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-Computational Aerodynamics

Rimon Arieli

Faculty of Aerospace EngineeringTECHNION Israel Institute of Technology

Haifa Israel 32000

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Outline

. ,

2. Modeling

3. Numerical methods

4. T es of CFD codes

5. CFD Process

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What is CFD?

modeling (mathematical physical problem formulation) andnumerical methods (discretization methods, solvers, numerical

, , .

Historically only Analytical Fluid Dynamics (AFD) andExperimental Fluid Dynamics (EFD).

CFD made possible by the advent of digital computer andadvancing with improvements of computer resources

500 flo s 194720 teraflo s 2003

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Why use CFD?

Analysis and Design1. Simulation-based desi n instead of build & test

More cost effective and more rapid than EFDCFD provides high-fidelity database for diagnosing flow field

2. Simulation of physical fluid phenomena that are difficult forexperimentsFull scale simulations (e.g., ships and airplanes)

Environmental effects (wind, weather, etc.)azar s e.g., exp os ons, ra a on, po u on

Physics (e.g., planetary boundary layer, stellar evolution)

Knowledge and exploration of flow physics

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Where is CFD used?

Where is CFD used? Aerospace

Aerospace Automotive

Biomedical

Biomedical Chemical Processing

F18 Store Separation

Hydraulics

Marine

ower enera on Sports

Temperature and naturalAutomotive

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convec on curren s n e eye

following laser heating.

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Where is CFD used?

Where is CFD used?Chemical Processing

Aerospacee Automotive

Polymerization reactor vessel - prediction of

flow separation and residence time effects. Chemical

Processing Hydraulics

Hydraulics

Marine

Power Generation Sports

HVAC

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Streamlines for workstation

ventilation

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Where is CFD used?

Where is CFD used?

Marine (movie) Sports

Aerospace Automotive

Biomedical

Chemical Processing

HVAC

H draulics

Marine

Power Generation

S orts

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Flow around coolingtowers

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Modeling

Modeling is the mathematical physics problem formulationin terms of a continuous initial boundary value problem

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

Modeling includes:

1. Geometry and domain.

3. Governing equations

4. Flow conditions

5. Initial and boundary conditions

6. Selection of models for different applications

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Modeling (geometry and domain)

Simple geometries can be easily created by few geometricparameters (e.g. circular pipe)

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, ,z), and spherical system (r, , ) should be appropriately chosen for abetter resolution of the eometr e. . c lindrical for circular i e .

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Modeling (coordinates)

z z z

(r,,z) (r,,)(x,y,z)Cartesian Cylindrical Spherical

z

x x xr r

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Coordinates

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Modeling (governing equations) -

+

+

+

=

+

+

+

2

2

2

2

2

2

z

u

y

u

x

u

x

p

z

uw

y

uv

x

uu

t

u

2 2 2

2 2 2

v v v v p v v vu v wt x y z y x y z

+ + + = + + +

222 wwwwwww

+

+

+

=

+

+

+ 222 zyxzz

wy

vx

ut

( ) ( ) ( )0=

+

+

+

wvu

Convection Piezometric pressure gradient Viscous termsLocalacceleration

zyxt

RTp =

Equation of state

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Modeling (initial conditions)

Initial conditions (ICS, steady/unsteady flows)

convergence path, i.e. number of iterations (steady)or time ste s unstead need to reach conver edsolutions.

More reasonable guess can speed up theconvergence ???

For complicated unsteady flow problems, CFD codes

are usually run in the steady mode for a fewiterations for getting a better initial conditions

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Modeling (boundary conditions)

Boundary conditions: No-slip or slip-free on walls, periodic, inlet velocit inlet mass flow rate constant ressure etc.

outlet (constant pressure, velocity convective, numerical beach, zero-gradient), and non-reflecting (for compressible flows, such asacoustics 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 airfoil

r

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Axisymmetric

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Modeling (selection of models)

Based on the physics of the fluids, CFD is distinguished intodifferent categories using different criteria, and designed forsolving certain fluid phenomenon by applying different models:

Viscous vs. inviscid (Re) Turbulent vs. laminar (Re, Turbulent models)

Incompressible vs. compressible (M, equation of state)

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

(Pr, , 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)

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Modeling (Turbulence and free surface models)

Turbulent models:

Turbulent flows at high Re usually involve both large and small scalevortical structures and very thin turbulent boundary layer (BL) near the wall

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.

