Jan Sotoˇsek Joˇzef Stefan Institute
Jamova cesta 39 1000, Ljubljana, Slovenia jan.sotosek@hotmail.com
Boˇstjan Konˇcar Joˇzef Stefan Institute
Jamova cesta 39 1000, Ljubljana, Slovenia
Bostjan.Koncar@ijs.si ABSTRACT
This study explores the accuracy of numerical simulations of vortex flow meters using an open-source software. Mesh of a vortex flowmeter with a specially designed bluff body was cre- ated with Salome-Meca and numerical simulations were performed with the open source code OpenFOAM. Simulations were carried out for different Reynolds numbers ranging from 5000 to 50000. Results were then compared to the experimental data at the same Reynolds numbers.
The main goal of the study is to accurately predict the vortex shedding frequency and to demon- strate its linear dependency on the mass flow rate. Flow simulations were performed using different simulation techniques such as Direct Numerical Simulation (DNS), Large Eddy Sim- ulation (LES) and Unsteady Reynolds Averaged Navier Stokes (URANS). However, at higher Reynolds numbers both DNS and LES simulations are computationally too demanding so only URANS was used at Re above 35000.
1 INTRODUCTION
Vortex flow meters are widely used in various industries, including nuclear, due to their excellent characteristics, such as high accuracy, linear output signal, wide measurement range, absence of moving parts and low cost of investment and maintenance [1]. The essential part of the vortex flow meter is an obstacle (bluff body) that is mounted transversely in the measuring pipe. Behind a bluff body, vortices are generated periodically, with a shedding frequency that is directly proportional to the volume flow rate. However, a clear linear dependence usually appears only at higher pipe Reynolds numbers above 20000.
In this study, the flow behind the bluff body is simulated numerically over a range of Re numbers, between 5000 and 50000. The simulations are compared and validated against the dedicated experiments performed in an air pipe with a specially designed prismatic bluff
Unsteady Reynolds Averaged Navier Stokes (URANS) [6] simulations with k-ωSST turbulence model is used.
The simulated vortex shedding frequencies are validated against the available experimen- tal data [7], where the frequency signals behind the bluff body were obtained using a hot-film probe. Several different analysis methods have been tested and compared, to obtain unique frequency characteristics from the simulations. Pressure, velocity, lift force and vorticity oscil- lations at three different points behind the bluff body have been recorded, fast Fourier transform analysis has been used to determine the frequency [8].
Likewise, a comprehensive verification of simulations has been performed to ensure suf- ficient accuracy and convergence of numerical results. Different numerical methods require different numerical schemes that have been tested, showing that DNS needs most accurate higher-order schemes.
2 VORTEX FLOW METER PRINCIPLE AND TEST SECTION
Vortex flow meters consist of a measuring pipe with a bluff body mounted transversely across it. When turbulent flow passes the bluff body, Karman vortex street is being formed [9]. Bluff bodies with sharp edges on either side ensure a consistent location, where the flow boundary layer is being separated and vortices are being generated periodically [10]. Periodic formation of vortices on those sharp edges, alternating between left and right side of the bluff body, is called vortex shedding. By measuring the frequency of vortex shedding the volumetric flow rate can be calculated. This frequency can be measured using many different methods.
One of the techniques for shedding frequency detection is a hot wire anemometer. A heated wire is placed in the stream behind the bluff body. Cooling of the wire depends on the direction and speed of the flow. This temperature fluctuations of the wire change its resistance and by monitoring that we can measure the vortex frequency. This method was used in the experiments performed by Konˇcaret al.[7] by placing a sensor at three points behind the bluff body as shown in Figure 1. Numerical simulations were performed for the same experimental setup and bluff body dimensions and the results were compared with experimental data.
Using the measured frequency the fluid velocity can be calculated as u= f·D
St , (1)
whereuis the fluid velocity,f is the vortex shedding frequency,Dis the pipe diameter and St is the Strouhal number, which is a dimensionless number describing oscillating flow mechanisms.
Vortex flow meters Strouhal number should be constant for a wide range of Reynolds numbers from around 2·104 to 2·106. This means that a vortex flow meter can be used to measure flows of different fluids over a large span of flow velocities. Range of vortex flow meter and the
Figure 1: Pipe and bluff body dimensions.
quality of vortices depends on the shape of the bluff body. Good bluff body decreases errors as the created vortices are stronger and hence easier to measure, it can also increase the range of Reynolds numbers where the vortex flow meter operates with acceptable accuracy.
3 COMPUTATIONAL MODEL AND MESH
Numerical simulations were performed on a vortex flow meter test section with air flow.
