Large Eddy Simulation of Shear-Free Interaction of Homogeneous Turbulence with a Flat-Plate Cascade

Transcription

1 Large Eddy Simulation of Shear-Free Interaction of Homogeneous Turbulence with a Flat-Plate Cascade Abdel-Halim Saber Salem Said Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Engineering Mechanics Dr. Saad A. Ragab, Chairman Dr. Muhammad R. Hajj Dr. William J. Devenport Dr. Demetri Telionis Dr. Surot Thangjitham July 23, 2007 Blacksburg, Virginia Keywords: Homogeneous Turbulence, Cascade Interaction, Large Eddy Simulation, Acoustic Radiation, High-Order Finite Difference c Copyright 2007, Abdel-Halim S. Salem Said

2 Large Eddy Simulation of Shear-Free Interaction of Homogeneous Turbulence with a Flat-Plate Cascade Abdel-Halim S. Salem Said (ABSTRACT) Studying the effects of free stream turbulence on noise, vibration, and heat transfer on structures is very important in engineering applications. The problem of the interaction of large scale turbulence with a flat-plate cascade is a model of important problems in propulsion systems. Addressing the problem of large scale turbulence interacting with a flat plate cascade requires flow simulation over a large number of plates (6-12 plates) in order to be able to represent numerically integral length scales on the order of blade-toblade spacing. Having such a large number of solid surfaces in the simulation requires very large computational grid points to resolve the boundary layers on the plates, and that is not possible with the current computing resources. In this thesis we develop a computational technique to predict the distortion of homogeneous isotropic turbulence as it passes through a cascade of thin flat plates. We use Large-Eddy Simulation (LES) to capture the spatial development of the incident turbulence and its interaction with the plates which are assumed to be inviscid walls. The LES is conducted for a linear cascade composed of six plates. Because suppression of the normal component of velocity is the main mechanism of distortion, we neglect the presence of mean shear in the boundary layers and wakes, and allow slip velocity on the plate surfaces. We enforce the zero normal velocity condition on the plates. This boundary condition treatment is motivated by rapid distortion theory (RDT) in which viscous effects are neglected, however, the present LES approach accounts for nonlinear and turbulence diffusion effects by a sub-grid scale model. We refer to this type of turbulence-blade interaction as shear-free interaction. To validate our calculations, we computed the unsteady loading and radiated acoustic pressure field from flat plates interacting with vortical structures. We consider two fundamental

3 problems: (1) A linear cascade of flat plates excited by a vortical wave (gust) given by a 2D Fourier mode, and (2) The parallel interaction of a finite-core vortex with a single plate. We solve the nonlinear Euler equations by a high-order finite-differece method. We use nonreflecting boundary conditions at the inflow and outflow boundaries. For the gust problem, we found that the cascade response depends sensitively on the frequency of the convected gust. The unsteady surface pressure distribution and radiated pressure field agree very well with predictions of the linear theory for the tested range of reduced frequency. We have also investigated the effects of the incident gust frequency on the undesirable wave reflection at the inflow and outflow boundaries. For the vortex-plate interaction problem, we investigate the effects of the internal structure of the vortex on the strength and directivity of radiated sound. Then we solved the turbulence cascade interaction problem. The normal Reynolds stresses and velocity spectra are analyzed ahead, within, and downstream of the cascade. Good agreement with predictions of rapid distortion theory in the region of its validity is obtained. Also, the normal Reynolds stress profiles are found to be in qualitative agreement with available experimental data. As such, this dissertation presents a viable computational alternative to rapid distortion theory (RDT) for the prediction of noise radiation due to the interaction of free stream turbulence with structures. iii

4 Dedication I would like to dedicate this dissertation to my parents, my wife, my daughters Noura and Menna, and my son Mahmoud. iv

5 Acknowledgments First of all, I would like to express my deep gratitude and love to my advisor, Dr. Saad Ragab, for his support, guidance and patience. I owe him a big debt. I would like to thank Dr. Muhammad Hajj for his kind support. I would like also to thank Dr. William Devenport, and Dr. Jon Larssen for sending us their experimental data, and Dr. Demetri Telionis, and Dr. Surot Thangjitham for their service on my committee. My thanks are due to Mr. Mohammad Elyyan and my daughter Noura for their help in the presentation and Tecplot. I would like to thank the Department of Engineering Science and Mechanics for supporting me financially through GTAs. My gratitude is due to the Egyptian embassy for supporting me financially throughout my stay in the USA. v

6 Contents 1 Introduction Motivation Technical Problem Approaches Numerical Simulation of Turbulent Flows Rapid Distortion Theory Experimental Work Flat Plates Interacting with Vortical Structures Objectives Accomplished Work Mathematical Model for Large Eddy Simulation Navier-Stokes Equations Navier-Stokes Equations in Nondimensional Form Equations For Large Eddy Simulation Subgrid-scale Model Numerical Method and Boundary Conditions Introduction Temporal Discretization Spatial Discretization Boundary Conditions vi

7 4 Response of a Flat-Plate Cascade to Incident Vortical Waves - 2D Calculations Introduction Governing Equations Glegg s Linearized Potential Flow Solution Two-Dimensional Euler Simulations CONCLUSIONS Vortex-Plate Interaction Introduction Vortex-Plate Interaction Conclusions Interaction of Homogeneous Turbulence with a Flat-Plate Cascade - Comparison with Experimental Data Inflow Turbulence Comparison with Experimental Data Spatially Decaying Isotropic Turbulence Computational Domain and Inflow Spectra Comparison of LES with Larsen Experimental Data Interaction of Homogeneous Turbulence with a Flat-Plate Cascade - Comparison with RDT Introduction Graham s RDT Solution Comparison of LES with Graham s RDT Six-Plate Cascade Three-Plate Cascade Conclusions and Recommended Future Work 116 vii

8 Bibliography 120 Appendices 126 A Graham s RDT 126 B Glegg s Linearized Solution 129 viii

9 List of Figures 4.1 Flat plate cascade and computational domain Unsteady lift response and sound power using Glegg s linear solution at Mach number M = Unsteady lift response and sound power using Glegg s linear solution at Mach number M = Comparison of unsteady lift response with Glegg s linear solution, M = Comparison of unsteady lift response with Glegg s linear solution, M = Test case 1: A snapshot of pressure contours Test case 1: Pressure amplitudes for propagating mode ν = 1 and decaying mode ν = Test case 1: Sensitivity of surface pressure jump to grid step sizes Test case 1: Sensitivity of surface pressure jump to streamwise domain length Test case two: A snapshot of pressure contours Test case two: Pressure amplitudes for propagating mode ν = 2 and decaying mode ν = Test case two: Sensitivity of surface pressure jump to grid step sizes Test case two: Sensitivity of surface pressure jump to streamwise domain length Test case 3: A snapshot of pressure contours Test case 3: Sensitivity of surface pressure jump to grid step sizes Test case 3: Sensitivity of surface pressure jump to streamwise domain length Test case 4: A snapshot of pressure contours ix

10 4.18 Test case 4: Pressure amplitudes for propagating mode ν = 1 and decaying mode ν = Test case 4: Sensitivity of surface pressure jump to grid step sizes Test case 4: Sensitivity of surface pressure jump to streamwise domain length Parallel vortex-plate interaction Flow properties of Oseen and Taylor vortices Oseen vortex, a snapshot of vorticity field, t = Oseen vortex, a snapshot of pressure filed, t = Oseen vortex, a snapshot of vorticity field, t = Oseen vortex, a snapshot of pressure field, t = Oseen vortex, a snapshot of vorticity field, t = Oseen vortex, a snapshot of pressure field, t = Oseen vortex, a snapshot of vorticity field, t = Oseen vortex, a snapshot of pressure field, t = Taylor vortex, a snapshot of vorticity field, t = Taylor vortex, a snapshot of pressure field, t = Directivity of pressure amplitude on a circle r = 6.2 centered at x = 3.1, z = 0 at time t = Pressure signature at x = 0.5, z = Lift coefficient Energy spectrum function, spatial LES, coarse grid Energy spectrum function, spatial LES, fine grid Streamwise variation of dynamic model coefficient in spatial decaying turbulence Flat plate cascade and computational domain Target and numerically generated energy spectra at inflow boundary Mid-passage distribution of normal Reynolds stresses and q 2 = u 2 + v 2 + w x

11 6.7 Reynolds stress profiles at x = Normalized Reynolds stress profiles at x = Reynolds stress profiles at x = Normalized Reynolds stress profiles at x = Spanwise vorticity contours for a single plate placed in isotropic turbulence, no-slip condition is applied Reynolds stress profiles at (x x LE )/c = 0.92 and (x x LE )/c = 1.53 for a single plate, no-slip boundary condition is applied Reynolds stress contours u 2 for a single plate Reynolds stress contours w 2 for a single plate Six-plate cascade and computational domain Case A: 3D-energy spectra of the incident turbulence, inflow (x=-4.836), and upstream of cascade (x=-0.269) A snapshot of the instantaneous upwash velocity contours(xz-plane) A snapshot of the instantaneous upwash velocity contours at plane x = A snapshot of the instantaneous streamwise velocity contours (xz-plane) A snapshot of the instantaneous spanwise velocity contours (xz-plane) A snapshot of the instantaneous velocity vectors (xz-plane) A snapshot of the instantaneous velocity vectors (yz-plane) x = A snapshot of the instantaneous pressure fluctuation contours (xz-plane) A snapshot of the instantaneous pressure fluctuation contours (yz-plane), x = A snapshot of the instantaneous density fluctuation contours (xz-plane) A snapshot of the instantaneous density fluctuation contours (yz-plane), x = Case A: A snapshot of the instantaneous pressure fluctuation contours (no plates) xi

12 7.14 Case A: A snapshot of the instantaneous density fluctuation contours (no plates) Contours of the averaged streamwise-reynolds stress component Contours of the averaged spanwise-reynolds stress component Contours of the averaged upwash-reynolds stress component Contours of the averaged square of the pressure fluctuations (pp) Mid-passage distribution of q Normalized q 2 profiles for different ratios of plate spacing to integral length scale s/l Averaged TKE, Reynolds stresses, and pressure fluctuation (read right) Normal Reynolds stress profiles at x = Normal Reynolds stress profiles at x = Normal Reynolds stress profiles at x = Normal Reynolds stress profiles at x = Normal Reynolds stress profiles at x = Normal Reynolds stress profiles at x = Normal Reynolds stress profiles at x = Case A: Profiles of q 2 /q0 2 at plane x = Case A: Profiles of q 2 /q0 2 at plane x = Case A: Profiles of q 2 /q0 2 at plane x = One dimensional energy spectra, Eww(k 1 ) at z/s = One dimensional energy spectra, Eww(k 1 ) at z/s = One dimensional energy spectra, Euu(k 1 ) at z/s = One dimensional energy spectra, Euu(k 1 ) at z/s = One dimensional energy spectra, Evv(k 1 ) at z/s = One dimensional energy spectra, Evv(k 1 ) at z/s = plate cascade and computational domain D-energy spectra of the incident turbulence xii

13 7.40 Case B: Snapshot of the instantaneous upwash velocity contours (xz-plane) Case C: Snapshot of the instantaneous upwash velocity contours (xz-plane) Case D: A snapshot of the instantaneous velocity vectors (xz-plane) Case D: Snapshot of the instantaneous upwash velocity contours (xz-plane) Case B: A snapshot of the instantaneous velocity vectors (xz-plane) Case C: A snapshot of the instantaneous velocity vectors (xz-plane) Case B: Snapshot of the instantaneous pressure contours (xz-plane) Case C: Snapshot of the instantaneous pressure contours (xz-plane) Case D: Snapshot of the instantaneous pressure contours (xz-plane) Case B: Snapshot of the instantaneous density fluctuation contours (xz-plane) Case C: Snapshot of the instantaneous density fluctuation contours (xz-plane) Case D: Snapshot of the instantaneous density fluctuation contours (xz-plane) Case B: Contours of the averaged streamwise Reynolds stress component Case C: Contours of the averaged streamwise Reynolds stress component Case D: Contours of the averaged streamwise Reynolds stress component Case B: Contours of the averaged spanwise Reynolds stress component Case C: Contours of the averaged spanwise Reynolds stress component Case D: Contours of the averaged spanwise Reynolds stress component Case B: Contours of the averaged upwash Reynolds stress component Case C: Contours of the averaged upwash Reynolds stress component Case D: Contours of the averaged upwash Reynolds stress component Case B: Contours of the averaged square of the pressure fluctuations Case C: Contours of the averaged square of the pressure fluctuations Case D: Contours of the averaged square of the pressure fluctuations Averaged streamwise decay of the turbulent kinetic energy Case B: Normal Reynolds stress profiles at x = Case C: Normal Reynolds stress profiles at x = Case D: Normal Reynolds stress profiles at x = xiii

14 7.68 Case B: Normal Reynolds stress profiles at x = Case C: Normal Reynolds stress profiles at x = Case D: Normal Reynolds stress profiles at x = Case B: Normal Reynolds stress profiles at x = Case C: Normal Reynolds stress profiles at x = Case D: Normal Reynolds stress profiles at x = Case B: Profiles of q 2 /q0 2 at plane x = Case C: Profiles of q 2 /q0 2 at plane x = Case D: Profiles of q 2 /q0 2 at plane x = Case B: Profiles of q 2 /q0 2 at plane x = Case C: Profiles of q 2 /q0 2 at plane x = Case D: Profiles of q 2 /q0 2 at plane x = Case B: Profiles of q 2 /q0 2 at plane x = Case C: Profiles of q 2 /q0 2 at plane x = Case D: Profiles of q 2 /q0 2 at plane x = xiv

15 List of Tables 3.1 The a m coefficients of fourth-order Runge-Kutta scheme [54] The b m coefficients of fourth-order Runge-Kutta scheme [54] Boundary-points fifth-order scheme coefficients Characteristics of the inflow turbulence and computational domain Values of q0 2 at the planes of comparison for different cases xv