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Examples of modeling (Turbulence and freesurface models)

URANS, Re=105, contour of vorticity for turbulent flowaround NACA-0012 with angle of attack 60 degrees

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

degrees

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Numerical methods

The continuous Initial Boundary Value Problems (IBVPs)are scret ze nto a ge ra c equat ons us ng numer camethods. Assemble the system of algebraic equations

Numerical methods include:.

2. Solvers and numerical parameters

3. Grid eneration and transformation

4. High Performance Computation (HPC) and post-processing

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Discretization methods

Finite difference methods (straightforward to apply, usually for regulargrid)

Finite volumes and finite element methods usuall for irre ularmeshes)

Each type of methods above yields the same solution if the grid is fineenough. However, some methods are more suitable to some cases

Finite difference methods for spatial derivatives with different order of

accuracies can be derived using Taylor expansions, such as 2nd

order

Higher order numerical methods usually predict higher order ofaccuracy for CFD, but more likely unstable due to less numericaldissipation

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

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Discretization methods (Contd)

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

efficiency. Usually, higher-order temporal discretization is used when the spatial

. Stability: A discretization method is said to be stable if it does not

magnify the errors that appear in the course of numerical solutionrocess. 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 rangeof Mach numbers, etc.

Selection of discretization methods should consider efficiency, accuracy and

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specia requirements, suc as s oc wave trac ing.

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Discretization methods (example)

D incompressi e aminar ow oun ary ayer

(L,m+1)

0=+ yv

xu

=

y

m=MMm=MM+1(L,m)(L-1,m)

2y

u

e

p

xy

uv

x

uu

+

=

+

m=0

L-1 Lx

(L,m-1)

l1l lm

m m

uuu u u

x x

= 2

1 12 22l l lm m m

uu u u

y y

+

= +

1l lmm m

vuv u uy y

+ = l

l lv

FD Sign( )0

es

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

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Discretization methods (example)

1

2l l ll l l lm m mFD

u v vyv u FD u BD u

+ + + +

B2 B3 B1

x y y y y yBD

y

1 /

ll lmu u e

= m m

x x B4( )11 1 2 3 1 4 /

ll l l l

m m m m mB u B u B u B u p e

x

+

+ + =

l14 1

12 3 1

1 2 3

0 0 0 0 0 0

0 0 0 0 0

l

lB u

B B x eu

B B B

o ve us ng

Thomas algorithm

1 2 3

1 2 1

0 0 0 0 0

0 0 0 0 0 0 l lmm l

B B B

B B u p

=

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4 mm

mmx e

To be stable, Matrix has to be

Diagonally dominant.

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Solvers and numerical parameters

Solvers include: tridiagonal, pentadiagonal solvers, solution-adaptivesolver, multi-grid solvers, etc.

can e e er:

direct (Cramers 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

Single/double precisions

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Numerical methods (grid generation)

Grids can either be structured (hexahedral)or unstructured (tetrahedral). Dependsupon type of discretization scheme and

structured

Scheme Finite differences: structured

n e vo ume or n e e emen :structured or unstructured

Applicationwith highly-stretched structuredgrids

unstructured

complex geometries Unstructured grids permit automatic

adaptive refinement based on the

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pressure gradient, or regionsinterested (FLUENT)

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Numerical methods (grid transformation)

y

Transform

x

o o

x x

f f f f f

= + = +

Transformation between physical (x,y,z)x x x

y y

f f f f f

y y y

= + = +

, , ,

important for body-fitted grids. The partialderivatives at these two domains have therelationshi 2D as an exam le

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Types of CFD codes

Commercial CFD code: FLUENT, Star-CD,CFDRC, CFX/AEA, ESI-Fastran, EZNSS, etc.

Other CFD software includes the Gridgeneration software (e.g. Gridgen, Gambit)

. .Tecplot, FieldView)

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CFDSHIPIOWA

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CFD Process

Purposes of CFD codes will be different for different applications:investigation of bubble-fluid interactions for bubbly flows, study ofwave induced massivel se arated flows for free-surface etc.

Depend on the specific purpose and flow conditions of the problem,different CFD codes can be chosen for different applications(aerospace, marines, combustion, multi-phase flows, etc.)

Once purposes and CFD codes chosen, CFD process is the steps toset up the IBVP problem and run the code:

1. Geometr

2. Physics

3. Mesh

.

