Implementation of Aeroacoustic Methods in OpenFOAM

Transcription

1 EXAMENSARBETE I TEKNISK MEKANIK 120 HP, AVANCERAD NIVÅ STOCKHOLM, SVERIGE 2016 Implementation of Aeroacoustic Methods in OpenFOAM ERIKA SJÖBERG KTH KUNGLIGA TEKNISKA HÖGSKOLAN SKOLAN FÖR TEKNIKVETENSKAP

2 TRITA TRITA-AVE 2016:01 ISSN

3 Abstract A general method is established for external low Mach-number flows where aeroacoustic analogies are used to decouple the sound generation from the sound propagation. The CFD solver OpenFOAM is used to compute the flow induced sound sources and Ffowcs-Williams and Hawkings acoustic analogy is implemented to calculate the propagation of sound. Incompressible and compressible source data is gathered for a test case and upon evaluation of the noise emission the assumption of incompressibility prove to be valid for a low Mach-number flow. Furthermore the advantage of non-reflecting boundary conditions in OpenFOAM is appraised and found to be effective. Lastly the method is tested on a more complicated test case in terms of a generic side mirror and results are found to agree well with previous studies. 3

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5 Acknowledgments I want to extend my warmest thank Creo Dynamics for giving me the opportunity to do my master thesis at their company. I have felt like a part of Creo from day one and could not have wished for better colleges; your help and expertise have made this thesis possible. Moreover I want to extend a special thanks to Johan Hammar who has guided me through this process and always put time aside for me no matter how busy of a schedule he has had. I also want to thank my examinator Ciarán O Reilly for his time and valuable input. Creo Dynamics, January 2016 Erika Sjöberg 5

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7 Contents Notation 9 1 Introduction Motivation Research Objective Outline Theory Methodology CFD Analysis - predicting the flowfield Direct Simulation Turbulence Modelling Reynolds averaged Navier-Stokes Equation Large Eddy Simulations Detached Eddy Simulation Spalart Allmaras Turbulence Model Wall functions Turbulence Modelling Boussinesq Approximation Hybrid Methods Lighthill s Acoustic Analogy Curle s Analogy Ffowcs-Williams and Hawkings Analogy CFD Software Spatial Discretization Temporal Discretization Pressure and Velocity Coupling SIMPLE Algorithm PISO Algorithm Validation Case Cylinder Test Case

8 8 Contents 4.2 Review of Flow around a Cylinder Test Cases Cylinder Setup Boundary Conditions Mesh Validation Flow Validation Acoustic Validation Results Incompressible Solution Compressible Solution Interim Conclusion Generic Side Mirror Case Review of Flow around a Side Mirror Test Cases Side Mirror Setup Boundary Conditions Mesh Flow Results Acoustic Results Interim Conclusion Concluding Remarks Future Work Bibliography 53

9 Notation Nomenclature Notation q i e o Heat flux Internal energy ρ density τ ij viscous stress tensor R ideal gas constant c v specific heat (constant volume) c p specific heat (constant pressure) µ molecular viscosity σ Prandtl number η Kolmogorov length scale Λ Kolmogorov large size eddies u i velocity U i time averaged velocity u fluctuating velocity P time averaged pressure p fluctuating pressure p pressure x i coordinate position δ ij Kronecker delta e 0 total internal energy e internal energy ν t kinetic eddy viscosity K kinetic energy S ij mean strain rate tensor T temperature a speed of sound SP L sound pressure level f frequency strouhal frequency f st 9

10 10 Notation Nomenclature Notation τ r G o T ij H(f ) Dimensionless quantities C D C L Re M St retarded time distance between source and observer Green s free field function Lighthill s stress tensor Heavyside step function drag coefficient lift coefficient Reynolds number Mach number Strouhal number

11 1 Introduction 1.1 Motivation Aeroacoustics is the study of flow induced sound and was pioneered by James Lighthill back in the 1950 s. Flow induced sound can be created by turbulent wakes, detached boundary layers, vortex structures in the flow and flow interaction with solid walls [Y.Khalighi, 2010]. Flow-generated sound is today a well known problem in the transport industry, and according to a World Health Organisation report side effects of noise can involve fatigue, stress, hearing loss and hormonal imbalance [Y.Khalighi, 2010]. Emphasis is today spent on noise control and to mitigate the aerodynamic noise a deeper understanding of the physics of the flow field and the sound field is necessary. Resolving both the sound and flow field presents an inherent scale problem where as Crighton [1993] so well described it... in the 45 seconds of take-off roll of terrifyingly loud Boeing 707, the total energy radiated as sound is only about enough to cook one egg. Due to the scale difficulties severe limitations are put on computational resources and the aeroacoustic field have been heavily dependent upon experimental methods. With the advance of technology and computational power acoustic analogies have proven to be an effective tool. Lighthill established aeroacoustics when he developed an equation for the propagation of sound waves for a turbulent flow. Since then many researches have followed in his footsteps and made extensions to Lighthill s original theory. Some of the more renowned are Curle s analogy that takes into account the presence of solid boundaries, and Ffowcs-Williams and Hawkings equation that also includes surfaces in motion. 1

12 2 1 Introduction Due to the limitation of computational resources aeroacoustic analogies are a very useful tool since it drasticly reduces the computational power needed when solving for sound propagation. Acoustic analogies reduces the computational domain to the near field where high resolution is required and the sources are computed. All non-linear effects are prescribed to the near field and a linear wave operator propagates the sound in the acoustic domain. 1.2 Research Objective The objective of the present thesis is to establish a method in OpenFOAM to calculate the sound field using aeroacoustic analogies. The results should be compared to previous studies done using commercial available software. To achieve the objectives this work is broken down into several sub-goals: Build a good test case. The test case should be simple enough that one can focus on the essential flow features but still capture the unsteady von Kármán vortex shedding. Evaluate both the aerodynamic (CFD) part as well as the acoustic field calculated using the Ffowcs-Williams and Hawkings analogy. Test the methodology on a more demanding case. Other aims behind the research is to evaluate the existing boundary conditions in OpenFOAM with the goal of minimizing reflections at the boundaries. 1.3 Outline The following structure will be used in the report. Chapter 2 will cover the general theory and governing equations behind the flow and aeroacoustic field. Chapter 3 briefly explains the OpenFOAM software, user settings, and discretization schemes. Chapter 4 and 5 presents the Validation case and a study of a Generic Side Mirror respectively. Conclusions and future work is found in chapter 6.

