The performance of CFD RANS models in predicting wind loads on flat plates: A comparative study with DNS

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IN THE FIELD OF TECHNOLOGY DEGREE PROJECT

ENERGY AND ENVIRONMENT AND THE MAIN FIELD OF STUDY MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2016 ,

The performance of CFD RANS

models in predicting wind loads on flat plates

a comparative study with DNS ELIN DAHLQVIST

KTH ROYAL INSTITUTE OF TECHNOLOGY

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.

Examensarbete: EGI 2016-071 MSC EKV1158 Hur CFD RANS modeller presterar i att prediktera

vindlaster p˚ a platta plattor

Elin Dahlqvist

Godk¨ant: Examinator: Handledare:

2016-09-27 Paul Petrie Repar Jens Fridh

Uppdragsgivare: Kontaktperson:

University of Calgary David Wood .

Sammanfattning

.

Att kunna f¨ orutsp˚ a vindlaster ¨ ar viktigt f¨ or att kunna optimera kostnaden f¨ or takmonterade solsystem, och en b˚ ade lovande och allt mer popul¨ ar metod f¨ or att g¨ ora detta ¨ ar att anv¨ anda sig av CFD. Denna studie unders¨ okte hur olika RANS modeller presterar i att f¨ orutsp˚ a ett normalt vinklat fl¨ ode mot en platt platta f¨ or fyra olika f¨ orh˚ allanden mellan plattans kortsida och l˚ angsida. De fyra turbulensmodeller studien testar var SST k-!, RSM, standard k-✏ och RNG k-✏; och de fyra olika f¨ orh˚ allandena p˚ a plattan var 3,2, 1,6, 1 och en o¨ andligt l˚ ang kortsida.

Parametrarna studien fokuserade p˚ a ¨ ar dragkoefficenten p˚ a plattan och recirkuleringsl¨ angden bakom den, simuleringarna utf¨ ordes i ANSYS CFX f¨ or ett fl¨ ode med Re = 1200.

Vid j¨ amf¨ orelse med en DNS studie och experimentell studie stod det klart att RSM hade presterat b¨ ast i att f¨ orutsp˚ a draget p˚ a plattan f¨ or de finita f¨ orh˚ allandena, och att det inte fanns n˚ agon modell som ensamt presenterat b¨ ast i att f¨ orutsp˚ a recirkuleringsl¨ angden. Generellt s˚ a presterade samtliga modeller b¨ attre i att f¨ orutsp˚ a draget j¨ amf¨ ort med att f¨ orutsp˚ a recirkuler- ingsl¨ angden, f¨ or vilken felmarginalen p˚ a predikteringarna var st¨ orre. ¨ Aven fast RSM presterade lika svagt som de andra modellerna i att f¨ orutsp˚ a recirkuleringsl¨ angden, s˚ a kunde den rela- tivt v¨ al f¨ orutsp˚ a l¨ aget f¨ or b˚ ade det h¨ ogsta v¨ ardet av den turbulenta r¨ orelseenergin och h¨ ogsta hastigheten bakom plattan, och ocks˚ a l¨ aget f¨ or virvelns centrum i str¨ omningslinjerna bakom plattan.

I fallet med o¨ andligt l˚ ang kortsida var ingen modell riktigt n¨ ara valideringsdatan varken f¨ or dragkoefficienten eller f¨ or recirkuleringsl¨ angden. Samtliga modeller underskattade draget och

¨

overskattade recirkuleringsl¨ angden, men n¨ armast valideringsdatan var standard k-✏ vilken ¨ ar

k¨ and f¨ or att fungera d˚ aligt f¨ or fl¨ oden likt det i denna studie. En f¨ orklaring till dessa ¨ overraskande

resultat kan vara att turbulensmodellerna inte enkelt kan hantera den ¨ okade turbulens och mer

komplexa virveldynamiker som tillh¨ or fl¨ odet f¨ or den o¨ andligt l˚ anga kortsidan.

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Thesis Work: EGI 2016-071 MSC EKV1158

The performance of CFD RANS models in predicting wind loads on flat plates

Elin Dahlqvist

Approved: Examiner: Supervisor:

2016-09-27 Paul Petrie Repar Jens Fridh

Commisioner: Contact Person:

University of Calgary David Wood .

Abstract

.

Being able to predict wind loads on solar cells is essential in order to optimize the cost for roof-top mounted solar systems, and a both promising and increasingly popular way of making these flow predictions is by using CFD. This study investigated the performance of four di↵erent RANS models in predicting a normal flow onto a thin flat plate for four di↵erent aspect ratios.

The four turbulence models were SST k-!, RSM, standard k-✏ and RNG k-✏; and the plate aspect ratios were 3.2, 1.6, 1 and an infinite span. The main parameters of interest were the mean drag on the plate and the mean recirculation length in the wake centerline, the simulations were carried out in ANSYS CFX for a flow with Re = 1200.

In a comparison with a DNS study and an experimental study, it was clear that RSM was the best model in the prediction of the drag for the finite aspect ratios, while there was no outstanding model that accurately predicted the recirculation length. Generally, all the models predicted the drag better than the recirculation length, for which the predictions had a quite significant margin of error for some of the aspect ratios. Although RSM was as deficient as the linear viscosity models in predicting the recirculation length, it still quite managed to predict the correct location of the peak turbulence kinetic energy and peak velocity behind the plate, and also the location of the recirculation vortex core in the mean streamlines behind the plate.

In the infinite span case, none of the models were close at predicting neither the drag nor the

recirculation length to be anywhere close to the DNS- and experimental results. All of the

models underestimated the drag and overestimated the recirculation length, and the best model

in this case was the standard k-✏ which is known to perform poorly for flows like in the present

study. One of the reasons for these results might be that the turbulence models cannot easily

handle the increased turbulence and more complex vortex dynamics pertinent to the infinite

span flow.

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Acknowledgements

First and foremost, I would like to express my gratitude to my supervisor David Wood at the University of Calgary. Not only did he give me the opportunity to conduct this exiting degree project within my area of interest, but he also provided invaluable support throughout the whole process by sharing his expert knowledge, and answering my many questions related to this sometimes difficult but intriguing topic.

In addition to the wonderful assistance provided by David Wood, I also received great support from Arman Hemmati who guided me through all the struggles that came up during the project, and got me back on track whenever I felt lost in my research. Moreover, he let me use the obtained results from his PhD thesis and reprint plots from the same. This accomplishment would not have been possible without his highly valued help.

