2d free jet simulations (fluent) - princeton...
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2D Free Jet Simulations (FLUENT)
Oct. 31 2013Yan Zhan
Oct. 31, 2013
1
Outline• 2D Laminar Plane Jet
– Theoretical Solutions– Numerical Calculations (FLUENT code)
• Mean Stream Velocity• Jet Half Width• Momentum Thickness
• 2D Laminar Round Jet– Theoretical Solutions– Numerical Calculations (FLUENT code)
• Mean Stream Velocity• Jet Half Width• Momentum Thickness
2
2D Laminar Plane Jet― Theore cal Solu ons
Air emerges into a still air from a 2D slot at x=0– Constant momentum flux (J) at any cross section– Spread at constant pressure (P = 1 bar)– Red = 400 ( ux = 0.64 m/s)– Temperature = 300 K– ρ = 1.225 kg/m3;ν = 1.46073×10‐5 m2/s;
dyuJJxwQR
xJdyuwmQxyJ
hx
Ju
hx
Ju
x
x
y
x
23/1
2
3/1
3/2
3/1
2
23/1
2
23/12
,)36(/
)36( ,)48
( where
)]tanh()(sec2[)6
(
)(sec)323(
2
2
0
yu
yuu
xuu
yu
xu
yxy
xx
yx
Laminar Plane Jet (Schlichting 1933)
Fig.1 Typical free 2D laminar plane jet streamline pattern
3
2D Laminar Plane Jet― Numerical Calcula ons
Inlet velocity
Wall
Pressure outlet
Symmetrical axis Interior
Halved geometry
4
• Centerline Velocity( )
2D Laminar Plane Jet― Numerical Calcula ons
.) as off drops ( : 3/1xuuvsxPlot cc
cu0.01 0.6388123
0.05 0.63888186
0.1 0.63220511
0.2 0.6074034
0.3 0.5727235
0.4 0.537284
0.5 0.5115343
0.6 0.49355775
0.7 0.477681
0.8 0.4557375
0.9 0.44144434x
u c
0 0.2 0.4 0.6 0.8
0.45
0.5
0.55
0.6
4217.0007174.0442.106.2
2686.006898.0234
xxxx
xu c
5
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
).(max is where,: xm
m
x uudyvs
uuPlot
xu
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
x/d = 10 x/d = 20
6
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
).(max is where,: xm
m
x uudyvs
uuPlot
xu
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
x/d = 30 x/d = 40
7
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
).(max is where,: xm
m
x uudyvs
uuPlot
xu
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
x/d = 50 x/d = 60
8
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
).(max is where,: xm
m
x uudyvs
uuPlot
xu
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
x/d = 70 x/d = 80
9
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
).(max is where,: xm
m
x uudyvs
uuPlot
xu
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
ux/umax
y/d
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
numericalTheoretical
x/d = 90 x/d = 100
10
• Jet Half Width ( )
2D Laminar Plane Jet― Numerical Calcula ons
.)ith linearly w changes ( : 2/12/1 xr
dxvs
drPlot
2/1r /d / /d
1 0.539526
5 0.596278
10 0.65713
20 0.733453
30 0.836392
40 0.925598
50 1.00985
60 1.15413
70 1.21262
80 1.32187
90 1.74276
100 2.17414
r1/2/d
x/d
0.5 1 1.5 20
20
40
60
80
100
11
• Jet Self‐similarity
2D Laminar Plane Jet― Numerical Calcula ons
).(max is where,: maxmax2/1
xx uu
uuvs
ryPlot
y/r1/2
u x/um
ax
-15 -10 -5 0 5 10 150
0.2
