3-70 Section 3 3.6 External Incompressible Flows Alan T. McDonald Introduction and Scope Potential ßow theory (Section 3.2) treats an incompressible ideal ßuid with zero viscosity. There are no shear stresses; pressure is the only stress acting on a ßuid particle. Potential ßow theory predicts no drag force when an object moves through a ßuid, which obviously is not correct, because all real ßuids are viscous and cause drag forces. The objective of this section is to consider the behavior of viscous, incompressible ßuids ßowing over objects. A number of phenomena that occur in external ßow at high Reynolds number over an object are shown in Figure 3.6.1. The freestream ßow divides at the stagnation point and ßows around the object. Fluid at the object surface takes on the velocity of the body as a result of the no-slip condition. Boundary layers form on the upper and lower surfaces of the body; ßow in the boundary layers is initially laminar, then transition to turbulent ßow may occur (points ÒTÓ). FIGURE 3.6.1 Viscous ßow around an airfoil (boundary layer thickness exaggerated for clarity). Boundary layers thickening on the surfaces cause only a slight displacement of the streamlines of the external ßow (their thickness is greatly exaggerated in the Þgure). Separation may occur in the region of increasing pressure on the rear of the body (points ÒSÓ); after separation boundary layer ßuid no longer remains in contact with the surface. Fluid that was in the boundary layers forms the viscous wake behind the object. The Bernoulli equation is valid for steady, incompressible ßow without viscous effects. It may be used to predict pressure variations outside the boundary layer. Stagnation pressure is constant in the uniform inviscid ßow far from an object, and the Bernoulli equation reduces to p¥ + 1 2 rV = constant 2 (3.6.1) where p¥ is pressure far upstream, r is density, and V is velocity. Therefore, the local pressure can be determined if the local freestream velocity, U, is known. © 1999 by CRC Press LLC 3-71 Fluid Mechanics Boundary Layers The Boundary Layer Concept The boundary layer is the thin region near the surface of a body in which viscous effects are important. By recognizing that viscous effects are concentrated near the surface of an object, Prandtl showed that only the Euler equations for inviscid ßow need be solved in the region outside the boundary layer. Inside the boundary layer, the elliptic Navier-Stokes equations are simpliÞed to boundary layer equations with parabolic form that are easier to solve. The thin boundary layer has negligible pressure variation across it; pressure from the freestream is impressed upon the boundary layer. Basic characteristics of all laminar and turbulent boundary layers are shown in the developing ßow over a ßat plate in a semi-inÞnite ßuid. Because the boundary layer is thin, there is negligible disturbance of the inviscid ßow outside the boundary layer, and the pressure gradient along the surface is close to zero. Transition from laminar to turbulent boundary layer ßow on a ßat plate occurs when Reynolds number based on x exceeds Rex = 500,000. Transition may occur earlier if the surface is rough, pressure increases in the ßow direction, or separation occurs. Following transition, the turbulent boundary layer thickens more rapidly than the laminar boundary layer as a result of increased shear stress at the body surface. Boundary Layer Thickness Definitions Boundary layer disturbance thickness, d, is usually deÞned as the distance, y, from the surface to the point where the velocity within the boundary layer, u, is within 1% of the local freestream velocity, U. As shown in Figure 3.6.2, the boundary layer velocity proÞle merges smoothly and asymptotically into the freestream, making d difÞcult to measure. For this reason and for their physical signiÞcance, we deÞne two integral measures of boundary layer thickness. Displacement thickness, d*, is deÞned as ¥ d* = d æ u ö æ yö ò è1 - U ø d è d ø (3.6.2) 0 FIGURE 3.6.2 Boundary layer on a ßat plate (vertical thickness exaggerated for clarity). Physically, d* is the distance the solid boundary would have to be displaced into the freestream in a frictionless ßow to produce the mass ßow deÞcit caused by the viscous boundary layer. Momentum thickness, q, is deÞned as q = d ò ¥ 0 y uæ u 1 - ö dæ ö U è U ø è dø (3.6.3) Physically, q is the thickness of a ßuid layer, having velocity U, for which the momentum ßux is the same as the deÞcit in momentum ßux within the boundary layer (momentum ßux is momentum per unit time passing a cross section). © 1999 by CRC Press LLC 3-72 Section 3 Because d* and q are deÞned in terms of integrals for which the integrand vanishes in the freestream, they are easier to evaluate experimentally than disturbance thickness d. Exact Solution of the Laminar Flat-Plate Boundary Layer Blasius obtained an exact solution for laminar boundary layer ßow on a ßat plate. He assumed a thin boundary layer to simplify the streamwise momentum equation. He also assumed similar velocity proÞles in the boundary layer, so that when written as u/U = f(y/d), velocity proÞles do not vary with x. He used a similarity variable to reduce the partial differential equations of motion and continuity to a single thirdorder ordinary differential equation. Blasius used numerical methods to solve the ordinary differential equation. Unfortunately, the velocity proÞle must be expressed in tabular form. The principal results of the Blasius solution may be expressed as d = x 5 Re x (3.6.4) and Cf = tw 0.664 = 1 Re x rU 2 2 (3.6.5) These results characterize the laminar boundary layer on a ßat plate; they show that laminar boundary layer thickness varies as x1/2 and wall shear stress varies as 1/x1/2. Approximate Solutions The Blasius solution cannot be expressed in closed form and is limited to laminar ßow. Therefore, approximate methods that give solutions for both laminar and turbulent ßow in closed form are desirable. One such method is the momentum integral equation (MIE), which may be developed by integrating the boundary layer equation across the boundary layer or by applying the streamwise momentum equation to a differential control volume (Fox and McDonald, 1992). The result is the ordinary differential equation t æ d* ö q dU dq = w 2 - ç + 2÷ dx rU è q ø U dx (3.6.6) The Þrst term on the right side of Equation (3.6.6) contains the inßuence of wall shear stress. Since tw is always positive, it always causes q to increase. The second term on the right side contains the pressure gradient, which can have either sign. Therefore, the effect of the pressure gradient can be to either increase or decrease the rate of growth of boundary layer thickness. Equation (3.6.6) is an ordinary differential equation that can be solved for q as a function of x on a ßat plate (zero pressure gradient), provided a reasonable shape is assumed for the boundary layer velocity proÞle and shear stress is expressed in terms of the other variables. Results for laminar and turbulent ßat-plate boundary layer ßows are discussed below. Laminar Boundary Layers. A reasonable approximation to the laminar boundary layer velocity proÞle is to express u as a polynomial in y. The resulting solutions for d and tw have the same dependence on x as the exact Blasius solution. Numerical results are presented in Table 3.6.1. Comparing the approximate and exact solutions shows remarkable agreement in view of the approximations used in the analysis. The trends are predicted correctly and the approximate values are within 10% of the exact values. Turbulent Boundary Layers. The turbulent velocity proÞle may be expressed well using a power law, u/U = (y/d)1/n, where n is an integer between 6 and 10 (frequently 7 is chosen). For turbulent ßow it is © 1999 by CRC Press LLC 3-73 Fluid Mechanics TABLE 3.6.1 Exact and Approximate Solutions for Laminar Boundary Layer Flow over a Flat Plate at Zero Incidence Velocity Distribution u æ yö = f ç ÷ = f ( h) è dø U f (h) = 2h Ð h2 f (h) = 3/2 h Ð 1/2 h3 f (h) = sin (p/2 h) Exact q d d* d 2/15 39/280 (4 Ð p)/2p 