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Aerodynamic Drag Prediction Using Unstructured Mesh Solvers. Dimitri J. Mavriplis National Institute of Aerospace Hampton, Virginia, USA. Overview. Introduction Physical model fidelity Grid resolution and discretization issues

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aerodynamic drag prediction using unstructured mesh solvers

Aerodynamic Drag Prediction Using Unstructured Mesh Solvers

Dimitri J. Mavriplis

National Institute of Aerospace

Hampton, Virginia, USA

VKI Lecture Series, February 3-7, 2003

overview
Overview
  • 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

overview1
Overview
  • 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

overview2
Overview
  • 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

VKI Lecture Series, February 3-7, 2003

overview3
Overview
  • Designing an efficient unstructured mesh solver for computational aerodynamics
    • Discretization
    • Solution Methodologies
    • Efficient Hardware Usage

VKI Lecture Series, February 3-7, 2003

overview4
Overview
  • 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

introduction
Introduction
  • 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

introduction1
Introduction
  • 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

cfd perspective on meshing technology
CFD Perspective on Meshing Technology
  • Sophisticated Multiblock Structured Grid Techniques for Complex Geometries

Engine Nacelle Multiblock Grid by commercial software TrueGrid.

cfd perspective on meshing technology1
CFD Perspective on Meshing Technology
  • Sophisticated Overlapping Structured Grid Techniques for Complex Geometries

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

unstructured grid alternative
Unstructured Grid Alternative
  • Connectivity stored explicitly
  • Single Homogeneous Data Structure

VKI Lecture Series, February 3-7, 2003

characteristics of both approaches
Characteristics of Both Approaches
  • 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

unstructured meshes for aerodynamics
Unstructured Meshes for Aerodynamics
  • 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

VKI Lecture Series, February 3-7, 2003

physical model fidelity
Physical Model Fidelity
  • 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
  • Situate RANS tools within this suite

VKI Lecture Series, February 3-7, 2003

physical model requirements unstructured mesh methods
Physical Model Requirements(Unstructured Mesh Methods)

VKI Lecture Series, February 3-7, 2003

unstructured mesh euler solvers
Unstructured Mesh Euler Solvers
  • 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

VKI Lecture Series, February 3-7, 2003

example euler solution of dlr f4 wing body configuration
Example: Euler Solution of DLR-F4 Wing-body Configuration
  • 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

example euler solution of dlr f4 wing body configuration1
Example: Euler Solution of DLR-F4 Wing-body Configuration
  • 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

example euler solution of dlr f4 wing body configuration2
Example: Euler Solution of DLR-F4 Wing-body Configuration
  • 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 vs rans solution
Euler vs. RANS Solution

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 vs rans solution1
Euler vs. RANS Solution
  • 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

coupled euler boundary layer approach
Coupled Euler-Boundary Layer Approach
  • 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

coupled euler boundary layer approach1
Coupled Euler-Boundary Layer Approach
  • 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 vs rans solution2
Euler vs. RANS Solution

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 vs rans solution
Euler-IBL vs. RANS Solution

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

coupled euler boundary layer approach2
Coupled Euler-Boundary Layer Approach

VKI Lecture Series, February 3-7, 2003

coupled euler boundary layer approach3
Coupled Euler-Boundary-Layer Approach
  • 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

les and des methods
LES and DES Methods
  • 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

VKI Lecture Series, February 3-7, 2003

les and des notable successes
LES and DES: Notable Successes
  • 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

grid resolution and discretization issues
Grid Resolution and Discretization Issues
  • 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

cell centered vs vertex based
Cell Centered vs Vertex-Based
  • 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

cell centered vs vertex based1
Cell Centered vs Vertex-Based
  • 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

cell centered vs vertex based2
Cell Centered vs Vertex-Based
  • 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

example dlr f4 wing body aiaa drag prediction workshop
Example: DLR-F4 Wing-body (AIAA Drag Prediction Workshop)

VKI Lecture Series, February 3-7, 2003

illustrative example dlr f4
Illustrative Example: DLR-F4
  • 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

cell versus vertex discretizations
Cell versus Vertex Discretizations
  • Similar Lift for both codes on cell-centered grid
  • Baseline NSU3D (finer vertex grid) has lower lift

VKI Lecture Series, February 3-7, 2003

cell versus vertex discretizations1
Cell versus Vertex Discretizations
  • 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

grid resolution and discretization issues1
Grid Resolution and Discretization Issues
  • 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

discretization
Discretization
  • 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

spatial discretization
Spatial Discretization
  • 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

upwind discretization
Upwind Discretization
  • First order scheme
  • Second order scheme
  • Gradients evaluated at vertices by Least-Squares
  • Limit Gradients for Strong Shock Capturing
matrix artificial dissipation
Matrix Artificial Dissipation
  • 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

entropy fix
Entropy Fix

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

entropy fix1
Entropy Fix
    • 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
  • Applicable to upwind and art. dissipation schemes

