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Aerodynamic Drag Prediction Using Unstructured Mesh Solvers

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Aerodynamic Drag Prediction Using Unstructured Mesh Solvers

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Aerodynamic Drag Prediction Using Unstructured Mesh Solvers

Dimitri J. Mavriplis

National Institute of Aerospace

Hampton, Virginia, USA

VKI Lecture Series, February 3-7, 2003

- Introduction
- Physical model fidelity
- Grid resolution and discretization issues
- Designing an efficient unstructured mesh solver for computational aerodynamics
- Drag prediction using unstructured mesh solvers
- Conclusions and future work

VKI Lecture Series, February 3-7, 2003

- Introduction
- Importance of Drag Prediction
- Suitability of Unstructured Mesh Approach

- Physical model fidelity
- Inviscid Flow Analysis
- Coupled Inviscid-Viscous Methods
- Large-Eddy Simulations (LES and DES)

VKI Lecture Series, February 3-7, 2003

- Grid resolution and discretization issues
- Choice of discretization and effect of dissipation
- Cell centered vs. vertex based
- Effect of discretization variations on drag prediction

- Grid resolution requirements
- Choice of element type
- Grid resolution issues

- Grid convergence

- Choice of discretization and effect of dissipation

VKI Lecture Series, February 3-7, 2003

- Designing an efficient unstructured mesh solver for computational aerodynamics
- Discretization
- Solution Methodologies
- Efficient Hardware Usage

VKI Lecture Series, February 3-7, 2003

- Drag prediction using unstructured mesh solvers
- Wing-body cruise drag
- Incremental effects: engine installation drag
- High-lift flows

- Conclusions and Future Work

VKI Lecture Series, February 3-7, 2003

- Importance of Drag Prediction
- Cruise: fuel burn, range, etc…
- High-lift: Mechanical simplicity, noise

- High accuracy requirements
- Absolute or incremental: 1 drag count
- Specialized computational methods
- Wide range of scales
- Thin boundary layers
- Transition

VKI Lecture Series, February 3-7, 2003

- Issues centric to unstructured mesh approach
- Advantages and drawbacks over other approaches
- Accuracy, efficiency
- State-of-the art in aerodynamic predictions

- De-emphasize non-method specific issues
- Validation/ verification
- Drag integration

VKI Lecture Series, February 3-7, 2003

- Sophisticated Multiblock Structured Grid Techniques for Complex Geometries

Engine Nacelle Multiblock Grid by commercial software TrueGrid.

- Sophisticated Overlapping Structured Grid Techniques for Complex Geometries

Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)

- Connectivity stored explicitly
- Single Homogeneous Data Structure

VKI Lecture Series, February 3-7, 2003

- Structured Grids
- Logically rectangular
- Support dimensional splitting algorithms
- Banded matrices
- Blocked or overlapped for complex geometries

- Unstructured grids
- Lists of cell connectivity, graphs (edge,vertices)
- Alternate discretizations/solution strategies
- Sparse Matrices
- Complex Geometries, Adaptive Meshing
- More Efficient Parallelization

VKI Lecture Series, February 3-7, 2003

- Computational aerodynamics rooted in structured methods
- High accuracy and efficiency requirements

- Unstructured mesh methods 2 to 4 times more costly
- Mitigated by extra structured grid overhead
- Block structured
- Overset mesh
- Parallelization

- Accuracy considerations
- Validation studies, experience

- Unstructured mesh solvers potentially more efficient than structured mesh alternatives with equivalent accuracy

- Mitigated by extra structured grid overhead

VKI Lecture Series, February 3-7, 2003

- State-of-the-art in drag prediction: RANS
- Entire suite of tools available to designer
- Useful to examine capabilities of other tools
- Lower fidelity – lower costs
- Numerous rapid tradeoff studies

- Higher fidelity – higher costs
- Fewer detailed analyses

- Lower fidelity – lower costs
- Situate RANS tools within this suite

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

- Inviscid flow unstructured mesh solvers well established – robust
- No viscous effects
- No turbulence/transition modeling

- Isotropic meshes
- Good commercial isoptropic mesh generators
- Good convergence properties

- No viscous effects

VKI Lecture Series, February 3-7, 2003

- 235,000 vertex mesh
- (ICEMCFD tetra)
- Fully tetrahedral mesh

- Convergence in 50 cycles
- (multigrid)

