(3) BIOMECHANICS of LOCOMOTION through FLUIDS

Transcription

1 (3) BIOMECHANICS of LOCOMOTION through FLUIDS Questions: - Explain the biomechanics of different modes of locomotion through fluids (undulation, rowing, hydrofoils, jet propulsion). - How does size influence the mode and speed of swimming? - What determines the energy cost of swimming and how does it compare to running and flying? Fluid Mechanics: - Fluid density, hydrostatic pressure, buoyancy, Bernoulli principle, viscosity. - Boundary layer, laminar and turbulent flow, Reynolds number, hydrodynamic drag and lift. Biomechanics of Locomotion though Fluids 3-1

2 3.1 FLUID STATICS Motion through gases and liquids such as air and water for animal locomotion. Density of fluid - Mass per unit volume ( kg / m 3 ) Water ρ = 1000 kg / m 3 ρ = Seawater ρ = 1026 kg / m 3 (Due to dissolved minerals) Air ρ = 1.21 kg / m 3 (At sea level and 20 C) m V Pressure in fluid - Force per unit area ( N / m 2 or Pa ) p = F A Hydrostatic pressure: - Increase with depth in fluid (The fluid must support its own weight) p = p 0 + ρgh Where p 0 = Pressure at surface (Atmospheric pressure). h = Depth (m). - Pressure in water increases (1 atmospere per 10.3m increase in depth). Biomechanics of Locomotion though Fluids 3-2

3 Atmospheric pressure: - Atmospheric pressure decreases with altitude. p p by a e - = Where y = Altitude (m) p a = Pressure at sea level (1.013 x 10 5 N / m 2 ). b = Constant (1.16 x 10-4 m -1 ). Biomechanics of Locomotion though Fluids 3-3

4 Buoyant force: - A body partially or fully immersed in a fluid experiences an upward buoyant force. - The magnitude of the force is equal to weight of the fluid displaced by the body (Archimedes principle). - Weight acts at the centre of gravity. - The buoyant force acts at the geometrical centre of body immersed in fluid (centre of buoyancy). Biomechanics of Locomotion though Fluids 3-4

5 3.2 FLUID DYNAMICS Steady flow: - Velocity, density and pressure at each point do not vary with time. Equation of continuity: - Assume an ideal fluid (incompressible, nonviscous and not turbulent). - Fluid flow through a pipe of uniform size. Av = constant Where A = Cross sectional area of pipe (m 2 ). v = Velocity of fluid ( m / s ). - The product of the area and fluid speed at all points along the pipe is constant for an incompressible fluid. SmallA High v Large A Low v - If the fluid is compressible ( ρ constant) ρ Av = constant Biomechanics of Locomotion though Fluids 3-5

6 Bernoulli equation: The conservation of mass and energy applied to a fluid. p ρv 2 + ρgy = constant Where y = Height of fluid (m). - The work done on a fluid per unit volume ( W = F x = p V ) is equal to the changes in kinetic energy (PE) and potential energy (PE) per unit volume. - The sum of the pressure ( p ), the kinetic energy per unit volume ( ρv 2 / 2 ) and the and gravitational potential energy per unit volume ( ρgy ) as the same value at all points along streamline. - The fluid in the section ( x 1 ) moves to the section of length ( x 2 ). The volume of fluid in the two sections are equal. Biomechanics of Locomotion though Fluids 3-6

7 Bernoulli principle: - In horizontal flow ( y = constant ) p ρv = constant - Pressure is less where velocity is high. - Pressure of an incompressible fluid can be measured with Venturi tubes. Biomechanics of Locomotion though Fluids 3-7

8 Flow around an asymmetrical body: - The Bernoulli effect states that the increased velocity of the air above the wing compared to the air beneath the wing causes the decreased pressure above the wing. - This is NOT the primary effect producing lift on wings. Viscosity: - Property of a real fluid due to inertial friction. - Consider the force between two plates (Area A, separation d, relative velocity of plates, v ). - The force required to overcome the viscosity of the fluid: F viscous = ηav d Where η = Coefficient of viscosity (N s / m 2, Temperature dependent). Air (20 C) η = 1.82 x 10-5 N s / m 2 Water (20 C) η = 1.00 x 10-3 N s / m 2 Biomechanics of Locomotion though Fluids 3-8

