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Introduction to Computational Fluid Dynamics (CFD) Tao Xing, Maysam Mousaviraad, Shanti Bhushan and Fred Stern IIHR —Hydroscience — Hydroscience & Engineering C. Maxwell Stanley Hydraulics Laboratory The University of Iowa 58:160 Intermediate Mechanics of Fluids http://css.engineering.uiowa.edu/~me_160/ August 29, 2012

Faces of Fluid Mechanics

Archimedes

Newton

(C. 287-212 BC)

(1642-1727)

Navier

Stokes

(1785-1836)

(1819-1903)

Leibniz (1646-1716)

Bernoulli (1667-1748)

Euler (1707-1783)

Reynolds

Prandtl

Taylor

Kolmogorov

(1842-1912)

(1875-1953)

(1886-1975)

(1903-1987)

Faces of Fluid Mechanics

Archimedes

Newton

(C. 287-212 BC)

(1642-1727)

Navier

Stokes

(1785-1836)

(1819-1903)

Leibniz (1646-1716)

Bernoulli (1667-1748)

Euler (1707-1783)

Reynolds

Prandtl

Taylor

Kolmogorov

(1842-1912)

(1875-1953)

(1886-1975)

(1903-1987)

Outline 1. What, why and where of CFD? 2. Modeling 3. Numerical methods 4. Types of CFD codes 5. CFD Educational Interface 6. CFD Process 7. Example of CFD Process 8. 58:160 CFD Labs

What is CFD? •

• •

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.) Historically only Analytical Fluid Dynamics (AFD) and Experime Experimental ntal Fluid Dynamics (EFD). CFD made possible by the advent of digital computer and advancing with improvements of computer resources (500 flops, 194720 teraflops, 2003 1.3 pentaflops, Roadrunner at Las Alamos National Lab, 2009.)

Why use CFD? • Analysis and Design 1. Simulation-based Simulation-based design instead of “build & test” 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

Where is CFD used? • Where is CFD used? • Aerospace • Automotive • Biomedical

Aerospace

Biomedical

• Chemical • • • • • •

F18 Store Separation

Processing HVAC Hydraulics Marine Oil & Gas Power Generation Sports Automotive

Temperature and natural convection currents in the eye follow ing laser heating.

Where is CFD used? Chemical Processing

• Where is CFD used? • • • • • • • • • •

Aerospacee Automotive Biomedical Chemical Processing HVAC Hydraulics Marine Oil & Gas Power Generation Sports

Polym erization reactor vessel - predictio n of flow s eparation and residence time effects.

Hydraulics

HVAC S t r ea m l i n e s f o r w o r k s t a t i o n ventilation

Where is CFD used? Marine

Sports

• Where is CFD used? • • • • • • • • • •

Aerospace Automotive Biomedical Chemical Processing HVAC Hydraulics Marine Oil & Gas Power Generation Sports

http://gallery.ensight.com/keyword/sport/1/701168838_KVnHn#!i=701168838&k=KVnHn

Oil & Gas Flow of lubricating m u d o v e r d r i l l b it

Power Generation Flow around cooling towers

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 domain 2. Coordinates 3. Governing equations 4. Flow conditions 5. Initial and boundary conditions 6. Selection of models for different applications

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).

Modeling (coordinates) z

z

Cartesian

Cylindrical

z

(r,,z)

(x,y,z)

Spherical (r,,)

z

y x

x

General Curvilinear Coordinates

r

y

r

y

x

General orthogonal Coordinates

Modeling (governing equations) •

Navier-Stokes equations (3D in Cartesian coordinates) 2u 2 u 2 u u u u u p u v w 2 2 2 t x y z x y z x ˆ

2v 2v 2v v v v v p u v w 2 2 2 t x y z y y z x ˆ

2w 2w 2w w w w w p u v w 2 2 2 t x y z z y z x ˆ

Local acceleration

Convection

Piezometric pressure gradient

Viscous terms

u v w 0 Continuity equation t x y z

p RT 2

R

D R 2

Dt

3 DR 2 ( ) 2 Dt

Equation of state

pv

L

p

Rayleigh Equation

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 (Ma) • Single- vs. multi-phase (Ca) • Thermal/density effects (Pr, , Gr, Ec) • Free-surface flow (Fr) and surface tension (We) • Chemical reactions and combustion (Pe, Da) • etc…

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

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 nonreflecting (for compressible flows, such as acoustics), etc.

