## Stenosis Qualitative

#### Straight to the Heart

• This reviews qualitative but complex physical aspects of flow through a restricted orifice.
•  Fluid flow through the stenotic orifice is associated with a pressure gradient that supplies the acceleration force.  Velocity reaches a maximum shortly downstream of the physical orifice at the vena contracta; the velocity there is significantly greater than would be predicted by a simplistic description of the geometry (area change).
• Downstream of the vena contracta, the fluid "decelerates" (acceleration opposite to the direction of flow) due to increasing pressure in the downstream direction ("pressure recovery") and dramatic (viscous) friction forces that are amplified by turbulence.
• Under appropriate circumstances, there is little energy lost on the center streamline upstream of the vena contracta; a stenosis is NOT a "resistance".  This allows us to apply the simplified Bernoulli equation on the central streamline within in this confined region.  Although there is a dramatic decrease in pressure across the stenosis ("gradient"), this energy is not lost but rather converted into kinetic energy, the mass and velocity of the fluid.  Energy loss due to friction occurs primarily in the pressure recovery zone where pressure is actually increasing in the downstream direction.
• A stenosis sets the stage for a drastic range in the hemodynamic environment that includes turbulence downstream of the constriction and large variations in pressure, velocity, and shear forces.  We see some fluid elements trapped near the stenosis in a "recirculation zone" and others that spin like a top ("vorticity").
• The Reynolds number is a ratio of inertial to viscous forces and is a necessary descriptor that determines the "shape" of a flow whenever there is a convective acceleration (change in fluid velocity with location).

I suspect that anyone reading this page (5 of you to date!) is likely quite familiar with the qualitative aspects of stenosis flow from daily experience in cardiology.  I hope this review will provide perspective by showing and analyzing the results of computational fluid dynamic solutions (CFD).  Go to the PREVIOUS PAGE for a link pathway to see how you can make your own images from the example computations.   Solutions for axis-symmetric stenosis flow will be presented recurrently where a two-dimensional representation of the flow is shown.  A picture's worth 1000 words, so I hope you get the idea from this figure:

In all that follows, flow enters from the left-hand side of the figure into a virtual cylindrical conduit with uniform radius until reaching an obstruction which consists of an orifice plate, essentially a wall with a circular hole in it at the center line of the parent tube.  The stenosis flow models here are simplistic of course, as representations of the geometry that occurs in the heart.  However the physical principles depicted apply both to stenosis and valvular regurgitation. All of the flows shown here are time invariant.  That isn't a limitation of the method, just a limitation of practicality for presentation of the concepts.

Flow through a stenosis necessitates acceleration of fluid elements; velocity increases on the upstream side of the constriction but also must decrease on the downstream side.  Both of these situations constitute acceleration, the latter being a vector quantity.  In the setting of fluid dynamics, acceleration is a vector field  - a vector quantity that typically varies with location in space (it also changes with time, but not in this discussion).  Acceleration in the direction of motion increases speed, opposite the direction of motion decreases speed. Acceleration perpendicular to the direction of motion does not change the speed of the fluid element, only the direction of motion. The acceleration vector can, and typically does have components that change both the magnitude and direction of the velocity vector. Without having any prior knowledge of how this occurs, we might naïvely guess the following figure as a first approximation of the flow representation:

Here is depicted a uniform velocity at each axial location – the same unvarying velocity over the entire tube cross section, but varying with axial location and the cross-sectional area; no radial velocities appear.  The numerical value of the uniform axial velocity is:

$$\Large \bar{u}_x(x) = \frac{q}{A(x)}$$

Here,$$\bar{u}_x(x)$$ is the average axial velocity at location $$x$$.  $$u$$ is used to represent velocity to avoid confusing it with voltage or volume; subscript $$x$$ is to indicate the component of velocity in the axial direction and the $$x$$ in parentheses indicates that the value can change with axial location.  $$q$$ is the volume flow rate (e.g. mL/sec) and $$A(x)$$ is the physical cross-sectional area (e.g. cm2) which also can vary with axial coordinate, $$x$$.   In the particular example case shown above and below, the stenosis orifice area is $$\frac{1}{4}$$ of the parent tube area ($$\frac{1}{2}$$ the diameter) so the average velocity there is $$4$$ times the value of the parent tube.  For incompressible flow in a rigid conduit (with no inlets or outlets), the flow rate is invariant with axial location.  Notice however that both the cross-sectional area, and consequently $$\bar{u}_x$$, can change abruptly - discontinuously - with axial location.

