Basic Fluid Mechanics

Transcription

1 Basic Fluid Mechanics Chapter 7B: Forces on Submerged Bodies 7/26/2018 C7B: Forces on Submerged Bodies 1 Forces on Submerged Bodies Lift and Drag are forces exerted on an immersed body by the surrounding fluid flowing around the body in the normal and streamwise directions. F pnda tda where are the normal and tangential unit vectors to the body surface. From a first order examination, these forces are a result of; i. the pressure distribution around the body ii. shear stress distribution on the body Therefore these forces are related to the character of the flow field (i.e., laminar or turbulent). 7/26/2018 C7B: Forces on Submerged Bodies 2 w 1

2 Forces on Submerged Bodies F net F lift F lift = (C l Av 2 )/2 F drag = (C d Av 2 )/2 F drag Direction of motion The Lift force always acts perpendicular to the drag force. Forces can be determined using appropriate force coefficients. 7/26/2018 C7B: Forces on Submerged Bodies 3 Drag Force in Subsonic Flows Drag on an object can be separated into two general groups: i) Lifting bodies - i.e., airfoils, turbine blades ii) Non-lifting bodies - i.e., structural members Note: 1 - The total drag for lifting bodies is different than for non-lifting bodies. 2 - Lifting bodies in supersonic flows also have wave drag. 3- Typically induced drag is > profile (parasitic) drag at takeoff, but less at cruise. 7/26/2018 C7B: Forces on Submerged Bodies 4 2

3 Drag Force in Subsonic Flows Skin Friction Drag - forces due to the shear stress distribution on the surface of the body - In laminar flow, shear stresses are due to viscous effects. - For a turbulent flow the shear stresses are more complex and are a result of both viscous effects and inertial interactions between fluid elements. Pressure Drag forces due to the pressure distribution around an object (changes depending on the state of the boundary layer, i.e., laminar or turbulent). Induced Drag - the drag component that is associated with the vortex motion about lifting surfaces Note: Another component of drag found in supersonic flows is Wave Drag, this force is a result of the normal stresses. 7/26/2018 C7B: Forces on Submerged Bodies 5 Forces on Submerged Bodies (General Rules of Thumb) When t/c << 1 : streamlined body C D,f >> C D,p When t/c ~1 : bluff or streamline separated body C D,p >> C D,f 7/26/2018 C7B: Forces on Submerged Bodies 6 3

4 Forces on Submerged Bodies (General Rules of Thumb) When the total drag is a result of pressure forces & shear stresses (laminar or turbulent) acting on the surface of the body: A) For bodies with relatively large separated regions, the viscous drag is usually a small part (< 10%) of the total drag. 7/26/2018 C7B: Forces on Submerged Bodies 7 Forces on Submerged Bodies (General Rules of Thumb) B) For streamlined bodies at small angles of attack or for bodies at low Re#, where the separated region is either small or non-existent, the viscous drag is the dominant contributor to the total drag. 7/26/2018 C7B: Forces on Submerged Bodies 8 4

5 Drag on a Sphere 7/26/2018 C7B: Forces on Submerged Bodies 9 Drag on a Sphere 7/26/2018 C7B: Forces on Submerged Bodies 10 5

6 Drag on a Sphere The Effect of Boundary Layer State on the wake size and Drag coefficient. 7/26/2018 C7B: Forces on Submerged Bodies 11 Drag on a Sphere 7/26/2018 C7B: Forces on Submerged Bodies 12 6

7 Effect of Roughness on Sphere C D 7/26/2018 C7B: Forces on Submerged Bodies 13 Pressure Distribution on a Smooth Sphere 0.5x10 5 Re 2.8x /26/2018 C7B: Forces on Submerged Bodies 14 7

8 Critical Points for Flow over a Cylinder: An Inviscid Analysis From the stream function formulation: E B D r=a 2 a u U ( 1 2 )sin ( 1a) r 2 a u U ( 1 2 )cos ( 1b) r At the surface of the cylinder: r u 2U sin ( 2a) u 0 ( 2b) r c p 2 1 4sin ( 3) 7/26/2018 C7B: Forces on Submerged Bodies 15 Inviscid Analysis: Continued E D p o = stagnation pressure = p A B r=a p = freestream static pressure = p D p = p E = p B is the inflection point p D > p C pressure increases across curved streamlines p s is the surface pressure c p If p = p E = p B p s q p 2 1 4sin ( 4) 2 c p 0 1 4sin 1 2 sin Note: a) when =0; c p =1 b) when =30; c p = 0 c) when =90; c p = -3 7/26/2018 C7B: Forces on Submerged Bodies 16 8

