[american institute of aeronautics and astronautics 43rd aiaa aerospace sciences meeting and exhibit...

43rd AIAA Aerospace Sciences Meeting and Exhibit, January 10–13, 2005/Reno, NV

Preliminary Study of the SGS Time Scales for

Compressible Boundary Layers using DNS Data

M. Pino Martin∗

Department of Mechanical and Aerospace Engineering

Princeton University, Princeton, NJ 08544

We use direct numerical simulation data of turbulent boundary layers and a priori testing

to estimate the time scales that are associated with the subgrid-scale terms in the conser-

vative form of the governing equations. The exact and modeled subgrid-scale terms are

gathered from the filtered direct numerical simulation fields for several time steps. The

model coefficients are evaluated using the Lagrangian averaging technique. We analyze the

time scales directly by performing temporal autocorrelations along fluid particle paths. To

assess the subgrid-scale models, we compare the time scales and the quantities given by the

exact and modeled representations. It is found that in general, mixed models give a larger

integral time scale than that of the exact terms and that the discrepancy is not related to

the prescribed memory length associated with the evaluation of the model coefficients.

I. Introduction

In the large-eddy simulation (LES) technique, the contribution of the large, energy-carrying structuresto momentum and energy transfer is computed exactly, and the effect of the smallest scales of turbulenceis modeled. While a substantial amount of research has been accomplished for LES of incompressible flows,applications to compressible flows have been significantly fewer. This is in part due to (1) the increasedcomplexity introduced by the need to solve an energy equation, which introduces additional (relative tothe incompressible flow formulation) unclosed terms; (2) the increased cost of the numerical simulation ofcompressible flows, and (3) the limited amount of accurate and detailed data at relevant conditions that canbe used to test and develop LES approaches and models.

The traditional LES approach is based on filtering the governing equations and modeling the correspond-ing subgrid-scale (SGS) terms.1, 2, 3, 4 The resulting SGS models are based on scaling arguments leadingto eddy viscosity models,5, 6 physically-based assumptions such as scale similarity,7 and practical add-hocapproximations leading to the monotonically integrated LES approach (MILES).8, 9, 10, 11 An alternate ap-proach has recently been developed, where the unfiltered velocity field is approximated and used to computethe SGS terms in their exact form. Such approach is used in the estimation model, where an estimate ofthe unfiltered velocity is obtained by generating subgrid-scales two times smaller than the grid scale throughthe nonlinear interactions among the resolved scales.12, 13, 14 Also, in the approximate deconvolution model(ADM), a mathematical approximation of the unfiltered solution is constructed and used to calculate thenonlinear terms in the filtered governing equations.15, 16, 17, 18

∗Assistant Professor, [email protected] c© 2005 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free

license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by thecopyright owner.

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American Institute of Aeronautics and Astronautics Paper 2005-0665

43rd AIAA Aerospace Sciences Meeting and Exhibit10 - 13 January 2005, Reno, Nevada

AIAA 2005-665

Copyright © 2005 by M. Pino Martin. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Early applications of the traditional LES to compressible flows used a transport equation for the internalenergy per unit mass6, 19 or for the enthalpy per unit mass.20, 21 In these equations the SGS heat flux wasmodeled in a manner similar to that used for the SGS stresses, while two additional terms, the SGS pressuredilatation and the SGS contribution to the viscous dissipation were neglected. More recent calculations usethe transport equation for the total energy, which is in conservation form including the treatment of the SGSterms. Vreman et al.22 perform a LES of weakly compressible temporal mixing layer neglecting all SGSterms in the total energy equation. Knight et al.23 perform a LES of isotropic turbulence on unstructuredgrids and compared the results obtained with the Smagorinsky5 model with those obtained when the energydissipation was provided only by the dissipation inherent in the numerical algorithm in a MILES approach.Comte and Lesieur24 proposed the use of an eddy viscosity model for the sum of the SGS heat flux and SGSturbulent diffusion, neglecting the SGS viscous diffusion. Rizetta and Visbal25 perform a LES of flow over acompression corner using a dynamic Smagorinsky model and compared their results to the DNS of Adams26