Lastly, I would like to thank my supervisor Jens Fridh at the Royal Institute of Technology for

his support and for giving me the freedom to shape this degree project how I wanted, although

he steered me in the right direction whenever I needed it.

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Acronyms

2D Two-dimensional.

3D Three-dimensional.

BSL Baseline.

CFD Computational Fluid Dynamics.

DNS Direct Numerical Simulation.

LES Large Eddy Simulation.

PV Photovoltaic.

RANS Reynolds Average Navier-Stokes.

RNG Re-Normalisation Group.

RSM Reynold Stress Model.

SST Shear Stress Transport.

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Nomenclature

C

_µ

, C

_1✏

, C

_2✏

k-✏ model coefficients

C

_d

Mean drag coefficient

D

_ij

Tensor in RSM

F

1

, F

2

Blending functions

h Plate height

k Turbulence kinetic energy

p Pressure

P

_kb

, P

_✏b

Production term for buyoancy forces

P

_k

Turbulence production term

S

_ij

Mean strain rate

U

_inf

Freestream velocity

U

_i

, U

_j

, U

_k

Mean velocity in tensor notation x

_i

, x

_j

, x

_k

Position vector

↵

₁

, ↵

₂

SST k-! model coefficients

′

,

₁

,

₂

SST k-! model coefficients

ij

Kronecker delta; 0 if i ≠ j, 1 if i = j

✏ Energy dissipation rate

µ Kinematic viscosity

⌫

_t

Eddy Viscosity

⇢ Density

✏

,

_k

k-✏ model coefficients

k1

,

_!1

,

_k2

,

_!2

SST k-! model coefficients

⌧

_ij

Reynolds stress tensor

u

_∗

Friction velocity

Y

₊

Dimensionless number describing the wall distance

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1 Introduction

Turbulence is a complex phenomena and a widely used term in several branches of engineering.

Despite turbulence being a well known concept, it is hard to find a single unique definition that shortly describes it; but based on several definitions, turbulence can be explained as a chaotic flow which is characterized by low momentum di↵usion, high momentum convection, and rapid changes of the flow properties both in space and time. Accordingly, turbulence is a three-dimensional, time dependent phenomena with a random behaviour. This irregularity is the main reason why turbulence is hard to predict, and remains a challenge for both scientists and engineers. The fact that the majority of the flows in engineering are turbulent in nature, makes knowledge in this area both useful and important. The more we know, the better we will be able to predict how di↵erent flows will act, and hence obtain information that can be used to avoid accidents and optimize constructions that interact with fluid flows.

This thesis seeks to investigate how di↵erent turbulence models perform in predicting turbulent flows on flat plates, with the intention to provide knowledge that could be useful in the appli- cation of using CFD to forecast wind loads on solar panels. This introduction chapter presents the background to this topic as well as previous work in this research area.

1.1 Background

Over the last years, renewable energy has become a highly relevant topic as a response to the recognition of global warming and other environmental issues caused by the energy sector. One of the most prominent technologies in the renewable energy field is solar energy, which can be used for both heating/cooling and power generation. Solar power can be generated in two ways;

either from heating water that generates steam which drives a steam turbine, or from the direct conversion of sunlight to electricity. Either way, sunlight has to be harnessed, which is why solar energy systems include solar panels. These panels can be placed at any place where the solar insolation is high enough to produce a fair amount of power or heat. In urban areas where the space is usually limited and big parts of the ground are shaded, it is convenient to place the solar panels on rooftops. However, rooftop mounted systems are not entirely unproblematic.

The sometimes heavy wind on urban rooftops demands the construction that holds the panels in place to be carefully dimensioned to withstand forces on the panels caused by extreme wind.

Thus, a robust construction is important, but robustness is not the only wanted feature for

the support structures. Simultaneously as being safe and durable, they must also be as cost

efficient as possible to reduce the total price for the solar system. Moreover, they must be

as light as possible since the roof has to hold the weight of the solar system, and strengthen

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1.Introduction

roofs is usually both expensive and time-consuming. The development of the photovoltaic (PV) technology has resulted in major cost reductions for the panels, thus making the cost for the support structure and the installation cost of the system more significant [1]. This holds true especially for locations at high latitudes where the panels have to be rather tilted in order to work efficiently. A key part in optimizing the support structure is to obtain knowledge about the potential extreme wind loads that the panels can be exposed to, and thereby have to be dimensioned to withstand. For instance, this can be investigated by traditional wind tunnel testing in combination with local wind data or, an increasingly popular method, computational fluid dynamics (CFD).

In a paper by Blocken [2], previous research is compiled and presented in a table that provides an overview of research topics covered in computational wind engineering (CWE) conference proceeding 1992, 1992, 2000, 2006 and 2010. One of the new topics, first time covered in 2010, is wind loads on solar panels, which points to an increased interest for this issue. Research in this area is not complete and further investigation of uncovered topics in this field of research could be beneficial for the development of the solar power industry.

1.2 Previous work

As mentioned earlier, CFD is an upcoming method for wind forecasting and is becoming in- creasingly common in the industry. This is a result of CFD having several advantages over wind tunnel testing, for example it is quicker, often cheaper, o↵ers the possibility to simulate the problems in full scale, and provides good tools for visualisation of the predicted flow. In addi- tion, the computers today are constantly improving which increases the potential for CFD to become an accepted industry standard by allowing for more complex problems to be simulated.

One type of problem that often falls in the complex category is wind flows on blu↵ bodies. The accuracy of numerical computation of this kind of problem has been studied for many di↵erent applications, two of them being flows on buildings and flows on solar panels. For example, Aly and Bitsuamlak [3] validated the use of CFD by investigating model scale e↵ects for numerical simulation and wind tunnel testing of wind loads on solar panels. One of the conclusions from the study was that CFD had two advantages over wind tunnel testing: (1) the CFD simulations performed better in the small model cases since no measuring instruments were required; (2) the CFD method allowed for full scale modeling which is not always possible for wind tunnel testing.

Nonetheless, there are still some drawbacks with CFD. For instance the results obtained from CFD investigations can be erroneous, which is often a consequence of insufficient turbulence modeling or poor convergence. In general, there are three ways of solving a fluid flow numerically;

these are direct numerical simulation (DNS), Large eddy simulation (LES) and Reynolds Average

Navier-Stokes (RANS) simulation. DNS is a very sophisticated method that fully resolves the

governing equations of the flow, and is almost exclusively used in research. This is mainly due to

its high computational cost, which also limits DNS to only be used for solving flows with a fairly

low Reynolds number. The other approaches, LES and RANS, do not completely resolve the

Navier-Stokes equations, hence they both involve turbulence modeling that reduces the accuracy

of the result but in turn lowers the computational cost. Since turbulence models are widely used

and involve many simplifications of how turbulence behaves in reality, it is important to validate

and evaluate the turbulence models in order to enhance CFD as a method.

