Grid Quality and Resolution Issues from the Drag

Prediction Workshop Series

The DPW Committee

Dimitri Mavriplis : University of Wyoming USAJ. Vassberg, E. Tinoco, M. Mani : The Boeing Company USAO. Brodersen, B. Eisfeld: DLR Braunschweig, GERMANYR. Wahls, J. Morrison: NASA Langley Research Center , USAT. Zickhur, D. Levy: Cessna Aircraft Co. USAM. Murayama: Japan Aerospace Exploration Agency, JAPAN

Motivation

• DPW Series – Assess State-of-art for Transonic Cruise Drag

Prediction using RANS methods• DPW I: Anaheim CA, June 2001• DPW II: Orlando FL, June 2003• DPW III: San Francisco CA, June 2006• DPW IV: June 2009

– Considerable scatter in results particularly for cases with flow separation (off-design)

– Emerging Consensus• Discretization errors are (a) dominant source of error

Motivation

• DPW focused increasingly on assessing discretization/grid induced errors– DPW I: Single grid study– DPW II: Grid convergence study (3 grids)– DPW III: All results examined in context of grid

convergence study (3 or 4 grids)

• Implications– Dominant discretization errors preclude accurate

assessment of other errors• Turbulence/transition modeling

Motivation

• DPW demonstrated grid convergence for some codes mostly for attached flow cases

• Separated flow cases much more difficult to obtain grid independent results

• Scatter often does not decrease with increasing grid resolution

• Contradictory grid convergence results– Different grid families converge to different results

Overview

• Overview of DPW test cases

• DPW Gridding Guidelines

• Discussion of gridding issues– Grid Resolution– Grid Convergence– Grid Quality

• Possible improvements

• Conclusions

DLRF4-F6 Test Cases (DPW I,II,III)

• Wing-Body Configuration• Transonic Flow• Mach=0.75, Incidence = 0 degrees, Reynolds number=3,000,000

DPW III Series Cases

• Designed fairing to suppress flow separation (Vassberg et al. AIAA 2005-4730)

DPW III Series Cases

• 2 closely related simple wing geometries– Well behaved flow– Enhanced grid refinement study (4 grids)

General Gridding Guidelines• Grid Resolution Guidelines

– BL Region• Y+ < 1.0, 2/3, 4/9, 8/27 (Coarse,Med,Fine,VeryFine)• 2 cell layers constant spacing at wall• Growth rates < 1.25

– Far Field: 100 chords – Local Spacings (Medium grid)

• Chordwise: 0.1% chord at LE/TE• Spanwise spacing: 0.1% semispan at root/tip• Cell size on Fuselage nose, tail: 2.0% chord

– Trailing edge base:• 8,12,16,24 cells across TE Base (Coarse,Med,Fine,Veryfine)

• Grid Convergence Sequences– X3 increase in resolution per refinement– Maintain same family of grids in sequence

Overset Meshes (DPW III)

Overset Meshes (DPW III)

Structured Multi-Block Wing-Body GridsConstructed with Boeing Zeus/Advancing Front Method

Typical Wing Grid H-H Topology

Embedded Blunt Trailing Edge Grid Block

VGRID : Wing Body (~40M pts)

VGRID : Wing Alone (~30M pts)

DPW Submitted Grids

• Wide variety of grid types and constructions

• Grid topology and type affects local resolution

• Compliance with guidelines not evaluated precisely

• Large data-base of high-quality aero grids made available

DPW I RESULTS (circa 2001)

• Drag polar for single grid resolution

DPW II RESULTS (circa 2003)

• Drag vs number of grid points (Wing-body alone)

DPW III RESULTS (2006)

• Idealized drag vs grid index factor (N-2/3)– Wing-body and Wing-body+fairing

Grid Related Experiences from DPW

• Grid Resolution

• Grid Convergence

• Grid Quality

Grid Resolution

• Always need more– DPW I: ~ 3M pts– DPW III: ~ 40M pts– Interim/Follow-on studies/DPW4: > 100M pts– Grid convergence studies point to need for > 109 pts

• Wide range of scales present in aerodynamics– Highly variable:

• Far field ~100 MAC• Trailing edge ~.01 MAC

– Anisotropic:– Boundary Layer Y+=1: ~ 10-6 MAC

Grid Resolution

• Wide range of scales requires:– Intuition or rule-based grid generation– Anisotropic in Boundary Layer (and spanwise)– Codified in DPW guidelines

• Effect of Grid Resolution is Complex– Direct effect on surface profiles is small– Indirect effect can be large

• Location of separation• Integration of small differences Lift, Drag, Moment

W1 Grid Convergence Study

• CP at station 5:

W1 Grid Convergence Study

• CP at station 5:

W1 Grid Convergence Study

• CP at station 5:

W1 Grid Convergence Study

• CP at station 5:

Effect of Normal Spacing in BL

• Inadequate resolution under-predicts skin friction– Direct influence on drag prediction– Indirect influence: Wrong separation prediction

Effect of Normal Resolution for High-Lift

(c/o Anderson et. AIAA J. Aircraft, 1995)

• Indirect influence on drag prediction• Easily mistaken for poor flow physics modeling

Grid Resolution

• Separated flow cases more demanding and often contradictory experiences

Grid Resolution• Side-of-Body Separation increases with grid resolution

– Boeing: Overset– Boeing: Unstructured– DLR: Unstructured

• Side-of-Body Separation constant with grid resolution– Boeing: Block Structured– JAXA: Block Structured, Unstructured

• Trailing edge separation grows with grid res:– UW : Unstructured (NSU3D)