and experimental data at higher Reynolds numbers. In the absence of detailed data at high Mach numbers,none of the existing SGS models and their multiple implementations regarding the evaluation of the modelcoefficients (constant, dynamic, and ensemble or Lagrangian averaged) have been calibrated or tested forrobustness and generality in a systematic way. Thus, it has been difficult to assess the success of the SGSmodels and impossible to develop the traditional LES methodology into a robust tool for the simulation ofsupersonic and hypersonic flows.

Recent developments in initialization techniques27 and simulation procedures28 allow for the DNS ofsupersonic and hypersonic wall-bounded flows at controlled conditions. Martin27 performs DNS of turbulentboundary layers in the range of Mach number 3 to 9 and varying the wall-temperature condition for wall toboundary-layer-edge temperature ratio of 2 to 5. The average time to compute a single boundary layer flowfield is roughly two days. Using the same procedures, Wu & Martin29, 30 perform DNS of shock turbulentboundary layer interactions. The DNS statistics for typical cases are gathered in five days and include Mach3 flow over a compression corner and the interaction of a turbulent boundary layer and a reflected shockconfiguration. It is expected that our ability to perform DNS of these canonical flows in a reasonable turnaround time will allow the parametric studies to further develop LES for supersonic and hypersonic flows.

In the present work, we focus on the continued development of traditional LES models using Martin’sDNS data of turbulent boundary layers. Favre-filtering31 of the mass, momentum and energy conservationequations leads to the LES equations. For compressible flows, these equations include five SGS terms:32

the SGS stresses; SGS heat flux; SGS viscous diffusion; SGS viscous dissipation; and the redistribution ofturbulent kinetic energy by the SGS scales or SGS turbulent kinetic energy diffusion. In previous work, weimplemented and validated SGS mixed models for compressible flows in isotropic turbulence and turbulentboundary layers.32, 33 These models combine the scale-similarity assumption7 with eddy viscosity models5

and include a dynamic evaluation of the model coefficients6 along homogeneous directions. More complexflows, however, are three dimensional and/or involve shockwaves. Thus, the evaluation of the model coef-ficients cannot be done using ensemble averages along spatial directions. Instead a Lagrangian approach ispreferred, where averages are taken in time along the fluid particle paths.34 For compressible flows, thisapproach involves specifying a time constant for each of the five subgrid-scale terms.

In this paper, we use DNS data to develop an understanding of the characteristic time scales of thedifferent mechanisms of turbulence. In turn, we test the Lagrangian implementation of SGS models forwall-bounded, compressible flows. The paper is organized as follows. The LES equations for compressibleturbulence and their closure using mixed models are reviewed in Sections II and III, respectively. Themethodology for the Lagrangian time scale study is described in Section IV. The DNS data and the filteringoperation are summarized in Section V. Results and conclusions are presented in Sections VI and VII,respectively.

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II. Conserved equations for LES

To obtain the equations governing the motion of the resolved eddies, we must separate the large fromthe small scales. LES is based on the definition of a filtering operation: a resolved variable, denoted by anoverbar, is defined as35

f(x) =

∫

D

f(x′)G(x,x′; ∆)dx′ , (1)

where D is the entire domain, G is the filter function, and ∆ is the filter-width associated with the wavelengthof the smallest scale retained by the filtering operation. Thus, the filter function determines the size andstructure of the small scales. Also in compressible flows, it is convenient to use Favre-filtering31 to avoid theintroduction of extra subgrid-scale terms in the equation for mass conservation. A Favre-filtered variable isdefined as f = ρf/ρ. Applying the Favre-filtering operation to the mass, momentum and energy conservationequations, we obtain their traditional LES form

∂ρ

∂t+

∂

∂xj

(ρ uj

)= 0 , (2)

