Report Final2

Modelling and Simulation CFD Laboratory Report

Introduction This laboratory exercise report introduces the aims of the studies conducted, the relevant discussion

and conclusion based on the obtained results.

Aims The laboratory exercise was designed to

1. Demonstrate how to set up and run a flow computation simulation using Ansys Fluent

2. Obtain results to investigate the effects and accuracy of different convection schemes and grid

densities

3. To make comparisons between predictions using a simple turbulence model and experimental

measurements

Investigations

a. Grid-Dependence and Accuracy of Discretization Schemes

A discretization scheme is used to approximately represent the differential equations used to model the

problem by a set of algebraic ones linking the variable nodal values. To investigate the effect of grid

density and discretization schemes on accuracy, a sensitivity study is conducted varying both

parameters.

Two discretization schemes are used in this investigation -

i. First Order Upwind

The first order upwind scheme uses the variable value of the node in the upwind direction to

approximate the value of the node downwind to it. As a first order scheme, this discretization technique

generally sees a linear drop in error as the grid length is reduced for uniform grids. For example, if the

grid length is halved, the error is also halved.

ii. QUICK

The Quadratic Upstream Interpolation for Convective Kinematics or QUICK scheme is a third order

upwind discretization technique that attempts to fit a parabola between three points to approximate

the fourth.

As QUICK is a third order discretization scheme, the drop in error is a function of the relative change in

grid length cubed. For example, if the grid length is halved, the error is reduced to an eighth (0.53).

However, the QUICK scheme is not bounded, and can produce undershoots and overshoots in regions of

steep gradients, as a result of where the maximum or minimum of the fitted parabola lies.

b. Turbulent Flow Calculations

Experimental data reported by Nakayama (1985) for a turbulent flow at a Reynolds number of 1.2×106

and angle of incidence of 0 is provided. The same flow is computer and the obtained results are

compared against experimental data.

Model Setup

a. Grid-Dependence and Accuracy of Discretization Schemes

A laminar flow model around an airfoil is set up for a Reynolds number of 240. The flow is computed at a

10° angle of incidence. Four case files with increasing grid densities are provided. The grids are all ‘C’

type meshes. The coarsest grid to be used for the laminar flow calculations has 40 nodes wrapped

around the airfoil and wake, and 10 from the airfoil to the outer boundary. The other grids have

densities of 80x20, 160x40 and 320x80 nodes, each obtained by successively dividing each cell

dimension by 2.

A solution is obtained on the coarsest grid using the first order upwind scheme for convection. The

values of the lift and drag coefficients, Cl and Cd, are recorded and the same simulation is run using the

QUICK scheme without re-initialization. This is done as it is faster to start this calculation from the

converged 1st order upwind scheme one, instead of re-initializing the solution. This is repeated for the

medium and fine grids and the results are recorded.

b. Turbulent Flow Calculations

The same airfoil geometry is used to conduct this section of the exercise as well. The incident angle

however is reduced to 0 and they Reynolds number is 1.2 x 10 6 (velocity = 17.5m/s). The pertaining grid

is provided in the ‘airf-turb.cas’ case file. This is a 180x80 grid, but possesses a much higher near-wall

grid density than the grids mentioned earlier. In order to capture the much steeper near-wall gradients

that are expected in turbulent flow, a finer grid is essential. To obtain as accurate numerical results as

possible, the QUICK scheme for convection of momentum is used. The 1st order upwind scheme is used

for the convection of turbulence quantities (k and ε) for stability. To account for the turbulence, the 2-

equation k-ε model is used. When applied on this grid with the Enhanced Wall Treatment selected the

near-wall viscous layer is not fully resolved by the simulation, but a wall-function approximation is

employed.

Results In this this section, the results from all the simulation runs are presented and discussed.

1. Grid-Dependence and Accuracy of Discretization Schemes

a. 40 x 10 Grid – First Order Upwind

Case File Grid Scheme Cl Cd Iterations

airf-10.cas 40x10 FOU 7.1091E-01

2.6364E-01

70

Grid

Figure 1: 40 x 10 Coarse C-Type Mesh

The grid shown above is the mesh used for this flow computation. Note that this is grid is very coarse.

