importance of the navier-stokes forces on the flow in a ... · importance of the navier-stokes...

Importance of the Navier-Stokes Forces on the Flow in aHydrodynamic Torque Converter

MATTHIAS WOLLNIK, WERNER VOLGMANN, HORST STOFFFluid-Energy Machines

Ruhr-UniversitaetUniversitaetsstr. 150, D-44780 Bochum

[email protected] http://www.rub.de/fem

Abstract: The importance of the various influences acting on the flow field of a hydrodynamic torque converteris investigated in detail by analysing the individual terms of the time-averaged Navier-Stokes equations. Forcesoriginating from pressure gradients, viscous and effective shear, Coriolis effect, convective and centrifugalacceleration are determined in several locations throughout the flow field. The velocity vector in magnitude anddirection solved by a commercially available 3D computer code is the basis for the analysis. With increasingspeed ratio of turbine-versus-pump the centrifugal force is dominant while shear forces become insignificant.

Key–Words: Hydrodynamic Torque Converter, Navier-Stokes Forces, Vane, Pump, Turbine

1 IntroductionThe flow within hydrodynamic torque converters ishighly complex. Although a lot of numerical and ex-perimental efforts have been made to predict and an-alyse the flow in an automotive torque converter [1,2, 3], it is still not fully understood how the operat-ing behaviour can be tuned by shaping the geometry.Therefore an effort is made to understand how forcesare acting on the fluid over a large range of operatingconditions. Not only the absolute values of the indi-vidual Navier-Stokes terms but also the turning (yawangle δ) and the slope (pitch angle γ) of the flow fieldhave to be analysed throughout the domain in order toget a comprehensive idea of the flow.

2 Problem FormulationThe forces acting within a hydrodynamic torque con-verter are analysed by solving the Navier-Stokes equa-tion in the following steady-state formulation.

convection force︷︸︸︷ρ · (~w ·∇) ~w =

pressure force︷︸︸︷−∇p

+ρ · νeff∇2 ~w︸︷︷︸

friction force

−2ρ · ~ω × ~w︸︷︷︸Coriolis force

−ρ · ~ω × (~ω × ~r)︸︷︷︸centrifugal force

(1)

For a representative example a well-documented typeof hydrodynamic torque converter is chosen to showhow the individual forces act on the fluid flow revolv-ing in a toroidal domain. In the past the Navier-Stokes

equation had been solved for the whole domain with-out investigating the importance of the individual con-tributions.

3 Problem Solution3.1 Torque converter CFD modelThe torque converter used in this study is the modelW240 by Fichtel & Sachs with the channel contourshown in figure 1. It is documented in the book ofFörster [4], its cross section is nearly spherical andhas a nominal outer diameter of D = 240 mm. It con-

Fig. 1: Section of a toroidal bladed channel [4].

tains 11 stator vanes, 31 blades in the impeller of thepump and 29 blades in the turbine. One blade passageis modelled for each component. Clearances betweenblades and shell as well as core are neglected and thepump domain was circumferentially positioned so thatthe exit of the pump blade points to the middle of aturbine blade passage. Each component has its own

Proceedings of the 4th WSEAS International Conference on Fluid Mechanics and Aerodynamics, Elounda, Greece, August 21-23, 2006 (pp168-173)

number of cells [ - ]

λ[-

]

0 200000 400000 600000 800000

2.36E-03

2.38E-03

2.40E-03

2.42E-03

2.44E-03

2.46E-03

2.48E-03

2.50E-03 λ grid studies

λ Förster

λ at chosen cell number

Fig. 2: Grid refinement studies showing λ = MPρω2D5 ,

the pump torque coefficient, versus the number of cellsat SR = 0.01.

frame of reference in order to analyse the character-istic operating lines with the speed ratio of turbine-versus-pump of SR = 0.01, 0.4 and 0.8 at a pumpspeed of 2000 min−1. Therefore the complete charac-teristics were interpolated corresponding to these re-sults.

At the interfaces between the components a frozenrotor model is used. A steady-state flow field is as-sumed as well as circumferential periodicity for eachof the components. Physical properties as for exam-ple density ρ, kinematic viscosity ν and temperaturesare assumed to remain constant for simulating the flowwith the commercial flow solver CFX10.1. The calcu-lated effective kinematic viscosity νeff in equation 1 isdue to effective shear and therefore contains molecu-lar and turbulent viscosity. Also the CFX - "SpecifyBlend Factor" is set to a value of 1.0 so that the flow issolved with Second Order differencing for the advec-tion terms.