∂ρ ui

∂t+

∂

∂xj

(ρ uiuj + pδij − σij

)= −∂τij

∂xj, (3)

∂ρ e

∂t+

∂

∂xj[(ρ e + p) uj − σij ui + qj ] = − ∂

∂xj

(γcvQj + 1

2Jj −Dj

), (4)

where the energy is given by

ρ e = ρ cvT +1

2ρuiui +

1

2τii , (5)

and γ is the ratio of the average specific heats; cv is the average specific heat at constant volume; and thediffusive fluxes are given by

σij = 2µSij −2

3µδij Skk, qj = −k

∂T

∂xj, (6)

where Sij = 12(∂ui/∂xi + ∂uj/∂xi) is the strain rate tensor, and µ and k are the viscosity and thermal

conductivity corresponding to the filtered temperature T .The effect of the subgrid scales appears on the right hand side of the governing equations through the

SGS stresses τij ; SGS heat flux Qj ; SGS turbulent kinetic energy diffusion Jj ; and SGS viscous diffusionDj . These quantities are defined as:

τij = ρ (uiuj − uiuj) , (7)

Qj = ρ(ujT − uj T

), (8)

Jj = ρ ( ˜ujukuk − ujukuk) , (9)

Dj = σijui − σij ui . (10)

The equation of state has been used to express the pressure-velocity correlation appearing in the total energyequation in terms of Qj . Notice that the filtered energy also includes τii.

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III. Mixed Models for Compressible Turbulence

The subgrid-scale terms that are given in equations 7-10 can be closed using the mixed-models describedin Martın et al.32 In the present work we focus on the treatment of the SGS stresses, heat flux and turbulentkinetic energy diffusion. We use the one-coefficient dynamic model is used to compute τij , the mixed modelis used to compute Qj , and the model proposed by Knight et al.23 is used to compute Jj , which relies onthe choice of model for τij . These models are composed of a scale-similar and an eddy-viscosity part. Inthis way, the models reproduce the local energy events that the grid cannot resolve, while providing thedissipation that is generally underestimated by purely scale-similar models.

A. SGS stress tensor

Various models have been devised to represent this term. In Martın et al.6, we find that a one-coefficientmodel gives the most accurate results. Such a model is given by

τij = −C2∆2ρ |S|

(Sij −

δij

3Skk

)+ ρ (˜uiuj − ˜ui

˜uj) (11)

= Cαij + Aij , (12)

where ∆ = (∆x ∆y ∆z)1/3, with ∆x, ∆y and ∆z as the grid spacings in the LES grid; |S| = (2Sij Sij)

1/2.The coefficient is adjusted dynamically according to

C =〈LijMij〉 − 〈NijMij〉

〈MklMkl〉(13)

with Lij =(

ρui ρuj/ρ)− ρui ρuj/ρ , (where

˘f = ρf/ρ , and the hat represents the application of the test

filter G of characteristic width ∆ =√

6 ∆); Mij = βij − αij with βij = −2∆2ρ | ˘S|(˘Sij − δij˘Skk/3); and

Nij = Bij − Aij with Bij = ρ (˘

˘ui˘uj −

˘˘ui

˘˘uj).

B. SGS heat flux

Speziale et al.20 proposed a mixed model of the form

Qj = −C∆

2ρ |S|

PrT

∂T

∂xj+ ρ

(˜uj T − ˜uj

˜T

), (14)

where C is given by Eq. 13 and PrT is adjusted dynamically according to

PrT = C〈HkHk〉

〈KjHj〉 − 〈VjHj〉, (15)

with

Hj = −∆2ρ | ˘S| ∂˘T

∂xj+ ∆

2

ρ |S| ∂T

∂xj, (16)

Kj =( ρuj ρT/ρ

)− ρuj ρT/ρ , (17)

Vj = ρ

(˘

˘uj˘T −

˘˘uj

˘˘T

)−

ρ

(˜uj T − ˜uj

˜T

). (18)

