CFD for Yacht Design

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CFD for Yacht Design

• Jonathan Binns

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Aims

• Remove some of the mysticism of CFD and relate
  possibilities to yacht design problems
• Learn the limitations, differences and
  similarities of different CFD methods
• Realise similarities to aerodynamic design
  problems and take advantage of vast body of aero
  information
• When somebody asks is it possible? be able to
  say YES with confidence.

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CFD Context

• CFD can be thought of as any prediction method
• Simplest form is F Cd ½ ? V2A
• Cd from experiments
• Currently most complex is direct numerical
  simulation

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To be an expert in CFD

• Expert in fluid mechanics
• Expert in 3D geometry
• Expert in numerical methods
• Expert in efficient software development
• Expert in efficient hardware use
• ?Nearly impossible for one person
• ?MUST take an engineering approach

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To be an expert in CFD
Try for these

• Expert in fluid mechanics
• Expert in 3D geometry
• Expert in numerical methods
• Expert in efficient software development
• Expert in efficient hardware use

Appreciate this
Leave these to the experts
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Cost / Benefit of CFD

• Time benefits
• Reduction in model iterations may be achieved,
  especially for unknown design spaces
• Quality benefits
• Model tests still most accurate, but careful CFD
  can give greater insight into flow
• Bertram (2000), pp 19-20

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Work classification for CFD

• Pre-processing 30-90
• Grid generation dictates result quality
• Absolute precision in setting up runs must be
  achieved
• Computation 1
• Once hardware and software is obtained, this
  reduces to just waiting around
• Post-processing 10-20
• Like experiments, this is where real value adding
  is achieved
• Bertram (2000), pp 20-21

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Part_I-gtCFD_Fundamentals

• Required skills
• Appreciation of differential equations
• Appreciation of numerical methods
• Knowledge of classical fluid dynamics

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Basic equations

• Conservation of mass
• Conservation of momentum, (Fma)
• Conservation of a scalar (eg temperature)

Transport (moving) by means other than convection
(eg diffusion), as well as sources and sinks
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JB Rules for Equations

• Memorise the origin
• Understand the derivation, ONCE
• Look for the intuitive ones

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Conservation of mass

• Leads to
• For incompressible fluid

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Conservation of momentum, incompressible fluid
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A few notes

• Differential equations can also be expressed as
  integral equations
• Extensive use of Gauss divergence theorem to
  swap between volume integrals and surface
  integrals
• See Ferziger and Peric (2002), pp 1-20 for an
  explanation of the derivation

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Equations for an incompressible, viscous fluid

• Providing u, v, w f(?ij)
• 4 equations, 4 unknowns (u, v, w, p), job done,
  from a physics point of view

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Equations for an incompressible, inviscid fluid
1)

• 4 equations, 4 unknowns (u, v, w, p), job done,
  from a physics point of view

2)
3)
4)
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Equations for an incompressible, inviscid,
irrotational fluid

• Introduce a velocity potential
• v ??
• 2 equations, 2 unknowns (?, p), job done, from a
  physics point of view

Conservation of mass becomes Laplace equation
1)
Conservation of momentum becomes unsteady
Bernoulli equation
2)
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Solution method

• For the simplest of geometries (eg a circle) the
  above equations can be solved analytically
• For anything more complicated numerical methods
  MUST be used

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Basic principle of numerical methods

• Discretise the fluid domain
• Establish field variables at each node
• ui, vi, wi, pi and/or ?i
• Relate differential equations to field variables
• This relationship dictated by the method

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Relating field variables to differential equations

• Boundary element methods (BEM)
• Finite element methods (FEM)
• Finite difference methods (FDM)
• Finite volume methods (FVM)
• See Bertram (2000), pp 14-15 for good summary

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Boundary element method (BEM)

• Volume integrals transformed to surface integrals
  (3D), surface integrals to line integrals (2D) by
  use of Gauss divergence theorem
• Most ship analysis codes based on this approach,
  eg ShipFlow, Seakeeper, Hydrostar
• Flow must be incompressible, inviscid and
  irrotational (although viscous fudges used
  frequently)
• Grid generation is so simple dynamic re-gridding
  is easy

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Finite element method (FEM)

• Relationship between field variables and
  differentials through shape functions within each
  control volume (assumed behaviour)
• Doesnt get used much for fluid dynamics,
  because?
• Errors are hard estimate in fluid application
  (Bertram, p 14)?
• Unstructured nature lends to hard to solve
  matrices (Ferziger and Peric, p 37)?

