Fluid Behavior in a Steady Flow Stenosis Model

Fluid Behavior in a Steady Flow Stenosis Model

Group M1

BE 310: Bioengineering Laboratory IV

May 3rd, 1999

Mukta Agrawal

Sonia Ahn

Bethany Gallagher

Catherine LaRocco

Adrian Shieh

Abstract

The purpose of the experiment was to characterize the steady flow behavior of Newtonian fluids through a model stenosis. The onset of turbulence was determined graphically; a plot of dimensionless pressure drop versus Reynolds number showed a linear relationship for Re < 1000, followed by a dramatic leveling. For Re > 1500, the dimensionless pressure drop is constant. A similar transition over the same Reynolds number range was seen for a plot of Pr2/Lv versus Reynolds number; for Re < 1500, the plot is approximately horizontal, showing that the flow is laminar, while for higher Reynolds numbers, the relationship is linear. Both plots strongly point to a turbulent transition at Reynolds numbers near 1500 in a stenosis. In comparison, the dye indicator was difficult to employ, and yielded inconsistent results for the critical Reynolds number, between 6292 and 10259 for trials run with water, and 5189 to 5962 for 5% sucrose.

For Reynolds numbers greater than 1000, the pressure drop was related to velocity by a power law relationship, approximately P  v2, predicted by Bernoulli’s equation. Fluid resistance, developed using a circuit analog for fluid flow, was calculated and plotted against Re, and showed a linear dependence for Re > 1000. Resistance was directly related to length and viscosity, and inversely related to stenosis diameter. Increasing the length from 45 mm to 90 mm increased resistance by 10-15%. A decrease in diameter from 6.5 mm to 5 mm resulted in a 250-300% increase in resistance. An increase of 29% in viscosity resulted in a 1200-1500% increase in resistance.

Background

The transition point is the point at which laminar flow turns to turbulent flow. Geometric parameters affect where this transition takes place. In a long, cylindrical tube, the critical Reynolds number is typically taken as 2300. Fundamental relationships derived from fluid mechanics can be used to determine this transition point via changes in the flow behavior. The governing equations are the dimensionless continuity and Navier-Stokes equations. The total velocity has dimensionless axial and radial components. The equations assume an incompressible and Newtonian fluid; water and glucose solutions are good examples, but blood is not. This theory does not model blood flow entirely accurately.

Laminar viscous flow is modeled by the Haygen-Poiseuille equation in a long circular tube of constant cross section, which states that the pressure drop is proportional to the flow rate (and therefore velocity):

P = (8L/r4) Q = (8L/r2) v[1]

The simplified Bernoulli’s equation, on the other hand, neglects viscous losses and is frequently used to model relatively inviscid fluids at higher Reynolds numbers. According to the simplified Bernoulli’s equation, assuming two points located at the same height, the pressure drop is proportional to the difference of the squares of the upstream and downstream velocities:

P =  (v22 – v12) /2[2]

Plotting the pressure versus velocity curve, it becomes apparent that the relationship is a square law correlation, suggesting that the fluid is not behaving in a laminar, viscous manner, but rather turbulently, following Bernoulli’s equation. A plot of P versus (v2) should yield a straight line with a slope of ½, provided that the fluid is following the simplified Bernoulli’s equation. Deviations can be attributed to the factor that there are entrance and exit effects at the portals of the stenosis, as well as viscous loss effects that are normally neglected in Bernoulli’s equation. If you include head losses in the equation, it takes the form:

P =  (v22 – v12) /2 + KV2/2[3]

K is a constant that depends on the nature of the losses and effects in the system. For a sharp orifice, K can range from 0.2 to 0.5(6)