- 3 minutes on 8 Pentiums
- 50 times faster than RANS

VKI Lecture Series, February 3-7, 2003

- 235,000 vertex mesh
- (ICEMCFD tetra)
- Fully tetrahedral mesh

- Convergence in 50 cycles
- (multigrid)

- 3 minutes on 8 Pentiums
- 50 times faster than RANS

VKI Lecture Series, February 3-7, 2003

- 235,000 vertex mesh
- (ICEMCFD tetra)
- Fully tetrahedral mesh

- Convergence in 50 cycles
- (multigrid)

- 3 minutes on 8 Pentiums
- 50 times faster than RANS
- 1.65 million vertices

VKI Lecture Series, February 3-7, 2003

Euler Solution (235,000 pts)

RANS Solution (1.65M pts)

- 235,000 vertex mesh
- (ICEMCFD tetra)
- Fully tetrahedral mesh

- Convergence in 50 cycles
- (multigrid)

- 3 minutes on 8 Pentiums
- 50 times faster than RANS

VKI Lecture Series, February 3-7, 2003

- Exclusion of viscous effects
- Boundary layer displacement
- Incorrect shock location
- Incorrect shock strength

- Supercritical wing sensitive to viscous effects
- Euler solution not useful for transonic cruise drag prediction

VKI Lecture Series, February 3-7, 2003

- Incorporate viscous effects to first order
- Boundary layer displacement thickness
- More accurate shock strength/location

- Retain efficiency of Euler solution approach
- Isotropic tetrahedral meshes
- Fast, robust convergence

VKI Lecture Series, February 3-7, 2003

- Stripwise 2-dimensional boundary layer
- 18 stations on wing alone
- Interpolate from unstructured surface mesh
- Transpiration condition for simulated BL displacement thickness

VKI Lecture Series, February 3-7, 2003

Euler Solution (235,000 pts)

RANS Solution (1.65M pts)

- 235,000 vertex mesh
- (ICEMCFD tetra)
- Fully tetrahedral mesh

- Convergence in 50 cycles
- (multigrid)

- 3 minutes on 8 Pentiums
- 50 times faster than RANS

VKI Lecture Series, February 3-7, 2003

Euler-IBL Sol. (235,000 pts)

RANS Solution (1.65M pts)

- 235,000 vertex mesh
- (ICEMCFD tetra)
- Fully tetrahedral mesh

- Convergence in 50 cycles
- (multigrid)

- 3 minutes on 8 Pentiums
- 50 times faster than RANS

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

- Vastly improved over Euler alone
- Correct shock strength, location
- Accurate lift
- Reasonable drag
- More sophisticated coupling possible

- 25 times faster than RANS
- Neglibible IBL compute time
- Convergence dominated by coupling
- Parameter studies
- Design optimization

VKI Lecture Series, February 3-7, 2003

- RANS failures for separated flows
- Good cruise design involves minimal separation
- Off design, high-lift

- LES or DES as alternative to turbulence modeling inadequacies
- LES: compute all scales down to inertial range
- Based on universality of inertial range

- DES: hybrid LES/RANS (near wall)
- Reduced cost

- LES: compute all scales down to inertial range

VKI Lecture Series, February 3-7, 2003

- European LESFOIL program
- Marie and Sagaut: LES about airfoil near stall

- DES for massively separated aerodynamic flows
- Strelets 2001, Forsythe 2000, 2001, 2003

- Two to ? Orders of magnitude more expensive than RANS
- Predictive ability for accurate drag not established

- RANS methods state-of-art for foreseeable future

VKI Lecture Series, February 3-7, 2003

- Choice of discretization and effect of dissipation (intricately linked)
- Cells versus points
- Discretization formulations

- Grid resolution requirements
- Choice of element type
- Grid resolution issues

- Grid convergence

VKI Lecture Series, February 3-7, 2003

- Tetrahedral Mesh contains 5 to 6 times more cells than vertices
- Hexahedral meshes contain same number of cells and vertices (excluding boundary effects)
- Prismatic meshes: cells = 2X vertices

- Tetrahedral cells : 4 neighbors
- Vertices: 20 to 30 neighbors on average

VKI Lecture Series, February 3-7, 2003

- On given mesh:
- Cell centered discretization: Higher accuracy
- Vertex discretization: Lower cost

