Digital Filter-based Turbulent Inflow Generation for Jet ...

AIAA 2014-1401 AIAA SciTech 13-17 January 2014, National Harbor, Maryland 52nd Aerospace Sciences Meeting

Digital Filter-based Turbulent Inflow Generation for Jet Aeroacoustics on Non-Uniform Structured Grids Nitin S. Dhamankar∗, Chandra S. Martha†, Yingchong Situ‡, Kurt M. Aikens∗, Gregory A. Blaisdell§, Purdue University, West Lafayette, IN-47907.

Anastasios S. Lyrintzis¶, Downloaded by PURDUE UNIVERSITY on May 13, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2014-1401

Embry Riddle Aeronautical University, Daytona Beach, FL-32114.

and Zhiyuan Lik Purdue University, West Lafayette, IN-47907.

Inclusion of the nozzle geometry with a turbulent inflow boundary condition is essential for realistic jet noise simulations. In the current study, a digital filter-based turbulent inflow condition, extended in a new way to non-uniform curvilinear grids, is implemented to achieve this. The proposed method has several key features desirable for jet noise simulations, with some limitations. To validate the method, a quasi-incompressible zeropressure-gradient flat plate turbulent boundary layer is simulated at a high Reynolds number. The boundary layer produced by the current method is shown to agree reasonably well with theory and a recycling-based turbulence injection method. The length of the adjustment region necessary for synthetic inlet turbulence to recover from modeling errors is estimated. A low Reynolds number wall-resolved jet simulation including a round nozzle geometry is performed. The method is shown to be effective in producing sustained turbulence on a non-uniform, non-Cartesian grid at a barely turbulent Reynolds number. The effect of variation of the inlet integral length scales on the recovery of turbulent fluctuations is studied and recommendations are made for choosing these parameters. A possible spurious noise source is identified near the turbulent inlet for the current method. It is shown that this spurious noise does not affect the acoustic field outside of the jet significantly, though it is recommended to attenuate this noise artificially by using a sponge zone.

Nomenclature Roman symbols DJ DF F , G, H H ILSi J Lr Ma

Exit diameter of the nozzle Damping factor for the sponge zone Flux vectors in curvilinear coordinates Boundary layer shape factor Integral length scale in i direction Determinant of the Jacobian of the coordinate transformation Reference length Local Mach number

∗ Graduate

Research Assistant, School of Aeronautics and Astronautics, AIAA Student Member. Research Assistant, School of Aeronautics and Astronautics; Currently, Software Engineer, Intel Corp, 3600 Mission College Blvd, Santa Clara, CA 95054. ‡ Graduate Research Assistant, Department of Computer Science. § Professor, School of Aeronautics and Astronautics, AIAA Associate Fellow. ¶ Distinguished Professor and Chair, Aerospace Engineering Department, AIAA Associate Fellow. k Professor, Department of Computer Science. † Graduate

1 of 35 American Institute of Aeronautics and Astronautics

Copyright © 2014 by Nitin Dhamankar, Chandra Martha, Yingchong Situ, Kurt Aikens, Gregory Blaisdell, Anastasios Lyrintzis, and Zhiyuan Li.. Published by the American Institute of Aeronautics and Astronautics,

Downloaded by PURDUE UNIVERSITY on May 13, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2014-1401

Mr Ni p Q Q R RJ Ruù` Re St T Tr t UC Ue Ur uτ u, v, w x, y, z

Reference Mach number Number of grid points in i direction Static pressure Second invariant of the velocity gradient tensor Vector of conservative flow variables Reynolds stress tensor Exit radius of the nozzle Two-point correlation function for u` fluctuations Reynolds number Strouhal number Static temperature Reference static temperature Time Jet exit centerline velocity Velocity at the edge of the boundary layer Reference velocity Friction velocity Cartesian velocity components Cartesian coordinates

Greek symbols αf γ δ99 δdisp △i △t θ µr ξ, η, ζ Π ρ ρr

Spatial filtering parameter Ratio of specific heats for air Boundary layer thickness, distance from wall where velocity is 99% of freestream velocity Boundary layer displacement thickness Grid spacing in i direction Time step size Boundary layer momentum thickness Reference molecular viscosity Generalized curvilinear coordinates Wake parameter for turbulent mean velocity profiles Density Reference density

Accent marks (¯) (~)

Averaged quantity Vector quantity

Superscripts ( )∗ ( )+ ( )′

Dimensional quantity Non-dimensional quantity in terms of wall units Perturbation quantity

Subscripts ( )inlet ( )0

Quantity at the inlet boundary Ambient quantity

I.

Introduction

Strict noise regulations at major airports and increasing environmental concerns have made prediction and attenuation of jet noise an active research topic. Designing quieter nozzle geometries which do not incur a significant thrust penalty is a challenging task. Experimental studies have been fundamental to the development of jet noise reduction devices such as chevrons. Since the underlying mechanism of this noise generation is still not completely understood, the design process is largely based on empirical data. Very high costs are associated with the trial-and-error based experimental examination of novel design ideas. Computational prediction tools which can simulate this complex flow realistically, efficiently, and with proven



reliability can complement the experimental research to reduce this cost. Even though direct numerical simulation (DNS) is the most accurate computational fluid dynamics (CFD) approach based on solving the Navier-Stokes equations, DNS of high Reynolds number jet flows from complex nozzle geometries would require computational resources of an order that is impossible to achieve in the near future. DNS involves resolution of all the relevant length and time scales of a turbulent flow under study. The wide range of length and time scales present at higher Reynolds numbers makes DNS an impractical method to simulate jet engine noise. Compared to DNS, large eddy simulation (LES) has shown promise to be a feasible and accurate way of computationally simulating jet flows. In LES, only the energetic large scales of a turbulent flow are resolved and the smaller less energetic eddies are modeled. In the last two decades, LES has been increasingly applied in conjunction with computational aeroacoustics (CAA) formulations1 to predict the far-field noise of a jet. Some examples of such studies can be found in a review of the topic by Bodony and Lele.2 A decade ago, it was the usual practice in such simulations to exclude the nozzle geometry from the computational domain in order to avoid the associated high computational cost. Inclusion of a nozzle requires proper resolution of the boundary layer inside it and such a wall-resolved simulation was considered prohibitively expensive. Instead, shear layers were introduced at the domain inlet (assumed to be downstream of the nozzle exit plane) and they were triggered by means of some artificial excitation to give turbulent shear layers downstream. As opposed to this “simulated” phenomenon, in high Reynolds number experiments on jet noise, the turbulent boundary layer on the inner surface of the nozzle evolves into a fully turbulent shear layer. This discrepancy in modeling the inlet conditions has been one of the major reasons preventing a reasonable comparison of the computed noise levels with experimental findings. Also, Uzun and Hussaini3 suggest that the high frequency noise sources located within a few nozzle diameters of the nozzle exit are absent in simulations which exclude the nozzle geometry. With continuing advances in computational technology and with the advent of petascale computer clusters, inclusion of the nozzle geometry has become feasible. Since it is essential for a realistic simulation, many of the studies in the last decade have been targeted towards this problem.3–11 Inclusion of the nozzle geometry itself requires a boundary condition for modeling the no-slip nature of the viscous walls. But this solves only a part of the problem. The crucial next step involves triggering the boundary layer on the inner nozzle wall into a turbulent state, which is present in a physical experiment. A turbulent boundary layer is important to give rise to a turbulent free shear layer, eventual jet breakup and the associated aerodynamic noise. In a computational domain, obtaining natural transition of a boundary layer from laminar to turbulent nature is prohibitively expensive at high Reynolds numbers.12 This makes a turbulent inflow condition necessary so that the boundary layer is fully turbulent, preferably from the nozzle inlet itself. Considering the high cost associated with jet noise simulations, it is necessary to seek efficient and effective ways to achieve this. The current study focuses on a particular class of turbulence generators, which is based on digital filters. By extending this class to curvilinear coordinates in an inexpensive and easy-to-implement way, it is aimed to explore its potential for jet noise studies. This paper is organized as follows. Section II reviews several turbulent inflow generators from the viewpoint of jet noise simulations. Section III gives a quick overview of the governing equations and the numerical methods used in the current study. Section IV describes the digital filter-based turbulent inflow and its proposed extension to curvilinear coordinates. Sections V and VI discuss the setup, results and observations for the test cases. Finally, concluding remarks are given in section VII.

II.

Review of Turbulent Inflow Methods

Due to increasing application of LES to a variety of flows, specifying unsteady turbulent inflow conditions has developed into an active area of research. Different ideas for turbulence injection with varying degree of accuracy and complexity have been proposed over the last 15 years. Some of them have also been applied in jet noise studies. Before reviewing some of these methods, it is important to understand the key features expected from a turbulence injection method when being applied to simulate jet noise. These can be given as follows: 1. A jet noise simulation with a nozzle usually involves a large number of time iterations. The time step is restricted due to the small grid spacings inevitable to resolve the boundary layer. Also, the flow has to be simulated for a long time to be able to gather statistics and acoustics data samples over an adequately long period. The turbulence injection method therefore has to be capable of producing instantaneous inlet conditions for a large number of iterations (on the order of a million). Certain 3 of 35 American Institute of Aeronautics and Astronautics

turbulence injection methods involve creating a three-dimensional turbulent field and then extracting slices from that field to be used at the inlet (using Taylor’s hypothesis). Such methods are impractical for jet noise simulations. Storing a dataset containing ≈ 105 − 106 inlet slices requires an unreasonable amount of data storage. Also, the added cost of file reading operations during the simulation is a serious drawback. Therefore, the method should be able to produce the instantaneous inlet turbulent fields on the fly.


2. The turbulence injection method should have a fairly low computational cost. In a large-scale jet noise simulation on a petascale platform, an expensive inlet boundary condition can result in a performance bottleneck. Usually, only a small proportion of the total processor cores are assigned to the inlet boundary. These processors should not spend an unreasonably long time for computing the inlet turbulent fields. 3. The method should not induce any artificial periodic behavior in the flow-field. Such artificial periodicity can be inherent in a method itself or can also result from technical aspects such as a low-period pseudorandom number generator. Jet noise study involves analysis of a wide range of frequencies, from low frequency noise in the low observer angles to the high frequency noise that propagates along the sideline directions. Any cyclic behavior introduced in the flow field will, therefore, have an adverse impact on the noise prediction. 4. The method should be applicable and effective in curvilinear coordinate systems to handle complex nozzle geometries. 5. Finally, the implicit requirement is the ability to produce sustained turbulence. The turbulence injected at the nozzle inlet should be sustained and ultimately result in physically realistic fluctuations in the free shear layer. For this, the generated turbulent boundary layer should match reasonably well with its physical counterpart under similar flow conditions. With the above requirements in mind, several approaches of turbulence injection are now reviewed. The simplest approach involves superimposing completely random (in space and time) fluctuations over a mean turbulent velocity profile at the inlet. The random fluctuations are limited in amplitude to appropriate values and can also be scaled to satisfy a given Reynolds stress tensor. Even though it is simple and cost-effective, such a method usually fails at producing sustained turbulence. The energy in such a random turbulent field is equally distributed over the entire wavenumber range. This disagrees with the physical turbulence in which the low wavenumber region (or the large scales of turbulence) has the higher energy content.13 The turbulence produced by this method gets dampened in a short distance away from the inlet plane due to this relative lack of energy in the large scales.14, 15 Despite this drawback, variations of this method have been applied in jet noise studies. Bogey, Marsden and Bailly9, 10 have used random divergence-free velocity fluctuations and Bogey and Bailly8 have used random pressure fluctuations to trip the boundary layer inside the nozzle. The random fluctuations method was modified by Lee et al.16 Using a Fourier transform-based approach, they were able to generate three-dimensional velocity fluctuations that satisfied a prescribed energy spectra. Although this method represents an improvement over the random fluctuations, various drawbacks associated with it have been pointed out by Klein et al.15 which also mark it as unsuitable for application in jet noise simulations. Some of the drawbacks include the inherent periodicity of the computed fluctuations, the requirement of a full three-dimensional reference energy spectra and the applicability to only uniform Cartesian grids. A significant advancement in the field of turbulence injection was the recycling-based approach by Lund et al.12 They used an auxiliary simulation which produced its own inflow conditions by rescaling the velocity field from a downstream location (based on the boundary layer scaling laws) and re-introducing it at the inflow. Instantaneous planes of velocity were extracted from this auxiliary simulation to be used at the inlet of the actual simulation. The boundary layer produced by it was shown to have the canonical features of a physical turbulent boundary layer. It also did not require a redevelopment region near the inlet, typical of some synthetic turbulence generators to be discussed later. The necessity of an auxiliary simulation adds to the cost of the method, which is sometimes reduced by including the “recycling box” in the actual computational domain. Even though the original formulation was only applicable to incompressible flows, it has been extended to compressible flows by Sagaut et al.17 using various rescaling methods for



