Compressible turbulent plane channel DNS datasets

The database contains detailed statistics of compressible turbulent plane channel (TPC) flow, obtained from direct numerical simulation (DNS), with a very-high-order massively parallel solver of the compressible Navier-Stokes equations. It contains datasets for 25 different flow conditions determined by the corresponding HCB friction Reynolds number and centerline Mach number, covering the ranges 100⪅Reτ★⪅1000 and 0.3⪅M¯CLx⪅2.5. All calculations are for strictly isothermal wall conditions at temperature Tw=298K in a medium-size (MB) computational box (8πδ×2δ×4πδ where 2δ is the channel-height). Statistics (moments and pdfs) were collected after the elimination of the transient, and post-processed to create the dataset, which contains only plain text (.txt) space-separated multicolumn files for ease of use. The dataset for each flow-condition is tagged by the values of (Reτ★,M¯CLx) and is organized in 4 directories: (0) global data files, (1) profiles and budgets (meanflow profiles, velocity-moments up to 6-order, budgets of Reynolds-stresses transport, turbulent fluxes appearing in transport equations for velocity-moments and thermodynamic quantities, correlation coefficients between thermodynamic variables, and skewness and flatness profiles) as a function of the wall-distance, (2) single-variable probability density functions (pdfs) for numerous flow quantities at selected wall-normal distances, and (3) two-variable joint pdfs for numerous couples of flow-quantities at the same selected wall-normal distances.

is the channel-height).Statistics (moments and pdfs) were collected after the elimination of the transient, and postprocessed to create the dataset, which contains only plain text (.txt) space-separated multicolumn files for ease of use.
The dataset for each flow-condition is tagged by the values of (Re τ , MCL x ) and is organized in 4 directories: (0) global data files, (1) profiles and budgets (meanflow profiles, velocitymoments up to 6-order, budgets of Reynolds-stresses transport, turbulent fluxes appearing in transport equations for velocity-moments and thermodynamic quantities, correlation coefficients between thermodynamic variables, and skewness and flatness profiles) as a function of the wall-distance, (2) single-variable probability density functions (pdfs) for numerous flow quantities at selected wall-normal distances, and (3) two-variable joint pdfs for numerous couples of flowquantities at the same selected wall-normal distances.

Value of the Data
• A particular effort was made to construct a matrix (Cartesian grid) of (Re τ , MCL x ) -values, allowing the investigation of Re τ -effects at constant MCL x , and of MCL x -effects at constant Re τ [8,9] .• The present datasets include a large number of turbulent correlations which are not available in existing datasets of compressible turbulent plane clannel (TPC) or turbulent boundarylayer (TBL) flows (i) profiles of the complete set of moments and correlations (orders 2 and 3) of turbulent fluctuations of thermodynamic variables [7] (ii) profiles of all velocity fluctuations moments and correlations (up to 6-order for Reynolds fluctuations and up to 4-order for Favre fluctuations) (iii) profiles of all fluxes for turbulent transport by the fluctuating velocities appearing in the transport equations of second moments of velocity and thermodynamic variables [6] (iv) single-variable and joint two-variable pdfs (probability density functions) of a large number of flow-quantities [8,9] in addition to profiles of meanflow, Reynolds-stresses (and the detailed budgets of their transport equations) and turbulent heatfluxes.
• The database includes Re τ 10 0 0 datasets for which a common inner/outer region becomes clearly visible in the profiles.
• The datasets can be easily used as reference data for the analysis and modeling of compressible aerodynamic wall-flows.

Background
High-speed aerodynamics in aerospace applications require highly accurate DNS data, both to enhance our understanding of the complex multiscale aerothermodynamic phenomena [18,21] observed in compressible wall-turbulence and to use as reference data for the development and validation of advanced turbulence models [15] .Data of compressible turbulent boundary-layers (TBL) [1,20] and compressible turbulent plane channels (TPC) [14,22] have largely contributed to this goal.
The principal parameters determining the flow physics of these canonical compressible turbulent wall-flows are the couple of Reynolds and Mach numbers and the wall-temperature (wall-heatflux) condition [22] .Notice that most of compressible wall-turbulence studies (including the present database) use a Sutherland's law for the dynamic viscosity [ 6 , p. 706], so that the wall-temperature (TPC) or the external flow temperature (TBL) in K is an additional parameter of flow conditions.In the case of canonical compressible TPC flow the wall-heatflux is determined by frictional heating [9,17] , so that the flow conditions are determined by the triplet (Re τ , MCL x , Tw ) .Computations which control the wall-heatflux by an artificial sink-term in the energy equation [21] are not considered in the present database.
The motivation for the creation of the present database is twofold: (a) the development of a (Re τ , MCL x ) -matrix of datasets whose need was felt in earlier work [6,14] , and (b) the accurate computation of a large number of correlations [7,15]  the centerline streamwise Mach number, the HCB friction Reynolds number and the nondimensional wall-distances in the 3 inner-units systems.

