Mixing Layer Report

8/2/2019 Mixing Layer Report

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Turbulent Flows (MA6622)

Mixing Layer Numerical Simulation Report

Prepared byLow Chee Meng

(G1002972A)

Introduction

Turbulent flows are distinguished from laminar flows by its chaotic and stochastic nature.They have been described as being diffusive, dissipative, 3-dimensional, and unsteady. Most flowsin nature are turbulent and there are many useful engineering applications for turbulent flows. Theeffects of turbulence can be advantageous or disadvantageous and it is thus important for engineersto understand the causes, effects and behaviour of turbulent flows so as to be able to predict andcontrol it.

In the study of turbulence, much progress have been made in both experimental andcomputational approaches. High-fidelity numerical simulations are able provide much insight intothe behaviour of fluid flows and are usually cheaper and quicker to implement than physical

experiments. However, the accuracy of the simulation depends on many factors and it crucial for theuser to exercise judgement in implementating the simulation and intrepreting the results.

The intent of this project was to gain some insight in the generation, evolution andbehaviour of turbulence in a mixing layer. A commercial simulation software Fluent was used andthe results are presented in this paper.

Overview of Simulation and Analysis Parameters

The simulation is of a plane mixing layer as shown in Figure 1. The domain is 15m in lengthwith a height of 2m. A splitter plate of 2m length is located at the height of 1m on the left hand sideof the domain. The fluid to be used in the computation is air. The simulation involves 2 air-streamsentering the domain from the left boundary and exiting on the right boundary. They are initiallyseparated by the splitter plate before joining up where the plate ends at 2m. Figure 2 summarizes thekey parameters of the simulation.

Figure 1. Schematic of Simulation Domain

The grid was a structured grid consisting of 66000 cells. The turbulence models used in the

1

Ul=0.73 m/s

Re = 50000

Uh=1.46 m/s

Re = 100000

1m

1m

15m

Splitter Plate

y

x

2m


2/14

simulation to close the RANS equations were the k-, Spalart-Allmaras, and Reynolds-Stressmodels. Each simulation was run for 1000 iterations to minimize the residuals, which is a measureof numerical errors.

Fluid Air with =1.225kg/m3

=1.7894x10-5 kg/msDomain Physical

Dimensions2D, 15m x 2m

Fluid Streams U1=0.730335 m/sU2= 1.46067 m/s

(note: 2D flow with V=0 m/s)

Splitter Plate 2m (length) x 1mm (thickness)

ComputationalGrid Size

66000 cells TurbulenceModels

k-, Spalart-Allmaras Reynolds-Stress

Figure 2. Key parameters of Simulation

In the analysis of the plane mixing layer, it is also useful to define several parameters which

will be used to describe the flow. For each turbulence model, data for the mean axial velocity ,turbulent kinetic energy k, Reynolds shear stress , variance of the fluctuating component of theaxial velocity was either directly taken from the data pool generated by Fluent, or otherwisemanually calculated. The results are shown in the following sections.

Parameter Definition

Mean axial velocity

k Turbulent kinetic energy (k = 1/2)

Axial-directional normal component of Reynolds-Stresses

Reynolds shear stress component

Uh, Ul Uh and Ul are respectively the higher and lowervelocities of the two fluid streams.

Characteristic ConvectionVelocity, Uc

Uc=1/2(Uh + Ul)

Characteristic VelocityDifference, Us

Us=Uh - Ul

y0.9, y0.1 Cross-stream locations at whichU(x,y0.1)=Ul(x)+ 0.1Us(x)U(x,y0.9)=Ul(x)+ 0.9Us(x)

Characteristic width offlow, (x)

(x) =1/2(y0.1+y0.9)

Reference lateralposition,

(x) = 1/2(y0.1(x)+y0.9(x))

Scaled cross-streamcoordinate,

=(y-(x))/(x)

Scaled Velocity, f() ( - Uc)/(x)

Figure 3. Mixing Plane Parameters

2


3/14

Reynolds-Stress Model Simulation

In the Reynolds-Stress Model (RSM), closure of the Navier-Stokes equations is achieved bysolving the transport equations for the Reynolds-Stresses. In contrast with k- and Sparlart-Allamaras model, the RSM model does not invoke the concept of turbulence viscosity.