• Trailing edge separation constant with grid res:– JAXA: Structured, Unstructured– Boeing: Overset

• Experimentation with much finer grids required to understand behavior…

Grid Convergence

• Increased focus of DPW Series

• For second-order accurate method, error should decrease as O(h2)– Define average cell size h as: N-1/3

• N=number of grid pts

– Drag vs N-2/3 should plot as straight line– Project to y-axis to get continuum value

Importance of Grid Convergence

Agreement on initial grid (DPW I) gets worse (Lee-Rausch et al. AIAA-2003-3400)

Grid Convergence

• Grids must come from same “family”– Self-similar topologically– Same relative variations of resolution

• Achieved through IJK factors for structured grids• Requires global grid spacing factor for unst. grids• Boundary layer growth must be taken into account

• Not clear how well all grids meet these requirements– Most likely represents state-of-art

• Perform grid convergence at fixed Lift or fixed incidence conditions ?

• Grid convergence for attached flow cases• Inconsistent behavior for separated flow case

– Separation bubble grows with grid resolution

Grid Convergence (Overflow)

• More consistent grid convergence at fixed CL

Grid Convergence (Wing Alone)

W1-W2 Grid Convergence Study(NSU3D Unstructured)

•Apparently uniform grid convergence

W1-W2 Results

• Discrepancy between results on 2 different families of grids (both generated with VGRID)

W1-W2 Results

• Removing effect of lift-induced drag : Results on both grid families converge consistently

– Consistent grid convergence at fixed CL instead of alpha

Grid Quality

• Distinguish grid quality from grid resolution– Relative distribution of resolution– Topology– Element type/shape– Aspect ratio– Orthogonality (BL, hybrid)

• Grid quality is (should be) constant for self-similar family of grids used for grid convergence study

Two Unstructured Grid Topologies

65 million pt grid72 million pt grid

High Resolution grids for DLR-F6 (DPW II) using NSU3D solver

Grid Convergence on Topology #1

• Drag is grid converging• Sensitivity to dissipation decreases as expected

65M pt mesh Results

• 10% drop in CL at AoA=0o: closer to experiment• Drop in CD: further from experiment• Same trends at Mach=0.3• Little sensitivity to dissipation

Grid Convergence• Grid convergence apparent using self-similar family of grids

• Large discrepancies possible across grid families– Sensitive areas

• Separation, Trailing edge• Pathological cases ?

• Would grid families converge to same result limit of infinite resolution ?– i.e. Do we have consistency ?– Due to element types ?, Aspect ratio ?

• Possible ways forward:– Higher order discretizations– Adjoint-based error estimation

hp-adaptive DG Li Wang and Dimitri Mavriplis

Adjoint-Based Spatial Error Estimation + AMRAdjoint-Based Spatial Error Estimation + AMR

Adjoint Solution : Green’s Function for Objective (Lift)

Change in Lift for Point sources of Mass/Momentum

Error in objective ~ Adjoint . Residual (approx. solution)

Predicts objective value for new solution (on finer mesh)

Cell-wise indicator of error in objective (only)

hp-adaptive DG Li Wang and Dimitri Mavriplis

hh-refinement for target functional of lift-refinement for target functional of lift

Fixed discretization order of p = 1

Final h-adapted mesh (8387 elements) Close-up view of the final h-adapted mesh

hp-adaptive DG Li Wang and Dimitri Mavriplis

Comparison between h-refinement and uniform mesh refinement

Error convergence history vs. degrees of freedom

Functional Values and Corrected Values

hh-refinement for target functional of lift-refinement for target functional of lift

Complex Geometry: Vehicle Stage Separation(CART3D/inviscid)

Top View

Side View

• Initial mesh contains only 13k cells

• Final meshes contain between 8M to 20M cells

Initial Mesh

Pressure Contours

M∞=4.5, α=0°

• Minimal refinement of inter-stage region

• Gap is highly refined

• Overall, excellent convergence of functional and error estimate

Cutaway view of inter-stage

Unsteady Problems

Total error in solution

Temporal error

(discretization/resolution)

Spatial error

(discretization/resolution)

Flow

Algebraic error

Mesh Other Flow Mesh Other Flow Mesh Other

•Solution of time-dependent adjoint: backwards integration in time•Disciplinary adjoint inner product with disciplinary residual

•Interaction of isentropic vortex with slowly pitching NACA0012

•Mach number = 0.4225

•Reduced frequency = 0.001

•Center of pitch is quarter chord

•Functional is

Time-integrated functional

8,600 elements

Unsteady Adjoint Error Estimation

Density contours of initial condition

Comparison of adapted temporal domain

Temporal Error Adaptation

Algebraic Error Adaptation

Adapted Flow/Mesh convergence tolerances:

Adjoint-Based Refinement Results

•Error in Lift versus CPU TimeUniform cost is only finest solution costAdaptive cost is all solutions (+ adjoint cost)Corrected value provides further improvement

Conclusions• Grid related issues are dominant error source in drag

prediction

• Grid and solver interaction is complex

• Drive toward much higher resolution

• Grid quality difficult to assess

• Inconsistent grid convergence results point to possible inconsistent errors : O(1)

• All these issues much more prevalent for separated flows

Conclusions• A posteriori error estimation

– Criteria for adaptive meshing– Gradient based– Adjoint based for specific outputs

• Looks promising– Extends to temporal, algebraic error from different disciplines– Combine spatial, temporal, algebraic…– Still a linearization about current solution– Slow to production– Inconsistent errors not resolvable with AMR

• A priori error estimation– Grid quality metrics– Approximation error of test functions