Basic Bluff-Body Aerodynamics
Flow around Bluff Bodies
Structures of interest in this book can generally be classified as bluff bodies with respect to the air flow around them, in contrast to streamlined bodies, such as aircraft wings and yacht sails (when the boat is sailing across the wind). Figure 4.1 shows the flow patterns around an airfoil (at low angle of attack), and around a two-dimensional body of rectangular cross section. The flow patterns are shown for steady free-stream flow; turbulence in the approaching flow, which occurs in the atmospheric boundary layer, as discussed in Chapter 3, can modify the flow around a bluff body, as will be discussed later.
It can be seen in Figure 4.1 that the flow streamlines around the airfoil follow closely the contours of the body. The free-stream flow is separated from the surface of the airfoil only by a thin boundary layer, in which the tangential flow is brought to rest at the surface. The flow around the rectangular section (a typical bluff body) in Figure 4.1 is characterized by a ‘separation’ of the flow at the leading edge corners. The separated flow region is divided from the outer flow by a thin region of high shear and vorticity, a region known as a free shear layer, which is similar to the boundary layer on the airfoil, but not attached to any surface. These layers are unstable in a sheet form and will roll up towards the wake, to form concentrated vortices, which are subsequently shed downwind.
In the case of the bluff body with a long ‘after-body’ in Figure 4.1, the separated shear layer ‘re-attaches’ on to the surface. Flowever, the shear layer is not fully stabilized and vortices may be formed on the surface, and subsequently roll along the surface.
Pressure and Force Coefficients
Bernoulli’s Equation
The region outside the boundary layers in the case of the airfoil and the outer region of the bluff-body flow are inviscid (zero viscosity) and irrotational
N7
Figure 4.1 Flow around streamlined and bluff bodies.
(zero vorticity) flows, and the pressure, p, and velocity, U, in the fluid are related by Bernoulli’s Equation:
Denoting the pressure and velocity in the region outside the influence of the body by p„ and U0, we have:
The surface pressure on the body is usually expressed in the form a non- dimensional pressure coefficient:
In the region in which Bernoulli’s Equation holds:
At the stagnation point, where U is zero, Equation (4.3) gives a pressure coefficient of one. This is the value measured by a total pressure or pitot tube pointing into a flow. The pressure (l/2)paU01 is known as the dynamic pressure. Values of pressure coefficient near 1.0 also occur at the stagnation point on a circular cylinder, but the largest (mean) pressure coefficients on the windward faces of buildings are usually less than this theoretical value.
In the regions where the flow velocity is greater than U0, the pressure coefficients are negative. Strictly, Bernoulli’s Equation is not valid in the separated flow and wake regions, but reasonably good predictions of surface pressure coefficients can be obtained from Equation (4.3), by taking the velocity, U, as that just outside the shear layers and wake region.
Force Coefficients
Force coefficients are defined in a similar non-dimensional way to pressure coefficients:
where F is the total aerodynamic force and A is a reference area (not necessarily the area over which the force acts). Often A is a projected frontal area.
In the case of long, or two-dimensional, bodies, a force coefficient per unit length is usually used:
where f is the aerodynamic force per unit length, and b is a reference length, usually the breadth of the structure normal to the wind.
Aerodynamic forces are conventionally resolved into two orthogonal directions. These may be parallel and perpendicular to the wind direction (or mean wind direction in the case of turbulent flow), in which case the axes are referred to as wind axes, or parallel and perpendicular to a direction related to the geometry of the body (body axes). These axes are shown in Figure 4.2.
Following the terminology of aeronautics, the terms ‘lift’ and ‘drag’ are commonly used in wind engineering for cross-wind and along-wind force components, respectively. Substituting ‘L’ and ‘D’ for ‘F’ in Equation (4.4) gives the definition of lift and drag coefficients.
The relationship between the forces and force coefficients resolved with respect to the two axes can be derived using trigonometry, in terms of the angle, «, between the sets of axes, as shown in Figure 4.3. « is called the angle of attack (or sometimes angle of incidence).
Figure 4.2 (a) Wind axes and (b) body axes.
Figure 4.3 Relationship between resolved forces.
Functional Dependence of Pressure and Force Coefficients
Pressure and force coefficients are non-dimensional quantities, which are dependent on a number of variables related to the geometry of the body and to the upwind flow characteristics. These variables can be grouped together into non-dimensional groups, using processes of dimensional analysis, or by inspection.
Assuming that we have several bluff bodies of geometrically similar shape, which can be characterized by a single length dimension (for example, buildings with the same ratio of height, width and length, and with the same roof pitch, characterized by their height, b). Then the pressure coefficients for pressures at corresponding points on the surface of the body may be a function of a number of other non-dimensional groups: jc„
CtC...
