A Falling Drop scale. Symbols for the various physical quantities - PDF document


 DNS of Multiphase Flows — Simple Front Tracking DNS of Multiphase Flows — Simple Front Tracking Direct Numerical Simulations of In this lecture we apply our code to a few problems and examine its performance. We will, Multiphase specifically, look at Flows-7 
 • A falling drop and its collision with a no-slip wall • A rising bubble and its interaction with a no-slip wall Results and Tests • The Rayleigh-Taylor instability in a domain with full slip vertical walls Gretar Tryggvason DNS of Multiphase Flows — Simple Front Tracking DNS of Multiphase Flows We usually do our simulations in arbitrary computations units but report the results in non-dimensional units. For multi fluid flows we often encounter the following non dimensional numbers, where d and U stand for a length and a velocity A Falling Drop scale. Symbols for the various physical quantities follow the usual convention. µ Hitting a Wall Re = ρ dU Oh = p ρσ d Ohnsorge: Reynolds: µ N = ρ ∆ ρ gd 3 Ca = µU Archimedes: Capillary: µ 2 σ M = ∆ ρ gµ 4 We = ρ dU 2 Weber: Morton: ρ 2 σ 3 σ Eo = Bo = ∆ ρ gd 2 Fr = ρ U 2 Eötvös: Froude: σ ∆ ρ gd (or Bond) DNS of Multiphase Flows — Simple Front Tracking DNS of Multiphase Flows — Simple Front Tracking For a falling drop, as well as a rising bubble, the velocity can be written as a function of the various parameters specifying the We can select other repeated variables to obtain other problem, as well as time relationships, but in all cases the problem is specified by two non-dimensional numbers, plus the ratio of the U = f ( ρ d , µ d , ∆ ρ g, σ , d, ρ o , µ o , t ) densities and viscosities. The particular non-dimensional Notice that we include gravity multiplied by the density difference, numbers selected usually depend on the various limiting cases we want to explore. sine that is the effective buoyancy force. Using the diameter d , drop density, and density difference times gravity, as the repeated Sometimes we can ignore the dependency on the viscosity variables we find that the non-dimensional relationship is: or surface tension, in which case the dynamics depends s ! ρ U 2 ρ ∆ ρ gd 3 , ∆ ρ gd 2 , ρ d , µ d t 2 ∆ ρ g only on one non-dimensional number. If both can be ∆ ρ gd = f , µ 2 σ ρ o µ o ρ d ignored the problem is even simpler and is described by one non dimensional number being equal to a constant. Or Fr = f ( N, Eo, r, m, τ ) where s Fr = ρ U 2 N = ρ ∆ ρ gd 3 Eo = ∆ ρ gd 2 r = ρ d m = µ d t 2 ∆ ρ g τ = ∆ ρ gd, , , , , µ 2 σ ρ o µ o ρ d

DNS of Multiphase Flows — Simple Front Tracking DNS of Multiphase Flows This is the problem that we have been using to test our code, except here Simulation of a we will take the density and viscosity ratios to be larger. The physical and drop that falls numerical parameters are specified in the first few lines of the code onto a rigid wall %=============================================================== and bounces Lx=1.0; Ly=1.0; gx=0.0; gy=-100.0; sigma=10; % Domain size and rho1=0.1; rho2=2.0; m1=0.01; m2=0.2; % physical variables slightly unorth=0; usouth=0; veast=0; vwest=0; time=0.0; rad=0.15; xc=0.5; yc=0.7; % Initial drop size and location %-------------------- Numerical variables ---------------------- nx=32; ny=32; dt=0.001; nstep=200; maxit=200; maxError=0.001; beta=1.5; Nf=100; This gives the following non-dimensional numbers: N=256.5 Galileo Number Eo=1.71 Property ratios N = ρ ∆ ρ gd 3 = 2 ⇥ 1 . 9 ⇥ 1 ⇥ 100 ⇥ 0 . 3 3 ρ b / ρ l =20 = 256 . 5 µ 2 0 . 2 2 = 2 . 0 µ b /µ l =20 r = ρ d 0 . 1 = 20 Eörtvös Number ρ o Eo = ∆ ρ gd 2 = 1 . 9 ⇥ 100 ⇥ 0 . 3 2 = 0 . 2 A 32 by 32 grid. m = µ d = 1 . 71 0 . 01 = 20 σ 10 µ o DNS of Multiphase Flows DNS of Multiphase Flows For initial checks of the code, we can use relatively benign Looking at how the velocity and the marker function evolve in parameters, where we do not expect numerical difficulties time is usually the first step in examining the results. In many and the resolution required for convergence is modest. cases, however, we desire a more quantitative description of Then we ask: the evolution. This is useful for Does it look right? • Assessing the convergence of the solution as the numerical parameters, such as the grid resolution, are varied Is the solution as symmetric as it should be? • Quantifying how the solution changes as the physical Does rotating or flip the problem give the same solution? parameters describing the problem are changed Can we test some aspects of the code using analytical The diagnostic variables, or the quantities of interest, can be solutions? defined in several ways, but here we focus only on the simplest ones, such as the area of the drop and the location and Does the solution converge under grid refinement? velocity of its centroid DNS of Multiphase Flows DNS of Multiphase Flows R Centroid of drop The area of the drop should be constant since the flow is Z ⇣ ∂ x 2 ∂ x , ∂ y 2 X C = 1 Z x da = 1 Z ( x, y ) da = 1 ⌘ incompressible, and monitoring the area serves as a check da A A 2 A ∂ y on the accuracy of the computations. = 1 Z da = 1 I ⇣ ⇣ x 2 + y 2 ⌘ x 2 + y 2 ⌘ r n ds 2 A 2 To compute the area as well s several other quantities of The velocity of the drop centroid can be found by Z interest it is often useful to convert the elementary differencing the location of the centroid definition as a volume or area integral to a surface integral V C = d X C since surface integrals can be found with a high degree of dt accuracy. Thus, the area is given by: The velocity of the drop centroid can also be found by by integrating over the boundary, but this is usually less accurate. Area of drop Other elementary quantities of interest include the interface Z Z 1 da = 1 Z ⇣ ∂ x ∂ x + ∂ y da = 1 Z r · x = 1 I ⌘ A = da = x · n ds length which is found by 2 ∂ y 2 2 I S = ds

