Transient Heat Transfer Effects from a Flapping Wing

Behrouz AbedianRobert Lind Tuesday, October 25, 2005

Azuma, p26

Presented at the COMSOL Multiphysics User's Conference 2005 Boston

Numerical Experimentation with a 2D Model

Subject: Warm wing flapping in cold air flying forward with speed U.

Purpose: Investigate heat transfer effects using:

The trailing temperature field.The rate of heat transfer.

Results to Date: Established the Numerical Model of a Flapping Wing

We demonstrate the heat transfer from a flapping wing Vary the wings flapping frequency, , and its forward speed, U: Generate various flow regimes and heat flows. Examples of temperature fields from two trial runs:

Future: Quantitative assessment of the heat transfer.

(2, 3200) (1, 1100)

Research Inspiration: Birds, Insects and FishFlapping wings and fanning tails

Wings for lift & thrust Tails for thrust only

Alexander

Azuma, p194

Flapping Wings: Focus on Birds

Symmetrical flapping: produces no net thrustDownstroke forward thrust.Upstroke reverse thrust.Exactly cancels !

Azuma, p28

Biomechanical Research Parameters

ULK =Reduced frequency:

Flapping wing parameters:

Reynolds frequency:

Length of wing, LAir viscosity Flapping angular frequency, Wing forward speed, U

2Re Lf =

Angular speedForward speed

Inertia forcesViscous forces

kUL fReRe ==

U

L

Dimensionless numbers used to scale and organize the wing motion

Dimensionless Numbers Categorize the Wing Motion of Flying Creatures

Parameters Small Wasp Locust Pigeon

Wing Length, L 0.0006 m 0.04 m 0.25 m

Angular frequency, 400 rads/s 20 rads/s 5 rads/s

Forward velocity, U 1 m/s 4 m/s 5 m/s

Reduced frequencyk= L/U 0.24 0.20 0.25

Reynolds frequencyRef= L2/ 10 2,300 22,500

Biomechanical dimensionless numbers are used to determine the wing angular frequencies, , and forward speeds U, in the numerical mode.

Model the Flapping Wing with Femlab

Femlabs multiphysics package:Fluid Dynamics Module: Incompressible Transient Analysis

Navier-Stokes equation: momentumHeat Transfer Module: Convection & Conduction Transient Analysis

Energy Equation for temperature field

Model assumptionsWing: Rigid wing, L=0.1m

Sinusoidal motion, (t)=(/2)sin(t)Forward speed, U=0.4Temperature, T = 275 K

Air: IncompressibleTemperature, T = 255 KViscosity, = 13.91e-6 m/s2

Flow: Laminar flow only

(t)=(/2)sin(t)

0.1m

Fluid Dynamics: Develop the Flapping Wing Motion Equations

jtvitudttdP )()()( +=

s ( ) ( )( ) ( )( )[ ]jtitLtP sincos +=( ) ( ) ( )tst sin2=

( )20 sLA =

( ) ( ) ( )[ ] ( )ttsAtu cossin2sin0 =( ) ( ) ( )[ ] ( )ttsAtv cossin2cos0 =

Velocity components of the flapping wing for the Navier-Stokes equation.

u(t) Horzontal velocity v(t) verticle velocity

Fluid Dynamics: Modify the Upstroke Horizontal Velocity

( ) ( ) ( )[ ] ( )ttsAtu cossin2sin0 =

Solution: Apply an on/off function to the horizontal velocity. A 7-term Fourier Series cancels the upstroke horizontal velocity.

Problem: Cancel the reverse thrust:

Fluid Dynamics: Apply the Motion Equations The wing is fixed in place

To simulate wing motion:The Equations of motion are applied to the fluid, at wing surface: Horizontal velocity, u(t) and Vertical velocity, v(t)

L

v(t)

u(t)

Heat Transfer: Apply the Energy Equation

+

=yTv

xTucQ p

Issue: Heat is convected from the wing (275 K) to the air (255 K) and is swept away downstream ~ a trailing temperature field.

Density, 1.265 kg/m3

Kinematic viscosity, 13.91e-6 m/s2

Specific heat, cp 1008 J/kg-K

Thermal conductivity, k 0.0255 W/m-K

Approach: 1st solve the Navier-Stokes fluid flow solution in the model.2nd apply the 2D heat conduction equation, (Convection is not specified)

Thermal properties of air at 255 K and atmospheric pressure.

Summary: The Model Geometry and Boundary Conditions

1.0m

0.5m

Fixed line as wingLength, L=0.1m, Temperature, T=275 K

Bulk fluid: AirVelocity, U= L/kTemperature, T=255 K

Outlet Pressure, p=0

Solving the model: 1st solve the fluid field. 2nd solve the heat field, using the flow field results.

The completed multiphysics model is ready for solving(Fluid dynamic subdomain and heat transfer subdomain)

Trial Runs: Numerical Experiments

K U

1.5 0

2 1.5 0.075

1 1.5 0.150

0.25 1.5 0.600

K U

4.5 0

2 4.5 0.225

1 4.5 0.450

0.25 * 3.1 1.240

Heat is transferred for various flow regimes.Vary flapping frequency, , and forward speed U,Trials organized by dimensionless parameters K and Ref

Dimensionless Parameters:k=L/U Ref= L2/

Results of Trial Set 2: Ref = 3200. K = , 2, 1, 0.25Videos of solutions, 12 seconds with output 0.05 seconds per frame.

Trial Set 1: Ref = 1100 Trial Set 2: Ref = 3200

* Ref = 2200 maximum attained

Biological Observations Can be Quantified with Modeling

Observed in 3D

Modeled in 2D

Alexander

(1, 3200)

Trailing Flow of a slow gait:

The Next Steps

Ongoing Research: Analyze the wing surface data: Viscous drag on wing: Skin friction No, Cf = /(1/2)U2L Heat convection from wing: Nusselt No, Nu = qL/k(Tw-Ta)

Correlate with: Reduced frequency k = L/UReynolds frequency Ref = L2/

(2, 1100) (1, 1100) (0.25, 1100)

Trials produced ranges of heat transference and temperature fields

(, 1100)

Key Issues for Ongoing Research

Post processing data acquisitionIntegrate data along length of wing, L, and over one cycle, T. Viscous forces, , and heat convection, Q.

Wing was modeled as a single lineThe net heat flux on top and bottom surfaces is combined.We need to separate the top and bottom surface heat fluxes.

Model a compliant wingFemlabs fluid structure interaction technique

Citations

1. Alexander, David E. Natures flyers: Birds, insects, and the biomechanics of flight. The John Hopkins University Press. (2002).

2. Azuma, Akira. The Biokinetics of flying and swimming. Springer-Verlag, Tokyo. (1994).

3. Comsols Femlab Multiphysics.Tufts School of Engineering.

Transient Heat Transfer Effects from a Flapping WingNumerical Experimentation with a 2D ModelResults to Date: Established the Numerical Model of a Flapping WingResearch Inspiration: Birds, Insects and FishFlapping Wings: Focus on BirdsBiomechanical Research ParametersDimensionless Numbers Categorize the Wing Motion of Flying Creatures Model the Flapping Wing with FemlabFluid Dynamics: Develop the Flapping Wing Motion EquationsFluid Dynamics: Modify the Upstroke Horizontal VelocityFluid Dynamics: Apply the Motion EquationsHeat Transfer: Apply the Energy EquationSummary: The Model Geometry and Boundary ConditionsTrial Runs: Numerical ExperimentsBiological Observations Can be Quantified with ModelingThe Next StepsKey Issues for Ongoing ResearchCitations