Numerical Simulation of the Flow Field
in a Friction-Type Turbine (Tesla Turbine)
Andrés Felipe Rey Ladino
Institute of Thermal Powerplants School of Engineering,
Vienna University of Technology National University of Colombia,
Getreidemarkt 9/313, A-1060 Wien Calle 57c No. 40-51 Ap. 419, Bogotá, Colombia,
Tel.: ++57/1/3150057, Email: e0326542@student.tuwien.ac.at
INTRODUCTION
In the context of research project at Institute for Powerplants at the Vienna University of
Technology in joint study with the National University of Colombia, the flow inside the Tesla
turbomachine is investigated. The Tesla turbine, an unconventional turbomachinery that uses
smooth disks instead of blades, is described principally by the loading coefficient curve, the
efficiency and degree of reaction vs. the flow rate parameter. In order to describe its behaviour,
the rotational speed is maintained constant and the flow rate is changed with the purpose of
simulate a virtual brake. Since the flow is characterized in a transitional regime, both laminar and
turbulent cases are simulated. The work presented represents an initial significant step towards
the analysis of this type of flow using CFD tool. Starting from a simple axi-symmetric model of
the flow between two co-rotating disks in two dimensions, the model is improved including the
outlet of the turbine with the casing, and at the end a 3D simulation of a single disk is performed
including the effects of the nozzles. A complete model of a Tesla turbine is restricted by
computer resources. The simulations were carried out in winter and summer semester 2003-2004.
The commercial computational fluid dynamics (CFD) code FLUENT as well as the grid
generation software GAMBIT were used for the investigation. Both codes are available on the
CFD-server COMPAQ SC45 (Fluid Dynamics and Finite Element Server) of the Computer
Center of Vienna University of Technology.
PREVIOUS WORK
Starting from the patent of Nikola Tesla [1], (Figure 1), extensive analytical work were made in
the 60´s, and 70´s. Most of literature consider the flow laminar and incompressible, but also some
turbulent approach were made. The flow is characterized by the Reynolds number and the Mach
number. It is found various regimes of flow and process as laminar, turbulent, forward transition
and relaminarization from turbulent to laminar.
Figure 1.: American Patent No. 1,061,206 of Tesla turbine [1]. Figure 2.: Schematic diagram of Tesla turbine [2].
GEOMETRY AND MESHING
Following the experimental work and geometry used by Rice [2], three models with similar
geometry are proposed and solved. The geometry is characterized by the ratio of radii: r1/r2=6,06
and the ratio of the gap to the outer radius: r1/b=200 (these geometrical parameters are the same
in the three models); the angle of the nozzle is =20°. The flow is simulated for
.
Reb = Å" b2 = 25.9
Figure 3.: Geometry for one gap 2D model. Figure 4.: 2D Model of the turbine with full Figure 5.: Mesh for the 3D slide model.
peripheral admission.
DESCRIPTION OF THE NUMERICAL METHOD
Governing equations.
The Navier Stokes equations are solved for steady flow, incompressible, laminar and turbulent
case, and 2D (Axisymmetric Swirl) and 3D approach. Fluent uses the Finite Volume Method and
in addition the double precision characteristic was used. The following options were selected:
Segregated and implicit solver, and for the discretization: PRESTO! (Pressure Staggering
Option), SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) for Pressure-Velocity
coupling, Second Order Upwind was selected for Momentum. For turbulent solution the standard
k- model with near wall treatment is used. The y+<5 relation near the wall must be
accomplished.
Boundary Conditions.
Inlet was specified with a fixed velocity-inlet boundary. For the turbulence simulations was
selected a value of 5% of turbulence intensity. The nondimensional turbulent kinetic energy is set
to k*=0.00375 [-], and the turbulent dissipation rate is *=2,6953x10-4 [-].
Outlet condition was specified with a pressure-outlet boundary, with a gauge pressure value of 0 [-].
Disks and housing employ no slip condition rotating at 0.005 [-]. The walls are adiabatic.
Convergence and computed time.
The solution of the first model (Figure 3) take 4 minutes to converge with 6.800 quadrilateral
cells, the second model (Figure 4) with 237,620 cells took 3h to reach a suitable solution and
finally the 3D model (Figure 5) with 2,412,000 cells consume 14 days after 5,000 iterations.
RESULTS
First Model: Flow between disks. Figure 6 shows the velocity profile at three radial stations,
inlet, middle and outlet. The strong acceleration between to frontier stations is noticed. Figure 7
presents one characteristic of the radial flow that appears only in the laminar case: the inflection
of the profile in the middle of the radius. Figure 8 depicts the change of pressure (total, dynamic
and static). As it can be seen, in the radial direction, the total pressure in the middle of the gap
does not show variation, meaning that the flow in this plane do not perform work, but in other
plane, at 5% of the gap from the wall, the variation is more appreciable.
Laminar solution
Figure 6.: Velocity magnitude, laminar Figure 7.:Inflection of radial velocity for Figure 8.: Pressure distribution along radius,
solution. different radial stations. laminar solution.
Characteristic curves of the Tesla turbine are presented in Figure 9 and Figure 10.
20
0.45

