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Computational Tools for Aircraft Design ITA – Aircraft Design Department V 18

# Computational Fluid Dynamics

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Computational fluid dynamics (CFD) is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the millions of calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. Even with high-speed supercomputers only approximate solutions can be achieved in many cases. Ongoing research, however, may yield software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is often performed using a wind tunnel with the final validation coming in flight test.

### Text of Computational Fluid Dynamics

Computational Tools for Aircraft DesignITA – Aircraft Design Department

V 18

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 2

• CFD What It is?• Overview on Mesh Technology• Turbulence Modeling

Contents

3

CFD What It Is?

4

Fluid (gas and liquid) flows are governed by partial differential equations (PDE) which represent conservation laws for the mass, momentum, and energy. Computational Fluid Dynamics (CFD) is the art of replacing such PDE systems by a set of algebraic equations which can be solved using digital computers. The object under study is also represented computationally in an approximate discretized form.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 5

Introduction Why use CFD?

Numerical simulations of fluid flow (will) enable

• architects to design comfortable and safe living environments• designers of vehicles to improve the aerodynamic characteristics• chemical engineers to maximize the yield from their equipment• petroleum engineers to devise optimal oil recovery strategies• surgeons to cure arterial diseases (computational hemodynamics)• meteorologists to forecast the weather and warn of natural

disasters• safety experts to reduce health risks from radiation and other

hazards• military organizations to develop weapons and estimate the

damage• CFD practitioners to make big bucks by selling colorful pictures :-)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 6

Introduction

Practice of engineering and science has been dramatically altered by the development of

Scientific computingMathematics of numerical analysisTools like neural networksThe Internet

Computational Fluid Dynamics is based upon the logic of applied mathematics

provides tools to unlock previously unsolved problemsis used in nearly all fields of science and engineering

Aerodynamics, acoustics, bio-systems, cosmology, geology, heat transfer, hydrodynamics, river hydraulics, etc…

What is?

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 7

Introduction – What It is?CFD is the simulation of fluids engineering systems using modeling (mathematical physical problem formulation) and numerical methods (discretization methods, solvers, numerical parameters, and grid generations, etc.)

CFD made possible by the advent of digital computer and advancing with improvements of computer resources (500 flops, 1947à20 teraflops, 2003)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 8

Introduction – Why Use CFD?

More cost effective and more rapid than EFD*

CFD provides high-fidelity database for diagnosing flow field

2. Simulation of physical fluid phenomena that are difficult for experiments

Full scale simulations (e.g., ships and airplanes)Environmental effects (wind, weather, etc.)Hazards (e.g., explosions, radiation, pollution)Physics (e.g., planetary boundary layer, stellar evolution)

Knowledge and exploration of flow physics* Experimental Fluid Dynamics

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 9

Introduction

“We are in the midst of a new Scientific Revolution as significant as that of the 16th and 17th centuries when Galilean methods of systematic experiments and observation supplanted the logic-based methods of Aristotelian physics”

“Modern tools, i.e., computational mechanics, are enabling scientists and engineers to return to logic-based methods for discovery and invention, research and development, and analysis and design”

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 10

IntroductionScientific method

Aristotle (384-322 BCE)Greek philosopher, student of PlatoLogic and reasoning was the chief instrument of scientific investigation; Posterior AnalyticsTo possess scientific knowledge, we need to know the cause of which we observe

Through their senses humans encounter facts or dataThrough inductive means, principles created which will explain the dataThen, from the principles, work back down to the facts

Example: Demonstration of the fact (Demonstratio quia) » The planets do not twinkle» What does not twinkle is near the earth» Therefore the planets are near the earth

Knowledge of Aristotle’s work lost to Europe during Dark Ages. Preserved by Mesopotamian (modern day Iraq) libraries.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 11

IntroductionScientific method

Galileo Galilei (1564-1642)Formulated the basic law of falling bodies, which he verified by careful measurements. He constructed a telescope with which he studied lunar craters, and discovered four moons revolving around Jupiter.Observation-based experimental methods: required instruments & tools ; e.g., telescope, clocks.Scientific Revolution took place in the sixteenth and seventeenth centuries, its first victories involved the overthrow of Aristotelian physics

Convicted of heresy by Catholic Church for belief that the Earth rotates round the sun. In 1992, 350 years after Galileo's death, Pope John Paul II admitted that errors had been made by the theological advisors in the case of Galileo.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 12

IntroductionMathematics

Isaac Newton (1643 – 1727)Laid the foundation (along with Leibniz) for differential and integral calculusIt has been claimed that the Principiais the greatest work in the history of the physical sciences. Book I: general dynamics from a mathematical standpoint Book II: treatise on fluid mechanicsBook III: devoted to astronomical and physical problems. Newton addressed and resolved a number of issues including the motions of comets and the influence of gravitation. For the first time, he demonstrated that the same laws of motion and gravitation ruled everywhere under a single mathematical law.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 13

IntroductionFluid Mechanics

Faces of Fluid Mechanics : some of the greatest minds of history have tried to solve the mysteries of fluid mechanics

Archimedes Da Vinci Newton Leibniz Euler

Bernoulli Navier Stokes Reynolds Prandtl

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 14

From mid-1800’s to 1960’s, research in fluid mechanics focused upon

Analytical methodsExact solution to Navier-Stokes equations (~80 known for simple problems, e.g., laminar pipe flow)Approximate methods, e.g., Ideal flow, Boundary layer theory

Experimental methodsScale models: wind tunnels, water tunnels, towing-tanks, flumes,...Measurement techniques: pitot probes; hot-wire probes; anemometers; laser-doppler velocimetry; particle-image velocimetryMost man-made systems (e.g., airplane) engineered using build-and-test iteration.

1950’s – present : rise of computational fluid dynamics (CFD)

IntroductionFluid Mechanics

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 15

IntroductionHistory of computing

Computing, 1945-1960Early computer engineers thought that only a few dozen computers required worldwideApplications: cryptography (code breaking), fluid dynamics, artillery firing tables, atomic weaponsENIAC, or Electronic Numerical Integrator Analyzor and Computer, was developed by the Ballistics Research Laboratory in Maryland and was built at the University of Pennsylvania's Moore School of Electrical Engineering and completed in November 1945

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 16

IntroductionHistory of computing

In the early 1930s Polish cryptographers first broke the code of Germany's cipher machine Enigma. They were led bymathematician Marian Rejewski and assisted by material provided to them by agents of French intelligence. For much ofthe decade, the Poles were able to decipher their neighbour's radio traffic, but in 1939, faced with possible invasion anddifficulties decoding messages because of changes in Enigma's operating procedures, they turned their information over tothe Allies. Early in 1939 Britain's secret service set up the Ultra project at Bletchley Park, 50 miles (80 km) north ofLondon, for the purpose of intercepting the Enigma signals, deciphering the messages, and controlling the distribution ofthe resultant secret information. Strict rules were established to restrict the number of people who knew about the existenceof the Ultra information and to ensure that no actions would alert the Axis powers that the Allies possessed knowledge oftheir plans.The incoming signals from the German war machine (more than 2,000 daily at the war's height) were of the highest level,even from Adolf Hitler himself. Such information enabled the Allies to build an accurate picture of enemy plans and ordersof battle, forming the basis of war plans both strategic and tactical. Ultra intercepts of signals helped the Royal Air Force towin the Battle of Britain. Intercepted signals between Hitler and General Günther von Kluge led to the destruction of alarge part of the German forces in Normandy in 1944 after the Allied landing.

