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## CFD for Yacht Design

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### Extensive use of Gauss' divergence theorem to swap between volume integrals and ... surface integrals to line integrals (2D) by use of Gauss' divergence theorem ... – PowerPoint PPT presentation

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Title: CFD for Yacht Design

1
CFD for Yacht Design
• Jonathan Binns

2
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.

3
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

4
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

5
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
6
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

7
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

8
Part_I-gtCFD_Fundamentals
• Required skills
• Appreciation of differential equations
• Appreciation of numerical methods
• Knowledge of classical fluid dynamics

9
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
10
JB Rules for Equations
• Memorise the origin
• Understand the derivation, ONCE
• Look for the intuitive ones

11
Conservation of mass
• For incompressible fluid

12
Conservation of momentum, incompressible fluid
13
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

14
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

15
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)
16
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)
Bernoulli equation
2)
17
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

18
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

19
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

20
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

21
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)?

22
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

23
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

24
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

25
Relating field variables to differential equations
• Boundary element methods (BEM)
• Finite volume methods (FVM)
• Finite element methods (FEM)
• Finite difference methods (FDM)

26
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
• Interface capturing
• A multiphase simplification where volume fraction
is transported
• See Notes and Ferziger and Peric (2002), pp
381-397

27
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!

28
• 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

29
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
30
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

31
Discretisation errors
• The difference between solving the differential
equations exactly or numerically

32
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)

33
Richardson extrapolation
• Systematically refine the grid to find the grid
independent solution (GIS)
• Even works for different discretisation schemes
• Azcueta (2002), p 42

34
Iteration errors
• Difference between full solution of equations and
iterated solution
• Direct matrix solvers are rarely used (unless you
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

35
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
36
A note on CFD errors IIParticipant Summary
• 22 participants many others (1st DPW 18)
• US 50 Govt 31
• Europe 29 Industry 46
• 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
37
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
40
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

41
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!!!!!!!!