0.133 1/3 3/8 (p Ð 2)/p 0.344 a= d Re x x 5.48 4.64 4.80 5.00 b = C f Re x 0.730 0.647 0.654 0.664 not possible to express shear stress directly in terms of a simple velocity proÞle; an empirical correlation is required. Using a pipe ßow data correlation gives d 0.382 = x Re1x 5 (3.6.7) and Cf = tw 0.0594 = 1 Re1x 5 rU 2 2 (3.6.8) These results characterize the turbulent boundary layer on a ßat plate. They show that turbulent boundary layer thickness varies as x4/5 and wall shear stress varies as 1/x1/5. Approximate results for laminar and turbulent boundary layers are compared in Table 3.6.2. At a Reynolds number of 1 million, wall shear stress for the turbulent boundary layer is nearly six times as large as for the laminar layer. For a turbulent boundary layer, thickness increases Þve times faster with distance along the surface than for a laminar layer. These approximate results give a physical feel for relative magnitudes in the two cases. TABLE 3.6.2 Thickness and Skin Friction CoefÞcient for Laminar and Turbulent Boundary Layers on a Flat Plate Reynolds Number 2E 5E 1E 2E 5E 1E 2E 5E + + + + + + + + 05 05 06 06 06 07 07 07 Boundary Layer Thickness/x Skin Friction CoefÞcient Turbulent/Laminar Ratio Laminar BL Turbulent BL Laminar BL Turbulent BL BL Thickness Skin Friction 0.0112 0.00707 0.00500 0.00354 0.00224 0.00158 0.00112 0.000707 0.0333 0.0277 0.0241 0.0210 0.0175 0.0152 0.0132 0.0110 0.00148 0.000939 0.000664 0.000470 0.000297 0.000210 0.000148 0.0000939 0.00517 0.00431 0.00375 0.00326 0.00272 0.00236 0.00206 0.00171 2.97 3.92 4.82 5.93 7.81 9.62 11.8 15.6 3.48 4.58 5.64 6.95 9.15 11.3 13.9 18.3 Note: BL = boundary layer. The MIE cannot be solved in closed form for ßows with nonzero pressure gradients. However, the role of the pressure gradient can be understood qualitatively by studying the MIE. Effect of Pressure Gradient Boundary layer ßow with favorable, zero, and adverse pressure gradients is depicted schematically in Figure 3.6.3. (Assume a thin boundary layer, so ßow on the lower surface behaves as external ßow on © 1999 by CRC Press LLC 3-74 Section 3 FIGURE 3.6.3 Boundary layer ßow with presssure gradient (thickness exaggerated for clarity). a surface, with the pressure gradient impressed on the boundary layer.) The pressure gradient is favorable when ¶p/¶x < 0, zero when ¶p/¶x = 0, and adverse when ¶p/¶x > 0, as indicated for Regions 1, 2, and 3. Viscous shear always causes a net retarding force on any ßuid particle within the boundary layer. For zero pressure gradient, shear forces alone can never bring the particle to rest. (Recall that for laminar and turbulent boundary layers the shear stress varied as 1/x1/2 and 1/x1/5, respectively; shear stress never becomes zero for Þnite x.) Since shear stress is given by tw = m ¶u/¶y)y=0, the velocity gradient cannot be zero. Therefore, ßow cannot separate in a zero pressure gradient; shear stresses alone can never cause ßow separation. In the favorable pressure gradient of Region 1, pressure forces tend to maintain the motion of the particle, so ßow cannot separate. In the adverse pressure gradient of Region 3, pressure forces oppose the motion of a ßuid particle. An adverse pressure gradient is a necessary condition for ßow separation. Velocity proÞles for laminar and turbulent boundary layers are shown in Figure 3.6.2. It is easy to see that the turbulent velocity proÞle has much more momentum than the laminar proÞle. Therefore, the turbulent velocity proÞle can resist separation in an adverse pressure gradient better than the laminar proÞle. The freestream velocity distribution must be known before the MIE can be applied. We obtain a Þrst approximation by applying potential ßow theory to calculate the ßow Þeld around the object. Much effort has been devoted to calculation of velocity distributions over objects of known shape (the ÒdirectÓ problem) and to determination of shapes to produce a desired pressure distribution (the ÒinverseÓ problem). Detailed discussion of such calculation schemes is beyond the scope of this section; the