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C. SGS turbulent diffusion

We use the simplest model for Jj , which is that of Knight et al.23 The main assumption for this model is

that ui ≈ ˜ui, givingJj ≈ ukτjk (19)

D. Evaluation of scale-similar terms

The scale-similar part of the model is based on the assumption that the most active SGS scales are thosecloser to the cutoff, and that the scales with which they interact are those immediately above the cutoff wavenumber.7 The smallest resolved scales are identified using multiple filtering operations. The two filteringoperations that we use are in accordance with a second-order discretization to minimize the commutationerror between the filtering and differencing operations,36 and are given by

f i =

1

4

(fi−1 + 2fi + fi+1

), (20)

f i =1

8

(fi−1 + 6fi + fi+1

), (21)

where the corresponding filter widths are37 ∆i =√

6∆i and ∆i =√

3∆i.

E. Model coefficients

To evaluate the model coefficients in equations 13 and 15 one must use the ensemble-averaging operation〈.〉. For boundary layers, this operation can be performed simply using spatial averaging along homogeneousdirections. In contrast for three-dimensional flows, this operation is performed using the Lagrangian-ensembleaveraging proposed by Meneveau et al.,34 where the averaging is performed along a fluid path-line, namely

If = 〈f〉 =

∫ t

−∞

f(t′) W (t − t′) dt′, (22)

where W (t) is an exponential weighting function chosen to give more weight to recent times. For example,to evaluate the mixed-model coefficient for the SGS stress in Eq. 13, we can write

C(~ξ, t) =In

LM

InMM

(~ξ, t) , (23)

where n indicates the time step, and

InLM (~ξ, t) =

∫ t

−∞

(Lij(~ξ, t

′) − Nij(~ξ, t′))

Mij(~ξ, t′) W (t − t′) dt′ , (24)

InMM (~ξ, t) =

∫ t

−∞

Mij(~ξ, t′) Mij(~ξ, t

′) W (t − t′) dt′ . (25)

Following the work by Meneveau et al.,34 if the weighted function is chosen as

W (t) = τ−1p exp(−t/τp) , (26)

where τp = 1.5∆(−8In−1

LM In−1MM

)−1/8controls the memory length of the Lagrangian averaging, then the

integrals given by equations 24 and 25 can be approximated as

InLM (~ξ) = H

(ε [(Lij − Nij) Mij ]

n(~ξ) + (1 − ε) In−1

LM (~ξ − ~un∆t))

, (27)

InMM (~ξ) = H

(ε [MijMij ]

n (~ξ) + (1 − ε) In−1MM (~ξ − ~un∆t)

), (28)

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where H(ξ) = ξ if ξ ≥ 0 and zero otherwise, and ε is given by

ε =∆t/τp

1 + ∆t/τp. (29)

The evaluation of the integrals at (~ξ − ~u∆t) can be performed by linear interpolation. If C(~ξ, t) = 0, InLM is

set to zero to avoid complex values of τp.

IV. Scope and methodology for the time scale study

In the Lagrangian-ensemble averaging procedure proposed by Meneveau et al.,34 the parameter thatcontrols the memory of the Lagrangian averaging, τp in Eqn. 26, is chosen based on practical arguments sothat the evaluation of Equations 24 and 25 can be approximated by Equations 27 and 28.

To evaluate the model coefficients for τij and Qj (and therefore that of Jj according to Eqn. 19), τp isgiven by

τpτij = 1.5∆(−8In−1

LM In−1MM

)−1/8, (30)

τpQj = 1.5∆(−8In−1

KH In−1HH

)−1/8, (31)

where IKH and IKK are approximated as

InKH(~ξ) = H

(ε [(Kj − Vj)Hj ]

n(~ξ) + (1 − ε) In−1

KH (~ξ − ~un∆t))

, (32)

InHH (~ξ) = H

(ε [HjHj ]

n(~ξ) + (1 − ε) In−1

HH (~ξ − ~un∆t))

. (33)