Preliminary investigations concerning the size of thegrid showed, that a grid with a total of about 620 000cells provides appropriate grid-independent numericalsolutions. The development of the computational so-lution with increasing numbers of cells are displayedin figure 2. Figure 3 gives a detailed view of the statorgrid near the hub.

Fig. 3: Computational grid at hub of stator.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

SR [ - ]0 0.2 0.4 0.6 0.8 1

0.0E+00

4.0E-04

8.0E-04

1.2E-03

1.6E-03

2.0E-03

2.4E-03

2.8E-03

3.2E-03

3.6E-03

4.0E-03

0

0.2

0.4

0.6

0.8

1ExperimentalComputational

λ [ - ]η [ - ]

µ [ - ]

η

λ

µ

Fig. 4: Operating performance of type W240 for pumptorque coefficient λ, torque conversion µ and effi-ciency η.

As shown in figure 4 the computational torque con-verter characteristics compare very well to the exper-imental results, considering the lack of informationconcerning the density, viscosity and temperature ofthe oil used during the measurements. The averagediscrepancy is less than 3.3% for the power number∆λλ and less than 3.1% for the torque conversion num-

ber ∆µµ and the efficiency ∆η

η , respectively.

3.2 Angle definitionsFor describing a vector ~X in a three-dimensional do-main it is necessary to have its modulus | ~X| and the di-rection of this vector expressed through the flow turn-ing (pitch angle γ) and the slope (yaw angle δ). Theseangles are used in the commonly employed NACA-definition parting from the centerline of the shaft likein bladings of the axial turbomachinery. Due to thefact that the machine axis is used as a reference, prob-lems occur in radial machines. Especially in the pumpand the turbine of automotive torque converters, wherethe principal flow direction changes by 360◦, the stan-dard NACA-definition fails.

Fig. 5: Meridional section


Fig. 6: Definition of yaw angle δ.

Therefore the flow direction, prescribed by the aero-foil, is represented by one polygon line at half-spanand pointing in streamwise direction (figure 5). Thisflow direction is used as the reference for the upcom-ing angle definitions.

At each constant streamwise location (SL) a quasi-tangential plane ~eθ~eb is created between the circum-ferential direction ~eθ and the tangent to the aerofoilmean line ~eb. In this plane the flow turning (yaw an-gle δ) between the vector ~X , representing the indi-vidual Navier-Stokes-Forces or the mean velocity, andthe tangent to the aerofoil mean line ~eb is measured.Therefore the yaw angle δ can be calculated using theprojection ~X~eθ~eb

of vector ~X onto the ~eθ~eb-plane:

δ = acos~eb · ~X~eθ~eb

|~eb| · | ~X~eθ~eb|

(2)

= acos~eb ·

(~X − ~X~ek

)|~eb| · |

(~X − ~X~ek

)|. (3)

An exemplary configuration for the stator as well asthe above mentioned polygon line can be seen in fig-ure 6.

According to figure 7 the slope (pitch angle γ) iscreated between the vector ~eb and the projection ~X~ek~eb

of vector ~X onto the quasi-tangential plane ~ek~eb. Thepitch angle γ is therefore defined by:

γ = acos~eb · ~X~ek~eb

|~eb| · | ~X~ek~eb|

(4)

= acos~eb ·

(~X~ek

+ ~X~eb

)|~eb| · |

(~X~ek

+ ~X~eb

)|. (5)

The vector ~ek = ~eθ × ~eb is perpendicular to the cir-cumferential direction ~eθ and the aerofoil direction ~eb

and points to the core of the torque converter’s com-putational domain (figure 5).

Fig. 7: Definition of pitch angle γ.

The combination of both angle definitions yieldsto a three-dimensional figure (figure 8), which showsboth the directions (~eθ, ~eb and ~ek) and the projectionsonto the planes ( ~X~ek~eb

and ~X~eθ~eb). The angles δ and

γ in this right-hand system ~eθ, ~eb and ~ek are definedstarting from the reference plains ~X~ek~eb

and ~X~eθ~eb, re-

spectively. A vector ~X , identical with the blade sur-face direction ~eb, refers to the values of δ = 0◦ andγ = 0◦.

