Surface Drag Modeling for Milled Surfaces
SURFACE DRAG MODELING FOR MILLED SURFACES
Division of Fluid Mechanics, KTH
Author: Lars Johansson
Supervisor at KTH: Jens Fransson
Supervisor at GKN:
Hans Mårtensson and Fredrik Wallin
Summary
One of the governing sources of energy loss in a modern day jet engine is attributed to surface drag. This energy loss can be divided into friction loss and to surface geometry loss. The friction loss is the shear stress the fluid experience due to a no slip condition at the wall, while the surface geometry loss is due to pressure drop when the fuel passes an obstacle.
The objective of this work is to study the drag coefficient of a plate for different types of milled tracks and for different kinds of flow conditions. The theories used to calculate the drag coefficient are based on the momentum thickness theory including shear stress- and pressure integration. The computations were carried out with ANSYS CFX assuming a Shear Stress Transport � − � turbulence model. The steady state flow conditions tested are varying boundary layer thicknesses, milled track heights, milled track widths, Reynolds numbers over the milled track height, Reynolds numbers over the plate length and free-stream angle of attack. By knowing what affects the drag coefficient for different types of milled tracks, more practical models can be developed making the prediction of surface drag inside the jet engine more accurate.
This report has resulted in a formula that predicts the drag coefficient for different types of milled surfaces. The formula is derived from the assumption that the CFD results on ANSYS CFX are correct. A physical test has not been made to verify those results, however this has to be done to prove that this formula is valid.
Preface
This master thesis in the field of fluid mechanics is done with the help of GKN Aerospace. I would like to thank the people involved in my master thesis and to thank all people around me at GKN for all the help I have received.
I would specially thank my two supervisors Hans Mårtensson and Fredrik Wallin at GKN who have given me the support and knowledge to make this thesis possible and for the help they have offered me during this project. I would also like to give my gratitude and appreciation to my supervisor Jens Fransson at KTH for all the help and support I have received throughout this project.
Trollhättan, June 2015
Lars Johansson
Nomenclature
U = Free stream velocity [m/s] ν = Kinematic viscosity [m2/s]
�= Shear stress by the wall [Pa] y = Height of plate [m]
� = Density [kg/m3] � = Length of plate [m] h = Boundary layer thickness [m] u = Velocity distribution along the stream line [m/s]
�!= Profile height (cusp height) [m] �! = Distance between the milled tracks [m]
� = Radius of the miller tool [m] �!= Total pressure [Pa]
� = Static pressure [Pa] � = Boundary layer thickness [m]
�! = Displacement thickness [m] �! = Momentum thickness [m]
∆�!= First layer thickness of the mesh �!"#$= Velocity at the disturbance body [m/s]
� = Cusp angle on the milled tracks [˚] �!"!#$%= Momentum loss [kg*m/s]
� = width of test plate [m] � = Total length of plate [m]
� = Cusp angle [˚] � = Free stream angle of attack [˚]
Contents Summary ...... 8 Preface ...... 9 Nomenclature ...... 10 1. Introduction ...... 1 1.1 Background ...... 1 1.2 Purpose ...... 1 1.2 Thesis question ...... 1 1.3 Limitations ...... 2 2. Theory ...... 3 2.1Milled tracks ...... 3 2.2 Momentum thickness �2 ...... 4 2.3 y+ determination ...... 6 2.4 SST k-� turbulence model ...... 8 2.5 Effect of pressure gradient ...... 8 2.6 Effect of flow separation ...... 9 2.7 Drag coefficient ...... 11 2.7.1 Momentum thickness �2 ...... 11 2.7.2 Force integration ...... 12 2.8 Effective dynamic pressure ...... 13 3 Method ...... 15 3.1 Geometry ...... 15 3.2 Mesh ...... 16 3.3 Pre-process ...... 18 3.3 Post process ...... 19 4. Result ...... 19 4.1 Varying � ...... 19 4.2 Varying ��� ...... 20 4.2 ��� dependence ...... 22 4.3 Varying free stream angle of attack � ...... 25 5. Conclusion ...... 29 6. Future work ...... 33 7. Reference ...... 34 8. Appendix ...... 35
8.1 Theory ...... 35 8.1.1 k-� model ...... 35 8.2 Method ...... 37 8.2.1 Boundary conditions ...... 37 8.3 Results ...... 42 8.3.1 Varying inlet boundary thickness � ...... 42 8.3.2 Varying � and ��� ...... 42 8.3.3 Varying ��� and ��� ...... 47
1. Introduction
1.1 Background One of the fundamental loss sources in a modern day jet engine is loss due to surface drag. Surface drag loss can be divided into friction loss and loss due to surface geometries. The friction loss is the shear stress the fluid experience due to a no-slip condition at the wall, while the surface geometry loss is due to pressure drop when the flow is passing a curved body. It is of great interest to be able to predict these drag losses as accurate as possible, in order to make the jet engine operate at optimal conditions.
Some parts of the jet engine are manufactured by milled cutters. The milled cutter uses a milling tool to carve out the material resulting in a part with a milled finish. For some cases the milled surface has a purpose such as guiding the flow in a certain way but for other cases they are just a by-product from the manufacturing process. Nevertheless, a milled surface will generate drag. Even though normally the milled surface is not that rough it will still generate pressure losses due the curved body flow. It is therefore important to study the drag coefficient of a milled surface to be able to predict the surface drag inside a jet engine.
In practice milled surfaces are often described as similar to sand grain roughness. However, a better understanding is necessary in how the roughness affects the flow and subsequently how this can be best used in design. It is known that smooth surfaces generate less drag than rough surfaces, i.e. when the flow is laminar compare to transitional or fully turbulent. However, for specific surface structures it is known that the drag decreases for a fully turbulent flow due to fact that turbulent coherent structures are less frequent. Furthermore, if there is a laminar flow over the surface it could be more prone to separate, resulting in high pressure drag loss. By accurately knowing how the drag coefficient inside a jet engines behaves at different flow conditions the possibility of running the jet engine at optimal conditions are substantially improved.
1.2 Purpose The purpose of this project is to investigate the drag coefficient for different milled surfaces at different flow conditions. By knowing what affects the drag coefficient it is possible to derive a formula that is able to predict the drag for a milled surface.
1.2 Thesis question
• What affects the drag coefficient for milled surfaces? • How can the drag coefficient for milled surfaces be modeled? • What effects has flow separation on the drag coefficient? • Can a sand grain roughness model be used to model the drag coefficient of a plate with milled tracks?
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1.3 Limitations
The flows that have been studied are all subsonic flows. Furthermore only low Reynolds numbers have been studied, ranging from ��� = 3.7�5 − 1�6, i.e. the Reynolds number defined over the milled track surface length L. The geometry of the milled tracks that has been tested ranges from cusp angles � = 0 − 135˚. Furthermore the free stream angle of attack � has been tested from 0 − 90ᵒ and the milled track height over the boundary layer thickness has been tested from !" = 0.02 − 0.09. Figure 1 describes the studied parameters. !