state of the art continues to progress rapidly. Drag Any object immersed in a viscous ßuid ßow experiences a net force from the shear stresses and pressure differences caused by the ßuid motion. Drag is the force component parallel to, and lift is the force component perpendicular to, the ßow direction. Streamlining is the art of shaping a body to reduce ßuid dynamic drag. Airfoils (hydrofoils) are designed to produce lift in air (water); they are streamlined to reduce drag and thus to attain high liftÐdrag ratios. In general, lift and drag cannot be predicted analytically for ßows with separation, but progress continues on computational ßuid dynamics methods. For many engineering purposes, drag and lift forces are calculated from experimentally derived coefÞcients, discussed below. Drag coefÞcient is deÞned as © 1999 by CRC Press LLC 3-75 Fluid Mechanics CD = FD 1 2 rV A 2 (3.6.9) where 1/2rV2 is dynamic pressure and A is the area upon which the coefÞcient is based. Common practice is to base drag coefÞcients on projected frontal area (Fox and McDonald, 1992). Similitude was treated in Section 3.3. In general, the drag coefÞcient may be expressed as a function of Reynolds number, Mach number, Froude number, relative roughness, submergence divided by length, and so forth. In this section we consider neither high-speed ßow nor free-surface effects, so we will consider only Reynolds number and roughness effects on drag coefÞcient. Friction Drag The total friction drag force acting on a plane surface aligned with the ßow direction can be found by integrating the shear stress distribution along the surface. The drag coefÞcient for this case is deÞned as friction force divided by dynamic pressure and wetted area in contact with the ßuid. Since shear stress is a function of Reynolds number, so is drag coefÞcient (see Figure 3.6.4). In Figure 3.6.4, transition occurs at Rex = 500,000; the dashed line represents the drag coefÞcient at larger Reynolds numbers. A number of empirical correlations may be used to model the variation in CD shown in Figure 3.6.4 (Schlichting, 1979). FIGURE 3.6.4 Drag coefÞcient vs. Reynolds number for a smooth ßat plate parallel to the ßow. Extending the laminar boundary layer line to higher Reynolds numbers shows that it is beneÞcial to delay transition to the highest possible Reynolds number. Some results are presented in Table 3.6.3; drag is reduced more than 50% by extending laminar boundary layer ßow to ReL = 106. Pressure Drag A thin ßat surface normal to the ßow has no area parallel to the ßow direction. Therefore, there can be no friction force parallel to the ßow; all drag is caused by pressure forces. Drag coefÞcients for objects with sharp edges tend to be independent of Reynolds number (for Re > 1000), because the separation points are Þxed by the geometry of the object. Drag coefÞcients for selected objects are shown in Table 3.6.4. Rounding the edges that face the ßow reduces drag markedly. Compare the drag coefÞcients for the hemisphere and C-section shapes facing into and away from the ßow. Also note that the drag coefÞcient © 1999 by CRC Press LLC 3-76 Section 3 TABLE 3.6.3 Drag CoefÞcients for Laminar, Turbulent, and Transition Boundary Layers on a Flat Plate Reynolds Number 2E 5E 1E 2E 5E 1E 2E 5E + + + + + + + + 05 05 06 06 06 07 07 07 Drag CoefÞcient Laminar BL 0.00297 0.00188 0.00133 0.000939 0.000594 0.000420 0.000297 0.000188 Turbulent BL 0.00615 0.00511 0.00447 0.00394 0.00336 0.00300 0.00269 0.00235 Transition Laminar/ Transition % Drag Reduction Ñ 0.00189 0.00286 0.00314 0.00304 0.00284 0.00261 0.00232 Ñ Ñ 0.464 0.300 0.195 0.148 0.114 0.081 Ñ Ñ 53.6 70.0 80.5 85.2 88.6 9.19 Note: BL = Boundary layer. TABLE 3.6.4 Drag CoefÞcient Data for Selected Objects (Re > 1000) Object Square prism 3 CD(RE* > ~ 10 ) Diagram b/h = ¥ 2.05 b/h = 1 1.05 Disk 1.17 Ring 1.20b Hemisphere (open end facing ßow) 1.42 Hemisphere (open end facing downstream) 0.38 C-section (open side facing ßow) 2.30 C-section (open side facing downstream) 1.20 a b Data from Hoerner, 