9 Inviscid Analysis: Continued For =30; point B, u 2U sin 2U sin30 U For =90; point C, u 2U sin 2U sin90 2U p o = s 30 and 2 c p 14sin therefore, p C < p D = p 7/26/2018 C7B: Forces on Submerged Bodies 17 c p p s p 3 q Critical Points for Flow over a Cylinder 7/26/2018 C7B: Forces on Submerged Bodies 18 9

10 Critical Points for Flow over a Cylinder 7/26/2018 C7B: Forces on Submerged Bodies 19 Downwash & Wing Tip Vortex Airflow over the top of the wing surface moves inward Airflow over the bottom of the wing surface moves outward Due to the pressure changes The pressure difference causes a secondary flow around the wing tip, from bottom to top. This results in a swirling motion off the ends of the wing (i.e., Wing Tip Vortices.) 7/26/2018 C7B: Forces on Submerged Bodies 20 10

11 Downwash & Wing Tip Vortex These trailing vortices produce what is referred to as downwash This region extends well above and below the wing When an aircraft is in the downwash region of another an increase in drag, induced drag will be realized When a wing enters a region of downwash, both lift and drag are affected Beyond the tip vortex there is second region referred to as the upwash zone 7/26/2018 C7B: Forces on Submerged Bodies 21 Downwash & Wing Tip Vortex This results in a pressure decrease at the center of the tip vortex and a corresponding decrease in temperature. If the surrounding humidity is high, this temperature decrease causes the vapor (i.e., the moisture in the air) to condense, making the helical tip vortex visible. 7/26/2018 C7B: Forces on Submerged Bodies 22 11

12 Downwash & Wing Tip Vortex 7/26/2018 C7B: Forces on Submerged Bodies 23 Downwash & Wing Tip Vortex Note: 1- Condensation of the moisture within the surrounding air makes visible the helical tip vortex. 2- This decrease in temperature is a result of the pressure decrease at the center of the tip vortex. 7/26/2018 C7B: Forces on Submerged Bodies 24 12

13 Upwash & Drag Reduction Downwash - causes an increased drag on trailing aircraft, thus requiring increased power to maintain conditions. Upwash - causes increased lift, and corresponding reduced drag and power Birds sense this condition and fly in a V-formation to make use of this upwash effect 7/26/2018 C7B: Forces on Submerged Bodies 25 Upwash & Drag Reduction of the V-formation b - wingspan s - spacing Note: Drag is minimal when s=0 7/26/2018 C7B: Forces on Submerged Bodies 26 13

14 Optimum V-formation for 9 Birds Note: The ideal spacing is not exactly a V, but instead a formation that is more swept back at the tips and less at the apex 7/26/2018 C7B: Forces on Submerged Bodies 27 Drag Force in Subsonic Flows There are two dominant types of Drag: 1) Surface drag - is a result of the shear stresses between the surface and the fluid. EX: Kyle (1989) reported that wearing loose clothing can increase surface drag from 2% to 8%. EX:Van Ingen Schenau (1982) reported a 10% reduction in surface drag when a speed skater wears a smooth body suit. 7/26/2018 C7B: Forces on Submerged Bodies 28 14

15 Drag Force in Subsonic Flows 2) Form drag - occurs when a fluid passes over an object and is diverted outward creating a low pressure region behind the object. Low Form Drag High Form Drag Note: The orientation of the object will affect the frontal area and will play an important role in the amount of form drag. 7/26/2018 C7B: Forces on Submerged Bodies 29 Lift and Drag Components The - sign accounts for the fact that the pressure force is always directed toward the surface. Rewriting in terms of the angle () between the axis and the surface normal, L psinda cosda A D pcosda sinda A Note: Since is ~ 90 deg over most of the airfoil; a) the lift force is mainly due to pressure, and b) drag is dominated by the shear stress in a streamlined body at small angle of attack. 7/26/2018 C7B: Forces on Submerged Bodies 30 A A w w î 15