9 The Motion of a Body through a Fluid - Is equivalent to the motion of a fluid around a body. - Drag force acts to oppose the motion of a body through a fluid. - The drag force depends on: - The velocity of the body relative to the fluid. - The properties of the fluid. - The size and shape of the body. Boundary layer: - Viscous fluid interacts with surface of body, sticks to surface forming a very thin "boundary layer that is carried along with the body. - Velocity of fluid gradually diminishes with distance from the body (velocity gradient). - Forms retarding drag force, called viscous drag (also called friction drag, surface drag). Biomechanics of Locomotion though Fluids 3-9

10 Types of Fluid Flow The types of fluid flow depends on: - The size, shape and roughness of the object. - The viscosity of the fluid and the fluid velocity. Fluid flow around a body is characterised by Reynolds number, Where v = Relative velocity of body and fluid (m / s). Low Re = Small and slow. l = Geometric length (m). High Re = Large and fast. ρ = Density of fluid (kg / m 3 ). η = Coefficient of viscosity of fluid (N s / m 2 ). Flow type depends on Reynolds number: Re < 1 Laminar flow (viscous drag only). Re 1 Transition to partially turbulent flow. Re = 1< Re <10 3 Turbulent wake grows (Viscous and pressure drag) < Re <10 6 Turbulent wake grows (Pressure drag dominates). Re 10 6 Transition to fully turbulent flow (pressure drag decreases). Re >10 6 Fully turbulent flow (pressure drag dominates). ρvl η Biomechanics of Locomotion though Fluids 3-10

11 Laminar (Stokes flow): - Fluid moves around body in uniform layers of differing speeds (boundary layer). - Viscous drag force exerted on sides of body due to viscosity of fluid. F v = κlηv Where v = Relative velocity of body and fluid (m / s). l = Characteristic length (m). η = Coefficient of viscosity of fluid (N s / m 2 ) (Measure of resistance of fluid to flow). κ = Constant, depends on shape of body. (κ = 3π for sphere). Note that F v v, independent of fluid density and area of body. Biomechanics of Locomotion though Fluids 3-11

12 Partially turbulent: - The fluid is unable to follow surface contours and the boundary layer separates from surface. - Bernoulli effect, wake forms (region of turbulence and low pressure behind object). Net pressure on the front of object. - Pressure drag is given by F p 1 = ρscdv 2 2 Where ρ = Density of fluid (kg / m 3 ). S = Characteristic area (m 2 ). C D = Drag coefficient (Depends on shape of body and Reynolds number). Note that F p v 2, again independent of fluid density and area of body. As result, pressure drag always dominates viscous drag. Biomechanics of Locomotion though Fluids 3-12

13 Fully turbulent flow: - Boundary layer becomes turbulent. - Reduces the tendency of the boundary layer to separate from body. - Separation point moves forward. Sharp decrease in drag force, then continues to increase. - The size of the wake decreases. - Characteristic areas: S w = Wetted area (total surface area of body) (m 2 ) S f = Frontal (cross sectional area) (m 2 ). S p = Planar area (hydrofoils and aerofoils). Biomechanics of Locomotion though Fluids 3-13

14 Drag Coefficient Wind tunnel measurements can empirically determine drag forces and coefficients. Drag coefficient vs Reynolds number (bluff bodies): When the drag is dominated by viscous drag, we say the body is streamlined, and when it is dominated by pressure drag, we say the body is a bluff body. - C D approximately constant over a wide range of Re (i.e. velocity) (Re ) - Minimum value of C D at Re 2 x x Abrupt drop in drag at critical velocity (change to turbulent boundary layer makes wake smallerand reduces pressure drag. Net drag decreases at onset of turbulence. For Re = 10 6 and: l = 0.1 m; v = 10 m/s in water and 150 m/s in air. l = 1.0 m; v = 1 m/s in water and 15 m/s in air. Biomechanics of Locomotion though Fluids 3-14

15 Streamlined body: - Rounded at front, tapers gradually to point at back. - Low drag coefficient (C D 0.05) (mostly surface friction). - Designed to reduced turbulence in wake. Surface roughness: - Shifts transition of partially turbulent to fully turbulent to lower Reynolds number (lower velocity). - Lower drag, C D. - e.g. Fuzzy tennis balls, dimples on golf balls. Biomechanics of Locomotion though Fluids 3-15

16 Surface Friction - Surface friction contributes to total drag. - Significant for streamlined bodies and flat surfaces (i.e. hydrofoils and aerofoils). F friction 1 = ρswc f v 2 2 Where C f = Surface friction coefficient. S w = Wetted surface area (m 2 ). - For laminar flow in boundary layer (Re < 10 6 ): - For turbulent flow in boundary layer (Re > 10 6 ): C f C f = = / Re 1/ Re Bluff bodies: - At onset of the turbulent boundary layer, increased friction drag (But reduced wake, Reduced pressure drag). Biomechanics of Locomotion though Fluids 3-16