Of course the actual flow cannot occur as suggested by the above figure.  The cross-sectional area can change discontinuously, but the actual velocities cannot since that would constitute an infinite acceleration.  The next figure is a more realistic approximation of the actual flow from the CFD solution showing velocity depicted as vectors but also by the color scale:

The figure depicts the calculated velocity vectors for this simple stenosis geometry with flow in the parent tube occurring at a prescribed Reynolds number of 1000. Flow enters at the left side of the figure with a prescribed uniform velocity of 1.0 (so the vectors at the far left also serve the purpose of a velocity scale legend). We can refer to the variation of velocity over the cross section as the "velocity profile".  If you look closely, you'll see that the velocity profiles are never entirely uniform; the velocity always approaches 0.0 near the wall of the conduit ( upper border of the flow figure). In fact the fluid at the wall always must move at the same velocity as the wall, zero in this particular situation (the "no slip" condition).   Since we know that the flow rate is the same at every axial location, slow moving fluid near the wall means that fluid elsewhere must move faster than the cross-sectional average.  The slow moving fluid near a solid boundary is called a boundary layer. A boundary layer will always occur when the fluid possesses viscosity and ALL biological fluids are viscous.  In effect, a boundary layer excludes high velocity flow from occurring immediately adjacent to a wall and for a variable distance away from the wall.  This also means that the cross-sectional area available for flow is effectively smaller than the physical area.  There will always be fluid elements moving faster than the computed average, $$\bar{u}_x(x)$$, at every axial location.

At the seventh velocity profile from the left, i.e. at the physical stenosis,  velocity has increased dramatically but in a way that might not have been anticipated. Clearly some of the velocity vectors are NOT directed axially only; the vectors include a significant radial component, in this case directed towards the centerline (lower edge of the figure) so that the flow is actual converging towards the centerline as it passes the physical stenosis.  The component of velocity that is not directed axially does not contribute to the axial flow rate.  Hence this too is velocity in excess of the average value.  Furthermore we see again that there are relatively slower velocity vectors at the very edge of the physical stenosis.  The specific velocities generated by the flow geometry contribute to the stenosis; the fluid flow itself is a component of the stenosis.

To delve a little further, you can see that it doesn't look like there's much of a radial velocity component at the centerline (there's actually NONE right at the centerline), yet velocity actually increases at the centerline due to the flow contraction. Flow velocity at the centerline is being enhanced from contributions by radial velocities farther away from the centerline.  The continuity equation for this situation (axis-symmetric flow in a polar geometry) looks like this:

$$\Large \frac{\partial u_x}{\partial x} + \frac{u_r}{r} + \frac{\partial u_r}{\partial r} = 0$$

$$u_x$$ is the velocity component in the axial direction, $$u_r$$ is velocity in the radial direction. The first term, $$\partial u_x/\partial x$$ has to do with how much the axial velocity is increasing as we move in the $$x$$ direction (downstream).  The equation shows that this is contributed to by radial velocity, $$u_r/r$$, but also by a radial velocity gradient, $$\partial u_r/\partial r$$.The fact that the radial component of velocity is changing in the radial direction also adds to the axial velocity.  (I just didn't want you to think that anything slipshod was going on here.)

The next figure represents the velocity again in the alternative format of a contour plot which allows us to quickly identify regions of high and low velocity; lost in this depiction is the directional information afforded by the vector format.  The figure clearly shows the location of the vena contracta, outlined by the red contour at the conduit centerline (lower edge of the plot), about half a diameter downstream from the physical  stenosis.   Qualitatively the vena contracta is always downstream of the physical stenosis although it's shape depends on the Reynolds number (see below) and the specific configuration of the stenosis geometry.  For example, I learned from these numerical experiments that the vena contracta does not necessarily reside at the centerline.  Picture a mild stenosis where the orifice plate does not protrude into the stream very far.  In that case the vena contracta could be displaced from the centerline, forming a toroid (donut shape) in a three-dimensional representation.

Shown on the far right-hand side of the figure is the color velocity scale which has been configured to represent the full range of velocities of the plot; the maximum velocity at the vena contracta is 5.865 in this case.  In this example, the velocity there is almost 1.5 times the average velocity at the physical stenosis ( 4.0 ).  There would be a substantial error in estimating the stenosis flow velocity from flow rate and stenosis cross-sectional area alone.  Note that the actual velocity at the vena contracta will always be greater than the value computed from the flow rate and physical stenosis area.

