## Lid Driven Cavity

Engineers and Mathematicians -- Get Lost!  This is a non-rigorous webpage intended to introduce principles to medical folks. You're welcome to images, videos etc. and numerical files are also available.  I'd also be happy to hear from you if I've made an obvious goof.

The video above depicts an animation of the fluid flow in a Lid Driven Cavity that results when the "lid" or upper side of the square cavity is drawn to the right at a constant velocity, thereby inducing the fluid into a circular motion, i.e. creating a vortex. This doesn't have much to do with cardiology, but is here to impart some principles of fluid dynamics for a flow situation that's a little more complicated than what you may be used to thinking about.

### Introduction

The lid-driven cavity (LDC) is a common test or bench-mark problem in computational fluid dynamics (CFD) particularly as one that critically tests the accuracy of the advection (convective acceleration) scheme used for the computations.   The figure that follows is an example calculation that illustrates the essential features of the computation which consists of a rectangular cavity, a square one in this case, where the (Newtonian) fluid inside is set in motion by dragging the upper edge of the cavity (the "lid") to the right.  The latter specification constitutes the nontrivial boundary condition of the problem and is represented in the figure by (poorly visible) green arrows at the upper edge of the square.  For a real (viscous) fluid, the fluid elements in contact with the lid move at the same velocity as the lid.  The right, left, and lower boundaries of the cavity are stationary so the fluid at those borders has a velocity of zero.  The test problem is invariably 2-dimensional as implied by the figure; it's as if we're taking a tomographic slice of a rectangular trough that extends infinitely into and out of the plane of the screen (paper?).   This particular solution was computed for a Reynolds number of 250; this number constitutes the overall characteristic relationship between the inertial (convective acceleration) and viscous forces governing the flow.

In the figure as shown, velocity of the fluid is represented by arrows that indicate both magnitude and direction of velocity associate with each cell; color also represents velocity magnitude.  The moving lid sets the fluid into a grossly circular motion, a vortex.  The figures were derived using computational fluid dynamics software where the velocity is calculated at a number of fixed locations (cells  within the computational domain.  If you look closely at the above figure you will see that the computational grid is also included in a light blue color. The particular solution above was determined for a 100 x 100 cell, 2-dimensional grid.  FYI, that means that the solution comes down to a set of equations with 10,000 variables!  (Actually 30,000 variables since we need to compute the $$x$$-velocity ($$u$$), $$y$$-velocity ($$v$$), and pressure ($$p$$) at each of the 10,000 cells.)  Note that only a fraction of the computed velocities are shown so that the figure isn't saturated with vectors. The next figure shows more of the vectors (still not all) and you can see how it starts to become more difficult to interpret.  Note: you can see a full-size rendition of any of the figures in this article by right clicking the figure and selecting "Open image in new tab".

The next 2 figures also show the velocity vectors, still colored by velocity magnitude, but in a format that might be easier to interpret. Once again not all of the vectors are being shown, but this format has its advantages for some purposes. ALL of the vectors are shown that originate at a number of selected slices through the cavity. However this format tends to obscure the fact that there is a vortex whirling around a particular location inside the cavity.

The next 2 figures are contour plots – essentially each is a "topographic map" showing you where the highs and lows are. The figure on the left shows velocity magnitude contours.  Note however that while the actual velocity in the solution ranges from 0.0 to 1.0, the color scale has been edited, ranging from 0.0 to 0.5 only.  This was done because many of the velocities in the solution are lower than 0.5 and so the majority of the figure would just be some shade of blue if the color scale weren't adjusted (I'm just adjusting the color scale on your color flow Doppler screen). Similarly pressure contours in the right-hand figure have also been adjusted. Obviously there is a big pressure high point in the upper right-hand corner of the cavity and a low point in the upper left-hand corner. There isn't a lot of variation in the pressure over the rest of the solution, but adjusting the color scale at least allows us to see that there is a relative pressure low at the center of the vortex.

We are not interested in the details / mathematics of CFD in this article, but rather in some principles of fluid mechanics that may be valuable for cardiology.