A stenosis is a narrowing of the lumen of a blood vessel, and some of the common causes are atherosclerosis, calcification or malformation of valves, and thrombosis caused by tissue trauma. Atherosclerosis, which is the most common form of arteriosclerosis, is the leading cause of non-valvular stenoses. In this disease, localized plaques, also known as atheromas, protrude in to the lumen of the artery and therefore reduce blood flow. The atheromas further impede blood flow to organs by serving as sites for thrombus formation. Thrombi are much more likely to develop with turbulent than with laminar flow. This is a viscous cycle for the body, as both turbulent flow and thrombi are to be avoided at all costs, yet one causes the other, to greater and greater degrees. Atherosclerosis is a disease that may lead to obstruction of coronary blood flow. This may result in a serious compromise of the heart’s ability to function as a pump as well as electrical properties essential to the body’s functioning. The cause of atherosclerosis is believed to result from damage to the endothelium caused by smoking, high blood cholesterol, and diabetes. There are many ways to treat atherosclerosis, including a rotational burr, balloon angioplasty, and stents, as shown in the picture below:

Another physiological application of the experiment may be seen in coarctation. Coarctation of the aorta is a constriction (narrowing) of a part of the aorta. This generally occurs close to the region where arteries to the head and neck arise. The constriction obstructs blood flow to the lower parts of the body. It causes blood pressure to increase above the coarctation, resulting in higher blood pressure in the upper part of the body as compared to the lower part of the body.

The left side of the heart works harder than usual when it tries to pump blood past the coarctation. This may cause the left ventricle to enlarge. If the coarctation is very severe, an infant's heart may fail resulting in rapid heart rate, rapid breathing and poor feeding. In less severe narrowing, the child may have no noticeable side effects except increased blood pressure in the arms and head. High blood pressure can damage other blood vessels in the body and head as well.

The age when the coarctation is repaired depends on the severity of the constriction and symptoms it causes the child. Children with coarctation require antibiotics (also called SBE prophylaxis) prior to any dental work or surgery on the month, bowel or bladder. This helps prevent the uncommon, but possible, occurrence of bacterial infection forming on the aortic lining in or near the coarctation(7).

Methods

Experimental Manometer

To analyze the pressure gradient across the stenosis models, it was necessary to construct a manometer that would allow for measurements of the relative pressure difference on either side of the narrowing. The stenosis model was set between two T-tubes that were connected to narrow catheter tubing. The tubes were then attached at the apex by a stopcock. This stopcock allowed for air to either be released or injected into the system. This facilitated the process of recording the pressure difference because the water level could be raised to avoid being obstructed by the opaque T-tubes. It also allowed for a greater range of measurements to be obtained, especially when there was a large pressure difference over the stenosis. The connections of the apparatus were made water tight through the use of parafilm and teflon tape. Unwanted air in the system gave rise to a greater measured pressure difference and increased error due to the volume of the trapped air in the tubing.

Stenosis Model

The stenosis models in the experiment were extremely idealized. The narrowing was uniformly cylindrical and cut from smooth, clear tubing. The cylindrical structure has a discrete length parameter, making it easy to calculate the Reynolds number. It was necessary for the tubing to be clear to see the behavior of the injected dye in the stenosis. Four different models were built using two different diameters of tubing, 5 mm and 6.5 mm, and two different lengths of 45 mm and 90 mm. The ends of the tubes of larger diameter model were then encased in a larger tube that fit over the connection with the T-tube. The models fit flush with the T-tube at the junction. The smaller diameter model was inserted into the T-tube. The interface between the T-tube and the stenosis model was also made water tight with the use of parafilm and tape.

Experimental Setup

The water tank was connected to the T-tube of the manometer by a 13mm diameter tube. Upstream from the stenosis was a Y-tube through which an extremely narrow catheter tube was run. The catheter tube was fitted onto a 200 L pipette tip. This tip was then joined to a stopcock and attached to a 20 cc syringe used to inject dye into the system. The catheter tube had a beveled tip that rested at the entrance of the stenosis model. Considerable effort was made to keep the catheter in the center of the tube to avoid having the ejected dye hit the walls of the model. The stenosis/manometer complex was inserted into the apparatus as previously mentioned. Downstream, the fluid emptied into a collection bucket.

