Pritom_6254999_first_asgn
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Florida International University *
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Aerospace Engineering
Date
Feb 20, 2024
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Uploaded by MajorIronJaguar34
Name:Pritom Kumar Saha PID:6254999 %%First: viscosity = (1.2*10^(-3)); %Ns/m^2 max_frw_bld_velo = 110/100; %m/s aorta_rad =(2.5/2)/100; %m aorta_rad_sq = ((2.5/2)^2)/10000; %m^2 dp_dz = (-max_frw_bld_velo*4*viscosity)/aorta_rad_sq; % wall_shear_strs = (-dp_dz/(2))*aorta_rad; Answer: dp_dz = -33.7920 wall_shear_strs = 0.2112 Justify your choice of boundary conditions in no more than two sentences. No slip and r = o at the center conditions. Initially we consider no change in velocity to theta, fully developed and Vr=0. velocity is zero at the outer most inner radius (r=R)).
%%Next: Pressure = -33.7; Velocity_over_time = []; Pressure_drop_over_time = []; Blood_density = 1060; Heart_rate = 2*pi*(50/60); for cardiac_cycle_t = 0:.01:1.2 k = (-Pressure/(Blood_density*Heart_rate))*sin(Heart_rate*cardiac_cycle_t); p = Pressure*cos(Heart_rate*cardiac_cycle_t) Velocity_over_time = [Velocity_over_time,k]; Pressure_drop_over_time = [Pressure_drop_over_time,p]; end
; figure(1) subplot(2,1,1) plot(0:.01:1.2,Velocity_over_time); title(
'velocity as function of time'
); subplot(2,1,2) plot(0:.01:1.2,Pressure_drop_over_time); title(
'pressure drop as function of time'
);
Use the Womersley number: Matlab code: womersley_number = .0125*sqrt(2*pi*(50/60)*(1060/.0012)) womersley_number = 26.8826 The Womersley value is more than 20 so it is inertia dominated. This number represents pulsatile flow. Alpha (womersley_number) is big so it is high pulsatility. So our assumptions was valid that inertia dominated.
Is the velocity a function of radius in this analysis? Why or why not Velocity is not the function of radius as we used harmonic oscillation. Which is related to the angular velocity. Angular velocity W = 2*pi*f So depended on frequency.
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What is the volumetric flow rate over one cardiac cycle:
Matlab approach: Velocity_over_time = [Velocity_over_time,k]; %% Volumetric Flow Rate radius = 0:.001:.0125; for x = 1:length(radius) Q(x) = (mean(Velocity_over_time))*(pi*radius(x)^2) end Total = sum(Q); figure(2) plot(radius,Q) title(
'Volumetric Flow over radius'
) xlabel(
'radius (m)'
) ylabel(
'Volumetric flow m^3/s'
) axis([0 max(radius) -.01 .01]) Y = sprintf(
'Integrate Volumetric flow: %d'
,Total); disp(Y) Output: Integrate Volumetric flow: 2.613428e-23 Which is close to zero. What does this tell you about the validity of the assumed pressure drop. Volumetric flow rate and pressure drop are dependent on each other. As volumetric flow is zero so pressure drop should be less. Our assumed pressure drop is not correct.
