Navier-Stokes solver

The Navier-Stokes solver controls the compressible flow equations in Flow360. It determines how the simulation solves for fluid velocity, pressure, and density throughout the domain. Proper configuration of these parameters is essential for achieving accurate and converged solutions.

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Available Parameters

Parameter	Description	Applicable
Absolute tolerance	Convergence criteria for NS residual	always
Relative tolerance	Relative residual reduction for unsteady cases	always
Kappa MUSCL	Parameter controlling the upwind/downwind/central scheme influence	always
Order of accuracy	Spatial discretization order (1st or 2nd)	Advanced
Equation evaluation frequency	Frequency of solving equations	Advanced
Linear solver	Linear solver selection: Normal (default) or Krylov	Advanced
Linear solver - max iterations	Maximum number of linear solver iterations or Krylov iterations	Advanced
Linear solver - max preconditioner iterations	Number of preconditioner sweeps per Krylov iteration	Advanced, Linear solver is Krylov
Linear solver - relative tolerance	Relative tolerance for the Krylov linear solver convergence	Advanced, Linear solver is Krylov
CFL multiplier	Factor applied to CFL specified in Time Stepping	Advanced
Numerical dissipation factor	Controls dissipation	Advanced
Limit velocity	Whether to apply limiters to velocity	Advanced, Linear solver is Normal
Limit pressure density	Whether to apply limiters to pressure/density	Advanced, Linear solver is Normal
Solve type	Compressible or incompressible formulation	Advanced
Low Mach preconditioner	Acceleration technique for low Mach flows	Advanced
Low Mach preconditioner threshold	Minimum Mach number for preconditioner scaling	Advanced, Low Mach preconditioner is On
Update Jacobian frequency	How often to update the Jacobian matrix	Advanced
Max force jacobian update physical steps	Controls Jacobian matrix update criteria	Advanced
Line search - residual growth threshold	Pseudotime nonlinear residual norm convergence ratio above which residual norm increase is allowed	Advanced, Linear solver is Krylov
Line search - max residual growth	Hard cap on the residual norm ratio — never allow the residual norm to grow beyond this factor over a single pseudotime step	Advanced, Linear solver is Krylov
Line search - activation step	Pseudotime step threshold before the max residual growth limit is activated	Advanced, Linear solver is Krylov

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Detailed Descriptions

Absolute tolerance

The primary convergence metric for how far to reduce the Navier-Stokes residuals.

• Default: 1.0e-10

• Example: 1.0e-8

Notes:

• Steady state: the simulation runs pseudo-steps until the residual falls below this threshold (or the maximum number of steps is reached).

• Unsteady: within each physical step, pseudo-steps are run until the residual falls below this threshold.

• For steady cases, at least 5 orders of magnitude reduction is recommended.

• Smaller values result in more accurate solutions but may require more iterations.

Relative tolerance

Relative residual reduction target for the Navier-Stokes equations per physical step.

• Default: 0.0

• Example: 1e-2

Notes:

• Applicable only to unsteady simulations.

• Specifies the required reduction relative to the initial residual of each physical step (e.g., 1e-2 means the residual must drop by two orders of magnitude).

• Within each physical step, pseudo-steps are run until this relative reduction is achieved, the absolute tolerance is met, or max_pseudo_steps is reached — whichever comes first. The solver then advances to the next physical step.

Kappa MUSCL

Controls the blending between upwind and central schemes in the spatial discretization.

• Default: -1 (second-order upwind scheme)

• Example: 0.33 (blended scheme for low dissipation)

Notes:

• Range: [-1, 1] (values above 0.33 not recommended)

• -1: Pure second-order upwind (most stable)

• 0.33: Blended upwind/central scheme (less dissipative, good for low subsonic flows)

• 1.0: Unstable full downwind scheme (avoid)

Advanced

Order of accuracy

Determines whether the solver uses 1st or 2nd order spatial discretization.

• Default: 2 (second order)

• Example: 1 (first order for initialization)

Notes:

• 1st order: More stable but more dissipative, less accurate

• 2nd order: More accurate but less stable

• For initial flow field development in unsteady cases with rotating components, running 1st order for 1-2 revolutions can help establish the flow before switching to 2nd order

• Should match the turbulence model’s order of accuracy

Equation evaluation frequency

Controls how often (in pseudo steps) the Navier–Stokes equations are solved.

