computational fluid dynamics applied to the design of aerospace vehicles
DESCRIPTION
An overview of computational fluid dynamics applied to the design of aerospace vehicles is carried out. This document is composed of the following sections: introduction, CFD process, CFD basics, Mesh technology, and turbulence modelingTRANSCRIPT
Computational Aerodynamics for Aircraft Design ITA – Aircraft Design Department
V 29
• Introduction
• A taste of history
• CFD Basics
• Overview on mesh technology
• Turbulence modeling
Introduction
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.
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
Analysis and Design 1. 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
* Experimental Fluid Dynamics
Practice of engineering and science has been dramatically altered by the development of
Scientific computing
Mathematics of numerical analysis
Tools like neural networks
The Internet
Computational Fluid Dynamics is based upon the logic of applied mathematics
provides tools to unlock previously unsolved problems
is used in nearly all fields of science and engineering
Aerodynamics, acoustics, bio-systems, cosmology, geology, heat transfer, hydrodynamics, river hydraulics, etc…
9. Fletcher, C. A. J. “Computational Techniques for Fluid Dynamics,” Springer
Series in Computational Physics, Vols. 1-2, 2nd Edition, 1991.
10. Versteeg, H.K. and Malalasekera, W., An introduction to Computational Fluid
Dynamics: The Finite Volume Method (2nd Edition), Prentice Hall, 1st Edition.
http://www.cfd-online.com
http://www.aoe.vt.edu/~mason/Mason_f/MRsoft.html
http://www.ensight.com/
http://www.metacomptech.com
Products
CFD++
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.
http://www.sai.msu.su/sal/sal1.shtml
http://gd.tuwien.ac.at/study/baum-lse/node2.html
Linux Software Encyclopedia
http://sourceforge.net/
Source Forge
http://www.cfdreview.com/
Source Forge
Top 500 computers in the world compiled: www.top500.org
Computers located at major centers connected to researchers via Internet
A taste of history
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, 194720 teraflops, 2003)
“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”
Aristotle (384-322 BCE)
Greek philosopher, student of Plato
Logic and reasoning was the chief instrument of
scientific investigation; Posterior Analytics
To possess scientific knowledge, we need to know the
cause of which we observe
Through their senses humans encounter facts or data
Through inductive means, principles created which will
explain the data
Then, 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.
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.
Isaac Newton (1643 – 1727) Laid the foundation (along with Leibniz) for differential and integral calculus It has been claimed that the Principia is the greatest work in the history of the physical sciences. Book I: general dynamics from a mathematical standpoint Book II: treatise on fluid mechanics Book 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.
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
From mid-1800’s to 1960’s, research in fluid mechanics focused upon
Analytical methods
Exact 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 methods
Scale models: wind tunnels, water tunnels, towing-tanks, flumes,...
Measurement techniques: pitot probes; hot-wire probes; anemometers; laser-doppler velocimetry; particle-image velocimetry
Most man-made systems (e.g., airplane) engineered using build-and-test iteration.
1950’s – present : rise of computational fluid dynamics
Konrad Zuse (1910-1995) was a construction engineer for the
Henschel Aircraft Company in Berlin, Germany at the beginning of
WWII. Konrad Zuse earned the semiofficial title of "inventor of the
modern computer" for his series of automatic calculators, which he
invented to help him with his lengthy engineering calculations.
Zuse has modestly dismissed the title while praising many of the
inventions of his contemporaries and successors as being equally if
not more important than his own.
One of the most difficult aspects of doing a large calculation with
either a slide rule or a mechanical adding machine is keeping track
of all intermediate results and using them, in their proper place, in
later steps of the calculation. Konrad Zuse wanted to overcome that
difficulty. He realized that an automatic-calculator device would
require three basic elements: a control, a memory, and a calculator
for the arithmetic.
The First Freely Programmable Computer invented by Konrad Zuse
Computing, 1945-1960
Early computer engineers thought that only a few dozen computers required worldwide
Applications: cryptography (code breaking), fluid dynamics, artillery firing tables, atomic weapons
ENIAC, 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
In the early 1930s Polish cryptographers first broke the code of Germany's cipher machine Enigma. They were led by mathematician
Marian Rejewski and assisted by material provided to them by agents of French intelligence. For much of the decade, the Poles were able
to decipher their neighbour's radio traffic, but in 1939, faced with possible invasion and difficulties decoding messages because of changes
in Enigma's operating procedures, they turned their information over to the Allies. Early in 1939 Britain's secret service set up the Ultra
project at Bletchley Park, 50 miles (80 km) north of London, for the purpose of intercepting the Enigma signals, deciphering the messages,
and controlling the distribution of the resultant secret information. Strict rules were established to restrict the number of people who knew
about the existence of the Ultra information and to ensure that no actions would alert the Axis powers that the Allies possessed knowledge
of their 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 orders of battle, forming the basis of
war plans both strategic and tactical. Ultra intercepts of signals helped the Royal Air Force to win the Battle of Britain. Intercepted signals
between Hitler and General Günther von Kluge led to the destruction of a large 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 this
code-breaking machine came from the Ultra project.
Ultra Project
Computer History
Year/Enter
Computer History
Inventors/Inventions
Computer History
Description of Event
1936 Konrad Zuse - Z1 Computer
First freely programmable computer.
1942
John Atanasoff & Clifford Berry
ABC Computer
Who was first in the computing biz is not always as easy as ABC.
1944
Howard Aiken & Grace Hopper
Harvard Mark I Computer
The Harvard Mark 1 computer.
1946
John Presper Eckert & John W. Mauchly
ENIAC 1 Computer
20,000 vacuum tubes later...
1948
Frederic Williams & Tom Kilburn
Manchester Baby Computer & The Williams Tube
Baby and the Williams Tube turn on the memories.
