This document describes a GPU-based omni-directional integration method for reconstructing 3D pressure fields from measured acceleration data. It validates the method using a direct numerical simulation database and applies it to experimental measurements of turbulent channel flow over a compliant surface. The key findings are that the 3D omni-directional integration method achieves high accuracy (<1% error) and performance (~1 minute runtime), and the reconstructed pressure fields from this method correlate well with the flow behavior.

1. GPU-based, Parallel-line, Omni-directional Integration of
Measured Acceleration Field to Obtain the 3D Pressure
Distribution
Jin Wang, Cao Zhang and Joseph Katz
Department of Mechanical Engineering
Johns Hopkins University
Baltimore, MD
Sponsored by ONR, Program Officer: D. Nalchajian

2. Reconstruction of pressure from velocity field:
1. Obtain distribution of material acceleration from PIV Data
2. Two main options for calculating the pressure:
a. Direct integration of material acceleration, with known reference pressure
at a certain point. (1.Liu, X. and Katz, J., Exp. Fluids,2006. 2.Liu, X. and Katz,J., Phys. Fluids, 2008. 3.
Liu,X. and Katz,J., J. Fluid Mech., 2013. 4. Pranav J. ,et. al., J. Fluid Mech., 2014.)
b. Solve the Pressure Poisson Equation (PPE):
(1. BW van Oudheusden, Meas. Sci. Technol.,2012. 2. Jaw, S.Y., et. al., Journal of Visualization, 2009. 3.
Sina Ghaemi ,et. al., Exp. Fluids, 2012. 4. John J Charonko, et. al., Meas. Sci. Technol., 2010.)
Dominant term
Material Acceleration
Negligible for high Re and in
flow away from boundaries
Pressure Reconstruction From Material Acceleration

3. An extension of the 2D method to 3D case.
(Liu & Katz, 2006,2008,2013)
The 3D GPU based Parallel-line, Omni-directional Integration Method
Grid on Sphere
Grid casted on
sample volume
𝛺~4𝜋/N
Icosahedron
Grid
System
Two parameters for virtual path
generation:
1. Angle. (number of grid points)
2. Parallel line spacing
Our Sample volume: 100x47x38
Number of triangular faces (N)
needed to resolve resolution of
sample volume grids:
𝛺*(L/2)2~0.5∆𝑥2
𝛺~4𝜋/N
L is length scale of the sample
volume
∆𝑥 is grid spacing of sample
volume
Number of paths: N~104
GPU Info:
Tesla K40c
Clock Rate: 745 MHz
Cuda Cores: 2880
Memory: 12GB
Run on GPU
~1 minute
Integrations: 108
Run on CPU ~3 hours

4. Validation of Pressure Reconstruction Methods with
JHU DNS Database
• JHU Turbulence DNS Database
• In database: Material acceleration is calculated via
Lagrangian method
• Selected region for testing is 64x64x64
Direct numerical simulation of forced isotropic turbulence, turbulent channel flow
Three-component velocity and pressure can be fetched from the database.
(http://turbulence.pha.jhu.edu)
Domain(Range of x, y and
z)
Lx × Ly × Lz = 8πh ×
2h × 3πh
Rec 2.2625e4
Total Number of grid points 2048×512×1536 Centerline velocity 1.1312 m/s
Grid size 2π/1024 Friction Velocity 4.9968e-2 m/s
viscosity 5e-5 kg·m-1·s-1 Viscous length scale 1.0006e-3 m
Simulation time step 0.0013 s Reτ 999.35
JHU DNS Database Parameters for Turbulent Channel Flow

5. Methods To Be Used
Omni3D
Application of the 3D GPU-based Omni-
directional integration
Omni2D
 Integrate the volumetric pressure
plane by plane (XY) via the 2D
virtual-boundary omni-directional
integration.
 2D pressure on XZ plane at y=ymax/2
is used for matching of planar
pressure
PPE-Omni2D
 Solving Pressure Poisson Equation
 Boundary conditions:
Dirichlet B.C from Omni2D
on top face
𝛻𝑝 = −𝜌
𝐷𝒖
𝐷𝑡
, on other faces
PPE-Bernoulli
 Solving Pressure Poisson Equation
 Boundary conditions:
Dirichlet B.C from Bernoulli Equation
on top face
𝛻𝑝 = −𝜌
𝐷𝒖
𝐷𝑡
, on other faces
(Scarano et al., 2012, 2013)