�� ��� = � �
�
�
� �
�
�� � �
Figur 1: ��Definitions of the milled track parameters �
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2. Theory
2.1Milled tracks A typical plate that would be tested will have a surface consisting of very tiny milled tracks. The different dimensions of a milled track are defined as:
Figure1: The milled track has a milled track height Hc, milled track width Ae and milled track radius R
The milled track height is defined as �!, the milled track width is defined as �! and the milled radius is denoted as �. To relate the different parameters with each other one should think about the setup in a geometric perspective, where a right triangle could be spotted in the middle of a milled track:
R
R-H c
Ae/2
Figure 2: Right triangle relating the milled track dimensions by the Pythagoras theorem
Pythagoras theorem gives:
! �! = � − � ! + !! → � = 2 �! − (� − � )! (1) ! ! ! !
One can also use the cusp angle � to calculate the wanted quantities. The cusp angle is defined geometrically in figure 3.
3
Figure 3: Figure describing how the cusp angle are defined
By using Pythagoras theorem the following equation holds:
! !!!! !! cos = → � = ! (2) ! ! !!!"# !
When the cusp height �! and the cusp angle � are given the desired radius of the milling tool can be calculated.
2.2 Momentum thickness �� One way of calculating drag is by calculating the momentum thickness �!. Calculation of the momentum thickness goes in several steps. The first step to calculate is the boundary layer thickness �. It is defined as the height of the velocity profile from where the boundary layer velocity is 99% of the local free stream velocity (Pijush K, Kundu, 1990, P.213). When � is known one can calculate the displacement thickness �!. The displacement thickness is the distance normal to the surface from which the free stream has to be displaced to be able to sustain the same mass flux. It is defined as:
! ! ��� = �(� − �∗) → � = 1 − ! �� (3) ! ! ! !
U denotes the free stream velocity and u corresponds to the velocity distribution in the boundary layer. δ
4
Figure4: Displacement thickness (Pijush K, Kundu, 1990, P.306)
Figure 2: Streamline displacement due to boundary layer. The letter A defines the inlet and the letter B defines the outlet (Pijush K, Kundu, 1990, P.306)
From the displacement thickness one can calculate the momentum thickness �!, which multiplied with the density times velocity square gives the momentum loss due to the boundary layer. The momentum balance between section A and B in figure 5 are given as:
! ��!ℎ − ��!�� − �� �! = ��!� (4) ! ! !
On the left side of equation (6) one has taken the inflow momentum minus the outflow momentum. The difference in momentum is then given as the momentum thickness �!times ! �� . By substituting the expression for the displacement thickness �! in equation (4) one can now solve for �!. The resulting equation then becomes:
! �!� = �(� − �)�� (5) ! !
The numerical solution for this equation assuming a Blasius boundary layer profile in the laminar case is. � � = 0.664 !" (6) ! !
For the turbulent case there are many different relations and where
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! ! � � = 0.036� !" ! (7) ! !
Is one relation (Schlichting, 1979, P.638). Here � is defined as the distance along the plate, � is the kinematic viscosity of air and U is the free stream velocity. Equation (7) is derived from the momentum thickness formulation above by using the 1/7-th-power distribution law which describes the velocity distribution over a flat plate (Schlichting, 1979, P.637).
! ! = ! ! (8) ! !
There are however limitations with the use of momentum thickness when calculating the drag over a plate and that is that it’s only valid for surfaces with no pressure gradient, i.e. only flat surfaces. However, the momentum thickness method can be used for milled tracks if the velocity profile is taken at some distance after the end of the milled track, if one can guarantee that no pressure gradient is present in the velocity profile.
2.3 y+ determination One way of measuring the quality of a mesh when a turbulent flow is present is by using y+ units (Pijush K, Kundu, 1990, P.448). Close to the wall in a turbulent flow the viscosity forces has a big impact. The theory that is used in this region is called Law of the wall, which states that the velocity profile near the wall does not depend on the free stream velocity or the thickness of the flow but instead only on the density, shear stress, kinematic viscosity and the height y. One can rewrite the velocity profile such that it only depends on length and mass, this is done by introducing the friction velocity �∗ (Pijush K, Kundu, 1990, P.450):
� = !! (9) ∗ !
By dividing the friction velocity times the height of the velocity profile with the dynamic viscosity the velocity profile equation can be written in a complete non dimensional form.
This non dimensional coordinate �! has the following relation.
! !!∗ �! = = � = �(�!) (10) !∗ !
This equation is called law of the wall and states that the fraction ! (which also is defined as !∗ � ) should be a universal function of !!∗ (Pijush K, Kundu, 1990, P.450). The function ! ! between y+ and u+ is different depending on what sort of layer with in the inner region of the boundary layer that is studied.
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Figure 3: This figure describes the different sub layers close to the wall for a turbulent flow , the region after � �! > �� is given by the free stream velocity(Pijush K, Kundu, 1990, P.450)
By studying figure 6 one concludes that to fully resolve a boundary layer one has to take into account that there are both a linear and a logarithmic dependency between �! and �! depending if one wants to study the viscous sublayer, buffer layer or the logarithmic layer. Resolving all the way to the viscous layer is the same as resolving the entire boundary layer.
By calculating the first mesh height at �! = 1 the entire inner region will be resolved. With a no-slip boundary assumption i.e. a linear dependence between �! and �! yields the following expression.
! �! = = 1 (11) !∗
By using the definition of �! one can solve for the first mesh layer (Pijush K, Kundu, 1990, P.450)
!!∗∆!! !!! �! = → ∆�! = (12) ! !!∗
Where ∆�!is defined as the appropriate height of the first mesh layer for a desired �!value. When �! is put to one, �!has to be equal to one as well due to the linear dependence. When �! is equal to one, the velocity profile �(�) equals the friction velocity at one viscous length scale above the surface. Equation (12) can therefore be rewritten as:
∆� = ! = ! (13) ! !" ! By using equation (13) when choosing the first layer thickness of the mesh it is guaranteed that all sublayers in the turbulent boundary layer are resolved.
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2.4 SST k-� turbulence model The main thing why one must have a turbulence energy model is that one cannot afford to directly capture every scale of motion in the flow (Pijush K, Kundu, 1990, P.420). Therefore as an effort of decreasing the computational power needed, the mean flow is calculated. During the years a lot of models have been developed to be able to model turbulence energy, some more advanced than others. Wilcox explains two different types of turbulence energy models, one-equation model and two-equation model (David C. Wilcox, 1994, p.77 & 83). One thing that these equations have in common is that they both use the Boussinesq eddy- viscosity approximation which says that turbulence decays unless there is shear in the flow. However, the one-equation model is incomplete because it only relates the turbulence length scale with a typical flow dimension while the two equation model is complete in a sense that it provides an equation for the length scale or its equivalent.
The turbulence model used in this thesis is the shear stress transportation k-� model. It is well adapted to near wall problems and it requires less computational power than the more advanced two equation models (David C. Wilcox, 1994, p.84-89). The shear stress transportation k-� model combines the k-� model and the k-� model, where the k-� model is used in the inner region of the boundary layer and the k-� model is used in the free shear region (David C. Wilcox, 1994, p.84-89). For a more detailed view on how the shear stress transportation model is derived see appendix 7.1.
2.5 Effect of pressure gradient A curved body in a free stream will induce a pressure gradient. When the flow encounters the front of the curved body the stream lines converges resulting in an increase of velocity and consequently an increase in static pressure (Pijush K, Kundu, 1990, P.317). When the flow passes the top of the curved body the stream lines starts to diverge resulting in a lower static pressure and a higher free stream velocity. Pijush K, Kundu describes this in more detail using the boundary layer equation with x taken along the surface of the body:
! �(! !) = !" (14) !!! !"## !"