16 Chapter 1 Introduction 1.1 Motivation Studying the problem of the effects of free stream turbulence on noise, vibration, and heat transfer on structures is very important in engineering applications. The problem of the interaction of large scale turbulence with a flat plate cascade is a model of important problems in propulsion systems. Some linearized solutions such as the rapid distortion theory (RDT) are used to predict the response to incident turbulence from a flat plate cascade. Large eddy simulation (LES) could be used as an alternative to RDT which can relax some of the limitations of RDT. Our goal is to develop an LES code which can be used as an alternative means for solving the problem of cascade response to incident turbulence. 1.2 Technical Problem The interaction of free stream incident turbulence with engineering structures is a significant source of vibration, noise, unsteady loading, and heat transfer. Bushnell [21] gave an exten- 1

17 sive review of body-turbulence interaction problems. A significant component of the noise radiated by shrouded propellers and turbofans is due to interactions of ingested turbulence with rotor blades and guide vanes. The incident turbulence is usually generated in boundary layers on surfaces upstream of the propeller such as the vessel hull, control surfaces as well as in the wakes of rotor blades upstream of guide vanes. It may also be present in the incident free stream due to environmental effects such as atmospheric turbulence or breaking of internal or surface gravity waves. The incident turbulence usually contains large-scale structures with integral length scales comparable to the blade-to-blade spacing or even larger. The interactions of these structures with rotating blades or guide vanes result in pressure fluctuations, unsteady lift, vibrations and hence noise radiation. Simultaneously, the turbulence length scales, intensities and wavenumber-frequency spectra are significantly modified by the interaction with the blades. It is highly desirable to have proper understanding of the flow characteristics through developing computer codes which can be used to predict the flow fields. 1.3 Approaches Numerical Simulation of Turbulent Flows Direct Numerical Simulation (DNS), solution of Reynolds Averaged Navier-Stokes equations (RANS), and Large Eddy Simulation (LES) are three approaches to numerical simulation of turbulent flows. To predict a turbulent flow, one may use a numerical method to solve the time dependent Navier-Stokes equations for the instantaneous flow variables without the use of a turbulence model. Such a solution is known as a direct numerical simulation (DNS). The mesh and time advancement must resolve all of the dynamically relevant turbulent scales from the largest scales down to the smallest scales. This resolution requirement puts an upper limit on the Reynolds number that can be successfully simulated on a given computer [38]. 2

18 A more wide spread utilization of the DNS is prevented by the fact that the number of grid points needed for sufficient spatial resolution scales as Re 9 4 and the CPU-time as Re 3 [10]. RANS is the most frequently used approach in engineering applications. In this method, all turbulent fluctuations are averaged over a long period of time, and their statistical effects on the mean flow are modeled. The turbulence models are usually complex, because they are required to consider all of the turbulent scales including the large scales which may not be the same in different flow fields. Because of the averaging procedure, no detailed information can be obtained about turbulent structures. On the other hand, DNS represents the other extreme where all of the dynamically significant eddies are computed and none are modeled [3]. The LES is a compromise between the DNS and the RANS approaches. In this approach, large-scale eddies are computed whereas small scales are modeled. Since it is assumed that small-scale eddies have an isotropic and homogeneous structure, simpler and more universal subgrid scale models than the models required for RANS can be used. As the large-scale turbulence is to be computed, the resolution requirement for the mesh is much more than in RANS, but not as demanding as in DNS because the small scales are modeled. LES is well suited for detailed studies of complex flow physics including massively separated unsteady flows, large scale mixing, or aerodynamic noise [3] Rapid Distortion Theory Because of its simplicity and efficiency, Rapid Distortion Theory (RDT) has been extensively used for investigating the interactions of turbulence with blades and prediction of radiated noise from rotors. Kullar and Graham [27] obtained an integral equation for the loading of a flat-plate linear cascade due to an incident three-dimensional gust composed of upwash velocity component superimposed on a uniform stream. They examined the effects of Mach number and three-dimensionality of gust on acoustic resonances between cascade blades. 3

19 Glegg [13] also obtained an integral equation for the loading (expressed as a jump in the velocity potential across the blades), and solved that equation by the Wiener-Hopf method. He obtained analytical expressions for the unsteady loading, acoustic mode amplitude, and sound power output of the cascade. One of his conclusions is that the primary effect of sweep on the radiated sound power is to cause the propagating acoustic modes to become cut off. This effect depends on the Mach number. Graham [23] used RDT with simplifying assumptions and obtained analytical solutions for the turbulence spectra downstream of a flat-plate linear cascade. He noted that... the turbulence flow field for these convective flows is inhomogeneous in the streamwise direction over a distance of order L [integral length scale]. This is the region within which there is a significant pressure field associated with the interaction between the turbulence and the leading edge. Therefore, a simplified RDT in which the turbulence is assumed to be homogeneous in the streamwise direction does not apply in this region. Also, RDT does not apply for streamwise distances much greater than L from the leading edge. Boquilion et al. [2] have also used RDT to analyze the interaction of turbulence with a linear flat-plate cascade. They used Glegg s theory which considers plates of finite chord. In their work as well as Graham s work, the transverse velocity is zero on the wake centerline. Because traditional RDT neglects inviscid nonlinear and viscous effects, the vorticity field of the incident turbulence is frozen as it convects with the uniform free-stream velocity. The vortex sheets on the plates and the wake induce an irrotational velocity field, and because of limitations, that field does not have an effect on stretching or tilting of the vorticity in the incident turbulence. Moreover, the infinitesimally thin trailing vortex sheets remain flat and parallel to the free stream, and hence the upwash velocity continues to be zero in the wake on the plates planes. However, these sheets may deform by self induction and induce transverse velocity perturbations. Majumdar and Peak [33] used RDT to predict the distortion of ingested free stream tur- 4

20 bulence by the strain field of non-uniform mean flow upstream of an open or ducted fan. They assumed that the incident turbulence is given by von Karman spectrum. They used a strip-theory to predict the unsteady forces on rotating fan blades, and determined the radiated sound by solving the convected wave equation with the help of Green s function. They found that the distortion of incident turbulence under static (zero forward speed) conditions produced high tonal noise levels, whereas the radiated sound is generally broadband under flight (aircraft approach) conditions. Atassi et al. [1] examined the effect of mean flow swirl on the acoustic and aerodynamic response of a set of guide vanes. The swirl is imparted to the incoming flow by a rotor upstream of the guide vanes which are modeled by an unloaded (zero-mean lift) annular cascade. They linearized the Euler equations around a non-uniform mean state and assumed time-harmonic disturbances. Because of the disturbance equations have variable coefficients, Atassi et al. used a finite-difference method and solved for the flow in a single blade passage assuming quasi-periodic conditions in the circumferential direction. They showed that the mean swirl changes the mechanics of the scattering of incident acoustic and vortical disturbances. They pointed to the importance of the radial phase of the incident disturbance in the scattering process Experimental Work The current investigation is motivated by the cascade experiments conducted by Larssen and Devenport [28] (see also Larssen [29]). The experimental setup consists of a six-blade linear cascade. They adopted a mechanically rotating active grid design in order to generate the large scale turbulence. They compared the experimental blade-blocking data to linear cascade theory (RDT) by Graham [23] and showed good qualitative agreement. In our investigation we did our simulation on a configuration that has geometric properties similar to their experimental setup for the purpose of comparison. 5

21 1.4 Flat Plates Interacting with Vortical Structures The Blade-Vortex Interaction problem is a fundamental problem in aeroacoustics. Researchers have formulated mathematical models of varying levels of fidelity, and obtained both analytical and numerical solutions. Howe [25] has presented a comprehensive analytical treatment of sound radiated by the interaction of line vortices with a flat plate, among other vortex sound problems. Glegg et al. [14] gave a recent review of theories for computing leading edge noise due to the interaction of a line vortex as it convects past an airfoil of finite thickness. In Computational Aeroacoustics (CAA), the field equations that describe the mechanisms of sound generation and propagation are solved numerically. Delfs et al. [15] solved the linearized Euler equations using a high-order finite difference method, and determined the noise radiated by the interaction of a finite-core vortex with a sharp edge. Delfs [16] also solved the same equations for the interaction problem and determined the sound radiated by a 2D airfoil with a rounded leading edge. Grogger et al. [17] also solved the linearized Euler equations, and determined the noise generated by the interaction of localized three-dimensional vorticity with the leading edge of an airfoil. They studied the effects of the airfoil s thickness ratio on the strength and directivity of radiated noise. A good review of experimental work on blade-vortex interaction is given by Wilder and Telionis [18]. They used Laser-Doppler velocimetry to experimentally investigate two-dimensional airfoil-vortex interaction. They used oscillating airfoil to generate the vortex which interacts with a NACA63 2 A015 airfoil at two different angles of attack; α = 0 (unloaded blade) and α = 10 (loaded blade). Vorticity fields were constructed and surface pressure fluctuations on the airfoil were determined. The flow Reynolds number is Re=19000 and Mach number is nearly zero. They found that a vortex skimming over a blade at zero incidence does not induce separation, and that the vortex quickly loses its strength because of the viscous effects. Casper et al [22]. predicted the loading noise from unsteady surface pressure measurements on a NACA0015 airfoil immersed in grid-generated turbulence. They predicted the far field noise by using the time-dependent surface pressure as input to formulation A of Farassat, 6

22 a solution of the Ffowcs Williams-Hawkings equation. Moreau et al [20] performed an experimental investigation of the turbulence-interaction noise presented on various bodies of different relative thickness. They found that the turbulence-interaction noise is reduced significantly by increasing the airfoil thickness. Polacsek et al. [35] have presented a numerical method for predicting turbofan noise due to rotor-stator interaction. Their computational model is composed of three components: (1) a 3D RANS code to estimate spatial distribution and strength of noise sources, (2) an Euler solver for near field acoustic propagation and (3) a Kirchhoff integral for far-field radiation. The source amplitude, obtained from post-processing RANS output, is over-predicted in comparison with experiments, and it has to be adjusted to match the in-duct measurements. This indicates that RANS codes are not suitable for predicting noise sources due to rotor-stator interaction and calls for more detailed modeling such as large-eddy simulations. Nevertheless, with the adjusted source strength, the predicted sound pressure level and directivity patterns are in fairly good agreement with experimental data. 1.5 Objectives The objective of our work is to develop an efficient computational method, based on large eddy simulation, to be used as an alternative to RDT in predicting the turbulence-cascade interaction, and to address different numerical issues, like, using high-order finite difference schemes, application of different wall boundary conditions, and use of non-reflecting boundary conditions at inflow and outflow boundaries. 1.6 Accomplished Work To validate the implementation of our code; we computed the unsteady loading and radiated acoustic pressure field from flat plates interacting with vortical structures. We considered 7

23 two fundamental problems: (1) A linear cascade of flat plates excited by a vortical wave (gust) given by a 2D Fourier mode, Ragab and Salem-Said ( [45], [46]). and (2) The parallel interaction of a finite-core vortex with a single plate. We solved the two-dimensional nonlinear Euler equations over a linear cascade composed of six plates for a range of discrete frequencies of the incident gust. We use Giles [12] nonreflecting boundary conditions at the inflow and outflow boundaries, and study their performance at different frequencies of the incident gust. These boundary conditions have been also investigated for the cascade problem by Hixon et al. [26] and used by Sawyer et al. [42] for aeroacoustic prediction of rotor-stator interaction noise. Giles conditions, being based on a Taylor series expansion for small ratio of tangential wavenumber to frequency, are approximately nonreflecting. Rowley and Colonius [39] (see also Colonius [7] for a review) have developed numerically nonreflecting conditions. Yaguchi and Sugihara [51] have also proposed new nonreflecting boundary conditions for multidimensional compressible flow. Prediction of radiated sound by a cascade of blades due to interaction with turbulence can benefit from these new non-reflecting conditions. For the three-dimensional case we used large-eddy simulation to investigate the interaction of homogeneous isotropic turbulence with a cascade of thin flat plates, and determine the distortion of turbulence as it passes through the cascade, Salem-Said and Ragab [44]. We consider a case in which the integral length scale of the incident turbulence is comparable to the cascade pitch, hence we have to solve the flow over many passages simultaneously. Periodicity of the instantaneous flow in the direction normal to the plates is determined by the need to represent the large scales of the incident turbulence rather than by the cascade pitch. The governing equations are based on the full Favre-filtered compressible Navier-Stokes equations but with special treatment of solid walls. Because of the large number of plates (six in the present investigation), it is not feasible to resolve the turbulence within the viscous regions over those surfaces, especially for practical high Reynolds numbers. Our approach aims at resolving the inviscid nonlinear mechanisms and the decay of the incident turbulence. Hence, we impose the zero normal velocity condition and relax the 8

24 no-slip conditions to milder zero-shear or slip-wall conditions. This treatment amounts to neglecting generation of wall turbulence but it will capture vorticity shedding from sharp edges particularly the trailing vortex sheets. Therefore, the distortion of turbulence will be dominated by the suppression of the normal velocity on the plates. In this way we will be able to simulate incident flow at high turbulence Reynolds number but at the expense of losing turbulence generation by mean shear in wall boundary layers and wakes. Our results for the Reynolds stresses and energy spectra downstream of the cascade agree with Graham s RDT theory in its region of validity. We refer to this type of turbulence-cascade interaction as shear-free interaction. Such boundary treatment have been used by Perot and Moin [40]. 9

25 Chapter 2 Mathematical Model for Large Eddy Simulation The equations governing the flow field in our simulation are: Compressible Navier-Stokes equations, the energy equation, and the equation of state. In this chapter, we present the governing equations of the flow field in dimensional and nondimensional forms, the equations for large eddy simulation, and finally, we discuss the subgrid scale model for the momentum and the energy equations. 2.1 Navier-Stokes Equations Navier-Stokes equations are written as: Continuity Equation: Momentum Equation: ρ t + (ρ u j) x j = 0 (2.1) 10