13 2Theory 2.1 Methodology Utilizing the continuity, momentum and energy equations in combination with initial and boundary conditions the flow field as well as the acoustic part can be found. Focusing on the sound generation and its propagation various methods can be used varying in computational effort and accuracy. Two main numerical Computational Aero Acoustic (CAA) approaches can be distinguished; 1. Direct Methods: A transient solution is calculated where the source and its propagation is resolved out to the far field. This method requires a very fine grid for the spatial and temporal resolution which places stringent demands on computational resources. X. Gloerfelt [2003] writes that direct methods can only predict good acoustic results for simple configurations at moderate Re numbers. 2. Hybrid methods: The source field is computed using CFD analysis and the propagation is predicted using a transport method. Hybrid methods significantly reduces the computational demand since only the nearfield needs to be spatially and temporally resolved. Fig. 2.1 shows a schematic layout of the various CAA methods available [C. Wagner, 2007]. 3

14 4 2 Theory Figure 2.1: Various CAA Methods A distinction will be made between calculating the flow field (1) and using the data from the flow field to predict the sound field (2). Sec 2.2 will present various methods available to solve for the flow field and sec 2.4 will focus on the implementation of acoustic analogies to calculate the sound radiation. 2.2 CFD Analysis - predicting the flowfield Direct Simulation Direct methods is the most exact and perhaps also the most straightforward methodology in CAA. In Direct Simulation (DS) the governing equations are solved without using physical simplifications therein resolving the physical phenomena without modelling. The governing equations used are the fundamental equations of fluid dynamics: the continuity, momentum and the energy equation. They are here written in conservative form for a compressible fluid using the index notation and Einstein convention. The continuity equation, ρ t + (ρu i) = 0 (2.1) x i The momentum equation, (ρu i ) t + (ρu iu j ) x j = p x i + τ ij x j (2.2)

15 2.2 CFD Analysis - predicting the flowfield 5 And the energy equation, (ρe 0 ) t + (ρe 0u j + pu j ) x j = (τ iju i q j ) x j (2.3) Where the energy e 0 is the total internal energy, e 0 = e u iu i (2.4) Further equations to close the system of equations are needed. The viscous stress tensor in eq. 2.2 and eq. 2.3 is for a Newtonian fluid defined as: ( ui τ ij = µ + u j 2 ) u k δ x j x i 3 x ij k (2.5) Where µ is the dynamic molecular viscosity. The heat flux, q i, can be modeled with Fourier s law: q i = µc p σ T x i (2.6) Here c p is the specific heat and σ is the Prandtl number. Under the assumption of a thermally perfect gas the Ideal gas law can be written as: p = ρrt (2.7) Where R is the ideal gas constant. For a calorically perfect gas the internal energy and enthalpy relations can be modeled as: e = c v T h = c p T c p = c v + R (2.8) DS can be compared to Direct Numerical Simulation in CFD and the sound emission can be evaluated anywhere in the flow field but at the price of daunting computer power. DS requires a computational grid resolving the entire span from the Kolmogorov micro scale η to the large size eddies, Λ. This presents an

16 6 2 Theory inherent multi scale problem where A. Johansson [2013] writes that the ratio of the smallest to the largest scales of turbulence can be estimated as: Λ η Re3/4 Λ (2.9) For a grid spanning in three directions and resolved down to the Kolmogorov scale this would account in a grid-point-growth of Re 9/4 Λ, or taking temporal resolution into account ReΛ Turbulence Modelling Reynolds averaged Navier-Stokes Equation Reynolds averaged Navier-Stokes equations (RANS) are a time averaged version of the instantaneous Navier-Stokes equations. To cut down on computational cost the velocity vector and the pressure are divided into a steady (time-averaged) and a fluctuating part. u i = U i + u i p = P + p (2.10) This decomposition is known as the Reynolds decomposition. Incorporating the fluctuating velocity and pressure components into the incompressible Navier- Stokes equation the resulting mean flow equation can after time averaging be written as: U i t + U j U i x j = 1 ρ P + ( ν U ) i u x i x j x i u j j (2.11) With the implementation of the RANS equations a new term, ρu i u j, is found on the RHS describing the relationship between fluctuating velocities. This term is often referred to as the Reynolds stress tensor and introduces a closure problem where six additional unknown turbulent stresses arise. To close the set of equations a wide variety of turbulence models are available: Algebraic models/zero equation models: Case specific and not very general. Works well for the scenario they were created for but need additional information such as velocity gradients or geometry specifications. One-equation models: Somewhat more general. Like the name a one-equation model usually solves a transport equation for one turbulent variable like

17 2.3 Turbulence Modelling 7 the turbulent kinetic energy K, or the eddy viscosity ν T. Somewhat more general but case specific input is still required. An example of a common one-equation model is the Spalart-Allmaras model. Two-equation models: Two transport equations are solved for two different variables. Isotropic turbulence is generally assumed and no additional information is needed; hence the model is "completely formulated in terms of local quantities" [A. Johansson, 2013]. Example of common two-equation models are k-ɛ and k-ω Large Eddy Simulations Large Eddy Simulation, or LES, is a model where instead of averaging equations as done in RANS, the equations are filtered. This is done on the Navier-Stokes equation and the result are variables that depend on both space and time. Filtering of variables is a part of a hybrid method where large eddies are resolved and small sub-grid scale information is modelled. A. Johansson [2013] writes that the small scale turbulence, or the Kolmogorov scale, is more isotropic and contains less energy than the large scale turbulence; hence errors introduced in the modelling process should be less critical. The small scales are usually modelled using the Boussinesque hypothesis Detached Eddy Simulation Detached Eddy Simulation (DES) was originally formulated for the Spalart Allmaras model and is a combination of Unsteady RANS (URANS) and LES. URANS is very similar to RANS since they both solve for the time-averaged flow but differs in the sense that URANS keeps the transient term [P.Spalart, 2007]. DES models the boundary layer using RANS while the outer eddies are resolved using LES. ρ ν t t + ρũ j ν t = ( ) µ + µt ν t + C b2ρ ν t ν t + P Ψ (2.12) x j x j σ ν t x j σ ν t x j x j ν t = ν t f i (2.13) The production term, P; ( P = C b1 ρ s + ν ) t k 2 d 2 f 2 ν t s = (2s ij s ij ) 1 2 (2.14) And the destruction term Ψ ; ( ) νt 2 Ψ = C w1 ρf w (2.15) d From the RANS SA model the distance, d, stems from the distance to the nearest wall, while in the DES model the distance d comes from the minimum of the cell length and the turbulent length scale d. Hence

18 8 2 Theory d = min(d, D des ) (2.16) This means that in the case where d < C des, which would occur in the bondary layer, the DES model switches to RANS mode [P.Spalart, 2007] Spalart Allmaras Turbulence Model The Spalart Allmaras (SA) model is a one-equation model that solves for ν. ν is referred to as nut in OpenFOAM but is often called the Spalart Allmaras variable. The SA model drops the last part in eq when solving for the eddy viscosity. Following relations are found in the SA model: where the viscous damping functions are: ν T = νf v1 (2.17) X 3 f v1 = X 3 + C 3 v1 And the transport equation is written as: X = ν ν (2.18) ν t + ũ ν j = x j where c b1 S ν }{{} production + 1 σ [ ( (ν + ν) ν ) ( ) ν ν ν 2 + c x j x b2 ] c j x j x w1 f w j d } {{ }} {{ } Diffusion destruction (2.19) And vorticity S is modelled to keep its log-layer characteristics [S Deck, 2002]: S = ν 2Ω ij Ω ij f v3 + k 2 d 2 f v2 (2.20) Ω ij = 1 ( ũi ũ ) j (2.21) 2 x j x j x f v2 = 1 f 1 + xf v3 = 1 (2.22) v1 Following the recommendations in P.Spalart [2007] for adequate ratio of turbulent kinematic viscosity to kinematic viscosity, ν t = 3 5, nut ( ν) was set to ν in the turbulent mirror case Wall functions To circumvent the need for a very fine mesh resolution wall functions can instead be used. Wall functions models the near-wall flow using empirical laws and information such as distance to the wall, pressure gradients, shear stress etc. Wall functions are based of von Kármáns law of the wall and for successful usage 30 < y + < 300 while in some cases the range is even larger; 11 < y + < 300.

