TRACKING OF VORTICES IN COMPUTATIONAL FLUID DYNAMICS

Transcription

1 TRACKING OF VORTICES IN COMPUTATIONAL FLUID DYNAMICS T. Nyers Student, Faculty of Mechanical Engineering, Budapest University of Technology and Economics P. Tóth Ph.D Student, Department of Fluid Mechanics, Budapest University of Technology and Economics H-1111 Budapest, Bertalan L. u Tel: (+36-1) , M. M. Lohász Assistant Professor, Department of Fluid Mechanics, Budapest University of Technology and Economics H-1111 Budapest, Bertalan L. u Tel: (+36-1) , Abstract: According to the coherent structure concept turbulent flow is not merely a random process but the movement of larger scales can be described as the interaction of vortices (coherent structures). Hence the more precise understanding of the evolution, merging, tearing, creation and disappearing of vortices could provide new information on controlling turbulent flows. The first important step in the numerical investigation of vortices is the detection of vortices, for which a larger number of methods exist. The second step is the tracking and observing of these structures. Our paper presents an algorithm which can select the vortices separately and so their interaction can be investigated quantitatively. The methodology is applied for an axisymmetric (2D) jet, where the creation merging and disappearing of vortices can easily be tracked. Keywords: Vortex, Tracking, Axisymmetric Shear Layer, Merging 1. INTRODUCTION Turbulence is still one of the most difficult questions in classical mechanics, although its research dates back to the nineteenth century [13]. During this long period both investigation tools and treatment have developed spectacularly contributing to its understanding. One important improvement in the tools is the availability of computers with increasing computing power. Present computers allow for solving the continuity and the Navier-Stokes equations in all details i.e. simulating turbulence. The problem associated with this simulation is that the spatial and temporal scales of turbulent motion depends on the typical Reynolds number of the flow, i.e. only flows with a moderate Reynolds number can be completely resolved by computers at present. To circumvent this limitation in the last decades of the last century it was proposed to focus only on the simulation of the large scales and model the small ones, this technique is called the Large-Eddy Simulation (LES) [7]. Besides this development in tools a new concept was also developed to treat turbulent motions. [9] proposed to consider turbulence not only statistically but to search for structures in the flow, since it is believed that turbulent fluctuations can be described as the movement of coherent structures. Coherent structures (CS) [10] are fluid regions which keep some of their properties for a relatively large spatial and/or temporal extent. Typical CS s are the big vortices in turbulent flows. The investigation of turbulence from this viewpoint needs the definition of the vortices i.e. a method for detecting vortices. From the huge amount of vortex detection criteria [2] the 1 / 12

2 Q criteria is a widely used one for investigating the result of LES [5]. Q is the second scalar invariant of the velocity derivative tensor and can be expressed for incompressible flows as: Q = (Ω ij Ω ji -S ij S ji )/2 (1) where S ij and Ω ij are the symmetric and the antisymmetric part of the velocity derivative tensor respectively. This expression highlights that positive regions of the Q field are associated with the fact that rotation is dominant over the shear. It is shown in [2] that such regions can be used to define vortices. Using this criterion, vortices in any LES can be detected and visualised by rendered isosurfaces of a given Q threshold. Furthermore their movement can be investigated by composing movies of these images (see for example [5, 12]). Despite the usefulness of this animation the vortices and their interaction are difficult to analyse since: 1) vortices are hiding each other, 2) it is impossible to track several hundreds of vortices simultaneously. This problem can be solved if the vortices are identified separately and only particular vortices are visualised at the same time. If the vortices are one by one identified they can be tracked [15], this is especially useful when investigating the interaction of adjacent vortices. Both the movement and the interaction of the vortices are very important when trying to understand and influence turbulent flows. Two such important interactions are the tearing of one vortex into two or more and the pairing or merging of vortices. The aim of the present paper is to present a method for this latter purpose and its implementation into a general purpose unstructured finite volume Computational Fluid Dynamics solver. At the end of the paper the application of the method for a 2D annular shear layer simulation will be presented. 2. FLOW AND SETUP OF THE TEST ENVIROMENT In Sections 3 and 4 algorithms to partition and track the vortices and to detect their interaction will be presented. In order to verify the algorithms and to present examples during the description of the methodology computation details of a simple flow will be introduced in this section. 2.1 DESCRIPTION OF THE JET SIMULATION An axisymmetric free jet simulation was carried out by the commercial Computational Fluid Dynamics code Fluent 6.3. The flow parameters correspond to the hot wire measurement of [4]. The flow is computed by using the incompressible unsteady solver of the code. The simulation was carried out at the discharge Reynolds number of Re = U 0 D/ν = , where U 0 is the maximum velocity at the inlet and D is the inlet diameter. 2.2 THE FLOW FIELD In this section the most important characteristics of an annular shear layer are summarised following the literature [8, 14]. In round jet flows an annular shear layer is formed between the fluid discharging from a nozzle and the ambient fluid moving at lower velocity. Due to its cross directional instability the shear layer rolls up immediately downstream of the nozzle discharge [8]. Kelvin-Helmholtz-type ring-shaped vortices are formed (hereafter K-H vortices). These ring shaped vortices can also be visualised by isosurfaces of Q (Figure 2). The K-H vortices are convected downstream and they may undergo vortex pairing procedure (two vortex rings merge). In an axisymmetric simulation the flow variables and gradients do not change in the tangential (azimuthal) direction, therefore the annular shear 2 / 12

