OpenFOAM Residuals: How to Read and Interpret Solver Convergence

Residuals are the primary signal of whether an OpenFOAM simulation is converging, stalling, or diverging. Understanding what they represent — and how to interpret their behaviour — is essential for diagnosing any simulation problem.

By CFDpilot · Updated April 2026

What residuals mean in OpenFOAM

A residual measures how well the discretised equation is satisfied at the current iteration. For a linear system Ax = b, the residual is |b - Ax| normalised by a reference value. A residual of 1e-3 means the equation is satisfied to 0.1% — not to be confused with a 0.1% error in the result.

In the log, each iteration prints the initial and final residual for each solved field:

smoothSolver:  Solving for Ux, Initial residual = 0.0234, Final residual = 1.2e-06, No Iterations 3
smoothSolver:  Solving for Uy, Initial residual = 0.0187, Final residual = 9.4e-07, No Iterations 3
GAMG:  Solving for p, Initial residual = 0.152, Final residual = 0.00741, No Iterations 4

The initial residual is what matters for assessing convergence — it measures the residual at the start of the iteration, before the inner solver runs.

Convergence criteria in fvSolution

Convergence tolerances are set in fvSolution:

solvers
{
    p
    {
        solver          GAMG;
        tolerance       1e-06;    // absolute convergence threshold
        relTol          0.1;      // relative: stop when residual drops by 10x
        smoother        GaussSeidel;
    }

    U
    {
        solver          smoothSolver;
        smoother        symGaussSeidel;
        tolerance       1e-07;
        relTol          0.1;
    }
}

SIMPLE
{
    residualControl
    {
        U       1e-04;    // stop outer iterations when U residual < 1e-4
        p       1e-04;
        k       1e-04;
        epsilon 1e-04;
    }
}

residualControl controls when the outer SIMPLE loop stops. The inner solver tolerances (tolerance, relTol) control how tightly each linear system is solved per iteration.

Under-relaxation factors and their impact on residuals

Under-relaxation in SIMPLE damps updates to prevent oscillations. The factors are set in the relaxationFactors block of fvSolution:

relaxationFactors
{
    fields
    {
        p           0.3;   // pressure: 0.2-0.4 is standard
    }
    equations
    {
        U           0.7;   // velocity: 0.5-0.8 typical
        k           0.5;
        epsilon     0.5;
        omega       0.5;
    }
}

Lower relaxation factors slow convergence but increase stability. If residuals spike or diverge, halving both U and p factors is the first corrective action. Never exceed 0.9 for U or 0.5 for p in a standard RANS case.

Non-orthogonal correctors and pressure residuals

On meshes with non-orthogonality above 40–50°, the pressure equation requires non-orthogonal corrections. Without these, the pressure residual plateaus at an elevated level:

SIMPLE
{
    nNonOrthogonalCorrectors  2;   // 1-3 for unstructured meshes
}

Each corrector adds a full pressure solve. For max non-orthogonality below 70°, 1–2 correctors suffice. Above 70°, prioritise mesh improvement over increasing correctors.

Residuals for transient simulations (pimpleFoam)

In transient cases, the outer PIMPLE loop runs multiple predictor-corrector cycles per time step. Convergence within each time step is controlled by:

PIMPLE
{
    nOuterCorrectors    3;
    nCorrectors         2;
    nNonOrthogonalCorrectors 1;

    outerCorrectorResidualControl
    {
        U   { tolerance 1e-05; relTol 0; }
        p   { tolerance 5e-04; relTol 0; }
    }
}

In transient runs, the outer PIMPLE loop should converge in 2–3 iterations per time step. If it always runs the maximum correctors, reduce the time step (lower CFL) or increase outer correctors.

Monitoring forces alongside residuals

Residuals alone are insufficient to declare convergence — forces or pressure drops can still oscillate even when residuals are low. Always monitor engineering quantities:

functions
{
    forces
    {
        type            forces;
        libs            (forces);
        patches         (bodyWall);
        rho             rhoInf;
        rhoInf          1.225;
        CofR            (0 0 0);
        writeControl    timeStep;
        writeInterval   1;
    }
}

If forces oscillate while residuals are low, the flow is likely inherently unsteady (vortex shedding, recirculation). Switch to a transient solver and time-average the results.

What good convergence looks like

For a steady-state simulation with simpleFoam:

• Residuals should drop monotonically over the first 100–300 iterations
• U, p residuals reaching 1e-4 to 1e-5 is typically sufficient for engineering quantities
• Turbulence quantities (k, epsilon, omega) typically converge later and to a higher residual level than U and p
• Residuals that plateau and oscillate slightly (e.g. between 1e-4 and 5e-4) after an initial drop are normal — the solution has converged

Diagnosing stalled or diverging residuals

Residuals stuck above 1e-2 and not decreasing:

• Under-relaxation too aggressive (U > 0.7, p > 0.3) — reduce both
• Unbounded div scheme (Gauss linear) — switch to linearUpwind
• Poor mesh quality — run checkMesh

Residuals that decrease initially then spike back up:

• Classic sign of numerical instability — reduce relaxation factors further
• Check for backflow at the outlet (pressureInletOutletVelocity on outlet)

p residual much higher than U residual:

• Pressure solve not tight enough — reduce p relTol or increase GAMG agglomeration levels
• For highly stretched meshes, try increasing nNonOrthogonalCorrectors

Plotting residuals

Use foamMonitor (built-in) or extract with a simple grep:

# Extract initial residuals for Ux from log
grep "Solving for Ux" log.simpleFoam | awk '{print $9}' | tr -d ','

# Extract all fields into logs/ directory for gnuplot
foamLog log.simpleFoam

# Live monitoring during a running simulation
foamMonitor -l postProcessing/residuals/0/residuals.dat

CFDpilot generates residual plots automatically from your uploaded log file.