In turn, these integrals are used to compute the model coefficient for the SGS heat flux as

Prt(~ξ, t) = C(~ξ, t)In

HH

InKH

. (34)

Here we use DNS data to estimate the time scales that are associated with τij , Qj , and Jj . Then,we compare these time scales with those of the SGS models. A method to evaluate SGS models is the a

priori test, where the velocity fields obtained from DNS are filtered to yield the exact SGS terms and thefiltered quantities on the LES grid. To gather Lagrangian statistics, we store and filter DNS fields for severalnumerical time steps. We evaluate the exact SGS terms from Equations 7-9 and their time scales directlyby performing temporal autocorrelations of the terms along fluid particle paths. In addition, we evaluatethe Lagrangian implementation of the model coefficients, Equations 23 and 34, and their corresponding timescales, as well as the model approximations, Equations 12, 14, and 19 and their time scales. To assess theSGS models, we compare the time scales and the quantities given by the exact and modeled representations.

V. Flow conditions and filtered DNS data

We use the DNS data of of turbulent boundary layers in the range of Mach 3 to 8.27 The flow conditionsare given in Table 1. Figures 1 through 2 plot the van-Driest transformed velocity profiles, the budget ofturbulent kinetic energy, and the comparison of the skin friction against empirical predictions, showing thegood accuracy of the DNS data.

To filter the DNS data, we apply a top-hat filter to a DNS variable along each of the three directionsusing

f i =1

2n

fi−n

2+ 2

i+ n

2−1∑

i−n

2+1

fi + fi+ n

2

, (35)

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Case Mδ ρδ (kg/m3) Tδ (K) Tw/Tδ Reθ θ (mm) H δ (mm)

M2 2.32 0.0962 145.02 1.91 4452 0.851 3.7 9.90

M3 2.98 0.0907 219.55 2.58 2390 0.430 5.4 6.04

M4 3.98 0.0923 219.69 3.83 3944 0.523 8.5 9.77

M5 4.97 0.0937 220.97 5.40 6225 0.657 12.2 14.82

M6 5.95 0.0952 221.49 7.32 8433 0.733 16.5 21.00

M7 6.95 0.0963 221.61 9.60 10160 0.778 22.3 28.60

M8 7.95 0.0973 221.46 12.25 13060 0.832 28.2 36.92

Table 1. Dimensional boundary layer edge and wall parameters for the DNS data.27

therefore ∆i = n ∆i, where ∆i and ∆i are the LES and DNS grid spacings, respectively, and n representsthe non-dimensional filter width. The trapezoidal rule is used in the wall-normal direction to account for thegrid stretching. We use n= 4, 2 and 2 in the streamwise, spanwise and wall-normal directions, respectively.The resulting grid resolution along the homogeneous directions are listed in Table 2. For these filter widths,the amount of turbulent kinetic energy in the subgrid scales is up to 16%, which is shown in Fig. 3.

To gather Lagrangian statistics and determine for what time interval we must gather DNS flow fields,we assume that energetic eddies convect with speed Ueddy = 0.8Uδ and that these eddies are of O(δ) insize. Thus, the effect of one energetic eddy influences the flow at a particular Eulerian point in space for aninterval of time τeddy = O(δ/Uδ). We gather DNS flow fields for a time t = 2τeddy.

To determine how far apart DNS flow fields should be, we consider the increment of time from onenumerical iteration to the next, dt. Because of numerical stability limits, dt is restricted to the smallest timescale in the flow field. For wall-bounded flows, dt is given by the smallest time scale near the wall and theDNS and LES dt are related as

dtDNS ≈ dtLES/nk , (36)

where nk is the wall-normal filter width that is used to filter the DNS data onto the LES grid (an equalsign in Eqn. 36 applies for the a priori test). The near-wall events, however, carry less than 10% of theSGS energy in the SGS. For this reason, we gather DNS flow fields that are separated dtfield = 10dtDNS ordtfield = 5dtLES. The tracking of fluid particles is done using trilinear interpolation. Figure 4 illustratesthe trajectory of a particle that is released at z+=8. The displacement along the spanwise and wall-normaldirections are 1% and 15% of a boundary layer thickness, respectively.

