Computational Fluids Mechanics
Part 1： Numerical methods
Part 2： Turbulence models
Part 3： Practice of numerical simulation
N.Oshima
A5-32
M.Tsubokura A5-33

oshima@eng.hokudai.ac.jp
mtsubo@eng.hokudai.ac.jp

Division of Mechanical and Space Engineering
Computational Fluid Mechanics Laboratory
http://www.eng.hokudai.ac.jp/labo/fluid/index_e.html

Computational Fluid Mechanics

Part1： Numerical Methods
Objective：

Numerical methods for fluids mechanics and their accuracy
1. Introduction
2. Numerical methods for fluid mechanics (1) ~ Basic equations
3. Numerical methods for fluid mechanics (2) ~Discretizing schemes
4. Numerical methods for fluid mechanics (3) ~Coupling algorism
5. Numerical methods for fluid mechanics (4) ~Additional problems
6. Reliability of numerical simulation
SummaryProf. Oshima

A5-32

oshima@eng.hokudai.ac.jp

Computational Fluid Mechanics

Part1： Numerical Methods (1a)1. Introduction– What do we want to know ?
– How to do numerical simulation ?
– Basic mathematics for Fluid Mechanics
2. Numerical methods for fluid mechanics (1) ~ Basic equations
3. Numerical methods for fluid mechanics (2) ~Discretizing schemes
4. Numerical methods for fluid mechanics (3) ~Coupling algorism
5. Numerical methods for fluid mechanics (4) ~Additional problems
6. Reliability of numerical simulation

What we want to know ？
• 3-D and unsteady features of fluid flow
– turbulence– vortex interactions• Realistic design of fluid machine– detail design (ex. parts design, flow conditions, etc.)– unsteady conditions (ex. cascade, sloshing, etc.)• Dynamic interaction in complex flows– reactive flows / multiphase flows– flow induced instability and acoustics problems

Instability of free shear layer
Transition
eddiesbladeLES result (Re=3900, case 2)
Instantaneous enstrophy distribution

Experiment (Corke Re=8000[13])Transition
eddiesStreaklines visualization(smoke wire)Vortex structures in wall share layer

Figure 4: Instantaneous eddy structures at the stagnation region of plane(left) and round (right) impinging jets when six waves are imposed at the inlet(left: lz/pD=1/6; right: lq/pD=1/6, iso-surface: Dp’=4.5)

Design of Turbomachine (1)- 3D Measuring of Torque Converter -shell sideLeadingedge

core sidelaser sheet

Input shaft rotating speed 400 [r/min]
Speed ratio
0.05 / 0.3 / 0.8

Y

Meridional flow direction(main flow direction)X

core sideZshell sidelaser sheet

Measurement Result
Speed Ration 0.05

Secondary Flow Velocity Vectors

Stream Linescore side

shell sidecore side* Starting points of stream lines
are located at 3 [mm] upstream
from leading edge of stator blade.

shell side

* These flow field planes are view
from upstream* 3 [ mm ] downstream from leadingedge of stator blade[m/s]core sideshell side

Meridional Flow Velocity Distribution[m/s]Design of Vehicle (1)Single Vehicle

LES Grid
1,000,000 points

stream line

Platoon runningUnsteady flow design of Formula car– LOLA B03/51– Formula NIPPON (2003~05)– Shape: tetrahedron– Element numbers: 117,060,909– Node numbers: 20,957,323

Design of Vehicle (2)Shinkan-sen (Bullet Train）
300km/h(2002)

Sources of Flow Noiseby Pantographs→ 360km/h(2012)

Prediction of Far-field Sound Spectra& Sound Source Distribution■ Div・(u x ω)

Far-Field Sound Pressure Spectra

MOVIE
Identified Instantaneous Sound Sources

Combustion flow in Engineering
Ex. Design of Gasturbine•

Technical Elements

compressorcombustor– spray– turbulence– heat transfer– material design– detail reaction– Ignition / extinction– radiation– noise / vibration

turbineSolution for Engineering problems•Lean premixed combustion•Lower emission / high efficiency

Strategy for turbulent combustion design
ex. combustion flow modeling
Structural design

Turbulent mixing

Structural analysis

Turbulence model

Flame propagation

Reaction model・Stress / vibration
・Thermal striping

・Swirl / Recirculation・Heat / Mass flux

・Premix / Diffusion flame
・Emission / Blow out

・Rarefied combustion
・Alternative fuels

Flame dynamics

Chemical reaction

Sound analysis
・Noise / instability

PDF approach／RANS103 Flamelet approach

1 Detail of chemical／LES

element reaction

Element reaction／DNSNOOHCO：：Exercise 1aConsider an application of computational fluid mechanics on to
your interest design or research problem, and tell its targeting
objective you want to know.

