Lecture Objectives:
Simple algorithm
Boundary conditions
Navier Stokes Equations
Continuity equation
v x v y v z


0
x
y z
This velocities that constitute advection coefficients: F=rV
Momentum x
v x
v x
v x
v x
p
2vx
 2vx
2vx
ρ(
 vx
 vy
 vz
)    μ 2  μ 2  μ 2  SM x
τ
x
y
z
x
x
y
z
Momentum y
v y
v y
v y
v y
2vy
2vy
2vy
p
ρ(
 vx
 vy
 vz
)    μ 2  μ 2  μ 2  SM y  ρ g (T  T )
τ
x
y
z
y
x
y
z
Momentum z
v z
v z
v z
v z
p
 2 vz
 2 vz
 2 vz
ρ(
 vx
 vy
 vz
)    μ 2  μ 2  μ 2  SM z
τ
x
y
z
z
x
y
z
Pressure is in momentum equations
which already has one unknown
In order to use linear equation solver we need to solve two problems:
1) find velocities that constitute in advection coefficients
2) link pressure field with continuity equation
Pressure and velocities in NS
equations
How to find velocities that constitute advection coefficients?
v x
v x
v x
v x
p
2vx
 2vx
2vx
ρ(
 vx
 vy
 vz
)    μ 2  μ 2  μ 2  SM x
τ
x
y
z
x
x
y
z
a P Vx,P  a E Vx, E  a WVx, W  aSVx,S  a NVx, N  a HVx, H  a LVx,P L  f
aP  6

x
2
x
Vx

,
a


W
x 2
x
x 2
................................
aE  

r
Vx  Vy  Vz
r
For the first step use Initial guess
And for next iterative steps use
the values from previous iteration
Pressure and velocities in NS
equations
How to link pressure field with continuity equation?
SIMPLE (Semi-Implicit Method for Pressure-Linked Equations ) algorithm
v x
v x
v x
v x
p
2vx
 2vx
2vx
ρ(
 vx
 vy
 vz
)   μ
 μ 2  μ 2  SM x
τ
x
y
z
x
x 2
y
z
x
W

p Pw – Pe (PW  PP )/2 – (PP  PE )/2 (PW – PE )/2



x
x
x
x
a P Vx P  a E Vx E  a W Vx W  a S Vx S  a N Vx N  a H Vx H  a L Vx L  f 
Aw
x
P
x
Ae
Aw=Ae=Aside
(PW – PE )/2
Aside
x
We have two additional equations for y and x directions
The momentum equations can be solved only when the pressure field is given or is
somehow estimated. Use * for estimated pressure and the corresponding velocities
E
SIMPLE algorithm
Guess pressure field: P*W, P*P, P*E, P*N , P*S, P*H, P*L
1) For this pressure field solve system of equations:
x:
a P Vx P  a E Vx E  a W Vx W  a S Vx S  a N Vx N  a H Vx H  a L Vx L  f 
y:
………………..
………………..
z:
Solution is:
(PW – PE )/2
Aside
x
V *x P , V *x E , V *x W , V *x S , V *x N , V *x H , V *x L
2) The pressure and velocity correction
P = P* + P’
V = V* + V’
P’ – pressure correction
V’ – velocity correction
For all nodes E,W,N,S,…
Substitute P=P* + P’ into momentum equations (simplify equation) and obtain
V’=f(P’)
V = V* + f(P’)
3) Substitute V = V* + f(P’) into continuity equation solve P’ and then V
4) Solve T , k , e equations
SIMPLE algorithm
start
p=p*
Guess p*
Step1: solve V* from momentum equations
Step2: introduce correction P’ and express V = V* + f(P’)
Step3: substitute V into continuity equation solve P’ and then V
Step4: Solve T , k , e equations
no
Converged
(residual check)
yes
end
Other methods
SIMPLER
SIMPLEC
PISO
variation of SIMPLE
COUPLED - use Jacobeans of nonlinear velocity functions to form
linear matrix ( and avoid iteration )
Surface boundaries
wall functions
Wall surface
Introduce velocity
temperature and concentration
Use wall functions to model the micro-flow in the vicinity of surface
Using relatively large mesh (cell) size.
Surface boundaries
wall functions
Course mesh distribution in the vicinity of surface
Y
Wall surface
Velocity in the first cell will depend on the distance y.
Surface boundary conditions and
log-wall functions
1/ 2
 
V   
r
1

log(
y
E)
*
y
E is the integration constant and
y* is a length scale
Friction velocity
dV
  t
dy
1/ 2
 
Vt   
r
u+=V/Vt
y*=(n/Vt)
y+=y/y*
- von Karman's constant
The assumption of ‘constant shear stress’ is used here.
Constants k = 0.41 and E = 8.43 fit well to a range of boundary layer flows.
Surface cells
Turbulent profile
Laminar sub-layer
K-e turbulence model
in boundary layer
Wall shear stress
dV
  t
dy
Eddy viscosity
k2
 t  rC
e
Wall function for e

e   
r
3/ 2
1
y
Wall function for k
1
k  1/ 2
C
 
 
r
V
Modeling of Turbulent Viscosity
in boundary layer

const antn t
MVM


forced
m ixinglength


One - Eq.





 Free











High
Re
wall




bounded






k - e 





Models based on μ t 


 Low Re
T KEMT wo 
 Buoyancy






Eq. 
Curvature


k - 





k - l





k - kl


k  f





......


convection
1  Layer

2  Layer
3  Layer

natural convection
Temperature and concentration
gradient in boundary layer
Depend on velocity field
• Temperature
q=h(Ts-Tair)
Tair
h = f(V) = f(k,e)
Ts
Into source term
of energy equation
• Concentration
F=hc(Cs-Cair/m)
m=Dair/Ds
m- segregation coefficient
Cair
Cs
hC = f(V, material prop.)