Data Description
The database is contained in directory GV_TPC_MB_AIR0 , where AIR0 denotes the flow model detailed in [ 6 , p. 706], MB denotes the computational-box-size ( 8 πδ × 2 δ × 4 πδ), TPC is the acronym for turbulent plane channel, and the prefix GV was used to identify the present database.The parent directory GV_TPC_MB_AIR0 contains 25 subdirectories Fig. 1 a, each corresponding to a different flow-condition ( Table 1 ), eg • 0_GD_global_data ( §3.1 ) Each individual datafile is tagged with the flow-conditions, to avoid mistakes (we will use the typical subdirectory Retaus_0965_MCLx_1p50_isoTw_0298_MB_AIR0 to describe the datasets).Each datafile contains a header (lines of comments starting with # ), followed by the actual numerical data in space-separated columns.The comments are quite detailed and describe the data contained in the file.The numbers of comment-lines, of data-columns and of data-lines are systematically given in line 3 of each file ( Fig. 2 ), as this can be helpful in reading the data by a script or program.All datafiles can be directly used for plotting [12] .

GD (global data)
The subdirectory 0_GD_global_data of each dataset contains 3 files of global data.Sampling parameters for averages, pdfs and joint pdfs [ Reτ is the friction Reynolds number in HCB-scaling (1) ; MCL x is the centerline Mach number (1) ; t is the computational time-step [8] ; ts , ts q , ts 2 q are the sampling time-steps for averages, pdfs, and joint pdfs; t OBS , t OBS q , t OBS 2 q are the sampling observation intervals for averages, pdfs and joint pdfs; N bins q , N bins 2 q are the sampling bins for the calculation of pdfs and joint pdfs; Ns , Ns q , Ns 2 q is the number of samples involved in the calculation of pdfs and joint pdfs; (•) + denotes wall-units] PBs; ts = t pdfs; ts q = t joint pdfs; These files contain the corresponding global data in human-readable format in the comments section ( Fig. 2 a), and are followed by a single line of space-separated data, which can be practical for extracting and assigning value(s), using awk [16] , in scripts or plotting commands.

PBs (profiles and budgets)
All the datafiles in subdirectory 1_PBs_profiles_and_budgets of each dataset contain profiles as a function of the wall-distance.Therefore, the first 4 columns in each file tabulate wall-distance in different scalings ( y/δ, y , y # , y + ).The following datafiles are available in subdirectory 1_PBs_profiles_and_budgets • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_budgets_rxx.txt tabulates the profiles of the terms in the transport-equations for the streamwise velocity variance r xx := ρu u • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_budgets_rxy.txt tabulates the profiles of the terms in the transport-equations for the velocity covariance r xy := ρu v • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_budgets_ryy.txt tabulates the profiles of the terms in the transport-equations for the wall-normal velocity variance r yy := ρv v • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_budgets_rzz.txt tabulates the profiles of the terms in the transport-equations for the spanwise velocity variance r zz := ρw w • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_MF_meanflow.txt tabulates profiles of meanflow-variables (dynamic and thermodynamic), both Reynolds-and when appropriate Favre-averaged, with different scalings • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_rms_skew_flat.txt tabulates the profiles of coefficients-of-variation CV q := q rms / q and higher-order moments (up to 6-order, skewness S q 3 := q 3 /q 3 rms , flatness F q 4 := q 4 /q 4 rms , superskewness sS q 5 := q 5 /q 5 rms and superflatness sF q 6 := q 6 /q 6 rms [4] ) for all flow-variables ( V , u , v , w , p, ρ, T , h , s , h t ) and Mach-numbers ( M, M x , M y , M z ); contrary to all of the other files in subdirectory 1_PBs_profiles_and_budgets , statistics in this file were obtained from single-variable standardized pdfs ( §3.3 ) tabulates fluxes which appear in different transport equations [6] (u i , ρu i u j , ρu i u j u k , ρu i u j u k u , ρ u i , ρ 2 u i , u i u j , ρ u i u j , p u i , p 2 u i , p u i u j , ρh u i , ρh 2 u i , ρh u i u j , ρs u i , ρs 2 u i , ρs u i u j ) • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_TTS_thrm_trblnc_strctr.txt tabulates the coefficients-of-variation (CVs) and correlation coefficients up to 3-order between thermodynamic variables ( p, ρ, T , h , s , a ); also CVs and skewness of inverses ( 1 /ρ, 1 /T , 1 /p, 1 /a ) and selected correlations including inverses • GV_TPC_0965_1p50_isoTw_0298_MB_AIR0_Vlct_HoMs.txt tabulates all the moments of velocity-components in (•) -units , up to 6-order for Reynolds fluctuations ( ūi , u i u j , u i u j u k , u i u j u k u , u i u j u k u u m , u i u j u k u u m u n ), and up to 4-order for Favre fluctuations ( ˜ u i , Notice that with the definitions used for the budgets, the Reynolds-stresses are −r i j .Statistics for correlations and budgets of many other quantities are available, and will be included in a future V2 version of the database.

