Implicit surface tension model for stimulation of interfacial flows

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1 surface tension model for stimulation of interfacial flows Vinh The University of Massachusetts Dartmouth March 3rd, 2011

2 About my project Project Advisor Dr. Mehdi Raessi Department of Mechanical Engineering Project Objective To study implicit of surface tension. To generate faster and better model that produce no spurious currents and has larger time step restriction. 2 / 23

3 Introduction What is interfacial flow? What are the applications? 3 / 23

5 Modeling of interfacial flow To accurately model the interfacial flow is challenging because: The discontinuity of fluid properties (such as density). The interfacial boundary condition (surface tension). 5 / 23

6 Modeling of surface tension in interfacial flow Currently there are two main interfacial flow models Explicit: Precise but has high computational cost due to small time step restriction. : Lower computational cost than explicit due to higher time-step restriction. Draw backs: appearing of nonphysical velocities (spurious current) Common flow solver: Continuum surface force (CSF) 6 / 23

7 Method used CSF method The implicit model studied in this project is based on Continuum Surface Force method This method was first proposed by Brackbill et al Drawbacks: This method generates unphysical velocities (spurious currents) The spurious current is caused by: Imbalance of the surface tension and pressure gradient. Error in computing the curvature. (this project) 7 / 23

8 (Francois et al. 2007) The pressure drop across the interface: σ is the surface tension coefficient k is the mean curvature p = p 2 p 1 = σk (1) k = 1 R I + 1 R II (2) δ is the delta function represent interface (Raessi et al. 2008) 8 / 23

9 My task Tasks: Study the CSF model Study the stability of the model (CFL condition) as time-step increases using different curvature solving method: Level set (LS) and Advecting Normal Compare the results with exact curvature. Challenges: To get used to the code and understand what s the function of each parts requires lots of trials errors. To manipulate it to do what I want also requires lots of trials and errors. 9 / 23

10 CFL number/ CFL condition The Courant-Friedrichs-Lewy condition (CFL condition) is a necessary condition for convergence while solving partial differential equations numerically. C= 1 CF L = u. t x C (3) 10 / 23

11 For a 2D interface between two fluids, depicted in Fig. 1a, the discretized VOF function representing fluid 1 is shown in Fig. 1b. As can be seen, the volume fractions vary sharply from zero to one across the interface. This discontinuous behavior makes it difficult to accurately evaluate the first and second derivatives of f, which leads to inaccurate interface normals and curvatures. Smoothing the f field prior to evaluating $f improves the values [12]. We assess the accuracy of ^n and j calculated from f in Section 2.3. But first, we briefly present the LS method, which is known to yield more accurate normals and curvatures. Level Set method 2.2. Level set method In the LS method, the interface is represented by a smooth function / called the LS function; for a domain X, / is defined [15] as a signed distance to the boundary (interface) ox j/ð~xþj ¼ minðj~x ~xijþ for all ~xi 2 ox ð6þ implying that /ð~xþ ¼0 on ox. Choosing / to be positive inside X, we then have 8 >< > 0; ~x 2 X /ð~xþ ¼ 0; ~x 2 ox >: < 0; ~x 62 X For the 2D interface depicted in Fig. 1a, the discretized LS function, defined at the center of each cell, is shown in Fig. 1c. The unit normal vector and curvature at any point on the interface are calculated from / by and ^n ¼ r/ jr/j r/ j ¼ r jr/j Since / is smooth and continuous across the interface (see Fig. 1c), $/ can be calculated accurately. In the LS method, the motion of the interface is defined by the following advection equation: (Raessi et al. 2007) o/ þ~u r/ ¼ 0 ot ð10þ When / is advected, the / = 0 contour moves at the correct interface velocity; however, contours of / 6¼ 0 do not necessarily remain distance functions. This can result in an irregular / field that in turn leads to problems ð7þ ð8þ ð9þ 11 / 23

12 interface velocity; however, contours of / 6¼ 0 do an irregular / field that in turn leads to problems Level Set method F function representing fluid 1 and (c) the level set function (Raessi et al. 2007) 12 / 23

