1. Energy of a Stable Bathtub Vortex
Aaron Meyer
April 26, 2016
Mathematical Modeling
University of Utah
This paper focuses on modeling the energy present in a specific type of vortex. Using
previously developed models, the parameters for optimizing the amount of kinetic
energy contained in a ‘bathtub’ type vortex are evaluated. Power generation is the
focus of this optimization, and as such quantities potentially effecting the amount of
kinetic energy in the vortex are analyzed in the context of power production via turbine.

2. 1 Introduction
Fluid dynamics is an extensively researched and constantly evolving field of mathematics.
In this report, we examine one specific phenomenon which occurs within fluids and attempt
to model its behavior. A vortex is a region within a body of fluid in which the fluid elements
have an angular velocity. In this paper, existing models for fluid flow within a vortex are
examined, with emphasis on their velocity fields. Following the initial research, the outline
for the problem we wish to answer was developed, along with assumptions which allow us
to put forth a reasonable model.
The goal of this paper specifically is to analyze a stable, irrotational, bathtub vortex and
find the kinetic energy associated with such a vortex. By developing a model for the kinetic
energy, we are able to optimize fluid characteristics and turbine placement in the context of
maximizing the energy of the system. The intent of this optimization is to create a model
accompanying the idea of energy harvesting from such a vortex, formed by a constant
azimuthal inflow of fluid into a stationary cylinder. The basic idea of such a system can
be seen in the following illustration.
Figure 1: A stable vortex power generation system
By developing a model for the kinetic energy of the vortex system, this paper aims to
provide a qualitative description for calculating the amount of power that could be produced
in a simplified, idealized turbine system.
2 Background
Extensive research into the governing mathematical and physics based issues was necessary
in order to fully understand the problem we are posing. This section details the reasons
for selecting this problem specifically, as well as important research which provides the
background necessary to understand the issues surrounding the proposed model.
2.1 Motivation
The ‘bathtub vortex’ is a common phenomenon which is familiar to most observant people.
What appears as an interesting whirl around drains in daily life has its roots in generalized
1

3. fluid dynamics problems which have been extensively researched. Coupled with a back-
ground in civil engineering, finding a way to model this phenomenon in a relatable way
was an important motivation. Using power generation as the main catalyst for learning
about vortices, we looked specifically at the energy present in a bathtub vortex and made
an attempt to model a larger system. This proved interesting as while these types of power
generation plants exist, there is little published research regarding the math governing
these structures. It is our hope that the models proposed create a knowledge base which
can be expanded to creating real, physical systems.
2.2 Key Research
Upon initial research, we discover that there are a multitude of types of vortices with
models describing their flow. In the interest of simplicity, the ‘bathtub’ vortex was chosen
as it is defined as an axisymmetric, stable, line vortex described by Andersen et al.[1]
The main issue present when modeling fluid behavior is finding and using solutions to the
Navier-Stokes equations. These equations represent the relationship between the viscous
forces present in a moving fluid and a pressure term. Solutions come in the form of a
velocity field, which in our analysis is necessary to find kinetic energy. The Navier-Stokes
Equations can be represented as the system:
∂u
∂t
+ u · u = −
P
ρ
+ ν 2
u, (1)
· u = 0 (2)
where u, P, ρ, and ν are the velocity, pressure, density, and kinematic viscosity, respec-
tively.
3 Model
3.1 Notation and Assumptions
The vessels in which vortices are analyzed here are axisymmetric cylindrical tanks, therefore
the use of cylindrical coordinates (r, θ, z) is employed. The tank bottom lies in the (r, θ)
plane with the drain hole in the center such that the central axis and floor of the vessel
correspond to r = 0 and z = 0 respectively. The velocity field within the tank is described
by the vector u = (ur, uθ, uz). The radius and height of the tank are R and H respectively,
the radius of the drain hole is d/2, and the density of the fluid in the vessel is denoted
ρ = (ρr, ρθ, ρz). The variable Re appears extensively in this analysis and represents the
Reynolds number, a dimensionless quantity describing the ratio of inertial forces to viscous
forces in the fluid such that:
2