1965. Based on ring area. for a two-dimensional object (long square cylinder) is about twice that for the corresponding threedimensional object (square cylinder with b/h = 1). © 1999 by CRC Press LLC 3-77 Fluid Mechanics Friction and Pressure Drag: Bluff Bodies Both friction and pressure forces contribute to the drag of bluff bodies (see Shapiro, 1960, for a good discussion of the mechanisms of drag). As an example, consider the drag coefÞcient for a smooth sphere shown in Figure 3.6.5. Transition from laminar to turbulent ßow in the boundary layers on the forward portion of the sphere causes a dramatic dip in drag coefÞcient at the critical Reynolds number (ReD » 2 ´ 105). The turbulent boundary layer is better able to resist the adverse pressure gradient on the rear of the sphere, so separation is delayed and the wake is smaller, causing less pressure drag. 400 100 10 CD 1 0.01 0.06 100 101 102 103 104 105 106 Re = VD v FIGURE 3.6.5 Drag coefÞcient vs. Reynolds number for a smooth sphere. Surface roughness (or freestream disturbances) can reduce the critical Reynolds number. Dimples on a golf ball cause the boundary layer to become turbulent and, therefore, lower the drag coefÞcient in the range of speeds encountered in a drive. Streamlining Streamlining is adding a faired tail section to reduce the extent of separated ßow on the downstream portion of an object (at high Reynolds number where pressure forces dominate drag). The adverse pressure gradient is taken over a longer distance, delaying separation. However, adding a faired tail increases surface area, causing skin friction drag to increase. Thus, streamlining must be optimized for each shape. Front contours are of principal importance in road vehicle design; the angle of the back glass also is important (in most cases the entire rear end cannot be made long enough to control separation and reduce drag signiÞcantly). Lift Lift coefÞcient is deÞned as CL = FL 1 2 rV A 2 Note that lift coefÞcient is based on projected planform area. © 1999 by CRC Press LLC (3.6.10) 3-78 Section 3 Airfoils Airfoils are shaped to produce lift efÞciently by accelerating ßow over the upper surface to produce a low-pressure region. Because the ßow must again decelerate, inevitably there must be a region of adverse pressure gradient near the rear of the upper surface (pressure distributions are shown clearly in Hazen, 1965). Lift and drag coefÞcients for airfoil sections depend on Reynolds number and angle of attack between the chord line and the undisturbed ßow direction. The chord line is the straight line joining the leading and trailing edges of the airfoil (Abbott and von Doenhoff, 1959). As the angle of attack is increased, the minimum pressure point moves forward on the upper surface and the minimum pressure becomes lower. This increases the adverse pressure gradient. At some angle of attack, the adverse pressure gradient is strong enough to cause the boundary layer to separate completely from the upper surface, causing the airfoil to stall. The separated ßow alters the pressure distribution, reducing lift sharply. Increasing the angle of attack also causes the the drag coefÞcient to increase. At some angle of attack below stall the ratio of lift to drag, the liftÐdrag ratio, reaches a maximum value. Drag Due to Lift For wings (airfoils of Þnite span), lift and drag also are functions of aspect ratio. Lift is reduced and drag increased compared with inÞnite span, because end effects cause the lift vector to rotate rearward. For a given geometric angle of attack, this reduces effective angle of attack, reducing lift. The additional component of lift acting in the ßow direction increases drag; the increase in drag due to lift is called induced drag. The effective aspect ratio includes the effect of planform shape. When written in terms of effective aspect ratio, the drag of a Þnite-span wing is C D = C D, ¥ + CL2 par (3.6.11) where ar is effective aspect ratio and the subscript ¥ refers to the inÞnite section drag coefÞcient at CL. For further details consult