Model Limitations


It's important to consider model limitations when comparing a mathematical solution to physical observations.  The models of this section incorporate an ideal fluid approximation to compute flow velocities using potential flow theory; the meaning of the latter will be explored in a later section.  An ideal fluid exhibits no viscosity whatsoever while almost all real fluids exhibit the property.  For the ideal fluid, the boundary condition at the surface of the cylinder is that the fluid must move tangentially to the surface; the boundary condition for a real fluid is that the fluid velocity at the surface is literally 0.  The latter is true no matter how small the viscosity or how large the Reynolds number; implications for the fluid motion solution are dramatic.

The above image is a recapitulation of the pressure solution surrounding the cylinder.  Without having to do any serious math, it's immediately apparent that the solution is symmetrical -- bisymmetrical in fact being symmetrical relative to either a vertical or horizontal line (plane) passing through the center of the cylinder.  The fluid exerts a force on the cylinder as a result of the pressure distribution.  Pressure is the only force in this model -- an ideal fluid exerts no forces due to sheer stress.  Apparently, the forces acting on the upstream side of the cylinder are identical to the forces acting on the downstream side.  Consequently,there is no net force acting on the cylinder resulting from the pressure distribution!   This is known as d'Alembert's paradox (Wikipedia link) and it is in contradiction to your everyday experience.  You would conclude from this calculation that a gale wind might have no effect on the objects in its path since the forces upstream are identical to those downstream.

This leads us to consider how even a small amount of viscosity could have a dramatic effect on the flow solution and the implications are of great importance in cardiology as well.  In the schematic below, more realistic velocity profiles are suggested in the vicinity of a solid boundary represented by the gray block.  The figure depicts what happens when a real fluid is subjected to an adverse pressure gradient -- a situation where a component of the pressure force is directed opposite to the direction of fluid motion.  The deceleration resulting from the pressure force causes fluid elements near the boundary to literally reverse their direction of motion, even though the main flow continues in the original direction.   This causes the thickness of the slow moving region near the solid wall to increase dramatically.  This is referred to as boundary layer separation, or sometimes just flow separation.  The latter may be misleading in that it seems to suggest that the fluid is no longer in contact with the boundary whereas what is meant is that the high velocity stream is no longer in contact with the boundary.

Flow reversal and boundary layer separation typically originate where they velocity profile exhibits an inflection point, a situation where the rate of change of velocity in a direction perpendicular to the surface is 0.  This is represented in the above by the turquoise colored velocity profile near the center of the figure.  It will be apparent that profiles to the left (upstream) side  of this one all exhibit velocities directed entirely downstream whereas profiles to the right (downstream) side include velocities that are directed upstream within the ever thickening boundary layer. 

If you are wondering whether this situation ever occurs in the cardiovascular system, the answer is absolutely -- particularly when there is pathology!  This very situation that is the most likely reason for flow to become turbulent -- an adverse pressure gradient exerts a strongly destabilizing effect on the flow. For cardiovascular purposes, we are essentially talking about a situation where the flow is decelerating.  Most of the situations you are familiar with that result in a heart murmur or turbulent flow patterns on a Doppler examination are specifically the result of decelerating flow in the presence of an adverse pressure gradient.  This includes regurgitation or stenosis of any valve, VSD, PDA, and flow past any intra-vascular obstruction (bruits).  An adverse pressure gradient destabilizes the flow and promotes the transition to turbulence.

Is an adverse pressure gradient ever normal?  Yes -- an adverse pressure gradient occurs with pulsatile flow in the large arteries during the second half of systole when the blood flow velocity is decreasing.  This happens all the time!  It occurs also in the vicinity of arterial bifurcations where the total vascular cross-sectional area increases by a small amount (usually about 15%) in transitioning from the parent to daughter vessels.  It is these regions of adverse pressure gradient and thickened boundary layer that are most predisposed to the initiation of atherosclerosis.  Boundary layer separation is of the greatest importance in cardiovascular health and disease.