As promised by Newton's laws, the fluid is accelerated in moving from the inlet at the left of the plot to the location of maximal velocity at the vena contracta.  This acceleration must be caused by a net force which is due principally to the pressure gradient in this instance.  In clinical terminology, pressure gradient is often reported as a single number with physical units of pressure.  More correctly however, pressure gradient is another vector field with a magnitude and direction at each location; it has physical units of pressure/distance (stress/length).  In any case, the figure below depicts the intense spatial variation of pressure in the vicinity of the stenosis that all cardiologists are familiar with from cath lab experience.  The figure suggests a structure to this pressure field, however, that is not readily appreciated from sliding a catheter back and forth across a valve.  Notice for example that the pressure gradient is not directed only axially in the immediate vicinity of the stenosis, but that there is marked pressure variation from the conduit centerline to the wall.   For the particular model shown, the pressure gradient force acts to accelerate the fluid towards the centerline on the upstream side of the stenosis and away from the centerline on the downstream side.

Comparison of this figure with the one above will allow you to appreciate the fact that the vena contracta (velocity maximum) coincides with the physical location of a pressure minimum, a region bounded in the above figure by the contours of the deepest blue color. Downstream of this location, the pressure begins to increase in the axial direction in the "pressure recovery" zone. A subsequent figure will allow you to better appreciate this latter aspect of the physics.  However it should come as no surprise to you that the fluid continuing downstream from the vena contracta must be subjected to a force opposite the direction of motion for the velocity to return to the more sedate values downstream of the vena contracta.  That force again derives from a pressure gradient acting to accelerate the fluid in the upstream direction, an adverse pressure gradient (synonymous with "pressure recovery" here).  Fluid deceleration at this location also derives from frictional forces to a much greater extent than was true upstream of the stenosis.

Before leaving this figure, it's worth noting that the pressure field settles down, from the location of the vena contracta and downstream, so that the pressure has little variation over the cross section; the pressure gradient force is directed essentially upstream or downstream with no appreciable radial component.  This is referred to sometimes as a "plane pressure" approximation (pretty much always assumed for clinical purposes).

The above figure is just a recap of the pressure field to show the extent of the pressure recovery zone (PRZ) starting at the vena contracta, where there is a local pressure minimum, extending a couple of diameters downstream to a local pressure maximum before the pressure begins to decrease in the downstream direction due to the usual mechanism of pressure loss (friction).   The extent of this PRZ will depend on many things relating to the geometry and flow parameters.  I know that it is a common source of confusion to clinicians that the pressure can be increasing in the downstream direction following a stenosis.  The clinical literature sometimes applies the term "resistance" for quantifying stenosis severity, but this is a misnomer that contributes to the confusion in my opinion. The term "resistance" in both electrical terminology and in physical hemodynamics has to do specifically with a loss of energy (pressure) due to friction.  This has little to do with the predominant physical processes occurring in the immediate vicinity of a stenosis.  The pressure drop across a stenosis can be thought of as a conversion of pressure "energy" into kinetic energy of the fluid (more technically the pressure does work on the fluid to increase its kinetic energy). There is little energy lost to friction there and, in fact, application of the simplified Bernoulli equation makes the approximation that NO energy is lost.  On the downstream side of the stenosis, some of the kinetic energy of the fluid is "recovered" and pressure increases (pressure does negative work on the fluid and decreases its kinetic energy). There's no one to stop you from applying the resistance formula in the region downstream of a stenosis, but this is inappropriate and you would find nonsensically that the resistance value is negative there if you do.

The next figure was generated by "labeling" fluid elements at the inlet of the conduit (far left) and ALSO at the vertical white line that transects the flow image; fluid paths were tracked through the stenosis and also as they orbit slowly in a recirculation zone. Color represents the speed of the fluid elements so you don't get bored.  A dividing streamline separates the flow into 2 regions that have no flow communication with each other in a strictly mathematical sense.  The dividing streamline also shows where boundary layer separation occurs, at the edge of the stenosis orifice in this case.  This recirculation zone is a region of slowly flowing fluid that is separated from the main stream by a boundary layer that is detached from the wall. The location of boundary layer reattachment is not included in the figure but would be off screen to the right.   In 3 dimensions, the dividing streamline is asurface that sequesters the recirculation zone from the rest of the flow although the rapidly moving jet is the source of energy to keep the recirculation zone moving

Here we begin to see how extremes in hemodynamic environment can occur in the vicinity of a physical disruption of the flow field.  The figure would imply that fluid in the recirculation zone is trapped there forever since there are no streamlines that traverse the zone – no pathway in to or out of it.  The observation of such circumstances led researchers to consider whether a location such as this might be predisposed to disease ( such as atherosclerosis ) as a result of increased time of residence (for fluid, particles, chemicals, etc.) or decreased flow rate.  Such circumstances do indeed relate to disease production and a principal contributor seems to be the decrease in shear stress at the endothelium resulting from the slower fluid flow; this alteration in physical environment results in endothelial dysfunction and diminished release of endothelial nitric oxide.  In reality, pulsatility, turbulence, and minor variations in flow geometry prevent the pictured recirculation zone from being entirely sequestered from the main stream.  Fully three-dimensional flows can result in some very interesting pathways by which fluid can both enter and exit these slow flow regions.

