Numerical Solution Algorithms for Compressible Flows

(1)

Numerical Solution Algorithms

for Compressible Flows

Lecture Notes

Hrvoje Jasak

Faculty of Mechanical Engineering

and Naval Architecture

University of Zagreb, Croatia

Academic Year 2006-2007

Course prepared for the Aerospace Engineering Program Tempus NUSIC Project JEP-18085-2003

c

(2)

(3)

Part I

(8)

(9)

Chapter 1 Introduction

Computational Fluid Dynamics

• Definition of CFD, from Versteeg and Malalasekera: “An Introduction to Computational Fluid Dynamics”

“Computational Fluid Dynamics or CFD is the analysis of sys-tems involving fluid flow, heat transfer and associated phenomena such as chemical reactions by means of computer-based simula-tion.”

• CFD is also a subset of Computational Continuum Mechanics: fundamen-tally identical numerical simulation technology is used for many sets of simular partial differential equations

– Numerical stress analysis

– Electromagnetics, including low- and high-frequency phenomena – Weather prediction and global oceanic/atmosphere circulation models – Large scale systems: galactic dynamics and star formation

– Complex heat and mass transfer systems

– Fluid-structure interaction and similar coupled systems

• In all cases, equations are very similar: capturing conservation of mass, momentum, energy and associated transport phenomena

(10)

(11)

Chapter 2 Introduction: CFD in

Aeronautical Applications

2.1 Modern Aircraft Design and CFD

In this section we will explore the role and history of Computational Fluid Dy-namics in the aerospace industry. Problems of aerospace design were leading the technological push for a long period in the 20th, dictating areas of research and, together with nuclear research, expanding the use of numerical modelling.

Even today, aerospace and related technology (e.g. rocket design) is considered sufficiently serious to limit the access to latest design by world powers to other governments: however, major parts of technology may be considered more than 50 years old. Example: Chuck Yeager and the first supersonic flight, 1947?

In NASA, the new push towards manned space exploration and reach-out towards a manned Mars mission involves mainly fluid dynamics challenges. The mission requirements beyond Earth’s orbit are more or less settled.

New work in aircraft design concentrates on optimising the existing technology with very few revolutionary new ideas. Main area of work still follws traditional “functional decomposition” of functions on an airplane: wings for lift, rudder for steering, body as useful volume. However, new simulation techniques allow us to re-visit functionally interesting alternatives: flying wind configuration has been revived recently in the B-2 bomber after 50 years from the first attempt.

Introduction

• Aerospace industry is the first and most prevalent in the use of numerical techniques, including Computational Fluid Dynamics (CFD)

• Early beginning of CFD in early 1960’s

(12)

• The creation of the CFD-service industry started in the 1980’s • The CFD industry expanded significantly in the 1990’s

• First fully computer-based design process for external aerodynamics design in a commercial aircraft: Airbus 380 in 2000’s

• In most phases of the process, it was the aerospace industry driving the CFD development to answer to its needs

Early adpotion of numerical modelling in aerospace applications has brought with it some interesting consequences: living with expensive computer hardware and limited memory space leads to simplified modelling techniques and carefully tuned solution algorithms for the set of problems under consideration. Example: 1-equation turbulence models for airfoil calculations, e.g. Baldwin-Lomax. Aerospace Industry and CFD

• Use of CFD is no longer in question: definitely used throughout the design process

• Questions on fidelity and accuracy: can we get sufficiently reliable results?

• Roll-out of CFD continues with more complex requirements, increase of the computer power and applicability of new methods (optimisation)

• Some problems still hit issues with level of performance increase: how much is the difference in results quality between steady 2-D RANS and 3-D LES for single airfoil design

(13)

2.1 Modern Aircraft Design and CFD 13 • Challenges in aircraft design moving elsewhere: systems integration, con-trol components (e.g. electro-hydraulics), packaging, computer support and “battlefield integration”, advanced materials (single-crystal turbine blades) etc.

The only truly revolutionary new technology on its way to (military) aircraft is a scram-jet engine: air-breading jet engine without a compressor or a tur-bine, where shock management in supersonic flow is used to create the necessary compression.

State of the market

• Boeing and Airbus totally dominate the commercial airliner market. Num-ber of smaller players on the edges and in business/regional jet business: ATR, Gulfstream, Raytheon etc.

• Military situation a bit more diverse: BAE Systems, Lockheed Martin, Sukhoi and a number of smaller manufacturers

• I also count missile systems and aircraft engine manufacturers (General Electric, Rolls Royce, Pratt and Whitney)

• NASA. In the latest budget statement, claims that all of its critical problems are associated with managing fluid flow: the space bit in the middle is not nearly that difficult.

• High-speed car aerodynamics: highly specialised, very rich, with very clear requirements. Often bundled with aerospace: wings are turned upside-down, creating down-force instead of lift; concerns about drag very differ-ent than in standard car industry. However, the aerodynamics problem is much more complex than in aircraft. Example: proximity to the ground: important boundary layer effects; trying to organise a much more complex flow pattern

• Aircraft design also includes other flow physics and auxiliary component simulations

In order to understand the requirements of various uses of CFD, let us in-troduce a simple flow classification, based on how tightly managed is the flow field.

My flow classification

• Smooth flows: engineering machinery specifically organises the flow for maximum stability and efficiency. Design conditions are clearly defined and

(14)

their variation is relatively small. Fractional changes in flow characteris-tics have profound performance effects (detached flow, small recirculations, turbulence. Example: aircraft at cruising speed, turbo-machinery blades, yacht design

• Rough flows: flow regime is uncertain, the main object of design is not flow management, but flow may still have critical effect on performance. Example: electronics cooling, passenger compartment comfort in aircraft, swimsuit of Olympic swimmers

• In aerospace, we mostly deal with smooth flows

Scope of Simulations

• Traditionally, experimental studies in aerospace are important, but full-scale models are more and more out of the question. This creates ideal scope for numerical studies

• Questions we look to answer with numerical simulation techniques range from simple lift and drag studies to extremely complex physical problems: stall characteristics, stability in manoeuvres, sensitivity and robust design, optimisation, aero-acoustic noise. A number of new techniques stem from use of CFD in aerospace and are still spreading through the rest of CFD and industry

• The “baseline” physics involved is relatively simple: compressible Newto-nian flows of an ideal gas

• . . . complications easy to add: incompressible to hypersonic flow regime

Speed Range Mach Number low subsonic < 0.3 high subsonic 0.3 − 0.6 transonic 0.6 − 1.1 supersonic 1 − 5 hypersonic > 5

(15)

2.1 Modern Aircraft Design and CFD 15 0.01 0.1 1 10 100 1000 0 1 2 3 4 5 6 7 8 9 general aviation jet aircraft model airplanes u, m/s log(Re) dust

insects gliders airships

• Even in simplest flows, we do not have an easy job: turbulence compli-cates the situation immensely! The problem of turbulence modelling for engineering applications is still unsolved; however, the physics is straight-forward and well understood

• Away from the baseline, physics can get considerably more complex: com-bustion, de-icing, multi-phase flow etc.