4. Re =
4Q
πνd2
(3)
where Q is the outflow rate and ν is the kinematic viscosity of the fluid.
We assume an incompressible fluid of constant density (water) meaning ρ = ρ. Additionally,
the model for velocity used is Burgers’ vortex, which solves the Navier-Stokes equations
exactly. Burgers’ vortex gives the velocity components as in the model used by Yokoyama
et al.[9]
ur = −
αr
2
, uθ =
Γc
2π
1 − exp(−Re ∗ r2/8)
r
, uz = αz (4)
where α > 0 is a constant and Γc is the circulation defined by Γc = R udr. We assume an
irrotational flow, therefore Γc is a constant by definition.[2]
Moreover, because we are specifically interested in the energy present to power a rotating
turbine, the axial (ur) and radial (uz) velocities are neglected. Therefore, the only velocity
considered in the kinetic energy calculations is uθ defined in equation (4).
By making all of the appropriate quantities dimensionless, we simplify the calculations by:
a) Minimizing the need to solve equations numerically, b) Making it easier to manipulate
quantities without the worry of unit discrepancies, c) Generalizing the problem to develop
equations which can be used to describe different dimensional inputs. To make quantities
dimensionless, we divide all dimensioned variables by characteristic quantities. In this
report, all dimensionless quantities will be shown with the prime (’) symbol. Lengths are
divided by characteristic length R (r = r
R ... etc) and velocities are divided by w, the
constant volume outflow velocity. Here, w = 4Q/(πd2), giving u = u/w.
Finally, because of the scope of this analysis, complicating factors such as boundary condi-
tions, surface tension, and pressure differences are neglected. See section 5.1 for how this
model may be refined.
3.2 Principal Equations
The principal equation describing the kinetic energy of particles in motion (kinetic energy
per unit mass) is KEparticle = 1
2ρ||u||2. Integrating over the volume of our cylinder and
converting to cylindrical coordinates gives:
KE =
V
ρ
2
||u||2
dV (x, y, z) =
ρ
2
V
||u||2
rdV (r, θ, z) (5)
Given our assumption of looking at only azimuthal velocity, this equation reduces to:
3

5. KEθ =
ρ
2
V
u2
θrdrdθdz (6)
where uθ is defined as in equation (4).
3.3 Method
Once equation (6) is derived, the optimization stage of the problem becomes our main focus.
The goal as stated earlier is to maximize this kinetic energy using the different variables
which may be subject to change. To begin this process, the kinetic energy must be made
dimensionless for the reasons outlined in section 3.1. This is carried out by transforming
the variables in the kinetic energy equation to dimensionless variables and extracting the
characteristics. Because of integral linearity, these quantities can be pulled out:
K ∗ KEθ = K ∗ (
1
K
)
ρ
2
V
u 2
θ r dr dθdz (7)
where K is composed of the characteristics such that K = ρH2O||w||2R3. Additionally,
by creating dimensionless differential lengths, the limits of integration over the cylinder
become 0 to H , 0 to 2π, and 0 to R = 1, in the r, θ, and z directions respectively.
Sustituting uθ from equation(4) gives the dimensionless kinetic energy expression:
KEθ =
H
0
2π
0
1
0
Γc
2π
1 − exp(−Re ∗ r 2/8)
r
2
r dr dθdz . (8)
Solving this integral provides the general expression for the azimuthal kinetic energy in
terms of the circulation, H/R ratio and the Reynolds number as follows:
KEθ =
H Γ2
c log Re
16 + 2Γ 0, Re
8 − Γ 0, Re
4
4π
(9)
where Γ(a, z) represents the incomplete gamma function. Clearly this integral result de-
pends on the constant H/R ratio, Γc and the Reynolds number. This result and its impli-
cations on the proposed optimization problem are discussed in the results section.
The next step in creating a complete model optimizing energy harvesting is describing the
amount of energy in the vortex with varying distances from the central axis. By integrating
over small r intervals between 0 and 1, it is possible to find an ideal turbine size.1 Using
1
Because the turbine is assumed to harness the kinetic energy perfectly (no friction) as well as have no
effect on the fluid motion, clearly a turbine with radius r = R would produce the most energy. However
the scope of this project is to show how we can relate increasing r to KE. Therefore, the results produced
are merely a representation of how turbine size COULD be optimized given information about its efficiency
and affect on the fluid flow.
4