26 (ρ u i ) + (ρ u t x i u j + p δ ij ) = σ ij j x j (2.2) Energy Equation: Equation of State: (ρ E ) + [ (ρ E + p )u t x j] = (σ j x ij u i qj ) (2.3) j p = ρ R T (2.4) where ρ, u i, p, T, and R are density, velocity components, pressure, temperature, and gas constant, respectively. E p = (γ 1)ρ u i u i (2.5) 2 ) u k δ ij (2.6) 3 σ ij = µ ( u i x j + u j x i q j = κ T x j x k (2.7) where γ, µ, and κ are the specific heat ratio, dynamic viscosity, and thermal conductivity of the fluid particle at position x i, respectively. 2.2 Navier-Stokes Equations in Nondimensional Form Using a length (C ) and the free stream flow variables as reference values, we define the following dimensionless variables: t = t U C, x i = x i C, u i = u i, ρ = ρ, p = p U ρ ρ U 2, T = T, E = E T U 2, µ = µ, κ = κ, c µ κ p = c p c p 11

27 Using the above dimensionless variables, we rewrite the Navier-Stokes equations in nondimensional form as follows: Continuity Equation: Momentum Equations: ρ t + (ρu j) = 0 (2.8) x j (ρu i ) t + (ρu iu j + pδ ij ) x j = 1 σ ij (2.9) Re x j Energy Equation: (ρe) t + [(ρe + p) u j] x j = 1 Re x j [ σ ij u i 1 P r (γ 1) M 2 q j ] (2.10) Equation of State: p = ρ R T (2.11) where E = p (γ 1)ρ u iu i (2.12) ( ui σ ij = µ + u j 2 x j x i 3 ) u k δ ij x k (2.13) q j = κ T x j (2.14) The Reynolds number Re, the Mach number M and the Prandtl number P r are given by Re = ρ U C µ (2.15) 12

28 M = U γr T (2.16) P r = µ c p κ (2.17) The nondimensional gas constant is given by R = R T U 2 = 1 γm 2 (2.18) Here, we assume that µ and κ are functions of temperature, but we assume that c p to be constant, hence c p = Equations For Large Eddy Simulation The Favre-filtered governing equations for the resolved scales are (see Erlebacher et al. [32] and Ragab and Sheen [37]) given as: Continuity Equation: ρ t + ( ρũ j) x j = 0 (2.19) Momentum Equation: ( ρũ i ) t + ( ρũ iũ j + pδ ij ) x j = 1 σ ij Re x j τ ij x j (2.20) Energy Equation: ( ρẽ) t + [( ρẽ + p)ũ j] x j = 1 Re x j [ σ ij ũ i ] 1 P r (γ 1) M q 2 j x j [c p Q j ] (2.21) Equation of State: 13

29 p = ρ R T (2.22) where Ẽ = p (γ 1) ρ + 1 2ũiũ i (2.23) ( ũi σ ij = µ + ũ j 2 x j x i 3 ) ũ k δ ij x k (2.24) q j = κ T x j (2.25) τ ij = ρ (ũ i u j ũ i ũ j ) (2.26) ) Q j = ρ (ũj T ũ j T (2.27) Here τ ij, and Q j represent the SGS (subgrid scale) stress tensor and heat flux, respectively. In order to close the system of equations we need to model these terms. 2.4 Subgrid-scale Model In large-eddy simulation, information from the resolved scales are used to model the stresses of the unresolved scales by a sub-grid-scale (SGS) model. There is only one unclosed term in the momentum equation, i.e., the SGS stress τ ij. The dynamic version of the Smagorinsky s eddy-viscosity model [43] introduced by Germano et al. [55] is used to model the SGS stress. The model is τ ij 1 ( 3 τ kkδ ij = 2C ρ 2 S S ij 13 ) S kk δ ij (2.28) 14

30 where S ij = 1 ( ũi + ũ ) j, (2.29) 2 x j x i S = (2 S ij Sij ) 1 2, (2.30) = ( ) 1 3, (2.31) = ( ) 1 3. (2.32) Here C is the Smagorinsky coefficient that is assumed to be independent of filter width. In the present study, i = 2 i is used, where ( ) implies test filtering, see Toh [47]. We let L ij = ρ ũ i ũ j ρ ũ i ρ ũj ρ, (2.33) M ij = 2 2 ρ S Sd ij ρ S Sd ij, (2.34) S ij d = Sij 1 S kk δ ij, (2.35) 3 S d ij = S ij 1 3 S kk δ ij. (2.36) Then Smagorinsky coefficient is given by C = ( Lij 1 3 L kkδ ij ) Mij M ij M ij (2.37) To avoid the large fluctuations in the values of C that may cause instability of numerical solutions, we follow Zhang and Chen [52] and determine C as follows: 15

31 C = L d ij M ij M ij M ij, L d ij = L ij 1 3 L kkδ ij (2.38) where the symbol represents double filtering, i.e., a grid filter ( ) is applied first followed by a test filter ( ). The unclosed term in the total energy equation is the SGS heat flux Q j. Using the Smagorinsky s eddy-diffusivity model, Q j is modeled as Q j = ρν T P r T T = C 2 ρ S x j P r T T x j (2.39) where C is the eddy-viscosity coefficient given by Equation (2.38) and P r T is the turbulent Prandtl number. Here, we assume P r T = 0.7. We used the dynamic model, only, for the simulations of homogeneous turbulence without the cascade, and used Smagorinsky model with constant coefficient C for turbulence cascade interaction where the value of C is an average value based on the simulations without the cascade. 16

32 Chapter 3 Numerical Method and Boundary Conditions 3.1 Introduction In large-eddy simulation; information from the resolved length scales are used to model the stresses of the unresolved length scales by a sub-grid-scale (SGS) model. Therefore, it is important that the numerical error be sufficiently small. Tolstykh [48] developed a fifth-order non-centered compact scheme in which artificial dissipation is controllable. This scheme is very attractive for large eddy simulation because it does not require explicit filtering as with centered compact schemes and numerical dissipation can be minimized. Ragab and El- Okda [36] have investigated the application of Tolstykh s scheme to LES of temporal decay of isotropic homogeneous turbulence and for flow over a surface-mounted prism. They found that a procedure that combines a compact sixth-order scheme with Tolstykh s compact upwind fifth-order scheme (C6CUD5) produced accurate LES results. Compact centered schemes are non-dissipative, and hence they require some kind of filtering for de-aliasing and preventing odd-even decoupling in inviscid flow regions. We use, for the spatial discretization, the compact sixth-order scheme which is nondissipative, and hence a filter is also used to 17

33 damp dispersed high wavenumbers. Our LES code contains two options; the first is to combine the sixth-order scheme with a compact upwind fifth-order scheme due to Zhong [53]. The second is to use a tenth-order compact spatial filter as given by Visbal and Gaitonde [49]. For the near-boundary points, we use successively lower even-order compact filters. The low-storage five-stage fourth-order Runge-Kutta scheme of Carpenter and Kennedy [54] is used to advance the solution in time. The details of the numerical model are given below. 3.2 Temporal Discretization Following Toh [47], consider the model equation dq dt = L(q) (3.1) where q is a flow variable, t is time and L is a spatial operator. The five-stage fourth-order explicit Runge-Kutta scheme [54] is used to advance the solution in time. To advance the solution from step n to n + 1, we use the algorithm q 0 = q n (3.2) H 0 = L(q 0 ) (3.3) Then for m = 1,..., 4, we use q m = q m 1 + b m th m 1 (3.4) H m q m + a m H m 1 (3.5) 18

34 m a m Table 3.1: The a m coefficients of fourth-order Runge-Kutta scheme [54] m b m Table 3.2: The b m coefficients of fourth-order Runge-Kutta scheme [54] and for the final step, we use q 5 = q 4 + b 5 H 4 (3.6) q n+1 = q 5 (3.7) 3.3 Spatial Discretization We use a sixth-order compact finite-difference scheme (Lele [30]) for spatial derivatives. In the nonperiodic x direction, the scheme is given by: 19

35 c ē c ē c ē c ē c ē c ē c ē c ē Table 3.3: Boundary-points fifth-order scheme coefficients αf i 1 + f i + αf i+1 = 1 h [a(f i+1 f i 1 ) + b(f i+2 f i 2 )] (3.8) where α = 1/3, a = 7/9, b = 1/36, and h is the grid step size. This scheme is applied for i = 3,..., n 2. Fifth-order explicit schemes (Carpenter et al. [6]) are used at the boundary points i = 1, 2, n and n 1: i = 1: f i = 1 h ( c 0f i + c 1 f i+1 + c 2 f i+2 + c 3 f i+3 + c 4 f i+4 + c 5 f i+5 + c 6 f i+6 + c 7 f i+7 ) (3.9) f i+1 = 1 h (ē 0f i + ē 1 f i+1 + ē 2 f i+2 + ē 3 f i+3 + ē 4 f i+4 + ē 5 f i+5 + ē 6 f i+6 + ē 7 f i+7 ) (3.10) i = n: f i = 1 h ( c 0f i + c 1 f i 1 + c 2 f i 2 + c 3 f i 3 + c 4 f i 4 + c 5 f i 5 + c 6 f i 6 + c 7 f i 7 ) (3.11) f i 1 = 1 h (ē 0f i + ē 1 f i 1 + ē 2 f i 2 + ē 3 f i 3 + ē 4 f i 4 + ē 5 f i 5 + ē 6 f i 6 + ē 7 f i 7 ) (3.12) 20

36 The coefficients are given by Carpenter et al. [6] and reproduced in Table(3.3). Because of the singularities at the leading and trailing edges, we do not apply the sixth-order compact scheme at points that straddle these edges on z planes that coincide with the plates. Instead, we use the explicit fifth-order schemes Eq (3.10) at the pints i = ile + 1 and ite + 1, and Eq (3.12) at the points i = ile 1 and i = ite 1, where ile and ite are the indices of the leading and trailing edges of the plate, respectively. The viscous and turbulent stresses terms are evaluated using a fourth-order central difference scheme. Our present LES code contains two options. In the first option, we follow the same procedure in Ragab and El- Okda [36] but we replace Tolstykh s fifth order scheme by a more efficient upwind scheme due to Zhong [53]. In the second option, we use a tenth-order compact spatial filter as given by Visbal and Gaitonde [50, 49]. For the near-boundary points, we use successively lower even-order compact filters with a filter parameter given by α f = m f, where m f is the order of the filter. For example, α f = 0.40 for the tenth-order filter, α f = for the eighth-order filter, and so on. The filter is applied to the conservative variables once every time step after the fifth stage of the Runge-Kutta scheme. 3.4 Boundary Conditions On the upper and lower surfaces of each plate, we have two options: a zero-shear wall and a slip-wall boundary conditions. For the zero-shear wall boundary condition, we simply assign the flow variables of the grid points near the wall to the wall points. In the slip-wall boundary condition, the Euler equations are solved on the upper and lower surfaces of each plate. Poinsot-Lele [34] slip wall boundary condition is used. The flux vector derivative parallel to the plate (x operator) is evaluated using the sixth-order compact scheme as shown above at the points ile + 2 i ite 2. The velocity derivatives normal to the wall (z direction) are evaluated by a one-sided explicit third-order scheme and the pressure derivative is evaluated by a first-order scheme. The flow variables at the leading and trailing edges are obtained by averaging the solutions at the two nearest field points above and below 21

37 the edges. At the inflow and outflow boundaries, we use Giles nonreflecting boundary conditions. Here, we present them for two-dimensional flow in the xz-plane. They have been generalized to 3D flow by Hagstrom and Goodrich [24] which we use in the simulations presented in chapter (7). At the inflow boundary, the incoming flow is composed of two contributions: A uniform flow which is parallel to the plates (U, W = 0, ρ and p ), and either a turbulent fluctuation, as in chapters (6,7), or a vortical gust with prescribed velocity components, as in chapter (4), (ũ, w) but without pressure or density variations. Additional perturbations at the boundary are due to the interaction of the turbulence (gust) with the cascade (u, w, ρ and p ). For the purpose of applying the boundary conditions only, we write u = U + ũ + u (3.13) w = W + w + w (3.14) p = p + p (3.15) ρ = ρ + ρ (3.16) In developing nonreflection boundary conditions, Giles [12] linearized the 2D Euler equations about a uniform basic state. The one-dimensional characteristic variables are related to the field perturbations (u, w, ρ and p ) by c 1 = p ρ c u (3.17) c 2 = c 2 ρ p (3.18) c 3 = ρ c w (3.19) (3.20) c 4 = p + ρ c u (3.21) At the inflow boundary, we determine the time derivatives of the incoming characteristic variables from Giles conditions: 22

38 c 2 t + W c 2 z = 0 (3.22) c 3 t + W c 3 z (U + c ) c 4 z 1 2 (U c ) c 1 z = 0 (3.23) c 4 t + W c 4 z 1 2 (U c ) c 3 z = 0 (3.24) where the z derivative is evaluated by a fourth-order central difference formula. We also determine the time derivative of the outgoing characteristic variable c 1 from c 1 t = (U c )( p x ρ c u x ) (3.25) where the x derivative is evaluated by a fifth-order one-sided explicit scheme [6] using information within the computational domain. The values of the perturbations and characteristic variables used to evaluate the x and z derivatives are updated at each of the five stages of the Ruge-Kutta scheme. As shown by Hixon et al. [26], the time derivative of the characteristic variables can be used to obtain the derivative of the conservative variables on the boundary. We take the time derivative of equations (3.17)- (3.21), and then use the known derivatives of the characteristic variables to determine the derivatives of the fluctuations. To take into account the gust at the inflow, we determine the time derivatives of the conservative variables by ρe t ρ t = ρ t (3.26) ρu = u ρ t t + ρ( u t + ũ t ) (3.27) ρw = w ρ t t + ρ( w t + w t ) (3.28) = 1 p γ 1 t + ρu( u t + ũ t ) + ρw( w t + w t ) (u2 + w 2 ) ρ t (3.29) At the outflow boundary, the time derivative of the incoming characteristic are given by 23

39 c 1 t + w c 1 z + u c 3 z = 0 (3.30) whereas the changes in the outgoing characteristics (c 2, c 3, c 4 ) are obtained from information within the computational domain. 24

