Quantum algorithm for the Navier–Stokes equations by using the streamfunction-vorticity formulation and the lattice Boltzmann method

This study investigates the prediction of two-dimensional lid-driven cavity fluid flow in a square domain using supervised machine learning techniques. The classical problem of lid-driven cavity flow is utilized to demonstrate the integration of the mesoscopic Lattice Boltzmann Method (LBM) with machine learning algorithms. The LBM is employed to generate macroscopic velocity and pressure data for varying Reynolds numbers (Re = 2000, 3200, 5000, and 7500). Machine learning-based regression models are then applied to forecast key macroscopic variables, including density, pressure, and velocity based on directional coordinate measurements. For all datasets, the Random Forest regressor consistently predicted velocity with high precision, with R² scores of 0.9998 (Re = 2000), 0.9997 (Re = 3200), 0.9996 (Re = 5000), and 0.9996 (Re = 7500), and RMSE values ranging between 0.0032–0.0042. For pressure prediction, XGBoost outperformed others with an R² score of 0.9250 and RMSE of 1.04×10⁻⁴, while Random Forest followed closely with R² = 0.9063 and RMSE = 1.02×10⁻⁴. On the other hand, Decision Tree (R² = 0.112) and Linear Regression (R² = 0.064) performed badly in terms of prediction, underscoring the superiority of ensemble-based approaches. The study further demonstrates that ensemble-based approaches like Random Forest and XGBoost outperform traditional regression methods, showing significant capability for accelerating the parametric research and enhancing design optimization in engineering domain.

In this paper, two-dimensional lid-driven cavity flow phenomena at steady state were simulated using two different scales of numerical method: the finite difference solution to the Navier–Stokes equation and the cubic interpolated pseudo-particle lattice Boltzmann method. The aspect ratio of cavity was set at 1, 2/3, 1/2 and 1/3 and the Reynolds number of 100, 400 and 1,000 for every simulation condition. The results were presented in terms of the location of the center of main vortex, the streamline plots and the velocity profiles at vertical and horizontal midsections. In this study, it is found that at the simulation of Reynolds numbers 100 and 400, both methods demonstrate a good agreement with each other; however, small discrepancies appeared for the simulation at the Reynolds number of 1,000. We also found that the number, size and formation of vortices strongly depend on the Reynolds number. The effect of the aspect ratio on the fluid flow behavior is also presented.

We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots.

The Lattice Boltzmann Method (LBM) which is derivative of the lattice gas automata (LGA) has recently studied as a numerical scheme for simulating fluid flows governed by the Navier-Stokes equation. The LBM is a discrete numerical method based on the Boltzmann equation. The dynamics of the LBM is divided into a propagation step and a collision step. We have proposed a model for weakly ionized plasma by introducing diffusion as a velocity to LBM. In our LBM model, ionization, recombination at the wall and ambipolar diffusion are considered. This model uses 2D9V (2 dimensions, 9 velocities) model and BGK collision operator. We have simulated various shape of discharge tubes by this model. MATLAB is suitable for matrix calculation and has a command for shifting matrix which is important in LBM. By this work, it is shown that our model simulates weakly ionized plasma well.

In recent years, the Lattice Boltzmann (LB) method has become a powerful numerical \ntool for simulating complex fluid flows and modeling physics and chemistry in \nfluids. Derived from the continuum Boltzmann equation used in statistical \nmechanics, the LB method has the advantage of describing non-equilibrium dynamics, \nespecially in fluid-flow applications involving interfacial dynamics and complex \nboundaries, without simplifying the physics. In addition, the parallel structure \ninherent in the LB method makes it extremely suitable for parallel computing. \nBecause of these features, the LB method affords the most comprehensive pore-scale \napproach to systematically investigate fundamental issues involving flow and \nreactive transport in porous media. In this paper, the state of the art of this \nmethod is discussed. Specifically, a multi-component LB model for simulating \nreactive transport in porous media at the pore scale is presented. In the model, a \nset of distribution functions is introduced to simulate fluid flow and solute \ntransport. The LB equation for flow recovers the correct pore-scale continuity and \nNavier-Stokes equations. The LB equations for solute transport are modified to \nrecover advection-diffusion equations for total concentrations at the pore scale. \nThe model takes into account advection, diffusion, homogeneous reactions among \nmultiple aqueous species, heterogeneous reactions between the aqueous solution and \nminerals, as well as changes in solid and pore geometry. Homogeneous reactions are \ndescribed through local equilibrium mass action relations. Mineral reactions are \ntreated kinetically through boundary conditions at the mineral surface. Simulation \nexamples presented include injection of carbon dioxide saturated brine into a \nlimestone rock with pore geometry derived from a thin section, crystal formation \nfrom a supersaturated solution without flow, and crystal formation during carbon \ndioxide sequestration in oceanic sediments.