1947/48
John Bardeen, Walter Brattain & Wiliam Shockley
The Transistor
No, a transistor is not a computer, but this invention greatly affected the history of computers.
1951
John Presper Eckert & John W. Mauchly
UNIVAC Computer
First commercial computer & able to pick presidential winners.
1953
International Business Machines
IBM 701 EDPM Computer
IBM enters into 'The History of Computers'.
1954
John Backus & IBM
FORTRAN Computer Programming Language
The first successful high level programming language.
1955 (in use 1959)
Stanford Research Institute, Bank of America, and General Electric ERMA and MICR
The first bank industry computer - also MICR (magnetic ink character recognition) for reading checks.
1958
Jack Kilby & Robert Noyce
The Integrated Circuit
Otherwise known as 'The Chip'
1962
Steve Russell & MIT
Spacewar Computer Game
The first computer game invented.
1964
Douglas Engelbart
Computer Mouse & Windows
Nicknamed the mouse because the tail came out the end.
1969 ARPAnet The original Internet.
1970 Intel 1103 Computer Memory
The world's first available dynamic RAM chip.
1971 Alan Shugart &IBM
The "Floppy" Disk Nicknamed the "Floppy" for its flexibility.
1973
Robert Metcalfe & Xerox
The Ethernet Computer Networking
Networking.
1974/75 Scelbi & Mark-8 Altair & IBM 5100 Computers
The first consumer computers.
1976/77 Apple I, II & TRS-80 & Commodore Pet Computers
More first consumer computers.
1978
Dan Bricklin & Bob Frankston
VisiCalc Spreadsheet Software
Any product that pays for itself in two weeks is a surefire winner.
1979 Seymour Rubenstein & Rob Barnaby
WordStar Software
Word Processors.
1981
IBM
The IBM PC - Home Computer
From an "Acorn" grows a personal computer revolution
CFD Applications
Where is CFD used?
Aerospace
Automotive
Biomedical
Chemical Processing
HVAC (heating, ventilation, and
air conditioning)
Hydraulics
Marine
Oil & Gas
Power Generation
Sports
F18 Store Separation
Temperature and natural
convection currents in the eye
following laser heating.
Aerospace
Automotive
Biomedical
Where is CFD used?
Polymerization reactor vessel - prediction of
flow separation and residence time effects.
Streamlines for workstation ventilation
Where is CFD used?
Aerospacee
Automotive
Biomedical
Chemical
Processing
HVAC
Hydraulics
Marine
Oil & Gas
Power Generation
Sports
HVAC
Chemical Processing
Hydraulics
Where is CFD used?
Aerospace
Automotive
Biomedical
Chemical Processing
HVAC
Hydraulics
Marine
Oil & Gas
Power Generation
Sports Flow of
lubricating mud
over drill bit
Cooling towers flow
Marine
Oil & Gas
Sports
Power Generation
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°
Comparison of Computed Wake Vortex Evolution Flowfield (OVERFLOW Code) with
Experiment (“2-pair”)
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.
CFD Basics
CFD Process Model Equations
Discretization
Grid Generation
Boundary Conditions
Solve
Post-Processing
Uncertainty Assessment
In mathematics, partial differential equations (PDE) are a type of
differential equation, i.e., a relation involving an unknown function
(or functions) of several independent variables and their partial
derivatives with respect to those variables. Partial differential
equations are used to formulate, and thus aid the solution of,
problems involving functions of several variables; such as the
propagation of sound or heat, electrostatics, electrodynamics, fluid
flow, and elasticity. Seemingly distinct physical phenomena may
have identical mathematical formulations, and thus be governed by
the same underlying dynamic. They find their generalization in
Stochastic partial differential equations. Just as ordinary differential
equations often model dynamical systems, partial differential
equations often model multidimensional systems.
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 waves
Parabolic equations Information travels downstream/forward in time;
directions at finite speeds; the solution can be constructed using a
marching/time-stepping method
Elliptic equations Information propagates in all directions at infinite speed;
describe equilibrium phenomena (unsteady problems are never elliptic
Getting Started: classification of PDEs
Viscous Model
Boundary
Conditions
Initial
Conditions
Convergent
Limit
Contours
Precisions
(single/
double)
Numerical
Scheme
Vectors
Streamlines Verification
Select
Geometry
Geometry
Parameters
Physics Mesh Solve Post-
Processing
Compressible
ON/OFF
Flow
properties
Unstructured
(automatic/
manual)
Steady/
Unsteady
Forces Report (lift/drag, shear
stress, etc)
XY Plot
Domain Shape
and Size
Heat Transfer
ON/OFF
Structured
(automatic/
manual)
Iterations/
Steps
Validation
Reports
• 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
• 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).
2
2
2
2
2
2ˆ
z
u
y
u
x
u
x
p
z
uw
y
uv
x
uu
t
u
2
2
2
2
2
2ˆ
z
v
y
v
x
v
y
p
z
vw
y
vv
x
vu
t
v
0
z
w
y
v
x
u
t
RTp
L
v pp
Dt
DR
Dt
RDR
2
2
2
)(2
3
Convection Piezometric pressure gradient Viscous terms Local
acceleration
Continuity equation
Equation of state
Rayleigh Equation
2
2
2
2
2
2ˆ
z
w
y
w
x
w
z
p
z
ww
y
wv
x
wu
t
w
• 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, g, Gr, Ec)
• Free-surface flow (Fr) and surface tension (We)
• Chemical reactions and combustion (Pe, Da)
• etc…
• 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
•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 airfoil o
r
x Axisymmetric
• 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, g, 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…
• 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
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
• 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 post-
processing
• 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 implicit method (e.g. Beam-Warming method)
• 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.
Explicit and implicit methods are approaches used in
numerical analysis for obtaining numerical solutions of
time-dependent ordinary and partial differential
equations, as is required in computer simulations of
physical processes.