I'll also reminder you here that these computations employ a turbulence model called the $$k-\epsilon$$ model.  In consequence the velocities and fluid element paths shown are a kind of average expectation, a mean velocity vector at each location.  A manifestation of turbulence is that it superimposes a seemingly random velocity onto the mean velocity vector; the magnitude of the random vector is related to the turbulence intensity (more below).  The turbulence is isotropic in the $$k-\epsilon$$  model meaning the magnitude of the turbulence vector is independent of direction; that's one of the limitations of the model.

The next figure focuses on the location of the boundary layer reattachment where the dividing streamline from the previous figure ends up touching the wall of the of the tube (not seen on the above figure).  This is included just so you know what we are looking for to identify boundary layer reattachment. On either side of the reattachment point ( upstream and downstream ) we can see that the velocity vectors adjacent to the wall point away from the spot.  This means that the fluid is flowing downstream to the right of this point and upstream to the left side of it.  The point of boundary layer separation  has the opposite characteristic – velocity vectors point towards the point of separation on both the upstream and downstream sides.

Why does boundary layer separation occur?  It often has to do with an adverse pressure gradient – i.e. the pressure recovery zone.  In such a region, pressure force acts opposite to the direction of flow. Since there will always be a region of slow moving fluid next to the wall, "deceleration" ( acceleration opposite the direction of flow ) affects this fluid by actually reversing the direction of motion.  The dotted line in the figure below suggests the location of the boundary layer and you can see why there will be a region next to the wall where the fluid flows upstream, just as it does in the computations shown. The boundary layer separation for the stenosis occurs right at the tip of the obstruction (the valve "leaflet") making it difficult to illustrate with the computations in this example.

The figure above (showing how the fluid velocity vectors relate to boundary layer reattachment) illustrates another interesting attribute of the flow, that of entrainment. We know that the flow rate for the rigid tube is invariant with location.  Yet the presence of the recirculation zone, with velocity vectors directed opposite to the main flow near the wall, means that there is a region near the wall where fluid flows upstream.  Consequently the recirculation zone is also a region in the conduit where there is an increase in the actual downstream flow rate.  The main jet entrains fluid  from the surroundings so that the total forward flow is actually increased.  The following figure quantifies that effect for the flow solution depicted, showing the forward flow as a ratio, i.e. total forward flow divided by the invariant flow rate.

The recirculation zone starts at the physical stenosis (axial coordinate 0.0) and increasing entrainment occurs to a maximum of almost 20%, i.e. nearly 20% of $$q$$ (the invariant flow rate) flows upstream with 120% of $$q$$ flowing downstream.  Does entrainment contribute to the enhanced forward flow velocity at the vena contracta?  Hmmmm....  I wouldn't say so myself.

Technical Note: The solution exhibits a "glitch" at the location of the physical stenosis that I believe is due to the physically implausible geometry specification. This stenosis has sharp corners on it that translate into an unrealistic "singular" boundary condition with a 0.0 velocity specification right next to a high velocity point. The small stencil pressure projection method used here does not enforce numerically exact mass conservation for the finite volume method.  While the numerical solution has converged, this discontinuity introduces the spike and wobble in the flow rate seen above.

The next 2 images depict the "Total Energy" of the fluid defined as $$p + \rho |u|^2/2$$ (also known as total pressure or stagnation pressure).  This is an important clinical concept since application of the Simplified Bernoulli Equation is dependent upon the result.  The first figure below shows contour lines that indicate the variation of total energy in the vicinity of the stenosis.  There is an inviscid core of fluid from the inlet and extending essentially to the location of the vena contracta.  The fluid here is not actually inviscid - it has the same viscosity as elsewhere.  What's inviscid is the way in which the fluid flows – the fluid layers do not shear against each other significantly.  When a fluid flows in this way there is little/no energy lost as the fluid element moves along a streamline.

As we all know, the clinically applicable guts of the Bernoulli equation appears as follows:

$$\Large p_1 + \frac{\rho}{2} |u_1|^2 = p_2 + \frac{\rho}{2} |u_2|^2 + L_{\mu}$$

The quantity $$L_{\mu}$$ here is used to represent an energy loss due to friction where $$\mu$$ refers to the viscosity of the fluid.  The figure depicts the inviscid core where $$L_{\mu}$$ is negligible; this is the only region of the stenosis flow where the Simplified Bernoulli Equation is applicable.