Figure 4 and 5 respectively tabulates the salient parameters for a plane mixing layer and

shows the mean axial velocity at five locations downstream of the splitter plate, where thetrailing edge of the splitter plate is x=0. The cross-stream coordinate is given in actual physicaldimensions in meters. Hence, the positions y=-1 and y=1 are thus respectively the upper and lowerwalls of the flow domain.

Figure 4. Salient Parameters at Various Axial Positions

From Figure 5, it can be observed that the velocities are zero at the walls due to the no-slipcondition. Away from the walls, the cross-stream variation of can be broadly separated into 3regions: Uh, Ul , Uh Ul.. The first two regions are simply where the influenceUh and Ul have on each other is relatively low. In the middle region, in which the cross-stream rateof change of becomes significant demarcates the region of mixing between the two initiallyseparated fluid streams. The width of this mixing region is characterised by (x) and from Figure 4,can be observed by noting the portion of the curves in which the slope d/dy is relativelysignificant.

3

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

Figure 5. Axial Variation of Profile

1m

2m

4m

5m

6m

Cross-Stream Position, y (m)

-(m/s)

Axial Position (x)Parameter 1m 2m 4m 5m 6m

1.46 1.46 1.44 1.44 1.44

0.7 0.69 0.67 0.66 0.65

0.76 0.76 0.77 0.78 0.78

1.08 1.07 1.06 1.05 1.040.1 0.17 0.3 0.36 0.39

0.02 0.02 0.03 0.03 0.04(x) 0.09 0.15 0.27 0.33 0.35 0.06 0.1 0.17 0.2 0.21

Ul

Uh

Us

UcY

0.1

Y0.9


4/14

The axial variation of (x), from the data shown in Figure 3, is plotted and shown in Figure6 below. It can be observed that (x) varies approximately linearly with x. In other words, fromx1m to x=8m, and actually beyond, d/dx is approximately constant. This corroborates welltheoretical analysis from which Equation 1, which introduces and defines the spreading rate S as anon-dimensional parameter, in terms d/dx, Us and Uc. S has been determined to be independent of

the flow velocities Us and Uc, just as the spreading rate of a round jet is independent of Re. From theslope of the curve in Figure 6 and the data for U s and Uc, for this present simulation S was found to

be 0.08. The value for S was down to be between 0.06 to 0.11 for a range of flows (Dimotakis1991).

S (Uc/Us)(d/dx) [1]

Another interesting effect worthy of note is that the mixing layer tends to spread towards thelow speed stream. In Figure 7, the axial variations of y0.9, y0.1 and are shown. They respectivelydemarcate the upper and lower boundaries and the mid-plane of the mixing layer. It can be observedall three planes vary linearly with x tend towards the upper, lower velocity flow region. FromFigure 8, which plots the variation of (y0.9- ) and (y0.1- ) with x, it can be observed that thegrowth of the mixing layer is symmetric about .

4

3m 4m 6m 7m 8m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f(x) = 0.06x + 0.03

Figure 6. Growth of (x)

(x)

Linear Regression for(x)

Axial Position (m)

3m 4m 6m 7m 8m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 7. Axial Variation of Y0.9, Y0.1 and Y-bar

Y0.9

Linear Regression forY0.9

Y-bar

Linear Regression forY-bar

Y0.1

Linear Regression forY0.1

Axial Pos ition (m)

Cross-StreamP

osition(m)


5/14

Figure 9 shows the scaled velocity profile of the mixing layer. In plotting the scaled velocityf() against the scaled cross-stream coordinate , it is observed that the velocity profiles are self-similar. It should be noted that the mixing layer will be self-similar only after it is developed, i.e.

the mixing layer is not self-similar immediately downstream of the splitter plate where x is small.