A three-dimensional (3D) lattice Boltzmann flux solver (LBFS) is presented in this paper for the simulation of both isothermal and thermal flows. The present solver combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. It applies the finite volume method (FVM) to solve the N-S equations. Different from the conventional N-S solvers, its viscous and inviscid fluxes at the cell interface are evaluated simultaneously by local reconstruction of LBE solution. As compared to the conventional LBE solvers, which apply the lattice Boltzmann method (LBM) globally in the whole computational domain, it only applies LBM locally at each cell interface, and flow variables at cell centers are given from the solution of N-S equations. Since LBM is only applied locally in the 3D LBFS, the drawbacks of the conventional LBM, such as limitation to uniform mesh, tie-up of mesh spacing and time step, tedious implementation of boundary conditions, are completely removed. The accuracy, efficiency and stability of the proposed solver are examined in detail by simulating plane Poiseuille flow, lid-driven cavity flow and natural convection. Numerical results show that the LBFS has a second order of accuracy in space. The efficiency of the LBFS is lower than LBM on the same grids. However, the LBFS needs very less non-uniform grids to get grid-independence results and its efficiency can be greatly improved and even much higher than LBM. In addition, the LBFS is more stable and robust.

Lattice Boltzmann (LB) methods are usually developed on cubic lattices that discretize the configuration space using uniform grids. For efficient computations of anisotropic and inhomogeneous flows, it would be beneficial to develop LB algorithms involving the collision-and-stream steps based on orthorhombic cuboid lattices. We present a new 3D central moment LB scheme based on a cuboid D3Q27 lattice. This scheme involves two free parameters representing the ratios of the characteristic particle speeds along the two directions with respect to those in the remaining direction, and these parameters are referred to as the grid aspect ratios. Unlike the existing LB schemes for cuboid lattices, which are based on orthogonalized raw moments, we construct the collision step based on the relaxation of central moments and avoid the orthogonalization of moment basis, which leads to a more robust formulation. Moreover, prior cuboid LB algorithms prescribe the mappings between the distribution functions and raw moments before and after collision by using a moment basis designed to separate the trace of the second order moments (related to bulk viscosity) from its other components (related to shear viscosity), which lead to cumbersome relations for the transformations. By contrast, in our approach, the bulk and shear viscosity effects associated with the viscous stress tensor are naturally segregated only within the collision step and not for such mappings, while the grid aspect ratios are introduced via simpler pre- and post-collision diagonal scaling matrices in the above mappings. These lead to a compact approach, which can be interpreted based on special matrices. It also results in a modular 3D LB scheme on the cuboid lattice, which allows the existing cubic lattice implementations to be readily extended to those based on the more general cuboid lattices. To maintain the isotropy of the viscous stress tensor of the 3D Navier–Stokes equations using the cuboid lattice, corrections for eliminating the truncation errors resulting from the grid anisotropy as well as those from the aliasing effects are derived using a Chapman–Enskog analysis. Such local corrections, which involve the diagonal components of the velocity gradient tensor and are parameterized by two grid aspect ratios, augment the second order moment equilibria in the collision step. We present a numerical study validating the accuracy of our approach for various benchmark problems at different grid aspect ratios. In addition, we show that our 3D cuboid central moment LB method is numerically more robust than its corresponding raw moment formulation. Finally, we demonstrate the effectiveness of the 3D cuboid central moment LB scheme for the simulations of anisotropic and inhomogeneous flows and show significant savings in memory storage and computational cost when used in lieu of that based on the cubic lattice.

A lattice Boltzmann method on nonuniform mesh is developed for the simulation of incompressible flows. The computation mesh is uncoupled from the lattice. The particle distribution function on the mesh is obtained using the Lagrange interpolation technique at each time step. Comparison with the finite difference solutions for the two-dimensional driven cavity flow and the experimental results for flow around a circular cylinder shows that the present scheme not only can predict the flow separation characteristic satisfactorily with much less mesh but also increase the computational stability. The influence of the mesh structure and interpolation on the results will also be discussed.