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Figure4: The change in boundary layer thickness for a flow around a curved body. The boundary layer thickness is compressed at the front of the body and it expands at the back of the body resulting in a rapid increase of drag (Pijush K, Kundu, 1990, P.317)
! When the free stream is accelerating the (! !) term becomes smaller than zero because !!! !"## the !" term is smaller than zero as can be seen in figure 7. From figure 7 one can also see that !" the flow behind the curved body experience a deceleration, resulting in a static pressure ! increase which according to equation (10) corresponds to a positive (! !) .When the flow !!! !"## switches from an accelerating to a decelerating flow a point of inflection is created. The point of inflection starts at the wall but as the pressure gradient increases the point of inflection moves higher up in the boundary layer which implies a deceleration of the fluid close to the wall, in other words the boundary layer increases in thickness (Pijush K, Kundu, 1990, P.319).
But why is the drag higher for a curved body than for a flat plate? Pijush K, Kundu explains it by saying that a plate with no pressure gradient increases its boundary layer thickness only by the viscous diffusion while the boundary layer thickness for a curved body increase both with the viscous diffusion and by advection away from the surface (point of inflection)resulting in a more rapid increase of boundary layer thickness.
2.6 Effect of flow separation An important phenomenon occurs when the adverse pressure gradient is so strong that the flow direction next to the wall changes, resulting in a region of backward flow. This phenomenon is called separation.
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Figure 5: Flow around a curved body. If the pressure gradient behind the curved body is high enough separation will occur. In this figure the separation starts at point S, resulting in a region of backward flow (Pijush K, Kundu, 1990, P.319)
It has been proven that the separation point is not only dependent on the geometry but also on the Reynolds number over the diameter of the body (��� = !" , � �� �ℎ� �ℎ������� �����ℎ) ! and if the boundary layer is laminar or turbulent (Pijush K, Kundu, 1990, P.319). For laminar boundary layer the separation point occurs earlier than for the turbulent boundary layer.
Figure 6: The left figure describes the separation point for a laminar boundary layer and the left figure describes the separation point for a turbulent boundary layer. The separation occurs earlier for a laminar boundary layer than for a turbulent boundary layer.
Pijush K, Kundu also mentions that for a boundary layer with a laminar flow the pressure wake downstream the separation points is approximately constant and lower than the upstream pressure, the drag coefficient is therefore constant in a large range of Reynolds numbers. However in the transitional region between laminar and turbulent flow the pressure inside the wake increases resulting in a drag coefficient decreases. For a fully turbulent flow the separation point slowly moves up stream as the Reynolds number increases resulting in a slowly increase of drag coefficient (Pijush K, Kundu, 1990, P.327).
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Figure 7: The drag coefficient for a circular cylinder with diameter d for different Reynolds numbers in a loglog scale.
In summary: For a high enough pressure gradient separation will occur, when separation is present in the boundary layer the velocity region close to the wall changes direction and a wake is created. It is difficult to calculate the drag using the boundary layer thickness method, instead the drag is calculated knowing the difference in pressure upstream the body and in the wake. At lower �� the �! values increase quite a lot due to the skin friction drag Pijush K, Kundu also mention that the drag coefficient is approximately constant for laminar flow (�� = 10! − 10!), decreasing in the transition region and slowly increasing in the turbulent region.
2.7 Drag coefficient
2.7.1 Momentum thickness �� There is a lot of ways to calculate the drag coefficient on a plate. The method used in this thesis will be focused on the momentum thickness and surface force integral (Pijush K, Kundu, 1990, P.313). The momentum thickens integral that has been used in this project is only valid for a plate with no pressure gradient. Even though this project is about calculating the surface drag for a plate with different surface geometries it can be used to validate the solution for a flat plate compare to the more accurate surface force integral (Pijush K, Kundu, 1990, P.312). A more exact solution of the boundary layer equation is possible by solving the Karman momentum integral (Pijush K, Kundu, 1990, P.315), which takes into account the pressure gradient. This is however a more complicated integral to discretize and that’s why the momentum thickness for a flat plate is used. However, the momentum thickness method can be used for milled tracks if the velocity profile is taken at some distance after the end of the milled track, if one can guarantee that no pressure gradient is present in the velocity profile.
As has been described earlier in the theory section, the momentum thickness is derived from when a momentum balance is taken before and after it has passed a plate, resulting in the equation:
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! ��!ℎ − ��!�� − ��∗�! = ��!� (15) ! !
One can also derive the momentum thickness formula form the definition of shear stress (David C. Wilcox, 1994, p.637):
� (�) = ��! !!! (16) ! !"
Which states that the shear stress over a plate is equal to the momentum gradient times density and velocity squared. The drag for an entire plate using the momentum thickness with no pressure gradient can therefore be derived as:
! ! ! � = � (�) �� = � � (�) �� = � ��! !!! �� = ���!� (17) !"#$ ! ! ! ! ! !" !
Where l is the length of the plate and b is the width of the plate. By normalizing equation 17 the drag coefficient can be calculated:
! !!" !"#$ !"#! !"! !! �!"#$ = ! = ! (18) !!!!" !!!!" ! !
It has also been shown that the drag coefficient for a plate with no pressure gradient can be written as (Schlichting, 1979, P.638):
� = 2 ∗ !! (19) !"#$ !
2.7.2 Force integration An alternative way of calculating the drag of a plate is instead of integrating over the boundary layer, integrate over the surface. There are two different types of drag forces acting on a plate, shear stress due to the zero velocity on the wall and pressure drag when obstacles are present (Schlichting, 1979, P.758-769). The drag due to shear stress and the drag due to pressure are given as:
�!"#$ = � � �! �� (20)
�!"##$% !"#$% = � − �! �!�� (21)
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�∞ ��! �
��! ��! � ��!
Figure 11: The pressure and shear stress directions on a milled track
Both the shear stress and the static pressure are in the x direction integrated over the surface of the milled track. Because the forces are integrated in the same direction they can be summed up together:
�!"#$ = �!"#$+�!"##$% !"#$%& = �! � �! �� + � − �! �!�� (22)
By normalizing equation (15) the total drag coefficient over the plate can be derived:
!!"!""#$!!!!"#$ !! ! !!!"! !!!! !!!" �! = ! = ! (23) !!!! !!!! ! !"# ! !"#
Where �!"# is the wetted area, i.e. the total area of the plate.
2.8 Effective dynamic pressure When calculating the drag coefficient from a small body placed in a boundary layer one has to consider the velocity distribution inside the boundary layer. This is done by introducing the effective dynamic pressure. What the effective dynamic pressure does is that it changes the velocity in the free stream dynamic pressure term such that it considers the height of the surface body and boundary layer thickness. According to Hörner(1975) the effective dynamic pressure for a turbulent boundary layer is given by:
! � ≈ �0.75 !" (24) !"" !
Where � is given as the dynamic pressure, �� as the cusp height and � as the boundary layer thickness. When the cusp height and boundary layer thickness ratio is high the effective pressure will increase and vice versa.