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Finite difference method (FDM)

• Truncated Taylor series expansion used to express
  differentials as differences between field
  variables
• Simple conceptually, difficult to implement in 3D
• Truncation errors can actually violate
  conservation of mass
• Bertram (2002), p 15
• There is more to life than finite differencing
• Press et al. (1992), p 833

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Finite volume method (FVM)

• Integral form of mass and momentum conservation
  equations are used in simple trapezoidal or
  midpoint rules
• Continuity is forced for each control volume
• Basis of nearly all commercial codes (all I know
  about)

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Levels of FVM

• Direct numerical simulation (DNS)
• Full Navier-Stokes equations solved over entire
  domain
• Problem of resolving every time scale and every
  spatial scale
• Large eddy simulation (LES)
• Small time and space scales are filtered out
• Requires careful selection of filter
• Reynolds averaged Navier-Stokes (RANS)
• Small time and space scales are averaged,
  fluctuating component is related to other field
  variables through Reynolds stress equations,
  energy production and dissipation is then
  transported through the flow
• Requires Reynolds stress equations
• Inviscid analysis

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Relating field variables to differential equations

• Boundary element methods (BEM)
• Finite volume methods (FVM)
• Finite element methods (FEM)
• Finite difference methods (FDM)

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Free surface methods

• Interface tracking
• Kinematic boundary condition
• A particle on the free surface stays on the free
  surface
• Dynamic boundary condition
• Momentum along the free surface is conserved, use
  the unsteady Bernoulli equation
• Interface capturing
• A multiphase simplification where volume fraction
  is transported
• See Notes and Ferziger and Peric (2002), pp
  381-397

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Boundary conditions

• Nearly all boats operate in a very large domain
  (the world)
• Impossible to model the world (massive time and
  space scales required)
• Boundaries must be placed, BCs dramatically
  effect the results!

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To be steady or un-steady

• Conservation of momentum equation has an unsteady
  term
• In time steady flows (eg aeroplane wing) can be
  ignored
• But in analysis can be used to march to a
  solution, like using a successive under
  relaxation factor

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Generalised errors in CFD

• Modeling errors
• Only by experimental comparison
• Discretisation errors
• Estimated by Richardson extrapolation
• Iteration errors
• From theoretical analysis of matrix solvers
• Ferziger and Peric (2002), pp 34-35

Mainly concerned with these
Software engineers concerned with these
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Modeling errors

• The errors that would be present even if the
  differential equations were solved analytically
• Can only be estimated once iteration and
  discretisation errors have been assumed small and
  experimental results are available

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Discretisation errors

• The difference between solving the differential
  equations exactly or numerically

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Discretisation errors

• Not as difficult as they sound
• Knowing the order of the truncation error, p, the
  error in field variable will be
• Ferziger and Peric (2002), pp 58-60
• Ferziger and Peric (1996)

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Richardson extrapolation

• Systematically refine the grid to find the grid
  independent solution (GIS)
• Even works for different discretisation schemes
• Azcueta (2002), p 42

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Iteration errors

• Difference between full solution of equations and
  iterated solution
• Direct matrix solvers are rarely used (unless you
  write your own code), instead numerical solutions
  are iterated from an approximate solution to
  another
• Should be taken care of by a tolerance, set to
  order of 0.001 but NEVER less than machine
  precision

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A note on CFD errors IGeometry Summary

• Diffference between CFD groups has been shown to
  be larger than all errors combined
• For the following geometry

From 2nd AIAA CFD Drag Prediction Workshop - Data
Summary and Comparison
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A note on CFD errors IIParticipant Summary

• 22 participants many others (1st DPW 18)
• US 50 Govt 31
• Europe 29 Industry 46
• Asia 21 Academia 21
• 20 codes, 30 data submittals
• Grid Types Turbulence Models
• 14 1-to-1 structured 16 Spalart-Allmaras
• 11 Unstructured 5 Menters SST
• 5 Overset 3 k-w
• 2 k-Wilcox, k-e, other

From 2nd AIAA CFD Drag Prediction Workshop - Data
Summary and Comparison
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A note on CFD errors IIIResult Summary
From 2nd AIAA CFD Drag Prediction Workshop - Data
Summary and Comparison
38
A note on CFD errors IVStatistical Summary

• 30 data submittals
• 16 complete sets for Case 1
• 30 partial data sets for Case 2
• 7 data sets for Case 3
• 3 data sets for Case 4
• 480 Total Solutions Computed!
• 1.25 years of CPU time!!!

What hope do we mortals have?!??
From 2nd AIAA CFD Drag Prediction Workshop - Data
Summary and Comparison
39
A note on CFD errors IVStatistical Summary

• For medium grid on wing-body, DPW II results are
  better than DPW I.
• Regarding grid convergence for the collective
• There is no reduction in spread
• There is no reduction in core scatter
• The medians MAY be converging, although it cant
  be proven with the present results.
• Increments tend to be considerably better in both
  scatter and median.
• Much work needs to be done to define what is
  meant by grid convergence, i.e. how to carry it
  out.

What hope do we mortals have?!??
Increments tend to be considerably better in both
scatter and median.
From 2nd AIAA CFD Drag Prediction Workshop - Data
Summary and Comparison
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CFD Concept

• CFD starts from a simple concept (ie no mass
  produced)
• Basis of method of solution is extremely complex
  (eg 1 million nodes with 4 variables all
  interelated)
• Intricacies of solution add complexity (eg
  iterative matrix solvers)
• Result is something like an experiment, a virtual
  unlimited number of unknown variables, all
  dependant in an unknown function
• ?Methodical approach must always be used

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CFD Concept

• Mathematically most fluid flow problems have
  infinite solutions, eg
• A free surface is stable for an infinitely small
  amount of time at any state
• RANS solutions are closed (same number of
  equations as unknowns) by approximate means
• ?Mathematically any solution is as good as the
  next, iteration errorsdiscretisation errors0.0,
  but modelling errors can be enormous!!!!!!!!