Supersonic Flow Over a Double Circular Airfoil

7/26/2019 Supersonic Flow Over a Double Circular Airfoil

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UNIVERSITT DUISBURG ESSEN

Supersonic flow over a double circular airfoil

Gas Dynamics Project 2013

Eray Inanc

Matriculation Number : 2235683

September 11, 2013

Department of Fluid Dynamics

Prof. Dr. Ing. Ernst Von Lavante

Dipl.Ing . Harun Kaya


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Contents1. Introduction ..................................................................................................................................... 3

1.1 Task .......................................................................................................................................... 3

1.2 Aim of the project.................................................................................................................... 4

1.3 Theory ...................................................................................................................................... 4

1.3.1 Shocks .............................................................................................................................. 4

1.3.2 Expansion waves.............................................................................................................. 5

1.3.3 Pressure and temperature .............................................................................................. 7

1.3.4 Mach number and angle of attack .................................................................................. 7

1.3.5 Lift and drag coefficients ................................................................................................. 7

1.3.6 Strength of the Shock Waves .......................................................................................... 8

1.4 Simulation Programs ............................................................................................................... 9

1.4.1 Gridgen ............................................................................................................................ 9

1.4.2 Star-CCM+ ........................................................................................................................ 9

2. Calculations ................................................................................................................................... 10

2.1 Geometrical Calculations and airfoil thickness ..................................................................... 10

2.2 Shockwave Calculations ........................................................................................................ 11

2.2.1 Area_1 to Area_2........................................................................................................... 12

2.2.2 Area_1 to Area_4........................................................................................................... 12

2.2.3 Area_2 to Area_3........................................................................................................... 13

2.2.4 Area_4 to Area_5........................................................................................................... 14

2.3 Shockwave Strength Calculations .......................................................................................... 15

2.4 Lift and Drag Coefficients ...................................................................................................... 16

3. Simulations .................................................................................................................................... 16

3.1 Mesh Generation ................................................................................................................... 16

3.2 Simulation with Star-CCM+ ................................................................................................... 17

3.3 Results of the simulation ....................................................................................................... 17

4. Comparison ................................................................................................................................... 19

4.1 Simulation Results of Location and Strength of the Shocks .................................................. 19

4.2 Simulation Results of Lift and Drag Coefficients ................................................................... 20

4.3 Final Remarks ........................................................................................................................ 20

5. Bonus Part Calculations for Double Edged Airfoil ......................................................................... 21

5.1 Geometrical Calculations and airfoil thickness ..................................................................... 21

5.2 Shockwave Calculations ........................................................................................................ 22


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5.2.1 Area_1 to Area_2........................................................................................................... 23

5.2.2 Area_1 to Area_4........................................................................................................... 23

5.2.3 Area_2 to Area_3........................................................................................................... 24

5.2.4 Area_4 to Area_5........................................................................................................... 25

5.3 Shockwave Strength Calculations .......................................................................................... 26

5.4 Lift and Drag Coefficients ...................................................................................................... 27

6 Simulations .................................................................................................................................... 27

6.1 Mesh Generation ................................................................................................................... 27

6.2 Simulation with Star-CCM+ ................................................................................................... 28

6.3 Results of the simulation ....................................................................................................... 28

7 Comparison ................................................................................................................................... 30

7.1 Simulation Results of Location and Strength of the Shocks .................................................. 30

7.2 Simulation Results of Lift and Drag Coefficients ................................................................... 31

7.3 Final Remarks ........................................................................................................................ 31


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Rlower

upper

lower

1.

Introduction

1.1

Task

Supersonic flow over a double circular arc airfoil is to be analyzed. The shape, given in figure 1,consists of two circles with radius of curvature R lower and Rupper. The oespodig thikess is

upper and lower. The cord length is L. The airfoil is flying at a Mah ue M ad agle of attak .

Given data is:

Ambient pressure pa= 0.101325 MPa (1 atm)

Temperature = 300K

Core length L = 1 meter

Mach number = 2.4

Agle of attak = -2

(R/L)lower = 2,5

(R/L)upper = 2,6

Questions:

1)

Working first with analytical methods, determine:

a. Thickness of the airfoil,

b. Location and strength of leading edge shocks,

c.

Lift coefficient,

d. Drag coefficient.

2) Carry out flow simulation of the flow field in task 1 using Star-CCM+

3)

Compare analytical result with simulation.

4) Additional task: Replace the biconical airfoil with double edge airfoil and repeat the same

tasks.