No-slip walls: u=0,v=0 Outlet, p=c

Inlet ,u=c,v=0 r

o

x

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

Axisymmetric

Periodic boundary condition in spanwise direction of an airfoil

Modeling (selection of models) • CFD codes typically designed for solving certain fluid

phenomenon 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, , 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…

Modeling (Turbulence and free surface models) • Turbulent flows at high Re usually involve both large and small scale vortical structures and very thin turbulent boundary layer (BL) near the wall

• 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.

Modeling (examples)

Wave breaking in bump flow simulation Deformation of a sphere.(a)maximum stretching; (b) recovered shape. Left: LS; right: VOF.

Two-phase flow past a surface-piercing cylinder showing vortical structures colored by pressure

Movie

Movie

Wedge flow simulation

Movie

Modeling (examples, cont’d) Air flow for ONR Tumblehome in PMM maneuvers

Movie

Waterjet flow modeling for JHSS and Delft catamaran

Broaching of ONR Tumblehome with rotating propellers

Movie

Modeling (examples, cont’d)

T-Craft (SES/ACV) turning circle in calm water with water jet propulsion (top) and straight ahead with air-fan propulsion (bottom)

Movie

Regular head wave simulation for side by side ship-ship interactions

Movie

Modeling (examples, cont’d)

Movie

Ship in three-sisters rogue (freak) waves

Damaged stability for SSRC cruiser with two-room compartment in beam waves

Movie

Vortical Structures and Instability Analysis Fully appended Athena DES Computation Re=2.9×108, Fr=0.25 Isosurface of Q=300 colored using piezometric pressure - Karman-like shedding from Transom Corner - Horse-shoe vortices from hull-rudder (Case A) and strut-hull (Case B) junction flow. - Shear layer instability at hull-strut intersection

DTMB 5415 at =20 DES Computation Re=4.85×106,Fr=0.28 Isosurface of Q=300 colored using piezometric pressure - The sonar dome (SDTV) and bilge keel (BK TV) vortices exhibits helical instability breakdown. - Shear-layer instabilities: port bow (BSL1, BSL2) and fore-body keel (K SL). - Karman-like instabilities on port side bow (BK ) . - Wave breaking vortices on port (FSBW1) and starboard (FSBW2). Latter exhibits horse shoe type instability.

Movie

Modeling (examples, cont’d) Movie (CFD) Movie (EFD at Iowa wave basin)

CFD simulations to improve system identification (SI) technique

Broaching simulation of free running ONR Tumblehome

Movie (CFD) Movie (EFD)

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 methods 2. Solvers and numerical parameters 3. Grid generation and transformation 4. High Performance Computation (HPC) and postprocessing

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 2 nd 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 implicit method (e.g. Beam-Warming method)

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 methods are 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.

Discretization methods (example) • 2D incompressible laminar flow boundary layer (L,m+1)

u v 0 x y

(L-1,m)

u u p 2u u v 2 x y x e y

u uml l um uml 1 u x x

u vml uml 1 uml v y y

vml

y m=MM+1 m=MM

(L,m) m=1 m=0

(L,m-1)

L-1

L

x

2u l l l 2 u 2 u u 1 m 1 m m 2 y y

FD Sign( vml )0 y m

2nd order central difference i.e., theoretical order of accuracy Pkest= 2.

