chapter 3 - a cfd theoretical analysis

7/26/2019 Chapter 3 - A CFD Theoretical Analysis

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Chapter 3 Numerical Methodology

As mentioned before the software Star-CCM+ (10.04.009-R8) will be used toperform all the calculation necessary to compute lift, drag, lift coefficient, drag coefficient,

pressure field, velocity field and other variables to perform a complete external

aerodynamic analysis of the Bumerangue EX27 Cross Country.

The software make use of different approaches regarding numerical computation.

It has two-dimensional and three-dimensional computational tools but in this study only

the three-dimensional approach will be used in the calculation once the target net

variables such as lift and drag depend of the three-dimensional geometry. The geometry

was drawn using CATIA V5 a technical software used in a variety of applications to

design parts of any geometry encountered at engineering.

It is imperative to notice that only half of the geometry is needed to calculate all

the variables since the airplane is symmetric at the longitudinal plane (xz at the figure).

The symmetry reduces the price of the simulation by allowing only half body calculations

that can be extrapolated to the full body of the aircraft.

Before starting any setup of the simulation the software need a mesh domain

where it will perform the calculation per volume using the finite volume method.

Figure 1 - Rendered Half Geometry at Catia V5


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3.1 The Finite Volume Method

Finite volume method is based on the discretization of the integral forms of the

conservation equations. Each physical equation is integrated into a control volume and

then discretized at each control volume centroid. The control volumes are commonly

called as cells. Each cells face interact with the other cells so, despite the calculation

occurs mainly to their centroids, they are then smoothed to follow contours and other

flow gradient properties such as pressure and velocity.

Figura 2 - Different geometries used to discretize control volumes

Figure 3 - Discretization method


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The conservation equations that were discretized at the software are listed. The

general equations wont have the unsteady term once that the simulation will be

performed assuming permanent regime and incompressible flow (no density variation).

This solving approach is commonly known as RANS (Reynolds Averaged Navier-

Stokes).

The continuity (or mass conservation equation):

( ) = 0 Eq. (1)

The Momentum equation:

= 1 Eq. (2)

Where the variables i and j relate to the coordinates x,y,z generating three equations

of moment. One for each direction.

Accordingly with the RANS methodology (Reynolds,1884), an average process is

applied to the equations (1 and 2) as follows:

= 1 Eq. (3)

The knowledge of the averaged field does not allow the calculation of the advective trip:

Eq. (4)

Hence, it is necessary the decomposition into two terms: the average () and thefluctuation () as follows:

= Eq. (5)


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The decomposition of the term originates an additional tensor:

= Eq. (6)

Where is the Reynolds tensor. To model it the Boussinesq hypothesis (1877), thatsays the momentum transfer caused by turbulent eddies can be modeled with eddy

viscosity:

= 2 23 Eq. (7)=1

2

Eq. (8)

Where is the turbulent viscosity, is the field average deformation tensor and isthe turbulent kinetic energy. Incorporating the equations (6,7 and 8) to the governing

equations (1 and 2) the final equations result:

( ) = 0 Eq. (9)

= 1 Eq. (10)

The energy equation will not be solved since the flow at this application have negligible

temperature variations hence their solution will not pay off the accuracy reached

comparing it with the increase of the computational cost.

After the discretization of the equations the software ensemble the equations related to

the cells centroids using a matrix, then resulting into a problem of linear equations. The

software solves the matrix iteratively, reaching a final solution after the residual of all

cells calculation converge.

3.2 Mesh Domain Size and Boundary Conditions Study

When simulating aerodynamics of any geometry it is important to certify that the

mesh domain drawn around the geometry does not have any interference in the result.

This analysis is the first point that has to be validated in order to achieve results with nodomain interference. The domain used for this study is a semi-spherical domain.


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The characteristic length of the airplane is = 4.6220 . Some authors say that inorder to have a result that does not have any domain interference the domain must have

a diameter of at least 10 times the characteristic length. The next figures show the

different domain sizes simulated to better comprehend and choose the best domain size

for this problem.

Figure 4 - Different Domain sizes schemesnot in scale.

A tridimensional mesh was created using the software ICEM CFD (Ansys Inc), a

commercial preprocessing software that creates the volumes that will be

simulated in the CFD software (Star-CCM+). The mesh parameters are described

in the table (1).


