Adapted Linear Forcing for Inlet Turbulent Fluctuations Generation and Application to a Conical Vortex Flow

Abstract

A method based on linear forcing is developed to generate realistic turbulent fluctuations at the inlet of a computational domain. The method uses a precursor computation of convected forced isotropic turbulence and variables are extracted from a plane to be imposed at the inlet of the main simulation. In the precursor computation, the turbulent fluctuations are generated and sustained using a synthetic generator and a linear forcing which have been adapted in order to allow the specification of the intensity and length scale. The method is then applied to the analysis of a conical vortex in interaction with a wall subjected to external turbulence using Improved Delayed Detached Eddy Simulation. Good agreement with experimental data is achieved both concerning the decay of the turbulent fluctuations in the upstream region and concerning the sensitivity of the vortex to the Free Stream Turbulence.

Alternative Strategies for the Forced Convected Fluctuations in the Precursor Simulation

As mentioned in Sect. 2.2, possibilities to avoid using synthetic fluctuations in the precursor simulation, and therefore simplify the method, are briefly discussed in this appendix. The first part deals with the use of simple white noise at the inlet and the second part deals with the use of a periodic precursor simulation. As will be seen in both cases, while not being strictly necessary, the use of synthetic fluctuations with a prescribed physical spectrum allows to reduce the length of the computational domain of the precursor simulation.

1.1 White Noise at the Inlet of the Precursor Simulation

A first alternative to the strategy presented in Sect. 2.2 is to replace the synthetic fluctuations with a prescribed spectrum by a simple white noise. First, it should be mentioned that initial tests have shown that simulations using a white noise at the inlet are less stable, and that simulations could be only be carried out using the forcing using filter average (simulations using the forcing with plane average were found to diverge). Figure 24 compares the evolution the turbulent kinetic energies and the skewness coefficient as a function of x obtained using synthetic fluctuations whith that obtained using a white noise (the energy was arbitrarily set to the same value of the resolved kinetic energy as for the synthetic fluctuations). The parameters are the same as in Sect. 2.2 ($k_0=0.0121$, $L_f=0.125$ $L_{mf}=5L_f$ and $\varDelta x =0.02$). The main conclusion is that the distance from the inlet at which $k_r$, $k_s$ and $S_u$ reach constant values is much larger using a white noise (while $x=6\simeq 3.5 U_0T_t$ has been estimated sufficient using the synthetic fluctuations, $x=12$ seems to be necessary using white noise). While no attempt has been made to analyse the effect of the intensity of the white noise, it is therefore thought that the use of synthetic fluctuations with a prescribed physical spectrum allows to minimize this transition length and therefore the length of the computational domain for the precursor simulation.

Fig. 24

Inluence of inlet conditions in the precursor simulations

1.2 Fluctuations Extracted from a Periodic Precursor Simulation

Fig. 25

Influence of $L_x$ in a periodic precursor simulation: Time velocity spectra of the convected forced fluctuations

Fig. 26

Influence of $L_x$ in a periodic precursor simulation: Velocity time correlation of the convected forced fluctuations

Another possibility to avoid using synthetic fluctuations is to use a periodic precursor simulation. To avoid spurious periodicities in the fluctuations extracted from a plane, the length $L_x$ of the domain in the direction of convection has to be large enough so the time for the fluctuations to flow through the length of the domain is large enough so the correlations in time are negligeable (i.e. $\frac{L_x}{U_0} \gg \frac{L_t}{k_t^\frac{1}{2}}$). A short comparison between both strategies is made in this section. To do this test, forced convected fluctuations are generated using triperiodic domain of width $L_y=L_z=1.2$ and of different length $L_x=1.2\text{, } 6 \text{ and } 12$. The flow is initialised using a random white noise superimposed to a mean velocity $U_0=1$ and the forcing is applied as described in Sect. 2 using $k_0=0.0121$, $L_f=0.125$ and $\varDelta x =0.02$. As the flow is homogeneous in space at any time, the average entering the forcing term can be taken as a space averaging over the whole domain. When a statistical steady state is reached, the fluctuations are extracted from a plane at a fixed position and prescribed at the inlet of the same test domain as in 2.3.

Fig. 27

Influence of $L_x$ in a periodic precursor simulation: Decay of the fluctuations

Figure 25 and 26 present the time velocity spectra and the time correlations of the velocities at the plane where variables are extracted for each $L_x$ and make a comparison with the fluctuations generated as described in Sect. 2.2 at $X_p=6$. As expected, when $L_x$ is short, a strong periodicity is obtained at the frequency $f_0=U_0 / L_x$ and longer $L_x$ lead to weaker periodicity. From the time correlation of v shown in Fig. 26 (right), it can be seen that even for the longest $L_x=12$, a small but non negligeable peak in the correlation is observed at $t\simeq \frac{L_x}{U_0}=12$. As in our case, the main purpose of the generation of the fluctuations is a study of the influence of FST on the dynamics of a conical vortex, especially concerning the spectral content of velocities and pressure, it has been considered that such a periodicity would lead to an ambiguity in the analysis. It can also be estimated from Fig. 26 that a length $L_x$ higher than 16, which corresponds approximately to $\frac{L_x}{U_0} \gtrsim 10 \frac{L_t}{k_t^\frac{1}{2}}$ would be necessary to obtain negligeable spurious periodicity.

Figure 27 however shows that the decays of the fluctuations, when prescribed at the inlet of the test domain, seems independent of $L_x$. From this short analysis, it can be concluded that such a strategy to generate fluctuations is a possible alternative. The main advantage is that the method is greatly simplified (no synthetic fluctuations is needed and the average can simply be taken as a global spatial average) and the main disadvantage is a higher cost of the precursor simulation, due to a required $L_x$ more than two time longer than with the method described in Sect.2.2.