DNS of Multiphase Flows DNS of Multiphase Flows Here we use integration over the front to compute the area Results for three grids and the centroids. The code to do so is: 1 1 1 0.9 0.9 0.9 32 by 32 64 by 64 128 by 128 0.8 0.8 0.8 %================= DIAGNOSTICS ============================= 0.7 0.7 0.7 Area(is)=0; CentroidX(is)=0; CentroidY(is)=0; Time(is)=time; 0.6 The front, the velocity field, and the marker function at time 0.2 0.6 0.6 0.5 0.5 0.5 for j=1:Nf, Area(is)=Area(is)+... 0.4 0.4 0.4 0.25*((xf(j+1)+xf(j))*(yf(j+1)-yf(j))-(yf(j+1)+yf(j))*(xf(j+1)-xf(j))); 0.3 0.3 0.3 CentroidX(is)=CentroidX(is)+... 0.2 0.2 0.2 0.125*((xf(j+1)+xf(j))^2+(yf(j+1)+yf(j))^2)*(yf(j+1)-yf(j)); 0.1 0.1 0.1 CentroidY(is)=CentroidY(is)-... 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.125*((xf(j+1)+xf(j))^2+(yf(j+1)+yf(j))^2)*(xf(j+1)-xf(j)); 0.08 0.7 4 end Y-centroid Y-velocity 0.078 Drop area 0.6 2 0.076 CentroidX(is)=CentroidX(is)/Area(is);CentroidY(is)=CentroidY(is)/Area(is); versus time versus time versus time 0.5 0 0.074 0.072 0.4 -2 % plot(Time,Area,'r','linewidth',2); axis([0 dt*nstep 0 0.1]); 0.07 0.3 -4 0.068 % set(gca,'Fontsize',18, 'LineWidth',2) 0.066 0.2 -6 % T1=Time;A1=Area;CX1=CentroidX;CY1=CentroidY; 0.064 0.1 -8 % T2=Time;A2=Area;CX2=CentroidX;CY2=CentroidY; 0.062 0.06 0 -10 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 DNS of Multiphase Flows DNS of Multiphase Flows A bubble is the inverse of a drop, where a light fluid blob moves in a heavy liquid. We will make the domain twice as long, so that the bubble will have time to reach an A Rising approximate steady state before hitting the top wall Lx=1.0;Ly=2.0;gx=0.0;gy=-100.0; sigma=10; % Domain size and Bubble rho1=2.0; rho2=0.05; m1=0.1; m2=0.005; % physical variables unorth=0;usouth=0;veast=0;vwest=0;time=0.0; rad=0.15;xc=0.5;yc=0.3; % Initial bubble size and location Colliding with a %-------------------- Numerical variables ---------------------- nx=32;ny=64;dt=0.00125;nstep=400; Nf=100; maxit=200;maxError=0.001;beta=1.5; Wall N = ρ ∆ ρ gd 3 = 2 × 1 . 95 × 100 × 0 . 3 3 = 1 . 053 × 10 3 Galileo Number µ 2 0 . 1 2 Eo = ∆ ρ gd 2 = 1 . 95 × 100 × 0 . 3 2 Eörtvös Number = 1 . 755 10 σ DNS of Multiphase Flows DNS of Multiphase Flows Simulation of a bubble rising in 2 a narrow domain 1.8 and colliding 1.6 The front, the velocity field, and the marker function with the top rigid 1.4 TODO on three different grids at time 0.5 wall 1.2 1 Grid 1: nx=32; ny=64; dt=0.00125; nstep=400; Nf=100; 0.8 Grid 2: nx=2*32; ny=2*64; dt=0.5*0.00125; nstep=2*400; Nf=2*100; 0.6 Grid 3: nx=4*32; ny=4*64; dt=0.125*0.00125; nstep=8*400; Nf=4*100; N=1.053 x 10 3 0.4 Eo=1.755 0.1 2 4 0.2 0.09 Area versus time Centroid 1.8 3.5 ρ b / ρ l =40 0.08 0 1.6 3 0 0.2 0.4 0.6 0.8 1 versus 0.07 Centroid µ b /µ l =20 1.4 2.5 0.06 1.2 2 time Velocity 0.05 1 1.5 0.04 1 versus 0.8 A 32 by 64 grid. 0.03 0.5 0.6 0.02 time 0 0.4 0.01 -0.5 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5