Y(t+t) = F(Y(t)),
while for an implicit method one solves an equation
G(Y(t), Y(t+t))=0,
to find Y(t + Δt).
It is clear that implicit methods require an extra computation (solving the above equation), and they can
be much harder to implement. Implicit methods are used because many problems arising in practice are
stiff, for which the use of an explicit method requires impractically small time steps Δt to keep the error
in the result bounded (see numerical stability). For such problems, to achieve given accuracy, it takes
much less computational time to use an implicit method with larger time steps, even taking into account
that one needs to solve an equation of the form (1) at each time step. That said, whether one should use
an explicit or implicit method depends upon the problem to be solved.
Explicit methods calculate the state of a system at a later time from the state of the system at the current
time, while implicit methods find a solution by solving an equation involving both the current state of
the system and the later one. Mathematically, if Y(t) is the current system state and Y(t + Δt) is the state
at the later time (Δt is a small time step), then, for an explicit method
Consider the ordinary differential equation
2 y 0,dy
y adt
1- The forward Euler method
21k kk
k
y ydyy
dt t
yields
2
1k k ky y t y Explicit!
Consider the ordinary differential equation
2 y 0,dy
y adt
2- The backward Euler method
211
k kk
k
y ydyy
dt t
yields
2
1 1k k ky t y y Implicit!
2- The backward Euler method
211
k kk
k
y ydyy
dt t
2
1 1k k ky t y y
This is a quadratic equation, having one negative and one positive root. The positive root is
picked because in the original equation the initial condition is positive, and then y at the next
time step is given by
n the vast majority of cases, the equation to be solved when using an implicit scheme is much
more complicated than a quadratic equation, and no exact solution exists. Then one uses root-
finding algorithms, such as Newton's method.
1
1 1 4
2
k
k
t yy
t
Finite difference methods
Analysis of trunctation errors
Approximation of second-order derivatives
Approximation of mixed derivatives
High-order approximations
Finite Volume Method Discretization
In the finite-volume approach, the integral form of the conservation equations are applied to
the control volume defined by a cell to get the discrete equations for the cell. The integral
form of the continuity equation for steady, incompressible flow is
0V ndS The integration is over the surface S of the control volume and is the outward normal at
the surface. Physically, this equation means that the net volume flow into the control volume
is zero. Consider the rectangular cell shown below.
n
The velocity at face i is taken to be i i iV u i v j
Applying the mass conservation equation to the control
volume defined by the cell gives
1 2 3 4 0u y v x u y v x
0
y
v
x
u
2
2
y
u
e
p
xy
uv
x
uu
• 2D incompressible laminar flow boundary layer
m=0 m=1
L-1 L
y
x
m=MM m=MM+1
(L,m-1)
(L,m)
(L,m+1)
(L-1,m)
1l
l lmm m
uuu u u
x x
1
ll lmm m
vuv u u
y y
1
ll lmm m
vu u
y
FD Sign( )<0 l
mv
l
mvBD Sign( )>0
2
1 12 22l l l
m m m
uu u u
y y
2nd order central difference
i.e., theoretical order of accuracy
Pkest= 2.
1st order upwind scheme, i.e., theoretical order of accuracy Pkest= 1
1 12 2 2
1
2
1
l l ll l l lm m mm m m m
FDu v vy
v u FD u BD ux y y y y y
BDy
1 ( / )l
l lmm m
uu p e
x x
B2 B3 B1
B4 1
1 1 2 3 1 4 /ll l l l
m m m m mB u B u B u B u p e
x
1
4 1
12 3 1
1 2 3
1 2 3
1 2 1
4
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
l
l
l
l lmm l
mm
mm
pB u
B B x eu
B B B
B B B
B B u pB u
x e
Solve it using
Thomas algorithm
To be stable, Matrix has to be
Diagonally dominant.
In numerical linear algebra, the tridiagonal matrix algorithm
(TDMA), also known as the Thomas algorithm (named after
Llewellyn Thomas), is a simplified form of Gaussian elimination
that can be used to solve tridiagonal systems of equations. A
tridiagonal system for n unknowns may be written as
1 1 1i i i i i ia x b x c x d 1 1 1 1
2 2 2 2 2
3 3 3 3
1
0
0
n
n n n n n
b c x d
a b c x d
a b x d
c
a b c x d
For such systems, the solution can be obtained in O(n) operations instead of O(n3)
required by Gaussian elimination.
Solver: Thomas Algorithm
The first step consists of modifying
the coefficients as follows, denoting
the new modified coefficients with
primes:
1
1'
'
1
1
1'
'
1
'
1
; 1
; 2,3,..., 1
; 1
; 2,3,..., 1
i
i
i i i
i
i i i
i i i
ci
bc
ci n
b c a
and
di
bd
d d ai n
b c a
The solution is then obtained by back substitution:
'
n nx d
' '
1 ; 1, 2,..., 2,1i i i nx d c x i n n
MATLAB Implementation
• 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
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 grids
Unstructured grids
Hybrid grids: some portions of flow field are
structured (viscous regions) and others are
unstructured
Overset (Chimera) grids
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 blockwise
Unstructured 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
2D Cell Types
3D Cell Types
Submarine
Moving Control Surfaces
Artificial Heart Chamber
Surface Ship Appendages
Branches in Human Lung
Structured-Unstructured Nozzle Grid
Unstructured surface mesh for external aerodynamics – PT cruiser – 12 millions cells
An example of a
course-fine
distribution in a
multi-zonal grid.
An example of chimera grid
An example of a multi-zonal grid
The most important advantage of using the chimera scheme of oversetting grids is to reduce substantially the time and effort to generate a
grid. This is especially true for three-dimensional configurations with increasing geometric complexity, such as a fully appended ship. The
time and effort required to plan and generate complicated grids using solely multi-zonal grid generation techniques quickly becomes
prohibitive.