VKI Lecture Series, February 3-7, 2003

discretization formulations
Discretization Formulations
  • 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

VKI Lecture Series, February 3-7, 2003

effect of artificial dissipation level
Effect of Artificial Dissipation Level
  • Increased accuracy through lower dissipation coef.
  • Potential loss of robustness
effect of entropy fix for artificial dissipation scheme
Effect of Entropy Fix for Artificial Dissipation Scheme
  • Insensitive to small values of d=0.1, 0.2
  • High drag values for large d and scalar scheme
effect of artificial dissipation
Effect of Artificial Dissipation

VKI Lecture Series, February 3-7, 2003

effect of low order dissipation blending for shock capturing
Effect of Low-Order Dissipation Blending for Shock Capturing
  • Lift and drag relatively insensitive
  • Generally not recommended for transonics
comparison of discretization formulation art dissip vs grad rec
Comparison of Discretization Formulation (Art. Dissip vs. Grad. Rec.)
  • Least squares approach slightly more diffusive
  • Extremely sensitive to entropy fix value
effect of limiters on upwind discretization
Effect of Limiters on Upwind Discretization
  • Limiters reduces accuracy, increase robustness
  • Less sensitive to non-monotone limiters
effect of discretization type
Effect of Discretization Type

VKI Lecture Series, February 3-7, 2003

effect of element type
Effect of Element Type
  • 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)
effect of element type in bl region
Effect of Element Type in BL Region
  • Little overall effect on accuracy
  • Potential differences between two codes
grid resolution issues
Grid Resolution Issues
  • 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

illustration of spanwise stretching vgridns c o s pirzadeh nasa langley
Illustration of Spanwise Stretching (VGRIDns, c/o S. Pirzadeh, NASA Langley)
  • Factor of 3 savings in grid size

VKI Lecture Series, February 3-7, 2003

effect of normal spacing in bl
Effect of Normal Spacing in BL
  • Inadequate resolution under-predicts skin friction
  • Direct influence on drag prediction
effect of normal resolution for high lift c o anderson et aiaa j aircraft 1995
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 convergence 2d euler
Grid Convergence (2D Euler)
  • Lift converges as h2
  • Drag vanishes in continuous limit
grid convergence
Grid Convergence
  • 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

designing an efficient unstructured mesh solver for aerodynamics
Designing an Efficient Unstructured Mesh Solver for Aerodynamics
  • Discretization
  • Efficient solution techniques
    • Multigrid
  • Efficient hardware utilization
    • Vector
    • Cache efficiency
    • Parallelization

VKI Lecture Series, February 3-7, 2003

discretization1
Discretization
  • 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

discretization2
Discretization
  • 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

spatially discretized equations
Spatially Discretized Equations
  • 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

multigrid methods
Multigrid Methods
  • 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

multigrid correction scheme linear problems
Multigrid Correction Scheme(Linear Problems)

VKI Lecture Series, February 3-7, 2003

multigrid for unstructured meshes
Multigrid for Unstructured Meshes
  • Generate fine and coarse meshes
  • Interpolate between un-nested meshes
  • Finest grid: 804,000 points, 4.5M tetrahedra
  • Four level Multigrid sequence
geometric multigrid
Geometric Multigrid
  • 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

agglomeration vs geometric multigrid
Agglomeration vs. Geometric Multigrid
  • 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
Agglomeration Multigrid
  • Agglomeration Multigrid solvers for unstructured meshes
    • Coarse level meshes constructed by agglomerating fine grid cells/equations

VKI Lecture Series, February 3-7, 2003

agglomeration multigrid1
Agglomeration Multigrid
  • Automated Graph-Based Coarsening Algorithm
  • Coarse Levels are Graphs
  • Coarse Level Operator by Galerkin Projection
  • Grid independent convergence rates (order of magnitude improvement)
agglomeration mg for euler equations
Agglomeration MG for Euler Equations
  • Convergence rate similar to geometric MG
  • Completely automatic

VKI Lecture Series, February 3-7, 2003

anisotropy induced stiffness
Anisotropy Induced Stiffness
  • 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

directional solver for navier stokes problems
Directional Solver for Navier-Stokes Problems
  • 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

multigrid line solver convergence
Multigrid Line Solver Convergence
  • DLR-F4 wing-body, Mach=0.75, 1o, Re=3M
    • Baseline Mesh: 1.65M pts

