COMPUTATION OF DRAG AND LIFT COEFFICIENTS FOR SIMPLE TWO-DIMENSIONAL OBJECTS WITH REYNOLDS NUMBER

The article deals with comparison of drag and lift coefficients for simple two-dimensional objects, which are often discussed in fluid mechanics fundamentals books. The commercial CFD software ANSYS/FLUENT 13 was used for computation of flow fields around the objects and determination of the drag and lift coefficients. The flow fields of the two-dimensional objects were computed for velocity up to 160 km per hour and Reynolds number Re = 420 000. Main purpose was to verify the suggested computational domain and model settings for further more complex objects geometries. The more complex profiles are used to stabilize asymmetrical ('z'-shaped) pantographs of high-speed trains. The trains are used in two-way traffic where the pantographs have to operate with the same characteristics in both directions. Results of the CFD computations show oscillation of the drag and lift coefficients over time. The results are compared with theoretical and experimental data and discussed. Some examples are presented in the paper.


INTRODUCTION
One of the elementary tasks in fluid mechanics in flow around objects is the determination of the drag and lift coefficients.The inception of this paper was inspired by work that tested the behavior of flow around prismatic objects designed for installation at the asymmetrical pantograph for high-speed trains where one of the tasks was to provide at least a basic specification of those coefficients.Therefore, the main goal was not to specify the accuracy of the CFD calculations, solver, or turbulence models using simple two-dimensional objects, but to test the credibility of simple "engineering" CFD models for the calculation of general, prismatic bluff-bodies or the settings of some of their parameters.The simulation of flow around bodies used the commercial CFD software ANSYS/FLUENT 13.The calculated values of the drag and lift coefficients for individual simple objects were compared to the table values which are presented in most basic books on fluid dynamics.The presented results of basic two-dimensional objects also contain charts with the development of drag and lift coefficients over time.Finally, the paper contains the values obtained for the profiles designed for use in the pantograph.

BOUNDARY CONDITIONS AND COMPUTATIONAL DOMAIN
In general, the calculation of flow around bluff bodies and determination of their drag coefficients is a problem that usually cannot be solved easily.A typical example is the flow around a cylinder object -a topic of many papers and the research continues.The problem of the cylinder primarily rises from the fact that the identification of the separation point using a simple model is more than less unlikely; at the same time, prediction of the separation point will have a major effect in the values of the drag and lift coefficient values.Furthermore, the task becomes the deeper with the threedimension turbulent effects.The situation is somewhat easier in objects where the flow separation point is defined by the domain of those bodies.Some bodies are on the border of the two cases.Yet still, in such cases the simulation of an object's behavior during flow is not simple as it always includes a non-stationary case with the generation of a time-dependent flow field with a vortex wake.Interesting phenomenological model for the vortex-shedding process behind bluff cylindrical bodies is presented in the article of Boyle et al. [1].The experimental measurement and CFD computations of drag for bluff bodies are presented in numerous publications, e.g., Igarashi [2], Liaw [3], Munson et al. [4], Gera et al. [5] or Lee [6].

Figure 1: Computational domain and grid
The computational domain and grid are made as a 2D task because the goal was to avoid a 3D simulation which, although more suitable, demands much more computation power and is therefore impractical for the conditions in question.The size of the computational domain was selected at 8000 x 3000 mm with basic 2D objects and profiles designed for the pantograph being continuously inserted.Profile height was unified at 120 mm, except for the profile design no. 2. The numerical simulation used a standard grid; the number of cells in the computational domain (see Figure 1) was approximately 200,000; the number slightly changed depending on the profile used.The solution itself used the ANSYS/FLUENT 13 software [7] with a preset segregated second-order implicit solver that is expanded with calculation of the energy equation.The EPJ Web of Conferences inlet boundary condition was selected as "velocity inlet" for input of the airflow velocity.The velocity of 44.44 m/s (corresponds to 160 kph) was set as unique for all the profiles.The velocity corresponds to Reynolds number Re = 420000; i.e. is turbulent flow.The outlet uses the boundary condition of "pressure-outlet".The profile walls are defined using the "wall" boundary condition; the side walls of the computational domain are defines as the "symmetry" boundary condition.Only two-equation turbulence models were selected: RNG k-H and "realizable"k-H (RKE); the original intent included a test of the RSM model, yet time constraints disallowed the test.
It is known and confirmed by the results that the time step length is a most important parameter in the tasks.Initially, the time step was estimated using assumed Strouhal number at t = 0.0005 s; however, the value was later modified to 0.00001 s for some cases, because the original value would not return satisfactory results at all times.