Examine now the situation surrounding the cylinder.  In the next image, the pressure solution is displayed in color (cool colors for low pressure, etc.) and also as vectors; the vectors depict the pressure force.  Streamlines are also shown to remind you of the fluid particle paths as they move past the cylinder left to right.  As fluid elements begin to move past the pole of the cylinder (see box), they begin to encounter an ever increasing adverse pressure gradient.

Specifics of the pressure field are suggested in the subsequent figure which depicts pressure contours instead of streamlines.  Notice how fluid elements traveling from the poles towards the downstream (right) side of the cylinder move against an intense adverse pressure gradient (pressure contours close together, moving from a low pressure to a high-pressure zone).  For this reason, boundary layer separation is expected for this geometry even at relatively low Reynolds numbers.  The simplified potential flow model gives a poor representation of flow around the cylinder since it doesn't accomodate such occurances.

For a more realistic representation, picture instead streamlines leaving the contour of the cylinder somewhere in the vicinity of the box inset in the above figures and continuing downstream in a pattern that may be quite complicated, i.e. a wake downstream of the cylinder.  Streamlines would not continue around the contour of the cylinder. Consequently pressure would not increase back to its upstream value, as suggested by the images, and the real pressure would be significantly lower on the downstream side of the cylinder than the upstream side.  With a higher pressure upstream than downstream, there is a net pressure force on the cylinder acting in the direction of the flow.  This description is more in keeping with your real life experiences -- you expect a wind hitting the cylinder from the left would act to force it to the right. 

There are some interesting and informative websites that depict more realistic flows past a cylinder.  These include computations or movies of vortex shedding.  The wake downstream of a cylinder in high Reynolds number flow can exhibit what's referred to as a von Karaman vortex street.

Fluid force that impedes the motion of an object to the fluid is called drag.  There are different forms of drag depending on the origin of the force.  The type of drag just described, due to boundary layer separation and a higher pressure on the upstream side of the object, is called form drag.

The following figure taken from a Wikipedia entry shows the pressure distribution for the potential flow (same as above) and also a more realistic pressure distribution from measurements made during high Reynolds number flow over a real cylinder.  Clearly there is quite a significant force acting in the direction of the fluid flow or, if you prefer, to impede the cylinder moving through the fluid medium.

(From Wikipedia) Pressure distribution for the flow around a circular cylinder. The dashed blue line is the pressure distribution according to potential flow theory, resulting in d'Alembert's paradox. The solid blue line is the mean pressure distribution as found in experiments at high Reynolds numbers. The pressure is the radial distance from the cylinder surface; a positive pressure (overpressure) is inside the cylinder, towards the centre, while a negative pressure (underpressure) is drawn outside the cylinder.


It may perhaps help to solidify the boundary layer separation concept by considering 2 engineering applications that relate to the discussion.  First, it is likely known to you that manufacturers of golf balls are quite keen on the fluid dynamics of cylindrical/spherical objects moving through a fluid medium, the air.  A great deal of research goes into designing the specifics of the dimple pattern on golf balls so that they will fly as far as possible, other things being equal.  The dimples act to promote turbulence in the boundary layer of air next to the golf ball surface.  This acts to shift the boundary layer separation from the golf ball to a location farther towards the downstream side of the ball -- in a sense it causes the flow to approximate the potential flow solution to a greater degree, i.e. farther around the downstream side.  In consequence, the pressure difference between the upstream and downstream sides is minimized and there is less form drag.

The airfoil used for the wings of an airplane is designed so that boundary layer separation does not occur under normal operating circumstances.  For the airfoil images shown in this section, the simplified potential flow solution gives a much more realistic approximation of the pressure distribution around a wing than for a cylinder.  In fact, it is imperative to the pilot and passengers of an airplane that the wings are operated within limitations so that boundary layer separation does not occur.  A wing where boundary layer separation has occurred is said to have stalled.  If a stall occurs, the airplane is quickly converted from a device that flies to a device that falls to the ground until/unless the pilot takes steps to bring about reattachment of the boundary layer to the wing.  

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