40 Chapter 4 Response of a Flat-Plate Cascade to Incident Vortical Waves - 2D Calculations 4.1 Introduction In order to verify the validity of our code and precisely examine the boundary conditions, we compute the unsteady lift and radiated sound from a flat plate cascade due to an incident vortical wave (gust). We solve the two-dimensional nonlinear Euler equations over a linear cascade by a high-order finite-different method. We use Giles nonreflecting boundary conditions at the inflow and outflow boundaries. We compare our results with Glegg s linearized potential flow solution. In the present simulations, we consider an unstaggered six-blade linear cascade as shown in Figure (4.1). The plates have zero thickness and are at zero incidence relative to the mean flow. The blade-to-blade spacing (s) is 0.806c, where c is the blade chord. This geometric 25

41 parameter is motivated by the cascade experiments conducted by Larssen and Devenport [28] (see also Larssen [29]). Periodic y=6s k S C Outflow Inflow Periodic x=0 x=c y=0 Figure 4.1: Flat plate cascade and computational domain. 4.2 Governing Equations Euler equations We solve the full nonlinear Euler equations in the conservative form on a uniform Cartesian grid. 26

42 ρ t + ρu j x j = 0, (4.1) ρu i t + (ρu iu j + pδ ij ) x j = 0, (4.2) ρe t + (ρe + p)u j x j = 0 (4.3) where E = p (γ 1)ρ u iu i. (4.4) A perfect gas with specific heat ratio γ = 1.4 is assumed. p = ρ R T (4.5) The coordinates and flow variables are made non-dimensional by using the plate chord, c, the free-stream velocity U, density ρ, and temperature T as reference values. The reference pressure is ρ U. 2 In this chapter we consider two-dimensional vortical waves only, hence the gust velocity field (ũ, w) is divergence free; it is given by: ũ = u o e i(k 1x+k 2 z ωt) + cc (4.6) w = w o e i(k 1x+k 2 z ωt) + cc (4.7) where u o = k 2 w o /k 1 and ω = k 1 U. In the above equations, cc stands for the complex conjugate of the preceding term. We assume that k 1 = k 2 = 2πn/(sn b ), where n is an integer and n b is the number of blades in the computational domain. All the results reported here are obtained for w o = 0.01U. 27

43 4.2.1 Glegg s Linearized Potential Flow Solution Glegg [13], see Appendix (B), has developed a complete solution to the response of a staggered flat plate cascade to incident 3D plane waves. He solved the compressible linearized potential flow equation, and accounted for the finite chord of the blades. He provided analytical expressions for the unsteady normal force and the upstream and downstream sound power; all as functions of the wavenumber and frequency of the incident gust and the geometric properties of the cascade. We have developed a linear-theory code based on Glegg s analytical solution, and checked it by reproducing the normal force and sound power spectra for all of the cases reported in Glegg [13]. We then use the linear code to provide data for comparison with the nonlinear Euler calculations. Following Glegg, we define the normal force coefficient CN by CN = L/πw o ρ U c, where L is the normal force per unit span; obtained as an integral along the chord of the pressure jump across the plate. The sound power per unit span, W ± is also normalized by woρ 2 U sn b /2. (The upper sign is for upstream radiation and lower sign is for downstream radiation.) The normal force magnitude and sound power for the cut-on modes at free stream Mach number M = 0.3 as function of the reduced frequency of the incident gust, κ = ωc/2u, are shown in Figure (4.2). The second (m=1) and third (m=2) acoustic modes are cut-on, each over a range of frequencies, but they do not overlap. There is a small window of frequencies where both modes are cut-off (κ = to κ = 5.915). The frequency for wavenumber n = 9 is κ = which falls in that window. Upstream sound power (given by dashed lines) is less than the downstream sound power (given by the solid lines). The normal force and sound power results at M = 0.5 are depicted in Figure (4.3). 28

44 Magnitude of Lift Coefficient Magnitude of Lift Coeff. m=1 Upstream Sound m=1 Downstream Sound m=2 Upstream Sound m=2 Downstream Sound m=1 m= Normalized Sound Power Reduced Frequency Figure 4.2: Unsteady lift response and sound power using Glegg s linear solution at Mach number M =

45 Magnitude of Lift Coefficient m=1 Magnitude of Lift Coeff. m=1 Upstream Sound m=1 Downstream Sound m=2 Upstream Sound m=2 Downstream Sound Normalized Sound Power m= Reduced Frequency Figure 4.3: Unsteady lift response and sound power using Glegg s linear solution at Mach number M = Two-Dimensional Euler Simulations We integrated the unsteady two-dimensional nonlinear Euler equations in time on a Cartesian uniform grid. First, we present the unsteady lift spectrum for a range of frequencies of the incident gust. We obtained numerical solutions for twelve separate runs corresponding to n = 1, 2,..., 12. In each run, the normal force on each of the six plates was computed by integrating the pressure jump on the plate, and its time history was recorded. Excluding a transient period, we decomposed the normal force coefficient into Fourier modes in time. The magnitude of the mode whose frequency is equal to the gust frequency was averaged over the six plates. The spectrum of the obtained normal force magnitude is depicted in 30

46 Figures( 4.4) and ( 4.5) at the two Mach numbers M = 0.3 and 0.5, respectively. Overall, the agreement between the present Euler calculations and Glegg s linear solution is very good. The sensitivity of the surface pressure distribution to grid resolution and domain length will be discussed next Linear Theory (Glegg) 2D Euler, M=0.3 Magnitude of Lift Coefficient Reduced Frequency Figure 4.4: Comparison of unsteady lift response with Glegg s linear solution, M =

47 Linear Theory (Glegg) 2D Euler, M=0.5 Magnitude of Lift Coefficient Reduced Frequency Figure 4.5: Comparison of unsteady lift response with Glegg s linear solution, M = 0.5. We investigate in more detail the pressure field and excited acoustic modes for three frequencies corresponding to n = 11, 8 and 9 at Mach number M = 0.3. As we shall see, each frequency results in a qualitatively different cascade response. We use five grids as shown in Table (1). Grids A, B, C and D are used to evaluate sensitivity to the step sizes ( x = z), whereas grids C, E, and F are used for sensitivity to the streamwise extent of the computational domain. In this table, n s and n c are the number of points on the cascade pitch s and on the plate chord c, respectively. L x is the streamwise extent of the computational domain and t is the time step. Test case 1: n=11 A snapshot of pressure contours (p 1/γM 2 ) for incident gust with mode number n = 11 is shown in Figure( 4.6). For this gust, the reduced frequency is κ = 7.146, for which the 32

48 Table 1 Grid parameters for test cases 1-3 Grid n s n c L x t A B C D E F third acoustic mode is cut-on. Pressure fluctuations propagate towards the lower left corner upstream of the cascade and towards the lower right corner downstream of the cascade. We decompose the pressure field p(x, z, t) into a double Fourier series in z and t, each mode is of the general form p(x, ν, ω l ) = ˆp lν (x)e iω lt+iνk 2 z (4.8) The cascade response at the forcing frequency ω l = ω includes only one propagating mode ν = 1 as shown in figure ( 4.2). The amplitude ˆp lν (x) for this mode is depicted in Figure( 4.7) upstream of the leading edge and downstream of the trailing edge for the four grids A, B, C, and D. The upstream radiated pressure agrees very well with predictions using Glegg s [13] linear theory. Reflection from the inflow boundary is negligible. As we refine the grid, downstream radiation converges to the linear theory prediction. However, a small wave reflection from the outflow boundary is evident by the weak variation in the wave amplitude. In addition to the propagating mode (ν = 1), there are other modes that decay exponentially upstream and downstream of the cascade. The dominant exponentially decaying mode is (ν = 7), which is also depicted in Figure( 4.7). This mode shows very little sensitivity to grid resolution and is not influenced by reflection from the inflow or outflow boundaries. The pressure jump across a plate p(x, t) is decomposed into Fourier series in time; of which 33

49 a mode is p(x, ω l ) = p l (x)e iω lt (4.9) At the gust frequency ω l = ω, the real and imaginary parts of pl (x) are compared to predictions of Glegg s linear theory in Figure( 4.8) for grids A, B, C and D. Grid convergence is shown. We note here that the coarse grid A gives 13 points per wave length whereas the fine grid D gives 105 points. The singularity in p(x, t) at the leading edge is difficult to resolve, nevertheless the predicted pressure distribution varies smoothly there. Near the trailing edge we also see a small glitch in the pressure. Sensitivity of the surface pressure distribution to the extent of the computational domain in the streamwise direction is a good indicator of the reflections from the inflow and outflow boundaries. With the leading edge at x = 0, the inflow boundary is placed at x = 2, 3, and -4 for the three grids C, E and F, respectively. The domain length L x is given in Table (1). The surface pressure distributions for the three domains are shown in Figure( 4.9) along with the linear theory prediction. For this frequency the effects of the domain length are negligible. Reflection from the boundaries is negligible because the wavenumber vector of the excited acoustic mode makes a small angle with the normal to the boundary, which is the right condition for the application of Giles nonreflecting boundary conditions. 34

50 Figure 4.6: Test case 1: A snapshot of pressure contours. 35

51 Linear Theory Euler (Grid ns=25) Euler (Grid ns=49) Euler (Grid ns=97) Euler (Grid ns=193) p Upstream Radiation Downstream Radiation Plate x/c Figure 4.7: Test case 1: Pressure amplitudes for propagating mode ν = 1 and decaying mode ν = 7. 36

52 Linear Theory Euler (Grid ns=25) Euler (Grid ns=49) Euler (Grid ns=97) Euler (Grid ns=193) (Complex) Dp Imaginary Part Real Part x/c Figure 4.8: Test case 1: Sensitivity of surface pressure jump to grid step sizes. 37

53 Linear Theory Euler (Lx=5.0375) Euler (Lx=7.0525) Euler (Lx=9.0675) (Complex) Dp Imaginary Part Real Part x/c Figure 4.9: Test case 1: Sensitivity of surface pressure jump to streamwise domain length. Test case 2: n=8 A snapshot of pressure contours is shown in Figure( 4.10) for incident gust with mode number n = 8. (Because the domain of six blades includes two wave lengths in the z direction, only half of the domain is shown in the figure.) For this gust, the reduced frequency is κ = 5.197, for which the second acoustic mode is cut-on. Upstream of the cascade, pressure fluctuations propagate towards the upper left corner, and downstream of the cascade they propagate towards the upper right corner. Reflection from the outflow boundary is evident and is more significant than from that at the inflow boundary. This is because the wavenumber vector has 38

54 larger tangential component at outflow. Reflection from the outflow boundary contaminates the pressure field causing significant dependence of the surface pressure distribution on the locations of the inflow and outflow boundaries. The cascade response at the forcing frequency ω l = ω includes only one propagating mode ν = 2. The amplitude ˆp lν (x) of radiated pressure for this mode is depicted in Figure( 4.11) upstream of the leading edge and downstream of the trailing edge for the three grids A, B, and C. With grid refinements, the upstream radiated pressure converges to the predictions using Glegg s [13] linear theory. The undulations in the pressure amplitude are about 5% of the mean value. However, stronger undulations are observed for the downstream radiated wave because of reflection from the outflow boundary. (Because of the significant reflection from the downstream boundary, we felt that it is not necessary to obtain results for the finest grid D.) In addition to the propagating mode (ν = 2), there are other modes that decay exponentially upstream and downstream of the cascade. The dominant exponentially decaying mode is (nu = 4), which is also depicted in Figure( 4.11). This mode shows very little sensitivity to grid resolution and is not influenced by reflection from the inflow or outflow boundaries. At the gust frequency ω l = ω, the real and imaginary parts of surface pressure jump p l (x) are compared to predictions of Glegg s linear theory in Figure( 4.12) for grids A, B, and C. Comparison with linear theory predictions is poor. And as shown in Figure( 4.13) the surface pressure distribution is very sensitive to the locations of the inflow and outflow boundaries. In this case n = 8 reflection from the boundaries is significant because the wavenumber vector makes a large angle with the normal to the boundary, for which Giles nonreflecting boundary conditions breakdown; especially at the outflow boundary. This is a challenging case for nonreflecting boundary conditions. 39

55 Figure 4.10: Test case two: A snapshot of pressure contours. 40

56 Linear Theory Euler (Grid ns=25) Euler (Grid ns=49) Euler (Grid ns=97) Upstream Radiation Downstream Radiation p Plate x/c Figure 4.11: Test case two: Pressure amplitudes for propagating mode ν = 2 and decaying mode ν = 4. 41

57 Linear Theory Euler (Grid ns=25) Euler (Grid ns=49) Euler (Grid ns=97) (Complex) Dp Imaginary Part Real Part x/c Figure 4.12: Test case two: Sensitivity of surface pressure jump to grid step sizes. 42

58 Linear Theory Euler (Lx=5.0375) Euler (Lx=7.0525) Euler (Lx=9.0675) (Complex) Dp Imaginary Part Real Part x/c Figure 4.13: Test case two: Sensitivity of surface pressure jump to streamwise domain length. Test case 3: n=9 Pressure contours for incident gust with mode number n = 9 are shown in Figure( 4.14). (Because the domain of six blades includes three wave lengths in the z direction, only one third of the domain is shown in the figure.) For this gust, the reduced frequency is κ = , which falls in the frequency range where no acoustic mode is cut-on as shown in figure ( 4.2). Pressure fluctuations are given by standing waves that are dominant in the near field and decay exponentially upstream and downstream of the cascade. The pressure field exhibits a node at x = from the plate leading edge. 43

59 At the gust frequency ω l = ω, the real and imaginary parts of surface pressure jump p l (x) are compared to predictions of Glegg s linear theory in Figure( 4.15) for grids A, B, C and D. Grid convergence is shown, and excellent agreement with the linear theory is obtained. The surface pressure distributions for different domain lengths are shown in Figure( 4.16) along with the linear theory prediction. It is evident that the effects of the domain length are negligible. Figure 4.14: Test case 3: A snapshot of pressure contours. 44

60 Linear Theory Euler (Grid ns=25) Euler (Grid ns=49) Euler (Grid ns=97) Euler (Grid ns=193) (Complex) Dp Real Part Imaginary Part x/c Figure 4.15: Test case 3: Sensitivity of surface pressure jump to grid step sizes. 45