### Effect of Reynolds Number on the LDC Solution

A fundamental aspect of fluid mechanics is that the characteristics of a flow are governed by the Reynolds number (Re).  The following sequence of figures is one of several locations of the website that illustrates and examines the fact.   The first group of 6 figures show solutions for increasing values of Re.  The figures are essentially polluted with vectors, but this particular format allows us to readily see the location of the vortex center (the "eye of the hurricane"), which moves from a location somewhat towards the upper right quadrant towards the center of the cavity with increasing Re.  The precise location of the vortex center is one aspect involved in the critical evaluation of the computational algorithm; I haven't looked at that for these solutions but there's a meager comparison with other solutions below.

Re = 100, 250, 500

Re = 1000, 2000, 5000

Re = 100

Re = 250

Re = 500

Re = 1000

Re = 2000

Re = 5000

### Understanding the Continuum Hypothesis and Continuity (LDC Example Version)

Cardiovascular clinicians typically are aware of something referred to as the continuity equation which appears in a form resembling the following:

$$\Large A_1 v_1 = A_2 v_2$$

This usually appears in the context of evaluating a stenosis where we can measure/estimate an upstream area and velocity ( subscript 1 ) so that the cross sectional area at station 2 can be evaluated after measuring the velocity there.  I will refer to this as the clinical continuity equation since there are some assumptions involved that may not always be appreciated. These include:

1. The blood is incompressible ( excellent assumption )
2. The structure between stations represented by the subscripts is rigid/noncompliant ( generally not a good assumption, but often adequate for the purpose )

It should also be clear that the velocities represented in the equation are averages over the cross-section and directed perpendicularly to the areas noted in the equation.  We are usually talking about making the velocity measurements with a Doppler interrogation which does not determine the averages for us.  Just letting you know there's a little bit of slop in the concept as presented.  For the next section, we'll consider a more advanced conceptualization of the "continuity" principle using the LDC simply as an example situation; the principles apply to all fluids, not just this particular flow.  Subsequently (following section), a more complete mathematical / conceptual representation of continuity, one that applies i.e. must be true at every location withing the fluid (the continuity principle applied to a point in space within the fluid).

Within the precepts of a continuum is a concept that each material descriptor (mass, velocity, etc.) is continuous (differentiable), i.e. that it changes smoothly from one location to the next.  We know that this isn't true if you look closely enough - all matter is particulate at some level.  But the continuum hypothesis often is an excellent approximation until we start looking very closely, i.e. submicroscopically.  Clearly a fluid (or solid) can "break" apart so that all of the parts are no longer connected; my understanding is that the continuum approximation is no longer valid at that point, although it still applies to the parts.  Engineers also consider a shock wave as an example where there can be a discontinuity (in pressure, density, velocity, etc.), but this too is a kind of approximation that isn't really true if you look closely enough.

We'll now explore the consequences of the continuum hypothesis as it applies to the LDC problem.  Consider first that the rectangular (square) cavity is closed with respect to the entry or exit of any fluid.  The size of the cavity is fixed and the fluid is incompressible.  Consequently the fluid must move in such a way that fluid elements continue to "get out of each others way" (2 fluid elements can't occupy the same space) but without "separating"; the fluid can't break apart.  In 2 dimensions, one of the resulting implications is that the path of each fluid element forms a ( simply ) closed loop.  This was implied by the video at the top of the webpage which tracks simultaneous trajectories of a number of fluid elements. After showing the velocity vectors obtained for the solution, the video starts by labeling some of fluid elements and tracking them over time.  Here's a freeze frame image from an animation similar to the one at the top of the page, where fluid elements at a given horizontal "surface" were labeled at the start and all of their trajectories were tracked over time.

This was a Re 250 example.  The next 2 figures show a Re 5000 example that I selected because someone put an example on the web of the Stream Function at that Re (My software doesn't compute stream functions because it's designed for 3D.)  Hopefully you'll appreciate the marked resemblance between the figures.

Although fluid trajectories may not form an exact closed line in the figure on the left, this would be due to an artifact of the solution method; fluid element paths were tracked numerically so that the streamline of a particular element doesn't necessarily join itself exactly after completing a cycle around the vortex.  The stream function plot on the right however is precise on this point; i.e. fluid element trajectories for this 2 dimensional problem must each form a closed loop (since fluid elements don't exit the domain).  The point of showing this aspect of the solution is for you to recognize that (in 2 dimensions and with time invariant flow) fluid element trajectories (streamlines) cannot cross or touch each other.   If we were to stick in another streamline between any 2, it would stay between the streamlines on either side of it.  Note: Technically I'm showing fluid path lines in the figure on the left. Streamlines are defined as lines that are tangent to the velocity field at a particular moment in time; they can change from moment to moment if the flow is not time-invariant.  However streamlines and path lines are the same thing for the steady flow (time-invariant) situation we're talking about (and so are streak lines).