Left. The Colossus computer at Bletchley Park,Buckinghamshire, England, c. 1943. Funding for thiscode-breaking machine came from the Ultra project.

Ultra Project

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 17

IntroductionHigh-performance computing

Top 500 computers in the world compiled: www.top500.orgComputers located at major centers connected to researchers via Internet

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 18

Outline – CFD What It is?

CFD ProcessModel EquationsDiscretizationGrid GenerationBoundary ConditionsSolvePost-ProcessingUncertainty Assessment

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 19

Where is CFD used?

Where is CFD used?

AerospaceAutomotiveBiomedicalChemical ProcessingHVACHydraulicsMarineOil & GasPower GenerationSports

F18 Store Separation

Temperature and natural convection currents in the eye following laser heating.

Aerospace

Automotive

Biomedical

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 20

Where CFD Is Used? Aircraft Design

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 21

Where CFD Is Used? Aircraft Design

Prediction of the wake vortices up to 6.5 wingspans generated by the DLR-F11 aircraft model making use of a 4th-order central scheme and the automatic mesh refinement technique. Inviscid simulation, M∞=0.2, α=10°

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 22

Where CFD Is Used? Aircraft Design

Comparison of Computed Wake Vortex Evolution Flowfield (OVERFLOW Code) with Experiment (“2-pair”)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 23

Where CFD Is Used? Aircraft Design

Stagnation pressure loss at the fan face of an air-intake at the fixed point submitted to crosswind. M∞ = 0.045, α = 0o,β = 9o, Re = 3.9x106. Airbus France, NSMB code.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 24

Where is CFD used?

Polymerization reactor vessel - prediction of flow separation and residence time effects.

Streamlines for workstation ventilation

Where is CFD used?AerospaceeAutomotiveBiomedicalChemical ProcessingHVACHydraulicsMarineOil & GasPower GenerationSports

HVAC

Chemical Processing

Hydraulics

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 25

Where is CFD used?

Where is CFD used?AerospaceAutomotiveBiomedicalChemical ProcessingHVACHydraulicsMarineOil & GasPower GenerationSports

Flow of lubricating mud over drill bit

Flow around cooling towers

Marine

Oil & Gas

Sports

Power Generation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 26

Where is CFD used?

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 27

Getting Started: CFD Notation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 28

Getting Started: Tensorial Quantities in Fluid Dynamics

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 29

Getting Started: Vector Multiplication Rules

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 30

Getting Started: Elementary Tensor Calculus

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 31

Getting Started: Divergence Theorem of Gauss

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 32

Getting Started: Governing Equations of Fluid Dynamics

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 33

Getting Started: Description of Fluid Motion

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 34

Getting Started: Flow Models and Reference Frames

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 35

Getting Started: Eulerian vs. Langrangian Viewpoint

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 36

Getting Started: governing equations

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 37

Getting Started: classification of PDEs

PDEs can be classified into hyperbolic, parabolic and elliptic ones• each class of PDEs models a different kind of physical processes• the number of initial/boundary conditions depends on the PDE ype• different solution methods are required for PDEs of different type

Hyperbolic equations Information propagates in certain directions at finite speeds; the solution is a superposition of multiple single wavesParabolic equations Information travels downstream/forward in time; directions at finite speeds; the solution can be constructed using a marching/time-stepping methodElliptic equations Information propagates in all directions at infinite speed; describe equilibrium phenomena (unsteady problems are never elliptic

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 38

Getting Started: classification of PDEs

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 39

Getting Started: classification of PDEs

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 40

CFD Process

Viscous Model

Boundary Conditions

Initial Conditions

Convergent Limit

Contours

Precisions(single/double)

Numerical Scheme

Vectors

StreamlinesVerification

Geometry

Select Geometry

Geometry Parameters

Physics Mesh Solve Post-Processing

CompressibleON/OFF

Flow properties

Unstructured(automatic/

manual)

Forces Report(lift/drag, shear stress, etc)

XY Plot

Domain Shape and

Size

Heat Transfer ON/OFF

Structured(automatic/

manual)

Iterations/Steps

Validation

Reports

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 41

Modeling

• Modeling is the mathematical physics problem formulation in terms of a continuous initial boundary value problem (IBVP)

• IBVP is in the form of Partial Differential Equations (PDEs) with appropriate boundary conditions and initial conditions.

• Modeling includes:1. Geometry and domain2. Coordinates3. Governing equations4. Flow conditions5. Initial and boundary conditions6. Selection of models for different applications

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 42

Modeling (geometry and domain)

• Simple geometries can be easily created by few geometric parameters (e.g. circular pipe)

• Complex geometries must be created by the partial differential equations or importing the database of the geometry(e.g. airfoil) into commercial software

• Domain: size and shape

• Typical approaches • Geometry approximation

• CAD/CAE integration: use of industry standards such as Parasolid, ACIS, STEP, or IGES, etc.

• The three coordinates: Cartesian system (x,y,z), cylindrical system (r, θ, z), and spherical system(r, θ, Φ) should be appropriately chosen for a better resolution of the geometry (e.g. cylindrical for circular pipe).

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 43

Modeling (governing equations) – Navier-stokes Equations

∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

2

2

2

2

2

2ˆzu

yu

xu

xp

zuw

yuv

xuu

tu µρρρρ

∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

2

2

2

2

2

2ˆzv

yv

xv

yp

zvw

yvv

xvu

tv

µρρρρ

( ) ( ) ( ) 0=∂

∂+

∂∂

+∂

∂+

∂∂

zw

yv

xu

tρρρρ

RTp ρ=

L

v ppDtDR

DtRDR

ρ−

=+ 22

2

)(23

Convection Piezometric pressure gradient Viscous termsLocalacceleration

Continuity equation

Equation of state

Rayleigh Equation

∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

2

2

2

2

2

2ˆzw

yw

xw

zp

zww

ywv

xwu

tw

µρρρρ

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 44

Modeling (flow conditions)

• Based on the physics of the fluids phenomena, CFD can be distinguished into different categories using different criteria

• Viscous vs. inviscid (Re)

• External flow or internal flow (wall bounded or not)

• Turbulent vs. laminar (Re)

• Incompressible vs. compressible (Mach number)

• Single- vs. multi-phase (Ca)

• Thermal/density effects (Pr, γ, Gr, Ec)

• Free-surface flow (Fr) and surface tension (We)

• Chemical reactions and combustion (Pe, Da)

• etc…

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 45

Modeling (initial conditions)

• Initial conditions (ICS, steady/unsteady flows)• ICs should not affect final results and only

affect convergence path, i.e. number of iterations (steady) or time steps (unsteady) need to reach converged solutions.