5. Reports

6. Post processing

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CFD Process

Contours

Geometry

Select

Physics Mesh Solve Post-

Processing

Unstructured Steady/ Forces ReportHeat Transfer

Reports

Vectors

Geometry

GeometryCompressible

ON/OFF

(automatic/

manual)

Unsteady (lift/drag, shearstress, etc)

XY Plot

ON/OFF

Structured

(automatic/

Iterations/

Steps

Convergent

Limit

StreamlinesVerification

arame ers

Flow propertiesDomain Shape

and Size

manual)

Viscous Model Precisions

(single/

double)

Validation

BoundaryConditions

NumericalScheme

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Conditions

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Geometry

Selection of an appropriate coordinate D rmin h m in iz n h

Any simplifications needed? What kinds of sha es needed to be used to best

resolve the geometry? (lines, circular, ovals, etc.)

For commercial code, geometry is usually createdus ng commerc a so ware e er separa e romthe commercial code itself, like Gambit, or

For research code, commercial software (e.g.Gridgen) is used.

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Physics

Flow conditions and fluid properties1. Flow conditions: inviscid, viscous, laminar, or

turbulent, etc.

2. Fluid properties: density, viscosity, andthermal conductivit etc.

3. Flow conditions and properties usually presentedin dimensional form in industrial commercial CFD

software whereas in non-dimensional variables forresearch codes.

Selection of models: different models usually fixedb codes o tions for user to choose

Initial and Boundary Conditions: not fixed bycodes, user needs specify them for different

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Mesh

Meshes should be well designed to resolve importantfl w f r whi h r n n n fl w

condition parameters (e.g., Re), such as the gridrefinement inside the wall boundary layer

Mesh can be generated by either commercial codes(Gridgen, Gambit, etc.) or research code (using

. , , .

The mesh, together with the boundary conditionsneed to be exported from commercial software in a

certain format that can be recognized by theresearch CFD code or other commercial CFD

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.

S l

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Solve

Setup appropriate numerical parameters oose appropr ate o vers Solution procedure (e.g. incompressible flows)

o ve e momen um, pressure o ssonequations and get flow field quantities, such as

, ,integral quantities (lift, drag forces)

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R t

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Reports

Reports saved the time history of the residualsof the velocity, pressure and temperature, etc.

Report the integral quantities, such as totalpressure drop, friction factor (pipe flow), lift and, .

XY plots could present the centerline

velocity/pressure distribution, friction factor,distribution (airfoil flow).

AFD or EFD data can be im orted and ut on

top of the XY plots for validation

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Post-processing

Analysis and visualization Calculation of derived variables

Vorticity

Wall shear stress Calculation of integral parameters: forces,

moments

Visualization (usually with commercial software)

Simple 2D contours 3D contour isosurface plots

Vector plots and streamlines (streamlines arethe lines whose tangent direction is the sameas e ve oc y vec ors

Animations

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Post-processing (Uncertainty Assessment)

imu ation error: the difference between a simulation result and thetruth (objective reality), assumed composed of additive modeling andnumerical errors:

Verification: process for assessing simulation numerical uncertainties and,w en con ons perm , es ma ng e s gn an magn u e o e simulation numerical error itself and the uncertainties in that error estimate

Validation: process for assessing simulation modeling uncertainty by usingbenchmark experimental data and, when conditions permit, estimating thesign and magnitude of the modeling error itself.

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Post-processing (UA, Verification)

onvergence s u es: onvergence s u es requ re a m n mum om=3 solutions to evaluate convergence with respective to inputparameters.

, ,

meshes(i). Monotonic convergence:

(ii). Oscillatory Convergence:

(iii). Monotonic divergence:

(iv). Oscillatory divergence:

.

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Post-processing (Verification: Iterative Convergence)

Typical CFD solution techniques for obtaining steady state solutionsinvolve beginning with an initial guess and performing time marching or

.

The number of order magnitude drop and final level of solution residualcan be used to determine stopping criteria for iterative solution techniques

(b)(a)

)(1

LUI SSU =

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Iteration istory: (a). So ution c ange ( ) magni ie view o tota resistance over ast two

periods of oscillation (Oscillatory iterative convergence)

Post processing (Verification RE)

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Post-processing (Verification, RE)

Generalized Richardson Extrapolation (RE): Formonotonic convergence, generalized RE is used toestimate the error and order of accuracy due to the

selection of the input parameter.

integer powers ofx as a finite sum.

The accuracy of the estimates depends on how manyterms are retained in the expansion, the magnitude(importance) of the higher-order terms, and the validity

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E l f CFD P (M h)

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Example of CFD Process (Mesh)

Grid need to be refined near the foil

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sur ace to reso ve t e oun ary ayer

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Example of CFD Process (Solve)

Residuals vs. iteration

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E l f CFD P (R t )

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Example of CFD Process (Reports)

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Example of CFD Process (Post-processing)

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Example of CFD Process (Post-processing)

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