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Optimization-based method for conjugate heat transfer problems

Liang Fang, Xiandong Liu, Lei Zhang

TL;DR

This paper addresses efficient partitioned solution of conjugate heat transfer problems by pairing a semi-implicit finite-volume Navier–Stokes solver with an optimization-based coupling across the fluid–solid interface. The Navier–Stokes equations are advanced with an explicit treatment of convection and implicit pressure/diffusion, ensuring stability under a CFL-like condition $0\le C_m\le 1$, while the interface heat transfer is enforced by solving a constrained optimization problem for the interface flux $\mathbf{g}$ via sequential quadratic programming. A reduced representation $\mathbf{g}=\boldsymbol{\Phi}\boldsymbol{\beta}$ using Laplace–Beltrami eigenfunctions accelerates the optimization, yielding a fast, parallelizable coupling. Numerical experiments across lid-driven cavity, diffusion, flow over a heated plate, and natural convection demonstrate improved efficiency and robustness over traditional Dirichlet-to-Neumann or SIMPLE-based approaches, with accurate temperature continuity at the interface and stable, time-dependent flow fields. The work offers a flexible framework applicable with various fluid solvers and points to future enhancements such as factorization strategies, nearly implicit schemes, and Robin-type interface conditions to further boost performance.

Abstract

We propose a numerical approach for solving conjugate heat transfer problems using the finite volume method. This approach combines a semi-implicit scheme for fluid flow, governed by the incompressible Navier-Stokes equations, with an optimization-based approach for heat transfer across the fluid-solid interface. In the semi-implicit method, the convective term in the momentum equation is treated explicitly, ensuring computational efficiency, while maintaining stability when a CFL condition involving fluid velocity is satisfied. Heat exchange between the fluid and solid domains is formulated as a constrained optimization problem, which is efficiently solved using a sequential quadratic programming method. Numerical results are presented to demonstrate the effectiveness and performance of the proposed approach.

Optimization-based method for conjugate heat transfer problems

TL;DR

This paper addresses efficient partitioned solution of conjugate heat transfer problems by pairing a semi-implicit finite-volume Navier–Stokes solver with an optimization-based coupling across the fluid–solid interface. The Navier–Stokes equations are advanced with an explicit treatment of convection and implicit pressure/diffusion, ensuring stability under a CFL-like condition , while the interface heat transfer is enforced by solving a constrained optimization problem for the interface flux via sequential quadratic programming. A reduced representation using Laplace–Beltrami eigenfunctions accelerates the optimization, yielding a fast, parallelizable coupling. Numerical experiments across lid-driven cavity, diffusion, flow over a heated plate, and natural convection demonstrate improved efficiency and robustness over traditional Dirichlet-to-Neumann or SIMPLE-based approaches, with accurate temperature continuity at the interface and stable, time-dependent flow fields. The work offers a flexible framework applicable with various fluid solvers and points to future enhancements such as factorization strategies, nearly implicit schemes, and Robin-type interface conditions to further boost performance.

Abstract

We propose a numerical approach for solving conjugate heat transfer problems using the finite volume method. This approach combines a semi-implicit scheme for fluid flow, governed by the incompressible Navier-Stokes equations, with an optimization-based approach for heat transfer across the fluid-solid interface. In the semi-implicit method, the convective term in the momentum equation is treated explicitly, ensuring computational efficiency, while maintaining stability when a CFL condition involving fluid velocity is satisfied. Heat exchange between the fluid and solid domains is formulated as a constrained optimization problem, which is efficiently solved using a sequential quadratic programming method. Numerical results are presented to demonstrate the effectiveness and performance of the proposed approach.

Paper Structure

This paper contains 16 sections, 2 theorems, 51 equations, 14 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Numerical methods
  4. Semi-implicit method for the Navier-Stokes equations
  5. Optimization-based method for heat transfer problems
  6. Sequential quadratic programming method for solving the optimization problem
  7. Reduced optimization-based method
  8. Optimization-based method for conjugate heat transfer problems
  9. Numerical results
  10. Lid-driven cavity
  11. Diffusion problem
  12. Flow over a heated plate
  13. Natural convection
  14. Conclusions
  15. Numerical stability
  16. ...and 1 more sections

Key Result

Theorem 1

The semi-implicit method, with Rhie-Chow interpolation for computing the face velocity, is stable if the CFL condition is satisfied: where $C_m = C_{m1} + C_{m2} = \frac{u_0\Delta t}{\Delta x}+\frac{v_0\Delta t}{\Delta y}$, $u_0$ and $v_0$ represent the characteristic flow velocities in the $x-$ and $y-$directions, respectively, $\Delta x$ and $\Delta y$ denote the spatial discretization sizes in

Figures (14)

  • Figure 1: computational domain for conjugate heat transfer.
  • Figure 2: illustration of fluid-solid interface with two adjacent cells.
  • Figure 3: lid-driven cavity: $x$-velocity field and streamline pattern for $\text{Re}=1000$.
  • Figure 4: lid-driven cavity: (a), (b), (c) $x$-velocity along vertical line through geometric center of cavity; (d), (e), (f) $y$-velocity along horizontal line through geometric center of cavity.
  • Figure 5: lid-driven cavity: evolution of the $x$-velocity at the center point.
  • ...and 9 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Remark 3.1
  • Remark 3.2
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • proof
  • Remark