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1.Introduction

This modeling issue has been addressed before by several studies, which investigate the ability of di↵erent turbulence models to predict wind loads on blu↵ bodies. The RANS approach in particular requires thorough validation since it leaves the real turbulence completely unresolved, and instead it is modeled by mean quantities. Hence the flow is described in statistical properties with time-averaged pressure and velocity fields, which cause inaccuracies since turbulence is a time dependent phenomena. Consequently, RANS models have some well known limitations in their capacity to predict certain types of fluid flows, and among those are flows on blu↵ bodies with sharp edges. The main deficiency in this case, is the incapability to accurately model transient features as separation, recirculation and vortex shedding [2]. These shortcomings can be overcome by using a more sophisticated method as LES, which was proven to perform better than RANS simulations already in the late 80s and early 90s [4, 5]. LES partially resolves the turbulence by solving the governing equations for large scale turbulence and model the small scale turbulence. However, a refined method as LES does not come without any drawbacks;

compared to RANS modeling it is computationally expensive, and is sensitive to errors in the applied inlet- and wall boundary conditions [2]. Nevertheless, the quality of the flow predictions is superior to those by RANS in most applications, as for example in simulating wind flows around sharp edge blu↵ bodies representing buildings. Although, engineering does not always require precise predictions as long as they are good enough for the application in question and the inbuilt errors in the results are known.

Aside from DNS, LES and RANS, there is a method called Detached Eddy Simulation (DES), which can be described as a mix between RANS and LES. Breuer, Joviˇci´c, and Mazaev [6]

conducted a study in which RANS, DES and LES are tested and evaluated by looking at the quality of their flow predictions for a flow on an inclined flat plate at Re = 20000. The results showed that the Unsteady RANS (URANS) simulations was not able to predict the unsteady characteristics of the separated flow, which the LES managed to do, and the DES almost did.

The study clearly showed, that the shortcoming of RANS become very clear when transient flows are modeled.

Nevertheless, there are cases where URANS has proven to not be entirely deficient. In a paper by Lasher [7] a two-dimensional blocked flow normal to a flat plate was studied by using several turbulence models to perform computations for both quasi-steady and transient cases. One of the goals with the study was to assess the ability to predict force coefficients for the di↵erent cases.

The study concluded that the drag was under predicted in the quasi-steady simulations, while the transient simulations predicted the actual value of the drag very well. Another interesting conclusion was that the quasi-steady simulations were un-sensitive to the choice of turbulence models, whereas the transient cases seemed to be sensitive in that aspect.

The above mentioned conclusions regarding the ability of RANS models to predict the drag correctly corresponds well with the findings of Jubayer and Hangan [8]. They recently conducted a study on wind loads on ground mounted PV systems using a SST-k! turbulence model. The study showed that despite deficiencies in predicting vortex shedding, the RANS model approach is able to give a a good prediction of the lift and drag coefficients, and could thereby be useful in cases where less computationally demanding approaches are necessary.

Although the amount of studies on RANS models ability to predict flows on flat plates are rather limited, the problem of describing a turbulent flow around a flat plate is not a new one.

Already in 1927 Fage and Johansen [9] conducted an experimental study on the air flow behind

an inclined flat plate with an infinite span. This work was later followed by a quite recent study

on a similar topic by Najjar and Balachandar [10] and Wu et al. [11] who both investigated the

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1.Introduction

wake flow behind a normal infinite span flat plate. These are two of the rather few studies, to the present author’s knowledge, that have been made on the characteristics of the wake behind a normal flat plate. Moreover, most of these investigated an infinite span case, but if the flat plate is suppose to represent a single solar panel it would be useful to investigate how the flow behaves for a finite aspect ratio as well.

Hemmati [1] addressed this topic by using DNS to investigate the characteristics of the wake

flow behind a flat plate for a number of aspect ratios (AR) at Re = 1200. Among many other

things, it was found that the wake characteristics changed for the di↵erent aspect ratios which

were defined as the ratio of the plate chord to span. The aspect ratios investigated were AR

3.2 corresponding to a solar array, AR 1.6 corresponding to a solar panel, AR 1 and an infinite

span case. To the authors knowledge, there are few studies that look into the ability of RANS

models to capture the flow features corresponding to di↵erent aspect ratios, for example those

representing a single solar panel and a solar array.

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2 Aims and Objectives

In light of the possible benefits of increasing the knowledge about the performance of di↵erent turbulence models in the application of simulating wind loads on a generic shape as a thin flat plate, this thesis aims to asses the performance of the standard k-✏ (SKE) model, Re- Normalisation (RNG) k-✏ model, Shear Stress Transport (SST) k-! model and Baseline (BSL) Reynold Stress Model (RSM) in how they predict a normal flow onto a infinitely thin flat plate for three finite aspect ratios and an infinite span case. The finite aspect ratios being investigated are AR 3.2, AR 1.6 and AR 1, where the aspect ratio is defined as the ratio between the chord and the span of the plate. The study will essentially focus on how well the models can predict the flow features in the wake, mainly the recirculation length and the drag on the plate. Hence this study is not fully comprehensive, and aims at giving a basic assessment of the performance rather than a thorough evaluation.

The objectives of the study are formulated to make the assessment based on both quantitative and qualitative grounds. The assessment of the performance will be done by comparing the results of the study to earlier studies which are based on experiments or a highly accurate numerical approach as DNS. Following three objectives have been established:

Compare the turbulence models quantitatively by assessing their accuracy in predicting the:

(1) mean drag on the plate; (2) and the recirculation length in the center line wake flow of the plate.

Compare the turbulence models qualitatively by evaluating mean streamline plots, velocity plots, and turbulence kinetic energy plots in the wake centerline.

Compare the turbulence model’s performance for di↵erent aspect ratios by studying the quan-

titative parameters and qualitative plots for four di↵erent aspect ratios: (1) AR = 3.2; (2)

AR = 1.6; (3) AR = 1; and an infinite span case.

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3 Turbulence model Theory

The purpose of this sub chapter is to briefly introduce the turbulence models used in this study.

