analysed in chapter 2.3 is documented. Then there are some practical comments to the simulations and uncertainties, and a short text about CFD simulations in general.
Two programs were used for the CDF analysis, namely OpenFOAM and ANSYS CFX.
The computational domain contained the runner, part of the draft tube and guide vanes for all 3-D simulations. The 2-D simulations were performed with only one runner channel and four guide vanes using periodic boundary conditions. The steady
simulations performed with OpenFOAM were compared to CFX simulations and gave equal results.
OpenFOAM: Boundary conditions were set as in table 1. At the interface between
rotating and stationary domain a GGI surface was used without any averaging. The
simulations performed with OpenFOAM were steady state. Transient simulations gave
too high pressure-oscillations arising from numerical instability. The case setup was
similar to the reference case ERCOFTAC Centrifugal Pump by Petit [22].
Table 1 Boundary conditions in OpenFOAM
Boundary Velocity Pressure Turbulence, k and ε
Inlet C=[C
r, C
u, C
z] zeroGradient values corresponding to 7%
turbulence intensity
Outlet zeroGradient meanValue=0 zeroGradient
Walls W=[0, 0, 0] zeroGradient zeroGradient
The 2D case included only one runner channel and had “cyclicGgi” boundary conditions at the periodic surface.
ANSYS CFX: CFX were used for transient analysis. The interface between rotating and stationary domain was set to “Transient rotor stator”. Boundary conditions were set as in table 2, and the numerical schemes were set to “higher resolution” for advection terms and first order for turbulence numerics. Time steps were set to 0.00015 second which corresponds to 0.7 degree rotation per time step.
Table 2 Boundary conditions in CFX
Boundary Condition
Inlet Mass flow rate and direction, medium
turbulence intensity
Outlet Opening , reference pressure
Walls no slip, smooth wall
Practical comments to the simulations in chapter 2.3:
The flow did not always converge to a constant state. Therefore the main output from the simulations, which is the head, was averaged over a number of iterations. This was done for both steady and unsteady simulations.
Grid convergence test were performed, but this only applies to the mean-flow since the boundary layer is modelled by a logarithmic law of the wall. With appropriate wall distance, i.e. y
+=30-100, all simulations with more than 70.000 cells in one channel gave equal results.
The two turbulence models K-epsilon and K-omega SST were tested in this work, showing no significant difference in the resulting mean flow.
Uncertainty and errors: Widely accepted definitions of uncertainties and errors have been listed in Versteeg [24]. Some elements of that list are presented here together with comments that apply to the simulations performed in this thesis.
¾ Iterative convergence errors:
i. Steady simulations: For some operation points the flow was not
steady-state and the solution did therefore not converge to a steady steady-state.
ii. Transient simulations: Time-steps were set to a value corresponding to approximately one degree of rotation which is a widely used time-step size in the literature. However, it might be too high to capture details of transient flow structures.
¾ Uncertainty due to approximate representation of geometry: The guide vanes opening was not exactly the same in the simulations as in the measurements
¾ Uncertainty due to boundary conditions: The inlet was placed at the guide vane entrance and the outlet just a small distance down in the draft tube.
¾ Physical model uncertainty: The turbulence models used do not capture effects of rotation and curvature.
About CFD
The CFD simulations solve the Navier-Stokes equations for incompressible flow. This flow is then uncoupled from the energy equation and the pressure has a pure elliptic behaviour. A direct simulation, DNS, of the flow is not possible due to the requirement of spatial and temporal resolution which grows with the Reynolds number to the power of 9/4. The smallest scales in the flow are therefore modelled by turbulence models in order to have a feasible amount of equations to solve. When a Reynolds averaging is performed on the Navier-Stokes equation, turbulent quantities called Reynolds stresses comes forth. These are modelled by the Boussinesq assumption, eq. (47), that links them to the turbulent viscosity, μ
t, and turbulent kinetic energy, k. The turbulent viscosity is a function of the two turbulent quantities modelled and represents an analogy between the physical processes of turbulent and molecular movements. The molecular movements exchange momentum regardless of the mean-flow direction and turbulence does the same but have a larger length scale.
The last term in eq. (47) is a normal stress that can be absorbed into a modified pressure in the CFD simulations, Pope [23]. The flow needs not to be resolved into the walls if a wall-law is applied. Such a wall-law can be used in the log-law layer were y
+takes values between 30 and 500, Versteeg [24], where y
+is the normalized wall distance eq.
(48). However it is beneficial to have y
+in the lower part of that range. During simulation of characteristics the flow rate ranges from nominal flow rate and down to zero. Since the wall distance depends on the velocity through u
t, it is impossible to have
“correct” y
+value for all simulations. Resolving the boundary layer is an option but
requires y
+=1 close to the wall, which for the model turbine in this thesis corresponds to
a wall distance of approximately y=0.005mm. This, together with the growth factors for
grid cells at maximum 1.2, results in a very high number of cells. The boundary
condition for the turbulent quantity epsilon is also an issue. Unlike the turbulent kinetic energy whose value is zero at the wall, the value and gradient of epsilon at the wall is unknown.
[ ] yut
y U
P
(48)
In the incompressible Navier-Stokes equation there is no description of the time evolution of the pressure. The pressure is therefore coupled to the flow field through various algorithms, e.g. SIMPLE and PISO, Versteeg [24]. These velocity-pressure couplings generally modify the continuity equation to solve for the pressure. The time-step is normally restricted by the CFL condition. By using the SIMPLE algorithm the time-step can be enlarged disregarding the CFL condition.
For the flow in the rotating part of the turbine a decomposition of the acceleration is performed, eq. (49), and the equations can be solved for the relative velocity instead of the absolute velocity. The last term represents the centrifugal acceleration, which can also be absorbed into a modified pressure, Alidoosti [25].
2 2
2 [ / ]
abs rel
a a w m s
r Z Z
u
(49)
5 Papers
5.1 Short Summary of papers
In document
Characteristics of reversible-pump turbines
(sider 48-52)