Additionally, figure 8 contains the projection of ~Xon the meridional plane, necessary for a modified def-inition of βNACA with the meridional flow direction in-stead of the machine axis as the reference. The def-inition used at the Ruhr-Universitaet is the peripheralflow turning β defined as the counter-clockwise anglebetween the circumferential direction ~eθ and the tan-gent to the aerofoil mean line ~eb.

Fig. 8: Yaw angle δ and pitch angle γ for deflectionand slope of the toroidal flow.

Contrary to this chosen approach of decomposingthe Navier-Stokes equation throughout the entire torqueconverter Marathe and Lakshminarayana analysed thepitch and yaw angles in a turbine and a stator [5] andin a pump [6] by using the NACA angle convention.However their analysis is limited to the inlet and exitplanes of these components, of which the surface vec-tors are almost parallel to the axis of rotation. There-


Streamwise Location [ - ]

cosγ

|X|[

N/m

3 ]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-8E+06

-6E+06

-4E+06

-2E+06

0

2E+06

4E+06

6E+06

8E+06

1E+07

1.2E+07

1.4E+07

1.6E+07

1.8E+07FrictionPressureConvection

W240 Stator at SR=0.01

Fig. 9: Streamwise forces by Navier-Stokes terms instator.

fore the NACA angle convention was convenient.

3.3 Navier-Stokes term averagingIn order to follow the forces by Navier-Stokes termsand the mean velocity along the streamwise directionthrough the entire torque converter, these forces areaveraged for each location along this stream path. Thusthe x-, y- and z-components are momentum-averagedon cross sections at constant streamwise locations.

As a result averaged vectors ~X for every Navier-Stokes term and the mean velocity, respectively, arecalculated at a certain streamwise location. The vector~X is used to examine the pitch angle and the yaw angleof each term in the Navier-Stokes equation throughoutthe domain applying equation 2 and equation 4, re-spectively.

3.4 Scaled absolute contributing termsAs mentioned earlier it is necessary to know the lengthof a vector in addition to the pitch and the yaw angle.In order to combine the information about size and di-rection, the absolute values of the individual Navier-Stokes terms are scaled. The factor cos γ is used forthe contribution to accelerate/decelerate the flow instreamwise direction while the factor sin δ is used forthe contribution to the peripheral momentum transmis-sion. For analysing the influence on the absolute meanvelocity, the cos γ-scaled forces are accumulated. Dueto the fact that three different domains are analysed,the Navier-Stokes equation (1) has to be solved inde-pendently for the individual torque converter compo-nents.

The graphical display in each following figure con-tains vertical dash-dotted lines, indicating the lead-ing edges and trailing edges of blades and vanes. Al-though the angle distribution through the entire torqueconverter was analysed for the 0.01, 0.4 and 0.8 speedratios, only the results for SR = 0.01 are shown.

3.4.1 StatorAs the stator domain is stationary, only the forces dueto convection, pressure and friction appear in figure 9


sinδ

|X|[

N/m

3 ]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-2E+06

0

2E+06

4E+06

6E+06

8E+06

1E+07

1.2E+07

1.4E+07

1.6E+07

1.8E+07

2E+07

FrictionPressureConvection


Fig. 10: Peripheral forces by Navier-Stokes terms instator.

to figure 11. Additionally the absolute mean veloc-ity values are shown in the last-mentioned figure. Thescaling of the friction force, which points almost inopposite direction of the mean-line vector ~eb, empha-sizes the influence in the cos γ-scaled figure 9 and low-ers the influence in the sin δ-scaled figure 10. For thespeed ratio of SR = 0.01 the absolute value of thescaled friction force is therefore about one order ofmagnitude smaller than the scaled absolute values ofthe convection force and the pressure force, which aredominating the flow at this speed ratio. These forcesare acting in flow direction up to a region in front ofthe trailing edge, where a change in signs shows a turnback in the acting direction (figure 9).



Σ(co

sγ|X

|)[N

/m3 ]

Abs

olu

teM

ean

Vel

oci

ty[m

/s]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-2.0E+07

-1.0E+07

0.0E+00

1.0E+07

2.0E+07

3.0E+07

4.0E+07

0

5

10

15

20

25

30

cos γ-scaledAbsolute Mean Velocity

Fig. 11: Accumulated streamwise forces by Navier-Stokes terms in stator.

