CFD Final Report - 3

24-718 Computational Fluid Dynamics MAY 11, 2012 Final Report / Page 1 of 25

Hans Mhelvin D. Cabrera

e-mail: [email protected]

Biomedical Engineering Department

Carnegie Mellon University

Pittsburgh, PA 15213

Blood Flow in the Aorta

at an Anastomosis with

Left Ventricular Assist

Device Outflow: My Two-

Dimensional Navier-Stokes

Solver vs. Published Results

Left ventricular assist devices (LVADs) are an option available for

people with heart muscle that can no longer support proper blood

circulation. The LVAD takes blood from the heart and injects it into an

anastomosis to the aorta. The hemodynamic environment created may

lead to serious medical complications, such as stagnant flow that can

lead to clot formation, which puts the patient at risk of subsequent

thromboembolism and stroke. Computational fluid dynamics (CFD) has

proved to be a valuable tool to assess the possible flow conditions that

would be created given a particular configuration of the LVAD outflow

graft to the aorta. I have developed a two-dimensional Navier-Stokes

solver based on the finite differencing with the Chorin projection

method to investigate blood flow at the anastomosis site, varying the

angle of entry for the LVAD cannula. Results have been demonstrated

to be qualitatively comparable with other LVAD anastomosis studies as

well as other CFD research perform on blood vessels.

Keywords: blood flow, aorta, anastomosis, left ventricular assist device,

Navier-Stokes, computational fluid dynamics

1 Introduction

For a patient with advanced heart failure, a left ventricular assist device (LVAD) can provide mechanical circulatory

support to augment or completely substitute for the pumping function of the weakened heart. The LVAD draws the

blood from the base of the left ventricle through an outflow cannula, and then delivers it into the aorta via a surgical

connection referred to as an anastomosis (See Fig. 1).


Fig. 1 Schematic of a LVAD anastomosis [Yang et al, 2009]

Although the LVAD anastomosis is effective in restoring sufficient blood flow for the body, it can also introduce

unnatural hemodynamic conditions in the aorta that then result in significant clinical dangers. Stagnant flow has

been associated with the formation of clots, which can obstruct the flow of blood and deprive downstream tissues of

oxygen [Folie et al, 1989]. Surgeons dealing with ventricular assist devices are concerned about the possibility of

those clots detaching from the blood vessel and travelling to smaller diameter arteries, such as those in the brain

(leading to stroke) or in the heart (leading to a heart attack) [Gross et al, 1999]. This relocation, known as

thromboembolism, becomes more likely in the presence of high wall shear stress [Folie et al, 1989]. On the

contrary, low shear stresses have been linked to atherosclerosis [Ku et al, 1985].

Computational fluid dynamics (CFD) simulations have been performed for a variety of flow scenarios in the

cardiovascular system to predict characteristics such as velocity, shear stress, and pressure. A subset of these studies

focuses on branched flow in blood vessels, where the branch can be another blood vessel, or a graft made of one or

more artificial materials [Perktold et al, 2002]. The largest of the arteries, the aorta, has been investigated with CFD

methods both in isolation [Liu et al, 2009] and in cases where there is a foreign source contributing flow to its

lumen, such as a cannula used in cardiopulmonary bypass [Pekkan et al, 2008]. Of the latter type of study, it is

possible to identify research that looks specifically at the hemodynamics around the site of anastomosis with an

LVAD outflow.

A common topic of these particular studies is to look at the effects of changing the position of the LVAD graft as it

is anastomosed onto the aorta. Kar et al performed a two-dimensional CFD study of aortic flow when comparing

LVAD outflow anastomosis to the descending aorta versus the ascending aorta, identifying regions of stagnant flow

[Kar et al, 2005]. May-Newman et al was interested in how flow in the aorta would be affected by changing the

angle of entry for the LVAD outflow graft [May-Newman et al, 2004]. These two works constitute the inspiration

and foundation for my individual study. Using a two-dimensional Navier-Stokes solver that I developed, I obtained

velocity, pressure, and shear stress results that were qualitatively compared to published results from groups

researching the same or similar problems.


2 My Two-Dimensional Navier-Stokes Solver

The 2D section of the aorta around the site of anastomosis with the LVAD graft is idealized as a rectangular domain

with rigid walls for both the LVAD cannula and the aorta. To facilitate comparison with published results, the

dimensions selected matched those in the work done by May-Newman et al, in which they used a rectangle that is 30

mm tall and 25 cm long, with a gap that is approximately 15 mm wide at the top wall, centered close to 10 cm from

the left end. The opening is where blood flow comes in from the anastomosed LVAD cannula. Furthermore, this

blood flow comes at a variable angle of entry θ, with the following values considered for this study: 30°, 45°, 60°,

and 90° [May-Newman et al, 2004]. Lastly, the rigid wall simplification ignores the compliance of the aorta, and

can result in higher shear stress results than are present in actuality, but it is a reasonable assumption for studies like

this which are qualitative and serve as a foundation for subsequent research.