It so happens that we can restate this fact in terms of the clinical continuity equation. The figure below is a close-up of some streamlines ( fluid element trajectories) from the LDC solution.  Consider now the region bounded by the 2 white line segments and the streamlines connecting the edges of the line segments. The streamlines are tangent  to the fluid motion; no fluid crosses the bounded region except at the 2 white line segments (where the vectors are shown).  The size of the bounded region is fixed; an incompressible fluid cannot accumulate within the region.  Consequently the rate that fluid flows into the region must exactly equal the rate that it flows out.  This means that the product of the "area" (actually "length" of the white line segment for the 2 dimensional case) and the ( average ) velocity at the lower left entry point must exactly equal the area-velocity product at the exit point: $$v_1 A_1 = v_2 A_2$$.  If we allowed the streamlines to cross, we would have a situation where the cross sectional area goes to zero ( and negative ); this would mean that the velocity would have to go to infinity which would be a meaningless, nonphysical result.

A simpler way of conveying the preceding paragraph is simply this: The flow rate between (any)2 streamlines is a constant.

A corollary of the above statement is that we can tell that the fluid is speeding up or slowing down by looking at how far apart the streamlines are because the area-velocity product  between 2 streamlines remains constant; the closer the streamlines are, the faster the fluid between them is moving.  With reference to the above figure, we see that the fluid enters ( crosses the white line segment ) at the lower left, slows down as it makes the bend to the right, and then increases in velocity, exiting the "stream tube" at a higher velocity than when it entered ( upper right white line segment).   The flow rate through this "channel" however is constant.

Consider the 2-D cavity again but remember that the walls of the cavity are rigid and that no flow passes into or out of the square.  In that case, ANY (simply connected) curve or line that starts and ends on a wall (red or blue curve examples below) partitions the cavity into TWO fixed areas.  Since neither of the spaces thus defined can expand or contract, the net flow across any/every line thus defined must  equal zero.

This may be intuitively obvious to you, but we'll be a little more rigorous.  We just proved that the flow rate occurring between any 2 streamlines remains constant. On careful consideration you'll also be able to surmise that any line that starts and ends on the surface of the cavity will transect each of the streamlines it encounters TWICE (or some multiple of 2) in such a way that there is both an inflow and outflow streamtube crossing the line

Finally we can consider ANY closed loop within the 2 dimensional flow (green closed loop above) and see that any streamline it encounters must also be cut TWICE or some multiple of 2.  A streamline pair defines a "channel without walls" through the fluid and any/every closed loop within the fluid will have both an inflow and outflow point for each channel.   No flow can accumulate in any region because the inflow of each must exactly equal the outflow.

A technical note on this subject; when specifying an incompressible CFD problem where the flow can enter / exit the domain (e.g. a conduit or tube flow of some kind), it's essential to specify that the net inflow and outflow rates are numerically identical and I do not mean "pretty close".  This may seem like a silly statement, but remember that we're talking about specifying a complicated computer problem involving thousands of equations.  It's entirely possible in this circumstance to tell the computer that the inflow rate is 1.00 and the outflow rate is 1.01.  This problem as stated does not have a solution!  These problems are typically solved iteratively, advancing closer and closer towards the correct answer until the solution error becomes acceptably negligible.   An error in problem specification like the one described means that the solution algorithm will simply hover around some unacceptable error level and never reach an answer (yep, I've done that).  You cannot have a rigid tube where more incompressible fluid is flowing in than out or vice versa.

For the next phase in our examination of fluid behavior in this flow, we'll work on visualizing how the fluid distorts / deforms in the process of flowing.  Upon first becoming acquainted with fluid dynamics, I found this aspect of the discipline actually a little shocking and am still fascinated by it.  In a virtual sense, we're going to "label" some of the fluid in the flow and track it as the flow evolves.  In a real flow, this can be done using a variety of flow visualization techniques.   A thin wire that traverses the domain, for example, can be made to release a line of bubbles by passing a current through the wire (the bubbles have to be the right size so they don't float and distort the flow).  Instead of focusing on the velocities as they vary within the domain (Eulerian frame of reference), we'll now focus on the material that comprises the fluid (material or Lagrangian frame of reference).  You are familiar with this method of course, using radio-opaque dyes or echo-contrast to track flow through the circulation.  However we're interested in this from a much more detailed perspective.