The subsequent figure also shows total energy represented by color; fluid element paths (streamlines) have been illustrated to show how the conduit center streamline ( lower edge of the flow figure ) lies within the inviscid core and extends from the inlet to the vena contracta along the tube.  Our ability to estimate the pressure difference between the inlet and the vena contracta is dependent on the velocity measurements obtained at these 2 locations as we all know.  Furthermore the value of $$\large u_1$$ is often negligible with respect to $$\large u_2$$so that the pressure difference is approximated by the well-known SBE:

$$\Large \Delta p = \frac{\rho}{2} |u_2|^2$$

The figure also illustrates streamlines originating near the wall of the conduit (upper left) that do not maintain the inlet energy level to the level of the vena contracta; this is apparent from yellow, green, and blue streamlines that extend from the edge of the stenosis.  Although pressure is essentially constant over the cross-section at the vena contracta, fluid elements following these streamlines have lost energy due to intense shears ( viscous friction ) that occurs as the fluid flows past the obstruction.  Although the fluid on the center streamline retains virtually all its energy up to the location of the vena contracta, the total stream issuing through the stenosis does not.  We'll explore this in mathematical detail in a subsequent article.

Technical Digression

This section relates to an issue that likely would only interest an engineer; it deals with an apparent discrepancy in the flow solution that may have some clinical relevance, but not much. I'd welcome any comments from engineers on the explanation given.

The "Total Energy" (or "Total Pressure") contour plot above depicts a situation that seems physically unrealistic - total energy increasing in the direction of flow along a streamline.  I have assumed for some time that this represents either a bug in my software, or that the model is inadequate. However close review of these solutions causes me to suspect that this may actually be a realistic aspect of the flow. For flow along a streamline, the simplified Bernoulli equation assumes that the effect of friction is always a decrease in energy:

$$\Large p_1 + \frac{\rho}{2} |u_1|^2 = p_2 + \frac{\rho}{2} |u_2|^2 + L_{\mu}$$

This is not entirely correct however.  Surrounding fluid that is moving faster than the elements on the streamline of interest can exert a drag force in the direction of flow.  In that situation, there is positive shear work done on the fluid element that causes an increase in total energy along the streamline. Furthermore, turbulence increases the transfer of energy across streamlines dramatically, i.e. through convection.  I believe that this may be the explanation for what appears to be a physically implausible aspect of the solution.   Certainly total energy of the closed system cannot increase without an internal source, but fluid elements can transfer energy to others by this mechanism.

The following figure depicts velocity vectors at the physical stenosis where a color scale adjustment illustrates the fact that the highest velocity fluid elements are not at the centerline at this axial position; elements with the highest velocity surround the centerline as a ring that is closer to rim of the obstruction.  These elements have the potential to transfer energy to streamlines closer to the centerline through shear stress (friction).  They can literally drag more centrally located fluid elements in the direction of flow, increasing their total energy.

The velocity contour plot, reprise below, helps conceptualize how a ring (in 3D) of relatively high velocity issues forward from the physical stenosis with lower velocities  nearer the centerline.

The above 2 figures illustrate (perhaps) an "obvious" situation where energy could be added to a streamline through drag forces.  However more subtle circumstances might also cause this result.  All that is necessary is a net positive shear force in the direction of the flow which could result from nuances of the velocity profile that aren't readily visualized.  Turbulent energy transfer may also contribute although turbulence intensity isn't particularly high at this location (see below).

The next figure depicts color adjusted streamlines with the lowest color value (blue) set to the total energy at the inlet (centerline pressure plus kinetic energy = 0.5). Consequently any color other than deep blue depicts where streamliine energy has risen above the inlet.  The figure includes the highest energy hot spots occurring in the flow.

I've observed this increase in Total Energy for many of my own numerical studies (it's consistent) and also on stenosis problems that were run on FLUENT (it's not my software only).  I think this may be realistic aspect of the flow.  Digression over.

The following 2 plots show the centerline velocity as a function of axial location and subsequently both pressure and "total energy" as previously defined.  Both the pressure and total energy plots assume an arbitrary datum of $$0.0$$ at the centerline of the tube inlet. The axial coordinate is indicated in terms of tube diameters with $$0$$ being the location of the physical stenosis.  For the velocity plot, there is a uniform velocity of $$1.0$$ at the inlet which increases markedly to the maximal value ~1/2 diameter downstream of the physical stenosis at the vena contracta.  Subsequently the centerline velocity gradually decreases with the flow deceleration.  The velocity profile never returns to the uniform velocity however, so the centerline velocity remains elevated above the cross-sectional average of $$1.0$$.

