COURSE NUMBER: ME 323 Fluid Mechanics II 3 credit hour External flows Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1
External flows The study of external flows is of particular importance to the aeronautical engineer in the analysis of airflow around the various components of an aircraft. In fact, much of the present knowledge of external flows has been obtained from studies motivated by such aerodynamic problems. Other examples of external flows: the flow of fluid around turbine blades, automobiles, buildings, smokestacks, spray droplets, bridge abutments, submarine pipelines, river sediment, and red blood cells, etc. Low Reynolds number flows (Re < 5) are called creeping flows or Stokes flows and rarely occur in engineering applications. Flow around spray droplets, lubrication in small gaps, and flow in porous media would be afew exceptions. We will direct our attention ti to high h Reynolds number flows (Re > 1000). 2
High Reynolds number flows can be subdivided d d into three major categories: (1) incompressible immersed flows involving such objects as automobiles, helicopters, submarines, low speed aircraft, take off and landing of commercial aircraft, buildings, and turbine blades; (2) flows of liquids that involve a free surface as experienced by a ship or a bridge abutment; and (3) compressible flows involving high speed objects (V > 100 m/s) such as aircraft, missiles, and bullets. We will focus our attention on the first category of flows in this chapter and consider cases in which the object is far from a solid boundary or other objects. 3
The flow becomes significantly influenced by the presence of a boundary or another object, as shown in Fig. 8.2; in part (d) the slender object must beatleastfivebody lengths belowthefreesurfacebefore free surface effects can be neglected. 4
High Reynolds number incompressible immersed flows around blunt bodies and flows around streamlined bodies. Separated region: A region of recirculating flow. Wake: A region of velocity defect that grows due to diffusion. The boundaries of the wake, the separated region, and the turbulent boundary layer are quite time dependent. Shear stresses due to viscosity are concentrated in the thin boundary layer, the separated region, and the wake; outside tid these regions the flow is approximated byan inviscidi id flow. 5
The separated region eventually closes; the wake keeps diffusing into the main flow and eventually disappears as its area becomes exceedingly large (the fluid regains the free stream velocity). Time average streamlines do not enter a separated region; they do enter a wake. The separated region is always submerged within the wake. Flow around a blunt object is usually treated empirically, as was done for a turbulent flow in a conduit. We are interested primarily in the drag, the force the flow exerts onthe body in the direction of the flow. the lift, which acts normal to the direction of flow. 6
We present the drag F D and lift F L in terms of dimensionless coefficients: the drag coefficient and lift coefficient, defined as where A is most often the projected area (projected on a plane normal to the direction of the flow); for airfoil shapes, the area is based upon the chord (see Fig. 8.4). 7
In a flow around a streamlined body, the separated region is insignificantlyi ifi small or nonexistent. A boundary layer that develops on a plane streamlined surface, such as an airfoil, is usually sufficiently thin that the curvature of the surface can be ignored and the problem can be treated as a boundary layer developing on a flat plate. If the flow in the boundary layer on a streamlined body can be determined, the drag can be calculated, since the drag is a result of the shear stress and pressure force acting on the body surface. Outside the boundary layer there exists an inviscid free stream flow. Before the velocity profile in the boundary layer can be determined, it is necessary that the inviscid flow solution be known. It is found by completely ignoring the boundary layer and solving the appropriate inviscidi id equations. 8
Flow separation and stall When separation occurs on a streamlined body near the forward portion of an airfoil, as it will with a sufficiently large angle of attack (the angle the oncoming flow makes with the chord, a line connecting the trailing edge with the nose), the flow situation is referred to as stall, as shown in Fig. 8.4. Stall is highlyhl undesirable onaircraft at cruise conditions and leads to inefficiencies when it occurs on turbine blades. It is used, however, to provide the high drag needed when landing an aircraft, or in certain maneuvers by stunt planes. 9
Flow separation and reattachment The location of the separation point is dependent primarily on the geometry of the body; if the body has an abrupt change ingeometry, separation will occur at, or near, the abrupt change. In addition, reattachment will occur at some location, as shown. 10
Downstream of the separation point the x component velocity near the wall is in the negative x direction and thus at the wall du/dy must be negative. Upstream of the separation point the x component velocity near the wall is in the positive x direction, demandingdi thatt du/dy at the wall be positive. Hence we conclude that the separation point is defined as that point where (du/dy) wall = 0. 11
Separation on the flat surface occurs as the flow is approaching a stagnation region where the velocity is low and the pressure is high. As the flow approaches the stagnation region the pressure increases, that is, dp/dx > 0; the pressure gradient is positive. Since separation is often undesirable, a positive i pressure gradient is called an adverse pressure gradient; a negative gradient is a favorable pressure gradient. In general, e the effect e of an adverse pressure e gradient ga results s in decreasing easi velocities in the streamwise direction; if an adverse pressure gradient acts on a surface over a sufficient distance, separation may result. This is true even if the surface is a flat plate, such as the wall of a diffuser. 12