• More reasonable guess can speed up the convergence

• For complicated unsteady flow problems, CFD codes are usually run in the steady mode for a few iterations for getting a better initial conditions

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 46

Modeling(boundary conditions)

•Boundary conditions: No-slip or slip-free on walls, periodic, inlet (velocity inlet, mass flow rate, constant pressure, etc.), outlet (constant pressure, velocity convective, numerical beach, zero-gradient), and non-reflecting (for compressible flows, such as acoustics), etc.

No-slip walls: u=0,v=0

v=0, dp/dr=0,du/dr=0

Inlet ,u=c,v=0 Outlet, p=c

Periodic boundary condition in spanwise direction of an airfoilo

r

xAxisymmetric

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 47

Modeling (selection of models)

• CFD codes typically designed for solving certain fluidphenomenon by applying different models

• Viscous vs. inviscid (Re)

• Turbulent vs. laminar (Re, Turbulent models)

• Incompressible vs. compressible (Ma, equation of state)

• Single- vs. multi-phase (Ca, cavitation model, two-fluid

model)

• Thermal/density effects and energy equation(Pr, γ, Gr, Ec, conservation of energy)

• Free-surface flow (Fr, level-set & surface tracking model) and

surface tension (We, bubble dynamic model)

• Chemical reactions and combustion (Chemical reaction

model)

• etc…

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 48

Modeling (Turbulence and free surface models)

• Turbulent models:

• DNS: most accurately solve NS equations, but too expensive

for turbulent flows

• RANS: predict mean flow structures, efficient inside BL but excessive

diffusion in the separated region.• LES: accurate in separation region and unaffordable for resolving BL

• DES: RANS inside BL, LES in separated regions.

• Free-surface models:

• Surface-tracking method: mesh moving to capture free surface,

limited to small and medium wave slopes

• Single/two phase level-set method: mesh fixed and level-set

function used to capture the gas/liquid interface, capable of

studying steep or breaking waves.

• Turbulent flows at high Re usually involve both large and small scale

vortical structures and very thin turbulent boundary layer (BL) near the wall

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 49

Examples of modeling (Turbulence and free surface models)

DES, Re=105, Iso-surface of Q criterion (0.4) for turbulent flow around NACA12 with angle of attack 60 degrees

URANS, Re=105, contour of vorticity for turbulent flow around NACA12 with angle of attack 60 degrees

URANS, Wigley Hull pitching and heaving

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 50

Numerical methods

• The continuous Initial Boundary Value Problems (IBVPs) are discretized into algebraic equations using numerical methods. Assemble the system of algebraic equations and solve the system to get approximate solutions

• Numerical methods include:1. Discretization methods2. Solvers and numerical parameters3. Grid generation and transformation4. High Performance Computation (HPC) and post-

processing

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 51

Discretization methods• Finite difference methods (straightforward to apply,

usually for regular grid) and finite volumes and finite element methods (usually for irregular meshes)

• Each type of methods above yields the same solution if the grid is fine enough. However, some methods are more suitable to some cases than others

• Finite difference methods for spatial derivatives with different order of accuracies can be derived using Taylor expansions, such as 2nd order upwind scheme, central differences schemes, etc.

• Higher order numerical methods usually predict higher order of accuracy for CFD, but more likely unstable due to less numerical dissipation

• Temporal derivatives can be integrated either by the explicit method (Euler, Runge-Kutta, etc.) or implicitmethod (e.g. Beam-Warming method)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 52

Discretization methods (Cont’d)

• Explicit methods can be easily applied but yield conditionally stable Finite Different Equations (FDEs), which are restricted by the time step; Implicit methodsare unconditionally stable, but need efforts on efficiency.

• Usually, higher-order temporal discretization is used when the spatial discretization is also of higher order.

• Stability: A discretization method is said to be stable if it does not magnify the errors that appear in the course of numerical solution process.

• Pre-conditioning method is used when the matrix of the linear algebraic system is ill-posed, such as multi-phase flows, flows with a broad range of Mach numbers, etc.

• Selection of discretization methods should consider efficiency, accuracy and special requirements, such as shock wave tracking.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 53

Discretization methods (Cont’d)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 54

Discretization methods (Cont’d)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 55

Discretization methods (Cont’d)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 56

Discretization methods (Cont’d)

Analysis of trunctation errors

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 57

Discretization methods (Cont’d)

Approximation of second-order derivatives

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 58

Discretization methods (Cont’d)

Approximation of mixed derivatives

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 59

Discretization methods (Cont’d)

High-order approximations

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 60

Discretization methods (example)

0=∂∂

+∂∂

yv

xu

2

2

yu

ep

xyuv

xuu

∂∂

+

∂∂

−=∂∂

+∂∂

µ

• 2D incompressible laminar flow boundary layer

m=0m=1

L-1 L

y

x

m=MMm=MM+1

(L,m-1)

(L,m)

(L,m+1)

(L-1,m)

1l

l lmm m

uuu u ux x

−∂ = − ∂ ∆

1

ll lmm m

vuv u uy y +

∂ = − ∂ ∆

1

ll lmm m

v u uy − = − ∆

FD Sign( )<0lmv

lmvBD Sign( )>0

2

1 12 2 2l l lm m m

u u u uy y

µµ + −

∂ = − + ∂ ∆

2nd order central differencei.e., theoretical order of accuracyPkest= 2.

1st order upwind scheme, i.e., theoretical order of accuracy Pkest= 1

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 61

Discretization methods (example)

1 12 2 2

12

1

l l ll l l lm m mm m m m

FDu v vyv u FD u BD u

x y y y y yBDy

µ µ µ+ −

− ∆+ − + + + − ∆ ∆ ∆ ∆ ∆ ∆ ∆

1 ( / )l

l lmm m

u u p ex x

− ∂= −

∆ ∂

B2 B3 B1

B4( )11 1 2 3 1 4 / ll l l l

m m m m mB u B u B u B u p ex

−− +

∂+ + = −

∂1

4 112 3 1

1 2 3

1 2 3

1 2 14

0 0 0 0 0 00 0 0 0 0

0 0 0 0 00 0 0 0 0 0

ll

l

l lmm l

mmmm

pB uB B x euB B B

B B BB B u pB u

x e

∂ − ∂ •• × =• • • • •• •• ∂ − ∂

Solve it usingThomas algorithm

To be stable, Matrix has to be Diagonally dominant.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 62

Solvers and numerical parameters• Solvers include: tridiagonal, pentadiagonal solvers,

PETSC solver, solution-adaptive solver, multi-grid solvers, etc.

• Solvers can be either direct (Cramer’s rule, Gauss elimination, LU decomposition) or iterative (Jacobi method, Gauss-Seidel method, SOR method)

• Numerical parameters need to be specified to control the calculation. • Under relaxation factor, convergence limit, etc.• Different numerical schemes• Monitor residuals (change of results between

iterations)• Number of iterations for steady flow or number of

time steps for unsteady flow• Single/double precisions

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 63

DiscretizationGrid Generation

Flow field must be treated as a discrete set of points (or volumes) where the governing equations are solved.Many types of grid generation: type is usually related to capability of flow solver.