thermodynamic variables. Such an approach has been used by Uzun and Hussaini3, 7 in jet noise studies. They also modified it for simulating a nozzle including chevrons11 with encouraging results. Even though this approach has shown good potential for use in jet noise simulations, it suffers from two inherent drawbacks. Based on the distance between the recycling station and the inlet plane, spurious low frequency components are introduced in the flow-field.11, 18–20 This hinders its application in jet noise simulations with complete confidence. A usual recommendation to alleviate this problem is to have a longer separation between the inlet plane and the recycling station.11, 19 However, this offsets the advantage offered by zero redevelopment region length. Morgan et al.19 have shown that certain improvements to the traditional recycling-based method can eliminate the spurious low frequency modes with a relatively smaller recycling length, although at the cost of added complexity. The second drawback is the necessity of an equilibrium region in which the scaling arguments can be applied.20 Such a region may not exist at all in certain nozzle geometries (a simple converging nozzle is an example). An additional region upstream of the actual inlet is then required where the scaling laws are valid. For example, Uzun and Hussaini11 used a long pipe (length = 4.8 × nozzle radius) upstream of the nozzle inlet. The high cost and possible contamination of the solution with low frequency modes associated with these drawbacks make search for a better alternative necessary. In the last decade, several turbulence generators have been proposed which are not based on the recycling approach. Generally categorized as “synthetic turbulence generators”, these methods focus on generating inlet fluctuations that satisfy prescribed length scales and Reynolds stress tensor components through certain processing of random numbers. Though successful in imposing the desired length scales and fluctuation levels, certain terms in the Reynolds-stress budgets (such as the pressure-strain term) obtained from such synthetic turbulence usually deviate from those present in physical turbulence.20 As explained by Keating et al.,20 this results in rapid decay of wall-normal fluctuations and gradual decay of Reynolds shear stress and streamwise fluctuations as well. However, at some distance downstream of the inlet plane, the eddies recover into physically realistic structures. The turbulent fluctuations are reestablished and the flow resembles physical turbulence. Thus, these methods require a redevelopment or adjustment region near inlet which converts into additional computational cost. Considering that comparable costs are also associated with the recycling-based approaches, it is worth assessing their potential for jet noise studies. Therefore, the following discussion reviews some synthetic turbulence generators. One such method was developed by Sandham et al.21 and extended to higher Mach numbers by Li.22 In this approach, specific inner and outer layer disturbances with associated phase information are introduced in the boundary layer. The modes are designed to emulate the physical features such as the inner layer nearwall streaks and their lift up.18 This method could not impose desired Reynolds stress tensor components and was found to generate a long transient.18 One of the few turbulence generators that can readily handle curvilinear geometries was proposed by Kempf et al.14 This method can produce a synthetic turbulent field featuring the required length scales by diffusing a random signal. As the diffusion process occurs in the actual physical space, it can deal with any curvilinear geometry. By properly adjusting the diffusion coefficient, it is possible to generate a field that has the desired smooth variation of the integral length scales. Despite its clear advantages, it suffers from certain shortcomings. The original method14 is based on generation of a three-dimensional turbulent field and then extracting slices from it for use in an actual simulation. This is impractical for reasons discussed earlier (item 1 on page 4). Kempf et al.14 suggest generating a finite number of slices and then convecting through them periodically. This could cause low frequency spurious modes similar to the recycling-based approach. It might be possible to adapt this method to generate instantaneous inlet planes on the fly. The approach proposed by Xie and Castro23 for another class of synthetic generators could be applied for this. Even then, the requirement of solving a two-dimensional diffusion equation for every time step of the actual simulation can turn into a serious performance bottleneck for the entire application. Another novel way of prescribing length scales in random fluctuations was proposed by Klein et al.15 Using mathematical properties of filtering, they designed digital filters capable of establishing a Gaussian two-point correlation function in random fluctuations. This filtered random signal features the required spatial length scales and can be further transformed to conform to a prescribed Reynolds stress tensor. The original method required expensive three-dimensional filtering and storage of the inlet slices. This drawback was eliminated by a modification proposed by Xie and Castro.23 The improved method requires only a two-dimensional filtering operation (which is fairly cost-effective) and can generate inlet planes on the fly. This approach is applicable to uniform Cartesian inlet planes only and interpolation based extensions have been proposed for application to curvilinear grids.23 The boundary layer generated by such a method has



been shown to require an adjustment region length on the order of 20 boundary layer thicknesses.18 Jarrin et al.24 proposed the synthetic eddy method (SEM) that introduces artificial eddies through the inlet plane. Each eddy is represented by a specific shape function which describes its spatial and temporal characteristics. The method is capable of prescribing specified length and time scales and a given Reynolds stress tensor. Although the original method requires an adjustment region, its length can be shortened to a few boundary layer thicknesses by the modifications proposed by Pamies et al.,25 and by Adamian and Travin.26 These modifications involve precise specifications for the shapes of the turbulent structures in different regions of the boundary layer, increasing the complexity of the method. Coupled with a good quality pseudorandom number generator, the synthetic turbulence methods discussed above are assured of not introducing any artificial periodicity in the flow-field. This is one distinct advantage of the synthetic turbulence generators over the traditional recycling-based approach. The computational costs of the two approaches can be considered equivalent since both require a certain additional domain length for different reasons. If a synthetic turbulence generator can be applied easily in curvilinear grids, it can be considered a good choice for jet noise simulations. One argument in favor of recycling-based methods can be that the boundary layer produced by them agrees very well with a physical turbulent boundary layer and this good agreement may not be achieved by a “synthetic” approach. The degree of realism required from the boundary layer inside a nozzle for a jet noise simulation can be a debatable topic. In most of the experimental studies on jet noise, the boundary layer or the turbulent conditions inside the nozzle are not analyzed in detail. Thus, there is a lack of reference physical values for matching the boundary layer inside the nozzle. Even if such data becomes available, simulating the thin boundary layers present in high Reynolds number experimental studies exactly is going to be prohibitively expensive for the near future. In view of the above discussion, it can be said that synthetic turbulence generators present a reasonable way of turbulence injection for jet noise simulations.

III.

Governing Equations and Numerical Methods

The governing Favre-filtered non-dimensionalized Navier-Stokes equations are solved in conservative form on generalized curvilinear coordinates employing a structured, uniform mesh in the transformed space. The non-dimensionalization is carried out in the following way: ρ=

ρ∗ u∗ p∗ t∗ x∗ , ui = i∗ , p = ∗ ∗2 , t = ∗ ∗ , xi = i∗ . ∗ ρr Ur ρr U r Lr /Ur Lr

In vector form, the governing equations can be written as 1 ∂Q ∂ Gi − Gv ∂ Hi − Hv ∂ Fi − Fv + + = 0, + J ∂t ∂ξ J ∂η J ∂ζ J

(1)

(2)

where t is time; ξ, η, and ζ are the generalized curvilinear coordinates of the computational space, and J = Det [ ∂ ( ξ, η, ζ ) / ∂ ( x, y, z ) ] is the determinant of the Jacobian of the coordinate transformation from Cartesian space to the transformed space. Q is the vector of conservative flow variables; F , G and H represent the flux vectors in generalized coordinates with subscripts i and v indicating the inviscid and viscous fluxes respectively. The implicit LES approach is used in the current study and no explicit subgrid scale modeling terms are employed. The dissipation provided by the grid and the numerical method is assumed to emulate the physics of the subgrid scales. The spatial derivatives at interior points of the computational grid are computed using the non-dissipative sixth-order compact differencing scheme of Lele.27 At near-boundary points, this sixth-order scheme cannot be used since the differencing stencil extends outside the computational domain. Hence, on the boundary points, a third-order one-sided compact scheme is applied. On points next to the boundaries, a fourth-order compact differencing scheme is used. Spatial filtering is used to suppress the high wavenumber numerical instabilities that can be caused by discrete treatment of the boundary conditions, unresolved scales and mesh non-uniformities.28 A sixthorder tri-diagonal filter by Visbal and Gaitonde29 is used at the interior points. At near-boundary points, a different formulation is necessary since the filter stencil extends outside the computational domain. Hence, at grid points one and two grid spacings away from the boundaries, sixth-order one-sided filters by Gaitonde and Visbal28 are used. The boundary points are left unfiltered. The spatial filtering operation is carried out in the uniformly spaced computational domain. The conservative flow variables are filtered in all spatial 6 of 35 American Institute of Aeronautics and Astronautics


directions at the end of every time step. Every time step, the sequence of directions in which the filtering is performed is changed in an alternating manner. This is done in order to eliminate any biasing effects.30 The filtering parameter αf is set to 0.47 in all the test cases performed in this study. Both the spatial differencing and spatial filtering operations require solution of large tri-diagonal systems of equations. The current implementation uses an efficient hybrid parallel solver to achieve this. This solver uses the truncated SPIKE algorithm by Polizzi and Sameh31 for diagonally dominant narrow-banded systems. Details of the implementation of this solver can be found in refs. 32, 33. In test cases involving cylindrical grids, a centerline treatment consisting of the method proposed by Mohseni and Colonius34 for the radial direction (to avoid the centerline singularity) and the method proposed by Bogey et al.35 for the azimuthal direction (to avoid the time-step restriction due to very fine azimuthal spacing near the centerline) is used. For multi-block topologies, a superblock-based approach by Martha et al.33 is used. Time advancement is performed using a standard fourth-order explicit Runge-Kutta scheme. More details about the current implementation of LES and CAA methodologies can be found in refs. 36, 37.

IV.

Current Digital Filter-based Turbulent Inflow Generation Method

The digital filter-based synthetic turbulence injection method was initially proposed by Klein et al.15 Since then, various modifications have been suggested to the original method. Some of these modifications increase the computational efficiency23, 38 and some change the physical modeling aspects, such as modeling of the two-point correlation function.23, 39 In the current work, the approach suggested by Xie and Castro23 is used for generating correlated random fluctuations on uniform Cartesian meshes. This approach avoids the three-dimensional filtering used in the original method, is computationally efficient, and allows generation of the inlet conditions on the fly. In the current study, a new approach is proposed to extend the usability to non-uniform curvilinear grids. Finally, the suggestions of Touber and Sandham18 are used to specify the inlet conditions for thermodynamic variables. IV.A.

