Basic fluids
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The document discusses fluid simulation techniques, beginning with the basic equations of incompressible fluid dynamics. It describes how to numerically solve these equations using operator splitting and a staggered grid approach. Key steps include advecting quantities using a semi-Lagrangian method, integrating body forces, and solving a Poisson equation to enforce incompressibility through pressure updates. Boundary conditions and techniques for simulating smoke and heat are also covered at a high level.
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Max Yuen
Mar 2020
(use g = 9.8m/s2 for all problems.)
Background
Many physics problems cannot be solved directly by hand or analytically. We resort to numerical
methods to give us approximations to the problem. In this lab you will learn the Euler method,
which allows you to solve Newton’s laws of motion. This is done by treating the velocity as a
piecewise linear function with many time intervals and during interval the acceleration is assumed
to be uniform. This allows us to use the kinematic equations we learned about in the first half
of the class to approximate the motion. If we choose to partition the motion into smaller time
intervals, the approximation becomes much better since the differences between adjacent intervals
become smaller. In this lab, this numerical analysis method will be applied to the motion of a
falling object under the influence of gravity and drag force. If you are adventurous, you can even
try to extend this to 2D and compute the realistic trajectory of a baseball. You might even try
some other problems, such as a mass attached to a spring.
Euler’s Method Foundations
This method is well suited for problems where the acceleration is a function of the velocity, as in
the case of a falling object under the influence of gravity and drag:
a = f(v) (1)
Falling object with drag force
The model for drag fits the prescription for using Euler’s method since the net force on a falling
object with drag is given by:
ma = −mg −FD (2)
ma = −mg −
1
2
ρairACDv
2 · sgn(v) (3)
a = −g
(
1 +
ρairACDv
2 · sgn(v)
2mg
)
(4)
a = f(v) ← Equation of Motion (5)
where m is the mass of the falling object, a is the acceleration of the object (which is positive when
pointed up), ρair is the density of air (about 1.29 ·10−3kg/m3), A is the cross-sectional area, CD is
the drag coefficient, v is the object’s velocity, and sgn(v) is the signum function which returns the
sign of the argument. The second signum function is there to guarantee that the direction of the
drag force is always in the opposite direction of the velocity function. Note that we see that the
acceleration is an explicit function of v, which sort of makes this a chicken or egg problem. This is
because we need a to get v, but to get a we need v, so which one do we compute first? Hold that
thought. We’ll talk more on how to program this in EXCEL or Google Sheets later.
1
Figure 1: FBD for an object falling under the pull of gravity and resistance by drag force
Terminal Velocity
In lecture, we talked about how after waiting for some time, if the object started at rest the
speed will increase and the drag force will also become larger and eventually balance out with the
gravitational force. When this happens, we have reached terminal velocity vterm = −v. This can
be solved by setting a = 0:
0 = −mg −
1
2
ρairACDv
2 · sgn(v) (6)
2mg = ρairACDv
2
term (7)
→ vterm =
√
2mg
ρairACD
(8)
Using this definition for the terminal ...
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