CFD-FVM-13: Errors and uncertainty in CFD modelling

Errors and uncertainty in CFD

The formulation of a number of guidelines for best practice in CFD, the most influential of which are the AIAA (1998) and ERCOFTAC (2000) guidelines.

Error

a recognisable deficiency in a CFD model that is not caused by lack of knowledge. Causes of errors, defined in this way, are:

• Numerical errors – roundoff errors, iterative convergence errors, discretisation errors
• Coding errors – mistakes or ‘bugs’ in the software
• User errors – human errors through incorrect use of the software

Uncertainty

a potential deficiency in a CFD model that is caused by lack of knowledge. The main sources of uncertainty are:

• Input uncertainty – inaccuracies due to limited information or approximate representation of geometry, boundary conditions, material properties etc.
• Physical model uncertainty – discrepancies between real flows and CFD due to inadequate representation of physical or chemical processes (e.g. turbulence, combustion) or due to simplifying assumptions in the modelling process (e.g. incompressible flow, steady flow)

Numerical errors

• Roundoff error
• Iterative convergence error
• Discretisation error

Roundoff errors

Roundoff errors are the result of the computational representation of real numbers by means of a finite number of significant digits, which is termed the machine accuracy.

Iterative convergence errors

In practice, the available resources of computing power and time dictate that we truncate the iteration sequence when the solution is sufficiently close to the final solution. This truncation generates a contribution to the numerical error in the CFD solution.

Discretisation errors

Temporal and spatial derivates of the flow variable, which appear in the expressions for the rates of change, fluxes, sources and sinks in the governing equations, are approximated in the finite volume method on the chosen time and space mesh.

Input uncertainty

• Domain geometry
• Boundary conditions
• Fluid properties

Domain geometry

It is impossible to manufacture the duct perfectly to the design specifications; manufacturing tolerances will lead to discrepancies between the design intent and a manufactured part

Boundary conditions

Simple assumptions, e.g. given temperature, given heat flux, adiabatic wall, are often made in the computations; the accuracy of these will affect the calculation result. A contribution to the input uncertainty is associated with the inaccuracy of all assumptions involved in the process of defining the boundary conditions.

Fluid properties

All fluid properties (e.g. density, viscosity, thermal conductivity) depend to a greater or lesser extent on the local value of flow parameters, such as pressure and temperature.

Physical model uncertainty

Limited accuracy or lack of validity of submodels

CFD modelling of complex flow phenomena, such as turbulence, combustion, heat and mass transfer, involves semi-empirical submodels. They encapsulate the best scientific understanding of complex physical and chemical processes.

Limited accuracy or lack of validity of simplifying assumptions

The accuracy and appropriateness of all simplifying assumptions for a given flow determine the size of their contribution to physical model uncertainty.

Verification and validation

• Verification: the process of determining that a model implementation accurately represents the developer’s conceptual description of the model and the solution to the model. Roache (1998) coined the phrase ‘solving the equations right’. This process quantifies the errors.
• Validation: the process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model. Roache (1998) called this ‘solving the right equations’. This process quantifies the uncertainty.

Verification

The process of verification involves quantification of the errors.

Validation

The process of validation involves quantification of the input uncertainty and physical model uncertainty

• Input uncertainty can be estimated by means of sensitivity analysis or uncertainty analysis. In sensitivity analysis the effects of variations in each item of input data is studied individually. Uncertainty analysis, on the other hand, considers possible interactions due to simultaneous variations of different pieces of input data and uses Monte Carlo techniques in the design of the programme of CFD test runs.
• quantitative assessment of the physical modelling uncertainty requires comparison of CFD results with high-quality experimental results.

Guidelines for best practice in CFD

• AIAA guide (1998)
• ERCOFTAC guidelines (2000)

Reporting / documentation of CFD simulation inputs and results

• Input documentation
• Result interpretation and reporting