Discretization methods (example) B

3 1 B1 B2 FD l l ul l v v 2 y l l l m m m FD um1 2 BD um1 2 um 2 vm 1 y x y y y y BD y l

l 1 m1

B u

Bu Bu l 2 m

l 3 m1

l 1 4 m

Bu

um

l 1

um

x l B4 p / em x

l ( p / e) m x

l p Solve it using l 1 B4u1 l x e 1 Thomas algorithm B2 B3 0 0 0 0 0 0 u1 B B B 0 0 0 0 0 1 2 3 0 0 0 0 0 B B B 1 2 3 l l 0 0 0 0 0 0 B1 B2 umm p l 1 B4umm x e mm To be stable, Matrix has to be Diagonally dominant.

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

Numerical methods (grid generation) • Grids can either be structured

structured

(hexahedral) or unstructured (tetrahedral). Depends upon type of discretization scheme and application • Scheme Finite differences: structured Finite volume or finite element: structured or unstructured • Application Thin boundary layers best unstructured resolved with highly-stretched structured grids Unstructured grids useful for complex geometries Unstructured grids permit automatic adaptive refinement based on the pressure gradient, or regions interested (FLUENT)

y

Numerical methods (grid transformation)

Transform

o

x Physical domain

•Transformation between physical (x,y,z)

o

Computational domain f f f f f x x x x x

and computational (,,z) domains, important for body-fitted grids. The partial f f f f f derivatives at these two domains have the y y y y y relationship (2D as an example)

High performance computing • CFD computations (e.g. 3D unsteady flows) are usually very expensive which requires parallel high performance supercomputers (e.g. IBM 690) with the use of multi-block technique. • As required by the multi-block technique, CFD codes need to be developed using the Massage Passing Interface (MPI) Standard to transfer data between different blocks. • Emphasis on improving:

• Strong scalability, main bottleneck pressure Poisson solver for incompressible flow. • Weak scalability, limited by the memory requirements.

Figure: Strong scalability of total times without I/O for CFDShip-Iowa V6 and V4 on NAVO Cray XT5 (Einstein) and IBM P6 (DaVinci) are compared with ideal scaling.

Figure: Weak scalability of total times without I/O for CFDShip-Iowa V6 and V4 on IBM P6 (DaVinci) and SGI Altix (Hawk) are compared with ideal scaling.

Post-Processing • Post-processing: 1. Visualize the CFD results (contour, velocity

vectors, streamlines, pathlines, streak lines, and iso-surface in 3D, etc.), and 2. CFD UA : verification and validation using EFD data (more details later) • Post-processing usually through using commercial software

Figure: Isosurface of Q=300 colored using piezometric pressure, free=surface colored using z for fully appended Athena, Fr=0.25, Re=2.9×10 8. Tecplot360 is used for visualization.

Types of CFD codes • Commercial CFD code: FLUENT, StarCD, CFDRC, CFX/AEA, etc. • Research CFD code: CFDSHIP-IOWA • Public domain software (PHI3D, HYDRO, and WinpipeD, etc.) • Other CFD software includes the Grid generation software (e.g. Gridgen, Gambit) and flow visualization software (e.g. Tecplot, FieldView)

CFDSHIPIOWA

CFD Educational Interface

Lab1: Pipe Flow

1. Definition of “CFD Process” 2. Boundary conditions 3. Iterative error 4. Grid error 5. Developing length of laminar and turbulent pipe flows. 6. Verification using AFD 7. Validation using EFD

Lab 2: Airfoil Flow

Lab3: Diffuser

Lab4: Ahmed car

1. Boundary conditions 2. Effect of order of accuracy on verification results 3. Effect of grid generation topology, “C” and “O” Meshes 4. Effect of angle of attack/turbulent models on flow field 5. Verification and Validation using EFD

1. Meshing and iterative convergence 2. Boundary layer separation 3. Axial velocity profile 4. Streamlines 5. Effect of turbulence models 6. Effect of expansion angle and comparison with LES, EFD, and

1. Meshing and iterative convergence 2. Boundary layer separation 3. Axial velocity profile 4. Streamlines 5. Effect of slant angle and comparison with LES, EFD, and RANS.

