Coupling of VOF-Based Solver with LPT for Simulation of Cavitating Flows

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1 Coupling of VOF-Based Solver with LPT for Simulation of Cavitating Flows Ebrahim Ghahramani Mechanics and Maritime Sciences, Chalmers University of Technology, Gothenburg, Sweden Ebrahim Ghahramani CHALMERS / 56

2 Cavitating Flows 1 Introduction Cavitating Flows Multi-scale solver 2 Multiphase Flow Methods Homogeneous mixture method Lagrangian Particle Tracking 3 Tutorial Adding LPT library to the solver Inject a bubble Implementing Rayleigh-Plesset equation Ebrahim Ghahramani CHALMERS / 56

3 Cavitating Flows Cavitation A common phenomenon in industrial hydraulic systems Cavitation erosion: the result of violent collapses of the flowing micro-bubbles on very short time scales Cavitating flows include an extensive range of cavity structures with different length scales From micro bubbles to large sheet cavities that may fully cover the surface of a hydraulic device, e.g. a hydrofoil Ebrahim Ghahramani CHALMERS / 56

4 Cavitating Flows Numerical Models Eulerian models Limited to modeling of structures larger than the computational cell Unable to resolve cavitation nuclei and bubbles Lagrangian models Unable to model large-scale non-spherical structures A hybrid multi-scale approach Ebrahim Ghahramani CHALMERS / 56

5 Multi-scale solver 1 Introduction Cavitating Flows Multi-scale solver 2 Multiphase Flow Methods 3 Tutorial Ebrahim Ghahramani CHALMERS / 56

6 Multi-scale solver Background Multi-scale solvers are popular in other applications as well; e.g. atomizing gas-liquid flows Aurélia Vallier developed a LPT-VOF solver before (OSCFD course 2011) The solver was developed by coupling of interfoam solver with Lagrangian library Today, we are going to develop a similar solver by adding the Lagrangian library to interphasechangefoam solver to make it more suitable for cavitating flows Ebrahim Ghahramani CHALMERS / 56

Multi-scale solver Large cavity structures are resolved by Eulerian models based on Volume of Fluid (VOF) approach Small sub-grid structures are treated as Lagrangian bubbles using the Lagrangian

7 Multi-scale solver Large cavity structures are resolved by Eulerian models based on Volume of Fluid (VOF) approach Small sub-grid structures are treated as Lagrangian bubbles using the Lagrangian Particle Tracking (LPT) method Aim: To couple interphasechangefoam solver with the Lagrangian library Improving the lagrangian Library to consider bubble size variation Ebrahim Ghahramani CHALMERS / 56

8 1 Introduction 2 Multiphase Flow Methods Homogeneous mixture method Lagrangian Particle Tracking 3 Tutorial Ebrahim Ghahramani CHALMERS / 56

9 Homogeneous mixture method Eulerian Model Transport equation based method is utilized here Multiphase fluid is treated as a single fluid mixture with average properties such as density and viscosity ρ m = αρ l + (1 α)ρ v µ m = αµ l + (1 α)µ v α: liquid volume fraction Mass transfer between the phases is defined by explicit source terms Continuity equation ṁ: rate of mass transfer u i x i = ( 1 ρ l 1 ρ v )ṁ Ebrahim Ghahramani CHALMERS / 56

10 Homogeneous mixture method Navier-Stokes equation (ρ m u i ) t + (ρ mu i u j ) x j = τ ij x j + ρ m g i + S st S st : surface tension force α is obtained by solving a scalar transport equation α t + (αu i) = ṁ x i ρ l ṁ is obtained from semi-empirical cavitation models Ebrahim Ghahramani CHALMERS / 56

11 Homogeneous mixture method Homogeneous mixture method in OpenFOAM interphasechangefoam solver similar to interfoam but with mass transfer between the two phase ls $WM_PROJECT_DIR/applications/solvers/multiphase/interPhaseChangeFoam interphasechangefoam.c alphacontrols.h alphaeqn.h alphaeqnsubcycle.h createfields.h peqn.h UEqn.H interphasechangedymfoam Make phasechangetwophasemixtures Ebrahim Ghahramani CHALMERS / 56