Computational Fluid Mechanics

Part1： Numerical Methods (1b)1. Introduction– What do we want to know ?
– How to do numerical simulation ?
– Basic mathematics for Fluid Mechanics
2. Numerical methods for fluid mechanics (1) ~ Basic equations
3. Numerical methods for fluid mechanics (2) ~Discretizing schemes
4. Numerical methods for fluid mechanics (3) ~Coupling algorism
5. Numerical methods for fluid mechanics (4) ~Additional problems
6. Reliability of numerical simulation

How to do numerical simulation?
• Basic elements of numerical simulation
physics

Governing eq.
ex. incompressible assumption
turbulence model ･･･

Boundary condition
ex. inlet / outlet
wall conditionmathematics

Numerical scheme
ex. upwind scheme
MAC algorithm

Numerical gridex. grid resolution
unstructured mesh

Get a numerical solution of differential equation••

Matrix solver ／ parallel computing
Post processing ／ visualization

globallocal

Governing eq.
Generalized convection-diffusion eq.
  u      j  St : variable＝velocity component, temperature, etc.
 u   : convection term，u : velociy vector   j : diffusion term，j : diffusion flux vector

S : production (dissipation) term gravity force，heat source, etc.

Ex. Thermal equation T(x,t)
TT   T  Q u ktx x  x  c pTvariable;   T ,,t
tConstant facorts; u, k , Q,  , c p

uT u  T    Tu  , Tu : convection fluxx
  T  j k , j  kT diffusion fluxx  x S  Q c p ,Q : heat source， : density，c p : capacity

Numerical solution of deferential eq.
Deferential eq. （+ B.C.）
Ex. dfdx

 2x→ Algebraic eq.
f x   x 2

x  0 , f 0  0

Approx. of equation

Approx. of solution

Equation system （incl. B.C.） → sequence of values
f 0   0f 1  f 0   2 f 0 f 2   f 1  2 f 1
 f 0, f 1, f 2, f 3, 0,1, 4,9, Ax  bNumerical solution of 2D/3D problems
Solution

Numerical Grid

Approximation

Conservation laws for
Mass, momentum &energy

Values on millions gird points should be solved！

Algorism for solutionEx. 1D incompressible flow
k 1
k 1u

ui

1i
Eq.1
0dx

kkuik 1  uik
uik1  2uik  uik1 1 pik 1  pik11
k ui  ui 1  ui

Eq.2
dtdx
dx 2
dx

k 1
k 1
k 1
kk
k
1
p

2p

p
D
F

Fi

1ii

1ii

1i
Eq.3

2

dx
dtdx

Time marching k → k+1
Algorism 1:

Solve u by eq.1 then solve p by eq.2 (successive)

Algorism 2:

Solve p by eq.3 then solve u by eq.2 (successive)

Algorism 3:

Solve u by eq.1 and solve p by eq.3 (parallel)Numerical solution of complex problems
Ex. Combustion flow by flamelet model

Distribution ofDistribution of

Levelset Scalar G

Mixture fraction

Feedback

Velocity field

Distribution of

Temperature &Density

Approximation & Error
Experiment
Real physics

Theoryapprox.

Numerical simulation

Differential eq.

approx.

Descretized eq.

Setting conditions

B.C.

Numerical grid

(Correct value)(Correct value)(Correct value)

proc.

proc.

Measured value

Solution

proc.

Num. solutionPrediction

Error ＝ Analyzable discrepancy from correct answer
due to approximation and procedures

Reliability of numerical solution
Problem: How to remove numerical errors from a correct solution ?
Mean velo.<U> experiment

Case1 2nd Central Scheme

WiggleCase2 QUICK

dwith numerical diffusion

・ Cubic obstacle in channel- Park ＆ Taniguchi (1995)- LES-WS (1995, Tegernsee)

Over-estimation ofReynolds stresses

Fluid vs. Solid
Fluid flow analysis

Solid structural analysis

・Analyze shape deformation rate・Arrangement of elements is
not fixed or unobservable・Analyze shape deformation
・Arrangement of elements is fixedand observable