Structured gridsUnstructured gridsHybrid grids: some portions of flow field are structured (viscous regions) and others are unstructuredOverset (Chimera) grids

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 64

DiscretizationGrid Generation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 65

DiscretizationGrid Generation

Block-structured meshes• Multilevel subdivision of the domain with structured grids within blocks• Can be non-matching, special treatment is necessary at block interfaces• Provide greater flexibility, local refinement can be performed blockwiseUnstructured meshes• Suitable for arbitrary domains and amenable to adaptive mesh refinement• Consist of triangles or quadrilaterals in 2D, tetrahedra or hexahedra in 3D• Complex data structures, irregular sparsity pattern, difficult to implement

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 66

DiscretizationGrid Generation: examples of cell types

2D Cell Types

3D Cell Types

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Structured Grids

Structured multi-block grid around a multi-element airfoil

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 68

Structured Grids

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 69

Structured Overset Grids

Submarine

Moving Control Surfaces

Artificial Heart Chamber

Surface Ship Appendages

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 70

Unstructured Grids

Branches in Human LungStructured-Unstructured Nozzle Grid

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 71

Unstructured Grids

Unstructured surface mesh for external aerodynamics – PT cruiser – 12 millions cells

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 72

DiscretizationAlgebraic equations

To solve NSE, we must convert governing PDE’s to algebraic equations

Finite difference methods (FDM)Each term in NSE approximated using Taylor series, e.g.,

Finite volume methods (FVM)Use CV form of NSE equations on each grid cell ! ME 33 students already know the fundamentals !Most popular approach, especially for commercial codes

Finite element methods (FEM)Solve PDE’s by replacing continuous functions by piecewise approximations defined on polygons, which are referred to as elements. Similar to FDM.

( )

( )( )

1

221 1

22

2

i i

i i i

U U U O xx xU U U U O xx x

+

+ −

∂ −= + ∆

∂ ∆∂ − +

= + ∆∂ ∆

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 73

Solve

Run CFD code on computer

2D and small 3D simulations can be run on desktop computers (e.g., FlowLab)Unsteady 3D simulations still require large parallel computers

Monitor ResidualsDefined two ways

Change in flow variables between iterationsError in discrete algebraic equation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 74

Post-processing Pressure Distribution

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 75

Post-processing Pathlines

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 76

Post-processing Pathlines

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 77

Post-processing Trajectory

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 78

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 79

Uncertainty Assessment

Process of estimating errors due to numerics and modeling

Numerical errors Iterative non-convergence: monitor residualsSpatial errors: grid studies and Richardson extrapolationTemporal errors: time-step studies and Richardson extrapolation

Modeling errors (Turbulence modeling, multi-phase physics, closure of viscous stress tensor for non-Newtonian fluids)

Only way to assess is through comparison with benchmark data which includes EFD uncertainty assessment.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 80

Limitations of Current Technology

For fluid mechanics, many problems not adequately described by Navier-Stokes equations or are beyond current generation computers.

TurbulenceMulti-phase physics: solid-gas (pollution, soot), liquid-gas (bubbles, cavitation); solid-liquid (sediment transport)Combustion and chemical reactionsNon-Newtonian fluids (blood; polymers)

Similar modeling challenges in other branches of engineering and the sciences

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 81

Conclusions

Because of limitations, need for experimental research is greatHowever, focus has changed

From Research based solely upon experimental observationsBuild and test (although this is still done)

ToHigh-fidelity measurements in support of validation and building new computational models.

Currently, the best approach to solving engineering problems often uses simulation and experimentation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 82

Applications

Capabilities of Current TechnologyComplex real-world problems solved using Scientific ComputingCommercial software available for certain problemsSimulation-based design (i.e., logic-based) is being realized.Ability to study problems that are either expensive, too small, too large, or too dangerous to study in laboratory

Very small : nano- and micro-fluidicsVery large : cosmology (study of the origin, current state, and future of our Universe)Expensive : engineering prototypes (ships, aircraft)Dangerous : explosions, fires

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 83

http://www.cfd-online.com

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http://www.fluent.com

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 85

http://www.aoe.vt.edu/~mason/Mason_f/MRsoft.html

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 86

http://www.ensight.com/

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 87

ProductsCFD++

CFD++ is a superset of the various CFD methodologies available and provides accuracy, robustness and ease of use over all flow regimes.

MIME

Multipurpose Intelligent Meshing Environment for CFD++, CAA++ and ED. Powerful mesh generation software, yet it is so simple to use.

CAA++

Computational Aeroacoustics Software Suite. Metacomp's cost effective solution to noise prediction.

ED Designer

Computational Electrostatic Paint Deposition tool from Metacomp. Developed in collaboration with Delight Inc., Japan.

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 88

http://www.sai.msu.su/sal/sal1.shtml

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http://gd.tuwien.ac.at/study/baum-lse/node2.html

Linux Software Encyclopedia

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 90

Literature

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 91

Literature

9. Fletcher, C. A. J. “Computational Techniques for Fluid Dynamics,” SpringerSeries in Computational Physics, Vols. 1-2, 2nd Edition, 1991.

92

Overview on Mesh Technology

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 93

Single Block Grids – Creating the Mapping

Conformal MappingTransfinite InterpolationSolving PDEs

EllipiticParabolic/Hyperbolic

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow 94

Single Block Grids – Creating the Mapping

Conformal Mapping

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Single Block Grids – Creating the Mapping

Conformal Mapping Transformations

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Single Block Grids – Creating the Mapping

Conformal Mapping – Schwarz Christoffel

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Single Block Grids – Creating the Mapping

Algebraic Mappings

• Construct mapping between the boundaries of the unit square (cube) and the boundaries of an “arbitrary” region which is topologically equivalent

• Combine 1 D interpolants using Boolean sums to construct mapping-Transfinite interpolation (TFI)

• Not guaranteed to be one-to-one• Orthogonality not guaranteed• Very fast• Quite General• Grid quality not always assured

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Single Block Grids – Creating the Mapping

Algebraic Mappings – 1D Interpolants

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Single Block Grids – Creating the Mapping

Algebraic Mappings – Transfinite Interpolation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow100

Single Block Grids – Creating the Mapping

Algebraic Mappings – Transfinite Interpolation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow101

Single Block Grids – Creating the Mapping

Algebraic Mappings – Example

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Single Block Grids – Creating the Mapping

Algebraic Mappings – Example

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Single Block Grids – Creating the Mapping

Algebraic Mappings – Example

Numerically generated airfoil transformation [(x, y) ↔ (ξ, η)] showing a “C” grid topology

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow104

Single Block Grids – Creating the Mapping

Algebraic Mappings – Example

Numerically generated wing transformation [(x, y, z) ↔ (ξ, η, ζ)] showing a “O” grid topology

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow105

Single Block Grids – Creating the Mapping

PDE Grid Generation

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CFD Perspective on Meshing Technology

CFD initiated in structured grid contextTransfinite interpolationElliptic grid generationHyperbolic grid generation

Smooth, orthogonal structured gridsRelatively simple geometries

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Unstructured meshes initially confined to FE community

Sparse matrix methods not competitiveMemory limitationsNon-linear nature of problems