CFD process • Purposes of CFD codes will be different for different applications: investigation of bubble-fluid interactions for bubbly flows, study of wave induced massively separated 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 to set up the IBVP problem and run the code: 1. Geometry 2. Physics 3. Mesh 4. Solve 5. Reports 6. Post processing

CFD Process Geometry

Physics

Mesh

Solve

Reports

PostProcessing

Select Geometry

Heat Transfer ON/OFF

Unstructured (automatic/ manual)

Steady/ Unsteady

Forces Report

Contours

Compressible ON/OFF

Structured (automatic/ manual)

Iterations/ Steps

XY Plot

Vectors

Flow properties

Convergent Limit

Verification

Streamlines

Viscous Model

Precisions (single/ double)

Validation

Boundary Conditions

Numerical Scheme

Geometry Parameters

Domain Shape and Size

Initial Conditions

(lift/drag, shear stress, etc)

Geometry • Selection of an appropriate coordinate • Determine the domain size and shape • Any simplifications needed? • What kinds of shapes needed to be used to best resolve the geometry? (lines, circular, ovals, etc.) • For commercial code, geometry is usually created using commercial software (either separated from the commercial code itself, like Gambit, or combined together, like FlowLab) • For research code, commercial software (e.g. Gridgen) is used.

Physics • Flow conditions and fluid properties

• •

1. Flow conditions: inviscid, viscous, laminar, or turbulent, etc. 2. Fluid properties: density, viscosity, and thermal conductivity, etc. 3. Flow conditions and properties usually presented in dimensional form in industrial commercial CFD software, whereas in nondimensional variables for research codes. Selection of models: different models usually fixed by codes, options for user to choose Initial and Boundary Conditions: not fixed by codes, user needs specify them for different applications.

Mesh • Meshes should be well designed to resolve important flow features which are dependent upon flow condition parameters (e.g., Re), such as the grid refinement inside the wall boundary layer • Mesh can be generated by either commercial codes (Gridgen, Gambit, etc.) or research code (using algebraic vs. PDE based, conformal mapping, etc.) • The mesh, together with the boundary conditions need to be exported from commercial software in a certain format that can be recognized by the research CFD code or other commercial CFD software.

Solve • Setup appropriate numerical parameters • Choose appropriate Solvers • Solution procedure (e.g. incompressible flows) Solve the momentum, pressure Poisson equations and get flow field quantities, such as velocity, turbulence intensity, pressure and integral quantities (lift, drag forces)

Reports • Reports saved the time history of the residuals of the velocity, pressure and temperature, etc. • Report the integral quantities, such as total pressure drop, friction factor (pipe flow), lift and drag coefficients (airfoil flow), etc. • XY plots could present the centerline velocity/pressure distribution, friction factor distribution (pipe flow), pressure coefficient distribution (airfoil flow). • AFD or EFD data can be imported and put on top of the XY plots for validation

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 are the lines whose tangent direction is the same as the velocity vectors) Animations

Post-processing (Uncertainty Assessment) • Simulation error: the difference between a simulation result S and the truth T (objective reality), assumed composed of additive modeling δSM and numerical δSN errors: Error: S S T SM SN

2

Uncertainty: U S

2 2 U SM U SN

• Verification: process for assessing simulation numerical

uncertainties U SN and, when conditions permit, estimating the sign and magnitude Delta δ*SN of the simulation numerical error itself and the uncertainties in that error estimate USN SN I G T P I

J

j

j 1

2 U SN U I 2 U G2 U T 2 U P 2

I: Iterative, G : Grid, T: Time step, P: Input parameters

• Validation: process for assessing simulation modeling

uncertainty U SM by using benchmark experimental data and, when conditions permit, estimating the sign and magnitude of the modeling error δ SM itself. 2 2 2 U U U V D SN E D S ( ) D

SM

D : EFD Data; U V : Validation Uncertainty

SN

E U V

Validation achieved

Post-processing (UA, Verification) • Convergence studies: Convergence studies require a minimum of m=3 solutions to evaluate convergence with respective to input parameters. Consider the solutions corresponding to fine S k 1 , medium S k 2 ,and coarse meshes

k 21 Sk 2 S k 1

S k 3

Monotonic Convergence

k 32 Sk 3 S k 2

Rk k 21 k 32 (i). Monotonic convergence: 0

View more...
Faces of Fluid Mechanics

Archimedes

Newton

(C. 287-212 BC)