Procedure for Generating Velocity Fluctuations on Uniform Cartesian Grids

The simple case of a two-dimensional domain is considered first. In this, the inlet boundary is a onedimensional line. This line is uniformly divided in N − 1 intervals with the uniform spacing being h. Each of the N points on this line is assigned a random value, r. Assume that r represents the initial value for the fluctuation in a particular velocity component u. This set of random numbers must satisfy the following properties: !, N X rk = N = 0, (3) rk k=1

rk rk =

N X

rk2

k=1

rk rk+p =

N X

!,

rk rk+p

k=1

N = 1,

(4)

!,

(5)

N = 0, for p 6= 0.

Thus, the random values must be picked from a set of normally distributed random numbers having zero mean and unit variance. Eq. (5) specifies that these are uncorrelated random numbers, which are referred to as “white noise” from now on. The filtering operation creates correlated length scales in this white noise. A one-dimensional filter function is written as follows: u `k =

W X

ba rk+a ,

(6)

a = −W

where ba are the filter coefficients and W is the filter stencil half-width which is related to the integral length `k = 0. scale to be prescribed. u` represents the filtered velocity fluctuations. It follows from Eq. (3) that u


Also, using Eqs. (4) and (5), the following relationships can be deduced: W X

u `k u`k =

b2a ,

(7)

ba ba−p .

(8)

a = −W W X

u `k u `k+p =

a = −W +p


The two-point correlation function for this velocity component is given by  , ! W W X X u `k u`k+p 2 Ruù` (ph) = = ba , ba ba−p  u `k u `k a = −W a = −W +p

(9)

where ph signifies the separation in terms of p intervals of the uniform spacing h. This filter has to be designed in such a way that the correlation established by Eq. (9) matches in form with the correlation observed in actual physical flows. Klein et al.15 assumed a Gaussian function for the two-point correlation. Based on experimental observations, Xie and Castro23 suggested that an exponential function form is more appropriate for turbulent shear flows. Xie and Castro23 state this correlation function as πr , (10) Ruù` (r) = exp − 2 ILS in which ILS is the integral length scale and r is the separation variable. Referring to the uniformly spaced one-dimensional inlet boundary, if ILS ≈ nh, the required correlation function can be written as π|p| Ruù` (ph) = exp − , (11) 2n

where the absolute value of p signifies the symmetric nature of the correlation function. The filter coefficients should be designed in such a way that the two-point correlation established by Eq. (9) matches with the expected behavior given by Eq. (11). This means that the following equation must be satisfied:  , ! W W X X π |p| 2  . (12) ba ba−p  ba = exp − 2n a = −W +p

a = −W

Analytical solution of this problem is not straightforward and certain approximations are necessary. It `k u `k = 1. Xie is necessary in a later step that the variance of the filtered fluctuations should be unity or u and Castro23 approximated a solution which satisfies this property by ba = ˜ba

,

W X

m = −W

˜b 2 m

!1/2

, where ˜bm ≃ exp

−π |m| n

.

(13)

The above expression is only approximately valid in the range n = 2, . . . , 200. The filter width has to be large enough to satisfy W ≥ 2 n for this approximation to hold.23 Thus, following the filtering operation, a one-dimensional inlet data with zero mean (` uk = 0), unit variance (` uk u `k = 1), and a prescribed integral length scale (ILS) is generated. It is to be noted that to be able to apply the filter function at the boundaries of this inlet line, the random number dataset should be larger than the number of points on the inlet line. In particular, for a filter half-width of W and an inlet boundary with N points, random numbers rk from k = 1 − W to k = N + W are needed. In three-dimensional domains, the inlet plane is two-dimensional. Consider such an inlet in the y − z plane. Suppose that it is uniformly discretized by Ny and Nz points in the y and z directions, respectively. Following Xie and Castro,23 a two-dimensional filter is obtained by convolution of two one-dimensional filters as follows: bc,d = bc bd , (14)


where bc refers to the filter coefficient in y direction and bd refers to the filter coefficient in z direction. Suppose Wy and Wz are the required filter half-widths in y and z directions respectively. Then, the filtering operation is written as follows: Wy Wz X X u `j,k = bc,d rj+c,k+d . (15)


c = −Wy d = −Wz

This filter is applied to a two-dimensional array of normally distributed random numbers. Again, to be able to apply the filter at the boundaries, the size of the random number array should be larger than the extents of the inlet plane. Particularly, it should be [1 − Wy : Ny + Wy , 1 − Wz : Nz + Wz ] in size. As with the one-dimensional case, the filter stencil half-widths should be chosen so as to satisfy the requirements of Wy ≥ 2 ny and Wz ≥ 2 nz . Here, nα corresponds to the number of grid intervals per integral length scale in the α direction. Such two-dimensional correlated random data is generated every time step. Three different sets are necessary for three different velocity components. Thus, a large number of normally distributed random numbers are required to be generated. In the current implementation, the Mersenne Twister pseudorandom number generator40 is used to generate these uniformly distributed random numbers. This generator has a period of 2 19,937 − 1. With such a long period, the inflow conditions are assured of not introducing any artificial periodicity in the flow-field. From this uniformly distributed set, normally distributed random numbers are calculated using the Box-Muller transform.41 Suppose U1 and U2 are two independent random numbers uniformly distributed in the interval (0, 1]. Then two independent random variables Z0 and Z1 , which are normally distributed, can be obtained using the Box-Muller transform as p p Z0 = −2 ln U1 cos(2πU2 ), Z1 = −2 ln U1 sin(2πU2 ) . (16)

To establish the streamwise correlation, the filtered random inlet data at an instant must be correlated to the previous inlet fluctuations (except on the very first time step of a simulation). This avoids the t computationally expensive three-dimensional filtering originally suggested by Klein et al.15 Suppose u`j,k are t−△t the two-dimensional filtered random fluctuations at the current time step. Let u˘j,k be the fluctuations from the previous time step. Then for the current time step, fluctuations u ˘tj,k are given by Xie and Castro23 as 1/2 π△t π△t t−△t t t +u `j,k 1 − exp − . (17) u ˘j,k = u ˘j,k exp − 2T T

In Eq. (17), △t is the computational time step and T refers to the Lagrangian time scale which is calculated as ILSx T = , (18) u in which ILSx is the integral length scale in the streamwise direction and u is the prescribed mean inlet velocity profile. Finally, before the correlated random fluctuations u ˘j,k are introduced at the inlet, they are passed through a transformation to ensure that the velocity fluctuations at the inlet will satisfy a prescribed Reynolds stress tensor. This transformation is based on the Cholesky decomposition of the Reynolds stress tensor and was originally proposed by Lund et al.12 Henceforth, this transformation is referred to as “Lund’s transformation”. Suppose u ˘j,k , v˘j,k and w ˘j,k represent the three independent sets of fluctuations in x, y and z velocities respectively, generated through the procedure described so far. Each set individually features the correlated length scales due to the two-dimensional filtering and also the streamwise correlation established by Eq. (17). Each of these sets satisfies the conditions of zero mean, unit variance and zero covariance with the other two sets. Let R represent the Reynolds stress tensor to be prescribed. Components of R are given by Rlm = u′l u′m where u′l indicates fluctuation in the l component of velocity. Then, R is the correlation ′ ′ matrix for the final velocity fluctuations u′j,k , vj,k and wj,k . These final fluctuations are calculated as 

  u′j,k a11  ′   =  vj,k  a21 ′ a31 wj,k

0 a22 a32

  u ˘j,k 0   0   v˘j,k  . a33 w ˘j,k


(19)

In Eq. (19), the matrix on the right hand side is called the “amplitude tensor” and is given by the Cholesky decomposition of the Reynolds stress tensor R. The components of this amplitude tensor are as follows:12 p a11 = R11 , (20) a21 = R21 / a11 , (21) q a22 = R22 − a221 , (22) a31 = R31 / a11 ,

(23)

a32 = ( R32 − a21 a31 ) / a22 , q a33 = R33 − a231 − a232 .

(24) (25)

uj,k = uj,k + u′j,k ,

(26)


Thus, the velocity fluctuations are calculated every time step and are added to the prescribed mean inlet velocity profile to get the instantaneous inlet velocity field, i.e.,

vj,k = wj,k =

′ v j,k + vj,k , ′ wj,k + wj,k .

(27) (28)

In the current implementation, the inlet conditions are generated at the start of each time step and then imposed during the sub-steps of the Runge-Kutta time advancement scheme. It is to be noted that Lund’s transformation is meant to be used on a set of uncorrelated random variables, which feature the correlation given by the Reynolds stress tensor after the transformation. In the digital filter-based methods, the fluctuations which are passed through Lund’s transformation already feature the correlated length scales established by filtering. Fathali et al.42 have shown that this decoupled approach of first creating length scales in individual velocity components and then transforming these fluctuations to prescribe a given Reynolds stress tensor can result in deviation of the actual length scales from the prescribed ones. Thus, the length scales can only be approximately imposed in the current approach. IV.B.

Extension to Non-Uniform Curvilinear Grids

The filtering operation is limited to uniform Cartesian grids and it cannot be readily applied to inlet planes in curvilinear geometries. In jet noise simulations, the ability to handle typical nozzle inlet shapes is desired. Another shortcoming of the filter-based techniques is the limitation of using only a single integral length scale in a particular direction, on the uniformly-spaced Cartesian inlet mesh. This is an issue mainly in wall-bounded flows where the integral length scale is supposed to reduce to zero towards the wall. For curvilinear grids, the usually recommended procedure is to have a uniformly spaced Cartesian grid, larger than or equal to the actual curvilinear inlet plane in size. The filtering operation is performed on the uniform grid and then interpolation is used to get the velocity fluctuations at the actual inlet. Such a procedure has been recommended by Rana et al.13 and by Xie and Castro.23 To vary the integral length scales over the inlet plane, one approach is to vary the filter coefficients spatially. But Klein et al.15 state that a strong variation through change of the filter coefficients results in deviation of the correlation function from its prescribed shape. Veloudis et al.43 proposed a zonal approach to allow integral length scale variation. In this method, the inlet plane is split into several zones which themselves are uniformly spaced Cartesian meshes. The filtering operation is applied in each zone separately with separate filter coefficients, thus resulting in different integral length scales. A stepwise variation in length scales is possible with this approach. But it suffers from discontinuities in the fluctuations at zonal boundaries. Also, in curvilinear geometries, the two-step procedure of producing fluctuations on several uniform zones and then interpolating the fluctuations to the actual inlet is both complicated and computationally expensive. In certain curvilinear geometries, this zonal approach cannot be readily applied to have length scale variation over the inlet plane. For example, in a pipe flow simulation, the length scales are required to decrease radially towards the the wall. A Cartesian zone-based approach cannot handle such requirements. In the current work, a new way of introducing the digital filter-based fluctuations in curvilinear geometries is suggested. This approach does not involve the excessive computational cost of interpolation and allows smoother variation in length scales, albeit with some restrictions. When the governing equations 10 of 35 American Institute of Aeronautics and Astronautics