- Equivalent Accuracy-Cost Comparisons Difficult
- Often based on equivalent numbers of surface unknowns (2:1 for tet meshes)
- Levy (1999)
- Yields advantage for vertex-discretization

VKI Lecture Series, February 3-7, 2003

- Both approaches have advantages/drawbacks
- Methods require substantially different grid resolutions for similar accuracy
- Factor 2 to 4 possible in grid requirements

- Important for CFD practitioner to understand these implications

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

- NSU3D: vertex-based discretization
- Grid : 48K boundary pts, 1.65M pts (9.6M cells)

- USM3D: cell-centered discretization
- Grid : 50K boundary cells, 2.4M cells (414K pts)
- Uses wall functions

- NSU3D: on cell centered type grid
- Grid: 46K boundary cells, 2.7M cells (470K pts)

VKI Lecture Series, February 3-7, 2003

- Similar Lift for both codes on cell-centered grid
- Baseline NSU3D (finer vertex grid) has lower lift

VKI Lecture Series, February 3-7, 2003

- Pressure drag
- Wall treatment discrepancies

- NSU3D : cell centered grid
- High drag, (10 to 20 counts)
- Grid too coarse for NSU3D
- Inexpensive computation

- USM3D on cell-centered grid closer to NSU3D on vertex grid

Concentrate exclusively on Vertex-Discretizations

VKI Lecture Series, February 3-7, 2003

- Choice of discretization and effect of dissipation (intricately linked)
- Cells versus points
- Discretization formulations

- Grid resolution requirements
- Choice of element type
- Grid resolution issues

- Grid convergence

VKI Lecture Series, February 3-7, 2003

- Governing Equations: Reynolds Averaged Navier-Stokes Equations
- Conservation of Mass, Momentum and Energy
- Single Equation turbulence model (Spalart-Allmaras)
- Convection-Diffusion – Production

- Vertex-Based Discretization
- 2nd order upwind finite-volume scheme
- 6 variables per grid point
- Flow equations fully coupled (5x5)
- Turbulence equation uncoupled

VKI Lecture Series, February 3-7, 2003

- Mixed Element Meshes
- Tetrahedra, Prisms, Pyramids, Hexahedra

- Control Volume Based on Median Duals
- Fluxes based on edges
- Single edge-based data-structure represents all element types

Fik = F(uL) + F(uR) + T |L| T-1 (uL –uR)

- Upwind discretization

- Matrix artificial dissipation

VKI Lecture Series, February 3-7, 2003

- First order scheme

- Second order scheme

- Gradients evaluated at vertices by Least-Squares
- Limit Gradients for Strong Shock Capturing

- First order scheme

- Second order scheme

- By analogy with upwind scheme:

- Blending of 1st and 2nd order schemes for strong shock capturing

VKI Lecture Series, February 3-7, 2003

L matrix: diagonal with eigenvalues:

u, u, u, u+c, u-c

- Robustness issues related to vanishing eigenvalues
- Limit smallest eigenvalues as fraction of largest eigenvalue: |u| + c
- u = sign(u) * max(|u|, d(|u|+c))
- u+c = sign(u+c) * max(|u+c|, d(|u|+c))
- u – c = sign(u -c) * max(|u-c|, d(|u|+c))

VKI Lecture Series, February 3-7, 2003

- u = sign(u) * max(|u|, d(|u|+c))
- u+c = sign(u+c) * max(|u+c|, d(|u|+c))
- u – c = sign(u -c) * max(|u-c|, d(|u|+c))
d = 0.1 : typical value for enhanced robustness

d = 1.0 : Scalar dissipation

- L becomes scaled identity matrix

- T |L| T-1 becomes scalar quantity
- Simplified (lower cost) dissipation operator

VKI Lecture Series, February 3-7, 2003

- Examine effect of discretization type and parameter variations on drag prediction
- Effect on drag polars for DLR-F4:
- Matrix artificial dissipation
- Dissipation levels
- Entropy fix
- Low order blending

- Upwind schemes
- Gradient reconstruction
- Entropy fix
- Limiters

- Matrix artificial dissipation

VKI Lecture Series, February 3-7, 2003

- Increased accuracy through lower dissipation coef.
- Potential loss of robustness