Figure 2: Residuals for analysis

The first order upwind scheme is a stable discretization technique that generally does not face problems

converging.

b. 40 x 10 Grid – QUICK


airf-10.cas

40x10 QUICK 5.8572E-01

2.1017E-01

80

The residuals plot for this analysis is shown below. Notice that the second peak is due to running the

calculation after obtaining results from the First Order Upwind calculation. The number of iterations

specified above is the number to achieve convergence sing QUICK after the first convergence using FOU.

Figure 3: Residuals Plot

It is seen that the values of Cl and Cd are not the same as those obtained by using the same grid but

with a FOU discretization scheme.

Figure 3.2: Pressure Contour Plot

c. 80 x 20 Grid – First Order Upwind


airf-20.cas 80x20 FOU 6.5953E-01

2.3267E-01

110

Grid

Figure 4: 80 x 20 C Type Mesh

Note that this is a medium grid and hence the number of iterations required to achieve convergence has

increased.


d. 80 x 20 Grid – QUICK


airf-20.cas

80x20 QUICK 5.8835E-01

1.9763E-01

110




Figure 6: Residuals plot

Note that the values of Cl and Cd obtained is not the same as those recorded using the same grid and

FOU scheme. It is also observed that areas of high pressures and gradients have been localised upon

refining the grid.

Figure 6.1: Pressure Contour Plot

e. 160 x 40 Grid – First Order Upwind


airf-40.cas 160x40 FOU 6.3777E-01

2.1313E-01

230

Grid



f. 160 x 40 Grid – QUICK


airf-40.cas

160x40 QUICK 5.9726E-01

1.9327E-01

200





Figure 8.1: Contour Pressure Plot

For this case as well, it is seen that areas of high pressure are localised and pressure gradients are better

defined as the grid density has been increased.

g. 320 x 80 – First Order Upwind


airf-80.cas 320x80 FOU 6.2158E-01

2.0287E-01

730

Grid


The fine 320 x 80 grid used for this part of the study is shown in the image above.

Figure 10: Residuals Plot showing instability

It is seen that for this model set up the solution does not achieve convergence and that the residuals for

continuity keep increasing. To control this behaviour, the under-relaxation factors were reduced by a

magnitude of 0.1 from the default settings. As there does not appear to be any issue with the quality of

the grid, the reason for instability is most likely due to strong coupling between the variables.

When there is strong coupling between variables in a flow, in an unbounded condition, this may cause a

variable to vary by an unreasonably large factor that in turn causes the second variable to change by a

huge factor as well. This is the mechanism through which instability is formed and to restrict the

variables from changing unreasonably, an under-relaxation factor is used. This tool allows the user to

specify the fraction to which the code is allowed to change a variable. This number varies between 0 and

1. An under-relaxation factor of 0 allows no change to a variables value whereas 1 (no under-relaxation

effect) allows the code to vary the value of the variable with

no hindrance.

The new under-relaxation factors are shown in the image to

the left. These have been obtained by reducing the default

setting by a magnitude of 0.1.

After the under-relaxation factors were reduced, this

allowed the solution to converge without experiencing

instability.

Figure 10.2

Figure 11: Residuals plot showing convergence

h. 320 x 80 – QUICK


airf-80.cas

320x80 QUICK 6.0190E-01

1.9229E-01

510


Discussion

Based on the results presented in the sections earlier, the data obtained has been tabulated below.