CFD Perspective on Meshing Technology

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Current State of Unstructured Mesh CFD Technology

Method of choice for many commercial CFD vendors

Enabling factorsMaturing grid generation technologyBetter Discretizations and solvers

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow109

Maturing Unstructured Grid Generation Technology (1990-2000)

Isotropic tetrahedral grid generationDelaunay point insertion algorithmsSurface recoveryAdvancing front techniquesOctree methods

Mature technologyNumerous available commercial packagesRemaining issues

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow110

Maturing Unstructured Grid Generation Technology (1990-2000)

Anisotropic unstructured grid generationExternal aerodynamics

Boundary layers, wakes: O(10**4)Mapped Delaunay triangulationsMin-max triangulationsHybrid methods ß

Advancing layersMixed prismatic – tetrahedral meshes

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow111

Anisotropic Unstructured Grid Generation

Hybrid methodsSemi-structured natureLess mature: issues

Concave regionsNeighboring boundariesConflicting resolutionConflicting Stretchings VGRIDns Advancing Layers

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Evolved to Sophisticated Multiblock and Overlapping Structured Grid Techniques for Complex Geometries

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

CFD Perspective on Meshing Technology

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow113

Enabling CFD Solver Developments (1990 – 2000)

Edge-based data structureBuilding block for all element typesReduces memory requirementsMinimizes indirect addressing / gather-scatterGraph of grid = Discretization stencil

Implications for solvers, Partitioners

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Multigrid solversMultigrid techniques enable optimal O(N) solution complexityBased on sequence of coarse and fine meshesOriginally developed for structured grids

Enabling CFD Solver Developments (1990 – 2000)

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Agglomeration Multigrid solvers for unstructured meshes

Coarse level meshes constructed by agglomerating fine grid cells/equations

Enabling CFD Solver Developments (1990 – 2000)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow116

Agglomeration Multigrid

•Automated Graph-Based Coarsening Algorithm

•Coarse Levels are Graphs

•Coarse Level Operator by Galerkin Projection

•Grid independent convergence rates (order of magnitude improvement)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow117

Enabling CFD Solver Developments

Line solvers for Anisotropic problemsLines constructed in mesh using weighted graph algorithmStrong connections assigned large graph weight(Block) Tridiagonal line solver similar to structured grids

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Graph-based Partitioners for parallel load balancing

Metis, Chaco, JostleEdge-data structure à graph of gridAgglomeration Multigrid levels = graphsExcellent load balancing up to 1000’s of processors

Homogeneous data-structures(Versus multi-block / overlapping structured grids)

Enabling CFD Solver Developments (1990 – 2000)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow119

NASA Langley Energy Efficient Transport

Complex geometryWing-body, slat, double slotted flaps, cutouts

Experimental data from Langley 14x22ft wind tunnel

Mach = 0.2, Reynolds=1.6 millionRange of incidences: -4o to 24o

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Boundary Conditions

Typical conditionsWall

No-slip (u = v = w = 0)Slip (tangential stress = 0, normal velocity = 0)With specified suction or blowingWith specified temperature or heat flux

InflowOutflowInterface Condition, e.g., Air-water free surfaceSymmetry and Periodicity

Usually set through the use of a graphical user interface (GUI) – click & set

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Initial Mesh Generation (VGRIDns)S. Pirzadeh, NASA Langley

Automatic switching from AL to AF based on:Cell aspect ratioProximity of boundaries of other frontsVariable height for advancing layers

Background Cartesian grid for smooth spacing controlSpanwise stretching

Factor of 3 reduction in grid size

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VGRID Tetrahedral Mesh

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

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Prism Merging OperationCombine Tetrahedra triplets in advancing-layers region into prisms

Prisms entail lower complexity for solver

VGRIDns identifies originating boundary point for ALR vertices

Used to identify candidate elementsPyramids required as transitional elements

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Prism Merging Operation

Initial mesh: 18.2M TetrahedraMerged mesh: 3.9M prisms, 6.6M Tets, 47K pyramids

64% of Tetrahedra merged

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Global Mesh Refinement

High-resolution meshes require large parallel machinesParallel mesh generation difficult

Current approachGenerate coarse (O(10**6) vertices on workstationRefine on supercomputer

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow126

Global Mesh Refinement

Refinement achieved by element subdivisionGlobal refinement: 8:1 increase in resolutionIn-Situ approach obviates large file transfersInitial mesh: 3.1 million vertices

3.9M prisms, 6.6M Tets, 47K pyramidsRefined mesh: 24.7 million vertices

31M prisms, 53M Tets, 281K pyramidsRefinement operation: 10 Gbytes, 30 minutes sequentially

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NSU3D Unstructured Mesh Navier-Stokes Solver

Mixed element gridsTetrahedra, prisms, pyramids, hexahedra

Edge data-structureLine solver in BL regions near wallsAgglomeration Multigrid accelerationNewton Krylov (GMRES) acceleration optionSpalart-Allmaras 1 equation turbulence model

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Parallel Implementation

Domain decomposition with OpenMP/MPI communication

OpenMP on shared memory architecturesMPI on distributed memory architecturesHybrid capability for clusters of SMPs

Weighted graph partitioning (Metis) (Chaco)Coarse and fine MG levels partitioned independently

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Computed Pressure Contours on Coarse Grid

Mach=0.2, α=10 degrees, Re=1.6M

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Computed Versus Experimental Results

Good drag predictionDiscrepancies near stall

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Multigrid Convergence History

Mesh independent property of MultigridGMRES effective but requires extra memory

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Parallel Scalability

Good overall Multigrid scalabilityIncreased communication due to coarse grid levelsSingle grid solution impractical (>100 times slower)

1 hour soution time on 1450 PEs

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow133

AIAA Drag Prediction Workshop (2001)

Transonic wing-body configurationTypical cases required for design study

Matrix of mach and CL valuesGrid resolution study

Follow on with engine effects (2003)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow134

Cases Run

Baseline grid: 1.6 million pointsFull drag polars for Mach=0.5,0.6,0.7,0.75,0.76,0.77,0.78,0.8Total = 72 cases

Medium grid: 3 million pointsFull drag polar for each mach numberTotal = 48 cases

Fine grid: 13 million pointsDrag polar at mach=0.75Total = 7 cases

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow135

Sample Solution (1.65M Pts)

Mach=0.75, CL=0.6, Re=3M2.5 hours on 16 Pentium IV 1.7GHz

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow136

Drag Polar at Mach = 0.75

Grid resolution studyGood comparison with experimental data

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow137

Cases Run on ICASE Cluster

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

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow138

Current and Future Issues

Adaptive mesh refinementMoving geometry and mesh motionMoving geometry and overlapping meshesRequirements for gradient-based designImplications for higher-order Discretizations

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow139

Potential for large savings through optimized mesh resolution

Well suited for problems with large range of scalesPossibility of error estimation / controlRequires tight CAD coupling (surface pts)

Mechanics of mesh adaptationRefinement criteria and error estimation

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow140

Various well know isotropic mesh methodsMesh movement

Spring analogyLinear elasticity

Local RemeshingDelaunay point insertion/RetriangulationEdge-face swappingElement subdivision