- Insensitive to small values of d=0.1, 0.2
- High drag values for large d and scalar scheme

VKI Lecture Series, February 3-7, 2003

- Lift and drag relatively insensitive
- Generally not recommended for transonics

- Least squares approach slightly more diffusive
- Extremely sensitive to entropy fix value

- Limiters reduces accuracy, increase robustness
- Less sensitive to non-monotone limiters

VKI Lecture Series, February 3-7, 2003

- Right angle tetrahedra produced in boundary layer regions
- Highly stretched elements for efficiency
- Non obtuse angle requirement for accuracy

- Semi-structured tetrahedra combinable into prisms
- Prism elements of lower complexity (fewer edges)
- No significant accuracy benefit (Aftosmis et. Al. 1994 in 2D)

- Little overall effect on accuracy
- Potential differences between two codes

- Possibly greatest impediment to reliable RANS drag prediction
- Promise of adaptive meshing held back by development of adequate error estimators
- Unstructured mesh requirement similar to structured mesh requirements
- 200 to 500 vertices chordwise (cruise)
- Lower optimal spanwise resolution

VKI Lecture Series, February 3-7, 2003

- Factor of 3 savings in grid size

VKI Lecture Series, February 3-7, 2003

- Inadequate resolution under-predicts skin friction
- Direct influence on drag prediction

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

- Lift converges as h2
- Drag vanishes in continuous limit

- Seldom achieved for 3D RANS
- Wide range of scales: 109 in AIAA DPW grid
- High stretching near wall/wake regions
- Good initial mesh required (even if adaptive)

- Prohibitive Cost in 3D
- Each refinement: 8:1 cost
- 4:1 accuracy improvement (2nd order scheme)

- Emphasis:
- User expertise, experience
- Verification, validation, error estimation

VKI Lecture Series, February 3-7, 2003

- Discretization
- Efficient solution techniques
- Multigrid

- Efficient hardware utilization
- Vector
- Cache efficiency
- Parallelization

VKI Lecture Series, February 3-7, 2003

- Mostly covered previously
- Vertex-based discretization
- Matrix-based artificial dissipation
- k2=1.0, d=0.1
- No low order blending of dissipation (k1 = 0.0)

- Hybrid Elements
- Prismatic elements in boundary layer
- Single edge based data-structure

VKI Lecture Series, February 3-7, 2003

- Edge-based data structure
- Building block for all element types
- Reduces memory requirements
- Minimizes indirect addressing / gather-scatter
- Graph of grid = Discretization stencil
- Implications for solvers, Partitioners

VKI Lecture Series, February 3-7, 2003

- Integrate to Steady-state
- Explicit:
- Simple, Slow: Local procedure

- Implicit
- Large Memory Requirements

- MatrixFreeImplicit:
- Most effective with matrix preconditioner

- Multigrid Methods

VKI Lecture Series, February 3-7, 2003

- High-frequency (local) error rapidly reduced by explicit methods
- Low-frequency (global) error converges slowly
- On coarser grid:
- Low-frequency viewed as high frequency

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

- Generate fine and coarse meshes
- Interpolate between un-nested meshes
- Finest grid: 804,000 points, 4.5M tetrahedra
- Four level Multigrid sequence

- Order of magnitude increase in convergence
- Convergence rate equivalent to structured grid schemes
- Independent of grid size: O(N)

VKI Lecture Series, February 3-7, 2003

- Multigrid methods:
- Time step on coarse grids to accelerate solution on fine grid

- Geometric multigrid
- Coarse grid levels constructed manually
- Cumbersome in production environment

- Agglomeration Multigrid
- Automate coarse level construction
- Algebraic nature: summing fine grid equations
- Graph based algorithm

VKI Lecture Series, February 3-7, 2003

- Agglomeration Multigrid solvers for unstructured meshes
- Coarse level meshes constructed by agglomerating fine grid cells/equations

VKI Lecture Series, February 3-7, 2003

- Automated Graph-Based Coarsening Algorithm
- Coarse Levels are Graphs
- Coarse Level Operator by Galerkin Projection
- Grid independent convergence rates (order of magnitude improvement)

