3 Introduction Resistances exerted by surfaces are a result of viscous stresses which create resistance (or drag) to motion as a body travels through a fluid. Aeronautical engineers and naval architects are vitally interested in the drag on an airplane or the surface. The phenomena responsible for shear stress are viscosity, and velocity gradients presented in Chapter 2. In addition the concepts of the boundary layer and separation, introduced in Chapter 4, will be further expanded. Will consider surface resistance in two flow situations: 1) uniform flows 2) Non-uniform flows; i.e., boundary-layer flows - Laminar boundary-layer flows - Turbulent boundary-layer flows
4 9.1 Surface Resistance with Uniform Laminar Flow A one-dimensional laminar flow with parallel streamlines occurs for example between two plates: one is stationary and the other moving and also between two stationary parallel plates. These flows are uniform and steady. These flows illustrate the connections between velocity gradient and shear stress. Differential Equation for Uniform Laminar flow Consider the control volume shown in Fig. 9.1, which is aligned with the flow direction s. The streamlines are inclined at an angle θ wi respect to the horizontal plane. The control volume has dimensions Ds Dy unity; that is, the control volume has a unit length into the page. By application of the momentum equation, the sum of the forces acting in the s-direction is equal to the net outflow of momentum from the control volume. The flow is uniform so the outflow of momentum is equal to the inflow and the momentum equation reduces to
5 Figure 9.1 Control volume for analysis of uniform flow with parallel streamlines.
6 There are three types of forces that act on the control volume: - Pressure forces - Shear forces - Weight The pressure forces in s direction, The net force due to shear stress is, The weight along the flow direction is, Summing the above forces and dividing by DsDy lead to,
7 Substituting for shear stress t = m du/dy, leads to (9.3) where μ is constant. This equation is now applied to a flow between two plates; one is moving and the other is stationary.
8 Flow Produced by a Moving Plate (Couette Flow) Consider the flow between the two plates shown in Fig The lower plate is fixed, and the upper plate is moving with a speed U. The plates are separated by a distance L. In this problem there is no pressure gradient in the flow direction (dp/ds = 0), and the streamlines are in the horizontal direction (dz/ds = 0),so Eq. (9.3) reduces to along with the two boundary conditions: Integrating this equation twice gives Applying the boundary conditions results in Figure 9.2 Flow generated by a moving plate (Couette flow).
9 The shear stress is constant and equal to This flow is known as a Couette flow after a French scientist, M. Couette, who did pioneering work on the flow between parallel plates and rotating cylinders. It has application in the design of lubrication systems. EXAMPLE 9.1 SHEAR STRESS I, COUETTE FLOW SAE 30 lubricating oil at T = 38 C flows between two parallel plates, one fixed and the other moving at 1.0 m/s. Plates are spaced 0.3 mm apart. What is the shear stress on the plates? Solution: This is significant.
10 Non-Uniform Flows: Boundary-Layer Flows 9.2 Qualitative Description of the Boundary Layer The boundary layer is the region adjacent to a surface over which the velocity changes from the free- stream value (with respect to the object) to zero at the surface. This region, which is generally very thin, occurs because of the viscosity of the fluid. The velocity gradient at the surface is responsible for the viscous shear stress and surface resistance. The boundary-layer development for flow past a thin plate oriented parallel to the flow direction shown in Fig. 9.4a. The thickness of the boundary layer, δ, is defined as the distance from the surface to the point where the velocity is 99% of the free-stream velocity. The actual thickness of a boundary layer may be 2% 3% of the plate length, so the boundary-layer thickness shown in Fig. 9.4a is exaggerated at least by a factor of five to show details of the flow field. Fluid passes over the top and underneath the plate, so two boundary layers are depicted (one above and one below the plate). For convenience, the surface is assumed to be stationary, and the free-stream fluid is moving at a velocity U o.
11 Figure 9.4 Development of boundary layer and shear stress along a thin, flat plate. (a) Flow pattern above and below the plate. (b) Shear-stress distribution on either side of plate.
12 The development and growth of the boundary layer occurs because of the no-slip condition at the surface; that is, the fluid velocity at the surface must be zero. As the fluid particles next to the plate pass close to the leading edge of the plate, a retarding force (from the shear stress) begins to act on the particles to slow them down. As these particles progress farther downstream, they continue to be subjected to shear stress from the plate, so they continue to decelerate. In addition, these particles (because of their lower velocity) retard other particles adjacent to them but farther out from the plate. Thus the boundary layer becomes thicker, or grows, in the downstream direction. The broken line in Fig. 9.4a identifies the outer limit of the boundary layer. As the boundary layer becomes thicker, the velocity gradient at the wall becomes smaller and the local shear stress is reduced. The initial section of the boundary layer is the laminar boundary layer. In this region the flow is smooth and steady. Thickening of the laminar boundary layer continues smoothly in the downstream direction until a point is reached where the boundary layer becomes unstable. Beyond this point, the critical point, small disturbances in the flow will grow and spread, leading to turbulence. The boundary becomes fully turbulent at the transition point. The region between the critical point and the transition point is called the transition region.