Case File Grid Upwind Quick

Iterations

Cl Cd Iterations

Cl Cd

airf-10.cas 40x10 70 7.1091E-01

2.6364E-01

80 5.8572E-01

2.1017E-01

airf-20.cas 80x20 110 6.5953E-01

2.3267E-01

110 5.8835E-01

1.9763E-01

airf-40.cas 160x40

230 6.3777E-01

2.1313E-01

200 5.9726E-01

1.9327E-01

airf-80.cas (Lab) 320x80

730 6.2158E-01

2.0287E-01

510 6.0190E-01

1.9229E-01

airf-80.cas (supplied info)

320x80

XX 6.2160E-01

2.0280E-01

XX 6.0190E-01

1.9220E-01

Table 1: Results of sensitivity study

The table below indicated the error from solution for each one of the cases. Assuming that the results

obtained from case airf-80 (fine grid) are extremely close to the real solution of the problem, the error

has been calculated for each case by subtracting the value obtained in that case from that of airf-80.cas.

Case File Grid Upwind QUICK

Cl Error Cd Error Cl Error Cd Error

airf-10.cas 40x10 8.9310E-02 6.0840E-02

-1.6180E-02 1.7970E-02

airf-20.cas 80x20 3.7930E-02 2.9870E-02

-1.3550E-02 5.4300E-03

airf-40.cas 160x40

1.6170E-02 1.0330E-02

-4.6400E-03 1.0700E-03

airf-80.cas 320x80

Table 2: Error Comparison

The graph below represents the information from the table above. The Y axis indicates the error

whereas the X axis indicated the number of times the grid length has been halved relative to the first

grid (40 x 10 grid). For example – For X = 2, this means that the grid length has been halved twice (or

made a quarter). This produced a grid of 160 x 40.

Graph 1: Error Comparison

Although the number of data points is not sufficient to establish a trend, it is clear from the graph that

the error from the QUICK scheme drops quicker up refining the grid than the FOU scheme. Even to start

off with, the errors in the QUICK scheme are considerably smaller than those corresponding to FOU.

-4.0E-02

-2.0E-02

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

0 0.5 1 1.5 2 2.5 3 3.5

Erro

r

Length Half Factor

Upwind - Cl Error

Upwind - Cd Error

QUICK - Cl Error

QUICK- Cd Error

It is also evident from the last two data points that QUICK scheme tends to achieve a grid independent

solution quicker than a first order upwind scheme.

In theory, for a uniform Cartesian grid, the error should reduce by a half each time the grid length is

halved for a first order upwind scheme and reduce to an eighth for a QUICK scheme. The table below

shows the values for errorn0/errorn+1 where n represents the case (ex – airf10) and n+1 is the following

case (for airf10, this is airf20). Upon calculating these values, we are able to compare these with the

expected values. A term Error factor is introduced to refer the value defined.

Case File Grid Upwind QUICK

Cl Error Factor Cd Error Factor Cl Error Factor Cd Error Factor

airf-10.cas 40x10 2.3546E+00 2.0368E+00 1.1941E+00 3.3094E+00

airf-20.cas 80x20 2.3457E+00 2.8916E+00 2.9203E+00 5.0748E+00

airf-40.cas 160x40

Table 2: Error Factors table

The error factors for the Upwind cases are quite close to the expected value of 2 (if errorn+1 is half of

errorn when grid length is halved). However, this is not the case for QUICK scheme. The expected factor

of change is 8. The values vary significantly, however it is evident that these values are much higher than

those of the Upwind scheme. This is an indication of how quickly the errors reduce as a function of grid

density also inferring that higher order schemes such as QUICK achieve grid-independent solutions with

a lesser grid density than a lower order scheme such as FOU. The reason for the error factors for the

QUICK scheme to be less than expected may be explained by the nature of the grid (C type mesh) and

that the model has achieved grid-independence.

Turbulent Flow Calculations

Grid

Figure 13: 180 x 80 Grid with high near wall resolution

The table shown below summarizes information on the model and the results obained.

Case File Grid Iterations Cl Cd

airf-turb.cas 180x80 182 1.58E-01 1.40E-02


For turbulent flow, the near wall gradients are quite drastic. To capture this information, a much finer

mesh is required close to the wall. The velocity vector below shows how steep the gradient in velocity

magnitude is. If the mesh were not as fine, this gradient wouldn’t have been captured resulting in poor

accuracy.