6. a simple Matlab code (see appendix A) the integral below was solved using equally spaced
limits of integration (a, b) going from 0 to 1 in the dr integral.
KEθ =
H
0
2π
0
b
a
Γc
2π
1 − exp(−Re ∗ r 2/8)
r
2
r dr dθdz . (10)
The results of the models developed above are summarized in the next section.
4 Results
The model developed above gives a value for kinetic energy which depends on three variable
quantities: H , Re, and r . The dependence on the first two values (H and Re) arises
from the kinetic energy integral over the entire tank where the integral in the r direction
is evaluated from 0 to 1. The third value presents itself in the iterated integration with
variable limits over the radius.
4.1 Optimizing H and Re
From equation (9) it is clear that fixing the Reynolds number and assuming a constant
circulation produces a linear relationship between H and the dimensionless kinetic energy.
Upon converting this back into the actual amount of kinetic energy by multiplying by K
from equation (7), this term simply becomes a linear function of tank size. This result is
expected as increasing the tank size (radius and height) and keeping all other parameters
equal will produce a linear increase in kinetic energy.
We anticipate increasing the Reynolds number will produce a similar result as a higher
Reynolds number corresponds to inertial forces taking precedence over viscous forces. How-
ever, because the Re terms are embedded within incomplete gamma functions in equation
(9), the relationship is less obvious. The following graph shows how changing the Reynolds
number affects the kinetic energy in the system.
Reynolds Number (Re)
0 500 1000 1500 2000 2500 3000
KEθ/Γ2
c
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Modified Kinetic Energy vs Reynolds Number of a Stable Bathtub Vortex
Figure 2: Relating Reynolds Number and Kinetic Energy
5

7. 4.2 Optimizing Turbine Size
By integrating over small, constant length limits from 0 to 1 of the r integral, the result is a
spike in kinetic energy at r ≈ 0.125 as shown in figure 3. This shows us that implementing
a turbine which captures the majority of the kinetic energy on the curve is optimal. As
stated above, while a turbine with radius r = R would produce the most power, our goal is
to make the turbine small while capturing as much of the energy present as possible. Figure
4 gives the amount of kinetic energy when iterating b from 0 to 1 in the following:
KEθ =
H
0
2π
0
b
0
Γc
2π
1 − exp(−Re ∗ r 2/8)
r
2
r dr dθdz . (11)
giving the total kinetic energy present we could capture with a turbine of radius r =
r ∗ R.
r' Integration Upper Limit
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
KEθ/Γ2
c
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Kinetic Energy vs Vortex Core Distance Integration Limits
Figure 3: Equal sized integral limits from 0 to 1
in the r direction
r'
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
KEθ/Γ2
c
0
1
2
3
4
5
6
7
8
9
10
Kinetic Energy vs Distance From Vortex Core
Figure 4: Kinetic energy present from the vortex
core to r
These figures tell us that were we to make a power generation system as shown in figure
1, it would be optimal to make the turbine at least 0.2 times the radius of the tank to
capture the “spike” seen in figure 3. While adjusting different parameters such as drain
hole size and flow rate would change the overall values for kinetic energy, this ratio of r/R
will remain the same and provides a good generalized model for the amount of energy in
relation to distance from the vortex core.
5 Analysis and Conclusions
5.1 Improvements
While the model proposed is a good start to analyzing the kinetic energy present in a
vortex, there is plenty of room for improvement. The next step is to transition into a more
realistic velocity model for use in the kinetic energy integration. While Burgers’ vortex is
6

8. a decent representation of the fluid flow present in vortices, it does not capture elements
such as boundary conditions and surface tension. By using a more general model which
takes these properties into consideration, developing a more accurate and general model
would be possible which would reduce the number of assumptions. Doing so would produce
results which match more closely to how the system behaves in reality.
As the motivation for this model is related to power generation, incorporating the prop-
erties of a turbine used to capture the kinetic energy in the vortex would lead to a better
model. Clearly no turbine exists as posed in this paper, so taking friction, efficiency and
especially the turbine’s effect on the fluid flow is an important consideration when mov-
ing this model forward. Creating a velocity profile which not only recognizes the fluid
properties mentioned above, but also relates the turbine system would be a substantial
development and a good modeling problem in itself.
5.2 Conclusions
The analysis carried out in this report provides a good base model for continuing education
regarding the energy contained in vortices. Substantial assumptions were made in order to
create a simple description for the energy harvesting problem posed. We find that provided
an ideal turbine system, we can generally predict the amount of kinetic energy within the
vortex, with the variable quantities of tank size, Reynolds number of fluid, and distance
from vortex core. While substantial improvements and additional work should be carried
out in expanding this model, we believe this paper presents an accurate description and
starting model with respect to the question posed.
7