• There is significant penetration of general purpose CFD tools into the aerospace companies, but this is still considered a massive untapped market from the commercial CFD point of view. It is unclear that general purpose tools will be sufficiently good to do the job.

Numerical simulation software

• You don’t do CFD without computers! Early efforts with pieces of paper and rooms of people date from UK Metheorological office, running large scale weather forecasting simulations

• In the last 10 years, CFD performance and use coming together

– Computers power is a cheap commodity. Massively parallel computers are commonplace today and can be easily handled in software

– In aerospace, understanding the physics is typically not a problem – Numerical methods cleaned up of systemic errors and gross failures – Sufficient experience in research departments

– Validation against “trusted” experimental data – Understanding of simplifications and assumptions

• In other industries, roll-out of numerical simulation tools limited by expe-rience. Phases of integration of CFD in the design process:

(16)

1. Research and development departments: validation and assessment of capabilities. Typically involves detailed study of old designs or production pieces and comparison with available measured data. 2. Pre-design: experimenting with early prototypes and new ideas away

from the current development line

3. Design and pre-production: new product development.

4. Production: optimisation of existing components and incremental de-velopment of the running design

• In aerospace design, it is no longer sufficient to make a plane fly – Economy, fuel consumption

– Government regulations: noise and pollution levels. Example: noise pollution caused by the supersonic shock wave on the ground killed supersonic flight! Simulation objective: dissipate the shock between the plane and the ground

– Passenger comfort. Includes both oscillatory and non-oscillatory flows around the aircraft, as well as cabin heating and air-conditioning. Ex-ample: Boeing 747-300 with a wiggly tail

– Military application requirements: agile manoeuvring system and un-stable aerodynamic configurations

– In some cases, aerodynamics design does not dominate: instead, it is necessary to make a bad aerodynamic shape fly. Example F-117 stealth bomber. This is also a good example of what happens when the numerical simulation software (in this case, simulation followed elec-tromagnetic signature) cannot handle a traditional engineering shape of an aircraft. Note that F-117 is an old aircraft: scheduled to be retired from US Air Force by the end of 2008.

Process of performing a CFD simulation has evolved through the years, with maturing numerical simulation tools and transition of the design work from the drafting board to a Computer-Aided Desegn (CAD) system. This is still ongoing: over the last few years, providers of CAD solutions have started talking about Product Lifetime Management (PLM) solutions, moving the complete cycle under a single IT-based system. Even in a relatively modern field like CFD, legacy practices still act as a stumbling block. Example: traditionally, a meshing tool and CFD solver are two separate software components; add to this the problem of transferring geometrical data from a CAD package into a mesher and attempting to run an optimisation simulation in this manner. From the moves in the market, it is expected that software convergence may happen over the next 5-10 years (one generation of CFD software tools. Note that the similar problem in structural

(17)

2.1 Modern Aircraft Design and CFD 17 analysis has already been overcome: compared to fluid flow, physics involved in structural simulation is significantly simpler.

Phases of a CFD Simulation

• Description of the geometry. Airfoil curve data, CAD surface or any-where in between. External aerodynamics = geometry of interest located in a large domain (atmosphere)

• Extraction of the fluid domain. In cases where a CAD description is given, a considerable amount of clean-up may be required. This is not easily done, no reliable automatic tools.

• Mesh generation. Based on the given fluid domain, a computational mesh is created. Tools range form manual (points and cells), semi-automatic (block splitting, template geometries, surface wrapping (adaptation of a mesh template to a given surface) to fully automatic (tetrahedral and hex-ahedral/polyhedral automatic mesh generators). Mesh generation is the most demanding and time-consuming process today. Significant push to automatic tools. In spite of automatic tools, there is room for engineer-ing judgement, as a quality solution can be obtained more cheaply by con-structing a quality mesh. A good mesh takes into account what the solution should look like.

• Physics setup. Select the governing equations and specify the material properties and boundary conditions involved. Second level of engineering judgement: how much does the knowledge of detailed material behaviour improve the final result. Example: specific heat capacity of water as a function of temperature; thermal expansion coefficient of water as a function of temperature T .

• Boundary condition setup. This includes both the location and type of boundary conditions used. The role of boundary conditions is to include the influence of the environment to the solution. In “big box” cases, this is easier than is other engineering simulations

• Solver setup and simulation. Choice of discretisation parameters and numerical solution procedure: differencing schemes, relaxation parameters, multigrid, convergence tolerance etc..

• Data post-processing and analysis of results. Not always straightfor-ward.

– Integral studies. In simple lift and drag studies, we could be looking at a small number of integral properties.

(18)

– Flow organisation, where global characteristics of the flow are con-trolled to achieve stability or a desired pattern

– Management of detailed flow structure. Example: remove the vortex depositing dirt on a part of the windshield

– Sensitivity and robust design studies. Usually cannot be seen in results without experience or require specialised simulations.

Advanced visualisation tools are a part of the game: provides a way of managing the wealth of data.

20 years ago, leading CFD tools were developed at Universities, centred around strong research groups and attracting significant funding from the indus-try. As a response to deployment problems, large aerospace companied develop their own research teams and in-house expertise.

Today, CFD software development at Universities is winding down signifi-cantly: the components a good research platform requires are substantial and very few groups can afford to finance the effort (there is little research value and publishable results in writing a new “known technology” CFD code. Majority of groups rely on commercial CFD software to do their research.

In-house software development in large companies suffers a similar fate: the work that can be done by commercial software is migrated to commercial CFD codes and sometimes even outsorced. Apart from financial pressures, this is related to software development, maintenance and validation work required to keep in-house codes working and up-to-date with technology.

The above pushes larg-scale software work to commercial organisation, which have grown from small companies in late 1980s and early 1990s to large organisa-tions. Current trend towards large packages that can satisfy all simulation needs to all customers under the same hood acts as a counter-weight for this state of affairs. Specialised software for specialised needs and technology components giving competitive advantage will be kept separate.

CFD Software Development

• Small experimental codes: playing around with physics and numerical meth-ods

• In-house “general” CFD solver development

• In-house custom-written software for specific purposes: e.g. wing-nacelle en-gine system, turbine blade optimisation, simulation of unstable manoeuvres in military jets, calculation of directional derivatives and solution stability, matching computations with measured data sets etc.

