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A.G.K. van Dijk
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As the share of wind energy in the global electricity mix increases, it becomes more important to accurately predict wind turbine wakes in order to optimise wind farm layouts and to reduce fatigue loads on downstream turbines. Wake models based on the Reynolds-averaged Navier–Stokes equations offer a good balance between accuracy and computational cost, but their performance depends strongly on the turbulence closure used. The standard k–ε model is widely used in wind energy, but it is known to overpredict the eddy viscosity in the near wake. This leads to an overestimation of turbulent mixing and wake recovery. To address this, researchers have proposed extended closures, which have only been validated in fully elliptic three-dimensional RANS solvers. The Forced Ainslie Wake Model has improved the physical consistency of parabolic wake modelling by reintroducing part of the actuator disk force. Its turbulence closure is currently limited to a constant eddy viscosity. A transport-based turbulence model that is compatible with the parabolic marching scheme of the FAWM is therefore needed.
This thesis develops and validates a parabolised, axisymmetric k–ε turbulence model that can be coupled to the Forced Ainslie Wake Model. Three turbulence closures are implemented: the standard k–ε model and two of its extensions. The k–ε–fP model limits the eddy viscosity in regions of high shear through a variable C*μ derived from the nonlinear eddy viscosity model framework. The extended k–ε model with a sink term Sk accounts for the extraction of turbulent kinetic energy by the actuator disk. The transport equations for k and ε are discretised on a staggered finite volume grid using a second-order upwind scheme for the convective terms and central differencing for the diffusive terms. The turbulence model is coupled with the FAWM through a Picard iteration scheme that exchanges velocity and eddy viscosity at each axial station until convergence. The implementation is verified through the decay of isotropic turbulence, the Method of Manufactured Solutions, and a self-similar co-flowing jet simulation. In all three tests, the solver achieves the formal order of convergence. This confirms the correct implementation of the discretisation and the coupling between the transport equations.
During the validation against LES data for a DTU 10 MW reference wind turbine across four operating conditions, numerical stability problems were encountered. In the original model formulation, only 5 out of 12 simulations converged. Some of the converged cases produced non-physical values of the turbulent kinetic energy. The source of the instabilities was traced to the axial gradient of the radial velocity, ∂V/∂x, which contributes to the strain-rate invariant and therefore the production term. Because the radial velocity is derived from the axial velocity through the continuity equation, ∂V/∂x depends on the second axial derivative of U. Any irregularity in the actuator disk forcing is transferred to the axial velocity through the momentum equation. In a parabolic solver, the axial diffusion and pressure terms that would normally dampen these gradients are absent, leading to an overestimation of ∂V/∂x. Neglecting ∂V/∂x in the computation of the production term restored convergence in 11 out of 12 cases. This also keeps all predicted values within physical bounds.
With the modified formulation, the three turbulence closures were compared against the LES reference data. For the axial velocity, the k–ε–fP model performs best in the near wake, with MAPE values between 1 and 4%, while the k–ε–Sk model produces the lowest errors in the far wake. The standard k–ε model has the largest velocity errors across all configurations. The velocity errors increase with increasing thrust coefficient and decreasing wind speed. For the turbulent kinetic energy, all three models overpredict the LES values by a large margin, with full-wake MAPE values ranging from 50 to 126%. The k–ε–Sk model produces the lowest TKE errors in every configuration, while the standard model and k–ε–fP model have comparable errors.
In summary, this thesis demonstrates that parabolised turbulence models can be coupled with the parabolic FAWM solver. It also shows that the extended closures improve predictions compared to the standard k–ε model. However, the original model formulation with ∂V/∂x included in the production term is not reliably stable, and the presented results were obtained with a modified formulation. No single closure is best for all flow variables simultaneously. Further work is needed to resolve the stability issue of the original formulation. ...
This thesis develops and validates a parabolised, axisymmetric k–ε turbulence model that can be coupled to the Forced Ainslie Wake Model. Three turbulence closures are implemented: the standard k–ε model and two of its extensions. The k–ε–fP model limits the eddy viscosity in regions of high shear through a variable C*μ derived from the nonlinear eddy viscosity model framework. The extended k–ε model with a sink term Sk accounts for the extraction of turbulent kinetic energy by the actuator disk. The transport equations for k and ε are discretised on a staggered finite volume grid using a second-order upwind scheme for the convective terms and central differencing for the diffusive terms. The turbulence model is coupled with the FAWM through a Picard iteration scheme that exchanges velocity and eddy viscosity at each axial station until convergence. The implementation is verified through the decay of isotropic turbulence, the Method of Manufactured Solutions, and a self-similar co-flowing jet simulation. In all three tests, the solver achieves the formal order of convergence. This confirms the correct implementation of the discretisation and the coupling between the transport equations.