・Unequilibrium physics･ Internal mechanism is statistical,
isotropic and homogeneous・Equilibrium physics
・Internal mechanism is structural,anisotropic and inhomogeneousExercise 1b
Solve the three algorisms for 1D incompressible flow under the following conditioni.c.

uik1  uik
D 
0dx

b.c.

u0  U ,ki

p1  p00dx

u N  u N 1
 0,dxpN  0at t=tk

at x=x0

at x=xN

Computational Fluid Mechanics

Part1： Numerical Methods (1c)1. Introduction– What do we want to know ?
– How to do numerical simulation ?
– Basic mathematics for Fluid Mechanics
2. Numerical methods for fluid mechanics (1) ~ Basic equations
3. Numerical methods for fluid mechanics (2) ~Discretizing schemes
4. Numerical methods for fluid mechanics (3) ~Coupling algorism
5. Numerical methods for fluid mechanics (4) ~Additional problems
6. Reliability of numerical simulation

Equations of flow phenomena
MassMomentumD   u         u Dt  ttDu t  u tp
  u   u t   Σ ft
Dt  tu t  u v v,Energy2Σ      u I  2S ,3De  e   u  e     q 
Dt  t
diffusion

pTDs
Dt

heatsource1 0 0 I  0 1 0
0 0 1

p

Dv
Dt

pressurework

Dv p Dp 2
    u  ,Dt  Dtv

1

Basic mathematics for fluid mechanics
Operations of vector and tensor variablesa b a   1  ， b   1  ， a t  a1
 a2 b2 a  b  a t b  a1

 a11 a12  a11t，Aa2  ， b t  b1 b2  ， A  a a21 a22  12b a2   1   a1b1  a2b2 b2 2a  a  a  at a

ab  A  b t A  b1 b2   11
a21a12  b1a11  b2 a21 b1 a12  b2 a22 a22 

a a bab t   1 b1 b2    1 1 a2  a2b1

a1b2 a2 b2 c  a b t  c1b ba t   1 a1b2 

a c2  1 b1 b2   c  ab t
a2 

 

aAb   11
a21a21 a22 a12  b1   a11b1  a12 b2 
a22  b2  a21b1  a22 b2 b a b a ta2    1 1 1 2   ab t
b2 a1 b2 a2  

Derivative operators for vector variables x u ，uv   y  ⇒ u f x u v
 ， f  x y
f y   x u x v x u t  uvu y v y  y 
A   x

   a11 a12   a11 a21
y  a21 a22   xyu   u  u  ut   u   v  u ， xy 

a12 a22 xy   uf  f   u  u  f

   u   u         u Convection of scalar u j    u j     u j 
x j
x j
x j

   uu t  u t   u   u    u t

u j ui   u j ui  ui  u j 
x j
x j
x j

Convection of vector uDerivative operators for vector variables(continued) x u ，uv  y  ⇒ u 

f x u v
 ， f  x y
f y  x u x v x 
u t  uvu y v y   y   f   x  f x   2 f  2 f2
f


22
y  f y  xy  u t   x  u x v x    2u  2u
 2  2y  u y v y   x yScalar Lapｌacian operator

 2v  2 v 2tux 2 y 2  

Vector Lapｌacian operator

Derivative operators for vector variables(continued)
u t  S  Ω u
 2 x1  ui u j  1  v uS 

2  x j xi  2  x y w u  x zu v
y xv
2yw vy zu w z x v w z y w 
2z 
 0

1  ui u j  1  v uΩ 

2  x j xi  2  x y w u  x zu v
y xu w z x v w  Voricity tensorz y  (anti-symmetric
0  component) 0  3
 20w v
y z3   2 
1 01  ， ω    u   2 3 
 10 Strain tensor(symmetric component)Other Eqs. of flow phenomena
Kinetic energy

1 2 1
1 2 2 2K  u  ukuk  u v w
222





u （Momentum eq.）tuupt u  u   u  u  u       u  ft
K   pu  p  u t u   K     u  Σ     u  fttui Σij ,   ui Σij  u  Σ  
x j
x jVorticity

ωu

u
pt

u

u




2

S


f
t


Exercise 1c
Complete the component formulation of vorticity equation

i


w i v i w i  ?
t
x
y
z

Confirm the following relation between Stress tensor formulations

 2   Σ        u I  2S  3
t   u   2u t3