2. K-epsilon Study: (Mohammed Zohair Uz Zaman: s204948)

K-epsilon is a turbulence model used to simulate flow characteristics around a body in

computational fluid dynamics. It is a two transport equation method model. The first transported

variable determines the turbulent kinetic energy (K), whereas the second transported equation

determines turbulent dissipation (ε).

For this model it is assumed that turbulent viscosity is isotropic, which basically means that the

ratio of Reynolds stress to mean rate of deformations is constant in all directions. K-epsilon

models are less sensible to adverse pressure gradients and boundary layer separation hence there

use in external aerodynamics is not recommended.

Under K-epsilon model there are two formulations: The Standard K-epsilon model and the

Realizable K-epsilon model. Though in fluent standard k-epsilon model is one of the most used

models, the use of k-epsilon realizable model in fluent is recommended. The difference between

the two being that:

The realizable k-epsilon model has been modified with an alternate formula for turbulent

viscosity.

And also the transport equation for the dissipation rate has been derived from an exact equation for transport of mean-square vorticity fluctuation.

Transport equations for standard k-epsilon model:

The co-efficients used are:

𝐶𝜇 𝐶𝜀1 𝐶𝜀2

0.09 1.44 1.92

Transport equations for Realizable k-epsilon model:

2.1 Analysis:

This analysis was conducted for a VA-2 supercritical air foil, with the following flow conditions:

Mach No:0.1

Reynolds No:180,000

Angle of attack=0*

Total Pressure (P0)=101325 Pa

Static Pressure (P)=100168.9 Pa

From,

Total Temperature (T0)=288.15 K

Static Temperature (T)=287.5 K

From,

Further Details of the analysis are as given below:

Mesh:

For this simulation two meshes were used as stated earlier, one with y+1 (fine mesh) having

80654 Total elements and 80086 Total nodes. Another with y+30 (coarse mesh) having 70102

Total elements and 69525 Total nodes.

This was done in order to obtain the trend by use of different meshes, where the fine mesh starts

within the viscous sublayer and the coarse mesh starts in the buffer layer, of the viscous region.

Solvers:

In case of both the flow models density solver was used.

Models:

Both the models of K-epsilon, K-epsilon standard and K-epsilon realizable were used, and the

data obtained by both these methods is compared.

Materials:

For the entire simulation air was used as the material, within the domain, it was set to the following

conditions:

Density: Ideal gas (kg/m3)

Specific Heat (Cp) :1006.43 (j/kg-k)

Thermal conductivity: 0.0242 (w/m-k)

Viscosity: Sutherland three co-efficient method (kg/m-s)

Solution Methods:

In case of both models, The flow was simulated with implicit formulation as it is much more

stable and will produce a converged solution faster Ref (fluent_13.0_workshop02-Air foil), with

least square cell based gradient, and flow set to second order, with both turbulent energy and

dissipation in first order.

All the data obtained from the above simulation was obtained for a reference length of 0.2 m and

an energy convergence of 10-4.

Since there was no experimental data available for Mach no 0.1 and Reynolds no 180000, for the

VA-2 air foil, all the data obtained from FLUENT has been compared with the data obtained from

VGK, for the same simulation. This is done in order to establish a standard of comparison, to

validate all the data obtained from FLUENT.

2.2 Brief introduction to VGK:

VGK is Full Potential CFD method, developed just for supercritical air foil, using an O-grid. VGK

has been extensively developed and typically provides a very high standard of prediction

accuracy. This program involves a method to calculate the supercritical flow past an air foil,

including the effects of boundary layer and wake. The inviscid solution takes account of the

viscous layers by using the equivalent source method of light hill.

2.3 Results and Conclusions:

2.3.1 Pressure Co-efficient Contours:

All the images listed below show the co-efficient of pressure of contours, taken from fluent for

different meshes and different k-epsilon models.

As can be inferred from the contours below, for the small Mach no. of 0.1 and Reynolds no. of

just 180,000 that the co-efficient of pressure distribution below does not vary much with the

different meshes (changing y+),as well as the different k-epsilon models.

But it can be seen clearly that the air foil is producing lift as the pressure on the upper surface of

the air foil is much lower, than the pressure on the lower surface of the air foil, creating a suction

effect, pulling the air foil up. Also there is a trailing edge pressure recovery, which is expected.

Figure 2.1 Pressure Coefficient contours for y+1 Realizable

Figure 2.2 Pressure Coefficient contours for y+1 Standard

Figure 2.3 Pressure Coefficient contours for y+30 Realizable

Figure 2.4 Pressure Coefficient contours for y+30 Standard

2.3.2 Mach no. Contours:

The various images listed below show the contours of Mach no, for the VA-2 air foil, taken from

Fluent. For different meshes and k-epsilon models.

