Systematic Errors

In Computational Fluid Dynamics (CFD), the numerical solutions are subject to three main types of systematic errors: modeling errors, discretization errors, and iteration (or convergence) errors. Each of these errors has distinct characteristics and implications for the accuracy and reliability of the simulation results.

1. Modeling Errors

Explanation:

  • Definition: Modeling errors arise from the simplifications and assumptions made during the formulation of the mathematical model used to describe the physical problem. These errors represent the difference between the real-world physical phenomena and the exact solution of the mathematical model.
  • Causes:
    • Simplified Physics: Simplifying the governing equations (e.g., using RANS instead of LES or DNS for turbulence modeling).
    • Geometry and Boundary Conditions: Simplifying the geometry of the domain or using approximate boundary conditions can also introduce modeling errors.
    • Physical Assumptions: Assumptions such as incompressibility, steady-state conditions, or isotropic turbulence may not fully capture the actual physics of the problem.

Minimization:

  • Refined Models: Use more accurate and sophisticated models that capture the relevant physics (e.g., using LES or DNS for turbulent flows).
  • Validation: Compare simulation results with experimental data or results from more accurate models to assess the impact of modeling assumptions.
  • Sensitivity Analysis: Perform sensitivity analyses to understand how different assumptions affect the results and refine the model accordingly.
  • Detailed Geometry and Boundary Conditions: Use detailed and accurate representations of the geometry and boundary conditions to minimize discrepancies.

2. Discretization Errors

Explanation:

  • Definition: Discretization errors occur when continuous partial differential equations (PDEs) are approximated by discrete algebraic equations on a computational grid. These errors represent the difference between the exact solution of the PDEs and the exact solution of the discretized equations.
  • Causes:
    • Grid Resolution: A coarse grid leads to larger discretization errors. The accuracy of the numerical solution depends on the fineness of the grid.
    • Order of the Numerical Scheme: The truncation error associated with the numerical scheme used (e.g., first-order vs. second-order schemes) affects the magnitude of discretization errors.
    • Numerical Diffusion: Low-order schemes can introduce numerical diffusion, smearing out sharp gradients in the solution.

Minimization:

  • Grid Refinement: Use a finer grid, especially in regions with steep gradients, to reduce discretization errors. Grid independence studies help ensure that the solution does not change significantly with further grid refinement.
  • Higher-Order Schemes: Employ higher-order numerical schemes that have lower truncation errors to improve accuracy.
  • Adaptive Mesh Refinement (AMR): Use adaptive mesh refinement techniques that automatically refine the grid in regions requiring higher resolution.
  • Consistency Checks: Compare results obtained with different grid resolutions and discretization schemes to ensure consistency and minimize errors.

3. Iteration (Convergence) Errors

Explanation:

  • Definition: Iteration errors, also known as convergence errors, occur when iterative methods are used to solve the discretized algebraic equations, and the solution has not fully converged to the exact solution of these equations. These errors represent the difference between the exact solution of the discretized equations and the solution obtained after a finite number of iterations.
  • Causes:
    • Insufficient Iterations: Stopping the iterative process too early before the solution has converged.
    • Poor Convergence Criteria: Inadequate or improperly set convergence criteria can result in large iteration errors.
    • Numerical Instabilities: Poorly chosen initial conditions, under-relaxation factors, or numerical methods can cause instability or slow convergence.

Minimization:

  • Adequate Iterations: Ensure that the iterative solver runs for a sufficient number of iterations until the residuals (differences between successive iterations) reach a satisfactorily low threshold.
  • Robust Convergence Criteria: Set strict convergence criteria based on the reduction of residuals or the stabilization of key quantities (e.g., mass flow rates, pressure).
  • Initial Guess and Relaxation: Provide a good initial guess and use appropriate relaxation factors to enhance convergence stability and speed.
  • Residual Monitoring: Continuously monitor the residuals and other relevant physical quantities to ensure that the solution is converging towards a steady state.

Summary

  • Modeling errors are minimized by refining the physical models and validating against experimental data.
  • Discretization errors are minimized by refining the grid, using higher-order schemes, and conducting grid independence studies.
  • Iteration errors are minimized by ensuring sufficient iterations, using proper convergence criteria, and monitoring residuals.

Understanding and managing these errors is crucial for obtaining accurate and reliable results in CFD simulations.