19 2.3 Turbulence Modelling Turbulence Modelling As previously mentioned the closure problem causes a need for turbulence modelling. Using time-averaging methods instead of looking at the instantaneous continuity and Navier-Stokes equations we are left with six unknowns (see sec ) called the Reynolds stresses and the scalar transport terms. To close this system of equations turbulence modeling is used Boussinesq Approximation One of the early attempts of modeling the turbulent shear stress was made by the French 19 th century scientist Boussinesq [A. Johansson, 2013]. Boussinesq described the Reynolds stresses using mean velocity gradients. ρu i u j = ρν T S ij 2 3 ρkδ ij (2.23) Here ν T is the kinetic eddy viscosity, K the kinetic energy and S ij the mean strain rate tensor: K = 1 2 u i u j S ij = 1 2 ( Ui + U ) j x j x i (2.24) Using the Boussinesq model the closure problem is reduced to model the eddy viscosity. Since the eddy viscosity is dominated by the length and velocity scale of the large turbulent eddies (Λ, V ) this assumption brings about a huge reduction in computational cost [A. Johansson, 2013].

20 10 2 Theory 2.4 Hybrid Methods The hydrodynamic phenomenon is the salient feature in the flow region with the energy of the acoustic field generally of the order of 1% as compared to the total energy [J.Larsson, 2002]. Since the sound is generated in the flow region and propagated through the far field hybrid methods decouple the flow generation from the acoustic propagation in the far field; consequently allowing for methods adapted for the various regions. Hybrid methods are only applicable for aeroacoustic problems that exhibit a one-way coupling between the flow and the acoustics. In the case of a one-way coupling the flow is independent of the acoustic part, hence no energy is fed back into the flow from the acoustic wave propagation. An advantage behind one-way coupled problems is that they can be separated into two different problems, one part being the flow induced source field while the other one is propagation of sources. Figure 3.2 shows the three different flow regions generally used when dealing with aeroacoustic flows. Far field Near field Source Region Figure 2.2: Flow and Acoustic Regions CFD is a prevalent tool when resolving the sound sources in the flow region. In a direct coupling scenario the sound sources can be computed at a much lower computational cost than direct methods thanks to a much smaller computational domain. In this case the computational region should cover all non-linear effects, which is usually only up to a few wave lengths [C. Wagner, 2007] Lighthill s Acoustic Analogy James Lighthill laid the groundwork for the models of sound generation and is today considered the father of aeroacoustics. To reduce the generated sound from jet engines Lighthill developed analogies uncoupling the sound field from the source field. Lighthill used the fundamental equations of fluid dynamics to model the source field as an inhomogeneous wave equation.

21 2.4 Hybrid Methods 11 Differentiating the continuity equation (eq. 2.1) with respect to time, and the momentum equation (eq. 2.2) with respect to space, while subtracting the latter from the differentiated continuity equation one gets: x i 2 2 ρ t 2 2 ρu i u j x i x j = 2 p x i 2 2 τ ij x i x j (2.25) Subtracting a 2 2 ρ from equation (eq. 2.25) it can be re-formulated as Lighthill s wave equation: 2 ρ t 2 2 ρ a2 x 2 = 2 T ij (2.26) i x i x j Where T ij is the Lighthill s stress tensor and defined as T ij = ρu i u j τ ij + (p a 2 ρ)δ ij (2.27) No assumptions has been made at this point and Eq is exact. A distinction has been made between the sound sources and the propagation of sound sources. The left hand side is an ordinary wave operator whereas the right hand side is the acoustic source terms. For Lighthill s equation to be applicable the right hand side should be known as well as decoupled from the acoustic field. By comparing the magnitude of the three terms in the stress tensor Lighthill deduced that the momentum flux tensor ρu i u j is the only significant contributor to sound production T ij for cold jets. The acoustic wave equation can be solved analytically if the right hand side is assumed known [J.Larsson, 2002]. A common approach is to integrate the sources using a free fields Green function. 1 ρ(x, t) ρ 0 = 4πa 2 Where τ is the retarded time; τ = t r a 1 r 2 T ij V (y) (2.28) y i y j Curle s Analogy Lighthill s theory was further extended by Curle to incorporate the presence of solid boundaries upon the aerodynamic sound. Curle s approach was to find a solution to the inhomogeneous wave equation where the double divergence of

22 12 2 Theory Eq can be taken outside the integral sign. The derivation is carried out in a similar fashion as Lighthill s original work but with two additional steps. To account for solid boundaries a surface integral is added through the Kirchoff- Helmholtz formula [Ask, 2008], then a transformation from source coordinates to observer coordinates is done. Starting with the general solution of the inhomogeneous wave equation previously mentioned but this time on a bounded domain. 1 ρ(x, t) ρ 0 = 4πa 2 V 1 r 2 T ij V (y) y i y j 1 ( 1 ρ 4π S r n + 1 r r 2 n ρ + 1 ) r ρ S(y) a r n t (2.29) Utilizing partial integration and symmetrical properties from the Green s function G 0 [S. Rienstra, 2015]: G 0 = G 0 G 0 x i y i τ = G 0 t (2.30) Yields: 1 2 ρ(x, t) ρ 0 = 4πa 2 x i x V j T ij r dv (y) 1 n j 4πa 2 x S i r [ρu iu j + pδ ij τ ij ] ds(y) (2.31) + 1 ρu i n i 4πa 2 t S r ds(y) The second surface integral in Equation 2.31 characterizes the monopole field created by fluid vibrations of the body [5] which in many cases can be neglected. In the case of solid surfaces where the velocity on the surface is zero Curle s final equation reads: 1 2 ρ(x, t) ρ 0 = 4πa 2 x i x V j 1 4πa 2 x S i T ij r dv (y) n j r (pδ ij τ ij ) ds(y) (2.32) A more detailed derivation can be found in J.Larsson [2002].