3 layer instabilities [8] can not be modelled. Thus vortex ring break-up can not be initiated by this procedure opposed to a three dimensional simulation case. As a consequence only a few vortex tearing procedures can be observed downstream in the simulation domain. However permanent vortex shedding due to the tearing effect can be observed with the Q criteria at the nozzle edge. The sizes of the vortices increase by the pairing (merging) procedure and they are convected out at the domain outflow or some of them disappear in the simulation domain. 2.3 COMPUTATIONAL DOMAIN AND NUMERICAL PARAMETERS The commercial code Fluent 6.3 [11] uses a cell-centred finite-volume method with collocated variable arrangement. For the axisymmetric jet simulation, the pressure based solver was applied solving the governing equations sequentially and coupling them using the PISO method in a non-iterative manner [11]. The time is advanced using Gear's two step second-order implicit method. The first order upwind scheme was used for spatial discretisation of the convective terms in order to have a robust and smooth simulation for testing. The second order scheme was used for the interpolation of the pressure. The spatial derivatives were evaluated cell based. Fig.1. Computational mesh and boundary conditions The computational domain and the grid can be seen in Figure 1. The domain extends up to 19D in the streamwise (denoted by x) direction and r=5d in the radial (r) direction. The cells of the grid are concentrated at r=0.5d up to x=6d, downstream from here the grid is stretched in order to provide an almost uniform radial cell distribution downstream in the middle of the simulation domain. The mesh is also non-uniformly distributed in the x direction with increasing cells size in the streamwise direction. The nozzle was modelled by an infinitely thin wall in the region -1D<x<0, r=0.5d. The internal flow of the nozzle was not included, the jet discharged at x= Boundary Conditions A mean velocity profile was prescribed for the region x=0, r<0.5d. A constant (tophat) mean velocity profile was imposed without any perturbation. The far-field condition was prescribed as constant, using the pressure outlet condition of [6]. This prescribes a static pressure if the flow leaves the domain and a total pressure if it enters the domain. Reverse flow is set to be perpendicular to the boundary. At the wall no-slip condition was imposed. Axis boundary condition was prescribed at r=0. 3 / 12

4 3. VORTEX DETECTION Vortex identification algorithm is necessitated by the idea of tracking the life of a single vortex in the flow field. This requires that not only the rotation dominated flow features (i.e. vortices) have to be selected but each of these types of regions needs a unique identifier to be assigned to. In addition, once the vortices are labelled they have to be tracked through the time domain and their interactions have to be recorded too. The aim of the algorithm presented below is to label and track the vortices and register the merging and tearing events in the flow field. According to the widely used Q criteria a positive scalar can be assigned to the rotation dominated vortices in the flow field. The flow regions are treated as a vortex if the Q value exceeds a predefined threshold value. This regions needs to be divided into disjunctive regions corresponding to individual vortices. Fig.2. Q contours for the Q> D 2 /U 0 2 regions. The task can be resolved easily through graph-search methods if graph theory is applied for the unstructured mesh topology. The computational grid can be interpreted as a looped graph, where the cell centres are the nodes of the graph and the neighbouring connection between the cells represented by the graph edges. The algorithm is implemented in Fluent s User Defined Function environment [6] for arbitrarily unstructured meshes. The core of the searching algorithm is a classic recursive method for traversing graphs called depth-first search (DFS). It involves the following of the graph edges from node to node, with the goal of systematically visiting every node and every edge in the graph. The algorithm starts at a root node; marks it as visited, and then visits all the nodes that are adjacent to it. The same recursive procedure is applied on any of the adjacent nodes which have not yet been marked [16, 17]. It deepens into the graph until it hits the boundary and then needs to backtrack. The implementation requires an interface (adjacency-matrix, adjacency-lists representation) through which it can access the graph s topology. Since the topology of the mesh is generated automatically, Fluent s built-in functions explicitly provide adjacent relationship between nodes. This method resembles the way of graphics editor applications detect the same-coloured areas of bitmap pictures. Thus, the well known algorithm of image processing tasks, the flood fill [1] is employed. Flood fill is a specialized DFS designed for splitting graphs into connected components. Flood fill performs the 4 / 12

