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The L2 norm of Q is commonly called the residual of the equation solution
and is computed as
where N, total number of grid points
M, number of elements in Q (5 for Navier-Stokes Equations)
• It represents the change in the solution over an iteration averaged over all
the grid points and equations.
• The L2 residual is usually displayed and plotted as its log values, [log(L2)].
• This shows the order-of-magnitude of the change in the solution.
• Generally desire the L2 residual to approach zero with iterations.
L2 Residual
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Reporting.ppt
- 2. 2 Objective is to achieve a high level of credibility and confidence in the results from CFD performed as part of the design and analysis of a propulsion system. Credibility and confidence are obtained by demonstrating acceptable levels of error and uncertainty as assessed through verification and validation. This discussion is focused on the question… How good are your CFD results? Determining this precisely is often difficult, but we can try to understand sources of error and ways to demonstrate quality CFD results. Objective
- 3. 3 The characteristics of flows of interest may include: • Mach number range (Static,Mach 0 to hypersonic, Mach 25) • Shock waves. • High Reynolds numbers (laminar transitioning to turbulent). • Turbulent, viscous flow. • Shock / boundary layer interactions. • Boundary layer separation. • High-temperature effects at hypersonic Mach numbers. • Compressor face effects. • Unsteady flows (hammershocks, unstart) Flow Characteristics
- 4. 4 The CFD analysis may provide: • Steady-state flow at cruise conditions. • Features of the shock system (positions, cowl stand-off). • Features of the boundary layers (thickness, separation). • Pressure recovery at throat and compressor face. • Spillage (side walls, cowl). • Bleed requirements (slots, bleed holes). • Distortion at compressor face. • Drag. • Off-design flow features. • Unsteady flow features (unstart, stability). • Aerodynamic and thermal loads. Objectives of a CFD Analysis
- 5. 5 1. Provide Qualitative Information. Provide features of flow field (conceptual, rough estimates of quantities). Accuracy requirements are low. 2. Provide Incremental Quantities, P. • P = Pbaseline + P • Errors partially cancel for P • P = (P2-P1)actual + (E2-E1) = Pactual + E • Moderate accuracy requirement. 3. Provide Absolute Quantities, P. Requires high level of accuracy. Levels of the Use of CFD Results
- 6. 6 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.” (AIAA-G-077-1998). Verification checks that there are no programming errors and that the coding correctly implements the equations and models. It also examines discretization errors in the CFD calculations. 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.” (AIAA-G-077-1998). Validation checks that the CFD results agree with reality as observed in experiments, flight tests, or applications. Verification and Validation
- 7. 7 Uncertainty “A potential deficiency in any phase or activity of the modeling process that is due to the lack of knowledge.” (AIAA-G-077-1998). The use of potential indicates that deficiencies may or may not exist. Error “A recognizable deficiency in any phase or activity and simulation that is not due to lack of knowledge.” (AIAA-G-077-1998). The deficiencies are knowable up examination. Acknowledged errors have procedures for identifying and removing them. Unacknowledged errors are undiscovered and no set procedures exist to find them. Uncertainty and Error
- 8. 8 Total Error = Physical Modeling Error + Discretization Error + Programming Error + Computer Round-off Error + Usage Error + Post-Processing Error Physical modeling errors are usually acknowledged and may be quantified. Discretization errors are grid dependent but a priori knowledge of the proper grid for a desired level of accuracy is generally lacking. Programming errors should have been minimized by programmers. Computer round-off errors are usually negligible or controllable. Usage errors are minimized by proper training. Post-processing errors are usually controllable. Errors in CFD Results
