Banks, Joseph

Hudson, Dominic A.

Phillips, Alexander B.

Turnock, Stephen R.

English

Southampton (e-Prints Soton)

Performing self propelled simulations of a kayak, using a Body-force paddle model  J. Banks*, A.B. Phillips, S.R. Turnock, D.A. Hudson *jb105@soton.ac.uk  1   Introduction  Understanding  the  interactions  between  a  hull and  a  propeller  is  a  common  problem  naval architects  have  to  solve  to  provide  efficient powering  solutions.  The world  of  elite  sport is becoming  increasingly  scientific  in  a  similar drive  for  increased  performance.  To  allow  a kayak’s  hull  and  paddle  to  be  optimised,  their interactions have to be considered to provide a realistic  race  scenario.  It  is  therefore  proposed that numerical techniques currently used by naval architects could be applied to the problem of a self propelled kayak. The  computational  cost  of  fully  resolving  the flow around a rotating propeller and hull inhibits the use of numerical simulations for commercial use. However, several groups have implemented simplified  body  force  propeller  models,  which accurately induce the accelerations produced by a propeller into the fluid (Phillips et al, 2010). A similar  body  force  methodology  is  adopted  to simulate the impact a paddle stroke has on the fluid around a moving kayak. This is done using the  open  source  CFD  package  OpenFOAM (OpenFOAM, 2009). 2  Theoretical approach A  finite  volume  method  is  adopted,  using  a Volume  of  Fluid  (VOF)  approach  for  the  free surface. This method is derived from the surface integration  of  the  conservative  form  of  Navier Stokes’  equations  over  a  control  volume.  The incompressible Reynolds averaged Navier-Stokes (RANS)  equations,  written  in  tensor  form,  are defined as  () ()''i j j iij i j j ji j ijUU U UU Pt x x x x xu u fx                      ‎ 2-1 and 0iiUx  ‎ 2-2, for momentum and mass continuity respectively. While the volume fraction transport equation is defined as ()0jjcU ctx   ‎ 2-3, where  c  is  the  volume  fraction  defined  as (Vair/Vtotal) (Peric and Ferziger, 2002).  The fluid density, ρ, and viscosity, µ, can then be calculated as (1 ) air water cc         ‎ 2-4 and  (1 ) air water cc         ‎ 2-5 respectively.  External  for ces  applied  to  the  fluid  are represented as fi, which include buoyancy forces and  momentum  sources  representing  the influence of the paddle. The effect of turbulence is represented in equation ‎ 2-1 by the Reynolds stress tensor  '' ij uu  and is modelled using the k-omaga  SST  turbulence  model  contained  within OpenFOAM-1.6 (OpenFOAM, 2009).  The  SST  model  blends  a  variant  of  the  k -ω model  in  the  inner  boundary  layer  and  a transformed version of the k-ε model in the outer boundary  layer  and  the  free  stream  (Menter, 1994).  This  has  been  shown  to  be  better  at replicating the flow around the stern of a ship, than simpler models such as k-ε, single and zero equation  models  (Larsson  et  al,  2000)(Hino, 2005).  3  Body force Paddle model 3.1  Simple mathematical force model To  start  with  the fluid  forces  generated  by  the paddle blade were calculated based on a simple mathematical  model  of  a  flat  plate  rotating around a point moving with an advance speed of U0 (see  Figure ‎ 3-1).  The angle  of rotation  θ is measured from the horizontal, in the direction of movement (i.e. increases throughout the stroke).   Figure ‎ 3-1 - free body diagram of paddle model. The normal velocity encountered by the blade at a radius r is given by   ‎ 3-1, where   is the unit vector normal to the blade calculated as  ‎ 3-2. It follows, therefore that the hydrodynamic force on a length of blade dr ,at a radius r  is given by  ‎ 3-3, where ρ is the density of water, CD is the drag coefficient and c is the chord of the blade at r. 3.2  Calculating  momentum source strengths from paddle force model To represent the impact of the paddle on the fluid, the  calculated  paddle  forces  are  applied  to  a propulsive domain located within the fluid. This domain represents the swept area of the paddle defined by the length (R) and the width (c).  An inner radius can also be defined to account for the length of the paddle handle. The propulsive domain is then divided up into sectors of radius dR  and  angle  dθ,  this  is  demonstrated  in Figure ‎ 3-2.    