15TH EUROPEAN TURBULENCE CONFERENCE, 25-28 AUGUST, DELFT,. THE NETHERLANDS
Effect of linear feedback control on the optimal transient growth in particle-laden channel flow Yang Song, Chunxiao Xu & Weixi Huang
AML, Department of Engineering Mechanics, Tsinghua University, Beijing, China
Abstract The optimal transient growth process in a particle-laden channel flow is studied under the influence of the linear feedback
control. The equilibrium Eulerian approach with the assumption that the particles are small and spherical is adopted. The effect of initial distribution of particles on the optimal transient growth of perturbations is discussed. The LQG control of the particle-laden flow system is considered and compared with the no control cases.
Keywords particle-laden, optimal transient growth, LQG
INTRODUCTION
Desertification and sandstorm have become noteworthy problems in the past several decades. Due to the large geometrical scale, the sand flow is always at high Reynolds numbers and turbulent. As we know, there are large-scale coherent structures dominate in high Reynolds number flows [1-2]. These structures have a significant influence on the sediment discharge. Since sand flow is a specific case of multiphase flow, the physical models for multiphase flow can be used in the analysis. Generally speaking, according to the property and concentration of particles there are several computational approaches-dusty gas model, equilibrium Eulerian model, Eulerian-Eulerian model, Lagrangian point particle model and fully resloved model [3]. To analyze the dynamics of small inertial particles, like sand, equilibrium Eulerian approach has been widely used in the previous studies. But only a few of those are concerned with the instability especially transient growth of perturbations [4], which has been fully discussed in pure fluids flow.
In this paper, the optimal transient growth process in a particle-laden channel flow is computed with the analysis of the effect of a feedback control. Here the particles are assumed to be spherical and small enough, so that the equilibrium assumption can be satisfied. The Stokes drag formula is used to model the interaction between fluid and particles. The base fluids flow and concentration of particles are selected to satisfy the mean momentum equations in turbulent channel flow.
MODEL FORMULATION AND RESULTS
Firstly, we consider the optimal transient growth process of perturbations with an initial uniform distribution of particles and the relaxation time τ of particles is sufficiently large. These assumptions lead to a dusty gas model, in other words, the velocity of particle is the same as that of the fluid. The equations of the system can be simplified. The results are compared with the previous studies [5]. It is seen that a uniform distribution of particles has the same effect as increasing the Reynolds number itself for the transient growth process of perturbations.
Figure 1. The transient growth of perturbations at (α,β)=(0,2) with (a) pure fluid and laminar base flow (Re=5000),
(b) pure fluid and turbulent base flow (Re=30000). The three lines stand for no control, wall information output feedback control and full information state feedback control.
Secondly, another validation is done for neglecting the sufficiently large relaxation time assumption. In this case, one should solve the full system of the equilibrium Eulerian approach [4]. Then the optimal transient growth
0 500 1000 1500 0 1000 2000 3000 4000 5000 t G m a x without control full information wall information 0 20 40 60 80 100 0 2 4 6 8 10 t G m a x
(a)
(b)
process of perturbations with an initial distribution of particles is performed. This computation model is close to the realistic sand flow naturally. Compared with the former results, the different optimal transient growth processes and optimal structures are provided and studied. These distinguished differences should be owing to the initial distribution of particles. Then this influence on the transient growth process and optimal structures is studied and discussed.
Figure 2. The maximum transient growth amplifications and associated optimal times are as a function of span-wise
wave number with turbulent base flow at Re=30000, α=0.
When considering the effect of feedback control on the transient growth process of perturbations, the opposite proportion control and the Linear Quadratic Gaussian (LQG) control are employed. The opposite proportion control, which is designed by weakening the quasi stream-wise vortices, works with a probe location at y+≈15 based on the previous studies of pure fluids flow [6]. On the other hand, the probe location of LQG control system is set at walls. The detected quantities are the wall frictions in two parallel directions. The actuators are the wall normal component of velocity at wall in both cases. The verification is done for plane Poiseuille flow and the results are compared with the previous studies by Bewley and Liu [7]. In figure 1, the transient growth processes of perturbations with laminar and turbulent base flow are shown. The LQG control has a sufficient effect on reducing the transient growth of perturbations even in the turbulent case. The influence on maximum transient growth amplifications of perturbations are shown in figure 2. Here we can see the control has a good performance in reducing the transient growth amplifications at large wave numbers. But things become different at small wave numbers.
Finally we extend the both cases to particle-laden channel flow. The transient growth processes under different controls will be shown in the conference.
References
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[5] P. G. Saffman. On the stability of laminar flow of it dusty gas. Journal of Fluid Mech, 13(1): 120–128, 1962.
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China Physics, Mechanics and Astronomy, 56 (6): 1053-1061, 2013.
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