For a given geometry a greater distance is required to reduce the velocity near the wall to zero, resulting in the movement of the separation point to the rear, as can be observed in Fig. 8.8, 8 where both spheres are moving with the same velocity (the sphere in (b) has sandpaper attached in the nose region). In Fig. 8.8a it is observed that separation occurs on the front half of the sphere, in a region of favorable pressure gradient. This separation is due to the centrifugal effects as the fluid moves around the sphere. This phenomenon of drag reduction is observed in the drop in the drag coefficient curves for a sphere and a cylinder. 13
Drag Coefficient The drag coefficient i curves for two bodies that do notexhibit sudden geometric changes will be presented; the drag coefficients for the smooth sphere and the long smooth cylinder are shown in Fig. 8.9 over a large range of Reynolds numbers. At Re < 1 creeping flow with no separation results. For the sphere, this creeping flow problem has been solved, with the result that 14
Separation is observed at Re = 10 over a very small area on the rear of the body. The separated area increases as the Reynolds number increases until Re = 1000, where the separated region ceases to enlarge; during this growth of the separated region the drag coefficient decreases. At Re = 1000, 95% of the drag is due to form drag (the drag force due to the pressure acting on the body), and 5% is due to frictional drag (the drag force due to the shear stresses acting on the body). 15
The drag coefficient curve is relatively flat for smooth bodies over the range 10 3 < Re < 2x 10 5.Theboundary layer before the point of separation is laminar and the separated region is as shown in Fig. 8.8a. If the surface is rough (dimples on a golf ball), the drop in the C D curve may occur at Re 8 x 10 4. Since a lower drag is usually desirable, surface roughness is often added; the dimples on the golf ball may increase the flight distance by 50 to 100%. After the sudden drop in drag, the C D curve is observed to again increase with increased Reynolds number. Experimental data are not readily available for Re >10 6 for a sphere and Re > 6 x 10 7 for a cylinder; however, a value of C D 0.2 for a sphere at large Reynolds number appears acceptable. 16
For cylinders of finite length and for elliptic cylinders, the drag coefficients are presented in Table 8.1. The finite length cylinders are assumed to have two free ends. If one end is fixed to a solid surface, its length must be doubled when using Table 8.1. 17
Blunt objects with sudden geometry changes have separated regions that are relatively insensitive to the Reynolds number; the drag coefficients for some common shapes are given in Table 8.2. 18
Blunt objects with sudden geometry changes have separated regions that are relatively insensitive to the Reynolds number; the drag coefficients for some common shapes are given in Table 8.2. 19
20
21
22
Vortex Shedding Long blunt objects, such as circular cylinders, exhibit a particularly interesting phenomenon when placed normal to a fluid flow; vortices or eddies (regions of circulating fluid) are shed from the object, regularly and alternately from opposite sides, as shown in Fig. 8.10. The resulting flow downstream is often referred to as a Kármán vortex street, named after Theodor von Kármán (1881 1963). The vortices are shed in the Reynolds number range 40 < Re < 10 000, and are accompanied by turbulence above Re=300. 23
For high Reynolds number flows, that is, flows with insignificant viscous forces, the shedding frequency f, in hertz, depends only on the velocity and diameter. Thus f = f(v, D). Using dimensional analysis we can show that fd/v = const. The dimensionless vortex shedding frequency, is called Strouhal number, 24
From the experimental results, we observe that the Strouhal number is essentially constant (0.21) over the range 300 < Re < 10000; hence, the frequency is directly proportional to the velocity over this relatively large Reynolds number range. When a vortex is shed, hd a small force is applied to the structure; if the frequency of shedding is close to the natural frequency (or one of the harmonics) of the structure, the phenomenon of resonance may occur in which the response se to the applied force is multiplied ie by a large agefactor. For example, when resonance occurs on a television tower, the deflection of the tower due to the applied force may become so large that the supporting cables fail, leading to collapse of the structure. t 25
26
Streamlining If the flow is to remain attached to the surface of a blunt object, such as a cylinder or a sphere, it must move into regions of higher and higher pressure as it progresses to the rear stagnation point. At sufficiently high Reynolds numbers (Re > 10) the slow moving boundary layer flow near the surface is unable to make its way into the high pressure region near the rear stagnation point, so it separates from the object. Streamlining reduces the high pressure at the rear of the object so that the slow moving flow near the surface is able to negotiate its way into a slightly higher pressure region. The fluid may not be able to make it all the way to the trailing edge of the streamlined object, but the separation region will be reduced to only a small percentage of the initialiti separated tdregion on the blunt object. The included angle at the trailing edge must not be greater than about 20 or the separation region will betoolarge andtheeffectof streamlining will be negated. 27