Rupper

Minf.

L

Figure 1.Double biconical airfoil in supersonic flow


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1.2 Aim of the project

With the given task, the numerical part and simulation part have to be completed separately and

after the results it has to be compared whether the commercial codes in Star-CCM+ is using working

turbulence models.

The numerical analysis part has to be started by understanding the given data. The theory about the

given data then have to be analyzed and calculated with right understanding of the formulas. The

sequent for the solving the task would be started by geometrical analysis of the airfoil. Thickness of

the airfoil could be found easily with geometry. After determining the thickness of the airfoil,

location of the leading edge shocks has to be determined. Strength of the shock could be determined

after the calculation of the pressure difference after the shock over entrance pressure of the shock.

Lift and drag coefficient could be calculated by only neglecting the expansion waves caused by the

biconical surface. As the area and pressure varies by the geometry, differential calculations are used

in order to calculate both drag and lift coefficient.

1.3

Theory

1.3.1 Shocks

As we are figuring out the biconical airfoils behavior, we have to understand the basics of the

shockwaves. A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries

energy and can propagate through a medium through a field such as the electromagnetic field. Shock

waves are characterized by an abrupt, nearly discontinuous change in the characteristics of the

medium. Across a shock there is always an extremely rapid rise in pressure,

temperature and density of the flow. In supersonic flows, expansion is achieved through

an expansion fan. The flow has the capability to bypass the speed barrier as a result of the viscosity

of the air. Thus, results moving shocks and oblique shocks. An oblique shock represents a moving

shock with a deflection angle.

An oblique shock wave, unlike a normal shock, is

formed with respect to the upstream flow direction.

It will occur when a supersonic flow encounters a

corner that effectively turns the flow into itself and

compresses. The upstream streamlines are uniformly

deflected after the shock wave. The most common

way to produce an oblique shock wave is to place a

wedge into supersonic, compressible flow. Similar to

a normal shock wave, the oblique shock wave

consists of a very thin region across which nearly discontinuous changes in the thermodynamic

properties of a gas occur. While the upstream and downstream flow directions are unchanged across

a normal shock, they are different for flow across an oblique shock wave. Oblique shocks results in

pressure, temperature and density increase which is used as an intake flow advantage, or

disadvantage of fast deformation [1].

Figure 2 Oblique Shock Wave


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The sharp edges must be used in the wedges in order to create an attached oblique shock wave. The

wedge angle determines the shock to be attached or detached shock.

Figure 3 Attached and detached oblique shock waves [2].

1.3.2 Expansion waves

Thus as it will also be calculated, in this project the edges are sharp enough to generate attached

oblique shock waves. However, as the first oblique shock occurs, the flow will be expanded because

of the biconical geometry. Thus, this makes the use of the PrandtlMeyer Expansion rules.


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Mint

Slip Line

upper

lower

3

1

2

4

1 23 4

Oblique

Shocks

Expansion

Waves

The exhaust outtake will be slipped in two regions. The density will be different in upper and lower

parts of the airfoil due to the angle of attack not being zero and the not symmetrical geometry. The

different densities will result as a slip line at the exhaust, thus this separation of the flows will result

in another compression angle. This angle will be the same as the angle of attack and two different

oblique shocks will be formed at the edge of the rare-end. So the leading edge oblique shocks and

expansion waves will be expected in the form of figure 5.

Figure 5 Biconical airfoil shock waves

As i the figue is the epesets the defletio agle of the oliue shokaes. This is calculated

by the NACA (National Advisory Committee by Aeronautics) tables which have been determined bymainly analytical calculations and experiments.

Figure 4 Prandtl - Meyer Expansion waves in smooth surfaces [2]


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The lift coefficient CLis calculated by the

following formula that includes all the

complex dependencies.

Where CLis the lift coefficient, is the

angle of attack and M is the Mach

number

The drag coefficientis a dimensionless quantity that is used to quantify the drag or resistance of an

object in a fluid medium. Where a lower drag coefficient indicates the object will have less

aerodynamic drag. The drag coefficient is always associated with a particular surface area.

The drag coefficient of any object comprises the effects of the two basic contributors to

fluid drag: skin friction and form drag. The drag coefficient of a lifting airfoil also includes the effects

of lift-induced drag.The drag coefficient of a complete structure such as an aircraft also includes the

effects of interference drag.

Drag coefficient CDcan be calculated by the insertion of the complex dependencies into a formula as:

Where Cd is the lift coefficient, t is the thickness of the airfoil, c is the length of the airfoil and M is

the Mach number.