Mixed elements (non-simplicial)Anisotropic subdivision required in transition regions

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow141

Subdivision Types for Tetrahedra

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow142

Subdivision Types for Prisms

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Subdivision Types for Pyramids

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Subdivision Types for Hexahedra

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Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow146

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow147

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow148

Large potential savings for 1 or 2D features

Directional subdivisionAssumes element faces to line up with flow featuresCombine with mesh motion

Mapping techniquesHessian basedGrid quality

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow149

Refinement Criteria

Weakest link of adaptive meshing methodsObvious for strong featuresDifficult for non-local (ie. Convective) features

eg. Wakes

Analysis assumes in asymptotic error convergence region

Effect of variable discretization error in design studies, parameter sweeps

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow150

Compute sensitivity of global cost function to local spatial grid resolutionKey on important output, ignore other features

Error in engineering output, not discretization errore.g. Lift, drag, or sonic boom …

Captures non-local behavior of errorGlobal effect of local resolution

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow151

Reproduced from Venditti and Darmofal (MIT, 2002)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow152

Reproduced from Venditti and Darmofal (MIT, 2002)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow153

Reproduced from Vendittiand Darmofal(MIT, 2002)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow154

Overlapping Unstructured Meshes

Alternative to moving mesh for large scale relative geometry motionMultiple overlapping meshes treated as single data-structure

Dynamic determination of active/inactive/ghost cells

Advantages for parallel computingObviates dynamic load rebalancing required with mesh motion techniquesIntergrid communication must be dynamically recomputed and rebalanced

Concept of Rendez-vous grid (Plimpton and Hendrickson)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow155

Overlapping Unstructured Meshes

Simple 2D transient example

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow156

Minimize Cost Function F with respect to design variables v, subject to constraint R(w) = 0

F = drag, weight, costv = shape parametersw = Flow variablesR(w) = 0à Governing Flow Equations

Gradient Based Methods approach minimum along direction :

vF

∂∂

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow157

Grid Related Issues for Gradient-based Design

Parametrization of CAD surfacesConsistency across disciplines

eg. CFD, structures,…Surface grid motionInterior grid motionGrid sensitivitiesAutomation / Parallelization

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow158

Preliminary Design Geometry X34 CAD Model

23,555 curves and surfaces

c/o J. Samareh, NASA Langley

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Launch Vehicle Shape Parameterization

c/o J. Samareh, NASA Langley

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow160

Sensitivity Analysis

• Manual differentiation

• Complex variables

• Finite-difference approximations

analysis codefield grid generator

surface grid generator

Gridv

GridGe

GeometryvGrid Gr m yid o etr

f

f s

sF x x xF ∂ ∂∂

∂=

∂∂∂ ∂∂ ∂

142431424 1 14243 3 4243v design variables

(e.g., span, camber)

objective function

(e.g., Stress, CD)

c/o J. Samareh, NASA Langley

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow161

Finite-Difference Approximation Error for Sensitivity Derivatives

ParameterizedHSCT Model

c/o J. Samareh, NASA Langley

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow162

Grid Sensitivities

Ideally should be available from grid/cad software

Analytical formulation most desirableBurden on grid / CAD softwareDiscontinous operations present extra challenges

Face-edge swappingPoint addition / removalMesh regeneration

vGeometry

Geometry Grid

Grid Grid

v Grid

∂∂

∂∂

∂∂

=∂

∂ xx s

s

ff

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow163

High-Order Accurate Discretizations

Uniform X2 refinement of 3D mesh:Work increase = factor of 82nd order accurate method: accuracy increase = 44th order accurate method: accuracy increase = 16

For smooth solutions

Potential for large efficiency gainsSpectral element methodsDiscontinuous Galerkin (DG)Streamwise Upwind Petrov Galerkin (SUPG)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow164

Higher-Order Accurate Discretizations

Transfers burden from grid generation to Discretization

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow165

Spectral Element Solution of Maxwell’s Equations

J. Hesthaven and T. Warburton

(Brown University)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow166

High-Order Discretizations

Require more complete surface definitionCurved surface elements

Additional element pointsSurface definition (for high p)

Chapter 15: Computational Fluid DynamicsME33 : Fluid Flow167

Combined H-P Refinement

Adaptive meshing (h-ref) yields constant factor improvement

After error equidistribution, no further benefitOrder refinement (p-ref) yields asymptotic improvement

Only for smooth functionsIneffective for inadequate h-resolution of featureCannot treat shocks

H-P refinement optimal (exponential convergence)Requires accurate CAD surface representation

168

Modeling Turbulent Flows

u Unsteady, aperiodic motion in which all three velocity components fluctuate ⌫ mixing matter, momentum, and energy.

u Decompose velocity into mean and fluctuating parts:Ui(t) ≡ Ui + ui(t)

u Similar fluctuations for pressure, temperature, and species concentration values.

What is Turbulence?

Time

U i (t)Ui

ui(t)

Why Model Turbulence?

u Direct numerical simulation of governing equations is only possible for simple low-Re flows.

u Instead, we solve Reynolds Averaged Navier-Stokes (RANS) equations:

where (Reynolds stresses)

(steady, incompressible flow w/o body forces)

jiij uuR ρ−=

j

ij

jj

i

ik

ik x

Rxx

Uxp

xUU

∂∂

+∂∂

∂+

∂∂

−=∂∂ 2

µρ

Is the Flow Turbulent?

External Flows

Internal Flows

Natural Convection

5105×≥xRe along a surface

around an obstacle

where

µρULReL ≡where

Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow.

L = x, D, Dh, etc.

,3002 ≥hD Re

108 1010 −≥Ra µαρβ 3

Pr TLgGrRa x∆

≡=

20,000≥DRe

∞−=∆ TTT s

Ts= temperature of the wallT∞= fluid temperature far from the surface of the object

Grashof Prandtl

How Complex is the Flow?

u Extra strain ratesl Streamline curvaturel Lateral divergencel Acceleration or decelerationl Swirll Recirculation (or separation)l Secondary flow

u 3D perturbationsu Transpiration (blowing/suction)u Free-stream turbulenceu Interacting shear layers

Turbulence Model&

Near-Wall Treatment

Flow Physics

AccuracyRequired

ComputationalResources

TurnaroundTime

Constraints

ComputationalGrid

Zero-Equation Models

One-Equation ModelsSpalart-Allmaras

Two-Equation ModelsStandard k-εRNG k-εRealizable k-ε

Reynolds-Stress Model

Large-Eddy Simulation

Direct Numerical Simulation

Turbulence Modeling Approaches

IncludeMorePhysics

IncreaseComputationalCostPer Iteration

Availablein FLUENT

RANS-basedmodels

u RANS equations require closure for Reynolds stresses.

u Turbulent viscosity is indirectly solved for from single transport equation of modified viscosity for One-Equation model.

u For Two-Equation models, turbulent viscosity correlated with turbulent kinetic energy (TKE) and the dissipation rate of TKE.

u Transport equations for turbulent kinetic energy and dissipation rate are solved so that turbulent viscosity can be computed for RANS equations.

