Implementation of a Wall-Modeled Sharp Immersed ...

AIAA 2016-0257 AIAA SciTech 4-8 January 2016, San Diego, California, USA 54th AIAA Aerospace Sciences Meeting

Implementation of a Wall-Modeled Sharp Immersed Boundary Method in a High-Order Large Eddy Simulation Tool for Jet Aeroacoustics Nitin S. Dhamankar∗, Gregory A. Blaisdell† Purdue University, West Lafayette, IN-47907

Downloaded by PURDUE UNIVERSITY on February 18, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-0257

Anastasios S. Lyrintzis‡ Embry-Riddle Aeronautical University, Daytona Beach, FL-32114 High-order finite-difference (FD) discretization-based solvers are ideal for studying jet noise due to their low dissipation and dispersion errors. However, they are inherently incapable of handling the geometrical complexities common in many noise reducing nozzle designs. The immersed boundary method (IBM) represents an effective way of eliminating this shortcoming of the high-order FD solvers. In addition, to be able to simulate experimental-scale high Reynolds number flows from such complex nozzle geometries at an affordable computational cost, the turbulent boundary layer on the inner nozzle wall must be treated using a wall model. In a wall-modeled simulation, the large flow gradients near an immersed wall are not explicitly resolved and therefore, an IBM methodology appropriate for wall-resolved simulations is no longer valid. In the present work, an IBM implementation that allows a selected portion of an immersed boundary to be treated using a wall model boundary condition is described. Numerical methods employed near such an immersed boundary are crucial for the success of the wall model, and a detailed description of the chosen methods is provided. The proposed methodology is tested on a zero-pressure-gradient, attached flat plate turbulent boundary layer. It is observed that the current method produces near-wall results that are dependent on the location of the first grid point off the immersed wall, which is undesirable. However, the method is demonstrated to produce a reasonably accurate outer region for a turbulent boundary layer. It is suggested that the proposed methodology is a highly economical, though approximate way to treat the turbulent boundary layer on the inner nozzle wall in jet noise studies.

Nomenclature Roman symbols Cf cp F , G, H h J L M N p Pr P rt Q

Skin friction coefficient Specific heat at constant pressure Flux vectors in curvilinear coordinates Uniform grid spacing on the transformed computational grid Determinant of Jacobian of the coordinate transformation Length Mach number Number of grid points Static pressure Prandtl number Turbulent Prandtl number Vector of conservative flow variables

∗ Graduate

Research Assistant, School of Aeronautics and Astronautics. School of Aeronautics and Astronautics, AIAA Associate Fellow. ‡ Distinguished Professor and Chair, Aerospace Engineering Department, AIAA Associate Fellow. † Professor,

1 of 32 American Institute of Aeronautics and Astronautics Copyright © 2015 by Nitin Dhamankar, Gregory Blaisdell, Anastasios Lyrintzis. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

r Re t T u u, v, w uτ x, y, z


Greek symbols αf γ δ99

Temperature recovery factor for a boundary layer, radial coordinate Reynolds number Time Static temperature Velocity vector Cartesian velocity components Friction velocity Cartesian coordinates

δdisp Δs Δt θ κ μ ξ, η, ζ Π ρ τ

Spatial filtering parameter Ratio of specific heats for air Boundary layer thickness based on a location where the streamwise velocity is 99% of the freestream velocity Boundary layer displacement thickness Grid spacing in direction s Time-step size Boundary layer momentum thickness Log-law von K´ arm´ an constant Dynamic viscosity Generalized curvilinear coordinates Coles’ wake strength parameter Density Viscous stress

Accent marks () ()

Averaged quantity, filtered quantity Vector quantity

Subscripts ( )i ( )n ( )r ( )tang ( )v ( )w

Inviscid quantity, quantity along i direction, quantity at the inlet of the domain Wall-normal quantity Reference quantity Wall-tangential quantity Viscous quantity Quantity at the wall

Superscripts ( ) ( )∗ ( )+

Differentiated quantity, fluctuation quantity Dimensional quantity Quantity in inner wall units

Acronyms BP CAA CFL DNS FD FTC FV GP IB IBM LES MP

Boundary Point Computational Aeroacoustics Courant-Friedrichs-Lewy condition number Direct Numerical Simulation Finite-Difference Flow-Through-Cycle Finite-Volume Ghost Point Immersed Boundary Immersed Boundary Method Large Eddy Simulation Matching Point

2 of 32 American Institute of Aeronautics and Astronautics

MPI N/A RANS RK4 rms SP

Message Passing Interface Not-Applicable Reynolds-Averaged-Navier-Stokes Fourth-order Runge-Kutta scheme Root Mean Square Surface Point


I.

Introduction

Increasingly severe airport noise regulations, adverse impact on community and the environment, and concern for crews working in the close proximity of fighter aircraft have made reduction of jet mixing noise an active research topic. Ingenious design of engine nozzles is one of the fundamental ways of modifying the jet flow dynamics and associated aerodynamic noise. Significant experimental and computational research efforts have been focused on this challenging problem of designing quieter nozzles without a collateral thrust penalty in the past two decades. Although experimental efforts have been the frontrunner in such design efforts, high-fidelity computational simulations are considered an important complementary research tool to provide insights into the complex flow-field and noise generation mechanisms. Large eddy simulation (LES) is an accurate and feasible way of computationally simulating high Reynolds number jet flows. In LES, only the energetic large scales of a turbulent flow (which are more problemdependent) are resolved and the smaller less energetic eddies (which are more universal in character) 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 Lele2 . Although rapid advances in computing technologies have made it feasible to simulate high Reynolds number jet flows, it is still challenging to realistically and accurately simulate the complex nozzle geometries investigated by experimentalists. For example, some design ideas that have received attention from the experimental research community for passenger and/or military aircraft include chevrons3 , lobed-mixers4 , beveling5 , and corrugations6 . Solvers based on finitevolume (FV) methods and unstructured-grids can inherently handle such geometrical complexities. However, the superior low-dissipation and low-dispersion characteristics of high-order finite-difference (FD) structuredgrid methods are attractive for jet noise studies. A simple high-order structured grid solver cannot handle the sharp corners encountered in some of the nozzle geometries, such as chevrons. Relatively less complicated and smooth geometrical features, such as beveling the nozzle exit, can be simulated with a high-order structured solver7 , although the effort involved in generating a 3-D body-conforming grid around a non-axisymmetric nozzle geometry is nontrivial. An immersed boundary method (IBM) represents an efficient solution to the problem of extending a simple high-order structured solver to be able to handle complex geometries. In an IBM, the solid body under study is immersed in a non-body-conforming background grid, and the effect of the immersed boundary (IB) on the surrounding flow is indirectly imposed8 . The advantages IBM has over alternatives such as the overset grid method or interface treatments were pointed out in Ref. 9. An IBM is very useful for jet noise simulations, especially when a number of different nozzle geometries are to be compared under identical operating conditions. A justification for the use of IBM in jet noise studies was provided in our previous work in Ref. 9. The important points from that discussion are briefly repeated below. 1. An IBM avoids the laborious and error-prone process of generating body-conforming 3-D grids, hence increasing the efficiency of the workflow. 2. Excluding the local region near an immersed nozzle geometry where spatial discretization may be varied, high-order FD schemes can be applied consistently elsewhere in the domain, allowing the jet and the generated acoustic waves to propagate without any change in the discretization or resolution properties of the numerical methods till they reach the physical domain boundaries. 3. A single axisymmetric non-body-conforming background grid can be used for a number of different nozzle geometries. Therefore, only a 2-D slice (to be rotated in the azimuthal direction) of the background grid needs to be worked on, giving due attention to the flow and acoustic wave resolution requirements in different parts of the domain. Obviously, the amount of nozzle geometry variation that can be handled using a single background grid is limited. However, a single background grid should 3 of 32 American Institute of Aeronautics and Astronautics

be usable for a family of nozzle shapes that exhibit only a gradual change in a certain geometrical parameter. When using such a single background grid for a family of nozzle shapes under identical operating conditions, the CAA integral surface location and grid resolution is identical for all nozzles. Thus, a fair comparison between their computed noise signatures is provided. This is difficult to achieve when body-conforming grids are prepared separately for all nozzle geometries.


4. A nozzle occupies only a small fraction of the total domain volume used for typical jet noise simulations. Therefore, the number of solid points on the background grid, and the associated overhead is relatively minor. A variant of IBM was implemented in a high-order FD-based LES tool in our recent work9 . With this IBM implementation, it is possible to simulate wall-resolved flows with complex nozzle shapes using curvilinear background meshes, thus allowing an efficient distribution of the background grid points. However, this IBM approach only facilitates the inclusion of complex nozzle geometries. The wall-resolved simulations of experimental-scale high Reynolds number jets issuing from such nozzles require computer resources of the order that is impossible to achieve for the foreseeable future. To allow the use of IBM to simulate jets from complex nozzles, the turbulent boundary layer on the inner nozzle wall must be treated using an approximate boundary condition that does not require the inner viscous sublayer to be spatially resolved. The boundary conditions that allow such modeling of the inner layer are referred to as wall models. Various wall modeling methodologies proposed and used for LES have been reviewed by Piomelli and Balaras10 and more recently by Piomelli11 . According to Choi and Moin12 , the number of grid points required to simulate a wall-resolved flow scale as Re13/7 , whereas for a wall-modeled flow, they scale as Re. Apart from reduction in the required number of grid points, wall models provide an additional cost saving in terms of turnaround times, especially when using explicit time-advancement schemes. Since the inner viscous sublayer of a boundary layer is not resolved when using a wall model, much larger grid spacings can be used, which imply a much larger allowable time-step without violating the Courant-Friedrichs-Lewy (CFL) stability condition. The drastic reduction in computational cost facilitated by wall models make them an inevitable component of any jet noise simulation methodology to be used as a practical noise prediction tool. IBM and wall-modeling for LES have both been active research areas since their introductions in 1972 by Peskin13 , and in 1970 by Deardorff 14 , respectively. However, efforts found in the literature in the direction of wall-modeling an immersed boundary are relatively recent. Some of these efforts have been tabulated in table 1, which have been shown to produce encouraging results. It is to be noted that these implementations are based on second-order spatial discretization schemes in the fluid region, which are not optimum when jet noise simulations are being considered. The details of how the IBM and wall model are implemented differ significantly between different references and a thorough comparison between different formulations and their results is outside the scope of the current work. Although many implementations are based on the same general idea (such as the use of some form of the law of wall15 ), the details of the actual implementation can vary significantly. For example, the enforcement of a given wall model formulation can be carried out by setting pseudo flow values at solid points inside an immersed boundary (ghost-point approach) or by modifying flow values at fluid points in the vicinity of an IB (off-wall approach). Combining standard wall models that work reasonably well on body-conforming meshes with an IBM is a challenging task, especially when high-order spatial discretization schemes are being used. Most of the standard equilibrium wall models used with body-conforming grids make an assumption about the wall-normal grid spacing and where the first grid point off the wall lies in terms of wall units10 . In an IBM, no such control over the placement of the first grid point off an immersed boundary (IB) is possible. Since an IB cuts across the background grid in an arbitrary manner, the first point off an IB can lie in the viscous sublayer or the buffer layer or the log layer. This is incompatible with conventional equilibrium wall models based on the assumption that the first point off the wall lies in a region where viscous effects are negligible10 . Also, compared to a wall-resolved IBM, a wall-modeled IBM needs to employ a different numerical methodology for indirect imposition of boundary conditions. The large flow gradients at the wall are not resolved by the coarse grid used in a wall-modeled simulation, and therefore, assumptions about the flow behavior in this region that work well for a wall-resolved IBM do not hold for a wall-modeled IBM. This is especially important when high-order spatial discretization schemes are used in the fluid region, since they imply a stronger coupling between the near-wall region and the outer flow. Giving due attention to these issues, an equilibrium wall model boundary condition is formulated and added to our IBM implementation9 in a high-order LES solver in the present work. The implementation is tested for a standard zero-pressure-gradient flat plate turbulent


