Mina R. Mankbadi, Manan A. Vyas, James R. DeBonis, and Nicholas J. Georgiadis

Glenn Research Center, Cleveland, Ohio

Evaluation of Infl ow Turbulence Methods in Large-Eddy 

Simulations of a Supersonic Boundary Layer

NASA/TM—2018-219966

August 2018

https://ntrs.nasa.gov/search.jsp?R=20180006518 2018-12-09T17:42:36+00:00Z
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Mina R. Mankbadi, Manan A. Vyas, James R. DeBonis, and Nicholas J. Georgiadis

Glenn Research Center, Cleveland, Ohio

Evaluation of Infl ow Turbulence Methods in Large-Eddy 

Simulations of a Supersonic Boundary Layer

NASA/TM—2018-219966

August 2018

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This work is supported by NASA’s Transformational Tools and Technology (TTT) project. Computational resources were provided 

by the NASA High-End Computing (HEC) program through the NASA Advanced Supercomputing (NAS) Division at Ames 

Research Center.

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This report is available in electronic form at http://www.sti.nasa.gov/ and http://ntrs.nasa.gov/

This work was sponsored by the 

Transformative Aeronautics Concepts Program.
NASA/TM—2018-219966 1

Evaluation of Inflow Turbulence Methods in Large-Eddy 

Simulations of a Supersonic Boundary Layer

Mina R. Mankbadi, Manan A. Vyas, James R. DeBonis, and Nicholas J. Georgiadis

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

Wall-resolved Large-eddy simulations (LES) were used to investigate inflow turbulence of a 

supersonic boundary layer. Two high-order structured Navier-Stokes solvers were used to calculate a 

Mach 2.29 boundary layer at RReeθθ = 4,640. The purpose of this work is to evaluate and compare two 

different synthetic inflow turbulence methodologies through careful examination of the Reynolds stress 

tensor, skin friction coefficient, and adjustment length. Boundary-layer inflow turbulence was simulated 

using both the Synthetic Eddy Method (SEM) and Digital Filtering (DF). The LES results showed that 

both methods reasonably account for the effects of inflow turbulence. They demonstrated good agreement 

with experimental data for the mean and fluctuating velocities. In particular for SEM, a length scale of

δδ/5 matched the experimental skin friction coefficient. Whereas, very small length scales such as δδ/10

or δδ/20 attenuated the overall turbulence levels. In particular for DF, the implementation was compared 

between two flow solvers. Although the Reynolds tensor profiles compared well with experiment, the 

values of the skin-friction coefficient differed by 15 percent and were tabulated. Both DF and SEM 

adjustment lengths were 6δδ and did not significantly differ between simulations. 

Nomenclature

CCff skin friction coefficient 

EE total energy per unit mass

ff frequency

kk turbulent kinetic energy

κ wave number

LL SEM length scale 

MM freestream Mach number

NN number of eddies

PP pressure

qqii heat flux

RR Reynolds number

TT temperature

tt time

UU mean velocity

uuii Favre-filtered velocity

uuττ friction velocity

uu+ dimensionless mean velocity

uuıı

′

uuȷȷ ������′/uuττ

2 dimensionless Reynolds streamwise normal stress

xxii Cartesian coordinate

yy+ wall coordinate 
NASA/TM—2018-219966 2

ττiiii stress tensor

ρρ density

δδ∗ displacement thickness

δδ boundary layer thickness

Θ momentum thickness

Subscripts:

0 total condition

ww wall properties

∞ freestream properties 

i index

j index

Superscripts:

+ wall coordinates

′ fluctuating

1.0 Introduction

In recent years, there has been interest in accounting for the effects of inflow turbulence when 

performing high-fidelity computational fluid dynamics simulations, such as those described in Klein

et al., (Ref. 1) Lund et al., (Ref. 2) and DiMare et al. (Ref. 3). This was prompted by awareness of the 

need for unsteady inflow boundary conditions in order to obtain an accurate scale-resolving solution. It is 

well known that simulating natural transition in LES is computationally impractical due to the large 

computational costs. Neither does artificial tripping address this need, since it produces mixed results that 

are imprecise and arbitrary (Ref. 1). Recycling and rescaling, proposed by Lund et al. (Ref. 2), was the 

first practical solution. But, it requires an ancillary LES known to be computationally expensive. This 

work is motivated by the need for a less-expensive and more user-friendly turbulent inflow technique. 

The purpose of this work is to evaluate and compare two different synthetic inflow turbulence 

methodologies through careful examination of the Reynolds stress tensor, skin friction coefficient, and 

adjustment length.

Specifying unsteady turbulence at inflow boundaries is an active area of research (Refs. 1 to 13).