The fluid of interest for this CFD study is blood, which is modeled as Newtonian, since in large arteries, shear rates

are sufficiently low such that viscosity can be deemed constant [Fung, 2005]. For simplicity, the fluid is also

considered isothermal and incompressible. A further simplification is made by assuming that blood flow is laminar,

and therefore, a Reynolds number of 1000 was used. As in the study performed by May-Newman et al, constant

velocities were prescribed at both inlets: 0.0589 m/s for the inflow directly from the heart, and 0.7368 m/s from the

LVAD pump. The former represents 2.5 l/min of blood flow from the aortic valve, while the latter represents 5

l/min from the LVAD outflow cannula, for a total of 7.5 l/min, a value that is in the upper part of the range of

normal cardiac outputs [Shahcheraghi et al, 2002; May-Newman et al, 2004]. This was chosen because it was

important to study the best-case scenario where the LVAD is providing a relatively high volumetric rate along with

some contribution from the heart; if there are any issues with the hemodynamics of this scenario, then it is very

likely that a change in design or surgical procedure is required. For clarity, it must be pointed out that the LVAD

outflow serves as an inflow to the fluid problem, since we are considering the aorta as the flow domain. Finally, the

no-slip condition was applied to all the walls of the tubes, also matching the conditions from the study by May-

Newman et al [May-Newman et al, 2004]. A summary diagram of the boundary conditions that were imposed in the

model is shown in Fig. 2.

Fig. 2 Summary of boundary conditions

2.1 Solution Procedure: Chorin Projection Method The objective of the study is to obtain values for velocities u

and v in the x- and y-directions, respectively, pressure p, and shear stress τyx across the flow domain. Towards this

end, the 2D, isothermal, incompressible Navier-Stokes equations are set up as follows (See Fig. 3):

(

)

( )

( )

( )

(

)

( )

( )

( )

( )


Fig. 3 The Navier-Stokes equations to be addressed by the solver. In order, the conservation of x-momentum,

conservation of y-momentum, and the continuity equations are listed.

My CFD solver is based on a finite difference scheme using the Chorin projection method, which marches in time

[Chorin, 1968]. In each iteration, the algorithm does the following: predicts the velocities un and v

n at the current

time step n, uses this velocity field to solve for the pressure pn+1

at the progressive time step n+1, and then uses this

pressure field as a correction to solve for the velocities un+1

and vn+1

at the progressive time step n+1. To avoid

oscillation of the pressure solution, a staggered grid is used, where the pressure is defined at the center of the cell,

the x-velocity u is defined at the midpoint of the eastern border of the cell, and the y-velocity v is defined at the

midpoint of the northern border of the cell. The numbering of the cells is selected to begin at 1, to facilitate

implementation of the solver into software, wherein matrix indices must begin at 1, not 0. Discretization of the

partial derivatives is based on formulae from Hirt et al, using a discretization mixture parameter of 0.9 to avoid

stability issues [Hirt et al, 1975]. The pressure is obtained by solving a Poisson (elliptical) equation using the Gauss-

Seidel with successive over-relaxation method that iterates with a relaxation factor of 1.7 and has a convergence

criterion where either the residuals are less than a tolerance of 0.001 or a maximum number of 100 iterations is

reached. A time step stabilization scheme is employed with a safety factor of 0.5, until the end time of 0.2 seconds

is reached. There were no body force terms included for either the x or y direction. The resultant velocities for a

regular grid were calculated using bilinear interpolation. My MATLAB script for the solver may be found in the

appendix.

2.2 Grid Convergence Study No qualitative difference in the velocity and shear stress results was detected when

doubling the number of grid points (from 43x13 to 86x26) in both the x and y directions (See Fig. 4). There was

however, a noticeable qualitative difference in pressure magnitude results between the coarse and fine meshes, with

the latter displaying a large hotspot with relatively high magnitude. The pressure for a given single cell of the coarse

mesh does not properly capture the pressures in its corresponding four cells in the fine mesh. Based on this finding,

it has been decided that only the pressure results from the finer mesh will be used for comparison with the published

results.

(a) coarse mesh results (b) fine mesh results

Fig. 4 Comparison of velocity vector, velocity magnitude, pressure magnitude and shear stress results

between a coarse mesh and a finer mesh.

3 Shear Stress Calculation

The shear stress in the direction of the length of the aorta on the face normal to its wall was calculated using

Stokes’ constitutive relations that rely on the assumption that the blood can be modeled as a Newtonian fluid, where


a constant dynamic viscosity μ of 0.003 kg/(m sec) was used (See Fig. 5). The derivatives in the calculation were

discretized using a second-order central difference.

(

)

Fig. 5 Stokes constitutive relation for shear stress used in this study

4 Simulation Parameters

The model was implemented in the MATLAB® programming environment. A simulation of 0.2 seconds using the

fine mesh lasted 2 h 45 min using an HP Pavilion dv7 Notebook PC with Intel® Core™ i5-2410M CPU @ 2.30

GHz with 6.00 GB RAM.

