ABSTRACT INTRODUCTION

Transcription

1 Numerical simulation of pulsating flow around a cube C. Dargent, D. Dartus, J. George Institut de Mecanique des Fluides de Toulouse, Avenue du Professeur Camille Soula, Toulouse, France ABSTRACT The purpose of this paper is to present numerical solutions for two-dimensional time dependent flow around a cube. A computational analysis is carried out using the commercially available fluid dynamics code PHOENICS for a solution of the Reynolds equation, the continuity equation and, for the turbulent closure, the classical k-e model is implemented. The variant parameter is the angle of attack that varies sinusoidally in time between 10 and 30 degrees. Thus, the inflow conditions and the boundary conditions vary in time in the same manner. We show the wake characteristics for the two mean velocity components and for some inflow conditions, the occurrence of vortex shedding. INTRODUCTION The study of oscillatory flow past bodies is very important to predict the forces induced, for example, on offshore structures and the drag, lift and pressure coefficients. However, recently new areas of interest have opened up for such flows, in particular for that of air pollution problems. Such flows influence the distribution of a pollutant in the wake of an obstacle, eg a cube in wind tunnel or a building in a simulated urban environment. In such flows, appears the classical von Karman vortex shedding phenomenon that has been documented in the experimental works of Mottram [1], Mottram and Robati [2], Patel [3] and Graham [4] as specified Bearman and Graham [5] and in the numerical work of Scanlon [6]. All have studied amplitude oscillating

2 168 Computer Simulation flows where the critical parameter is the pulsation frequency. In the highly turbulent range of Reynolds numbers, the Strouhal number is found to be f FT independent of the Reynolds number and is defined by Str = with fy$ vortex shedding frequency, H bluff body width and U mean velocity. In periodic pulsating flow, there exist an interference with vortex shedding process [1,2]. The extend of this interference is found to be a function of the pulsation frequency with the vortex shedding process synchronizing itself at half the pulsating frequency : a phenomenon known as locking-in. In this paper, we consider a two dimensional oscillating flow where the angle of attack is dependent on time while the amplitude stays constant and two frequencies have been considered. We inlet will try to explain the both frequencies that appear in the variations of different variables (velocity, turbulent kinetic energy,...). One frequency is the frequency of the incident flow, the other can be attributed to the vortex shedding for convenient incident frequencies. NUMERICAL MODEL The fluid flow was modelled by partial differential equations describing the conservation of mass and the conservation of momentum in two rectangular Cartesian coordinate directions : conservation of momentum : d / \ d ( \ _ d ( du, (1) ^7vPU,J + -T \PU.UJJ- 1 +S Jt gxj ^ gx^ 0XjJ conservation of mass : f_p + JL_(QU ) - 0 (2) '' Turbulence is modelled using the classical k-e eddy-viscosity turbulence model, of which a description is to be found in Rodi [7] : (3X at hu,,-,, " k

3 Computer Simulation 169 For the solution algorithm, a control volume-finite difference formulation is used. The discretized linear algebraic equations are solved by the SIMPLEST algorithm. The code uses a staggered grid arrangement [8] for the discretization of the momentum equations. Implicit temporal differencing [8] is employed and, for the discretization of convective transport, the hybrid scheme [8] is the default scheme within the code. For the geometrical configuration, we have considered a cube which has the same dimensions as that used by Castro and Robins [9] : H=0.2m. A nonuniform computational mesh of 63X by 93Z was employed. The mean velocity is equal to lom/s and the angle of attack varies sinusoidally in time : f Jirt\ a = 20+10cod. \ T ) For the values of the turbulent kinetic energy and the dissipation, we have taken the mean experimental values of Castro and Robins [9] : and respectively. Three tests were simulated : one with a permanent flow, one other with a low frequency oscillating flow (T=2s) and the last with a high frequency oscillating flow (T=0.02s). For the flow, the Reynolds number based on the mean velocity and the cube width is equal to ^ and is sufficiency high to have a constant Strouhal number. All boundary conditions were implemented by the inclusion of additional source and/or sink terms in the finite volume equations for computational cells at the boundaries. They have been validated in the permanent case with the experimental results of Castro and Robins [9] in a previous work [10]. DISCUSSION OF RESULTS We will present the results for the both cases (low frequency: T=2s, f=0.5hz and high frequency: T=0.02s, f=50hz) for the point number 11 (at 2.25H behind the cube in the longitudinal axis and at 0.5H from the side face in the transversal axis). Uref and Wref represent the reference velocity with Uref=Wref=10m/s. We show the temporal evolution of the transversal U and longitudinal W velocities at the observation point and the corresponding spectrum obtained by a Fast Fourier Transform.