5 searches for unmarked vortex cells [17] and deploys DFS from that root node until all of the nodes of the graph have been marked as visited. In this case, depth-first search traverses nodes which satisfy the vortex criteria, and by recording the nodes visited by DFS, the domain can be separated into different regions. The flood fill method seemed to be very useful throughout the project; it can be used in many effective ways to handle difficulties related to connected structures NUMBERING AND TRACKING OF THE VORTICES At the beginning of vortex tracing, without any data from the previous stage, vortices can be named completely arbitrarily with the help of flood fill. The algorithm examines the whole domain, marks every single vortex with a unique vortex identifier. An identifier is assigned to each vortex by simply keeping track of the number of the DFS calls, thus every structure gets different indices based on the call number. The outcome is a field, which contains the labelled vortices (see Figure 3). Fig.3. Contours of labelled vortices This labelling is applied only once during the vortex tracking procedure. Every vortex of further time steps has to be indexed considering the previous state of the flow [15], i.e. the vortices in the previous time step and in the present time step has to be assigned to each other in such a way that the probability of they correspondence is as high as possible. Supposing small time intervals between successive results, if a cell fulfils the Q criterion in both time steps, it can be assumed that it belongs to the same structure. This condition can be satisfied by using a CFL number strictly below one (if we assume that vortices can move maximum with the speed of the underlying fluid). By starting searches from the boundary of the structures, existing at the actual time step, the vortex development can be identified. The final step is to add these cells to the structures from where the search has started. This vortex identification between successive time steps can be divided into seven steps. These steps are the same for the 4 special event cases (described in the next section) except that in the last step different events are detected. An example of the seven steps with different events is provided in Figure 4-6, where the subfigures are denoted in correspondence with the step numbers. These steps of the vortex tracking are the following: 1) Identified vortices at time step n has their index (colour corresponds to index number) 5 / 12

6 2) Vortex structures are detected at time step n+1 and marked 3) Intersection of time step n and n+1 are evaluated 4) Intersection cells inherit indices from time step n 5) DFS deploying started from the previously indexed cells to attach the unindexed cells 6) The result of Step 5: The cells at time step n+1 are indexed 7) Event checking, final result At step 6 a list containing the numbers and labels of the actual vortices is evaluated. During the step 7 this list is compared with the one containing the previous time step information and is utilized to detect events in the flow. 3.2 VORTEX EVENT DETECTION Since the event detection, (the last step of the aforementioned seven ones) covers different events to be identified; this section is devoted to explain the methodology of the assessment. Moving The simplest event is that the vortex just changes its position i.e. it moves. An example for this phenomenon is given in Figure 4. If in the last step no special other event was detected the vortex only moved. Tearing If after step 6 two separate vortices exist with the same index a tearing occurred during the time step. An example is provided in Figure 5. In this case one of the structures has to renamed, this renumbering in the present version of the code is ad-hoc. Merging If after step 6 vortices with different indices are adjacent to each-other it means that a merging occurred (if two vortices are merged this event used to be called pairing as well), an example is provided in Figure 6. After the detection of occurrence of the event the vortex has to be renamed to one of their parent vortices. In the present version of the code it is ad-hoc which part vortex will give its name to the resulting vortex. Disappearing and appearing If any of the vortices present in time-step n is missing from the list of vortices in timestep n+1 this vortex has disappeared. If vortex regions (detected by Q criteria) exist in time-step n+1 where it is impossible to find intersection with any vortex in time-step n a vortex has been created. 6 / 12

