Transcript Flow over an Obstruction

Flow over an Obstruction
MECH 523
Applied Computational Fluid
Dynamics
Presented by
Srinivasan C Rasipuram
Applications
Chip cooling
 Heat sinks
 Use of fire extinguishers at obstructions
 Though there are no significant applications
for flow over obstructions, this model is the
bench work for researchers to compare their
work and findings.

Case 1
u
T
4 cm
Tw
2 cm
0.5 cm
10 cm
Case 2
u
T
Tw
4 cm
0.5 cm
0.5 cm
0.5 cm
0.5 cm
0.5 cm
10 cm
Navier-Stokes Equations

Continuity
  ui 
 xi

0
No mass source has been assumed.

Momentum

 xj
 u
i
uj   -
p
 xi

ij
 xj
where  i j is the Stress tensor

 i j  

  u i  u j  2  u k

 - 

ij
x

 x i  3  x k
j

: i, j, k  1,2
 is the molecular viscosity of the fluid.

Energy

T 
u i e  p  
k eff
 Sh


 xi
 xi 
 xi 


Turbulent thermal conductivity keff = k + kt
Sh – Volumetric heat source

Brinkman Number
Br 
U
2
e
k T
where Ue is the velocity of undisturbed free stream
 Viscous heating will be important when Br
approaches or exceeds unity.

Typically, Br ≥ 1 for compressible flows.

But viscous heating has been neglected in the
simulations as Segregated solver assumes
negligible viscous dissipation as its default
setting.

Viscous dissipation – thermal energy created
by viscous shear in the flow.

In solid regions, Energy equation is

 xi
  T
k

 xi   xi 

u i  h  
Heat Flux due to conduction
 - density of the material
sensible enthalpy h 
T

Tref

C p dT
298.15 K
k - conductivi ty
T - Temperatu re
q  - Volumetric heat source
 q 
Standard k-є Turbulence Model
k - Turbulent Kinetic energy
є - rate of dissipation of turbulent kinetic
energy
k and є equations

k and є are obtained from the following
transport equations:

μ  k 
ρ

 μ  t 
  G k  G b - ρ ε - YM
D t  x 
σ x 
i 
k
i
Dk

2

μ  ε 
ε
ε
ρ

 μ  t 
  C1ε G k  C3ε G b  - C 2ε ρ
D t  x 
σ x 
k
k
i 
ε
i
Dε

where
 Gk represents the generation of turbulent kinetic
energy due to mean velocity gradients
 Gb is the generation of turbulent kinetic energy due
to buoyancy
 YM represents the contribution of the fluctuating
dilatation in compressible turbulence to the overall
dissipation rate
 C1є, C2є, C3є are constants
 k and є are the turbulent Prandtl numbers for k
and є respectively

Eddy or Turbulent viscosity
t   C
k
2

where C  is a constant.

The model constants
C1 = 1.44, C2 = 1.92, C = 0.09,
k = 1.0,  = 1.3
(Typical experimental values for these constants)
Turbulence Intensity

Turbulence Intensity
I
rms of the velocity fluctuatio ns
mean flow veloc ity, u avg
Discretization
 F
 x

 G
 y
 P
 u


2


 u  p -  xx



u
v



yx


F  
  T 


e

p
u
k

eff 

  x 


terms with k




terms with 


 v


0




 u v  p -  xy


0


2


 v -  yy


0




G  

  T  , P  
100000


e

p
v
k


eff




 y 





0
x terms with k




0




y terms with 


Discretization
…continued



 

1  
Fi,j .S 1  Fi -1,j .S 1  Fi,j .S 1  Fi,j-1 .S 1   P
i
i
j
j

Vij 
2
2
2
2 




 S i  1  S i  1  S j  1  S j  1 

2
2 
2
2
Vij 



2
2



Ideal gas model for Density calculations and
Sutherland model for Viscosity calculations

Density is calculated based on the Ideal gas equation.
p RT

Viscosity calculations
μ  C1

T
3
2
T  C2
C1 and C2 are constants for a given gas. For air at moderate
temperatures (about 300 – 500 K),
C1 = 1.458 x 10-6 kg/(m s K0.5)
C2 = 110.4 K
Reynolds Number calculation

For flow over an obstruction,
Re 
ρVD
μ
 is the density of the fluid
V is the average velocity (inlet velocity for internal
flows)
D is the hydraulic diameter
 is the Dynamic viscosity of the fluid
Re for V = 0.5 m/sec
For this problem, V = 0.5 m/sec,
air = 1.225 kg/m3, air = 1.7894 e–5 kg/m-sec

D  3.5 cm (for these cases)
Re  1198  2300  Laminar
Solver and Boundary conditions
Solver – Segregated

Inlet Boundary

Outlet boundary
– Velocity at inlet
– Gage Pressure at outlet
0.5 m/sec.
– Temperature at inlet
300 K
– Turbulence intensity
10%
– Hydraulic diameter
3.5 cm
0 Pa
– Backflow total
temperature – 300 K
– Turbulence intensity
10%
– Hydraulic diameter
3.5 cm
Wall boundary conditions
Heat sources
No heat flux at top and bottom walls
 Stationary top and bottom walls


Volumetric heat source for the (solid)
obstruction – 100,000 W/m3
Under relaxation factors
Pressure
Momentum
Energy
k

Viscosity
Density
Body forces
0.3
0.7
1
0.8
0.8
1
1
1
Convergence criteria
Continuity
0.001
x – velocity
0.001
y – velocity
0.001
Energy
1e-6
k
0.001

0.001
Case 1 – Grids
Number of nodes - 4200
Number of nodes - 162938
Number of nodes - 208372
Case 1 – Velocity contours
Case 1 – Temperature contours
Case 1 - Velocity Vectors
Case 1 –Contours of Stream function
Case 1 – Plot of Velocity Vs X-location
Case 1 – Plot of Temperature Vs X-location
Case 1 – Plot of Surface Nusselt number Vs X-location
Case 2 - Grids
4220 nodes
79984 nodes
42515 nodes
Case 2 – Contours of Velocity
Case 2 – Contours of
Temperature
Case 2 – Contours of Stream function
Case 2 - Velocity vectors
Case 2 – Plot of Velocity Vs X -location
Case 2 – Plot of temperature Vs X-location
Case 2 – Plot of Surface Nusselt number Vs x-location