- Convergence rate similar to geometric MG
- Completely automatic

VKI Lecture Series, February 3-7, 2003

- Convergence rates for RANS (viscous) problems much slower than inviscid flows
- Mainly due to grid stretching
- Thin boundary and wake regions
- Mixed element (prism-tet) grids

- Use directional solver to relieve stiffness
- Line solver in anisotropic regions

VKI Lecture Series, February 3-7, 2003

- Line Solvers for Anisotropic Problems
- Lines Constructed in Mesh using weighted graph algorithm
- Strong Connections Assigned Large Graph Weight
- (Block) Tridiagonal Line Solver similar to structured grids

VKI Lecture Series, February 3-7, 2003

- DLR-F4 wing-body, Mach=0.75, 1o, Re=3M
- Baseline Mesh: 1.65M pts

VKI Lecture Series, February 3-7, 2003

- High-Lift Case: Mach=0.2, 10o, Re=1.6M

- Intersected edges resolved by ghost vertices
- Generates communication between original and ghost vertex
- Handled using MPI and/or OpenMP
- Portable, Distributed and Shared Memory Architectures
- Local reordering within partition for cache-locality

- Graph partitioning must minimize number of cut edges to minimize communication
- Standard graph based partitioners: Metis, Chaco, Jostle
- Require only weighted graph description of grid
- Edges, vertices and weights taken as unity

- Ideal for edge data-structure

- Require only weighted graph description of grid
- Line solver inherently sequential
- Partition around line using weighted graphs

VKI Lecture Series, February 3-7, 2003

- Contract graph along implicit lines
- Weight edges and vertices
- Partition contracted graph
- Decontract graph
- Guaranteed lines never broken
- Possible small increase in imbalance/cut edges

VKI Lecture Series, February 3-7, 2003

- 32-way partition of 30,562 point 2D grid
- Unweighted partition: 2.6% edges cut, 2.7% lines cut
- Weighted partition: 3.2% edges cut, 0% lines cut

VKI Lecture Series, February 3-7, 2003

24.7M pts, Cray T3E

177K pts, PC cluster

- Moderate additional multigrid communication on coarse levels
- Large Multigrid convergence benefit

3M pts, Origin 2000

- Near equivalence of MPI and OpenMP on Shared Mem Arch.

- Absolute drag for transonic wing-body
- AIAA drag prediction workshop (June 2001)

- Incremental effects
- DLR engine installation drag study

- High lift flows
- Large scale 3D simulation (NSU3D)
- Experience base in 2D

VKI Lecture Series, February 3-7, 2003

- Transonic wing-body configuration
- Typical cases required for design study
- Matrix of mach and CL values
- Grid resolution study

- Follow on with engine effects (2003)

- Baseline grid: 1.6 million points
- Full drag Polars for Mach=0.5,0.6,0.7,0.75,0.76,0.77,0.78,0.8
- Total = 72 cases

- Medium grid: 3 million points
- Full drag polar for each Mach number
- Total = 48 cases

- Fine grid: 13 million points
- Drag polar at mach=0.75
- Total = 7 cases

VKI Lecture Series, February 3-7, 2003

- Mach=0.75, CL=0.6, Re=3M
- 2.5 hours on 16 Pentium IV 1.7GHz

VKI Lecture Series, February 3-7, 2003

- Separation in wing root area
- Post shock and trailing edge separation

VKI Lecture Series, February 3-7, 2003

- Y+ < 1 over most of wing surfaces
- Multigrid convergence < 500 cycles

VKI Lecture Series, February 3-7, 2003

- Lift values overpredicted
- Increased lift with additional grid resolution

VKI Lecture Series, February 3-7, 2003

- Grid resolution study
- Good comparison with experimental data

VKI Lecture Series, February 3-7, 2003

- Grid Drag Values
- Incidence Offset for Same CL

- Aft shock location results in lift overprediction
- Matching CL condition produces low suction peak
- Adverse effect on predicted moments

- Grid resolution study
- Discrepancies at Higher Mach/CL Conditions

- Grid resolution study
- Discrepancies at Higher Mach/CL Conditions

- Similar predictive ability for both approaches
- More scatter for structured methods
- More submissions/variations for structured methods

