Drag-reduction for Marine Vehicles: Learning from the Dolphin?

by: A(Tony).D. Lucey

Dedicated to, and acknowledging the work of, Professor P.W. Carpenter (R.I.P April 2008)

. Joint Technical Session of the Mechanical Panel of Engineers Australia, WA, The Institution of Mechanical Engineers, and American Society of Mechanical Engineers. 26th November, 2008, Perth, WA

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Contents

1. Gray’s paradox (1936)

2. Laminar and turbulent boundary layers

3. Kramer’s pioneering experiments (1957, 1960)

4. Theoretical predictions of transition delay

5. Theory verified - the Gaster Experiments (1987)

6. Hydro-elastic instabilities of compliant coatings

7. Design of artificial dolphin skins

Technical conclusions

8. Gray’s paradox re-assessd…

9. What have we learned from the dolphin?

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1. GRAY’S (1936) PARADOX

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The Paradox

Following contemporary naval engineering practice, Gray (1936) modelled the dolphin body as a flat plate

– skin friction only and no dynamic effects –to estimate drag assuming transition occurred at

Rex = 2 x 106

POWER = DRAG x SWIMMING SPEED

He found that to swim at 10 m/s the specific muscle power output required was

7 x mammalian norm ( of 40 W/kg)

Gray proposed that “if the flow is free from turbulence… power agrees closely…” - i.e. dolphins maintain laminar flow over their entire length

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2. Laminar and turbulent boundary layersBoundary layers over a flat plate (Van Dyke 1990)

Laminar profile – low friction

Turbulent profile – high friction

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Boundary-layer transition from laminar to turbulent – 2D, low disturbance environment – ‘natural transition’

Amplifying Tollmien-Schlichting wave

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Tollmien-Schlichting waves in natural transition

Schubauer & Skramstadt (1947)

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Hypothesis for dolphin’s maintenance of laminar (boundary- layer) flow… hence skin-friction reduction

Dolphin’s skin is able to ‘damp out’ Tollmien-Schlichting waves and thereby

postpone transition

(a) Longitudinal cross-section; (b) horizontal section through AA’;(c) Lateral cross-section.

Key: a, cutaneous ridges (or microscales); b, dermal papillae; c, dermal ridge; d, upper epidermal layer; e, fatty tissue.

Carpenter, Davies & Lucey (2000)

Structure of dolphin’s epidermis

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3. Kramer’s pioneering experiments (1957, 1960) Sea-based towing tests of slender body with a compliant

coating

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Kramer’s design of ‘artificial dolphin skin’

Up to 60 % drag reduction at 18 m/s.

Kramer believed that damping fluid eliminated Tollmien-Schlichting waves… not true!

But… laminar-flow properties were confirmed theoretically Carpenter & Garrad (1985, 1986), Lucey & Carpenter (1995)

c.f. dolphin’s epidermis

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Experimental attempts to emulate Kramer’s results

The 1960’s saw a flurry of ill-fated experiments that, overall, seemed to demonstrate that compliant coatings increased drag.

e.g. Puryear (1962) – ridge-formation on coated test specimen

By 1970, Bushnell (NASA) effectively concluded that compliant-coating was ill-founded as a technology.

However, what had been lacking was a proper theoretical foundation for the design of experiments

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4. Theoretical predictions of transition delay

Compliant-wall models

Fluid-structure interaction: Solve flow equations and wall equations concurrently linked by interfacial conditions

Flexible-plate plus spring foundation – simple one-dimensional (surface –

based) model

Single and two-layer (visco-) elastic slab(s) – two-dimensional (volume–

based) model

(From Carpenter 1991)

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Wave frequency

Reynolds number

damped amplifying (inside loop)

Compliant wall

Results for:

Rigid

Compliant

More Compliant

1ω

Range of T-S wave amplification

flow

Schematic stability diagram for 2D Tollmien-Schlichting waves in a boundary layer over rigid and compliant walls

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Using classical hydrodynamic stability approach (Orr-Sommerfeld equation) – suppression of T-S waves theoretically possible – Benjamin (1960), Landahl (1962), Carpenter & Garrad (1985), Lucey & Carpenter (1995) – hence transition delay

Attenuation occurs because wall compliance disrupts the energy- production mechanism of the growth of Tollmien-Schlichting waves

A sufficiently compliant wall can eliminate Tollmien-Schlichting waves entirely!