The next video starts by labeling multiple material lines of fluid.   I'll start by stating that material fluid elements remain distinct.  See if you can imagine how that's going to happen before viewing the video.

We see that each material line deforms.  Each can shorten and/or lengthen as time goes by, and they certainly don't remain straight.  However the material lines don't cross each other.   While they can get pretty darn close to each other, they begin distinct and remain distinct.

The software suggests that some of the lines actually break, particularly when they get close to the lid where they get stretched dramatically!  That's due to a program criterion that line segments reaching about 100 times their original length are no longer shown.  From a practical standpoint, material lines in a real fluid also get stretched to the point that the line is no longer distinct. No doubt you're familiar with the fact that you can always mix up a fluid to an extent that there's no hint that the material lines have not been interrupted.  It may be more constructive to think of bread / pizza dough with a line of food coloring in it. As long as you don't physically cut the dough, the line will remain intact as you stretch and knead material.

For a final visit to this continuum / continuity issue, consider the next video which starts by labeling the entire 2 dimensional domain with a square grid. Try to imagine how the grid will deform before watching the video; remember that the only thing different between this video and the last is simply that the material lines have been identified from the start in both the $$\large x$$ and $$\large y$$ directions.

Although there are numerical limitations in tracking the fluid elements, this video reveals a fact that I find striking.  Fluid elements remain touching the same ones they've (always) been touching!  An element can deform dramatically, but it can't get away from the fluid element that's (always) been right next to it.

Consider the line that defines the border between any two fluid elements.  At each point on the line there is a single velocity; the velocity can vary along the line but we can't have 2 velocities coexisting at the same point ( that would be an example of a discontinuity ).  So the border between 2 fluid parcels is a single border; it can't separate into 2 borders that move away from each other.  But this statement applies to any / every border between fluid elements, down to the level at which the continuum hypothesis is no longer adequate.

In the 2 dimensional incompressible flow shown, there is another instructive aspect.  At the start of the video, each fluid parcel is labeled as a square.  Although these squares subsequently deform dramatically, the cross sectional area of each /every must remain constant.  This may not be mathematically true for the numerically limited demonstration shown, but it's absolutely true for a true continuum.  Similarly in three dimensions, a bounded parcel of fluid cannot change its volume in an incompressible flow.  The surface of the parcel remains intact and remains touching the surrounding parcels – forever!

One reason that the video exhibits limitations for conveying the statements I've made above is that the solution has 2 singular cells where the boundary velocity specifications actually are physically unfeasible. These occur at the junctions of the lid with the cavity, i.e. at the upper right and upper left corners.  If you look closely at these cells, it's apparent that we've actually said that 2 different velocities exist at a single point. The lid is moving to the right at a specified velocity, but at the corner itself we are also saying that the fluid is stopped.  Exactly what happens at these corners in a real-life  situation could be pretty complicated and we won't delve into it here. Suffice it to say that the lid driven cavity problem has these limitations that we know about and the solution can't be expected to be highly accurate as we get too close to the singular points.

Understanding the Continuum Hypothesis and Continuity (Math Version)

The last section started out with the clinical continuity equation; it had some shortcomings. In the field of fluid dynamics, the continuity equation looks a little different:

$$\Large \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \equiv \nabla \cdot \mathbf{u} = 0$$

where $$\large u$$, $$\large v$$, and $$\large w$$ are the $$\large x$$, $$\large y$$, and $$\large z$$ components of the velocity vector $$\large \mathbf{u}$$ respectively.  That's if the fluid is incompressible.  For folks who deal with fluids flowing at significant Mach numbers, fluid compressibility become significant and the continuity equation looks scarier:

$$\Large \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} = \frac{\partial \rho}{\partial t} + \nabla ( \rho \mathbf{u} ) = 0$$

These continuity equations depict the fact that mass is conserved at each and every location within the fluid domain, but take into account the possibility that velocity can vary from point to point within the fluid.  With fluid density constant, the incompressible version indicates that there is no net flow into (or out of) any location; the compressible version states that the rate of accumulation of mass at a location is equal to the rate of mass inflow minus outflow.