(1642-1727)

Navier

Stokes

(1785-1836)

(1819-1903)

Leibniz (1646-1716)

Bernoulli (1667-1748)

Euler (1707-1783)

Reynolds

Prandtl

Taylor

Kolmogorov

(1842-1912)

(1875-1953)

(1886-1975)

(1903-1987)

Faces of Fluid Mechanics

Archimedes

Newton

(C. 287-212 BC)

(1642-1727)

Navier

Stokes

(1785-1836)

(1819-1903)

Leibniz (1646-1716)

Bernoulli (1667-1748)

Euler (1707-1783)

Reynolds

Prandtl

Taylor

Kolmogorov

(1842-1912)

(1875-1953)

(1886-1975)

(1903-1987)

Outline 1. What, why and where of CFD? 2. Modeling 3. Numerical methods 4. Types of CFD codes 5. CFD Educational Interface 6. CFD Process 7. Example of CFD Process 8. 58:160 CFD Labs

What is CFD? •

• •

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.) Historically only Analytical Fluid Dynamics (AFD) and Experime Experimental ntal Fluid Dynamics (EFD). CFD made possible by the advent of digital computer and advancing with improvements of computer resources (500 flops, 194720 teraflops, 2003 1.3 pentaflops, Roadrunner at Las Alamos National Lab, 2009.)

Why use CFD? • Analysis and Design 1. Simulation-based Simulation-based design instead of “build & test” 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

Where is CFD used? • Where is CFD used? • Aerospace • Automotive • Biomedical

Aerospace

Biomedical

• Chemical • • • • • •

F18 Store Separation

Processing HVAC Hydraulics Marine Oil & Gas Power Generation Sports Automotive

Temperature and natural convection currents in the eye follow ing laser heating.

Where is CFD used? Chemical Processing

• Where is CFD used? • • • • • • • • • •

Aerospacee Automotive Biomedical Chemical Processing HVAC Hydraulics Marine Oil & Gas Power Generation Sports

Polym erization reactor vessel - predictio n of flow s eparation and residence time effects.

Hydraulics

HVAC S t r ea m l i n e s f o r w o r k s t a t i o n ventilation

Where is CFD used? Marine

Sports

• Where is CFD used? • • • • • • • • • •

Aerospace Automotive Biomedical Chemical Processing HVAC Hydraulics Marine Oil & Gas Power Generation Sports

http://gallery.ensight.com/keyword/sport/1/701168838_KVnHn#!i=701168838&k=KVnHn

Oil & Gas Flow of lubricating m u d o v e r d r i l l b it

Power Generation Flow around cooling towers

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 domain 2. Coordinates 3. Governing equations 4. Flow conditions 5. Initial and boundary conditions 6. Selection of models for different applications

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).

Modeling (coordinates) z

z

Cartesian

Cylindrical

z

(r,,z)

(x,y,z)

Spherical (r,,)

z

y x

x

General Curvilinear Coordinates

r

y

r

y

x

General orthogonal Coordinates

Modeling (governing equations) •

Navier-Stokes equations (3D in Cartesian coordinates) 2u 2 u 2 u u u u u p u v w 2 2 2 t x y z x y z x ˆ

2v 2v 2v v v v v p u v w 2 2 2 t x y z y y z x ˆ

2w 2w 2w w w w w p u v w 2 2 2 t x y z z y z x ˆ

Local acceleration

Convection

Piezometric pressure gradient

Viscous terms

u v w 0 Continuity equation t x y z

p RT 2

R

D R 2

Dt

3 DR 2 ( ) 2 Dt

Equation of state

pv

L

p

Rayleigh Equation

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 (Ma) • Single- vs. multi-phase (Ca) • Thermal/density effects (Pr, , Gr, Ec) • Free-surface flow (Fr) and surface tension (We) • Chemical reactions and combustion (Pe, Da) • etc…

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

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 nonreflecting (for compressible flows, such as acoustics), etc.