The equations presented for each model and their corresponding coefficient values are those used in the CFX-solver, as reported by the ANSYS CFX help manual [12]. Turbulence models are used to close the Reynolds Averaged Navier Stokes (RANS) equations, which is a simplified version of the Navier-Stokes equations. The Navier-Stokes equations are the governing equations for any viscous incompressible flow, and can be expressed as [13]:

t

u ̃

_i

+ ̃ u

_{j j}

u ̃

_i

= − 1

⇢

ⁱ

̃p+ ⌫ ▽

²

u ̃

_i

−

ⁱ

u

_j

u

_i

,

i

u ̃

_i

= 0

(3.1)

Where p is the mean pressure, ⇢ is fluid density and ⌫ is the kinematic viscosity of the fluid. By representing the instantaneous variables into a mean (time-averaged) and a fluctuating compo- nent according to Reynolds decomposition, the equation can be rewritten as [13]:

t

(U

ⁱ

+ u

ⁱ

)+(U

^j

+ u

^j

)

^j

(U

ⁱ

+ u

ⁱ

) = − 1

⇢

ⁱ

(P + p)+⌫ ▽

²

(U

ⁱ

+ u

ⁱ

)−

ⁱ

(u

^j

u

i

) ,

ⁱ

(U

ⁱ

+ u

ⁱ

) = 0 (3.2) By definition, the fluctuation component has zero average. Implementing this to the above expression gives the steady Reynolds Average Navier-Stokes equations:

t

(U

ⁱ

) + (U

^j

)

^j

(U

ⁱ

) = − 1

⇢

ⁱ

(P) + ⌫ ▽

²

(U

ⁱ

) −

ⁱ

(u

^j

u

_i

),

i

(U

ⁱ

) = 0

(3.3)

which is the underlying equations for the turbulence models used in this study. In order to close these equations, turbulence models are used to approximate the Reynolds stresses and the tur- bulent scalar length terms. There is a wide range of turbulence models available, which all di↵er in their applicability to di↵erent flows, accuracy, complexity and computational cost. Among the most common RANS based models we find the ones this study aims to investigate:

1. Two equation linear eddy viscosity models

• k-✏ models

– Standard k-✏ model

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3.Turbulence model Theory

– RNG k-✏ model

• k-! models

– SST k-! model 2. Reynolds stress models

• BSL RSM

The two types of models di↵er primarily in how the Reynolds stresses are evaluated. The linear viscosity models are all based on the assumption that there is a correlation between the viscous stresses and the Reynolds stresses on the mean flow. The assumption estimates the turbulent viscosity to be isotropic, which means that the ratio between the Reynolds stresses and the mean strain rate is the same in all directions [14]. This is called the Boussinesq linear eddy viscosity assumption, and can be expressed as [13]:

⌧

_ij

= 2µ

^t

S

_ij

− 2

3 ⇢k

_j

(3.4)

Where S

_ij

is the strain rate defined as:

S

_ij

= 1

2 (

i

U

_j

+

j

U

_i

) (3.5)

In contrast, the RSM type of model is not based on the eddy viscosity assumption. Instead, the Reynolds stresses are directly calculated by using di↵erential transport equations to model each Reynold stress component [14].

3.1 Standard k-✏ model

The k-✏ model is a widely used model known for being stable, robust and fairly accurate in predicting many types of flows. The standard version was developed already during the 1970s by Launder and Spalding [15], and has since then been further developed and modified to all the di↵erent versions it exists in today. In common for all of them, is that they are based on two di↵erential transport equations which describe the turbulent kinetic energy k and the energy dissipation rate ✏. As mentioned in the previous section, the k-✏ model is based on the eddy viscosity concept, where the eddy viscosity is related to the turbulent kinetic energy and dissipation rate by:

µ

t

= C

^µ

⇢ k

²

✏ (3.6)

The two transport equations for k and ✏ reads:

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3.Turbulence model Theory

(⇢k) t +

x

_j

(⇢U

j

k ) =

x

_j

�(µ µ

_t

k

x

_j

� + P

k

− ⇢✏ + P

kb

(3.7) (⇢✏)

t +

x

_j

(⇢U

j

✏ ) =

x

_j

�(µ µ

_t

✏

✏ x

_j

� + ✏

k (C

✏1

P

_k

− C

✏2

⇢✏ + C

✏1

P

_✏b

) (3.8) where the coefficient values are:

C

✏1

= 1.44 C

^✏2

= 1.92

k

= 1

^✏

= 1.3 C

^µ

= 0.09

P

_kb

and P

_✏b

describe the influence from buoyancy forces, and P

_k

represent the turbulence pro- duction due to viscous forces. P

_k

is modelled in CFX as:

P

_k

= µ

t

� U

_i

x

j

+ U

_j

x

i

� U

_i

x

j

− 2

3 U

_k

x

_k

�3µ

t

U

_k

x

_k

+ ⇢k� (3.9)

Although the standard k-✏ model o↵ers rather good flow predictions for a wide range of cases, there are some types of flows where the model has proven to be deficient. According to ANSYS [12], some of these are: (1) flows with boundary layer separation; (2) flows with rapid changes in the mean strain rate; (3) flows in rotating fluids; and (4) flows over curved surfaces.

3.2 RNG k-✏ model

In an attempt to account for smaller scale motions, Yakhot et al. [16] developed the RNG k-✏

model by re-normalizing the Navier-Stokes equations. The equations for the eddy viscosity, turbulent kinetic energy and dissipation rate is the same as for the SKE model; but the closure coefficients are di↵erent and in CFX they are defined according to:

C

_✏2

≡ ̃ C

_✏2

+ C

_µ ³

(1 − �

0

)

1 +

³

(3.10)

≡ k

✏

� 2S

_ij

S

_ji

(3.11)

the remaining closure coefficients are:

C

✏1

= 1.42 C ̃

✏2

= 1.68 C

µ

= 0.085

k

= 0.72

^✏

= 0.72 = 0.012

⁰

= 4.38

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3.Turbulence model Theory 3.3 SST k-! model

A remaining issue in turbulence modeling is the prediction of flow separation from a smooth surface [12]. Accurate flow prediction is vital in many technical applications; for example in the airplane aerodynamics where ✏-based models are unsuitable because of their inability to correctly predict flow separation, resulting in overestimation of the performance of an airfoil.