VKI Lecture Series, February 3-7, 2003

multigrid insensitivity to mesh size
Multigrid Insensitivity to Mesh Size
  • High-Lift Case: Mach=0.2, 10o, Re=1.6M
implementation on parallel computers
Implementation on Parallel Computers
  • 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
partitioning
Partitioning
  • 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
  • Line solver inherently sequential
    • Partition around line using weighted graphs

VKI Lecture Series, February 3-7, 2003

partitioning1
Partitioning
  • 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

partitioning example
Partitioning Example
  • 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

parallel scalability mpi
Parallel Scalability (MPI)

24.7M pts, Cray T3E

177K pts, PC cluster

  • Moderate additional multigrid communication on coarse levels
  • Large Multigrid convergence benefit
parallel scalability
Parallel Scalability

3M pts, Origin 2000

  • Near equivalence of MPI and OpenMP on Shared Mem Arch.
drag prediction using unstructured mesh solvers
Drag Prediction Using Unstructured Mesh Solvers
  • 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

aiaa drag prediction workshop 2001
AIAA Drag Prediction Workshop (2001)
  • 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)
cases run
Cases Run
  • 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

sample solution 1 65m pts
Sample Solution (1.65M Pts)
  • Mach=0.75, CL=0.6, Re=3M
  • 2.5 hours on 16 Pentium IV 1.7GHz

VKI Lecture Series, February 3-7, 2003

observed flow flow details mach 0 75 c l 0 6
Observed Flow Flow Details Mach = 0.75, CL=0.6
  • Separation in wing root area
  • Post shock and trailing edge separation

VKI Lecture Series, February 3-7, 2003

typical simulation characteristics
Typical Simulation Characteristics
  • Y+ < 1 over most of wing surfaces
  • Multigrid convergence < 500 cycles

VKI Lecture Series, February 3-7, 2003

lift vs incidence at mach 0 75
Lift vs Incidence at Mach = 0.75
  • Lift values overpredicted
  • Increased lift with additional grid resolution

VKI Lecture Series, February 3-7, 2003

drag polar at mach 0 75
Drag Polar at Mach = 0.75
  • Grid resolution study
  • Good comparison with experimental data

VKI Lecture Series, February 3-7, 2003

comparison with experiment
Comparison with Experiment
  • Grid Drag Values
  • Incidence Offset for Same CL
surface cp at 40 9 span
Surface Cp at 40.9% Span
  • Aft shock location results in lift overprediction
  • Matching CL condition produces low suction peak
  • Adverse effect on predicted moments
drag polars at other mach numbers
Drag Polars at other Mach Numbers
  • Grid resolution study
  • Discrepancies at Higher Mach/CL Conditions
drag rise curves
Drag Rise Curves
  • Grid resolution study
  • Discrepancies at Higher Mach/CL Conditions
structured vs unstructured drag prediction aiaa workshop results
Structured vs Unstructured Drag Prediction (AIAA workshop results)
  • Similar predictive ability for both approaches
    • More scatter for structured methods
    • More submissions/variations for structured methods
absolute drag prediction aiaa dpw 2001
Absolute Drag Prediction (AIAA DPW 2001)
  • 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

timings on various architectures
Timings on Various Architectures

VKI Lecture Series, February 3-7, 2003

cases run on icase cluster
Cases Run on ICASE Cluster
  • 120 Cases (excluding finest grid)
  • About 1 week to compute all cases

VKI Lecture Series, February 3-7, 2003

incremental effects
Incremental Effects
  • 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

dlr f6 configuration
DLR-F6 Configuration
  • 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

dlr tau unstructured solver
DLR tau Unstructured Solver
  • 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

dlr tau unstructured solver1
DLR tau Unstructured Solver
  • 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

computed absolute values
Computed Absolute Values
  • Overprediction of lift for all cases
  • Under-prediction of drag for all cases

VKI Lecture Series, February 3-7, 2003

computed incremental values
Computed Incremental Values
  • Absolute drag underpredicted by 10-20 counts
  • Installation drag accurate to 1 to 4 counts
    • Similar to variations between wind-tunnel campaigns
effect of adaptive grid resolution
Effect of (Adaptive) Grid Resolution
  • Absolute drag correlation decreases as grid refined
  • Incremental drag correlation improves as grid refined
prediction of installation drag
Prediction of Installation Drag
  • 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

high lift flows
High-Lift Flows
  • Complicated flow physics
  • High mesh resolution requirements
    • On body, off body
  • Complex geometries
    • Original driver for unstructured meshes in aerodynamics
high lift flows1
High-Lift Flows
  • 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

nasa langley energy efficient transport
NASA Langley Energy Efficient Transport
  • 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

vgrid tetrahedral mesh
VGRID Tetrahedral Mesh
  • 3.1 million vertices, 18.2 million tets, 115,489 surface pts
  • Normal spacing: 1.35E-06 chords, growth factor=1.3
computed pressure contours on coarse grid
Computed Pressure Contours on Coarse Grid
  • Mach=0.2, Incidence=10 degrees, Re=1.6M
spanwise stations for cp data
Spanwise Stations for Cp Data
  • Experimental data at 10 degrees incidence

