Shear Stress In A Steady Confined Jet Biology Essay

Published:

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

Research has been conducted previously on the blood pumping action of the heart organ in the human body. The heart is responsible for the rhythmic contractions which pump the blood throughout the body. There are four major valves in the heart, two of which are based in the left heart (aortic and mitral valves) and two which are based in the right heart (pulmonary and tricuspid valves). These valves help the blood to be pumped in one particular direction, and thereby prevent any backflow or leakage of the blood.

Cardiovalvular regurgitation however, is a condition of the heart where the blood leaks in the wrong direction, and is ejected backwards across a ruptured heart valve. This improper functioning of the heart valve leads to a turbulent jet behind the valve. This regurgitant jet can damage the blood cells and platelets if the resulting shear stress exceeds 200N/m2.

Previous experiments have managed to set up laboratory apparatus that simulate the pulsatile beating action of the human heart. A plunger has been used to provide the periodic beating motion that would affect the fluid flowing through the heart. The fluid used in this laboratory setup has been air at a fixed pressure from a compressor, which manages to successfully simulate blood in its fluidic nature. Using apparatus and tools, the setup has allowed the measurement of velocities of the confined jet flow in the experiment. With this as the basis of readings, previous theses have employed hot wire anemometry techniques to obtain velocity profiles of the confined jet, which have then been used to calculate the shear stress due to the fluid jet.

A new aspect explored in this Final Year Project is the direct measurement of shear stress in the confined jet. This involved the more complicated technique of X-wire probe in the hot wire anemometry. The resulting voltage signals and their subsequent resolution into velocity and hence shear stress, allowed a comparison of the velocity and shear stress values obtained by this method, and those acquired from the single hot wire method utilized in the previous theses.

To establish a connection with the previous year theses, and hence enable a proper comparison of results, the single hot-wire readings were repeated for the steady jet flow, the values of which were processed and the following graphs were plotted and compared: U/Um (y-axis) against r/Ru (x-axis), Um/Uo (y-axis) against x/D (x-axis). Using the orifice as d = 7mm, and a steady jet at a constant pressure of p = 1.5 bar, the centreline mean velocity and radial velocities were measured along the jet at four distinct points; x = 8d, x = 10d, x = 12d, x = 15d.

Once a level of consistency in the results had been determined, the cross wire probe hot wire anemometry experiments were proceeded with, the results of which were then compared to those obtained previously. The results gave fairly accurate match

CHAPTER 1 - INTRODUCTION

Background

The heart is the organ responsible for pumping blood to the rest of the body by repeated, rhythmic contractions. The heart can be classified as two halves, a left and a right. Each half consists of two chambers-an atrium (low-pressure receiving chamber) and a ventricle (high-pressure pumping chamber. There are four major valves in the heart, two of which are based in the left heart (aortic and mitral valves) and two which are based in the right heart (pulmonary and tricuspid valves). Figure 1.1 shows the structure of the heart and the course of blood flow through the heart chambers (Burleson, 1993). Each heart cycle consists of four phases: the systolic (pumping) and diastolic (resting) phases, and two other shorter phases in which all the four valves are closed. When closed, these valves prevent any leakage or backflow of the blood.

However, any tear in or the improper closure of one of these valves leads to a heart disorder known as cardiovascular regurgitation (or heart valve leakage). Under this condition, blood is ejected backwards into the left atrium across a ruptured or improperly closed heart valve during the systolic phase (refer to Figure 1.2). Both cases are referred to as the lesion. As a result, one of the various types of turbulent jets will be formed. This jet can damage the blood cells and platelets if the resulting shear stress due to the regurgitating jet exceeds 200N/m2. Thus it is of great importance to be able to quantify the severity of this condition.

The amount of blood leaking backward through such a lesion during one cardiac cycle is called the regurgitant volume. The regurgitant volume is a quantity of interest in the assessment of cardiovalvular regurgitation as it represents a potential loss of cardiac output. Throughout the years, many methods have been developed in order to quantify the severity of mitral regurgitation. These include angiography, Colour Doppler flow mapping and Doppler Ultrasound (Low, 2005)

In angiography, the severity of mitral regurgitation is quantified by the amount of opacification of the left atrium produced by radiographic contrast medium injected into the left ventricle. Four different grades ranging from 1+ to 4+ (mere to severe regurgitation) exist. These grades depend on the degree of opacification of the left atrium, the apparent size of the left atrium, and the number of cardiac cycles required for maximal opacification. However, this method only measures the appearance of the flow instead of the regurgitant flow. Furthermore, this method is invasive, time-consuming, semi-quantitative and subjective (Burleson, et al., 1993).