The k-! model is a two-equation model developed by Wilcox [17] in 1998. Despite major improvements of the separation prediction compared to the k-✏ models, the k-! model had some shortcomings related to freestream sensitivity and prediction of adverse pressure gradients. In an attempt to overcome these deficiencies, Menter [18] successfully developed first the BSL k-!

model, and later the SST k-! model in 1994. The latter model has through various validation studies proven to give the most accurate flow separation predictions of all the linear viscosity models [12].

The SST k-! model is a mix between the Wilcox k-! model and the k-✏ model. Through a blending function F1, the Wilcox k-! is activated near walls and switches to the k − ✏ model in the freestream. In CFX, the SST model consists of two equations; one for the turbulent kinetic energy and one for the dissipation rate:

(⇢k)

t + x

_j

(⇢U

^j

k ) = x

^j

��µ + µ

_t

k3

� k

x

_j

� + P

k

−

^′

⇢k! + P

kb

(3.12)

(⇢k)

t + x

_j

(⇢U

j

! ) = x

_j

��µ + µ

_t

!3

� !

x

_j

�+(1−F

1

)2⇢ 1

!2

! k x

_j

! x

_j

+↵

3

!

k P

_k

−

3

⇢!

²

+P

!b

(3.13) the CFX values of the model coefficients are:

′

= 0.09 ↵

1

=

⁵₉ ¹

= 0.075

k1

= 2

^!1

= 2

↵

₂

= 0.44

2

= 0.0828

k2

= 1

!2

=

_0.856¹

In order to accurately describe the transport of the turbulent shear stress, the SST model in- cludes a limiter to the formulation of the eddy viscosity [12]:

⌫

_t

= a

₁

k

max (a

¹

!, SF

2

) (3.14)

where ⌫

t

=

^µ_⇢^t

, F

2

is a blending function that restricts the limiter to the boundary wall layer, and S is an invariant measure of the strain rate. The production term of ! is described by:

P

_!

= ↵

₃

µ

t

⋅ P

k

(3.15)

(25)

3.Turbulence model Theory 3.4 BSL RSM model

In contrast to the linear viscosity models, the RSM models close the RANS equations by using transport equations to solve every individual Reynolds stress component. The BSL RSM model is !-based, which results in a more accurate near wall treatment through an automatic switch between a wall function and low-Reynolds formulation based on the grid spacing. In CFX, the equations for the Reynold stresses are given by [12]:

⇢u

_i

u

_j

t + x

_k

(U

k

⇢u

i

u

j

) = P

^ij

− 2 3

′

⇢!k

ij

+

^ij

+ P

ij,b

+ x

_k

��µ + µ

c

k

� u

_i

u

_j

x

_k

� (3.16) where the pressure-strain correlation

_ij

is expressed by:

ij

=

^′

C

₁

⇢! �−u

ⁱ

u

_j

+ 2

3 k

_ij

� − ˆ↵ �P

^ij

− 2

3 P

_ij

� − ˆ�D

^ij

− 2

3 P

_ij

� − ˆ⇢kS

^ij

− 1

3 S

_{kk ij}

(3.17) in which the tensor D

_ij

and the production tensor P

_ij

of Reynold stresses are described by:

D

_ij

= −⇢u

i

u

_k

U

_k

x

j

− ⇢u

j

u

_k

U

_k

x

j

(3.18)

P

_ij

= −⇢U

i

U

_j

U

_j

x

_k

− ⇢u

j

u

_k

U

_i

x

_k

; P = 0.5P

kk

(3.19)

the coefficients expressed as:

′

= 0.09 ↵ ˆ = �

⁸^+C₁₁²

� ˆ =

^(8C₁₁²⁻²⁾

ˆ =

^(60C₅₅²⁻⁴⁾

C

₁

= 1.8 C

₂

= 0.52

The equation for ! has two sets of constants, one corresponding to the ! zone and the other to the ✏ zone. The !-formulation is given by:

(⇢!)

t + (U

k

⇢! ) = ↵

3

!

k P

_k

+P

!b

−

3

⇢!

²

+

x

_k

��µ + µ

_t

!3

� !

x

_k

�+(1 −F

1

)2⇢ 1

2

! k x

_k

!

x

_k

(3.20) where the constants in the ! zone are:

1∗

= 2

1

= 2

1

= 0.075 ↵

₁

=

′

−

₍^′²)^0.5

= 0.553

(26)

3.Turbulence model Theory

and the constants in the ✏ zone are:

∗2

= 1

2

= 0.856

2

= 0.0828 ↵

₁

=

^′

−

₍^^′²₎0.5

= 0.44 and the turbulent viscosity in given by:

µ

T

= ⇢ k

! (3.21)

(27)

4 Method

The methodology used in this study was mainly of quantitative character, with a main focus to obtain parameters that could be used to compare the results both within the study, but also to results from previous studies on the same topic. The whole CFD procedure can be divided into seven major steps: (1) defining modeling goals, (2) identifying computational domain and create a solid model, (3) designing and creating the mesh, (4) setting up the solver and run the simulation, (5) performing a mesh independence test, (6) post processing the obtained simulation data, and lastly (7) comparing and validating the results.

The first item on the list was carried out after a literature review on the topic, primarily targeting studies on the wake flow of thin flat plates. The literature review revealed that a majority of these previous studies had used the mean drag coefficient, C

_D

, and the recirculation length, L

_w

as descriptive parameters of the flow. Both these parameters were used in the work of Hemmati [1], which served as a main source of validation data for this study. Hence, obtaining these two coefficients was central. The definition of the mean drag coefficient is C

_d

= D�2⇢U

_inf²

, where D is the drag on the plate, ⇢ is the fluid density, U

_inf

is the free stream velocity and h is the plate chord. The recirculation length is simply defined as the distance where the stream-wise velocity component is equal to zero in the wake of the plate.

As mentioned previously, the Reynolds number of the investigated flow is 1200 and was chosen to match the study by Hemmati, with the Reynolds number being defined as Re = U

inf

h �⌫, where ⌫ is the kinematic viscosity of the fluid. This is a fairly low Reynolds number compared to what is common for fluid flows in engineering, for example Toja-Silva et al. [19] used Reynolds number equal to 7 ⋅10

⁴

in their study on wind loads on roof-top mounted solar panels. However, the Reynolds number in this particular study should not be critical since the separation points on flat plates are fixed at the edges, which should make the problem fairly independent of the Reynolds number.