VKI Lecture Series, February 3-7, 2003

comparison of surface cp at middle station
Comparison of Surface Cp at Middle Station

VKI Lecture Series, February 3-7, 2003

computed versus experimental results
Computed Versus Experimental Results
  • Good drag prediction
  • Discrepancies near stall
multigrid convergence history
Multigrid Convergence History
  • Mesh independent property of Multigrid

VKI Lecture Series, February 3-7, 2003

parallel scalability1
Parallel Scalability
  • 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

two dimensional high lift
Two-Dimensional High-Lift
  • 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

typical agreement for nsu2d solver
Typical Agreement for NSU2D Solver
  • Good CP agreement in linear region of CL curve

(Lynch, Potter and Spaid, ICAS 1996)

typical agreement for nsu2d solver1
Typical Agreement for NSU2D Solver
  • CLmax overpredicted
  • CLmax Incidence overpredicted by 1 degree

(Lynch, Potter and Spaid, ICAS 1996)

effect of grid resolution and dissipation
Effect of Grid Resolution and Dissipation
  • Wake capturing requires fine off-body grid
  • Enhanced by low dissipation scheme
  • More difficult further downstream
  • Slat wake deficit consistently overpredicted
prediction of incremental effects
Prediction of Incremental Effects
  • Adequate Reynolds number effect prediction
    • Provided no substantial transitional effects
    • Transition is important player
      • Transition models

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

prediction of gap overlap effects
Prediction of Gap/Overlap Effects
  • 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)

status of high lift simulation
Status of High Lift Simulation
  • 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

VKI Lecture Series, February 3-7, 2003

conclusions and future work
Conclusions and Future Work
  • 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

VKI Lecture Series, February 3-7, 2003

adaptive meshing and error control
Adaptive Meshing and Error Control
  • 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
  • Mechanics of mesh adaptation
  • Refinement criteria and error estimation

VKI Lecture Series, February 3-7, 2003

mechanics of adaptive meshing
Mechanics of Adaptive Meshing
  • 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

VKI Lecture Series, February 3-7, 2003

subdivision types for tetrahedra
Subdivision Types for Tetrahedra

VKI Lecture Series, February 3-7, 2003

subdivision types for prisms
Subdivision Types for Prisms

VKI Lecture Series, February 3-7, 2003

subdivision types for pyramids
Subdivision Types for Pyramids

VKI Lecture Series, February 3-7, 2003

subdivision types for hexahedra
Subdivision Types for Hexahedra

VKI Lecture Series, February 3-7, 2003

adaptive tetrahedral mesh by subdivision
Adaptive Tetrahedral Mesh by Subdivision

VKI Lecture Series, February 3-7, 2003

adaptive hexahedral mesh by subdivision
Adaptive Hexahedral Mesh by Subdivision

VKI Lecture Series, February 3-7, 2003

adaptive hybrid mesh by subdivision
Adaptive Hybrid Mesh by Subdivision

VKI Lecture Series, February 3-7, 2003

refinement criteria
Refinement Criteria
  • 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

adjoint based error prediction
Adjoint-based Error Prediction
  • 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 …
  • 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

adjoint based mesh adaptation criteria
Adjoint-based Mesh Adaptation Criteria

Reproduced from Vendetti and Darmofal (MIT, 2002)

VKI Lecture Series, February 3-7, 2003

adjoint based mesh adaptation criteria1
Adjoint-based Mesh Adaptation Criteria

Reproduced from Vendetti and Darmofal (MIT, 2002)

VKI Lecture Series, February 3-7, 2003

high order accurate discretizations
High-Order Accurate Discretizations
  • 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)
higher order methods
Higher-Order Methods
  • Most effective when high accuracy required
  • Potential role in drag prediction
    • High accuracy requirements
    • Large grid sizes required
higher order accurate discretizations
Higher-Order Accurate Discretizations
  • Transfers burden from grid generation to Discretization

VKI Lecture Series, February 3-7, 2003

spectral element solution of maxwell s equations
Spectral Element Solution of Maxwell’s Equations

J. Hestahaven and T. Warburton (Brown University)

combined h p refinement
Combined H-P Refinement
  • 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

conclusions
Conclusions
  • 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

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