We're now in a position to consider and derive one of the continuity equations shown above.  The clinical continuity equation was for a rather specific circumstance relating to flow across a stenosis.  We'll now look at an arbitrary parcel of fluid within the domain.  We'll do this for a small square of fluid in 2 dimensions; I assure you that the square is arbitrary enough and that the concept extends readily to 3 dimensions.

The center of the small square below is an arbitrary parcel of fluid within the domain. We've defined location $$\large x,y$$ at the center of the square with $$\large x$$ being the horizontal coordinate, $$\large y$$ the vertical.  The square has dimensions $$\large \Delta x$$ by $$\large \Delta y$$, the lengths of the horizontal and vertical directions.  $$\large \mathbf{u}(x,y) = (u,v)$$ is the velocity vector as a function of position and has $$\large x$$ and $$\large y$$ components, $$\large u$$ (in red) and $$\large v$$ (in blue) respectively.  The velocity changes with location within the domain and the vectors have been drawn is such a way to bring this to your attention. We designate the velocity at location $$\large x,y$$ as $$\large \mathbf{u}|_{(x,y)}$$ or $$\large (u,v)|_{(x,y)}$$.

At the left hand border of the square, the velocity is $$\large (u,v)|_{x-\Delta x/2,y}$$ and similarly designated velocities are shown at the other borders.  Then a mathematical expression for the rate at which mass crosses the left hand border is:

$$\large (\rho u)_{(x-\Delta x/2,y)} \Delta y$$ : left border

$$\large \rho$$ is the density of the fluid, which is also allowed to change with location (for the moment); $$\large \Delta y$$ is the surface "area" of the left border.  It's an important point here that although the velocity at the left border has both $$\large x$$ and $$\large y$$ components, only the $$\large x$$ component of the velocity contributes to the transport of fluid (mass) across the border.  More generally, only the component of velocity that is perpendicular to the border contributes to transport across the border.  The component of velocity perpendicular to the transport surface is the flux velocity  where flux refers to the fact that something is crossing the border.

To continue with the development, the transport of mass across the 4 borders can be written as:

$$\large -(\rho u)_{(x-\Delta x/2,y)} \Delta y$$ : left border

$$\large +(\rho u)_{(x+\Delta x/2,y)} \Delta y$$ : right border

$$\large -(\rho v)_{(x,y-\Delta y/2)} \Delta x$$ : lower border

$$\large +(\rho v)_{(x,y+\Delta y/2)} \Delta x$$ : upper border

Again only the $$\large y-$$ directed component of velocity ($$\large v$$) at the upper and lower borders contributes to mass flux across the border.  $$\large +$$ and $$\large -$$ signs have now been added to indicate the whether the fluid (mass) is entering ($$\large -$$) or leaving ($$\large +$$) the square.

And now we're ready to express a physical fact, one of mass conservation, using these preliminary results.  We combine the statements for fluxes to indicate that the  net rate of mass accumulation within the square is equal to the rate that mass is entering minus the rate that it is leaving the square:

$$\Large \frac{d \rho}{dt}|_{(x,y)} \Delta x \Delta y = - \left[ (\rho u)_{(x+\Delta x/2,y)} -(\rho u)_{(x-\Delta x/2,y)} \right] \Delta y - \left[ (\rho v)_{(x,y+\Delta y/2)} -(\rho v)_{(x,y-\Delta y/2)} \right] \Delta x$$

The stuff on the right hand side is the sum of the fluxes derived above -- all the mass entering and leaving the square.  The left hand side is rate of change of the density multiplied by $$\large \Delta x \Delta y$$, the "volume" (area) of the square; it's the rate of change of mass inside the square.  Now we divide the whole by $$\large \Delta x \Delta y$$:

$$\Large \frac{\partial \rho}{\partial t} = \LARGE - \frac{ (\rho u)_{(x+\Delta x/2,y)} - (\rho u)_{(x-\Delta x/2,y)}}{\Delta x} - \frac{ (\rho v)_{(x,y+\Delta y/2)} -(\rho v)_{(x,y-\Delta y/2)}}{\Delta y}$$

And for the final step we allow the size of the square to shrink, taking the limit of the expressions as both $$\large \Delta x$$ and $$\large \Delta y$$ approach zero.

$$\Large \frac{\partial \rho}{\partial t} = - \frac{\partial (\rho u)}{\partial x} - \frac{\partial (\rho v)}{\partial y}$$

or

$$\Large \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$

This is the continuity equation for a compressible fluid, a statement that mass is neither created or destroyed anywhere inside the fluid domain (so we're assuming that there is no nuclear physics going on here).  For an incompressible fluid, the fluid density $$\large \rho$$ is a constant and the equation reduces to:

$$\Large \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \nabla \cdot \mathbf{u} = 0$$