Main calculation in our task would require a complex calculation of both lift and drag, since the

surface area A is biconical and does not have an equal density distribution. Thus, the equation for the

lift and drag would be different then the subsonictransonic double edge, lift and drag coefficients.

1.3.6 Strength of the Shock Waves

A shock wave propagates through an undisturbed gas at the supersonic speed v o > ao, where a0is the

speed of sound in the undisturbed gas. The value of v0increases with the strength of the shock wave,the strength being defined here as (p1p0)/p0. As the strength of the shock wave approaches zero,

the speed of propagation approaches a0. The speed of a shock wave with respect to the compressed

gas behind it is subsonic: V1 > a1, where a1is the speed of sound in the compressed gas behind the

shock wave. As the oblique shock wave fits the description of the vo > ao since it is supersonic flow the

strength of an oblique shock is calculated by the following formula:

Figure 6 Drag and Lift Coefficients [3]


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1.4 Simulation Programs

1.4.1 Gridgen

Gridgen is a complete meshing toolkit used to generate two or three-dimensional grids for complex

geometries in a production environment, often where CFD is mission critical. The software's originsare in the demanding U.S. aerospace industry, where Gridgen continues to earn its reputation for

usability and high quality grids, both of which are vital for reliable simulations. Today Gridgen is used

in aerospace, automotive, power generation, chemical process and other industries for which CFD is

an integral part of the design process [5].

Where the simulations were simulated in Star-CCM+, the meshes which needed for it were designed

in Gridgen V16.04R2.

1.4.2 Star-CCM+

Star-CCM+ is one of the most practical, complete and easy-to-use engineering simulation program in

the market. The main usage of this program is mainly computational fluid dynamics (CFD). This

pogas ai adatages ae;

CAD & PLM integration, fully integrated with most common design programs such as Pro-

Engineer, SolidWorks and Gridgen.

Built-in meshing technology, polyhedral meshing, which also will be used later to see the

results.

Intuitive simulation user environment Multi-disciplinary solutions

Engineering analysis, especially for fluid mechanics department [6].

CFD (Computational Fluid Dynamics) is an essential part in fluid mechanics.. It represents a vast area

of ueial aalysis i the field of fluids flo pheoea. Headay i the fi eld of CFD simulations

is strongly dependent on the development of computer-related technologies and on the

advancement of our understanding and solving ordinary and partial differential equations (ODE and

PDE). Since direct numerical solving of complex flows in real-like conditions requires an

overwhelming amount of computational power success in solving such problems is very much

dependent on the physical models applied. These can only be derived by having a comprehensive

understanding of physical phenomena that are dominant in certain conditions.


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upper

L/2

Rupper 1

1

2.

Calculations

2.1 Geometrical Calculations and airfoil thickness

Upper part of the airfoil:

With circular arcs geometry:

And with the Pythagoreans theorem:

Where;

L = 1 m, Rupper = 2,6*L so 1=~2,55*L

upper= 2,6 *L2,55*L = 0,05 * L => 0,05 m

tan(1 = L/ / 1=> 1= ~ 11,09o

(1.1)


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lowerL/2

Rlower

2

2

Area_1

Area_2

Area_3

Area_5

Area_4

Lower part of the airfoil:



Where;

L = 1 m, Rlower = 2,5*L so 2= ~ 2,45*L

lower= 2,5 *L2,45*L = 0,05 * L => 0,05 m

tan(2) = L/ / 2=> 2= ~ 11,53o

(1.2)

Thus making the thickness:

(1.3)

2.2

Shockwave Calculations


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Area_1

Area_21

1

M1

M1

1

11

Mn1

M2n

M2

Area_1

Area_4

4 4

M1

Due to the attack of angle and irregular upper and lower bodies, the airfoil will have different shocks

at different areas. The area_1 will be the initial conditions, where the area_2 and area_4 is where the

first oblique shocks happen. Area_3 and area_4 will have another couple of oblique shocks due to

the flow having different densities at the upper part and lower part, thus forming a separation line.

The separation line will have the same angle as the angle of attack.

2.2.1 Area_1 to Area_2

We ill all the oespodig opessio agle 1and the result compression will be:

From this point on, the formulas in the NACA reports will be used.