16 Lift and Drag Components Surface Pressure and Shear Stress Distribution on a streamlined body immersed within a flow. ĵ î where î and ĵ are parallel and normal to the incoming velocity vector 7/26/2018 C7B: Forces on Submerged Bodies 31 Generic Airfoil Pressure Distribution Dark Blue represents + pressure and Light Blue represents c p 7/26/2018 C7B: Forces on Submerged Bodies 32 16

17 Lift and Drag Components Airfoils are shaped in a way to manipulate air pressure based on Bernoulli s principle. Air moves faster over the upper surface of the wing, which decrease the local pressure (or increasing suction). Air adjacent to the lower surface of the wing, moves at a lower speed and creates a higher pressure, or pushing force. 7/26/2018 C7B: Forces on Submerged Bodies 33 Lift and Drag Coefficients S S Note: In general the force coefficients from dimensional analysis are equal to a f(, Re, c/l, t/l, /L, TI) 7/26/2018 C7B: Forces on Submerged Bodies 34 17

18 Lift and Drag Coefficients C L and C D profiles for a conventional NACA 2415 and a laminar boundary layer type airfoil. C D 7/26/2018 C7B: Forces on Submerged Bodies 35 The Effect of Finite Aspect Ratio on the Lift and Drag Coefficients AR = span/cord = w/c The lift-to-drag ratio is critical (i.e. the larger the ratio, the more effective the airfoil is in flight). L/D ratio is dependent on the angle that the airfoil makes with the incoming air, this is called the ANGLE OF ATTACK (AoA). Increasing the AoA increases the L/D ratio to a point; beyond that point the angle becomes too steep, the boundary layer separates and the airfoil stalls. 7/26/2018 C7B: Forces on Submerged Bodies 36 18

19 Typical Airfoil Designations NACA designation 2415 profile has the following characteristics: o 2 - Maximum chamber is 2% of chord o 4 - Maximum chamber 40% of chord o 15 - Maximum thickness 15% of chord 7/26/2018 C7B: Forces on Submerged Bodies 37 The Magnus Effect The Magnus effect describes the curved path that is observed by spinning projectiles. Explained by Bernoulli s principle and the pressure differences caused by relative differences in flow velocities. Examples: Baseball: curveball, slider Golf: slice, hook Tennis: top-spin forehand Auto Racing: downforce Soccer: bender Volleyball: top-spin jump serve 7/26/2018 C7B: Forces on Submerged Bodies 38 19

20 Forces due to Rotating Bodies 7/26/2018 C7B: Forces on Submerged Bodies 39 Forces due to Rotating Bodies actual direction of flight intended direction of flight low pressure zone high pressure zone 7/26/2018 C7B: Forces on Submerged Bodies 40 20

21 Forces due to Rotating Bodies 7/26/2018 C7B: Forces on Submerged Bodies 41 Forces due to Rotating Bodies A golf club imparts backspin on the golf ball and increases the length of the drive. Dimples on a golf ball delay boundary layer separation and can dramatically influence the length of a drive. Depth of Dimple Carry Length of Drive (mm) (m) (m) The Mechanics of Sport, E. Bade. 7/26/2018 C7B: Forces on Submerged Bodies 42 21

22 Terminal Speed An object falling through a fluid reaches its terminal speed (V T ) when the drag force is equal to its weight. This results in a net force on the object equal to zero and from Newton s Law, a and therefore no further acceleration can occur. weight drag force 7/26/2018 C7B: Forces on Submerged Bodies 43 Estimated Terminal Speeds of Spheres Ball Weight Diameter K Terminal V T (pounds) (inches) (Drag Factor) (mi/hr) 16-lb shot Baseball Golf ball Basketball Ping-Pong ball Weight Diameter K Terminal VT Ball (N) (cm) (Drag Factor) (m/s) 16-lb shot Baseball Golf ball Basketball K 2 C D D g 8 W Adapted from Sport Science by Peter J. Brancazio. V T C D : coefficient of drag : fluid density D: sphere diameter W: weight of sphere V T : terminal speed 7/26/2018 C7B: Forces on Submerged Bodies 44 g K 22