A plot of the pressure distribution along the axial centerline clearly demonstrates the dramatic decrease in pressure across the stenosis to a local minimum at the vena contracta that we're all familiar with.  For this particular situation, pressure recovery occurs from the vena contracta to ~2 diameters downstream of the stenosis. A transition occurs from there to ~3 diameters at which point the pressure approaches a stable decline rate due to viscous dissipation.  In a model with turbulence, such as this, the rate of pressure loss is increased above the value predicted by the Poiseuille solution of $$8 \mu q / (\pi r_0^4)$$ per unit of conduit length.

The centerline total energy plot depicts constant total energy in the region upstream of the stenosis, then a seemingly implausible increase in energy near the physical stenosis (See Technical Digression above).  The important aspect of this plot is the fact that energy remains essentially at the inlet level, within the inviscid core flow, until energy losses begin downstream of the vena contracta.  Further downstream, there is rather dramatic energy loss in the vicinity of the pressure recovery zone; energy decline actually approaches maximal in a region where pressure is increasing in the downstream direction.  So this is an interesting region where kinetic energy is being dissipated (lost) as heat as well as being returned to "pressure energy".  If we were to calculate the "resistance" in this zone ($$\Delta p/q$$), the value would be negative despite the fact that energy loss is greatest here.  (I'm not recommending that you do this. I'm telling you it's not applicable.)  Energy loss gradually transitions downstream to parallel the pressure tracing where energy decline is essentially reflected entirely by the pressure decline.  Total energy on the centerline remains elevated relative to the pressure tracing since the kinetic energy on the centerline is greater than it was at the inlet (see the velocity plot above).

We'll consider now some nuances of the flow; these may have importance relative to clinical problems such as endothelial damage and predisposition to endocarditis.  Downstream of the obstruction, there is a region where fluid velocity changes rapidly over a short distance; there is a marked velocity gradient perpendicular to the direction of flow.  In the velocity contour plot below, the square bracket implies where this region is occurring. It originates at the edge of the obstruction and the velocity gradient gradually decreases downstream from that location, i.e. the variation of velocity becomes more gradual. This flow field implies a circumstance where each fluid element experiences a higher velocity on the centerline side than on the side towards the conduit wall.  Each fluid element is spinning in this region.  There is a well-defined means to express this fact in a quantitative sense call the vorticity that has to do with the velocity gradients.  Vorticity is another vector field quantity that indicates both the magnitude and direction of spin for each fluid element in accordance with the right-hand rule.  The circle-with-arrow on the plot suggests the direction of spin for the fluid elements in the shear layer.  If you use your right hand with your fingers curled around the direction of spin, your thumb points in the direction of the vorticity vector, i.e. directly out of the computer screen towards you in this case.

The software used to create these images ( downloadable at this site ) allows for vorticity vectors stemming from the CFD solutions to be displayed.  However all of the vectors are either directed straight into the screen or straight out of it for these two-dimensional flows.  Consequently we can simply represent the vorticity magnitude with another contour plot (below).

This shows what we had already surmised from the previous plot, that the fluid in the vicinity of the obstruction (e.g. valve leaflet ) is spinning like crazy.  I'll make no claim as to the accuracy of the values shown which occur in a singular region of the solution (not very accurate); we are only trying to gain a qualitative understanding of how flow occurs in the vicinity of the stenosis.

While I'm at it, there is another location of high vorticity at the upper left-hand corner. This one is due to an artifact caused by the way the flow at the inlet was specified, i.e. uniform velocity all the way across. This specification actually implies a velocity discontinuity - another numerical singularity; the velocity is zero at the wall but full speed ahead right next to the wall.  This flow velocity specification is unrealistic and just a lazy man's way of indicating how much flow is entering the system. We know ahead of time that this is an implausible specification and simply accept the fact that the flow field in the immediate vicinity of this location is not going to be very useful/accurate.  Interestingly, vorticity behaves as a conserved quantity that is transferred along with the flow, just like heat, green dye, or any other material that can be transported and defused in a fluid.

The next contour plot depicts the turbulence intensity generated by the $$k-\epsilon$$ turbulence model.  As noted previously this model is likely rather inaccurate for the complex flows generated by a stenosis, but the plot affords a qualitative understanding of the flow physics.  The turbulence intensifies in the shear layer downstream of the edge of the obstruction and expands to include much of the cross sectional area of the conduit.  While cardiologists look at Doppler ultrasound depictions of turbulence on a daily basis, these do not necessarily coincide with more rigorous definitions and analyses.  I believe the plot likely affords a better understanding of the turbulence surrounding a stenosis than clinical determinations allow.  There are more detailed observations available in the hemodynamics literature.