From table 2, the values for M=141:



1st

Oblique Shock

3rd

Oblique Shock


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From table 2, the values for M=1.29:

2.2.3

Area_2 to Area_3

The expansion due to the biconical airfoil is calculated by the following formula:

[ ] Where the biconical airfoil angular length from start to end is 2*1,thus this will result the maximum

angle which M2

will turn and that is:

The v2(M*2) is the maximum turning angle during the biconical expansion waves. From table.2 we

can check the corresponding angle results ate the M*2=2.79 ( ~ 2.8) and 2=21o. The pressure ratios

for M2=1.89 M*2=2.79 are also:

An expansion wave is isentropic, hence Po2=P02*


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Area_2Area_3

1

2

M2

1

The oblique shock wave will happen just at the rear edge of the airfoil because of the separation line.

And the separation line has to continue form of the angle of attack with the same angle. Thus thecalculations are:

We will call the corresponding compressio agle 3and the result compression will be:




The expansion due to the biconical airfoil is calculated by the following formula:

[ ] Where the biconical airfoil angular length from start to end is 2*1,thus this will result the maximum

angle which M2 will turn and that is:

M*2

2nd

Oblique Shock

Separation Line


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Area_4 Area_5

4

5

2

The v4(M*4) is the maximum turning angle during the biconical expansion waves. From table.2 we

can check the corresponding angle results ate the M*4=3.03 and 2=19,27o. The pressure ratios for

M4=2.03, M*4=3.03 are also:

An expansion wave is isentropic, hence Po4=P04

*


And the separation line has to continue form of the angle of attack with the same angle. Thus the

calculations are:




2.3

Shockwave Strength Calculations

1st

Oblique Shockwave Strength:

2nd


M*4

4th

Oblique Shock

Separation Line


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3rd


4

thOblique Shockwave Strength:

2.4

Lift and Drag Coefficients

Lift coefficient hee = -2o= -0,0349 Rad:

Drag coefficient where t=0,1m (from 1.3) and c= 1m:

3. Simulations

3.1

Mesh Generation

The required mesh to use in the

simulations was generated with the

help of the program which was already

mentioned in the theory section.

Gridgen v16 has the option to

automatically generate the required

domain after generating connectors.

The main mesh is a giant circle with

the airfoil in the middle. The left side is

defined to be the inlet and the right

side is the outlet. The generated mesh

has the Cartesian coordinate system

where the origin is the beginning of

the airfoil.Figure 7 Mesh in Gridgen v16


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3.2 Simulation with Star-CCM+

As discussed at the theory section, the simulation was done in Star-CCM+ in order to compare the

results with the numerical and experimental results. The following properties were selected in the

options of Star-CCM+.

Simulation Properties Chosen Variables

Models

Gas-Air

Ideal Gas

Implicit Unsteady

K-Epsilon Turbulence

Coupled Flow

Two-Dimensional

Realizable K-Epsilon Two-Layer

RANS

Stationary

Reference ValuesReference Pressure: 1 bar

Initial Conditions

Velocity Composite

X: 2837.57 kph

Y: -102.54 kph

3.3

Results of the simulation

Figure 8 Residuals of the Simulation of the Biconical Airfoil


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Figure 9 Mach Number

Figure 10 Lift Coefficient

Figure 11 Drag Coefficient


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4.

Comparison

After calculation numerically and carrying out a flow simulation in Star-CCM+ the following results

occur. First of all, numerical calculations are mainly analytical assumptions. This results for a varying

error. In simulation, the calculations are done by the Reynolds stress models which are done by

Reynolds Averaging of Navier Strokes Equations. Especially very high Reynolds number and usingRANS modeling the errors are considerably high. The results of the simulation also depends on the

quality of the mesh and test variables. By the variables, initial conditions and flow properties are in a

crucial role.

4.1 Simulation Results of Location and Strength of the Shocks

Figure 12 Oblique Shock Waves and Expansion Waves

As it can be seen from the simulation scene, the shock locations and angles are as the same as the

numerical calculations. To compare the results for the strength the pressures on the exact solutions

are gotten from the simulation results.

1st

Oblique Shockwave Strength Simulation Result:

2ndOblique Shockwave Strength Simulation Result:

3rd


12

4 5

UPPER EXPANSION WAVES

LOWER EXPANSION WAVES


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4th


4.2 Simulation Results of Lift and Drag Coefficients

Lift coefficient: Drag coefficient: 4.3

Final Remarks

The location and the strength of the shocks are roughly similar with an error of 2-3o

in angle deviation

and 0,5 error in the strength. However, the main topic is that the location and the strength of the

shocks are at the trailing edges and there are several expansion waves at the surface of the airfoil

due to divergent of the flow. Strength of the shock waves are high at the intake and low at the rear

edge, with this the simulation results are confirming the numerical calculations.