5 Results and Discussion

5.1 My Velocity Results vs. Published Results My velocity results (vector and magnitude plots) for all four

simulated angles of entry were qualitatively comparable with the study on LVAD insertion angle performed by

May-Newman et al [May-Newman et al, 2004]. In their 3D study, flow recirculation was most prominent at 90°,

decreasing as the angle of entry decreased. This trend was captured in my 2D study, where it was demonstrated that

the region of recirculating flow (shown as a light blue tail pointing to the left, back to the heart) kept getting larger

as one increased the angle of entry from 30° to 90° (See Figs. 6 and 7).

It must be noted that the results that they presented are for a prescribed velocity of 0 m/s directly from the heart and

0.7368 m/s from the LVAD. In contrast, I used a prescribed velocity of 0.0589 m/s directly from the heart (with the

same prescribed velocity that they used for the LVAD). Nevertheless, this difference between the two studies does

not compromise attempts at qualitative comparison, since it is evident from the velocity magnitude images at all four

angle studies that my prescribed velocity directly from the heart (marked in dark blue at the left wall) made a

negligible contribution to the overall velocity field anyway; the LVAD velocity is the dominant factor for

consideration.


Fig. 6 Comparison of published velocity vector field from May-Newman et al (top rows) with my velocity

vector arrow (middle rows) and magnitude (bottom rows) plots, across the four different angles of entry.

Fig. 7 A refinement of the mesh (doubling the nodes in both the x and y direction) for the velocity results at a

90° LVAD graft angle of entry for enhanced visualization.

5.2 My Shear Stress Results vs. Published Results Fig. 8 shows that my results for shear stress are qualitatively

comparable with the shear stress results from a study on coronary artery bypass graft (CABG) anastomosis

[Sankaranarayanan et al, 2005]. In their study, instead of an LVAD graft anastomosed onto the aorta, a bypass graft

is anastomosed to a coronary artery. They compared an in-plane surgical configuration with an out-of-plane

configuration, which is similar to the comparison made in our study between a 45° and 90° angle of entry. The

shear stress results for both studies are comparable because they both demonstrate an increase in maximum shear

stress when the graft inlet direction is more perpendicular to the main flow domain (i.e., the coronary artery in their

study and the aorta in our study).

One way that the two studies differ is in the relative size of the branch compared to the main stream. In our study,

the LVAD branch is a smaller diameter than the aortic flow domain, whereas in their study, the bypass branch is a

larger diameter than the coronary arterial flow domain. This meant that the outlet of their branch was partially

pinched, resulting in an elliptical shear stress region, whereas our branch outlet is more circular.


Another difference between the two studies is that in their study, the overall high shear stress distribution region

shrunk when they used the out-of-plane configuration. This was why the group ended up recommending the out-of-

plane configuration to surgeons, despite the higher maximum shear stress associated with it. Unlike theirs, our study

did not reveal a significant difference in shear stress region size when going from 45° to 90°. It is postulated that

their 3D CFD simulation is able to capture phenomena related to the concave geometry of the vessel wall that our

2D study is not.

Fig. 8 Comparison between the shear stress results of 45° LVAD entry (top left) with the in-plane CABG

anastomosis (bottom left) and between the 90° LVAD entry (top right) with the out-of-plane CABG

anastomosis (bottom right).

4.3 My Pressure Results vs. Published Results Fig. 9 demonstrates that my pressure results are qualitatively

comparable to pressure results from a study performed by Marsden et al on their new design of a Y-shaped Fontan

graft [Marsden et al, 2009]. For comparison with the Fontan graft, we look at one of the branches from a bifurcation

of the inferior vena cava before it connects to the pulmonary artery. The blood (travelling upwards) from the

inferior vena cava branch to the horizontal pulmonary artery represents a branched flow scenario similar to an

LVAD graft anastomosed at a 45° angle to the aorta, which was investigated in our CFD study (see velocity

distribution in top of figure). In the case of the inferior vena cavae anastomosed to the pulmonary artery, the

pressure distribution has a hot-spot concentrated around the wall directly opposite the side of flow entry. This result

is analogous to what was observed in the case of the LVAD graft anastomosed to the aorta; the hot-spot was also

focused at the wall across from the hole to where the LVAD was connected. In both cases, the maximum pressure

magnitude is due to incoming flow that makes contact with the opposing wall.


Fig. 9 (Top half): Comparison between the velocity magnitude plots of the inferior cavae anastomosed to the

pulmonary artery versus the LVAD anastomosed to the aorta at an angle of 45°. (Bottom half): Comparison

between the pressure magnitude plots of the inferior cavae anastomosed to the pulmonary artery versus the

LVAD anastomosed to the aorta at an angle of 45°. To properly make comparisons, consider that the two

visualizations are mirror images across the horizontal and vertical axes. Arrows have been placed to

facilitate comparison: the white arrow represents a horizontal inlet, the red arrow represents a branched,

angled inlet, and the blue arrow represents an outlet.