13 The turbulent boundary layer is characterized by intense crossstream mixing as turbulent eddies transport high-velocity fluid from the boundary layer edge to the region close to the wall. This cross-stream mixing gives rise to a high effective viscosity, which can be three orders of magnitude higher than the actual viscosity of the fluid itself. The effective viscosity, due to turbulent mixing is not a property of the fluid but rather a property of the flow, namely, the mixing process. Because of this intense mixing, the velocity profile is much fuller than the laminar-flow velocity profile as shown in Fig. 9.4a. This situation leads to an increased velocity gradient at the surface and a larger shear stress. The shear-stress distribution along the plate is shown in Fig. 9.4b. It is easy to visualize that the shear stress must be relatively large near the leading edge of the plate where the velocity gradient is steep, and that it becomes progressively smaller as the boundary layer thickens in the downstream direction. At the point where the boundary layer becomes turbulent, the shear stress at the boundary increases because the velocity profile changes producing a steeper gradient at the surface.
14 9.3 Laminar Boundary Layer - Boundary-Layer Equations In 1904 Prandtl, 1 first stated the essence of the boundary-layer hypothesis, which is that viscous effects are concentrated in a thin layer of fluid (the boundary layer) next to solid boundaries. In 1908, Blasius, one of Prandtl's students, obtained a solution for the flow in a laminar boundary layer on a flat plate with a constant free-stream velocity. One of Blasius's key assumptions was that the shape of the nondimensional velocity distribution did not vary from section to section along the plate. That is, he assumed that a plot of the relative velocity, u/u 0, versus the relative distance from the boundary, y/δ, would be the same at each section. With this assumption and with Prandtl's equations of motion for boundary layers, Blasius obtained a numerical solution for the relative velocity distribution, shown in Fig In this plot, x is the distance from the leading edge of the plate, and Re x is the Reynolds number based on the free-stream velocity and the length along the plate (Re x = U 0 x/ν). In Fig. 9.5 the outer limit of the boundary layer (u/u 0 = 0.99) occurs at approximately,
15 Figure 9.5 Velocity distribution in laminar boundary layer.
16 Since y = δ at this point, the following relationship is derived for the boundary-layer thickness in laminar flow on a flat plate The Blasius solution also showed that which can be used to find the shear stress at the surface. The velocity gradient at the boundary becomes (at a section; x =constant)
17 Shear Stress The shear stress at the boundary is obtained from Surface Resistance Because the shear stress at the boundary, t o, varies along the plate, it is necessary to integrate this stress over the entire surface to obtain the total surface resistance, F s. For one side of the plate, (9.13)
18 EXAMPLE 9.3 LAMINAR BOUNDARY-LAYER THICKNESS AND SHEAR STRESS Crude oil at 70 F(n = 10-4 ft2/s, S = 0.86) with a free-stream velocity of 1 ft/s flows past a thin, flat plate that is 4 ft wide and 6 ft long in a direction parallel to the flow. The flow is laminar. Determine and plot the boundary-layer thickness and the shear stress distribution along the plate.
19 The results for Example 9.3 are plotted in the accompanying figure and listed in Table 9.1. Solution Reynolds-number variation with distance Boundary-layer thickness Shear-stress distribution
20 Table 9.1 RESULT δ AND τo FOR DIFFERENT VALUES OF x x = 0.1 ft x = 1.0 ft x = 2 ft x = 4 ft x = 6 ft x1/ τ0, psf δ, ft δ, in
21 To find the resistive force, substitute in Eq. (9.13) for t 0 and integrate, giving (9.14)
22 Shear-Stress Coefficients It is convenient to express the shear stress at the boundary, t 0, and the total shearing force F s in terms of π-groups involving the kinetic pressure of the free stream,. The local shear-stress coefficient, c f, is defined as Substituting the value for t 0 in the above gives c f as a function of Reynolds number based on the distance from the leading edge. The total shearing force, given by Eq. (9.13), can also be expressed as a π-group
23 where A is the plate area. This π-group is called the average shear-stress coefficient. Substituting Eq. (9.14) into the definition of C f, gives EXAMPLE 9.4 RESISTANCE CALCULATION FOR LAMINAR BOU#DARY LAYER ON A FLAT PLATE Crude oil at 70 F (ν = 10-4 ft2 /s, S = 0.86.) with a free-stream velocity of 1 ft/s flows past a thin, flat plate that is 4 ft wide and 6 ft long in a direction parallel to the flow. The flow is laminar. Determine the resistance on one side of the plate.