No-slip walls: u=0,v=0 Outlet, p=c

Inlet ,u=c,v=0 r

o

x

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

Axisymmetric

Periodic boundary condition in spanwise direction of an airfoil

Modeling (selection of models) • CFD codes typically designed for solving certain fluid

phenomenon 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, , 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…

Modeling (Turbulence and free surface models) • Turbulent flows at high Re usually involve both large and small scale vortical structures and very thin turbulent boundary layer (BL) near the wall

• 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.

Modeling (examples)

Wave breaking in bump flow simulation Deformation of a sphere.(a)maximum stretching; (b) recovered shape. Left: LS; right: VOF.

Two-phase flow past a surface-piercing cylinder showing vortical structures colored by pressure

Movie

Movie

Wedge flow simulation

Movie

Modeling (examples, cont’d) Air flow for ONR Tumblehome in PMM maneuvers

Movie

Waterjet flow modeling for JHSS and Delft catamaran

Broaching of ONR Tumblehome with rotating propellers

Movie

Modeling (examples, cont’d)

T-Craft (SES/ACV) turning circle in calm water with water jet propulsion (top) and straight ahead with air-fan propulsion (bottom)

Movie

Regular head wave simulation for side by side ship-ship interactions

Movie

Modeling (examples, cont’d)

Movie

Ship in three-sisters rogue (freak) waves

Damaged stability for SSRC cruiser with two-room compartment in beam waves

Movie

Vortical Structures and Instability Analysis Fully appended Athena DES Computation Re=2.9×108, Fr=0.25 Isosurface of Q=300 colored using piezometric pressure - Karman-like shedding from Transom Corner - Horse-shoe vortices from hull-rudder (Case A) and strut-hull (Case B) junction flow. - Shear layer instability at hull-strut intersection

DTMB 5415 at =20 DES Computation Re=4.85×106,Fr=0.28 Isosurface of Q=300 colored using piezometric pressure - The sonar dome (SDTV) and bilge keel (BK TV) vortices exhibits helical instability breakdown. - Shear-layer instabilities: port bow (BSL1, BSL2) and fore-body keel (K SL). - Karman-like instabilities on port side bow (BK ) . - Wave breaking vortices on port (FSBW1) and starboard (FSBW2). Latter exhibits horse shoe type instability.

Movie

Modeling (examples, cont’d) Movie (CFD) Movie (EFD at Iowa wave basin)

CFD simulations to improve system identification (SI) technique

Broaching simulation of free running ONR Tumblehome

Movie (CFD) Movie (EFD)

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 methods 2. Solvers and numerical parameters 3. Grid generation and transformation 4. High Performance Computation (HPC) and postprocessing

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 2 nd 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 implicit method (e.g. Beam-Warming method)

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 methods are 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.

Discretization methods (example) • 2D incompressible laminar flow boundary layer (L,m+1)

u v 0 x y

(L-1,m)

u u p 2u u v 2 x y x e y

u uml l um uml 1 u x x

u vml uml 1 uml v y y

vml

y m=MM+1 m=MM

(L,m) m=1 m=0

(L,m-1)

L-1

L

x

2u l l l 2 u 2 u u 1 m 1 m m 2 y y

FD Sign( vml )0 y m

2nd order central difference i.e., theoretical order of accuracy Pkest= 2.

Discretization methods (example) B

3 1 B1 B2 FD l l ul l v v 2 y l l l m m m FD um1 2 BD um1 2 um 2 vm 1 y x y y y y BD y l

l 1 m1

B u

Bu Bu l 2 m

l 3 m1

l 1 4 m

Bu

um

l 1

um

x l B4 p / em x

l ( p / e) m x

l p Solve it using l 1 B4u1 l x e 1 Thomas algorithm B2 B3 0 0 0 0 0 0 u1 B B B 0 0 0 0 0 1 2 3 0 0 0 0 0 B B B 1 2 3 l l 0 0 0 0 0 0 B1 B2 umm p l 1 B4umm x e mm To be stable, Matrix has to be Diagonally dominant.