(19)

2.1 Modern Aircraft Design and CFD 19 – Simplified physics, e.g. potential flow and boundary layer codes – Hooked-up mesh generation and parametrisation

– Special purpose codes: sensitivity, aero-acoustics etc.

– In-house development kept secret: competitive advantage. Example: Pratt & Whitney material properties databases

• Government-sponsored (National Labs) developments

• General-purpose CFD packages: from a fridge to a stealth plane • University research codes; public-domain software

• “Write-your-own” CFD solver

• Software getting increasingly complex: you need a PhD to join the game Market situation: Aerospace CFD

• Aerospace atypical for the general CFD picture: early adopter with lots of experience in-house and specific tools targeted to applications

• In-house codes extremely important and integrated into the design process. However, currently approaching “vintage” status

• Example: Boeing dominated by multi-block structured solvers, which cur-rently hinders development. Airbus came in later and developed unstruc-tured solvers in-house, with the massive competitive advantage

• There are problems with in-house codes: development effort more complex, people with knowledge move on, process of acceptance and validation very long

• Simulation software needs to become more user-friendly and closer to the CAE line. This implies extra work apart from “raw” solver capability which is not easily handled in-house.

• Additional CAD-related requirements and cost of keeping CFD develop-ment teams in-house opens the room for commercial general-purpose CFD packages

• There also exists a number of consortium or government-sponsored codes. Example: NASA (USA), DLR (Germany)

(20)

Remaining Challenges

• Mesh generation, especially parallel mesh generation • Handling massively parallel simulations

• Integration into the CAD-based design process • Fluid-structure interaction and aeroacoustics

• On the cusp between two generations of general-purpose CFD solvers: pro-cedural programming, Fortran and C against object orientation

• The push for bigger, faster, more accurate simulations in external aerody-namics not so strong in the aerospace market: meshes are already suffi-ciently large. Also, extensive experience of the required size of the model, mesh resolution and locally fine meshes from the days when computer power was expensive

• In aircraft engine design, the opposite is the case. ASC Project (Advanced Simulation and Computing), US Dept of Energy, Los Alamos, Livermore, Sandia, Stanford University and other partners

http://www.stanford.edu/group/cits/research/index.html http://www.llnl.gov/PAO/news/asc/

– Tip-to-toe simulation of a turbo-fan aircraft engine, including fan, turbo compressor, combustion chambers and turbine. Preferred mod-elling technique: Large Eddy Simulation

– Integrated Multicode Simulation Framework

– As a part of the project, world’s biggest parallel computers have been built:

∗ ASC Red, Sandia 1996

∗ ASC Blue Livermore, Los Alamos, 1998 ∗ ASC White, Livermore, 2001

∗ ASC Q, Los Alamos, 2003

∗ ASC Red Storm, Sandia, 2004, 40-TeraOps ∗ ASC Purple, Livermore, 2005, 100-TeraOps

∗ ASC Blue Gene Livermore, Los Alamos, : 130 000 CPUs and 360 TeraFlops performance.

– For comparison, ASC Linux, 960 node-linux box with 1920 processors and 3.8 TB produces peak performance of 9.2 TeraFlops/s

– The idea of doing a complete engine is somewhat abandoned: not enough power for LES on compressor or turbine. Using combined RANS/LES simulation approach with coupling on interfaces.

(21)

2.2 Scope of Computational Efforts 21

2.2 Scope of Computational Efforts

The level and fidelity of numerical simulation is tailored to the design process: it will cover everything form preliminary design tools running in 1-2 seconds to full transient CFD studies for complex physics simulations. The use of analytical and “pedestrian” methods in early design phases cannot be ignored: laying out the initial set-up of a jet engine compressor is done using precisely the techniques taught in University turbomachinery courses. Once, the basic design is laid down, more detailed tools will be used to satisfy design requirements and optimise the performance.

Aerodynamic Drag

• Drag varies with the velocity squared: major influence at aerospace speed. Narrow improvements in drag lead to considerable advances:

A 15% drag reduction on the Airbus A340-300B would yield a 12% fuel saving, other parameters being constant.

(Mertens, 1998)

• Chasing drag improvements in highly optimised shapes is only of marginal interest Cd = 0.47 Sphere Cd = 0.50 Half sphere Cone Cube Cd = 0.42 Cd = 1.05 Angled cube Long cylinder Short cylinder Cd = 0.80 Cd = 0.82 Cd = 1.15 Cd = 0.04 Streamlined body

• Simulations include functional subset cases, e.g. airfoils, wings, tails config-uration, nacelle-to-wing assembly, but also full aircraft models

• Subjects of interest include shock-boundary layer interaction: effects of shocks on standard turbulence model prediction is still in question.

(22)

High-Lift Aerodynamics

• High-lift wing configuration very important: lower take-off and landing speed, higher pay-load etc.

• Study of multi-element airfoil configuration: high flow curvature, flow sepa-ration, wakes from upstream elements, laminar-to-turbulent boundary layer transition etc.

• High-lift devices added to wings include flaps and slats (common), but also leading edge extensions, vortex generators and blown flaps

• The subject of control is boundary layer management and flow stability (avoiding stall)

(23)

2.2 Scope of Computational Efforts 23 • Looking at Formula 1 aerodynamics, many similar devices can be found Unsteady Aerodynamics

• In most cases, aerodynamic flow are considered steady-state: flight at cruis-ing speed, steady-state lift-off configuration etc.

• Unsteady effects are sometimes critical, both in oscillatory and non-oscillatory regime

• Oscillatory instability: dynamic stall on helicopter rotor blades in forward flight; vortex shedding behind bluff bodies

• Non-oscillatory flows: flow separation at the high angle of attack. Turbu-lence effects are critical for accurate modelling

• Unsteady transonic effects, moving or oscillating shock studies: significant effect on the performance, especially in cases of high-speed helicopter rotor blades

• Unsteady aerodynamics is closely related to aero-elasticity. Sources of un-steadiness are mechanically generated: flutter

Rotary Aerodynamics

• Simulation of helicopter rotor blades usually considered a specialised area of research: special assumptions and modelling regime

• Study of dynamic stall, blade-vortex interaction, blade-to-blade interaction, blade tip effects and transonic flow effects

• Similar effects, but at lower speeds can be found in other devices, e.g. wind turbines, propeller design, turbo-machinery

High-Speed Aerodynamics

• At high speed, the equation of state and ideal gas assumptions break down. In other aspects, the flow is becoming easier to handle. Generally refers to speed of Ma = 5 and above

• For high speed, and due to the real gas effects we speak of aerothermody-namicsrather than aerodynamics.