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 ! 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 UO x
x x
U U U UO x
x x
Run CFD code on computer
2D and small 3D simulations
can be run on desktop
computers (e.g., Fluent)
Unsteady 3D simulations still
require large parallel
computers
Monitor Residuals
Defined two ways
Change in flow variables
between iterations
Error in discrete algebraic
equation
CBU-115 separation from F-16 GBU-38 separation from B1B
JASSM jettison
Wake visualization of HART II rotor with blade-vortex interaction.
Predicted ground plane acoustic sound pressure levels.
Sikorsky UH-60A rotor-hub-fuselage interaction.
CFD/CSD coupled solution.
Steady Maneuver: Horizontal Turn
Process of estimating errors due to
numerics and modeling Numerical errors
Iterative non-convergence: monitor residuals
Spatial errors: grid studies and Richardson extrapolation
Temporal 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.
For fluid mechanics, many problems not
adequately described by Navier-Stokes
equations or are beyond current generation
computers. Turbulence
Multi-phase physics: solid-gas (pollution, soot), liquid-gas
(bubbles, cavitation); solid-liquid (sediment transport)
Combustion and chemical reactions
Non-Newtonian fluids (blood; polymers)
Similar modeling challenges in other branches of
engineering and the sciences
Because of limitations, need for experimental research is great
However, focus has changed From
Research based solely upon experimental observations
Build and test (although this is still done)
To High-fidelity measurements in support of validation and building new computational models.
Currently, the best approach to solving engineering problems often uses simulation and experimentation
Capabilities of Current Technology
Complex real-world problems solved using Scientific
Computing
Commercial software available for certain problems
Simulation-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-fluidics
Very large : cosmology (study of the origin, current state, and
future of our Universe)
Expensive : engineering prototypes (ships, aircraft)
Dangerous : explosions, fires
Overview on
Mesh Technology
Conformal Mapping
Transfinite Interpolation
Solving PDEs
Elliptic
Parabolic/Hyperbolic
Conformal Mapping
Conformal Mapping Transformations
Conformal Mapping – Schwarz Christoffel
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
• Orthogonally not guaranteed
• Very fast
• Quite General
• Grid quality not always assured
Algebraic Mappings – 1D Interpolants
Algebraic Mappings – Transfinite Interpolation
Algebraic Mappings – Transfinite Interpolation
Algebraic Mappings – Example
Algebraic Mappings – Example
Algebraic Mappings – Example
Numerically generated airfoil transformation [(x, y) ↔ (ξ, η)] showing a “C” grid topology
Algebraic Mappings – Example
Numerically generated wing transformation [(x, y, z) ↔ (ξ, η, ζ)] showing a “O” grid topology
PDE Grid Generation
Full-potential Equation Remapped
0t
U V W
J J J J
U, V, and W are the contravariant velocities and are given by
1 4 5
4 2 6
5 6 3
U A A A
V A A A
W A A A
with
2 2 2
1
2 2 2
2
2 2 2
3
4
5
6
x y z
x y z
x y z
x x y y z z
x x y y z z
x x y y z z
A
A
A
A
A
A
CFD Perspective on Meshing Technology
CFD initiated in structured grid context
Transfinite interpolation
Elliptic grid generation
Hyperbolic grid generation
Smooth, orthogonal structured grids
Relatively simple geometries
Unstructured meshes initially confined to FE
community
CFD Discretizations based on directional splitting
Line relaxation (ADI) solvers
Structured Multigrid solvers
Sparse matrix methods not competitive
Memory limitations
Non-linear nature of problems
Method of choice for many commercial CFD vendors
Fluent, StarCD, CFD++, …
Advantages Complex geometries
Adaptivity
Parallelizability
Enabling factors Maturing grid generation technology
Better Discretizations and solvers
Isotropic tetrahedral grid generation Delaunay point insertion algorithms
Surface recovery
Advancing front techniques
Octree methods
Mature technology Numerous available commercial packages
Remaining issues Grid quality
Robustness
Links to CAD
Anisotropic unstructured grid generation
External aerodynamics
Boundary layers, wakes: O(10**4)
Mapped Delaunay triangulations
Min-max triangulations
Hybrid methods
Advancing layers
Mixed prismatic – tetrahedral meshes
Solutions to three-dimensional convection-dominated problems often contain lower
dimensional features, the resolution of which is critical to the accuracy of the method, eg. a
plane shock wave, or a boundary layer. Resolving this feature numerically is computationally
expensive if an isotropic structured or unstructured mesh is used. It has become common to
use anisotropic meshes for these problems, however a rigorous justification of the appropriate
mesh, or a reliable measure of the numerical error on such a mesh is not yet apparent. For
many problems the presence of singular features such as shock-waves, boundary layers and
cracks requires a mesh that is highly stretched in a particular direction. Although there is still
some controversy about the stability and accuracy of finite element computations with highly
stretched tetrahedra, we believe that tetrahedral meshes are very satisfactory for many
applications exhibiting anisotropy provided there are no large angles. To achieve a good
quality stretched mesh that avoids large dihedral angles, we combine a point placement
strategy in physical space with an affine transformation of the metric used for the Delaunay
in-sphere test. The placement of points in physical space at locations normal to the boundary
of the singularity allows one to maintain precise control over the position of the mesh nodes.
On the other hand, the use of a modified metric in the Delaunay test ensures good connectivity
with a layered arrangement of tetrahedra in the region of anisotropy.