4 170 Computer Simulation S(f> t/t A f (Hz > Fig 1 : Temporal evolution of the transversal velocity for the low frequency case and the corresponding spectrum s(f) i f (Hz > Fig 2 : Temporal evolution of the longitudinal velocity for the low frequency case and the corresponding spectrum From Figures 1 and 2, we can see that the velocities behind the cube are always periodic, but do not remain sinusoidal. This deformation of the incident flow is due perhaps to numerical inaccuracies and also to the fact that our case is an asymetrical problem. By a spectral analysis, we can also see an harmonic for the longitudinal velocity, but not for the transversal velocity. This harmonic is due to the FFT calcul, then the Fourier transformation is taken on a periodic non sinusoidal signal, which leads to the appearance of the harmonics. This does not correspond to a physical phenomenon! like vortex shedding. We do not show the temporal evolution of the turbulent kinetic energy and the

5 Computer Simulation 171 dissipation, but we have observed the same behavior. The velocity field shows that the wake has the same form as the permanent case, but its length is a little longer. It appears like a superposition of equilibrium states. 0.3 T S(f> O.OOOO30 - O O.OOOO1Ot/T Fig 3 : Temporal evolution of the transversal velocity for the high frequency case and the corresponding spectrum S(f) o.ooo10 - t/t o '. o i n o. f < Fig 4 : Temporal evolution of the longitudinal velocity for the high frequency case and the corresponding spectrum From Figures 3 and 4, two characteristic frequencies appear, one (50Hz) involved in the incident flow frequency, and an other (5Hz) probably involved in the vortex shedding frequency.

6 172 Computer Simulation >; I0.56m/s Fig 5 : The flow field at three different times (4.5T, 7.5T, 10.5T) for an angle of attack equal to 10 The evolution of the flow field (Fig 5) would confirm this assumption. As the Reynolds number is large enough to have a constant Strouhal number, we observe that the simulated frequency is of the order of the experimental vortex shedding frequency for flows past cylinders [11] where Str=0.2. The difference in the computed Strouhal number (Str=0.1) may be attributed to the non-regular shape of the body at the angles of attack considered

7 Computer Simulation 173 However, this phenomenon is lightly damped with time (Fig 3 and 4). To try to explain that, we must consider that the Upwind discretization scheme leads to a certain degree of false diffusion [6]. It can also be seen that the same damping mechanism exists for the temporal evolution of the turbulent kinetic energy and the dissipation, however this is not shown in this paper. CONCLUSIONS It is very important to understand all of the phenomenons which exist in a flow around a body with a regular geometry, before the study of flows around complex obstacles such as those which can be seen in reality. Therefore, we are interested by a two-dimensional flow around a cube. This work constitutes a first approach to the problem. We have demonstrated that there exists a vortex shedding frequency and, in the next stage, we will attempt to verify that the found frequency is really the vortex shedding with the following simulation : fixed angle of attack and transient simulation. An other exercice is to reduce the amount of false diffusion. For this, we will use an other discretization scheme (SUCCA) [6]. Finally, an experiment in a wind tunnel is under construction. Experimental results will permit us to valid the numerical results. REFERENCES 1. Mottram, R.C. The measurement of pulsating flow', Conference on the Basic Principles and Practice of Flow Measurement, National Engineering Laboratory, East Kilbride, Glasgow, Mottram, R.C. and Robati, B The effect of pulsation on vortex flow meters', Proceedings of the International Conference on the Metering of Petroleum, Oyez Scientific and Technical Services Ltd., London, March Patel, M.H. The influence of vortex shedding on the roll motions of a flatbottomed barge', European Mechanics Colloquium n JI9, Imperial College, July Graham, J.M.R. 'Vortex shedding from single edge in oscillating flow', European Mechanics Colloquium n 119, Imperial College, July 1979.

8 174 Computer Simulation 5. Bearman, P.W. and Graham, J.M.R. 'Vortex shedding from bluff bodies in oscillatory flow : A Mechanics, Vol. 99, Part 2, pp , report on Euromech 119', Journal of Fluid 6. Scanlon, T. 'Vortex shedding flowmeter - Pulsating flow CFD studies', PhD dissertation, Department of Mechanical Engineering, University of Strathclyde, Glasgow, Dec Rodi, W, Turbulence models and their applications in hydraulics, A.I.R.H., Delft, Patankar, S.V., Numerical heat transfert and fluid flow. Hemisphere, New York, Castro, IP and Robins, A.G. The flow around a surface-mounted cube in uniform and turbulent streams', Journal of Fluid Mechanics, Vol. 79, Part2, pp , Dargent, C. 'Etude numerique du contournement d'un obstacle naturel ou artificiel par le vent', Rapport de DEA, Institut de Mecanique des Fluides, Toulouse, Sept Kourta, A. Boisson, H.C. Chassaing, P. and Ha Minh, H. 'Nonlinear interaction and the transition to turbulence in the wake of a circular cylinder', Journal of Fluid Mechanics, Vol. 181, pp , 1987.