Comparison between reduced-channel simulations based on POD with primitive variables and with conservative variables

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The numerical approach for solving the governing equations

Simulations conducted in this study were performed by using the in-house parallel (MPI) DNS solver, named CHORUS (Compressible High-Order Unsteady Simulation), that has been developed at LIMSI for unsteady compressible ow simulations. The ability of the CHORUS software to compute high Reynolds compressible ows has been demonstrated on various test-cases in previous studies (see for instance [22]).
The resolution of the governing equations (2.1) is based on a nite volume approach. An operator splitting procedure is employed that splits the resolution into the Euler part and the viscous problem: @Qc @t + r FEuler = r Fvisc.

Numerical conguration : the compressible turbulent channel ow

We consider a compressible turbulent channel ow conguration where the air ows in between two horizontal solid walls maintained at the same constant temperature that limit the channel at the lower and upper bounds. The conguration is shown in Figure 2.1. The streamwise, spanwise, and wall-normal directions of the ow are respectively denoted by x, y, and z. The dimensions of the channel are (LxLy Lz) = (2 4 3 2)H, where H is the half of the space between the two horizontal solid walls and is taken as the reference length scale (H = 1). The corresponding velocity components in the space directions are respectively denoted u, v, and w. The mesh in the streamwise and spanwise direction has a constant grid spacing. On the opposite, in the wall-normal direction, the grid is tightened near the solid wall by using a hyperbolic tangent function to ensure that the rst point above the wall satises the constraint z+ < 1 (expressed in the wall units).

Boundary and initial conditions:

In the streamwise (x) and the spanwise (y) directions, we suppose periodic developments and periodic boundary conditions are prescribed. In wall normal direction z, a no slip boundary condition is prescribed on the solid walls and the temperature is prescribed.
The density is then calculated by solving the continuity equation to ensure the conservation of mass. At the initial state, the streamwise velocity prole in the normal to the wall direction is dened as: ut=0 = 3 U0 z(2H 􀀀 z) 4H2.

Treatment of the periodic boundary condition in the streamwise direction.

In our model (Figure 2.1), to simplify the simulation, the streamwise direction is assumed to be periodic. In fact, one must notice that the pressure does not evolve periodically, however the pressure gradient which appears in the Navier-Stokes equations (2.1) can be considered as periodic. To enforce a periodic ow motion and compensate looses of the mass, the momentum and the total energy due to viscous eect in the boundary layers, a macroscopic pressure gradient is added to the channel ow that could be viewed as an external force applied to the ow. The magnitude of this macroscopic pressure gradient must be calculated at each time step. To do so, we average the equations of the momentum components over an horizontal plane, in the x and y directions. For this, we dene an operator, noted < >xy applied to a variable : < >xy= 1 LxLy Z Lx 0 Z Ly 0 dx dy.

DNS results of the subsonic channel ow

We consider a periodic plane channel ow bounded by two isothermal solid walls separated by a gap of Lz = 2 H. The Mach number is Ma = 0:5. The Reynolds number based on the bulk velocity and the channel half-height is ReH = 3000, corresponding to a friction Reynolds number equal to Re = 180: Present simulations have been conducted by using the unlimited OS7 scheme, described previously. This test-case have largely been studied through both DNS and LES. Although incompressible, the DNS reference results from J. Kim et al. [38] are generally taken as a reference solution. The present results will be compared with these reference results.
The mesh of this test-case is 9797129 in the xyz directions, respectively. As the initial state corresponds to the solution for a laminar ow, a transition towards turbulence occurs during the simulation after a long time integration. After this transition, statistics are calculated over a very long time.
The mean streamwise velocity prole (< u >), non-dimensionalized by the friction velocity, is plotted versus the wall normal direction expressed in wall units (z+) in Figure 2.3. The convergence of the statistics of mean values in the present DNS can be judged on the perfect t obtained between the proles from the upper and the lower half of the channel.
A logarithmic law is clearly obtained in the present results which ts the classical log-law distribution (Figure 2.3).

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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 Inuence 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 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 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 

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