When a body is streamlined, the surface area is increased substantially. This eliminates the majorityof the form or pressure drag but increases the shear drag on the surface. To minimize drag, the idea is to minimize the sum of the form or pressure drag and the shear drag. Consequently, the streamlined body cannot be so long that the shear drag is larger than the pressure drag plus the shear drag for a shorter body. An optimization procedure is required. Obviously, for a low Reynolds number flow (Re < 10) the drag is due primarily to shear drag and thus streamlining is unnecessary. Finally, it should be pointed out that another advantage of streamlining is that the periodic shedding of vortices is usually eliminated. The vibrations produced by vortex shedding are often undesirable, so streamlining not only decreases drag but can eliminate the vibrations. 28
29
Cavitation Cavitation is a very rapid change of phase from liquid to vapor which occurs in a liquid whenever the local pressure is equal to or less than the vapor pressure. The first appearance of cavitation is at the position of lowest pressure in a field of flow. Four types of cavitation have been identified: 1. Traveling cavitation, ti which h occurs when vapor bubblesbbl or cavities are formed, are swept downstream, and collapse. 2. Fixed cavitation,, which occurs when a fixed cavity of vapor exists as a separated region. The separated region may reattach to the body, or the separated region may enclose the rear of the body and be closed by the main flow, in which case it is referred to as supercavitation. 3. Vortex cavitation, which is found in the high velocity, and thus low pressure, core of a vortex, often observed in the tip vortex leaving a propeller. 4. Vibratory cavitation, which may exist when a pressure wave moves in a 30
liquid. A pressure wave consists of a pressure pulse, which has a high pressure followed by alow pressure. The low pressure part of the wave(orvibration) can result in cavitation. The Traveling cavitation, in which vapor bubbles are formed and collapse, is associated with potential damage. The instantaneous pressures resulting from the collapse can be extremely high (perhaps 1400 MPa) and may cause damage to stainless steel components, as happens on the propellers of ships. Cavitation occurs whenever the cavitation number σ, defined by is less than the critical cavitation number σ crit, which depends on the geometry of the body and the Reynolds number. Here p is the absolute pressure in the undisturbed free stream and p v is the vapor pressure. As σ decreases below σ crit, the cavitation increases in intensity, moving from traveling cavitation to fixed cavitation to supercavitation. 31
The drag coefficient of a body is dependent on the cavitation number and for small cavitation numbers is given by C D (σ) = C D (0)(1 + σ) (8.3.4) where some values of C D (0) for common shapes are listed in Table 8.3 for Re 10 5. 32
The hydrofoil, an airfoil typebodythatisusedtoliftavesseloutofthewater, is a shape that is invariably associated with cavitation. Drag and lf lift coefficients and critical cavitation numbers are given in Table 8.4 for a typical hydrofoil with 10 5 <Re<10 6,wheretheReynoldsnumberisbasedonthe chord length and the area used with C D and C L is the chord times the length. 33
34
LIFT AND DRAG ON AIRFOILS An airfoil if il is a streamlined body designed d to reduce the adverse pressure gradient so that separation will not occur, usually with a small angle of attack, as shown in Fig. 8.12. Without separation the drag is due primarily to the wall shear stress, which results from viscous effects in the boundary layer. The boundary layer on an airfoil is very thin, and thus it can be ignored when solving for the flow field surrounding the airfoil. Hence the lift on an airfoil can be approximated by integrating the pressure distribution as given by the inviscid flow solution on the wall. 35
Empirical results for the drag For a typical airfoil if ilthe lift and drag coefficients i aregiven in Fig. 8.13. For a specially designed airfoil the drag coefficient may be as low as 0.0035, but the maximum lift coefficient is about 1.5. The design lift coefficient (cruise condition) is about 0.3, which is near the minimum drag coefficient condition. This corresponds to an angle of attack of about 2, far from the stall condition of about 16. 36
The drag coefficient presented may seem quite low compared with the coefficients of the preceding section. For airfoils a much larger projected area is used, namely, the plan area, which h is the chord c (see Fig. 8.12) times the length L of the airfoil. Thus the drag and lift coefficients are defined as Conventional airfoils are not symmetric; hence there is a positive lift coefficient at zero angle of attack. To take off and land at relatively low speeds, it is necessary to attain significantly higher lift coefficients than the maximum of 1.7 of Fig. 8.13. Or if a relatively low lift coefficient is to be accepted, the area c x L must be enlarged. Both are actually accomplished. Flaps are moved out from a section of each airfoil, resulting in an increased chord, and the angle of attack of the flap is also increased. 37
Slots are used to move high pressure air from the underside into the relatively low momentum boundary layer flow on the top side, as shown above; this prevents separation from the flap, thereby maintaining high lift. The lift coefficient can reach 2.5 with a single slotted flap and 3.2 with a doubleslotted flap. On some modern aircraft there may be three flaps in series with three slots along with a nose flap, to ensure that the boundary layer does not separate from the upper surface of the airfoil. The effective length of the airfoil when calculating the lift is taken to be the tip to tip distance, the wingspan. The fuselage acts to produce lift on the midsection of the aircraft. The drag calculation must include the shear acting on the airfoil, the fuselage, and the tail section. 38
39
40