Examples of relevant non-dimensional groups are as follows:
- • b/za (Jensen Number)
- (where z0 is the roughness length of the ground surface, as discussed in Section 3.2.1)
- • f„, the turbulence intensities in the approaching flow
- • [tjb), (CJb), (tjh) representing ratios of turbulence length scales in the approaching flow, to the characteristic body dimension
- • (Ub/v), Reynolds Number, where v is the kinematic viscosity of air
Equation (4.6) is relevant to the practice of wind-tunnel model testing, in which geometrically scaled models are used to obtain pressure (or force) coefficients for application to full-scale prototype structures (see Section 7.4). The aim should be to ensure that all relevant non-dimensional numbers (jc,, it2, n3, etc.) should be equal in both model and full scale. This is difficult to achieve for all the relevant numbers, and methods have been devised for minimizing the errors resulting from this. Wind-tunnel testing techniques are discussed in Chapter 7.
Reynolds Number
Reynolds Number is the ratio of fluid inertia forces in the flow to viscous forces, and is an important parameter in all branches of fluid mechanics. In bluff-body flows, viscous forces are only important in the surface boundary layers and free shear layers (Section 4.1). The dependence of pressure coefficients on Reynolds Number is often overlooked for sharp-edged bluff bodies, such as most buildings and industrial structures. For these bodies, separation of flow occurs at sharp edges and corners, such as wall-roof junctions, over a very wide range of Reynolds Numbers. However, for bodies with curved surfaces such as circular cylinders or arched roofs, the separation points are dependent on Reynolds Number, and this parameter should be considered. Surface roughness has significant effects on flow around circular cylinders (see Section 4.5.1) and other bodies with curved surfaces. This may sometimes be used to advantage to modify the flow around these shapes on wind-tunnel models to approximate the flow around the full- scale body (see Section 7.4.4). The addition of turbulence to the flow also reduces the Reynolds Number dependence for bodies with curved surfaces.
In most references to Reynolds Number in this book, the breadth of the body, b (i.e. the diameter in the case of a circular cylinder), is used to form the Reynolds Number, denoted by Reb. However, the average height of roughness on the body, k (not to be confused with ground roughness length, z0), is also used as a length scale in Section 4.5.1 - forming the ‘roughness Reynolds Number’ Rek, equal to Uk/u.
Flat Plates and Walls
Flat Plates and Walls Normal to the Flow
The flat plate, with its plane normal to the air stream, is representative of a common situation for wind loads on structures. Examples are elevated hoardings and signboards, which are mounted so that their plane is vertical. Solar panels are another example, but, in this case, the plane is generally inclined to the vertical to maximize the collection of solar radiation. Free-standing walls are another example, but the fact that they are attached to the ground has a considerable effect on the flow and the resulting wind loading. In this section, some fundamental aspects of flow and drag forces on flat plates and walls are discussed.
For a flat plate or wall with its plane normal to the flow, the only aerodynamic force will be one parallel to the flow, i.e. a drag force. Then if pw and pi are the average pressures on the front (windward) and rear (leeward) faces, respectively, the drag force, D, will be given by:
where A is the frontal area of the plate or wall.
Then dividing both sides by (1/2)pllU2A, we have:
In practice, the windward wall pressure, pw, and pressure coefficient, Cp w, vary considerably with position on the front face. The leeward (or ‘base’) pressure, however, is nearly uniform over the whole rear face, as this region is totally exposed to the wake region, with relatively slow-moving air.
The mean drag coefficients for various plate and wall configurations are shown in Figure 4.4. The drag coefficient for a square plate in a smooth, uniform approach flow is about 1.1, slightly greater than the total pressure in the approach flow, averaged over the face of the plate. Approximately 60% of the drag is contributed by positive pressures (above static pressure) on the front face, and 40% by negative pressures (below static pressure) on the rear face (ESDU, 1970).
The effect of free-stream turbulence is to increase the drag on the normal plate slightly. The increase in drag is caused by a decrease in leeward, or base, pressure, rather than an increase in front face pressure. A hypothesis for this is that the free-stream turbulence causes an increase in the rate of entrainment of air into the separated shear layers. This leads to a reduced radius of curvature of the shear layers, and to a reduced base pressure (Bearman, 1971).
Figure 4.4 also shows the drag coefficient on a long flat plate with a theoretically infinite width into the paper - the ‘two-dimensional’ flat plate. The drag coefficient of 1.9 is higher than that for the square plate. The reason for the increase on the wide plates can be explained as follows. For a square plate, the flow is deflected around the plate equally around the four sides. The extended width provides a high-resistance flow path, thus forcing the flow to travel faster over the top edge, and under the bottom edge. This faster flow results in more entrainment from the wake into the shear layers, thus generating lower base, or leeward face, pressure and higher drag.
Figure 4.4 Drag coefficients for normal plates and walls.
Rectangular plates with intermediate values of width to height have intermediate values of drag coefficient. A formula given by ESDU (1970) for the drag coefficient on plates of height/breadth ratio in the range, 1/30
In the case of two-dimensional plate, strong vortices are shed into the wake alternately from top and bottom, in a similar way to the bluff-body flow shown in Figure 4.1. These contribute greatly to the increased entrainment into the wake of the two-dimensional plate. Suppression of these vortices by a splitter plate has the effect of reducing the drag coefficient to a lower value, as shown in Figure 4.4.
