1. “Solve the equations right” Mathematical problem Validation :
Definitions
Verification :
“Solve the equations right”
Mathematical problem
Validation :
“Solve the right equations”
Science/engineering activity
Roache, 1998 1

2. Verification includes two different activities: - Code Verification
Definitions
Verification includes two different activities:
- Code Verification
Guarantee that the computer code does not
contain errors (bugs). Error evaluation that
requires the knowledge of the exact solution
- Solution/calculations Verification
Estimate the numerical uncertainty for a
numerical solution that has an unkown
exact solution (Error estimation) 2

3. - Assessing the modeling uncertainty of
Definitions
Validation
- Assessing the modeling uncertainty of
(a selected aspect of) a mathematical model
Needed:
the definition of the quantities of interest
the selection of a metric to quantify the
modeling uncertainty (on-going topic of
research) 3

4. An error is defined as the difference between
Definitions
− Error vs Uncertainty
An error is defined as the difference between
a given solution and its “true/exact” value. It has
a sign and it requires the knowledge of the “truth”
An uncertainty defines an interval that should
contain the “true/exact” value with a certain
degree of confidence. It is defined with a ∓ 4

5. - Discretization error Three components of the numerical error behave
- Round-off error
- Iterative error
- Discretization error
Three components of the numerical error behave
differently with the increase of the number of
degrees of freedom (grid refinement) 5

6. - Due to the finite precision of computers
Numerical Error
Round-off error:
- Due to the finite precision of computers
- Decreased by the use of double precision
- May be dominant in ill-conditioned problems
- It grows with the increase of the number of
degrees of freedom (grid refinement) 6

7. − Round-off error, example: Polynomial interpolation in 2-D (or 3-D)
Numerical Error
− Round-off error, example:
Polynomial interpolation in 2-D (or 3-D)
(n+1)2 coefficients determined from (n+1)2 points
where fi(xi,yi) is known 7

8. − Round-off error, example: Polynomial interpolation in 2-D (or 3-D)
Numerical Error
− Round-off error, example:
Polynomial interpolation in 2-D (or 3-D)
Coefficients of the polynomial obtained from
the solution of a linear systems of algebraic
equations
First term of the main diagonal : 1
Last term of the main diagonal : xnyn 8

9. - Non-linear equations (convection in momentum equations)
Numerical Error
Iterative error:
- Non-linear equations (convection in momentum
equations)
- Decoupling of the equations
(turbulence model solved for a given mean
velocity field and momentum equations
solved with a “known” eddy-viscosity) 9

10. - Discretization schemes with explicit corrections
Numerical Error
Iterative error:
- Discretization schemes with explicit corrections
for the high-order terms
- Solution of the systems of algebraic equations
with iterative methods
(Jacobi, Gauss-Seidel, Conjugate Gradients,
GMRES, ...) 10

11. - In principle, it can be reduced to machine
Numerical Error
Iterative error:
- In principle, it can be reduced to machine
accuracy (if there are no problems with the
round-off error)
- The reduction of the iterative error becomes
hardest with the increase of the number of
degrees of freedom (grid refinement). Multigrid
techniques may avoid problems with the size of
the system of algebraic equations to solve 11

12. - It is important to define (know) the meaning of one iteration
Numerical Error
Iterative error:
- It is important to define (know) the meaning of
one iteration
- Iterative error estimates based on differences
or normalized residuals of the last iteration
performed are not reliable
- For iterative error estimates, the L∞ norm is
a more appropriate choice than the L1and L2
norms 12

13. - RANS (with eddy-viscosity model) calculation
Numerical Error
Iterative error:
- RANS (with eddy-viscosity model) calculation
of the turbulent flow in a channel
- Initial velocity field obtained from the specified
profiles at the inlet
- Solution converged to machine precision (10-14) 13

14. - Convergence criteria based on the maximum
Numerical Error
Iterative error:
- Convergence criteria based on the maximum
difference between consecutive iterations, et
- Iterative error obtained from the difference to
the solution converged to machine accuracy
- Example of iterative errors for the horizontal
velocity component, U1 14

15. Iterative error: Numerical Error
Largest iterative error is 2 orders of magnitude larger than et! 15

16. Discretization error:
Numerical Error
Discretization error:
- Consequence of the transformation of the
continuum equation(s) into a system of
algebraic equations
- It may have a geometric component, which
may even be the dominant contribution in
domains bounded by surfaces with high
curvature 16

