An analytical model for rotors in confined flow across operating regimes

Abstract

Rotors operating in confined flows, or blockage, are commonly encountered in wind and water tunnels, as well as in shallow or dense deployments of hydrokinetic turbines. Confinement induces a streamwise pressure gradient in the channel, modifying the rotor induction, thrust, and power. To account for these effects, physics-based or empirical blockage corrections are used as a transfer function between the dynamics of an object operating in confined and unconfined settings. However, existing blockage models are largely only applicable to rotors operating at relatively low thrust coefficients, such that the assumptions of classical momentum theory are valid. Further, rotors are often partially misaligned with the inflow, which modifies both the geometric blockage and the thrust force, whereas existing blockage models assume perfectly aligned flow conditions. We develop a generalised engineering model for an actuator disk operating in confined flow at arbitrary misalignment angles and thrust coefficients, termed the Unified Blockage Model. The analytical model shows excellent agreement with large eddy simulations of an actuator disk and elucidates the coupled interactions between thrust, misalignment, and blockage. To predict bladed rotor dynamics, the Unified Blockage Model is incorporated into a blade element momentum (BEM) model framework and validated against blade-resolved simulations across a wide range of tip-speed and blockage ratios. Finally, a blockage correction method is developed based on the Unified Blockage Model and validated against a suite of numerical and experimental data.

Figures (14)

• Figure 1: Control volume for analysis of a confined, misaligned turbine. The streamwise and spanwise directions are $x$ and $y$, respectively. The turbine is yaw/tilt misaligned at angle $\gamma$, where positive misalignment is a counter-clockwise rotation. The sides of the control volume are rigid so there is no mass flux through them. As in classical momentum theory, we consider four stations for the analysis, and the flow variables are labeled with the corresponding station as subscript numbers. The cross-sectional areas, streamwise velocities, spanwise velocities, and pressures are denoted as $A$, $u$, $v$, and $p$, respectively. The pressure at the outlet is $p_{4,w}$ within the streamtube and $p_{4}$ outside of the streamtube. The unit vector normal to the misaligned turbine is $\hat{n}$.
• Figure 2: Comparison between the Unified Blockage Model (Eqs. \ref{['eq:final_1']}-\ref{['eq:final_6']}) and LES for (a) rotor-normal induction, (b) thrust coefficient, and (c) power coefficient across local thrust coefficients $C_T'$ and blockage ratios $\beta$ with rotor-aligned inflow ($\gamma=0$). The predictions of classical one-dimensional momentum theory and the blockage model of steiros2022analytical are plotted for reference.
• Figure 3: Comparison of (a) bypass velocity, (b) initial streamwise wake velocity, (c) spanwise initial wake velocity, (d) outlet wake area, (e) bypass pressure drop, and (f) streamtube pressure drop between the Unified Blockage Model developed in §\ref{['sec:model_derivation']} and LES of an actuator disk for various local thrust coefficients $C_T'$ and rotor-aligned inflow ($\gamma = 0$).
• Figure 4: Comparison between the Unified Blockage Model (Eqs. \ref{['eq:final_1']}-\ref{['eq:final_6']}) and LES for (a) induction factor, (b) thrust coefficient and (c) power coefficient across misalignment angles $\gamma$ and blockage ratios $\beta$ with $C_T'=2$. Classical momentum theory and the blockage model of steiros2022analytical are not shown as they assume perfect alignment between the rotor and inflow.
• Figure 5: Comparison of (a) bypass velocity, (b) initial streamwise wake velocity, (c) spanwise initial wake velocity, (d) outlet wake area, (e) bypass pressure drop, and (f) streamtube pressure drop, between the Unified Blockage Model developed in §\ref{['sec:model_derivation']} and large eddy simulations for various misalignment angles.