7 Fig.4. Phases of vortex tracking without any special event Fig.5. Phases of vortex tracking with tearing event 7 / 12

8 Fig.6. Phases of vortex tracking with pairing event 3.3. POSSIBLE PROBLEMS Since our method was only tested for a simple 2D flow with a single numerical settings which will be presented in the following section, it is anticipated that some problems will arise when applying to more complex vortex motion phenomena or even for 3D flows [12]. Greater time steps significantly impair the efficiency of the algorithm. Subtle details of smaller vortices cannot be tracked with large cells, so cell size is a key factor as well. The implemented DFS is based on a recursive method [17] that can lead to branching problems and stack overflow in case of greater amount of cells. A more in-depth analysis is needed to scrutinise the influence of these factors over the overall performance of the code. 4. APPLICATION EXAMPLE The previously described method has been applied for the shear layer simulation presented in Section 2. Some information on the vortex flow field is also evaluated by extending the capabilities of the algorithm. In this application the first step is to set the threshold of Q for defining vortices. Since no easily applicable theory have been proposed till now in order to determine this value, it was set by visual comparison of the results at different thresholds. Finally the value of Q= D 2 /U 0 2 was used, detecting the vortices as presented in Figure 2. The second step is to run the simulation with the methodology presented in Section 3 applied for each time-step. In every time-step the vortices are identified, they are numbered following the rules described in Section 3. If an event occurs it will be stored in a file for post processing purposes, by describing the type of the event and the vortices participated in it. This demonstration example has to be considered with caution, since the results have not been validated and the numerical methodology is not appropriate for this high Reynolds number flow (i.e. the flow would not be axisymmetric but completely 3D and turbulent, higher order scheme should be used for the momentum advection terms). 8 / 12

9 More detailed information can be obtained concerning the vortex field if some predefined properties of the vortices are also evaluated during the event tracking. We propose here to define some quantities. For the simple treatment of these vortex property definitions the particular vortex selection function I(i), was introduced. This is 1 for the vortex #i and 0 for the others. Therefore the streamwise position actually the mass centre of every vortex can be defined as: The area of the vortices: xi( i) QdA x( i) = (2) I( i) QdA A ( i) = I( i) da A length scale for a vortex can be defined for 2D vortices as follows: A( i) r ( i) = π (4) This would be the radius of a vortex for an exactly circular structure. In a following step this file can be analysed. The file directly enables to make statistics about the events. Furthermore detailed information on the flow field can be gained by rerunning the simulation using this file. In this second run the interaction of the vortices are a- priori known and statistics and plots can be generated in a local coordinate system of the events. For example a local time coordinate can be defined by setting the origin at the time of a pairing and the evolution of the vortices before their merging can be analysed in detail. (3) Fig.7. Visualisation of the merging of four selected vortices, the contours show the pressure, the solid lines on the contour plots show the vortex. The main diagram shows the evolution of these vortices in time and their interactions. The abscissa is the streamwise position and thickness of the lines represents the diameter (Eq. 4.) of the vortices. The instantaneous snapshots (t*) correspond to time instances at the ticks of the ordinate. In Figure 7 on the ten pressure contour plots four selected vortices are visualised using the above mentioned methodology: by analysing the a-priori generated events database, three merging events were selected in such a way that the last merging is the successive merging of 9 / 12

10 the previous two. In the subfigure t1 the two downstream vortices merge and in the subfigure t4 the two upstream. In subfigure t7 the merging of the two resulted larger vortices is visualised. The contour plots of Figure 7 also highlight the widely observed phenomena that vortices coincide with local pressure minima [5, 10]. The merging processes are also plotted in the diagram between the contour plots, where the streamwise position (x) of these selected vortices is plotted at every time-step of the simulation in terms of non-dimensional simulation time. Besides the position the length scale of the vortices (2r) is also highlighted by the width of the corresponding line. To be able to distinguish between different vortices these lines are coloured corresponding to the index of the corresponding vortices. This kind of representation allows the more detailed analysis of the pairing processes. By investigating the slope of the vortex position lines in the diagram it can be deducted that the vortices are moving approximately with the local advection velocity which is the half of the maximum velocity U 0, since the ambient fluid is not moving. The sizes of the vortices grow when they move downstream. This increase in size enables that the border of the adjacent vortices near to each other and finally they undergo merging. After merging the size of the resulting vortex is higher than those of their parents. In the future we shall analyse the merging process in a more detailed way and compared the results to the theory given in [18, 3]. The authors found that the merging process can be divided in two distinct phenomena. The first period is when the vortices are growing without interacting and by diffusing their vorticity. In the second phase after reaching a critical distance to vortex size ratio they start to interact and merge. The previously proposed tools enable such investigation for a complete spatially developing shear layer configuration. In Figure 8 similar diagram is presented as the one in Figure 7 with the difference that in this chart all vortices are plotted and the timeframe and spatial domain is larger. From this figure it emerges that the velocity of the vortices next to the nozzle is lower than the approximately constant velocity far downstream in the domain. Fig.8. The same as the diagram in Fig. 6, but for the complete computational domain, the colours are used to distinguish between the individual vortices 10 / 12