The sin δ-scaled forces in figure 10 indicate the dis-tribution of the torque converting forces within the sta-tor. Due to the large incidence angles the peaks for thescaled friction and pressure forces are near the lead-ing edge. The well channeled flow further down thestream explains the decrease in sin δ-scaled forces.

In figure 11 the accumulated cos γ-scaled forces areshown with the absolute mean velocity in the stator.The influence of the forces in streamwise direction onthe acceleration/deceleration of the absolute mean ve-locity takes effect with a short delay.


3.4.2 TurbineIn addition to the forces already described in the pre-vious subsection, the Coriolis force and the centrifu-gal force occur due to the rotating frame of reference.At the speed ratio SR = 0.01 these additional forces,like the shear force, play only a minor role, though.Therefore the flow in the turbine is dominated by thepressure force and the convection force.


cosγ

|X|[

N/m

3 ]

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-6E+06

-4E+06

-2E+06

0

2E+06

4E+06

6E+06

8E+06

1E+07

CoriolisFrictionCentrifugalPressureConvection

W240 Turbine at SR=0.01

Fig. 12: Streamwise forces by Navier-Stokes terms inturbine.

As well as in the stator, the shear force vector pointsin opposite directions of the blade-metal vector ~eb inalmost the whole turbine, therefore emphasizing thecos γ-scaled values (figure 12) and lowering the sin δ-scaled contributions (figure 13). Nevertheless, at thisspeed ratio of SR = 0.01 the shear force is about oneorder of magnitude smaller than the dominating scaledpressure and convection forces. As shown in figure 12the scaled pressure and convection forces are actingopposite to the tangent direction of the aerofoil meanline between the leading edge and 60% of the turbine’sstreamwise location (1.03 ≤ SL ≤ 1.61). Opposedto this these force act in streamwise direction in theother parts of the turbine domain. The large positivevalues for the scaled pressure and convection forcesindicate an large incidence angle in front of the leadingedge. This high incidence angle is also responsiblefor high negative values of the sin δ-scaled pressureand convection forces near the inlet in figure 13. Dueto the slowly turning turbine the incidence angles arehigh at the exit of the domain, causing jumps in thelines. Between the edges of the turbine the trends forthe dominant forces run almost parallel, reaching theirminimum in the middle of the domain.


sinδ

|X|[

N/m

3 ]

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

-6E+06

-4E+06

-2E+06

0

2E+06

4E+06

6E+06

8E+06

1E+07

1.2E+07



Fig. 13: Peripheral forces by Navier-Stokes terms inturbine.



Σ(co

sγ|X

|)[N

/m3 ]

Abs

olu

teM

ean

Vel

oci

ty[m

/s]

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-2.0E+07

-1.0E+07

0.0E+00

1.0E+07

2.0E+07

3.0E+07

0

5

10

15

20

25


Fig. 14: Accumulated streamwise forces by Navier-Stokes terms in Turbine.

Like in the stator, the absolute mean velocity fol-lows the accumulated force’s trend. Negative forcevalues slow down the flow until SL = 1.61 in the tur-bine domain. The flow accelerates again with positivescaled forces to the end of the domain.

3.4.3 PumpContrary to the turbine at the speed ratio of SR =0.01, the rotational velocity of 2000 min−1 generatesCoriolis and centrifugal forces which are no more neg-ligible.


cosγ

|X|[

N/m

3 ]

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9-6E+06

-4E+06

-2E+06

0

2E+06

4E+06

6E+06

8E+06

1E+07CoriolisFrictionCentrifugalPressureConvection

W240 Pump at SR=0.01

Fig. 15: Streamwise forces by Navier-Stokes terms inpump.

Like in the other two domains the friction force ispointed against the blade-metal vector ~eb and the abso-lute value is smaller than the values of the other forces.Due to the large incidence angles at the leading edgeof the pump, positive cos γ-scaled values appear (fig-ure 15). After 10% of the domain the flow is wellchanneled with centrifugal force values that acceleratethe flow. While the cos γ-scaled centrifugal force stayspositive, changes in signs occur for the other cos γ-scaled forces. Up to SL = 2.2 and after SL = 2.42the pressure force contributes to the acceleration of theflow. Inbetween these SL the scaled pressure force de-celerates the flow. Due to the averaging on surfaces atconstant streamwise locations, the Coriolis force hasa component pointing against the tangent to the aero-foil mean line ~eb. This component reaches its max-imum absolute value at about 35% of the pump do-main. The scaled convection force is again parallel tothe scaled pressure force though the distance is largerthan in the other domains at the SR = 0.01. Like inthe other domains at this speed ratio, the sin δ-scaled



sinδ

|X|[

N/m

3 ]