23 2.4 Hybrid Methods Ffowcs-Williams and Hawkings Analogy Ffowcs-Williams and Hawkings (FW-H) extended the work that Curle had published by taking into account the sound generation from arbitrary motion of a body in a turbulent flow. FW-H equation is a generalization of Curle s analogy where the governing equations are re-written in such a way that the source terms will account for boundary effects. The result is an equation valid for a continuous infinite space. 1 2 ρ(x, t) ρ 0 = 4πa 2 x i x V j T ij r(1 l jv j a ) dv (y ) 1 4πa πa 2 x i S t S S F i r(1 l jv j a ) Q r(1 l jv j a ) ds(y ) ds(y ) (2.33) Where the source terms in a moving reference frame are: T ij = ρ(u i + v i)(uj + v j) τij + (p a2 (ρ ρ )) δ ) ij Fi (ρ(u = i + v i) uj + pδ ij τij n j (2.34) Q = (ρ v i + ρu i )n i The three source terms T ij, F i, Q are associated with quadrupole, dipole, and monopole source mechanisms respectively. The quadrupole source mechanism is due to fluctuating stresses in the fluid (unsteady Reynolds stress) and dipole sources are created by external unsteady forces, or fluid pressure on a solid boundary. Monopole sources are due to volume flow or fluctuating mass injection. In the case of an impermeable surface simplifications can be made. Focusing on the dipole term which is of most importance to this work, ui is by definition equal to zero on the surface, and Fi can therefore be written as: F i = (pδ ij τ ij ) n j (2.35) Numerical implementation of Eq can be difficult due to the combination of spatial and temporal derivatives with respect to the observer frame of reference [Williamson, 1996]. Later formulations by Bretner and Farassat resolves the Ffowcs-Williams and Hawkings analogy in the frequency domain to circumvent

24 14 2 Theory the problem of emission time. With the implementation of the FW-H analogy isotropic wave propagation is considered, and can hence only be expected to provide good results in flows with zero or low mean motion. To incorporate mean flow the convected FW-H analogy can be used where Gloerfelt et al. introduces the concept of having the observer move with the mean flow. The derivation can be found in X. Gloerfelt [2003] and the convected FW-H equation for the frequency domain is written as following: { 2 x 2 i + k 2 2iM i k x i M i M j 2 x i x j } [H( f )c 2 ρ (x, ω)] [ = 2 T x i x ij (x, ω)h(f)] 2 j xi 2 [F i (x, ω)δ(f )] iωq(x, ω)δ(f ) (2.36) From the integral solution by X. Gloerfelt [2003] the source terms can be written as: T ij = ρ(u i Ui )(u j Uj ) + (p c ρ)δ 2 ij τ ij F i = [ρ(u i 2Ui ) u j + pδ ij τij ] n j (2.37) Q = ρu i n i It should here be noted that instead of using the free-space Green s function the convective Green s function should be used.

25 3 CFD Software OpenFOAM is an open source CFD software package and was created by Henry Weller in It is written using C++ as programming language, structured as a library, and released under the GNU Public License. One of the core ideas of OpenFOAM is to share the source code with its users. Pre-built solvers for a wide variety of applications are available in OpenFOAM, but also the opportunity of customizing a solver, or building one s own solver. OpenFOAM can easily be run in parallel. The general structure of OpenFOAM is divided into three main directories; a System, Constant and a Time directory. OpenFOAM Case System controldict fvschemes fvsolution Constant polymesh properties Time Directories U,P,nuTilda Piso Algorithm Figure 3.1: OpenFOAM case structure Set boundary conditions Solve momentum eq. 15 compute intermediate velocity field (v )

26 16 3 CFD Software The System directory contains at minimum the controldict, fvschemes, and fv- Solution. The controldict controls parameters such as start/end time, step size, when and what files to output etc. fvschemes and fvsolution dictates what discretization schemes to use, equation solvers and tolerances respectively. The Constant directory contains the mesh and other physical properties needed for the calculations. For a turbulent case the turbulent properties would be specified here [Greenshields, 2015]. 3.1 Spatial Discretization Much time can be spent discussing and testing the numerous schemes in Open- FOAM. Since this is out of the scope of this thesis only a few things will be said about the chosen settings. The schemes used are second order, and well tested at Creo Dynamics. The GAMG solver, short for Geometric Agglomerated Algebraic Multigrid solver, is a linear solver used for the pressure. SnGradSchemes is a user defined variable in OpenFOAM that allows the user to chose what surface normal grad scheme to use. Limited, is used for this work, which means that a limited nonorthogonal correction is to be used. Gaussian integration is a second order discretization scheme and is defined in the fvschemes dict together with the choice of interpolation scheme. In the Gaussian integration values from cell centers need to be interpolated to face centers. A linear interpolation scheme is used for this. 3.2 Temporal Discretization OpenFOAM provides a wide range of temporal discretization schemes varying in accuracy and computational cost. During a steady state simulation the SteadyState option can be specified in the fvschemes for the time scheme, and the time derivative will be switched off. For a transient problem the solution is time dependent and a solution will be found by a time-marching method. The Crank-Nicolson (CN) method averages properties between time steps where both the old and new values are used. CN uses a weighted average between spatial steps where for an equal contribution close resemblance can be found to central-differencing schemes. The Crank-Nicolson method is often used for parabolic equations and can for the heat conductivity equation T t = α 2 T be written as [Anderson, 1995]: T n+1 i T n i t = α 1 2 (T n+1 i+1 + T n i+1 ) ( 2T n+1 i 2T n i ) (T n+1 i 1 + T n i 1 ) ( x) 2 (3.1)

27 3.3 Pressure and Velocity Coupling 17 Where α is the thermal diffusivity constant. Being an implicit method Eq. 3.1 can not be solved for a particular node point but has to be solved for all grid points simultaneously. Crank-Nicolson is conditionally bounded and comes with a time constraint. Since it is not a pure explicit method CN does not have as strict of a time step for stability. The Courant- Friedrichs-Levy (CFL) condition is used to find a suitable time step for temporal accuracy as well as stability. CFL = u t x 1 (3.2) The CFL condition is a necessary condition for stability but does not ensure a stable solution. Based on the CFL value the computational time-step for the time advancement for this study s two cases is found: t cylinder = s t mirror = s (3.3) With a resulting max CFL value of around 0.48 for the cylinder and 1.5 for the mirror. 3.3 Pressure and Velocity Coupling PISO and SIMPLE are two algorithms commonly used in OpenFOAM to solve the equations for velocity and pressure. PISO is a semi-implicit method that stands for Pressure-Implicit Split Operator and is developed for transient problems. SIMPLE on the other hand, or Semi-Implicit Method for Pressure-Linked Equations, is a steady-state algorithm. PIMPLE is a merged version of the PISO- SIMPLE algorithms. Both SIMPLE and PISO are discretized using a staggered velocity field. This is done so that in the case of a checkerboard pressure field the discretised solution will not exhibit a non-physical uniform pressure distribution. H.K Versteeg [1995] presents a grafic figure of the discretized volume where scalar variables are defined on the nodes (black dots) while velocities are defined between nodes in fig. 3.2.