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� �
≠ � �(�) ≠ �
�� �� � �
�
EquationFigure 8: Explains(14) basically how the defines surface thebody dynamic with height pressure Hc relates over to thea surface velocity body distribution in such in athe way boundary that layer, i.e. depending on the height Hc of the surface body and the boundary layer height it will experience different kinds of drag,
Relating equation (24) to the drag coefficient will result in the following relationship:
! �� = �� 0.75 !" (25) ! !
Where ��! is given as the drag coefficient the surface body experience when there is no boundary layer, i.e. when it is experiencing the free stream dynamic pressure. However for cases when the surface body is inside a boundary layer the drag coefficient can be predicted
! using the hörner correction factor 0.75 !". !
One other way of describing the behavior of the drag coefficent inside a boundary layer is by using the 1/7-power distribution law which is given as:
! ! = !" ! (26) ! !
The dynamic pressure over the milled track can then be described as:
! ! ! ! ! ! ! !" ! ! ! !" ! !" ! �! = � � = � � = � � = � (27) ! ! ! ! ! ! !
This will correspond to the drag coefficient:
! �� = �� !" ! (28) ! !
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3 Method
3.1 Geometry The goal with this project is to get a more detailed understanding on how different flow conditions and different kinds of milled tracks affects the drag coefficient. To test this, a geometric model has been created using MATLB. This model is then imported to the CFD program ANSYS CFX from where different kinds of milled tracks has been tested with different inlet conditions. The 3D-model created in MATLAB is divided into three parts an inlet part, a middle part and an outlet part, see figure 1.
Inlet section Test section Outlet section U
y
z x Figure 1: The CAD model used in ANSYS CFX
The inlet section is used to prevent calculation singularities in the beginning of the test section including laminar-turbulence transition behavior. The test section is where the milled track is located and tested and the outlet section is there to give the flow a more natural downstream behavior such that the calculation will not be affected by the boundary condition at the outlet.
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3.2 Mesh When the geometry is done it is time to mesh. The thing one should think about when doing the mesh is the calculation of the first layer thickness to be sure that the viscous layer is resolved which is done by fulfilling the relation:
∆� = ! (1) ! !
That is dependent on which velocity resp. kinematic viscosity that is present in the flow, the first layer thickness of the mesh has to be adapted. The growth rate of the mesh in the vertical direction is calculated depending on how high the domain is and how many nodes it should be at the height. It was found that for a domain height of 0.05 meter a growth rate of 1.2 is sufficient to have a good resolution of the entire boundary layer. Due to the two dimensional milled tracks the resolution in the lateral direction is also important. By doing some trial and error it was found that 33 nodes per milled track will be sufficient:
Figure 2: Mesh resolution in x and y direction for a milled track
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-4 x 10 8
7.5
7
6.5 10 15 20 25 30 35 40 45 50 Pressure drag per milled trac milled per drag Pressure Nodes per milled trac x-direction -3 x 10 1.44
1.42
1.4
1.38
1.36
Friction drag per milled trac milled per Friction drag 10 15 20 25 30 35 40 45 50 Nodes per milled trac x-direction
Figure 3: The drag convergence for different milled track node resolutions. The top graph describes the pressure drag convergence and the bottom graph describes the friction drag convergence.
As can be seen in figure 3 the slope decreases continuously when the milled track node resolution increases. 33 nodes per milled track was used because the difference in the drag output relative to a 49 node resolution is minimal and the computational power needed to solve for a 33 milled track node resolution was found to be a fraction of what it took for the computer to solve for a 49 milled track node resolution. To further minimize the computational power needed the vertical resolution of 33 nodes per milled track was only done for the test section because a growth rate of the vertical resolution was put at the inlet and outlet from the test section making the mesh decrease in a direction away from the test section, see figure 4.
Figure9
In the z direction the width of the plate is resolved by two nodes, this is due to the symmetry around the y axis. For more information about mesh in general see appendix 7.2.1.
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3.3 Pre-process The next step in the solution process is to define the turbulence model, the heat transfer model and boundary conditions to the meshed CAD-model. First and foremost the fluid used in the domain is air at room temperature.The turbulence model that has been used is the Shear stress transport (SST) k-� turbulence model. It is used due to its ability to accurately calculate the turbulence close to the wall with the minimum use of computational power. Furthermore, due to the low velocities of the flow an isothermal heat transfer model is used, i.e. the temperature of the flow is assumed to be constant throughout the fluid domain. At last it is time to define the boundary conditions. There are 6 surfaces in the CAD-model inlet, outlet, ceiling, floor and two side walls.
Ceiling
Outlet Inlet
Sides
Floor
At the inlet a velocity and turbulence profile is imported. They are imported to be able to simulate different types of boundary layer thicknesses without changing the geometry of the model. They are imported from another model which consists of a long flat plate from which the velocity profile including the turbulence profile has been exported at different distances and at different free stream velocities with the free stream turbulence put as:
• 2% turbulence intensity • 1 mm eddy length
Attempts have been made to generate a velocity profile by the use of the 1/7 velocity power distribution law with a corresponding k-� turbulence profile using MATLAB but it did not work.
A translation periodicity condition was put on the two sides. By using the translational periodicity condition the flow will not be disturbed by close wall effects, because it can be seen as an infinity wide plate from where measurement is made in the middle of it. Furthermore the boundary condition at the floor was put to a no slip wall, i.e. by setting the velocity to zero. On the ceiling and outlet an outlet condition with the atmospheric pressure as reference pressure were applied. The outlet boundary condition is similar to the opening boundary condition but the user does not need to specify a temperature but only the pressure. This makes the transition between the flow volume and outside the flow volume more natural
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i.e. in a sense that the temperature does not need to be specified. For more detailed information regarding alternative boundary conditions used see appendix 7.2.2.
3.3 Post process When the model is solved the next step is to post process the results. The method used to calculate the drag at the test section is by the use of force integration. In ANSYS CFX there is a function called force which integrates the shear stress and pressure force over the plate. To verify the result from ANSYS CFX build in force integration the shear stress and pressure over the test section are also exported from the model and integrated using MATAB:s build in trapezoidal integration function.
4. Result The variables that has been varied during the CFD simulations are the cusp angle �, the Reynolds number along the plate ��� = !", the !" relationand the free stream angle of attack ! ! �. These parameters were chosen because they were found to affect the drag coefficient the most. For each simulations only one of these four parameters are varied while the other parameters are fixed.
Once again, the different parameters are defined as follows:
�
� �
� � � � �� �
�� � Figure 10: Definition of the variables that are going to be varied in the result section. The variables that are going to �� be varied are �, ���, and �. �
4.1 Varying � In this section � has been varied with constant ���=1E6, !"=5.3% and � = 0˚. This resulted ! in the following graph:
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Varying α 0,005
0,0045
Cd 0,004
0,0035
0,003 0 10 20 30 40 50 60 α
�� Figure 11: The drag coefficient ��for different cusp angles� with constants ��� = ���, =5.3% and� = �˚. �
For low cusp angles (� < 10˚) the drag coefficient is approximately constant and takes the value of the drag coefficient of a flat plate. For cusp angles higher than 14 degrees the drag coefficient starts to increase, it increases until it reaches a cusp angle of about 48 degrees and then it starts to decrease. By studying the post process plots from the CFD simulations it was found that separation occurred between the cusp angles 14 and 22 degrees which may be one explanation on why the drag coefficient increases. The decrease in drag coefficient after 48 degrees may be that the wake behind the milled track is so big that it starts to cover the milled track behind it which will result in smaller relative pressure difference between the front side and back side of the milled track. For more details about the behavior of the drag coefficient for a varying � see appendix7.3.