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

Numerical methods (grid generation) • Grids can either be structured

structured

(hexahedral) or unstructured (tetrahedral). Depends upon type of discretization scheme and application • Scheme Finite differences: structured Finite volume or finite element: structured or unstructured • Application Thin boundary layers best unstructured resolved with highly-stretched structured grids Unstructured grids useful for complex geometries Unstructured grids permit automatic adaptive refinement based on the pressure gradient, or regions interested (FLUENT)

y

Numerical methods (grid transformation)

Transform

o

x Physical domain

•Transformation between physical (x,y,z)

o

Computational domain f f f f f x x x x x

and computational (,,z) domains, important for body-fitted grids. The partial f f f f f derivatives at these two domains have the y y y y y relationship (2D as an example)

High performance computing • CFD computations (e.g. 3D unsteady flows) are usually very expensive which requires parallel high performance supercomputers (e.g. IBM 690) with the use of multi-block technique. • As required by the multi-block technique, CFD codes need to be developed using the Massage Passing Interface (MPI) Standard to transfer data between different blocks. • Emphasis on improving:

• Strong scalability, main bottleneck pressure Poisson solver for incompressible flow. • Weak scalability, limited by the memory requirements.

Figure: Strong scalability of total times without I/O for CFDShip-Iowa V6 and V4 on NAVO Cray XT5 (Einstein) and IBM P6 (DaVinci) are compared with ideal scaling.

Figure: Weak scalability of total times without I/O for CFDShip-Iowa V6 and V4 on IBM P6 (DaVinci) and SGI Altix (Hawk) are compared with ideal scaling.

Post-Processing • Post-processing: 1. Visualize the CFD results (contour, velocity

vectors, streamlines, pathlines, streak lines, and iso-surface in 3D, etc.), and 2. CFD UA : verification and validation using EFD data (more details later) • Post-processing usually through using commercial software

Figure: Isosurface of Q=300 colored using piezometric pressure, free=surface colored using z for fully appended Athena, Fr=0.25, Re=2.9×10 8. Tecplot360 is used for visualization.

Types of CFD codes • Commercial CFD code: FLUENT, StarCD, CFDRC, CFX/AEA, etc. • Research CFD code: CFDSHIP-IOWA • Public domain software (PHI3D, HYDRO, and WinpipeD, etc.) • Other CFD software includes the Grid generation software (e.g. Gridgen, Gambit) and flow visualization software (e.g. Tecplot, FieldView)

CFDSHIPIOWA

CFD Educational Interface

Lab1: Pipe Flow

1. Definition of “CFD Process” 2. Boundary conditions 3. Iterative error 4. Grid error 5. Developing length of laminar and turbulent pipe flows. 6. Verification using AFD 7. Validation using EFD

Lab 2: Airfoil Flow

Lab3: Diffuser

Lab4: Ahmed car

1. Boundary conditions 2. Effect of order of accuracy on verification results 3. Effect of grid generation topology, “C” and “O” Meshes 4. Effect of angle of attack/turbulent models on flow field 5. Verification and Validation using EFD

1. Meshing and iterative convergence 2. Boundary layer separation 3. Axial velocity profile 4. Streamlines 5. Effect of turbulence models 6. Effect of expansion angle and comparison with LES, EFD, and

1. Meshing and iterative convergence 2. Boundary layer separation 3. Axial velocity profile 4. Streamlines 5. Effect of slant angle and comparison with LES, EFD, and RANS.