24 Solution Reynolds number. Value for C f Total shear force.
25 9.4 Boundary Layer Transition Transition is the zone where the laminar boundary layer changes into a turbulent boundary layer as shown in Fig. 9.4a. As the laminar boundary layer continues to grow, the viscous stresses are less capable of damping disturbances in the flow. A point is then reached where disturbances occurring in the flow are amplified, leading to turbulence. The critical point occurs at a Reynolds number of about 10 5 (Re cr 10 5 ) based on the distance from the leading edge. Vortices created near the wall grow and mutually interact, ultimately leading to a fully turbulent boundary layer at the transition point, which nominally occurs at a Reynolds number of (Re tr ). For purposes of simplicity in this text, it will be assumed that the boundary layer changes from laminar to turbulent flow at a Reynolds number 500,000. Transition to a turbulent boundary layer can be influenced by several other flow conditions, such as free-stream turbulence, pressure gradient, wall roughness, wall heating, and wall cooling. With appropriate roughness elements at the leading edge, the boundary layer can become turbulent at the very beginning of the plate. In this case it is said that the boundary layer is tripped at the leading edge.
26 9.5 Turbulent Boundary Layer In the majority of practical problems the boundary layer is turbulent which is primarily responsible for surface shear force, or surface resistance. Velocity Distribution The velocity distribution in the turbulent boundary layer is more complicated than the laminar boundary layer. The turbulent boundary has three zones of flow that require different equations for the velocity distribution in each zone, as opposed to the single relationship of the laminar boundary layer. Figure 9.6 shows a portion of a turbulent boundary layer in which the three different zones of flow are identified. The zone adjacent to the wall is the viscous sublayer; the zone immediately above the viscous sublayer is the logarithmic region; and, finally, beyond that region is the velocity defect region. Each of these velocity zones will be discussed separately. Figure 9.6 Sketch of zones in turbulent boundary layer. Viscous Sublayer The zone
27 The three zones in a turbulent boundary layer. Figure 9.6 Sketch of zones in turbulent boundary layer.
28 Boundary-Layer Thickness and Shear-Stress Correlations Unlike the laminar boundary layer, there is no analytically derived equation for the thickness of the turbulent boundary layer. It can be shown that the thickness of the turbulent boundary layer is where x is the distance from the leading edge of the plate and Rex is U 0 x/ν. Many empirical expressions have been proposed for the local shear-stress distribution for the turbulent boundary layer on a flat plate. One of the simplest correlations is
29 The corresponding average shear-stress coefficient is The variation of C f with Reynolds number is shown by the solid line in Fig This curve corresponds to a boundary layer that begins as a laminar boundary layer and then changes to a turbulent boundary layer after the transition Reynolds number. This is the normal condition for a flat-plate boundary layer. Table 9.3 summarizes the equations for boundary-layer-thickness, and for local shear-stress and average shear-stress coefficients for the boundary layer on a flat plate. The boundary layer, however, usually consists of laminar and turbulent. To account for this combined effect, it can be shown that the average shear-stress coefficient for this case is,
30 Figure 9.12 Average shear-stress coefficients.
31 Table 9.3 SUMMARY OF EQUATIONS FOR BOU"DARY LAYER ON A FLAT PLATE Laminar Flow Rex, Turbulent Flow Rex, ReL < ReL Boundary-Layer Thickness, δ Local Shear-Stress Coefficient, c f Average Shear-Stress Coefficient, C f
32 If the boundary layer is tripped by some roughness or leadingedge disturbance (such as a wire across the leading edge), the boundary layer is turbulent from the leading edge. This is shown by the dashed line in Fig For this condition the boundary layer thickness, local shear-stress coefficient, and average shear-stress coefficient are fit reasonably well by the following equations, which are valid up to a Reynolds number of 10 7.
33 EXAMPLE 9.6 LAMINAR/TURBULENT BOUNDARY LAYER ON FLAT PLATE Assume that air 20 C and normal atmospheric pressure flows over a smooth, flat plate with a velocity of 30 m/s. The initial boundary layer is laminar and then becomes turbulent at a transitional Reynolds number of The plate is 3 m long and 1 m wide. What will be the average resistance coefficient C f for the plate? Also, what is the total shearing resistance of one side of the plate, and what will be the resistance due to the turbulent part and the laminar part of the boundary layer? Air properties at 20 C:
34 Solution Reynolds number based on plate length Average shear-stress coefficient Total shear force Transition point
35 Laminar average shear-stress coefficient Laminar shear force Turbulent shear force
36 EXAMPLE 9.7 RESISTANCE FORCE WITH TRIPPED BOUNDARY LAYER Air at 20 C flows past a smooth, thin plate with a free-stream velocity of 20 m/s. Plate is 3 m wide and 6 m long in the direction of flow and boundary layer is tripped at the leading edge. Solution Reynolds number Reynolds number is less than Average shear-stress coefficient
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