boundary layer. The results from these tests are scrutinized to gauge the effectiveness of the implementation in modeling the wall shear stress for an immersed wall and in obtaining realistic fluctuations in the outer region of the boundary layer.


Table 1: A list of some of the research efforts combining IBM and wall-modeling.

Reference

Spatial discretization

Tessicini et al.16

2nd -order FD

Based on solving the turbulent boundary layer equations on a refined mesh near an IB

Kalitzin and Iaccarino17

2nd -order FV

Based on a lookup table for friction velocity using the local Reynolds number

Choi et al.18

2nd -order FD for viscous fluxes

Based on reconstructing velocity near an IB using the power law

Roman et al.19

2nd -order FV

Based on reconstructing velocity near an IB and enforcing a modeled eddy viscosity at an IB

Capizzano20

2nd -order FV

Based on solving thin boundary layer equations near an IB, used for Reynolds-Averaged-Navier-Stokes (RANS) equations

Chen et al.21

2nd -order FV

Based on solving turbulent boundary layer equations on a refined mesh near an IB

Kang22

2nd -order FD

Based on solving the turbulent boundary layer equations and modifying the sub-grid scale eddy viscosity near an IB

Nam and Lien23

2nd -order FV

Based on the law of wall for computation of the wall shear stress

Yang et al.24

2nd -order FD

Multiple approaches, based on the thin boundary layer equations and the law of wall

Yang et al.25

2nd -order FD in wall-normal direction

Based on a model velocity profile that accounts for the inner viscous layer as well as for pressure gradients and inertia effects

Brief description

This paper is organized as follows. Section II gives an overview of governing equations and numerical methods used in the current study. Section III describes the addition of a wall model boundary condition in our previously implemented IBM methodology. Section IV discusses the setup, results and observations for the test cases. Finally, concluding remarks are given in section VI.

II.

Governing Equations and Numerical Methods

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

ρ∗ , ρ∗r

ui =

u∗i , Ur∗

p=

p∗ , ρ∗r Ur∗2

t=

t∗ , L∗r /Ur∗

xi =


x∗i . L∗r

(1)

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

(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 flux vectors in the 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 sub-grid scale modeling terms are employed. The dissipation provided by the grid and the numerical method is assumed to emulate the dissipation of the sub-grid scales. The spatial derivatives at interior points of the computational grid are computed using the non-dissipative sixth-order compact FD scheme of Lele26 . It is given by Downloaded by PURDUE UNIVERSITY on February 18, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-0257

αfi−1 + fi + αfi+1 = a

fi+2 − fi−2 fi+1 − fi−1 +b , 2h 4h

(3)

for a variable f , with α = 1/3, a = 14/9, b = 1/9, and h being the grid spacing on the uniformly-spaced transformed grid. At near-boundary points, this sixth-order scheme cannot be used since the differencing stencil extends outside the computational domain. Therefore, on points next to the boundaries, a fourthorder compact FD scheme is used, which is given by the following expression for the second grid point away from a boundary: 1 1 3 f1 + f2 + f3 = (f3 − f1 ). (4) 4 4 4h On the boundary points, a 3rd-order one-sided compact FD scheme is applied, which is given by the following expression for the first grid point:

f1 + 2f2 =

1 (−5f1 + 4f2 + f3 ). 2h

(5)

These compact FD schemes are denoted by “Cn”, where “C” stands for compact implicit nature of the scheme and “n” is the order of accuracy of the scheme. For example, the sixth-order compact FD scheme is denoted by “C6”. Compact FD schemes are implicit schemes, in which the derivative at a given grid point depends not only on the function values at nearby grid points, but also on the derivatives at nearby grid points. In addition to these compact schemes, certain standard explicit FD schemes have also been utilized in the current study in the vicinity of immersed boundaries (IB). This is explained in section III. These are denoted by “En”, with “E” representing the explicit nature of the scheme, followed by the order of accuracy “n”. Spatial filtering is used to suppress the high wavenumber numerical instabilities that can be caused by discrete treatment of boundary conditions, unresolved scales, and mesh non-uniformities27 . Implicit tridiagonal central filtering schemes by Gaitonde and Visbal27 are utilized in the present work, which are given by N an (fi+n + fi−n ). αf f¯i−1 + f¯i + αf f¯i+1 = (6) 2 n=0 It is possible to obtain a 2N th -order formula with a 2N + 1 point stencil, with N + 1 coefficients a0 , . . . , aN derived in terms of αf . A sixth-order filter is used at the interior points of a grid. This filter has a 7-point stencil. At near-boundary points, a different formulation is necessary since the sixth-order filter stencil extends outside the computational domain. In the vicinity of an IB, lower-order central tri-diagonal filtering schemes are used. More details about this setup are provided in section III. The central tri-diagonal filtering schemes are designated by “In” in this work, where “I” stands for the implicit nature of the filter, followed by the order “n”. At grid points one and two grid spacings away from the other physical boundaries (excluding any IB), sixth-order one-sided-biased filters by Gaitonde and Visbal27 are used. The boundary points are left unfiltered. The spatial filtering operation is carried out in the uniformly-spaced transformed computational domain. The conservative flow variables are filtered in all spatial directions at the end of every time-step (unless mentioned otherwise). At 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 6 of 32 American Institute of Aeronautics and Astronautics


effects28 . The spatial filtering parameter αf is set to 0.47, except for the lower-order central tridiagonal schemes applied in the close vicinity of an IB (see section III for more details). For multi-block topologies, a superblock-based approach by Martha et al.29 is used. Non-overlapping superblocks are formed by logical decomposition of the computational domain of the problem under investigation. Each superblock is further partitioned into multiple smaller blocks, each of which is mapped to a particular computer core. This multi-block approach has unique advantages such as a fixed number of grid/solution files for variable core counts, flexible computational domain description, and efficient parallel file input/output operations. Communication is handled through the message passing interface (MPI) standard for distributed-memory parallel computer architectures. Both the spatial differencing and spatial filtering operations require solution of large tri-diagonal linear 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 Sameh30 for diagonally dominant narrow-banded systems. Due to limiting communication only to the nearest neighbors, this solver exhibits excellent parallel efficiency. It has been shown29 to yield about 74% efficiency at 91,125 cores on the Kraken supercomputer (with respect to a baseline case using 2,744 cores). Details of the implementation of this solver can be found in Refs. 29, 31, 32. For cylindrical grids common in jet noise simulations, a centerline treatment consisting of the method proposed by Mohseni and Colonius33 for the radial direction (to avoid the centerline singularity) and the method proposed by Bogey et al.34 for the azimuthal direction (to avoid the time-step restriction due to very fine azimuthal spacing near the centerline) is used. To enable simulations of off-design supersonic jets, shock capturing methods based on characteristic filters are available in the current implementation7, 35 . Time-advancement is performed using a standard fourth-order explicit Runge-Kutta scheme (RK4). Various boundary conditions are available in the solver, which include a far-field radiation boundary condition36, 37 , characteristic-based (partially non-reflecting) inflow and outflow boundary conditions38, 39 , different viscous wall boundary conditions40 , an equilibrium wall model for body-conforming meshes41 , and a digital filter-based turbulent inflow boundary condition42 . A sharp immersed boundary method (IBM) appropriate for wall-resolved simulations was implemented in the code recently9 , and in the present work, it is extended to include an optional wall model boundary condition. More details about the current implementation of the LES and CAA methodologies can be found in Refs. 9, 40–45.

III.

Implementation of a Wall-Model for Immersed Boundaries

In an IBM, the surface geometry of a solid body is immersed in a non-body-conforming background grid. The boundary conditions required at this “immersed” boundary (IB) are indirectly imposed such that the outer flow is affected in approximately the same manner as it would have been on a body-conforming grid with a direct imposition of boundary conditions. Various ways in which an IBM can be implemented are reviewed by Mittal and Iaccarino8 . A sharp IBM based on the idea of ghost points was previously implemented in our base LES solver9 . In this approach, artificial flow values are imposed at ghost points lying inside an IB, such that the required boundary conditions at the IB are approximately satisfied. A ghost point (GP) is a background grid point that itself lies inside the solid body; however, it has at least one neighboring background grid point belonging to the fluid region. This method allows a “sharp” representation of the IB and is considered suitable for high Reynolds number flows over rigid solid bodies8 . Our IBM implementation described in Ref. 9 is appropriate for a wall-resolved simulation, in which the flow near an IB is sufficiently resolved. However, when a wall model boundary condition is desired, multiple changes to the formulation are necessary, which are discussed in this section. This methodology is an addition to our previous implementation, discussed in Ref. 9. Therefore, a short overview of the previously-implemented wall-resolved IBM is given below to provide context for the following discussion. For a detailed discussion of the previous implementation, please refer to Ref. 9. The wall-resolved IBM implementation9 involves two separate stages. The first stage is called the preprocessing stage, which classifies the background grid points into points belonging to the fluid region, to the solid region, and to points lying on the IB itself called the surface points (SP). An IB is represented using a triangulated surface geometry of the body under consideration. An illustration of this classification is provided in figure 1. The preprocessor then identifies ghost points (GP) in each grid coordinate direction separately. A GP is a point belonging to the solid region which has one nearest background grid point



belonging to the fluid region in a given grid direction. The preprocessor also identifies boundary points (BP) on the IB that are closest to the corresponding GP, such that the line segment joining a GP to its BP is normal to the IB. The information generated by the preprocessor is written to a file which is later read in by the LES solver. The LES solver casts probes in the normal direction to the IB from each GP into the fluid region. Image points are placed along this probe in the fluid region. The image points are not coincident with any background grid point in general, and to obtain the flow variable values at these image points, trilinear interpolation is employed. During the LES solver stage, pseudo flow values are set at the GP by using linear extrapolation of data using values at the image points and the required boundary condition at the BP. Such linear extrapolation is sufficient for a wall-resolved flow since the grid is assumed to be fine enough in the vicinity of an IB to accurately resolve the gradients in flow variables. The numerical schemes are modified in the vicinity of an IB and the schemes used in the current implementation for such wall resolved IBs are indicated in figure 1 (consult section II for more information on nomenclature used for the numerical schemes). Basically, lower-order explicit FD schemes are used near an IB before switching to the high-order compact FD schemes of the base solver away from an IB. The same is applicable to filtering schemes as well. By inclusion of the GP into the derivative system of the outer flow, the effect of an IB on the outer flow is emulated. The resulting formulation was found to reduce the sixth-order spatial accuracy of the base solver to about third-order in the vicinity of an IB9 .