• Regimes of hypersonic flow: separation is done based on the choice of equa-tion of state

(24)

– Perfect gas. Flow regime still Mach number independent, but there are problems with adiabatic wall conditions

– Two-temperature ideal gas. Rotational and vibrational motion of the molecules needs to be separated and leads to two-temperature models. Used in supersonic nozzle design

– Dissociated gas. Multi-molecular gases begin to dissociate at the bow shock of the body.

– Ionised gas. The ionised electron population of the stagnated flow becomes significant, and the electrons must be modelled separately: electron temperature. Effect important at speeds of 10 − 12km/s Rudder and Steering Diagrams

• In automated steering/targeting systems, the aircraft/missile is controlled by a computer: given target or flight path

• Automatic control systems rely on the diagrams showing the response on steering commands: in practice, large look-up tables or fitted functional data. Consider a case of a rotating missile with 2 × 4 control surfaces. • The steering data created by computation: combinations of control

con-figurations with lift, drag, pitch, yaw orientation and force response. This typically involves 5-10 thousand simulations, done automatically on mas-sively parallel computers. Automatic mesh generation, massive parallelism and controlled accuracy are essential.

http://people.nas.nasa.gov/ aftosmis/home.html

Internal Flows and Auxiliary Devices

• Internal flows: incompressible, low speed, aerodynamics forces typically of no consequence

• Example: passenger compartment comfort, heating, cooling and ventila-tion. Closer to “standard” CFD and usually handled by general-purpose CFD packages

Stability and Robust Design

• Stability analysis takes into account the effects of uncertainly (noise) in the input parameters. Example: how much will the lift coefficient on the airfoil change with a 5% change in the angle of attack?

(25)

2.2 Scope of Computational Efforts 25 – At stall: catastrophic change

– What about a NACA 0012 (symmetric airfoil profile) at zero angle of attack?

• Stability of the solution on small perturbations can be examined in different ways:

– Lots of simulations: detailed analysis, lots of work

– Special numerical techniques: forward derivatives, adjoint equations (continuous and discrete), Proper Orthogonal Decomposition methods • All of the above are extensively used in aerospace simulations. However,

looking at results is not easy: need to understand the meaning • Robust design studies

– Under normal circumstances, looking to maximise the performance of a device in absolute terms. Example: maximum lift in multi-element airfoils

– In reality, requirements are different: consider aircraft landing in a storm, where angle of attack is not constant. Thus, the optimisation process should account for uncertainty of the input parameters and provide stable performance across the range.

– Such effects typically lead to different optimisation results: envelope of performance instead of maximum lift

• Matching of computations with experimental data in combined experimen-tal and numerical studies. Example: unknown flow pattern at the entry of the jet engine combustor, but measured pressure and temperature data available at the outlet.

Fluid-Structure Interaction

• The first step in modelling is to choose the domain of interest. In simple situations, this will cover only a single material or a single governing law. Unfortunately, this is not always the case

• Example: wing flutter

– Aerodynamic forces from fluid flow determine the load on the wing. Wing itself is an elastic structure and deforms under load

– Deflection of the elastic wing changes the flow geometry: a new solu-tion produces different surface load

(26)

• Fluid-structure simulations involve both the fluid and solid domain. Care must be given to the coupling methods and stability of the algorithm

2.3 Finite Volume or Finite Element?

Two sets of numerical techniques handling computational continuum mechanics dominate the field: the Finite Volume Method (FVM) and the Finite Element Method. Once can clearly show both are based on the same principles and are closely mathematically related. Various variants and generalisations can also be devised, but so far their impact has been limited. Some deserve a mention:

• Discontinuous Galerkin discretisation provides a common framework for the FVM and FEM. It combines the conservative flux formulation which is a basis of the FVM with the elemental shape function and a weak formula-tion of the FEM. One of interesting uses would be a formal higher-order extension beyond second-order integrals. So far, the most important use is the generalisation of mathematical machinery underpinning both methods • Lattice Gas and Lattice Boltzmann methods claim to simulate the flow equations from basic principles of molecular dynamics instead of using the continuum equations. Clearly, averaging over sufficient number of latice operations will yield the original PDEs and producing the required solution. Attractions of this method follow from simplifications of latice operations to very primitive accuracy (e.g. 3 velocity levels) and simplifications in complex geometry handling

Numerical Techniques in Aerospace Simulations: Spatial Techniques • Finite Difference Method (FDM): really appropriate only for structured

meshes; no conservation properties. Not used commercially. Important use of FDM is in aero-acoustic simulations, where high-order discretisation is essential (e.g. 6th order in space and 10th order in time). Problems with high-order boundary conditions.

• Finite Volume Method: dominates the fluid simulation arena

• Finite Element Method. No particular reason why it cannot be used; how-ever, the bulk of the numerical method development targeted to FVM. As a result, some techniques and solution methodology not suitable for fluid flow. I do not know any FEM fluid flow aerospace solvers, but FEM dominates the structural analysis arena

• Discontinuous Galerkin: a formal unification of the FEM and FVM ideas. Strongly conservative and consistent, but extensions are still impractical

(27)

2.3 Finite Volume or Finite Element? 27 (control of matrix properties, solution techniques etc.). Consider it work-in-progress

• Monte Carlo Methods: extensively used in low-density high-speed aerody-namics (Space Shuttle re-entry). Techniques are specialised for high effi-ciency

• Spectral techniques: special purposes only. Extremely efficient and accurate for “box in a box” and cyclic matching simulations, e.g. DNS

Handling Temporal Variation

• Steady state: no temporal discretisation required • Time domain: bulk of transient flow simulations

• Frequency domain: special purposes. Example: in turbo-machinery simu-lations, it is possible to extract the dominant frequencies. Instead of solving a time-dependent problem, a series of steady simulations is set up, each for a selected frequency (effects of the temporal derivative now convert into a source/sink term). The time-dependent behaviour is recovered from the combination of frequency solutions.

Simplified Flow Solvers in Industrial Use

• It is not always necessary to run a full Navier-Stokes solver to obtain usable results. Also, the simulation time is sometimes critical: approximate result now.