17. Discretization error: - Usually, it is the main contribution to the
Numerical Error
Discretization error:
- Usually, it is the main contribution to the
numerical error
- Can only be determined with the knowledge of
the exact solution
- Tends to diminish with the increase of the
number of degrees of freedom (grid refinement)
- Estimate of the discretization error may be
obtained from grid refinement studies 17

18. Its goal is: check that the code is bug-free
Code Verification
Its goal is: check that the code is bug-free
Does the numerical error tend to zero when
the grid refinement / time step tends to zero?
Is the observed order of convergence in
agreement with the “theoretical order”
of the code? M
- If not, find out why?!

19. Error described by a Power Series Expansions
Code Verification
Error described by a Power Series Expansions
fi → Local or functional flow quantity for grid i
fexact → Exact solution of local or functional flow
quantity
eo → Error for cell size zero (supposed to be “0”)
a → Constant
hi → Typical cell size
p → Observed order of grid convergence If

20. Geometrical Similarity - Typical cell size definition, hi
Code Verification
Geometrical Similarity
- Typical cell size definition, hi

21. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
- Definition of a typical cell size (single
parameter) requires sets of geometrically
similar grids
- Consider a 1-D problem for
- Geometrically similar grids have a single
definition of the dimensionless distance (s)
as a function of the “dimensionless” node
counter (x)

22. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Sets of 31 grids with 5 ≤ Ncells ≤320

23. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Similar grids

24. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Unsimilar grids

25. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Similar grids

26. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Unsimilar grids

27. Geometrical Similarity (a 1-D example) - Error norms:
Code Verification
Geometrical Similarity (a 1-D example)
- Error norms:
- eo(f), p and a obtained in the least squares
sense from the data of the six finest grids

28. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Similar grids

29. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Unsimilar grids

30. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Unsimilar grids

31. Geometrical Similarity (a 1-D example)
Code Verification
Geometrical Similarity (a 1-D example)
Unsimilar grids

32. - In some practical applications, there are no
Code Verification
Code Verification
- In some practical applications, there are no
exact solutions available. For example, the
Reynolds-averaged Navier-Stokes equations
supplemented by a turbulence model do not have
any known exact solutions
− The Method of Manufactured Solutions (MMS)
is a viable alternative to perform Code Verification
when analytical solutions are not available. 32

33. Method of Manufactured Solutions: Choose the calculation domain
Code Verification
Method of Manufactured Solutions:
Choose the calculation domain
Choose the analytical solution for the
dependent variables of the problem
Substitute the chosen solution in the
differential equations to determine source
terms that guarantee that the selected exact
solution satisfies the “new” system of
differential equations 33

34. Solution Verification
Solution/Calculations Verification
− Exact solution is not available
− Numerical error estimation usually assumes
that the discretization error is dominant (this
requires an iterative error at least two orders
of magnitude smaller)
− Methods based on grid refinement studies are
one of the alternatives for the estimation of the
discretization error/uncertainty
− Mathematical problem 34

35. Solution Verification
− Estimate the uncertainty, U, of a numerical
solution of a flow quantity f for which the
exact solution is not known
Objective:
with a confidence level of 95%
Safety factor
Error estimate 35

36. Solution Verification
Error described by a Power Series Expansions
fi → Local or functional flow quantity for grid i
fo → Estimate of the exact solution of local or
functional flow quantity
dRE → Estimate of the error
a → Constant
hi → Typical cell size
p → Observed order of grid convergence 36

37. Solution Verification
At least 3 grids are required to determine fo, a, p 37

38. Solution Verification
Apparent convergence or divergence for a grid
triplet with h2/h1=h3/h2!
Convergence ratio :
Previous equation can only be used for
Monotonic Convergence
0 < R <1  Monotonic Convergence
-1 < R <  Oscillatory Convergence
R >  Monotonic Divergence
R <  Oscillatory Divergence 38

39. Solution Verification
Example I
Sets of 31 grids with 5 ≤ Ncells ≤320
Geometrically similar grids require L
A simple 1-D example

40. Solution Verification
Example I
- Determination of solution at x=0.5
- Three grid sets of 21 grids with 5 ≤ Ncells ≤80
a) Equally-spaced geometrically similar
grids, Equal
b) Stretched geometrically similar grids, Stret
c) Stretched non-geometrically similar
grids, StretN L
Different grid node distributions requiring interpolation (Stret) and with no interpolation (Equal)