GPc

SPa

GPb Derivative C6 Filtering I6

C6 I6

GPa

C4 E2 E2 E2 I6 I4 I2 No filter

Immersed boundary

Solid point

Probe

Ghost point

Boundary point One-sided scheme

Surface point

Image point 1

Bad point

Image point 2

Central scheme

Figure 1: Schematic of the wall-resolved IBM (reproduced from Ref. 9). The wall model boundary condition implemented in the present work is now discussed. The currently chosen wall model formulation is based on an equilibrium wall model implemented previously41 in our LES solver for body-conforming meshes. This formulation uses the law of wall15 for a fully-turbulent attached boundary layer in order to specify an appropriate wall shear stress, assuming that the inner viscous region of the boundary layer is not explicitly resolved. Although the wall model formulation used in the current study is analogous to that described in Ref. 41, its current implementation is significantly different due to the involvement of immersed boundaries. Note that the model formulation is valid only for attached



turbulent boundary layers. It is not applicable when flow phenomena such as boundary layer separation or flow through sharp corners is involved. In the current study, the wall model is not supplemented with any sensor formulation to detect the existence of a turbulent boundary layer and to apply the model only near such a selected portion of an IB. It is assumed that the information about where an attached turbulent boundary layer exists is known a priori. In an IBM, a wall model would typically be required for only a portion of an IB where a turbulent boundary layer is attached to the IB. The other parts of the IB should be treated in a wall-resolved manner. Therefore, the preprocessor stage of our previous implementation9 is modified to perform a classification of ghost points into wall-resolved and wall-modeled ghosts. An example of this classification is shown in figure 2 for the IB of a converging nozzle immersed in a background grid. The complete 3-D geometry would be obtained by rotating this plane through 360° about the centerline along the x axis. A turbulent boundary layer exists on the inner nozzle wall, whereas a laminar boundary layer develops over the outer wall due to flow-entrainment by the downstream free jet (assuming no co-flow exists and ambient turbulence is negligible). It is desirable to apply the wall model boundary condition only on the inner wall. In the current implementation, the classification is performed using a reference direction and a tolerance angle. From each GP, an outward normal is cast on the IB. If the angle made by this outward normal with the reference direction is within the specified tolerance, the GP is classified to be a wall-modeled ghost. Otherwise, it is a wall-resolved ghost to be treated with the method described in Ref. 9. For the converging nozzle example depicted in figure 2, the reference direction is the local radially-inward direction, and a suitable tolerance angle is 45°. Note that in case of multiple IBs, the wall model reference direction and the tolerance angle can be specified separately for each IB. Also, in certain special cases, a turbulent boundary layer exists on the majority of the surface of an IB. In such cases, a reference direction can be specified for the wall-resolved GPs, and all the other GPs can be treated using the wall-modeled IBM. For example, consider the addition of an immersed lobed-mixer geometry, coaxial with and inside the converging nozzle shown in figure 2, to simulate a two-stream flow. For such an inner lobed-mixer, turbulent boundary layers exist on both inner and outer surfaces, where a wall model boundary condition would be appropriate. At the lip of such a mixer, the wall-resolved IBM treatment would be more appropriate. Assuming that this lip is flat and perpendicular to the x direction, the wall-resolved reference direction can be specified as the x axis direction and a small tolerance angle of 5° can be used. Thus, only the GPs at the lip of the mixer will be treated using the wall-resolved IBM, whereas all the other GPs will be treated using the wall-modeled IBM. The flow extrapolation to GPs during the solver stage is now discussed. For a wall-modeled ghost, the lower spatial resolution in the vicinity of an IB indicates that the extrapolation procedure and boundary conditions will need to be different than those used in a wall-resolved IBM. A simple linear extrapolation of flow data to wall-modeled GPs is not appropriate considering the large gradients in flow variables that are not captured on the coarse background grid typical of wall-modeled simulations. For example, the streamwise velocity at the 1st fluid point off an IB can be as high as 70% of the freestream velocity in a wall-modeled simulation. Through experimentation, a methodology that works reasonably well for such cases was formulated and is described below. In figure 3, a schematic of the current wall-modeled IBM is depicted. Note that henceforth, all the references to lengths l1 , l2 , l3 , and l4 are with respect to figure 3. A wall-modeled GP is indicated and fluid points away from it are numbered in an increasing order starting from 1. Note that here the intersection of the grid line with the IB is marked to be the boundary point (BP). This is different from the BP definition used in wall-resolved IBM9 , where a BP is defined as the point on the IB that is closest to the corresponding GP, such that the line segment joining a GP to its BP is normal to the IB (see figure 1). Referring to figure 3, a probe normal to the IB is extended in the fluid region from the BP. A sampling point is placed along this probe at approximately 3 grid spacings away from the BP. The grid spacing used for this purpose is chosen to be the minimum grid spacing in any direction in the vicinity of the IB. Usually the background mesh is prepared with a smaller spacing in the wall-normal direction to better resolve the boundary layer. The sampling point provides the input to the wall model. Following the convention used in Ref. 41, this sampling point is called the wall model matching point (MP) in the current work. The reasoning behind its placement is discussed later in this section. Note that in general, the MP is not coincident with any background grid point, and the flow values at the MP are obtained using trilinear interpolation. In the current implementation, the 1st layer of fluid points close to the wall-modeled portion of an IB is completely skipped during the simulation (the reasoning behind this exclusion is explained below). Therefore, the 2nd fluid point off the IB is the first fluid point where a solution is sought. In a wall-resolved IBM, the flow extrapolation to the GPs is performed using a linear extrapolation along a probe normal to the


Nozzle

Oute


Turbulent inflow boundary condition

r wa

Inner

ll

wall

Centerline of cylindrical nozzle Immersed boundary of a converging nozzle Wall-modeled ghost point Wall-resolved ghost point Wall model reference direction - radially inward direction Outward normal probe making less than 45 degrees angle with the radially inward direction Outward normal probe making more than 45 degrees angle with the radially inward direction Figure 2: Example of classification of ghost points into wall-modeled and wall-resolved points for a converging nozzle geometry immersed in a non-body-conforming background grid.

IB9 . In a wall-modeled IBM, the extrapolation is performed along individual grid lines, with a polynomial of the same order as that of the FD scheme that is used at the first fluid point away from the IB where a solution is sought (which is the 2nd fluid point away from an IB in the current implementation). As described later in this section, a second-order explicit central FD scheme is used at the 2nd fluid point away from the IB on an artificially coarsened grid, thus skipping a point on either side of the 2nd fluid point. Therefore, the flow extrapolation to the GP is performed using a second-order polynomial passing through known values at the 2nd and 4th fluid points off the IB and a desired boundary condition at the BP. The lengths l1 , l2 , and l3 indicated in figure 3 are relevant for this extrapolation. This ensures that when the second-order explicit central FD scheme is used at the 2nd fluid point off the IB, with the scheme stencil containing the GP, the 2nd fluid point and the 4th fluid point, the scheme approximates the slope of a curve that passes through the desired boundary value at the IB. To perform this extrapolation, the flow values at the BP must be known. These are obtained from the required boundary conditions for different flow variables. For primitive flow variables, these boundary conditions are as follows. All three velocity components are zero by definition at the BP due to the no-slip boundary condition. The IB is assumed to be an adiabatic wall. The pressure at the BP is obtained through a linear extrapolation from the values at the 2nd and 4th fluid points off the IB. For an adiabatic wall, a boundary condition of the form ∂(p/ρ)/∂n = 0 should ideally be enforced in a wall-resolved simulation9 , with n representing the outward normal direction at an IB. However, this condition is valid only very near the wall (in the viscous sublayer), and this region is not resolved in a grid used for a wall-modeled simulation. The following adiabatic Crocco-Busemann relation46 is invoked to obtain the static temperature at the BP, using the known flow values at the matching point MP: TBP = TM P +

r | uM P |2 (γ − 1) Mr2 , 2

(7)

where r is the temperature recovery factor, and Mr is the reference Mach number for a simulation. The recovery factor is approximated with r = P r1/3 , which is appropriate for a turbulent flat plate boundary layer46 , with P r = 0.7 for air. Once pBP and TBP are known, ρBP is found using the ideal gas law. Finally, using the data at the BP, 2nd fluid point off the IB and 4th fluid point off the IB, a second-order extrapolation 10 of 32 American Institute of Aeronautics and Astronautics

is performed to obtain pseudo flow values at the GP. When derivatives of the primitive variables are computed for the viscous terms, these extrapolated pseudo flow values for ghosts in a particular direction are enforced before the derivatives in that direction are computed.

