Introduction:
There is a number of interesting content in the Internet about water solving a maze. As a fluid dynamics expert I was intrigued by this phenomenon and always wanted to study the physics behind this phenomenon from a fluid dynamic point of view. This study will focus on the fluid dynamic parameters that are responsible for this interesting phenomenon. We will be using CFD simulations to study and visualize and understand this fluid phenomenon.Preparing for CFD Simulation:
So to start a CFD simulation we will be needing a geometry, in this case a maze, then we will need to fix some parameters (boundary conditions and equations) and assumptions for the simulation.
Geometry:
We will be using a 2D mace which is placed in X Y plane, where X being the horizontal axis and Y being the vertical axis and gravity will be along negative Y axis. To reduce the dynamic pressure from the inlet fluid velocity we have considered a reservoir above the mace and have also considered a reservoir below the mace to prevent back pressure or reverse flow. The flow inside the maze will be along the direction of the gravity in other words the flow inside the mace will be aided by gravity.
Equations:
In order to accurately capture the physics we will need to take into consideration all the factors that would affect the flow in the maze. There are number of fluid dynamic equations that represent different characteristics of the fluid flow. Hence it is imperative that we choose the right equations that influence our fluid flow of interest.
The first step will be to choose the fluid that will solve the mace, in our case we can choose water. But water will not be alone it has to be accompanied by air. If we consider water only(without air), that means the mace is in a vacuum. This is not a realistic scenario hence we have to consider air also in the simulation. Hence we have to consider a multiphase simulation where air will be the primary phase and water will be the secondary phase. Appropriate multiphase equations has to be considered that will take into account the interaction between the two fluids.
Second, we need to consider the gravitation forces, in our case we will consider gravity along the negative Y axis. The simulation has to take into account the interaction between the fluids and the wall and also the interaction between the two fluids. Hence fine meshing of the fluid domain is
required to accurately capture the frictional losses that takes place because of fluid interaction with the walls of the maze and also to capture the surface tension effects of water. Glass is considered the material for the mace walls. The flow is considered to be a viscous flow.
Third, of all the factors that can affect the flow of the fluid the most important factor will be turbulence. Capturing the fluid turbulence accurately will be the difference between a stimulation that closely captures a realistic flow scenario and an inaccurate simulation that loosely resembles a fluid flow. Fluid flow is complex and this complexity is derived from the turbulence, hence simulating fluid flow without the turbulence equation and considering the simulation to be laminar will be easy but will be inaccurate and hence misleading. Considering the simulation to be laminar is not practical especially after taking consideration of the size of the geometry and after calculating the Reynolds number of the flow it can be easily concluded that the flow will be turbulent. Hence appropriate turbulence equations has to be considered in the simulation.
“Turbulence adds the complexity to the fluid flow and since turbulence is the most important factor in simulating a fluid flow we have named our company after this fascinating fluid phenomenon. Flowturb is short for flow turbulence.”
Fourth, the flow is considered to be highly dynamic and varies with respect to time. Hence we need to consider a transient incompressible NS(Navier-Stoke) equations. This will help us to study the flow dynamics with respect to time and the time step for this simulation must be very small in order to capture the smallest flow variations. This equation is the momentum equation that evaluates the variation of fluid momentum inside the area of interest, by accurately calculating the velocity, pressure and density variations.
Fifth, there could be a possibility of phase change, due to localized variation of temperature and pressure of fluids. In the event of phase change either water could vaporize and become moisture in the air or moisture in the air can condense to become water. This phenomenon can be accounted for by considering a species transport equation along with temperature equation.
Finally, continuity equation will be accounted to account for the mass of fluids that enter, stay and exit the fluid domain. This equation evaluates the principle of conservation of mass by making sure all the mass inside the area of interest has been accounted for.
These equations we have noted above are the governing equations for this flow regime and solving all of these equations will help us predict, simulate, visualize and help understand our flow of interest.