4.2 Varying ��� In this section the Reynolds number over the plate ��� has been varied for different cusp angles � with constant !"and �. !
20
Varying α 0,005
0,0045
Cd 0,004 ReL=1E6
ReL=3.6E5 0,0035
0,003 0 10 20 30 40 50 60 α
�� Figure 3: Drag coefficient plotted for different cusp angles and Reynolds numbers for a constant and �. �
By looking at figure 3 the drag coefficient decreases with increasing Reynolds number for cusp angles smaller than 28 degrees. But for cusp angles higher than 28 degrees the drag coefficient increases with increasing Reynolds number. One explanation for this behavior may be that the viscosity forces is higher than inertial forces for cusp angles smaller than 28 degrees which will result in a decline of drag coefficient with an increase in Reynolds number (Pijush K, Kundu, 1990, P.315). However, for cusp angles higher than 28 degrees the inertial forces dominates the flow regime resulting in a higher drag coefficient for a higher Reynolds number as can be seen in figure 4:
21
Varying α 0,0045 0,004 0,0035 0,003 0,0025 Cp, ReL=3.6E5 Cd 0,002 Cf, ReL=3.6E5 0,0015 Cp, ReL=1E6
0,001 Cf, ReL=1E6 0,0005 0 0 10 20 30 40 50 60 α
Figure4: Pressure- and skin friction drag coefficient plotted for two different Reynolds number and for different cusp angles.
Where �� and �� is defined as:
! �� = !"#$$%"# ��� �� = !!!!"# !"#$!! (1) !!!"# !!!"#
Comparing figure 3 with figure 4 one can see that the �� and �� lines intersect where the separation starts, i.e. where the inertial force becomes the predominant one. The skin friction is always bigger for the plate with lower Reynolds number as can be seen in figure 4, which is the case for smooth flat plates (Schlichting, 1979, P.638). However, for larger milled track angles the skin friction decreases while the pressure drag increases and due to earlier separation the pressure drag coefficient increase faster for higher Reynolds number which will result in higher drag for higher Reynolds numbers. For further investigation see appendix 7.3.
4.2 �� dependence � In this section the !" dependence will be investigated. For this simulation the Reynolds ! number is set to ��� = 1�6 and the cusp angle � = 31˚. By varying !" the drag coefficient ! had the behavior as depicted in figure 5.
22
CFD 0,0052
0,0051
0,005
0,0049
Cd 0,0048
0,0047
0,0046
0,0045
0,0044 0,06 0,07 0,08 0,09 0,1 0,11 0,12
��/�
Figure12
According to figure 5 the drag coefficient increases when !" increases. This can be explained ! by the effective dynamic pressure that takes into account the velocity distribution over a cusp (Hörner, 1965). According to Hörner the drag coefficient for a body inside a boundary layer is given as:
! �� = �� 0.75 !" (2) ! !
! Where �� is given as the drag coefficient for zero boundary layer and the 0.75 !" term ! ! corrects the zero boundary layer drag coefficient such that it includes the velocity distribution inside a boundary layer. By looking at equation (2) there should be a higher drag coefficient for a milled track with higher !" ratio then with a lower !" ratio. For example, imagine two ! ! milled tracks with the same cusp height but with different boundary layer height, it is logic to say that the milled track experience a lower boundary layer height �, has a higher drag coefficient than a milled track experiencing a higher boundary layer height because the milled track with a lower boundary layer height is affected with a higher velocity.
To test whether the Hörner relation holds for the CFD simulation in figure 2 simulations has to be made on a cusp with zero boundary layer height, to be able to get the ��! term. Unfortunately that sort of simulation has not been made, but by approximating a power function to the CFD curve in figure 2 it is possible to estimate the value of ��!:
23
! �� = 0.0148 ∗ 0.75 !" !.! (3) !
This is the function of the line in figure 5. By assuming the same ��! coefficient the corresponding hörner drag coefficient is given as:
! �� = 0.0148 ∗ 0.75 !" (4) !
The same process is also done for the 1/7 distribution law correction factor which resulted in the following drag coefficient approximation:
! �� = 0.013 !" ! (5) !
Plotting equations (3)-(5) in the same graph gives the results shown in figure 6
CFD, 1/7 law and Hörner correc on 0,0054
0,0053
0,0052
0,0051
0,005 CFD Cd 0,0049
0,0048 1/7 law
0,0047 Hörner correc on
0,0046
0,0045
0,0044 0,06 0,07 0,08 0,09 0,1 0,11 0,12
��/�
Figure13
As one can see from figure 6 the drag coefficient with the 1/7 power distribution law correction factor gives a better approximation than the Hörner correction factor.
24
4.3 Varying free stream angle of attack � Within the drag coefficient study different free stream angles of attack has been investigated. The free stream angle is defined such that for 0 degrees the milled track are transverse against the free stream and for 90 degrees the milled tracks are oriented along the plate.
For free stream angles between 0˚ and 35˚ the free stream angle of attack is defined from the domain with 0˚ free stream angle and for the free stream angles between 45˚-90˚ the free stream angle of attack is defined from the 90˚ domain. These two different domains with the corresponding angle ranges were chosen because it gave stable simulation results. See figure 7 for an illustration of the two domains.
When one varies the direction of the flow the distance traveled for a fluid particle is longer than if the flow has a 0˚ velocity vector. This will generate a higher Reynolds number for higher free stream angles and result in a higher drag coefficient. In order to make the Reynolds number constant along the surface the absolute value of the velocity is increased when the free stream angle of attack is varied.
� = 0˚ − 35˚ � = 45˚ − 90˚
� � U � U
� � x 0˚������ 90˚������
Figure 14: Two different domains are used when varying the free stream angle of attack 0˚ domain and 90˚ domain. These two different domains with the corresponding angle ranges was chosen because it gave stable simulation results
The �� result from varying the free stream angle of attack � is shown in figure 8. In figure 9 !" is plotted as a function of the angle of attack �, i.e. �� in relation to the drag coefficient at !"! zero degrees ��0.
25
Varying � 0,005
0,0045
0,004
Cd 0,0035
0,003
0,0025
0,002 0 10 20 30 40 50 60 70 80 90 100 �
Figure 15: Calculating the drag coefficient Cd while varying the free stream angle of attack �
Varying � in for Cd/Cd0 1
0,95
0,9
0,85 Cd/Cd0
0,8
0,75
0,7 0 10 20 30 40 50 60 70 80 90 100 �
Figure 9: Varying � for the procental difference in Cd from 0̊ to 90̊.