17 Hydrofoils and Aerofoils Hydrofoils / aerofoils: - Produce lift when moving through the water / air from the asymmetrical motion of body through fluid. Hydrodynamic forces: - Hydrodynamic force acts at angle to direction of motion of body. - Resolve force on body into components: Drag - Force acting backwards along direction of motion. F D 1 = ρs pcdv 2 2 Where C D = Drag coefficient. Lift - Force acting perpendicular to direction of motion. F L 1 = ρs pclv 2 2 Where C L = Lift coefficient. Biomechanics of Locomotion though Fluids 3-17

18 Lift and drag depend on angle of attack, α Drag - F D increases with increasing α - F D 0 at α = 0. Lift - F L 0 for α = 0. - F L reaches maximum, then decreases rapidly (stalling). Aspect ratio: - Aspect ratio, - For foils with same planar area ( S p = span.chord ), high aspect ratio give the same lift for less drag. Wind tunnel testing: A = span chord - Hydrofoils and aerofoils of same shape and angle of attack produce same lift at same Reynolds number. ( Re = ρvl / η, l is foil chord) Test scale model of aerofoil in wind tunnel with appropriate wind speed. Biomechanics of Locomotion though Fluids 3-18

19 3.3 BUOYANCY Animal density: - Swimming animals without special adaptations for buoyancy are more dense than water. Most fish ρ 1080 kg / m 3 Water ρ = 1000 kg / m 3 Seawater ρ = 1026 kg / m 3 Muscle ρ = 1060 kg / m 3 Bone ρ = 2000 kg / m 3 Dense animals avoid sinking by - Swimming upwards (e.g. plankton). - Swimming horizontally with fins at +ve angle of attack to produce lift. (must exceed minimum swimming speed to stay afloat) Biomechanics of Locomotion though Fluids 3-19

20 Low density animals: - Part of animal consists of low density material. Animal density water density. - Fat, blubber (ρ 930 kg / m 3 ) (e.g. seals, whales) - Wax esters (ρ 860 kg / m 3 ). - Low density body fluids (ion depleted fluids (e.g. deep sea squids). - Gas (ρ 0) - Air-filled lungs (swimming mammals and reptiles). - Swim bladder (many bony fish). - Gas-filled floats (cuttlefish, nautilus). Swimming animals: - Buoyancy of animal essentially negates effect of gravity. - Swim by - Undulation (fish, eels, spermatozoon) - Rowing (water beetle, duck). - Hydrofoils (dolphin, penguin). - Jet propulsion (squid). Biomechanics of Locomotion though Fluids 3-20

21 3.4 SWIMMING by UNDULATION Swim with wave-like motion. - Waves travel backwards along the body. - Pushes the organism forward. Biomechanics of Locomotion though Fluids 3-21

22 High Reynolds Number High Reynolds number (Re = ρvl / η >> 1) - Large and fast. - Movement through the water controlled by inertia (Neglect viscosity of surrounding water). - If the animal stops undulating, forward motion Examples - Fish, eels v 12 cm/s, l 15 cm η water 1 x 10-3 N s / m 2 ρ water 1 x 10 3 kg / m 3 Re 1.8 x 10 4 ( >> 1 ) Swimming (acceleration) - Fish of mass m accelerates to velocity v by driving mass of water M backwards at velocity V. - Fish required to do work W to accelerate (from rest). - Fish will attain a higher velocity for the same work if it pushes a larger mass of water backwards. 1 1 mv MV = 2 m mv M 2 W = + 2 Biomechanics of Locomotion though Fluids 3-22

23 Swimming (constant velocity): - Tail mostly moves water sideways (transversely) (The net momentum of the water in the transverse direction = 0). - Fish with tail area A tail swimming at velocity v swim - Water is given by sideways velocity v water Power output of tail: P = F tail v tail Where F tail = p water t Since p water = momentum of water. F tail = F = tail A M tail water ρv v swim water t v water P = ( A tail ρv swim v water )( v tail ) Biomechanics of Locomotion though Fluids 3-23