While we all have a notion about what it is, turbulence is quite a difficult topic at the analytical level.  Nobel laureate Richard Feynman said that turbulence is "the most important unsolved problem of classical physics.".   Horace Lamb reportedly addressed the British Association for the Advancement of Science saying, "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."The solutions you see here show a grid where values of the velocity and pressure are computed at each cell. These CFD problems were solved on my desktop PC.  Calculations for turbulent flows without using a "turbulence model" require a MUCH finer grid than you see and a

Turbulence is markedly predisposed to by the presence of an adverse pressure gradient, a circumstance we have already seen is in effect in this region of the flow.  I'm sure I've emphasized it elsewhere, but it's worth saying again that virtually all the clinical circumstances that result in turbulence and a murmur are associated with the emergence of flow into an expanded region and the presence of an adverse pressure gradient.  This circumstance results in a much lower critical Reynolds number than the 2000-2300 for a straight cylindrical tube with smooth walls.

### What Do You Mean by "Reynolds Number"?

This section touches on one of the most fundamental aspects of fluid dynamics – dynamic similarity.  It is necessary to understand the meaning and impact of this concept before we can begin to discuss the topic of stenosis "flow dependence".  The Reynolds number is a ratio of inertial  (specifically the fluidconvective acceleration) to viscous forces that qualify how a flow will occur.  Being a ratio of 2 quantities having the same physical units, the Reynolds number is a dimensionless number ( no physical units ). When comparing flowing fluids from one situation to another, we MUST take the Reynolds number into account and the fluid engineering literature ALWAYS includes this specification since results cannot be interpreted otherwise.  You'll also see references to the "flow regime" for a particular real life situation ( like aortic stenosis for example ) which is the range of Reynolds numbers, and possibly/likely other nondimensional flow parameters that govern the situation.

I think many cardiologists are familiar with the formula for the Reynolds number of a cylindrical conduit and I even saw it on a board question once.

$$\Large Re =\LARGE \frac{\rho\;d\;|u|}{\mu}$$

Other folks might spit back a number when the topic comes up, 2000-2300?  This would be the "critical" Reynolds number ($$Re_{crit}$$) at which the transition to turbulence typically occurs in a straight cylindrical tube with smooth walls.  The size of the convective acceleration of the general problem is approximated as $$\rho |u|^2$$ where $$|u|$$ is a velocity that characterizes the situation. The viscous friction term relates to the viscosity of the fluid and a velocity gradient,  it's approximated as $$\mu |u|/d$$. The appearance of the diameter ($$d$$) in the formula has to do with the fact that the velocity gradient relates to a velocity change over distance.   If it weren't a tube, we would replace the $$d$$ ( diameter ) in the formula with a more general characteristic length ($$l$$); $$l$$ might refer to the thickness of a wing or the length of a ship under other circumstances.  Divide the $$\rho |u|^2$$ stuff by $$\mu |u|/d$$ and out comes the Reynolds number.

ALL CFD problems are calculated by solving for nondimensional variables  right from the start.  In part, this is done to isolate the effect of the Reynolds number on the solution.  In the following  nondimensional quantities are denoted with a tilde (~) over the symbol.  All the velocities in the solution are velocity ratios:

$$\Large \tilde{u} \equiv \LARGE \frac{u}{|\bar{u}|}$$

For our situation, $$|\bar{u}|$$ is the cross sectional mean of the parent conduit as before. All the distances and sizes are in terms of length ratios (measured in diameters in this case):

$$\Large \tilde{x} \equiv \LARGE \frac{x}{d}$$

All the pressures are ratios too:

$$\Large \tilde{p} \equiv \LARGE \frac{p}{\rho |\bar{u}|^2}$$

If we want to apply the CFD solution to a real-life situation, we need to reverse the above process; multiply $$\tilde{u}$$ by $$|\bar{u}|$$, $$\tilde{x}$$ by $$d$$, and $$\tilde{p}$$ by $$\rho |\bar{u}|^2$$, to convert to CFD solution values to the distances, velocities, and pressures for the actual aorta, airplane, spacecraft, etc.  This is considered a trivial step in the engineering world (and believe me – it is trivial compared to calculating the CFD solution).   But this same solution is applicable to every situation with the prescribed relative geometry and Reynolds number.  We don't have to recalculate a solution just because we're dealing with an elephant versus a mouse if the Reynolds number is the same (it isn't).   In more familiar terms, it doesn't bother you if you're given a drug dosage in mg/kg (per kilogram of body mass) and you have to do a multiplication to determine the dose for a specific individual.