To explain this behavior one has to study the direction of the velocity
26
Ux � U � Uz � U Ux=cos(�)U � �
Figure 16: The velocity free stream angle of attack is varied to simulate different milled tracks angles
The drag force over and obstacle is defined as:
! � ˚_!"#$ = �� ˚_!"#$�!"#0.5� � (6)
When the velocity direction changes the direction of the force will also change: ! ! � ˚_!"#$ = �� ˚_!"#$�!"#0.5� � → �! = ��!�!"#0.5 ���� � � → �! = ! � ˚_!"#$ cos � (7)
With this knowledge the corresponding drag coefficient for different free stream angles of attack � can be presented as:
! !! ! ˚!"#$ !"# ! ��! = = (8) !!!"# !!!"#
Equation (8) will be zero at � = 90˚. In reality at � = 90˚ the drag coefficient will be equal to the CFD simulation at � = 90˚, in other words;
! ��! = � ˚!"#$ + (� ˚!"#$ − � ˚!"#$) cos � (9)
We can now plot equation (9) together with the CFD solutions in figure 8:
27
Varying � 1,2
1
0,8
0,6 CFD Cd/Cd0
0,4 Equa on
0,2
0 0 10 20 30 40 50 60 70 80 90 100 �
Figure17: CFD simulation verse the cos(�)^2 dependence
In conclusion: The ���(�)^2 dependence is a fair approximation to the simulated CFD result.
28
5. Conclusion What relation can be used when predicting the dragcoefficientfor different milled surfaces for different fluid environment? By using the result gained from the different CFD simulations an approximation of the drag coefficient dependence can be made as follow:
! �� = � + � = �� !" ! cos � ! + � (10) !"#$ !"#$ ! ! !"#$
The drag coefficient is divided into one pressure �!"#$ and one friction term �!"#$. ��!is ! given as the drag coefficient over the cusp when there is no boundary layer. The term !" ! ! take into account the present of a boundary layer and the cos � ! term takes into account the free stream angle of attack.
From the result section regarding varying ��� and � it was found that the drag coefficient is strongly dependent on these two factors. By using the knowledge of the 1/7 power distribution law factor it is possible to derive an expression for the ��! term.
! !" ! !" �� = ��! → ��! = ! (11) ! !" ! !
��! can be calculate as a function of the milled track angle � and ���. To make it easier to use in the formula the milled track angle is substituted to !", i.e. the milled track height over !" the milled track width. Because �� has already been calculated for different !" and ��� by !" the use of equation (11) ��! can be calculated:
29
Cd for different ReL and for varying ��/�� 0,012
0,011
0,01 ReL=1E6
Cd 0,009 ReL=3.6E5 0,008
0,007
0,006 0 0,02 0,04 0,06 0,08 0,1 ��/��
Figure 18
Due to a singularity point for milled track with a zero milled track angle that point is not plotted in the figure. The point corresponding the decline in drag coefficient is not plotted either because it will lead to an over complex ��! formula, this formula will therefore over predict the drag for milled track angles before separation and for milled track angles when the wake behind the milled track is so big that it starts to cover the milled track behind it.
By looking at figure 9 one can see that there is a linear behavior between �� and !" which is !" denoted ��! in this report:
�� ∝ �� = � !" (12) ! ! !"
Where �! determines the slope of the linear curve, which by looking at figure 9 is dependent on the Reynolds number ���.For high Reynolds numbers the slop is steeper resulting in a lager value of�! than for a small Reynolds number. Because only two Reynolds number has been tested it is difficult to derive an exact formula for �! as a function of the Reynolds number having. Having this said, �! for ��� = 1�6 and ��� = 3.6�5 are calculated to
�! = 0.076 ��� ��� = 1�6 �! = 0.033 ��� ��� = 3.6�5
The full drag coefficient equation can now be derived:
! �� = �1 !" !" ! cos � ! + � (13) !" ! !"#$
30
Just to be clear, this equation will over predict drag for milled surfaces where separation has not occurred and for milled track angles when the wake behind the milled track is so big that it starts to cover the milled track behind it.
To be absolutely sure that this drag coefficient formula is conservative towards the CFD it is tested for a CFD simulation from where the Reynolds number is set as one million and for where the milled track height is varied:
Cd as a func on of ��/�� for CFD and the derived 0,018 formula 0,016
0,014
0,012
0,01 Equa on Cd 0,008 CFD
0,006
0,004
0,002
0 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 ��/��
Figure 19: �� is varied with a constant Ae for ��� = ���
As can be seen from figure 24the figure the drag coefficient is conservative towards the CFD. Because this equation does not consider the slip stream tendency for high milled angles it is not valid for that range. However, for the range that we are interested in !" = 0 − 0.1 the !" equation will give a good approximation to the CFD results. Also worth mentioning is that the drag on the surface with milled tracks is very low and for !" < 0,05 is almost equal to the !! drag coefficient of a smooth surface.
The result from simulation different types of milled surfaces can also be related to sand grain roughness models. The plot below relates different cusp angles with different sand grain roughnesses:
31
Varying δ/L
0,0044 0,0042 0,004 α=22˚ 0,0038 α=0˚ Cd 0,0036 α=10˚ 0,0034 k=6.4 μm 0,0032 0,003 k=12.8 μm 0,0028 0 0,02 0,04 0,06 0,08 0,1 0,12 δ/L
Figure 20: Delta/L symbolizes the boundary layer inlet height over the total length of the plate. This plate has a Reynolds number of ReL=3.6E5.
A cusp angle of � = 22ᵒ corresponds to a �� = 0,1�� = 100µ� which is equivalent to a sand grain roughness of � = 6,4µ�. This is an interesting observation which says that a sand grain roughness with an equivalent sand grain roughness on � = 100µ� will give a much higher drag losses than a milled surface with a milled track height of �� = 100µ�.
It is very important to take in consideration that this formula is derived from the assumption that the CFD results on ANSYS CFX are correct. A physical test has not been made to verify those result, however this has to be done to prove that this formula is valid.
32
6. Future work
The obvious next step in this project is to build a test rig to be able to tell whether the derived equation is valid when physical measurements are made. Further investigations on the Reynolds number dependence for the drag coefficient formula can be furthered investigated. By testing a range of Reynolds numbers one may be able to derive a drag coefficient equation that is also dependent on the Reynolds number.
Apart from investigating the Reynolds number, efforts on including the slip stream tendency for high milled track angles in the drag coefficient formula can also be made, doing so the formula will be valid for a higher range of milled track angles.
33
7. Reference
Books
David C. Wilcox, 1994, Turbulence modeling for CFD, 2nd edition, p.77-89 & 637, DCW Industries
Pijush K. Kundu, 1990, Fluid Mechanics, 1st edition, p.213-450,Academic Press
Schlichting, 1979, Series in Mechanical Engineering, 7th edition, p. 638-769,McGraw Hill Higher Education
S. F. Hörner, 1965, Fluid dynamic drag: Practical Information on Aerodynamic Drag and Hydrodynamic Resistance,2nd edition, p. 138,HoernerInc
Web
DORMER, Fräsning, dbvjpegzift59.cloudfront.net/16753/151859-z0nFf.pdf, (Received 2014- 09-28)
Tom Benson, 2014, The Drag Equation, http://www.grc.nasa.gov/WWW/k- 12/airplane/drageq.html (Received 2014-11-25)
Andre Bakker, 2012, Applied Computational Fluid Dynamics,http://www.bakker.org/dartmouth06/engs150/07-mesh.pdf(Received 2014-11-20)
M. S. Kilic, G. Haller, A. Neishtadt, 2005, Unsteady fluid flow separation by the method of averaging, http://torroja.dmt.upm.es/pubs/2011/rgm_jj_philtrans11.pdf (Received 2015-03- 30)
R.A.C.M Slangen, 2009, Experimental investigation of artificial boundary layer transition, http://www.lr.tudelft.nl/fileadmin/Faculteit/LR/Organisatie/Afdelingen_en_Leerstoelen/Afdel ing_AEWE/Aerodynamics/Contributor_Area/Secretary/M._Sc._theses/doc/090421_Thesis_R oel_Slangen.pdf(Received 2014-10-24)
Ricardo Garcia-Mayoral and Javier Jimenez , 2011, Drag reduction by riblets,http://torroja.dmt.upm.es/pubs/2011/rgm_jj_philtrans11.pdf (Received 2014-12-15)
ElieBou-Zeid, Marc B. Parlange, and Charles Meneveau, 2007: On the Parameterization of Surface Roughness at Regional Scales, http://journals.ametsoc.org/doi/pdf/10.1175/JAS3826.1(Received 2014-04-14)
34
8. Appendix
8.1 Theory
8.1.1 k-� model
The shear stress transportation k-� model has a similar form to the standard k-� model with a difference in that the SST k-� incoporates a cross-diffusion derivative term in the � equation, it also includes an blending functionwich acts as a switch between the k-� and k-� model. Furthermore the turbulent viscosity term is modified to account for the transported turbulent shear stress, the modeling constant are also different (ANSYS Help Viewer, SST k-� model). The two main equations are given as.