- Unstructured mesh capabilities comparable to other methods
- Lift overprediction tainted assessment of overall results
- Absolute drag prediction not within 1 count
- 10 to 20 counts
- Poorer agreement at high Mach, CL (separation)
- Grid convergence not established
- Better results possible with extensive validation
- Potentially better success for incremental effects

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

- 120 Cases (excluding finest grid)
- About 1 week to compute all cases

VKI Lecture Series, February 3-7, 2003

- Absolute drag prediction to 1 count not yet feasible in general
- Incremental effects potentially easier to capture
- Cancellation of drag bias in non-critical regions
- Important in design study tradeoffs
- Pre-requisite for automated design optimization

- Engine installation drag prediction
- DLR study (tau unstructured grid code)
- (Broderson and Sturmer AIAA-2001-2414)

VKI Lecture Series, February 3-7, 2003

- Similar to DLR-F4
- Wing aspect ratio: 9.5
- Sweep: 27.1 degrees
- Twin engine (flow through nacelles)
- Test as wing-body alone
- Test 3 different nacelle positions
- Two nacelle types (not included herein)

- Subject of 2nd AIAA Drag Prediction Workshop (June 2003)

VKI Lecture Series, February 3-7, 2003

- Similar to NSU3D solver
- Vertex discretization
- Artificial dissipation
- Scaled scalar dissiption

- Agglomeration multigrid

- Spalart Allmaras turbulence model
- Productionalized adaptive meshing capability

VKI Lecture Series, February 3-7, 2003

- Productionalized adaptive meshing capability
- 3 levels of adaptive meshing employed
- Refinement based on flow-field gradients

- Wing-body grids
- Initial: 2.9 million points
- Final: 5.5 million points

- Wing-body nacelle-pylon grids
- Initial: 4.5 million points
- Final: 7.5 million points

VKI Lecture Series, February 3-7, 2003

- Overprediction of lift for all cases
- Under-prediction of drag for all cases

VKI Lecture Series, February 3-7, 2003

- Absolute drag underpredicted by 10-20 counts
- Installation drag accurate to 1 to 4 counts
- Similar to variations between wind-tunnel campaigns

- Absolute drag correlation decreases as grid refined
- Incremental drag correlation improves as grid refined

- Accuracy of absolute drag not sufficient
- Accurate installation drag (incremental)
- Changes in drag due to nacelle position detectable to within 1 to 2 counts
- Enables CFD design-based decisions
- Design optimization

- Results from careful validation study
- More complete study at AIAA DPW 2003

VKI Lecture Series, February 3-7, 2003

- Complicated flow physics
- High mesh resolution requirements
- On body, off body

- Complex geometries
- Original driver for unstructured meshes in aerodynamics

- Prediction of surface pressures
- Separation possible at design conditions(landing)

- Lift, drag and moments
- CLmax, stall

- Large 3D high-lift case
- 2D experience base

VKI Lecture Series, February 3-7, 2003

- Complex geometry
- Wing-body, slat, double slotted flaps, cutouts

- Experimental data from Langley 14x22ft wind tunnel
- Mach = 0.2, Reynolds=1.6 million
- Range of incidences: -4 to 24 degrees

VKI Lecture Series, February 3-7, 2003

- 3.1 million vertices, 18.2 million tets, 115,489 surface pts
- Normal spacing: 1.35E-06 chords, growth factor=1.3

- Mach=0.2, Incidence=10 degrees, Re=1.6M

- Experimental data at 10 degrees incidence

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

- Good drag prediction
- Discrepancies near stall

- Mesh independent property of Multigrid

VKI Lecture Series, February 3-7, 2003

- Good overall Multigrid scalability
- Increased communication due to coarse grid levels
- Single grid solution impractical (>100 times slower)

- 1 hour solution time on 1450 PEs

VKI Lecture Series, February 3-7, 2003

- Large body of experience in 2D
- High resolution grids possible
- 50,000 pts required for Cp on 3 elements
- Up to 250,000 pts required for CLmax
- Effect of wake resolution

- Rapid assessment of turbulence/transition models
- Ability to predict incremental effects
- Reynolds number effects
- Small geometry changes (gap/overlap)

VKI Lecture Series, February 3-7, 2003

- Good CP agreement in linear region of CL curve
(Lynch, Potter and Spaid, ICAS 1996)

- CLmax overpredicted
- CLmax Incidence overpredicted by 1 degree
(Lynch, Potter and Spaid, ICAS 1996)