VI. Results

We consider the analysis for the Mach 5 boundary layer data. To monitor the temporal evolution of theSGS terms, we compute the autocorrelation function. If X is a statistically stationary flow quantity, theautocorrelation of X is given by

ρX(τ) =

⟨(X(t◦) − X)(X(t◦ + τ) − X)

⟩⟨(X(t◦) − X)2

⟩1/2 ⟨(X(t◦ + τ) − X)2

⟩1/2, (37)

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Case δ+ Lx/δ Ly/δ Lz/δ ∆x+ ∆y+ Nx Ny Nz

M3 325 9.1 2.3 13.8 32.0 12.0 96 128 53

M4 368 7.9 2.0 15.4 30.4 11.2 96 128 55

M5 382 7.4 1.8 14.0 29.6 11.2 96 128 55

M6 396 7.0 1.7 15.3 28.8 10.0 96 128 56

M7 414 6.4 1.6 14.8 28.0 10.4 96 128 56

M8 430 6.0 1.5 15.6 27.2 10.0 96 128 57

Table 2. Grid resolution and domain size on the LES grid, where x, y, and z are the streamwise, spanwise andwall-normal directions, respectively.

where X is the mean value of X and brackets indicate ensemble average. Figure 5a plots the autocorrelationcoefficient for τ11, Q1, and J1 given by the exact and modeled quantities for a particle that is released atz+ = 15. We observe that the exact quantities have a faster decay rate. Figures 5b and 5c plot the samequantities for particles released at z/δ = 0.2 and 0.4, respectively. The model terms lag the exact terms.The slow decay rate seen in Fig. 5c shows that the memory of τ11 is longer than the time for which we havegathered statistics.

Figure 6 plots the autocorrelation coefficient for τ13, Q3, and J3 given by the exact and modeled quantitiesfor particle released at z+ = 15, z/δ = 0.2, z/δ = 0.4, and z/δ = 1.0. All correlations go to zero within thet = 2τδ time interval. Consistently, the autocorrelation for the exact terms is higher.

To measure the length of memory for the SGS terms, we evaluate the integral time scale given by

τX =

∫∞

0

ρX(τ)dτ . (38)

Figure 7 shows the normalized time scales for the exact and modeled terms, indicating that the time scalesgiven by the modeled quantities are one and a half to two times larger than those for the exact terms.

Figure 8a plots contours of τ11 on spanwise-wallnormal planes for the exact and modeled. The resultsare shown at t = 0 and 1.98τδ. For the initial and final times, the model represents the local events of theexact SGS quantity. Figure 8b plots the correlation coefficient between the exact and modeled quantitiesas an index of the overall agreement. For both temporal realizations, the correlation coefficient is about0.8 throughout the boundary layer indicating the good accuracy of the model representation. Correlationsof 0.5 and up are considered to be accurate results. Figures 9 and 10 show contour plots and correlationscoefficients for τ13 and Q3, illustrating the same good performance of the models. The same results arefound for the SGS turbulent kinetic energy diffusion, which relies on the model for the SGS stresses.

VII. Conclusions

We use direct numerical simulation data of turbulent boundary layers and a priori testing to estimatethe time scales that are associated with the subgrid-scale terms in the conservative form of the governingequations. The exact and modeled subgrid-scale terms are gathered from the filtered DNS fields for severalnumerical time steps. The model coefficients are evaluated using the Lagrangian averaging technique. We

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analyze the time scales directly by performing temporal autocorrelations along fluid particle paths. To assessthe subgrid-scale models, we compare the time scales and the quantities given by the exact and modeledrepresentations.