This suppression of vortex shedding is nearly complete when a flat plate is attached to a ground plane, and becomes a wall, as shown in the lower sketch in Figure 4.4. In this case, the approach flow will be of a boundary- layer form with a wind speed increasing with height as shown. The value of drag coefficient, with U taken as the mean wind speed at the top of the wall, Ui„ is very similar for the two-dimensional wall, and finite wall of square planform, i.e. a drag coefficient of about 1.2 for an infinitely long wall. The effect of finite length of wall is shown in Figure 4.5. Little change in the mean drag coefficient occurs, although a slightly lower value occurs for an aspect ratio (length/height) of about 5 (Letchford and Holmes, 1994).
The case of two thin normal plates in series, normal to the flow, as shown in Figure 4.6, is an interesting one. At zero spacing, the two plates act like a single plate with a combined drag coefficient (based on the frontal area of one plate) of about 1.1, for a square plate. For spacings in the range of 0-2b, the combined drag coefficient is actually lower than that for a single plate, reaching a value of 0.8 at a spacing of 1.5b, for two square plates. As the spacing increases, the combined drag coefficient then increases, so that, for very high spacings, the plates act like individual plates with no interference
Figure 4.5 Mean drag coefficients on walls in boundary-layer flow.
Figure 4.6 Drag coefficients for two square plates in series.
with each other, and a combined drag coefficient of 2.2. The mechanism that produces the reduced drag at the critical spacing of 1.5^ has not been studied in detail, but clearly there is a large interference in the wake and in the vortex shedding, generated by the downstream plate.
The drag forces on two flat plates separated by small distances normal to the flow is also a relevant situation in wind engineering, with applications for clusters of lights or antennas together on a frame, for example. Experiments by Marchman and Werme (1982) found increases in drag of up to 15% when square, rectangular or circular plates were within half a width (or diameter) from each other.
If uniform porosity is introduced, the drag on a normal flat plate or wall reduces as some air is allowed to flow through the plate, and reduces the pressure difference between front and rear faces. The reduction in drag coefficient can be represented by the introduction of a porosity factor, Kp, which is dependent on the solidity of the plate, <5, being the ratio of the ‘solid’ area of the plate, to the total elevation area, as indicated in Equation (4.9):
For two-dimensional plates (height to breadth approaching infinity), normal to the flow, with circular perforations, Kp is approximately equal to 8 for values of solidity between 0.4 and 0.8 (Castro, 1971).
However, for plates and walls of finite aspect ratio, Kp is not linearly related to the solidity. An approximate expression for Kp, which fits the data quite well for plates and walls with ratios of height to breadth between 0.2 and 5, is given by Equation (4.10):
Equation (4.10) has the required properties of equalling one for a value of 8 equal to 1, i.e. an impermeable plate or wall, and tending to zero as the solidity tends to zero. For very small values of <5 (for example an open-truss plate made up of individual members), Kp tends to a value of 28, since, from Equation (4.10):
noting that 81 is negligible in comparison with 28 for very small <5.
Considering the application of this to the drag coefficient for an open- truss plate of square planform, we have the following equation from Equations (4.9) and (4.10),
where CD At denotes that the drag coefficient, defined as in Equation (4.4), is with respect to the total (enclosed) elevation area of A,. With respect to the elevation area of the actual members in the truss A„„ the drag coefficient is larger, being given by:
In this case of a very open plate, the members will act like isolated bluff bodies with individual values of drag coefficient of 2.2.
Cook (1990) has discussed in detail the effect of porosity on aerodynamic forces on bluff bodies.
Flat Plates and Walls Inclined To the Flow
Figure 4.7 shows the case with the wind at an oblique angle of attack, a, to a two-dimensional flat plate. In this case, the resultant force remains primarily at right angles to the plate surface, i.e. it is no longer a drag force in the direction of the wind. There is also a tangential component, or ‘skin friction’ force. However, this is not significant in comparison with the normal force, for angles of attack greater than about 10°.
For small angles of attack,a, (less than 10°), the normal force coefficient, CN, with respect to the total plan area of the plate, viewed normal to its surface, is given approximately by:
Figure 4.7 Normal force coefficients for an inclined two-dimensional plate.
where a is measured in radians, not in degrees.
Equation (4.11) comes from a theory used in aeronautics. The ‘centre of pressure’, denoting the position of the line of action of the resultant normal force, is at, or near, one quarter of the height h, from the leading edge, again a result from aeronautical theory.
As the angle of attack, «, increases, the normal force coefficient, CN, progressively increases towards the normal plate case («=90°), discussed in Section 4.3.1, with the centre of pressure at a height of 0.5h. For example, the normal force coefficient for an angle of attack of 45° is about 1.5, with the centre of pressure at a distance of about OAb from the leading edge, as shown in Figure 4.7. The corresponding values for a equal to 30° are 1.2 and 0.38b (ESDU 1970).
Now, consider finite length walls and hoardings, at or near ground level, and hence in a highly sheared and turbulent boundary-layer flow. The mean net pressure coefficients at the windward end of the wall, for an oblique wind blowing at 45° to the normal, are quite high due to the presence of a strong vortex system behind the wall. Some values of area-averaged mean pressure coefficients are shown in Figure 4.8; these high values are usually the critical cases for the design of free-standing walls and hoardings for wind loads.