CFD Simulation:
Now that we have completed the preparation we can proceed with the simulation. The simulation will be a transient simulation and hence we start with time zero seconds. At zero seconds water from the inlet which is placed on the extreme left side top corner will start to flow inside the area of interest at the velocity of 1.5 m/s the length of the inlet opening is 10m. The water will first fill the reservoir and then over flow into the maze. There are pressure relief openings present on both the upper and lower reservoirs to prevent pressure buildup which may affect the inlet velocity for water entering the maze. Once the water enters the maze it will try to fill the closest chamber and then try to move on to the next chamber. Water will choose which chamber to move next based on the pressure difference. We need to remember that both the reservoirs and the mace are already filled with air so when water enters a chamber it displaces the air out of that chamber. If a chamber is enclosed, water occupying more and more volume would result in reduction in volume would result in increased pressure. On the other hand if a chamber is not encloses, air will have the opportunity to escape to other chambers and hence avoid pressure build up. This process of flow distribution based on pressure difference will continue until the water reaches the other side of the maze where the exit is located. It will take some time for the water to reach the other side of the mace. Once the water reaches the lower reservoir the simulation is complete.
“The governing equations will be solved for every mesh element in the fluid domain and for every time step. In our case we are going to consider uniform time step of 0 .001 seconds to ensure accuracy. Hence 1000 time steps have to be calculated in order to accurately capture 1 second of real time.”
Higher order discretization methods has to be used to ensure accuracy and appropriate under relaxation factor has to be considered for faster convergence. Numerical convergence has to be achieved for every time step to prevent inaccuracies.
Result Observations:
The simulation ran from 0 seconds to 250 seconds. The initial 50 seconds water fills the top reservoir. After 50 seconds, water enters the maze and starts filling the first chamber (C1) below the entrance. Within 59sec the chamber is filled and the water needs to move to the next chamber. By 60sec
water starts to fill R1 chamber on the right side and by 100sec it is filled and chamber R2 is half filled. But on the contrary, the chamber in the left side L1 has only been partially filled despite the proximity to the C1 chamber. By 224sec, the L1 chamber is full and small quantities of water has overflown but on the right side water has filled multiple large chambers and has reached the exit point. Of the total volume of water that entered the maze only a small insignificant volume of water has taken the wrong path whereas the remaining large volume of water has moved towards the right direction and has completed the maze in
224 seconds.
The reason for this behavior is as follows,
1. When water enters a chamber, it displaces air which was previously occupying the space. Now there are two scenarios that can happen to a displaced air,
a. Scenario 01: Air is in an enclosed space and it is getting displaced by water.
There is nowhere for air to go and hence it is getting compressed by the incoming volume of water. This results in an increased air pressure in this region. This rise in pressure reduces the flow rate of water entering this enclosed area.
b. Scenario 02: Air is not in an enclosed space and it is getting displaced by water.
Now air has the option to escape and it moves to the next chamber, If the next chamber is also filled by water it can go to the next chamber and hence it can go on until it reaches the last chamber. And even if the last chamber is filled with water, air can exit the mace. Or air can get lodged in portions of a chamber that water could not reach. Hence there is no buildup of pressure and there is no force excreted on the water that can reduce its flow rate.
2. So water has no restriction when it flows towards the right hand side chambers (R01, R02,..etc) but there is a substantial resistance when it tries to move towards the left hand side chambers and this resistance will increase as it occupies more space inside the left side chambers.
3. As water enters the maze and fills the C1 chamber it would be observed that the dynamic pressure of water falling down helps it to move towards the right side chamber easily because of the geometry this also adds to the reason why water has effortlessly filled the right side chambers.
4. As water enters the right side chamber air gets trapped and it has nowhere else to go. Hence there is a pressure buildup. The more water enters these chambers the more pressure builds up but after the pressure reaches a particular value some amount of air escapes to the other side and this relieves the pressure.
Closing Words:
The simulation results has clearly illustrated the complex but interesting fluid flow characteristics behind the phenomenon of water solving maze. It fascinating to see how nature when seen form the outside seems very chaotic and random, but on closer study expands into complex web of algorithmic instructions programed into the very fabric of nature and runs with unparalleled precision. Now that we have understood the complex beauty of this flow phenomenon, we can now appreciate the elegance of the pressure correcting flow behavior of water. I sincerely hope this study will refresh our sense of admiration, respect and appreciation for nature and would motivate us to preserve and conserve our truly magical planet.
Author
Antony
Director – CFD Operations