Various methods over the past two decades have been presented to the community with differing degrees 

of cost, accuracy, and complexity. The simplest inflow turbulence methodology imposes random 

fluctuations that are limited by amplitude to satisfy a given Reynolds stress tensor. Although it is costeffective, it often fails to produce sustained turbulence (Refs. 6 and 7). The turbulent kinetic energy is 

distributed equally across all frequencies. This is contrary to real turbulence where low frequency 

turbulence (large length scales) have more energy than high frequency turbulence (small length scales). 

Consequently, the turbulence from this method is not consistently correlated and decays rapidly after the 

inflow plane due to the absence of large length scale turbulence (Ref. 7).

The recycling and rescaling method uses an auxiliary boundary layer simulation and boundary layer 

scaling laws to generate the inflow profile (Ref. 2). To reduce the cost of the auxiliary simulation, the 

outflow profile is recycled by rescaling it to the inflow conditions. There are two limitations to this 

method, firstly, a spurious low frequency mode is introduced based on the distance between the recycling 

station and inflow plane (Refs. 8 and 9). Secondly, the boundary layer scaling laws require an equilibrium 

region. The high computational cost and possible contamination of the solution with low frequency 

modes inspired a class of synthetic turbulence methods. In turbulence, most of the energy is contained in 
NASA/TM—2018-219966 3

the large scales and then energy is transferred to smaller scales via a cascade. Some synthetic turbulence 

methods attempt to mimic this energy-exchange mechanism (Ref. 13).

This work will evaluate in particular the synthetic turbulence methods of Synthetic Eddy Method

(SEM) and Digital Filtering (DF) (Refs. 1 and 12). They are stochastic algorithms that generate an 

instantaneous velocity profile from a specified time-averaged Reynolds stresses. The assumption that 

synthetic turbulence methods make is that a turbulent flow can be approximated by reproducing a set of 

low-order statistics such as mean velocities and Reynolds stresses. The artificially generated structures 

obtained from SEM or DF do not necessarily satisfy the Favre-fitered Navier-Stokes equations being 

solved in an LES. Consequently, these methods require a redevelopment or adjustment region. An 

elongated adjustment region has an added computational cost that may be comparable to the 

rescaling/rescaling method, hence it is worth investigating this adjustment region.

This work uses the test problem of a supersonic boundary layer to assess the effectiveness of 

synthetic turbulence methods. The experiments of Dussauge et al. (Ref. 14) were used for validation and 

as a measure of the effectiveness of: (1) Synthetic Eddy Method; and (2) Digital Filtering. The length of 

the adjustment region necessary for synthetic inlet turbulence to recover from modelling errors will be 

estimated herein. The effect of varying the integral length scales of the turbulent fluctuations will be 

studied and recommendations will be made for choosing these parameters. Both SEM and DF inflow 

turbulence methodologies are evaluated within the Wave-Resolving LES (WRLES) code, henceforth 

referred to as WRLES-SEM and WRLES-DF. Another flow solver FDL3DI is used to establish 

confidence in the robustness of DF with respect to numerical scheme, henceforth referred to as FDL3DI-DF.

This paper is organized as follows. Section 1.0 gives a quick overview of the governing equations of 

Implicit LES and synthetic turbulence methodologies. Section 2.0 describes the supersonic boundary 

layer and experiment. Section 3.0 describes the flow solvers, boundary conditions, and wall-resolved grid. 

The results of this work presented in Section 5.1 will examine the effect of the SEM length scale on the 

Reynolds stresses, skin friction coefficient, and adjustment length. Section 5.2 will be a comparison of 

SEM and DF inside the WRLES code. Section 5.3 will compare the DF implementation across two LES 

solvers to establish confidence in our results.

2.0 Formulation

2.1 Navier-Stokes

In Large eddy simulations (LES), large-scale structures are resolved and a subgrid scale model is used 

to represent the scales that cannot be captured due to insufficient grid. Implicit LES (ILES) calculates the 

large-scale structures but does not use a subgrid scale model. Instead, the numerical dissipation present in 

the simulation, serves as the sub-grid closure. Since the numerical dissipation scales with the grid size, 

ILES provides a seamless transition to DNS as the grid is refined. All of the work herein uses the ILES 

approach. The Navier-Stokes equations are solved while the viscosity and thermal conductivity are 

allowed to vary with temperature. The continuity, momentum, and energy equations are:

∂∂ρρ

∂∂ +

∂∂

∂∂xxii

(ρρuuii) = 0, (1)

∂∂(ρρuuii)

∂∂ +

∂∂

∂∂xx �ρρuuiiuu � +

∂∂PP

∂∂xxii

= ∂∂ττiiii

∂∂xx , (2)

∂∂

∂∂ +

∂∂

∂∂xx �uu EE + uu PP� = ∂∂∂∂xx �uuiiττiiii� +

∂∂qq

∂∂xx . (3)

The velocity is denoted as uuii, while the density and pressure are ρρ and PP, respectively. The viscous stress 

tensor is ττiiii. The effect of the numerical filter on the flow solution was explored in a previous study 

