Maßgeschneiderte Werkstoffe gegen Kavitations- Erosion

Transcription

1 Maßgeschneiderte Werkstoffe gegen Kavitations- Erosion (Tailored Materials against Cavitation-Erosion) Authors: K. Ioakimidis, M. Mlikota Institute of Fluid Mechanics and Hydraulic Machinery (IHS) Institute for Materials Testing, Materials Science and Strength of Materials (IMWF) University of Stuttgart

2 page 2 of 17 Content 1 Introduction General Information Cavitation erosion and bubble collapse The Rayleigh Plesset equation Bubble collapse Approach and methodology Hydrodynamic point of view Grid generation and boundary conditions Solving the problem Mechanical point of view FEM techniques Material model Pressure impingement loading Project results and outlook Discussion FE erosion model I FE erosion model II Conclusions Future work References...17

3 page 3 of 17 1 Introduction Cavitation is one of the most severe problems in hydraulic machinery. It reduces the range of operation of the turbine and it can even destroy the machine. Therefore it is very important to be able to accurately predict the cavitation behavior and the resulting loading on the material. It is also very important to know the mechanisms of the destruction of the material. This allows in a long term to develop materials which are more resistant to cavitation erosion. This would lead to a severe reduction of the maintenance costs. The primary objective of the project is to provide a preliminary work for the description and prediction of all flow phenomena arising in the two-phase system formed by a collapsing cavitation bubble near a solid boundary as well as material response to it. It is well known that the problem of a bubble collapse near a wall involves complicated unsteady flow phenomena combined with the material subtraction from the solid surface. Obviously, the phenomenon is very complex since it includes both hydrodynamic and material aspects. To improve materials which could resist such an aggressiveness, a collaboration between engineers from the field of fluid mechanics and material mechanics is essential. In the preceding paragraphs we discuss the hydrodynamic mechanisms of bubble collapse (IHS contribution) and the material aspects (IWMF contribution). 2 General Information Christopher Brennen [1] describes clearly and understandable to everyone the amazing world of bubbles. He gives a brief description of the cavitating phenomena and shows that cavitation is a phenomenon with a wide range of applications. The phenomenon does not occur only in hydraulic machinery, but one can find it in the field of medicine, naval engineering, journal bearing engineering, rocket science, aerospace engineering, etc. There is a wide variety of types of cavitation and the main types occurring as instances in hydraulic machinery are the following: sheet cavitation or attached cavities traveling bubble cavitation cavitation clouds cavitating vortices As an example how cavitation looks like in reality see Figure 1.

4 page 4 of 17 Figure 1: Sheet / Cloud Cavitation created by a NACA 0012 hydrofoil at an angle of attack (IHS) There are also other types of cavitation and it is thinkable, from the combination of the applications variety and the types of cavitation, that there is a big research potential in cavitating phenomena for both fluid mechanics and structural engineers. In the literature cavitation refers as the formation or the development of vapor structures in an originally liquid flow. The phase change takes place at almost constant temperature in the regions where a local drop in pressure occurs and is generated by the flow itself. Practically, constant temperature and pressure drop under the liquid saturation pressure cause cavitation. In contrast, it is known that boiling of a liquid occurs if the liquid is heated at constant pressure. Bubble formation, which is a small pocket of vapor inside a liquid, is common occurrence of both, cavitation and boiling phenomena. Bubbles come in all sizes, shapes and forms and have different dynamical behavior. As an instance, bubbles produced by boiling collapse very slowly and relatively gently, but bubbles produced by cavitation, in most cases collapse violently and are dangerous and noisy. The most important feature arises when a bubble collapses near a wall or essentially on any solid surface, which could be a hydro turbine or a pump or even a teeth or a kidney stone [1]. There is no chance for the solid to survive; an explanation will be given afterwards. In the following we pay attention to the dynamics of a vapor-filled cavitation bubble collapse on a solid metal surface, such those of Kaplan turbine, Francis turbine and we concentrate on hydrodynamic cavitation. The following figures show the damage on various hydraulic machinery components.