6. x/h0
0.2
0.4
0.6
y/h0.2
0.4
z/h
0
0.2
0.4
x/h0
0.2
0.4
0.6
y/h
0.2
0.4
z/h
0
0.2
0.4
x/h0
0.2
0.4
0.6
y/h
0.2
0.4
z/h
0
0.2
0.4
x/h0
0.2
0.4
0.6
y/h
0.2
0.4
z/h
0
0.2
0.4
x/h0
0.2
0.4
0.6
y/h
0.1
0.2
0.3
0.4
2
0.4
0.008
0.0064
0.0048
0.0032
0.0016
0
-0.0016
-0.0032
-0.0048
-0.0064
-0.008
x/h0
0.2
0.4
0.6
y/h
0.1
0.2
0.3
0.4
2
0.4
0.005
0.004
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
-0.005x/h0
0.2
0.4
0.6
y/h
0.1
0.2
0.3
0.4
z/h
0
0.2
0.4
x/h0
0.2
0.4
0.6
y/h
0.1
0.2
0.3
0.4
z/h
0
0.2
0.4
x/h0
0.2
0.4
0.6
y/h
0.2
0.4
z/h
0
0.2
0.4
Comparison of Four Different Methods
turbulent channel flow
P(Pa)
Relative
Error
x/h0 0.2 0.4 0.6
y/h
0.2
0.4
z/h
0
0.2
0.4
Omni3D
RMS Error= 0.46%
Omni2D
RMS Error= 3.61%
PPE-Omni2D
RMS Error= 0.43%
PPE-Bernoulli
RMS Error= 4.33%

7. SyntheticPIVDataDNSdata Contributors of Pressure Error in Turbulent Channel Flow
𝛻2
𝑝 = −𝜵 ∙
𝐷𝒖
𝐷𝑡
𝑃𝑡𝑜𝑝 = 𝑃𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖
𝜵𝑝| 𝑜𝑡ℎ𝑒𝑟 = −
𝐷𝒖
𝐷𝑡
𝑷 𝟏
𝑷 𝟐
𝑷 𝟑
𝛻2
𝑝 =0
𝑃𝑡𝑜𝑝 = 𝑃𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖
𝜵𝑝| 𝑜𝑡ℎ𝑒𝑟 = 0
𝛻2
𝑝 = −𝜵 ∙
𝐷𝒖
𝐷𝑡
𝑃𝑡𝑜𝑝 = 𝑃 𝐷𝑁𝑆
𝜵𝑝| 𝑜𝑡ℎ𝑒𝑟 = −
𝐷𝒖
𝐷𝑡
1. Evaluate error in
PPE-Bernoulli
2. Evaluate error
from boundary
3. Evaluate error
from acceleration
Elevation: 0.35h
Elevation: 0.2h

8. Error Propagation
Omni3D PPE-Omni2D PPE-Omni2D
p
Relative
Error
• 300 % random noise is added to 10
central x-y planes.
• The error will decay quickly and then
remain a constant.
• PPE method depends on the Dirichlet
B.C.
Dirichlet
B.C.
ymax
zmax

9. x
Path 1
Path 2
Path 3
Selected Path
Further Improvement of Omni3D
Integration path is
weighted inversely
proportional to the
circulation of
material acceleration
Weighted Path
LargeErrorZone
• Requirement of known
separation between large and
small error zone
• Accuracy of pressure field
away from the wall is
improved
𝛻𝑝 = −𝜌
𝐷𝒖
𝐷𝑡
𝛻 ×
𝐷𝒖
𝐷𝑡
represents the error of
material acceleration
Selected Path