5 Conclusion

My velocity results for varying the angle of entry for the LVAD outflow graft as it inserts into the aorta have been

demonstrated to be qualitatively comparable to a similar study performed by May-Newman et al. Likewise, my

pressure results have been shown to be comparable to a study by Marsden et al performed on a Fontan graft, where

the anastomosis is between the inferior vena cava and the pulmonary artery. Finally, reasonable comparisons were

drawn between my shear stress results and those presented in a study by Sankaranarayanan et al on the anastomosis

involved in a coronary artery bypass graft. My 2D Navier-Stokes solver holds the promise of being useful not only

for further studies of the LVAD anastomosis with the aorta, but of potentially a variety of branched vessel

applications, similar to those used in comparison in this study [Bara et al, 2008; El Zahab et al, 2010; Dur et al,

2011; Ku et al, 2002, Lei et al, 2010; Hofer et al, 1996].

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Appendix

Matlab Script for My 2-D N-S Solver

%+24-718 CFD Final Project %Blood flow in Aorta at Anastomosis with LVAD Outflow %My 2-D N-S Solver %Overall plan: To march in (n+1) time steps %Code-friendly numbering used (i.e. i starts at 1, j starts at 1)

%Clear everything clear all close all clc

%Read Parameters t_end = 0.2; %in seconds i_max = 43; j_max = 13;


Xlength = 0.25; %x-length of domain (not counting boundary strip), in meters Ylength = 0.03; %y-length of domain (not counting boundary strip), in meters delta_x = Xlength/(i_max-1); %in meters delta_y = Ylength/(j_max-1); %in meters delta_t = 0.02; %for 1st time step, in seconds v_aorta = 0.0589; %aortic velocity, in meters/second v_LVAD = 0.7368; %LVAD pump velocity, in meters/second LVAD_angle = 90*pi/180; %lvad outflow graft angle of entry, in radians tau = 0.5; %safety factor for modifying time step (use after 1st time step) u_lid = 1; %normalized velocity of lid (for lid driven cavity problems) TOL = 0.001; %SOR tolerance eps = TOL; %same as TOL omega = 1.7; %SOR relaxation lambda = 0.9; %discretization mixture parameter in Hirt et al. 1975 iter_max = 100; %SOR max iterations, if TOL is not reached g_x = 0; %body force in x g_y = 0; %body force in y Re = 1000; %Reynolds # n_offset = 1; %need this because matrix indices start at 1, not 0

%Set t (time value) = 0, n (time index) = 0 t = 0; n = 0 + n_offset;

%Assign initial values to u,v,p UI = 0; %initial value for u VI = 0; %initial value for v PI = 0; %initial value for p

%Initialize u matrix (n = 0) u = zeros(i_max,j_max+1,1); for i = 1:i_max for j = 1:j_max+1 u(i,j,0+n_offset) = UI; end end

%Initialize v matrix (n = 0) v = zeros(i_max+1,j_max,1); for i = 1:i_max+1 for j = 1:j_max v(i,j,0+n_offset) = VI; end end

%Initialize p matrix (n = 0) p = zeros(i_max+1,j_max+1,1); for i = 1:i_max+1 for j = 1:j_max+1 p(i,j,0+n_offset) = PI; end end

%Define domain: solid (obstacle) = 0 vs. fluid = 1 % domain = zeros(i_max+1,j_max+1);


% for i = 2:i_max % for j = 2:j_max % if i == j % if i == 2 % domain(i,j) = 1; % elseif i == 3 % domain(i,j) = 1; % domain(i,j-1) = 1; % else % domain(i,j) = 1; % domain(i,j-1) = 1; % domain(i,j-2) = 1; % end % end % end % end

%Algorithm to calculate u,v at n+1 while t < t_end

%Step 1: Compute F(n), G(n) %according to (3.36-3.37) and (3.42, at the boundaries) %from the velocities at u(n), v(n)

% Set Boundary Conditions for u,v

%Set boundary values for u and v, assuming walls are stationary

%>>vertical walls for j = 2:j_max

%inflow on left wall u(1,j,n) = v_aorta; v(1,j,n) = 0;

%outflow on right wall u(i_max,j,n) = u(i_max-1,j,n); v(i_max+1,j,n) = v(i_max,j,n);

end

%>>horizontal walls for i = 2:i_max

hole_left_edge = floor((13/33)*i_max); hole_right_edge = floor((15/33)*i_max); %inflow hole and no slip on top wall if i >= 2 && i < hole_left_edge %no-slip u(i,j_max+1,n) = -u(i,j_max,n); v(i,j_max,n) = 0; elseif i >=hole_left_edge && i < hole_right_edge %inflow hole u(i,j_max+1,n) = v_LVAD*cos(LVAD_angle); v(i,j_max,n) = -v_LVAD*sin(LVAD_angle);


elseif i >=15 && i <= i_max %no-slip u(i,j_max+1,n) = -u(i,j_max,n); v(i,j_max,n) = 0; end