the references. The lift coefÞcient must increase to support aircraft weight as speed is reduced. Therefore, induced drag can increase rapidly at low ßight speeds. For this reason, minimum allowable ßight speeds for commercial aircraft are closely controlled by the FAA. Boundary Layer Control The major part of the drag on an airfoil or wing is caused by skin friction. Therefore, it is important to maintain laminar ßow in the boundary layers as far aft as possible; laminar ßow sections are designed to do this. It also is important to prevent ßow separation and to achieve high lift to reduce takeoff and landing speeds. These topics fall under the general heading of boundary layer control. Profile Shaping Boundary layer transition on a conventional airfoil section occurs almost immediately after the minimum pressure at about 25% chord aft the leading edge. Transition can be delayed by shaping the proÞle to maintain a favorable pressure gradient over more of its length. The U.S. National Advisory Committee for Aeronautics (NACA) developed several series of proÞles that delayed transition to 60 or 65% of chord, reducing drag coefÞcients (in the design range) 60% compared with conventional sections of the same thickness ratio (Abbott and von Doenhoff, 1959). © 1999 by CRC Press LLC Fluid Mechanics 3-79 Flaps and Slats Flaps are movable sections near the trailing edge of a wing. They extend and/or deßect to increase wing area and/or increase wing camber (curvature), to provide higher lift than the clean wing. Many aircraft also are Þtted with leading edge slats which open to expose a slot from the pressure side of the wing to the upper surface. The open slat increases the effective radius of the leading edge, improving maximum lift coefÞcient. The slot allows energized air from the pressure surface to ßow into the low-pressure region atop the wing, energizing the boundary layers and delaying separation and stall. Suction and Blowing Suction removes low-energy ßuid from the boundary layer, reducing the tendency for early separation. Blowing via high-speed jets directed along the surface reenergizes low-speed boundary layer ßuid. The objective of both approaches is to delay separation, thus increasing the maximum lift coefÞcient the wing can achieve. Powered systems add weight and complexity; they also require bleed air from the engine compressor, reducing thrust or power output. Moving Surfaces Many schemes have been proposed to utilize moving surfaces for boundary layer control. Motion in the direction of ßow reduces skin friction, and thus the tendency to separate; motion against the ßow has the opposite effect. The aerodynamic behavior of sports balls Ñ baseballs, golf balls, and tennis balls Ñ depends signiÞcantly on aerodynamic side force (lift, down force, or side force) produced by spin. These effects are discussed at length in Fox and McDonald (1992) and its references. Computation vs. Experiment Experiments cannot yet be replaced completely by analysis. Progress in modeling, numerical techniques, and computer power continues to be made, but the role of the experimentalist likely will remain important for the foreseeable future. Computational Fluid Dynamics (CFD) Computation of ßuid ßow requires accurate mathematical modeling of ßow physics and accurate numerical procedures to solve the equations. The basic equations for laminar boundary layer ßow are well known. For turbulent boundary layers generally it is not possible to resolve the solution space into sufÞciently small cells to allow direct numerical simulation. Instead, empirical models for the turbulent stresses must be used. Advances in computer memory storage capacity and speed (e.g., through use of massively parallel processing) continue to increase the resolution that can be achieved. A second source of error in CFD work results from the numerical procedures required to solve the equations. Even if the equations are exact, approximations must be made to discretize and solve them using Þnite-difference or Þnite-volume methods. Whichever