Characteristic boundary condition for thermal lattice Boltzmann methods

Lattice Boltzmann model for incompressible flows through porous media with time-fractional effects

Mixed convective characteristics of the combustion of a nanoparticles-laden fuel (n-butanol nanofluid) in a vented cavity are investigated. The nanofluid and the oxidizer enter the cavity through the inlets on the left and right vertical walls, respectively. However, the resulting product produced from the oxidation process of the fuel exits the cavity through the outlet at the bottom wall. Heat generated from the oxidation process causes natural convection within the cavity. The conjugate effect of natural and forced convection finally gives rise to mixed convection phenomena. In this regard, a mathematical model for mixed convection flow in a vented cavity is formulated with no-slip and isothermal boundary conditions. Having transformed the model into a dimensionless form, the stream function-vorticity formulation is used. The resulting equations are then solved numerically using the finite difference method. Numerical results are illustrated with the streamlines, isotherms, and isolines of fuel and oxidizer concentrations. The maximum values of the stream function (ψmax) and the temperature (θmax) are found to increase with an increase in the Frank–Kamenetskii number (Λ), volume fraction of nanoparticles (φ), and stoichiometric ratio (χ). On the contrary, they decrease with the increase in the Reynolds number (Re). When the Grashof number (Gr) is increased, ψmax increases and θmax decreases. The remaining concentrations of fuel, (CF)min, and oxidizer, (CO)min, are higher for an increase in Gr, whereas the opposite is recognized for increasing Λ. With the increase in Gr and Λ, the steady-state flow in the cavity tends to be oscillating and then chaotic.

Lattice Boltzmann simulation for unsteady shock wave/boundary layer interaction in a shock tube

A subgrid turbulence model for the lattice Boltzmann method is proposed for high Reynolds number fluid flow applications. The method, based on the standard Smagorinsky subgrid model and a single-time relaxation lattice Boltzmann method, incorporates the advantages of the lattice Boltzmann method for handling arbitrary boundaries and is easily implemented on parallel machines. The method is applied to a two-dimensional driven cavity flow for studying dynamics and the Reynolds number dependence of the flow structures. The substitution of other subgrid models, such as the dynamic subgrid model, in the framework of the LB method is discussed.

This paper presents a simulation of incompressible viscous flow within a two-dimensional square cavity. The objective is to develop a method originated from Lattice Gas (cellular) Automata (LGA), which utilises discrete lattice as well as discrete time and can be parallelised easily. Lattice Boltzmann Method (LBM), known as discrete Lattice kinetics which provide an alternative for solving the Navier–Stokes equations and are generally used for fluid simulation, is chosen for the study. A specific two-dimensional nine-velocity square Lattice model (D2Q9 Model) is used in the simulation with the velocity at the top of the cavity kept fixed. LBM is an efficient method for reproducing the dynamics of cavity flow and the results which are comparable to those of previous work.

The lattice Boltzmann method often involves small numerical time steps due to the acoustic scaling (i.e., scaling between time step and grid size) inherent to the method. In this work, a second-order dual-time-stepping lattice Boltzmann method is proposed in order to avoid any time-step restriction. The implementation of the dual time stepping is based on an external source in the lattice Boltzmann equation, related to the time derivatives of the macroscopic flow quantities. Each time step is treated as a pseudosteady problem. The convergence rate of the steady lattice Boltzmann solver is improved by implementing a multigrid method. The developed solver is based on a two-relaxation time model coupled to an immersed-boundary method. The reliability of the method is demonstrated for steady and unsteady laminar flows past a circular cylinder, either fixed or towed in the computational domain. In the steady-flow case, the multigrid method drastically increases the convergence rate of the lattice Boltzmann method. The dual-time-stepping method is able to accurately reproduce the unsteady flows. The physical time step can be freely adjusted; its effect on the simulation cost is linear, while its impact on the accuracy follows a second-order trend. Two major advantages arise from this feature. (i) Simulation speed-up can be achieved by increasing the time step while conserving a reasonable accuracy. A speed-up of 4 is achieved for the unsteady flow past a fixed cylinder, and higher speed-ups are expected for configurations involving slower flow variations. Significant additional speed-up can also be achieved by accelerating transients. (ii) The choice of the time step allows us to alter the range of simulated timescales. In particular, increasing the time step results in the filtering of undesired pressure waves induced by sharp geometries or rapid temporal variations, without altering the main flow dynamics. These features may be critical to improve the efficiency and range of applicability of the lattice Boltzmann method.