Matrix Product State Simulation of Reacting Shear Flows

TL;DR

This work introduces a matrix product state (MPS) tensor-network framework to simulate a two-dimensional compressible reacting shear flow described by six coupled PDEs, offering a scalable alternative to direct numerical simulation (DNS) for turbulent combustion. By encoding transport fields into an MPS and evolving with matrix product operators (MPOs) under a MacCormack-like scheme, the approach achieves DNS-grade accuracy with substantially fewer degrees of freedom, enabling memory reductions on the order of tens of percent and substantial compression at higher Reynolds numbers. The study systematically analyzes MPS truncation, bond-dimension allocation, and the impact of compressibility and heat release, showing that truncation error can be controlled and may decrease as system size grows, while dynamic bond-dimension strategies promise further gains. The results demonstrate that MPS-based simulations capture key physical phenomena such as reduced mixing due to compressibility and exothermicity, and the formation of eddy shocklets, offering a path toward scalable simulations of complex turbulent combustion on large scales and, potentially, on quantum hardware.

Abstract

Direct numerical simulation (DNS) of turbulent reactive flows has been the subject of significant research interest for several decades. Accurate prediction of the effects of turbulence on the rate of reactant conversion, and the subsequent influence of chemistry on hydrodynamics remain a challenge in combustion modeling. The key issue in DNS is to account for the wide range of temporal and spatial physical scales that are caused by complex interactions of turbulence and chemistry. In this work, a new computational methodology is developed that is shown to provide a viable alternative to DNS. The framework is the matrix product state (MPS), a form of tensor network (TN) as used in computational many body physics. The MPS is a well-established ansatz for efficiently representing many types of quantum states in condensed matter systems, allowing for an exponential compression of the required memory compared to exact diagonalization methods. Due to the success of MPS in quantum physics, the ansatz has been adapted to problems outside its historical domain, notably computational fluid dynamics. Here, the MPS is used for computational simulation of a shear flow under non-reacting and nonpremixed chemically reacting conditions. Reductions of 30% in memory are demonstrated for all transport variables, accompanied by excellent agreements with DNS. The anastaz accurately captures all pertinent flow physics such as reduced mixing due to exothermicity & compressibility, and the formation of eddy shocklets at high Mach numbers. A priori analysis of DNS data at higher Reynolds numbers shows compressions as large as 99.99% for some of the transport variables. This level of compression is encouraging and promotes the use of MPS for simulations of complex turbulent combustion systems.

Figures (18)

• Figure 1: The TDJ velocity profile characterized by two fluid streams moving with counter streamwise velocities.
• Figure 2: (a) Sketch of an unstructured tensor $\mathcal{T}$ and the corresponding MPS. Indices $i_j$ are physical indices. Indices $\alpha_j$ are a result of the MPS construction and are called bond indices. (b) Three site MPO. $\beta_i$ is used to denote the bond indices to distinguish them from the MPS bond indices.
• Figure 3: Example of singular value spectrum for the MPS of scalar $c_1$ with $\mathrm{Da}{} = 0$, $\mathrm{Re}{} = 2500$, and $\mathrm{Ma}_o{} = 0.2$. The size of the MPS is $N = 14$ which corresponds to a $2^7 \times 2^7 = 128 \times 128$ grid. The colors indicate the relative strength of correlations at each physical scale.
• Figure 4: MPS algorithm validation case with $\chi = r{} = 128$ showing contours of $c_1$. The black arrows show the velocity field $\vec{V}$. (a) and (c) are the DNS solution at $t = 0.8$ and $t = 1.2$, respectively. (b) and (d) are the MPS solution at $t = 0.8$ and $t = 1.2$, respectively.
• Figure 5: Non-reacting ($\mathrm{Da}{} \!=\! 0$) MPS simulation with $\chi = 34$ at $t {\sim}{} 2.7$. (a), (c), and (e) are $c_1$ contours at $\mathrm{Ma}_o{} =\numlist{0.2;0.4;0.6}$, respectively. (b), (d), and (f) are $\omega$ at $\mathrm{Ma}_o{} =\numlist{0.2;0.4;0.6}$, respectively. The black arrows show the velocity field $\vec{V}$.