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Table 1 ICEM Set up for Meshing

Mesh Type Tetrahedral/Pyramidal/Prisms

Max Element Size (Farfield) 10 7mm

Max Element Size Wing & Canard 50

Max Element Size Winglet 60

Number of Density Entities 21

Number of Prism Layers 22

Height of First Prism Layer 0.2

Total Prism Layer Height 54 mm

Prism Layer Growth ratio 1.2

Minimum Quality (ICEM) Pr-Prism Layers 0.35

Figure 5 Mesh Example (Blue plane Symmetry plane; Grey plane Auxiliaryplane)

The mesh was separated into six regions (Symmetry Plane, Wing, Fuselage,

Canard, Winglet and Far-field). Each of the regions has different boundary conditions.

On the symmetry plane.

The solution that is obtained with a symmetry plane boundary

is identical to the solution that would be obtained by mirroring the mesh about the

symmetry plane (in half the resulting domain). The wing, fuselage, winglet and canard

regions were separated so they could have different element sizes on the mesh, allowing the

mesh to better follow the leading edge and wing profile curves improving the quality of the

mesh. Their boundary condition was set to Wall or no-slip condition. The Far-field region or

the outside shell of the semi-spherical domain was set to Velocity Intake condition i.e. the

hall shell had the velocity set to a constant, the cruise speed in the boundary conditions.

The software set up for the study of size and boundary conditions was:


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Table 2 - STAR-CCM+ Set up

Three-Dimensional

Steady (RANS)

Constant density

Segregated Flow

Turbulent

K-Omega (SSTMenter)

(default Eq. constants)

All y+ Wall Treatment

= 200 = 87.45556 /

= 8.745556 / = 10

3.3 Refinement Study

Although the meshes built to check the domain influence in the result had the a close

number of cells a further study is required regarding the increase of the cell number oncethat more cells would provide better information (calculation) about the smaller scales of

the flow hence have influence at the final result.

Four meshes were made in order to check the number of cells influence. They were

made using the same ICEM set up shown in the table 1. One mesh density was modified

to increase the number of cells closer to the aircraft.

Figure 6 Mesh Density configuration


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The number of cells of each mesh is listed in the table (3):

Table 3 - Cell number of Refined meshes

Mesh n N of Cells

1 3965144

2 6138069

3 7654387

4 9700551

The simulations set ups used to calculate the influence and the convergence of the final

result were the same used to analyze the domain influence (Table 1). The final mesh to

be used in the simulations for all the aircraft conditions will be chosen based on the

convergence of each mesh and the time taken to finish the calculations.

3.3 Turbulence Models Study

Given the transient nature of the problem related to the flow instability the solution

has to take into account the influence of the flow variations related to the turbulence.

The main parameter to check if the flow has or not turbulent behavior is the Reynolds

number. It relates the degree of influence of inertial forces over the viscous forces i.e,

whether the flow suffers great or few influence of its viscosity. For the domain, boundary

conditions and refinement study, one flow condition was used in order to standardize the

analysis. The cruise condition Reynolds number is:

= =87.45564.6221.460410 = 2.7610A reference value largely used at the literature for external aerodynamic flow is

that air flows with Reynolds numbers higher than 5 10 can be considered fullyturbulent. Hence, the flow at cruise condition is turbulent and can be modeled as such.Three turbulence models were compared using the same flight condition (cruise)

and the same software set up only changing the turbulence model to one of the three

shown in the table(4).


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Table 4 - Turbulence models Simulated

Turbulence ModelN of

Equations

Spalart-Allmaras 1

(SST)Menter 2 (realizable two layer) 2

Each of the three models gives an equation (or more) that models the influence of the

Turbulent viscosity (at the calculation of flow moments given by equation (10).3.3.1 Spalart-Allmaras Model

The Spalart-Allmaras (1992), at first, emerged due to the necessity of complexflow simulations mainly in the external aerodynamics for aerospace applications area.

Accordingly with Lorin et al (2006) this model consists in an empiric modeling of the

production, trip, diffusion and destruction terms of the turbulent viscosity in the flow.

This model was design to external aerodynamic applications where there is free-

stream, walls interaction with high Reynolds Number and laminar regions. It is defined

as a turbulence model of one equation seen that only one equation models the trip term

for the auxiliary variable

, modeled by:

= 1 1 [ ] [ ] Eq. (11)

Where the right hand terms represent respectively: production of turbulent

viscosity, diffusion, dissipation, destruction and the transition to turbulence. The turbulent

viscosity is defined depending of the auxiliary variable along with a dumping functionfor close wall regions of flows (

as follows:

= ; = ; = 7.1 Eq. (12,13,14)

The production term is composed by a constant (, a dumping factor for wall regions(and for a modified deformation rate defined by:

=

; = 1

1 ; = 0.41; = 0.1355 Eq. (15,16,17,18)


