Verification & Validation

A CFD simulation produces numbers. But how do you know those numbers are right? The answer lies in a rigorous process called Verification and Validation (V&V). Without V&V, CFD results are just colorful pictures — potentially misleading and dangerous for engineering decisions.

The Fundamental Distinction

Verification: "Are we solving the equations right?"

Verification checks that:

• The code correctly implements the mathematical model
• Numerical errors are quantified and controlled
• The solution converges to the mathematical solution as the mesh is refined

Validation: "Are we solving the right equations?"

Validation checks that:

• The mathematical model represents the physical reality
• Predictions agree with experimental data
• The model is appropriate for the application

The V&V Hierarchy

Reality (Physical World)
    ↓ ← Validation (compare to experiments)
Mathematical Model (PDEs, turbulence models)
    ↓ ← Verification (check numerics)
Discrete Solution (CFD result)

Sources of Error

Model Errors

Arise from physics assumptions:

• Turbulence model limitations
• Laminar vs. turbulent assumption
• Incompressible approximation
• Boundary condition idealizations
• Geometry simplifications

Numerical Errors

Arise from discretization:

• Truncation error (finite differences)
• Iterative convergence error
• Round-off error (floating point)
• Grid-induced errors

Input Errors

Arise from uncertain inputs:

• Material properties
• Boundary condition values
• Initial conditions
• Geometry tolerances

Code Verification

Method of Manufactured Solutions (MMS)

• Choose an exact solution (any smooth function)
• Substitute into the governing equations
• Compute the source term needed to make it exact
• Run the code with that source term
• Compare computed vs. manufactured solution

Substitute into the advection equation $\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = S$

Compute the required source: $S = c\pi\cos(\pi x)\cos(\pi y)$

If the code is correct, refining the mesh should reduce the error at the expected rate.

Order of Accuracy Verification

For a scheme with formal order $p$:

$$\epsilon \propto h^p$$

Log-log plot of error vs. mesh size should have slope $p$.

• Halving mesh size should reduce error by factor of 4
• Observed slope $\neq$ 2 indicates a bug

Solution Verification

Grid Convergence Study

• Create at least 3 meshes with refinement ratio $r > 1.3$
• Compute the solution on each mesh
• Observe how key quantities change with mesh refinement
• Estimate the error in the finest mesh solution

Refinement Ratio

$$r = \frac{h_{coarse}}{h_{fine}}$$

For unstructured meshes:

$$h = \left(\frac{V_{domain}}{N_{cells}}\right)^{1/3}$$

Observed Order of Convergence

Given solutions on three grids ($f_1$ finest, $f_3$ coarsest):

$$p = \frac{\ln\left(\frac{f_3 - f_2}{f_2 - f_1}\right)}{\ln(r)}$$

If solutions are oscillating or $p < 0$, you're not in the asymptotic range.

Richardson Extrapolation

The Concept

If we know the order of convergence, we can extrapolate to zero mesh size:

$$f_{exact} \approx f_1 + \frac{f_1 - f_2}{r^p - 1}$$

This estimate is more accurate than any individual grid solution.

Requirements

• Solutions must be in the asymptotic range (smooth convergence)
• Observed $p$ should be close to formal order
• Same geometry, BCs, solver settings across grids

Limitations

• Assumes smooth solution
• Doesn't account for model errors
• Can be optimistic if criteria violated

Grid Convergence Index (GCI)

The Standard Approach

Roache's GCI provides a confidence interval for the discretization error:

$$GCI_{fine} = \frac{F_s \cdot |f_2 - f_1|}{r^p - 1}$$

Where:

• $F_s$ = safety factor (1.25 for 3+ grids, 3.0 for 2 grids)
• $f_1, f_2$ = solutions on fine and medium grids
• $r$ = refinement ratio
• $p$ = observed order of convergence

Interpretation

$$f_1 \pm GCI_{fine}$$

Grid Convergence Check

For three grids, verify:

$$GCI_{medium} \approx r^p \cdot GCI_{fine}$$

If this ratio differs significantly from expected, you may not be in the asymptotic range.

Practical Grid Study Procedure

Step 1: Create Grids

Use $r = 2$ for clear separation.

Step 2: Ensure Consistency

• Same turbulence model, BCs, schemes
• Same convergence criteria (residuals, iterations)
• Identical post-processing

Step 3: Extract Quantities

Focus on integral quantities (drag, mass flow) rather than local values.

Step 4: Calculate GCI

• Compute observed order $p$
• Apply Richardson extrapolation
• Calculate GCI for uncertainty band
• Verify grid convergence check

Step 5: Report Results

• Fine grid drag: $C_D = 0.285$
• GCI: 2.3%
• Reported result: $C_D = 0.285 \pm 0.007$

Validation

Validation Hierarchy

Start with simple cases, progress to complex:

• Unit problems — Single physics (e.g., laminar channel flow)
• Benchmark cases — Well-documented experiments
• Subsystem tests — Component-level validation
• System validation — Full application

Classic CFD Benchmarks

Experimental Uncertainty

Experiments have errors too:

$$E_{model} = S - D$$

Where:

• $E_{model}$ = model error (what we want to know)
• $S$ = simulation result
• $D$ = experimental data

The validation uncertainty is:

$$U_{val}^2 = U_{num}^2 + U_{input}^2 + U_{exp}^2$$

Only if $|S - D| < U_{val}$ can we claim the model is validated.

When Validation Fails

Best Practices

Documentation Requirements

A credible CFD study should report:

• Grid details — Cell counts, types, y+, quality metrics
• Numerical schemes — Spatial, temporal discretization
• Convergence evidence — Residual plots, monitors
• Grid study results — GCI, Richardson extrapolate
• Validation comparison — Against relevant experiments
• Uncertainty statement — Confidence interval on results

Common Mistakes

ASME V&V Standards

The ASME V&V 20-2009 standard provides formal procedures:

• Defines error and uncertainty terminology
• Specifies validation metrics
• Recommends hierarchical validation
• Required for nuclear and safety-critical applications

Uncertainty Quantification (UQ)

Beyond Deterministic CFD

Real-world inputs have uncertainty:

• Inlet velocity: $U = 10 \pm 0.5$ m/s
• Material properties: $\mu = 0.001 \pm 0.0001$ Pa·s
• Geometry: Manufacturing tolerances

Methods

Why UQ Matters

A drag coefficient of $C_D = 0.30$ is meaningless without context:

• Is it $0.30 \pm 0.03$ (10% uncertainty)?
• Or $0.30 \pm 0.001$ (0.3% uncertainty)?

The decision-making changes dramatically.

Case Study: Airfoil Validation

Setup

• NACA 0012 at $\alpha = 10°$, $Re = 6 \times 10^6$
• k-omega SST turbulence model
• Three grids: 50k, 200k, 800k cells

Grid Study Results

Validation

Key Takeaways

• Verification ≠ Validation — Verification checks numerics; validation checks physics
• Grid convergence studies are mandatory for credible CFD
• GCI provides a quantified uncertainty band for discretization error
• Richardson extrapolation estimates the grid-independent solution
• Observed order near formal order confirms asymptotic convergence
• Validation requires experimental data with known uncertainty
• Hierarchical validation builds confidence from simple to complex
• Report uncertainties — numbers without error bars are meaningless

What's Next

With a solid foundation in CFD theory and practice, the final lesson brings it all together: Practical CFD Workflow — a complete guide to running successful simulations, common pitfalls to avoid, and career paths in CFD.