Structural damping in the wall undermines the beneficial effects of wall compliance – hence, is destabilising.

Summary: effect of compliant coatings on Tollmien-Schlichting waves

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5. Theory verified - the Gaster Experiments (1987)

Very careful (3 year program) experiments in a towing tank at National Physical Laboratories, UK

Measured the growth in amplitude of excited Tollmien-Schlichting waves at the leading edge of a compliant panel

[Gaster’s (1987) paper entitled “ Is the dolphin a red herring?”]

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Rigid-wall results

Compliant-wall results

Experiment Theory

Experiment: compliant

Experiment: more compliant

Equivalent scale

Theoretical prediction of wall-based instability (Lucey & Carpenter 1995)

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Principal outcomes of the Gaster experiments

1. Wall compliance does attenuate Tollmien-Schlichting waves as predicted by theory – hence transition delay is possible.

2. The softer the wall, the greater the effect…

3. But… if wall is too soft, a different instability – Travelling-Wave Flutter (TWF) – sets in and this triggers premature transition

Neutral stability loops – waves of given frequency are unstable within each

loop

(Lucey & Carpenter 1995)

TWF Critical Reynolds number

TWF Critical Reynolds

number with wall

damping

Reynolds number or distance from leading edge

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6. Hydro-elastic instabilities of compliant coatings:

6.1 Travelling-wave flutter (TWF)

Numerical simulations of boundary- layer flow over plate-spring type compliant wall using (grid-free)

discrete-vortex method and boundary-element method for flow

solution (Pitman & Lucey 2004)

Increasing time

Initial condition

Key point: downstream propagating wave amplifies,

upstream attenuates

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6.2 Divergence instability

View from above compliant panel

Flow

Nonlinear divergence waves appear as quasi-two-dimensional ridges – with very slow downstream travel – Gad-el-Hak, Blackwelder & Riley (1985)

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Prediction of divergence from linear instability to

saturated nonlinear waves

Numerical simulations of potential flow over plate-

spring type compliant wall using boundary-element method for flow solution

(Lucey et al. 1997, Pitman 2007)

Initial condition: yery low amplitude deformation applied

at panel mid-point

Increasing time

Flow

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Three dimensional simulation of divergence linear instability – plate-spring wall. (Lucey 1998)

Flow

At time T At time 2T

At time 3T At time 4T

Note: emergence of quasi two-dimensional unstable waves from a three-dimensional form of initial excitation

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Comparison of TWF and divergence instabilitiesIn

crea

sing

tim

e

TWF – amplifies only as wave travels downstream of

source of excitation – convective instability

Divergence – downstream travelling wave but amplifies

both upstream and downstream of source of excitation –

absolute instability

Flow

(Carpenter, Lucey & Davies 2001)

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Summary of main instabilities in flow over compliant walls

Instability Effect of wall compliance

Effect of structural

damping in wall on onset

Wave character

Instability type Prediction methods

Tollmien- Schlichting waves

Stabilising DestabilisingModest downstream travelling

ConvectiveNegative energy wave (Class A)

Orr-Sommerfeld Eqn.Tailored spectral methods

Travelling- wave flutter Destabilising Stabilising

Fast downstream travelling

ConvectivePositive energy wave (Class B)

Rayleigh Eqn. Asymptotic methodsTailored spectral methodsNumerical simluation

Divergence Destabilising No effect

Static at onset – slow downstream after onset

AbsoluteK-H type (Class C)

Laplace Eqn.Special numerical methods; boundary- element, discrete- vortex

Plus others – e.g. coalescence of T-S waves and TWF – and nonlinear effects.