Hybrid methods
Semi-structured
nature
Less mature: issues
Concave regions
Neighboring
boundaries
Conflicting resolution
Conflicting
Stretchings VGRIDns Advancing Layers
NSC2KE solver
Bamg mesh
generator and mesh
adapter
NSC2KE solver
Bamg mesh
generator and mesh
adapter
NACA0012 - Freestream Mach number = 1.4
Evolved to Sophisticated Multiblock and Overlapping
Structured Grid Techniques for Complex Geometries
Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)
Edge-based data structure
Building block for all element types
Reduces memory requirements
Minimizes indirect addressing / gather-
scatter
Graph of grid = Discretization stencil
Implications for solvers, Partitioners
Multigrid solvers Multigrid techniques enable optimal O(N) solution complexity
Based on sequence of coarse and fine meshes
Originally developed for structured grids
Agglomeration Multigrid solvers for unstructured
meshes
Coarse level meshes constructed by agglomerating fine
grid cells/equations
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)
Line solvers for Anisotropic problems Lines constructed in mesh using weighted graph algorithm
Strong connections assigned large graph weight
(Block) Tridiagonal line solver similar to structured grids
Graph-based Partitioners for parallel load
balancing
Metis, Chaco, Jostle
Edge-data structure graph of grid
Agglomeration Multigrid levels = graphs
Excellent load balancing up to 1000’s of
processors
Homogeneous data-structures
(Versus multi-block / overlapping structured grids)
Complex geometry
Wing-body, slat, double slotted flaps, cutouts
Experimental data from Langley 14x22ft wind
tunnel
Mach = 0.2, Reynolds=1.6 million
Range of incidences: -4o to 24o
Typical conditions Wall
No-slip (u = v = w = 0)
Slip (tangential stress = 0, normal velocity = 0)
With specified suction or blowing
With specified temperature or heat flux
Inflow
Outflow
Interface Condition, e.g., Air-water free surface
Symmetry and Periodicity
Usually set through the use of a graphical user interface (GUI) – click & set
Combined advancing layers- advancing front Advancing layers: thin elements at walls
Advancing front: isotropic elements elsewhere
Automatic switching from AL to AF based on: Cell aspect ratio
Proximity of boundaries of other fronts
Variable height for advancing layers
Background Cartesian grid for smooth spacing control
Spanwise stretching Factor of 3 reduction in grid size
3.1 million vertices, 18.2 million tets, 115,489 surface pts
Normal spacing: 1.35E-06 chords, growth factor=1.3
Combine 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 elements
Pyramids required as transitional elements
Initial mesh: 18.2M Tetrahedra
Merged mesh: 3.9M prisms, 6.6M Tets, 47K
pyramids
64% of Tetrahedra merged
High-resolution meshes require large parallel
machines
Parallel mesh generation difficult
Complicated logic
Access to commercial preprocessing, CAD tools
Current approach
Generate coarse (O(10**6) vertices on workstation
Refine on supercomputer
Refinement achieved by element subdivision
Global refinement: 8:1 increase in resolution
In-Situ approach obviates large file transfers
Initial mesh: 3.1 million vertices
3.9M prisms, 6.6M Tets, 47K pyramids
Refined mesh: 24.7 million vertices
31M prisms, 53M Tets, 281K pyramids
Refinement operation: 10 Gbytes, 30 minutes
sequentially
NSU3D Unstructured Mesh Navier-Stokes Solver
Mixed element grids
Tetrahedra, prisms, pyramids, hexahedra
Edge data-structure
Line solver in BL regions near walls
Agglomeration Multigrid acceleration
Newton Krylov (GMRES) acceleration option
Spalart-Allmaras 1 equation turbulence model
Domain decomposition with OpenMP/MPI
communication
OpenMP on shared memory architectures
MPI on distributed memory architectures
Hybrid capability for clusters of SMPs
Weighted graph partitioning (Metis)
(Chaco)
Coarse and fine MG levels partitioned
independently
Mach=0.2, α=10o, Re=1.6M
Good drag prediction
Discrepancies near stall
Mesh independent property of Multigrid
GMRES effective but requires extra memory
Good overall Multigrid scalability Increased communication due to coarse grid levels
Single grid solution impractical (>100 times slower)
1 hour soution time on 1450 PEs
AIAA Drag Prediction Workshop (2001)
Transonic wing-body configuration
Typical cases required for design
study
Matrix of mach and CL values
Grid resolution study
Follow on with engine effects (2003)
Baseline grid: 1.6 million points Full drag polars for Mach=0.5,0.6,0.7,0.75,0.76,0.77,0.78,0.8
Total = 72 cases
Medium grid: 3 million points Full drag polar for each mach number
Total = 48 cases
Fine grid: 13 million points Drag polar at mach=0.75
Total = 7 cases
Mach=0.75, CL=0.6, Re=3M
2.5 hours on 16 Pentium IV 1.7GHz
Grid resolution study
Good comparison with experimental data
Adaptive mesh refinement
Moving geometry and mesh motion
Moving geometry and overlapping meshes
Requirements for gradient-based design
Implications for higher-order
Discretizations
Potential for large savings through
optimized mesh resolution
Well suited for problems with large range of
scales
Possibility of error estimation / control
Requires tight CAD coupling (surface pts)
Mechanics of mesh adaptation
Refinement criteria and error estimation
Various well know isotropic mesh methods
Mesh movement
Spring analogy
Linear elasticity
Local Remeshing
Delaunay point insertion/Retriangulation
Edge-face swapping
Element subdivision
Mixed elements (non-simplicial)
Anisotropic subdivision required in transition regions
Large potential savings for 1 or 2D
features
Directional subdivision
Assumes element faces to line up with flow
features
Combine with mesh motion
Mapping techniques
Hessian based
Grid quality
Weakest link of adaptive meshing methods
Obvious for strong features
Difficult for non-local (ie. Convective) features
eg. Wakes
Analysis assumes in asymptotic error convergence
region
Gradient based criteria
