows the drag coefficient as a function of Re_{D}.
In the low Reynolds number range, C_{D} decreases significantly with increasing Reynolds number, mostly due to the skin friction contribution. In the subcriticai regime, the drag coefficient changes insignificantly with Re_{D}. As the critical flow regime is reached, the drag coefficient decreases sharply with increasing Re_{D}. The wake becomes narrower by the turbulent boundary layer and the associated delayed separation. At higher Re_{D}, C_{D} is increasing again probably due to the development of a turbulent boundary layer already close to the front stagnation point.
The crossflow past the single tube is affected by freestream turbulence, surface roughness, compressibility of the fluid and some other factors. So, for instance, the position of the transition from laminar to turbulent flow in the boundary layer depends on the turbulence level. An increase in the freestream turbulence level leads to an earlier (lower Re_{D}) onset of the critical flow regime and corresponding changes in the drag coefficient. Surface roughness causes an earlier onset of the critical regime and a higher drag coefficient.
REFERENCES
žukauskas, A. A. and Ziugzda, J. (1985) Heat Transfer of a Cylinder in Cross Flow, Hemisphere Publishing Corporation, 1985.
Blevins, R. D. (1990) Flow-Induced Vibration, 2nd edn., Van Nostrand Reinhold, 1990.
Batchelor, G. K. (1970) An Introduction to Fluid Dynamics, Cambridge University Press, 1970.
Les références
- Å¾ukauskas, A. A. and Ziugzda, J. (1985) Heat Transfer of a Cylinder in Cross Flow, Hemisphere Publishing Corporation, 1985.
- Blevins, R. D. (1990) Flow-Induced Vibration, 2nd edn., Van Nostrand Reinhold, 1990.
- Batchelor, G. K. (1970) An Introduction to Fluid Dynamics, Cambridge University Press, 1970.