12 Homogeneous mixture method vi createfields.h Info<< "Creating phasechangetwophasemixture\n" << endl; autoptr<phasechangetwophasemixture> mixture = phasechangetwophasemixture::new(u, phi); volscalarfield& alpha1(mixture->alpha1()); volscalarfield& alpha2(mixture->alpha2()); const dimensionedscalar& rho1 = mixture->rho1(); const dimensionedscalar& rho2 = mixture->rho2(); const dimensionedscalar& psat = mixture->psat(); phasechangetwophasemixture is a class to correct the mixture viscosity and return the source term of the continuity and volume fraction (α) equations. Ebrahim Ghahramani CHALMERS / 56

13 Homogeneous mixture method ls phasechangetwophasemixtures phasechangetwophasemixture lninclude Make SchnerrSauer Merkle Kunz The source terms are calculated using a cavitation model. Available models in OF are SchnerrSauer, Merkle and Kunz models A new class is defined for each model which inherits phasechangetwophasemixture class. Ebrahim Ghahramani CHALMERS / 56

14 Homogeneous mixture method vi createfields.h Info<< "Creating phasechangetwophasemixture\n" << endl; autoptr<phasechangetwophasemixture> mixture = phasechangetwophasemixture::new(u, phi); volscalarfield& alpha1(mixture->alpha1()); volscalarfield& alpha2(mixture->alpha2()); const dimensionedscalar& rho1 = mixture->rho1(); const dimensionedscalar& rho2 = mixture->rho2(); const dimensionedscalar& psat = mixture->psat(); alpha1 & alpha2 are the volume fractions of phases 1 & 2 rho1 & rho2 are the densities psat is the saturation pressure of the fluid; used to calculate mass transfer rate. The properties of each phase as well as the saturation pressure should defined by user as an input in the transportproperties dictionary Ebrahim Ghahramani CHALMERS / 56

15 Homogeneous mixture method vapour transport equation α t + (αu i) = ṁ +α u i α u i x i ρ l x i x i u i = ( 1 1 )ṁ x i ρ l ρ v α t + (αu i) = α u i + ( 1 α( 1 1 ))ṁ x i x i ρ l ρ l ρ v V = ( 1 ρ l α( 1 ρ l 1 ρ v ))ṁ V = α V v + (1 α) V c This is the actual term calculated by phasechangetwophasemixture Ebrahim Ghahramani CHALMERS / 56

16 Homogeneous mixture method vi alphaeqn.h Pair<tmp<volScalarField>> vdotalphal = mixture->vdotalphal(); const volscalarfield& vdotcalphal = vdotalphal[0](); const volscalarfield& vdotvalphal = vdotalphal[1](); const volscalarfield vdotvmcalphal(vdotvalphal - vdotcalphal); tmp<surfacescalarfield> talphaphi; if (MULESCorr) { fvscalarmatrix alpha1eqn ( fv::eulerddtscheme<scalar>(mesh).fvmddt(alpha1) + fv::gaussconvectionscheme<scalar> ( mesh, phi, upwind<scalar>(mesh, phi) ).fvmdiv(phi, alpha1) - fvm::sp(divu, alpha1) == fvm::sp(vdotvmcalphal, alpha1) + vdotcalphal ); Ebrahim Ghahramani CHALMERS / 56

17 Homogeneous mixture method vi peqn.h Pair<tmp<volScalarField>> vdotp = mixture->vdotp(); const volscalarfield& vdotcp = vdotp[0](); const volscalarfield& vdotvp = vdotp[1](); while (pimple.correctnonorthogonal()) { fvscalarmatrix p_rgheqn ( fvc::div(phihbya) - fvm::laplacian(rauf, p_rgh) - (vdotvp - vdotcp)*(mixture->psat() - rho*gh) + fvm::sp(vdotvp - vdotcp, p_rgh) ); Ebrahim Ghahramani CHALMERS / 56

18 Homogeneous mixture method vi UEqn.H if (pimple.momentumpredictor()) { solve ( UEqn == fvc::reconstruct ( ( interface.surfacetensionforce() - ghf*fvc::sngrad(rho) - fvc::sngrad(p_rgh) ) * mesh.magsf() ) ); } Ebrahim Ghahramani CHALMERS / 56

19 Lagrangian Particle Tracking 1 Introduction 2 Multiphase Flow Methods Homogeneous mixture method Lagrangian Particle Tracking 3 Tutorial Ebrahim Ghahramani CHALMERS / 56