Empirical criteria
Effect of variable discretization error in design
studies, parameter sweeps
Compute sensitivity of global cost function to local spatial grid resolution
Key on important output, ignore other features Error in engineering output, not discretization error
e.g. Lift, drag, or sonic boom …
Captures non-local behavior of error Global effect of local resolution
Requires solution of adjoint equations Adjoint techniques used for design optimization
Reproduced from
Venditti and
Darmofal (MIT,
2002)
Reproduced from Venditti and
Darmofal (MIT, 2002)
Reproduced
from Venditti
and Darmofal
(MIT, 2002)
Overlapping Unstructured Meshes
Alternative to moving mesh for large scale relative geometry motion
Multiple overlapping meshes treated as single data-structure
Dynamic determination of active/inactive/ghost cells
Advantages for parallel computing Obviates dynamic load rebalancing required with mesh motion techniques
Intergrid communication must be dynamically recomputed and rebalanced
Concept of Rendez-vous grid (Plimpton and Hendrickson)
Overlapping Unstructured Meshes
Simple 2D transient example
Minimize Cost Function F with respect to design
variables v, subject to constraint R(w) = 0
F = drag, weight, cost
v = shape parameters
w = Flow variables
R(w) = 0 Governing Flow Equations
Gradient Based Methods approach minimum
along direction :
v
F
Grid Related Issues for Gradient-based Design
Parametrization of CAD surfaces
Consistency across disciplines
eg. CFD, structures,…
Surface grid motion
Interior grid motion
Grid sensitivities
Automation / Parallelization
23,555 curves and surfaces c/o J. Samareh, NASA Langley
c/o J. Samareh, NASA
Langley
• Manual differentiation
• Automatic differentiation tools (e.g., ADIFOR and ADIC)
• Complex variables
• Finite-difference approximations
analysis code
field grid generator
geometry modeler (CAD)
surface grid generator
Grid
v
Grid
Ge
Geometry
vGrid Gr m yid o etr
f
f s
sFx x x
F
v design variables
(e.g., span, camber)
objective function
(e.g., Stress, CD)
Finite-Difference Approximation Error for Sensitivity Derivatives
Parameterized
HSCT Model
c/o J. Samareh, NASA Langley
Grid Sensitivities
Ideally should be available from grid/cad software
Analytical formulation most desirable
Burden on grid / CAD software
Discontinous operations present extra challenges Face-edge swapping
Point addition / removal
Mesh regeneration
v
Geometry
Geometry
Grid
Grid
Grid
v
Grid
xx
s
s
ff
Uniform X2 refinement of 3D mesh:
Work increase = factor of 8
2nd order accurate method: accuracy increase = 4
4th order accurate method: accuracy increase = 16
For smooth solutions
Potential for large efficiency gains
Spectral element methods
Discontinuous Galerkin (DG)
Streamwise Upwind Petrov Galerkin (SUPG)
Transfers burden from grid generation to Discretization
J. Hesthaven and T.
Warburton
(Brown University)
Require more complete surface definition
Curved surface elements
Additional element points
Surface definition (for high p)
Adaptive meshing (h-ref) yields constant factor
improvement
After error equidistribution, no further benefit
Order refinement (p-ref) yields asymptotic
improvement
Only for smooth functions
Ineffective for inadequate h-resolution of feature
Cannot treat shocks
H-P refinement optimal (exponential convergence)
Requires accurate CAD surface representation
197
Modeling Turbulent Flows
Unsteady, aperiodic motion in which all three velocity components
fluctuate mixing matter, momentum, and energy.
Decompose velocity into mean and fluctuating parts:
Ui(t) Ui + ui(t)
Similar fluctuations for pressure, temperature, and species
concentration values.
What is Turbulence?
Time
U i (t)
Ui
ui(t)
Why Model Turbulence?
Direct numerical simulation of governing equations is only possible for
simple low-Re flows.
Instead, we solve Reynolds Averaged Navier-Stokes (RANS) equations:
where (Reynolds stresses)
(steady, incompressible flow w/o body forces)
jiij uuR
j
ij
jj
i
ik
ik
x
R
xx
U
x
p
x
UU
2
Why Model Turbulence?
Direct numerical simulation of governing equations is only possible for
simple low-Re flows.
Instead, we solve Reynolds Averaged Navier-Stokes (RANS) equations:
where (Reynolds stresses)
j
ij
jj
i
ik
ik
x
R
xx
U
x
p
x
UU
2
(steady, incompressible flow w/o body forces)
jiij uuR
The left hand side of this equation represents the change in mean momentum of fluid element
owing to the unsteadiness in the mean flow and the convection by the mean flow. This change
is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the
viscous stresses, and apparent stress owing to the fluctuating velocity field,
generally referred to as the Reynolds stress. i ju u
Is the Flow Turbulent?
External Flows
Internal Flows
Natural Convection
5105xRe along a surface
around an obstacle
where
ULReL 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
3
PrTLg
GrRa x
20,000DRe
TTT s
Ts= temperature of the wall
T∞= fluid temperature far from the surface of the object
Grashof Prandtl
How Complex is the Flow?
Extra strain rates
Streamline curvature
Lateral divergence
Acceleration or deceleration
Swirl
Recirculation (or separation)
Secondary flow
3D perturbations
Transpiration (blowing/suction)
Free-stream turbulence
Interacting shear layers
Choices to be Made
Turbulence Model
&
Near-Wall Treatment
Flow
Physics
Accuracy
Required
Computational
Resources
Turnaround
Time
Constraints
Computational
Grid
Zero-Equation Models
One-Equation Models
Spalart-Allmaras
Two-Equation Models
Standard k-e
RNG k-e
Realizable k-e
Reynolds-Stress Model
Large-Eddy Simulation
Direct Numerical Simulation
Turbulence Modeling Approaches
Include
More
Physics
Increase
Computational
Cost
Per Iteration Available
in FLUENT
RANS-based
models
RANS equations require closure for Reynolds stresses.
Turbulent viscosity is indirectly solved for from single transport
equation of modified viscosity for One-Equation model.