• Panel method. Combination of source, sinks, doublets and vortex el-ements used to assemble a “zero streamline” form which represents the body. Extremely fast and capable of producing indicative solutions with experience.

http://www.engapplets.vt.edu/fluids/vpm/

• Potential Flow Solvers. Incompressible formulation considered too ba-sic. However, the compressible potential formulation, or even a transient compressible potential can be very useful. The main effect missing in the simplified form is the viscosity effect in the boundary layer: effective change of shape for the potential region. Potential flow solver can be used to ac-celerate the solution to steady-state for more complex solver: initialisation of the solution

• Potential Flow with Boundary Layer Correction. Here, a combina-tion of the compressible potential and boundary layer correccombina-tion takes into account the near-wall effect: the geometry is corrected for displacement thickness in the boundary layer

(28)

• Euler Flow Solver. Neglects the viscous effects but the compressibility physics can be handled in full.

(29)

Chapter 3 CFD in Automotive Applications

CFD Methodology

• Numerous automotive components involve fluid flow and require optimi-sation. This opens a wide area of potential of CFD use in automotive industry

• CFD approaches the problem of fluid flow from fundamental equations: no problem-specific or industry-specific simplification

• A critical step involves complex geometry handling: it is essential to capture real geometrical features of the engineering component under con-sideration

• Traditional applications involve incompressible turbulent flow of Newtonian fluids

• While most people think of automotive CFD in terms of external aerody-namics simulations, reality of industrial CFD use is significantly different

Automotive CFD Today

• In numbers of users in automotive companies, CFD today is second only to CAD packages

(30)

– Engine coolant jackets

– Under-hood thermal management – Passenger compartment comfort

• In comparison with CFD, experimental studies are expensive, carry limited information and it is difficult to achieve sufficient turn-over

• The biggest obstacle is validation: can CFD results be trusted?

(31)

CFD in Automotive Applications 31 – Required accuracy is beyond the current state of physical modelling

(especially turbulence modelling)

– Simulation cost is prohibitive or turn-around is too slow

– Flow physics is too complex: incomplete modelling or insufficient un-derstanding of detailed physical processes

– In some cases, combined 1-D/3-D studies capture the physics without resorting to complete 3-D study

• Examples:

– Prediction of the lift and drag coefficient on a car body – In-cylinder simulations in an internal combustion engine

– Complete internal combustion engine system: air intake, turbo-charger, engine ports and valves, in-cylinder flow, exhaust and gas after-treatment • CFD can still contribute: parametric study (trends), reduced experimental

work etc.

• Numerical modelling is particularly useful in understanding the flow or looking for qualitative improvements: e.g. optimisation of vehicle soiling pattern on windows

(32)

(33)

CFD in Automotive Applications 33

• CFD is used across the industry, at various levels of sophistication

• Impact of simulations and reliance on numerical methods is greatest in areas that were not studied in detail beforehand

• Considerable use in cases where it is difficult to quantify the results in simple terms like the lift and drag coefficient

– Flow organisation, stability and optimisation

– Detailed look at the flow field, especially in complex geometry – Optimisation of secondary effects: fuel-air mixture preparation

(34)

CFD Capabilities in 1980s: Early Adoption in Aerospace Industry • Historically, early efforts in CFD involve simplified equations and

simula-tions relevant for aerospace industry

• Experience in achieving best results with limited computational resources: attention given to solution acceleration techniques

• Application-specific physical models

– Linearised potential equations, Hess and Smith, Douglas Aircraft 1966 – 3-D panel codes developed by Boeing, Lockheed, Douglas and others

in 1968

– Specific turbulence models for aerospace flows, e.g. Baldwin-Lomax – Coupled boundary layer-potential flow solver, Euler flow solver • Capabilities beyond steady-state compressible flow were very limited

(35)

CFD in Automotive Applications 35

Early Automotive CFD Simulations

• First efforts aimed at simplified external aerodynamics (1985-1988) • . . . but airfoil assumptions are not necessarily applicable

• Joint numerical and experimental studies: validation of numerical tech-niques and simulation tools, qualitative results, analysis of flow patterns and similar

• It is quickly recognised that the needs of automotive industry and (poten-tial) capabilities of CFD solvers are well beyond contemporary experimental work

• Focus of early numerical work is on performance-critical components: in-ternal combustion engines and exin-ternal aerodynamics

• Geometry and flow conditions are simplified to help with simulation set-up Example: Intake Valve and Manifold

• 2-D steady-state incompressible turbulent fluid flow

(36)

• Simulation by Peri´c, Imperial College London 1985

Automotive of CFD in 1990s: Expanding Computer Power and Vali-dated Models

• Numerical modelling is moving towards product design

– Improvements in computer performance: reduced hardware cost, Moore’s law

– Improved physical modelling and numerics: fundamental problems are with flow, turbulence and discretisation are resolved

(37)

CFD in Automotive Applications 37 • Notable improvement in geometrical handling: realistic 3-D geometry • Graphical post-processing tools and animations: easier solution analysis • Mesh generation for complex geometry is a bottle-neck: need better tools

Expansion of Automotive CFD

• Increase in computer performance drives the expansion of CFD into new areas by reducing simulation turn-over time

• Massively parallel computers provide the equivalent largest supercomputers at prices affordable in industrial environment (1000s of CPUs)

(38)

Physical Modelling

• New physical models quickly find their use, e.g. free surface flows

• Looking at more complex systems in transient mode and in 3-D: simulation of a multi-cylinder engine, with dynamic effects in the intake and exhaust system

• Computing power brings in new areas of simulation and physical modelling paradigms. Example: Large Eddy Simulation (LES) of turbulent flows Integration into a CAE Environment

• Computer-Aided Design software is the basis of automotive industry • Historically, mesh generation and CFD software are developed separately

and outside of CAD environment, but the work flow is CAD based! • Current trend looks to seamlessly include CFD capabilities in CAD Summary: Automotive CFD Today

• CFD is successfully used across automotive product development

• Initial “landing target” of external aerodynamics and in-cylinder engine simulation still not reached (!) – sufficient accuracy difficult to achieve Lessons Learned

• The success of CFD in automotive simulation is based on providing indus-try needs rather than choosing problems we may simulate: find a critical broken process and offer a solution

• Numerical simulation tools will be adopted only when they fit the product development process: robust, accurate and validated solver, rapid turn-over • Experimental and numerical work complement each other even if sufficient

accuracy for predictive simulations cannot be achieved

– Validation of simulation results ↔ understanding experimental set-up – Parametric studies: speeding up experimental turn-over

• True impact of simulation tools is beyond the obvious uses: industry will drive the research effort to answer its needs

(39)

Part II

(40)

(41)

Chapter 4 Mesh Handling

4.1 Introduction

When presenting a continuum mechanics problem for computer simulation, one needs to establish not only the mathematical model but also the computational domain. While the choice of physics is relatively general, numerical description of the domain of interest is considerably more complex. Looking at the area of external aerodynamics in aerospace, compressible Navier-Stokes equations for an ideal gas with typically suffice, while the wealth of geometrical shapes defies even basic classification.