SIMPLE (BASIC) TWO-DIMENSIONAL OBJECTS
The models named above tested the computations of the flow field as well as the drag and lift coefficients of seven simple prismatic bodies -see Table 1.The drag coefficient values of those bodies are generally available in the literature.Textbooks [7] [8] [9] were selected as the resource for this study; the values provided by those resources were shown in the Column 2 of Table 1.Columns 3 to 5 contain the values obtained through computation.

RKE t=0.0001s
The table suggests that while in some bodies, the publication values were equal (3), they did differ in other bodies by ca. ± 5% (2, 4, 6, 7); the difference amounted to 20% in Body 1.The deviations were likely based on the variety of resources used by the respective publications and from the various conditions under which the values were measured.Looking at the simulation results, the models used returned results of various degree of applicability in various bodies as expected.The spread of simulation results was rather large and most probably depended on the respective model being able to simulate wake behavior behind the body.A good degree of compliance with the literature was seen in the models used in Body 7 and 4, even 6.In all other cases, the numerical model would return the drag coefficient value by 20% higher than the value mentioned in the publications (Bodies 1, 3, and 5still acceptable for the intended use); the biggest difference was 50% (Body 2).The table presents the calculated lift coefficient values.Given the symmetrical nature of the bodies, the value should be nearly zero.This is not the case in several situations as the non-stationary simulation ran too long.However, no results with a higher c l value are credible.This discrepancy was eliminated through major reduction of the time step, as demonstrated in the values of Bodies 1 and 3. Examples of immediate velocities within the flow field of Body 7 are provided in Figure 2 and the development of drag and lift coefficients with representation of the mean values are provided in Figure 3.

DESIGNED PROFILE TWO-DIMENSIONAL OBJECTS FOR PANTOGRAPHS
As the results obtained in the testing of the models were not quite compelling in all cases and several questions remained unanswered, it was decided to verify the profiles for use at the asymmetrical pantograph (referred to as "Z-shaped") for high-speed trains.These trains are often used for operation in both directions; therefore, the pantographs need to have operational modes suitable for both directions.The aim of the designed pantograph profiles is to stabilize collection of electric current from the overhead lines in speeds over 160 kph that corresponds to Reynolds number Re=420000.Two profiles were designed with two directions of flow on the component; as such, there are four examples of flow around bodies, see Table 2.

RNG-ke t=0.0005s RKE t=0.0001s
corresponded with the model bodies.The results obtained are shown in Column 3 and 4 in Table 2. Rather than the absolute coefficient values, their relative behavior patterns are interesting.The design profiles differ greatly in all parameters.What is more crucial in terms of application of the shapes is that in both profiles, the values for the two directions differ greatly; therefore, the required properties cannot be achieved.All values and results of this initial study have been submitted to the industrial partner for further review.

CONCLUSIONS
A study task of simulating simple 2D bodies using two-equation models was performed upon requests for profile flow simulation.The values obtained from the simulations were compared to the results presented in the literature.The results have indicated that the accuracy of the methods, especially in several cases, is limited; however, the results are acceptable for the intended use if specific conditions are followed.The application of calculations to the profiles has indicated that the choice was not ideal, that their application in a two-way pantograph might result in non-compliance with the adherence pressure request in one of the directions.This initial study is assumed to continue.The prospective solutions include replacement of the profiles with fixed symmetrical profiles or, better, with a symmetric profile that tilts depending on the direction of travel.

Figure 3 :
Figure 3: Unsteady drag and lift coefficients for Body 7