20 Lagrangian Particle Tracking Discrete bubble method Tracking individual particles (bubbles) Particle p: x p, d p, u p,i and ρ p particle equation of motion m p du p,i dt dx p,i = u p,i dt (u f,i u p,i ) = m p τ p + (ρ p ρ f )g i τ p = 4 ρ p d 2 p 3 ρ f C D u f,i u p,i C D : Particle drag coefficient Ebrahim Ghahramani CHALMERS / 56

21 Lagrangian Particle Tracking Rayleigh-Plesset equation Variation of bubble radius in incompressible flows R(t) R(t) + 3 2Ṙ2 (t) = P B P l ρ l P B is the bubble pressure P B = P v + P g0 ( R0 R(t) Ṙ(t) 4ν l R(t) 2σ st ρ l R(t) ) 3k P g0 equilibrium pressure of dissolved gas and R 0 Equilibrium radius Equilibrium State R 0 3γ + 2σ st P 0 P v R 0 3γ 1 + R3γ P 0 P v ( P v 2σ ) st R P l = 0 Ebrahim Ghahramani CHALMERS / 56

22 Lagrangian Particle Tracking LPT in OpenFOAM ls $WM_PROJECT_DIR/src/lagrangian/solidParticle/ solidparticle.c solidparticlecloud.h solidparticle.h solidparticleio.c solidparticlecloud.c solidparticlecloudi.h solidparticlei.h lninclude Make Ebrahim Ghahramani CHALMERS / 56

23 Lagrangian Particle Tracking LPT in OpenFOAM solidparticlecloud: a class that contains a cloud of solid particles and track them in the flow field Subclass of Cloud<solidParticle> class Cloud is a Base cloud calls templated on particle type. Subclass of cloud class A cloud is a collection of lagrangian particles. Cloud has some general functions for tracking a collection of particles solidparticle : Subclass of the particle with additional member data and functions to track each particle Ebrahim Ghahramani CHALMERS / 56

24 Lagrangian Particle Tracking LPT in OpenFOAM In solidparticle class a new particle (object) is created Particles are tracked using the move function. ls solidparticle.h //- Construct from components inline solidparticle ( const polymesh& mesh, const vector& position, const label celli, const label tetfacei, const label tetpti, const scalar d, const vector& U );... //- Move bool move(trackingdata&, const scalar); Ebrahim Ghahramani CHALMERS / 56

25 Lagrangian Particle Tracking LPT in OpenFOAM In the move function the particle equations of motion are solved m p du p,i dt dx p,i = u p,i dt (u f,i u p,i ) = m p τ p + (ρ p ρ f )g i So, 2 types of arguments are needed 1 Equation time step: the "const scalar" in the argument list 2 Flow field data, e.g. u f, p, ρ, etc The flow field data are passed to move function via an object of the trackingdata class. The class is defined in the solidparticle class with the follwoing constructor: Ebrahim Ghahramani CHALMERS / 56

26 Lagrangian Particle Tracking LPT in OpenFOAM ls solidparticle.h inline trackingdata ( solidparticlecloud& spc, const interpolationcellpoint<scalar>& rhointerp, const interpolationcellpoint<vector>& UInterp, const interpolationcellpoint<scalar>& nuinterp, const vector& g ); The constructor arguments are some interpolationcellpoint to interpolate flow values at arbitrary points, a vector that refers to gravity vector (g), and a solidparticlecloud object (spc). Ebrahim Ghahramani CHALMERS / 56

27 Lagrangian Particle Tracking LPT in OpenFOAM ls $WM_PROJECT_DIR/src/lagrangian/solidParticle/ solidparticle.c solidparticlecloud.h solidparticle.h solidparticleio.c solidparticlecloud.c solidparticlecloudi.h solidparticlei.h lninclude Make Ebrahim Ghahramani CHALMERS / 56

28 Lagrangian Particle Tracking LPT in OpenFOAM A solidparticlecloud object is a container for particles ls solidparticlecloud.h //- Construct given mesh solidparticlecloud ( const fvmesh&, const word& cloudname = "defaultcloud", bool readfields = true ); An object is constructed in the Eulerian solver to have access to fvmesh solidparticlecloud has a different move function to pass flow data to particle group and track individual particles Ebrahim Ghahramani CHALMERS / 56