Figure ‎ 3-2 - Schematic of stroke propulsive domain divided into sectors. When the paddle blade passes through a sector, the paddle force (Fn) is calculated for a section of blade, with length and chord equal to the sector dimensions, located at the centre of that sector. The  momentum  source  term  for  that  sector  is then calculate as . Obviously as dr and dθ get smaller the propulsion model  better  represents  a  paddle  moving smoothly through the water. 3.3  Applying source terms within OpenFOAM The  standard  multiphase  solver  interFoam  was modified  to  accommodate  momentum  source terms  and  renamed  mom_interFoam.  A  new module  within  the  solver  was  created ‘createBodyForce.H’ which is called every time step  from  within  the  top  level  solver  program ‘mom_interFoam.C’. The parameters that define the propulsive model are defined within a dictionary located within the case files. The propulsion domain is defined by the paddle dimensions,  centre  of  rotation  and  unit  vectors providing the direction of forward motion and the roll angle. The polar coordinates of the centre of each  cell  within  the  mesh  are  then  calculated relative to the paddle centre of rotation. These are then used to determine which cells are within a given sector of the propulsive domain providing an accurate sector volume.  The run time of the current time step is then used to calculate the position of the paddle within the propulsive  domain,  based  on  prescribed  paddle angles  throughout  a  single  stroke  cycle  and  a defined stroke rate. This is used to calculate the angular velocity of the paddle using the paddle position form the previous time step. For  each  cell,  within  a  sector  containing  the paddle, the momentum source term is calculated using the paddle force calculated for that sector divided by the sector volume. The source term for all other cells are set to zero. These source terms  are  stored  within  a  volume  vector  field which is then added to the momentum equation (Ueqn.H  within  openFoam).  The  total  paddle force  is  determined  for  each  time  step  by multiplying each cell’s source term with its cell volume  and  summating  over  the  propulsive domain. The instantaneous thrust is then acquired by  resolving  this  force  into  the  direction  of movement.  3.4  Experimental data Experimental  data  for  a  rotating  paddle  was obtained as part of a student research project at the University of Southampton (Ellison, 2010). A kayak paddle was mounted on an instrumented pivot  mechanism  attached  to  a  towing  tank dynamometer  (Figure  ‎ 3-3).  This  allowed  lift, drag  and  rotation  angle  to  be  recorded  against time. A constant torque was applied to the paddle via  a  dropping  weight  allowing  the  dynamic forces  generated  by  the  blade  to  be  measured against time. Various angles of attack were tested with a range of torques.  Figure ‎ 3-3 - Experimental setup for paddle test. The experimental data for the blade at 90 degrees to the flow can be seen in Figure ‎ 3-4. Due to the blade  starting  out  of  the  water,  it  accelerates quickly  at  first,  entering  the  water  at  a  high velocity resulting in a peak in the thrust at this point.  A  small  blip  in  the  angle  data,  at approximately 110 degrees from the horizontal, is thought to be due to the rotary potentiometer and  not  the  flow  physics.  This  error  gets amplified when the angular velocity is calculated.    Figure ‎ 3-4 - Experimental data for a rotating paddle. 3.5  Validation against experimental data The paddle that was used in the experiments was represented  as  a  flat  plated  of  the  same  length and  chord  (0.54  and  0.2m  respectively).  A  flat plate drag coefficient of 1.2 was used as an initial approximation  (Hoerner,  1965)  whilst  the propulsion domain was divided into 18 angular and  8  radial  divisions,  with  an  inner  radius  of 0.1m. To start with the centre of rotation was placed on the  surface  of  the  water,  so  as  to  remove  the complications  of  paddle  entry.  The  angular velocity  was  maintained  at  a  constant  value throughout the stroke, determined as the average angular  velocity  from  the  experimental  paddle stroke.  Figure ‎ 3-5 - Thrust generated by propulsive model compared with experimental data. Out of a crude mesh of 40,000 cells, 740 were contained within the propulsive domain. It  can  be  seen  in  Figure  ‎ 3-5  that  the  constant velocity  propulsion  model  significantly underestimates  the  thrust  measured  in  the experiment. However it is not just the magnitudes which  do  not  match,  the  general  shape  of  the curves  differ  significantly.  