Reynolds Stress Terms in RANS-based Models

Turbulent Kinetic Energy:

Dissipation Rate of Turbulent Kinetic Energy:

ερµ µ

2kCt ≡Turbulent Viscosity:

Boussinesq Hypothesis:(isotropic stresses)

∂∂

+∂∂

+−=−=i

j

j

itijjiij x

UxUkuuR µδρρ

32

2/iiuuk ≡

∂+

∂∂

∂∂

≡i

j

j

i

j

i

xu

xu

xu

νε

u Turbulent viscosity is determined from:

u is determined from the modified viscosity transport equation:

u The additional variables are functions of the modified turbulent viscosity and velocity gradients.

One Equation Model: Spalart-Allmaras

( ) 21

2

2~

1

~~~~1~~~

dfc

xc

xxSc

DtD

wwj

bjj

ρν

ρν

νρµσ

νρν

ρν

∂∂

+

∂∂

+∂∂

+=

( )( )

+= 3

13

3

/~/~~

ννννννρµ

ct

ν~

Generation Diffusion Destruction

One-Equation Model: Spalart-Allmaras

u Designed specifically for aerospace applications involving wall-bounded flows.l Boundary layers with adverse pressure gradientsl turbomachinery

u Can use coarse or fine mesh at walll Designed to be used with fine mesh as a “low-Re” model, i.e., throughout

the viscous-affected region.l Sufficiently robust for relatively crude simulations on coarse meshes.

Two Equation Model: Standard k-ε Model

Turbulent Kinetic Energy

Dissipation Rate

εεεσσ 21, ,, CCk are empirical constants

(equations written for steady, incompressible flow w/o body forces)

Convection Generation DiffusionDestruction{ρεσµµρ −

∂∂

∂∂

+∂

∂∂

+∂

∂=

∂∂

444 3444 21444 3444 2143421 ikt

ii

j

j

i

i

jt

ii x

kxx

UxU

xU

xkU )(

DestructionConvection Generation Diffusion43421444 3444 2144444 344444 2143421

∂∂

∂∂

+∂∂

∂∂

+∂

=

∂∂

kC

xxxU

xU

xU

kC

xU

it

ii

j

j

i

i

jt

ii

2

21 )( ερ

εσµµ

εερ εεε

Two Equation Model: Standard k-ε Model

u “Baseline model” (Two-equation)l Most widely used model in industryl Strength and weaknesses well documented

u Semi-empiricall k equation derived by subtracting the instantaneous mechanical energy

equation from its time-averaged valuel ε equation formed from physical reasoning

u Valid only for fully turbulent flowsu Reasonable accuracy for wide range of turbulent flows

l industrial flowsl heat transfer

Two Equation Model: Realizable k-ε

u Distinctions from Standard k-ε model:l Alternative formulation for turbulent viscosity

where is now variable

n (A0, As, and U* are functions of velocity gradients)

n Ensures positivity of normal stresses;

n Ensures Schwarz’s inequality;

l New transport equation for dissipation rate, ε:

ερµ µ

2kCt ≡

ε

µ kUAAC

so

*1

+=

0u2i ≥

2j

2i

2ji u u)uu( ≤

bj

t

j

Gck

ck

cScxxDt

Dεε

ε

ενε

ερερεσµµερ 31

2

21 ++

−+

∂∂

+

∂∂

=

GenerationDiffusion Destruction Buoyancy

u Shares the same turbulent kinetic energy equation as Standard k-εu Superior performance for flows involving:

l planar and round jetsl boundary layers under strong adverse pressure gradients, separationl rotation, recirculationl strong streamline curvature

Two Equation Model: Realizable k-ε

Two Equation Model: RNG k-ε

Turbulent Kinetic Energy

Dissipation Rate

Convection DiffusionDissipation

{ {ρεµαµρ −

∂∂

∂∂

+=∂∂

44 344 2143421 ik

it

ii x

kx

SxkU eff

2

Generation

∂∂

+∂

∂≡≡

j

i

i

jijijij x

UxU

SSSS21,2

where

are derived using RNG theoryεεεαα 21, ,, CCk

(equations written for steady, incompressible flow w/o body forces)

Additional termrelated to mean strain& turbulence quantities

Convection Generation Diffusion Destruction

{RkC

xxS

kC

xU

iit

ii −

∂∂

∂∂

+

=

∂∂

4342144 344 21443442143421

2

2eff2

ρε

µαµεε

ρ εεε

Two Equation Model: RNG k-ε

u k-ε equations are derived from the application of a rigorous statistical technique (Renormalization Group Method) to the instantaneous Navier-Stokes equations.

u Similar in form to the standard k-ε equations but includes:l additional term in ε equation that improves analysis of rapidly strained flowsl the effect of swirl on turbulencel analytical formula for turbulent Prandtl numberl differential formula for effective viscosity

u Improved predictions for:l high streamline curvature and strain ratel transitional flowsl wall heat and mass transfer

Reynolds Stress Model

k

ijkijijij

k

jik x

JP

xuu

U∂∂

+−Φ+=∂

∂ερ

Generationk

ikj

k

jkiij x

UuuxU

uuP∂∂

+∂∂

∂+

∂∂′−≡Φ

i

j

j

iij x

uxup

k

j

k

iij x

uxu

∂∂

∂∂

≡ µε 2

Pressure-StrainRedistribution

Dissipation

TurbulentDiffusion

(modeled)

(related to ε)

(modeled)

(computed)

(equations written for steady, incompressible flow w/o body forces)

Reynolds StressTransport Eqns.

Pressure/velocityfluctuations

Turbulenttransport

)( jikijkkjiijk uupuuuJ δδ +′+=

Reynolds Stress Model

u RSM closes the Reynolds-Averaged Navier-Stokes equations by solving additional transport equations for the Reynolds stresses.l Transport equations derived by Reynolds averaging the product of the

momentum equations with a fluctuating propertyl Closure also requires one equation for turbulent dissipationl Isotropic eddy viscosity assumption is avoided

u Resulting equations contain terms that need to be modeled.u RSM has high potential for accurately predicting complex flows.

l Accounts for streamline curvature, swirl, rotation and high strain ratesn Cyclone flows, swirling combustor flowsn Rotating flow passages, secondary flows

Large Eddy Simulation

u Large eddies:l Mainly responsible for transport of momentum, energy, and other scalars,

directly affecting the mean fields.l Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly

dependent on flow configuration, boundary conditions, and flow parameters.u Small eddies:

l Tend to be more isotropic and less flow-dependentl More likely to be easier to model than large eddies.

u LES directly computes (resolves) large eddies and models only small eddies (Subgrid-Scale Modeling).

u Large computational effortl Number of grid points, NLES ∝l Unsteady calculation

2Reτu

Comparison of RANS Turbulence Models

Model Strengths WeaknessesSpalart-Allmaras

Economical (1-eq.); good track recordfor mildly complex B.L. type of flows

Not very widely tested yet; lack ofsubmodels (e.g. combustion,buoyancy)

STD k-εRobust, economical, reasonablyaccurate; long accumulatedperformance data

Mediocre results for complex flowsinvolving severe pressure gradients,strong streamline curvature, swirland rotation

RNG k-εGood for moderately complexbehavior like jet impingement,separating flows, swirling flows, andsecondary flows

