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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 Grad. Rec.)

    • Limiters reduces accuracy, increase robustness

    • Less sensitive to non-monotone limiters


    Effect of discretization type
    Effect of Discretization Type Grad. Rec.)

    VKI Lecture Series, February 3-7, 2003


    Effect of element type
    Effect of Element Type Grad. Rec.)

    • 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 Grad. Rec.)

    • Little overall effect on accuracy

    • Potential differences between two codes


    Grid resolution issues
    Grid Resolution Issues Grad. Rec.)

    • 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 ( Grad. Rec.)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 Grad. Rec.)

    • 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 Grad. Rec.)(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) Grad. Rec.)

    • Lift converges as h2

    • Drag vanishes in continuous limit


    Grid convergence
    Grid Convergence Grad. Rec.)

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics(Linear Problems)

    VKI Lecture Series, February 3-7, 2003


    Multigrid for unstructured meshes
    Multigrid for Unstructured Aerodynamics 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • Convergence rate similar to geometric MG

    • Completely automatic

    VKI Lecture Series, February 3-7, 2003


    Anisotropy induced stiffness
    Anisotropy Induced Stiffness Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

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


    Implementation on parallel computers
    Implementation on Parallel Computers Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • 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) Aerodynamics

    24.7M pts, Cray T3E

    177K pts, PC cluster

    • Moderate additional multigrid communication on coarse levels

    • Large Multigrid convergence benefit


    Parallel scalability
    Parallel Scalability Aerodynamics

    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 Aerodynamics

    • 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) Aerodynamics

    • 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 Aerodynamics

    • 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) Aerodynamics

    • 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 Aerodynamics 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 Aerodynamics

    • 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 Aerodynamics

    • 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 Aerodynamics

    • Grid resolution study

    • Good comparison with experimental data

    VKI Lecture Series, February 3-7, 2003


    Comparison with experiment
    Comparison with Experiment Aerodynamics

    • Grid Drag Values

    • Incidence Offset for Same CL


    Surface cp at 40 9 span
    Surface Cp at 40.9% Span Aerodynamics

    • 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 Aerodynamics

    • Grid resolution study

    • Discrepancies at Higher Mach/CL Conditions


    Drag rise curves
    Drag Rise Curves Aerodynamics

    • 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) results)

    • 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 results)

    VKI Lecture Series, February 3-7, 2003


    Cases run on icase cluster
    Cases Run on ICASE Cluster results)

    • 120 Cases (excluding finest grid)

    • About 1 week to compute all cases

    VKI Lecture Series, February 3-7, 2003


    Incremental effects
    Incremental Effects results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • Absolute drag correlation decreases as grid refined

    • Incremental drag correlation improves as grid refined


    Prediction of installation drag
    Prediction of Installation Drag results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

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


    Spanwise stations for cp data
    Spanwise Stations for Cp Data results)

    • 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 results)

    VKI Lecture Series, February 3-7, 2003


    Computed versus experimental results
    Computed Versus Experimental Results results)

    • Good drag prediction

    • Discrepancies near stall


    Multigrid convergence history
    Multigrid Convergence History results)

    • Mesh independent property of Multigrid

    VKI Lecture Series, February 3-7, 2003


    Parallel scalability1
    Parallel Scalability results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    • 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 results)

    VKI Lecture Series, February 3-7, 2003


    Subdivision types for prisms
    Subdivision Types for Prisms results)

    VKI Lecture Series, February 3-7, 2003


    Subdivision types for pyramids
    Subdivision Types for Pyramids results)

    VKI Lecture Series, February 3-7, 2003


    Subdivision types for hexahedra
    Subdivision Types for Hexahedra results)

    VKI Lecture Series, February 3-7, 2003


    Adaptive tetrahedral mesh by subdivision
    Adaptive Tetrahedral Mesh by Subdivision results)

    VKI Lecture Series, February 3-7, 2003


    Adaptive hexahedral mesh by subdivision
    Adaptive Hexahedral Mesh by Subdivision results)

    VKI Lecture Series, February 3-7, 2003


    Adaptive hybrid mesh by subdivision
    Adaptive Hybrid Mesh by Subdivision results)

    VKI Lecture Series, February 3-7, 2003


    Refinement criteria
    Refinement Criteria results)

    • 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 results)

    • 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 results)

    Reproduced from Vendetti and Darmofal (MIT, 2002)

    VKI Lecture Series, February 3-7, 2003


    Adjoint based mesh adaptation criteria1
    Adjoint-based Mesh Adaptation Criteria results)

    Reproduced from Vendetti and Darmofal (MIT, 2002)

    VKI Lecture Series, February 3-7, 2003


    High order accurate discretizations
    High-Order Accurate Discretizations results)

    • 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 results)

    • 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 results)

    • 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 results)

    J. Hestahaven and T. Warburton (Brown University)


    Combined h p refinement
    Combined H-P Refinement results)

    • 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 results)

    • 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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