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Flow Measurement: An Inverse Problem Formulation

Jiwei Li, Lingyun Qiu, Zhongjing Wang, Hui Yu

TL;DR

This paper develops a PDE-based framework for flow measurement by casting it as an inverse source problem for the wave equation with data collected on a sparse boundary surface. The forward model uses a moving-particle density $f_t$ driven by a flow via push-forward and solves $(1/c^2)\partial_t^2 U-\Delta U=\lambda(x,t)f_t(x)$ to relate particle motion to acoustic measurements. A well-posed inverse problem is formulated with a gradient-based minimization, its Fréchet derivative given by $D\mathcal{J}(f)=\mathcal{F}^*(\mathcal{F}(f)-U_{data})$, and an adjoint-based time-reversal approach to compute the gradient efficiently. Extensive 2D numerical experiments verify accurate reconstruction of stationary and moving sources, demonstrate robustness to noise, and explore different receiver layouts and source frequencies; comparisons with virtual ADCPs indicate improved accuracy. The approach offers high-resolution, multi-point velocity field estimation in complex flows and provides a flexible framework for simulating measurement scenarios and instrument configurations.

Abstract

This paper proposes a new mathematical formulation for flow measurement based on the inverse source problem for wave equations with partial boundary measurement. Inspired by the design of acoustic Doppler current profilers (ADCPs), we formulate an inverse source problem that can recover the flow field from the observation data on a few boundary receivers. To our knowledge, this is the first mathematical model of flow measurement using partial differential equations. This model is proved well-posed, and the corresponding algorithm is derived to compute the velocity field efficiently. Extensive numerical simulations are performed to demonstrate the accuracy and robustness of our model. Our formulation is capable of simulating a variety of practical measurement scenarios.

Flow Measurement: An Inverse Problem Formulation

TL;DR

This paper develops a PDE-based framework for flow measurement by casting it as an inverse source problem for the wave equation with data collected on a sparse boundary surface. The forward model uses a moving-particle density driven by a flow via push-forward and solves to relate particle motion to acoustic measurements. A well-posed inverse problem is formulated with a gradient-based minimization, its Fréchet derivative given by , and an adjoint-based time-reversal approach to compute the gradient efficiently. Extensive 2D numerical experiments verify accurate reconstruction of stationary and moving sources, demonstrate robustness to noise, and explore different receiver layouts and source frequencies; comparisons with virtual ADCPs indicate improved accuracy. The approach offers high-resolution, multi-point velocity field estimation in complex flows and provides a flexible framework for simulating measurement scenarios and instrument configurations.

Abstract

This paper proposes a new mathematical formulation for flow measurement based on the inverse source problem for wave equations with partial boundary measurement. Inspired by the design of acoustic Doppler current profilers (ADCPs), we formulate an inverse source problem that can recover the flow field from the observation data on a few boundary receivers. To our knowledge, this is the first mathematical model of flow measurement using partial differential equations. This model is proved well-posed, and the corresponding algorithm is derived to compute the velocity field efficiently. Extensive numerical simulations are performed to demonstrate the accuracy and robustness of our model. Our formulation is capable of simulating a variety of practical measurement scenarios.
Paper Structure (21 sections, 3 theorems, 35 equations, 12 figures, 2 tables)

This paper contains 21 sections, 3 theorems, 35 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical model for the flow measurement
  3. Physical process of the flow measurement
  4. Reformulation of the flow measurement
  5. Direct and inverse source problems for the wave equations
  6. Mathematical model of the flow measurement
  7. Direct source problems for the wave equations
  8. Inverse source problems for the wave equations
  9. Numerical schemes and experiments
  10. Statement of the minimization problem
  11. Reconstruction of the stationary source term
  12. Example 1: Different frequencies of source wave
  13. Example 2: Different layouts of receivers
  14. Example 3: Different noise levels
  15. Reconstruction of the moving source term and the flow field
  16. ...and 6 more sections

Key Result

Proposition 1

Assume that $f_t(x)\in C^1(\Omega\times[0,T])$ and $T_t(x)\in C^2(\Omega\times[0,T])$. Let $U_1(x,t),U_2(x,t)$ be the solutions of oriWaveEqns and appWave_sys respectively. Then $U_1(x,t),U_2(x,t)$ are sufficiently close provided that $T$ is small enough, namely for any $\epsilon>0$, there exists $\

Figures (12)

  • Figure 1: Observed zone at the junction of two streamsbestFlowDynamicsRiver1987.
  • Figure 2: Measurement instruments
  • Figure 3: The whole physical process of detection with ADCP. (a) The acoustic source wave is emitted by the transmitter and received by the moving particles in the flow during an extremely short period of time. (b) The acoustic wave scattered by the moving particles induces the response of the receivers.
  • Figure 4: Received data $U_\mathrm{data}$ in the settings of different central frequencies $q_0$ of the source wave. Colors represent the intensities of received data. The horizontal and vertical axes represent the time step when solving the forward problem and the position of receivers, respectively.
  • Figure 5: The contours of $f(x)$, i.e., reconstructions of the source term $f(x)$ for different central frequencies $q_0$ of the source wave. The red points represent the receivers. (a) is the exact locations of particles. (b-d) are reconstructions of the source term when the frequency of the source wave is 100 kHz, 10 kHz, and 1 kHz, respectively.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Remark 2
  • Proposition 3
  • proof
  • Remark 3