61 Linear Theory Euler (Lx=5.0375) Euler (Lx=7.0525) Euler (Lx=9.0675) (Complex) Dp Real Part Imaginary Part x/c Figure 4.16: Test case 3: Sensitivity of surface pressure jump to streamwise domain length. Test case 4: n=5 Next, we present results for the benchmark problem considered by Hixon et al. [26]. The cascade is made of four blades with pitch s = 1. The convected vortical gust is given by Eqs ( 4.6) and ( 4.7) for n = 5, and the Mach number is M = 0.5. Table (2) gives the parameters for the five grids used to investigate sensitivity to step sizes and domain length. A snapshot of pressure contours is shown in Figure( 4.17). For this gust, the reduced frequency is κ = 7.854, for which the second acoustic mode is cut-on. Upstream of the cascade, pressure fluctuations propagate towards the upper left corner, and downstream of 46

62 the cascade they propagate towards the upper right corner. The cascade response at the forcing frequency ω l = ω includes only one propagating mode ν = 1. The amplitude ˆp lν (x) for this mode is depicted in Figure( 4.18) upstream of the leading edge and downstream of the trailing edge for the four grids A, B, C and D. The upstream radiated pressure is 5% higher than that predicted by using Glegg s [13] linear theory. Reflection from the inflow boundary is negligible. As we refine the grid, downstream radiation converges to the linear theory prediction. However, the weak undulations in the wave amplitude indicate a small wave reflection from the outflow boundary. In addition to the propagating mode (ν = 1), there are other modes that decay exponentially upstream and downstream of the cascade. The dominant exponentially decaying mode is (ν = 3), which is also depicted in Figure( 4.18). This mode shows very little sensitivity to grid resolution and is not influenced by reflection from the inflow or outflow boundaries. It dominates the the radiated pressure in the near field. At the gust frequency ω l = ω, the real and imaginary parts of pl (x) are compared to predictions of Glegg s linear theory in Figure( 4.19) for grids A, B, C and D. Grid convergence is shown. Sensitivity of surface pressure distributions to domain length is shown in Figure( 4.20) for the grids C, E, and F, and as shown the effects of the domain length are negligible. In this case, reflection from the boundaries is much smaller than in case 2. The streamwise wavenumbers for the acoustic waves radiated upstream (k x + ) and downstream (kx ) are shown in Table 3. Also shown is cosθ ±, where θ is the angle that the wavenumber vector makes with the outward normal to the boundary. The outflow boundary in Case 2 for which (cosθ = 0.375) suffers the most wave reflection as we have shown, and it calls for applications of higher order nonreflecting boundary conditions such as those developed by Hagstrom and Goodrich [24]. 47

63 Table 2 Grid parameters for test case 4 Grid n s n c L x t A B C D E F Figure 4.17: Test case 4: A snapshot of pressure contours. 48

64 Linear Theory Euler (Grid ns=25) Euler (Grid ns=49) Euler (Grid ns=97) Euler (Grid ns=193) p Upstream Radiation Downstream Radiation Plate x/c Figure 4.18: Test case 4: Pressure amplitudes for propagating mode ν = 1 and decaying mode ν = 3. 49

65 Linear Theory Euler (Grid ns=25) Euler (Grid ns=49) Euler (Grid ns=97) Euler (Grid ns=193) (Complex) Dp Imaginary Part 0 Real Part x/c Figure 4.19: Test case 4: Sensitivity of surface pressure jump to grid step sizes. 50

66 Table 3 Wavenumbers for upstream and downstream propagating modes Case n ν k z k x + kx cosθ + cosθ Linear Theory Euler (Lx=5) Euler (Lx=7) Euler (Lx=9) 0.03 (Complex) Dp Imaginary Part 0 Real Part x/c Figure 4.20: Test case 4: Sensitivity of surface pressure jump to streamwise domain length. 51

67 4.3 CONCLUSIONS We considered the response of a flat plate cascade to two-dimensional vortical waves (gust). We solved the two-dimensional nonlinear Euler equations over a linear cascade composed of six plates for a range of frequencies of the incident gust. The cascade is unstaggered and the pitch to chord ratio is We use Giles [12] non-reflecting boundary conditions at the inflow and outflow boundaries. We analyzed the cascade response in terms of unsteady normal force, surface pressure distribution and radiated acoustic pressure field for three discrete frequencies of the incident gust. The lift spectrum agrees very well with Glegg s [13] solution to the linearized potential flow equation for the tested range of reduced frequency (0 < ωc/2u < 8 ). Since Giles boundary conditions are approximately nonreflecting, we have investigated undesirable wave reflection at the inflow and outflow boundaries and its variation with gust frequency. Certain frequencies excite acoustic modes whose wavenumber vectors are nearly normal to the boundary. In such cases, minor reflection at the boundary is obtained and the results are insensitive to the location of the computational domain boundaries (cases 2 and 4 of this chapter). Other frequencies may still excite acoustic modes whose wavenumber vectors deviate considerably from the normal direction to the boundary resulting in major reflection that contaminates the pressure field (case 2). In such a case, the numerical solution depends sensitively on the locations of the inflow/outflow boundaries. If the gust frequency is such that all acoustic modes are cut off, the pressure field decays exponentially towards the boundaries, and boundary treatment poses no problem (case 3). These observations are consistent with the basic assumption in Giles derivation of the approximately nonreflecting boundary conditions, which is based on a Taylor series expansion for small ratio of tangential wavenumber to frequency. Rowley and Colonius [39] (, Colonius [7] for a review) and Hagstrom and Goodrich [24] have developed more accurate numerically nonreflecting conditions. Prediction of radiated sound by a cascade of blades due to interaction with turbulence which includes a spectrum of frequencies can greatly benefit from these new nonreflecting boundary conditions. 52

68 Chapter 5 Vortex-Plate Interaction 5.1 Introduction The Blade-Vortex Interaction problem is a fundamental problem in aeroacoustics. Researchers have formulated mathematical models of varying levels of fidelity, and obtained both analytical and numerical solutions. Howe [25] has presented a comprehensive analytical treatment of sound radiated by the interaction of line vortices with a flat plate, among other vortex sound problems. Glegg et al. [14] gave a recent review of theories for computing leading edge noise due to the interaction of a line vortex as it convects past an airfoil of finite thickness. In Computational Aeroacoustics (CAA), the field equations that describe the mechanisms of sound generation and propagation are solved numerically. Delfs et al. [15] solved the linearized Euler equations using a high-order finite difference method, and determined the noise radiated by the interaction of a finite-core vortex with a sharp edge. Delfs [16] also solved the same equations for the interaction problem and determined the sound radiated by a 2D airfoil with a rounded leading edge. Grogger et al. [17] also solved the linearized Euler equations, and determined the noise generated by the interaction of localized three-dimensional vorticity with the leading edge of an airfoil. They studied the 53

69 effects of the airfoil s thickness ratio on the strength and directivity of radiated noise. In this chapter we solve the two-dimensional nonlinear Euler equations to study the effects of the internal structure of a vortex (i.e. the radial distribution of vorticity) on the sound generated by the interaction of a finite-core vortex with a flat plate of zero thickness. 5.2 Vortex-Plate Interaction The Incident Vortex A 2D vortex flow that is an exact solution to the steady compressible Euler equations can be constructed in polar coordinates by assuming v r = 0 and v θ = f(r) (5.1) ρ = g(r) (5.2) where f(r) and g(r) are arbitrary functions of r. It can be shown that the above fields satisfy the continuity and θ-momentum equations. The pressure p(r) is obtained by integrating the r momentum equation, dp dr = ρv2 θ r (5.3) and the temperature is obtained from the equation of state. Because pressure and density are functions of r only, it follows that any other thermodynamic property, such as entropy S, is also a function of r only. Hence the energy equation DS Dt = 0 (5.4) is satisfied. Instead of choosing the density g(r) arbitrarily, we choose the entropy to be uniform in space, S = S o, and use the isentropic equation ( ) γ p ρ = (5.5) p ρ 54

70 The final results are where ( p = p 1 γ 1 F (r) c 2 ) γ γ 1 (5.6) T = 1 γ 1 F (r) (5.7) T c 2 ( ρ = 1 γ 1 ) 1 γ 1 F (r) (5.8) ρ c 2 F (r) = r f 2 dr (5.9) r and c is the speed of sound at the reference conditions (c 2 = γp /ρ ). Oseen Vortex The velocity profile v θ (r) = f(r) = B r (1 e βr2 ) (5.10) is known as the Oseen vortex [4], for which we obtain F (r) = βb 2 [ (1 e ζ ) 2 2ζ + E 1 (ζ) E 1 (2ζ)] (5.11) where ζ = βr 2, and E 1 (ζ) is the exponential integral E 1 (ζ) = e ζt 1 t dt (5.12) The parameters β and B are related to the core radius r o and the corresponding maximum circumferential velocity v o by where c o = , and B = β = c o 2r 2 o (5.13) v or o 1 e co (5.14) We can convect the vortex by superposing a uniform flow (U, W ) onto the velocity field of the vortex. If the vortex center is at (x c, z c ), the cartesian components of velocity will be u(x, z) = U (z z c ) B r 2 (1 e βr2 ) (5.15) 55

71 where r 2 = (x x c ) 2 + (z z c ) 2. w(x, z) = W + (x x c ) B r 2 (1 e βr2 ) (5.16) Taylor Vortex The velocity profile v θ (r) = f(r) = Are αr2 (5.17) is known as the Taylor vortex [4], for which we get F (r) = A2 4α e 2αr2 (5.18) The parameters α and A are related to the core radius r o and the corresponding maximum circumferential velocity v o by and α = 1 2r 2 o Similarly, the Cartesian components of velocity will be (5.19) A = v o r o e (5.20) u(x, z) = U A(z z c )e αr2 (5.21) w(x, z) = W + A(x x c )e αr2 (5.22) The circumferential velocity v θ (r), circulation Γ(r) and vorticity ω y (r) for the Oseen and Taylor vortices are shown in Figure (5.2). We note that the Taylor vortex has a core region of positive vorticity (clockwise) and an outer region of negative vorticity, and that the circulation approaches zero as r. The Oseen vortex has one sign vorticity and its circulation is finite as r, and hence its far field decays like 1/r whereas that of the Taylor vortex decays exponentially. The coordinates and flow variables are made non-dimensional by using the plate chord, c, the free-stream velocity U, density ρ, and temperature T as reference values. The reference 56

72 10 z U v/vo Oseen-Velocity Oseen-Circulation Oseen-Vorticity Taylor-Velocity Taylor-Circulation Taylor-Vortcity C x 4 ro=0.125c vo=0.01u r/ro Figure 5.1: Parallel vortex-plate interaction. Figure 5.2: Flow properties of Oseen and Taylor vortices. pressure is ρ U. 2 The vortex radius is 0.125, where the maximum circumferential velocity is The free stream Mach number is M = 0.5. The two-dimensional nonlinear compressible Euler equations are solved on a Cartesian grid in the x z plane. The computational domain is the square ( 9 x 11, 10 z 10), and a uniform grid is used where the step sizes are x = z = The time step is t = The flow field is initialized by the superposition of the vortex field and a uniform flow, such a field is an exact solution to the Euler equations. Initially, the plate and vortex axis are contained in one plane with the axis being parallel to and upstream of the plate leading edge as shown in Figure (5.1). Therefore, the plate may split the vortex along its axis as it convects past the leading edge. Vortex shedding from the sharp leading and trailing edges is captured by the numerical solution to the nonlinear Euler equations. As the vortex convects with the uniform free stream along the x axis from left to right, the flat plate is suddenly introduced with its leading edge at x = 0 and trailing edge at x = 1. At this instant the vortex axis is upstream of the leading edge at x = 3. 57

73 Figure 5.3: Oseen vortex, a snapshot of vorticity field, t = 2. Figure 5.4: Oseen vortex, a snapshot of pressure filed, t = 2. Figure 5.5: Oseen vortex, a snapshot of vorticity field, t = 3. Figure 5.6: Oseen vortex, a snapshot of pressure field, t = 3. 58

74 Snapshots of vorticity and pressure coefficient fields [c p = 2(p p )/ρ U ] 2 at different stages of the interaction of the vortex with the plate are analyzed. The results for the Oseen vortex are shown in Figures (5.3) to (5.10). Vorticity contours at time t = 2 (vortex center is at x = 1) show a vortex sheet, that is of the same sign as the incident vorticity, emanating from the trailing edge as shown in Figure (5.3) (actually vorticity in the wake spreads over a few grid lines, but we will refer to it as a vortex sheet). Because the velocity field of the Oseen vortex decays slowly, the sudden application of the wall boundary conditions generates two transient pulses that propagate above and below the plate. These are shown by the two outermost pressure arcs in Figure (5.4). As the vortex convects towards the leading edge, it induces a downwash velocity on the plate which generates compression and rarefaction waves from the upper and lower sides, respectively. At time t = 3, the vortex center is now at the leading edge (x = 0), and the thin flat plate splits the vortex core as shown in Figure (5.5). At time t = 4, the split vortex is now at the trailing edge of the plate (x = 1), a vortex sheet of the same sign vorticity as the core vorticity is visible in the wake as shown in Figure (5.7). A vortex sheet of opposite vorticity also commences at the trailing edge. Immediately after the vortex center passes the leading edge it induces an upwash velocity there, and acoustic pressure waves, opposite in phase to the earlier waves, propagate above and below the plate as shown in Figure (5.8). At time t = 6, a vortex sheet opposite in sign to the core vorticity is sandwiched between the two parts of the split vortex as shown in Figure (5.9). By this time the interaction is complete, and radiated pressure filed is fully formed as shown in Figure (5.10). The interaction of a Taylor vortex with the plate shows similar characteristics as the interaction with Oseen vortex, but it is much delayed. The interaction remains negligible until the vortex center passes by x = 0.25, which is one vortex diameter upstream of the leading edge. Before this time the vorticity and pressure fields are dominated by the near field of the vortex flow. Vorticity and pressure fields are shown in Figures (5.11) and (5.12), respectively. In Figure (5.11), it is interesting to note that because the vorticity of the incident vortex is of mixed sign (positive in the core and negative outside) the vortex sheet separating the 59