In most cases, shape of the spatial domain is of primary interest: capturing it in all relevant detail is essential. In transient simulations, handling the temporal axis is considerably simpler. Due to uni-directional nature of interaction, it is sufficient to split the time interval into a finite number of time-steps and march the solution forward in time.

It quickly becomes clear that fidelity of geometrical description of an engi-neering object plays an important role. For example, in a heat exchanger, it is necessary to capture active surface area with some precision in order to correctly calculate the total heat transfer. At the same time, it is a question of engineering judgement to decide which geometrical features are important for the result and which may be omitted.

A computational mesh splits the space into a finite number of elements (cells, control volumes or similar), bounded by faces and supported by points. Compu-tational locations are located in the cells or on the points in a regular manner. The idea of mesh support is to discretise the governing equations over each cell and handle cell-to-cell interaction. Some mesh validity criteria follow directly from the above:

• Computational cells should not overlap;

(42)

Every discretisation method bring its own mesh validity criteria and measures of mesh quality. In general terms, a mesh that visually pleasing is also likely to support a quality solution. Our second concern is the interaction between the mesh resolution and (known or implied) solution characteristics. Features such as shocks, boundary layers and mixing planes require higher resolution that a “far field” section of the domain. Construction of a quality mesh is usually a question of experience and use of quality mesh generation tools. An ideal mesh would be the one uniformly distributing the discretisation error in the solution volume and producing “user-independent” (or, more precisely, user-experience-independent) result. The quest for fast and robust automatic mesh generators iteratively sensitised to the solution is still ongoing.

4.2 Complex Geometry Requirements

Computational Mesh

• A computational mesh represents a description of spatial domain in the simulation: external shape of the domain and highlighted regions of interest, with increased mesh resolution

• Mesh-less methods are possible (though not popular): the issue of describ-ing the domain of interest to the computer still remains

• Mesh generation is the current bottle-neck in CFD simulations. Fully au-tomatic mesh generators are getting better and are routinely used. At the same time, requirements on rapid and high-quality meshing and massively increased mesh size are becoming a problem

• Routinely used mesh size today

– Small mesh for model experimentation and quick games: 100 to 50k cells. Fast turn-around and qualitative results. Note that a number of flow organisation problems may be solved on this mesh resolution – 2-D geometry: 10k to 1m cells. Low-Re turbulent simulations may

require more, due to near-wall mesh resolution requirements – 3-D geometry: 50k to several million cells

– Complex geometry, 3D, industrial size, 100k to 10-50 million cells. Varies considerably depending on geometry and physics, steady/transient flow etc.

– Large Eddy Simulation (LES) 3-D, transient, 1-10 million cells. LES requires very long transient runs and averaging (20-50k time steps), which keeps the mesh resolution down

(43)

4.2 Complex Geometry Requirements 43 – Full car aerodynamics, Formula 1: 20-200 million cells for routine use.

Large simulations under discussion: 1 billion cells!

• On very large meshes, problem swith the current generation of CFD soft-ware becomes a limiting factor: missing parallel mesh generation, data file read/write, post-processing of results, hardware and software prices

Handling Complex Geometry

• In aerospace applications, geometrical information is usually available be-fore the simulation. In general, this is not the case: for simple applications, a mesh may be the only available description of the geometry

• Domain description is much easier in 2-D: real complications can only be seen in 3-D meshes

• Geometrical data formats

– 2-D boundary shape: airfoils. Usually a detailed map of x−y locations on the surface. Sometimes defined as curve data

http://www.ae.uiuc.edu/m-selig/ads/coord database.html

– Stereo Lithographic Surface (STL): a surface is represented by a set of triangular facets. Resolution can be automatically adjusted to cap-ture the surface curvacap-ture or control points. Creation of STL usually available from CAD packages

– Native CAD description: Initial Graphics Exchange Specification (IGES), solid model etc. In most cases, the surface is represented by Non-Uniform Rational B-Splines or approximated by quadric surfaces. Typ-ically, both are too expensive for the manipulations required in mesh generation and either avoided or simplified

• Geometry clean-up. Very rarely is the CAD description built specifically for CFD – in most cases, CAD surfaces (wing, body, nacelle) are assembled from various sources, with varying quality and imperfect matching. Surface clean-up is time-consuming and not trivial. In some cases, the mesh gener-ator may be less sensitive to errors in surface description, which simplifies the clean-up

• Feature removal. CAD description or STL surface may contain a level of detail too fine to be captured by the desired mesh size, causing trouble with 3-D mesh generation. Feature removal creates an approximation of the original geometry with the desired level of detail

(44)

Surface Mesh Generation

• In cases where the surface description is not discrete, a surface mesh may be created first

• STL surface is already a mesh. It may be necessary to additionally split the surface for easier imposition of boundary conditions: inlet, outlet, sym-metry plane etc.

• Surface mesh is usually triangular or quadrilateral. There are potential issues with capturing surface curvature: surface mesh will be considered “sufficiently fine”

Volume Mesh Generation

• The main role of the volume mesh is to capture the 3-D geometry

• The cells should not overlap and should completely fill the computational domain. Additionally, some convexness criteria (FVM) or a library of pre-defined cell shapes (FEM) is included.

• Computational mesh defines the location and distribution of solution points (vertices, cells etc.). Thus, filling the domain with the mesh is not sufficient - ideally some aspects of the solution should be taken into account.

• A-priori knowledge of the solution is useful in mesh generation. Trying to locate the regions of high mesh resolution (“fine mesh”) to capture critical parts of the solution: shocks, boundary layers and simular

• Quality of the mesh critical for a good solution and is not measured only in mesh resolution

• Mesh quality measures depend on the discretisation method – Cell aspect ratio

– Non-orthogonality – Skewness

– Cell distortion from ideal shape – . . . etc.