0.4
0.6
0.8
1
x/d = 20x/d = 30x/d = 40x/d = 50x/d = 60x/d = 70x/d = 80x/d = 90
12
• Momentum Thickness ( )
2D Laminar Plane Jet― Numerical Calcula ons
.:dxvs
dPlot
dyuu
uu xx )1(
0.01 1.97590e‐4
0.05 9.08948e‐4
0.1 1.67433e‐3
0.2 2.57365e‐3
0.3 4.28040e‐3
0.4 3.33453e‐3
0.5 4.26769e‐3
0.6 5.51115e‐3
0.7 6.22347e‐3
0.8 7.93727e‐3
0.9 1.01272e‐2
1.0 9.29760e‐3
/d
x/d
1 1.5 2
20
40
60
80
100
13
Outline• 2D Laminar Plane Jet
– Theoretical Solutions– Numerical Calculations (FLUENT code)
• Mean Stream Velocity• Jet Half Width• Momentum Thickness
• 2D Laminar Round Jet– Theoretical Solutions– Numerical Calculations (FLUENT code)
• Mean Stream Velocity• Jet Half Width• Momentum Thickness
14
2D Laminar Round Jet― Theore cal Solu ons
Air emerges into a still air from a circular orifice at x=0– Constant momentum flux (J) at any cross section– Spread at constant pressure (P = 1 bar)– Red = 400 ( u = 0.64 m/s)– Temperature = 300 K– ρ = 1.16 kg/m3;ν = 1.6×10‐5 m2/s;
Narrow Axisymmetric Jet (Schlichting 1933)
)(
0)(1
rur
rrruv
xuu
rvr
rxu
. and ,2 ,)
163( where
)4
1(83
22/12
222
xrrdruJJC
Cx
Ju
r
15
2D Laminar Round Jet ― Numerical Calcula ons
Inlet velocity
Wall
Pressure outlet
Axial symmetirc Interior
Halved geometry
16
• Centerline Velocity( )
2D Laminar Round Jet― Numerical Calcula ons
.) as drops ( vs: 1xuuxPlot cc
cu0.01 0.63920647
0.05 0.63778848
0.1 0.610317
0.2 0.5230199
0.3 0.4403066
0.4 0.372688
0.5 0.32611621
0.6 0.2895635
0.7 0.260005
0.8 0.2357024
0.9 0.2153695x
u c
0 0.2 0.4 0.6 0.8 10.2
0.3
0.4
0.5
0.6 04816.03009.0
03038.02337.02
xxxu c
17
• Mean Stream Velocity ( )
2D Laminar Round Jet― Numerical Calcula ons
x/d = 10 x/d = 20
u
).(max is where,: uudyvs
uuPlot m
m 18
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
x/d = 30 x/d = 40
u
).(max is where,: uudyvs
uuPlot m
m 19
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
x/d = 50 x/d = 60
u
).(max is where,: uudyvs
uuPlot m
m 20
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
x/d = 70 x/d = 80
u
).(max is where,: uudyvs
uuPlot m
m 21
• Mean Stream Velocity ( )
2D Laminar Plane Jet― Numerical Calcula ons
x/d = 90 x/d = 100
u
).(max is where,: uudyvs
uuPlot m
m 22
• Jet Half Width ( )
2D Laminar Round Jet― Numerical Calcula ons
.: 2/1
dxvs
drPlot
2/1r /d / /d
1 0.462644
5 0.397387
10 0.603436
20 0.691654
30 0.822712
40 0.977359
50 1.07118
60 1.20262
70 1.33417
80 1.46478
90 1.59214
100 1.69765
r1/2/d
x/d
0.4 0.6 0.8 1 1.2 1.4 1.6
20
40
60
80
100
23
• Jet Self‐similarity
2D Laminar Round Jet― Numerical Calcula ons
).(max is where,:2/1
xmm
x uuuuvs
rrPlot
24
• Momentum Thickness ( )
2D Laminar Round Jet― Numerical Calcula ons
dyuu
uu )1(
.:dxvs
dPlot
/1 4.879917e‐2
5 1.429468e‐1
10 2.109622e‐1
20 2.877028e‐1
30 3.523778e‐1
40 4.257012e‐1
50 4.583403e‐1
60 5.047484e‐1
70 5.455564e‐1
80 5.808257e‐1
90 6.108572e‐1
100 6.309687e‐1
/d
x/d
0.1 0.2 0.3 0.4 0.5 0.6
20
40
60
80
100
25