Efficiency
Loading
[%]
Coefficient
0.40
15
[-]
Pshaf t

m

0.35
poTotal

10
T
0.30

m

2
ro
0.25
5
0.20
0
0.15
0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.20 0.25 0.30 0.35 0.40 0.45 0.50
Flow Parameter [-] Uo Flow Parameter [-]
Uo
=

ro
Å" ro
Figure 9.: Characteristic curve, loading coefficient, from laminar case. Figure 10.: Efficiency of an isolated disk, laminar, non-dimensional
solution.
Turbulent solution
From turbulent case, the acceleration of the flow between inlet and outlet is lower (Figure 11) with a
higher torque (Figure 14) and also higher efficiency; More energy is extracted with higher
turbulence, the drag is higher. The total pressure at inlet is higher in comparison to the laminar case.
Figure 11.: Total velocity, turbulent case. Figure 12.:Inflection of radial velocity for Figure 13.: Pressure distribution along radius,
different radial stations. turbulent solution.
0.9
20

Loading
Coefficient
Efficiency
0.8
[-]
[%]
15
Pshaf t
0.7

m

poTotal
T
0.6
10
m

2
ro
0.5
5
0.4
0.3 0
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Flow Parameter [-] Uo Flow Parameter [-]
Uo
=

ro
ro
Figure 14.: Loading coefficient for turbulent solution, isolated disk. Figure 15.: Efficiency for turbulent solution, isolated disk.
Rotor: This model includes the outlet of the turbine, Figure 17 presents the difference of computed
efficiency of only the rotor and the rotor with the outlet, with a pipe length of 57 [-] from the rotor
outlet. The losses due to the outlet are considerable due to change of area and direction.
The low efficiency value is also comparable to the recently experimental work of Schmidt [3].
20
0.9

·
Loading
Efficiency
Coefficient
[%]
[-] 0.8
15
0.7
T

m
= 10
2
0.6
ro
0.5
5
Rotor Turbina Tesla
0.4
flow parameter vs Efficiency_rotor
flow parameter vs efficiency_57
0
0.3
0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.20 0.25 0.30 0.35 0.40 0.45 0.50
Uo
Flow Parameter

Uo
Flow Parameter ro
=
ro
Figure 16.: Loading coefficient for turbulent solution, turbine. Figure 17.: Efficiency for turbulent solution, turbine.
3D rotor slide model:
Figure 18.: Pathlines colored by velocity magnitude. Figure 19.: Contours of total pressure at the middle of the gap.
3.0 25