(45)

4.3 Mesh Structure and Organisation 45

4.3 Mesh Structure and Organisation

Influence of Mesh Structure

• Some numerical solution techniques require specific mesh types. Example: Cartesian meshes for high-order finite difference method

• Supported mesh structure may severely limit the use of a chosen discreti-sation method

• With mesh generation as a bottle-neck, it makes sense to generalise the solver to be extremely flexible on the meshing side, simplifying the most difficult part of the simulation process

Cartesian Mesh

• x − y − z mesh aligned with the coordinate system. May be defined by 2 points and resolution in 3 directions

• Mesh addressing (cells to neighbour cells, cells to points, points to neighbour points etc.) can be calculated on the fly given the mesh dimension

• Simple to define, efficient and can be used with any type of discretisation • Severe limitation on the geometry that can be handled: a box within a box • Extensions may include blocked-out cells or staircase boundaries

Structured Body-Fitted Mesh

• Body-fitted meshes originate from the non-orthogonal curvilinear coordi-nate system approach. The case-specific coordicoordi-nate system is created to fit the boundary

(46)

• The mesh is hexahedral and regularly connected. Real geometry can be captured but with insufficient control over local mesh resolution

• The use of contravariant coordinates for the solution vectors was quickly abandoned

Multi-Block Mesh

• Mesh created as a combination of multiple body-fitted blocks. All block and cells are still hexahedral

• In FVM, special coding is done on block interfaces, where the mesh con-nectivity cannot be implicitly established

• Much more control over mesh grading and local resolution. However, mesh generation in 3-D for relatively complex shapes is still hard and time-consuming: meshes need to match

(47)

4.3 Mesh Structure and Organisation 47

Unstructured Shape-Consistent Mesh

• At this stage, all meshes are hand-built. A complex 3-D mesh could take 2-3 months to construct

• Block connectivity above introduces the concept of storing mesh connec-tivity rather than calculating it: unstructured mesh

• Loose definition of connectivity allows more freedom: hexahedral and de-generate hexahedral meshes: prisms, pyramids, wedges etc. allow easier meshing

(48)

• From the numerical simulation point of view, this is a major step forward. Geometries of industrial interest can now be tackled with a detailed de-scription, which satisfies the design engineer

• At this stage, numerical simulation in an industrial setting really takes off. Handling airfoils and single wing or even wing-fuselage assembly is not too difficult. Hand-built meshes for a complete aircraft are still quite difficult

Tetrahedral and Hybrid Tet-Hex Meshes

• Tetrahedral mesh are not good from the numerics point of view • . . . but they could be generated automatically!

• In a solver can support tetrahedral meshes, mesh generation time for com-plex geometry reduces from weeks to hours.

• Great saving in mesh generation effort, faster turn-around of simulations and geometrical variation, mesh sensitivity studies can be performed on realistic geometries

• Tetrahedra are particularly poor in boundary layers close to walls. A hy-brid mesh is built by creating a layered hexahedral mesh next to the wall.

(49)

4.3 Mesh Structure and Organisation 49 The rest of the domain is filled with tetrahedra. A combined tet-hex mesh is a great improvement in quality

• On the negative side, cell count for a tetrahedral mesh of equivalent res-olution is higher than for hexahedra. A part of the price is paid in lower accuracy of the solver on tetrahedra: limited neighbourhood connectivity.

• Tetrahedral mesh generation techniques

– Advancing front method: starting from the boundary triangulation, insert tetrahedra from the live front using priority lists

– Delaunay triangulation: point insertion and re-triangulation. The ini-tial mesh is created by triangulating the boundary. New points are added in a way which improves the quality of the most distorted tri-angles and creates a convex hull around each point

Overset and Chimera Meshes

• Used for cases where a simple solver is used for complex cases or parts of geometry move relative to each other

• Each part is meshed in a simple manner and over-set on a background mesh. In regions of overlap, special discretisation practices couple the solution • Chimera approach is numerically problematic: issues of coupling,

(50)

Polyhedral Mesh Support

• In spite of automatic generation techniques, tetrahedral meshes are not of sufficient quality for industrial use. On the other hand, automatic hexahe-dral mesh generation has proven to be extremely challenging

• Finite Volume discretisation is not actually dependent on the cell shape: unlike FEM, there are no pre-defined shape functions and transformation tensors. This brings the possibility of polyhedral mesh support

• Finite Volume discretisation algorithm is reformulated into loops over cells and faces (still doing the same job)

• Polyhedral meshes are considerably better than tetrahedra, can be manip-ulated to be predominantly hexahedral, orthogonal and regular and can be created automatically

4.4 Manual Meshing: Airfoils

Mesh Structure for 2-D Airfoils

• Manual meshing of airfoil profiles really belongs to the past; it is still in-dicative to show how mesh handling governs the use of CFD

• O-mesh: NACA0012 example

• C-mesh: NACA32012 example, prettier in raeProfile • H-mesh

(51)

4.4 Manual Meshing: Airfoils 51 • Hybrid mesh structure: triangular mesh with prismatic layers: twoElement • Adapting to the geometry: transfinite mapping techniques

• Adapting to the solution: shock capturing with r-refinement • Meshing multi-element airfoil configurations

Mesh Generation by Partial Differential Equation

• Transfinite mapping operation can be viewed as a solution of the Laplace equation. Thus, a mesh can be created by solving an equation

• Mesh grading can be controlled by sizing functions: Laplace equation with variable coefficients

• An equivalent formulation exists for controlling mesh orthogonality

• This approach to mesh generation is useful in parametric studies, where a large number of similar geometries needs to be simulated. An initial template mesh is built and adjusted to the correct shape

e1 e e3 e e4 e e5 e e6 e e7 e e8 e e9 e e10 ee11ee12e e13 e e15 e e16 e e17 e e19 e e20 e e25 e e26 e e27 e e28 e e29 e e30 e e31 e e32 e e33 e e34 e e35 e e37 e e38 e e39 e e40 e e41 e e42 e e66 e e68 e e69 e e76 e e77 e e1 e e3 e e4 e e5 e e6 e e7 e e8 e e9 e e10 ee11ee12e e13 e e15 e e16 e e17 e e19 e e20 e e22 e e25 e e26 e e27 e e28 e e29 e e30 e e31 e e32 e e33 e e34 e e35 e e37 e e38 e e39 e e40 e e41 e e42 e e66 e e68 e e69 e e76 e e77 e

Polyhedral Mesh Generation • Tessalated mesh

– The Delaunay triangulation algorithm introduces points on proximity rules. During the creation of the mesh, a dual mesh of convex polyhe-dra is created and can be extracted by a post-processing operation – Interaction on the tessalated mesh and the boundary needs to be

re-covered after polyhedral mesh assembly

– Local control of mesh size achieved in the same way as in tetrahedral meshes

(52)

P 1 Pi V1 Voronoi vertex X Z Y

(53)

4.5 Adaptive Mesh Refinement 53 • Cut hexahedral and cut polyhedral mesh

– Most of mesh generation is straightforward: filling space with non-overlapping cells. Even close to boundaries, it is easy to build high quality layered structure

– Problematic parts of mesh generation are related to interaction of ad-vancing generation surfaces or boundary interaction in complex corners of regions where the mesh resolution dos not match the level of detail on the boundary description.