pdfsq (single variable pdfs)
Probability density functions (pdfs) were acquired at every wall-normal station of the computational halfchannel grid.In subdirectory 2_pdfsq_single_variable_pdfs of each dataset we include pdf-data at selected wall-normal stations and Mach-numbers ( M, M x , M y , M z ).The number of pdf-bins is the same at each wall-normal station but the range [ q min , q max ] of sampling (bin width q ) was adjusted for each variable and wall-normal station to include extreme events ( §4.2 ).

pdfs2q (two-variable joint pdfs)
Joint 2-variable pdfs were also acquired at every wall-normal station of the computational halfchannel grid.In subdirectory 3_pdfs2q_two_variable_joint_pdfs of the datasets we include joint-pdf-data at the same selected wall normal stations as for the single-variable pdfs ( §3.3 ).

Experimental Design, Materials and Methods
Direct numerical simulation (DNS) ( §4.1 ) was used to compute the flow ( Fig. 3 ) for each (Re τ , MCL x ) -condition ( Table 1 ).After calculating the flow for an initial time-interval t CNVRG , sufficiently long to eliminate the transient ( Fig. 4 ), the computations were continued for an additional time-interval t OBS with activated on-board sampling ( §4.2 ) of dynamic and thermodynamic flow-variables for the acquisition of averages (moments) and of pdfs.Fully automated post-processing was then applied to these data, to calculate correlations and moments of the fluctuating field and standardized pdfs, and to computer-generate the datasets included in the present database ( §4.3 ).The entire procedure was scripted to avoid human errors.

Solver and convergence
Following standard practice in the field [14,17,18,20,21] , introduced by Coleman et al. [3] , the flow was made streamwise invariant by the introduction of a body-acceleration f V x (t) (space invariant but time-dependent) replacing the streamwise pressure-gradient necessary to counter skin-friction (budgets of numerous second-moment equations [6] and the Reynolds-stresses budgets in the subdirectories 1_PBs_profiles_and_budgets of the present datasets demonstrate negligible influence of the time-variation of f V x ).The computational box ( Fig. 3 ) is delimited by the lower ( y = 0 ) and upper ( y = 2 δ) walls and the streamwise ( x ) and spanwise ( z) periodic boundaries.
The DNS computations were run using a very-high-order accurate DNS solver [5] , integrating the compressible Navier-Stokes equations with air thermodynamics and transport coefficients [ 6 , p. 706].The solver is massively parallel using hybrid MPI/OpenMP programming with verified scalabilty (up to 64736 cores) on several supercomputers [2,11,13,19] .The computation and onboard statistics acquisition of the 25 flow-conditions included in the database required ∼ 250 × 10 6 corehours .Results with this solver have been carefully verified both by consistency checks [9] and by comparison with data by other authors [5,8] .( t CNVRRG ) and the subsequent acquisition of statistics ( t OBS and t OBS q ) time-intervals ( Table 1 ).
Define the bulk-averaging operator and the average bulk Mach and Reynolds numbers [3,10] where V is the volume of the computational box with dimensions L x ×L y ×L z in the corresponding space directions ( Table 1 ).The numerical procedure described in [ 5 , (47,48)