The lift and drag coefficients have also are similar to the numerical calculations, where the error is

mainly lower than 0,5. This also proves that the angle of attack greatly affects the lift coefficient but

the sharp edges affect the drag coefficient. This proves that the supersonic flow required sharp

edged design at the airfoil.

Even with a very thin airfoil, the expansion waves are inevitable. This wave increase Mach number by

far and requires a lot more attention for the rear end oblique shocks, where those shocks would be

much more powerful due to the increased velocity. Even with this thin airfoil setup, the expansion

waves could carry the Mach number after the first shock, all the way to the initial Mach number.

And the last concerns is the oblique shock waves and the expansion waves are forced to crash at the

far out of the airfoils, increasing the turbulences, hence increase the disturbances at the separate

parts of the machine which uses this airfoil.


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upper

L/2

Rupper 1

1

5.

Bonus Part Calculations for Double Edged Airfoil

5.1 Geometrical Calculations and airfoil thickness

Upper part of the airfoil:



Where;

L = 1 m, Rupper = 2,6*L so 1=~2,55*L

upper= 2,6 *L2,55*L = 0,05 * L => 0,05 m

tan(1) = upper/ (L/2) => 1= ~ 5,71o

(2.1)


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lowerL/2

Rlower

2

2

Area_1

Area_2

Area_3

Area_5

Area_4

Lower part of the airfoil:



Where;

L = 1 m, Rlower = 2,5*L so 2= ~ 2,45*L

lower= 2,5 *L2,45*L = 0,05 * L => 0,05 m

tan(2) = lower/ (L/2) => 2= ~ 5,71o

(1.2)

Thus making the thickness:

(1.3)

5.2

Shockwave Calculations


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Area_1

Area_21

1

M1

M1

11

Mn1

M2n

M2

Area_1

Area_4

4 4

M1

Due to the attack of angle and irregular upper and lower bodies, the airfoil will have different shocks

at different areas. The area_1 will be the initial conditions, where the area_2 and area_4 is where the

first oblique shocks happen. Area_3 and area_4 will have another couple of oblique shocks due to

the flow having different densities at the upper part and lower part, thus forming a separation line.

The separation line will have the same angle as the angle of attack.







1st

Oblique Shock

3rd

Oblique Shock


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5.2.3

Area_2 to Area_3

Where the double edge airfoil angular turn is 2*1,thus this will result the maximum angle which M2


From table.2, M*2=2.53 and 2=23,28

o. The pressure ratios for M2=2.07, M*2=2.53 are also:



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Area_2 Area_3

1

2

M2

1



calculations are:





Where the double edge airfoil angular turn is 2*1,thus this will result the maximum angle which M2


From table.2 we can check the corresponding angle results ate the M*4=2.63 and 2=22,35. The

pressure ratios for M4=2.16, M*4=2.63 are also:


M*2

2ndOblique Shock

Separation Line


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Area_4 Area_5

4

5

2



calculations are:




5.3 Shockwave Strength Calculations

1st


2nd


3rd


M*4

4th

Oblique Shock

Separation Line


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4th


5.4

Lift and Drag Coefficients

Lift coefficient:

Drag coefficient where:

6 Simulations

6.1 Mesh Generation

The required mesh to use in the simulations was generated with the help of the program which was

already mentioned in the theory section. Gridgen v16 has the option to automatically generate the

required domain after generating connectors. The main mesh is a giant circle with the airfoil in the

middle. The left side is defined to be the inlet and the right side is the outlet. The generated mesh

has the Cartesian coordinate system where the origin is the beginning of the airfoil.

Figure 13 Double Edge Airfoil Mesh in Gridgen v16


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6.2 Simulation with Star-CCM+

As discussed at the theory section, the simulation was done in Star-CCM+ in order to compare the

results with the numerical and experimental results. The following properties were selected in the

options of Star-CCM+.

Simulation Properties Chosen Variables

Models

Gas-Air

Ideal Gas

Implicit Unsteady

K-Epsilon Turbulence

Coupled Flow

Two-Dimensional

Realizable K-Epsilon Two-Layer

RANS

Stationary

Reference ValuesReference Pressure: 1 bar

Initial Conditions

Velocity Composite

X: 2837.57 kph

Y: -102.54 kph

6.3 Results of the simulation

Figure 14 Residuals of the Simulation of the Double Edge Airfoil


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7

Comparison

After calculation numerically and carrying out a flow simulation in Star-CCM+ the following results

occur. First of all, numerical calculations are mainly analytical assumptions. This results for a varying

error. In simulation, the calculations are done by the Reynolds stress models which are done by

Reynolds Averaging of Navier Strokes Equations. Especially very high Reynolds number and usingRANS modeling the errors are considerably high. The results of the simulation also depends on the

quality of the mesh and test variables. By the variables, initial conditions and flow properties are in a

crucial role.