is chosen, the solver must guard against introducing numerical instability, round-off errors, and numerical diffusion (Hoffman, 1992). Role of the Wind Tunnel Traditionally, wind tunnel experiments have been conducted to verify the design and performance of components and complete aircraft. Design veriÞcation of a modern aircraft may require expensive scale models, several thousand hours of wind tunnel time at many thousands of dollars an hour, and additional full-scale ßight testing. New wind tunnel facilities continue to be built and old ones refurbished. This indicates a need for continued experimental work in developing and optimizing aircraft conÞgurations. Many experiments are designed to produce baseline data to validate computer codes. Such systematic experimental data can help to identify the strengths and weaknesses of computational methods. CFD tends to become only indicative of trends when massive zones of ßow separation are present. Takeoff and landing conÞgurations of conventional aircraft, with landing gear, high-lift devices, and © 1999 by CRC Press LLC 3-80 Section 3 ßaps extended, tend to need Þnal experimental conÞrmation and optimization. Many studies of vertical takeoff and vectored thrust aircraft require testing in wind tunnels. Defining Terms Boundary layer: Thin layer of ßuid adjacent to a surface where viscous effects are important; viscous effects are negligible outside the boundary layer. Drag coefÞcient: Force in the ßow direction exerted on an object by the ßuid ßowing around it, divided by dynamic pressure and area. Lift coefÞcient: Force perpendicular to the ßow direction exerted on an object by the ßuid ßowing around it, divided by dynamic pressure and area. Pressure gradient: Variation in pressure along the surface of an object. For a favorable pressure gradient, pressure decreases in the ßow direction; for an adverse pressure gradient, pressure increases in the ßow direction. Separation: Phenomenon that occurs when ßuid layers adjacent to a solid surface are brought to rest and boundary layers depart from the surface contour, forming a low-pressure wake region. Separation can occur only in an adverse pressure gradient. Transition: Change from laminar to turbulent ßow within the boundary layer. The location depends on distance over which the boundary layer has developed, pressure gradient, surface roughness, freestream disturbances, and heat transfer. References Abbott, I.H. and von Doenhoff, A.E. 1959. Theory of Wing Sections, Including a Summary of Airfoil Data. Dover, New York. Fox, R.W. and McDonald, A.T. 1992. Introduction to Fluid Mechanics, 4th ed. John Wiley & Sons, New York. Hazen, D.C. 1965. Boundary Layer Control, Þlm developed by the National Committee for Fluid Mechanics Films (NCFMF) and available on videotape from Encyclopaedia Britannica Educational Corporation, Chicago. Hoerner, S.F. 1965. Fluid-Dynamic Drag, 2nd ed. Published by the author, Midland Park, NJ. Hoffman, J.D. 1992. Numerical Methods for Engineers and Scientists. McGraw-Hill, New York. Schlichting, H. 1979. Boundary-Layer Theory, 7th ed. McGraw-Hill, New York. Shapiro, A.H. 1960. The Fluid Dynamics of Drag, Þlm developed by the National Committee for Fluid Mechanics Film (NCFMF) and available on videotape from Encyclopaedia Britannica Educational Corporation, Chicago. Further Information A comprehensive source of basic information is the Handbook of Fluid Dynamics, edited by Victor L. Streeter (McGraw-Hill, New York, 1960). Timely reviews of important topics are published in the Annual Review of Fluid Mechanics series (Annual Reviews, Inc., Palo Alto, CA.). Each volume contains a cumulative index. ASME (American Society of Mechanical Engineers, New York, NY) publishes the Journal of Fluids Engineering quarterly. JFE contains ßuid machinery and other engineering applications of ßuid mechanics. The monthly AIAA Journal and bimonthly Journal of Aircraft (American Institute for Aeronautics and Astronautics, New York) treat aerospace applications of ßuid mechanics. © 1999 by CRC Press LLC