(3) BIOMECHANICS of LOCOMOTION through FLUIDS …€¦ · Bernoulli equation: The conservation of...

(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

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 / m3 )

Water ρ = 1000 kg / m3

Seawater ρ = 1026 kg / m3 (Due to dissolved minerals)Air ρ = 1.21 kg / m3 (At sea level and 20°C)

Pressure in fluid

- Force per unit area ( N / m2 or Pa )

Hydrostatic pressure:

- Increase with depth in fluid (The fluid must support its own weight)

Where p0 = Pressure at surface (Atmospheric pressure) .h = Depth (m).

- Pressure in water increases (1 atmospere per 10.3m increase in depth).

Vm

ρ =

AF

p =

ghρpp += 0


Atmospheric pressure:

- Atmospheric pressure decreases with altitude.

Where y = Altitude (m)pa = Pressure at sea level (1.013 x 105 N / m2) .b = Constant (1.16 x 10-4 m-1) .

byaepp -=


Buoyant force:

- A body partially or fully immersed in a fluidexperiences 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).


Small A Large AHigh v Low v

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 (m2).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.

- If the fluid is compressible ( ρ ≠ constant)

ρ Av = constant


Bernoulli equation:

The conservation of mass and energy applied to a fluid.

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 inkinetic energy (PE) and potential energy (PE)per unit volume.

- The sum of the pressure ( p ), the kinetic energyper unit volume ( ρv2 / 2 ) and the andgravitational potential energy per unit volume( ρgy ) as the same value at all points alongstreamline.

- The fluid in the section ( ∆x1 ) moves to the section of length ( ∆x2 ). The volume of fluid in the two sections are equal.

constant=+21

+ 2 gyρvρp


Bernoulli principle:

- In horizontal flow ( y = constant )

- Pressure is less where velocity is high.

- Pressure of an incompressible fluid can be measured with Venturi tubes.

constant=21

+∴ 2vρp


Flow around an asymmetrical body:

- The Bernoulli effect states that the increased velocity of the air above the wingcompared 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:

Where η = Coefficient of viscosity (N s / m2,Temperature dependent).

Air (20°C) η = 1.82 x 10-5 N s / m2

Water (20°C) η = 1.00 x 10-3 N s / m2Biomechanics of Locomotion though Fluids 3-8

dAvη

Fviscous =

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).


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 / m3).η = Coefficient of viscosity of fluid (N s / m2).

Flow type depends on Reynolds number:

Re < 1 Laminar flow (viscous drag only).

Re ≈ 1 Transition to partially turbulent flow.

1 < Re < 103 Turbulent wake grows (Viscous and pressure drag).

103 < Re < 106 Turbulent wake grows (Pressure drag dominates).

Re ≈ 106 Transition to fully turbulent flow (pressure drag decreases).

Re > 106 Fully turbulent flow (pressure drag dominates).


ηvlρ

=Re

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.

Where v = Relative velocity of body and fluid (m / s).l = Characteristic length (m).η = Coefficient of viscosity of fluid (N s / m2)

(Measure of resistance of fluid to flow).κ = Constant, depends on shape of body.

(κ = 3π for sphere).

Note that Fv ∝ v, independent of fluid density and area of body.

vηlκFv =


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

Where ρ = Density of fluid (kg / m3).S = Characteristic area (m2).CD = Drag coefficient (Depends on shape of

body and Reynolds number).

Note that Fp∝ v2, again independent of fluid density and area of body.

As result, pressure drag always dominates viscous drag.

2

21

= vSCρF Dp


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:

Sw = Wetted area (total surface area of body) (m2)Sf = Frontal (cross sectional area) (m2).Sp = Planar area (hydrofoils and aerofoils).


Drag Coefficient

Wind tunnel measurements can empirically determine drag forces and coefficients.

Drag coefficient vs Reynolds number (bluffbodies):

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 isa bluff body.

- CD approximately constant over a wide range of Re (i.e. velocity) (Re ≈ 103 → 105)

- Minimum value of CD atRe ≈ 2 x 105 - 2 x 106.

- Abrupt drop in drag at critical velocity(change to turbulent boundary layermakes wake smallerand reduces pressure drag.

∴ Net drag decreases at onset ofturbulence.

For Re = 106 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.


Streamlined body:

- Rounded at front, tapers gradually to point at back.

- Low drag coefficient (CD ≈ 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, CD.

- e.g. Fuzzy tennis balls, dimples on golf balls.


Surface Friction

- Surface friction contributes to total drag.

- Significant for streamlined bodies and flat surfaces (i.e. hydrofoils and aerofoils).

Where Cf = Surface friction coefficient.

Sw = Wetted surface area (m2).

- For laminar flow in boundary layer (Re < 106):

- For turbulent flow in boundary layer (Re > 106):

Bluff bodies:- At onset of the turbulent

boundary layer, increased friction drag (But reduced wake, ∴ Reduced pressure drag).