5 4 3 Immersed boundary (IB)


2 1

Ghost point (GP)

1 2

3 4

Skipped fluid point

5 Included fluid points Matching point for wall model (MP) Boundary point (BP)

Figure 3: Schematic of wall-modeled IBM. One of the major issues in applying wall models that work well on body-conforming grids in an IBM is the lack of control over where the 1st grid point off the wall lies. In a general IBM setup, the 1st point off an IB can lie in the viscous sublayer, the buffer layer, or the log layer. The second-order polynomial extrapolation of primitive flow variables discussed above can result in very large values of negative streamwise velocity at a GP if the 1st fluid point away from an IB is very close to the IB. To avoid the associated numerical errors, the 1st layer of fluid points away from an IB is excluded from the simulation. This artificial grid coarsening ensures that the first fluid point away from an IB where a solution is sought is in the log-law region of the boundary layer (this assumes that the background grid spacing is appropriately tailored, for example, a value of Δr+ ≈ O(30) for the setup shown in figure 2). The purpose of the proposed wall model methodology is to approximately account for the effect of the inner layer without actually resolving it. Therefore, the exclusion of the 1st layer of fluid points is justified. The solution in this near wall region is not desired when a wall model is being used, and it is omitted in order to obtain a more stable and accurate solution in the outer layer. In the wall-resolved IBM implementation9 , the flux vectors (F , G, and H in Eq. 2) are formed at the GPs using the extrapolated primitive flow variables and derivatives of the primitive flow variables at those GPs. This treatment is different for the wall-modeled IBM implementation. The main purpose of the wall model boundary condition is to allow the outer flow to experience the same high wall shear stress that is present in a wall-resolved turbulent boundary layer, while the defining feature of this shear stress – the large velocity gradient at the wall, is not explicitly resolved by the grid. In the current method, an equilibrium wall model provides a boundary condition for the wall shear stress at the BP. Details about how this shear stress is calculated are given later in this section. Using this modeled shear stress, flux vectors at the BP are formed. Then, similar to the procedure used for the primitive flow variables, extrapolation of flux vectors to the GP is performed using a second-order polynomial interpolating the flux vectors at the BP, and the 2nd and 4th fluid points away from the IB. The estimation of the wall shear stress at the BP is now discussed. Any suitable wall model can be chosen to provide an approximate value for the wall shear stress. In the current work, an equilibrium wall model formulation previously implemented41 in our LES solver for body-conforming meshes is used as a reference. This model assumes the existence of a fully-developed, attached, zero-pressure-gradient turbulent boundary layer over the wall-modeled boundary. This assumption allows invocation of the law of wall for the time-averaged mean wall-tangential velocity profile, given for the viscous sublayer and the log-law region


by the following expression by Reichardt47 : u+ tang =

+ + ln (1 + 0.4y + ) + 7.23486 1 − e−y /11 − (y + /11) e−0.33 y , κ

(8)

∗ ∗ + = y ∗ u∗τ ρ∗w / μ∗w = Rer y uτ ρw / μw (Rer is the Reynolds with κ = 0.41. Note that u+ tang = utang / uτ , y number based on the reference quantities for a simulation), with u∗τ = τw∗ /ρ∗w , subscript tang denoting the component of velocity tangential to the wall, and subscript w denoting values at the wall. In this equation, y is to be considered a coordinate locally normal to the wall. Note that in Eq. 8, coefficients from Reichardt’s47 original expression are modified to better match the log-law section with the following expression


u+ tang =

log(y + ) + B, with κ = 0.41, and B = 5, κ

(9)

that is commonly employed for the log region in zero-pressure-gradient incompressible flat plate turbulent boundary layers.45 Note that Eq. 8 does not include the wake portion of the velocity profile. The wake region is excluded since flow data is sampled from locations close to a wall for use with a wall model (usually from the log-law region) and not from the outer wake region. For compressible flows, u+ tang in Eq. 8 is replaced with an effective van Driest velocity u+ . For an adiabatic wall, the relationship between the actual velocity VD and the van Driest transformed velocity is given by48 u∗V D 1 = sin−1 (B), u∗tang B 2 c∗p Tw∗ u∗tang and b∗ = , with B = √ P rt b∗

(10)

where Tw∗ is the dimensional wall temperature, c∗p is the dimensional specific heat for air at constant pressure, and P rt is the turbulent Prandtl number. Non-dimensionalizing with respect to the reference velocity Ur∗ and reference static temperature Tr∗ , the expression for B can be simplified to

(γ − 1) P rt Mr2 utang . (11) B= √ 2 Tw Note that Tw is TBP in the current context and is calculated with Eq. 7. The current study employs a value of 0.7 for P rt . For an incompressible zero-pressure-gradient attached turbulent boundary layer, the log-law region + 15 + (with δ99 denoting the local boundary estimated by Eq. 9 is considered to hold well for 30 < y + < 0.3 δ99 layer thickness in wall units). When used for a wall model though, it is suggested that Eq. 9 be invoked + 49 within a more conservative range of 50 < y + < 0.1 δ99 . The Reichardt’s expression given in Eq. 8 describes the viscous sublayer, the buffer layer, and the log layer, and therefore it is used in the current study instead of Eq. 9. This provides some margin of safety in the design of the background grid. The use of a wall model for LES necessitates use of sufficiently coarse meshes such that Eq. 8 can be assumed to be satisfied instantaneously near the wall10 . The currently used wall model implementation is based on this assumption. It is assumed that Eq. 8 and the compressibility-related transformations (Eqs. 10 and 11) hold in the simulated turbulent boundary layer instantaneously near the wall. Now, given an accurate input for utang from a point at a wall-normal height y, Eq. 8 can be solved iteratively for the local uτ at the wall. Using the uτ value from the previous time-step as an initial guess, very few iterations (usually < 5) of Newton’s method are necessary to obtain an estimate for uτ at a given time-step. Once uτ is known, the wall shear stress can be computed as (12) τw = ρw u2τ . Note that ρw is ρBP in the current context. The input for utang has conventionally been taken from the 1st point off the wall on body-conforming meshes. The 1st point off the wall is intentionally placed in the log layer when designing such a mesh41 and used as the matching point. However, using a grid point higher than the 1st point off the wall as the matching point has been shown to improve the predictions of an equilibrium wall model by Kawai and Larsson49 . This practice avoids sampling flow data from the near-wall region which is under-resolved and relatively inaccurate in such wall-modeled simulations. From the previous tests using the body-conforming 12 of 32 American Institute of Aeronautics and Astronautics

equilibrium wall model implemented in our solver41 , placing the matching point about 3 grid spacings away from the wall has been found to work well. Therefore, in the current wall-modeled IBM, the wall model matching point (MP) is placed along a wall-normal probe extended from each BP at about 3 grid spacings at the MP is decomposed away from the BP (length l4 in figure 3). Figure 4 shows how the velocity vector u along the wall-tangential direction utang and the wall-normal direction un . The magnitude utang is utilized for the computation of τw . Once τw is known at the BP, the flux vectors need to be formed at the BP. This requires the knowledge of the Cartesian viscous stress tensor at the BP. In order to translate the known τw to the Cartesian viscous stress tensor, an orthonormal Cartesian basis is first defined at each BP, consisting of unit vectors { n, l, m} (see figure 4). Vector n is normal to the IB and points towards the fluid region, whereas vectors l and m are tangential to the IB at the BP. Vectors l and m can be oriented randomly in the wall-tangential plane at the BP as long as { n, l, m} are orthogonal. Then, the distribution of the wall shear stress along the tangential directions l and m is computed using


τnl = τln = τw τnm = τmn = τw

u tang • l , | utang | u tang • m . | utang |

(13) (14)

For an incompressible boundary layer, the wall-normal derivative of the wall-normal velocity is identically zero, giving τnn = 0. This condition is also utilized for compressible flows in the current implementation, following Ref. 41. Therefore, except the components given in Eqs. 13 and 14, all the other components of the viscous stress tensor in basis { n, l, m} are set to zero. The Cartesian viscous stress tensor is then obtained through the change of basis transformation for second-order tensors50 as follows: ⎡ ⎤ ⎤ ⎡ τxx τxy τxz τnn τnl τnm ⎢ ⎥ ⎥ ⎢ −1 −1 T , (15) ⎣τyx τyy τyz ⎦ = T τll τlm ⎦ T ⎣ τln τzx τzy τzz τmn τml τmm ⎤ ⎡ n x ny nz ⎥ ⎢ with T = ⎣ lx (16) ly lz ⎦ . mx my mz The heat flux at the BP is set to zero when forming the flux vectors since the IB is assumed to be an adiabatic wall. The computation of flux vectors at the BP also requires grid metric terms at the BP. Since BP is not coincident with a background grid point, the grid metrics are not directly available at the BP and they are approximated by linear interpolation between the metrics at the corresponding GP and the 1st fluid point off the wall. Note that in the above discussion, only wall-modeled ghost points are addressed. Wall-modeled surface points (SP) can also exist, which would lie on an IB itself (and not inside it). For such SPs, the required boundary conditions are directly enforced. It can be considered to be the limiting case in which the GP in figure 3 coincides with the BP. Since the treatment for such surface points is analogous to that described for ghost points, it is not given in detail here for brevity. The treatment of multi-direction ghost/surface points remains similar to that described in Ref. 9. Note that a ghost/surface point can be wall-modeled along one grid direction and wall-resolved in other grid direction/s. The pseudo flow values for a ghost/surface point are maintained separately for each grid direction along which it is a ghost/surface point. Therefore, fluid points away from such a ghost/surface point in a given grid direction are affected by the appropriate boundary condition. The numerical methods used for spatial discretization and filtering in the vicinity of a wall-modeled portion of an IB are now discussed. In an IBM, the boundary conditions are enforced at an IB in an approximate manner. To avoid a strong coupling of this relatively less accurate flow-field with the outer flow, lower-order explicit FD schemes are used near an IB in our wall-resolved IBM implementation9 . This argument is even more justified when using an IBM with a wall model. With a wall model, the large gradients in the flow-field very near the wall are not resolved and applying high-order compact FD schemes to such low-resolution data is not justifiable. Near such a wall, the use of lower-order explicit FD schemes has been found to produce better and less grid-dependent results compared to higher-order compact FD schemes with body-conforming meshes41 . Therefore, in the present work, numerical schemes are chosen to achieve a 13 of 32 American Institute of Aeronautics and Astronautics

Immersed boundary (IB) Boundary point (BP)


Matching point (MP)

Figure 4: The orthonormal basis vectors { n, l, m} formed at the BP.