As can be inferred from the contours below that for given low velocity and Reynolds no. , That

there is very little variation between the contours for various meshes (changing y+) and between

the k-epsilon solvers.

What can be seen from the contours clearly is that there is a flow stagnation point at the leading

edge and the trailing edge of the air foil, where velocity is zero. The wake coming of the air foil

can also be seen clearly, the wake extends for meters behind the air foil up to the point at which

the velocities reach the Fairfield value.

Also it can be observed that the velocity increases from the point of stagnation at the leading edge

to maximum, at the point where the air foil thickness is maximum both on the upper and lower

surface, after which the velocity of the flow starts decreasing, this change in the velocity profile,

creates an adverse pressure gradient, which can be seen from the increasing thickness of the

boundary layer, over the upper and lower surface of the air foil.

Figure 2.5 Mach Number contours for y+1 Realizable

Figure 2.6 Mach Number contours for y+1 Standard

Figure 2.7 Mach Number contours for y+30 Realizable

Figure 2.8 Mach Number contours for y+30 Standard

2.3.3 Cp Distribution:

All Cp distributions obtained from fluent for various meshes and k-epsilon models have been

compared with Cp distribution, obtained for the same flow conditions from VGK, which is a

proven CFD method (2.2).As there is no experimental data available for Mach 0.1 and Reynolds

no 180,000. All conclusions below have been drawn on the basis of this comparison.

Figure 2.9 Pressure Co-efficient of y+1 Realizable vs VGK

Figure 2.9 Pressure Co-efficient of y+1 Standard vs VGK

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Y+1 Realizable VGK

Cp

X/C

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Y+1 Standard VGK

Cp

X/C

Figure 2.9 Pressure Co-efficient of y+30 Realizable vs VGK

Figure 2.9 Pressure Co-efficient of y+30 Standard vs VGK

From all the above figures it can be inferred that, the pressure co efficients obtained from fluent

over the upper surface are more or less similar when compared to the ones obtained from VGK,

with slight variations as we near the trailing edge, as these variations are very small this analysis

is still useful in obtaining the co-efficient of pressure distributions over the air foil.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Y+30 Realizable VGK

Cp

X/C

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Y+30 Standard VGK

Cp

X/C

But the co efficients of pressure obtained over the lower surface of the air foil matches almost

exactly with the co efficients obtained from VGK, with almost no variation.

Both the above mentioned trends, for the upper and lower surface of the air foil, seem to be the

same for all meshes and k-epsilon models, except in the case of Y+30 (Coarse Mesh) k-epsilon

standard (fig 2.9), where the contours for the co-efficient of pressure obtained from fluent follow

the contours of co-efficient of pressure obtained from VGK, both for upper and lower surface of

the air foil, but the co-efficient of pressure values obtained from fluent are over estimated, but not

by much, hence this can still be considered a good method to obtain Cp distribution over the air

foil.

As the Mach no. and Reynolds no. are very low, there are not many significant changes through

the flow field and hence any of the above methods can be used to obtain the pressure co-efficient

distribution over the air foil, with little error.

Note:

In all the cases it can be seen that there is increasingly negative pressure at the trailing edge which

does not follow the general pressure co-efficient distribution trend, from this it can be said that

both the fine and coarse mesh, using both models of k-epsilon fail to capture exactly the pressure

co-efficient distribution at the trailing edge. But in the general scheme of things these increasingly

negative values at the trailing edge can be ignored, to obtain pressure co-efficient distribution

values over the air foil with reasonable accuracy.

2.3.4 Lift and Drag

Table 2.1 CL and CD

As earlier the primary method of comparison has been done by comparing the lift, drag and lift

to drag ratio obtained by the various methods in fluent to the same values obtained from VGK

(2.2).

y+1 (Fine Mesh) with k-epsilon realizable model, over estimates the lift obtained from

VGK by 7%, and the drag by 13%,the lift to drag ratio vary by 7.7%,hence it can be said

that this method is reasonably accurate to predict lift and drag values, as the percentage

variation is not much.

Solver CL CD CL/CD

y+1 REALIZABLE 0.23187239 0.021053 11.01382

y+1 STANDARD 0.35224469 0.012443 28.30947

y+30 REALIZABLE 0.255925 0.018914 13.53078

y+30 STANDARD 0.255925 0.018914 13.53078

VGK 0.21494 0.01812 11.86203

y+1 (Fine Mesh) with k-epsilon standard model, over estimates the lift obtained from

VGK by 38%, and underestimates the drag by 31%, giving a very optimistic and high,

but wrong value of lift to drag ratio of 28, which varies by 58% from the value obtained

from VGK, hence it can be said that this is not a sound method for predicting lift and drag

values.