- 9. 9 Many physical models are used in CFD and we must understand the limits and possible errors with each model: • Spatial Dimensions. (2D, axisymmetric, 3D) • Temporal Dimensions. (steady-state, unsteady) • Equations. (Navier-Stokes, approximations) • Turbulence Model. • Thermodynamic and Transport Property model. • Chemistry Model. • Flow Boundary Conditions (inflow, outflow). • Compressor Face Model. • Bleed / Blowing Model. • Vortex Generator Model. Physical Models in CFD
- 10. 10 Errors that exist because continuum flow equations and physical models are represented in a discrete domain of space (grid) and time. Level of error is function of interactions between the solution and the grid. Since solution is unknown before-hand, the grid generation may not produce the optimum grid. Discretization error is quantified through verification methods that examine grid convergence. Discretization error is controllable. Discretization Error
- 11. 11 Convergence used in two ways: Iterative Convergence Simulations should demonstrate iterative convergence. As the algorithm iterates the solution, the simulation results approach a fixed value (residuals drop and level off). This applies to both steady and unsteady flow solutions. In the case of unsteady flows, the iterative convergence applies to iterations over a time step. Grid Convergence As the grid spacing is reduced, the simulation results become insensitive to the grid and approach the continuum results. Iterative and Grid Convergence
- 12. 12 Iterative Convergence can be demonstrated in several ways: 1. Value of the largest change in the solution over an iteration, Qbig. 2. Iteration history of the L2 norm of the equation residual. 3. Examine conservative variables. 4. Iteration history of a local or global quantity of the flow. Usually some measure of all of these are used to determine iterative convergence. The largest change in the solution over an iteration is simply Iterative Convergence m i Big Q Q , ˆ max ˆ
- 13. 13 The L2 norm of Q is commonly called the residual of the equation solution and is computed as where N, total number of grid points M, number of elements in Q (5 for Navier-Stokes Equations) • It represents the change in the solution over an iteration averaged over all the grid points and equations. • The L2 residual is usually displayed and plotted as its log values, [log(L2)]. • This shows the order-of-magnitude of the change in the solution. • Generally desire the L2 residual to approach zero with iterations. L2 Residual N M Q L N i M m m i * ˆ 2 1 1 2 ,
- 14. 14 Residual Plots Order-of-magnitude drop (about 5) Leveling off of residual psuedo-unsteadiness Navier-Stokes Equations The behavior of the residual is not always so smooth. Unsteadiness (real or pseudo) in the solution (typical of turbulent and separated flows) may limit how far the residual drops. Often a couple of orders-of-magnitude drop is all one achieves.
- 15. 15 Residual Plots (continued) Spalart-Allmaras turbulence model equations
- 16. 16 The solution of the conservation equations must satisfy those conservative principles. Iterative convergence can be demonstrated through examination of the conservative variables: For example, one can examine the conservation of mass through a duct: Examine Conservative Variables Variation of the mass flow through the duct. Statistical analysis of the variation. This indicates the level of iterative convergence error in determining the mass flow through the duct.
- 17. 17 The change in a local or global property is perhaps the best measure of iterative convergence since it typically directly monitors the quantity that is important for the engineering analysis. Change in Local or Global Quantity Lift on the ONERA M6 wing with number of cycles. Local quantity is point value. Global quanity is integrated over a portion of the flow field. Examples include: • Lift on a wing • Drag on a wing • Inlet recovery • Heat flux Global quantities usually show a smooth convergence.