Where the first one calculates the turbulent kinetic energy and the second one calculates the specific energy dissipation (David C. Wilcox, 1994, p.84-89). The corresponding variables are given by:
35
The variables above describe the effective diffusivity for the SST k-� model (ANSYS Help Viewer, SST k-� model). The parameters that describe the turbulent production are given by:
Furthermore the turbulence dissipation of k and � are given by:
Because the SST k-� models is based on both k-� model and k-� model a cross-diffusion term has to be included and it is defined as:
The SST k-� model has the following modeling constants (ANSYS Help Viewer, SST k-� model):
36
8.2 Method
8.2.1 Boundary conditions
The boundary is the major part of building a fluid model. It is where the solution starts and ends, if they are set wrongly the entire model will be invalid. There are a lot of boundary conditions one can choose from, in our case we want to study several boundary conditions to be able to see which one that is the most appropriate when measure the induced surface drag, for example should we study a closed flow with pressure gradients or an open canal flow with no pressure gradient. Another example can be if the inlet should have a developed velocity profile, a straight velocity profile or a flow volume with a slip condition before the inlet. This section is divided into five different boundary condition subsections, stretching from the inlet to the outlet. Each section will involve how the different boundary conditions work and how they are implemented in ANSYS Workbench CFX.
8.2.1.1 Inlet Several inlet conditions have been tested. The first inlet condition consists of a straight velocity profile. In an attempt to make the boundary conditions more consistent, instead of having only velocity the total pressure was used because it also involves static pressure. With and opening condition this resulted in the following graph:
Figure 21: Opening boundary condition at the ceiling with a straight inlet velocity profile The second inlet condition used in this test rig was an air volume with a slip wall condition at the bottom of the rectangular volume. By doing this a more natural flow will pass the inlet and maybe then decrease the shear stress error in the beginning of the plate.
37
Slip bc No slip bc
The third inlet condition that has been used is to import a velocity profile to the inlet. The advantage of importing a developed velocity profile is that it decreases the length of the plate needed to get a high enough boundary layer thickness which makes measurement of the surface drag much easier. This velocity profile is then exported, the values exported are total pressure, total temperature, velocity direction, turbulence kinetic energy and turbulence dissipation.
8.2.1.2 Walls
To minimize the computation power needed for this simulation the width of the plate is such that it can be meshed by one mesh cell. And by inserting symmetry commands on the walls the plate becomes theoretically infinite long, in the case with a flat plate a slip wall condition will be equivalent. By using symmetry/slip wall boundary condition and not no slip wall condition the user does not have to bother about wall effects (Pijush K, Kundu, 1990, P.450). The disadvantage with this method is that it kills all the transverse velocity components.
One alternative way of treating the wall condition is by using a translational periodicity boundary condition. By inserting this condition the width of the plate will be infinite. The advantage with the translation periodicity compare to the symmetry condition is that it will not cancel the transverse velocity components.
8.2.1.3 Test plate At the floor of the test domain a no slip wall condition is added. i.e. the velocity at the wall is zero.
8.2.1.4 Ceiling Three boundary conditions have been used at the ceiling, symmetry, opening and outlet boundary condition.
The idea with the symmetry condition is to simulate a pipe flow, which is a type of flow within a closed conduit.
38
Opening was the second boundary condition used. With an opening boundary condition the user defines the opening pressure and the opening temperature. This boundary condition makes the flow into an open canal flow.
The outlet boundary condition is similar to the opening boundary condition but here the user does not need to specify a temperature just only a pressure. This makes the transition between the flow volume and outside the flow volume more natural i.e. in a sense that the temperature does not need to be specified.
8.2.1.5 Outlet The outlet condition uses the same boundary conditions as the third ceiling condition, i.e. at the outlet a static outlet pressure is set.
8.2.1.6 CFD results for varying boundary layer conditions
The first plot describes the shear stress on the plate when a symmetry boundary condition is set at the ceiling of the rectangular volume and a straight velocity profile is set as inlet boundary condition:
Figure 22: Shear stress distribution along the plate with symmetry as boundary on the ceiling
Due to the symmetry boundary, the flow will develop into a proper pipe flow, if the length of the plate was longer the shear stress of the simulation will go over the shear stress of the analytical solution, due to that the analytical solution uses a no pressure gradient assumption. The pipe flow has an higher shear stress and it may be due to it will get a contribution by the shear stress of the ceiling.
39
When an opening boundary condition is applied (with a straight velocity profile inlet boundary condition) on the ceiling the following plot was obtained.
Figure 3: Shear stress distribution along the plate with opening as boundary on the ceiling
The shear stress error is a bit smaller than when symmetry boundary condition is used. But it seems to be a constant offset error with a few Pascal. The flow in this test setup can be approximated by a proper canal flow, the error can be described as if the local friction coefficient has a permanent offset error.
The next simulation done is with an opening condition at the ceiling with an air volume in the beginning of the plate
40
Figure 4: Shear stress distribution along the plate with opening as ceiling boundary with an air volume in the beginning of the test plate
The shear stress did not decrease in the beginning it actually increased. One explanation can be that the numerical simulation in ANSYS assumes a laminar flow in the beginning of the plate and after a few centimeters in to the plate the turbulence approximation becomes valid. While the shear stress error increase the momentum thickness error stays the same.
8.2.1.7 The boundary conditions used in the final domain
Figure23: The boundary condition used in the final domain
41
8.3 Results
8.3.1 Varying inlet boundary thickness � Simulation on a smooth flat plate, two different types of milled tracks plate and two different types of sand roughness plates has been tested for varying inlet boundary layer thickness. The velocity is put to � = 37.16 m/s. The sand roughness models have been tested to give an understanding on what sort of sand roughness models that gives approximately the same drag coefficient as a plate with milled track. The different CFD simulations are plotted below:
Varying δ/L
0,0044 0,0042 0,004 α=22˚ 0,0038 α=0˚ Cd 0,0036 0,0034 α=10˚ 0,0032 k=6.4 μm
0,003 k=12.8 μm 0,0028 0 0,02 0,04 0,06 0,08 0,1 0,12 δ/L
Figure 24: Delta/L symbolizes the boundary layer inlet height over the total length of the plate. This plate has a Reynolds number of ReL=3.6E5.