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The modified deformation rate depends on the distance which is the distance to theclosest wall and is the absolute value of the deformation rate calculated with filteredvariables of the field, ie:

=2 Eq. (19)

The destruction term is calibrated by an non-dimensional function . This function isdefined by an unitary value for the logarithmic boundary layer, intensified the term of

destruction, as the calculation closes to the wall region, and tend to zero in the far

distances. This function is given by:

= 1 ; = ; ; = 0.3 ; = 2.0 Eq. (20,21,23,24,25)

The model is implemented on STAR-CCM+ and will be the primarily base of comparison

for aerodynamic forces once proven that the calculation can give closest to experimental

results regarding this forces.

3.3.2 (SST Menter) ModelThis model combines various desired characteristics of a model of two equations.

The two greatest characteristics of this model are the ponderation of the wall distances

through constants and the growing limitation of the turbulent viscosity at highly shear

regions of flows. The modeling by regions of this model make use of the Wilcox (1988) at the boundary layer regions and the model from Launder and Sharma(1974) for far from boundary layer regions.

An advantage of this model is that it is capable of modify the turbulent viscosity

to better predict detaching close to wall (boundary layer) regions. The disadvantage of

two equation models is that they cannot predict detaching in boundary layers. The SST-

Menter model use blending functions to eliminate this problem.

Menter (1992), presented the SST model (Shear Stress Transpor), from the formulation

proposed by Menter, called New Baseline Model. This model is used for general

applications. The model is also implemented on STAR-CCM+ and will be the secondary

base of comparison for aerodynamic forces.


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3.3.3 (Realizable) ModelAs mentioned, the traditional model is not well suited in resolving flows in

near-wall regions (Two equation model). At or near the viscous sublayer, the viscous

forces are dominant over the turbulent forces, effectively damping any effects of

turbulence. The two-layer approach applies a modified model in the viscous sublayer

region, allowing to be applied in meshes that could otherwise be unsuitable for sucha model. The two-layer approach is expected to produce better results with a mesh that

is otherwise not optimized for a scheme. One drawback of the two-layer approachis the fact that, as mentioned, more traditional schemes could be expected toperform poorly. The model is also implemented on STAR-CCM+ and will be the third

base of comparison for aerodynamic forces.

3.4 Wall Treatment

All of the three turbulence models require a mathematical treatment of regions close

to wall since the viscous sublayer depends on very fine mesh in order to achieve

acceptable results.

The wall regions present higher rate of deformation in flows, which makes the region

with greater effects of turbulence, therefore harder to model and achieve results close to

the physics in this matter. For this reason wall functions were developed in order to model

the interaction between the viscous sublayer of the boundary layer and the far from wall

regions of the flows. The close to wall flow is generally divided into three other regions.

One inner layer so called linear layer, where the flow can be approximated as laminar,

and the velocity grows linearly with the increase of the wall distance. A more external

layer called logarithmic layer where the turbulence effects are more predominant and

one intermediary layer so called dumping layer that lays between linear and logarithmic

layer.


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Figure 1 General Wall treatment Law for turbulent flow at smooth surfaces.

Making use of non-dimensional parameters +and +, known as viscous scales givenby:

+= ; += Eq. (26)

Where is the averaged flow velocity, y is the wall distance, and are the density andviscosity of the flow respectively, and is the friction velocity defined as:

=

Eq. (26)

The software STAR-CCM+ has different wall treatment for each turbulence model. For

two turbulence models ( SST and Spalart-Allmaras) All +wall treatment was used.For the Realizable the two layer all +was used.The All y+ Wall Treatment is a hybrid model. It provides a more realistic modeling than

either the low-Re or the high-Re treatments, for when the wall-cell centroid falls in the

buffer region. This wall treatment is meant to give results similar to the low-y+ treatment

as + 0and to the high-Reynolds treatment for +> 30This treatment uses damping


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functions for the source terms in the transport equation, but the source terms of the model

in the wall cell are suitably modified using the blended wall laws.

The Two-Layer All y+ Wall Treatment is a hybrid approach that seeks to recover thebehaviors of the other two wall treatments in the limit of very fine or very coarse

meshes. It contains a wall boundary condition for that is consistent with the two-layerformulation. Apart from the specification of the Reynolds stresses in the wall cell. It is adesign goal that this wall treatment should give results similar to the low-y+ treatmentas + 0and to the high-y+ treatment for +> 30. It will also give reasonable resultsfor intermediate meshes where the cell centroid falls in the buffer layer.

chapter 3 - a cfd theoretical analysis

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