Figure 10 shows the profiles of the turbulent kinetic energy, k, at various axial positions. Acomparison with Figure 5 suggests that the peak of each curve is where the mean velocity gradientsare highest in the mixing layer. The d/dy curve has an inflection point which leads to instabilityand turbulence. The Kelvin-Helmholtz instability is common feature in shear layer flows in whichvelocity gradients are high such as the one in this simulation. Near the walls, the k also increasesalong with the velocity gradients.

Figures 11 and 12 show the profiles for the Reynold's stresses and , and uponcomparion with Figure 5, it can be inferred that the Reynold's stresses are associated with highvelocity gradients. Indeed, Equation 3 shows the assumed relationship between the Reynolds-Stresses and velocity gradients.

= 2/3kij T (d/dxj + d/dxi) [3]

Since the components of k include , the close similarity between Figure 10 and 11should be expected. Upon closer inspection of the relative magnitudes of k and , it can be also

inferred that is a large component of k for this present simulation. Other observations are that

5

3m 4m 6m 7m 8m

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

f(x) = -0.04x - 0.01

f(x) = 0.04x + 0.01

Figure 8. Axial Varation of Y0.9- and Y0.1-

y0.9

Linear Regressionfor y0.9

y0.1 Linear Regressionfor y0.1

Axial Position, x (m)

Y

-

-1.5 -1 -0.5 0 0.5 1 1.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Figure 9. Scaled Velocity Profile

1m

2m

4m

5m

6m

(-Uc)/Us


6/14

the axial variation of k and is observed to be very small compared to the streamwisevariations, and also that both are higher at higher flow velocities or Reynolds numbers.

6

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

Figure 11. Axial Variation of Profiles

1m

2m

4m

5m

6m

y (m)

-(m2/s2)

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

Figure 10. Axial Variation of Turbulent Kinetic Energy

1m

2m

4m

5m

6m

Crosstream Position (m)

TurbulentKineticEnergy(m2

/s2)

-1.5 -1 -0.5 0 0.5 1 1.5

-0.010000

-0.005000

0.000000

0.005000

0.010000

0.015000

0.020000


1m

2m

4m

5m

6m

Crosstream Position (m)


7/14

Figure 12 shows the profiles of the Reynolds shear stress . It is observed that ishigh where the velocity gradients are high, similar with . It can also be seen that at the upperwall (y=1) is positive whereas at the lower wall (y=-1) it is negative. This can be explained bynoting the sign of Equation 3 below. Since turbulent viscosity T is positive, it can be seen that is negatively related to d/dy. According to the coordinate system shown in Figure 1, it can beshown that at the lower wall (y=-1), d/dy is positive, and thus is negative, whereas the

opposite holds for upper wall where d/dy is negative.

= - T(d/dy) [3]

Figure 13 and 14 respectively show /Us2 and /Us2 plotted against the scaled cross-stream coordinate . It can be observed that the profile shapes of the reynolds stresses are generallyrather similar. Thus is can be said that Reynolds-Stresses also exhibit self-similarity when scaledwith the velocity difference Us and the local layer thickness (x). It is further observed that thescaled profiles at 1m and 2m seem to be marginally separated from the profiles at 4m,5m, and6m. For the scaled profiles, the profile at 1m seems to be separated from the remaining

profiles. This seems to indicate that the mixing layer is still not fully developed below 2m in termsof the of Reynolds-Stresses.

7

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

Figure 13. Scaled profiles

1m

2m

4m

5m

6m

/(Us)^2

-1.5 -1 -0.5 0 0.5 1 1.5-0.005000

0.000000

0.005000

0.010000

0.015000

0.020000

0.025000

0.030000

Figure 14. Scaled Profiles

1m

2m

4m

5m

6m

/(Us)^2


8/14

k- Model Simulation

The k- model is a two equation model, in which model transport equations for k and , thedissipation rate are solved in order to determine T, which will be used to model the Reynolds-Stresses. Equation 4 shows the relationship between T, k and . The k- model is stable and robustand less demanding on computational resources compared to RSM models, albeit with a

compromise in accuracy. It is also not suitable for flows with boundary layer separation, or wherethere are sudden changes in the mean strain rate. However, for simple shear flows such the presentsimulation, the model is said to be capable and reliable.