As can be seen in figure 6 the drag coefficient decreases with increasing boundary layer inlet height. Every line seems to have the same drag- / boundary thickness dependencies but depending on the size of the cusp angle or sand grain size the line either shifts up or down, if the grain size is high and/or if the cusp angle is high the line shifts up. For very small cusp angles the drag coefficient is the same as for a flat plate, as can be seen in figure 6. One can from this plot also see similarities in the sand grain models and the milled tracks. Figure 6 shows that a sand grain model with a sand grain size of 6.4 [µm] is equivalent to a milled track plate with a � = 22˚. In the simulation above � = 22̊ corresponds to a �� = 0,1�� = 100µ� which isequivalent to a sand grain roughness of� = 6,4µ�.
8.3.2 Varying � and �� � To calculate the drag coefficient for a higher range of milled track angle the milled track angle has been varied by varying the milled track height �� and having the milled track width �� put as constant. This result in a varied � and a varied !" for each calculated drag ! coefficient. However it was found that the milled track angle has a larger effect on the drag
42 coefficient than the fraction !" which makes this study somewhat feasibly, i.e. the drag ! coefficient curve is determined mainly by the milled track angle. Varying α and ��/� 0,008
0,007
0,006
0,005
Cd 0,004 Milled tracks
0,003 Flat plate 0,002
0,001
0 0 20 40 60 80 100 120 140 160 α
Figure 25: Varying cusp angle and varying Hc/delta
For low cusp angles (� < 17˚) the drag coefficient is constant and takes the value of the drag coefficient of a flat plate. For cusp angles higher than 17 degrees the drag coefficient starts to increase, it increases until it reaches a cusp angle of about 76 degrees and then it starts to decrease. By studying the post process plots from the CFD simulations it was found that separation occurred between the cusp angles 10 and 21 degrees. To clarify on what is happening in the graph the pressure drag coefficient �� and the skin friction coefficient �� will be presented for the cusp angles 0, 17, 40, 75 and 135 degrees.
43
-3 x 10 α=0 14 α=17
12 α=40 α=75 10 α=135 Cusp 8
6
4 Cf
2
0
-2
-4
-6 0.085 0.0855 0.086 0.0865 0.087 0.0875 0.088 0.0885 0.089 0.0895 0.09 x [m]
Figure 26: Skin coefficient plot for different cusp angles. The milled track of a � = �� cusp has also been plotted, it is only there for the reader to refer he result to the cusp geometry.
By studying the skin friction coefficient over the milled tracks in figure 8 one can draw the conclusion that for higher cusp angles the skin friction coefficient decreases. The change in shape of the Cf curves after � = 17˚ is due to separation, i.e. separation for this setup occurs after 17˚. While the skin friction coefficient decreases for increasing cusp angle the pressure drag coefficient increases with increasing cusp angle:
44
0.1 α=0 α=17 α=40 α=75 0.05 α=135 Cusp
0 Cp
-0.05
-0.1
0.085 0.0855 0.086 0.0865 0.087 0.0875 0.088 0.0885 0.089 0.0895 0.09 x [m]
Figure 27: Pressure coefficent plot for different cusp angles. The milled track of a � = �� cusp has also been plotted, it is only there for the reader to refer he result to the cusp geometry.
Between cusp angles 0-17 the drag coefficient is approximately the same, this may be explained by looking at the average value of the skin fiction- and pressure coefficient and see that it is approximately the same as for a flat plate. For cusp angles between 17 degrees to 75 degrees the pressure drag coefficient increases more than what the skin friction coefficient decreases resulting in a higher drag coefficient. However for cusp angles between 75-135 degrees the drag coefficient decreases, an explanation can be that the separation line for a milled track is far enough upstream such that the milled track behind is lying in the wake of that milled track. By looking in figure 9 the Cp difference for a cusp with � = 75 is higher than for a cusp with � = 135 which result in a lower drag coefficient for the higher cusp angle which may explain the decrease in drag coefficient. To explain this matter in a more detailed view the static pressure has been plotted before and after a milled track to make it easier to see the difference in pressure:
45
-4 x 10 0
-1
-2 Cusp height
-3 Before cusp After cusp
1.0128 1.0129 1.013 1.0131 1.0132 1.0133 1.0134 1.0135 1.0136 Pressure 5 x 10 Figure 28: The static pressure at the front and backside of a milled track with cusp angle � = ��˚. The difference in pressure is correlated with the pressure drag, higher pressure difference higher pressure drag
-4 x 10 1 Before cusp 0 After cusp
-1
-2
-3
Cusp height -4
-5
-6
-7 1.0129 1.013 1.0131 1.0132 1.0133 1.0134 1.0135 1.0136 1.0137 1.0138 Pressure 5 x 10 Figure 29:The static pressure at the front and backside of a milled track with cusp angle α=135˚. The difference in pressure is correlated with the pressure drag, higher pressure difference higher pressure drag
Comparing figure 10 with figure 11 one can tell that the pressure difference is higher for the milled track with the low cusp angle than for the milled track with the higher cusp angle, which is due to the increasing size of the pressure wake resulting in equal pressures on the both sides (displayed as vertical lines in figure 18-19 above).
46
�� 8.3.3 Varying ��� and � In this section the Reynolds number over the plate ��� has been varied for different cusp angles �.
Varying ReL 0,008
0,007
0,006
0,005 ReL=3.7E5 Cd 0,004
0,003 ReL=7.5E5
0,002 ReL=1E6
0,001
0 0 20 40 60 80 100 120 140 160 α
Figure 30: Drag coefficient plotted for different cusp angles and Reynolds numbers. The black line is the separation line which points out for which cups angles the separation starts, i.e. the first time where the different ��� lines intersect with the black line.
By looking at figure 12 the drag decreases with increasing Reynolds number for cusp angles smaller than 23 degrees. But for cusp angles higher than 23 degrees the drag coefficient increases with increasing Reynolds number. One explanation for this behavior may be that the viscosity forces is higher than inertial forces for cusp angles smaller than 23 degrees which will result in a decline of drag coefficient with an increase in Reynolds number (Pijush K, Kundu, 1990, P.315). However, for cusp angles higher than 23 degrees the inertial forces dominates the flow regime resulting in a higher drag coefficient for a higher Reynolds number as can be seen in the figure below:
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Varying ReHc 0,008
0,007
0,006
0,005 Cf, ReL=1E6 0,004 Cp, ReL=1E6
0,003 Cf, ReL=3.7E5
0,002 Cf, ReL=3.7E5 Pressure/firc on drag
0,001
0 0 20 40 60 80 100 120 140 160 -0,001 α
Figure 31: Pressure- and skinfriction drag coefficent plotted for two different reynolds number and for different cusp angles.
Where �� and �� is defined as:
! �� = !"#$$%"# ��� �� = !!!!"# !"#$!! (1) !!!"# !!!"#
Comparing figure 12 with figure 13 one can see that the �� and�� lines intersects where the separation starts, i.e. where the inertial force becomes is the predominant one. Before the separation the skin friction coefficient is higher for the one with the lower Reynolds number and approximately equal pressure drag coefficient which causes the decrease in drag coefficient for high Reynolds numbers.
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