29 Lagrangian Particle Tracking LPT in OpenFOAM ls solidparticlecloud.c void Foam::solidParticleCloud::move(const dimensionedvector& g) { const volscalarfield& rho = mesh_.lookupobject<const volscalarfield>("rho"); const volvectorfield& U = mesh_.lookupobject<const volvectorfield>("u"); const volscalarfield& nu = mesh_.lookupobject<const volscalarfield>("nu"); interpolationcellpoint<scalar> rhointerp(rho); interpolationcellpoint<vector> UInterp(U); interpolationcellpoint<scalar> nuinterp(nu); solidparticle::trackingdata td(*this, rhointerp, UInterp, nuinterp, g.value()); } Cloud<solidParticle>::move(td, mesh_.time().deltatvalue()); Cloud<solidParticle>::move includes a loop over all particles which calls solidparticle::move function to track particles individually Ebrahim Ghahramani CHALMERS / 56

30 Lagrangian Particle Tracking LPT in OpenFOAM In summary, to create and track solidparticle in a solver: 1 A solidparticlecloud object should be created in the solver 2 Arbitrary number of particles (solidparticle object) are added to the solidparticlecloud object 3 The flow data are passed to the particles by creating a trackingdata object in the solidparticlecloud move function 4 Individual particles are tracked by calling the Cloud<solidParticle>::move in solidparticlecloud::move Ebrahim Ghahramani CHALMERS / 56

31 1 Introduction 2 Multiphase Flow Methods 3 Tutorial Adding LPT library to the solver Inject a bubble Implementing Rayleigh-Plesset equation Ebrahim Ghahramani CHALMERS / 56

32 Adding LPT library to the solver Solver directory cd $WM_PROJECT_USER_DIR/applications mkdir myhybridsolver cd myhybridsolver cp -r $WM_PROJECT_DIR/applications/solvers/multiphase /interphasechangefoam/*. cp $WM_PROJECT_DIR/src/lagrangian/solidParticle/*. Ebrahim Ghahramani CHALMERS / 56

33 Adding LPT library to the solver vi Make/files interphasechangefoam.c solidparticle.c solidparticleio.c solidparticlecloud.c EXE = $(FOAM_USER_APPBIN)/myHybridSolver vi Make/options -I$(LIB_SRC)/lagrangian/basic/lnInclude -I$(LIB_SRC)/finiteVolume/lnInclude -lfinitevolume -llagrangian -lsolidparticle Ebrahim Ghahramani CHALMERS / 56

34 Adding LPT library to the solver main function vi interphasechangefoam.c #include "phasechangetwophasemixture.h" #include "solidparticlecloud.h" solidparticlecloud bubbles(mesh); turbulence->validate(); bubbles.move(g); Info<< "Cloud size= "<< bubbles.size() <<endl; runtime.write(); Ebrahim Ghahramani CHALMERS / 56

35 Adding LPT library to the solver Compile the solver wmake Ebrahim Ghahramani CHALMERS / 56

36 Inject a bubble 1 Introduction 2 Multiphase Flow Methods 3 Tutorial Adding LPT library to the solver Inject a bubble Implementing Rayleigh-Plesset equation Ebrahim Ghahramani CHALMERS / 56

37 Inject a bubble Adding LPT library to interphasechangefoam Goal: Inject A buble B. We will give in a dictionary (particleproperties): position (posb) diameter (db) velocity (UB) time when injection starts and ends. And the constructor needs to know position vector cell lable diameter velocity vi solidparticle.h //- Construct from components inline solidparticle ( const polymesh& mesh, const vector& position, const label celli, const label tetfacei, const label tetpti, const scalar d, const vector& U ); Ebrahim Ghahramani CHALMERS / 56

38 Inject a bubble vi solidparticlecloud.h scalar mu_; vector posb_; scalar db_; vector UB_; scalar tinjstart_; scalar tinjend_; void move(const dimensionedvector& g); //- Inject particles according to the dictionnary particleproperties void inject(solidparticle::trackingdata &td); Ebrahim Ghahramani CHALMERS / 56