13 The objective, then, of this work was to devise a method to calculate second-order accurate, or at least with second-order accuracy; however, such methods will fail to exactly conserve mass. verging, The objective, curvatures. then, Such of this a method work was is to presented devise a method next, in to which calculate interface second-order normals accurate, are advected or at least with converging, curvatures curvatures. are calculated Such a method directly is presented from the next, advected in whichnormals. interface normals are advected with the flow, the and and curvatures are calculated directly from the advected normals. 3. Advecting normals: a new method for calculating interface normals and curvatures 3. Advecting normals: a new method for calculating interface normals and curvatures 3.1. Mathematical fundamentals 3.1. Mathematical fundamentals Advecting Normal method As reviewed earlier, the evolution of the LS function is governed by Eq. (10) As reviewed earlier, the evolution of the LS function is governed by Eq. (10) o/ o/ þ~u r/ 0 ot þ~u r/ ¼ 0 ot Defining ~N ¼ r/ as asthe thevector normal normal to to the the contours contours of /, of the /, above the above equation equation can be can rewritten be rewritten as as o/ ot þ~u ~N ¼ 0 Taking the gradient of ofeq. Eq. (14), (14), we we obtain obtain o~n o~n ot ~u ot þ r ~u ~N ¼ 0 ð15þ ~N 0 Eq. (15) is the advection equation for normals. In 2D Cartesian coordinates, Eq. (15) results in the following Eq. equations: (15) is the advection equation for normals. In 2D Cartesian coordinates, Eq. (15) results in the follo equations: on x on þ o x þ o ðun ot ox ðun x þ vn yþ ¼0 ot ox ð16þ x þ vn y Þ¼0 and and on y þ o on y þ o ðun x þ vn yþ ¼0 ot oy ð17þ ot oy ðun x þ vn y Þ¼0 Next, consider the following lemma [28]: (Raessi Let u n ¼ et ~u r/ al. be the 2007) normal velocity of each level set, and set /ð~x; 0Þ to be the signed distance function. Then / Next, remains consider a signedthe distance following function lemma if and[28]: only if ru n r/ ¼ 0. The Letcondition u n ¼ ~u r/ ru n be r/ the¼ normal 0 can be velocity also expressed of each as level set, and set /ð~x; 0Þ to be the signed distance function. / remains a signed distance function if and only if ru n r/ ¼ 0. ~ ~ 13 / 23 ð14þ

14 Case study: Static drop Static water drop in zero gravity. ρ1 = ρ2 = 10 3 Kg/m 3 µ1 = µ2 = 0.05 g = 0 Surface tension time-step restriction t ST = 0.03 (Raessi et al. 2008) 14 / 23

15 What I did 15 / 23

16 Results - Level Set method vs. Exact Curvature 8 x 10!4 CFL vs. t for COMPUTED CURVATURE =0.015s =0.03s 7 =0.06s 6 7 x 10!8 CFL vs. t for Exact CURVATURE =0.015s =0.03s CFL 4 CFL t (s) t (s) Level set Exact curvature The timestep was increase as : t = 0.5, 2, 4 t ST 16 / 23 Student Version of MATLAB

17 Result- Level set at dt = 4, 8dt ST 4 x 107 CFL vs. t for Exact CURVATURE 70 CFL vs. t for COMPUTED CURVATURE =0.12s 3.5 =8*delta t ST 60 =0.24s CFL limit= CFL 2 CFL t (s) Level set dt = 4, 8 t ST t (s) Exact curvature dt = 8 t ST 17 / 23 Student Version of MATLAB

18 Result- Level set vs. Exact Curvature for maximum timestep 4.5 =0.096s CFL vs. Time for Computed Curvature 1.4 CFL vs. Time for Exact Curvature CFL CFL CFL@dt = 0.2s=6.67dtST CFL limit Time s Level set dt = 3.167, 3.2 t ST Time s Exact curvature dt = 6.67, 7 t ST Student Version of MATLAB 18 / 23 Student Version of MATLAB

19 Result- Level set vs Advecting Normal CFL vs. Time for Exact Curvature AN AN LS LS 3 CFL Time s 19 / 23

20 Question 20 / 23

21 Conclusion Proved: To accurately compute the curvature is crucial and it can increase the stability of the solution for implicit model 21 / 23

22 Future Research Continue to study the implicit models. Starting with the simulation. 22 / 23

23 References Mehdi Raessi Modeling surface tension-dominant, large density ratio, two-phase flow, 2008 University of Toronto J. U. BRACKBILL, D. B. KOTHE, AND C. ZEMACH A Continuum Method for Modeling Surface tension Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico,87545, July M. Raessi, J. Mostaghimi, M. Bussmann Advecting normal vectors: A new method for calculating interface normals and curvatures when two-phase ßows Department of Mechanical and Industrial Engineering, University of Toronto, Canada, April 28, Marianne M. Francois, James M. Sicilian, CCS-2; Douglas B. Kothe Modeling Interfacial Surface Tension in Fluid Flow Oak Ridge National Laboratory, / 23

Investigating The Stability of The Balance-force Continuum Surface Force Model of Surface Tension In Interfacial Flow

Investigating The Stability of The Balance-force Continuum Surface Force Model of Surface Tension In Interfacial Flow Vinh The Nguyen University of Massachusetts Dartmouth Computational Science Training