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7. Design of artificial dolphin skinsOverall strategy: Make wall sufficiently flexible to maximise suppression of Tollmien-Schlichting waves…. But not succumb to hydroelastic instabilities

Optimise!

5 key parameters:

Lower-layer Elastic modulus

Lower-layer thickness

Lower-layer damping coefficient

Upper-layer flexural rigidity

Upper-layer damping coefficient

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Results of optimisation…

Transition length can be extended by a factor of 5.7

(Carpenter & Morris (1990), Dixon, Lucey & Carpenter (1994) and others since)

All such optimisations suggest for compliant coatings optimized for transition delay:

Surface-wave speed = 0.7 x Flow speed

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Illustrative cases of skin- friction drag reduction

Based on slender body theory

1. Length 6 m, speed 36 m/s – 1% drag reduction

2. Length 2 m, speed 1.54 m/s – 76% drag reduction

(From: Klinge, Lucey & Carpenter 2000)

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Illustrative cases of skin- friction drag reduction…

…continued

1. Length 2.6 m, speed 23.1 m/s – 5% (14%)* drag reduction

2. Length 7 m, speed 2.6 m/s – 17% (25%)* drag reduction

Based on slender body theory

(From: Klinge, Lucey & Carpenter 2000)

*Note: Some account included here for beneficial effect of compliance on the

turbulent boundary layer

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Turbulent skin-friction reduction using compliant coatings

Choi et al. (1997) conducted experiments that demonstrated up to a 6% decrease in drag for turbulent boundary-layer over a

compliant coating.

However, elastic modulus of coating was much higher

(although thickness similar) than that required for

maintaining laminar flow

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The way forward: multiple compliant panels

Evolution of Tollmien-Schlichting waves over a rigid-complaint-rigid surface

Carpenter, Lucey & Davies (2001) based on Davies & Carpenter (1997)

Attenuation of Tollmien- Schlichting waves possible for very short compliant panels… that are much less susceptible to hydro- elastic instability

Design Strategy: use streamwise arrays of compliant panels with properties optimised to local Reynolds number

Indefinite postponement of

transition possible?

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Conclusions… so far

• Demonstrated, both experimentally and theoretically, that compliant coatings are capable of extending laminar flow… hence skin-friction drag reduction.

• Design of effective compliant coatings requires a sound knowledge of the diversity of waves that can exist within the flow-structure system.

• There is the prospect of indefinitely postponing transition by using arrays of compliant panels each of which is optimised for local flow conditions.

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8. Gray’s paradox reassessed

Major assumptions in Gray’s modelling

Effects of dolphin’s swimming motions neglected (assumed a rigid body)

Laminar-to-turbulent transition of boundary layer occurs at Re= 2 x 10^6 (flat plate value)

Swimming speed = 10 m/s

Mammalian muscle power = 40 W/kg

How good are these assumptions?

“…specific muscle power output required was 7 x mammalian norm…”

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Is drag increased by the unsteady flow associated with propulsion?

Lighthill (1971) suggested a 4- to 5-fold rise in drag due to boundary-layer thinning in the unsteady flow… but this neglects effect of re-laminarization due to flow acceleration.

Anderson et al. (2001) observed the predicted BL thinning on fish but found that the drag was only 1.5 times the rigid-body value.

Barrett et al. (1999) actually observed a 50 % drag reduction on their swimming robot tuna.

Hence… probably OK to assume there is no drag rise for an actively swimming dolphin.

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Streamwise pressure distribution exists along dolphin body due to its

shape – favourable pressure gradient reduces T-S wave growth.

consistent with observations by Romanenko (2002) on dolphins.

Hence:

From Aleev (1977)

Ret = 0.5 x ReL

What is the appropriate transitional Reynolds number?

i.e. about 10,000,000 for a 2 m long dolphin swimming at 10 m/s

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What is the swimming speed of a dolphin?

Most reliable aircraft-based observations of a school of common dolphins.Grushanskaya & Korotkin (1973) measured ultimate speed of c. 11 m/s.

But typical sustained speed in the range 3.5 to 5 m/s

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Power output of muscles?