Both y+30 (Coarse Mesh) with k-epsilon standard and k-epsilon realizable give the same

values for lift, drag and lift to drag ratio, they overestimate the lift value obtained from

VGK by 16% and drag value by 4%,the lift to drag ratio is also over estimated by 12%,

hence it is not a very good method to obtain lift and drag accurately, especially when

compared to y+1 (Fine mesh) with k-epsilon realizable which has lesser error especially

when it comes to predicting the lift to drag ratio.

From the above discussion it can be concluded that use of fine mesh with K-epsilon realizable

will give the Lift and Drag values with reasonable accuracy.

2.3.5 Wake velocity (Mach no.) in the X-direction

Figure 3.0 U/at vs X (Variation of wake Mach no. in X-direction)

The above figure shows the variation in Mach no. in the x-direction, in the wake behind the air

foil ,this was obtained by means of a rake placed behind the air foil in fluent, as there is no VGK

data available for comparison for Mach no. variation in the wake of the air foil, and also due to

non-availability of experimental data for the same, all the result discussion has been done by

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.08 0.085 0.09 0.095 0.1 0.105

Y+1 Realizable Y+1 STANDARD Y+30 REALIZABLE y+30 STANDARD

U/at

X

comparing the data obtained with the coarse (y+30) and fine (y+1) mesh, for k-epsilon standard

and realizable.

From the above figure we can see that for k-epsilon realizable for both the coarse and fine mesh,

produces the maximum variation in the Mach number in the X-direction in the X-direction behind

the air foil, indicating maximum loss in the flow energy caused by the presence of air foil, they

indicate a high value of viscous drag. Interestingly both k-epsilon realizable for coarse and fine

mesh produce exactly same results, with no variation at all, as can be seen in the above figure,

where both the k-epsilon data points completely overlap each other.

Both the K-epsilon standard for coarse and fine mesh, produce similar results, but not to degree

similarity as the K-epsilon realizable methods, The K-epsilon standard for coarse mesh show

higher values for Mach no. variation in the air foil wake, but both the k-epsilon standard produce

a smaller variation in wake Mach no. when compared to K-epsilon realizable, indicating lower

value of viscous drag.

But as there is no standard data present to which comparison to be made, so on the basis that both

K-epsilon Realizable for coarse and fine mesh produce same results, with absolutely no variation

in data, I will conclude that k-epsilon realizable is the best method to obtain the variation in Mach

no. in the wake in the x-direction.

2.3.5 Wake velocity (Mach no.) in the Y-direction

Figure 3.1 V/at vs Y (Variation of wake Mach no. in Y-direction

-0.0015

-0.0014

-0.0013

-0.0012

-0.0011

-0.001

-0.0009

-0.0008

-0.0007

-0.0006

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Y+1 STANDARD Y+1 REALIZABLE Y+30 REALIZABLE Y+30 STANDARD

V/at

Y

Aerospace Dynamics MSc

Introduction to CFD

The above figure shows the variation in Mach no. in the Y-direction, in the wake behind the air foil

,this was obtained by means of a rake placed behind the air foil in fluent, as there is no VGK data

available for comparison for Mach no. variation in the wake of the air foil, and also due to non-

availability of experimental data for the same, all the result discussion has been done by comparing the

data obtained with the coarse (y+30) and fine (y+1) mesh, for k-epsilon standard and Realizable.

From the above figure we can see that K-epsilon realizable for both the coarse and fine mesh, produces

the maximum variation in the Mach number in the Y-direction behind the air foil, indicating maximum

loss in the flow energy caused by the presence of air foil, they indicate a high value of viscous drag.

Interestingly both K-epsilon realizable for coarse and fine mesh produce exactly same results, with no

variation at all, as can be seen in the above figure, where both the K-epsilon data points completely

overlap each other.

Both the K-epsilon standard for coarse and fine mesh, produce similar results, but not to degree

similarity as the K-epsilon realizable methods, The K-epsilon standard for coarse mesh show higher

values for Mach no. variation in the air foil wake, but both the K-epsilon standard produce a smaller

variation in wake Mach no. when compared to K-epsilon realizable, indicating lower value of viscous

drag.

But as there is no standard data present to which comparison to be made, so on the basis that both K-

epsilon Realizable for coarse and fine mesh produce same results, with absolutely no variation in data,

I will conclude that k-epsilon realizable is the best method to obtain the variation in Mach no. in the

wake in the Y-direction.