Statistics acquisition and observation times
Semi-analytical velocity and temperature profiles were used to determine initial conditions for each new (Re τ , MCL x ) -configuration [ 5 , p. 790].Instead of applying random perturbations of the velocity field [ 5 , p. 790], the spatial organization of { ρ, u i , T } was interpolated from an existing simulation (at the nearest available conditions), with appropriate rescaling to fit the new target meanflow conditions.The initial part of the calculation ( t CNVRG ) served to eliminate the transient perturbation associated with the approximate initialization ( Fig. 4 ).After this initial transient, on-board sampling was activated and computations were continued to acquire converged statistics for an observation time-interval t OBS ( Fig. 4 ).
A large number for moments q 1 • • • q n txz (t OBS , y ) were acquired at each wall-normal station by a moving-averages approach where t s is the sampling time-step (in the present datasets sampling was performed at each iteration, ie t s = t), (•) xz denotes averaging in the homogeneous xz-directions, (•) txz denotes averaging in the homogeneous txz-directions, and generic q -notation is used for flow quantities.Simultaneously, extreme events were identified for a large number of dynamic and thermodynamic quantities.Using again generic q -notation for flow quantities symmetric q (y ) = q (2 δ − y ) : The sampling of extreme events allows the determination of the range covered by the pdfbins, for acquisition of pdfs, which starts after an initial observation time ( Fig. 4 ), required for a reliable evaluation of extreme events.The upper/lower halfchannel symmetrization in (4) is necessary to ensure symmetry of the lower/upper pdf-bins.
Joint 2-variable pdfs (subdirectory 3_pdfs2q_two_variable_joint_pdfs ) were sampled on a restricted range [ q1 − 5 q 1rms , q1 + 5 q 1rms ] × [ q2 − 5 q 2rms , q2 + 5 q 2rms ] , to reduce memory requirements.This range allows for an accurate description of the pdfs and for the calculation of the covariance, but not for higher-order correlations which are obtained directly by averaging (3) .

Post-processing and datasets construction
At the end of the computations, all statistical data ( §4.2 ) are assembled from the MPIprocesses in a single data-structure, and upper/lower halfchannel symmetrization or antisymmetrization is performed, both for averages and pdf-bins.Standard Reynolds-and Favreaveraging relations are applied on moments ( eg q 1 • • • q n or ρq 1 • • • q n ) to obtain turbulent correlations ( eg q 1 • • • q n or ρq 1 • • • q n ).Similarly, hit-data in the pdf-bins are readily transformed to pdfs by normalization of the integral to 1, q and q rms are computed by integration, and used to obtain the standardized pdfs which are included in the datasets.This procedure is performed by a post-processing module, which outputs automatically all the datafiles (comments and data) included in the present database.
a b s t r a c t The database contains detailed statistics of compressible turbulent plane channel (TPC) flow, obtained from direct numerical simulation (DNS), with a very-high-order massively parallel solver of the compressible Navier-Stokes equations.It contains datasets for 25 different flow conditions determined by the corresponding HCB friction Reynolds number and centerline Mach number, covering the ranges 100 Re τ 10 0 0 and 0 .3 MCL x 2 .5 .All calculations are for strictly isothermal wall conditions at temperature T w = 298 K in a mediumsize (MB) computational box ( 8 πδ × 2 δ × 4 πδ where 2 δ

(Fig. 1 .
Fig. 1.View of (a) the dataset directories for each (Reτ , MCL x , Tw ) contained in the database directory GV_TPC_MB_AIR0 , (b) the subdirectories contained in each dataset, and (c) the plain-text (.txt) files contained in each of these subdirectories (the typical dataset Retaus_0965_MCLx_1p50_isoTw_0298_MB_AIR0 will be used to describe the datasets).

Fig. 2 .
Fig. 2. Typical datafile headers (a) directory 0_GD_global_data global datafiles containing 1 line of data, (b) directories 1_PBs_profiles_and_budgets and 2_pdfsq_single_variable_pdfs datafiles containing several lines of data, and (c) directory 3_pdfs2q_two_variable_joint_pdfs datafiles containing several blocks of data with each block containing several lines of data (line 3 of each datafile indicates the number of lines of comments before the numerical data, the number of space-separated columns in each dataline, and the lines of data or the lines of data per block and number of blocks).

Table 1
, p. 791] fixes target (Re B w , M B w ) -conditions.These target values were carefully chosen during the construction of the database to built a (Re τ , MCL x ) -matrix instead (data for the relation between (Re τ , MCL x ) and (Re B w , M B w ) were interpolated/extrapolated from previously calculated datasets, learning as more datasets were made available).