10. Performance of Omni3D
To achieve high performance integration :
1. Grid points should be uniform.
2. Number of grid points should be huge enough to resolve the resolution of
sample volume.
3. The choice of line spacing is a trade off between efficiency and accuracy.
Type of grids Connectivity Vertex Interval Face Area
Subdivision of icosahedron 175.35 15.22 11.35
Comsol Multiphysics 4.4 16.71 13.14 8.34
Homogeneity of Grids (σ/µ)
Total Computing Time for one realization~60 s

11. • Transparent acrylic channel
• Cross-section: 203.2  50.8 mm2
• Sodium Iodide (NaI) solution:
ρ = 1.8103 kg/m3
ν = 1.110-6 m2/s
• Centerline speed: U0 = 2.5 m/s
• Friction velocity: uτ= 0.102 m/s
• Half-channel height: h=25.4 mm
• Reh = U0h/ν = 57000
• Reτ = uτh/ν = 2300
Upstream Inserts
Downstream Inserts
11
Application of Omni3D method
Turbulent Channel Flow Over a Compliant Surface

12. Illumination:
• High speed laser (527 nm)
• A thick 30 x 10 mm² laser “slab”
Image Acquisition:
• Four High speed digital cameras
• 6000 fps @ 600 x 1200 pixels.
Sample Volume:
• 30 (x) × 10 (y) ×10 (z) mm3.
Reconstructed volume:
• 1380 × 638 × 611 voxels
• The size of a single voxel is 18.8 µm
3D, multi-pass, direct cross- correlation:
• 48 x 48 x 48 voxels with 75% overlap
• The vector spacing is 0.23 mm.
Calibration, reconstruction and correlation
are performed using LaVision DaVis 8.1
Experimental Setup: TPIV
12

13. • M1 leaks 0.1% of the light (reference beam)
• M3 leaks 0.1% of the light (object beam)
• A fifth camera records interference fringes.
• Sampling rate:
3000 fps @ 1584 × 1024 pixels.
• Sample area:
17 (x) × 10 (z) mm2
• Fringe image enhancement and phase
evaluation procedures have been
developed and calibrated in-house using
both experimental and synthetic tests.
Experimental Setup: Mach-Zehnder Interferometry
13

14. P(Pa)
Instantaneous Pressure Field
Omni3D Omni2D
PPE-Omni2D PPE-Bernoulli
Material acceleration calculated following Liu & Katz, 2013
Red lines: contour plot of 𝜆2

15. Pressure-deformation correlations
Omni3D Omni2D
PPE-Omni2D PPE-Bernoulli
Peak Value: -0.15 Peak Value: -0.10
Peak Value: -0.11 Peak Value: -0.13
p-d correlation

16. x/h
y/h
-0.4 -0.2 0 0.2 0.40
0.1
0.2
0.3
0.4
p'
100
90
80
70
60
50
40
30
20
10
0
-10
Conditional Averaged Flow, Condition on Pressure
• Positive pressure at the interface of sweep
ejection transition.
• Considering the w’, it’s a 3D splat-flow.
• Negative pressure is under the negative spanwise
vortex
streamlines u’ at ∆𝑧/ℎ = 0 v’ at ∆𝑧/ℎ = 0 w’ at ∆𝑥/ℎ =0
w’ at ∆𝑥/ℎ =0streamlines u’ at ∆𝑧/ℎ = 0 v’ at ∆𝑧/ℎ = 0
Conditional averaged flow condition on p(∆𝑥 = 0, 𝑦 = 0.02ℎ, ∆𝑧 = 0)<-𝜎 𝑝
Conditional averaged flow condition on p(∆𝑥 = 0, 𝑦 = 0.02ℎ, ∆𝑧 = 0) > 𝜎 𝑝
Sweep Ejection Transition
Shear Layer
Splat Flow
x/h
y/h
-0.4 -0.2 0 0.2 0.40
0.1
0.2
0.3
0.4
p'
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
High pressure

17. 17
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Conclusions
 A GPU-based parallel-line omni-directional integration method is developed
for reconstruction of 3D pressure field.
 To achieve high performance of the method, the grid points should be
uniform and number of grid points should be huge enough.
 The Omni3D method is applied to turbulent channel flow over a complaint
surface, and reconstructed pressure is correlated with the flow