(Refs. 4 and 5).
NASA/TM—2018-219966 4

2.2 The Synthetic Eddy Method (SEM)

The Synthetic Eddy Method (SEM) by Jarrin et al. (Refs. 12 and 13) is applied to the boundary layer 

under consideration to simulate the effects of inflow turbulence. It requires four inputs: (1) a Reynolds 

stress tensor, (2) the mean flow, (3) a turbulent length scale, and (4) a convective velocity. The SEM 

method represents the turbulent flow field as a collection of eddies, and convects these eddies through a 

box around the inflow plane. Each individual eddy has a contribution to the instantaneous velocity

component at a given point, based on its distance from that point and the turbulent length scale. This 

ensures that fluctuations being added to the mean flow are correlated. The instantaneous component is 

then added to the mean flow to obtain the total instantaneous velocity. SEM scales the velocity 

fluctuations according to the Reynolds stress tensor in such a way, that when the flow is time-averaged, 

the original Reynolds stress tensor is retrieved at the inflow plane. A detailed formulation can be found in 

Reference 12.

The original work of Jarrin et al. (Refs. 12 and 13) does not cover compressible cases where the 

temperature and density profile are varying within the boundary layer. For this work, the mean density 

and pressure profiles were specified at the SEM inflow boundaries based on the precursor RANS 

simulation. No fluctuating pressure or density signal was added to the Reynolds averaged mean values of 

pressure and density. The method appeared to be sensitive to the length scale especially when the length 

scale was too large (Refs. 4 and 5). In these instances, it failed to replicate the Reynolds stress tensor at 

both the inflow plane and downstream stations. The original formulation adjusted the length scale in 

regions where the grid was too coarse. Here, the length scale was not adjusted based on the maximum 

grid size, instead it was best to ensure that the grid was fine enough to resolve the specified length scale. 

The convective velocity was set to half of the freestream value, UUcccccccc = UU∞/2.

2.3 Digital Filtering (DF)

The Digital Filtering (DF) method was originally proposed by Klein et al. (Ref. 1). The DF method is 

applied to the boundary layer under consideration to simulate the effects of inflow turbulence. It requires 

four inputs: (1) a Reynolds stress tensor, (2) the mean flow, and (3) a turbulent length scale in each 

direction, and (4) a convective velocity. In DF, a filter with a width corresponding to the specified length 

scale is applied to a set of random numbers distributed on the inflow plane. The filtering process provides 

structure to the random signal, which serves as the artificially generated turbulent structure. This signal is 

scaled by the input Reynolds stresses to obtain the instantaneous velocity components, that when 

averaged provide the specified Reynolds stresses. The data at the current time step is correlated with 

previous time steps using a time correlation function. Originally, a Gaussian two-point correlation 

function was used. 

The implementation herein follows the work of Xie and Castro (Ref. 11). The original method 

employed 3D random data and filtering, whereas, the method of Xie et al. (Ref. 11) requires only 2D 

filtering operations on the inflow plane. In addition, an exponential correlation function rather than a 

Gaussian is used to advance the solution in time. A further modification is made in this work. Here, the 

filtering process is performed on the computational grid, rather than a uniform grid, avoiding the need for 

interpolation onto the computational grid. The filter width is scaled with the local grid size in order to 

produce the correct length scale. The mean density and pressure profiles were specified, based on the 

precursor RANS simulation. No fluctuating pressure or density signal was added to the Reynolds 

averaged mean values of pressure and density. The integral length scale of the filter chosen was 1/2 of the 

boundary layer height in all three directions. More details about the 2D implementation are provided in 

Vyas et al. (Ref. 15).
NASA/TM—2018-219966 5

TABLE 1.—FLOW CONDITIONS

Parameter Value

MM∞ 2.29

UU∞, mm/ss 545.0

PP∞ , PPPP 50663

TT0,KK 300

δδ99, mmmm 10.0

ΘΘ, mmmm 0.87

δδ∗, mmmm 3.0

RReeδδ 53,420

RReeθθ 4,640

CCff 2.1E-3

3.0 Boundary-Layer Description

The boundary layer simulated here corresponds to the experiments of Dussauge et al., (Ref. 14) which 

were carried out in the turbulent supersonic wind tunnel at the Institut Universitaire des Systemes 

Thermiques Industriels (IUSTI). Table 1 shows the experimental flow conditions used for the LES. Since 

the Reynolds number is moderate at 53,420, we were able to match it without exhausting numerical 

resources. A precursor 2D RANS simulation was used to develop a boundary layer thickness of 10 mm 

matching these operating conditions. The resulting Reynolds stress tensor and mean flow profiles were 

used as inputs for the synthetic turbulence methodologies under examination. The experimentalists 

gathered Reynolds stress information at their xx = 280 mmmm station which corresponds to the LES station 

of xx/δδ = 10. Mean flow and stress profiles are compared with the experimental PIV data for validation 

purposes.