For Two-Equation models, turbulent viscosity correlated with turbulent
kinetic energy (TKE) and the dissipation rate of TKE.
Transport equations for turbulent kinetic energy and dissipation rate are
solved so that turbulent viscosity can be computed for RANS equations.
Turbulent
Kinetic Energy: Dissipation Rate of
Turbulent Kinetic Energy:
e
2kCt Turbulent Viscosity:
Boussinesq Hypothesis: (isotropic stresses)
i
j
j
itijjiij
x
U
x
UkuuR
3
2
2/iiuuk
i
j
j
i
j
i
x
u
x
u
x
ue
Turbulent viscosity is determined from:
is determined from the modified viscosity transport equation:
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
Dt
Dww
j
b
jj
b
3
1
3
3
/~/~
~
ct
~
Generation Diffusion
Destruction
One-Equation Model: Spalart-Allmaras
Designed specifically for aerospace applications involving wall-
bounded flows.
Boundary layers with adverse pressure gradients
turbomachinery
Can use coarse or fine mesh at wall
Designed to be used with fine mesh as a “low-Re” model, i.e., throughout
the viscous-affected region.
Sufficiently robust for relatively crude simulations on coarse meshes.
Two Equation Model: Standard k-e Model
Turbulent Kinetic Energy
Dissipation Rate
eee 21, ,, CCk are empirical constants
(equations written for steady, incompressible flow w/o body forces)
Convection Generation Diffusion
Destruction
e
i
kt
ii
j
j
i
i
j
t
i
ix
k
xx
U
x
U
x
U
x
kU )(
Destruction Convection Generation Diffusion
kC
xxx
U
x
U
x
U
kC
xU
i
t
ii
j
j
i
i
j
t
i
i
2
21 )(e
e
ee
eee
Two Equation Model: Standard k-e Model
“Baseline model” (Two-equation)
Most widely used model in industry
Strength and weaknesses well documented
Semi-empirical
k equation derived by subtracting the instantaneous mechanical energy
equation from its time-averaged value
e equation formed from physical reasoning
Valid only for fully turbulent flows
Reasonable accuracy for wide range of turbulent flows
industrial flows
heat transfer
Distinctions from Standard k-e model:
Alternative formulation for turbulent viscosity
where is now variable
(A0, As, and U* are functions of velocity gradients)
Ensures positivity of normal stresses;
Ensures Schwarz’s inequality;
New transport equation for dissipation rate, e:
e
2kCt
e
kU
AA
C
so
*
1
0u2
i
2
j
2
i
2
ji u u)uu(
b
j
t
j
Gck
ck
cScxxDt
Dee
e
e
e
ee
e
e 31
2
21
Generation Diffusion Destruction Buoyancy
Shares the same turbulent kinetic energy equation as Standard k-e
Superior performance for flows involving:
planar and round jets
boundary layers under strong adverse pressure gradients, separation
rotation, recirculation
strong streamline curvature
Two Equation Model: Realizable k-e
Two Equation Model: RNG k-e
Turbulent Kinetic Energy
Dissipation Rate
Convection Diffusion
Dissipation
e
i
k
i
t
i
ix
k
xS
x
kU eff
2
Generation
j
i
i
j
ijijijx
U
x
USSSS
2
1,2
where
are derived using RNG theory eee 21, ,, CCk
(equations written for steady, incompressible flow w/o body forces)
Additional term
related to mean strain
& turbulence quantities Convection Generation Diffusion Destruction
Rk
Cxx
Sk
Cx
Uii
t
i
i
2
2eff
2
1
e
e
ee eee
Two Equation Model: RNG k-e
k-e equations are derived from the application of a rigorous statistical
technique (Renormalization Group Method) to the instantaneous Navier-
Stokes equations.
Similar in form to the standard k-e equations but includes:
additional term in e equation that improves analysis of rapidly strained flows
the effect of swirl on turbulence
analytical formula for turbulent Prandtl number
differential formula for effective viscosity
Improved predictions for:
high streamline curvature and strain rate
transitional flows
wall heat and mass transfer
k
ijk
ijijij
k
ji
kx
JP
x
uuU
e
Generation k
ikj
k
j
kiijx
Uuu
x
UuuP
i
j
j
iij
x
u
x
up
k
j
k
iij
x
u
x
u
e 2
Pressure-Strain
Redistribution
Dissipation
Turbulent
Diffusion
(modeled)
(related to e)
(modeled)
(computed)
(equations written for steady, incompressible flow w/o body forces)
Reynolds Stress
Transport Eqns.
Pressure/velocity
fluctuations
Turbulent
transport
)( jikijkkjiijk uupuuuJ
Reynolds Stress Model
RSM closes the Reynolds-Averaged Navier-Stokes equations by
solving additional transport equations for the Reynolds stresses.
Transport equations derived by Reynolds averaging the product of the
momentum equations with a fluctuating property
Closure also requires one equation for turbulent dissipation
Isotropic eddy viscosity assumption is avoided
Resulting equations contain terms that need to be modeled.
RSM has high potential for accurately predicting complex flows.
Accounts for streamline curvature, swirl, rotation and high strain rates
Cyclone flows, swirling combustor flows
Rotating flow passages, secondary flows
Large Eddy Simulation
Large eddies:
Mainly responsible for transport of momentum, energy, and other scalars,
directly affecting the mean fields.
Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly
dependent on flow configuration, boundary conditions, and flow parameters.
Small eddies:
Tend to be more isotropic and less flow-dependent
More likely to be easier to model than large eddies.
LES directly computes (resolves) large eddies and models only small
eddies (Subgrid-Scale Modeling).