T = Ck2/where C = 0.09

[4]

Figure 15 shows the axial variation of the profile. It is largely similar to that generatedby RSM. Figure 16 shows the scaled velocity profile of and is shows the self-similarity of themixing layer. Comparing Figures 15 and 16 to Figures 5 and 9, it is observed that the resultsobtained with the k- model do not show a kink in the vicinity of y=0 =0.6. Therefore the resultsfor the k- model seem to more accurately reflect other experimental and computational resultscompared with the RSM model.

8

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

Figure 15. Axial Variation of

1m

2m

4m

5m

6m

Cross-steam Position, y (m)

-(m/s)

-1.5 -1 -0.5 0 0.5 1 1.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Figure 16. Scaled Velocity Profiles of

1m

2m

4m

5m

6m

f()


9/14

Figures 17, 18, and 19 respectively show the axial variation of the mixing layer thickness(x), the direction of its movement, and the development of the boundaries of the mixing layerabout its mid-plane. From the Figure 17, d/dx is observed to be 0.05, which give a spreading ratevalue Sk- = 0.07 compared with SRSM=0.08. However, the trends are similar, with Figure 18 showingthe tendency of the mixing layer to move towards the lower speed stream, and its symmetric growthabout the midplane .

Figures 20, 21, and 22 show the variations of the profiles of k, , and in the axial

direction. The date for turbulent kinetic energy k was directly extracted from Fluent, while

9

3m 4m 5m 6m 7m 8m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

f(x) = 0.05x + 0.06

Figure 17. Axial Variation of (x)

(x)

Linear Regressionfor (x)

Axial Pos ition (m)

(x)-(m)

3m 4m 5m 6m 7m 8m

0

0.050.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 18. Axial Variation Y0.1, Y0.9 and Y-bar

Y0.9

Linear Regressionfor Y0.9

Ybar

Linear Regressionfor Ybar

Y0.1Linear Regressionfor Y0.1

Axial Position (m)

Y(m)

3m 4m 5m 6m 7m 8m

-0.25

-0.2

-0.15

-0.1

-0.05

00.05

0.1

0.15

0.2

0.25

Figure 19. Axial Variation of (y0.9-) and (y0.1-)

Y0.9-

Linear Regression

for Y0.9-Y0.91-

Linear Regressionfor Y 0.91-


10/14

was calculated from k via the relationship ===2/3k. was calcuated byextracting the data for T and du/dy using the relationship between them shown in Equation [3].

From Figure 20, it can be observed that the profile for k is generated by the k- modeldiffers rather significantly from that generated by the RSM model. Firstly, the magnitudes of kcalculated by the k- model is one order of magnitude larger than that calculated by RSM. The areaunder the curve is also fuller which suggests that a broader region of the flow has higher k. The data

also shows that k increases with increasing downstream distance, which is contrary to the datashown in Figure 10. The

10

-0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800

0.000

0.020

0.040

0.060

0.080

0.100

0.1200.140

0.160

0.180

0.200

Figure 20. Axial Variation of k

1m

2m

4m

5m

6m

Cross-stream Position (m)

k(m2/s2)

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

-0.060

-0.040

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0.120Figure 22. at Various Axial Positions

3m

4m

6m

7m

8m


--(m2/s2)

-0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

Figure 21. Axial Variation of

1m

2m

4m

5m

6m

Cross-Stream Position (m)

-(m2/s2)

-0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.180

0.200

Figure 20. Axial Variation of k

1m

2m

4m

5m

6m


k

(m2/s2)


11/14

Figure 23 and 24 show the scaled profiles of and . It can be observed that whilethe general shapes of the curves seem to resemble each other, the separation between them aresignificant. In Figure 23, it can be observed that the profiles at 1m and 2m seem to form a pair,and the profiles for 5m and 6m form another pair, while the the 3m profile is clearly separated from

both pairs. In Figure 24, the profiles for 1m and 2m form a pair, while the profiles for 4m, 5m, and6m form another group. Hence, for the Reynolds-Stresses, it is observed that while the shapes of the

scaled curves seem to be similar, the placement of the curves are somewhat irregular. This seems tocontradict the the concept of self-similarity. Unlike for the scaled velocity profile, the k- does notseem to generate data that will provide for self-similar , , and profiles.