41. Solution Verification
Example I
Determination of solution at x=0.5 L
Disturbance to grid similarity

42. Solution Verification
Example I
Determination of solution at x=0.5 L
The effect of grid node distribution. The “asymptotic range”

43. Solution Verification
Example I
Determination of solution at x=0.5 L
The effect of grid node distribution. The asymptotic range

44. Solution Verification
Example I
Determination of solution at x=0.5 L
Disturbances from lack of geometric similarity

45. Solution Verification
Example I
Determination of solution at x=0.5 L

46. Solution Verification
Example I
Determination of solution at x=0.5 M
Disturbances from interpolation

47. Solution Verification
Example I
Determination of solution at x=0.5 M
Disturbances from interpolation

48. Solution Verification
Example II
- Flow around the KVLCC2 tanker at model and
full scale Reynolds number (double-body)
- PARNASSOS with SST k-w model
- Sets of 6 nearly geometrically similar grids with
direct application of no slip condition with y2+<0.6
- Blending between third (b=1) and first-order
(b=0) upwind L
A “real test case”
- Selected flow quantities are resistance coefficients
and nominal wake fraction

49. Solution Verification
Example II L
High quality grid sets (3 Workshops with the same test case!)

50. Solution Verification
Example II L

51. Solution Verification
Example II
Rn=4.6×106 L
Numerics, numerics and numerics...
Explain the figure
KVLCC2 tanker
SST k-w model

52. Solution Verification
Example II
Rn=2.03×109 L
Numerics, numerics and numerics...
KVLCC2 tanker
SST k-w model

53. Solution Verification
Example II
Rn=4.6×106 L
Numerics, numerics and numerics...
KVLCC2 tanker
SST k-w model

54. Solution Verification
Example II
Rn=2.03×109 L
Numerics, numerics and numerics...
KVLCC2 tanker
SST k-w model

55. Solution Verification
Example II
Rn=4.6×106 L
Numerics, numerics and numerics...
KVLCC2 tanker
SST k-w model

56. Solution Verification
Example II
Rn=2.03×109 L
Numerics, numerics and numerics...
KVLCC2 tanker
SST k-w model

57. Solution Verification
Example II M
An example of what careless CFD can do...
40 minutes
KVLCC2 tanker
SST k-w model

58. Solution Verification
Example II
Rn=4.6×106 M
An example of what careless CFD can do...
KVLCC2 tanker
SST k-w model

59. Solution Verification
Example II
Rn=4.6×106 M
An example of what careless CFD can do...
KVLCC2 tanker
SST k-w model

60. Goal: Estimate an interval that contains the modeling error,
Validation (ASME V&V 20)
ASME V&V 20 procedure
Goal: Estimate an interval
that contains the modeling error,
with 95% confidence
Needed:
a numerical simulation result with uncertainty estimate
a corresponding experimental data set with uncertainty estimate L
the parameter uncertainty

61. Goal: Estimate an interval that contains the modeling error,
Validation (ASME V&V 20)
ASME V&V 20 procedure
Goal: Estimate an interval
that contains the modeling error,
with 95% confidence
Comparison error,
Validation uncertainty, L

62. Analysis of the outcome:
Validation (ASME V&V 20)
ASME V&V 20 procedure
Analysis of the outcome:
If too big:
“Houston, we have a problem...” L
If is acceptable:
The model is good enough, for this “noise level”

63. Flow at the propeller plane of a tanker at model scale
Validation (ASME V&V 20)
Flow at the propeller plane of a tanker at model
scale
- Double body (no gravity waves)
Comparison of axial velocity at the propeller
plane
RANS equations with an eddy-viscosity model
Parameter uncertainty equal to zero 63

64. Validation (ASME V&V 20)
Experimental
Numerical
Numérico 64

65. Validation (ASME V&V 20)
Usual
Comparison 65

66. Validation (ASME V&V 20)
Introduction of
experimental
uncertainty 66

67. Validation (ASME V&V 20)
Introduction
of numerical
uncertainty 67

68. Comparasion of validation error with validation uncertainty
Validation (ASME V&V 20)
Comparasion of validation error with validation uncertainty 68

69. Validation (ASME V&V 20)
What we want to avoid! 69