gradual transition from lower-order explicit FD schemes near an IB to the higher-order compact FD schemes of the base solver away from an IB. These numerical schemes are specified in figure 5 with reference to the GP shown in figure 3. The GP is used only to provide an appropriate boundary condition for the outer flow and the 1st fluid point off the IB is skipped during the simulation. Therefore, differentiation and filtering operations are not applicable (N/A) at these points. To set pseudo flow values at the GP, a second-order polynomial extrapolation is employed using data at the 2nd and 4th fluid points with the known boundary conditions at the IB. To ensure that the differentiation scheme applied at the 2nd fluid point estimates the slope of this same second-order curve, a second-order central explicit FD scheme is applied at the 2nd point using data at the GP and the 4th fluid point. Thus, the differentiation stencil at the 2nd fluid point is applied on an artificially coarsened grid to have twice the grid spacing of the actual grid. This scheme is denoted as “E2c”, with “c” representing the coarsened grid. For the 3rd fluid point, a third-order one-sided-biased explicit FD scheme (“E3”) is applied with a stencil that encompasses fluid points 2–5. Starting with the 4th fluid point, compact FD schemes are used. The fourth-order compact central FD scheme (“C4”) is used at the 4th fluid point. The sixth-order compact central FD scheme (“C6”) is used at fluid points 5 and higher. Filtering is performed with lower-order central tridiagonal schemes near an IB before switching to sixthorder tridiagonal central filter (“I6”) of the base solver. The pseudo flow values specified at the GP only serve to achieve appropriate flow derivatives at the 2nd fluid point, and they are not relevant for the filtering operation. Therefore, the solution at the GP and 2nd fluid point is not filtered. The 3rd fluid point is filtered using the “I2” scheme and the 4th fluid point is filtered using the “I4” scheme. The “I6” filtering scheme of the base solver is applied for fluid points 5 and higher. It is to be noted that the filter parameter αf is set to a value of 0.47 for the “I6” scheme, whereas a value of 0.48 is used for the “I2” and “I4” schemes. For a given scheme, a higher αf indicates a less dissipative filter. For a given αf value, higher-order schemes are less dissipative than lower-order schemes. The higher αf value is chosen for the lower-order “I2” and “I4” schemes to reduce the dissipation caused by them very near an IB. The use of sixth-order one-sided-biased filters by Gaitonde and Visbal27 would have been more appropriate at fluid points 3 and 4. However, in our present implementation, allowing the use of sixth-order one-sided-biased filters in the interior of a domain (as would be necessary in a general IBM setup) would have required considerable code modification and additional computational cost due to the increase in the stencil size it implies. Therefore, lower-order “I2” and “I4” schemes are used with a higher αf value as a compromise.


I6 I4

Filtering Schemes

I2 N/A N/A N/A 1

2

3

5

4

N/A N/A


E2c E3

Derivative Schemes

C4 C6

Figure 5: Numerical methods used for wall-modeled IBM (see figure 3 for legend).

IV.

Test Cases

It is necessary to validate the implementation described in section III before its application in a jet noise study. A fully-developed, zero-pressure-gradient, attached flat plate turbulent boundary layer is a canonical test case to assess performance of equilibrium wall models due to its simplicity and availability of reference data from direct numerical simulations (DNS) and experiments. In this section, the current implementation of an equilibrium wall model in the framework of an IBM solver is tested for such a flat plate boundary layer. Although no IBM would be necessary in practice for such a simple geometry, these tests provides important insights in the ability of the current method in maintaining a turbulent boundary layer over a wall-modeled portion of an IB. The equilibrium wall model formulation used in the current study is similar to a body-conforming wall model implemented and validated previously by Aikens et al.41 . Therefore, behavior of the wall model formulation with respect to different grid resolutions or wall-model matching point heights is not studied in the current work. Rather, findings from Ref. 41 are used as a reference to select a background mesh and a wall-model matching point height (about 3 grid spacings away from an immersed wall, as mentioned in section III). The main purpose of the tests performed here is to assess the impact of applying the wall model on an immersed boundary, instead of applying it using a body-conforming mesh. It is desirable that results obtained from the IBM wall model be independent of how the immersion takes place, i.e., the results should be independent of how far the first point off the wall lies from the wall. The test cases presented in this section are designed to evaluate this aspect of the current implementation. A.

Setup of the Test Cases

All the simulations presented in this section are performed for a zero-pressure-gradient, quasi-incompressible (M∞ = 0.25), attached flat plate turbulent boundary layer. The reference length scale for the simulations is the boundary layer thickness (δ99 , based on a location where the streamwise velocity u is 99% of the freestream velocity U∞ ) at the inlet of the domain (δ99,i ). The domain used for the simulations is a simple rectangular parallelepiped with dimensions of 50 δ99,i × 15 δ99,i × 2.95 δ99,i in the streamwise (x), wall-normal (y), and spanwise (z) directions, respectively. The setup of the test cases, including the boundary conditions applied at different faces, is shown qualitatively in figure 6. The boundary conditions used are: a digital filter-based synthetic turbulent inflow42 at the streamwise domain inlet, a characteristic-based outflow38 15 of 32 American Institute of Aeronautics and Astronautics

Background grid

Characteristic-based outflow boundary condition

Stress-free symmetry boundary condition Turbulent inflow boundary condition


with viscous corrections39 at the streamwise domain outlet, a stress-free symmetry boundary condition42 at the upper surface, and flow periodicity in the spanwise direction. At the lower boundary, a flat plate wall is immersed inside the background mesh, and is treated with the wall-modeled IBM described in section III. A large streamwise domain extent is used to allow a sufficient adaptation distance51 for the synthetic turbulence fed in at the domain inlet to recover, as well as to be able to study the streamwise evolution of the boundary layer over a large distance. The wall-normal extent is larger than the 3 δ99,i to 5 δ99,i that should ideally be required for such simulations52, 53 . This is related to the turbulent inflow condition as well. The digital filter-based synthetic turbulence inflow method used in the present study produces spurious acoustic waves at the inlet42 . A turbulent boundary layer produces noise on its own as well. In a compressible flow solver such as the one used in the present study, proper treatment of these acoustic waves at the upper boundary of the domain is crucial. The stress-free symmetry boundary condition used at the upper surface is not appropriate for letting such waves propagate freely. The waves reflect from the upper boundary and the resulting interference of such acoustic waves can severely contaminate the solution. This can also lead to an irrecoverable divergence of the solution. A sponge zone could be used to damp these acoustic waves near the upper boundary41 . However, in the present study, a sponge zone is not used to avoid any possible adverse impact it could have on the boundary layer evolution. Instead, a large wall-normal domain extent is used with aggressive grid stretching in the wall-normal direction towards the upper boundary. Due to grid stretching, the acoustic waves originating at the inflow and those produced by the turbulent fluctuations in the boundary layer are increasingly weakened as they approach the upper boundary. Although this approach increases the simulation cost, it ensures a clean treatment of acoustic waves for maintaining solution stability. The spanwise domain extent is larger than the extent of (π/2) δ99,i that is typically considered sufficient for such simulations53, 54 .

Immersed boundary Figure 6: Setup of the flat plate turbulent boundary layer test case for wall-modeled IBM. The digital-filter based turbulent inflow condition42 used in this study can enforce a desired mean velocity profile, Reynolds stress profiles and correlated length scales in the velocity fluctuations it generates. These quantities are required as an input to the method. The mean y and z direction velocities are set to zero at the inflow. For the mean streamwise velocity profile, an expression for the incompressible law of wall, comprising of Reichardt’s expression47 for the viscous sublayer and the log layer, and a wake portion suggested by Coles55 is used. It is given by u+ =

2Π π + + ln (1 + 0.4y + ) + 7.23486 1 − e−y /11 − (y + /11) e−0.33 y sin2 η , + κ κ 2

(17)

with η = y/δ99,i , κ = 0.41, and Π = 0.5. Note that in Eq. 17, coefficients from Reichardt’s47 original expression are modified to better match the log-law section with Eq. 9. Although the test cases presented in this study are conducted at a low, quasi-incompressible Mach number, the solver is intended to be used at high compressible and even supersonic Mach numbers for jet noise studies. For compressible flows, u+ in Eq. 17 is replaced with an effective van Driest velocity u+ V D . For an adiabatic wall, the relationship between the actual velocity and the van Driest transformed velocity is given by Eq. 10. Using the definition ∗ ∗ and T∞ , the expression for B of adiabatic wall temperature, and non-dimensionalizing with respect to U∞ 16 of 32 American Institute of Aeronautics and Astronautics

in Eq. 10 can be simplified to


B = u M∞

(γ − 1) P rt , 2 2 + (γ − 1) r M∞

(18)

where r is the temperature recovery factor, and is commonly approximated with P r1/3 for turbulent flat plate boundary layers46 . The current study employs a value of 0.7 for both P r and P rt . The law of wall given in Eq. 17, combined with the van Driest transformation for compressible flows, is used to specify the mean streamwise velocity profile in the present study. The Reynolds number based on the boundary layer ∗ ∗ θi / μ∗∞ ) is chosen to be about momentum thickness at the inlet and freestream quantities (Reθ,i = ρ∗∞ U∞ 4,000. This allows the downstream flow to be compared with the detailed DNS data available from Sillero et al.52, 56 for an incompressible, zero-pressure-gradient, attached flat plate turbulent boundary layer in the Reynolds number range of Reθ = 2,780–6,650. The Reynolds number based on friction velocity at the inlet ∗ / μ∗∞ ) is about 1,360. The Reynolds number based on δ99,i and freestream quantities (Reτ,i = ρ∗∞ u∗τ,i δ99,i ∗ ∗ ∗ (Reδ99,i = ρ∞ U∞ δ99,i / μ∗∞ ) is about 34,285. For the chosen mean inflow velocity profile, a distance of 1 wall unit is equivalent to about 7.5 × 10−4 δ99,i and the friction velocity is uτ,i ≈ 0.0397 U∞ . Townsend’s laws are used as the reference input profiles for the normal Reynolds stresses57 . These profiles, as well as the reference Reynolds shear stress profile (from DNS results at Reθ = 1, 410) are taken from Ref. 58. The reference inputs for turbulent length scales are chosen to be {1.28, 0.25, 0.25} δ99,i in the {x, y, z} directions, respectively. In the previous wall-resolved studies of the currently used turbulent inflow method40, 42 , it has been observed that the downstream flow evolution is relatively independent of the length scales prescribed at the inlet, as long as the Reynolds number is large enough (about Reθ > 1,000) and the prescribed length scales are at least approximately close to the integral length scales in a physical flow. Note that although detailed information about length scales and turbulence statistics is available at Reθ = 4,000 from the DNS of Sillero et al. 52, 56 , no attempt is made to match the inflow conditions with the reference DNS. In jet noise studies, the turbulent inflow conditions at the inlet of a nozzle are rarely known from experimental measurements and some approximate inflow conditions have to be relied upon. It is important that the inflow condition and the wall model allow development of realistic turbulence in spite of beginning with relatively unphysical turbulence at the inlet, in order to be applicable in problems of practical interest. A triangulated surface of a rectangular parallelepiped, serving as the immersed boundary (IB), is placed such that its upper surface coincides with the y = 0 plane (see figure 6). The background mesh is placed such that its ymin plane is either at y = 0 (giving a body-conforming mesh) or below y = 0 (giving a body-non-conforming mesh). Therefore, every point in the ymin plane of the background mesh is a wallmodeled surface point or ghost point, depending on where the ymin plane of the background mesh is located. As shown in figure 6, the distance between the immersed wall surface at y = 0 and the first background grid point away from the immersed wall is denoted by dy1 . As mentioned earlier, it is desirable that the results be independent of how far the first point off the wall lies, i.e., independent of dy1 . To study the performance of the current implementation, a background mesh is constructed and it is placed at different offsets from the y = 0 plane to obtain different values of dy1 . Various grids used in the current study and the distances dy1 used for each of them are listed in table 2. Note that all the distances mentioned in wall units in this table are calculated with respect to the friction velocity based on the mean streamwise velocity profile at the inlet (uτ,i ). All the grids are uniformly-spaced in the x and z directions. The grids are uniformly-spaced in the y direction up to a height of 2 δ99,i , beyond which grid stretching is employed in the y direction. The Δy values tabulated in table 2 are the y grid spacings in the uniformly-spaced section near the ymin end of the domain. Grids 1 to 5 are identical in all regards, except in their y-direction placements, which introduces a different dy1 for different grids. When using an equilibrium wall model, the wall normal grid spacing is dictated mainly by the Reynolds number, the boundary layer thickness and the chosen location of the wall model matching point41 . The wall model matching point location is desired to be in the log layer of the mean streamwise velocity profile. It is suggested that the matching point height + + (see figure 3), where δ99 is the local δ99 in wall units41 . It has should be within the range 50 < l4+ < 0.1 δ99 also been found that having the matching point location a few grid spacings away from the wall (and not being the grid point right next to the wall) reduces the error in estimation of wall shear stress inherent in wall-modeled simulations due to under-resolution of the turbulence near a wall49 . In the current method, the matching point height (l4 in figure 3) is chosen to be 3 grid spacings away from an IB. At the inlet, for the currently chosen inflow conditions, the suggested range for wall-normal spacing near the wall can be + + found using 50 < l4+ = 3 Δ+ y < 0.1 δ99,i = 133.3, which gives 16.7 < Δy < 44.4. All the grids used in the