%no-slip on bottom wall u(i,1,n) = -u(i,2,n); %"averaged" v(i,1,n) = 0;

end

%define specific domain conditions

% Compute F,G

F = zeros(i_max,j_max+1); G = zeros(i_max+1,j_max);

%F & G boundary values

for j = 2:j_max F(1,j) = u(1,j,n); F(i_max,j) = u(i_max,j,n); end

for i = 2:i_max G(i,1) = v(i,1,n); G(i,j_max) = v(i,j_max,n); end % % [u,v,F,G] = BoundaryCellValues2(i,j,u,v,F,G,domain,i_max,j_max);

%Compute F(n)

d2u_dx2 = zeros(i_max,j_max+1); d2u_dy2 = zeros(i_max,j_max+1); du2_dx = zeros(i_max,j_max+1); duv_dy = zeros(i_max,j_max+1);

for i = 2:i_max-1 for j = 2:j_max

d2u_dx2(i,j) = (... u(i+1,j,n) ... - 2*u(i,j,n) ... + u(i-1,j,n) ... )... /(delta_x^2);

d2u_dy2(i,j) = (... u(i,j+1,n) ... - 2*u(i,j,n) ... + u(i,j-1,n)...


)... /(delta_y^2);

du2_dx(i,j) = (1/delta_x)*(... (((u(i,j,n) + u(i+1,j,n))/2)^2) ... - (((u(i-1,j,n) + u(i,j,n))/2)^2) ... ) ... + (lambda/delta_x)*(... ((abs(u(i,j,n) + u(i+1,j,n)))/2)... *((u(i,j,n) - u(i+1,j,n))/2) ... - ((abs(u(i-1,j,n) + u(i,j,n)))/2)... *((u(i-1,j,n) - u(i,j,n))/2) ... );

duv_dy(i,j) = (1/delta_y)*(... ((v(i,j,n) + v(i+1,j,n))/2)... *(((u(i,j,n) + u(i,j+1,n))/2)) ... - (((v(i,j-1,n) + v(i+1,j-1,n))/2))... *((u(i,j-1,n) + u(i,j,n))/2) ... ) ... + (lambda/delta_y)*(... ((abs(v(i,j,n) + v(i+1,j,n)))/2)... *((u(i,j,n) - u(i,j+1,n))/2) ... - ((abs(v(i,j-1,n) + v(i+1,j-1,n)))/2)... *((u(i,j-1,n) - u(i,j,n))/2) ... );

F(i,j) = u(i,j,n) ... + delta_t*(... (1/Re)*(d2u_dx2(i,j) + d2u_dy2(i,j)) ... - (du2_dx(i,j)) ... - (duv_dy(i,j)) ... + g_x ... ); end end

%Compute G(n)

d2v_dx2 = zeros(i_max+1,j_max); d2v_dy2 = zeros(i_max+1,j_max); duv_dx = zeros(i_max+1,j_max); dv2_dy = zeros(i_max+1,j_max);

for i = 2:i_max for j = 2:j_max-1

d2v_dx2(i,j) = (... v(i+1,j,n) ... - 2*v(i,j,n) ... + v(i-1,j,n)... )... /(delta_x^2);


d2v_dy2(i,j) = (... v(i,j+1,n) ... - 2*v(i,j,n) ... + v(i,j-1,n)... )... /(delta_y^2);

duv_dx(i,j) = (1/delta_x)*(... ((u(i,j,n) + u(i,j+1,n))/2)... *(((v(i,j,n) + v(i+1,j,n))/2)) ... - (((u(i-1,j,n) + u(i-1,j+1,n))/2))... *((v(i-1,j,n) + v(i,j,n))/2) ... ) ... + (lambda/delta_x)*(... ((abs(u(i,j,n) + u(i,j+1,n)))/2)... *((v(i,j,n) - v(i+1,j,n))/2) ... - ((abs(u(i-1,j,n) + u(i-1,j+1,n)))/2)... *((v(i-1,j,n) - v(i,j,n))/2) ... );

dv2_dy(i,j) = (1/delta_y)*(... (((v(i,j,n) + v(i,j+1,n))/2)^2) ... - (((v(i,j-1,n) + v(i,j,n))/2)^2) ... ) ... + (lambda/delta_y)*(... ((abs(v(i,j,n) + v(i,j+1,n)))/2)... *((v(i,j,n) - v(i,j+1,n))/2) ... - ((abs(v(i,j-1,n) + v(i,j,n)))/2)... *((v(i,j-1,n) - v(i,j,n))/2) ... );

G(i,j) = v(i,j,n) ... + delta_t*(... (1/Re)*(d2v_dx2(i,j) + d2v_dy2(i,j)) ... - (duv_dx(i,j)) ... - (dv2_dy(i,j)) ... + g_y ... ); end end

%Step 2: Solve the Poisson ("pressure") equation (3.32) %for the pressure p(n+1). This is an elliptic equation.

