DNS results of the subsonic channel ow

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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).

DNS results for the supersonic channel ow.

We here consider the same geometrical conguration with however a higher Mach number (Ma = 1:5). The Reynolds number based on the bulk velocity and the channel halfheight is ReH = 3000, corresponding to a slightly higher friction Reynolds number equal to Re = 220: This test-case has largely been studied through both DNS and LES, and the reference solution generally considered is from G. N. Coleman et al. [20]. The present results will be compared with these reference results. The mesh of the supersonic test-case is 257 161 161 in the x y z directions, respectively. The mesh dimenion is supersonic ow is larger than that in subsonic ow to well capture the sturucture of turbulence. In the supersonic case, the transition towards turbulence occurs after a very long integration time, much longer than in the supersonic test case. Once the turbulent ow is fully developed, statistics are calculated over an integration time equal to tend 􀀀 tstart = 75 time units.
Mean proles of the density, the static temperature and the streamwise velocity component are plotted versus the normal to the wall direction in Figure 2.6. The streamwise velocity component is dimensionless by the friction velocity and z is expressed in wall units, and the proles of u in both bottom half channel and top half channel are the same.
A logarithmic law is clearly obtained in the present results which ts the classical log-law distribution (Figure 2.6, bottom); its extent is however weaker than in subsonic case. Wall normal proles of the temperature and the density (Figure 2.6) recover classical distributions. The mean density at the wall reaches 1.355 while its value at the channel center is 0.98. These values are in complete agreements with the reference values given in Table 2.2. The mean temperature is unity at the wall since it is the value taken as reference for the dimensionless variables, and reaches 1.37 at the channel center, which is weakly underestimated compared with the reference value (Table 2.2).

Comparison between results of subsonic ow and supersonic ow

To compare results of the subsonic channel ow with those of the supersonic channel ow, mean proles of the density, the viscosity and the streamwise velocity are presented in Figure 2.8. The present results can be compared with results of [27] in Figure 2.9. who explored the compressibility eects on turbulence in channel ow by studying four congurations: Ma = 0:3 with Re = 181; Ma = 1:5 with Re = 221; Ma = 3:0 with Re = 556; and Ma = 3:0 with Re = 1030. The two rst congurations approximately correspond to the present simulations. The present DNS results favorably compare with results coming from the literature and the eects of compressibility are very well reproduced.

Inlet Synthetic boundary conditions: rescaling approaches

Determining appropriate boundary conditions is a key issue for the simulation of ows, as on the one hand it must describe the physics and on the other hand it must be dened in a suitable way for numerical resolution. In this section, we ask the question: how is it possible to dene a boundary condition that mimics the physics of the ow?
The simplest answer would be to superpose random uctuations on a desired mean pro- le, where Q =< Q > +QR. This method has been implemented with some success. Lee et al [45] used this method for direct numerical simulation of compressible turbulence. Le and Moin [43] generated anisotropic turbulence for the generation of inlet boundary conditions for a backward facing step. The amplitudes of the uctuations can be set to satisfy some statistics such as the Reynolds stresses but typically these statistics are second-order and computed at one point only. It is very dicult to impose phase relationships between the uctuations and higher-order correlations cannot be satised and the ow lacks turbulent structure. As a result, the success of these methods has been limited. In particular such characteristics of the turbulent boundary layer as the momentum thickness or wall friction are not always recovered. This led Aksevoll and Moin [4] to introduce a auxiliary or precursor simulation, from which they could select a location where these characteristics were close to the desired values. A simple auxiliary simulation is that of a periodic channel ow in both horizontal directions. Such a simulation was used by Kaltenbach [34] to generate inlet ow conditions for LES of a plane diuser. This idea can be extended to a turbulent boundary layer. However the boundary condition at the top of the domain has to be modied to reproduce that of a free-stream ow, which is done with a symmetry condition at the top of the channel (which ensures that the vertical velocity is zero). However, unlike the boundary layer, a velocity eld in a channel is characterized by a non-zero mean advection. To address this issue, Spalart and Leonard [76] added source terms to the Navier-Stokes equations which led to an equilibrium spatially evolving boundary layer with a correct momentum thickness and wall friction. A considerable simplication of the method was provided by Lund et al [49] who only modied the boundary conditions of the equations. The idea is to use as an inlet boundary condition the velocity eld extracted at the outlet and properly rescaled using Spalart and Leonard’s ideas.
The mean ow is rescaled according to the law of the wall in the inner region and the defect law in the outer region. The velocity uctuations are rescaled with the local turbulent intensity. In order to avoid over-determination of the problem, the momentum thickness is directly imposed at the inlet from the computation of the wall friction using a correlation similar to the Ludwig-Tillmann correlation [49]. This recycling method has been popular over the years [43] and several variants have been proposed [87].

The reconstruction procedure

A pre-requisite for the procedure (step 0) is the POD basis for the reference ow. Which means the POD basis is computed from reference simulation with snapshot methods.
Then at each time step, the reconstruction procedure can be divided into several stages:
1. Estimation of the amplitudes using linear estimation.
2. Reconstruction of the velocity eld.
3. Rescaling of the eld.
4. Determination of the characteristics and construction of the boundary condition.
We will focus on the following aspects:
the choice of the POD variables (step 0):
– primitive or conservative.
– selection of variables to aggregate.
the denition of the rescaling (step 3).
the association of the variables with characteristics (step 4).
We now describe the last two steps.

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 

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