5 page 5 of 17 Figure 2: Cavitation damage of a pump impeller (IHS) Figure 3: Close up of the cavitation damage of Figure 2

6 page 6 of 17 Figure 4: Cavitation damage of a Kaplan blade (IHS) Figure 5: Close up of a Kaplan blade Figure Cavitation erosion and bubble collapse It is well known that the violent collapse of such bubbles can severely cause material damage or in other words, is responsible for the material abstraction from a solid

7 page 7 of 17 surface. Another important issue that may result from that collapse is noise emission. Responsible for that phenomena are high velocities, pressures and temperatures that may result from that collapse [2]. We consider spherical bubbles, since the spherical analysis represents the maximum possible consequences of bubble collapse [2], although collapsing bubbles do not remain spherical. Numerous experimental studies worldwide were done in order to improve understanding of the phenomenon; while in contrast, less computational work is done. However, some recent results show that it is possible to compute a single bubble collapse [4], [5] The Rayleigh Plesset equation Under the main consideration of spherical bubbles, Brennen [2] explains the work done by Rayleigh (1917) and Plesset (1949) and he represents the derivation of the Rayleigh-Plesset Equation (RPE), which describes the bubble growth and collapse. Jean-Pierre Franc [2] discusses why the RPE is a useful tool for understanding the mechanism of bubble radius growth and collapse. According to the RPE both Brennen and Franc give an extended discussion of the various aspects of cavitation. Although the RPE is a powerful tool, the derivation has been done under simplifications, thus it cannot describe all the dynamic phenomena occurred by a single bubble collapse. The case of a bubble cloud collapse is more uncomfortable situation, while there is interaction between the bubbles and this factor mitigates the results. The considerations we do to derive the RPE are: spherical bubble radius the liquid is considered to be incompressible, constant density the dynamic viscosity of the liquid is assumed to be constant and uniform the temperature and pressure within the bubble are assumed to be always uniform thermodynamically, considerations of the bubble contents are necessary Under these considerations we derive the RPE: + + = (1) We count seven terms for this complex nonlinear equation form the left: is the instantaneous tension (driving term) determined by the conditions far from the bubble is the thermal term, and depending on the magnitude of that term the bubble is expected to have different dynamical behavior bubble content thermodynamic term the fourth and fifth term are the inertial terms

8 page 8 of 17 viscous or damping term at last we find the bubble surface tension term Because of this complexity, many cavitation models based on the RPE have been developed. If we neglect thermal effects, viscous effects, the surface tension and the second order term we reduce the RPE to: = 2 3 (2) which describes the rate of bubble radius growth. This is a very useful tool to understand how a cavitation bubble behaves in liquid cavitating conditions. Though, the above equations fail to describe all the phenomena occur a bubble collapse near a solid boundary. The two phase incompressible flow cavitating models developed yet, as the Schnerr Sauer model [6], Kunz model, Merkle model [8] and the simplified Rayleigh-Plesset model [9] are based on the assumptions of equation (1). These models are able to predict accurately where cavitation occurs and the main characteristics of cavitating phenomena. However, they are unable to predict cavitation erosion Bubble collapse The subject is particularly important because of the reasons we already mentioned, damage and noise. Despite the bubbles does not remain spherical, it is often argued that the spherical analysis represents the maximum possible consequences. Thus we take sphericity into account. Numerous studies have shown that a bubble collapse near a solid boundary is characterized by high velocities, temperatures and pressures instantaneously by the end of the collapse. This collapse is followed by pressure or shock waves of high intensity. Fujikawa and Akamatsu [10] measured very high levels of pressure of about 100 MPa at the instant of the impact. Brennen [2] in his book according the RPE refers that the collapse will begin if the bubble has reached its maximum radius in order of 100 the original radius and more specific. "...if the original partial pressure of gas in the nucleus was about 1 bar the value of that pressure at the start of collapse would be about 1 µbar. If the typical pressure depression in the flow yields a value for of say 0.1 bar it would follow from the RPE that the maximum pressure generated would be about bar (note 1 bar = 100 kpa) and the maximum temperature would be about 4x10 4 times the ambient temperature!" Furthermore, during the collapse a micro jet is produced which is directed towards the wall. Current estimates of the micro jet velocity normal to the wall give values in order at least of 100 m/s which values are easy to obtain with taking into account the water hammer formula from Joukowski and Allievi: = (3) The duration of the pressure pulse is fixed by the jet diameter which is about 1/10 of the original diameter of the bubble. That means if the bubble has 1 mm diameter, the

