Objective Theoretical Background Method I - Prediction of Drag from Wake Measurements (4) Introducing the total pressure as Method II - Measurement of the Surface Pressure distribution on the body Figure 2 - Pitot-static tube Cylindrical Test Model Figure 3 - Pressure port locations on the test model Experimental Procedure Questions to be Answered Figure 4 -Drag coefficient of a circular cylinderExperiment 7 Aerodynamics of Flow Around a Cylinder Objective The objective of this experiment is to determine the aerodynamic lift and drag forces, FL and FD, respectively, experienced by a circular cylinder placed in a uniform free-stream velocity, U∞. Two different methods will be used to determine these forces. Theoretical Background The total drag on any body consists of skin friction drag and form drag. Skin friction drag is a result of the viscous forces acting on the body while the form drag is due to the unbalanced pressure forces on the body. The sum of the two is called total or profile drag. There are different methods that can be used to determine the drag forces, two of which will be used in this experiment and are discussed in detail below. Method I - Prediction of Drag from Wake Measurements By measuring the velocity profiles in the wake of the cylinder and using conservation of linear momentum, the drag force on the cylinder can be determined, provided that the flow is steady. Figure 1 shows a control surface around the body, where Section II, the region where wake measurements will be obtained, is located a short distance behind the body, as shown. The static pressure at this location, which is relatively close to the cylinder, is different from the free-stream pressure, p∞. Hence there will be a net contribution to the momentum balance due to this pressure difference. In order to account for and minimize this effect, another section, Section I (an imaginary section), is chosen far behind the body such that the pressure is equal to the free-stream pressure. Therefore, the net pressure forces acting on this new control surface will be zero. The general conservation law can then be written without considering the effects contributed by pressure, as follows: (1) ∫−=∝ 111)( dyuUuWFDρwhere W is the width of the body and u1 denotes the velocity profile in the wake measured at Section I. Unfortunately, the flow cannot relax to the freestream pressure, where p1 = p∞ unless the measuring station is placed very far downstream of the cylinder, usually more than 100 diameters! It is therefore not realistic to obtain measurements at section I. Nevertheless, it is possible to establish the relationship between the flow quantities measured at the hypothetical section I and those measured at actual section II which is located close to the cylinder. The procedure is as follows: Figure 1 - Sketch of the test configuration.In order to find the relation between FD, the drag force, and the velocity profile at the measured section II, we apply the continuity equation, equation 2, along a stream tube between I & II 2211dyudyuρρ= (2) By substituting, eq. 2 in 1, it follows that: ∫−=∝ 112)( dyuUuWFDρ (3) Furthermore, we make the assumption that the flow moves from Section II to I without pressure losses, i.e., the total pressure remains constant along each streamline between I and II. Using Bernoulli’s equation, this can be written as: 2221221211upupρρ+=+ (4) Introducing the total pressure as 221upPtρ+= (5) We can rewrite equation 3 by using equations 4 and 5. Hence, we have: [ ][ ]dyppppppWFtttD∫∝∝∝−−−−=2222 (6) where the integral extends over cross-section II. Introducing a dimensionless drag coefficient, CD, as follows : WdUFCDD221ρ= (7) where Wd (width x diameter) is the reference area of the body, Equation 6 ca be written as: ⎟⎠⎞⎜⎝⎛⎥⎥⎦⎤⎢⎢⎣⎡−−−=∫∝∝∝dydqppqppCttD22212 (8) where 221∞∝∝∝=−= Uppqtρ (9) Method II - Measurement of the Surface Pressure distribution on the body When the Reynolds number is sufficiently large, Re > 103, the skin friction drag of a bluff body is relatively negligible compared to its form drag. Then the measurement of the drag forces due to surface pressures acting on (and normal to) the body will be a good approximation to the total drag. For a cylinder, the body lift and drag per unit length due to the pressure distribution only are given by: ∫∝−= dxppFL)( (10) ∫∝−= dyppFD)( (11) where the integrations are taken around the contour of the cylinder. Using the cylindrical coordinate system one can rewrite equation 11 as: (12) ∫∝−=πθθ20cos)( drppFDwhere r is the radius of the cylinder, p is the pressure, θ is the angular position, and the integration is taken around the cylinder, starting from the stagnation point.Similarly, the lift force can be estimated as: (13) ∫∝−=πθθ20sin)( drppFL Apparatus The following components will be used: 1. Wind Tunnel. 2. Pitot-static tube (see description below). 3. A cylindrical test model with circumferential pressure ports (see description below). 4. Scanivalve and scanivalve digital interface unit. 5. Computer-controlled vertical drive. Pitot-Static Tube A pitot tube can be used in the wind tunnel to measure the velocity of the tunnel. The assumption we have to make is that the static pressure is constant everywhere in a uniform free-stream inside the wind tunnel. This is a reasonable assumption considering that there is no pressure loss, therefore, no pressure gradient, in the tunnel. However, the situation will be very different for measurements taken inside a wake behind a bluff body where a significant amount of pressure variation exists across the wake profile. In order to accurately determine the velocity profile in the wake, a pitot-static tube should be used. The pitot-static tube, a sketch of which is shown in Figure 2, is a combination of the static tube and the pitot tube, which works in the following manner. Assume that the tube is properly aligned with the flow direction. If we further assume that the flow is steady, one-dimensional, incompressible and inviscid, all of which