39 Inject a bubble vi solidparticlecloud.c mu_(dimensionedscalar(particleproperties_.lookup("mu")).value()), posb_(dimensionedvector(particleproperties_.lookup("posb")).value()), db_(dimensionedscalar(particleproperties_.lookup("db")).value()), UB_(dimensionedVector(particleProperties_.lookup("UB")).value()), tinjstart_(dimensionedscalar(particleproperties_.lookup("tinjstart")).value()), tinjend_(dimensionedscalar(particleproperties_.lookup("tinjend")).value()) Cloud< solidparticle>::move(td); if(mesh_.time().value()> td.cloud().tinjstart_ && mesh_.time().value()< td.cloud().tinjend_) {this->inject(td);} Ebrahim Ghahramani CHALMERS / 56

40 Inject a bubble vi solidparticlecloud.c void Foam::solidParticleCloud::inject(solidParticle::trackingData &td) { label celli = -1; label tetfacei = -1; label tetpti = -1; mesh_.findcellfacept(td.cloud().posb_, celli, tetfacei, tetpti); if(celli > 0){ solidparticle* ptr1 = new solidparticle(mesh_, td.cloud().posb_, celli, tetfacei, tetpti,td.cloud().db_, td.cloud().ub_); Cloud<solidParticle>::addParticle(ptr1); } } // ************************************************************************* // Ebrahim Ghahramani CHALMERS / 56

41 Inject a bubble Compile the solver wmake Go to the test case directory : cd $WM_PROJECT_USER_DIR/run/testCase Have a look at the injector dictionary vi constant/particleproperties rhop rhop [ ] ; e e [ ] 0.2; mu mu [ ] 0.05; posb posb [ ] ( ); db db [ ] ; UB UB [ ] (0 0 0); tinjstart tinjstart [ ] 0.075; tinjend tinjend [ ] 0.085; psat psat [ ] 2340; sigma sigma [ ] 0.07; gamma gamma [ ] 1.4; Ebrahim Ghahramani CHALMERS / 56

42 Inject a bubble Run the solver myhybridsolver > log& Have a look at the log file to check if particles were injected. Load the results in paraview touch testcase.foam paraview File Open testcase.foam Visualize the bubble: Mesh regions: lagrangian/defaultcloud Glyph: Type Sphere, Mode Scalar,Scalar d, Edit Set Scalar Factor 100 Ebrahim Ghahramani CHALMERS / 56

43 Implementing Rayleigh-Plesset equation 1 Introduction 2 Multiphase Flow Methods 3 Tutorial Adding LPT library to the solver Inject a bubble Implementing Rayleigh-Plesset equation Ebrahim Ghahramani CHALMERS / 56

44 Implementing Rayleigh-Plesset equation Rayleigh-Plesset equation Variation of bubble radius in incompressible flows R(t) R(t) + 3 2Ṙ2 (t) = P B P l ρ l P B is the bubble pressure For simplicity P B = P v + P g0 ( R0 R(t) P g0 = 0 R 0 = R t=0 Ṙ t=0 = 0 Ṙ(t) 4ν l R(t) 2σ st ρ l R(t) ) 3k Ebrahim Ghahramani CHALMERS / 56

45 Implementing Rayleigh-Plesset equation New member data In solidparticle class Description bubble radius temporal derivative of bubble radius equilibrium radius dissolved gas pressure equilibrium dissolved gas pressure bubble inside pressure a numerical variable in solving the equation a numerical variable in solving the equation In the solver R_ Rdt_ R0_ pg_ pg0_ pb_ F0Old_ F1Old_ Ebrahim Ghahramani CHALMERS / 56

46 Implementing Rayleigh-Plesset equation New member data In solidparticlecloud class Description liquid saturation pressure liquid surface tension coefficient In the solver psat_ sigma_ In trackingdata class Description interpolator for surrounding pressure In the solver pinterp_ Ebrahim Ghahramani CHALMERS / 56

47 Implementing Rayleigh-Plesset equation vi solidparticle.h vector U_; scalar R_; // -radius scalar Rdt_; // -r_dot scalar R0_; // initial equilibrium radius scalar pb_; // bubble inside pressure scalar pg0_; // dissolved gas pressure at equilibrium scalar pg_; // dissolved gas pressure scalar F0Old_; scalar F1Old_; Ebrahim Ghahramani CHALMERS / 56

48 Implementing Rayleigh-Plesset equation vi solidparticle.h const interpolationcellpoint<scalar>& nuinterp_; const interpolationcellpoint<scalar>& pinterp_; const interpolationcellpoint<scalar>& nuinterp, const interpolationcellpoint<scalar>& pinterp, inline const interpolationcellpoint<scalar>& nuinterp() const; inline const interpolationcellpoint<scalar>& pinterp() const; bool move(trackingdata&, const scalar); void initialequilibriumcondition(); Ebrahim Ghahramani CHALMERS / 56