28 18 3 CFD Software Figure 3.2: Staggered grid for velocity components SIMPLE Algorithm SIMPLE, or the Semi-Implicit Method for Pressure-Linked Equations, is a steadystate algorithm based on work by Patankar and Spalding (1972). The SIMPLE algorithm uses a guess-and-correct method [H.K Versteeg, 1995] where in its initial stage SIMPLE approximates the velocity field using the momentum equation and a guessed pressure field p. The discretised u- and v-momentum equations are: a i,j u i,j = a nb u nb + (p l 1,J p l,j )A i,j + b i,j (3.4) a I,j v I,j = a nb v nb + (p I,J 1 p I,J )A I,J + b I,J (3.5) where a i,j and a nb are coefficients, for a more detailed explanation on how to calculate these see the work by H.K Versteeg [1995]. The correct pressure field p, relates to the guessed pressure field p and the correction, p, in the following way; p = p + p (3.6) Many times under-relaxation is used to reduce the risk of divergence. Underrelaxation reduces the amount the guessed pressure field is corrected by p, and the new pressure field can be calculated in a following way p new = p + α p p (3.7) Here α p is the under-relaxation factor and it is for the present case set to 0.33.

29 3.3 Pressure and Velocity Coupling 19 From the guessed pressure field, and the discretized momentum equation u and v velocities are solved for. Where for a converged solution PISO Algorithm u = u + u v = v + v (3.8) p = p u = u v = v (3.9) The PISO algorithm is similar to the SIMPLE algorith but has another corrector step. The main steps of the PISO loop are found in Fig. 3.3 below [Greenshields, 2015]. Figure 3.3: Main steps of Piso The PISO algorithm entails two corrector steps and will hence solve for the pressure equation twice. Due to this additional storage and computational power is needed when running PISO.

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31 4 Validation Case To test the methodology a validation case is essential. The test case should fill certain criteria such as well tested and documented in literature and limited geometrical complexity. A more straightforward validation case allows for more time to be spent on the paramount characteristics of the problem. 4.1 Cylinder Test Case Flow past a circular cylinder is chosen as the test case on the basis of these criteria. It is an area that has been intensively studied due to its fundamental importance and high applicability. Williamson [1996] categorises the vortex formation for various flow regimes into different groups based of their base suction coefficient: 21

32 22 4 Validation Case Figure 4.1: Vortex Dynamics in the Cylinder Wake Regime Figure 4.1 is a plot of the base suction coefficients for a wide variety of Reynolds numbers. The base suction coefficient is the negative value of base pressure coefficients (-Cp). Williamson points towards its usefulness when distinguishing between different flow regimes since the base suction coefficient is highly effected by vortex formation in the immediate wake. From Fig. 4.1 the laminar vortex shedding regime is found for Reynolds numbers ranging between 49 to The 3-D wake transition regime is found between B and C in Fig Fig. 4.1 and with increased Reynolds number the 3-D flow features become increasingly important. With Williamsons research in mind a Reynolds number of 150 was chosen for the study. Re = ρu D µ (4.1) The Reynolds number is based on the oncoming free-stream velocity and the diameter of the cylinder. The laminar flow past a cylinder and its vortex shedding regime is a well tested case both experimentally and numerically which makes it a well suited test-case.

33 4.2 Review of Flow around a Cylinder Test Cases 23 A hybrid method using Ffowcs-Williams and Hawkings method will be implemented in OpenFOAM where the flow field is solved using CFD and the fluctuating wall pressure on the cylinder will be used to calculate the far field sound radiation. This hybrid method only requires a spatially and temporally high resolution around the cylinder and is therefore beneficial in terms of computational resources. Evaluating the sound field in the frequency domain will circumvent the costly time integration, and utilizing the convected FW-H equation will account for the mean flow. The FW-H results will be compared to directly computed results as well as to previous studies. 4.2 Review of Flow around a Cylinder Test Cases Lixia Qu [2013] studied the intermediate wake for a 2D flow around a cylinder at a Reynolds number of 100. They found from their sensitivity study for the domain size that a physical domain height of 120*D (where D=diameter) caused less than a 0.5% discrepancy for factors such as drag and lift when compared to a domain height of 200D. Shair et al displayed that the stability of the wake and the critical Reynolds number is closely related to increases in blockage for a flow past a solid body. B. Kumar [2006] continued on the same lines as Shair et al s work and investigated the effect of blockage on critical parameters for the onset of wake instabilities. Their work found the Strouhal number to be highly effected by blockage, but that the effect of blockage is negligible when the lateral boundaries are positioned more than 100 D from the cylinder. 4.3 Cylinder Setup Choosing a physical domain for the numerical simulation is a balance between computational cost and accuracy. A larger domain reduces the effect of boundaries and blockage but can be computationally heavy. Since this study is done in two-dimensions (2-D) the problem is not as pertinent. For 2-D cylinder simulations, rectangular, polar, or O-grid domains are most common. A rectangular domain is selected which allows for a higher resolution in the wake and more flexibility regarding boundary conditions. The cylinder is placed with its centre at the origin of a Cartesian coordinate system.

34 24 4 Validation Case u H D z y x L Figure 4.2: Representation of the solution domain The diameter of the cylinder, D, is 1 and located at the center of the domain (L/2, H/2). The overall length is 200D and the height of the domain is 100D. While running a 2-D simulation OpenFOAM does not support strictly 2-D cases and the depth of the domain size is therefore specified as 1. This additional volume created by a front and back plane will later on be defined as empty and yield a 2-D domain. 4.4 Boundary Conditions A free-stream Dirichlet condition, u = u, v = o is assigned to the inlet, outlet, and freestream boundaries (top and bottom). The outlet and external walls are placed far enough from the cylinder that the disturbances caused by the cylinder are assumed to be small enough that no velocity gradients in the flow are present. Table 4.1: Cylinder Boundary Conditions. Velocity Pressure Inlet u vel specified (v=w=0) Wave Transmissive Outlet u vel specified (v=w=0) Wave Transmissive Wall No slip condition (u=v=w=0) Zero gradient Symmetry u vel specified (v=w=0) Wave Transmissive

35 4.5 Mesh 25 At the cylinder a no-slip boundary condition is applied. To compute the sound field the source terms will be modeled from the wall pressure terms. Since the acoustic energy often is less than 1% of the total energy of the flow it is of high importance to reduce the influence of boundaries reflecting back energy into the computational domain [J.Larsson, 2002]. If caution is not taken the effect of boundaries can have a larger impact on the sound field than the sound itself. A wave transmissive pressure boundary condition is therefore applied in OpenFOAM to all four boundaries. 4.5 Mesh Implementing FW-H s analogy to compute the sound region assumes that the generation of sound can be decoupled from its propagation. The aerodynamic quantities are transiently recorded using a CFD mesh. Since sound is the propagation of unsteady, small, pressure fluctuations it is of paramount importance that the mesh is fine enough to capture this phenomena. The geometry and the mesh are created using Beta CAE s pre-processor ANSA. For the purpose of this work one mesh was used for both the CFD as well as the acoustic computation. The dual purpose of the mesh calls for a higher grid resolution in the source region as well as downstream of the cylinder. In the cylinder wake the ANSA Size Box entity is used for local mesh control. Four boxes are placed around the cylinder stretching downstream, and allowing for refinement in the near wake to capture the periodic vortex shedding. Figure 4.3: Computational Mesh created in ANSA and used for the cylinder simulations in OpenFOAM To capture the high velocity gradients in the boundary layer 10 prism layers are used with a growth factor of 1.1 and a first layer height of D.