Spalart-Allamaras Model Simulation

The Spalart-Allamaras model is a one equation model which solves the transport equation

for the turbulent viscosity to provide closure for the RANS equations. The model is designed for

11

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

-0.100

-0.050

0.000

0.050

0.100

0.150

0.200Figure 24. Scaled Profiles

1m

2m

4m

5m

6m

/^2

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.180

0.200

Figure 23. Scaled

1m

2m4m

5m

6m

/^2


12/14

use with unstructured grids and is particularly suited for attached wall bounded flows but weak forfree shear flows.

The velocity profiles and scaled velocity profiles are shown in Figure 25 and 26respectively. That the velocity profiles display self-similarity is apparent from the Figure 25.

Figure 27 and 28 respectively show the profile of in x-y coordinates and the scaled. Similar to the date provided by the RSM and k- models, the magnitude of is largestwhere the velocity gradients are the largest near the wall and in the mixing layer. The scaled curves show strong self-similarity between the curves for the 4m, 5m and 6m axial positions, withthe curves at 1m and 2m somewhat separate from the others. The k profiles are shown in Figure 29.

12

-1.5 -1 -0.5 0 0.5 1 1.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Figure 26. Scaled

1m

2m

4m

5m

6m

(-Uc)/Us

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

Figure 25. Variation of with Axial Position

1m

2m

4m

5m

6m

y (m)

(m/s)


13/14

Figure 29 and 30 respectively show the rate of growth of the mixing layer and trajectory ofthe mixing layer downstream. Both sets of data agree with the data provided by RSM and k-models. It can observed that (x) grows linearly and that the mixing layer tends to move towards thelower velocity region downstream.

13

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

-0.005

0.000

0.005

0.010

0.015

0.020


1m

2m

4m

5m

6m

y (m)

-(m

2/s2)

-1.5 -1 -0.5 0 0.5 1 1.5

0.00000

0.00500

0.01000

0.01500

0.02000

0.02500

0.03000

Figure 28. Scaled Profiles

1m

2m

4m

5m

6m

/^2

3m 4m 6m 7m 8m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f(x) = 0.05x + 0.06

Figure 29. Growth of (x)

(x)

Linear Regressionfor (x)

Axial Pos ition (m)

(x)-(m)


14/14

Conclusion

In the present simulations, a commercial software was used to analyze the development of aplane mixing layer. Three different turbulence models were used in the computations and the resultswere compared.

It was clear that each turbulence model provided slightly different results. In plotting thecurves for k, and , it was clear that there were significant difference in the data sets mostdirectly related to turbulence. The difference was smallest when comparing the velocity profiles andscaled velocity profiles. The scaled profiles for all three models indicated self-similarity. Also,all 3 models showed that the mixing layer tend to move towards the lower velocity region of theflow, which agrees well with other simulations and experimental evidence. All 3 models alsoshowed that the Reynolds-Stresses and turbulent kinetic energy are high where the velocitygradients are high.

The choice of turbulence model would then depend on their comparative computationalexpense, as well as their proven suitability for the simulation problem. Two of the models invokethe turbulent viscosity hypothesis. The k- model has proven to be robust, stable, economical and it

provides relatively good predictions for many flows. It is however weak in simulating flows withhigh swirl and rotation, flows with high pressure gradients and separated flows. The Spalart-Allamaras model is relatively cheap and simple to implement and is suited for flows which areattached to mild mild separated. It is weak for free shear flows and for predicting turbulence decay.The Reynolds Stress Model does not invoke the turbulent viscosity hypothesis and solves thetransport equations for the Reynolds-Stresses directly. It is computationally expensive but also mostsuited for complex flows and flows with separation. Hence, the user has to exercise judgement in

the selection of turbulence models.

14

3m 4m 6m 7m 8m

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 30. Axial Variation of y0.9, y0.1 and

Y0.9


Ybar

Linear Regressionfor Ybar

Y0.1


Axial Pos ition (m)

Cross-Stream

Position(m)

Date post:	06-Apr-2018
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