9 page 9 of 17 jet diameter should be about 0.1 mm, which leads to a very small value for the duration of the pressure pulse of about 0.03 µs. Figure 6 shows schematically the micro jet. Figure 6: The micro jet and the impact on the solid boundary In order to capture this phenomenon computationally we have to take into account the compressibility of the liquid. Keller and Kolodner [11] proposed a modification of the RPE including compressibility of the liquid because as we explained the speed of sound and liquid compressibility cannot be neglected in the case of a bubble collapse near a wall. Figure 7: Micro jet experimental data inherited from [2] 3 Approach and methodology 3.1 Hydrodynamic point of view To simulate the bubble collapse we have chosen the commercial software ANSYS CFX [9], since it has been well tested and we have made positive experiences in simulating cavitating hydrofoils. However, in that cases compressibility of both fluid phases was not taken into account. Thus, it was challenge to test CFX in compressible flow. The general process to do a numerical simulation is the Pre- Processing, solve the problem and Post-Processing. Finally, the goal is to provide the computational fluid dynamics (CFD) results as initial conditions for the structural finite element analysis.

10 page 10 of Grid generation and boundary conditions This is the first step in doing CFD and generally the most time expensive, if it is needed to deal with complex geometries. In our case the geometry is quite easy to handle, so the grid generation is not time expensive. The grid is generated with ANSYS ICEMCFD. Various grid qualities in 2D and 3D have been generated in order to check the solver grid dependency in our case. The geometry is shown below as an instance. Figure 8: The 3D computational domain Figure 9: Close up on the bubble

11 page 11 of 17 Furthermore, the bubble was initialized and was filled with air. The blue region (see Figure 8) was set as an opening boundary, whereas the green region and the region close to the bubble was set to wall boundaries. While the fluid are initially steady at opening the pressure is varied as a time step dependent function in order to produce pressure pulsations in the liquid Solving the problem The flow is modeled as two phase compressible flow for reasons we explained above and the coupling between the two phases is achieved via Volume-Of-Fluid method which is an approach to couple the Navier-Stokes for both phases. This means that the Navier-Stokes equations are solved simultaneously for both phases water and air, and both phases are coupled via a transport equation. Firstly, computations are done without any modeling for the bubble growth and collapse, while such a model for compressible flow does not exist in ANSYS CFX. An attempt to implement the Keller modification of the Rayleigh-Plesset equation as User Defined Function (UDF) has been done. An explanation why the problem is not trivial to solve computationally will be given in the results discussion below. 3.2 Mechanical point of view FEM techniques To simulate the cavitation induced erosion process by means of continuum mechanics on the macroscopic level, the finite element method (FEM) provides two promising possibilities: The first possibility is based on the application of the Arbitrary Lagrangian-Eulerian (ALE) adaptive meshing technique (Figure 10, middle) and the ABAQUS user subroutine UMESHMOTION [12]. ALE adaptive meshing algorithm relocates the mesh by an amount equal to a computed value this feature could be used for simulating erosion where the mesh is relocated in accordance to the computed erosion depth. The erosion depth would be applied to each node based on the erosion rate calculated in the UMESHMOTION subroutine. The displacement of the nodes contains the material deformations as well as the displacements due to mesh motion. The partial model variable VOLC measures the volume loss due to adaptive mesh constraints. This variable could be used to quantify the simulated erosion process and to compare numerical and experimental results. The second possibility is based on the element removal technique [12] (Figure 10, right). It is assumed that damage is characterized by the progressive degradation of the material stiffness, leading to material failure. For each simulation increment and each finite element, information about specified output variable values is requested and compared to a damage criterion. If the retrieved values reach or overtake the damage criterion, the element is removed from the mesh - by this the erosion process could be simulated. For this purpose, two finite element erosion models will be introduced in order to achieve a better understanding of the mechanisms that are associated with the cavitation erosion process.