36 26 4 Validation Case Figure 4.4: Close up on the prism layers constructed to capture the high velocity gradients from the cylinder walls. Two different meshes are created ranging in how fine the wake is resolved. Table 4.2: Set up mesh independence study. Cylinder Mesh 1 Mesh 2 Nr. Cells Dist 1 st y + prism layer Largest cell size Box Largest cell size Box Largest cell size Box For validation the drag and lift force, as well as the Strouhal number, on the cylinder are compared between the two different meshes as well as to previous results. C d = F D 1 2 ρ U 2 D C L = F L 1 2 DρU St = fl 2 U (4.2) Where F D and F L are the forces in the longitudinal and lateral direction respectively. The drag and lift forces are per unit width being a 2-D geometry. Table 4.3: Mesh Independence Study. C L C d St Fine mesh Coarse mesh Comparing force parameters very slight differences are captured between the fine and the coarse mesh indicating that the fine mesh captures the flow field accurately.

37 4.6 Validation Validation To verify the accuracy of the results a two step validation process should be carried out. Where initially a flow assessment should be implemented followed by an acoustic validation. For trustworthy results it is critical that both the hydrodynamic flow calculations are calculated correctly and that the FW-H acoustic analogy is implemented accurately. Moreover simplifying assumptions made during the evaluation of the FW-H source terms should be assessed and if possible verified Flow Validation Thanks to a vast amount of published research from similar cylinder studies flow results are available for comparison. Table 4.5 presents some of them. Table 4.4: Comparison to previous studies. Re C L C D C pb St Present study Qu et al Qu et al (larger domain) Inoue and Hatakeyama The base suction coefficient, C pb at the base of the cylinder, can be compared to Lixia Qu [2013] study of the intermediate wake region for flow past a cylinder at a Reynolds number of 150. A close correlation is found. The drag coefficient also agrees well with previous studies and is the same as O.Inoue [2002] found in their study. The Strouhal number for the present cylinder simulations can be compared with the Strouhal-Reynolds-number relationship for the vortex shedding from a circular cylinder proposed by U. Fey [1998]. St = (Re) 1 2 (4.3) Equation 4.3 is developed from experimental data and would yield a Strouhal number of (Re = 150); which is within 0.2% of the Strouhal number obtained in the present study. A graphical representation of Eq. 4.3 and previous results are displayed in Fig 4.5.

38 St 28 4 Validation Case Fey Qu et al. Park et al. present Re Figure 4.5: Strouhal vs. Reynolds number comparison of similar 2-D flow past a cylinder studies. The Strouhal number from the present case, in Fig 4.5, is right on the trace of what is to be expected for a free stream external flow for a cylinder with Re = Acoustic Validation For the implementation of the FW-H analogy in Eq the quadrupole source term, T ij, is considered negligible in comparison to the dipole source term, F i, under the assumption of a low Mach number flow. Moreover the viscous shear stress, τ ij, is believed to have such small contribution to the dipole source term that only the pressure fluctuations are taken into account when calculating the sound field. To assess the validity of previous assumptions results soon to be published from Creo Dynamics is used. The data from Creo Dynamics is achieved using the commercial solver StarCCM + and an identical setup. Fig. 4.6 shows a directivity comparison between including all source terms, or only taking the pressure fluctuations into account when evaluating the FW-H sound field.

39 4.6 Validation Figure 4.6: Directivity plot comparison using StarCCM + ; based off FW-H wall pressure fluctuations ( blue line) and the inclusion of T ij and τ (red line). 300 Fig. 4.6 shows that only taking the fluctuating surface pressure into account during the evaluation of the sound field in the FW-H analogy gives a good representation at a Mach number of 0.2. Since the significance of viscous shear stress is less for higher Reynolds number flows one can expect an even better agreement for higher Reynolds cases. To validate the acoustic calculations a directivity comparison is made between the results from Creo Dynamics using StarCCM +, and the results of the present study. Only the fluctuating pressure in the dipole source term is considered Figure 4.7: Directivity plot comparison at f shed based of FW-H wall pressure fluctuations using StarCCM + ( red line) and OpenFOAM (blue line).

40 30 4 Validation Case Comparing the calculated directivity from StarCCM + and OpenFOAM at a diameter of 10D a close agreement is found. A distinguished latitudinal directionality of the sound is also evident from the directivity plot of the shedding frequency in Fig Results A steady state solution is primarily run and the converged results are used to initiate the incompressible and compressible simulations. Data from pressure fluctuations on the cylinder is captured once the von Kármán shedding shows a time independent periodic behaviour. A snap shot of the vorticity is presented in Fig 4.8. Figure 4.8: Vorticity cylinder 0 < ω D/ L < 2 Fig 4.8 captures von Kármán vortex shedding where vortices are shed from the upper and lower part of the cylinder Incompressible Solution The assumption of incompressibility is often assumed valid for low Mach number flows; Ma < 0.3. A Mach number of 0.2 is used in the present case and to investigate if this is a reasonable supposition a comparison of the directivity using an incompressible as well as a compressible approach is carried out.

41 4.7 Results Figure 4.9: Comparison compressible vs. incompressible results at a distance of 5D. (blue) compressible (- red) incompressible Fig 4.9 shows a slightly higher directivity in the compressible solution, but overall both solutions concur well. The Strouhal number for shedding frequency is the same. Lift (C L ), is calculated using r.m.s values and (C d ) is calculated from the time mean drag value. Table 4.5: Comparison Incompressible vs. Compressible results Re C L C d Compressible Incompressible C L in the compressible case is 5% larger than in the incompressible case and the same trend is seen for C d where a 2.3% greater C d is measured in the compressible solution Compressible Solution Fig 4.10 shows the cylinder pressure coefficient. As expected the pressure coefficient is close to unity in the frontal stagnation point at the rear (θ = 180) while a minimum is reached at θ of 95 deg.

42 Cd Cd Cp 32 4 Validation Case Figure 4.10 It is critical that the boundaries do not affect the solution since reflected sound waves will propagate through the domain. With the implementation of a wave transmissive boundary condition in OpenFOAM a large impact on the drag coefficient (C d ) is measured Time # Time #10-4 Figure 4.11: C d without wave transmissive boundary conditions Figure 4.12: C d with wave transmissive boundary conditions Fig exhibits a larger periodic behaviour increasing with time while Fig displays no such trend. This discrepancy can also be seen in the instantaneous pressure distribution in Fig 4.13, where an almost checker-board pressure propagation is captured in the left figure.