CFD process • Purposes of CFD codes will be different for different applications: investigation of bubble-fluid interactions for bubbly flows, study of wave induced massively separated 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 to set up the IBVP problem and run the code: 1. Geometry 2. Physics 3. Mesh 4. Solve 5. Reports 6. Post processing

CFD Process Geometry

Physics

Mesh

Solve

Reports

PostProcessing

Select Geometry

Heat Transfer ON/OFF

Unstructured (automatic/ manual)

Steady/ Unsteady

Forces Report

Contours

Compressible ON/OFF

Structured (automatic/ manual)

Iterations/ Steps

XY Plot

Vectors

Flow properties

Convergent Limit

Verification

Streamlines

Viscous Model

Precisions (single/ double)

Validation

Boundary Conditions

Numerical Scheme

Geometry Parameters

Domain Shape and Size

Initial Conditions

(lift/drag, shear stress, etc)

Geometry • Selection of an appropriate coordinate • Determine the domain size and shape • Any simplifications needed? • What kinds of shapes needed to be used to best resolve the geometry? (lines, circular, ovals, etc.) • For commercial code, geometry is usually created using commercial software (either separated from the commercial code itself, like Gambit, or combined together, like FlowLab) • For research code, commercial software (e.g. Gridgen) is used.

Physics • Flow conditions and fluid properties

• •

1. Flow conditions: inviscid, viscous, laminar, or turbulent, etc. 2. Fluid properties: density, viscosity, and thermal conductivity, etc. 3. Flow conditions and properties usually presented in dimensional form in industrial commercial CFD software, whereas in nondimensional variables for research codes. Selection of models: different models usually fixed by codes, options for user to choose Initial and Boundary Conditions: not fixed by codes, user needs specify them for different applications.

Mesh • Meshes should be well designed to resolve important flow features which are dependent upon flow condition parameters (e.g., Re), such as the grid refinement inside the wall boundary layer • Mesh can be generated by either commercial codes (Gridgen, Gambit, etc.) or research code (using algebraic vs. PDE based, conformal mapping, etc.) • The mesh, together with the boundary conditions need to be exported from commercial software in a certain format that can be recognized by the research CFD code or other commercial CFD software.

Solve • Setup appropriate numerical parameters • Choose appropriate Solvers • Solution procedure (e.g. incompressible flows) Solve the momentum, pressure Poisson equations and get flow field quantities, such as velocity, turbulence intensity, pressure and integral quantities (lift, drag forces)

Reports • Reports saved the time history of the residuals of the velocity, pressure and temperature, etc. • Report the integral quantities, such as total pressure drop, friction factor (pipe flow), lift and drag coefficients (airfoil flow), etc. • XY plots could present the centerline velocity/pressure distribution, friction factor distribution (pipe flow), pressure coefficient distribution (airfoil flow). • AFD or EFD data can be imported and put on top of the XY plots for validation

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 are the lines whose tangent direction is the same as the velocity vectors) Animations

Post-processing (Uncertainty Assessment) • Simulation error: the difference between a simulation result S and the truth T (objective reality), assumed composed of additive modeling δSM and numerical δSN errors: Error: S S T SM SN

2

Uncertainty: U S

2 2 U SM U SN

• Verification: process for assessing simulation numerical

uncertainties U SN and, when conditions permit, estimating the sign and magnitude Delta δ*SN of the simulation numerical error itself and the uncertainties in that error estimate USN SN I G T P I

J

j

j 1

2 U SN U I 2 U G2 U T 2 U P 2

I: Iterative, G : Grid, T: Time step, P: Input parameters

• Validation: process for assessing simulation modeling

uncertainty U SM by using benchmark experimental data and, when conditions permit, estimating the sign and magnitude of the modeling error δ SM itself. 2 2 2 U U U V D SN E D S ( ) D

SM

D : EFD Data; U V : Validation Uncertainty

SN

E U V

Validation achieved

Post-processing (UA, Verification) • Convergence studies: Convergence studies require a minimum of m=3 solutions to evaluate convergence with respective to input parameters. Consider the solutions corresponding to fine S k 1 , medium S k 2 ,and coarse meshes

k 21 Sk 2 S k 1

S k 3

Monotonic Convergence

k 32 Sk 3 S k 2

Rk k 21 k 32 (i). Monotonic convergence: 0

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