4.0 Computations

4.1 Flow Solver: WRLES

Both the Synthetic Eddy Method and Digital Filtering are built into the high-order Wave-Resolving 

LES (WRLES) code (Refs. 16 and 17) SEM and DF are compared within the same framework to ensure a 

consistent and fair assessment. WRLES is a finite difference code which contains central-differencing 

schemes that employ up to 13 point stencils. A Dispersion Relation Preserving (DRP) scheme developed 

by Bogey & Bailly (Ref. 18) was used here. The stencil size is 11 points and is formally fourth-order 

accurate. While a traditional fourth-order scheme uses fewer points in the stencil, the additional points in 

the DRP scheme are used to minimize the dispersion error. It is an explicit central scheme similar to the 

classic tenth-order central scheme but with weights carefully chosen to minimize the dispersion error. 

Since it is a central differencing scheme, it is not inherently dissipative. A spatial filter, developed 

specifically to match the resolution of the differencing scheme, is used to preserve numerical stability. A 

coefficient multiplying the effect of the filter has been added, so that the magnitude of the dissipation can 

be controlled. The DRP filter coefficient used was 0.5. Temporal discretization is performed using a lowdissipation and low-dispersion forth-order Runge-Kutta algorithm summarized in Berland et al. (Ref. 19).

WRLES uses the Message Passing Interface (MPI) standard to divide the domain across nodes and 

the Open Multi-Processing (OpenMP) directive to parallelize work across the cores on each node. In 

order to maintain order accuracy across interfaces, WRLES uses a 10-point overlapping, point-matched 

zone coupling.
NASA/TM—2018-219966 6

4.2 Flow Solver: FDL3DI

Only digital filtering was investigated within the FDL3DI code (Rizzetta et al., (Ref. 20) Visbal & 

Gaitonde, (Ref. 21) Visbal & Rizzetta (Ref. 22). FDL3DI is a finite difference code that employs a 

compact scheme to discretize the spatial derivatives in the governing equations. For the present 

computation, the viscous fluxes are computed using the 6th order compact scheme. Near the boundaries 

the order of the scheme is reduced to fourth order compact. The time marching is accomplished by 

incorporating the second-order accurate backward-implicit method of Beam and Warming (Ref. 20). An

8th order Pade-type filer is used for filtering, that requires a 9-point stencil. It is a nondispersive spatial 

filter and has been shown to be superior to the use of artificial dissipation. FDL3DI also uses the parallel 

MPI standard. The order is preserved across blocks through the utilization of an 11-point overlap.

4.3 Boundary Conditions

The boundary condition (BC) imposed on the top-wall is a nonreflecting characteristic freestream that 

allows shocks to leave the domain. On the inflow plane, digital filtering is applied for the entire extent of 

the transverse domain, whereas SEM is applied up to yy/δδ~1.2. After which a supersonic inflow boundary 

condition specifying the Mach number, pressure, and temperature is used. Both flow solvers use a 

supersonic outflow boundary condition that extrapolates values from the interior. An adiabatic viscous 

wall boundary condition was used. A periodic boundary condition was applied in the spanwise-direction

for all simulations.

4.4 Grid

The grid was kept fixed for both flow solvers and inflow turbulence methodologies. Grid details such 

as the number of points, wall spacing, domain size, and minimum/maximum grid spacing are provided in 

Table 2. The grid used is structured as shown in Figure 1. The coordinates are nondimensionalized by the 

boundary layer height, δδ. The width of the spanwise domain was 5δδ with uniform spacing. The total 

length of the section is 30δδ with grid-stretching at 20δδ towards the outflow boundary condition. The 

mesh uses 67.7 million points to resolve the boundary layer. The smallest grid spacing in the wall normal 

direction corresponds to a yy+ value of 0.7 to ensure that the viscous sub-layer is resolved. About 187 

points in yy were used within the boundary layer itself.

TABLE 2.—GRID PROPERTIES OF THE BOUNDARY LAYER MESH

Parameter Grida

xx yy zz

Number of points 1025 257 257

Wall spacing (ΔΔxx+, ΔΔyy+ΔΔzz+) 15 0.68 13

Length of domain 30 4 5

Minimum spacing 0.021 0.001 0.0195

Maximum spacing after xx/δ = 20 0.465 0.078 0.0195

 aNondimensionalized by δδ

Figure 1.—Boundary conditions and dimensions of the LES supersonic boundary layer simulation

(only every 8th point is plotted).

4δδ

30δδ

yy

xx Supersonic 

Outflow

Viscous Wall

Non-reflecting Freestream

Inflow 

Turbulence 
NASA/TM—2018-219966 7

5.0 Results

Three sub-sections of results will be presented: (a) The effect of the SEM length scale on the 

boundary layer will be investigated. (b) Compares the Synthetic Eddy Method to Digital Filtering within 

the same code. (c) A code-to-code comparison of digital filtering will be conducted. LES best practices 

such as time-averaging and spanwise-averaging described in Georgiadis et al. (Refs. 24 and 25) were 

implemented herein. The solution was initialized to freestream conditions, then run for 30 flow-through 

periods to allow the inflow turbulence boundary condition to establish a turbulent boundary layer. After

which data for time-averaging was gathered for 15 flow-through periods. Turbulent statistics were 

collected over the entire domain, after which they were spanwise averaged. Both FDL3DI and WRLES 

used the ILES approach. A precursor RANS was used as an input for SEM and DF.