9. References
[1] Andersen, Bohr. “The Bathtub Vortex in a Rotating Container.” Journal of Fluid
Mechanics 556 2012.
[2] Acheson, D.J. Elementary Fluid Dynamics. 1st Edition. Oxford: New York: Claredon
Press, 1990
[3] Bohling, L; Andersen, A. “Structure of a Steady Drain-Hole Vortex in a Viscous Fluid.”
Journal of Fluid Mechanics 656 2010
[4] Cristofano, L; Nobili, M. “Velocity Profiles in Bathtub Vortices: Validation of Analyt-
ical Models.” 32nd UIT Heat Transfer Conference, Pisa, 2015 pg. 1-9.
[5] Lundgren, T.S. “The Vortical Flow Above the Drain-Hole in a Rotating Vessel.” Journal
of Fluid Mechanics 155 (1985): 381-412.
[6] “Sepp Hassleberger: Water Vortex Drives Power Plant.” Accessed April 21, 2016.
[7] Trivellato, F., Bertolazzi, E., and Firmani, B. “Finite Volume Modelling of Free Sur-
face Draining Vortices”. Journal of Computational and Applied Mathematics, Applied
and Computational Topics in Partial Differential Equations, 103, no. 1 (March 15,
1999):175-85.
[8] Tryggeson, Henrik. “Analytical Vortex Solutions to the Navier-Stokes Equation.”
V ¨axj¨o, Acta Wexionensia, 114 (2007).
[9] Yokoyama, Naoto, Yuki Maruyama, and Jiro Mizushima. “Origin of the Bathtub Vortex
and its Formation Mechanism.” Journal of the Physical society of Japan 81 (2012).
8

10. A
Matlab Code
1 % Aaron Meyer
2 % April 15, 2016
3 %
4 % Function Name: tripint
5 % Inputs: bins -number of iterations to be performed between 0
and 1
6 % hrratio -ratio of height to radius of tank
7 % re -reynolds number
8 % Outputs: r-array of r values at each bin iteration from 0 to
1
9 % ke -kinetic energy at each r value
10 %
11 % Plots: Plot of ke vs r, the kinetic energy versus the
distance from the
12 % vortex core , where each is the upper limit of the
integral
13
14 function [r,ke]= tripint(bins ,hrratio ,re)
15 fun =@(x,y,z) ((1-exp(-re*x.^2/8))/x).^2*x; % Create the
integrand
16 ref=linspace (0,1,bins +1);
17 r1=linspace (0,ref(bins),bins); % Lower set of integration
limits
18 r2=linspace(ref (2) ,1,bins); % Upper set of integration
limits
19 ke=zeros (1, length(r1)); % Intitialize KE array
20 r=r1;
21 for i=1: length(r1)
22 ke(i)=triplequad(fun ,r1(i),r2(i) ,0,2*pi ,0, hrratio); %
Integrate over varying limits
23 end
24 plot(r,ke ,'LineWidth ' ,3)
25 title('Kinetic Energy vs Vortex Core Distance Integration
Limits ');
26 xlabel('r'' Integration Upper Limit ');
27 ylabel('$KE_theta / Gamma_c ^2$ ','Interpreter ','LaTex '); %
Plot and format
28 end
9

11. 1 % Aaron Meyer
2 % April 15, 2016
3 %
4 % Function Name: tripint2
5 % Inputs: bins -number of iterations to be performed between 0
and 1
6 % hrratio -ratio of height to radius of tank
7 % re -reynolds number
8 % Outputs: r-array of r values at each bin iteration from 0 to
1
9 % ke -kinetic energy at each r value
10 %
11 % Plots: Plot of ke vs r, the kinetic energy versus the
distance from the
12 % vortex core
13
14 function [r,ke]= tripint2(bins ,hrratio ,re)
15 fun =@(x,y,z) ((1-exp(-re*x.^2/8))/x).^2*x; % Create the
integrand
16 r=linspace (0,1,bins); %Create the r array from 0 to 1 with
(bins) number
17 ke=zeros (1, length(r)); %Initialize ke array
18 for i=1: length(r)
19 ke(i)=triplequad(fun ,0,r(i) ,0,2*pi ,0, hrratio); %
triple integrate over each r value
20 end
21 plot(r,ke ,'LineWidth ' ,3)
22 title('Kinetic Energy vs Vortex Core Distance ');
23 xlabel('r''');
24 ylabel('$KE_theta / Gamma_c ^2$ ','Interpreter ','LaTex ');
%Plot and format
25 end
10