Multiple Reference Frame (MRF)

• Multiple Reference Frame (MRF) is a computational method that uses distinct local coordinate frames to simplify the analysis of systems with rotating and stationary regions.
• It partitions complex domains—such as in CFD for rotating machinery—into separate zones to incorporate rotation-induced forces and improve simulation efficiency.
• MRF techniques extend to cooperative localization and quantum key distribution, where accurate transformation between local frames is critical for performance and security.

Multiple Reference Frame (MRF) denotes the explicit treatment of a system through distinct local frames rather than a single global frame. In steady Reynolds-Averaged Navier–Stokes simulations of rotating machinery, the computational domain is split into rotating and stationary regions; in cooperative localization, each agent maintains a body-fixed coordinate frame; and in reference-frame-independent quantum key distribution, rapidly varying frame misalignment must be estimated or compensated during post-processing (Reid et al., 5 Aug 2025, Tran et al., 2019, Tang et al., 2021). This suggests a common technical theme: the model is organized around transformations between local frames, and performance depends on how those transformations are represented, coupled, and validated.

1. Rotating and stationary subdomains in fluid mechanics

In stirred-tank and turbomachinery CFD, the MRF technique is a steady-state approach often used in Reynolds-Averaged Navier–Stokes simulations to model rotating machinery. The computational domain is split into two (or more) regions: a rotating frame encompassing the impeller and its immediate vicinity, and a stationary frame comprising the rest of the tank. Within the rotating region, the governing equations account for additional apparent forces due to rotation, while the impeller itself is fixed in space and the effect of rotation is included via modified equations. This avoids modeling transient impeller–baffle interactions explicitly and results in computational savings compared to transient sliding-mesh approaches (Reid et al., 5 Aug 2025).

In the stationary frame, the formulation is written as

$\nabla \cdot \mathbf{U} = 0,$

$$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$

In the rotating region, the relative velocity is

$$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$

and the momentum equation becomes

$$\nabla \cdot (\mathbf{U}_r \mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) - \mathbf{\Omega}\mathbf{U} + \nabla \cdot \mathbf{R}.$$

Here the frame transformation is embedded directly in the PDEs rather than in mesh motion. In practice, this is the defining feature of the classical MRF approximation for rotating equipment.

2. Relative and absolute velocity formulations

A more detailed MRF treatment distinguishes between absolute and relative velocities in the rotating region. In that formulation,

$$\boldsymbol{u}^a = \boldsymbol{u}_\omega + \boldsymbol{u}^r, \qquad \boldsymbol{u}_\omega = \boldsymbol{\omega} \times \boldsymbol{r},$$

where $\boldsymbol{u}^a$ is the absolute velocity, $\boldsymbol{u}^r$ is the velocity relative to the rotating frame, and $\boldsymbol{u}_\omega$ is the solid-body rotational contribution. When the governing equations are written directly for $\boldsymbol{u}^r$, the solution exhibits a discontinuity across the interface between rotating and stationary regions, which leads to numerical instabilities. To avoid this, the rotating-solid study reformulates the problem in terms of absolute velocity and derives a unified equation valid in both fluid and solid regions (Liu et al., 15 Jan 2026).

The unified absolute-velocity equation is

$$\frac{\partial u_i^a}{\partial t} + u_j^r \frac{\partial u_i^a}{\partial x_j^r} = -\frac{\partial p}{\partial x_i^r} + \frac{1}{Re} \frac{\partial^2 u_i^a}{\partial x_j^r \partial x_j^r} - \eta \phi \left( u_i^a - \psi \varepsilon_{ijk} \omega_j r_k^r \right) - \psi \varepsilon_{ijk} \omega_j u_k^a,$$

with associated continuity equation

$$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$0

Here $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$1 is 1 in the rotating region and 0 elsewhere, while $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$2 is the phase indicator used in the level-set-based volume penalization. The same source also states a core limitation of the classical MRF approximation: MRF is steady-state and neglects unsteady rotor-stator interactions, whereas sliding mesh is needed for capturing full transient effects (Liu et al., 15 Jan 2026).

3. MRF-zone geometry, sensitivity, and artifacts

For stirred-tank simulations, the diameter and thickness of the MRF region are not incidental implementation details but primary modeling parameters. In the Rushton turbine study, five diameters of the MRF region were examined, ranging from $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$3 to $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$4, with a thickness primarily of $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$5 and one additional case at $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$6. The results showed limited differences in velocity profiles and generally good agreement with available experimental data, but significant differences in the predicted turbulent field. For the largest diameters, a considerable amount of artificially generated turbulence appeared at the boundary of the MRF region, and the resulting mixing times changed by up to a factor of three (Reid et al., 5 Aug 2025).