Experimental Procedure

The procedure of this experiment was quite simple. For each stenosis model, the velocity of the flowing water was varied. For each velocity, the dye was injected into the system. The dye was then observed to determine whether the fluid exhibited laminar or turbulent behavior. The water was collected in a graduated cylinder for approximately 10 seconds to evaluate the actual flow rate. A wide range of flow rates was tested and the critical Reynolds number at the transition of flow behavior calculated. In addition to measuring the flow rates, the relative pressure difference was measured by recording the difference in height of the two columns of water in the manometer. The procedure was then repeated using a solution of 5% sucrose.

Results

In order to observe the flow behavior across a stenosis, dimensionless pressure was plotted against Reynolds number for all data. The dimensionless pressure drop was found by dividing P by v2, where v represents the velocity of fluid within the stenosis. The Reynolds number was found using the diameter of the stenosis as the length parameter. As shown in Figure 1, all the data points collapsed into one curve. The steeply sloped region at low Reynolds numbers corresponds to the laminar flow, and the flat area corresponds to the turbulent flow. Based on this plot, the transition from laminar to turbulent seems to occur between Reynolds number of 1000 – 2000.

Figure 1: Dimensionless P vs. Reynolds number

In order to see whether our data obeyed the Poiseuille’s law, Pr2/Lv vs. Reynolds number was plotted; if the flow is laminar, Pr2/Lv ought to be independent of Re and the plots would show a flat line (Figure 2). Our data points collapsed into two distinct curves representing different lengths. There seems to be a flat region at low Reynolds number, and as the Reynolds numbers increases beyond 2000, the curves display positive slopes. Therefore the transition point from laminar to turbulent flow can be approximated to be around 1000-2000 based on this plot.

Figure 2: Plot of Pr2/mLv vs. Reynolds number representing the Poiseuille’s law.

We also used the dye detection method to determine the critical Reynolds number for each stenosis model (Figure 3). However, the critical Reynolds numbers were found to be significantly greater that the values obtained from the dimensionless plot. The critical Re values for water were rather inconsistent varying from 6292 to 10259, but the critical Re values for 5% sucrose solution were more consistent and varied from 5189 to 5962.

(6.5x45) / (6.5x90) / (5x45) / (5x90)
Re critical (water) / 6292 / 7483 / 10259 / 8215
Re critical
(5% sucrose) / 5189 / 5962 / 5904 / 5617

Figure 3: Critical Reynolds numbers determined by the dye method

Next, the relationship between the Reynolds number and the resistance across the stenoses was studied. The resistance (kg/m3-s) was determined through dividing the pressure drop across the stenosis by the flow rate. When data points were plotted, the resistance was found to be linearly proportional to the Reynolds number for all cases (see Figure 4). However, the data points with low Reynolds number (~1500) deviated from the overall linear trend. By comparing different stenoses, the resistance positively related to the length of the stenoses and the viscosity of the fluid, and inversely related to the diameter. When the length was doubled from 4.5 to 9 cm, the resistance across the stenosis increased by 10-15%. The change in resistance due to change in the length of the stenoses was relatively small but statistically significantly different. When the diameter of stenoses was decreased 25% from 6.5 to 5mm, the resistance increased by 250-300%. The viscosity of the fluid was increased from 0.89 cps of water to 1.15 cps of 5% sucrose solution, and this 29% increase in viscosity led to 12-15 times greater resistance. The viscosity has the greatest influence on the resistance across the stenosis in our model and this shows that the fluid flow might be laminar.

Figure 4: Resistance vs. Reynolds number

The relationship between the pressure drop across the stenoses and the change velocity of the fluid was also analyzed. The pressure drop was found to be approximately proportional to the square of the velocity (see Figure 5). This demonstrates that our model follows the Bernoulli's equation. Equations for pressure drop in terms of velocity within the stenosis were determined through regression analysis and are shown in Figure 6. The pressure drop was greater for a given velocity when the length of stenosis was longer and the diameter was smaller.