current tests utilize Δ+ y = 33.3 (or Δy = 0.025 δ99,i ) near the wall, which lies in the suggested range. Notice that in a wall-resolved simulation, the minimum wall-normal spacing is required to be Δy + ≤ 1. The use of a wall model allows a much larger spacing to be used, which reduces the number of grid points as well as reduces the computational cost since a larger timestep can be used without causing instabilities. The streamwise and spanwise resolutions are discussed next. The recommended range of grid spacings to resolve the outer region of a boundary layer is 0.04 δ99 ≤ Δx, Δy, Δz ≤ 0.067 δ99 11 . The Δy chosen from the wall model considerations is already lower than the requirements for resolving the outer layer. For grids 1 to 5, Δx and Δz are both chosen to be 0.05 δ99,i , which is within the recommended range to resolve the outer layer. For grid 6, Δx and Δz are coarsened by a factor of 2, which means that they are larger than the recommended maximum spacings. Note that all grids (1–6) utilize identical y direction grid spacings. The test with the coarse grid 6 is performed to assess the additional impact of using inadequate grid resolution, which is sometimes inevitable in jet noise studies to limit the computational cost. Note that for all the grids, the streamwise and spanwise grid resolutions in wall units (Δx+ and Δz + , respectively) are much finer than the values recommended by Piomelli and Balaras10 (Δx+ > 1,000 and Δz + > 500 ) when using a wall model for LES. They argue that when using a mesh much finer than their recommendations, the inner layer cannot be assumed to be governed by the Reynolds-averaged Navier-Stokes equations, an assumption inherent in equilibrium wall models. However, it has to be noted that having such a coarse mesh in the x and z directions does not allow resolution of the outer region of the boundary layer at relatively low Reθ values. Thus, their recommendation can only be followed at extremely high Reynolds numbers. For moderate to high Reynolds numbers of practical interest, it is necessary to apply an equilibrium wall model on meshes that are fine to an extent that the assumptions inherent in the wall model formulation are no longer completely valid. However, it is desirable to have a stable and relatively accurate solution in such situations as well. Therefore, the current tests are performed on meshes that do not strictly obey the recommendations of Piomelli and Balaras10 . Note that for grid 1, dy1 = Δy, which means that the grid is body-conforming with respect to the wall, and there is no immersion. Then, from grid 2 to grid 5, the value of dy1 is successively reduced by translating the grid in the −y direction. The lowest value of dy1 is encountered in grids 5 and 6, for which dy1+ = 1, i.e., the first point off the wall is located in the region that would have been the viscous sublayer in a wall-resolved simulation. Note that in the current simulations, the inner layer is not resolved, and the first grid point off the wall is excluded from the solution in the current method (see section III). Thus, the value of dy1 is varied over the allowable range for the chosen y direction grid resolution to study its impact on the results. It is important to note that in the current method, the wall model matching point is located at an identical height (at y = 0.075 δ99,i ) from the wall for all the grids. Table 2: Grid details for the flat plate turbulent boundary layer tests.

Grid

Nx

Ny

Nz

Δx δ99,i

Δx+

Δy δ99,i

Δy +

Δz δ99,i

Δz +

dy1 δ99,i

dy1+

1 2 3 4 5 6

1,000 1,000 1,000 1,000 1,000 500

128 128 128 128 128 128

60 60 60 60 60 30

0.05 0.05 0.05 0.05 0.05 0.1

66.7 66.7 66.7 66.7 66.7 133.3

0.025 0.025 0.025 0.025 0.025 0.025

33.3 33.3 33.3 33.3 33.3 33.3

0.05 0.05 0.05 0.05 0.05 0.1

66.7 66.7 66.7 66.7 66.7 133.3

0.025 0.01875 0.0125 6.25 × 10−3 7.5 × 10−4 7.5 × 10−4

33.3 25 16.7 8.3 1 1

The solution domain is initialized using fluctuations generated by the digital filter-based synthetic inflow method42 . This inflow generator can produce filtered 2-D slices (y-z planes) of random fluctuations that feature the required transverse length scales. The fluctuations at a given time-step are correlated to fluctuations at the preceding time-step to establish streamwise length scales42 . When generating initial conditions, a time-step of Δx/(0.8 U∞ ) is used to generate successive slices from the digital filter-based generator, which are then copied to streamwise planes of the domain, beginning at the last plane and moving towards the inflow plane. This assumes the validity of Taylor’s hypothesis with a mean convection speed of 0.8 U∞ throughout the boundary layer. Although this is a crude approach for generating initial 18 of 32 American Institute of Aeronautics and Astronautics


conditions, it has been found to result in lower transient times and a more stable start-up in our tests. The wall model boundary condition expects a turbulent boundary layer to be present and applies the appropriate wall shear stress. If a simple laminar flow-field is used as an initial condition, the imposition of a high wall shear stress predicted by a wall model can cause instability issues for such an initially laminar boundary layer. The time-step used for the simulation is Δt = 6 × 10−3 δ99,i /U∞ , which corresponds to a CFL number of about 1 in the wall-normal direction (calculated as max{(v + c)Δt/Δy}). The simulation is run for 50,000 time-steps to allow removal of transients resulting from the unphysical initial conditions and to allow synthetic turbulence fed in at the inlet to pass through the entire streamwise domain extent. A flow-through-cycle (FTC) can be defined as the amount of time required for a particle moving at the reference freestream speed (U∞ ) to travel the streamwise extent of the domain once. The interval allowed for transients corresponds to 300 δ99,i /U∞ , therefore consisting of 6 FTCs. The simulation is then continued for another 50,000 time-steps to gather flow statistics. Thus, the statistical averaging time period is equal to 300 δ99,i /U∞ or 6 FTCs as well. B.

Discussion of Results

All the statistical quantities discussed in this section are time-averaged, as well as averaged along the statistically homogeneous spanwise direction z (except for the calculation of two-point correlations, for which only time-averaging is employed). First, evolution of several boundary layer parameters is discussed. Figures 7a to 7d show the streamwise evolution of boundary layer thickness (δ99 ), displacement thickness (δdisp ), Reynolds number based on the momentum thickness (Reθ ), and skin friction coefficient (Cf ) for the different grids used in the current study. The Cf is estimated for the simulations by solving the log law expression (Eq. 9) for uτ , using the mean streamwise velocity at y = 0.09375 δ99,i which lies in the log region throughout the domain. The evolution estimated by a momentum integral analysis53 based on the mean velocity profile given by Eq. 17 is included in these plots, which indicates the expected behavior of a fully-developed turbulent boundary layer. For Cf , the corresponding evolution from the DNS by Sillero et al.52, 56 is also included. As a boundary layer grows in the downstream direction, its δ99 , δdisp , and θ increase, whereas Cf drops. The synthetic turbulence input at the inflow boundary is not physically realistic, but only an approximation that can trigger the development of realistic turbulence downstream51 . Therefore, a certain adaptation distance is required near the inflow boundary within which the boundary layer parameters can exhibit unphysical evolution trends, such as θ decreasing with the downstream distance. In the current plots, a domain length of about 10 δ99,i near the inlet should be ignored for this reason. This initial adjustment region can also introduce offsets in boundary layer parameters from the expected evolution42 . However, the evolution curve slopes in the fully-developed region (beyond about 10 δ99,i ) should agree favorably with the slopes of the expected evolution from the momentum integral analysis or the DNS data. Examining figures 7a to 7d, it is clear that the evolutions obtained on all the grids in the current simulations qualitatively follow the correct trend for all boundary layer parameters. However, the evolutions of δdisp , θ, and Cf show a significant dependence on the grid being used. A major source of this grid-based variability is the location of the first point off the wall or dy1 . Note that grids 1 to 5 have the same resolution but different dy1 , whereas grids 5 and 6 have the same dy1 but different resolutions (see table 2). In figures 7b to 7d, the variation in evolutions from grid 1 to grid 5 is very significant, however, the variation from grid 5 to grid 6 is relatively small. Therefore, the current method is unsuccessful in achieving a boundary layer growth that is independent of dy1 and it is not a suitable approach for performing a detailed analysis of turbulent boundary layers or as a predictive tool for skin friction coefficient. However, in jet noise studies, the boundary layer properties on the inner nozzle wall are rarely available from experimental studies as a reference. The essential purpose of the wall model, when being applied in such jet noise simulations, is not to accurately predict the boundary layer properties within the nozzle, but to carry realistic fluctuations in the outer region of the boundary layer that will give rise to a turbulent shear layer in the free jet downstream. It may be necessary to obtain a desired shear layer thickness for the jet at a specific downstream location. Although the boundary layer thickness growth (or decay in a favorable pressure gradient section of a nozzle) may not be predicted very accurately within the nozzle by the current method, a desired free shear layer thickness can still be achieved by tuning the boundary layer thickness at the inflow of the nozzle appropriately. Therefore, the current method can be used in jet noise studies as an approximate but very economic way of treating the boundary layer on the inner nozzle wall, provided it carries physically realistic fluctuations in the outer layer. Examination of turbulence statistics within the boundary layer is required to asses this aspect of the 19 of 32 American Institute of Aeronautics and Astronautics

current method, and it is performed next. 0.32

1.8

0.3 0.28 δdisp /δ99,i

δ99 /δ99,i

1.6

1.4

1.2

0.26 0.24 0.22 0.2

1

0.16 0

10

20 30 x/δ99,i

40

50

0

(a) Evolution of δ99 .