4.1 Numerical setup

With the modeling goals clearly defined, the next step was to create both a computational

domain and a solid model that corresponded to the problem. The size of the domain is an

important specification to make since a poor definition of it most likely will result in inaccurate

results or inefficient use of computational power. A too small domain can give rise to numerical

influences from the boundary conditions, while a too large domain requires an unnecessary high

use of computational resources. The sensitivity of the size of the computational domain have

been carefully evaluated in the work by Hemmati, in which he investigates the wake flow for flat

(28)

4.Method

plates with aspect ratio 3.2, 1.6 and infinite span. The computational domain in the present study was the same as Hemmati [1] used.

The domain was based on a 3D Cartesian coordinate system for all the aspect ratios including the infinite span case. The shape of the domain was a rectangular box, with the one short side representing the inlet, and the other short side representing the outlet. The length of the domain was set to 25h, where h is the plate chord. The plate is placed 5h downstream from the inlet in the middle of the rectangle with respect to the z- and y- position. The width of the plate is h/AR, and in the infinite span case it was set to 2⇡h according to the infinite span study by Hemmati [1]. The height and width of the domain are both set to +/-8h with origo defined in the center of the plate back side. The computational domain for the finite aspect ratios is shown in fig. 4.1, and the one for the infinite span in fig. 4.2.

x y

z y

5h 20h 16h

16h Plate

Inlet Outlet

h

Figure 4.1. Schematic picture of the computational domain used for the finite aspect ratio simulations.

x y

z y

5h 20h 2⇡h

16h Plate

Inlet Outlet

h

Figure 4.2. Schematic picture of the computational domain used in the infinite span simula- tions.

The meshes used in the present study was created in ANSYS ICEM, which o↵ers the opportunity

to use a blocking strategy to refine the mesh in regions of interest. In short, the blocking strategy

makes it possible to divide the computational domain into several blocks, where each di↵erent

block in the model can be individually defined by various mesh parameters. Among these were:

(29)

4.Method

bunching law, spacing (size of the first cell), expansion ratio and the amount of nodes. This study required 16 di↵erent grids since four di↵erent aspect ratios were studied, and four di↵erent grids were used in the mesh independence test for each aspect ratio.

In common for all the grids was the use of Bigeometric as bunching law, with an expansion ratio of 1.02 between the cells. The blocking procedure was the same for each mesh, but the resulting blocking changed for each mesh according to the di↵erent geometries. The blocking was carried out by cutting out the plate shape from the fluid body through the three planes, the yz-plane, the xy-plane and the xz-plane. Each block was then defined individually, which the aim to achieve a higher mesh density close to the plate and in the wake.

The volume elements in the mesh consisted entirely of hexahedrals, which were small close to the plate and grew larger as the distance from the plate increased. The mesh density must be high enough to capture all the necessary details for an accurate solution, simultaneously as being as low as possible to save computational resources. Furthermore, it has to be designed carefully to ensure a high mesh quality that will result in a converged solution. A high mesh quality was achieved by verifying that the mesh parameters Orthogonal angle, Expansion Factor and Aspect Ratio were all within acceptable values before running the solver.

The solver input was specified in CFX-Pre by importing the mesh from ICEM and adjust the settings and boundary conditions for each simulation. For the most part, the boundary conditions were the same for all simulations and are presented in table 4.1, but there are two exceptions between the simulations that should be mentioned. The first di↵erence between the simulations was the choice mesh which changed for each aspect ratio and had di↵erent densities for each simulation in the mesh independence test. The second di↵erence was the choice of turbulence model and in one case also the turbulence numerics which was set to first order for standard the k-✏ model and high resolution for all the others. The convergence criteria for the simulations was defined by the residual momentum root mean square (RMS) value, which was to 10

⁻⁶

. The fluid material was set to air at 25 °C, with all its properties defined by the default value in the CFX fluid library.

Part Boundary Condition

Inlet Uniform velocity: U

_inf

= 1,01 (m/s)

Outlet Pressure outlet: Average static pressure equal to zero

Sides Symmetry

Top and Bottom Free slip wall

Plate No slip wall: Smooth wall

Table 4.1. Boundary conditions for the simulations.

As seen in table 4.1, the plate is modelled as a no-slip wall, which means that there are strong

gradients in the dependent variables and viscous e↵ects at the plate are large. Consequently,

modelling the flow near the plate raises two questions: how to account for viscous e↵ects at

the wall, and how to estimate the fluctuation of flow variables within the boundary layer re-

gion.

(30)

4.Method

According to experiments and mathematical analysis, the region near the free slip wall can be divided into two sub-layers. The layer closest to the wall is called the viscus sublayer, in which the flow is characterized by laminar-like behaviour and the viscosity is dominant in the momentum transport. In contrast, the layer farthest away from the wall has a turbulent-like flow where turbulence dominates the mixing process, and is called the logarithmic layer. In between these two sub-layers, there is a layer called the bu↵er layer, where the flow is equally influenced by turbulence and viscosity.[12]

By assuming that the logarithmic profile reasonably approximates the velocity distribution near the wall, the fluid shear stress can be numerically computed as a function of the velocity at a given distance from the wall. This can be done in two ways, one of them being by using a wall- function. An advantage of using wall-functions is that it reduces the need of computational power by using empirical formulas to model the boundary layer. All turbulence models in CFX work well with wall-functions, however, they are not the same for all turbulence models.[12]

The k-✏-based turbulence models use the scalable wall-function approach, which permits sim- ulations on arbitrary fine near-wall grids. The !-based models, on the other hand, have an automatic wall treatment, which works by switching between a wall-function approach and a low Re-formulation depending on the Y + value. Y+ is a dimensionless parameter that describes the distance to the wall according to:

Y + ≡ u

_∗

y

⌫

where u

_∗

is the friction velocity and ⌫ is the kinematic viscosity. The low Re-formulation solves the details of the boundary layer profile by using very small mesh length scales in the direction normal to the wall. Therefore this approach requires a low Y + close to the plate, thus requiring more computation resources than the wall-function approach. Turbulence models based on the omega-equation, such as the SST and BSL RSM are suitable for a low-Re approach. [12].

However, a correct near-wall treatment is not enough to obtain an accurate solution from a numerical simulation. The convergence criteria for the RMS residuals together with velocity monitoring points in the wake both served as indicators of a converged and correct solution.

The mesh resolution can also a↵ect the simulation outcome, and therefore a mesh independence study was conducted to ensure that the simulation results were mesh independent.