2/1Re33.1

=fC

5/1Re075.0

=fC

2

21

= vCSρF fwfriction


Hydrofoils and Aerofoils

Hydrofoils / aerofoils:

- Produce lift when moving through the water /air from the asymmetrical motion of bodythrough fluid.

Hydrodynamic forces:

- Hydrodynamic force acts at angle todirection of motion of body.

- Resolve force on body into components:

Drag - Force acting backwards along direction of motion.

Where CD = Drag coefficient.

Lift - Force acting perpendicular todirection of motion.

Where CL = Lift coefficient.

2

21

= vCSρF DpD

2

21

= vCSρF LpL


Lift and drag depend on angle of attack, α

Drag - FD increases with increasing α- FD ≠ 0 at α = 0°.

Lift - FL ≈ 0 for α = 0°.- FL reaches maximum, then

decreases rapidly (stalling).

Aspect ratio:

- Aspect ratio,

- For foils with same planar area( Sp = span.chord ), high aspectratio give the same lift for less drag.

Wind tunnel testing:

- Hydrofoils and aerofoils of same shape and angle of attack produce same lift at ( Re = ρvl / η, l is foil chord)same Reynolds number.

∴ Test scale model of aerofoil in wind tunnelwith appropriate wind speed.

chordspan

=A


3.3 BUOYANCY

Animal density:

- Swimming animals without special adaptations for buoyancy are more dense than water.

Most fish ρ ≈ 1080 kg / m3

Water ρ = 1000 kg / m3

Seawater ρ = 1026 kg / m3

Muscle ρ = 1060 kg / m3

Bone ρ = 2000 kg / m3

Dense animals avoid sinking by

- Swimming upwards (e.g. plankton).

- Swimming horizontally with fins at +veangle of attack to produce lift.

(must exceed minimum swimming speed to stay afloat)


Low density animals:

- Part of animal consists of low density material.∴ Animal density ≈ water density.

- Fat, blubber (ρ ≈ 930 kg / m3)(e.g. seals, whales)

- Wax esters (ρ ≈ 860 kg / m3).- 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).


3.4 SWIMMING by UNDULATION

Swim with wave-like motion.

- Waves travel backwards along the body.

- Pushes the organism forward.


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 cmηwater ≈ 1 x 10-3 N s / m2

ρwater ≈ 1 x 103 kg / m3 ∴ Re ≈ 1.8 x 104 ( >> 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 ifit pushes a larger mass of water backwards.

22

21

21 MVmvW +=

⎟⎠⎞

⎜⎝⎛ +=

Mmmv 1

21 2


Swimming (constant velocity):

- Tail mostly moves water sideways(transversely) (The net momentum ofthe water in the transverse direction = 0).

- Fish with tail area Atail swimming at velocity vswim

- Water is given by sideways velocityvwater

Power output of tail:

P = Ftailvtail

Where

Since pwater = momentum of water.


tp

F watertail =

))((=∴ tailwaterswimtail vvvρAP

waterswimtailtail vvρAF =∴t

vMF water

watertail =

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 ( vswim )

- 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.



Power vs speed

Cost of transport versus mass

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 / sηwater ≈ 1 x 10-3 N s / m2 ≈ 20 body length / s ρwater ≈ 1 x 103 kg / m3 ∴ Re ≈ 0.006 ( < 1 ). Wave frequency ≈ 50 Hz


Forces generated during undulation of tail:

- Viscous drag force on sperm tail (pressure drag is negligible)

for Re < 1

- Model sperm tail as a cylindrical rod.

Power required for swimming with a flagellum:

- Consider only work done to overcome drag on tail.

- Longitudinal motion of flagellum:

P = Fdragv= κaηlν2 ≈ ηlv2 (κa ≈1 for a long slender rod).

- Side to side motion of the flagellum:

side to side velocity >> forward velocity

P ≈ 50ηlν2 >> longitudinal motion power.

vηlκFv =


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.


Mechanics of rowing a boat on surface of thewater:

- Oars used to drive mass of waterbackwards.

- Produces a wake of forward moving waterbehind boat (Water in the boundary layer isdragged along by the hull).

- KE left behind in wake supplied by work ofrowing.

- Streamlined hull is designed to reducewake.

Rowing at constant velocity:

rate of transmission rate of transmissionof 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.



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

For duck:

- Webbed feet are spread during power stroke when rowing.

- Hull length l = 0.33 m.- Maximum speed of swimming (Fr ≈ 0.16), vmax= 0.7 m / s.

glv 16.0=

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 itswings).

Up / down component cancels over onecomplete cycle.

∴ COG moves forward only.

Upstroke different sign of angle of attack toflying.

- Not required to generate vertical force upstroke downstroketo overcome gravity. -ve angle of attack +ve angle of attack

∴R forward and down ∴R forward and up- Only require forward force for

propulsion.


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.


Leaping dolphins and penguins:

- Less drag in air than water.

- Avoids the high drag at surface (bow wave) when breathing.


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).


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