5.1 SEM Length Scale Study: WRLES-SEM

The purpose of this subsection is to assess the effect of the SEM length scale on the streamwise 

velocity, Reynolds stresses, and adjustment length. All of these simulations are done in the WRLES-SEM 

framework. Three length scales, LL = δδ/5, δδ/10, δδ/20, were examined. For the length scale of LL = δδ/5, 

Figure 2 depicts the mean velocity profiles in wall scaled coordinates. The flow adjusts from the artificial 

structures imposed at the boundary to realistic structures supported by the LES equations. The variation in 

the boundary layer profiles are caused by the variations in the local friction velocity. As the flow moves 

downstream from xx/δδ = 0 to 4 the scaled boundary layer edge velocity (uummmmmm

+ ) adjusts from 24 to 22. 

All of the profiles downstream of xx/δδ = 4 are no longer changing significantly and are a good match 

with the experimental data in both the outer and inner portions of the boundary layer.

Figure 3 depicts a similar trend for the Reynolds stress profiles at downstream stations x/δ = 2, 4,

6, 8, 10, and 12. In Figure 3(b) to (d), the normal stresses at xx/δδ = 0 peak at 3.2. These values come from 

the 2D precursor RANS simulation which does not differentiate between the normal stresses. As such, the 

turbulent kinetic energy is distributed equally across the streamwise (Figure 3(b)), transverse

(Figure 3(c)), and spanwise stresses (Figure 3(d)). This is clearly a limitation of the RANS methodology. 

LES does not have this limitation since it computes these quantities directly. SEM artificially generates 

turbulence, and although it matches the input Reynolds tensor, it does not necessarily satisfy the NavierStokes equations instantaneously. Consequently, there is an adjustment region between where the peak 

Reynolds stress adjusts from the RANS imposed values at the inflow to satisfy the LES equations.

Figure 2.—Time averaged streamwise velocity for 

the SEM LL = δδ/5 case.
NASA/TM—2018-219966 8

In Figure 3(a), the Reynolds shear stress is shown to converge after xx/δδ = 6. Figure 3(b) depicts the 

streamwise stress which has a well predicted peak of 8.28 at x/δ = 6 that matches the experimental peak of 

8.44. Figure 3(c) illustrates the transverse stress which is slightly over predicted by the simulation.

Figure 3(d) depicts the spanwise stress which continuously drops with downstream axial location. 

Comprehensively, the flow is adjusting to the synthetic turbulence between xx/δδ = 0 to 6. 

The former length scale of LL = δδ/5 is compared with two new simulations of length scales LL = δδ/10

and LL = δδ/20. The mean velocity profiles at x/δ = 10 are shown in Figure 4. There is an observable 

difference in the boundary layer profiles. The LL = δδ/5 case matches with experiment very well. In 

contrast, the LL = δδ/10 and LL = δδ/20 cases over predict velocity magnitude outside of the laminar 

sublayer.

Figure 3.—Reynolds stresses: (a) shear stress; normal stresses (b) streamwise, (c) transverse, and

(d) spanwise, for the SEM LL = δδ/5 case.

(a) (b)

(d) (c)
NASA/TM—2018-219966 9

Figure 4.—Time averaged streamwise velocity for 

the SEM length scale study.

In Figure 5(a), the shear stress profiles at x/δ = 10 are shown. The LES solution with LL = δδ/20

under-predicts the shear stress levels in the outer portions of the boundary layer. The LES solution with 

LL = δδ/5 is the closest match between simulation and experiment. The peak shear stress predictions in the 

outer portions of the boundary layer gradually improve as the length scale increases from LL = δδ/20 to

LL = δδ/5. Figure 5(b) depicts the streamwise normal stress. Again the LL = δδ/5 LES is the closest match 

between simulation and experiment. The experimental peak value of uu′uu ������′/uuττ

2 is 8.44, which matches 

closely with the 8.28 prediction of the LL = δδ/5 simulation. Although the peak value predictions match 

the experiment well, the stress levels in the outer portions of the boundary layer after the peak between a 

yy+of 30-100 are slightly under-predicted. The improvement in the outer portions of the boundary layer 

with increasing length scale in Figure 5(a) and (b) is consistently seen in Figure 5(c) and (d). The 

transverse stress is shown in Figure 5(c). The peak value is not significantly affected by the length scale; 

however, the outer portions of the boundary layer agree best with the LL = δδ/5 simulation. Figure 5(d)

depicts the spanwise stress. Increasing the length scale from the LL = δδ/20 to LL = δδ/10 resulted in a 

noticeable increase in the peak value predicted. However, there is good saturation of the peak value 

between the LL = δδ/10 and LL = δδ/5 simulations. Note that the LL = δδ/5 LES consistently predicts the 

highest turbulence values in the outer boundary layer.