Figure 5: Pressure drop versus velocity in the stenosis

(6.5x45) / (6.5x90) / (5x45) / (5x90)
(water) / 741.98v 1.89 / 882.67v 1.93 / 956.3v 1.94 / 1186.4v 1.82
(5% sucrose) / 750.89v 1.84 / 840.35v 1.90 / 936.1v 1.71 / 1232.3v 1.83

Figure 6: Equations for pressure drop (P) in terms of velocity

Based on above analysis, the pressure drop vs. (V2) was plotted to quantify the head losses (Figure 7). Slopes from the pressure drop vs. V2 were greater than ½, demonstrating that the head losses do exist in our model, and are significant. The loss was greater with smaller diameter stenoses due to the prominent entrance and exit effect, and the longer stenoses had greater loss due to the friction along the wall. The loss due to viscosity was negligible in our model. Experimental K was determined to range from 0.28 to 0.63.

Figure 7: Pressure drop vs. (V2)

Discussion

The goal of this experiment was to develop a simplified model of an arterial stenosis. The purpose of the experimental model was two-fold: (1) to analyze the resistance to flow generated by the stenosis, and (2) to predict the transition from laminar to turbulent flow, which causes numerous clinical complications for patients suffering from a stenosis. The use of dye to determine the critical Reynolds number, the method employed in the saccular aneurysm model, was applied. In addition, plotting the results and noting transitions in the flow behavior were also used to determine a transition region. As the results show, there is a significant variation in behavior at Reynolds numbers between 1000 and 2000. A plot of dimensionless pressure versus Re shows that, for Reynolds numbers less than 1500, the pressure is linearly related to Reynolds number; around Re = 1000, there is dramatic leveling, resulting in a nearly constant dimensionless pressure, regardless of Reynolds number (Figure 1). This behavior is indicative of a laminar to turbulent transition. Similar behavior was observed in a study modeling coronary artery stenosis (3). In addition, Pr2/Lv was plotted versus Reynolds number (Figure 2). The dimensionless term is derived from Poiseuille’s Law; if the fluid is behaving in a laminar and viscous manner, that term ought to be constant across a range of Reynolds numbers. Again, as the results support, the dimensionless parameter is relatively constant until it reaches Re between 1000 and 2000, when it becomes dependent on Reynolds number. This is a significant deviation from laminar flow behavior, which, in conjunction with Figure 1, strongly points to a critical Reynolds number range of 1000 to 2000. Discussing a critical Reynolds number is not instructive, since turbulence develops over an entire range of values, rather than suddenly occurring at a discrete value.

Another indicator that the fluid within the stenosis is not behaving in a laminar fashion is seen in a plot of pressure drop versus velocity. According to Hagan-Poiseuille flow, pressure should be proportional to flow rate, and therefore to velocity as well. However, the results clearly show that pressure drop exhibits an approximate square law dependence on velocity (Figures 5 & 6). This result can be derived from Bernoulli’s equation, which assumes inviscid flow. For turbulent flow conditions, the viscous losses can typically be neglected (5). The square of velocity dependence of pressure drop further indicates turbulent flow in the stenosis. Similar results have been found in other research studies (5). Interestingly, data points under Re = 1000 do not fit the trend, suggesting that behavior under these conditions is different. This would corroborate our initial assertion that there is a laminar to turbulent transition at Reynolds numbers near 1000 to 2000.

The dye results did not demonstrate turbulent flow within the stenosis until very high Reynolds numbers, exceeding 5000. One of the problems with the method was that we found it very difficult to see the dye and make a judgement to its behavior. Previous research done in this field has shown that turbulence generally occurs downstream, at the exit of the stenosis. Since our dye stream was focused in the stenosis, rather than downstream, it is quite plausible that turbulent flow had not developed in the stenosis proper. Our pressure-flow measurements, however, were taken in such a way that the data could apply to the flow behavior immediately at the exit, where turbulence tends to develop.