The numerical approach for solving the governing equations

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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 effects
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 flow 
2.1 The governing equations for compressible flows
2.2 The numerical approach for solving the governing equations
2.3 Numerical conguration : the compressible turbulent channel flow
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 flow
2.4.3 DNS results for the supersonic channel flow
2.4.4 Comparison between results of subsonic flow and supersonic flow
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 flow fields 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 flow fields
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: Definition of POD variables
5.2.1 Proper Orthogonal Decomposition
5.2.2 Results without rescaling
5.2.3 Results with rescaling
5.2.4 Influence of the type of decomposition: summary
5.3 Influence 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 Influence of the boundary condition characteristics
5.4.1 Results for different choices of Riemann invariants
5.4.2 Correction step in the estimation procedure of the POD amplitudes
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 flow 
6.1 Mesh interpolation for POD
6.2 Comparison between instantaneous fields 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 flow
7 Conclusions and perspectives 
7.1 Conclusion
7.2 Perspectives
A Viscous ux 
B Macroscopic Pressure gradient 
References