• Default: 1

• Example: 2–4

Notes:

• 1 provides the most robust convergence

• Higher values can reduce time cost but may slow down convergence

Linear solver

Selects the linear solver used at each pseudo-step of the Navier-Stokes equations.

Possible selections:

• Normal (default) — Standard iterative linear solver. Suitable for all simulation types including unsteady.

• Krylov — Krylov iterative solver. Offers improved convergence speed and reduced iteration counts for steady-state simulations.

⚠️ Krylov solver restrictions:

• Steady simulations only. The Krylov solver cannot be used with Unsteady time stepping.

• Incompatible with velocity and pressure/density limiters. Do not enable Limit velocity or Limit pressure density when using the Krylov solver.

• Kappa MUSCL = 0.33 may cause convergence stalling. If residuals stall at final convergence levels, try Kappa MUSCL values of 0 or -0.33 instead.

Linear solver - max iterations

Maximum number of linear solver iterations per pseudo-step.

• Default: 30 (Normal solver) / 15 (Krylov solver)

• Range: 1–50 (Krylov solver); no upper bound for Normal solver

• Example: 50 (Normal) / 20 (Krylov)

Notes:

• Normal solver: Typical range is 25–35 for the NS solver. Increase to 50–55 if the linear residual reduction ratio is insufficient.

• Krylov solver: Controls the number of the Krylov iterations used per pseudo-step. Start with the default of 15. If convergence is stalled, try increasing this value. If the case converges well, try reducing it for better performance.

Linear solver - max preconditioner iterations

Number of preconditioner iterations applied during each Krylov iteration.

Applicable when Linear solver is Krylov.

• Default: 25

• Range: 10–35 (typical)

• Example: 30

Notes:

• More iterations improve preconditioner quality but increase cost per iteration.

• Try increasing this value before increasing max iterations if convergence is insufficient.

Linear solver - relative tolerance

Target relative convergence tolerance for the Krylov linear solver. The solver considers the linear system converged when the residual drops below this fraction of the initial residual.

Applicable when Linear solver is Krylov.

• Default: 0.05

• Example: 0.01

Notes:

• Tighter tolerances (smaller values) improve accuracy but require more iterations.

• The adaptive CFL strategy uses this tolerance to decide when to increase or limit the CFL. Setting this value too low may prevent CFL from ramping up effectively.

• For low Mach number cases, values as low as 1e-3 may be needed for good convergence.

CFL multiplier

Scaling factor applied to the CFL number defined in Time Stepping.

• Default: 1.0

• Example: 0.5 (for better stability)

Notes:

• Reducing this value can help stability for challenging cases

• Effective CFL = Time Stepping CFL × CFL Multiplier

Numerical dissipation factor

Factor that controls the reduction in numerical flux dissipation for the low-dissipation Roe scheme.

• Default: 1.0 (standard scheme)

• Example: 0.2 (low dissipation)

Notes:

• Range: [0.01, 1.0]

• Recommended value: 0.2 for low-dissipation

Limit velocity

Applies limiters to velocity to enhance stability in transonic/supersonic flows.

• Default: False

• Example: True (for transonic flows)

• Applicable: when Linear solver is Normal

Notes:

• Enable for transonic or supersonic cases

• Can help prevent divergence in regions with strong shocks

Limit pressure density

Applies limiters to pressure and density to enhance stability in transonic/supersonic flows.

• Default: False

• Example: True (for transonic flows)

• Applicable: when Linear solver is Normal

Notes:

• Enable for transonic or supersonic cases

• Can help prevent negative pressures or densities near shocks

Solve type

Selects the formulation used for the Navier-Stokes equations.

Possible selections:

• Compressible (default) — Standard compressible Navier-Stokes formulation. Suitable for all Mach number regimes.

Notes:

• For most simulations, the default Compressible setting is appropriate

Low Mach preconditioner

Improves solver convergence for low Mach number flows by addressing the stiffness in the equation system.

• Default: False

• Example: True (for flows with \(M <\) 0.3)

Notes:

• Recommended for simulations with Mach number below 0.3

• Particularly beneficial for nearly incompressible flows

• Can significantly accelerate convergence for low-speed aerodynamics

Low Mach preconditioner threshold

Minimum Mach number used in the preconditioner scaling.