11 In this diagram many small vortices exist only for a very short time period, they are represented by small differently coloured regions next to large vortices. Obviously these spots should not be considered as a vortex and also their creation by tearing or merging as real vortex event. This noise in the registered events is a drawback of the algorithm. In Figure 9 a zoom of the previously demonstrated Figure 8 is presented in the proximity of the nozzle. The generation of the vortices can be seen here: the vortices are created by tearing from the vortex permanently present at the lip of the nozzle. The phenomenon is similar to the formation of drops at a water tap, when the tap is not closed properly. Fig.9. The zoom of Fig.8. highlights mainly the vortex tearing phenomena close to the nozzle 5. CONCLUSION In the present paper the first version of a code developed for vortex tracking is presented. The first problem of separating the vorticity dominated regions defined by a range of the Q scalar into individual vortices is solved by the flood fill algorithm, which is a common technology for image processing to find regions of same colour. The vortices being distinguished by their indices are tracked by comparing the vortex field to the one present at the previous time-step. During the tracking the adjacent vortices interact with each other which are called an event. Simple methods have been proposed to detect events which are especially important for understanding turbulent flow. The presented method is applied for the case of an annular shear layer. This case is suitable for benchmarking since the vortex development is well know from the literature and the vortex motion and the pairing events can also be tracked without any special tool (by watching the sequence of instantaneous flow fields). It was found that the events detected and the expectations agreed in most of the cases. The problems of the method are related to the occurrence of very small structures which are believed to be noises compared to the big vortices. Further research is needed to exclude this type of structures and to enable the use of the method for the investigation of complex 3D flows. 11 / 12

12 ACKLOWLEDGEMENT The authors would like to thank to the reviewer (L. Kullmann) for his suggestions to improve the manuscript. REFERENCES [1] André P. Intelligent Flood Fill or: The Use of Edge Detection in Image Object Extraction 2005 [2] Chakraborty P, Balachandar S, Adrian RJ. On the relationships between local vortex identification schemes. Journal of Fluid Mechanics 2005;535: [3] Cerretelli C, Williamson CHK. The physical mechanism for vortex merging. Journal of Fluid Mechanics 2003;475: [4] Crow SC, Champagne FH. Orderly structure in jet turbulence. Journal of Fluid Mechanics 1971;48(3): [5] Dubief Y, Delcayre F. On coherent-vortex identification in turbulence. Journal of Turbulence 2000;1:1-22. [6] Fluent, Inc. Fluent 6.3 User s Guide. January [7] Geurts B. Elements of Direct and Large-Eddy Simulation. R.T. Edwards 2003 [8] Ho C-M, Huerre P. Perturbed free shear layers. Annual Review of Fluid Mechanics 1984;16: [9] Hussain AKMF. Coherent structures - reality and myth. Physics of Fluids 1983;26(10): [10] Jeong J., Hussain F. On the identification of a vortex. Journal of Fluid Mechanics 1995;285: [11] Kim S-E. Large Eddy Simulation Using Unstructured Meshes and Dynamic Subgrid- Scale Turbulence Models. In 34th AIAA Fluid Dynamics Conference and Exhibit, Portland Oregon [12] Lohász MM, Rambaud P, Benocci C. Flow Features in a fully developed ribbed Duct Flow as a Result of MILES. Flow, Turbulence and Combustion 2006;77:59-76 [13] Lumley JL, Yaglom AM. A Century of Turbulence. Flow, Turbulence and Combustion 2001;66: [14] Mitchell BE, Lele SK, Moin P. Direct computation of the sound generated by vortex pairing in an axisymmetric jet. Journal of Fluid Mech 1999;383: [15] Post FH, Vrolijk B, Hauser H, Laramee RS, Doleisc H. The State of the Art in Flow Visualisation: Feature Extraction and Tracking Computer Graphics Forum 2003;22(4): [16] Rónyai L, Ivanyos G, Szabó R. Algoritmusok. TypoTEX 1998 [17] Sedgewick R. Algorithms in Java, 3rd Ed, Part 5 Graph Algorithms. Addison Wesley 2003 [18] de Sousa PJSAF, Pereira JCF. Reynolds number dependence of two-dimensional laminar co-rotating vortex merging Theoretical and Computational Fluid Dynamics 2005;19(1): / 12