12 page 12 of 17 Figure 10: Left: Mesh without application of the ALE adaptive meshing technique, Centre: Mesh with application of the ALE adaptive meshing, Right: Mesh after removal of elements Material model The Johnson-Cook (J-C) [13] plasticity formulation, which defines the flow stress as a function of equivalent plastic strain and temperature, was employed to model the mechanical behavior of the target material. The model is suitable for erosion-induced high-strain-rate deformations of many materials, including most metals. The Johnson-Cook material model expresses the von Mises flow stress as: σ = "+#$ &' ( )1, + - (4), + where A and B are the yield stress constant and the strain hardening coefficient, respectively, $ &' is the equivalent plastic strain, T material temperature, T r room temperature, T m material melting temperature, n strain hardening exponent and m the thermal softening exponent Pressure impingement loading Under the assumption that impingements of pressures caused by cavity implosions are randomly distributed over the impinged area, a random number generator was applied to numerically simulate the cavitation erosion loading. Figure 11 represents equivalent von Mises stresses caused by random pressure impingements. The loading conditions are based on the initial CFD results where pressure is equal to 10 MPa, duration of the impingement to 0.03 µs and the area experiencing the pressure to about 0.3 mm 2. [MPa] von Mises stresses caused by current random impact Residual stresses from previous random impact Figure 11: Equivalent von Mises stresses caused by random droplet impingements

13 page 13 of 17 The random loading can be easily switched to the repeated perpendicular impingements at the same point. The benefits of this type of the loading are lower computational times of simulations, which allowed more time effective model development. Figure 12 shows a cubic FEM model with von Mises stresses caused by repeated impingements at the same position with otherwise the same loading conditions as random impingements. Further simulations were performed using perpendicular impingement loading at the same position. [MPa] von Mises stresses caused by repeated impacts Figure 12: von Mises stress field caused by repeated impingements at the same point The model is prepared to change the frequency of impingements as well as nonuniform spatial and temporal pressure distribution for a single impingement (Figure 13). However, non-uniformity of spatial and temporal pressure distribution was not applied to the loading description due to absence of data. Impact pressure a) b) Time Figure 13: a) Non-uniform pressure distribution and b) time dependent pressure distribution for the single droplet impingement The water droplet impingement loadings were developed in the frame of the project KW21 BWL32DT: Influence of droplet size on erosion processes [14], and was adjusted to the needs of the current project. 4 Project results and outlook As already mentioned, CFD computations and numerical investigations were performed using proposed models. Both CFD and structural results are presented in the following subsections.

14 page 14 of Discussion The main difficulty is that the duration of the phenomenon is about 0.01 ms and the micro jet or the pulsation duration is about 0.03 µs. Thus in order to compute sec of the phenomenon very small time step is needed. Consequently, and to provide stable solution a fine computational mesh is needed respectively. With appropriate initial conditions and with a time step of 10-8 s a computation of 10 4 time steps is needed which results 600 hours of computation. Some of the results are shown below. These results are not in good agreement with the theory but they surely show the correct way of the approach. For that reason it was essential to estimate the pressures on the wall, the duration and the bubble interface velocity in order to go on with the structural computations with ABAQUS. Comparing the experimental data from the literature and after analyzing the RPE in MATLAB we estimated a pressure peak of about 10 MPa with the duration µs and a pitting surface of about 0.3 mm 2. Figure 14: CFD results, four time steps FE erosion model I The cavitation erosion process was simulated with the application of the FE model based on the ALE adaptive meshing and subroutine UMESHMOTION. Figure 15 shows the cubic model with the result of the simulation where displacements of nodes caused by applied external loading are equal to the calculated erosion damage depth. Generally, the erosion is activated when a predefined criterion is

15 page 15 of 17 reached. In this case, the first occurrence of the equivalent plastic strain of PEEQ crit = is used as the activation criterion of erosion. The FE erosion model I was developed in the frame of the project KW21 BWL32DT: Influence of droplet size on erosion processes [14], and was adjusted to the needs of the current project. [mm] Figure 15: Displacements caused by repeated impingements perpendicular to the model surface (FE erosion model I) FE erosion model II The additional erosion model II has been developed partially. Figure 16 shows that after a certain number of repeated impingements the damage evolution criterion was reached in impinged elements and therefore these elements got removed from the mesh. Similar to the first erosion model wherein the first occurrence of equivalent plastic strain PEEQ crit = was used as an activation criterion of erosion, in this model the same value represented the criterion for the removal of elements. [MPa] Figure 16: Elements get removed from the mesh after the damage evolution criterion PEEQ crit = is reached (FE erosion model II) When the elements from the first layer of elements get removed, the loading pressure caused by a single impact needs to be transferred to the elements lying under those already removed elements. Up to now, this transfer of loading is

