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TurbulentMixingLengthDissipationRateInlet 2 8

The widely used boundary condition for 2 8 in rotating machinery inlet is turbulentMixingLengthDissipationRateInlet.

omega:

 boundaryField
 {
".*_inlet"
    {
        type            compressible::turbulentMixingLengthDissipationRateInlet;
        mixingLength    0.1;
        phi             phi;
        k               k;
        value           $internalField;
    }
 }

For internal flows the parameter mixingLength is recommended to be equal to hydrostatic diameter of the pipe.

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Running Simulation - Convergence Monitoring - Torques

Torques are located in directory postProcessing/$region according to forces function object setup in system/forces.fun. Simple script torques.gp can plot the torques during the computation:

 

set grid
set xlabel "iterations"
set ylabel "torque [N.m]"
set key box above
set yrange [-10:5]

plot "< cat ./postProcessing/rotor/forcesBlade/0/forces.dat | grep -v '#' | sed 's/[(,)]/ /g' | awk '{print $11}' " with lines title "torque from pressure - x", \
     "< cat ./postProcessing/rotor/forcesBlade/0/forces.dat | grep -v '#' | sed 's/[(,)]/ /g' | awk '{print $12}' " with lines title "torque from pressure - y", \
     "< cat ./postProcessing/rotor/forcesBlade/0/forces.dat | grep -v '#' | sed 's/[(,)]/ /g' | awk '{print $13}' " with lines title "torque from pressure - z", \
     "< cat ./postProcessing/rotor/forcesBlade/0/forces.dat | grep -v '#' | sed 's/[(,)]/ /g' | awk '{print $14}' " with lines title "torque from viscous - x", \
     "< cat ./postProcessing/rotor/forcesBlade/0/forces.dat | grep -v '#' | sed 's/[(,)]/ /g' | awk '{print $15}' " with lines title "torque from viscous - y", \
     "< cat ./postProcessing/rotor/forcesBlade/0/forces.dat | grep -v '#' | sed 's/[(,)]/ /g' | awk '{print $16}' " with lines title "torque from viscous - z"  
     
pause 10
reread

Run the script:

# gnuplot torques.gp

openfoam example torque water turbine

Figure: Water turbine CFD, torques monitoring plot example.

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Water turbines - notes on calculations

Figure [*] represents a physical setting of a turbine casing with respect to a dam. Let us denote by $ h$ the head, i.e. the difference between heights of water levels in front of the dam and at the back of it. Vertical distance between the center of mass of the inlet surface of volute and the high water level is denoted by $ h_{In}$. Vertical distance between the center of mass of the outlet surface of draft tube and the low water level is denoted by $ h_{Out}$. Finally, vertical distance between the centres of mass of inlet and outlet surfaces is denoted by $ h_{IO}$.
sketch water turbine levels

Figure: General water turbine sketch. Physical setting of a turbine casing with respect to a dam. Water levels indicated.

We can see there are several simple relations among quantities just defined

img214img215

img216img217

img218img219

To simplify calculations we usually considers water an incompressible fluid with no phase changes possible and we solve the well known equations, the conservation laws the fluid obeys.

Coservation of linear momentum is expressed by the Navier-Stokes equation. In this case (incompressible fluid, steady state) it can take the form

img220

where $ \vec{U}$ is a velocity field, $ \nu$ kinematic viscosity, $ \varrho$ density, $ p$ pressure field and $ \vec{f}$ is a force density of a gravitational field, given by ([*]).

Conservation of mass is expressed by the continuity equation. In this case it takes the simple form

img222

emains to discuss boundary conditions for the equations mentioned above. For we usually do not know the inlet velocity field, we are about to prescribe pressure field for both inlet and outlet surface.

At the inlet surface a total pressure is known and it is equal to hydrostatic one. With the use of formula ([*]) we can write

img223

where $ \vec{r}_{0,H}$ is a position vector of an arbitrary point of the high water level (plane). Since total pressure is defined as a sum of static and dynamical pressure

img225

we can, by means of this definition ([*]) and formula ([*]), express $ p$ at the inlet

img226

At the outlet the situation is a little bit complicated. After the flow (with high total pressure) exits the draft tube it mixes with water (with low total pressure) surrounding the draft tube. We usually take simplifying steps and assume the static pressure of exiting flow equals the hydrostatic pressure of surrounding water, i.e. we write

img227

where $ \vec{r}_{0,L}$ is a position vector of an arbitrary point of the low water level.

Described steps correspond to real physical setting. In a task like this we need to know not only the $ h$, but even $ h_{In}$ (optionally $ h_{Out}$), i.e. a position of the turbine with respect to the high or low water level. Quantity $ h_{IO}$ can be obtained from the given turbine casing geometry, of course.

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