are solved on a structured curvilinear geometry, the physical mesh is mapped onto a uniformly spaced Cartesian computational mesh. The filtering operation can be performed on this mapped mesh which is by definition uniformly-spaced. Parallelizing this operation is relatively easy compared to the implementation of interpolation-based methods. Once the specified length scales are established on the mapped grid, the velocity fluctuations are simply mapped back onto the physical grid. Digital filtering is simply a mathematical operation which is used to generate three individual sets of correlated random numbers on the mapped grid. These three sets are considered as the filtered correlated random fluctuations in the u, v and w velocity components at the physical inlet plane. The orientation of the inlet plane and its curvilinear nature come into the picture only when Lund’s transformation is to be applied, to transform the three filtered sets into the final velocity fluctuations; which satisfy a prescribed Reynolds stress tensor. At this stage, the correct Cartesian stress tensor components at each point on the inlet must be used. If the curvilinear stress tensor components are known, they need to be transformed into their Cartesian counterparts necessary for Lund’s transformation. A change-of-basis transformation for second-order tensors44 can be used to achieve this. The use of this approach has the following consequences: 1. Since filtering takes place on the mapped uniformly spaced Cartesian grid, the integral length scales are effectively specified in terms of grid intervals. 2. Since the integral length scales at the inlet plane are specified in terms of grid intervals, the length scales can be varied by means of grid stretching. For example, grids are stretched away from the wall and are relatively fine near the wall to capture the boundary layer in wall-resolved simulations. On such a grid, a specific number of grid intervals in the wall-normal direction will cover less physical distance near the wall (where the grid is fine) and cover larger distances away from the wall (where the grid is stretched). Thus, a smoother variation in length scales in a particular direction can be achieved by appropriately designing the grid. 3. On the downside, if the grid itself has any peculiarities, those will be reflected on the length scales as well. Again considering the pipe flow example, it is seen that grid stretching near the wall results in expected behavior of the radial length scales near the wall. But if the azimuthal grid spacing is constant, then the azimuthal length scale will not decrease towards the wall. Also, in a typical cylindrical grid, the azimuthal spacing gets smaller and smaller towards the center. Hence, if turbulent inflow over the entire inlet plane is desired, this approach will give non-physical length scales towards the center where radial grid spacing is the largest and azimuthal grid spacing is the smallest. In the current study, this method is being proposed for jet noise simulations. Inside the nozzle, the turbulence is expected to be the strongest in the boundary layer and it is a reasonable approximation to add fluctuations only in the boundary layer. Such approximations have been used in jet noise simulations by Uzun and Hussaini3, 7 and Bogey et al.8, 9 The proposed method is readily applicable to curvilinear grids for the most common nozzle inlet shape (cylindrical), if only the fluctuations inside the boundary layer are required. Therefore, this proposed method is used as a turbulent inflow condition only for the boundary layer at the inlet. In particular, the digital filtering operation is carried out only in a zone near the wall. The rest of the inflow plane only enforces the prescribed mean inlet flow quantities. The thickness of this zone corresponds to the 99 % boundary layer thickness. Reference Reynolds stress profiles and mean velocity profile for a turbulent boundary layer can be obtained from direct numerical simulation or experimental results available in the literature. Including the core turbulence outside of the boundary layer is a topic for future research. IV.C.

Treatment of Thermodynamic Variables

Once the velocity fluctuations at the inlet plane at an instant are known, only the thermodynamic variables (density, temperature, and pressure) remain to be specified. In the current study, the suggestions of Touber and Sandham18 are followed to specify these conditions. The strong Reynolds analogy (SRA) is applied using the previously determined streamwise velocity fluctuations. The relationship between the temperature fluctuations and the streamwise velocity fluctuations is given by T′ u′ u2 = −(γ − 1)Ma2 , with Ma2 = Mr2 , u T T


(29)

where T is the local mean temperature, Mr is the reference Mach number and Ma is the local mean Mach number based on the local speed of sound. In the SRA, pressure fluctuations are considered to be negligible compared to the density, velocity and temperature fluctuations. The density fluctuations are given by ρ′ T′ =− . ρ T

(30)


Using Eqs. (29) and (30), the density fluctuations can be calculated. The pressure is maintained at a constant value. A constant pressure inlet forms an acoustically reflective surface. As opposed to a characteristic-based non-reflecting laminar inflow boundary condition,45 this boundary will reflect any acoustic waves incident on it. This is an undesirable property for a boundary condition meant to be used in jet noise simulations. A better formulation for modeling the pressure fluctuations in the turbulent boundary layer at the inlet, while achieving non-reflectivity of the inflow plane is highly desirable.

V. V.A.

Test Cases

Zero-Pressure-Gradient Quasi-Incompressible Flat Plate Turbulent Boundary Layer

The simplicity of the grid, availability of semi-empirical theoretical results, and feasibility of performing high Reynolds number simulations make this test case an attractive choice for validating a turbulence injection method. Lund et al.12 simulated a spatially evolving incompressible flat plate boundary layer to validate the recycling-based turbulent inflow method. Similar tests have been performed with variations of the synthetic eddy method.25, 26 Touber and Sandham18 simulated a Mr = 2 and Reδ99,inlet = 2, 500 flat plate turbulent boundary layer using a digital filter-based turbulent inflow boundary condition and reported a redevelopment region length of ≈ 20 δ99,inlet based on skin friction evolution. Since the proposed method differs from the reference digital filter-based methods, it is necessary to assess its capability in producing a turbulent boundary layer. For this purpose, a spatially evolving quasi-incompressible turbulent boundary layer spanning a Reynolds number range of Reθ ≈ 1, 410 − 2, 200 (based on the velocity at the edge of the boundary layer Ue and the boundary layer momentum thickness θ) is simulated at a reference Mach number of Mr = 0.2. The term quasi-incompressible refers to the low Mr of the simulation, at which the compressibility effects are minimal. The current results can therefore be compared with the results presented by Lund et al.12 using the recycling-based method. The recycling-based approach is supposed to require no redevelopment or adjustment region and has been shown to result in turbulent boundary layers that follow the canonical behavior. It is used as a reference dataset to evaluate the performance of the current method. V.A.1.

Simulation details

A single superblock topology resembling a parallelepiped is used. The grid extents are Lx (streamwise) × Ly (wall-normal)× Lz (spanwise) = [ 50 × 15 × π/2 ] δ99,inlet . The number of grid points in the three directions are Nx × Ny × Nz = 1, 344 × 160 × 128. The corresponding domain extents used by Lund et al.12 were [ 24 × 3 × π/2 ] δ99,inlet. The reasons behind using relatively larger streamwise and wall-normal domain lengths in the current study will be clear in sections V.A.2 and VI, respectively. The mean turbulent inlet velocity profile is chosen to give Reθ,inlet ≈ 1, 410. The mean velocity shape is assumed to obey the universal turbulent velocity profile given by the law of wall. One representation of this law is given by Reichardt’s formula,46 which includes the viscous sublayer, the buffer region, and the log layer. It is given by + + , (31) u+ = 2.5 ln 1 + 0.4y + + 7.8 1 − e−y /11 − (y + /11) e−0.33 y

where u+ and y + are the non-dimensional velocity and wall-normal distances in wall units. The wake portion is not added to the inlet profile. Additional tests were performed in which the inlet profile included a wake portion, but it was not found to have a significant effect on the downstream flow evolution for this case. For the current inlet profile, Reδ99,inlet = 14, 927. A total of 83 grid points are used in the wall-normal direction to resolve the boundary layer at the inlet. All the distances reported in wall units are based on the friction velocity at the inlet (uτ,inlet). The minimum 12 of 35 American Institute of Aeronautics and Astronautics


wall-normal spacing at the inlet is 1.476 × 10−3 δ99,inlet , which corresponds to △y + of 1. The maximum wall-normal spacing in the boundary layer at the inlet is △y + = 17. In the streamwise and the spanwise directions, constant spacings of △x+ = 25.22 and △z + = 8.37 are used. In the simulation of Lund et + al.,12 the corresponding spacings were △x+ ≈ 64, △z + ≈ 15 and △ymin ≈ 1.2. Therefore, the current grid resolution is considered adequate. The inlet reference Reynolds stresses are taken from Spalart’s DNS results47 for a zero-pressure-gradient incompressible turbulent flat plate boundary layer at Reθ = 1, 410. Hutchins and Marusic48 have presented two-point correlations of the streamwise velocity fluctuations at different wall-normal locations inside ∗ ∗ turbulent boundary layers for Reτ = ρ∗r δ99 uτ /µ∗r values ranging from 1,120 to 19,960. For this wide range of Reynolds numbers, they observed a good collapse in the outer-scaled two-point correlations. The outer scaling corresponds to the use of outer variables Ue and δ99 to scale the velocity and distance respectively. For the current inlet conditions, the value of Reτ at the inlet is ≈ 677.43. Even though this value is lower, the two-point correlations for the higher Reτ values can still be used as a reference for estimating the length scales. This is justified, mainly because of the collapse observed by Hutchins and Marusic,48 but also because the length scales are imposed only approximately in the current method, as has been explained towards the end of section IV.A. Based on the correlation function formula used in the current method, the integral length scale corresponds to a value of ≈ 0.2078 of the two-point correlation (on substitution r = ILS in Eq. (10)). The integral length scales are estimated based on this value of the reference two-point correlations presented by Hutchins and Marusic.48 With Ruù` = 0.2078 as the definition for the integral length scale, the variation in the streamwise integral length scale at different locations inside the boundary layer is very small and is neglected. The streamwise integral length scale is taken to be ≈ 0.8 δ99,inlet . In the spanwise direction, the grid is uniform and only one length scale can be imposed. The spanwise length scale has been reported48 to increase linearly with y/δ99 and hence a value of ≈ 0.2 δ99,inlet that is observed at the middle of the boundary layer is chosen. No reference was found for the wall-normal length scale and it is set to be equal to the spanwise length scale at the middle of the boundary layer. The wall-normal length scale decreases towards the wall and increases towards the outer portion of the boundary layer due to grid stretching. Both the wall-normal and spanwise length scales are specified in terms of grid intervals in the current method. In some synthetic turbulence methods, more intense turbulence has been observed downstream of the inlet plane with an increase in the inlet integral length scales.14, 49 The inlet integral length scales can have an effect on the redevelopment region length. In order to determine whether the recovery region length can be reduced, this test case has been simulated with two different length scales in two separate runs. The first run uses the boundary layer reference length scales mentioned earlier. The second run uses length scales larger by a factor of 2 than the boundary layer reference length scales. These two cases are designated as FPBL-1x and FPBL-2x, respectively. Identical length scales are used for all three velocity components (u, v and w) in a particular run for a particular direction. A characteristic-based adiabatic viscous wall boundary condition50 is used on the wall at the y = 0 plane. A characteristic-based outflow condition45 is applied at the outflow plane. In the spanwise direction, periodicity is assumed. At the top face of the domain, the following stress-free boundary conditions12, 51 are applied: ∂w ∂u = = 0, p = p0 , ρ = ρ0 , (32) ∂y ∂y where p0 and ρ0 are the ambient pressure and density. The v velocity component is kept free to adjust to the solution. The mean inlet profiles for primitive variables are used as initial conditions throughout the domain. The solution is time-advanced for a total of 3 × 106 steps with △t = 2.2 × 10−4 δ99,inlet /Ue , giving a maximum Courant-Friedrichs-Lewy (CFL) number of ≈ 0.89. The total simulation time is 660 δ99,inlet/Ue and the initial 99 δ99,inlet/Ue time period is allowed for transients. The statistical averaging interval corresponds to ≈ 11 flow-through-cycles of the domain (≈ 561 δ99,inlet/Ue ). V.A.2.