We find that for all subgrid-scale quantities, the model representation has a larger integral time scalethan the exact quantity. This result is consistent with that of He et al.38 They perform DNS and LES withan eddy viscosity model to compute homogeneous isotropic turbulence. From two-point, two-time Eulerianvelocity correlations, they find that the integral time scales for the LES velocity field are larger than thosefor the DNS velocity field.

In our simulations, the Eulerian correlation factors between the exact and model quantities show that themodels are locally accurate, being equally accurate at the early and final temporal flow realizations. It is alsoobserved that the decay rate for the Lagrangian autocorrelation coefficients are larger for the exact termsearly in the time history. These are indications that the integral time scale discrepancy is not due to thechoice of weighting function that is used to control the memory length in the evaluation of the Lagrangiancoefficients. Rather, the larger integral time scale is due to the mixed model representation. In addition, wefind that for the Mach 5 turbulent boundary layer, the degree of discrepancy in the time scales is not enoughto give inaccurate results during the a priori test.

VIII. Acknowledgments

This work was supported by the Air Force Office of Scientific Research under grant AF/F49620-02-1-0361and the National Science Foundation under grant # CTS-0238390.

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Journal , Vol. 39, 2001.26Adams, N., “Direct Simulation of the Turbulent Boundary Layer Along a Compression Ramp at M=3 and Reθ = 1685,”

Journal of Fluid Mechanics, Vol. 420, 2000.27Martin, P., “DNS of Hypersonic Turbulent Boundary Layers. Part I: Initilization and Comparison with Experiments,”

Submitted to Journal of Fluid Mechanics. Also AIAA Paper No. 2004-2337 , 2004.28Xu, S. and Martin, P., “Assessment of Inflow Boundary Conditions for Compressible Turbulent Boundary Layers,”

Physics of Fluids, Vol. 16, No. 7, 2004.29Wu, M. and Martin, P., “Direct numerical simulation of two shockwave/turbulent boundary layer interactions at Mach

2.9 and Reθ = 2400,” Submitted to AIAA Journal. Also AIAA Paper No. 2004-2145 , 2004.30Wu, M. and Martin, P., “Analysis of shockwave/turbulent boundary layer interaction using DNS and experimental data,”

AIAA Paper No. 2005-0310 , 2005.31Favre, A., “Equations des gaz turbulents compressible. I. Formes generales.” Journal Mecanique, Vol. 4, 1965.32Martin, P., Piomelli, U., and Candler, G., “Subgrid-Scale Models for Compressible LES,” Theoretical and Computational

Fluid Dynamics, Vol. 13, No. 5, 2000.33Martin, P., Weirs, G., Candler, G., Piomelli, U., Johnson, H., and Nompelis, I., “Toward the large-Eddy Simulation Over

a Hypersonic Elliptical Cross-Section Cone,” AIAA Paper No. 2000-2311 , 2000.34Meneveau, C., Lund, T., and Cabot, H., “A Lagrangian Dynamic Subgrid-scale Model of Turbulence,” Journal of Fluid

Mechanics, Vol. 319, 1996.35Leonard, A., “Energy cascade in large-eddy simulations of turbulent fluid flows,” Advances in Geophysics, Vol. 18A,

1974.36Ghosal, S. and Moin, P., “The basic equations for the large-eddy simulation of turbulent flows in complex geometry,”

Journal of Computational Physics, Vol. 118, 1995.37Lund, T., “On the use of discrete filters for large-eddy simulation,” Center for Turbulence Research Briefs, 1997.38He, G.-W., Rubinstein, R., and Wang, L.-P., “Effects of subgrid-scale modeling on time correlations in large eddy

simulation,” Physics of Fluids, Vol. 14, 2002.

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+++++++++++++++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

+++++

+ + + + + + + + + ++++++++++++++

z+

⟨uV

D⟩

10-1 100 101 102 1030

5

10

15

20

25

30M3M4M5M6M7M82.44 log z+ + 5.2z+

+

+

(a)

z+

Bud

gett

erm

s

0 10 20 30 40

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5M3M4M5M6M7M8Guarini

Dissipation

Production

Pressurediffusion

Viscousdiffusion

Turbulentdiffusion

(b)

Figure 1. DNS data from Martin (2004). (a) Mean velocity profiles and (b) turbulent kinetic energy budget.Variables are non-dimensionalized with uτ τw/zτ , τw = ρwu2

τ .