49 Implementing Rayleigh-Plesset equation vi solidparticlei.h const interpolationcellpoint<scalar>& nuinterp, const interpolationcellpoint<scalar>& pinterp, nuinterp_(nuinterp), pinterp_(pinterp), inline const Foam::interpolationCellPoint<Foam::scalar>& Foam::solidParticle::trackingData::pInterp() const { return pinterp_; } // ************************************************************************* // Ebrahim Ghahramani CHALMERS / 56

50 Implementing Rayleigh-Plesset equation vi solidparticle.c scalar nuc = td.nuinterp().interpolate(cpw); scalar pc = td.pinterp().interpolate(cpw); U_ = (U_ + dt*(dc*uc + (1.0 - rhoc/rhop)*td.g()))/(1.0 + dt*dc); #include "RP_Solver.H" // ODE solution algorithm for R-P equation void Foam::solidParticle::initialEquilibriumCondition(){ Rdt_ = 0.0; R_ = 0.5*d_; R0_ = R_; pg0_ = 0.0; pg_ = 0.0; } // ************************************************************************* // Ebrahim Ghahramani CHALMERS / 56

51 Implementing Rayleigh-Plesset equation vi solidparticlecloud.h scalar tinjend_; scalar psat_; // saturation pressure scalar sigma_; // surface tenstion inline scalar mu() const; inline scalar psat() const; inline scalar sigma() const; Ebrahim Ghahramani CHALMERS / 56

52 Implementing Rayleigh-Plesset equation vi solidparticlecloudi.h inline Foam::scalar Foam::solidParticleCloud::pSat() const { return psat_; } inline Foam::scalar Foam::solidParticleCloud::sigma() const { return sigma_; } // ************************************************************************* // Ebrahim Ghahramani CHALMERS / 56

53 Implementing Rayleigh-Plesset equation vi solidparticlecloud.c tinjend_(dimensionedscalar(particleproperties_.lookup("tinjend")).value()), psat_(dimensionedscalar(particleproperties_.lookup("psat")).value()), sigma_(dimensionedscalar(particleproperties_.lookup("sigma")).value()) const volscalarfield& nu = mesh_.lookupobject<const volscalarfield>("nu"); const volscalarfield& p = mesh_.lookupobject<const volscalarfield>("p"); interpolationcellpoint<scalar> nuinterp(nu); interpolationcellpoint<scalar> pinterp(p); td(*this, rhointerp, UInterp, nuinterp, pinterp, g.value()); ptr1->initialequilibriumcondition() Cloud<solidParticle>::addParticle(ptr1); Ebrahim Ghahramani CHALMERS / 56

54 Implementing Rayleigh-Plesset equation Compile the solver wmake Go to the test case directory : cd $WM_PROJECT_USER_DIR/run/testCase Have a look at the injector dictionary vi constant/particleproperties rhop rhop [ ] ; e e [ ] 0.2; mu mu [ ] 0.05; posb posb [ ] ( ); db db [ ] ; UB UB [ ] (0 0 0); tinjstart tinjstart [ ] 0.075; tinjend tinjend [ ] 0.085; psat psat [ ] 2340; sigma sigma [ ] 0.07; gamma gamma [ ] 1.4; Ebrahim Ghahramani CHALMERS / 56

55 Implementing Rayleigh-Plesset equation Run the solver myhybridsolver > log& Have a look at the log file to check if particles were injected. Load the results in paraview touch testcase.foam paraview File Open testcase.foam Visualize the bubble: Mesh regions: lagrangian/defaultcloud Glyph: Type Sphere, Mode Scalar,Scalar d, Edit Set Scalar Factor 0.5 Ebrahim Ghahramani CHALMERS / 56

56 Implementing Rayleigh-Plesset equation Problem description Channel dimension: 10m*1m Inlet pressure: 2.2 kpa Outlet pressure: 1.5 kpa Bubble initial diameter: 4 mm Ebrahim Ghahramani CHALMERS / 56

57 Implementing Rayleigh-Plesset equation Result Ebrahim Ghahramani CHALMERS / 56

CFD with OpenSource software

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