Testing Theories in Fluid Dynamics
Wilka Carvalho (SUNY)
Introduction
Objective
Many problems in mechanical engineering, even seemingly simple problems, cannot be solved
analytically. Analytical solutions are only as accurate as the theories they are grounded in so
because the physical world is currently not perfectly understood, analytical solutions can often be
faulty. This is often the case in fluid dynamics, as it is necessary to experiment in order to gain an
accurate solution which may improve the theory and allow for a accurate future analytical solutions.
We used Bernoulli's Principe and the continuity equation to analyze the flow of water and verify the
analytical solutions for the behavior of water under numerous conditions. We tested solutions
regarding the flow of water in regards to: Reynolds Number, Boundary Layer conditions, Laminar
and Turbulent flow, etc. Our ultimate goal was to verify the theoretical solutions with numerical and
analytical solutions.
Theory and Methods
The experiments done focused mainly on water running through a pipe. For all the cases, because
we dealt with water that had little variation, it was assumed that the overall fluid was inviscid,
incompressible, and irrotational. The water’s behavior was governed primarily by Bernoulli’s
Principle and the continuity equation. Together the two allowed us to define the behavior of the
fluid at different points along the pipe, as well as with different influences along the pipe acting on
it.
Bernoulli’s Principle, in essence, is a principle of conservation of mechanical energy. It states that
the sum of the energy of a system always remains equal, so the sum of its components must remain
𝜌𝑣 2
+
2
equal and can be expressed as: 𝑃 + 𝜌𝑔ℎ
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, where P is pressure, 𝜌 is density, g is
gravity, h is height, and 𝑣 is velocity. For the purpose of our experiments it was interpreted as: the
mechanical energy at one section of a pipe is equal to the mechanical energy at another section of a
pipe.
In fluid dynamics, the continuity equation states that flow through a tube of varying cross section
ⅆ𝜌
must be equivalent at all points and can be expressed as:
+ 𝛻 𝜌𝑢 = 0, where 𝑢 is the flow
ⅆ𝑡
velocity vector field. It is based on the Law of Conservation of Mass, which states that mass can
neither be created nor destroyed. In a steady state process (a process where a variable remains
constant) of a fluid flowing through a pipe, the rate at which mass enters the system is equal to the
rate at which it exits the system. We dealt only with incompressible fluids, which by definition of
the continuity equation meant that the local change in volume of the fluid was 0. This meant that at
any two sections along the pipe, the rate of change of Volume was equivalent, which led to the
simplification: 𝐴1 𝑣1 = 𝐴2 𝑣2 .
Figure 2: This
graph depicts
the inverse
relationship
pressure has
with height and
velocity under
Bernoulli’s
Principle when
applied to a pipe
of varying cross
section.
Figure 1: This
diagram
depicts the
application of
Bernoulli’s
Principle and
the Continuity
Equation to a
pipe with
varying cross
section and
height
Examples
References
Munson et al. (2009). Fundamentals of Fluid Mechanics. Jefferson
City: Don Fowley.
Turns, S. R. (2000). An Introduction to Combustion : Concepts
and Applications. Singapore: McGraw-Hill Higher Education.
Acknowledgements