Discussion of Results

Very long meandering positive and negative streamwise velocity fluctuations are known to exist in the log and lower wake regions of a turbulent boundary layer. These are referred to as “superstructures”48 and numerical, as well as experimental evidence of their existence has been presented by Hutchins and Marusic.48 Figure 1 shows the streamwise velocity contours at y = 9.61 × 10−2 δ99,inlet . This plane corresponds to △y + ≈ 65 13 of 35 American Institute of Aeronautics and Astronautics


based on the inlet mean velocity profile and falls in the log region over the entire domain. Several long meandering structures are present in both FPBL-1x and FPBL-2x. All the statistical quantities reported here are both time-averaged and spanwise-averaged. First, the evolution of several boundary layer properties is discussed. The reference results of Lund et al.12 using the recycling-resampling based turbulence injection method are designated as Lund-RR. The evolution is also compared with estimates using the momentum integral analysis, computed using a procedure analogous to that described by Lund et al.12 These estimates are computed assuming a velocity profile shape that has a wake portion,52 but keeping the Reθ,inlet value the same as the one used in the simulations. The complete profile is given by 2Π π + + + u+ = 2.5 ln 1 + 0.4y + + 7.8 1 − e−y /11 − (y + /11) e−0.33 y sin2 η , (33) κ 2

with η = y/δ (assuming δ = δ99 ), κ = 0.41, and Π = 0.5. This discrepancy in the velocity profile (between Eq. (31) and Eq. (33)) used at the inlet for the simulation and for the momentum integral estimates results in different absolute values for the displacement thickness (δdisp ), the momentum thickness (θ), the skin friction coefficient (Cf ), and the shape factor (H = δdisp /θ) for the two at the inlet. These inlet properties were different in the simulation of Lund et al.12 (they used Reθ,inlet = 1530), compared to the current simulations. Also, Lund et al.12 have computed the momentum integral estimates using a velocity profile containing only the log and the wake portion; and assuming δ (in the variable η for the wake term) to be the distance from the wall where the velocity equals the edge velocity Ue , and not 0.99 Ue as is done here. The absolute values of momentum integral estimates are sensitive to the velocity profile expression and the definition of δ used, and are therefore different here from those presented by Lund et al.12 (figure 3 provides one example). In consideration of these issues, the relevant comparison here is the slope of the evolution curves, and not the absolute values. This is applicable to all the evolution plots. Figure 2 shows the evolution of the momentum thickness (θ) with downstream distance. In both FPBL-1x and FPBL-2x, θ initially drops near the inlet where synthetic turbulence is being input. This corresponds to the initial adjustment region necessary for synthetic turbulent inflow methods. After a distance of about 20 δ99,inlet downstream, the growth rate of the momentum thickness recovers, though it is still lower than the growth rate observed in Lund-RR. The deviation after x ≈ 40 δ99,inlet is suspected to be due to the effect of the outflow boundary condition. A similar decrease in slope can also be observed for Lund-RR, though it is limited to a very short streamwise distance. Nearly identical behavior is observed in FPBL-1x and FPBL-2x, indicating that the larger integral length scales at the inlet were ineffective in reducing the redevelopment region length for this case. The streamwise distance lost due to the initial development region and the relatively slow growth rate of θ are the reasons behind using a larger streamwise domain length than what was used in Lund-RR. This allows for the comparison of the evolution with respect to Reθ over a larger interval. Figure 3 shows evolution of the skin friction coefficient with respect to Reθ . It goes through an adjustment region, but recovers by Reθ ≈ 1, 530 and the slope is maintained thereafter. The location where Reθ ≈ 1, 530 is at x ≈ 11.5 δ99,inlet . Therefore, based on the skin friction coefficient evolution, the recovery region length can be estimated to be ≈ 11.5 δ99,inlet . As mentioned earlier, Touber and Sandham18 have reported a redevelopment region length of ≈ 20 δ99,inlet based on the skin friction evolution for a particular implementation of the digital filter-based turbulent inflow. The evolution of the displacement thickness and the 99% boundary layer thickness with respect to Reθ are shown in figures 4 and 5, respectively. As the momentum thickness decreases initially in FPBL-1x and FPBL-2x, the evolution with Reθ exhibits an unusual behavior near the inlet. But after about 11.5 δ99,inlet (corresponding to Reθ ≈ 1, 530), the slopes are matched reasonably well with the momentum integral estimates. The evolution of the shape factor (H = δdisp /θ) is shown in figure 6. Contrary to the theoretically expected behavior, the shape factor increases in both FPBL-1x and FPBL-2x. The slower growth rate of the momentum thickness combined with the relatively correct growth rate of the displacement thickness is the reason behind this discrepancy. Still, it is to be noted that the increase is not large enough to indicate laminarization of the boundary layer. The value of the shape factor towards the exit is still close to the values expected from a turbulent boundary layer (∼ 1.3 − 1.4). A drawback of the current method is the shift in the values of boundary layer properties from what is specified at the inlet, due to the adjustment region. Even after the flow recovers, the integral thicknesses and Cf exhibit a shift from the theoretically expected values. This can be of concern in applications where exact values of these quantities are required to be imposed at a certain downstream location. But in jet noise simulations, this is not a serious drawback due to the lack of inner-nozzle reference flow parameters 14 of 35 American Institute of Aeronautics and Astronautics


and the prohibitive costs of exactly matching the experimental thin boundary layers, as discussed in section II. The mean streamwise velocity profiles at several downstream locations are now studied. Figure 7 plots the mean velocity profile at x = 11.5 δ99,inlet . At this location, Reθ values attained by the boundary layers in FPBL-1x and FPBL-2x are 1, 534 and 1, 528, respectively. This allows for comparison with the profile reported by Lund et al.12 at Reθ = 1, 530. The viscous sublayers obtained in FPBL-1x and FPBL-2x agree with Lund-RR. In the logarithmic region, the velocity is slightly over-predicted in FPBL-1x and FPBL2x. The profile shape is similar to what is observed in Lund-RR though, with none of the profiles exactly matching the log law. According to Lund et al.,12 an overprediction of velocity in the log region is an artifact of using finite-difference methods on relatively coarse meshes. Since the current grid is finer than what was used by Lund et al.,12 it is not clear why a relative overprediction should be observed. Similar behavior is observed in figure 8, which plots the mean velocity profiles at x = 39 δ99,inlet. At this location, Reθ values are 2, 054 and 2, 047 in FPBL-1x and FPBL-2x, respectively. These are compared with the profile reported by Lund et al.12 at Reθ = 2, 050. Again, a good agreement in the viscous sublayer and a slight overprediction in the log region are observed. The mean velocity profiles at four different streamwise locations are shown in figure 9 for the FPBL-1x run. Starting from x = 11.5 δ99,inlet , the velocity profiles remain identical in the viscous sublayer and in the log region. Thus, the mean velocity profiles reach a self-similar state (excluding the wake region) by x = 11.5 δ99,inlet , and there is no further adjustment. To examine whether the outer part of the velocity profile attains a self-similar state as well, the velocity defect (Ue − u)/uτ is plotted against the outer-scaled wall normal distance (y/δ99 ) in figure 10 for three downstream locations (from FPBL-1x ). The velocity defect law53 given by ( ) y π y 1 Ue − u 2 − ln + 2 Π 1 − sin , (34) = uτ κ δ99 2 δ99 is also plotted in the same figure (using κ = 0.41, and Π = 0.5). Focusing only on the wake region (beyond y/δ99 ≈ 0.2), it is clear that the outer velocity profile takes longer to attain self-similarity, and to match with the velocity defect law. The agreement is reasonably good by the station at x = 25 δ99,inlet . Next, root mean square (rms) velocity fluctuations and Reynolds shear stress profiles at two downstream locations are discussed. The Reynolds stresses are expected to recuperate only after the eddies have redeveloped into physically realistic structures some distance downstream of the inlet plane. This analysis will help estimate the distance required for complete recovery of the stresses. The first location is at x = 11.5 δ99,inlet, corresponding to Reθ ≈ 1, 530. The rms velocity fluctuations and the Reynolds stresses at this location are compared with those obtained in Lund-RR at the same Reθ value in figure 11. Very good agreement is observed for the peak value and the profile shape for all the rms velocity fluctuations and the Reynolds shear stress. This indicates that all the stresses have already recovered by this location and they match with what a recycling-based approach would give at the same Reθ value. Similar behavior is observed at x = 39 δ99,inlet, corresponding to Reθ ≈ 2, 050 shown in figure 12. Except the slightly slower decay of turbulence in the outer part of the boundary layer, the profiles match very well with those of Lund-RR. To summarize, the turbulent boundary layer developed by the current method requires an adjustment region near the inlet. Based on the growth rate of the momentum thickness, this region can be estimated to be about 20 boundary layer thicknesses long. Based on the evolution of Cf , δdisp , and δ99 ; establishment of a self-similar mean streamwise velocity profile in the viscous sublayer and the log layer; and the recovery of the Reynolds stresses; the adjustment region can be estimated to be about 11.5 boundary layer thicknesses long. In jet noise simulations, a quick recovery of turbulent fluctuations is arguably the most important property required from the inflow generator. Considering that the evolution properties are reasonable after 11.5 boundary layer thicknesses, it is recommended to allow at least a domain length of 11.5 δ99,inlet downstream of the nozzle inlet for high Reθ simulations. Imposing larger inlet integral length scales is not found to have any significant impact on the redevelopment region length or the recovery of Reynolds stresses in this high Reynolds number case. V.B.

Jet Flow From a Straight Round Nozzle

It is important to ensure that the turbulence generated by the current method is sustained inside a nozzle for successful application in jet noise simulations. As the most frequently encountered nozzle inlet shape is circular, a simple cylindrical round nozzle geometry is chosen for this test. It is referred to as a “straight 15 of 35 American Institute of Aeronautics and Astronautics

nozzle” indicating the absence of any converging/diverging section. Simulating the thin boundary layer inside a nozzle is prohibitively expensive and therefore the Reθ of the boundary layer is often limited to very low values in jet noise studies.3, 7 In the current test, a value of Reθ,inlet = 300 is used. According to Preston,54 the lowest Reθ value at which fully developed turbulence can occur is 320. It is based on the argument that at Reθ < 320, the inner and outer regions of the velocity profile overlap and there is no log layer. But according to Spalart,47 even though the log layer is not observed at low Reθ values (285-300), the turbulence is still sustained in the absence of a stabilizing effect. This stabilizing effect is commonly due to a favorable pressure gradient. In a straight nozzle simulation, there is only a very weak favorable pressure gradient inside the nozzle. Hence, it is considered appropriate to simulate this problem at Reθ = 300 and expect sustained turbulence inside the nozzle. However, it should be noted that due to the lower value of Reθ , the turbulence can not be expected to agree with the canonical behavior of turbulent boundary layers. Matching the boundary layer inside the nozzle with a fully developed turbulent boundary layer that may exist in an experimental setup is not targeted here.


V.B.1.