16 page 16 of 17 accomplished (Figure 17) but further improvements are needed in order to get better control over the removal of elements. Figure 17: Transfer of the loading pressure from one layer of elements to another The FE erosion model II was further developed and adjusted from the model presented in [14]. 5 Conclusions Concerning hydrodynamic aspects of the study, it can be concluded that a CFD simulation of a single bubble collapse near a wall boundary is possible. The solution to the problems arising in the CFD approach is probably the implementation of the Rayleigh-Plesset equation as it is without simplifications in order to capture the dynamical characteristics of the bubble collapse. Moreover, concerning material aspects, the goals set at the beginning of the project were achieved by adjusting a FE erosion model I based on ALE adaptive meshing technique and by further developing an additional second model based on the element removal technique. These models provide a good basis for future sophisticated modeling of the cavitation induced erosion process. Different types of loading were applied. Initial simulations were performed with random widespread impact loading, which mimics conditions present in the real cavitation erosion process to a good approximation. In order to reduce the computational times, a simplified repeated impact loading scheme at the same position was developed subsequently. The additional model based on the element removal technique has a high prospective for further cavitation erosion process modeling since it provides the possibility to model crack formation and fracture using the removal of elements. This feature could give improved insights into what happens inside the material during the process of erosion caused by repeated water droplet impingements. The model provides another interesting possibility, which is to investigate the influence of the material microstructure on the erosion process (e.g. different phases, different grain sizes). From mechanical point of view, the two numerical models, proposed in this project, can describe the erosion behavior of a material in qualitative terms. With further development of the models and enhancement of the applied damage criteria, the behavior of the cavitation erosion process could also be described in quantitative terms in the future.

17 page 17 of 17 6 Future work As a continuation of this project, a proposal for a long-term project is presently under preparation and intention is to submit it in the near future in the frame of AiF funding program as aimed for in the present project. 7 References [1] Christopher E. Brennen: The amazing world of bubbles. Engineering and Science No. 1, 2007 [2] Christopher E. Brennen: Cavitation and Bubble Dynamics, Oxford Engineering Science Series 44, Oxford University Press, 1995 [3] Jean-Pierre Franc. The Rayleigh-Plesset equation: a simple and powerful tool to understand various aspects of cavitation. CISM Courses and Lectures, Vol. 496 [4] A. Osterman et al. (2009), Numerical Simulation of a Near-Wall Bubble Collapse in an Ultrasonic Field, Journal of Fluid Science and Technology 4(1): [5] Vincent Minsier: Numerical simulation of cavitation-induced bubble dynamics near a solid surface. Ecole Polytechnique De Louvain. PhD Thesis, 2009 [6] Weixing Yuan, Jürgen Sauer, Günther Schnerr, Modeling and computation of unsteady cavitation flows ininjection nozzles, Mec. Ind. (2001) 2, [7] Merkle, C. L., Feng, J., and Buelow, P. E. O., Computational modeling of the dynamics of sheet cavitation, Proceedings of 3rd International Symposium on Cavitation, Grenoble, France (1998) [8] Kunz, R. F., Boger, D. A., Stinebring, D. R., Chyczewski, T. S., Lindau, J. W., Gibeling, H. J., Venkateswarn, S., and Govindan, T. R., A preconditioned Navier Stokes method for two-phase flows with application to cavitation prediction, Journal of Computers and Fluids, Vol. 29, pp (2000) [9] ANSYS CFX 14.0, Theory Guide. [10] Fujikawa S., Akamatsu T., Effects of non-equilibrium condensation of vapor on the pressure wave produced by collapse of a bubble in a liquid. J. Fluid Mech. 97, part 3, [11] Keller J.B., Kolodner I.I., Damping of underwater explosions bubble oscillations. J. Appl. Phys [12] Abaqus Analysis User s Manual [13] Wang, Y.-F.; Yang, Z.-G.: Finite element model of erosive wear on ductile and brittle materials, Wear (2008), Vol. 265, Pages [14] Mansoor, A.; Mlikota, M.: Tropfengrößeneinfluss auf Erosionsvorgänge, KW21 BWL32DT Abschlussbericht (2012)