75 Figure 5.7: Oseen vortex, a snapshot of vorticity field, t = 4. Figure 5.8: Oseen vortex, a snapshot of pressure field, t = 4. two parts of the split vortex is also of mixed sign. The pressure field in Figure (5.12) shows that an observer in the far field above or below the plate receives two consecutive pulses. The pulses above the plate are out of phase relative to those below the plate. The phase is determined by the sense of rotation of the vortex, which is clockwise in this case. Maximum acoustic pressure is radiated downstream relative to a line normal to plate at the leading edge. No radiation in the streamwise direction is detected in the plane of the plate. 60

76 Figure 5.9: Oseen vortex, a snapshot of vorticity field, t = 6. Figure 5.10: Oseen vortex, a snapshot of pressure field, t = 6. Howe [25] shows that the linear theory of the low Mach number, two-dimensional interaction of a line vortex with a flat plate predicts a dipole directivity pattern on a circle in the far field with center at the midpoint of the plate. In the present simulations, the Mach number is 0.5, and hence the effect of convection on the directivity cannot be neglected. At t = 6, the first pressure pulse falls on a circle with radius r = 6.2 and center at x = 3.1 and z = 0. as a result of convection. Directivity of the pressure amplitude on this circle for the two vortices is shown in Figure (5.13). The dipole character of the directivity pattern is clear. However, in the case of Oseen vortex the pattern is inclined towards the upstream. The pressure signature at a point above the plate (x = 0.5, z = 3) is shown in Figure (5.14) for the two finite core vortices and for a point vortex, the later is a prediction of an approximate low Mach number linear theory (Equation 8.1.8, [25]) (Note that [t] in the abscissa is given by [t] = t r/a ). For the Oseen vortex, the pressure signature shows an early pulse which represents the passage of the transient response caused by the sudden introduction of the plate in the vortex filed. The Taylor vortex shows more compact pressure signature as 61

77 Figure 5.11: Taylor vortex, a snapshot of vorticity field, t = 6. Figure 5.12: Taylor vortex, a snapshot of pressure field, t = p Oseen Taylor p p Figure 5.13: Directivity of pressure amplitude on a circle r = 6.2 centered at x = 3.1, z = 0 at time t = 6. 62

78 Oseen (Euler) Taylor (Euler) Point Vortex (h/c=0) Point Vortex (h/c=0.07) Oseen Vortex Taylor Vortex Cp Lift Coefficient Plate U[t]/C Plate Ut/C Figure 5.14: Pressure signature at x = 0.5, z = 3. Figure 5.15: Lift coefficient. compared to the Oseen vortex with qualitative differences in the second pulse that is generated after the vortex center passes the plate leading edge. The results of the linear theory would be obtained for an Oseen vortex in the limit of zero core radius. The unsteady lift is shown in Figure (5.15), the effects of the internal structure of the vortex on the lift is evident. Taken together, Figures (5.13) to (5.15) show that the radiated sound and plate response in the vortex-plate interaction problem depend on the internal structure of the vortex. 5.3 Conclusions We have simulated the parallel interaction of a finite-core vortex with a zero-thickness flat plate. We have investigated the effects of the internal structure of the vortex (radial variation of vorticity) on the strength and directivity of radiated sound. We considered two vortices: (1) The Oseen vortex, whose vorticity distribution is monotone and its circulation at infinity 63

79 is finite, and (2) The Taylor vortex whose vorticity distribution is of mixed sign and its circulation decays to zero at infinity. The core radius and maximum circumferential velocity are the same for the vortices. The simulations indicate that there are qualitative differences in the radiated sound and unsteady lift produced by the two vortices, and hence one needs to consider the internal structure of the vortex when studying blade-vortex interactions. 64

80 Chapter 6 Interaction of Homogeneous Turbulence with a Flat-Plate Cascade - Comparison with Experimental Data In this chapter, we are interested in the comparison of our numerical simulation with the available experimental results. The current investigation is motivated by the cascade experiments conducted by Larssen and Devenport [28] (see also Larssen [29]). They adopted a mechanically rotating active grid design in order to generate large scale turbulence. The experimental setup consists of a six-blade linear cascade. In our simulation, we use the same configuration as that of the experiment but with thin flat plates. The numerical results of this chapter are based on using the options of zero-shear wall boundary conditions and the sixth-order scheme combined with a compact upwind fifth-order scheme due to Zhong [53]. 65

81 6.1 Inflow Turbulence At the inflow boundary we want to have control over the incident velocity fluctuations. Therefore, we specify the three velocity components and temperature. The inflow boundary is a plane of constant x. The velocity components on that plane are prescribed by Fourier series in time: N u(y, z, t) = U + A n u(y, z)cosω n t + Bu(y, n z)sinω n t (6.1) v(y, z, t) = w(y, z, t) = n=1 N A n v (y, z)cosω n t + Bv n (y, z)sinω n t (6.2) n=1 N A n w(y, z)cosω n t + Bw(y, n z)sinω n t (6.3) n=1 where the coefficients of the series are functions of y and z. To generate the Fourier coefficients, we first generate a box of incompressible (divergence free) isotropic random field in the wavenumber space (k x, k y, k z ) using a method given by Durbin and Rief [9] (page 241). The modified wavenumber of the sixth-order compact scheme [30] is used so that the inflow velocity field is divergence free if the divergence is evaluated by that scheme. The 3D energy spectrum function, E(k), is given by von Karman spectrum where E(k) = q 2 LC vk (kl) 4 [1 + (kl) 2 ] p (6.4) C vk = Γ(p) Γ( 5 2 )Γ(p 5 2 ) (6.5) We use p = 17/6 and hence C vk = And q 2 is twice the turbulence kinetic energy, and L is a length scale that is related to the integral length scale, L 1 11 by: 66

82 L = 4(p 1)(p 2) 3πC vk L 1 11 (6.6) Moet et al. [19] generated a similar box of homogeneous turbulence and used it as initial condition in the investigation of the ambient turbulence effects on vortex evolution. the cascade problem, we use q 2 = 0.02 (The reference velocity is the free stream velocity). The numerical value of the integral length scale is L 1 11 = 280 mm (s/l 1 11 = 0.943), which is suggested by the cascade experiments of Larssen [29]. The next step in specifying the inflow velocity is to take the inverse Fourier transform in the y and z plane and interpret the wave number in the x-direction as a frequency in time by invoking Taylor s hypothesis, ω n = U k xn. Over the duration of a simulation, the box of turbulence is repeatedly fed at the inflow boundary, and hence the incident turbulence is perfectly periodic. To reduce the effects of this periodicity as well as the anisotropy of the incident turbulence, we conduct six independent simulations using six different boxes at the inflow boundary. Then we take ensemble average of the statistics of the six simulations. For 6.2 Comparison with Experimental Data Spatially Decaying Isotropic Turbulence Before we present results for the interaction problem, it is important to establish credibility of the simulations for spatially decaying isotropic turbulence without plates. Here, we compare LES results with experimental data for grid-generated turbulence: Comte-Bellot and Corrsin [8] experiments (referenced to as CBC in this chapter). Case a of CBC with grid size M = 5.08 cm is considered. The reference length is M = 5.08 cm, and reference velocity is U = 10 m/s. The Reynolds number is Re = 34000, and the free stream Mach number in the simulation is assumed to be 0.4. For spatial simulation, the energy spectrum at the inflow boundary matches the experimental data at station x/m = 42 in the CBC experiments 67

83 E / U^2 M Coarse grid (449,33,33) CBCM2 (x/m=42) CBCM2 (x/m=98) CBCM2 (x/m=171) Inflow (x/m=42) C6CUD5 (x/m=98) C6CUD5 (x/m=171) E / U^2 M Fine grid (897,65,65) CBCM2 (x/m=42) CBCM2 (x/m=98) CBCM2 (x/m=171) Inflow (x/m=42) C6CUD5 (x/m=98) C6CUD5 (x/m=171) km km Figure 6.1: spatial LES, coarse grid. Energy spectrum function, Figure 6.2: spatial LES, fine grid. Energy spectrum function, (usually referred to as Ut/M = 42 in temporal simulations), and the outflow boundary is taken at x/m = The cross plane (yz-plane) is a square of side 10.8M. Periodic boundary conditions are applied in the y and z directions. Comparison with CBC experimental data is done at stations x/m = 98 and x/m = 171. We have tested two grid resolutions. A coarse grid of (nx, ny, nz) = (449, 33, 33) points and a fine grid of (nx, ny, nz) = (897, 65, 65) points. The 3D energy spectrum function E(k) is shown in Figures (6.1) and (6.2) for the coarse and fine grids, respectively. The symbols in these figures are the experimental data as listed in table 3 of CBC. About 2/3 of the modes are well resolved but there is dissipation of the high wavenumbers. In the dynamic SGS model, Smagorinsky s constant, C, is a function of space and time. In the present simulation, we average C over the yz-plane, and hence C is a function of x and t. We depict C in Figure (6.3) as a function of x. On average, the value of C decays slightly with the decay of turbulence. The rapid decay near the outflow boundary is caused by the enhanced artificial damping near the boundary. Based on the variation of C with x as shown 68

84 0.1 Z A 0.08 Y X C B SGS model, C Coarse grid (449,33,33) Fine grid (897,65,65) S Outflow B x/m-42 B Inflow B=4.836 C A=4.911 C S=0.806 C Figure 6.3: Streamwise variation of dynamic model coefficient in spatial decaying turbulence. Figure 6.4: Flat plate cascade and computational domain in Figure (6.3), a constant value of 0.03 for C may be used with the classical Smagorinsky model which is more efficient than the dynamic model. We used the classical model with C = 0.03 in the cascade simulations presented in the next section and in chapter (7) Computational Domain and Inflow Spectra The current investigation is motivated by the cascade experiments conducted by Larssen and Devenport [28] (see also Larssen [29]). The experimental setup consists of a six-blade linear cascade. The geometric properties of the plate are: chord (c) = mm and thickness = 6.35 mm. The plate has a semi-circular leading edge and sharp trailing edge. The blade-toblade spacing (s) is 264 mm. In our simulations, we also consider a six-blade linear cascade as shown in Figure (6.4). The plate has zero thickness, and the chord (c) is the same as in the experimental setup. We use the chord c as a reference length. The pitch s is and the span B is The mean free stream velocity (U) is used as the reference velocity. 69

85 The Reynolds number based on c and U is and the Mach number is assumed to be 0.3. The cascade is unstaggered and the plates are at zero incidence relative to the mean flow. A uniform Cartesian grid of (nx, ny, nz) = (321, 145, 145) points is used in the present cascade simulations. At the inflow boundary we specify the three velocity fluctuations. The target 3D energy spectrum function E(k) is specified by von Karman spectrum as given by Eq (6.4). The integral length scale is specified by L 1 11 = 280 mm as given by Larssen [29] based on his experimental data. The cascade pitch s = 264 mm, and hence s/l 1 11 = However, the low-wavenumber end of the numerically generated fluctuations is limited by the transverse dimension of the computational domain (B) which includes only six blades (B = 6s = 5.66L 1 11). As a result of this limitation, wavenumbers smaller than the wavenumber where E(k) attains its maximum are not activated in the inflow fluctuations. The highest wavenumber of the spectrum of the generated fluctuations is also limited by the grid resolution, k max = π/ z = πnz/b. The target and numerically generated spectra are shown in Figure (6.5). The roll off of the 1D energy spectra at high wavenumbers is because we zero out the spherical shells with radius greater than k max in the 3D wavenumber space Comparison of LES with Larsen Experimental Data We compare LES results with experimental data provided by private communication with Larssen (see also [29]). The streamwise decay of the normal Reynolds stresses at midpassage (z/s = 0.5) is depicted in Figure (6.6). The experimental data show higher levels of the normal stresses and turbulence kinetic energy than those of LES. This is because the experimental data contain energy from the full spectrum (unfiltered) whereas the LES results represent only the resolved scales. Experimental data and LES results show that the reduction of turbulence kinetic energy (q 2 /2) at the mid-passage is almost complete a short distance into the passage (x = 0.65). Thus the reduction happens over a distance on the order of one integral length scale L 1 11/c = Downstream of the station x = 0.65, the 70

86 E, E11, E E(k)- von Karman E11(k1)- von Karman E22(k1)- von Karman E(k)-Inflow Boundary E11(k1)-Inflow Boundary E22(k1)-Inflow Boundary k, k1 uu, vv, ww, qq LE z/s = 0.5 uu-les vv-les ww-les qq-les uu-exp vv-exp ww-exp qq-exp x TE Figure 6.5: Target and numerically generated energy spectra at inflow boundary. Figure 6.6: Mid-passage distribution of normal Reynolds stresses and q 2 = u 2 + v 2 + w 2. decay rate of the kinetic energy is nearly the same as the decay rate upstream of the cascade. The normal Reynolds stress profiles are shown in Figure (6.7) at a station near the trailing edge (x = 0.840). The large discrepancy between LES results and experimental data in the streamwise u 2 and spanwise v 2 components is due to the more energetic incident turbulence in the experiments as compared to simulations. The close agreement in the upwash component w 2 is fortuitous, but it serves the purpose of showing that the shape is well predicted. A more meaningful comparison is obtained if we normalize each profile by the respective values of Reynolds stresses upstream of the cascade. At the point (z/s = 0.5 and x = 0.95), the experimental data are (u 2 1 = , v1 2 = , w1 2 = ) whereas the LES results are (u 2 1 = , v1 2 = , w1 2 = ). These values are then used to normalize the respective Reynolds stress profiles, which are depicted in Figure (6.8). The tangential profiles (u 2 and v 2 ) predicted by LES are in good agreement with the experimental data over most of the passage, whereas the upwash profile (w 2 ) is over-predicted by LES. We 71