Applicable when Low Mach preconditioner is On.

• Default: Freestream Mach number (if not specified)

• Example: 0.1

Notes:

• For flow regions with Mach numbers smaller than this threshold, the threshold value is used

• Only relevant when low Mach preconditioner is enabled

Update Jacobian frequency

Controls how often the Jacobian matrix is updated during the solution process.

• Default: 4 (every 4 pseudo-steps)

• Example: 1 (every pseudo-step for challenging cases)

Notes:

• Smaller values improve robustness but increase computational cost

• Reducing to 1 may help convergence for difficult cases

• May slow the solver by up to approximately 30% when set to 1

Max force jacobian update physical steps

If the current physical step is lower than this value, jacobian matrix will be updated at every pseudo step.

• Default: 0 (disabled)

• Example: 10

Note: Useful for cases where more stability is needed at the beginning

Line search - residual growth threshold

Pseudotime nonlinear residual norm convergence ratio above which residual norm increase is allowed. This ensures the residual norm only increases when the pseudotime residual has decreased by at least the specified ratio, preventing non-physical steps from degrading convergence.

• Default: 0.85

• Range: 0.5–1.0

• Applicable: when Linear solver is Krylov

Note: The line search works together with the adaptive CFL strategy — CFL is only increased when full steps are accepted and is reduced when steps need to be scaled back.

Line search - max residual growth

Hard cap on the allowed residual growth ratio per step. The line search ensures the residual does not grow by more than this factor in a single step, rejecting or scaling back updates that exceed it.

• Default: 1.1

• Range: ≥ 1.0

• Applicable: when Linear solver is Krylov

Line search - activation step

Pseudo-step number after which the max residual growth limit is enforced.

• Default: 100

• Example: 200

• Applicable: when Linear solver is Krylov

Notes:

• Delays enforcement of the max residual growth limit during early pseudo-steps.

• Particularly useful for angle-of-attack sweeps with restart, where residuals typically climb at the beginning of the simulation before settling down.

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💡 Tips

• For steady-state simulations:

  • Focus on absolute tolerance for convergence criteria

  • Consider using low Mach preconditioner for low-speed flows

  • Use 2nd order accuracy for final results

• For unsteady simulations:

  • Use relative tolerance (typically 1e-2 or 1e-3)

  • For cases with rotating components, consider starting with 1st order for initialization

  • Use AdaptiveCFL instead of ramp CFL strategy to automatically balance stability and convergence speed

• For low-dissipation schemes:

  • Start from a converged steady or unsteady solution

  • Aim for at least a two-order-of-magnitude reduction in nonlinear residuals

  • Consider halving the time step size

  • Use Kappa MUSCL = 1/3 for optimal accuracy

  • Set update Jacobian frequency = 1 for better stability

• For transonic or supersonic flows:

  • Enable limiters for velocity and pressure/density (not available with Krylov linear solver)

  • Use a more dissipative scheme (Kappa MUSCL = -1)

• For steady simulations where convergence is slow or stalls:

  • Consider switching the linear solver from Normal to Krylov

  • Start with the default Krylov settings (max iterations = 15, max preconditioner iterations = 25, relative tolerance = 0.05)

  • If residuals stall when using Kappa MUSCL = 0.33 with the Krylov solver, try values of 0 or -0.33

  • For low Mach number cases with the Krylov solver, try lowering the relative tolerance (e.g., to 1e-3) if convergence is difficult

• For Krylov cases that struggle to converge, adjust the line search parameters:

  • If the CFL is being cut too aggressively, increase the residual growth threshold (e.g., from 0.85 toward 1.0) to allow more residual growth

  • If residuals stall or oscillate excessively, lower the residual growth threshold to enforce stricter convergence per step

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❓ Frequently Asked Questions

• When should I use 1st order vs. 2nd order accuracy?

  Use 1st order for initial flow field development or for very challenging cases with strong discontinuities. Use 2nd order for final production runs to achieve accurate results. For unsteady simulations with rotating components, consider starting with 1st order for 1-2 revolutions before switching to 2nd order.

• How do I know if my simulation has converged properly?

  For steady simulations, look for at least 5 orders of magnitude reduction in residuals and stable force coefficients. For unsteady simulations, ensure residuals drop by 2-3 orders of magnitude within each physical time step.