The understand how the algebra turned into a partial differential equation, it's necessary for you to understand that an expression like

$$\Large _{\lim_{\Delta x \rightarrow 0}} \frac{f(x+\Delta x/2) - f(x-\Delta x/2)}{\Delta x}$$

is actually a definition of the derivative in calculus, i.e. the slope of a curve defined by function $$\large f(x)$$ at  point $$\large x$$.  These definitions have been replaced with partial derivatives to yield the continuity equations for either a compressible or incompressible fluid.  Depending on the fluid (or more specifically the Mach number), one of these mathematical expressions must be true everywhere inside the domain and so one of these 2 equations must be solved (in some sense) for each individual cell of fluid parcel of the computational domain.  When solving CFD problems, we don't actually take the last step where the algebra was converted to a differential equation.  Rather the equation is left in the algebraic form to be solved numerically (several algebraic equations at each cell of the domain).

### More about the Lid Driven Cavity

The LDC problem is one that is simple in its definition, yet requires a rather sophisticated computational algorithm to seek a solution.  These are numerical experiments where exact answers in terms of mathematical expressions are either unavailable or so complicated as to be impractical.  As noted at the start of the article, the LDC problem is often used as a benchmark problem to help evaluate whether a CFD algorithm is working properly. The figure below (from the web-link at the top of the article) depicts a plot of the normalized x-directed velocity ($$u/U$$) at the center ($$x= 0.5$$) of the cavity (Re 100).  An accepted standard (Ghia et al 1982) is shown along with the corresponding solution computed using well-respected commercially available fluid dynamics software (FLUENT, red).  Also superimposed is the corresponding solution from my home grown CFD software (blue).  I'm not in the fluid dynamics business, and this graph is hardly the proof of the pudding, but I'd like you to have a little confidence that the information I'm showing isn't pulled out of a hat. Note: the FLUENT solution was computed using a 32x32 square grid whereas mine was 100x100 so liable to be more accurate (IF the algorithm actually does a good job of modelling the physics of the flow).  The folks who wrote the FLUENT code are a couple orders of magnitude smarter than me, so they "cheated".

### Evolution of the Solution (Result) over Time

When we solve a fluid dynamics problem of this nature, the computer algorithms are designed to represent physics, not just make pretty pictures.  To get the fluid spinning and flowing in the cavity, the problem actually starts with the fluid at rest and evolves over time, just as it would in the real world.  This has major implications for how long it takes to solve one of these problems.  In the figures that follow, we see some of the intermediate solutions that occur in the process of this time evolution. At time zero, all the fluid in the cavity has velocity zero. While we could choose some other designation, the solutions here were created by "impulsively" starting the lid motion. The lid is at zero velocity at time zero, but is moving to the right at full speed ($$v= 1.0$$) at the very first time step ($$t=\Delta t$$).  That begins to drag the fluid in contact with the lid to the right and eventually sets all of the fluid in the cavity in motion.  If you bother to look through some of the intermediate solutions, you'll see that it takes a while for the vortex to develop and drift to its final location (Think of a hurricane picking up energy and moving across the ocean.)  "Time" in this Situation is "problem time" and is actually a time ratio.  One time unit for the problem is equal to the time it takes for the lid to move the length of the edge of the cavity.  For this particular problem it took about 20.78 time units for the solution to "converge", i.e. for the velocities to stop changing according to some preset criterion.  In this case that preset was 1.0 x 10-6; i.e. the maximal change in velocity from one time step to the next was one millionth of the characteristic velocity of the problem (1.0).  Typically there are hundreds or thousands of time steps involved  for a problem like this to converge (10392 for this particular example; the actual number increases with the number of cells in the domain and with Reynolds number ); this was a problem in 30,000 variables at each of those thousands of time steps.  It's very computationally intensive.

$$t = 1.0$$

$$t=2.0$$

$$t=3.0$$

$$t=5.0$$

$$t=20.78$$

### 3-Dimensional LDC

I don't think it's standard fare in the fluid dynamics literature, but I decided to look at results for a 3-dimensional LDC when developing my CFD software.  The results are interesting and allow further insights into the intricacies of fluid dynamics.