Validation of the Pressure Code using JHU DNS Database JHU turbulence DNS database (http://turbulence.pha.jhu.edu ) The data is from a direct numerical simulation of forced isotropic turbulence on a 1024 3 periodic grid, with energy injected at low wave number modes; The database has 1,024 instantaneous realizations. Data of the 3 components of the velocity as well as the pressure are available. Three adjacent planar areas with 256 256 grids are selected from the 1024 3 cube for the pressure code validation. Material acceleration is calculated in the Eulerian form (i.e., with the unsteady and convective terms calculated explicitly) based on 3-D plus t velocity data, using central difference scheme. Three adjacent planes selected from the DNS cube JHU DNS Database Parameters Domain (range of x, y and z) [0,2π] Dissipation rate 0.0928 Total number of grid 1024 3 Rms velocity 0.681 Grid size 0.00614192 Taylor Micro. Scale 0.118 Viscosity 0.000185 Taylor-scale Reynolds number 433 Simulation time-step 0.0002 Kolmogorov time scale 0.0446 Time interval between stored samples 0.002 Kolmogorov length scale 0.00287 Duration of samples stored [0, 2.048] Integral scale 1.376 Total kinetic energy 0.695 Large eddy turnover time: 2.02 Thanks to Mr. Yunke Yang, Dr. Zuoli Xiao and Dr. Minping Wan for their help in retrieving the DNS data from the database.
Pressure Reconstruction by Integration of the DNS Pressure Gradient DNS Pressure Distribution Pressure Distribution integrated from the DNS pressure gradient Error of Reconstruction p = p Cal - p DNS Relative error mostly below 0.2% (maximum is about 3%) DNS Pressure distribution overlapped with the in-plane material acceleration Big error is associated with large acceleration, i.e., large pressure gradient Same plot with zoomed scale Conclusion: If the pressure gradient is right, the reconstructed pressure is right. * : DNS pressure gradient is obtained using central difference scheme.
(a) Pressure Reconstructed from the Material Acceleration and the Influence of the Viscous Term integration of material acceleration only Error of Reconstruction p = p Cal -p DNS is about 6% Pressure difference between methods (a) and (b) p a - p b integration of material acceleration and the viscous terms (b) Error of Reconstruction p = p Cal -p DNS is about 6% Relative difference mostly below 0.3% (maximum is about 3%) Conclusion: Contribution of the viscous term to pressure is negligible.
Forcing Term Plus Error: The Origin of the 6% Error in the Pressure Reconstructed from the Material Acceleration Same Difference between the two (Relative difference mostly below 0.3%) is about 6% -0.09 Apply the reconstruction code Vector Map of the Forcing Term νδ Vector Map of the DNS Pressure Gradient -0.005 Conclusion: The 6% error is due to the forcing term plus the error. Reconstruction does not introduce error.
Durability of Pressure Reconstruction Code with Artificially Introduced Noise Random error with amplitude of 200% of maximum acceleration introduced to a vertical strip in the DNS pressure gradient field DNS Pressure distribution integration of the noise-added DNS pressure gradient DNS Pressure distribution overlapped with noise-added DNS pressure gradient Error: p = p Cal -p DNS Conclusions: Error is local; Relative error mostly below 0.2% (Maximum is about 17%, i.e., at least one order of magnitude smaller than the error in acceleration). Reconstruction suppresses the error.
Durability of Pressure Reconstruction Code with Artificially Introduced Noise Random error with amplitude of 40% of maximum acceleration introduced to the entire DNS pressure gradient field DNS Pressure distribution integration of the noise-added DNS pressure gradient DNS Pressure distribution overlapped with noise-added DNS pressure gradient Error distribution: Perror=pCal-pDNS is about 7%
Conclusions The pressure reconstruction code is validated using a DNS database of forced isotropic turbulence. If the pressure gradient (measured acceleration) is right, the reconstructed pressure distribution is right. Reconstruction does not introduce error. Error propagation from acceleration to pressure is slow and mostly remained local. The reconstructed pressure error is al least one order of magnitude smaller than the acceleration error. Reconstruction suppresses the error.