Similarly, we have to look at a solution of appropriate Reynolds number if we want to understand how a flow occurs in the real-life situation we are studying. Suppose the aorta has a 2.75 cm diameter, an average velocity during ejection of 60 cm/sec, blood density and viscosity of 1.0 and 0.035 Poise respectively, etc. (Poise is a unit of viscosity having physical dimensions of  gm cm-1sec-1) The Reynolds number would be 1.0 (gm/cm3) x 2.75 (cm) x 60 (cm/sec) / 0.035 (gm cm-1 sec-1)  ~ 4714. For pulsatile flow it turns out there is at least one other nondimensional parameter we have to concern ourselves with that essentially is a nondimensional frequency ($$\alpha = r \sqrt{\rho \omega /\mu}$$ where $$r$$ is tube radius, $$\omega = 2\pi f$$ or angular frequency. More specifically, $$\alpha$$ is proportional to the square root of frequency).

Up until here, this article has described the results for just one flow solution for a specific geometry at a single Reynolds number of 1000.  The following figures depict CFD solutions for thesamestenosis geometry as for the above figures, but at a range of Reynolds numbers from 25 to 5000.  This is a rather wide range that doesn't necessarily occur for mammalian stenosis flows; I wanted the differences to be readily visible to the naked eye.   ALL the CFD solutions have the same prescribed nondimensional inlet velocity ratio (1.0), dimensions (parent tube diameter = 1.0), and flow rate ($$\pi/4$$). In essence, fluid velocity and geometry are the same for all the solutions. All of the following examples have the same "flow rate"; the only thing that varies is the Reynolds number.

The next 4 conjoined figures illustrate the velocity contour plots; the Reynolds number is shown in the text for each plot.  All are displayed with the same velocity color scale with contour spacing of 0.2 (no dimensions).  All I'm trying to get across here is that the Reynolds number affects the shape of the flow.  This includes all of the attributes discussed above such as the size and location of the vena contracta, recirculation zone, velocity gradient of the shear layer, etc.  We see that the peak velocity at the vena contracta is similar between the solutions, but there will turn out to be quantitative differences.

The next 4 figures depict the actual velocity (ratio) vectors and this allows us more insights as to how the flows differ. A low Reynolds number flow has a relatively high level of viscous friction.  Note for example the seventh velocity profile from the left - at the physical constriction.  The velocity profile of the low Reynolds number flow exhibits a rounded appearance with a gradually increasing velocity in the progression from the edge of the stenosis to the centerline.  Hence there is a relatively thick boundary layer in this example.  An effectively thicker boundary layer excludes high velocity flow for a greater distance from the wall, effectively narrowing the stenosis and increasing the velocity at the vena contracta. With the progression of increasing Reynolds number, the velocity profile at the physical constriction exhibits a much steeper velocity gradient at the edge of of the physical constriction (thinner boundary layer) but also a significantly larger component of the velocity that is not directed axially; i.e. there is a greater component of the velocity directed towards the centerline.  This convergent flow was noted above as another mechanism by which vena contracta velocity is increased above the simple area ratio prediction ($$4.0$$).  Consequently the vena contracta velocity is increased above the area ratio prediction at higher Reynolds numbers, but for a different reason than the lower Reynolds number case.

The final plots trace the pressure and velocity along the centerline of the conduit for 4 values of the Reynolds number.  The pressure drop (clinical "gradient") and change in velocity are similar for all 4, but with quantitative differences that can be difficult to predict without the calculations or a controlled bench top experiment.  (The cath lab is a poor venue for learning the essentials of fluid dynamics; too many uncontrolled variables.)

The centerline pressure plot depicts the local pressure minimum at the vena contracta, a location that varies somewhat with Reynolds number. Variations in pressure gradient, pressure recovery, and rate of pressure decline downstream are all evident.  In general, the higher Reynolds number results in more gradual spatial changes for both pressure and velocity.

From these plots, there isn't a clear trend for the peak velocity at the vena contracta as dependent on Reynolds number.  Downstream, centerline velocity approaches the expected Poiseuille value ($$2.0$$) for the low Reynolds number examples; turbulence causes the velocity profile to flatten with increasing Reynolds number resulting in a lower downstream centerline velocity in the examples.

The changes in the flow physics that occur with changing Reynolds number alter the effective orifice area, more specifically the quantitative relationship between physical stenosis area and the effective area.   The literature uniformly indicates that the effective area increases with increasing Reynolds number with greater dependence on Reynolds number for relatively mild stenoses.  The effective area cannot, however, be any larger than the physical area.  The point is that "flow dependence" is an expected aspect of stenosis flow that need not have anything to do with changing physical stenosis area.  For the lowest Reynolds number example here (Re = 25), a non-negligible component of the pressure gradient is due to energy lost to friction as evidenced by a significantly greater pressure gradient despite a similar value for $$u$$ at the vena contracta.  This would be a case where the SBE can't be applied accurately since it assumes negligible losses along the centerline up to the vena contracta.