– Cut cell technology creates a rough mesh background mesh, either uniform hexahedral or capturing major features of the geometry. The mesh inside of the domain is kept and the one interacting with the boundary surface is adjusted or cut by the surface

– In some cases, the background mesh resolution can be automatically adjusted around the surface to match the local resolution requirements – Meshes are good quality and can be generated rapidly. Prismatic boundary layers may also be added. In some cases, background mesh adjustment or concave cell corrections are required.

Examples

• 3rd AIAA CFD Drag Prediction Workshop

http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw/

4.5 Adaptive Mesh Refinement

From the above examples it can be seen how the structure and quality of the mesh influences the solution. In first approximation, the number and distribu-tion of computadistribu-tional points determines out picture of the soludistribu-tion even in the absence of computational errors. In places where the solution varies rapidly or

(54)

complex physical processes occur, it is advisable to locally increase the density of computational points.

Putting the resolution requirement on a firmer basis, ona may postulate that every discretisation method aimed at continuum mechanics postulates a local variation of the solution between the computational points. A largest source of discretisation error is a discrepancy between the postulated and actual field variation. Grouping computational points closer together relaxes the difference between the prescribed and actual variation in the solution, reducing the discreti-sation error.

Mesh Resolution

• Mesh structure specifies where the computational points are located. Dis-cretisation practice postulates the shape of solution between the computa-tional points, which is the main source of discretisation error

• A sensible meshing strategy requires high resolution in regions of interest instead of uniformly distributing points in the domain. This implies some knowledge of the solution during mesh generation.

• The same can be achieved in an iterative way 1. Create initial mesh and initial solution

2. Examine the solution from the point of view of accuracy or resolution in “regions of interest”

3. Based on the available solution, adjust mesh resolution in order to improve the solution in the selected parts of the domain

4. Repeat until sufficient accuracy is achieved or computer resources are exhausted

• Performing mesh improvement by hand is tedious and time-consuming. For an automatic procedure, two questions need to be answered:

– Where to refine the mesh (adjust resolution)?

– How to change the mesh to achieve the required accuracy Types of Mesh Refinement

• Global refinement: mesh sensitivity studies

• h-refinement: introducing new computational points in regions of interest • r-refinement: re-organise the existing points such that more points fall into

(55)

4.5 Adaptive Mesh Refinement 55 • p-refinement: enriching the space of shape functions in order to capture the

solution more closely

• Mesh refinement cannot be done indiscriminately: locally refined meshes typically introduce increased mesh-induced errors as well. The trick it to locate the regions of poor mesh away from the regions of interest

CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC CCCCCCCCCCCCCCC BBBBB BBBBB BBBBB BBBBB BBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBB

Error- or Indicator-Driven Adaptivity

• In strongly shocked flows, it is relatively easy to identify regions of interest: shocks, boundary layer, contact discontinuities. In more complex situations or in presence of flow features of different strength, this is much more diffi-cult. Mesh-induced discretisation errors (poor mesh quality or insufficient resolution) also needs to be taken into account.

• A region of interest can usually be recognised by high gradients: rapidly varying solution

(56)

• Error indicators: highlight regions of interest. Example: magnitude of the second pressure gradient, Mach number distribution etc.

• Error estimates: apart from the spatial information (error distribution), they provide guidance on the absolute error level

Adjusting to Original Boundary Shape

• Traditionally, mesh adaptation was a part of the CFD solver instead of mesh generator. In cases where the refinement algorithm resorts to cell splitting, we may end up with a faceted surface representation instead of a smooth surface, which compromises the results.

• Solution: geometrical description of the boundary needs to be available from the solver instead of trying to recover the data from the original (coarse) mesh

• A further step is related to the specification of boundary conditions. In, for example, wind tunnel simulations, the velocity and turbulence at the inlet plane in shown from the measured data and interpolated onto the inlet patch of the mesh. Ideally, the boundary condition should be associated with space or with the boundary description, avoiding problems with in-terpolation. This leads to issues of CAD integration, which is beyond our scope

Examples of Automatic Meshing and Adaptivity • Supersonic flow, h-refinement

(57)

4.6 Dynamic Mesh Handling 57

0.0 0.5 1.0 1.5 2.0 2.5 3.0

4.6 Dynamic Mesh Handling

Many relevant simulations in continuum mechanics involve the cases where the shape of computational domain changes during the simulation, either in a man-ner prescribed up front or as a function of the solution. As we will show later, handling such cases generalises the discretisation practice to some form of “Ar-bitrary Lagrangian-Eulerian” practice, combining the view from the Lagrangian and Eulerian reference frame. This is usually terms dynamic mesh handling, coming in a number of different guises.

From the point of view of mesh handling, we can recognise two distinct situ-ations:

• Mesh deformation, where the structure and connectivity of the mesh remains unchanged, but the position of points supporting its shape changes. Mesh deformation is characterised by the fact that the number of point, faces, cells and boundary faces remains constant, as does the connectivity between the shapes;

• In a topologically changing mesh, the number of points, faces and cells or their connectivity varies during the simulation.

It will be shown that standard discretisation methods handle cases of mesh deformation without loss of accuracy, while topological changes may (depending

(58)

on the algorithm) involve solution re-mapping, with associated interpolation or data redistribution errors. Thus, mesh deformation is usually preferred, unless it implies excessive mesh-induced discretisation errors.

Typical examples of dynamic mesh handling in aerospace application include moving flap and slat simulation, aircraft landing, bomb or missile release (opening of the ordonnance bay), multi-stage turbomachinery simulations with rotor-stator interaction etc.

Moving Deforming Mesh

• There exist cases where the shape of the domain varies during the calcula-tion. Boundary motion may be prescribed in advance as a part of the case setup or be a part of the solution itself

• Internal mesh influences mainly the discretisation error: it is the external shape of the domain which carries the major influence. Moving deform-ing mesh algorithm will allow the domain to change its shape durdeform-ing the simulation and preserve its validity

• Shape changes are performed by point motion: the connectivity and struc-ture of the mesh remains unchanged

Topological Mesh Changes

• In cases of extreme shape change, moving deforming mesh is not sufficiently flexible: deforming the mesh to accommodate extreme boundary deforma-tion would introduce high discretisadeforma-tion errors

• Mesh motion can be accommodated by adding or removing computational cells to accommodate the boundary deformation. This is associated with higher discretisation errors and complications in the algorithm, but is some-times essential

• Common types of topological changes: – Attach/detach boundary

– Cell layer addition/removal – Sliding interface

(59)

4.6 Dynamic Mesh Handling 59

• Typically, a combination of several topological changes will be used together to achieve mode complex mesh changes

(60)

Numerical Solution Algorithms for Compressible Flows