• What can I do if my simulation is diverging?

  Try the following: (1) Use AdaptiveCFL instead of manual CFL settings to automatically adjust for stability, (2) Reduce the convergence limiting factor for complex cases with strong gradients or discontinuities, (3) Reduce the CFL number if using ramp CFL settings, (4) Switch to 1st order accuracy temporarily, (5) Enable limiters for transonic/supersonic flows, (6) Update the Jacobian more frequently (set update_jacobian_frequency = 1), (7) Check the mesh quality, particularly in high-gradient regions.

• When should I use the low-dissipation scheme?

  Use it for unsteady simulations where capturing fine flow structures is important, especially for aeroacoustics, turbulent flows, and vortex-dominated flows. It works best for low Mach number flows and should be started from a well-developed solution.

• How does the Kappa MUSCL parameter affect my solution?

  Kappa_MUSCL = -1 (default) gives a stable upwind scheme with higher numerical dissipation. Kappa MUSCL = 0.33 gives a less dissipative scheme that preserves flow features better but may be less stable. For low-speed flows without strong shocks, Kappa MUSCL = 0.33 often provides better results.

• What’s the difference between absolute and relative tolerance?

  Absolute tolerance sets a fixed residual threshold and applies to both steady and unsteady simulations. For steady simulations it is the sole convergence driver: pseudo-steps run until the residual drops below this value. Relative tolerance applies only to unsteady simulations and specifies a required reduction relative to the initial residual of each physical step (e.g., 1e-2 → two orders of magnitude drop). Within each physical step, pseudo-steps run until either tolerance is met or max_pseudo_steps is reached, then the solver advances to the next physical step.

• When should I use the Krylov linear solver instead of the Normal solver?

  The Krylov solver offers improved convergence speed and reduced iteration counts for steady-state simulations. It automatically enables an enhanced adaptive CFL strategy and adjusts equation evaluation frequencies for optimal performance. It is only available for Steady simulations and cannot be combined with Limit velocity or Limit pressure density. For unsteady cases, continue using the Normal solver.

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🐍 Python Example Usage

Below is a Python code example showing how to configure the Navier-Stokes solver:

import flow360 as fl

# Basic configuration for the Navier-Stokes solver
ns_solver = fl.NavierStokesSolver(
    absolute_tolerance=1e-10,
    order_of_accuracy=2
)

# Configuration for a low-speed simulation
ns_solver_low_mach = fl.NavierStokesSolver(
    absolute_tolerance=1e-10,
    low_mach_preconditioner=True,
    low_mach_preconditioner_threshold=0.1,
    kappa_MUSCL=0.33
)

# Configuration for an unsteady simulation with low dissipation
ns_solver_unsteady = fl.NavierStokesSolver(
    absolute_tolerance=1e-10,
    relative_tolerance=1e-3,
    numerical_dissipation_factor=0.2,
    update_jacobian_frequency=1,
    linear_solver=fl.LinearSolver(max_iterations=50)
)

# Configuration for a transonic/supersonic flow
ns_solver_transonic = fl.NavierStokesSolver(
    absolute_tolerance=1e-10,
    limit_velocity=True,
    limit_pressure_density=True,
    CFL_multiplier=0.5
)

# Configuration using the Krylov linear solver for steady simulations
# NOTE: Krylov solver is only valid with Steady time stepping
# NOTE: Do NOT combine with limit_velocity=True or limit_pressure_density=True
ns_solver_krylov = fl.NavierStokesSolver(
    absolute_tolerance=1e-10,
    linear_solver=fl.KrylovLinearSolver(
        max_iterations=15,
        max_preconditioner_iterations=25,
        relative_tolerance=0.05,
    ),
)

# Krylov solver with optional line search for additional stability during early pseudo-steps
ns_solver_krylov_with_line_search = fl.NavierStokesSolver(
    absolute_tolerance=1e-10,
    linear_solver=fl.KrylovLinearSolver(
        max_iterations=15,
        max_preconditioner_iterations=25,
        relative_tolerance=0.05,
    ),
    line_search=fl.LineSearch(
        residual_growth_threshold=0.85,
        max_residual_growth=1.1,
        activation_step=100,
    ),
)