2D Free Jet Simulations (FLUENT) - Princeton...

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)


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( )


.） 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 ( )


).(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


x/d = 10 x/d = 20

6




m

x uudyvs

uuPlot

xu

ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


x/d = 30 x/d = 40

7




m

x uudyvs

uuPlot

xu

ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


x/d = 50 x/d = 60

8




m

x uudyvs

uuPlot

xu

ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


x/d = 70 x/d = 80

9




m

x uudyvs

uuPlot

xu

ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


ux/umax

y/d

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


x/d = 90 x/d = 100

10

• Jet Half Width ( )


.）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


).(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 ( )


.: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)


• 2D Laminar Round Jet– Theoretical Solutions– Numerical Calculations (FLUENT code)


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



x/d = 10 x/d = 20

u

).(max is where,: uudyvs

uuPlot m

m 18



x/d = 30 x/d = 40

u


uuPlot m

m 19



x/d = 50 x/d = 60

u


uuPlot m

m 20



x/d = 70 x/d = 80

u


uuPlot m

m 21



x/d = 90 x/d = 100

u


uuPlot m

m 22

• Jet Half Width ( )


.: 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


).(max is where,:2/1

xmm

x uuuuvs

rrPlot

24

• Momentum Thickness ( )


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

2D Free Jet Simulations (FLUENT) - Princeton...

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