With the advent of non-invasive Doppler ultrasound techniques, two-dimensional Colour Doppler echocardiography became the most common echocardiographic method to diagnose the severity of valvular regurgitation. This method provides real-time tomographic views of both the cardiac structures and the regurgitant jet, correlating maximal jet area and length with angiographic grade (Cape, 1989). However, subsequent studies showed that it is inaccurate to extrapolate regurgitant volume from either the visualised size of the jet or from an integrated jet volume. Although both the extrapolated volume and actual regurgitant volume occurs for certain settings, they are fundamentally unequal because the jet volume also includes the fluid entrained by the jet. Furthermore, both the jet size and the entrainment volume depend mainly on the following factors which include the driving pressure for a given regurgitant volume, the chamber constraint and the instrument settings which may vary among patients (Lee, 2002).

Using the basic principles of turbulent jet flow, Cape el at. (1989, 1990) developed an equation that obtains the regurgitant volume and the corresponding lesion size of the jet from the quantities measurable by Doppler echocardiography. The derived equation is a function of the maximal velocity(U0), a distal velocity(Um), and the distance between the two measured locations(x) (Cape et al., 1990). This equation is useful for calculating the regurgitant flow rates and volumes for free jets, seen in 60% of valvular regurgitation cases, as it is based on the principle of conservation of momentum.

Objectives and Scope

The main objective of this project is to directly measure the shear stress in a steady confined jet using velocity measurements made within the jet issued from the orifice, and compare the readings to the kinematic shear stress estimated from the equation based on the mean velocity at various values of x/D throughout the jet.

This involves two parts; the first being the measurement of velocity of the jet at various values of x/D using single hot wire, calibrated according to the King's Law, such that the kinematic shear stress can be calculated from the velocity readings. The second part involves obtaining values at the same x/D points in the steady confined jet, but by using a cross-wire hot probe, thereby enabling the conversion of the readings into the u and v components, hence resulting in the shear stress values for the steady confined jet.

Hot wire anemometry is the main experimental technique used for measuring the velocity of the confined flow in an accurate manner. Single hot wire is used to obtain the velocity profile, however further research has been conducted using X-wire in the hot wire anemometry for direct shear stress measurements.

CHAPTER 2 - LITERATURE SURVEY

2.1 Turbulent Free Jets

Figure 1 in the appendix exhibits that the region located near the orifice has a potential core where there is an undiminished mean velocity, U0. This core interacts with the surrounding stagnant fluid in the test section, thereby creating a turbulent shear layer that permeates the flow until a fully turbulent jet is developed (Joel, 2010).

Turbulent free jets have two characteristics which are crucial in the derivation of the equation to predict orifice size and turbulent shear stress (Jean, 1998). These characteristics are conservation of axial momentum and dynamic similarity of velocity profiles at different axial locations.

Conservation of axial momentum means that, the mass flow increases while the velocity decreases along the jet in such a way that axial momentum is conserved. The rate of axial momentum is constant at all streamwise axial locations of x.

Dynamic similarity implies that, the velocity profiles at different axial locations past the laminar core are self-preserved. This means that a normalised plot of U/Um vs r/Ru will coincide for all positions of x, where Um denotes the centre line velocity at a distance x from the orifice, and Ru refers to the radius where the velocity equals Um/2 (Joel, 2010).

The regurgitant jet can be modelled as a free jet, as about 30% of the cases of cardiovalvular regurgitation appear to be free jets (Burleson, 1993).

A jet is considered free when its cross-sectional area at the origin is less than one-fifth of the total cross-sectional flow area of the region through which is flowing (Burleson et al., 1995). It is not constrained by solid boundaries.

As shown in Figure 2.1,the region located near the orifice or the flow development region, has a potential core where there is an undiminished mean velocity, U0 (Rajaratnam, 1976). This core interacts with the surrounding stagnant fluid, creating a turbulent shear layer that permeates the flow until a fully turbulent jet is developed (Cape et al., 1990).

Turbulent free jets have two characteristics which are crucial in the derivation of the equation to predict orifice size and turbulent shear stress (Jean, 1998). These characteristics are conservation of axial momentum and dynamic similarity of velocity profiles at different axial locations.

Conservation of axial momentum means that, the mass flow increases while the velocity decreases along the jet in such a way that axial momentum is conserved. The rate of axial momentum is constant at all streamwise axial locations of x.

Dynamic similarity implies that, the velocity profiles at different axial locations past the laminar core are self-preserved. This means that a normalised plot of U/Um vs r/Ru will coincide for all positions of x, where Um denotes the centre line velocity at a distance x from the orifice, and Ru refers to the radius where the velocity equals Um/2 ( Cape et al., 1990).