43 SPL (db) 4.7 Results 33 Figure 4.13: Implementation of different boundary conditions. Left figure: Zero gradient pressure condition. Right figure: Wave Transmissive pressure condition A probe is positioned in the domain at a radius of 5D from the cylinder [- 3.5D,-3.6D]. Pressure fluctuations are measured and used to compute the radiated sound. The sound emission is also computed in the same point using the pressure fluctuations on the cylinder and FW-H analogy. Equation 4.4 is used to calculate the Sound Pressure Levels (SPL) in Fig. 4.14: SP L = 10log 10 [ p 2 p 2 ref ] (4.4) Where p ref is Frequency (Hz) Figure 4.14: SPL at sensor 10 (red ) Measured, (black) FW-H

44 34 4 Validation Case From Fig the dominant node is found at the shedding frequency, f = The directly computed noise and the FW-H analogy results agree reasonable well for the first and second harmonics while at higher frequencies the FW-H analogy results are clearly underpredicting. This could be due to the neglection of the volume integral and hence the omission of the quadrupole term. Fig shows the directivity at the shedding frequency calculated from direct probe measurements as well as FW-H analogy Figure 4.15: Directivity comparison at a radius of 5 D from the cylinder, ( ) Measured, ( ) FW-H calculated. The longitudal dipole shape is expected at even multiples of the shedding frequency which is caused by the drag force on the cylinder [Y.Khalighi, 2010]. X. Gloerfelt [2003] s formulation is used where the convective effect is accounted for, however the influence of viscous forces is not taken into account in the CFD calculation of the sources. The slight left-lean in the directionality for the directly measured pressure fluctuations is caused by the quadrupole term. The quadrupole term is calculated using a volume integral and is very computationally heavy since the entire flow field needs to be stored. Based of the results achieved in Sec the quadrupole term was neglected. Had this not been done one would have expected the two impediance plots to agree better. Fig shows a closer look of the force coefficients, C L and C d over time.

45 4.7 Results C d and C L tu/d Figure 4.16: (black ) C L, (red) C d It should be noted here that the mean drag coefficient is subtracted from C d for an easier amplitude comparison. Comparing the magnitude of the lift and drag coefficients the amplitude of C L is much larger, indicating that the lift force has a greater contribution to the dipole sound. It can also be noted that C L has the frequency of vortex shedding, T = whereas C d oscillates at T /2. An overall comparison can be made between the implementation of wave transmissive boundary conditions and its exclusion, and a compressible versus incompressible approach Figure 4.17: Directivity comparison (red) non wave transmissive solution, (- black) wave transmissive BC implemented, (green) incompressible solution. Fig shows clearly that reflections at outer boundaries contributes to a

46 36 4 Validation Case larger error than assuming a constant density. 4.8 Interim Conclusion The calculation of source terms in OpenFOAM can with success be used for simpler geometries when using the Ffowcs-Williams and Hawkings equation to predict the flow induced noise field. The assumption of incompressibility can be made for a low Mach-number flow without loosing much accuracy in the sound field computation. The importance of reducing non-physical reflections at domain boundaries

47 5 Generic Side Mirror Case It is today known that exposed components of a moving object can create flow structures that will generate noise. Ask and Davidson [2009] writes that at a velocity of around 120 km/h the exposed components are the major noise contributor for a vehicle. With this in mind the external aerodynamic design of vehicles has changed immensely during the previous decades. 5.1 Review of Flow around a Side Mirror Test Cases The side mirror is one area of vehicles that has been studied extensively in the past. With the advances of computational resources CFD are becoming a prevalent tool and resources are shifted from experimental methods towards computer modelling. Ask and Davidson [2009] investigated flow past a generic side mirror on a flat plate at a Reynolds number of Re D = based of the mirror diameter. They found a horseshoe vortex forming from flow stagnation as one approaches the front side of the mirror. Ask and Davidson s study was carried out using FLUENT commercial solver and the results were compared with experimental data from Daimler Chrysler. Upon comparison the flow trends were found to agree perfectly with the experimental data from Daimler Chrysler (DC) in the wake and shear layer; however the fluctuation values were under predicted by 5dB. Moreover the laminar flow separation over the curvature of the mirror reported in experiments from DC was not captured in the simulations. B. Lockhande [2013] studied the same identical side mirror as Ask and Davidson but at a higher Reynolds number of and hence at a higher velocity (200km/h). Lockhande implemented FW-H acoustic analogy in Fluent and his results were found to mainly agree within 5dB. 37

48 38 5 Generic Side Mirror Case For the present side mirror study a similar approach is taken to calculate the radiated sound as was chosen for the cylinder. Due to the presence of walls FW- H analogy is used where the flow field is primarily solved for, and the fluctuating wall pressures are used to compute the sound field. Simulating a low Machnumber flow the quadrupole terms are considered negligible in comparison to the dipole terms and the assumption of incompressibility is considered reasonable. 5.2 Side Mirror Setup The mirror used for the present study is a simplified side mirror and is built up by half a cylinder with a diameter D of 0.2 m, merged with a quarter sphere on the top. The Reynolds number is based on the diameter of the mirror, and a free stream velocity of 39m/s. The mirror is mounted on a flat plate and Fig. 5.1 shows the geometry of the mirror while Figure 5.2 illustrates its position on the plate. The implemented setup is a direct replication of Ask and Davidson [2009] study. Figure 5.1: Side view and frontal view of the generic side mirror The computational domain is bounded by a 50D wide 25D high inlet 25D upstream of the mirror, and an outlet located 37.5D m downstream of the mirror. Employing a large computational domain the blockage ration is < 1% to minimize the effect of external walls.

49 5.3 Boundary Conditions m 0.9 m 1.5 m Figure 5.2: Setup of mirror and base plate 5.3 Boundary Conditions A free-stream Dirichlet condition, u = u = 39m/s, v = w = o is assigned to the inlet, outlet, and free stream boundaries (top and bottom). Running an incompressible simulation zero pressure gradients are used for all boundaries but the outlet since OpenFOAM requires at least one boundary with a specified value. Table 5.1: Boundary conditions generic side mirror. Velocity Pressure Inlet u vel = 39m/s, (v=w=0) zero gradient Outlet u vel = 39m/s, (v=w=0) fixed value Mirror no slip condition (u=v=w=0) zero gradient Bottom plate no slip condition (u=v=w=0) zero gradient External walls u vel = 39m/s, (v=w=0) zero gradient The choice of Solver and Solution settings in OpenFOAM is far from trivial and one could spend a vast amount of time trying to find the optimal combination. Due to time constraints the decision was made to move forward using similar settings to those used at Creo Dynamics. Table 5.2 lists some of the main settings specified in OpenFOAM s System dict.

50 40 5 Generic Side Mirror Case Table 5.2: Solver specifications. Function setting Time stepping CrankNicolson Grad Schemes Cell limited Gauss Linear Interpolation schemes Default Linear Laplacian Schemes Gauss Linear The Crank-Nicolson scheme was used with an off-centering coefficient of Ψ = 0.5. This off-centering coefficient blends the CrankNicolson scheme with the Euler scheme, where a Ψ = 1 would yield a pure Crank-Nicolson scheme and vice versa. Since the Euler method is of first order accuracy, while the Crank-Nicolson method is of second order, a blending of the two allows for an improved stability [Greenshields, 2015]. 5.4 Mesh OpenFoam s snappyhexmesh utility is utilized to generate a mesh. SnappyHexMesh builds a 3-dimensional mesh by starting of with a very coarse background blockmesh that will snap to the surface. Reduced computational cost can be achieved by keeping the mesh as coarse as possible in areas where refienement is unnecessary. This is achieved by utilizing four refinement boxes enclosing the mirror and its wake leaving the rest of the computational domain fairly coarse. Table 5.3: Grid density for the different boxes in the mesh. Level length (mm) Box Box Box Box To ensure a high mesh quality certain mesh quality controls can be set in SnappyHexMesh. The more critical ones are: maxnonortho = 65 minvol = 1e 13 mindeterminant = minfaceweight = 0.05

