Flow of a fluid normal to objects or groups of objects such as cylinders is referred to as crossflow. It is to be differentiated from *longitudinal flow* along the axis of symmetry, the term *inclined crossflow* being used to describe the intermediate condition. A crossflow is usually considered an External Flow, though for objects inside a duct, or groups of objects, there will be an overall pressure gradient and the flow may have some features of an *internal flow*. The objects could be *bluff* or *streamlined*, i.e., the cylinders need not be circular.

The flow of a *perfect fluid* (potential flow) past a cylinder bifurcates at the front edge of the body, φ = 0°, where static pressure is a maximum, accelerates to the pressure minimum at 90°, and decelerates in the presence of the adverse pressure gradient, re-uniting at φ = 180°. For a *real fluid*, the flow is substantially influenced by viscous effects, and the Streamlines are concentrated in a Boundary Layer near the wall (Figure 1). The solid wall also exerts *drag* on the fluid. Such a flow is characterized by the ratio of inertia to viscous forces, or Reynolds Number, Re

where ρ is fluid density, η is fluid viscosity, and u is a mean or bulk fluid velocity. D is a diameter or other suitable length, such as a hydraulic diameter.

For a very small Re or Creeping Flow, the streamlines are symmetric and drag is mainly due to skin-friction. In the range 1 ≤ Re ≤ 40, symmetry is lost and *boundary-layer separation* occurs at φ = 82°. Downstream, a *wake* forms with two stationary Vortices being observed at the rear of the cylinder. The flow is Laminar. For Re > 40, the flow becomes periodic with the vortices being shed alternately at a frequency, f, corresponding to a Strouhal Number, Sr = fD/η of 0.2. As Re further increases, the behavior becomes less coherent and turbulent eddies are observed in the wake. At the *critical Re*, Re_{c} = 2 × 10^{5}, the separation point suddenly moves downstream, φ > 90°, and the boundary layer becomes *turbulent*. There may be a separation bubble. The presence *of free-stream turbulence* or *surface-roughness* will decrease Re_{c}. In the *supercritical* range, a turbulent periodic behavior is apparent with Sr around 0.3.

Drag is due to two components: a) *skin-friction* or the viscous shear-force, F; and b) *form drag* due to the normal force, P, i.e., Pressure. The overall drag coefficient, C_{D}, is defined as:

where F_{x} and P_{x} are integrated values projected in the main-flow direction (see Figure 1) and A_{p} is the area (usually projected normal to the flow for a bluff body, and parallel to the flow for a streamline body.) Figure 2 shows c_{D} vs. Re for a single cylinder. At low Re, drag is due mainly to skin-friction and c_{D} decreases with Re. By Re = 200, drag is almost entirely pressure-related and c_{D} is fairly constant; the slight rise in the range 2 × 10^{3} - 2 × 10^{5} is a result of wake turbulence and narrowing of the wake due to upstream movement of the separation-point. The so-called drag crisis occurs at Re_{c} with c_{D} falling abruptly as the wake visibly narrows.

This description is qualitatively true for other similar geometries such as spheres (see Spheres, Flow Around and Drag). For other shapes, there are ignificant differences in both flow and pressure fields. For streamlined bodies such as a rectangular fin (flat plate) aligned with the flow, form drag and hence, c_{D}, will be much reduced. Conversely, for a baffle-plate placed across the flow, pressure will dominate, the wake will be of uniform width, and c_{D} will be a relatively constant value of around 2 over a wide Re-range. Other effects such as blockage, free-stream turbulence, surface roughness, and secondary-flows in complex geometries such as finned tubes, will significantly alter the nature of the flow.

For flow past *banks of objects*, there is an overall pressure gradient. The *friction coefficient*, f, is typically defined as:

where τ = (F_{x} + P_{x})/A is an equivalent stress due to drag on *all* surfaces in the bank (which may or may not include walls, entrance effects, etc.).
is the mean pressure drop along distance L; A_{c} is the *minimum* free cross-sectional area; A is the total (unprojected) surface area in contact with the fluid; and r_{h} = A_{c}L/A is a hydraulic radius [(Kays and London (1984)]. The term radius is actually a misnomer which has persisted for historical reasons; since for a circular cylinder r_{h} = r/2. Numerous variations on f exist, e.g., the Euler Number, Eu (see Tube banks, crossflow over). Charts of f, C_{D}, or Eu as a function of Re, present important information to the fluids engineer and are available for a wide variety of geometries and flow conditions

#### REFERENCES

Kays, W. M. and London, A. L. (1984) *Compact Heat Exchangers*, McGraw-Hill, New York.