2.2 Prediction Equation for Orifice Diameter

Cape et al., (1990) considered the turbulent Navier-Stokes equations for an incompressible fluid flow. By assuming no swirl and neglecting the body force terms, the governing Navier-Stokes equations are left with the axial and radial components only. In addition, the following conditions were assumed:

Velocity and stress gradients in the radial direction are much larger than those in the axial direction.

Viscous stresses are much smaller than the corresponding turbulent shear stresses.

Axial velocity fluctuations exceed radial velocity fluctutations.

Axial velocities are much higher than radial velocities.

The axial pressure gradient outside the jet is negligible.

The developing velocity profiles inside the jet at various axial positions are dynamically similar.

From the reduced equations, Cape et al., (1990) derived an equation for a free turbulent jet flow through a circular orifice as a function of Doppler measurable quantities, that is:

(2.1)

where Q0 is the regurgitant flow rate, and U0 and Um are the flow velocity at the orifice and distal centreline velocity at a distance x from the orifice respectively. The three quantities are all measurable with current available Doppler ultrasound equipments.

By equating the rate of transfer of axial momentum at the orifice to that at some distal position in the self-preserved region of the jet in which the curve Um/U0 vs r/x is independent of x, in accord with the conservation of momentum produces an equation as given in Cape et al., (1990):

(2.2)

where C is referred to as velocity constant and d the diameter of circular orifice, which is unknown (Winoto et al., 1991). Rajaratnam (1976) semi-empirically recommended a value of 6.3 for the constant C, hence the unknown orifice size can be predicted by:

(2.3)

2.3 Confined Jet

In many cases, a regurgitant flow can be considered as a turbulent free jet as the blood is forced through a small valve lesion by a high pressure gradient. However, it is more appropriate and practical to consider such regurgitant jets as confined rather than free turbulent jets as the atrial wall effects can be significant (Liu et al., 1997).

A jet is said to be a confined jet when its cross-sectional area is greater than one fifth of the cross-sectional area of the chamber into which it is flowing (Burleson et al., 1995).

When a confined jet is issued into a still fluid, it is constrained by the confining solid boundaries. Its characteristics differ from that of a free jet due to these confinements and presence of recirculating zones.

By considering its application to valvular regurgitation, the confined jet considered here is without a secondary stream (Figure 2.1). From the diagram, we can see that the development of the jet in the axial direction is separated into three distinct sections. The first region starts from the origin (i.e. orifice) to the end of the potential core (≈5d), where the flow velocity is nearly equal to the orifice velocity. The second region begins at the end of the potential core until where the jet width is almost equal to the duct diameter. This region is similar to the self-preserving region of the free turbulent jet, so normalised velocity profiles are dynamically similar. In the third region, the jet width equals to the duct diameter, and hence the confining walls become more significant and eventually pipe flow results.

The recirculating zone is formed outside the jet up to the end of the second region. This zone causes a loss in the momentum of the jet stream, which in turn, decreases the centreline axial velocity. This characteristic differentiates confined jet flow to free jet flow (Liu et al., 1997).

Generally, the flow pattern of confined jets depends on the ratio of orifice diameter to the pipe diameter, d/D and Reynolds number (Dealy, 1964). However, since the second region has a jet width smaller than the pipe diameter, the pipe wall boundary layer can be neglected. Therefore, the dependence on the Reynolds number can also be ignored. Hence, the following general equation is valid for measurements taken within this region:

where f is the velocity loss function to be determined.

A linear dependence for d/D was introduced based on the experiments done by Liu et al. (1997) in order to predict the orifice size for the confined jet.

After modification, the equation becomes:

Where C1 and C2 are empirical constants with generally accepted values of 6.3 and -1.4 respectively, and the above equation is valid for d/D<0.25.

Hence the general equation for predicting unknown orifice size in a fully developed turbulent jet becomes:

Or, in terms of regurgitant flow rate Q0:

(2.7)

2.4 Normalised Kinematic Turbulent Shear Stress

An expression is used to relate the distributions of the normalised kinematic turbulent shear stress ( with the measured mean velocities which is:

(2.8)

where f(

Rajaratnam et al. (1976) noted that the viscous term can be safely neglected for Reynolds number greater than 5000. As such, the following assumptions have been made in the derivation of this equation, which are the assumption of self-preservation and the neglect of normal stress and viscous diffusion terms.