The spectra of the turbulent kinetic energy are plotted in Figure 6. The data for the plots were 

obtained on the inflow plane at yy+ = 12. The location was chosen to correspond to the location of the 

peak shear and streamwise stresses. The spectra were computed for each point in zz, then spanwise 

averaged. Peaks in the spectra are seen at the inflow plane (xx/δδ = 0) beyond 200 kHz for all three cases. 

The first peaks in the spectra are at 272, 545, and 1087 kHz, for LL = δδ/5, δδ/10, δδ/20 respectively. The 

LL = δδ/5 and δδ/10 simulations contain harmonics of the primary frequency. It was theorized that these 

peaks represent artificial modes imposed on the flow by the SEM method. This was confirmed by 

computing the representative frequency of the SEM process, by using the eddy convection velocity and 

turbulent length scale; ffSSSSSS = UUcccccccc/LL. In the SEM method, the synthetic eddies travel the distance, LL, at 

the velocity, UUcccccccc, before being replaced in the simulation. The resulting SEM induced frequencies were 

272, 545, and 1090 kHz for LL = δδ/5, δδ/10, δδ/20 respectively, confirming that these modes are 

artificially introduced by the SEM.
NASA/TM—2018-219966 10

Figure 5.—Reynolds stresses: (a) shear stress; normal stresses (b) streamwise, (c) transverse, and

(d) spanwise, for the SEM length scale study.

(aa) LL = δδ/5 (bb) LL = δδ/10 (cc) LL = δδ/20

Figure 6.—Energy spectra for the SEM length scale study, shifted down by two orders of magnitude.

(a) (b)

(c) (d)
NASA/TM—2018-219966 11

The spectra show that the distribution of energy at the inflow plane is clearly related to the length 

scale specified. The larger the length scale, the more energy in the lower frequencies. As the length scale 

decreases more energy is placed at the higher frequencies. This can explain the reduction in the peak 

Reynolds stresses with decreasing length scale, because the simulation dissipates the energy at the highest 

frequencies. This is further seen in Figure 7 where the δδ/20 simulation contains the least energy. The 

downstream spectra show that the energy is redistributed to satisfy the LES equations and at xx/δδ = 10, 

the three simulations show very similar energy distributions. The downstream spectra also show that the 

artificial modes have been eliminated.

In Figure 8, the axial evolution of skin friction coefficient for all three length scales is shown. It is 

interesting to note that the CCffvalue of the LL = δδ/10 and LL = δδ/20 are the same, whereas the LL = δδ/5

simulation is higher by about fifteen percent. Recall the experimental CCff value is 2.10E-3, it matches 

closely the value of 2.16E-3 predicted by the LL = δδ/5 LES (Table 3). In contrast, the simulations with 

LL = δδ/10 and LL = δδ/20 both predict a CCffvalue of 1.80E-3, a 15 percent difference from experiment. 

Figure 9 depicts the derivative of the skin friction coefficient (ddCCff/dd ), which is a metric used to discern 

the extent of the adjustment region. The slope between xx/δδ = 0 and 6 varies significantly in the plot, 

hence the flow is adjusting within this region. However, it flat lines to be an infinitesimal negative 

number between xx/δδ = 6 to 12. Consequently, the adjustment length for all three SEM length scales is 

6δδ based upon the mean flow profiles, Reynolds stress profiles, skin friction coefficient, and as clearly 

seen by ddCCff/dd . Hence forth this comprehensive methodology is used to discern the length of the 

adjustment region objectively.

The key findings of this subsection are that although the choice of length scale did not adversely 

affect the adjustment length, it had a noticeable effect on the Reynolds stress profiles. Increasing the 

length scale from LL = δδ/20 to LL = δδ/5 improved the prediction of CCff by 15 percent resulting in a good

match with the experimental value of 2.10E-3. The choice of length scale of LL = δδ/5 resulted in a closer 

Reynolds stress profile match to experiment. 

Figure 7.—Comparison of Energy spectra at xx/δδ = 10 for the SEM 

length scale study.
NASA/TM—2018-219966 12

Figure 8.—Skin friction coefficient for the SEM length scale study.