51 5.4 Mesh 41 The final mesh satisfy these criteria and consists of cells, where of them are used to describe the mirror surface. To accurately capture the high velocity gradients at the surface of the mirror and its bottom plate 5-7 prism layers are used. Figure 5.3: OpenFOAM Side mirror mesh generated using snappyhexmesh utility in The y + value is a non-dimensional distance used as a tool when selecting the appropriate mesh configuration for turbulent flows. y + = u τairy v air (5.1) Where v air is the kinematic viscosity and u τair the frictional velocity. The y + value indicates how much of the turbulent boundary layer is resolved. Fig 5.4 and Fig 5.5 shows the suction and side view y + value for the first wall cells on the mirror. Figure 5.4: Frontal view y + values Figure 5.5: side view y + values

52 Cd 42 5 Generic Side Mirror Case For a y + < 5 the viscous sublayer is resolved. The mesh created for the side mirror simulation is fine close to the mirror with a y + value ranging between [ ] and an average of However, the mesh is not fine enough to accurately capture the viscous sublayer and wall functions are implemented in OpenFOAM. For this Spalart Allmaras DDES is used as LES turbulence modell. The time step used in the calculation is chosen on the basis of a low CFL value. The time step t mirror = s and the resulting max CFL value is 1.5. Every 25 th time step is recorded and the sampling frequency is 8 [khz]. The total sampling interval, T, is 40 U/D (0.206 sec) and extraction of data is started after 120 D/U. From the instantaneous drag coefficient no real statistical convergence of forces is achieved over time. This is believed to be due to large unsteady resolved structures in the near wake of the mirror [Ask and Davidson, 2005]. However, the mean C d of 0.44 agrees well with Ask and Davidson [2009] study of a generic side mirror where C d was found to be Both C d values were calculated using mean values and the projected frontal area of the mirror tu/d Figure 5.6: Mean and instantaneous drag coefficient For validation the sound pressure is calculated on certain receivers located at same positions as those used in Ask and Davidson [2005] study. The results are presented in the acoustic section.

53 5.5 Flow Results 43 Table 5.4: Position of Microphones in the domain Nr x y z D D 2.23D D D 4 0.5D 2.5D 1D The microphone positions are also depicted for visual inspection in the xy plane in Fig 5.7 M4 M1 M3 y x Figure 5.7: Microphone positions 5.5 Flow Results For flow comparison a snap shot of vorticity at y = 0.5D can be compared to Ask and Davidson [2005] study. Figure 5.8: Vorticity slice y = 0.5D present study Figure 5.9: Vorticity slice y = 0.5D Ask and Davidson [2009]

54 44 5 Generic Side Mirror Case The general flow configuration is similar in the both cases, with large trailing edge vortices generated on each side down stream of the mirror. Keeping these in mind the root mean square pressure, (P RMS ), values should be considered. 1 N ( 2 P RMS = p(t) p) (5.2) N Where a comparison between the suction and pressure side of the mirror as well as the near wake is found in Fig i=1 Figure 5.10: 10 < p RMS < 35 Figure 5.11: 10 < p RMS < 35 Figure 5.12: 10 < p RMS < 35 Figure 5.13: 15 < p RMS < 225 It is evident from Fig 5.10 through Fig 5.13 that the base plate encompassing the mirror wake experiences the largest pressure fluctuations, and that the trailing vortices in the near wake is the most significant sound source. From Fig 5.11, of the pressure suction side of the mirror, a high assymetric fluctuating pressure distribution is captured. This fluctuation might contribute to the unsteadiness in C d mentioned earlier.

55 5.5 Flow Results 45 Q, or the second invariant of the velocity gradient tensor [Greenshields, 2015], captures the horseshoe vortex in front of the mirror. The horseshoe vortex is created by flow stagnation and causes high pressure fluctuations on the side of the mirror which is seen in previous figures. Figure 5.14: Iso surface of Q =

56 SPL (db) 46 5 Generic Side Mirror Case 5.6 Acoustic Results A common assumption is that the volume integral in the FW-H equation can be regarded as negligible in comparison to the surface integrals for low Mach number flows. To numerically investigate if this seems reasonable the intensity of the dipole and the quadrupole terms can be approximated. I D ρu 6 c 3 l 2 I Q ρu 8 c 5 l 2 (5.3) And looking at the ration of the two for the present set up of a velocity of 39 m/s (Mach = 0.11) [Y.Wang, 2010]: ( ) I D u 2 = (5.4) I Q c The quadrupole intensity would thus only be about 2.5 % in comparison to the dipole source. Considering the computational cost of evaluating the volume integral the trade off seems reasonable in this case. Sampling the signal in the time domain presents challenges in terms of spectral leakage as well as capturing a long enough signal to obtain converged statistics. To minimize frequency leakage when truncating the discontinuous signals a power spectral density function is implemented in Matlab through the pwelch function. In this case a 50 percent overlap is used where each section is windowed using a Hamming window. The energy in the signal is conserved for the discrete time signal. The SPL is calculated in microphone 1, 3 and 4 (see Fig. 5.7 for positioning) Frequency (Hz) Figure 5.15: SPL at microphone no. 1

57 SPL (db) 5.6 Acoustic Results Frequency (Hz) Figure 5.16: SPL at microphone no SPL (db) Frequency (Hz) Figure 5.17: SPL at microphone no. 4 A higher SPL is measured for microphone 1 than microphone 3, which is reasonable due to its position in the wake if compared with fluctuation levels in Fig Due to the dipole directivity of the sound one would expect the highest sound pressure level to be calculated in microphone 4, which is also the case. A SPL of 75.3 [db] is found at a frequency of 20 [Hz]. SPL at microphone 3 and 1 can be compared to the experimental and numerical results from Ask and Davidson [2005] study.

58 48 5 Generic Side Mirror Case Figure 5.18: SPL at microphone no. 3 and 1; present study (red), Ask and Davidson [2005] (black), Daimler Chrysler experimental data (triangles) And microphone 4 positioned right above the mirror in the y-plane. Figure 5.19: SPL at microphone no. 4; present study (red), Ask and Davidson [2005] (black), Daimler Chrysler experimental data (triangles) In Fig it is evident that SPL for frequencies < 40 [Hz] is under predicted in microphone 1 and 3, when compared to Ask and Davidson [2005] study and experimental results from Daimler Chrysler presented in the same study. Microphone 4 in Fig predicts higher SPL then results from Ask and Davidson [2005] study but still under predicts the experimental results from DC. It should be noted that present data is processed using a pwelch function with an energy preserved spectra as mentioned earlier; reducing the peaks by 50%. If an amplitude corrected spectra would have been used one could expect the peaks to increase by roughly 2 [db], which would bring present results closer to experimental values. It has also been noted in previous studies by Y.Wang [2010] and Ask and