During the validation against LES data for a DTU 10 MW reference wind turbine across four operating conditions, numerical stability problems were encountered. In the original model formulation, only 5 out of 12 simulations converged. Some of the converged cases produced non-physical values of the turbulent kinetic energy. The source of the instabilities was traced to the axial gradient of the radial velocity, ∂V/∂x, which contributes to the strain-rate invariant and therefore the production term. Because the radial velocity is derived from the axial velocity through the continuity equation, ∂V/∂x depends on the second axial derivative of U. Any irregularity in the actuator disk forcing is transferred to the axial velocity through the momentum equation. In a parabolic solver, the axial diffusion and pressure terms that would normally dampen these gradients are absent, leading to an overestimation of ∂V/∂x. Neglecting ∂V/∂x in the computation of the production term restored convergence in 11 out of 12 cases. This also keeps all predicted values within physical bounds.
With the modified formulation, the three turbulence closures were compared against the LES reference data. For the axial velocity, the k–ε–fP model performs best in the near wake, with MAPE values between 1 and 4%, while the k–ε–Sk model produces the lowest errors in the far wake. The standard k–ε model has the largest velocity errors across all configurations. The velocity errors increase with increasing thrust coefficient and decreasing wind speed. For the turbulent kinetic energy, all three models overpredict the LES values by a large margin, with full-wake MAPE values ranging from 50 to 126%. The k–ε–Sk model produces the lowest TKE errors in every configuration, while the standard model and k–ε–fP model have comparable errors.
In summary, this thesis demonstrates that parabolised turbulence models can be coupled with the parabolic FAWM solver. It also shows that the extended closures improve predictions compared to the standard k–ε model. However, the original model formulation with ∂V/∂x included in the production term is not reliably stable, and the presented results were obtained with a modified formulation. No single closure is best for all flow variables simultaneously. Further work is needed to resolve the stability issue of the original formulation. ...
As the share of wind energy in the global electricity mix increases, it becomes more important to accurately predict wind turbine wakes in order to optimise wind farm layouts and to reduce fatigue loads on downstream turbines. Wake models based on the Reynolds-averaged Navier–Stokes equations offer a good balance between accuracy and computational cost, but their performance depends strongly on the turbulence closure used. The standard k–ε model is widely used in wind energy, but it is known to overpredict the eddy viscosity in the near wake. This leads to an overestimation of turbulent mixing and wake recovery. To address this, researchers have proposed extended closures, which have only been validated in fully elliptic three-dimensional RANS solvers. The Forced Ainslie Wake Model has improved the physical consistency of parabolic wake modelling by reintroducing part of the actuator disk force. Its turbulence closure is currently limited to a constant eddy viscosity. A transport-based turbulence model that is compatible with the parabolic marching scheme of the FAWM is therefore needed.
This thesis develops and validates a parabolised, axisymmetric k–ε turbulence model that can be coupled to the Forced Ainslie Wake Model. Three turbulence closures are implemented: the standard k–ε model and two of its extensions. The k–ε–fP model limits the eddy viscosity in regions of high shear through a variable C*μ derived from the nonlinear eddy viscosity model framework. The extended k–ε model with a sink term Sk accounts for the extraction of turbulent kinetic energy by the actuator disk. The transport equations for k and ε are discretised on a staggered finite volume grid using a second-order upwind scheme for the convective terms and central differencing for the diffusive terms. The turbulence model is coupled with the FAWM through a Picard iteration scheme that exchanges velocity and eddy viscosity at each axial station until convergence. The implementation is verified through the decay of isotropic turbulence, the Method of Manufactured Solutions, and a self-similar co-flowing jet simulation. In all three tests, the solver achieves the formal order of convergence. This confirms the correct implementation of the discretisation and the coupling between the transport equations.
During the validation against LES data for a DTU 10 MW reference wind turbine across four operating conditions, numerical stability problems were encountered. In the original model formulation, only 5 out of 12 simulations converged. Some of the converged cases produced non-physical values of the turbulent kinetic energy. The source of the instabilities was traced to the axial gradient of the radial velocity, ∂V/∂x, which contributes to the strain-rate invariant and therefore the production term. Because the radial velocity is derived from the axial velocity through the continuity equation, ∂V/∂x depends on the second axial derivative of U. Any irregularity in the actuator disk forcing is transferred to the axial velocity through the momentum equation. In a parabolic solver, the axial diffusion and pressure terms that would normally dampen these gradients are absent, leading to an overestimation of ∂V/∂x. Neglecting ∂V/∂x in the computation of the production term restored convergence in 11 out of 12 cases. This also keeps all predicted values within physical bounds.