The same study makes clear that mean-flow agreement is not sufficient to validate an MRF setup. The smallest zone offered the best match to experimental double-peak turbulent kinetic energy profiles just off the impeller tips, whereas larger zones produced artificial rings of high turbulent kinetic energy near the MRF interface. The frozen-flow passive-scalar calculations further showed global mixing times as low as $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$7 for the largest tested MRF region, versus $$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$8 for the smallest. By contrast, varying the vertical thickness of the MRF region had only minor effects on predicted velocities and turbulent kinetic energy, mostly at the vertical ends of the impeller (Reid et al., 5 Aug 2025).

The methodological consequence is explicit in the recommendations: the MRF interface should be placed away from areas of steep velocity and turbulence gradients, should not cross regions near baffles or vessel walls, and should not enclose the entire tank. The same source recommends treating MRF-region sensitivity as a standard practice, analogous to mesh sensitivity, particularly when turbulence quantities or mixing times are outputs of interest (Reid et al., 5 Aug 2025).

4. Immersed-boundary and volume-penalization variants

MRF can also be embedded in immersed-boundary formulations. In the volume-penalization study, the volume penalization method was combined with MRF and sliding mesh to develop immersed-boundary approaches for simulating flows around a rotating solid. A level-set function represents arbitrary geometries embedded in Cartesian grids, and the VPM body-forcing terms in the momentum equation are proposed for MRF and sliding mesh so as to build unified governing equations for both fluid and solid regions. The corresponding MRF body-force term is

$$\nabla \cdot (\mathbf{U}\mathbf{U}) = -\frac{1}{\rho}\nabla P + \nabla \cdot (\nu \nabla \mathbf{U}) + \nabla \cdot \mathbf{R}.$$9

which forces the absolute velocity within the solid region to match the prescribed rotating-solid velocity (Liu et al., 15 Jan 2026).

The rotating-cuboid validation compared VPM with MRF and VPM with sliding mesh against corresponding body-fitted simulations. The reported result is that the relative deviations of predicted pressure drop and torque between the present VPM and the body-fitted method are around $$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$0, demonstrating the validity of the present VPM (Liu et al., 15 Jan 2026). The same source also notes a diffuse-interface limitation: the no-slip boundary condition is enforced over a finite thickness, so there is always a residual small velocity at the solid surface. This suggests a characteristic trade-off of immersed MRF formulations: substantial simplification of geometry handling and meshing, at the cost of near-interface modeling error that must be quantified.

5. Time-varying and distributed reference frames

Outside rotating-flow solvers, multiple reference frames arise as alignment and estimation problems. In reference-frame-independent quantum key distribution, rapidly and randomly drifted reference frames shorten the link distance and decrease the secure key rate of realistic systems, while actively or inappropriately implemented calibration schemes increase complexity and may open security loopholes. The free-running RFI QKD scheme addresses this by classifying measurement events into multiple slices with the same misalignment variation of reference frames and performing the post-processing procedure individually for each slice. The frame relation in the $$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$1 and $$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$2 bases is

$$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$3

and the experiment reported a 100 km fiber link, more than 29 full $$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$4 periods of reference-frame variation in a 50.7-hour test, and an average secure key rate of about 734 bps with a total loss of 31.5 dB. Under realistic noise, the maximum tolerable total loss increased from 32.6 dB for the original RFI protocol to 43.3 dB for the free-running protocol (Tang et al., 2021).

In cooperative localization, each agent maintains a body-fixed coordinate frame, and its actual frame transformation from the global coordinate system is unknown. The objective is to determine the trajectories of rigid-body motions, or equivalently the frame transformations, with respect to the global frame up to a common frame transformation, using local measurements and information exchanged with neighbors. The relative transformation between agents is

$$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$5

and the paper proposes asymptotic and finite-time localization laws. Under both localization laws, the estimates of the frame transformations converge almost globally and up to an unknown constant transformation bias, with the limiting relation

$$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$6

Here the multiple-reference-frame problem is not a steady approximation but an observability and consensus problem on $$\mathbf{U}_r = \mathbf{U} - \mathbf{\Omega} \times \mathbf{r},$$7 (Tran et al., 2019).

6. Terminology and acronym ambiguity

The acronym MRF is overloaded across arXiv literatures. In magnetic resonance imaging it denotes Magnetic Resonance Fingerprinting, as in “Balanced multi-shot EPI for accelerated Cartesian MRF” and “GFB-MRF” (Benjamin et al., 2018, Arberet et al., 2020). In computer vision it denotes Markov Random Field, as in “Superpixelizing Binary MRF for Image Labeling Problems” and “CNN in MRF” (Wang et al., 2015, Bao et al., 2018). In fluid mechanics, by contrast, MRF denotes Multiple Reference Frame, and in adjacent coordination and communication problems the phrase refers more generally to the presence of several local frames whose transformations are unknown, drifting, or only partially observable. Context therefore determines whether MRF denotes a rotating-domain CFD approximation, a frame-alignment problem, Magnetic Resonance Fingerprinting, or a Markov Random Field.