%>>Used my own Gauss-Seidel SOR (edited solver from HW 5, Problem 1)

%Poisson Step 1: Mesh set up N = iter_max; %max # of iterations, it_max TOL = eps; %tolerance

%Initialize [A],{g},{w},[u]

num_int_pts = (i_max-1)*(j_max-1); %number of internal points


A = zeros(num_int_pts,num_int_pts); g = zeros(num_int_pts,1); w = zeros(num_int_pts,1); p(:,:,n+1) = zeros(i_max+1,j_max+1);

%Incorporate BCs

%Pressure Field BC's for j = 2:j_max p(1,j,n+1) = p(2,j,n+1); %left wall p(i_max+1,j,n+1) = p(i_max,j,n+1); %right wall end

for i = 2:i_max p(i,1,n+1) = p(i,2,n); %bottom wall p(i,j_max+1,n+1) = p(i,j_max,n+1); %top wall end

% %Poisson Step 2: For i = 1,...,(n-1) -> x(i) = a+i*delta_x % x = zeros(1,n-1); % for i = 1:n-1 % x(i) = a+i*delta_x; %x values for internal nodes % end % % %Solver Step 3: For j = 1,...,(m-1) -> y(i) = c+i*delta_y % y = zeros(1,m-1); % for j = 1:m-1 % y(j) = c+j*delta_y; %y values for internal nodes % end

%Poisson Step 4: For i = 1,...,(n-1), For j = 1,...,(m-1), w(i,j) = 0; %(Commented out because w(i,j) is already initialized as w(l) in InitMod) % w = zeros(n-1,m-1); % for i = 1:n-1 % for j = 1:m-1 % w(i,j) = 0; % end % end

%Poisson Step 5: Set q = (delta_x/delta_y)^2 & mu = 2*(1+q) (from Diff.

Eq. (4B))

q = (delta_x/delta_y)^2; mu = 2*(q+1);

%Poisson Step 6: Assemble [A] & {g} (using #5)

%Build [A] and {g} using LHS of difference equation (4B) %LHS = mu*w(i,j)-(w(i+1,j)+w(i-1,j))-q*(w(i,j+1)+w(i,j-1))

for j = j_max-1:-1:1

for i = 1:i_max-1


point_num = (i) +(j_max-1-(j)) *(i_max-1); %Point number

%initialize LHS, a 4x5 matrix describing the 5 w terms on the %the left hand side of the difference equation (4B) LHS = zeros(4,5);

%Row 1: The coefficient of each of the 5 w terms on the LHS % of the difference equation (4B), from left to right LHS(1,1) = mu; LHS(1,2) = -1; LHS(1,3) = -1; LHS(1,4) = -q; LHS(1,5) = -q;

%Row 2: The i index of each of the 5 w terms on the LHS % of the difference equation (4B), from left to right LHS(2,1) = i; LHS(2,2) = i+1; LHS(2,3) = i-1; LHS(2,4) = i; LHS(2,5) = i;

%Row 3: The j index of each of the 5 w terms on the LHS % of the difference equation (4B), from left to right LHS(3,1) = j; LHS(3,2) = j; LHS(3,3) = j; LHS(3,4) = j+1; LHS(3,5) = j-1;

%Row 4: The l index of each of the 5 w terms on the LHS % of the difference equation (4B), from left to right LHS(4,1) = (i) +(j_max-1-(j)) *(i_max-1); LHS(4,2) = (i+1)+(j_max-1-(j)) *(i_max-1); LHS(4,3) = (i-1)+(j_max-1-(j)) *(i_max-1); LHS(4,4) = (i) +(j_max-1-(j+1))*(i_max-1); LHS(4,5) = (i) +(j_max-1-(j-1))*(i_max-1);

%Examine the i and j indices of each of the 5 w terms on the LHS for w_term = 1:5

coeff = LHS(1,w_term); %the coefficient of this w term i_index = LHS(2,w_term); %the i index of this w term j_index = LHS(3,w_term); %the j index of this w term l_index = LHS(4,w_term); %the l index of this w term

%if this w term's indices represent an interior point if i_index >=1 && i_index <= i_max-1 && ... j_index >=1 && j_index <= j_max-1

%the coefficient of this w term should be placed in A A(point_num,l_index) = ... A(point_num,l_index) + coeff;