Simulation details

This test case is simulated at a Reynolds number of ReRJ = 50, 000 (based on the nozzle radius RJ and the jet centerline velocity UC ) and a reference Mach number of Mr = 0.9. A five-superblock topology is used for the grid. Specifications of the grid are given in table 1. This topology is representative of the typical topologies that will be used in full-scale jet noise simulations. However, the domain size is limited since the main focus is on studying the boundary layer inside the nozzle. The nozzle length is twice the nozzle radius (2RJ ). The nozzle inlet is at x = −2RJ and the nozzle exit is at x = 0. The domain length downstream of the nozzle is 10RJ . The grid is fine enough inside the nozzle to resolve the boundary layer. The grid in the downstream superblocks is not designed to resolve the free shear layer downstream of the nozzle exit very well. On the contrary, grid stretching is applied in the downstream superblocks to damp much of the vorticity before it reaches the outflow. The non-nozzle superblocks are included only to qualitatively assess the ability of the current method to give a turbulent free shear layer. A three-dimensional view of the topology, cut in half along the z = 0 plane, is presented in figure 13. A plane of the mesh from the nozzle superblock is presented in figure 14 in which every fifth point is shown to retain clarity. The entire nozzle grid is obtained by revolving this plane through 360◦ . Table 1: Grid details for simulations of jet flow through a straight nozzle Superblock 1 2 3 4 5

Description Straight Nozzle Downstream of nozzle exit Downstream of nozzle lip Outside the nozzle Outermost downstream

Nx × Nr × Nθ 608 × 128 × 1024 96 × 128 × 1024 96 × 32 × 1024 64 × 96 × 1024 96 × 96 × 1024

Approx. total grid points (×106 ) 79.7 12.58 3.15 6.29 9.44

The mean turbulent inlet velocity profile (given by Eq. 31) is chosen to give the Reynolds number based on the momentum thickness of the inlet boundary layer of Reθ,inlet ≈ 300. To achieve this, the value of the boundary layer thickness is set to δ99,inlet = 5.86×10−2RJ . The mean inlet velocity profile has a shape factor of 1.77, which is higher than the values expected from fully turbulent boundary layers (H ≈ 1.3 to 1.4). In Spalart’s DNS47 of a turbulent boundary layer at Reθ = 300, the shape factor was found to be 1.67. The higher H values are due to the low Reθ of the boundary layer. The mean velocities in y and z directions are set to zero at the inlet. The mean inlet pressure is kept constant at the reference ambient value (p0 ). The mean density profile is based on the following Crocco-Buseman relation for isothermal jets:55 −1 u γ−1 2 u ρ 1− = 1+ Mr , (35) ρ0 2 UC UC in which ρ0 is the reference ambient value for density. The number of points used to resolve the boundary layer at the inlet is 52. The first point off the wall is at a distance of △r = 3.68 × 10−4 RJ , corresponding to △r+ = 1 in terms of wall units (based on uτ,inlet). The 16 of 35 American Institute of Aeronautics and Astronautics


maximum radial spacing inside the boundary layer at the inlet is △r+ = 8.14 at the outermost portion of the boundary layer. The grid has 1024 points in the azimuthal direction. This corresponds to △θ+ = 16.667, in terms of wall units on the nozzle wall (at r = RJ ). In the streamwise direction, the grid spacing △x+ starts at a value of 1, is stretched to a value of about 15 in the middle section of the nozzle and is reduced back to a value of 3 towards the nozzle exit. The relative fineness near the inlet and exit of the nozzle is to allow the synthetic turbulence to develop, and for proper separation into a free shear layer, respectively. The fineness near the inlet is excessive and tests on grids with coarser streamwise resolution near the inlet (with △x+ ≈ 15, not presented here) have shown that such high streamwise resolution is not necessary for sustained turbulence. The inlet reference Reynolds stresses are taken from Spalart’s DNS47 results at the same Reθ value of 300. Even though no significant difference was observed with the larger inlet integral length scales for the flat plate boundary layer simulations (section V.A.2), two different sets of length scales are still tested in this case. According to Mare et al.,39 the development of a turbulent boundary layer is more sensitive to the inlet conditions at low rather than at high Reynolds numbers. Since this test case is simulated at a much lower Reθ and in a different coordinate system, the effect of the length scales imposed at the inlet can be different. In addition to the digital filter-based turbulent inflow method described in the current paper, this test case is also simulated using a simpler method of turbulence injection, using uncorrelated random fluctuations in the velocity at the inlet. Referring to section IV, this is equivalent to skipping the step in which “white noise” is filtered in the two-dimensional inlet plane and simply passing the random fluctuations through Lund’s transformation. Thus, the velocity fluctuations at the inlet still satisfy the prescribed Reynolds stresses, but there are no prescribed correlations for the length scales. This test will determine whether the twopoint correlations established by filtering are really important in producing sustained turbulence. The first run (designated as Nozzle-1x ) corresponds to using the length scales based on the boundary layer reference values, which are discussed in section V.A.1. The second run (designated as Nozzle-1.6x ) corresponds to using length scales larger by a factor of 1.6 than the boundary layer reference values. The final run (designated as Nozzle-WN ) corresponds to using the random fluctuations-based inflow in which no correlated length scales are imposed at the inlet. “WN” corresponds to the uncorrelated “White Noise” fluctuations. Figure 15 shows the inlet plane v velocity fluctuations at an instant for all three runs. The difference in the length scales between correlated velocity fluctuations and white noise is readily apparent. A characteristic-based adiabatic viscous wall boundary condition50 is used on the inner/outer nozzle walls, and on the nozzle lip. On the lip edges, the solution is computed as an average of the solutions from the two constituent faces. Tam and Dong’s56 radiation boundary condition, extended to three-dimensional problems by Bogey and Bailly,57 is applied to the xmin face in the fourth superblock (which is outside the nozzle, upstream of the nozzle exit) and on the rmax faces of both the fourth and fifth (which is the outermost superblock downstream of the nozzle exit) superblocks. At the outflow face, Tam and Dong’s56 outflow boundary condition is applied. In addition to grid stretching, a sponge zone3 is applied from x ≈ 6.8RJ to x = 10RJ to facilitate damping of the vorticity in the flow-field before it reaches the outflow plane. Rotational periodicity is assumed at the superblock boundaries in the azimuthal direction. The centerline singularity is handled using the treatment mentioned in section III. The mean inlet profiles for primitive variables are used as initial conditions throughout the domain including the portion downstream of the nozzle. Outside the jet radius; ambient pressure, ambient density, and zero velocity are used as initial conditions. The solution is time-advanced for a total of 3.4×105 steps with △t = 1.4 × 10−4 RJ /UC (giving a maximum CFL of ≈ 0.84). The total simulation time is 47.6 RJ /UC and the initial 10 RJ /UC time period is allowed for the transients. The statistical averaging interval corresponds to ≈ 19 flow-through-cycles of the nozzle (≈ 37.6 RJ /UC ). V.B.2.

Discussion of Results

Figure 16 shows iso-surfaces of the Q-criterion (second invariant of the velocity gradient tensor) at a value of 5(UC /RJ )2 in the boundary layer inside a section of the nozzle. The Q-criterion is used to identify coherent vortex structures in wall-bounded and free-shear flows.58 The turbulence fed by the current method (Nozzle1x and Nozzle-1.6x ) evolves into physically realistic structures downstream in the nozzle. The nozzle is populated with hairpin and other vortices in both Nozzle-1x and Nozzle-1.6x. In contrast, the vorticity fed by the random fluctuations based-method (Nozzle-WN ) dies down within about a quarter of the nozzle length and never recovers. Figure 17 illustrates the free shear layer at a distance of 1RJ downstream of the nozzle exit, through a contour plot of the instantaneous streamwise velocity at this plane (x = 1RJ ). The 17 of 35 American Institute of Aeronautics and Astronautics


shear layer is completely laminar in case of Nozzle-WN. However, for both Nozzle-1x and Nozzle-1.6x, the shear layer qualitatively features turbulent fluctuations, and is also noticeably thicker than the one observed in Nozzle-WN. This indicates that both Nozzle-1x and Nozzle-1.6x have sustained turbulence inside the nozzle which resulted in turbulent fluctuations in the free shear layer. On the contrary, the “white noise” fluctuations in Nozzle-WN have died down completely inside the nozzle, and the shear layer is similar to what would be observed if only mean quantities are imposed at the inlet without any fluctuations. Next, mean flow and turbulence statistics inside the nozzle are studied for the three runs. All the mean quantities reported here are both time-averaged and azimuthally-averaged. The mean turbulent velocity profiles and Reynolds stresses at two different locations (−1.2RJ , −0.2RJ ) inside the nozzle are studied. The location x = −1.2RJ is at 40% of the nozzle length. It is ≈ 13.5 δ99,inlet away from the inlet plane. Figures 18, 19, and 20 show the mean velocity profile, the rms velocity fluctuations, and the Reynolds shear stress at this location, respectively. Each figure indicates the Reθ value of the boundary layer in each run at this location. Also plotted are the reference values from Spalart’s DNS47 at Reθ = 300 (which are imposed at the inlet at x = −2RJ ). All the Reynolds stresses are close to zero in Nozzle-WN, indicating that the boundary layer is laminar at this location. The mean velocity profile in the case of Nozzle-WN exhibits a large overprediction above the log law, which is not expected in a turbulent boundary layer.3 The mean profiles for both Nozzle-1x and Nozzle-1.6x exhibit such overpredictions as well, but those are considerably smaller. Uzun and Hussaini3 have reported that such a profile is representative of the transitional nature of a boundary layer. In Nozzle-1x and Nozzle-1.6x, it is observed that the streamwise fluctuations have recovered enough to give a maximum streamwise rms fluctuating velocity comparable to what is imposed at the inlet at the lower Reθ value of 300. However, azimuthal and radial fluctuations, as well as the Reynolds shear stress are still underpredicted. The radial (wall-normal) fluctuations are the slowest to recover. It is clear that the Reynolds stresses have not fully recovered by this location. It is to be noted that the larger length scales used in Nozzle-1.6x have helped in a quicker recovery than what is observed in Nozzle-1x. The second location is closer to the nozzle exit, at x = −0.2RJ . This corresponds to 90% of the nozzle length and is ≈ 31 δ99,inlet away from the inlet plane. Figures 21, 22 and 23 show the mean velocity profile, the rms velocity fluctuations, and the Reynolds shear stress at this location, respectively. Again, each figure indicates the Reθ value of the boundary layer in each run at this location. It is clear that the momentum thickness grows more quickly with correlated inlet fluctuations, with Nozzle-1.6x giving the greatest growth rate. In Nozzle-WN, all Reynolds stresses are close to zero, which reveal the laminar nature of the boundary layer. In Nozzle-1x and Nozzle-1.6x, the velocity profile again indicates that the boundary layer is transitional. The shape factors attained by the velocity profiles at this location are 2.04, 1.933 and 2.435 for Nozzle-1x, Nozzle-1.6x and Nozzle-WN, respectively. The value for Nozzle-WN is clearly closer to that expected from laminar boundary layers (H ≈ 2.6), whereas the values for Nozzle-1x and Nozzle1.6x fall under the transitional regime.3 In both Nozzle-1x and Nozzle-1.6x, the rms velocity fluctuations and the Reynolds shear stress have recovered and are close to the values imposed at the inlet at the lower Reθ = 300, with better recovery in Nozzle-1.6x. But the streamwise fluctuations are overpredicted and azimuthal/radial fluctuations are still underpredicted. Such behavior has been noted with finite difference schemes on relatively coarse meshes.12, 59 Thus, the current method is found to be capable of producing sustained turbulence for low Reθ boundary layers on a non-Cartesian, non-uniform grid. In low Reθ simulations, imposing larger integral length scales at the inlet is found to give a faster recovery of the Reynolds stresses. Still, a nozzle length of about 20 δ99,inlet is recommended for reasonable fluctuations to be present in the boundary layer near the nozzle exit at such low Reynolds numbers.

VI.