TABLE 3.—SKIN FRICTION COEFFICIENT

Simulation methodology CCff

Experiment 2.100 E-3

Input (xx/δδ = 0) 1.845 E-3

WRLES-SEM, LL = δδ/5 2.161 E-3

WRLES-SEM, LL = δδ/10 1.806 E-3

WRLES-SEM, LL = δδ/20 1.806 E-3

WRLES-DF 1.920 E-3

FDL3DI-DF 1.880 E-3

Figure 9.—Skin friction coefficient rate of change, ddCCff/dd , for the SEM length scale study.
NASA/TM—2018-219966 13

5.2 Evaluation of Inflow Turbulence Methodologies: WRLES-SEM and WLRES-DF

The purpose of this section is to compare WRLES-SEM with WRLES-DF results. The SEM length 

scale used is δδ/5 . The length scale used for the DF simulation was δδ/2, based on previous work

(Ref. 15). Figure 10 compares the supersonic boundary layer’s mean flow profile both at the inflow 

plane (xx/δδ = 0) and downstream at xx/δδ = 10 for both the LES results and the experiment. Both 

simulations produce classic law-of-the-wall behavior. The SEM result shows good agreement with the 

experiment. The DF simulation over-predicts the scaled velocity (u+), indicating that the friction velocity, 

and hence wall shear stress is too low.

Figure 11 depicts the Reynolds stresses at stations of xx/δδ = [0, 10] for both WRLES-DF and 

WRLES-SEM. For WRLES-DF, the normal stress peak values adjusted from the inflow’s 3.2, 3.2, and 

3.2 (xx/δδ = 0) to 9.4, 1.3, and 2.6 at xx/δδ = 10 for the streamwise, transverse, and spanwise normal 

stresses, respectively. Note that the inflow values of the normal turbulent stresses are the same because 

the RANS model is a linear two equation model with no differentiation in uu′uu ������′/uuττ

2, vv′

vv ������′/uuττ

2, ww���′

�ww���′/uuττ

2

from kk. For WRLES-SEM, the normal stress peak values adjusted from the inflow’s 3.2 (xx/δδ = 0) in the 

three normal directions to 8.1, 1.1, and 2.0 at xx/δδ = 10 for the streamwise, transverse, and spanwise 

normal stresses, respectively. For WRLES-DF, the shear stress peak value adjusted from about 0.75

(xx/δδ = 0) to 0.92 (xx/δδ = 10). For WRLES-SEM, the shear stress peak value adjusted from about 0.81

(xx/δδ = 0) to 0.77 (xx/δδ = 10). Note that the yy+ location of the peak differs between each stress 

component.

Overall, the Reynolds stresses are predicted reasonably well by the simulations. WRLES-DF 

consistently over predicts, WRLES-SEM consistently under predicts, and the experiment lies between the 

two. In Figure 11(a)’s shear stress at xx/δδ = 10, the WRLES-DF solution is a closer match to the 

experimental data than the WRLES-SEM solution in the outer portions of the boundary layer. In contrast

Figure 11(c) shows the peak transverse stress of WRLES-SEM solution is a closer match to the 

experimental data than the WRLES-DF solution. In Figure 11(b), there is good agreement in the peak 

streamwise stress predicted by simulations and the experiment. Again, we see a closer match between 

WRLES-DF and the experiment for the transverse stress. There is no experimental data for the spanwise 

stress. 

Figure 10.—Time averaged streamwise velocity for the 

comparison between WRLES-SEM and WRLES-DF.
NASA/TM—2018-219966 14

Figure 11.—Reynolds stresses: (a) shear stress; normal stresses (b) streamwise, (c) transverse, and

(d) spanwise, for the comparison between WRLES-SEM and WRLES-DF.

Skin friction coefficient is plotted versus downstream distance in Figure 12. The experimental value 

of the skin-friction coefficient is 2.10E-3 as shown in Table 1. The value of CCff adjusts from the inflow 

value by dropping for several boundary layer heights before recovering. The value of CCff predicted is 

1.92E-3 and 2.16E-3 for WRLES-DF and WRLES-SEM, respectively (Table 3). The adjustment length is 

6δδ for both WRLES-DF and WRLES-SEM based upon the mean flow, Reynolds stress profiles, skin 

friction coefficient, and ddCCff/dd . 

The key finding is that both SEM and DF produced reasonable results for the supersonic boundary 

layer. The impact of this sub-section means that either method can accurately replicate inflow turbulence 

provided the mesh is wall-resolved. Flow problems involving separation and shock-boundary layerinteractions can be simulated using SEM or DF. There was a negligible change in computational time

between the two methods of about 3 percent. For SEM, the computational time required can vary slightly 

depending on the number of eddies used, which is controlled by the length scale.

(a) (b)

(c) (d)
NASA/TM—2018-219966 15

Figure 12.—Skin friction coefficient, for the comparison between WRLES-SEM 

and WRLES-DF.

5.3 Code-to-Code Comparison of Digital Filtering: WRLES-DF and FDL3DI-DF

The purpose of this subsection is to investigate sensitivities that may arise due to either the 

implementation and inputs of DF or the flow solver itself. In both WRLES-DF and FDL3DI-DF, the 

digital filter’s integral length scale was set to be δδ/2 in all three directions (Ref. 15). The mean velocity

profiles in wall coordinates at xx/δδ = 10 are shown in Figure 13. The scaled mean velocities predicted by 

WRLES-DF and FDL3DI-DF are higher than the experiment.