%else this w term's indices do not represent an interior

point else

%this is a BC that should be moved to the g vector g(point_num,1) = ... g(point_num,1) + (-coeff)*p(i_index+1,j_index+1,n+1);

end

end

end

end

%Finish building {g} using RHS of difference equation %RHS = -((delta_x)^2)*f(x(i),y(j))

for j = j_max-1:-1:1 for i = 1:i_max-1

l_value = (i) +(j_max-1-(j)) *(i_max-1);

inhomog_term = (1/delta_t)*(... ((F((i+1),(j+1))-F((i+1)-1,(j+1)))/(delta_x)) ... + ((G((i+1),(j+1))-G((i+1),(j+1)-1))/(delta_y)) ... );

g(l_value,1) = g(l_value,1) +... (-(delta_x^2))*(inhomog_term);

end end

%Poisson Step 7: Solve [A]{w} = {g} using Gauss-Seidel method with SOR

%Gauss-Seidel Step 1: Set iteration = 1 iter = 1;

%Gauss-Seidel Step 2: While iteration <= N, do Steps 3-6 w_old = zeros(num_int_pts,1);

while iter <= N

%Gauss-Seidel Step 3: For i = 1,l, apply Gauss-Seidel formula

for i = 1:num_int_pts %l = num_int_pts

A_w = 0; A_w_old = 0;


for j = 1:i-1 A_w = A_w + A(i,j)*w(j,1); end

for j = i+1:num_int_pts A_w_old = A_w_old + A(i,j)*w_old(j,1); end

w(i,1) = (1-omega)*w_old(i,1)+ ... (omega/A(i,i))*(g(i,1) - A_w - A_w_old);

end

%Gauss-Seidel Step 4: If norm of difference between current and old %numerical solution is less than the prescribed tolerance, %then stop iterating sum_w_wold = 0; for index = 1:size(w,1) sum_w_wold = sum_w_wold + (w(index)-w_old(index))^2; end

norm_w_wold = (sum_w_wold)^0.5;

if norm_w_wold < TOL break %Gauss-Seidel Step 7: stop the while loop and then output

{w} end

%Gauss-Seidel Step 5: Increment the iteration counter iter = iter+1;

%Gauss-Seidel Step 6: For i = 1,...,l set wold = wi for i = 1:num_int_pts w_old(i,1) = w(i,1); end

end

%Fill in internal nodes of u using w

w_row = 1; for u_col = j_max:-1:2 for u_row = 2:i_max % p(u_row,u_col,n+1) = 0; p(u_row,u_col,n+1) = w(w_row,1); w_row = w_row+1; end end

%%%

% p(:,:,n+1) = ... % p_progressive; %pressure at next time step (3D Matrix)


%Step 3: Compute the new velocity field u(n+1), v(n+1) %using (3.34), (3.35) %with the pressure values p(n+1) from Step 2.

%>>>Compute u for n+1 for i = 2:i_max-1 for j = 2:j_max u(i,j,n+1) = F(i,j) - ... (delta_t/delta_x)*(p(i+1,j,n+1)-p(i,j,n+1)); end end

%>>>Compute v for n+1 for i = 2:i_max for j = 2:j_max-1 v(i,j,n+1) = G(i,j) - ... (delta_t/delta_y)*(p(i,j+1,n+1)-p(i,j,n+1)); end end v_current = v(:,:,n+1);

%Step 3b: Stability condition: adjust the time step delta_t %according to Eq. (3.50)

u_max_pre = max(u(:,:,n+1)); u_max = max(u_max_pre'); v_max_pre = max(v(:,:,n+1)); v_max = max(v_max_pre');

val1 = (Re/2)*(((1/(delta_x^2))+(1/(delta_y^2)))^-1); val2 = delta_x/abs(u_max); val3 = delta_y/abs(v_max);

delta_t = tau*min([val1 val2 val3]) %delta_t = 0.05

t = t + delta_t

n = n + 1 %march

end

%Interpolate u,v for regular grid (from staggered grid)

%x as a function of i for the u nodes x_u = zeros(1,i_max); for i = 1:i_max x_u(i) = (i-1)*delta_x; end


%y as a function of j for the u nodes y_u = zeros(1,j_max+1); for j = 1:j_max+1 y_u(j) = ((j-1)-(1/2))*delta_y; end

%Calculate u for each x,y that is in the middle of a cell in the domain u_plot = zeros(i_max+1,j_max+1,1); for n_index = 1:n for i_index = 2:i_max for j_index = 2:j_max

%Regular grid coordinate of interest x = x_u(i_index); y = y_u(j_index);

%For a coordinate (x,y) compute the coordinates of the cell

corners %by first determining the appropriate (i,j) of the upper-right

cell %corner, encompassing (x,y) i = ceil(x/delta_x)+1; j = ceil((y+(delta_y/2))/(delta_y))+1;

%coordinates of encompassing cell corners x1 = (i-1)*delta_x; x2 = (i)*delta_x; y1 = ((j-1)-(1/2))*delta_y; y2 = (j-(1/2))*delta_y;

%four staggered x-velocities u1 = u(i-1,j-1,n_index); u2 = u(i,j-1,n_index); u3 = u(i-1,j,n_index); u4 = u(i,j,n_index);

