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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

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

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

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

Overview

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

VKI Lecture Series, February 3-7, 2003

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

- 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

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

- Sophisticated Multiblock Structured Grid Techniques for Complex Geometries

Engine Nacelle Multiblock Grid by commercial software TrueGrid.

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

- Connectivity stored explicitly
- Single Homogeneous Data Structure

VKI Lecture Series, February 3-7, 2003

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

- 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

- 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)

VKI Lecture Series, February 3-7, 2003

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

- 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 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 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 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 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

- 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 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 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 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 Approach

VKI Lecture Series, February 3-7, 2003

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

- 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

- 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

- 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

- 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-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-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)

VKI Lecture Series, February 3-7, 2003

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

- 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 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 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

- 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

- 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

- First order scheme

- Second order scheme

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

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

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 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

- 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

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

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

VKI Lecture Series, February 3-7, 2003

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.)

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

Effect of Limiters on Upwind Discretization

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

Effect of Discretization Type

VKI Lecture Series, February 3-7, 2003

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

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

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)

- Factor of 3 savings in grid size

VKI Lecture Series, February 3-7, 2003

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)

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

Grid Convergence (2D Euler)

- Lift converges as h2
- Drag vanishes in continuous limit

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

- Discretization
- Efficient solution techniques
- Multigrid
- Efficient hardware utilization
- Vector
- Cache efficiency
- Parallelization

VKI Lecture Series, February 3-7, 2003

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

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

- 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

- 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)

VKI Lecture Series, February 3-7, 2003

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

- 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

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

VKI Lecture Series, February 3-7, 2003

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

- Convergence rate similar to geometric MG
- Completely automatic

VKI Lecture Series, February 3-7, 2003

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

- 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

- 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

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

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

- 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

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

- 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)

24.7M pts, Cray T3E

177K pts, PC cluster

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

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)

- 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

- 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)

- 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, CL=0.6

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

VKI Lecture Series, February 3-7, 2003

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 values overpredicted
- Increased lift with additional grid resolution

VKI Lecture Series, February 3-7, 2003

Drag Polar at Mach = 0.75

- Grid resolution study
- Good comparison with experimental data

VKI Lecture Series, February 3-7, 2003

Comparison with Experiment

- Grid Drag Values
- Incidence Offset for Same CL

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

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

Drag Rise Curves

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

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)

- 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

VKI Lecture Series, February 3-7, 2003

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

- 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

- 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

- 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 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

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

VKI Lecture Series, February 3-7, 2003

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

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

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

- Complicated flow physics
- High mesh resolution requirements
- On body, off body
- Complex geometries
- Original driver for unstructured meshes in aerodynamics

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

- 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

- 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

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

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

VKI Lecture Series, February 3-7, 2003

Computed Versus Experimental Results

- Good drag prediction
- Discrepancies near stall

Multigrid Convergence History

- Mesh independent property of Multigrid

VKI Lecture Series, February 3-7, 2003

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

- 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

- Good CP agreement in linear region of CL curve

(Lynch, Potter and Spaid, ICAS 1996)

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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

VKI Lecture Series, February 3-7, 2003

Subdivision Types for Prisms

VKI Lecture Series, February 3-7, 2003

Subdivision Types for Pyramids

VKI Lecture Series, February 3-7, 2003

Subdivision Types for Hexahedra

VKI Lecture Series, February 3-7, 2003

Adaptive Tetrahedral Mesh by Subdivision

VKI Lecture Series, February 3-7, 2003

Adaptive Hexahedral Mesh by Subdivision

VKI Lecture Series, February 3-7, 2003

Adaptive Hybrid Mesh by Subdivision

VKI Lecture Series, February 3-7, 2003

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

- 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

Reproduced from Vendetti and Darmofal (MIT, 2002)

VKI Lecture Series, February 3-7, 2003

Adjoint-based Mesh Adaptation Criteria

Reproduced from Vendetti and Darmofal (MIT, 2002)

VKI Lecture Series, February 3-7, 2003

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

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

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

J. Hestahaven and T. Warburton (Brown University)

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

- 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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