DNS results for the supersonic channel ow

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Table of contents

1 Introduction 
1.1 Wall turbulence
1.2 Simulating turbulence: DNS, LES and RANS
1.3 Wall models
1.4 Compressibility eects
1.5 Approximate boundary conditions
1.5.1 Slip boundary condition
1.5.2 Control-based strategies
1.5.3 Synthetic wall boundary conditions
1.6 Outline of the thesis
2 DNS of the compressible channel ow 
2.1 The governing equations for compressible ows
2.2 The numerical approach for solving the governing equations
2.3 Numerical conguration : the compressible turbulent channel ow
2.3.1 Boundary and initial conditions:
2.3.2 Treatment of the periodic boundary condition in the streamwise direction
2.4 Results of the Direct Numerical Simulation
2.4.1 Statistical treatments of simulation data
2.4.2 DNS results of the subsonic channel ow
2.4.3 DNS results for the supersonic channel ow
2.4.4 Comparison between results of subsonic ow and supersonic ow
3 Reconstruction of synthetic boundary conditions 
3.1 Proper Orthogonal Decomposition
3.1.1 Direct Method
3.1.2 Method of snapshots
3.1.3 Symmetry
3.1.4 Convergence
3.1.5 Results
3.2 Linear Stochastic Estimation
3.2.1 General denition
3.2.2 Application
3.2.3 Results
3.3 Reconstruction method
3.3.1 Inlet Synthetic boundary conditions: rescaling approaches
3.3.2 Inlet synthetic boundary conditions: Structure-based decompositions
3.3.2.1 The synthetic eddy method (SEM)
3.3.2.2 POD-based reconstructions
3.3.3 Wall Synthetic boundary conditions
3.3.3.1 Current approaches
3.3.3.2 The reconstruction procedure
3.3.3.3 Step 3: Rescaling
3.3.3.4 Step 4: Implementation of the reconstruction
3.3.3.5 First test: Reduced simulation using reference ow elds as boundary conditions
3.3.3.6 Computational basis
4 Synthetic boundary condition on one wall 
4.1 Results at height h+0 = 18 (h0 = 0:1) with primitive variables
4.2 Results at altitude h+0 = 18 with conservative variables
4.3 Comparison between primitive and conservative variables in reduced channel
4.3.1 Instantaneous ow elds
4.4 Results at height h+0 = 54 (h0 = 0:3)
4.4.1 Results at height h+0 = 54 with primitive variables
4.4.2 Results at height h+0 = 54 for POD based on conservative variables
4.4.3 Comparison between reduced-channel simulations based on POD with primitive variables and with conservative variables
4.5 Summary
5 Synthetic boundary conditions on both walls 
5.1 Fourier-based reconstruction
5.1.1 Synthetic boundary conditions at h+0 = 18 (h0 = 0:1)
5.1.2 Unrescaled boundary conditions
5.2 Reduced simulation at h+0 = 18: Denition of POD variables
5.2.1 Proper Orthogonal Decomposition
5.2.2 Results without rescaling
5.2.3 Results with rescaling
5.2.4 Inuence of the type of decomposition: summary
5.3 Inuence of the snapshot basis
5.3.1 Evolution of the amplitude of the dominant mode
5.3.2 Results with new POD basis for altitude h+0 = 18
5.4 In uence of the boundary condition characteristics
5.4.1 Results for dierent choices of Riemann invariants
5.4.2 Correction step in the estimation procedure of the POD amplitudes109
5.5 Spectra in the reduced channel at h+0 = 18
5.6 Results at h+0
= 54 (h0 = 0:3)
5.7 Conclusion
6 Simulations in supersonic ow 
6.1 Mesh interpolation for POD
6.2 Comparison between instantaneous elds in reduced channel and reference
6.3 Statistics in reduced channel
6.3.1 Simulation with POD reconstruction of rst 35 samples
6.3.2 Simulation with POD reconstruction using new 30 samples
6.4 Spectra in the supersonic ow
7 Conclusions and perspectives 
7.1 Conclusion
7.2 Perspectives
A Viscous ux 
B Macroscopic Pressure gradient 
References