Figure 14 shows the Reynolds stress profiles for the WRLES-DF and FDL3DI-DF, the solutions 

agree with experimental data especially in the outer portions of the boundary layer. Both the shear stress 

and streamwise stress are matched well and the peak is well predicted. The difference between both 

solutions although small is still observable. WRLES-DF consistently predicted higher peak values than 

FDL3DI. For instance, the peak shear stress is 0.93, 0.88, and 0.74 for WRLES-DF, FDL3DI-DF, and 

experiment respectively. Likewise, the peak streamwise stress is 9.5, 9.5, and 8.5 for WRLES-DF, 

FDL3DI-DF, and experiment respectively. Consistent with previous findings, Likewise, the peak 

streamwise stress is 1.3, 1.28, and 0.80 for WRLES-DF, FDL3DI-DF, and experiment respectively. The 

predicted peak value of spanwise stress is 2.6 (WRLES-DF) and 2.3 (FDL3DI-DF). 

In Figure 15, the skin friction coefficient is shown. The value of both solvers match closely at

1.88E-3 and 1.92E-3 for FDL3DI-DF and WRLES-DF, respectively. Recall the experimental value is at 

2.10E-3. Table 3 is a summary of all the cases examined. The adjustment length is 6δδ for both WRLESDF and FDL3DI-DF. The computational time between FDL3DI-DF and WRLES-DF are similar. The 

outcome of this comparison between two flow solvers is that both WRLES-DF and FDL3DI-DF 

reasonably agree on the value of CCff and Reynolds stress profiles, while providing good agreement with 

experiment.
NASA/TM—2018-219966 16

Figure 13.—Time averaged streamwise velocity, for the 

comparison between WRLES-DF and FDL3DI-DF.

Figure 14.—Reynolds stresses: (a) shear stress; normal stresses (b) streamwise, (c) transverse, and

(d) spanwise, for the comparison between WRLES-DF and FDL3DI-DF.

(a) (b)

(c) (d)
NASA/TM—2018-219966 17

Figure 15.—Skin friction coefficient, for the comparison between WRLES-DF 

and FDL3DI-DF

6.0 Conclusions

Wall-resolved Implicit Large-eddy simulations were performed for a supersonic boundary layer to 

evaluate inflow turbulence methodologies. The first contribution of this work is a length scale study for 

the Synthetic Eddy Method (SEM) documenting its effects on the inflow spectra, downstream sustained 

Reynolds tensor, and CCff. The input normal stresses do not satisfy the LES equations. Hence an adjustment 

region downstream of the inflow, where the synthetic turbulence gives rise to physical turbulence. The 

skin friction coefficient was found to be a good descriptor of the adjustment region. The adjustment 

length was found to be unaffected by the SEM length scale. SEM imposes an artificial mode at the inflow 

plane that quickly vanishes. A small length scale attenuated the Reynolds stress profiles. In contrast, a 

large length scale resulted in the closest match between simulation and experiment for both the Reynolds 

stress profiles and the predicted skin friction coefficient. The turbulent spectra plots showed that large 

length scales resulted in more energy at lower frequencies, whereas small length scales resulted in energy 

at the higher frequencies, where it was dissipated. The adjustment length did not vary with SEM length 

scale and was discerned to be 6δδ.

The second major contribution of this work is a comparison of two approaches within the WRLES 

code: (1) the Synthetic Eddy method; (2) the Digital Filtering (DF) method. A comparison with the 

experimental Reynolds stress data 10 boundary layer heights downstream of the inflow was used to 

validate the methods. The simulated values were in good agreement with the experimental values. 

Although digital filtering Reynolds stress predictions matched the experiment closely in the outer portions 

of the boundary layer, it predicted a 9 percent lower CCff value than experiment. The SEM LL = δδ/5 case 

accurately predicted the 2.1E-3 CCff value of the experiment. Despite the differences between SEM and DF,

both methods provided adequate inflow turbulence to the boundary layer and had the same adjustment 

length of 6δδ. One method is not more robust than the other. The third contribution of this work is a 

comparison between two different flow solvers using Digital Filtering. The predictions made by both 

FDL3DI and WRLES are comparable. Both simulations predicted almost the same skin friction 

coefficient and had the same adjustment length.

The significance of this overall work is that both methods, Synthetic Eddy Method and Digital 

Filtering, accurately replicate the supersonic boundary layer after a finite adjustment length. This study 

points to the importance of investigating the downstream behavior of synthetic inflow turbulence. In 

particular, it points to the possibility of having the Reynolds stress profiles accurately replicated but still 

have a noticeable variation in skin-friction. The scaled mean flow profiles, Reynolds stresses, skinfriction coefficient, and ddCCff/dd were comprehensively used to discern the adjustment length. The DF and 

SEM adjustment lengths were all 6δδ. 
NASA/TM—2018-219966 18

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