87 uu-les vv-les ww-les uu-exp vv-exp ww-exp 0.4 uu-les vv-les ww-les uu-exp vv-exp ww-exp z / s z / s (uu, vv, ww) / U^2 Figure 6.7: Reynolds stress profiles at x = uu/uu1, vv/vv1, ww/ww1 Figure 6.8: profiles at x = Normalized Reynolds stress note that the discrepancies in u 2 and v 2 near the wall is caused by the zero-shear boundary condition which is directly enforced on the velocity near the wall. The normal Reynolds stress profiles at a streamwise station (x = 1.948) in the cascade wake are depicted in Figure (6.9), and the normalized profiles are shown in Figure (6.10). The normalized profiles show the correct trend except near the wake centerline. The spanwise profile (v 2 ) is closer to the measured one near the wake centerline whereas the u 2 profile is not. The experimental data for the streamwise profile (u 2 ) show very strong maximum above the centerline (z/s = 0.04), whereas the upwash profile has its maximum on the wake centerline (z/s = 0). These local maxima are generated by the instability of the mean shear of the wake profile. Thus they are not a part of the incident turbulence, although the instability may have been enhanced by the interaction with that turbulence. The mean shear in the boundary layers and wakes are not resolved by the current LES cascade simulations. To verify that the maxima in the fluctuations near the wake centerline are induced by the mean shear instability, we consider interaction of homogeneous turbulence with a single flat 72

88 uu-les vv-les ww-les uu-exp vv-exp ww-exp 0.4 uu-les vv-les ww-les uu-exp vv-exp ww-exp z / s z / s (uu, vv, ww) / U^2 Figure 6.9: Reynolds stress profiles at x = uu/uu1, vv/vv1, ww/ww1 Figure 6.10: Normalized Reynolds stress profiles at x = <uu> at x/c=0.92 <ww> at x/c=0.92 <uu> at x/c=1.53 <ww> at x/c=1.53 z 0 z x Figure 6.11: Spanwise vorticity contours for a single plate placed in isotropic turbulence, no-slip condition is applied uu, ww Figure 6.12: Reynolds stress profiles at (x x LE )/c = 0.92 and (x x LE )/c = 1.53 for a single plate, no-slip boundary condition is applied. 73

89 High Levels of uu by wake instability 0.2 Suppression of ww by plate surface High Levels of ww by wake instability z 0 z Plate Trailing Edge Plate Trailing Edge x x Figure 6.13: Reynolds stress contours u 2 for a single plate. Figure 6.14: Reynolds stress contours w 2 for a single plate. plate. A schematic of the plate and incident flow (from left to right) is shown in Figure (6.11). In this case, the plate leading edge is at (x = x LE = 1.0) and the plate chord (c = 1.0). A zero-thickness flat plate is placed in a uniform stream at Mach number M = 0.6 and Reynolds number based on chord of Re = At the inflow boundary, velocity fluctuations are superimposed on the uniform flow. The zero-shear condition used in the cascade simulations is abandoned, and instead we apply the usual no-slip wall boundary conditions on the plate. Reynolds stress profiles at two streamwise stations, one near the trailing edge (x x LE )/c = 0.92 and another in the wake (x x LE )/c = 1.53 are shown in Figure (6.12). While the transverse component w 2 is totaly suppressed on the plate, it attains a maximum on the wake centerline. The streamwise component u 2 increases slightly near the plate, but it has a very strong magnitude in the wake below and above the centerline. The local maxima of (u 2 ) and (w 2 ) are also shown by the contour plots in Figures (6.13), and (6.14), respectively. 74

90 Chapter 7 Interaction of Homogeneous Turbulence with a Flat-Plate Cascade - Comparison with RDT 7.1 Introduction In this chapter, we study the distortion of homogeneous isotropic turbulence as it passes through unstaggered cascade of thin flat plates. Spatial large eddy simulation (LES) is conducted for two linear cascades: a six-plate cascade, and a three-plate cascade. Because suppression of the normal component of velocity is the main mechanism of distortion, we neglect the presence of mean shear in the boundary layers and wakes, and allow slip velocity on the plate surfaces. We enforce the zero normal velocity condition on the plate and relax the no-slip condition to a zero-shear or slip wall condition. This boundary condition treatment is motivated by rapid distortion theory (RDT) in which viscous effects are neglected, however the present LES approach accounts for nonlinear and turbulence diffusion effects by a subgrid scale model. We tested two different wall boundary conditions; zero-shear boundary and 75

91 slip-wall boundary. We have presented LES results for zero-shear conditions in chapter (6). All the LES results in this chapter are based on the option of slip-wall boundary and the use of tenth-order filter compined with the compact six-order finite-difference scheme. To test the applicability of Graham s RDT solution [23], we introduce homogeneous turbulence of different spectral content and different intensity to the computational domain and compare with LES results. The normal Reynolds stresses and velocity spectra are analyzed ahead, within, and downstream of the cascade. 7.2 Graham s RDT Solution Using rapid distortion (RDT), Graham [23], see Appendix (A), has developed analytic solutions for the spectra of isotropic turbulence downstream of a linear cascade of thin flat plates. Because viscous and nonlinear effects are neglected, the disturbance produced by the cascade is an irrotational velocity field induced by flat vortex sheets that coincide with the plates and extend to infinity downstream of the trailing edge. Graham [23] assumes the incident turbulence velocity field to be homogeneous, and represents it by 3D Fourier integrals, Equation (7.1). He also assumes the turbulence after the introduction of the cascade to remain homogeneous in planes parallel to the plates, and obtains the Fourier coefficients of the distorted turbulence in terms of those of the incident turbulence. The incident turbulence velocity field is given by: u (x, t) = û (k)e i(ωt k jx j ) dk, j = 1, 2, 3 (7.1) where ω = k 1 U, and k is the wave number vector. Each Fourier component of the total flow field can be expressed as: u(x, t; k) = û (k)e i(ωt k jx j ) + ˆφ(k, x 3 )e i(ωt k 1x 1 k 2 x 2 ), (7.2) 76

92 where φ is the velocity potential due to the blocking effect of the cascade. Solution to Laplace s equation (subjected to zero normal velocity on the plates and Kutta condition at the trailing edge) for each Fourier component in the region between any two plates is given by: û 1 = û 1 + ik 1 τ cosh[τx 3 ]e ik3s cosh[τ(s x 3 )] û 3 e ik 3x 3 (7.3) sinh(τ s) û 2 = û 2 + ik 2 τ cosh[τx 3 ]e ik3s cosh[τ(s x 3 )] û 3 e ik 3x 3 (7.4) sinh(τ s) where τ = (k k 2 2 ) 1 2. û 3 = û 3 sinh[τx 3]e ik3s + sinh[τ(s x 3 )] û 3 e ik 3x 3 (7.5) sinh(τ s) Graham s solution is valid for a streamwise distance on the order of the integral length scale x = O(L 1 11) downstream of the leading edge. Assumptions of RDT imply that the solution is not valid for shorter or much greater distance than the integral length scale. 7.3 Comparison of LES with Graham s RDT In comparing LES with RDT, we found it necessary to account for the decay of turbulence, and hence the input spectra to RDT should be the spectra that would exist at the streamwise location of comparison but in the absence of the cascade. Therefore, to compare LES results with Graham s RDT results, we conduct simulations but without the cascade under identical conditions (grid, inflow spectra, etc.) as those of the six simulations conducted in the presence of the cascade. The simulations without the cascade provide the Fourier representation of the incident turbulence that is required by RDT. In other words, the input to RDT is determined by simulations of spatially decaying homogeneous turbulence. The velocity field in planes 77

93 Case No.of plates Domain size Grid size u U A B C D Table 7.1: Characteristics of the inflow turbulence and computational domain of constant x are stored as functions of time (t), and then Fourier transform is obtained in yz-plane and time (t). Taylor s hypothesis is then used to replace frequency by streamwise wavenumber. The characteristics of the inflow turbulence and the computational domain of the two considered cascades are shown in Table (7.1) Six-Plate Cascade For case (A), the integral length scale of the incident turbulence is L 1 11 = 0.940s and the turbulence intensity is u /U = , see Table (7.1). The cascade geometry and the computational domain are shown in Figure (7.1). The plate is represented by 30 grid points and the number of grid point in z-direction in the passage between two plates is 24 points. The plate leading edge is located at x = 0 and the plate chord c = 1. The time step t = The 3D-energy spectrum of the incident turbulence is depicted in Figure (7.2). The turbulence intensities shown in this table should be considered nominal values. They are used in generating boxes of homogeneous turbulence according to Von Karman spectrum. However, because of filtering and inflow boundary condition treatment, we noted a fast drop in the turbulence kinetic energy near the inflow boundary. A more reliable measure of the actual turbulence kinetic energy is given by its streamwise distributions in the absence of the cascade which is shown in Figure (7.19) for the 6-plate cascade and Figure (7.64) for the 3-plate cascade. 78

94 Z B Z Y Y X Periodic s Inflow Outflow Periodic Periodic Periodic X Six-Plate cascade Figure 7.1: Six-plate cascade and computational domain. 79

95 E(k) x= x= Case A k Figure 7.2: Case A: 3D-energy spectra of the incident turbulence, inflow (x=-4.836), and upstream of cascade (x=-0.269). Instantaneous Flow Field A snapshot of the instantaneous upwash velocity contours (w w), (where, w is the local mean value.), in xz-plane is shown by Figure (7.3). Large scale fluctuations upstream of the cascade are suppressed as the turbulence passes through the cascade. It is clear that the instantaneous flow is non-periodic from one passage to the next, and that the computational domain must include multiple blades in order to correctly capture the interaction of large scale turbulence with the cascade. Figure (7.4) is a snapshot of the instantaneous upwash velocity contours in the yz-plane at x = The plates break the large scale structures into smaller scales. Figures (7.5) and (7.6) show snapshots of the instantaneous streamwise and spanwise velocity contours, respectively. The spacial structure of these contours does not show significant 80

96 Figure 7.3: A snapshot of the instantaneous upwash velocity contours(xz-plane). Figure 7.4: A snapshot of the instantaneous upwash velocity contours at plane x = distortion by passing through the cascade plates. Figures (7.7) and (7.8) show instantaneous velocity fluctuation vectors in a region around the plates in xz- and yz-planes, respectively. The highly turbulent flow field structure is clear in the figures. As we expect, the velocity vectors are tangent to the plate surfaces because of the zero-normal velocity boundary condition applied on the plates. A snapshot of the instantaneous pressure fluctuations (p p), (where p is the local mean value.), in xz and yz planes are shown in Figures (7.9) and (7.10), respectively. The pressure contour levels (which could include acoustic pressure waves) are totally different upstream and downstream the cascade. The plate surfaces are under varying pressure amplitudes in the spanwise direction. The periodic flow structure in the spanwise direction is clear in Figure (7.10). The instantaneous density fluctuation contours (ρ ρ), (where ρ is the local mean value), are 81

97 Figure 7.5: A snapshot of the instantaneous streamwise velocity contours (xzplane). Figure 7.6: A snapshot of the instantaneous spanwise velocity contours (xzplane). also shown as snapshots in Figures (7.11) and (7.12). The incoming flow field is supposed to be divergence free, (i.e. no density fluctuations). (Figures (7.13) and (7.14) show the corresponding pressure and density contours for the case without plates) But, because of the presence of the cascade plates which interacts with the incoming turbulence it produces density fluctuations in the flow field. The density fluctuations also show different spatial structures upstream than downstream of the cascade. The periodic flow structure in the spanwise direction is clear in Figure (7.12). Normal Reynolds Stresses Figures (7.15), (7.16), and (7.17) show the y-averaged normal Reynolds stress contours. These contours are the average of six independent runs and are also averaged in the z- direction by taking the average over the halves of the passages above and below each plate. The figures show the spatial decay of the turbulence as it convects downstream. 82

98 Case A Velocity vector Case A Velocity vector x= z z x y Figure 7.7: A snapshot of the instantaneous velocity vectors (xz-plane). Figure 7.8: A snapshot of the instantaneous velocity vectors (yz-plane) x = Figure (7.15) shows the distortion of the streamwise component (uu) by the introduction of the cascade plates. The streamwise component shows higher values in the vicinity of the leading edge. It also shows high values in the passage near the plate surfaces and extends downstream the trailing edge. The streamwise component shows lower values in a thin region in the wake of each plate in planes coincident with the plane of the plates. To capture the correct structure of the wake of the plates we need to use much finer grid to represent the vortex sheet emanating from the trailing edge. However, this does not affect the solution in the passage outside this thin region. The incoming turbulence is homogeneous upstream of the cascade. As the turbulence passes through the cascade it becomes no longer homogeneous. However, because of the spacial decay of the turbulence it starts to become homogeneous again at distances far from the trailing edge. Figure (7.16) shows the distortion of the spanwise component (vv) by the introduction of the cascade plates. It shows almost the same configuration as the streamwise component 83

99 Figure 7.9: A snapshot of the instantaneous pressure fluctuation contours (xzplane). Figure 7.10: A snapshot of the instantaneous pressure fluctuation contours (yzplane), x = except that there is no higher values at the leading edges. Figure (7.17) shows the distortion of the upwash component (ww) by the introduction of the cascade plates. The turbulence starts homogeneous upstream of the cascade. As the turbulence passes through the cascade the cascade suppresses the upwash component. The suppression of the upwash component in the passages starts at the plane of the plates where the upwash component is zero and continues to affect the entire passage. The upwash component continues to have almost zero values in the wake of the plates for a distance about 1.5 chord but then starts to build up which could be because of the viscosity, the roll up of the vortex sheet, and the nonlinear effects. The averaged contours of the square value of the pressure fluctuations (pp) are shown in Figure (7.18). It is interesting to notice that high pressure fluctuations occur only around the leading edge and extend short distance downstream stream of the leading edge. It shows no pressure fluctuations from the trailing edge. The leading edge is the main source of the 84