Spurious Noise at the Inlet

In the current turbulent inflow generation method, all five variables are imposed at the inlet (even for subsonic flows, as has been the usual practice in jet noise simulations3, 7, 10 ), and the pressure over the inflow surface is kept constant. Therefore, the physical pressure fluctuations associated with the presence of vortices in a turbulent boundary layer are not simulated correctly. When velocity fluctuations are introduced at the inlet, the pressure in the flow-field near the inlet starts to exhibit the corresponding fluctuations, but the pressure at the inflow plane does not adjust accordingly. Such a constant pressure inlet constitutes an acoustically reflecting boundary. The net result is the generation of small amplitude pressure waves near the inflow plane, which then travel downstream. These are illustrated in figure 24. These waves necessitated



an increase in the wall-normal domain extent for the flat plate boundary layer simulations in section V.A. The stress-free boundary condition at the upper face could not allow these waves to propagate properly, resulting in instabilities. A longer wall-normal domain length with grid stretching was employed to damp this noise away from the wall. Even though a turbulent boundary layer will generate noise of its own, these waves appear to originate at the inlet boundary as if a line source is placed along the edge where the inflow plane meets the lower wall. The turbulent boundary layer simulations performed by Lund et al.12 using a recycling-based method were stable with a much smaller wall-normal domain length, although a similar stress-free boundary condition was used at the upper face. It is therefore believed that the noise produced at the current turbulent inlet boundary is spurious. This artificial noise is of concern for jet noise simulations. In order to assess its impact on jet noise prediction, two wall-resolved isothermal jet simulations are conducted using a converging round nozzle. The nozzle has a 5◦ inner contraction angle, similar to the baseline round nozzle (designated SMC000) from the small metal chevron (SMC) series of the experiments60 using the Small Hot Jet Acoustic Rig (SHJAR) at NASA Glenn Research Center. The final 1 radius (RJ , the nozzle exit radius) long section of the nozzle is the converging part, whereas the initial 1 RJ long section is of uniform circular cross-sectional area. In one of these simulations, a sponge zone is employed inside the nozzle to force attenuation of the spurious noise generated at the inlet. This sponge zone is based on the formulation used by Khaligi et al.61 to damp out vorticity near the outflow boundary in simulation of flow over a circular cylinder. Currently, it is applied in a cylindrical zone near the inlet, extending from xmin to xmax axially, and from r = 0 to r = rend radially. Here, xmin is the location of the inlet plane and xmax is about 10 δ99,inlet distance away from the inlet. If rmax is the maximum radius in the initial uniform section of the nozzle, then rend is chosen to be rmax − 1.2 δ99,inlet. The sponge zone formulation is based on the assumption that a known mean flow exists in the selected zone. In the current study, the mean flow quantities enforced at the inlet are used as the known mean flow values throughout the sponge zone, and therefore it is necessary to ensure that xmax is reasonably smaller than the location where the convergent portion of the nozzle begins. Also, it is important to ensure that the zone does not include any of the boundary layer (by limiting rend ). It should be possible to include spatial variation of the mean flow values in the sponge zone, or to have a dynamic sponge zone using mean flow values computed on the fly. Inside this selected sponge zone, all the primitive flow variables are damped towards their mean values by multiplying the difference between the actual and the mean values by a damping factor (DF ). These damped fluctuations are added to the mean values to be used for the next time advancement stage. The damping factor can be designed to have a desired spatial behavior. In the current study, it is set to vary only axially according to the following equations: xmin + xmax , 2 x − xmin ,0 , A = max xmid − xmin x − xmax ,0 , B = max xmid − xmax

xmid =

DF = exp( −2.28 A3 − 6.21 A11 ) + exp( −2.28 B 3 − 6.21 B 11 ).

(36)

This damping factor has a symmetric shape about xmid . It damps out disturbances approaching the zone from both of its axial ends with increasing strength towards the center of the zone. The effectiveness of such a zone is apparent in figure 25, in which the dilatation (or the divergence of velocity) is used as a means to visualize the low-amplitude pressure waves. The sponge zone appears to block the spurious noise generated at the turbulent inflow boundary significantly inside the nozzle. Therefore, a comparison of noise levels generated by these two jets will help quantify the effect of the spurious noise. Arrays of pressure probes are used in these two simulations to examine the noise levels and the spectral content of noise at different locations of interest. These arrays are shown in figure 26(a). For each probe shown in figure 26(a), there are 8 equally-spaced probes in the azimuthal direction. The results presented here are averaged over these 8 probes at each location. The array of probes placed closest to the expected location of an acoustic control surface is indicated. The solution near these locations is of higher priority since it is used for far-field propagation of sound using a suitable surface integral method. The jets are simulated at a Reynolds number of ReRJ = 50, 000 (based on the nozzle exit radius RJ and the jet exit centerline velocity UC ) and a reference Mach number of Mr = 0.9. A five-superblock topology, similar to the one used in section V.B, is used for the grid. The streamwise length of the 19 of 35 American Institute of Aeronautics and Astronautics


domain is 80 RJ beyond the nozzle exit. A total of about 125 million nodes are used to discretize the domain. The mean turbulent velocity profile (given by Eq. 31) enforced at the turbulent inlet boundary has Reθ,inlet ≈ 300. Various jet statistics such as the mean streamwise velocity along the jet centerline, and the rms fluctuating streamwise velocity along the centerline and the lipline are found to be nearly identical for these two simulations (not presented here). Therefore, the mean flow field and the turbulence statistics are unaffected by the spurious noise. At each probe location, pressure is recorded at a sampling Strouhal number of ≈ 66.67. The Strouhal number is the non-dimensional frequency given by St = f ∗ DJ∗ /UC∗ , with DJ being the nozzle exit diameter. The complete duration for which the samples are collected corresponds to 10 periods of a wave of St ≈ 0.3263. Since a minimum of 10 periods of a given wavelength are recommended to be included in the record for proper resolution,62 any frequencies below Stmin = 0.3263 are excluded from the spectra plots and from the computation of the overall sound pressure levels (OASPLs). This minimum Strouhal number is dictated by the length of the sample collected, and it had to be limited due to the high computational cost. It should be noted that the lowest significant frequency is about an order of magnitude lower than Stmin for this jet flow, based on the experimental noise spectra maps.60 The frequencies lower than Stmin become important for observer angles lower than about θ = 60◦ (measured from the downstream jet axis). It is shown later in this section that the spurious noise generated within the nozzle is dominant primarily in the high frequency range (St > 1). Therefore, in the following discussion, emphasis is put on the high frequency content of noise, which is captured reasonably well, and is important for larger observer angles. The OASPLs recorded at each probe located outside of the nozzle during the two simulations are within 2 dB of each other, with majority of the readings being in agreement within a dB. The only probes to record any major discrepancy are the ones inside the nozzle. Figure 26(b) provides a closer look at the probes in the vicinity of the nozzle. The noise recorded by probes marked with letters A-H is now examined. Table 2 lists the OASPLs recorded at these locations. The simulation employing a sponge zone inside the nozzle has an OASPL lower by about 10 dB at location A inside the nozzle, indicating the effectiveness of the sponge zone in blocking the spurious noise. Since the spurious noise is expected to propagate mainly in the downstream direction, the increasingly diminishing differences in the OASPLs recorded at locations A, B, and C between the two simulations indicate that the spurious noise does not have a significant impact outside the nozzle. This observation is further supported by the fact that the OASPLs recorded at all the probes located outside the jet (probes D-H) are in agreement within 2 dB for the two simulations. This agreement is important due the aforementioned proximity of these probes to the acoustic control surface. Table 2: Overall sound pressure levels for selected probe locations Probe location A B C D E F G H

OASPL (dB, re 20 µPa) Without sponge zone With sponge zone 138.06 128.28 137.82 134.76 152.47 152.89 129.44 127.77 126.71 125.78 127.77 126.07 132.76 132.40 132.71 133.17

Figure 27 presents the one-third octave spectra of noise recorded at locations A, E, F , and G. All the spectra plots are jagged towards the lower frequency end, although only the “well-resolved” frequencies have been included. Obtaining a larger sample length, splitting it into multiple smaller records, and averaging the spectra over these intervals would have helped achieve smoother plots. This has not been performed due to the high computational cost involved in collecting more samples, and the available record was not divided to retain the current Stmin . At location A, the largest discrepancies in SPLs are observed at frequencies larger than St = 1. The sponge zone has attenuated much of the higher frequency spurious noise inside the nozzle. The frequencies below St = 1 seem to be in a relatively better agreement for the two simulations,



though jaggedness does not allow for a fair comparison. Nevertheless, it is clear that when a sponge zone is not used, the high frequencies (St > 1) are dominant in the spurious noise, and the current Stmin is reasonably low for proper examination of spectra at other locations. The spectra results at locations E and F support this conclusion as the simulation not using a sponge zone exhibits relatively higher SPLs only for higher frequencies (St > 2) at these locations. Even though the OASPLs for the two simulations match within 2 dB at both E and F , it is clear from the spectra that not using a sponge zone produces high SPLs for higher frequencies. If the grid is fine enough to carry this high frequency noise to an acoustic control surface that extends upstream of the nozzle exit plane, it could have an impact on the prediction of the far-field spectra in the upstream observer angles (θ ≥ 90◦ ), though the impact on overall sound pressure levels is expected to be small. It is to be noted that a similar behavior is observed in spectra plot at the other upstream (with respect to the nozzle exit) probe location D (not presented here). On the other hand, the spectra plot at location G exhibits very small differences at high frequencies for the two simulations, and the smaller discrepancies at lower frequencies can be attributed to the jaggedness of the spectra. A similar behavior is observed in spectra plots at other probes along the inner array outside the jet (not presented here), with differences in the high-frequency noise content for the two simulations becoming increasingly indistinguishable as θ decreases. This indicates that these locations are not affected by the spurious noise significantly. It should be noted that as the grid coarsens away from the nozzle lip in the axial and radial directions, the grid supported maximum Strouhal number (Stg,max ) decreases, even though the time sampling-based Nyquist Strouhal number is much higher (≈ 33.33). The grid supported maximum Strouhal number can be estimated by r 1 2 T0 , (37) Stg,max = Nλ (△s/RJ ) MJ TJ where Nλ is the minimum number of grid spacings necessary to properly resolve a wavelength, △s/RJ is the relevant grid spacing non-dimensionalized by the nozzle exit radius, MJ is the jet exit Mach number based on the local speed of sound, and T0 /TJ is the ratio of ambient temperature to jet exit temperature (which equals 1 for the current isothermal jets). Probes D, E, and F lie closer to the nozzle exit compared to probes G and H. It is important to ensure that the discrepancy in the high frequency content observed at upstream observer angles is not only due to the better resolution of higher frequencies at these locations. In other words, it should be verified that the better agreement in spectra observed at G, H, and the later probes along the inner array is not simply because the grid at these locations is not fine enough to support the high frequencies contaminated by the spurious noise. The grid spacing in the y direction increases by a factor of ≈ 1.9 going from probe E to probe G. Assuming that a minimum of 6 grid spacings are necessary to resolve a given wavelength (Nλ = 6), Stg,max can be estimated to be ≈ 12.77 at location E, and to be ≈ 6.71 at location G, based only on the grid spacing in the y direction. Thus, frequencies up to Stg,max ≈ 6.71 are resolved at location G, although no significantly high SPLs are noticeable at higher frequencies in the corresponding spectra plot when a sponge zone is not used (plot 27(d) does not include frequencies St > 12 to highlight only the significant portion of the spectra). This supports the previous conclusion that even when a sponge zone is not used, the grid-resolved high frequencies are not significantly affected at θ < 90◦ by the spurious noise. To summarize, the spurious noise generated at the current turbulent inflow is observed to significantly affect only the acoustic field within the nozzle, and the high frequency content of noise at upstream observer angles. The former is usually of no interest, but the latter can be of importance for accurate prediction of farfield noise spectra at these high angles (θ ≥ 90◦ ). Therefore, until an acoustically non-reflective formulation that can also inject turbulence at the inlet is available, it is recommended to use a sponge zone such as the one suggested in this section to damp this spurious noise within the nozzle.

VII.