Subjected to limitations due toisotropic eddy viscosityassumption

Realizablek-ε

Offers largely the same benefits asRNG; resolves round-jet anomaly

Subjected to limitations due toisotropic eddy viscosityassumption

ReynoldsStressModel

Physically most complete model(history, transport, and anisotropy ofturbulent stresses are all accountedfor)

Requires more cpu effort (2-3x);tightly coupled momentum andturbulence equations

Near-Wall Treatments

u Most k-ε and RSM turbulence models will not predict correct near-wall behavior if integrated down to the wall.

u Special near-wall treatment is required.l Standard wall functionsl Nonequilibrium wall functionsl Two-layer zonal model

Boundary layer structure

Standard Wall Functions

ρτµ

/

2/14/1

w

PP kCUU ≡∗

( )

>

+

<= ∗

)(ln1Pr

)(Pr**

**

Tt

T

yyPEy

yyyT

κ

µρ µ PP ykC

y2/14/1

≡∗

qkCcTT

T PpPw

′′−

≡&

2/14/1)(* µρ

Mean Velocity

Temperature

where

where and P is a function of the fluid and turbulent Prandtl numbers.

thermal sublayer thickness

( )∗∗ = EyU ln1κ

Nonequilibrium Wall Functions

u Log-law is sensitized to pressure gradient for better prediction of adverse pressure gradient flows and separation.

u Relaxed local equilibrium assumptions for TKE in wall-neighboring cells.

u Thermal law-of-wall unchanged

=

µρ

κρτµµ ykCEkCU

w

2/14/12/14/1

ln1/

~

+

−+

−= ∗∗ µρκρκ

ykyy

yy

ky

dxdpUU vv

v

v2

2/12/1 ln21~where

Two-Layer Zonal Model

u Used for low-Re flows or flows with complex near-wall phenomena.

u Zones distinguished by a wall-distance-based turbulent Reynolds number

u High-Re k-ε models are used in the turbulent core region.u Only k equation is solved in the viscosity-affected region.u ε is computed from the correlation for length scale.u Zoning is dynamic and solution adaptive.

µρ ykRey ≡

200>yRe

200<yRe

Comparison of Near Wall Treatments

Strengths Weaknesses

Standard wallFunctions

Robust, economical,reasonably accurate

Empirically based on simplehigh-Re flows; poor for low-Reeffects, massive transpiration,∇p, strong body forces, highly3D flows

Nonequilibriumwall functions

Accounts for ∇p effects,allows nonequilibrium:

-separation-reattachment-impingement

Poor for low-Re effects, massivetranspiration, severe ∇p, strongbody forces, highly 3D flows

Two-layer zonalmodel

Does not rely on law-of-the-wall, good for complexflows, especially applicableto low-Re flows

Requires finer mesh resolutionand therefore larger cpu andmemory resources

Computational Grid Guidelines

Wall Function Approach

Two-Layer Zonal Model Approach

l First grid point in log-law region

l At least ten points in the BL.

l Better to use stretched quad/hex cells for economy.

l First grid point at y+ ≈ 1.

l At least ten grid points within buffer & sublayers.

l Better to use stretched quad/hex cells for economy.

50050 ≤≤ +y

Estimating Placement of First Grid Point

u Estimate the skin friction coefficient based on correlations either approximate or empirical:

l Flat Plate-

l Pipe Flow-

u Compute the friction velocity:

u Back out required distance from wall:

l Wall functions • Two-layer model

u Use post-processing to confirm near-wall mesh resolution

2.0Re0359.02/ −≈ Lfc2.0Re039.02/ −≈ Dfc

2// few cUu =≡ ρττ

y1 = 50ν/uτ y1 = ν/ uτ

Setting Boundary Conditions

u Characterize turbulence at inlets & outlets (potential backflow)l k-ε models require k and εl Reynolds stress model requires Rij and ε

u Several options allow input using more familiar parametersl Turbulence intensity and length scale

n length scale is related to size of large eddies that contain most of energy.n For boundary layer flows: l ≈ 0.4δ99

n For flows downstream of grids /perforated plates: l ≈ opening sizel Turbulence intensity and hydraulic diameter

n Ideally suited for duct and pipe flowsl Turbulence intensity and turbulent viscosity ratio

n For external flows:

u Input of k and ε explicitly allowed (non-uniform profiles possible).10/1 << µµ t

GUI for Turbulence Models

Define⌫Models⌫ Viscous...

Turbulence Model options

Near Wall Treatments

Inviscid, Laminar, or Turbulent

Example: Channel Flow with Conjugate Heat Transfer

adiabatic wallcold airV = 50 fpmT = 0 °F

constant temperature wall T = 100 °F

insulation

1 ft

1 ft

10 ft

P

Predict the temperature at point P in the solid insulation

Turbulence Modeling Approach

u Check if turbulent ⌫ ReDh= 5,980

u Developing turbulent flow at relatively low Reynolds number and BLs on walls will give pressure gradient ⌫ use RNG k-ε with nonequilibrium wall functions.

u Develop strategy for the gridl Simple geometry ⌫ quadrilateral cellsl Expect large gradients in normal direction to horizontal walls ⌫

fine mesh near walls with first cell in log-law region.l Vary streamwise grid spacing so that BL growth is captured.l Use solution-based grid adaption to further resolve temperature

Velocitycontours

Temperaturecontours

BLs on upper & lower surfaces accelerate the core flow

Prediction of Momentum & Thermal Boundary Layers

Important that thermal BL was accurately resolved as well

P

Example: Flow Around a Cylinder

wall

wall

1 ft

2 ft

2 ft

airV = 4 fps

Compute drag coefficient of the cylinder

5 ft 14.5 ft

u Check if turbulent ⌫ ReD = 24,600

u Flow over an object, unsteady vortex shedding is expected, difficult to predict separation on downstream side, and close proximity of side walls may influence flow around cylinder ⌫ use RNG k-ε with 2-layer zonal model.

u Develop strategy for the gridl Simple geometry & BLs ⌫ quadrilateral cells.l Large gradients near surface of cylinder & 2-layer model ⌫ fine mesh near surface & first cell at y+ = 1.

Turbulence Modeling Approach

Grid for Flow Over a Cylinder

Prediction of Turbulent Vortex Shedding

Contours of effective viscosity µeff = µ + µt

CD = 0.53 Strouhal Number = 0.297

UDSt

τ≡where

Summary: Turbulence Modeling Guidelines

u Successful turbulence modeling requires engineering judgement of:l Flow physicsl Computer resources availablel Project requirements

n Accuracyn Turnaround time

l Turbulence models & near-wall treatments that are availableu Begin with standard k-ε and change to RNG or Realizable k-ε if

needed.u Use RSM for highly swirling flows.u Use wall functions unless low-Re flow and/or complex near-wall

physics are present.

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