Large computational effort
Number of grid points, NLES
Unsteady calculation
2Reu
Model Strengths Weaknesses
Spalart-
Allmaras
Economical (1-eq.); good track record
for mildly complex B.L. type of flows
Not very widely tested yet; lack of
submodels (e.g. combustion,
buoyancy)
STD k-eRobust, economical, reasonably
accurate; long accumulated
performance data
Mediocre results for complex flows
involving severe pressure gradients,
strong streamline curvature, swirl
and rotation
RNG k-e
Good for moderately complex
behavior like jet impingement,
separating flows, swirling flows, and
secondary flows
Subjected to limitations due to
isotropic eddy viscosity
assumption
Realizable
k-eOffers largely the same benefits as
RNG; resolves round-jet anomaly
Subjected to limitations due to
isotropic eddy viscosity
assumption
Reynolds
Stress
Model
Physically most complete model
(history, transport, and anisotropy of
turbulent stresses are all accounted
for)
Requires more cpu effort (2-3x);
tightly coupled momentum and
turbulence equations
Near-Wall Treatments
Most k-e and RSM turbulence
models will not predict correct
near-wall behavior if integrated
down to the wall.
Special near-wall treatment is
required.
Standard wall functions
Nonequilibrium wall functions
Two-layer zonal model
Boundary layer structure
Standard Wall Functions
/
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PP kCUU
)(ln1
Pr
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**
**
Tt
T
yyPEy
yyy
T
PP ykCy
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kCcTTT
PpPw
2/14/1)(*
Mean Velocity
Temperature
where
where
and P is a function of the fluid
and turbulent Prandtl numbers.
thermal sublayer thickness
EyU ln1
Nonequilibrium Wall Functions
Log-law is sensitized to pressure gradient for
better prediction of adverse pressure gradient
flows and separation.
Relaxed local equilibrium assumptions for
TKE in wall-neighboring cells.
Thermal law-of-wall unchanged
ykCE
kCU
w
2/14/12/14/1
ln1
/
~
y
k
yy
y
y
k
y
dx
dpUU vv
v
v
2
2/12/1ln
21~
where
Two-Layer Zonal Model
Used for low-Re flows or
flows with complex near-wall
phenomena.
Zones distinguished by a wall-
distance-based turbulent
Reynolds number
High-Re k-e models are used in the turbulent core region.
Only k equation is solved in the viscosity-affected region.
e is computed from the correlation for length scale.
Zoning is dynamic and solution adaptive.
yk
Rey
200yRe
200yRe
Comparison of Near Wall Treatments
Strengths Weaknesses
Standard wall
Functions
Robust, economical,
reasonably accurate
Empirically based on simple
high-Re flows; poor for low-Re
effects, massive transpiration,
p, strong body forces, highly
3D flows
Nonequilibrium
wall functions
Accounts for p effects,
allows nonequilibrium:
-separation
-reattachment
-impingement
Poor for low-Re effects, massive
transpiration, severe p, strong
body forces, highly 3D flows
Two-layer zonal
model
Does not rely on law-of-the-
wall, good for complex
flows, especially applicable
to low-Re flows
Requires finer mesh resolution
and therefore larger cpu and
memory resources
Wall Function
Approach
Two-Layer Zonal
Model Approach
First grid point in log-law region
At least ten points in the BL.
Better to use stretched quad/hex
cells for economy.
First grid point at y+ 1.
At least ten grid points within
buffer & sublayers.
Better to use stretched quad/hex
cells for economy.
50050 y
Estimate the skin friction coefficient based on correlations either
approximate or empirical:
Flat Plate-
Pipe Flow-
Compute the friction velocity:
Back out required distance from wall:
Wall functions • Two-layer model
Use post-processing to confirm near-wall mesh resolution
2.0Re0359.02/
Lfc
2.0Re039.02/
Dfc
2// few cUu
y1 = 50/u y1 = / u
Setting Boundary Conditions
Characterize turbulence at inlets & outlets (potential backflow)
k-e models require k and e
Reynolds stress model requires Rij and e
Several options allow input using more familiar parameters
Turbulence intensity and length scale
length scale is related to size of large eddies that contain most of energy.
For boundary layer flows: l 0.499
For flows downstream of grids /perforated plates: l opening size
Turbulence intensity and hydraulic diameter
Ideally suited for duct and pipe flows
Turbulence intensity and turbulent viscosity ratio
For external flows:
Input of k and e explicitly allowed (non-uniform profiles possible).
10/1 t
Example: Channel Flow with Conjugate Heat Transfer
adiabatic wall
cold air
V = 50 fpm
T = 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
Check if turbulent ReDh= 5,980
Developing turbulent flow at relatively low Reynolds number and
BLs on walls will give pressure gradient use RNG k-e with
nonequilibrium wall functions.
Develop strategy for the grid
Simple geometry quadrilateral cells
Expect large gradients in normal direction to horizontal walls
fine mesh near walls with first cell in log-law region.
Vary streamwise grid spacing so that BL growth is captured.
Use solution-based grid adaption to further resolve temperature
gradients.
Velocity
contours
Temperature
contours
BLs on upper & lower surfaces accelerate the core flow
Important that thermal BL was accurately resolved as well
P
Example: Flow Around a Cylinder
wall
wall
1 ft
2 ft
2 ft
air
V = 4 fps
Compute drag coefficient of the cylinder
5 ft 14.5 ft
Check if turbulent ReD = 24,600
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-e with 2-layer zonal model.
Develop strategy for the grid
Simple geometry & BLs quadrilateral cells.
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 eff = + t
CD = 0.53 Strouhal Number (St) = 0.297
U
DSt
where
Summary: Turbulence Modeling Guidelines
Successful turbulence modeling requires engineering judgement of:
Flow physics
Computer resources available
Project requirements
Accuracy
Turnaround time
Turbulence models & near-wall treatments that are available
Begin with standard k-e and change to RNG or Realizable k-e if
needed.
Use RSM for highly swirling flows.
Use wall functions unless low-Re flow and/or complex near-wall
physics are present.