Muscle performance depends on type of fibre

• Slow oxidative fibresMainly aerobic metabolismSlow sustained activityRelatively slow contraction rates

• Fast glycolic fibresMainly anaerobic metabolismShort-burst activityHigh power outputVery high intrinsic contraction speedsPower output is 2 to 17 times that of slow fibres

Dolphin muscle has both types of fibreSustained aerobic output = 40 W/kg (Parry 1949)

Short-duration anaerobic output = 110 W/kg(Weis-Fogh & Alexander 1977)

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So… new estimates of swimming speed in m/s

Case CD Sustained Power output (40W/kg)

Maximum 110 W/kg

Ret = 2 ×106

(Gray)

0.0025

5.6

7.9

Ret = 13.75×106

0.0015

6.6

9.3

Laminar 0.00025 12.08 16.9

Gliding dolphin 0.0023* (0.0015) - -

*based on body area only.

Thus there is no paradox!Hence… no need to invoke special laminar flow properties of skin!

(from Babenko & Carpenter 2003)

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So… does the dolphin have a need for laminar-flow control?

Most scientists have followed Gray and focused on maximum sustained swimming speed.

Laminar flow needed to reach 10 m/s but not to reach commonly observed speeds. In any case the ‘porpoising’ swimming mode is used at high speeds….

Laminar-flow control more likely required for conserving energy during:Slower long-duration swimming‘Gliding’ during deep diving

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Energy efficient deep diving

From Williams et al. (2000)

Red: gliding; Black: powered swimming

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Evidence that the dolphin does have a laminar-flow control capability

In vivo measurement of dolphin-skin properties Madigosky et al. (1986)

Optimised double-layer ‘artificial dolphin skins’ have a free-wave speed of 6.5 m/s for a design flow speed of 9 m/s.

Results imply that dolphin skin optimized for ca. 9 m/s

Locations of surface wave-speed

measurements

Grey circles: 6-7 m/s; Open circles: no measurement.

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Skin folding – hydro-elastic instability (divergence) at high swimming speeds

From Essapian (1955)

Optimised double-layer ‘artificial dolphin skins’ experience divergence instability with a wavelength 40 mm at 9 m/s.

Wavelength of divergence on dolphin measured at 35-40 mm

This suggests that dolphin skin is an optimized compliant coating (that fails beyond its design flow speed)

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Cutaneous ridges - aligned normal to the streamwise direction

from Ridgway & Carder (1993)

Oblique Tollmien-Schlichting waves grow fastest over compliant walls.

Numerical simulation of Ali & Carpenter (2002) show that their growth rate is reduced by c. 25 % when cutaneous ridges are present.

Implies that dolphin skin has evolved for laminar-flow control

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Note that dolphins also have dermal ridges Lateral spacing is ca. 10-15 mm.

Therefore not adapted through evolution into riblets for turbulent flow.

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And… Other direct evidence of laminar flow over dolphin body

Hot-film & pressure-sensor measurementsKozlov et al. (1974), Pyatetskii et al. (1982)Romanenko (1986)

Overall conclusion is that fluctuation level in boundary layer over dolphin is much lower than comparable rigid body.

Can conclude that while Gray’s paradox has probably been resolved, there is much evidence that dolphin skin does possess the capability for

laminar-flow control

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9. What have we learned from the dolphin?

• How to design a technology capable of significant transition delay and thus reduced skin-friction drag… with the prospect of indefinite postponement of laminar-to-turbulent transition

• Vastly increased scientific knowledge of the physics of fluid-structure interaction… with applications in, for example:

Biomechanics – blood flow, respiration…Structural acoustics (in the presence of mean flow)Industry – sails, parachutes, convertible car roofs…Fluid-energy harvesting devices (where flow-induced instability can be exploited)

• New theoretical and computational techniques that have a multiplicity of applications.

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Acknowledgments to: Prof Peter Carpenter, Dr Mark Pitman, A/Prof Chris Davies, Dr Gerard Cafolla and the many others with whom I

have had the pleasure to work on compliant coatings