With the modified formulation, the three turbulence closures were compared against the LES reference data. For the axial velocity, the k–ε–fP model performs best in the near wake, with MAPE values between 1 and 4%, while the k–ε–Sk model produces the lowest errors in the far wake. The standard k–ε model has the largest velocity errors across all configurations. The velocity errors increase with increasing thrust coefficient and decreasing wind speed. For the turbulent kinetic energy, all three models overpredict the LES values by a large margin, with full-wake MAPE values ranging from 50 to 126%. The k–ε–Sk model produces the lowest TKE errors in every configuration, while the standard model and k–ε–fP model have comparable errors.
In summary, this thesis demonstrates that parabolised turbulence models can be coupled with the parabolic FAWM solver. It also shows that the extended closures improve predictions compared to the standard k–ε model. However, the original model formulation with ∂V/∂x included in the production term is not reliably stable, and the presented results were obtained with a modified formulation. No single closure is best for all flow variables simultaneously. Further work is needed to resolve the stability issue of the original formulation.
This thesis develops and validates a parabolised, axisymmetric k–ε turbulence model that can be coupled to the Forced Ainslie Wake Model. Three turbulence closures are implemented: the standard k–ε model and two of its extensions. The k–ε–fP model limits the eddy viscosity in regions of high shear through a variable C*μ derived from the nonlinear eddy viscosity model framework. The extended k–ε model with a sink term Sk accounts for the extraction of turbulent kinetic energy by the actuator disk. The transport equations for k and ε are discretised on a staggered finite volume grid using a second-order upwind scheme for the convective terms and central differencing for the diffusive terms. The turbulence model is coupled with the FAWM through a Picard iteration scheme that exchanges velocity and eddy viscosity at each axial station until convergence. The implementation is verified through the decay of isotropic turbulence, the Method of Manufactured Solutions, and a self-similar co-flowing jet simulation. In all three tests, the solver achieves the formal order of convergence. This confirms the correct implementation of the discretisation and the coupling between the transport equations.
During the validation against LES data for a DTU 10 MW reference wind turbine across four operating conditions, numerical stability problems were encountered. In the original model formulation, only 5 out of 12 simulations converged. Some of the converged cases produced non-physical values of the turbulent kinetic energy. The source of the instabilities was traced to the axial gradient of the radial velocity, ∂V/∂x, which contributes to the strain-rate invariant and therefore the production term. Because the radial velocity is derived from the axial velocity through the continuity equation, ∂V/∂x depends on the second axial derivative of U. Any irregularity in the actuator disk forcing is transferred to the axial velocity through the momentum equation. In a parabolic solver, the axial diffusion and pressure terms that would normally dampen these gradients are absent, leading to an overestimation of ∂V/∂x. Neglecting ∂V/∂x in the computation of the production term restored convergence in 11 out of 12 cases. This also keeps all predicted values within physical bounds.
With the modified formulation, the three turbulence closures were compared against the LES reference data. For the axial velocity, the k–ε–fP model performs best in the near wake, with MAPE values between 1 and 4%, while the k–ε–Sk model produces the lowest errors in the far wake. The standard k–ε model has the largest velocity errors across all configurations. The velocity errors increase with increasing thrust coefficient and decreasing wind speed. For the turbulent kinetic energy, all three models overpredict the LES values by a large margin, with full-wake MAPE values ranging from 50 to 126%. The k–ε–Sk model produces the lowest TKE errors in every configuration, while the standard model and k–ε–fP model have comparable errors.
In summary, this thesis demonstrates that parabolised turbulence models can be coupled with the parabolic FAWM solver. It also shows that the extended closures improve predictions compared to the standard k–ε model. However, the original model formulation with ∂V/∂x included in the production term is not reliably stable, and the presented results were obtained with a modified formulation. No single closure is best for all flow variables simultaneously. Further work is needed to resolve the stability issue of the original formulation.