By assuming that the maximum normalised kinematic turbulent shear stress is constant throughout the whole jet, the maximum turbulent shear stress can be estimated by multiplying the square value of the peak blood velocity with the density of blood to the present maximum /Um2 (Winoto et al., 1994), which corresponds to the following equation:

(2.9)

where Ub is the peak blood velocity, ρb is the density of blood(≈1056kg/m3).

Generally,Ub ranges from 2m/s to 5m/s. The jet peak blood orifice velocities depend on the type of regurgitation. For example, mitral regurgitation has a peak velocity of approximately 5m/s, tricuspid regurgitation has an initial peak velocity ranging from 2m/s to 5m/s (Burleson, 1993).

According to Winoto (1994), the above equation can estimate the distributions of /Um2 independent of the size of the orifice and the Reynolds number of the jet.

The estimation of the resulting turbulent shear stress is necessary as the resulting turbulent shear stress can cause sublethal or lethal damage to the red blood cells if it exceeds a certain threshold level. This threshold level usually ranges from 200-400N/m2 (Winoto et al., 1994).

Although the above equation is for a self-preserved free circular jet, it is used to estimate the turbulent shear stress for pulsatile jet flow in this experiment as well.

2.5 Hot Wire Anemometry

Hot wire anemometry is the

CHAPTER 3 - DESCRIPTION OF EXPERIMENT

3.1 Experimental Setup

The setup for the experiment is the same as the one used for the previous theses and research work on this topic. The experimental setup comprised of the air jet supply system, the confining pipes forming the test section, the traversing mechanism, the measuring equipment formed by the pitot-static tube and the hot wire anemometer, and the data acquisition module. The equipment was designed such that the measurements at various points in the test section were performed with ease, without affecting the other factors in the apparatus. The photographs of the laboratory equipments and the setup are shown in Figures 3. to figure 3.

3.1.1 Air Jet Supply System

The working fluid for this experiment was air, which was supplied to the experimental setup in the form of an air jet by a compressor. This compressor had the range of 0 - 10 bars, and a pressure regulator was used to control the pressure of the jet in the test section from 0 - 2.5 bars. The flow passes through a honeycomb filter before arriving at the test section.

3.1.2 Test Section

The test section is made of separated Perspex tubes, one of which is fixed and forms a passage for the jet to flow through a honeycomb filter. The other is an interchangeable section, which can be fixed along the fitting, thereby ensuring the orifice plate which forms the beginning of the test section, is also held in position for the experiment. The interchangeable section allows the orifice plates of various diameter circular holes, and various shapes, to be inserted into the test section and observe the effect on the flow. However for this project, the orifice plate with a diameter d = 7mm was used throughout the duration of the project.

3.1.3 Traversing Mechanism

To allow an accurate measurement of the jet, a traversing mechanism was employed in the experimental setup. The linear traverse was used to move the measuring equipments; pitot-static tube and the hot wire probe, in the streamwise direction of the flow. The accuracy of this traverse is 0.5mm. A second part of the traverse mechanism is the height gauge, which provided the degree of freedom in the radial direction of the jet flow, thereby presenting an opportunity to measure the velocity at various radial points in the jet. The accuracy of the height gauge was 0.01mm. The setup and its components are further illustrated in the photographs in Appendix A.

3.1.3 Measuring Equipment

The equipments used to measure the velocity in the jet flow at various points were the pitot-static tube, and the hot-wire

Writing Services

Essay Writing
Service

Find out how the very best essay writing service can help you accomplish more and achieve higher marks today.

Assignment Writing Service

From complicated assignments to tricky tasks, our experts can tackle virtually any question thrown at them.

Dissertation Writing Service

A dissertation (also known as a thesis or research project) is probably the most important piece of work for any student! From full dissertations to individual chapters, we’re on hand to support you.

Coursework Writing Service

Our expert qualified writers can help you get your coursework right first time, every time.

Dissertation Proposal Service

The first step to completing a dissertation is to create a proposal that talks about what you wish to do. Our experts can design suitable methodologies - perfect to help you get started with a dissertation.

Report Writing
Service

Reports for any audience. Perfectly structured, professionally written, and tailored to suit your exact requirements.

Essay Skeleton Answer Service

If you’re just looking for some help to get started on an essay, our outline service provides you with a perfect essay plan.

Marking & Proofreading Service

Not sure if your work is hitting the mark? Struggling to get feedback from your lecturer? Our premium marking service was created just for you - get the feedback you deserve now.

Exam Revision
Service

Exams can be one of the most stressful experiences you’ll ever have! Revision is key, and we’re here to help. With custom created revision notes and exam answers, you’ll never feel underprepared again.