To  try  and  improve the paddle model the blade angle data that was recorded  during  the  experiment  was  used  to prescribe  the  paddle  motion  during  the  stroke. This  modified  thrust  data  can  also  be  seen  in Figure ‎ 3-5. Despite the magnitude of the thrust being approximately half that of the experimental data it can now be seen that the general trends aligns much more closely. This is easier to in the final thrust trace, where the drag coefficient has been doubled to 2.4. There  are  many  improvements  that  need  to  be made to the body-force model that could account for  the  discrepancies  between  the  experimental data.  Predominantly  these  would  focus  on including added mass terms to the mathematical model, so as to include unsteady flow features, and  to  remove  the  step-like  variation  in  thrust through  a  smoother  implementation  of  source terms within the fluid domain.   3.6  Applying two propulsion models within the same simulation A second paddle model was easily added to the modified solver by having two identical modules that independently calculate momentum sources. The only difference is that the second paddle has a different centre of rotation and applies a phase shift to the prescribed paddle motion so that they are out of phase (typically 180 degrees).   4  Self propelled Simulation 4.1  Numerical model The  solver  settings  and  simulation  parameters can be found in Table ‎ 4-1.  Table ‎ 4-1 - Numerical settings Property   Mesh  Type of mesh   Unstructured (Hexahedral)  No. of elements  Approximately 1.2M  y+ on the hull   10-15  Domain Physics  Homogeneous Water/Air multiphase, kOmegaSST turbulence model, Automatic wall function  Boundary physics:  Inlet   Free stream velocity of 2m/s  Outlet   Zero gradient  Bottom/side wall  Wall with free stream velocity  Top   Opening  Hull   Wall with no slip condition  Solver settings:  Transient scheme  1st order Euler  Grad (U) Scheme  Gauss linear  Div (U)   Gauss limitedLinearV 1  Pressure coupling  PISO  Convergence criteria   P 1e-7, U 1e-6, k 1e-8, omega 1e-8  Multiphase control  Volume fraction coupling  Timestep control  max Courant No = 0.4  Processing Parameters:  Computing System  Iridis 3 Linux Cluster (University of Southampton)  Run type  Parallel (9 - 24 Partitions run on 5x8 core nodes each with 23 Gb RAM)  4.2  Meshing Technique An  unstructured  hexahedral  mesh  around  the kayak  was  created  using  the  snappyHexMesh utility within OpenFOAM. Firstly a coarse block mesh  of  hexahedral  cells  is  created,  using  the blockMesh  utility,  defining  the  size  of  the domain and the initial cell size in each direction. Specific  areas  within  the  domain  are  then specified  for  mesh  refinement  in  progressive layers.  For  each  layer  of  refinement  conducted each cell within the specified region is split into 8 equal  parts,  doubling  the  mesh  density  in  all directions.  However  uni-directional  refinement was used across the free surface to provide good wave  pattern  resolution,  whilst  minimising computational cost. Boundary layer element are also  grown  out  from  the  kayak  surface  mesh. This  localised  refinement  process  results  in  a general mesh structure that can be seen in Error! Reference source not found.. It should be noted that the images of the mesh are generated using Paraview  which  currently  displays  hexahedral cells  as  two  tetrahedral  cells.  The  mesh  is actually fully hexahedral.   Figure ‎ 4-1 - Kayak mesh structure. 4.3  Naked hull resistance The  naked  hull  resistance  of  the  kayak  was determined  to  be  22.68  N.  The  resulting  free surface deformation can be seen in Figure ‎ 4-2. Figure ‎ 4-2 - Free surface deformation for naked hull kayak simulation 4.4  Determining the self propelled stoke rate The naked hull resistance case files could then be used as an initial condition for the self propelled simulation. For this first attempt a self propelled kayak a fixed stroke rate was selected that would provide a thrust approximately equal to the naked hull  resistance.  