VIEWS: 73 PAGES: 9 POSTED ON: 4/4/2010
REYNOLDS NUMBER AND DRAG ON IMMERSED BODIES – SUPPLEMENTAL LECTURE In Lab 1 there were two components: 1) the Reynolds Apparatus for determination of turbulent flow; and 2) the Stokes Apparatus for determining the drag forces on submerged bodies. We will review both concepts below. Reynolds Number and Turbulence Osborne Reynolds performed a simple experiment to observe types of flow that was very similar to the experiment conducted during Lab 1 in this course. In the lab a small filament of die was introduced into the flow field in a narrow tube. As the flow rate was increased one could observe the evolution from laminar to transitional to fully turbulent flow conditions ( Figure 1). X X 1 Figure 1- Reynolds Experiment In laminar flow all particles move in a uniform, steady fashion in the direction of flow (left-to-right in Figure 1). In turbulent flow X X the time-averaged velocity is still in the direction of flow (left-to- right), but there is also a random component to the flow that moves both in the direction of flow and in axes normal to the direction of flow (up-and-down and in-and-out of the page). These random fluctuations in flow with time are a result of flow turbulence. To quantify turbulence the Reynolds number was introduced which represents a dimensionless relationship between inertial and viscous forces. ρVD VD Re = = μ ν Where, Re = Reynolds Number, ρ = fluid density, V = fluid velocity, D = characteristic length, μ = dynamic viscosity of fluid, and ν = kinematic viscosity of fluid. 2 Reynolds number and Pipe Flow The interpretation of the Reynolds number will vary under different flow environments, but with pipe flow one can say that flow is laminar if Re is less than about 2000 and is turbulent if it is greater than about 4000. This has ramifications for determining head (or energy) loss in pipe flow. During laminar flow the head loss per length of pipe is proportional to the velocity. hL ∝V l In turbulent flow conditions the head loss per length of pipe is proportional to the square of the velocity. hL ∝V 2 l There will be much more discussion on these relationships later in the course. 3 Flow over Immersed Bodies As an object moves through a fluid, forces imparted by the fluid will act on the object to slow it down. These forces are called drag forces. The total drag force represents a combination of the pressure forces acting on the body (pressure drag) and the wall shear stress (friction drag). Pressure Drag Pressure drag is the part of the total drag force that is due to the pressure applied to the object as it moves through the fluid. The total difference of the sum of pressure forces on the upstream side and the downstream side of the body will result in a net drag force. This is often called “form drag” as the pressure difference between upstream and downstream has a great deal to do with the shape of the body itself. As fluids move past bodies they will lose energy. If a body has a sharp edge on the downstream side the fluid may not have enough energy to maintain contact with the surface of the body – in this case the fluid may separate from the body. This is known as the separation point, and a turbulent wake is formed beyond this point. If a body has a streamlined edge on the downstream side the pressure rises more gradually and the separation point happens much further along the body. Beyond the separation point a wake is produced. Within the wake is a relatively low pressure environment. In basic terms, the larger the wake, the greater the difference between the upstream and downstream pressure and the greater the pressure drag. 4 Figure 2 - Pressure Drag on “blunt” and “streamlined” bodies Friction Drag Friction drag is a force that acts on a submerged body moving within a fluid due to the viscous forces acting along the surface of the body. The viscous forces produce a shear stress that will act on the surface area of the body within the fluid. If the shear stress distribution is known the total friction drag can be calculated by summing the shear stresses over all incremental areas of the body. Figure 3 - Friction Drag on a wing 5 Total Drag Force In practice it is very difficult to separate the values of pressure drag and friction drag on an object. Often these two types of drag force are combined into a total drag force, FD. The drag force is represented by the following equation: 1 FD = C D ρV 2 A 2 Where, FD = drag force, CD = drag coefficient, ρ = the density of the fluid, V = the velocity of the body within the fluid, and A = the cross sectional area. However, the drag coefficient is not a constant value, and may depend on the flow conditions (Reynolds Number), the shape of the body, and the skin roughness of the body, among other things. Variation of CD with Reynolds Number As a body moves through a fluid, the drag coefficient (CD) will change under different flow conditions. For incompressible fluids it is especially important to consider the degree of turbulence or the Reynolds number of the flow field. Considering spheres, as with Lab 1, we can see the variation in the drag coefficient with the Reynolds number in the figures below. These sorts of coefficients are derived experimentally and tables for the determination of CD for different shapes in different flow conditions can be found in most fluid mechanics textbooks. 6 Figure 4 - Drag Coefficient as a function of Reynolds number Figure 5 - Flow patterns past a Cylinder Here we can see that at a low Reynolds number CD varies linearly with Re. At very low values of Re we can substitute CD for 24/Re. 7 24 1 FD = ρV 2 A Re 2 12 μ FD = VA D It is shown that for low Reynolds numbers that represent completely laminar flow that the drag force is linearly proportional to the velocity. Once fully turbulent flow develops the value for CD will remain relatively constant. In this case the drag force will be proportional to the square of the velocity. The sudden change observed in Figure 4 at point E represents a X X transition from a laminar boundary layer to turbulent boundary layer. At this point the drag coefficient for a sphere will drop significantly. This is due to the reduction in the size of the wake produced ( Figure 5 ) – reducing suddenly the pressure drag on the X X object. A change from a laminar to turbulent boundary layer can produce different changes in the values for CD for bodies of different shapes. Determining CD from Terminal Velocity When an object is allowed to fall within a fluid it will initially accelerate from a zero velocity. However, eventually the drag and buoyancy forces will be balanced by the weight and the object will fall with a constant or terminal velocity. If one knows the geometry and weight of the object, some details about the fluid, and the terminal velocity of the object one may calculate the drag coefficient of that object for that flow condition. 8 The weight and buoyant forces can be determined from the density of the fluid and the object and the volume of the object. The drag force will balance the difference. FD = FB − W Also knowing that the following equation is applicable for drag forces, 1 FD = C D ρV 2 A 2 And the value for A can be determined from the cross-sectional area of the body. The value for CD can be solved for explicitly. 9