4.2 Mesh Independence Study

A mesh independence test was carried out for each turbulence model by running four simulations for each aspect ratio, using four di↵erent grids with distinct resolution; the grids used for each aspect ratio are presented in table 4.2, 4.3, 4.4 and 4.5. As seen in the tables, the grids di↵ered in the amount of mesh elements, plate thickness and amount of nodes in each direction from the plate. The reason why the plate thickness is decreasing for each aspect ratio is because the plate is assumed to be infinitely thin to avoid reattachment e↵ects. To ensure no reattachment e↵ects, the plate is only one element thick, and in order to keep a good quality mesh with a smooth growth ratio, the plate thickness had to decrease as the surrounding mesh was refined.

In other words, a thinner plate made it possible to decrease the size of the cells closest to the

(31)

4.Method

and the one representing the thickness. In addition to making the elements close to the plate smaller, the mesh was refined by adding more nodes in each direction from the plate surfaces to the domain boundaries, as well as adding more nodes on the plate itself. The total amount of elements, and how much the amount of nodes increase in each directions are displayed in table 4.2, 4.3, 4.4 and 4.5.

Grid Elements Plate thickness (m) Nodes in X+ Nodes in X- Nodes in YZ

G1 497 540 0.02 51 13 21

G2 1 159 358 0.015 68 17 27

G3 3 862 668 0.01 100 25 40

G4 6 573 922 0.0085 120 30 48

Table 4.2. Characteristics of the grids used in the grid independence test for AR 3.2.

Grid Elements Plate thickness (m) Nodes in X+ Nodes in X- Nodes in YZ

G5 695 244 0.02 51 13 21

G6 1 615 468 0.015 68 17 27

G7 5 384 712 0.01 100 25 40

G8 9 170 692 0.0085 120 30 48

Table 4.3. Characteristics of the grids used in the grid independence test for AR 1.6.

Grid Elements Plate thickness (m) Nodes in X+ Nodes in X- Nodes in YZ

G9 930 968 0.02 51 13 21

G10 2 165 483 0.015 68 17 27

G11 7 235 040 0.01 100 25 40

G12 12 235 732 0.0085 120 30 48

Table 4.4. Characteristics of the grids used in the grid independence test for AR 1.

(32)

4.Method

Grid Elements Plate thickness (m) Nodes in X+ Nodes in X- Nodes in YZ

G13 529 600 0.02 51 13 21, 64

G14 1 030 425 0.015 68 17 27, 72

G15 2 551 500 0.01 100 25 40, 84

G16 4 252 266 0.0085 120 30 48, 96

Table 4.5. Characteristics of the grids used in the grid independence test for an infinite span flat plate. The nodes in the Z-direction is the total amount of nodes over the whole span, unlike in the X- and Y-direction where the amount of nodes is counted from the plate edge.

The mesh independence study focused on the same two parameters the study seeks to obtain, the mean drag coefficient and the mean recirculation length. These two were calculated for each simulation and compared between each other to find out for which mesh further refinement could be considered redundant. The solutions were considered mesh independent when the relative change was less than 3 percent, the relative change being defined as:

Relative Change = x

₁

− x

0

x

0

× 100 %

The results from the mesh independence study are presented in table 4.6, 4.7, 4.8 and 4.9, where

each table displays the turbulence model, grid used in the simulation, mean drag coefficient,

relative change of the mean drag coefficient, mean recirculation length, the relative change of

the mean recirculation length, and lastly the Y + value. The displayed value of Y + is the average

value at the plate surface. Furthermore, it should be mentioned that the relative change does not

correspond to the rounded o↵ table values of the mean drag and the recirculation length; they

are calculated using the values of C

_D

and L

_w

when they had 8 significant numbers. Hence the

value of the parameters can be the same for two di↵erent grids, and still have a relative change

not equal to zero. In addition to the tables, there are mesh convergence plots corresponding to

each aspect ratio shown in figure 4.3, 4.4, 4.5 and 4.6. These contain the tabulated values of C

_D

and L

_w

for each aspect ratio plotted against the amount of elements used in the corresponding

mesh. Based on the outcome of the mesh independence test, it was decided that the solutions

were mesh independent for G3, G7, G11 and G15.

(33)

4.Method

Turbulence

model Grid Cd Relative

change (%) Lw Relative

change (%) Y +

SST k-! G1 1.24 - 1.23 - 1.57

G2 1.25 0.386 1.31 6.22 1.25

G3 1.24 0.345 1.39 5.88 0.877

G4 1.24 0.245 1.41 1.87 0.759

BSL RSM G1 1.14 - 1.61 - 1.41

G2 1.15 0.331 1.74 8.56 1.1

G3 1.14 0.0729 1.87 7.0 0.769

G4 1.14 0.0303 1.9 1.77 0.665

RNG k-✏ G1 1.24 - 1.43 - 0.683

G2 1.27 1.70 1.47 3.32 0.532

G3 1.30 2.62 1.47 0.089 0.37

G4 1.31 1.22 1.46 1.17 0.37

Standard k-✏ G1 1.45 - 1.1 - 0.813

G2 1.46 0.98 1.16 6.13 0.63

G3 1.49 1.79 1.20 3.35 0.44

G4 1.5 0.8 1.20 0.26 0.44

Table 4.6. Grid independence test for a normal flow on a flat plate with AR 3.2, and Reynolds number equal to 1200.

0 2 4 6 8

Elements

×10⁶ 1

1.2 1.4 1.6 1.8

CD

SST k-ω RSM RNG k-ϵ k-ϵ

(a) Mesh independence plot showing CD convergence for AR 3.2.

0 2 4 6 8

Elements

×10⁶ 1

1.5 2 2.5

Lw

(b) Mesh independence plot showing CD convergence for AR 3.2.

Figure 4.3. Mesh independence convergence plots fro AR 3.2.

(34)

4.Method

Turbulence

model Grid Cd Relative

change (%) Lw Relative

change (%) Y +

SST k-! G5 1.19 - 1.81 - 1.5

G6 1.21 1.15 1.89 4.91 1.17

G7 1.21 0.111 1.99 5.41 0.815

G8 1.20 0.235 2.04 2.13 0.702

BSL RSM G5 1.11 - 2.46 - 1.24

G6 1.12 0.971 2.67 8.81 0.966

G7 1.12 0.0938 2.87 7.55 0.671

G8 1.12 0.0782 2.92 1.55 0.578

RNG k-✏ G5 1.19 - 2.07 - 0.649

G6 1.21 1.65 2.17 4.87 0.495

G7 1.23 2.07 2.21 1.71 0.338

G8 1.25 0.886 2.20 0.22 0.291

Standard k-✏ G5 1.33 - 1.71 - 0.79

G6 1.35 1.5 1.79 4.62 0.61

G7 1.38 1.82 1.84 2.44 0.42

G8 1.39 1.01 1.83 0.23 0.42

Table 4.7. Grid independence test for a normal flow on a flat plate with AR 1.6, and Reynolds number equal to 1200.