In  time  this  will  become  an iterative process varying stroke rate to match the self propelled resistance. Without access to real stroke path data, the rate of  angular  velocity  throughout  the  stroke  was modelled as being sinusoidal, with zero angular velocity on paddle entry and exit. This provides a crude  approximation  of  how  an  athlete  might vary the velocity of the paddle to take account of the  kayak’s  forward  speed.  The  resulting  force trace for a single paddle operating at a stroke rate of 40 can be seen in Figure ‎ 4-3. This happened to provided an average thrust of 11.3 N, so with two paddles operating out of phase with each other the  average  thrust  would  be  22.6  N.  This  was assumed  to  be  close  enough  to  the  naked  hull resistance of for the purposes of this study.  Figure ‎ 4-3 - thrust generated by a single paddle, with a sinusoidal angular velocity, against time 4.5  Paddle-hull interactions The  paddle  induced  velocities  alongside  the kayak  can  be  seen  in  Figure  ‎ 4-4,  while  their effect on the pressure field over the hull can be seen in Figure ‎ 4-5. Although this simulation far from  replicates  a  realistic  paddle  stroke  the interaction between the paddle and the hull an be clearly seen.  To see the impact this change in pressure field has  on  the  kayak  you  have  to  look  at  the hydrodynamic forces and moments acting on the hull. In Figure ‎ 4-6 you can see how the side force varies  throughout  the  stroke.  As  the  paddle passes the right hand side of the hull, the pressure drops due to the increase in velocity, pulling the kayak to the right hand side (positive side force). The same phenomenon is observed on the left. Likewise the paddle hull interaction can clearly be  seen  in  all  three  moments.  Although  only initial,  un-validated  self  propulsion  data  is presented, the  potential  benefits  of this type  of analysis is clear.     Figure ‎ 4-4 - Paddle induced velocities throughout a single paddle stroke, viewed on a plane placed through the centre of rotation of the paddle.  Figure ‎ 4-5 - hydrodynamic pressure field on the bottom of the kayak (naked hull above, as paddle blade passes below).  Figure ‎ 4-6 - Self propelled hydrodynamic forces acting on the kayak  Figure ‎ 4-7 - Moments induced by the self propelled forces. 5  Conclusions A simplified mathematical model of a paddle has been  used  to  simulate  a  paddle  stroke  using  a body force method. Experimental data has been used  to  validate  the  model  and  refine  the  drag coefficients  used  until  improvements  to  the mathematical model can be included. An  initial  self  propelled  kayak  simulation  has been  performed  using  the  developed methodology,  which  highlights  the  significant impact the paddle stroke has on the kayak’s fluid dynamic forces. 6  References Ellison,  W.,  Turnock,  S.R.,  (2010)  Investigation  into  the effect of the kayak paddle stroke on elite kayak performance, Ship  Science  Individual  Project,  School  of  Engineering Sciences, University of Southampton, UK. Hino  T.  (2005)  CFD  Workshop  Tokyo  2005.  In:  The Proceedings of CFD Workshop Tokyo. Hoerner, S.F., (1965) Fluid-Dynamic Drag. 3-16. Published by the author, Midland Park, NJ Larsson  L,  Stern  F,  Bertram  V.  (2003)  Benchmarking  of Computational  Fluid  Dynamics  for  Ship  Flows:  The Gothenburg  2000  Workshop.  Journal  of  Ship  Research 2003;47:63–81(19). Menter,  F.R.,  (1994)  Two-equation  eddy-viscosity turbulence  models  for  engineering  applications.  AIAA Journal 32(8):1598 –605. OpenFOAM®,  (2009)  OpenFOAM  –  The  Open  Source CFD Toolbox- User Gide, Version 1.6. Peric, M., Ferziger, J.H., (2002) Computational Methods for Fluid Dynamics, Springer, 3rd edition. Phillips,  A.B.,  Turnock,  S.R.  and  Furlong,  M.E.  (2010) Accurate  capture  of  rudder-propeller  interaction  using  a coupled  blade  element  momentum-RANS  approach.  Ship Technology Research (Schiffstechnik), 57, (2), 128-139. 

Accurate capture of rudder-propeller interaction using a coupled blade element momentum-RANS approach.

CFD Workshop Tokyo 2005. In:

Fluid-Dynamic Drag. 3-16. Published by the author,

Investigation into the effect of the kayak paddle stroke on elite kayak performance,

OpenFOAM – The Open Source CFD Toolbox- User Gide, Version 1.6. Peric,

Two-equation eddy-viscosity turbulence models for engineering applications.

Performing self propelled simulations of a kayak, using a body-force paddle model

Performing self propelled simulations of a kayak, using a body-force paddle model

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