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X

Theoretical value of Cf

Cf

0.0005 0.0010 0.0015 0.0020 0.0025

0.0005

0.0010

0.0015

0.0020

0.0025

TheoryMach 3Mach 4Mach 5Mach 6Mach 7Mach 8X

(a)

Figure 2. Skin friction coefficients from the DNS data of Martin (2004). Error bars show a 7% error wherethe error in the empirical predictions is up 10%.

z / δPer

cent

age

ofT

KE

inS

GS

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30Mach 3Mach 4Mach 5Mach 6Mach 7Mach 8

(a)

Figure 3. SGS turbulent kinetic energy profiles for the filtered DNS data.

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0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0.050.100.150.20

XY

Z

x/δ

(a)

Figure 4. Trajectory of a particle released at z+ =8.

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t / τδ

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0τ11 exact

Q1 exact

J1 exact

τ11 model

Q1 model

J1 model

(a) z+ = 15

t / τδ

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) z/δ = 0.2

t / τδ

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

(c) z/δ = 0.4

Figure 5. Autocorrelation coefficients for different particles released at different wall-normal locations.

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t / τδ

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

τ13 exact

Q3 exact

J3 exact

τ13 model

Q3 model

J3 model

(a) z+ = 15

t / τδ

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) z/δ = 0.2

t / τδ

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

(c) z/δ = 0.4

t / τδ

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

(d) z/δ = 1.0

Figure 6. Autocorrelation coefficients for different particles released at different wall-normal locations.

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z / δ

τ L/τ

δ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

τ11 exact

Q1 exact

J1 exact

τ11 model

Q1 model

J1 model

(a)

z / δ

τ L/τ

δ

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

τ13 exact

Q3 exact

J3 exact

τ13 model

Q3 model

J3 model

(b)

Figure 7. Normalized integral time scales.

16 of 19


Spanwise (2δ)w

alln

orm

al(2

δ)

Exact

t / τδ=0

9.58.88.07.36.65.95.14.43.72.92.21.50.70.0Spanwise (2δ)

τ11

Model

Spanwise (2δ)

Model

Spanwise (2δ)

Exact

wal

lnor

mal

(2δ)

t / τδ=1.98

(a)

z / δ0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.001.98

t / τδ

(b)

Figure 8. (a) Contours of the normalized SGS stress tensor τ11 for the exact and model representations and(b) corresponding Eulerian correlation factor.

17 of 19


Spanwise (2δ)w

alln

orm

al(2

δ)

Exact

t / τδ=01.0000.8180.6360.4550.2730.091

-0.091-0.273-0.455-0.636-0.818-1.000

Spanwise (2δ)

τ13

Model

Spanwise (2δ)

Model

Spanwise (2δ)

Exact

wal

lnor

mal

(2δ)

t / τδ=1.98

(a)

z / δ0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.001.98

t / τδ

(b)

Figure 9. (a) Contours of the normalized SGS stress tensor τ13 for the exact and model representations and(b) corresponding Eulerian correlation factor.

18 of 19


Spanwise (2δ)w

alln

orm

al(2

δ)

Exact

t / τδ=00.0210.0170.0130.0100.0060.002

-0.002-0.006-0.010-0.013-0.017-0.021

Spanwise (2δ)

Q3

Model

Spanwise (2δ)

Exact

wal

lnor

mal

(2δ)

t / τδ=1.98

Spanwise (2δ)

Model

(a)

z / δ0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.001.98

t / τδ

(b)

Figure 10. (a) Contours of the normalized SGS stress tensor Q3 for the exact and model representations and(b) corresponding Eulerian correlation factor.

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