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9-6E+06

-4E+06

-2E+06

0

2E+06

4E+06

6E+06

8E+06



Fig. 16: Peripheral forces by Navier-Stokes terms inpump.

shear force has the lowest influence on the torque con-version. Although the cos γ-scaled centrifugal force isa main contributor in figure 15, the sin δ-scaled cen-trifugal force play only a minor role in figure 16. Thesin δ-scaled Coriolis force, which is almost constantover the entire pump domain, is one of the dominantsin δ-scaled forces. Together with the pressure force,these two sin δ-scaled forces act the same direction,while the sin δ-scaled convection force acts in the op-posite direction between SL = 2.1 and SL = 2.75.



Σ(co

sγ|X

|)[N

/m3 ]

Abs

olu

teM

ean

Vel

oci

ty[m

/s]

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9-2.0E+07

-1.0E+07

0.0E+00

1.0E+07

2.0E+07

0

5

10

15


Fig. 17: Accumulated streamwise forces by Navier-Stokes terms in pump.

The trend that the absolute mean velocity followsthe accumulated cos γ-scaled forces is confirmed forthe pump (figure 17). Only in the regions with nega-tive sums for the cos γ-scaled forces, the flow is decel-erated.

4 ConclusionThe steady-state distribution of forces due to the fiveNavier-Stokes terms has been determined. These forcesare described in detail for the absolute values scaledwith cos γ (streamwise) and sin δ (peripheral), respec-tively. With an increasing rotational velocity the cen-trifugal force becomes dominant in pump and turbine.Contrary to the centrifugally accelerated flow in thepump the flow is decelerated through the centrifugalforce in the turbine. The shear force is decreasingwith increasing speed ratios due to lower velocitiesand mass flow rates. Therefore the influence on the av-eraged velocity along the streamwise direction is lim-ited. In this investigation the Coriolis force has com-

ponents in streamwise direction because of the averag-ing on surfaces at constant streamwise locations. Thepressure force and the convection force have almostthe same characteristic lines for the stator. The dif-ference is caused by the friction force. In the turbinethe pressure force and convection force lines tend toseparate the more the speed ratio rises. The reasonfor this is the growing influence of the forces depend-ing on the rotational velocity while the friction forcestays at a low level. Though the pump is rotating at aconstant 2000 min−1, an influence of the speed ratiois noticeable by changing incidence angles and vary-ing mass flow rates. The pressure force as well as theconvection force show the influence of the large inci-dence angles behind the leading edge by jumps in thecorresponding distributions.

5 AcknowledgmentsThe authors would like to express their acknowledge-ment to Deutsche Forschungsgemeinschaft (DFG) forthe financial support of the project.

References:

[1] Ackermann J.: Experimentelle und nu-merische Strömungsuntersuchung eines Trilok-Drehmomentwandlers mit schmaler Kreislauf-form, Vol. 421 of Reihe 7 Strömungstechnik. VDIVerlag, Düsseldorf, 1st edn., 2001.

[2] Habsieger A., Flack R.: Flow Characteristics atthe Pump-Turbine Interface of a Torque Converterat Extreme Speed Ratios. International Journal ofRotating Turbomachinery, Vol. 9, 2003, pp. 419–426.

[3] Schweitzer J., Gandham J.: Computational FluidDynamics in Torque Converters: Validation andApplication. International Journal of RotatingTurbomachinery, Vol. 9, 2003, pp. 411–418.

[4] Förster H.J.: Automatische Fahrzeuggetriebe :Grundlagen, Bauformen, Eigenschaften, Beson-derheiten. Springer-Verlag, Berlin et al., 1990.

[5] Marathe B.V., Lakshminarayana B.: ExperimentalInvestigation of Steady and Unsteady Flow FieldDownstream of an Automotive Torque ConverterTurbine and Stator. International Journal of Ro-tating Turbomachinery, Vol. 2, No. 2, 1995, pp.67–84.

[6] Marathe B.V., Lakshminarayana B.: ExperimentalInvestigation of Steady and Unsteady Flow FieldDownstream of an Automotive Torque ConverterPump. International Journal of Rotating Turbo-machinery, Vol. 5, No. 2, 1999, pp. 99–116.


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