0 2 4 6 8 10

Elements

×10⁶ 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

CD

(a) Mesh independence plot showing CD convergence for AR 1.6.

0 2 4 6 8 10

Elements

×10⁶ 1.5

2 2.5

3 3.5

Lw

(b) Mesh independence plot showing CD convergence for AR 1.6.

Figure 4.4. Mesh independence convergence plots for AR 1.6.

(35)

4.Method

Turbulence

model Grid Cd Relative

change (%) Lw Relative

change (%) Y +

SST k-! G9 1.19 - 2.29 - 1.45

G10 1.21 1.11 2.39 4.62 1.13

G11 1.21 0.0383 2.52 5.17 0.779

G12 1.20 0.296 2.58 2.43 0.667

BSL RSM G9 1.11 - 3.21 - 1.16

G10 1.12 0.92 3.47 7.98 0.91

G11 1.12 0.0959 3.73 7.66 0.63

G12 1.12 0.0694 3.78 1.27 0.54

RNG k-✏ G9 1.19 - 2.63 - 0.631

G10 1.21 1.47 2.78 5.71 0.474

G11 1.22 1.59 2.87 3.38 0.32

G12 1.23 0.822 2.86 0.225 0.276

Standard k-✏ G9 1.31 - 2.24 - 0.77

G10 1.33 1.81 2.34 4.59 0.59

G11 1.35 1.53 2.41 4.45 0.41

G12 1.36 0.93 2.40 0.17 0.41

Table 4.8. Grid independence test for a normal flow on a flat plate with AR 1, and Reynolds number equal to 1200.

0 5 10 15

Elements

×10⁶ 1

1.1 1.2 1.3 1.4 1.5 1.6

CD

(a) Mesh independence plot showing CD convergence for AR 1.

0 5 10 15

Elements

×10⁶ 2.5

3 3.5

4 4.5

Lw

(b) Mesh independence plot showing CD convergence for AR 1.

Figure 4.5. Mesh independence convergence plots for AR 1.

(36)

4.Method

Turbulence

model Grid Cd Relative

change (%) Lw Relative

change (%) Y +

SST k-! G13 1.77 - 5.0 - 1.13

G14 1.77 0.427 5.46 9.37 0.875

G15 1.76 0.565 6.04 10.6 0.593

G16 1.75 0.515 6.29 4.09 0.505

BSL RSM G13 1.64 - 8.24 - 1.01

G14 1.66 0.917 9.04 9.69 0.786

G15 1.66 0.272 9.77 8.05 0.541

G16 1.66 0.116 10.03 2.66 0.464

RNG k-✏ G13 1.68 - 9.46 - 0.319

G14 1.71 1.39 9.46 0.0005 0.312

G15 1.73 1.44 9.53 0.771 0.21

G16 1.74 0.641 9.26 2.89 0.21

Standard k-✏ G13 1.83 - 5.54 - 0.55

G14 1.85 0.929 6.0 8.26 0.42

G15 1.88 1.34 6.24 3.93 0.29

G16 1.89 0.310 6.25 0.13 0.29

Table 4.9. Grid independence test for a normal flow on a flat plate with infinite span, and Reynolds number equal to 1200.

0 1 2 3 4 5

Elements

×10⁶ 1.6

1.7 1.8 1.9 2 2.1

CD

(a) Mesh independence plot showing CD convergence for an infinite span plate.

0 1 2 3 4 5

Elements

×10⁶ 6

8 10 12 14

Lw

(b) Mesh independence plot showing CD convergence for an infinite span plate.

Figure 4.6. Mesh independence convergence plots for the infinite span.

(37)

4.Method 4.3 Post-processing

The last step of the methodology was to post-process the data from the simulations, and validate it against the DNS- and experimental results from previous studies. The quantitative outputs from the post-processing were the mean drag coefficient and the centerline recirculation length.

The post-processing was carried out in MATLAB by using simulation data which was exported

from CFD-post. The drag coefficient was calculated according to its definition in the introduc-

tion. The pressure data was exported from the plate surface together with the corresponding

surface area, and the drag was calculated by multiplying the front pressure data with the front

surface area and subtract the product with the back pressure data multiplied by the back surface

data. The data for the recirculation length and turbulence kinetic energy plots was exported

from a line along the wake center, staring from the plate backside and ending at the domain

outlet. The whole line of data was not included in the plots, instead the focus was on the area

in the near wake of the plate. The velocity data for the mean streamline plots was exported

from an XY-plane along the centerline.

(38)

(39)

5 Results

This chapter contains the results from the study starting with C

_D

and L

_w

which are the two quantitative flow parameters this study has a special focus on. Next section presents the mean velocity component u plotted along the wake centerline for each aspect ratio, plotted together with DNS results. The third section shows mean streamline plots for each aspect ratio from the present study as well as reprinted plots from the DNS study by Hemmati. The last section of the chapter contains plots of the turbulence kinetic energy along the wake centerline for each aspect ratio, also plotted together with DNS results.

5.1 Mean drag and recirculation length

The result for AR 3.2 is displayed in table 5.1, and as can be seen, the RNG k-✏ model gives the best prediction of the recirculation length, while RSM gives the best prediction of the mean drag. The drag is second best predicted by SST k-! and third best predicted by the RNG k-✏, and the poorest prediction is given by standard k-✏.

As is evident from table 5.1, the mean drag is under predicted by RSM and over predicted by the linear viscosity models compared to the DNS results. For the recirculation length it is the other way around, meaning that the RSM model over predict the recirculation length while the linear viscosity models under predict the same. The most accurate prediction is provided by RNG k-✏, followed by SST k-! and RSM. The poorest prediction is, as for the mean drag, given by the standard k-✏.

Author Method Re C

_D

L

_w

Present study SST k-! (Steady, 3D) 1200 1.24 1.39 Present study BSL RSM (Steady, 3D) 1200 1.14 1.87 Present study RNG k-✏ (Steady, 3D) 1200 1.30 1.47 Present study Standard k-✏ (Steady, 3D) 1200 1.49 1.20

Hemmati [1] DNS (3D) 1200 1.16 1.6

Table 5.1. The mean drag coefficient and recirculation length for for AR 3.2.

table 5.2 demonstrates that the trend of how the models tend to over- and under predict C

_D