7.1 Simulation Results of Location and Strength of the Shocks

Figure 14 Oblique Shock Waves and Expanison Waves

As it can be seen from the simulation scene, the shock locations and angles are as the same as the

numerical calculations. To compare the results for the strength the pressures on the exact solutions

are gotten from the simulation results.

1stOblique Shockwave Strength Simulation Result:

2

ndOblique Shockwave Strength Simulation Result:

3rd


1

24

5

UPPER EXPANSION WAVES

LOWER EXPANSION WAVES


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4

thOblique Shockwave Strength Simulation Result:

7.2 Simulation Results of Lift and Drag Coefficients

Lift coefficient: Drag coefficient: 7.3

Final Remarks for Double Edge Airfoil

The location and the strength of the shocks are roughly similar with an error of 2-3o in angle deviation

and 0,5 error in the strength. However, the main topic is that the location and the strength of the

shocks are at the trailing edges and there are several expansion waves at the surface of the airfoil

due to divergent of the flow. Strength of the shock waves is high at the intake and low at the rear

edge, with this the simulation results are confirming the numerical calculations.

The lift and drag coefficients have also are similar to the numerical calculations, where the error is

mainly lower than 0,5. This also proves that the angle of attack greatly affects the lift coefficient but

the sharp edges affect the drag coefficient. This proves that the supersonic flow required sharp

edged design at the airfoil.

Even with a very thin airfoil, the expansion waves are inevitable. This wave increase Mach number by

far and requires a lot more attention for the rear end oblique shocks, where those shocks would be

much more powerful due to the increased velocity. Even with this thin airfoil setup, the expansion

waves could carry the Mach number after the first shock, all the way to the initial Mach number.

And the last concerns is the oblique shock waves and the expansion waves are forced to crash at the

far out of the airfoils, increasing the turbulences, hence increase the disturbances at the separate

parts of the machine which uses this airfoil.

As seen from these results, the sharp edges cause less lift. Therefore it is wiser to use biconicalairfoils. However this type of airfoils has little less drag coefficient.


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References:

[1] [Shapiro, Ascher H. The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1.]

[2]Chinese Journal of AeronauticsVolume 26, Issue 3,June 2013, Pages 501513

[3] The Dreese Airfoil Primer Copyright 2001-2007 John Dreese

[4] Landau, L. D., and E. M. Lifshits. Mekhanika sploshnykh sred, 2nd ed. Moscow, 1953

[5][Poinwise Products; http://www.pointwise.com/archive/faq-V13.shtml]

[6][Miloa Pei, Futue Deelopets of ta-CD and Star-CCM+, p.-27]

Tabel of Figures:

Figure 15.Double biconical airfoil in supersonic flow

Figure 16 Oblique Shock Wave

Figure 17 Attached and detached oblique shock waves [2].

Figure 18 Prandtl - Meyer Expansion waves in smooth surfaces [2]

Figure 19 Biconical airfoil shock waves

Figure 20 Drag and Lift Coefficients [3]

Figure 21 Mesh in Gridgen v16

Figure 22 Residuals of the Simulation of the Biconical Airfoil




Figure 26 Oblique Shock Waves and Expansion Waves

Figure 13 Double Edge Airfoil Mesh in Gridgen v16

Figure 14 Double Edge Airfoil Residuals of the Simulation of the Biconical Airfoil

Figure 15 Double Edge Airfoil Mach Number

Figure 16 Double Edge Airfoil Lift Coefficient

Figure 27 Double Edge Airfoil Drag Coefficient

Figure 28 Double Edge Airfoil Oblique Shock Waves and Expansion Waves
http://www.sciencedirect.com/science/journal/10009361http://www.sciencedirect.com/science/journal/10009361/26/3http://www.sciencedirect.com/science/journal/10009361/26/3http://www.sciencedirect.com/science/journal/10009361/26/3http://www.sciencedirect.com/science/journal/10009361/26/3http://www.sciencedirect.com/science/journal/10009361

Supersonic Flow Over a Double Circular Airfoil

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