%approximate the desired velocity u(x,y) at position (x,y) using %bilinear interpolation u_plot(i_index,j_index,n_index) = (1/(delta_x*delta_y))*(... (x2-x)*(y2-y)*u1... +(x-x1)*(y2-y)*u2... +(x2-x)*(y-y1)*u3... +(x-x1)*(y-y1)*u4... );

end end end

%x as a function of i for the v nodes x_v = zeros(1,i_max+1); for i = 1:i_max+1 x_v(i) = ((i-1)-(1/2))*delta_x; end


%y as a function of j for the v nodes y_v = zeros(1,j_max); for j = 1:j_max y_v(j) = ((j-1))*delta_y; end

%Calculate v for each x,y that is at the middle of a cell in the domain v_plot = zeros(i_max+1,j_max+1,1); for n_index = 1:n for i_index = 2:i_max for j_index = 2:j_max

%Regular grid coordinate of interest x = x_v(i_index); y = y_v(j_index);

%For a coordinate (x,y) compute the coordinates of the cell

corners %by first determining the appropriate (i,j) of the upper-right

cell %corner, encompassing (x,y) i = ceil((x+(delta_x/2))/(delta_x))+1; j = ceil(y/delta_y)+1;

%coordinates of encompassing cell corners x1 = ((i-1)-(1/2))*delta_x; x2 = (i-(1/2))*delta_x; y1 = (j-1)*delta_y; y2 = (j)*delta_y;

%four staggered y-velocities v1 = v(i-1,j-1,n_index); v2 = v(i,j-1,n_index); v3 = v(i-1,j,n_index); v4 = v(i,j,n_index);

%approximate the desired velocity v(x,y) at position (x,y) using %bilinear interpolation v_plot(i_index,j_index,n_index) = (1/(delta_x*delta_y))*(... (x2-x)*(y2-y)*v1... +(x-x1)*(y2-y)*v2... +(x2-x)*(y-y1)*v3... +(x-x1)*(y-y1)*v4... ); end

end v_now = v(:,:,n_index); %debug v_plot_now = v_plot(:,:,n_index); %debug end

%2D velocity plot over domain

%Coordinates for x and y plot


x_plot = zeros(1,i_max+1); for i = 1:i_max+1 x_plot(i) = ((i-1)-(1/2))*delta_x; end

y_plot = zeros(1,j_max+1); for j = 1:j_max+1 y_plot(j) = ((j-1)-(1/2))*delta_y; end

%Coordinates for x and y plot x_plot = zeros(1,i_max+1); for i = 1:i_max+1 x_plot(i) = ((i-1)-(1/2))*delta_x; end

y_plot = zeros(1,j_max+1); for j = 1:j_max+1 y_plot(j) = ((j-1)-(1/2))*delta_y; end

%Quiver plot and save for gif animation for n_index = 1:n

figure(1) scale = 1; quiver(x_plot,y_plot,(u_plot(:,:,n_index))',(v_plot(:,:,n_index))',scale) title(['Interpolated 2-D velocity over x-y domain, @ n = '

num2str(n_index)]) xlabel('x coordinate') ylabel('y coordinate') axis([0 Xlength 0 Ylength]) axis image

%save each plot as a .gif file (for use in creating an animation) saveas(1,'testing1.pgm'); w1=imread('testing1.pgm'); Wg1(:,:,1,n_index)=uint8(w1); S1=['CFD Project1 ' num2str(n_index) '.gif']; imwrite(Wg1(:,:,1,n_index),S1);

figure(2) scale = 1; z_plot = ( ( ((u_plot(:,:,n_index)).^2) + ((v_plot(:,:,n_index)).^2) )

.^0.5 ); surf(x_plot,y_plot,z_plot') title(['Interpolated 2-D velocity over x-y domain, @ n = '




%Plot of shear stress magnitude scale = 1; uu = u_plot(:,:,n_index); vv = v_plot(:,:,n_index);

figure(3) dv_dx = zeros(i_max-1,j_max-1); du_dy = zeros(i_max-1,j_max-1);

for i = 2:i_max for j = 2:j_max dv_dx(i-1,j-1) = (vv(i+1,j)-vv(i-1,j))/(2*delta_x); du_dy(i-1,j-1) = (uu(i+1,j)-uu(i-1,j))/(2*delta_y); end end visc = 0.003; tau_yx = visc.*(dv_dx+du_dy); z_plot2 = tau_yx;

x_plot2 = x_plot(2:i_max); %exclude ends y_plot2 = y_plot(2:j_max); %exclude ends

surf(x_plot2,y_plot2,z_plot2') title(['Interpolated shear stress over x-y domain, @ n = '



%Plot of pressure figure(4) scale = 1; z_plot3 = p(:,:,n_index); surf(x_plot,y_plot,z_plot3') title(['Interpolated pressure over x-y domain, @ n = ' num2str(n_index)]) xlabel('x coordinate') ylabel('y coordinate') axis([0 Xlength 0 Ylength])


axis image


end

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CFD Final Report - 3