The air flow is assumed as an incompressible horizontal flow without gravity effects. The mathematical model of the flow is described by Navier-Stokes equations that solve the fluid motion [11]. The incompressible Navier-Stokes equations consist of continuity equation
∇ ·u= 0, (2)
that describes the mass conservation over the fluid domain and of the momentum conservation equation
ρDu
Dt =−∇p+µ∇2u, (3)
whereρis the fluid density,uis fluid velocity,tis time,pis the fluid pressure andµis the fluid dynamic viscosity.
Simulations were performed on a three dimensional pipe domain with a bluff body. At the inlet cyclic boundary conditions were used. Constant pressure was defined at the outlet and no-slip condition was used on the walls of the pipe and on the bluff body.
Most accurate numerical simulations can be obtained using the conformal structural mesh with hexahedral elements. Improved mesh quality typically requires increased number of smaller elements, with adjacent elements having similar sizes, aspect ratio of elements should be close to 1, mesh should have low non-orthogonality and low skewness. A poor mesh quality can lead to inaccurate results and slower simulation convergence. In this study three different simulation methods were used, DNS,LES and URANS. The size and number of mesh elements may vary depending on the method used.
DNS method requires the finest mesh to resolve even the smallest scales of turbulence dissipation. The smallest length scale is defined as Kolmogorov length scaleη
η= ν3 ε
!1/4
, (4)
Unlike the previous two methods, URANS models all turbulence length scales. It uses special wall functions to model the boundary layer near the walls. Therefore it can have a much coarser mesh.
The comparison between dimensions of the mesh elements at different methods is given by dimensionless element sizes∆x+in stream-wise direction,∆z+in span-wise direction and
∆y+ in cross-wise deirection for boundary region near the walls. The values ∆x+,∆y+,∆z+ were first set as recommended by Tiseljet al. [12]. Then the meshes have been coarsened as much as possible while still achieving the same simulation results. Mesh density (in terms of dimensionless element size) and number of mesh elementsnfor different methods are listed in Table 1 and Table 2. At lower Re numbers the same meshes were used for LES and URANS methods. The parameter∆y+min describes the length of the first near-wall cell and∆ymax+ de- scribes the cells in the free stream. In the regions near the inlet and outlet, mesh cells can be elongated in the stream-wise direction in order to reduce the total number of cells in the mesh.
In this study cells close to the inlet and outlet have ∆x+ = 43 for DNS and ∆x+ = 150 for LES and URANS meshes. A two dimensional mesh slice can be seen in Figure 2.
Table 1: ∆x+,∆y+,∆z+values.
∆x∆x∆x+++ ∆y∆y∆yminmin+++min ∆y∆y∆ymaxmax+++max ∆z∆z∆z+++
DNS 8 0.5 5 4
LES 15 0.6 12 12
URANS 28 18 18 28
Table 2: Number of mesh elementsn.
n5000
nn50005000 nnn840084008400 nnn134001340013400 nnn240002400024000 nnn365000365000365000 nnn490004900049000 DNS 1.4·106 2.7·106 7.7·106 / / / LES 2.0·105 4.1·105 2.0·106 2.4·106 / / URANS 2.0·105 4.1·105 2.0·106 2.4·106 2.1·106 5.0·106
4 NUMERICAL METHODS
Numerical simulations were performed with the open source code Open-source Field Op- eration And Manipulation (OpenFOAM). DNS simulations require a third order gradient nu- merical schemes, while in the case of LES and URANS second order numerical schemes have been used. IcoFoam solver was used for DNS simulations as it is a transient solver without
Figure 2: 2D slice of a mesh for DNS atRe= 13400 at the top. Close-up of the mesh around the bluff body below.
hence it was used for LES and URANS simulations. To increase the accuracy, Courant number was set below 1 for all simulations. At the inlet cyclic boundary conditions were used to ensure a fully developed velocity profile. Velocity perturbations were introduced at the inlet, however they didn’t have any significant effect on the results.
4.1 DNS
DNS method numerically solves the Navier-Stokes equations for every mesh cell. It ac- counts for all length scales and as such does not need an additional turbulence model. In this study we were able to use DNS method to simulate flows at Re numbers5000,8400,13400.
4.2 LES
Like DNS, Large Eddy simulation resolves all larger length and time scales but uses a turbulence model for the smallest dissipating scales. In this study a Wall-Adapting Local Eddy- viscosity (WALE) model was used. This model describes subgrid-scale turbulence effects with a sub-grid eddy viscosity. Because the smallest length scales are modelled, LES meshes can be somewhat coarser than DNS meshes. This means that simulations run somewhat faster but they are also slightly less accurate than DNS simulations. Good wall resolved LES simulations can provide almost as accurate results as DNS, but they are not much faster. In this study we have used wall resolved LES method to simulate flows at Re numbers5000,8400,13400,24000.