- 18. 18 Verification assessment has two aspects: Verification of a Code Verifies that code has no programming errors and that the coding correctly implements the equations and models. The methods involve examining modules of code, checking basic assumptions (mass conservation), and comparing results to analytic results. Verification of a Calculation Verifies that a calculation (simulation) demonstrates a certain level of accuracy. The primary method for the verification of a calculation is the grid convergence study. Verification Assessment
- 19. 19 Quantifies: 1. “ordered” discretization error band (related to grid size by order p) 2. p, order of grid convergence (order-of-accuracy) 3. continuum or “zero grid spacing’’ value of observed quantity Approach: • Assumed or demonstrated that all other error terms are negligible, minimized, accepted, or under control. • Perform CFD solution on two or more grids of increased refinement. • Solutions must be in the “asymptotic range of convergence’’. • If three solutions, can then compute order of convergence, p. • Use Richardson extrapolation to compute continuum value. • Compute Grid Convergence Indices (GCI) as discretization error band. Grid Convergence Study
- 20. 20 • Viscous wall spacing set for y+ < 1.0 for fine grid. • Flow field resolution set to resolve shock system. • Grid density to keep grid spacing ratio below 15%. • Use ICEM CFD to construct and GRIDGEN to improve. • Subset grid to remove some sub-layer grid points for wall function. • Best if grid refinement ratio, r = h2 / h1 = 2.0 • h is grid spacing. • Minimum, r > 1.1 • WIND allows for grid sequencing (r = 2). Grid Generation Considerations
- 21. 21 • Example grid convergence study with three grids. • Mach 2.35. • r = h2 / h1 = 2. • Asymptotic range observed in plot. • Richardson extrapolation. Example: Supersonic Diffuser
- 22. 22 • Order of grid convergence, p. • Errors reduce as grid is refined. • h is the grid spacing. • C is a constant. • Use three values to solve for order of convergence. • For this example, p = 1.786 Order of Convergence . . . T O H h C f h f E p exact r f f f f p ln ln 1 2 2 3
- 23. 23 • Extrapolation of two quantities to continuum value at “zero grid spacing”. • Provides estimate of error. • Extrapolate is of higher order. • For this example, (p2/p0)h=0 = 0.97130 1 2 1 1 0 p h r f f f f Richardson Extrapolation
- 24. 24 • Standardized method for reporting discretization error. • Based on Richardson extrapolation. • Error band on fine grid solution. • Expressed as percentage. • FS is factor of safety (FS = 1.25 if three or more grids) • For this example, – GCI12 = 0.1031% (0.0010) – GCI23 = 0.3562% (0.0035) • “Pressure recovery is 0.971 with grid error of 0.001.” 1 p s fine r F GCI 1 1 2 f f f Grid Convergence Index (CGI)
- 25. 25 • Solutions are in range in which errors decrease at rate denoted by order of convergence, p. • Check that constant C is indeed constant over solutions, • Check with GCI values over three solutions, • For this example, ratio = 1.002 p h E C 12 23 GCI r GCI p Asymptotic Range of Convergence
- 26. 26 • Validation assessment addresses if CFD code agrees with reality through comparison to experiments. • Experiment must also address its own errors. • Unit Benchmark Subsystem Complete System. • Set of validation cases are being collected and studied that cover the range of flow features present in inlets (turbulent boundary layers, shocks, shock / boundary layer interactions, subsonic ducts, supersonic diffusers). • NPARC Alliance Validation Archive: www.grc.nasa.gov/www/wind/valid Validation Assessment
- 27. 27 In summary, for a CFD analysis, report the following: • Objective of CFD analysis • Geometry simplifications • Grid resolution • Boundary and initial conditions • Equations and physical models • Algorithm settings • Iterative convergence criteria • Grid convergence criteria • Results (values, plots, visualizations) • Errors that can be quantified • Sensitivity to models and parameters (turbulence, chemistry) Reporting CFD Results
- 28. 28 A sample of books and resources for learning more on CFD: Anderson, D.A., Tannehill, J.C., and Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer, McGraw-Hill, 1984. Anderson, J.D., Computational Fluid Dynamics, McGraw-Hill, 1995. Hirsch, C., Numerical Computation of Internal and External Flows (2 Volumes), John Wiley & Sons, 1988 & 1990. Hoffmann, K.A. and S.T. Chiang, Computational Fluid Dynamics (3 Volumes), EES Books, 2000. (also AIAA coarse). Wilcox, D.C., Turbulence Modeling for CFD, DCW Industries, 1998. AIAA Courses (CFD, Turbulence) Grid Generation training (ICEM, Pointwise, VGRID) Further Information on CFD