10

20 30 x/δ99,i

40

50

(b) Evolution of δdisp .

·10−3 3.4

7,000

3.2 6,000 Cf

Reθ


0.18

5,000

3 2.8 2.6

4,000 0

10

20 30 x/δ99,i

40

50

4,000

5,000

6,000

7,000

Reθ

(c) Evolution of Reθ .

Grid 1 Grid 3 Grid 6

2.4

(d) Evolution of Cf .

Grid 1 (with body-conforming wall model41 ) Grid 4 Momentum integral estimates

Grid 2 Grid 5 DNS52, 56

Figure 7: Comparison of streamwise evolution of boundary layer parameters. The mean streamwise velocity profiles at two streamwise locations are now discussed. Figures 8a and 8b show the velocity profiles in inner viscous wall units at streamwise stations corresponding to Reθ = 4,500, and Reθ = 5,500, respectively. The profiles from the DNS by Sillero et al.52, 56 as well as the log law curve from the law of wall are also included for reference. Note that at these Reynolds numbers, the extent of the log region is very small. The grid-based variability present in Cf (shown in figure 7d) introduces a significant variability in these inner-scaled velocity profiles as well. To remove the dependence on Cf and to focus more on the outer layer, the profiles are scaled with respect to outer parameters (U∞ and δ99 ) 20 of 32 American Institute of Aeronautics and Astronautics


25

25

20

20

u+

u+

and plotted in figures 9a and 9b. From these figures, it is clear that qualitatively, the profiles on all the grids follow the correct behavior and there is less scatter in the profiles compared to those in figures 8a and 8b. However, there are grid-based differences in the mean streamwise velocity at grid points closer to the wall. These differences tend to persists for about 50% to 60% of δ99 beyond which they are increasingly diminished towards the edge of the boundary layer. These near-wall differences are believed to stem from the dependence of wall shear stress imposed by the current method on dy1 .

15

10 1 10

15

102

10 1 10

103 y

+

103 y

(a) At locations corresponding to Reθ = 4,500.

Grid 1 Grid 2 Grid 4 Grid 6 DNS52, 56

102 +

(b) At locations corresponding to Reθ = 5,500.

Grid 1 (with body-conforming wall model41 ) Grid 3 Grid 5 log(y + )/0.41 + 5

Figure 8: Comparison of inner-scaled mean streamwise velocity profiles at different streamwise locations. The turbulence intensities in the x, y, and z directions, as well as the Reynolds shear stress (nondimensionalized with respect to the local mean streamwise velocity) are now discussed. Figures 10 and 11 plot these quantities at streamwise stations corresponding to Reθ of 4,500 and 5,500, respectively. In all the plots, a good qualitative agreement is observed for solutions from all the grids with the reference DNS52, 56 . Quantitatively, the turbulence intensities and the Reynolds shear stress are overpredicted close to the wall. However, they agree better with the DNS results beyond about 40% of δ99 . Again, grid-based variability is dominant in regions close to the wall and diminishes towards the outer edge of the boundary layer. This is linked to the aforementioned grid-based variability in the mean streamwise velocity (¯ u) close to the wall, since u ¯ is used to scale the turbulence intensities and the Reynolds shear stress. However, the good agreement present in the outer region of the boundary layer, observed in figures 10 and 11 for different grids and also with respect to the DNS results, is encouraging. In figures 7 to 11, results obtained using the equilibrium wall model implemented previously41 in our solver for body-conforming meshes are also included. These results are obtained using grid 1 since it is the only body-conforming mesh used in the current study. The results using the body-conforming wall model were expected to agree well with those using grid 5 with the current wall-modeled IBM. In grid 5, the first point where a solution is sought with the current wall-modeled IBM is approximately at a distance Δy from the IB, similar to that in grid 1 with the body-conforming wall model41 . This is due to the fact that on grid 5, the first point off the IB (which is skipped) is extremely close to the IB (at dy1+ = 1). However, the results obtained using the body-conforming wall model appear to agree better with those using grids 3 and 4 with the current wall-modeled IBM. The differences in the implementations between the body-conforming


0.8

0.8


u ¯/U∞

1

u ¯/U∞

1

0.6

0.6

0.4

0.4 0

0.5

1

1.5

0

y/δ99

1 2 4 6

1

1.5

y/δ99

(a) At locations corresponding to Reθ = 4,500.

Grid Grid Grid Grid

0.5

(b) At locations corresponding to Reθ = 5,500.

Grid 1 (with body-conforming wall model41 ) Grid 3 Grid 5 DNS52, 56

Figure 9: Comparison of outer-scaled mean streamwise velocity profiles at different streamwise locations.

wall model and the current IBM wall model are believed to be responsible for the differences in the results. In general, the body-conforming wall model matches the Cf evolution better with the momentum integral estimates (figure 7d), and therefore deviates less from the inner-scaled mean streamwise velocity profiles from the DNS (figure 8). For the turbulence intensities and the Reynolds shear stress, the results from the current method in general agree well with the body-conforming wall model in the outer region of the boundary layer. To better assess the flow features in the outer region of the simulated turbulent boundary layers, twopoint correlations are calculated for all three velocity fluctuations at a streamwise location corresponding to Reθ ≈ 4, 850 and at different wall normal heights of y/δ99 ≈ {0.3, 0.6, 0.9}. The DNS data by Sillero et al.52, 56 provide two-point correlations at these selected locations for comparison. Since the dy1 -based variability is prominent only near the wall, only grids 5 and 6 are considered in the two-point correlation analysis of the outer layer, which have a common dy1 but different grid resolutions (see table 2). The simulations with grids 5 and 6 were continued for an additional 300 δ99,i /U∞ to collect samples for computation of two-point correlations. The two-point correlation for streamwise velocity fluctuations with respect to a point at {x, y, z} is calculated as u (x, y, z) u (x + dx, y + dy, z + dz) . (19) Cu u (dx, dy, dz) = u (x, y, z) u (x, y, z) Analogous definitions are used for Cv v and Cw w . The resulting two-point correlations are plotted in figures 12, 13, and 14, corresponding to wall normal locations of y/δ99 ≈ {0.3, 0.6, 0.9}, respectively. These correlations provide insights about how the length scales present in the simulated boundary layers compare to those in the reference DNS. Integral length scales are defined based on the area integral under these correlation curves. In most of the plots, the results from grid 5 agree well with those from the DNS data. The most significant differences between the results on grid 5 and the DNS occur at y = 0.9 δ99 along the y direction (see figures 14d, 14e, and 14f). It is not clear why the DNS result show a nonzero correlation for a long distance in the +y direction at this location, which is not seen in the correlations from the currently simulated boundary layers. The y direction domain extent in the DNS studies52, 56 is 3.5 δ99,mid-domain , whereas it is 15 δ99,i in the current simulations. This difference in the wall-normal domain extents may be the reason behind the discrepancies seen in the correlations along 22 of 32 American Institute of Aeronautics and Astronautics


the y direction at a relatively large wall-normal height of y = 0.9 δ99 . However, from the good agreement observed in a majority of correlations, it is clear that grid 5 is carrying physically realistic large length scales in the outer region of the boundary layer. The two-point correlations obtained from grid 6 compare well with the DNS data qualitatively in all the plots. However, they usually decay slowly in comparison, implying that the large length scales present in the outer layer on grid 6 are larger than those present in the DNS or on grid 5. As mentioned earlier, grid 5 has adequate grid resolution in all directions to capture the flow structures in the outer layer, whereas grid 6 is coarser in the the x and z directions than the grid spacing limits recommended to resolve the outer layer. Therefore, the trend observed in the two-point correlations is not unexpected. However, it is important to note that resorting to coarser meshes to limit computational cost can result in deviation of large length scales in the outer layer from physically realistic values. To summarize, the results presented in this section indicate that the boundary layer growth and skin friction evolution obtained from the current wall-modeled IBM methodology depend significantly on the location of the first point off the wall, i.e., on dy1 . Therefore, this methodology is not accurate for detailed analysis of turbulent boundary layers or as a predictive tool for skin friction coefficient on a surface. Improving the formulation and implementation of the current method to bring the results closer to those obtained using the body-conforming wall model implemented in our solver previously41 , and reducing the dependence of the results on dy1 are goals for future research. However, the important feature necessary for jet noise simulations is the ability to carry realistic turbulent fluctuations in the outer region of the boundary layer that forms a turbulent shear layer in the jet plume downstream of the nozzle exit. From examination of the mean streamwise velocity profiles, turbulence intensities and Reynolds shear stress profiles, and two-point correlations for different velocity fluctuations at different wall-normal locations, it is shown that the current method performs reasonably well in maintaining a turbulent outer region for a boundary layer. Therefore, the current method can be considered to be a highly economic, although only approximate boundary condition for the inner immersed wall of a nozzle in jet noise studies.


·10−2

0.3 8 0.25 6 vrms /¯ u

u urms /¯

0.2 0.15

4

2

5 · 10−2 0

0

0.2

0.4

0.6 0.8 y/δ99

1

0

1.2

0

0.5

1

1.5

y/δ99

(a) Comparison of urms /¯ u.

(b) Comparison of vrms /¯ u.

0

0.25

0.2

0.15

u v /¯ u2

wrms /¯ u


0.1

−0.1

0.1

5 · 10−2

0

−0.2

0

0.2

0.4

0.6 0.8 y/δ99

1

1.2

(c) Comparison of wrms /¯ u.

Grid Grid Grid Grid

1 2 4 6

0

0.2

0.4

0.6 0.8 y/δ99

1

1.2

(d) Comparison of u v /¯ u2 .


Figure 10: Comparison of turbulence statistics at streamwise locations corresponding to Reθ = 4,500.


·10−2

0.3 8 0.25 6 vrms /¯ u

u urms /¯

0.2 0.15

4

2

5 · 10−2 0

0

0.2

0.4

0.6 0.8 y/δ99

1

0

1.2

0

0.5

1

1.5

y/δ99

(a) Comparison of urms /¯ u.