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25 Metabolic Energy Cost Metabolic energy cost of transport (swimming) - Determined from oxygen consumption (subsurface swimming of fish). Energy cost of transport ( J / kg.m ) depends on - Swimming speed ( v swim ) - Body size (The cost of transport decreases with increasing mass). - Environmental temperature. (cost of transport is lower at higher temperatures). Net cost of transport decreases with increasing size. Biomechanics of Locomotion though Fluids 3-25

26 Cost of transport versus mass Power vs speed Biomechanics of Locomotion though Fluids 3-26

27 Low Reynolds Number Low Reynolds number (Re = ρvl / η < 1) - Small and slow. - Movement through the water controlled by the viscosity of the fluid η (Neglect inertia of surrounding water). - If the animal stops undulating forward motion stops (almost) instantaneously. Examples: - Spermatozoon - Flagellates v 100 µm/s, l 60 µm Wave velocity 12 µm / s η water 1 x 10-3 N s / m 2 20 body length / s ρ water 1 x 10 3 kg / m 3 Re ( < 1 ). Wave frequency 50 Hz Biomechanics of Locomotion though Fluids 3-27

28 Forces generated during undulation of tail: - Viscous drag force on sperm tail (pressure drag is negligible) F v = κlηv - Model sperm tail as a cylindrical rod. for Re < 1 Power required for swimming with a flagellum: - Consider only work done to overcome drag on tail. - Longitudinal motion of flagellum: P = F drag v = κ a ηlν 2 ηlv 2 (κ a 1 for a long slender rod). - Side to side motion of the flagellum: side to side velocity >> forward velocity P 50ηlν 2 >> longitudinal motion power. Biomechanics of Locomotion though Fluids 3-28

29 3.5 ROWING Rowing underwater or on the surface of the water. - Use drag on oars to provide forward thrust. Water beetle has middle and hind legs with hinged hair-like bristles. - Spread for the power stroke. - Trail for the recovery stroke. Biomechanics of Locomotion though Fluids 3-29

30 Mechanics of rowing a boat on surface of the water: - Oars used to drive mass of water backwards. - Produces a wake of forward moving water behind boat (Water in the boundary layer is dragged along by the hull). - KE left behind in wake supplied by work of rowing. - Streamlined hull is designed to reduce wake. Rowing at constant velocity: rate of transmission rate of transmission of momentum = of forward momentum to water by oars to water in wake by hull - Oars with large blades are the most efficient. - Less power is used to accelerate a large mass of water at low speed than accelerate a small mass of water at high speed. Biomechanics of Locomotion though Fluids 3-30

31 Surface Waves Surfaces waves (bow and stern waves): - Water is given PE when raised in a wave since energy must be imparted from an external force, i.e. muscles. - Additional drag on body (wave drag). - Limits the speed of swimming on the surface of the water. Gravity is important in dynamics of water waves. - For dynamically similar wave patterns, body must travel with equal Froude number (Fr = ρvl / η ), where l = hull length). Power required becomes large when Fr 0.16, i.e. at v = 0. 16gl For duck: - Webbed feet are spread during power stroke when rowing. - Hull length l = 0.33 m. - Maximum speed of swimming (Fr 0.16), v max = 0.7 m / s. Biomechanics of Locomotion though Fluids 3-31

32 3.6 HYDROFOILS Types of hydrofoils - Wings (penguins). - Flippers (turtles). - Flukes (whales, dolphins). - Tails (tuna). Use lift on hydrofoil to generate thrust. - Mainly large and fast animals (Re >> 1). Example: Penguin (swims by beating its wings). Up / down component cancels over one complete cycle. COG moves forward only. Upstroke different sign of angle of attack to flying. - Not required to generate vertical force upstroke downstroke to overcome gravity. -ve angle of attack +ve angle of attack R forward and down R forward and up - Only require forward force for propulsion. Biomechanics of Locomotion though Fluids 3-32

33 Hydrofoils: - Greatest lift (for same drag and area) for high aspect ratio ( A = span / chord ). Long and narrow tails. - Some fish use tails as vertical hydrofoils. - Most fish use a combination of undulation and hydrofoil motion for locomotion. - Dolphins and whales use tails as horizontal hydrofoils. Biomechanics of Locomotion though Fluids 3-33

34 Leaping dolphins and penguins: - Less drag in air than water. - Avoids the high drag at surface (bow wave) when breathing. Biomechanics of Locomotion though Fluids 3-34

35 3.7 JET PROPULSION Locomotion is performed by squirting water out of a cavity in their body. - Not a steady velocity (series of jerks). Examples - Squids - Jellyfish - Scallops (Open and close their hinged shells). Biomechanics of Locomotion though Fluids 3-35