How could you approximate a better pressure drop while still assuming a simple harmonic oscillator Adding steady term we approximate a better pressure drop while still assuming a simple harmonic oscillator. Discuss how your analyses might change if you were analyzing blood flow in an arteriole In arteriole Womersley number is small and have small radius. So, when Womersley number is small the viscous force dominates over inertia force.So there will be a change in the pressure and velocity relationship. %%Finally use the analytical harmonic analysis that we discussed in class to plot the velocity profile over the course of one complete cardiac cycle Also plot the pressure drop and wall shear stress over this same cycle Has been calculated previously Matlab approach: density = 1060; viscosity = .0012 ; n = 1; %i =1 radius = .0125 ; f = 2*pi*(50/60); VQR = pi*15/100*radius^2; lambda = sqrt(i^3*n*f*(1060/viscosity));
t = 0:0.01:1.2; radius_chnage = linspace(0,radius,length(t));
%psi_store = []; for x = 1:length(t) psi(x) = real(((VQR*(viscosity*(lambda*radius)^2))/(pi*radius^4*exp(i*n*f*t(x))))/(1-
(2*besselj(1,lambda*radius))/(lambda*radius*besselj(0,lambda*radius)))); V_change(x) = 110/100 + (real((psi(end)/(viscosity*lambda.^2))*(1-
(besselj(0,lambda*radius_chnage(x))./besselj(0,lambda*radius)))*exp(i*n*f*t(x)))); Pressure(x)= -33.7 + real(psi(end))*exp(i*n*f*t(x)) strs(x)= real(abs(psi(end))/lambda*(besselj(1,lambda*radius))/(besselj(0,lambda*radius))*exp(i
*n*f*t(x))) end figure(1) plot(0:0.01:1.2,V_change) title(
'Velocity Profile (analytical)'
) figure(2) plot(0:0.01:1.2,Pressure) title(
'Pressure drop profile (analytical)'
) figure(3) plot(0:0.01:1.2,strs) title(
'wall stress (analytical)'
)
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What is the average wall shear stress over time for this analysis? How does this compare to the wall shear stress for the Poiseuille flow case? Average shear stress is :1.2873e-03 Which less than the one I got from the initial pouiselli calculation. Pouiselli is derive the value in a simplified way ignoring other parameters and it measure close to wall while analytical harmonic is more of close to center. In one sentence, how could you more accurately model blood flow in the patient? Realistical adding parameters for more geometric complexity, radial velocity and other direction as well as we can think about Casson model(describe the flow of visco-elastic fluids. This model has a more gradual transition from Newtonian to. the Yield region) Grad addition (
10 points
) First of all
, due to less availability of MRI module in LLMICs and contradiction due to MRI value as data acquisition modality, this paper includes a patient-specific CFD study(helps to visualize flow and pressure field) of a single coarctation of the aorta patient and data was collected using CT Angiography (CTA) Doppler echocardiography ( broadly available in LLMICs). Secondly, as there was not post repair CTA data
so they take the pre repair CTA data to segment 3D aorta geometry and approximate the post-repair geometry state and a hypothetical “healthy” state with
the coarctation removed. Thirdly
, using the velocity plot from the doppler they calculate volumetric flow rate (Q= A*V) for three conditions and as there was no post CTA so diameter measurement of the coarctation in the post repair dataset served as geometry data points. Then
after meshing to create 3D geometry using ansys and performing analysis they showed in figure 8 CFD derived velocity and Echo derived velocity comparison and have almost some similarities. Finally
, as the pressure difference across the coarctation is an important and Doppler echocardiography still recorded a pressure difference in excess of the 20-mmHg standard for indicating the need for intervention however in case 2 predicted pressure less than measured value. Boundary condition: The Doppler TTE data was used to define the domain’s boundary conditions
and give a point of reference for the maximum velocity and pressure difference at the coarctation to which the CFD results could be compared. The inlet, at the aortic sinus, was assumed to have a circular cross section in order to simplify the application of the inlet boundary condition. They conduct a simplified zero-pressure outlet and peak systolic volumetric flow rate inlet boundary conditions, a grid independence study was conducted for each case using meshes that ranged from circa 0.5, 2, and 4 million cells. Steady-state, peak systolic volumetric flow rate boundary conditions were applied in the inlet and outlet patches(table 1).They also consider steady state and no slip conditions. constitutive properties
were that the flow was, laminar, incompressible, Newtonian. Computation analysis: In pre condition, they collect CTA and doppler data but for post conditions there was not CTA data. Pre condition CTA data used for geometry and boundary conditions and then parameters are applied in ansys for modelling 3D geometry. The pre- and post-repair Doppler echocardiography investigations provided velocity data over several cardiac cycles at locations of interest in the domain and was processed using simplified Bernoulli equation
for applying as boundary conditions. In accordance to equation(
Q =VA
), the velocity data at each inlet and outlet patch was multiplied by the respective patch area to calculate the relevant volumetric flow
rates which would later be used for boundary condition definition. They consider three case and each case geometry was meshed in ANSYS ICEM-CFD using tetrahedral cells and five prism boundary layers
. Finally they analyze and segmented three models and compare data for three cases compare in Table 1 and Table 2. In short from table 2 ratios indicate the severity of the coarctation in those region and, although there is significant improvement post-repair, there still existed a small stenosis.