(b) Comparison of vrms /¯ u.

0

0.25

0.2

0.15

u v /¯ u2

wrms /¯ u


0.1

−0.1

0.1

5 · 10−2 −0.2 0

0

0.2

0.4

0.6 0.8 y/δ99

1

1.2

(c) Comparison of wrms /¯ u.

Grid Grid Grid Grid

1 2 4 6

0

0.2

0.4

0.6 0.8 y/δ99

1

1.2

(d) Comparison of u v /¯ u2 .


Figure 11: Comparison of turbulence statistics at streamwise locations corresponding to Reθ = 5,500.


1

0.5

0.5

0.5

0

Cw w

1

Cv v

C u u

1

0 −1

0

0 −0.4−0.2 0

1

0.2 0.4

dx/δ99

(b) Cv v vs. dx/δ99 .

(c) Cw w vs. dx/δ99 .

1

1

1

0.5

0.5

0.5

Cw w

(a) Cu u vs. dx/δ99 .

Cv v

0 0

0.5 dy/δ99

0 −0.2

1

0

0.2

−0.2

0.4

dy/δ99

0

0.2

(e) Cv v vs. dy/δ99 .

(f) Cw w vs. dy/δ99 .

1

1

1

0.5

0.5

0.5

Cv v

(d) Cu u vs. dy/δ99 .

0 −1 −0.5

0 0

0.5

dz/δ99 (g) Cu u vs. dz/δ99 .

1

0.4

dy/δ99

Cw w

C u u

dx/δ99

0

C u u


dx/δ99

−0.4−0.2 0

0.2 0.4

0 −0.4−0.2 0

0.2 0.4

dz/δ99 (h) Cv v vs. dz/δ99 .

−0.4−0.2 0

0.2 0.4

dz/δ99 (i) Cw w vs. dz/δ99 .

Grid 5 Grid 6 DNS52, 56 Figure 12: Two-point correlations for different velocity fluctuations with respect to a location at Reθ ≈ 4,850, at mid-span, and at y/δ99 ≈ 0.3.


1

0.5

0.5

0.5

0

Cw w

1

Cv v

C u u

1

0 −1

0

0 −0.4−0.2 0

1

0.2 0.4

dx/δ99



1

1

1

0.5

0.5

0.5

−0.5

Cw w


Cv v

0 0

0.5

0

−0.5

1

dy/δ99

0

0.5

−0.5

1

dy/δ99

0

0.5

dy/δ99



1

1

1

0.5

0.5

0.5

Cv v


0 −1 −0.5

Cw w

C u u

dx/δ99

0

C u u


dx/δ99

−0.4−0.2 0

0.2 0.4

0 0

0.5

dz/δ99 (g) Cu u vs. dz/δ99 .

1

0 −0.4−0.2 0

0.2 0.4


−0.4−0.2 0

0.2 0.4




1

0.5

0.5

0.5

0

Cw w

1

Cv v

C u u

1

0 −1

0

0 −0.4−0.2 0

1

0.2 0.4

dx/δ99



1

1

1

0.5

0.5

0.5

Cw w


Cv v

0 0 dy/δ99

0

1

0 dy/δ99

1

0 dy/δ99



1

1

1

0.5

0.5

0.5

Cv v


0

Cw w

C u u

dx/δ99

0

C u u


dx/δ99

−0.4−0.2 0

0.2 0.4

0 −1

0 dz/δ99

(g) Cu u vs. dz/δ99 .

1

1

0 −0.5

0

0.5


−0.5

0

0.5





V.

Considerations for Jet Simulations

The wall-modeled IBM implementation described in section III is intended to be applied on the inner nozzle wall in jet noise simulations, as previously shown in figure 2. The current wall-modeled IBM is based on skipping a layer of fluid points next to the wall-modeled ghost or surface points. This introduces certain issues that have to be considered when jet noise studies are performed. The first issue concerns the design of the background grid. In our wall-resolved IBM implementation9 , the background mesh can be completely independent of the immersed boundary geometry. A wall-resolved ghost/surface point can be a wall-resolved ghost/surface point in multiple grid directions. However, the wall-modeled IBM described in the current work is not as flexible and requires a wall-modeled ghost/surface point to be a wall-modeled ghost/surface point only in a single grid direction (although such a point can additionally be a wall-resolved ghost/surface point in other grid direction/s). If a wall-modeled ghost/surface point is allowed to be a wall-modeled ghost/surface point in more than one grid direction, it can give rise to multiple layers of skipped fluid points near an IB, which is not allowed in the current implementation. The configuration shown previously in figure 2 could give rise to this issue. To see why such a setup could fail with the current method, a closer look near the immersed surface is provided in figure 15. In this figure, the wall-modeled ghost point on the right is a wall-modeled ghost point in both ξ and η directions. This causes two successive skipped fluid points to occur along the left η grid line, which is not allowed.

Immersed boundary Wall-modeled ghost point Skipped fluid point

Figure 15: Example configuration where the current wall-modeled IBM implementation would fail (related to figure 2). To avoid this issue, the background mesh has to be designed to conform to the inner boundary of a nozzle. Figure 16 shows one such example in which each wall-modeled ghost point near the inner nozzle wall is a wall-modeled ghost point only along a single direction (η). Note that the outer nozzle surface, as well as the nozzle lip are treated with the wall-resolved IBM9 , and therefore there are no restrictions on the background grid to conform to the outer shape of a nozzle. Requiring background grid lines to align with the inner nozzle wall nullifies the point of using an IBM to some extent. However, the method allows inclusion of moderately sharp corners (such as the one in figure 16 where the converging portion begins) with smooth grid lines, since the sharp corner can be immersed between two smooth grid lines. Also, it allows a wallmodel boundary condition to be applied on irregularly-shaped inner nozzle surfaces (such as in nozzles with chevrons) without having to create a multi-block grid separately for the nozzle. A body-conforming wallmodel boundary condition cannot allow inclusion of sharp corners with high-order structured-grid solvers and would require significantly more grid generation effort to generate a body-conforming mesh for the interior of a complex nozzle (e.g., involving chevrons). When using a configuration such as the one in figure 16, one additional issue is encountered. Since a layer of fluid points next to the wall-modeled ghost/surface points is skipped from the simulation, the fluid points lying outside the nozzle and neighboring such skipped fluid points require a special boundary condition at the near-nozzle end. In figure 16, a skipped line of fluid points is shown in red color. The fluid point lying next to these skipped points in the ξ direction is shown in green color. A one-sided second-order explicit FD scheme is applied at such a fluid point, with the derivative stencil extending in the fluid region in the +ξ direction. The solution at such a fluid point has to be supplemented with a special boundary condition to avoid divergence. In the current implementation, after every sub-step of the RK4 time-advancement, the solution at such a fluid point is reset using an average of the solutions from all the fluid points that are immediately next to it in all directions and which do not require such a special boundary condition themselves. This simple procedure has been found to produce a stable solution at such points.


Immersed Nozzle


Skipped fluid points Fluid point requiring special boundary condition Figure 16: Example configuration where the current wall-modeled IBM implementation would work correctly (related to figure 2).

VI.

Summary

A sharp immersed boundary method, previously implemented in a high-order structured-grid FD-based LES solver9 , is extended to incorporate a wall model boundary condition for a specified portion of an immersed boundary. The simple linear extrapolation approach used in the ghost-point-based IBM for wallresolved simulations is not valid for a wall-modeled IBM, since the large gradients in flow variables near an IB are not resolved on a coarse grid typical of wall-modeled simulations. Therefore, the treatment of flow extrapolation to ghost points is changed significantly compared to that used for the wall-resolved IBM in Ref. 9. In the present work, the extrapolation treatment adopted for the wall-modeled portion of an IB ensures that the FD scheme applied at the first fluid point away from an IB where a solution is sought (which is the 2nd fluid point off the IB in the current method) approximates the slope of a curve that passes through the required boundary condition at the IB. The first layer of fluid points close to an IB is excluded from the simulation to avoid the adverse effects due to the possibility of the first point off the IB lying very close to the wall (in the viscous sublayer). One of the necessary boundary conditions is the instantaneous wall shear stress at the IB. In the present work, an equilibrium wall model formulation based on the law of wall41 is adopted for this purpose. Use of a wall model implies insufficient resolution of the inner region of a turbulent boundary layer and the use of high-order FD schemes is not appropriate near the immersed boundaries. Therefore, lower-order explicit FD schemes are employed near an IB and high-order compact FD schemes are used away from an IB. The same reasoning is applied to the selection of spatial filtering schemes as well. The resulting implementation is tested using LES of a quasi-incompressible, zero-pressure-gradient, attached flat plate turbulent boundary layer. The results indicate that the boundary layer growth and skin friction evolution obtained from the current wall-modeled IBM methodology are dependent on the location of the first point off the wall (dy1 ). Therefore, this methodology cannot be used for a detailed study of turbulent boundary layers. In the future, the formulation and/or the implementation will be reconsidered to identify ways to reduce the dependency on dy1 and to bring the results closer to those obtained using the body-conforming wall model implemented in our solver previously41 . However, quantities such as the skin friction coefficient for the boundary layer on the inner nozzle wall are rarely available from experimental jet noise studies, and therefore an accurate imposition of these quantities is not possible even with wall-resolved simulations. The important flow phenomenon for jet noise simulations can be considered to be the existence of realistic turbulent fluctuations in the outer region of the boundary layer, since those significantly influence the turbulent shear layer in the jet plume downstream of the nozzle exit. From examination of various turbulence statistics such as the mean streamwise velocity profiles, turbulence intensities and Reynolds shear stress profiles, and two-point correlations for different velocity fluctuations at different wall-normal locations, it is shown that the current method performs reasonably well in maintaining a turbulent outer region for a boundary layer. Therefore, the current method is considered a useful and low-cost, although


approximate alternative to explicitly resolving the turbulent boundary layer on the inner immersed wall of a nozzle in jet noise simulations. Additional issues that must be considered when applying the current method in jet simulations have been discussed.

Acknowledgments


We sincerely thank Chandra Martha of Intel Corp., Yingchong Situ of Google Inc., and Kurt Aikens of Houghton College, whose prior implementation of a large eddy simulation code was used as the basis for the current work. We also thank Zhiyuan Li of the Computer Science Department at Purdue University for his support. This material is based upon work supported by the National Science Foundation (NSF) under grant number OCI-0904675. The Carter and Rice clusters of the Rosen Center for Advanced Computing (RCAC) were utilized for the presented simulations, supported by an agreement through Information Technology at Purdue (ITAP), West Lafayette, Indiana.

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