Figure 15: Velocity Vector Plot showing high near wall velocity gradient

Predicted Values - Experimental Data Comparison

The final task of the laboratory

exercise is to plot the Coefficient of

Pressure against position in x and

U/Ufree values at different Y values on

the airfoil’s surface. The experimental

values and the predicted values from

the simulation are compared.

The four lines of interest are

1. x/C =0.83

2. x/C = 1.01

3. x/C = 1.2

4. x/C = 2.19

Figure 16: The image above shows the lines of interest

x/C = 1 marks the trailing tip (tail) of the airfoil.

Comparison of Coefficient of Pressure

Figure 17: The plot shows the overlay of experimental data and predicted values for Cp

The image shown above present an overlay of the experimental data provided and the values of Cp

predicted by the simulation. The experimental data is represented by red dots whereas the predicted

values of Cp are plotted using a white line. Upon studying the graph it is clear that there is very good

correlation between the two sets of data.

Comparison of Mean Velocity (U/Ufree)

Line 1: x/C = 0.83

From the plot shown below it appears that the predicted data diverges from the experimental values as

the y position increases past ~ 2.5 e-02. The experimental values record higher mean velocities than

predicted by the calculation. This may be attributed to the wall approximation function that is used as

opposed to fully resolving for variables in the near wall viscous layer. It can be said that the model

assumes a higher viscosity near the wall than that is expected in reality. This may also be due to a less

than sufficient mesh refinement to the grid close the wall. A drastic gradient in the boundary layer

demands a much finer mesh. A higher velocity in experimental data suggests that the magnitude of

velocity in fact rises from 0 (at the wall) to the Uy much faster than the simulation can predict.

Figure 18: Data at x/C = 0.83, Experimental: Red dots, Predicted: White line

Figure 19: Velocity Vector Plot

Line 2: x/C = 1.01

Figure 19: Data at x/C = 1.01 Experimental: Red dots, Predicted: White line

Similar to the plot seen for the previous line, the velocity predicted by the simulations seems to be

diverging from the values recorded during the experiment after steady correlation till Y = 1 e-02m. The

simulation, as seen earlier, seems to be under predicting the velocity magnitudes. However this is at a

position in X that is past the airfoil geometry. But it may be the case that the divergence from the

correct velocity values until x/C =1 has affected the accuracy of the values after x/C = 1. It also appears

that the simulation over predicts the velocities under Y ~ 0.2 e-02. This trend amplifies in the following

plots.

Line 3: x/C = 1.2

Data at x/C = 1.2 Experimental: Red dots, Predicted: White line

It appears that as the velocity is plotted at increasing values of x/C, the gradient of the velocity

magnitude is reducing. The change is not as sharp and abrupt as seen in earlier plots of the same type. It

is also seen that from this plot at x/C much farther away, the simulation is over predicting the velocity

magnitude under Y ~ 1.0 e-02m and under predicting after this point. The divergence from the

experimental values increases as Y increases. A slight hint of this trend was noticed in the previous plot.

Line 4: x/C = 2.19

At x/C = 2.19, the data sets do not correlate as well as they did at earlier x/C lines though the general

trends of the data sets are similar. Here it can be seen very evidently that the simulation under predicts

the velocities until Y~ 1.75e-02m and then begins to over predict. This trend is the opposite of what was

observed in plots seen earlier. As x/C has increased around two fold, since the last plot, the trend cannot

be predicted with confidence.

Data at x/C = 2.19 Experimental: Red dots, Predicted: White line

Reasons for deviation

The reasons for deviation may be

1. Near wall approximation function is used and the viscous wall layer is not resolved fully during

the calculation which might give rise to some errors although it is not expensive on

computational power

2. The grid may not be fine enough to capture the gradient. This may lead to inaccuracies that

affect the values of the velocities downwind of the airfoil’s trailing edge.

3. It is also important to consider the possibility of error in experimental data. It cannot be said

with certain where the error arises until more information on control data set is available.

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