- Wake capturing requires fine off-body grid
- Enhanced by low dissipation scheme
- More difficult further downstream
- Slat wake deficit consistently overpredicted

- Adequate Reynolds number effect prediction
- Provided no substantial transitional effects
- Transition is important player
- Transition models

(Valarezo and Mavriplis, AIAA J. Aircraft, 1995)

- Change due to 0.25% chord increase in flap gap
- CL increase at low/high incidences captured
- CL decrease at intermediate incidence missed
- Flap separation not captured by turb model

(Lynch, Potter and Spaid, ICAS 1996)

- Two-dimensional cases
- Good predictive ability provided flow physics are captured adequately
- Turbulence, transition
- Grid resolution

- Three dimensional simulations coming of age
- Grid resolution from 2D studies
- Extensive validation required

- Good predictive ability provided flow physics are captured adequately

VKI Lecture Series, February 3-7, 2003

- Cruise drag prediction requires improvement
- Incremental effects (cruise) to wind tunnel accuracy are feasible
- High-lift simulations in initial development
- Higher accuracy, efficiency, reliability
- Adaptive meshing
- Error estimation

- Higher-order methods

- Adaptive meshing

VKI Lecture Series, February 3-7, 2003

- Potential for large savings trough optimized mesh resolution
- Error estimation and control
- Guarantee or assess level of grid convergence
- Immense benefit for drag prediction
- Driver for adaptive process

- Error estimation and control
- Mechanics of mesh adaptation
- Refinement criteria and error estimation

VKI Lecture Series, February 3-7, 2003

- Various well know isotropic mesh methods
- Mesh movement
- Spring analogy
- Linear elasticity

- Local Remeshing
- Delaunay point insertion/Retriangulation
- Edge-face swapping
- Element subdivision
- Mixed elements (non-simplicial)
- Require anisotropic refinement in transition regions

- Mesh movement

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

VKI Lecture Series, February 3-7, 2003

- Weakest link of adaptive meshing methods
- Obvious for strong features
- Difficult for non-local (ie. Convective) features
- eg. Wakes

- Analysis assumes in asymptotic error convergence region
- Gradient based criteria
- Empirical criteria

- Effect of variable discretization error in design studies, parameter sweeps

VKI Lecture Series, February 3-7, 2003

- Compute sensitivity of global cost function to local spatial grid resolution
- Key on important output, ignore other features
- Error in engineering output, not discretization error
- e.g. Lift, drag, or sonic boom …

- Error in engineering output, not discretization error
- Captures non-local behavior of error
- Global effect of local resolution

- Requires solution of adjoint equations
- Adjoint techniques used for design optimization

VKI Lecture Series, February 3-7, 2003

Reproduced from Vendetti and Darmofal (MIT, 2002)

VKI Lecture Series, February 3-7, 2003

Reproduced from Vendetti and Darmofal (MIT, 2002)

VKI Lecture Series, February 3-7, 2003

- Uniform X2 refinement of 3D mesh:
- Work increase = factor of 8
- 2nd order accurate method: accuracy increase = 4
- 4th order accurate method: accuracy increase = 16
- For smooth solutions

- Potential for large efficiency gains
- Spectral element methods
- Discontinuous Galerkin (DG)
- Streamwise Upwind Petrov Galerkin (SUPG)

- Most effective when high accuracy required
- Potential role in drag prediction
- High accuracy requirements
- Large grid sizes required

- Transfers burden from grid generation to Discretization

VKI Lecture Series, February 3-7, 2003

J. Hestahaven and T. Warburton (Brown University)

- Adaptive meshing (h-ref) yields constant factor improvement
- After error equidistribution, no further benefit

- Order refinement (p-ref) yields asymptotic improvement
- Only for smooth functions
- Ineffective for inadequate h-resolution of feature
- Cannot treat shocks

- H-P refinement optimal (exponential convergence)

VKI Lecture Series, February 3-7, 2003

- Drag prediction is demanding, specialized task
- Unstructured mesh approach offers comparable accuracy, efficiency with future potential for adaptive meshing advantages
- Major impediments:
- Grid convergence
- Flow physics modeling

- Continued investment in extensive validation verification required for useful capability

VKI Lecture Series, February 3-7, 2003