Pre-Analysis & Start-Up
Pre-Analysis
In the Pre-Analysis step, we'll review the following:
- Governing Equations: We will review the governing equations that need to be solved in this problem.
- Boundary Conditions: We will go into more details about the boundary conditions that are applied in this problem.
Governing Equations
Before starting a CFD simulation, it is always good to take a look at the governing equations underlying the physics. In this case, although we have additional complexities such as pulsatile flow and non-newtonian fluids, the governing equations are the same as any other fluids problem. The most fundamental governing equations are the continuity equation and the Navier-Stokes equations. Here, let's have a quick review of the equations.
Continuity Equation:
However, as blood can be regarded as an incompressible fluid, the rate of density change is zero, thus the continuity equation above can be further simplified in the form below:
The Navier-Stokes Equation:
Unable to find DVI conversion log file.Boundary Conditions
Wall:
The easiest boundary condition to determine is the artery wall. We simply need to define the wall regions of this model and set it to “wall”. From a physical viewpoint, the “wall” condition dictates that the velocity at the wall is zero.
Inlet:
As we know, mammalian blood flow is pulsatile and cyclic in nature. Thus the velocity at the inlet is not set to be a constant, but instead, in this case, it is a time-varying periodic profile. The pulsatile profile within each period is considered to be a combination of two phases. During the systolic phase, the velocity at the inlet varies in a sinusoidal pattern. The sine wave during the systolic phase has a peak velocity of 0.5m/s and a minimum velocity of 0.1m/s. Assuming a heartbeat rate of 120 per minute, the duration of each period is 0.5s. This model for pulsatile blood flow is proposed by Sinnott et, al. [3] A figure of the profile within two periods is given below:
To describe the profile more clearly, a mathematical description is also given below:
Outlets:
The systolic pressure of a healthy human is around 120 mmHg and the diastolic pressure of a healthy human is around 80 mmHg. Thus taking the average pressure of the two phases, we use 100 mmHg (around 13332 Pascal) as the static gauge pressure at the outlets.