1. One of the major challenges in simulating multiphase flows is - PowerPoint PPT Presentation


 1. One of the major challenges in simulating multiphase flows is the advection of the marker function DNS of Multiphase Flows — Simple Front Tracking Direct Numerical that identifies the different fluids. There are two main ways to do so: We can update the marker function directly by solving an advection equation, or we can move the interface separating the different fluids Simulations of and then reconstruct the marker function from the location of the interface. Here we do the latter by Multiphase placing connected marker points at the interface, and then moving them with the flow. Flows-5 
 Advecting the Marker Function using Front Tracking (1 of 2) Gretar Tryggvason 2. When we use connected marker points to advect an interface for complex three-dimensional flows, it DNS of Multiphase Flows — Simple Front Tracking is important to use a data structure for the interface that allows for maximum flexibility in updating the Here we focus on a finite location of the points and their connectivity, yet is sufficiently simple so that it can be easily understood. region bounded by a Indeed, for such flows, the data structure can determine how easily the code is developed and new closed curve, representing a bubble or capabilities are added. For two-dimensional flows, on the other hand, essentially any data structure can a drop. t+ Δ t " be made to work, relatively easily. Here we will use a particularly simple data structure and represent the We will start by assuming interface by ordered marker points. While this results in a loss of flexibility, it works well for simple that the surface tension is zero and the viscosity of problems and provides a straightforward introduction to the use of connected marker point to advect both fluid is the same. t " interfaces. We will start by implementing the approach for flow where the only difference between the As for the flow solver, we Surface tension and unequal different fluid is their density and where surface tension is zero. We will also use a first order time will start using an explicit viscosities will be added later, integration, as we used for the flow solver. Surface tension and different viscosities, as well as higher first order method for the along with second order time time integration. integration order time integration, will be added later. 3. We will start by assuming that we are only dealing with one interface and that it is closed, thus DNS of Multiphase Flows — Simple Front Tracking representing a bubble or a drop embedded in a different fluid. Since the points are ordered, we use an Considered an ordered set of index L that takes on the values 1, 2, 3, and so on until we have enough points to discretize the entire connected points enclosing a closed region interface. We will take this number to be Nf. To simplify dealing with the first and the last points, we l-1 ! x f ( l ) = ( x ( l ) , y ( l )) , l = 1 , ...., N f introduce two ghost points on either end, so the total number of points is Nf + 2 and the interface is l ! 5 ! l+1 ! 4 ! To simplify the computations 3 ! resolved by points 2 to Nf + 1. The coordinate of point number 1 is set equal to the coordinate of the we introduce two ghost point 2 ! 1 ! so that the last point (N f +1) is last point resolving the interface, point Nf + 1, and the coordinate of point number Nf + 2 is set equal to N f ! the same as the first point and N f -1 ! point 2, the first point used to resolve the interface. By using the ghost points we can treat all the points N f -2 ! (N f +2) is equal to the second on the interface in the same way and then simply update the ghost points in a separate step. point Since the points are ordered, their distance p ( x ( l ) − x ( l − 1)) 2 + ( y ( l ) − y ( l − 1)) 2 ∆ s l,l − 1 = and other quantities are easily found

4. If the initial shape of the interface is a circle, its construction is particularly easy. Once we have decided DNS of Multiphase Flows — Simple Front Tracking where the center of the circle, xc and yc, is, where we start and in what direction to go, the coordinates Nf+1 Here we construct a circle, Nf+2 are easily found. We can obviously start anywhere and go either clockwise or counter clockwise, but here starting at the top and going 1 2 Nf counter clockwise. 3 we start directly above the center and go in the counter clockwise direction. The x coordinate is given as Nf-1 rad θ the x coordinate of the center minus the radius times the sine of the angle from the vertical axis, and the Notice that there are ghost points at both ends so that we run over (xc,yc) y coordinate is then given by the y coordinate of the center plus the radius times the cosine of the angle the points by: from the vertical axis. The angle increment is two pi divided by the number of points used to resolve the for l=2:Nf+1 …. end and then set the values at interface, Nf-1, and we increment it by l-1, where l is the point number, since we want to start with a zero 1 and Nf+2 angle so that the first point is directly above the center. %================== SETUP THE FRONT =================== Nf=100; xf=zeros(1,Nf+2);yf=zeros(1,Nf+2); uf=zeros(1,Nf+2);vf=zeros(1,Nf+2); tx=zeros(1,Nf+2);ty=zeros(1,Nf+2); for l=1:Nf+2, xf(l)=xc-rad*sin(2.0*pi*(l-1)/(Nf)); yf(l)=yc+rad*cos(2.0*pi*(l-1)/(Nf)); end 5. Since we are using two grids, the fixed regular structured two-dimensional grid where we solve the DNS of Multiphase Flows — Simple Front Tracking fluid equations and the one-dimensional moving grid, which we use to mark the interface and construct the density and the surface tension, we need to transfer information between those two grids. Transferring information between the front and the grid 6. The easiest way is to start with a point on the interface and ask what point on the fixed grid is closest. DNS of Multiphase Flows — Simple Front Tracking This, as we will emphasize again later, is a much simpler problem than asking what interface point is To transfer information between the front and the fixed grid, closest to a given grid point! Thus we try to always start with an interface point and find the closest point we need to find the grid point closest to the front on the fixed grid. How this is done is perhaps best explained by assuming a regular structured one- Left Boundary Front Location Right Boundary dimensional fixed grid with evenly spaced grid points. Here, the grid points, or the location where we 0 1 2 3 4 5 nx Control store each value are denoted by small circles and short vertical lines indicate the boundaries of the volume Horizontal Velocity (u) boundary control volumes. For simplicity we start by identifying the first point, located on the left boundary, as grid The index of the point to the left point number zero. For a one dimensional problem the front is a point and we can find the fixed grid Grid points of the front location is given by p where the point to the left of it by dividing the location of the front by the distance between each grid points and velocity is stored i = FLOOR( x f ( l ) / ∆ x ) + 1 taking the integer value. The standard way of converting a floating point value to an integer takes the The velocity at the front is interpolated to value to the left for positive values and the value to the right for negative values. We want the value to the points on the left and the right the left for all cases and therefore use the FLOOR command instead of the INT command.