4.3 URANS
URANS uses complex modeling for all turbulent scales and solves time-averaged mass and momentum equations. For turbulence modeling the k-ω SST model has been used [13].
This model introduces turbulent kinetic energy k, turbulent dissipation rate ε and specific tur- bulent dissipation rate ω. It resolves additional transport equations for these three variables to model all, also non-isotropic turbulence effects. As all turbulent scales are being modelled, a much coarser mesh can be used. This means much less computational power is required to run the URANS simulations. However the simulations are also less accurate therefore URANS is used mainly in engineering applications where very high accuracy is not required.
were just as accurate as those obtained from pressure data at points 1 and 3. Pressure oscilla- tions at point 1 was then selected for further analyses since the data were recorded at the same location as in the experiment. A fast Fourier Transform (FFT) analysis was used to extract the frequencies from the simulated transient data. This is an algorithm that converts the simulated time signal (in this case pressure oscillations) to a representation in the frequency spectrum. Re- sults contain all frequencies hidden in the data. For the considered case, the frequency with the highest peak (amplitude) in the frequency spectrum represents the vortex shedding frequency.
Figure 3 shows pressure oscillations over time (a) and the extracted frequency spectrum (b) at the point 1 behind the bluff body.
(a) (b)
Figure 3: (a) Pressure oscillations from DNS at Re = 5000. (b) FFT on pressure data from DNS atRe= 5000.
Residuals from numerical calculations were checked to examine the convergence of sim- ulations and conservation of mass and momentum. Mass and momentum error remain below 0.0085%for all simulated cases. Simulated dominant frequencies at different Reynolds num- bers were then compared with the measured frequencies [7]. Figure 4 shows a comparison of frequencies and Strouhal numbers at different Reynolds numbers.
A good agreement between the simulated and the measured frequencies can be observed.
In this case all three methods DNS, LES and URANS provide similar results.
In the range of measured Re numbers (8400 to 49000), the deviation in St numbers be- tween the experimental and simulation results is practically within the experimental uncertainty (±5%) [7]. Strouhal number is around St = 0.24 over the entire range of measured data.
Strouhal number is expected to be almost constant at Re higher thanRe ≈10000, but the pre- sented simulation results and the experimental data show that this characteristics may exist even
(a) (b)
Figure 4: (a) Graph of frequencies at different Reynolds numbers. (b) Graph of Strouhal num- bers at different Reynolds numbers.
(12.15) then the LES method (9.20) and consequently a higher Strouhal number. This is un- expected since all DNS simulations use the same setup and mesh quality. DNS simulation at Re= 5000was run several times using different modifications like finer mesh, smaller Courant number and others. All simulations provided similar results.
Different turbulence methods in principle provide similar results however the simulated flows have different levels of detail. Figure 5 shows the difference between the flow pressure fields atRe= 13400simulated by different methods.
Figure 5: The same time instants of the pressure field at Re=13400, simulated by DNS, LES and URANS.
Most details in the flow are resolved by the DNS method, where also the randomness of turbulence can be observed. Flow simulated by the LES method shows somewhat less de- tail compared to DNS. In the image of the flow simulated with the URANS method, only the smeared regions of the low pressure where the vortices appear can be seen, but without any details.
The flow evolution at two different Re numbers (8400 and 24000) as simulated by the LES method is shown in Figure 6. It can be seen that the turbulence structures are smaller and more pronuonced at the higher Re number.
Figure 6: Pressure fields at Re=8400 and Re=24000 simulated by the LES method.
In Figure 7, the simulated vorticity region behind the bluff body can be observed. This image shows very clearly that the vortices are formed periodically from the left and right side of the bluff body. It can be seen that vortices have different spinning directions depending on the side of the bluff body where they are generated. Blue and red colour define clockwise and counter-clockwise rotation of vortices, respectively.
Figure 7: Vorticity around the bluff body in vortex flow meter simulated by DNS atRe= 13400.
6 CONCLUSION
Numerical simulations seem to be a very good alternative to the experiments in determin- ing the vortex shedding frequency for vortex flow meters. All simulation methods produced very good results. It seems that URANS is a sufficiently accurate and fast method for deter- mining the vortex shedding frequency. However the URANS predicts less turbulence details in the flow compared to the LES and DNS methods. LES provides accurate shedding frequency results and a detailed flow simulations, so it is much better method for lower Reynolds numbers.
DNS simulations are computationally very demanding and are only viable at lower Reynolds numbers below15000. As LES provides comparable flow simulation accuracy to DNS, it should be a preferred choice for most of the cases. More data is needed in the range of lower Reynolds numbers where all three methods can be run to make a more comprehensive comparison be- tween the three methods. A detailed research is needed for DNS aroundRe= 5000in order to
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support provided by the Slovenian Re- search Agency, through the grants P2-0405 and J2-9209.
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