Loading
Coefficient 2.8 Efficiency
[-] [%]
20
2.6
15
2.4
2.2
10
2.0
5
1.8
1.6
0
0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.20 0.25 0.30 0.35 0.40 0.45 0.50
Figure 20.: Loading coefficient for turbulent solution, turbine. Figure 21.: Efficiency for turbulent solution, turbine.
The 3D model includes the effects due to the nozzle and the losses present in this zone are not
very high when loading coefficient and efficiency are compared (Figure 20 and Figure 21).
Figure 22.: Vectors of velocity magnitude and total pressure contours at the middle of the gap.
SUMMARY
The flow in the Tesla turbine was simulated with different geometrical models and laminar and
turbulent approach. Since the flow itself is found in a transition regime with multiple processes -
as transition, relaminarization, recirculation, between other phenomena- an exact simulation that
full fills all the physical requirements is quite difficult to achieve with CFD tool. Nevertheless,
both approximations are valid and they would show different features, characteristics and
behaviour typical for each case and for the Tesla turbine. One of these differences is that the
inflection point for laminar flow disappears in the turbulent case, for which the turbulent effects
act as a mechanism of balanced in the axial direction. In contrast, experimental research cannot
measured velocities profiles, only static pressure as it was reported by Adams [5], due to the fact
that the gap between disks is very thin; CFD provide solution to this problem and furthermore to
micro turbines. The flow hat also high swirling velocity components with significant gradients of
acceleration, which makes the solution more difficulty and need extra iterations in order to
achieve a good convergence solution. Convergence is an important issue in CFD because of the
iterative nature of the solution, and it can give evidence of a well posed model indicating some
physical facts. Extended details and results can be found in the diploma thesis of Rey A.F. [6].
The results show that multiple disk turbines are workable and feasible, in the engineering sense,
but they present a low efficiency in both laminar and turbulent cases; lower for laminar case; the
difference between these cases is significant. Moreover, the simulation of the gap and the results
of 3D slide model show that the efficiency was not as high as it was suggest by experimental
reports in literature that implying the low efficiency due to the presence of the nozzle. The
efficiencies are similar in the three models. The machine is feasible for a low range of power
where the efficiency of conventional turbomachinery is not very high and it can handle working
fluid with particles, contaminants and high viscosity.
Some conclusions of each model are:
2D two disks model: For the laminar solution, the profile of the radial velocity component
shows an inflection at about the middle radius; The flow presents strong acceleration especially at
the outlet; The gap can be reduced.
2D turbine model:The effects of the walls on the overall performance are not much significant
when the efficiency is compared with 2D two disks model; The losses due to the outlet are high,
where the flow presents high swirls and change of area and direction; The leading edge and
trailing edge of each disk can be improved.
3D turbine model: The high velocity of the flow increases the torque even though with the
presence of zones of ventilation; The losses owed to the nozzles are not very high; The tangential
component of velocity remains nearly constant from the outer to the inner region of the rotor,
because of the high velocity after the nozzle.
Future Work
Some of the following topics would be interesting for further investigation: 1. Compressible
analysis of the Tesla turbine; 2. Effects of the number of nozzles on performance of the turbine;
3.Optimization of the geometry, following analytical and the experimental work of Rice [2], and
recently by Schmidt [3], the analytical work of Lawn and Rice [4] is useful for optimization;
5.Improvement of the numerical method in order to handle transitional flows as well as
evaluation of other turbulence models to use in the simulations.
LITERATURE
1. Tesla, N.:  Turbine United States Patent No. 1061206, May 6, 1913.
2. Rice, W.:  An Analytical and Experimental Investigation of Multiple-Disk Turbines , Journal of
Engineering for Power, Trans. ASME, series A, Vol. 87, No. 1 Jan. 1965, pp. 29-36.
3. Schmidt, D. D.:  Biomass Boundary Layer Turbine Power System , California Energy Commission (CEC),
EISG PROGRAM [online] Available from Internet: http://eisg.sdsu.edu/Fullsums/00-06.htm > http://eisg.sdsu.edu/shortsums/shortsum0006.htm>, 1991, California, USA.
4. Lawn, Jr. J., Rice, W.:  Calculated Design Data for the Multiple Disk Turbine using Incompressible fluid ,
Journal of Fluids Engineering, Trans. ASME, Vol. 96 No. 3, September 1974, pp. 252-258.
5. Adams, R., Rice, W.:  Experimental Investigation of the Flow Between Corotating Disks , Journal of
Applied Mechanics, Vol. 37, Trans. ASME, Vol. 92 series E, No. 3, September 1970, pp. 844-849.
6. Rey A. F.:  Numerical Simulation of the Flow Field in a Friction-Type Turbine (Tesla Turbine) Diploma
thesis at the Institute of Thermal Powerplants, TUWien, Viena, Austria, July 2004.


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