Bachelor thesis
(2021)
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J.F. Bramlage, V. Catalán Pastor, M.A. Dabrowski, A.G.K. van Dijk, Leonhard Xaver Driever, S.E. Lubach, Øyvind Pettersen, K.A. Roman, A.M.A. Tournoy, Joe Verbist, R.N.H.W. van Gent, C.P. van Dercreek, K. Masania
Bush planes are general aviation aircraft, that enable transportation to remote areas, where there is no infrastructure supporting regular aviation. Their main features are the taildragger configuration, a short take off and landing distance (STOL) and they offer the ability to land on rough terrain. Paradoxically, although they are the aircraft most directly related to nature, bush planes are often old, polluting and loud, and thus far from being environmentally friendly. To partially overcome these disadvantageous characteristics, Group 12 designed a stateoftheart bush plane, using the principle of distributed propulsion, called the Twin Puffin. In order to design a bush plane, first an understanding is required of the needs and desires of the stakeholders. For this, a market analysis is performed and from this it can be concluded that the aircraft will serve for three main purposes: transport, medical emergency missions and tourism. After obtaining the insight into the market of bush planes, all possible design options are listed. Pruning of unfeasible, unrealistic and inapplicable options is done to end up with seven aircraft concepts. From those concepts, the most suitable and promising is then selected. The aircraft is chosen to be a twin boom concept, therefore the name Twin Puffin was chosen for the design. Following, the design is worked out in detail, where all the subsystems are designed. The fuselage, the structure of the plane, the energy source, the wing, the propulsion system, the empennage, landing gear and electrical systems are designed and optimised, so the final aircraft design is finalised. Inspired by Nature, the bush plane is named the Twin Puffin. ’Twin’ following the distinctive twinboom empennage, and the ’Puffin’, from the bird with a stubby display and a master of short takeoff and landing on the ocean cliffsides, a real inspiration for a STOL aircraft. The featured twin boom empennage make aft loading of cargo or a medical stretcher easy. Furthermore, the distributed propulsion is placed on the wing’s leading edge, allowing unobstructed view during all flight phases, solving the typical visibility issues of a traditional bush plane. The distributed propellers are powered by a hybrid engine using both electricity from batteries and power generated by an internal combustion engine that can run on diesel, jet fuel, and suitable types of biofuels. This allows for an increase in available power and a local reduction in the emissions and noise during electricallypowered takeoff and landing. Furthermore, the distributed electric propulsion lead to excellent STOL characteristics, as the blown air over the wing allow for a large increase in lift at low speeds. Moreover, the Twin Puffin is primarily built of the sustainable material flax fibre composite, making the aircraft more environmentally friendly. The Twin Puffin is estimated to produce 70% less noise and 50% emission, compared to competing aircraft and is thereby a modern, impressively performing bush plane design.
...
Bush planes are general aviation aircraft, that enable transportation to remote areas, where there is no infrastructure supporting regular aviation. Their main features are the taildragger configuration, a short take off and landing distance (STOL) and they offer the ability to land on rough terrain. Paradoxically, although they are the aircraft most directly related to nature, bush planes are often old, polluting and loud, and thus far from being environmentally friendly. To partially overcome these disadvantageous characteristics, Group 12 designed a stateoftheart bush plane, using the principle of distributed propulsion, called the Twin Puffin. In order to design a bush plane, first an understanding is required of the needs and desires of the stakeholders. For this, a market analysis is performed and from this it can be concluded that the aircraft will serve for three main purposes: transport, medical emergency missions and tourism. After obtaining the insight into the market of bush planes, all possible design options are listed. Pruning of unfeasible, unrealistic and inapplicable options is done to end up with seven aircraft concepts. From those concepts, the most suitable and promising is then selected. The aircraft is chosen to be a twin boom concept, therefore the name Twin Puffin was chosen for the design. Following, the design is worked out in detail, where all the subsystems are designed. The fuselage, the structure of the plane, the energy source, the wing, the propulsion system, the empennage, landing gear and electrical systems are designed and optimised, so the final aircraft design is finalised. Inspired by Nature, the bush plane is named the Twin Puffin. ’Twin’ following the distinctive twinboom empennage, and the ’Puffin’, from the bird with a stubby display and a master of short takeoff and landing on the ocean cliffsides, a real inspiration for a STOL aircraft. The featured twin boom empennage make aft loading of cargo or a medical stretcher easy. Furthermore, the distributed propulsion is placed on the wing’s leading edge, allowing unobstructed view during all flight phases, solving the typical visibility issues of a traditional bush plane. The distributed propellers are powered by a hybrid engine using both electricity from batteries and power generated by an internal combustion engine that can run on diesel, jet fuel, and suitable types of biofuels. This allows for an increase in available power and a local reduction in the emissions and noise during electricallypowered takeoff and landing. Furthermore, the distributed electric propulsion lead to excellent STOL characteristics, as the blown air over the wing allow for a large increase in lift at low speeds. Moreover, the Twin Puffin is primarily built of the sustainable material flax fibre composite, making the aircraft more environmentally friendly. The Twin Puffin is estimated to produce 70% less noise and 50% emission, compared to competing aircraft and is thereby a modern, impressively performing bush plane design.