Conclusions

A digital filter-based turbulent inflow method is implemented for application in jet noise studies. A new cost-effective way is proposed to extend digital filter-based methods to non-uniform curvilinear structured grids. The proposed method is tested on two problems. The first test involves simulation of a zero-pressuregradient flat plate turbulent boundary layer at Reθ = 1, 410 to 2, 200. Like most synthetic turbulence generators, the current method requires an adjustment region near the inlet for the turbulence to recover. From the high Reynolds number boundary layer simulations, the length of this region is estimated to 21 of 35 American Institute of Aeronautics and Astronautics


be ≈ 11.5 boundary layer thicknesses, based on recovery of the Reynolds stresses and establishment of a self-similar mean streamwise velocity profile (excluding the wake region). The streamwise growth rate of momentum thickness is not exactly matched with theory by this location. Nonetheless, the agreement is reasonable for jet noise studies, since wall-resolved simulations of high Reynolds number experimental jets with exact matching of the boundary layer properties is forbiddingly expensive. The second test involves application of the method in cylindrical coordinates, on a setup typical of jet-noise simulations including a nozzle. In spite of the very low Reθ value (Reθ,inlet = 300) of the boundary layer for this test, the method is shown to be successful in producing sustained turbulence in the straight round nozzle. A random fluctuation-based method is applied to the same problem and its failure in generating a turbulent free shear layer underlines the necessity of correlated turbulent fluctuations provided by the current method. Specifying larger integral length scales at the inlet is found to result in faster recovery of turbulent fluctuations only for low Reθ values. For high Reynolds number boundary layer simulations, the inlet integral length scale variation had no significant effect on the results. Since wall-resolved jet noise studies will probably be limited to very low Reθ values for the near future, it is recommended to specify inlet length scales larger by a factor of 1.5 to 2 than the available turbulent boundary layer reference length scales. Also, it is recommended to allow at least 20 boundary layer thicknesses for recovery in low Reθ simulations. The combination of not modeling the turbulent pressure fluctuations and imposition of a constant pressure over the entire inlet surface is found to give rise to low-amplitude pressure waves near the turbulent inflow boundary. This spurious noise is shown to have no significant impact on the acoustic field of a jet, except for the high frequency content at the upstream observer angles, that can be alleviated through the use of a sponge zone.

Acknowledgments This material is based upon work supported by the National Science Foundation (NSF) under grant number OCI-0904675. The Carter cluster of Rosen Center for Advanced Computing (RCAC) was utilized for the presented simulations, supported by an agreement through the Information Technology at Purdue (ITAP), West Lafayette, Indiana. Resources for some simulations were provided by the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF grant number OCI-1053575. The Kraken cluster at the National Institute for Computational Sciences (NICS) was used under allocation TG-ASC040044N.

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(a) FPBL-1x

(b) FPBL-2x

Figure 1: Instantaneous streamwise velocity contours at y = 9.61 × 10 the inlet mean velocity profile)

−2

δ99,inlet plane (△y + ≈ 65 based on


FPBL - 1x FPBL - 2x Lund - RR Momentum integral estimates

0.2

0.0042 0.0039

0.18

0.16

Cf

0.0033 0.0030

0.14

FPBL - 1x FPBL - 2x Lund - RR Momentum integral estimates Momentum integral estimates using velocity profile shape and δ definition assumed by Lund et al.

0.0027 0.12

0.0024 0.0021

0.1 0

10

20

x / δ 9930, inlet

40

50

Figure 2: Evolution of the momentum thickness in flat plate turbulent boundary layer simulations

0.0018

1400

1600

1800 Re 2000 θ

2200

Figure 3: Evolution of the skin friction coefficient in flat plate turbulent boundary layer simulations

1.8


0.28


1.6

0.24

δ 99 / δ 99 , inlet

δ disp / δ 99 , inlet


θ / δ 99 , inlet

0.0036

0.2

1.4

1.2

0.16 1

0.12

1400

1600

1800 Re 2000 θ

2200

1400

Figure 4: Evolution of the displacement thickness in the flat plate turbulent boundary layer simulations

1600

1800 Re 2000 θ

2200

Figure 5: Evolution of the 99% boundary layer thickness in flat plate turbulent boundary layer simulations


1.5

1.45

20

1.4

u+

H

15

1.35

10


1.25

1.2

1400

1600

1800 Re 2000 θ

FPBL - 1x FPBL - 2x Lund - RR Reichardt’s formula ( law of wall )

5

100

2200

Figure 6: Evolution of the shape factor in flat plate turbulent boundary layer simulations

101

y+

102

103

Figure 7: Comparison of the mean velocity profiles at Reθ ≈ 1, 530 in flat plate turbulent boundary layer simulations (Streamwise location is at x = 11.5 δ99,inlet for FPBL-1x and FPBL-2x )

25

20

15

u+


1.3

10

FPBL - 1x FPBL - 2x Lund - RR Reichardt’s formula ( law of wall )

5

100

101

y+

102

103

Figure 8: Comparison of the mean velocity profiles at Reθ ≈ 2, 050 in flat plate turbulent boundary layer simulations (Streamwise location is at x = 39 δ99,inlet for FPBL-1x and FPBL-2x )


25

10

x x x x

0 δ 99 , inlet 11.5 δ 99 , inlet 25 δ 99 , inlet 39 δ 99 , inlet

u+

15

100

101

+

y

102

0

103

Figure 9: Comparison of the mean velocity profiles at different streamwise locations in the flat plate turbulent boundary layer simulation FPBL-1x

+

u’rms

+

u’rms

3

FPBL - 1x FPBL - 2x Lund - RR

y / δ99

0.6

0.8

1

3

FPBL - 1x FPBL - 2x Lund - RR

+

2

w’rms

+

1.5

+

1 0.5

0.5 0

-0.5

|

-1 0

0.5

y / δ 99

( u’ v’ )

+

+

0

1.5 1

v’rms

w’rms

0.4

2.5

2

+

0.2

3.5

2.5

v’rms

0

Figure 10: Comparison of the mean velocity defect profiles at different streamwise locations in the flat plate turbulent boundary layer simulation FPBL1x

3.5

( u’ v’ )

4

2

5 Downloaded by PURDUE UNIVERSITY on May 13, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2014-1401

6

|

10

|

Velocity defect law x = 11.5 δ 99 , inlet x = 25 δ 99 , inlet x = 39 δ 99 , inlet

8

( Ue - u ) / uτ

20

= = = =

1

-0.5 -1 0

Figure 11: Comparison of rms fluctuating velocities (scaled by uτ ) and Reynolds shear stress (scaled by u2τ ) at Reθ ≈ 1, 530 in flat plate turbulent boundary layer simulations (Streamwise location is at x = 11.5 δ99,inlet for FPBL-1x and FPBL-2x )

0.5

y / δ 99

1

Figure 12: Comparison of rms fluctuating velocities (scaled by uτ ) and Reynolds shear stress (scaled by u2τ ) at Reθ ≈ 2, 050 in flat plate turbulent boundary layer simulations (Streamwise location is at x = 39 δ99,inlet for FPBL-1x and FPBL-2x )


1

0.75

y / RJ


Figure 13: 5-superblock topology used for the simulations of jet flow through a straight nozzle

0.5

0.25

-2

-1.5

-1 x / RJ

-0.5

Figure 14: Streamwise plane of the grid for the nozzle superblock (every fifth point is shown)



(a) Nozzle-1x

(b) Nozzle-1.6x

(c) Nozzle-WN

Figure 15: Instantaneous v - velocity fluctuations at the inlet plane in the simulations of jet flow through a straight nozzle



(a) Nozzle-1x

(a) Nozzle-1x

(b) Nozzle-1.6x

(b) Nozzle-1.6x

(c) Nozzle-WN

(c) Nozzle-WN

Figure 16: Iso-surfaces of the Q-criterion inside nozzle, Q = 5 (UC /RJ )2 in the simulations of jet flow through a straight nozzle

Figure 17: Instantaneous streamwise velocity contours at x = 1RJ plane in the simulations of jet flow through a straight nozzle


Nozzle - 1x ( Reθ = 344.66 ) Nozzle - 1.6x ( Reθ = 353.38 ) Nozzle - WN ( Reθ = 318.27 ) Spalart DNS ( Reθ = 300 ) Reichardt’s formula ( law of wall )

30

7

(u’rms )+

35

25

Nozzle - 1x ( Reθ = 344.66 ) Nozzle - 1.6x ( Reθ = 353.38 ) Nozzle - WN ( Reθ = 318.27 ) Spalart DNS ( Reθ = 300 )

6 5 4

(v’θ rms )+

u+

20 15

3 2

(v’R rms )+

5 0 0 10

101

102

103

∆ r+ Figure 18: Comparison of mean velocity profile at x = −1.2RJ plane in the simulations of jet flow through a straight nozzle

1

1 0 0

25

50

75

100

125

150

∆ r+ Figure 19: Comparison of rms fluctuating velocities (scaled by uτ ) at x = −1.2RJ plane in the simulations of jet flow through a straight nozzle. (The ′ ′ values of (vθ rms )+ and (urms )+ are offset by +1.5 and +3 respectively to retain clarity)

40


0.8

35

Nozzle - 1x ( Reθ = 403.96 ) Nozzle - 1.6x ( Reθ = 421.8 ) Nozzle - WN ( Reθ = 346.88 ) Spalart DNS ( Reθ = 300 ) Reichardt’s formula ( law of wall )

30 0.6

25

u

+

(u v )

’ + R

|

’


10

0.4

20 15

0.2

10 5

0 0

0.2

0.4

0.6

∆ r / δ99

0.8

1

1.2

Figure 20: Comparison of Reynolds shear stress (scaled by u2τ ) at x = −1.2RJ plane in the simulations of jet flow through a straight nozzle

0 0 10

101

102

103

∆ r+ Figure 21: Comparison of mean velocity profile at x = −0.2RJ plane in the simulations of jet flow through a straight nozzle



6

1

0.8

5 0.6

4

(v’θ rms )+


|

3

( u’ v’R )+

(u’rms )+

7

0.4

2

(v’R rms )+


0.2

1 0

0 0

25

50

75

100

125

0

150

∆ r+ Figure 22: Comparison of rms fluctuating velocities (scaled by uτ ) at x = −0.2RJ plane in the simulations of jet flow through a straight nozzle. (The ′ ′ values of (vθ rms )+ and (urms )+ are offset by +1.5 and +3 respectively to retain clarity)

0.2

0.4

0.6

∆ r / δ99

0.8

1

1.2

Figure 23: Comparison of Reynolds shear stress (scaled by u2τ ) at x = −0.2RJ plane in the simulations of jet flow through a straight nozzle

Figure 24: Spurious small amplitude pressure waves generated at the turbulent inflow boundary (FPBL-1x )



(a) Without a sponge zone inside the nozzle

(b) With a sponge zone inside the nozzle (green lines indicate the boundaries of the sponge zone)

Figure 25: Instantaneous dilatation contours in the simulations of jet flow through a converging nozzle

(a) Arrays of probes

(b) Closer look near the nozzle with probe designations

Figure 26: Pressure probes placed in the simulations of jet flow through a converging nozzle (indicated by orange circles)


135 130

120

Without sponge zone With sponge zone

115 110

SPL ( dB )

SPL ( dB )

120 115 110

105 100 95 90

105

85

100

St

5

10


80

15 20 2530

St

(a) Location A

5

10

15 20 2530

(b) Location E

120

125

115 120 110 105

SPL ( dB )

SPL ( dB )


125

100 95


115

110


90 105 85 80

5

St

10

100

15 20

2

St

(c) Location F

(d) Location G

Figure 27: One-third octave noise spectra at selected probe locations


4

6

8 10 12

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