100 Figure 7.11: A snapshot of the instantaneous density fluctuation contours (xzplane). Figure 7.12: A snapshot of the instantaneous density fluctuation contours (yzplane), x = sound radiated from the cascade. The streamwise decay of q 2 = u 2 + v 2 + w 2 at mid-passage (z/s = 0.5) is depicted in Figure (7.19). The decay of the same quantity without the cascade is also shown. The LES results show that the reduction of turbulence kinetic energy (q 2 /2) at the mid-passage starts upstream of the cascade (x = 0.2) and is complete a short distance into the passage (x = 0.65), afterwards the decay rate is nearly the same as the case without the cascade. The RDT results is in full agreement with LES downstream of the streamwise location x = The RDT does not apply near the leading edge of the plate or farther downstream of the trailing edge. The reduction in the turbulence kinetic energy in the passage persists to the plate surface. In other words, although the turbulence kinetic energy increases towards the wall, but it is still reduced relative to its free stream value. It is interesting to note that Graham showed that in the limiting case in which the ratio of cascade pitch to integral length scale of the incident turbulence (s/l 1 11) approaches zero, the turbulence kinetic energy 85

101 Figure 7.13: Case A: A snapshot of the instantaneous pressure fluctuation contours (no plates). Figure 7.14: Case A: A snapshot of the instantaneous density fluctuation contours (no plates). Figure 7.15: Contours of the averaged streamwise-reynolds stress component. Figure 7.16: Contours of the averaged spanwise-reynolds stress component. 86

102 Figure 7.17: Contours of the averaged upwash-reynolds stress component. Figure 7.18: Contours of the averaged square of the pressure fluctuations (pp). is reduced by a half once the cascade is introduced into the flow. Turbulence kinetic energy profiles, normalized by the free stream value (q1) 2 in the passage, as predicted by Graham s RDT analytic solution for s/l 1 11 = 11.7, 0.94 and are shown in Figure (7.20). The profile predicted by LES at x = is also shown and indicates excellent agreement with RDT. As the ratio s/l 1 11 approaches zero, the theoretical limit of 50% is obtained. The streamwise decay of the averaged (over the yz-plane) turbulence kinetic energy (TKE) and normal Reynolds stresses are shown in Figure (7.21). The reduction of the TKE is almost complete a short distance downstream the leading edge of about (x = 0.65). The TKE shows a further reduction at the trailing edge. Such reduction at the trailing edge could be minimized by using a finer grid to represent more accurately the vortex sheet emanating from the trailing edge. The reduction of the TKE happens just at the introduction of the plates and mainly because of the suppression of the normal component (upwash) of velocity fluctuation. The decay rate of both the TKE and the normal component (ww) has different values upstream the leading edge and downstream the trailing edge. However the 87

103 no cascade cascade-les cascade-rdt 0.4 s/l11= RDT s/l11= RDT s/l11= LES s/l11= RDT s/l11= 0.0 -RDT q^ z/s LE x TE q 2 2 /q 1 Figure 7.19: Mid-passage distribution of q 2. Figure 7.20: Normalized q 2 profiles for different ratios of plate spacing to integral length scale s/l streamwise and spanwise components, (uu) and (vv), have the same decay rate upstream and downstream the cascade. Figure (7.21) shows also the averaged square values of the pressure fluctuation (pp) which has almost constant values upstream the leading edge and rapid increase just ahead the leading edge. The high values of the pressure fluctuation continues from the leading edge until the trailing edge. The drop of the TKE could be converted as an increase in the pressure fluctuation and radiated as sound waves. However, a more detailed investigation is needed to understand the mechanism by which the kinetic energy drops. 88

104 E-03 Average (uu,vv,ww)/2, TKE TKE uu/2 vv/2 ww/2 pp TKE-noplates 6-pates 2.5E E E E E-04 Average (pp) 0 LE TE x 0.0E+00 Figure 7.21: Averaged TKE, Reynolds stresses, and pressure fluctuation (read right) Normal Reynolds stress profiles (normalized by the inlet turbulence intensity u 2.), see table(7.1), are compared in Figures (7.22), (7.23), (7.24) and (7.25) for planes in the passage at x = 0.067, 0.201, and 0.840, respectively. There is excellent agreement between RDT and LES results for the plane x = 0.840, which is in the region of applicability of RDT, but there is some deviation which increases at planes which is closer to the leading edge where RDT does not apply. The profiles for three planes downstream of the trailing edge are shown in Figures (7.26), (7.27) and (7.28) at x = 1.578, and 2.787, respectively. The LES results continue to agree with Graham s RDT formulation across the passage except near the wake center line, where the no penetration condition is relaxed in the LES simulation and hence the turbulence structures are free to cross the wake center line plane. The LES solution accounts for viscosity and nonlinear effects and hence the generation of upwash velocity fluctuations there. We recall that in Graham s RDT formulation, the trailing vortex sheets remain flat and parallel to the plates, and hence the upwash velocity is zero on the plate 89

105 uu-les vv-les ww-les uu-rdt vv-rdt ww-rdt uu-les vv-les ww-les uu-rdt vv-rdt ww-rdt plates u = x= plates u = x=0.201 Z/S Z/S (uu,vv,ww)/u (uu,vv,ww)/u Figure 7.22: Normal Reynolds stress profiles at x = Figure 7.23: Normal Reynolds stress profiles at x = and on the wake vortex sheets. Therefore, the linearized inviscid rapid distortion theory correctly predicts the behavior of Reynolds stresses for x = O(L 1 11), but it does not capture changes in these stresses due to nonlinear effects and viscosity. We note that if the incident turbulence is perfectly isotropic, then Graham s RDT formulation predicts identical profiles for the two tangential components u 2 and v 2. The small difference between the u 2 and v 2 profiles presented here is caused by a small anisotropy of the simulated turbulence. This anisotropy is present in the two simulations with and without the cascade. For the comparison with RDT to be quantified and to have a reasonable comparison between different cases, we normalize the quantity q 2 within the passage by the corresponding value q0 2 in the case without plates and plot the distribution of q 2 as function of z/s at different streamwise locations. Table (7.2) lists the magnitude of q0 2 in the case without cascade at planes x = 0.201, 0.840, and for different cases. 90

106 uu-les vv-les ww-les uu-rdt vv-rdt ww-rdt uu-les vv-les ww-les uu-rdt vv-rdt ww-rdt plates u = x= plates u = x=0.840 Z/S Z/S (uu,vv,ww)/u (uu,vv,ww)/u Figure 7.24: Normal Reynolds stress profiles at x = Figure 7.25: Normal Reynolds stress profiles at x = uu-les vv-les ww-les uu-rdt vv-rdt ww-rdt uu-les vv-les ww-les uu-rdt vv-rdt ww-rdt plates u = x= plates u = x=1.948 Z/S Z/S (uu,vv,ww)/u (uu,vv,ww)/u Figure 7.26: Normal Reynolds stress profiles at x = Figure 7.27: Normal Reynolds stress profiles at x =

107 uu-les vv-les ww-les uu-rdt vv-rdt ww-rdt 6-plates u = x=2.787 Z/S (uu,vv,ww)/u Figure 7.28: Normal Reynolds stress profiles at x = x-plane Case A Case B Case C Case D Table 7.2: Values of q0 2 at the planes of comparison for different cases 92

108 The profiles of the normalized values q 2 /q0 2 at planes x = 0.201, 0.840, and2.787 are depicted in Figures (7.29), (7.30), and (7.31), respectively. The comparison of these plots indicates that both RDT and LES results show a reduction of the TKE within the passage that takes place from the center of the passage to the plate surface. Figure (7.29) shows a significant discrepancy between the RDT and the LES solutions at plane x = The RDT solution predicts that the reduction to be complete instantaneously, over the entire passage, once the cascade is introduced. While, LES solution allows for the reduction to develop gradually in the passage as the turbulence convects downstream. Figure (7.30) shows excellent agreement between the RDT and the LES solutions. According to the assumptions of the RDT; the RDT is applicable at distances of the order of the integral length scale measured from the leading edge. The RDT solution is not valid for shorter or much greater distance than the integral length scale. Figure (7.31) shows disagreement between the RDT and the LES solutions, specially, near the wake center line. RDT solution assumes the vortex sheet emanating from the trailing edge is extended downstream as if we have semi-infinite plate. Nevertheless, excellent agreement is obtained between RDT and LES over the rest of the passage. Both RDT and shear-free LES do not predict the correct physical behavior in the wake of the plate. Reynolds stresses in the wake of the plate and the associated instabilities are not captured by the two models. The distortion of the turbulence spectra downstream of the cascade is of interest to hydro/aeroacoustic predictions of noise radiated by rotors or guide vanes. The one-dimensional energy spectra of the upwash velocity component Eww(k 1 ) are depicted in Figures (7.32) and (7.33) on the mid-passage (z/s = 0.5) and near the plate surface at (z/s = ), respectively, for different streamwise locations. The major change in the large scales happens between the station x = upstream of the leading edge and the station x = downstream of the leading edge. The spectra predicted by Graham s RDT formulation are in good agreement with the LES spectra for x > The spectra near the plate surface at (z/s = ) are shown in Figure (7.33). Because of the suppression of the upwash velocity by the plate surface there is a significant reduction of the energy at low wavenumbers. Most 93

109 RDT LES RDT LES plates u = x= plates u = x= z/s z/s q 2 2 /q q 2 2 /q 0 Figure 7.29: Case A: Profiles of q 2 /q 2 0 at plane x = Figure 7.30: Case A: Profiles of q 2 /q 2 0 at plane x = RDT LES plates u = x= z/s q 2 2 /q 0 Figure 7.31: Case A: Profiles of q 2 /q 2 0 at plane x =

110 10-2 x= LES x= LES 10-3 x= LES x= RDT x= LES x= RDT plates u = Z/S= x= LES x= LES 10-3 x= LES x= RDT x= LES x= RDT plates u = Z/S= Eww 10-6 Eww k k1 Figure 7.32: One dimensional energy spectra, Eww(k 1 ) at z/s = 0.5. Figure 7.33: One dimensional energy spectra, Eww(k 1 ) at z/s = of the reduction again happens between the stations x = and x = The spectra according to RDT agree very well with LES results at x = However, downstream of the trailing edge at station x = 2.787, the LES results indicate a build up of large scales transverse fluctuations whereas RDT results predict insignificant changes. The one-dimensional energy spectra of the streamwise and spanwise velocity components Euu(k 1 ), and Evv(k 1 ) are depicted in Figures (7.34) and (7.36) on the mid-passage (z/s = 0.5), respectively, and Figures (7.35) and (7.37) near the plate surface at (z/s = ), respectively, for different streamwise locations. Both the streamwise and spanwise energy spectra components have almost the same trend. The change in the large scales, on the mid passage, of the streamwise and spanwise energy spectra components (Figures (7.34) and (7.36)) is smaller than the corresponding reduction in the upwash energy spectra component. The major change in the large scales happens also between the stations x = upstream of the leading edge and the station x = downstream of the leading edge. The RDT results are in good agreement with the LES results. Figures (7.34) and (7.35) show 95

111 10-2 x= LES x= LES 10-3 x= LES x= RDT x= LES x= RDT plates u = Z/S= x= LES x= LES 10-3 x= LES x= RDT x= LES x= RDT plates u = Z/S= Euu 10-6 Euu k k1 Figure 7.34: One dimensional energy spectra, Euu(k 1 ) at z/s = 0.5. Figure 7.35: One dimensional energy spectra, Euu(k 1 ) at z/s = significant reduction in the streamwise and spanwise energy spectra components in all the turbulence scales between the streamwise locations x = and Good agreement over all scales near the passage center line. But near the wake center line, the disagreement between LES and RDT is expected as discussed before Three-Plate Cascade One of the main assumptions of the rapid distortion theory is that (u /U )(x 1 /L 1 11) 1, where x 1 is distance downstream of the leading edge. To fulfill such condition; the turbulence should be weak and the region of applicability of the RDT is at distances from the leading edge of the order of the integral length scale of the turbulence. For more investigation of the applicability of Graham s RDT we need to contradict this assumption by both increasing the incoming turbulence intensity and decreasing the integral length scale of the turbulence. To decrease the integral length scale of the turbulence we choose to simulate the flow field 96

112 10-2 x= LES x= LES 10-3 x= LES x= RDT x= LES x= RDT plates u = Z/S= x= LES x= LES 10-3 x= LES x= RDT x= LES x= RDT plates u = Z/S= Evv 10-6 Evv k k1 Figure 7.36: One dimensional energy spectra, Evv(k 1 ) at z/s = 0.5. Figure 7.37: One dimensional energy spectra, Evv(k 1 ) at z/s = of a 3-plate cascade of the same pitch as the previous 6-plate cascade. The domain length is reduced to half of that of the 6-plate cascade. Then we simulated three different levels of the incoming turbulence intensity. The different characteristics of the turbulence and the computational domain are listed for all cases in Table (7.1). The geometry and the computational domain of the 3-plate cascade are shown in Figure (7.38). 97

113 Z Y X Outflow Periodic 1.5 Inflow 1 Z Y Periodic Periodic 1 0 Periodic X Three-Plate cascade Figure 7.38: 6-plate cascade and computational domain. The 3d-energy spectra of the incident turbulence for cases A, B, C, and D are depicted in Figure (7.39). Instantaneous Flow Field Snapshots of the instantaneous upwash velocity contours in xz-plane for the cases B, C, and D are shown by Figures (7.40), (7.41), and (7.42). The integral length scale of the turbulence in cases B, C, and D is smaller than that in case A. The large scale fluctuation structures upstream of the cascade of cases B, C, and D are smaller than those of case A, see figure (7.3). Most of the large scale structures passes through the cascade without break down because the large scale structures upsteam of the cascade are smaller than those of case A while we keep the same pitch (B) for cases B, C, and D as that of case A. It is clear that the magnitude of the upwash velocity contour levels is becoming larger from cases B, 98

114 C, to D Case A Case B Case C Case D E(k) k Figure 7.39: 3D-energy spectra of the incident turbulence Figures (7.43), (7.44) and (7.45) show the instantaneous velocity fluctuation vectors in a region around the plates in the xz-plane for cases B, C, and D, respectively. The highly turbulent flow field structure is clear in the figures. The magnitude of the velocity vectors are getting higher from case B, C, to D. 99

115 Figure 7.40: Case B: Snapshot of the instantaneous upwash velocity contours (xzplane). Figure 7.41: Case C: Snapshot of the instantaneous upwash velocity contours (xzplane). 2.4 Case D velocity vector z x Figure 7.45: Case D: A snapshot of the instantaneous velocity vectors (xz-plane). 100