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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Towards a Quantum Algorithm for the Incompressible Nonlinear Navier-Stokes Equations

In this work, we present novel concepts for quantum algorithms to solve transient, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the incompressible Navier-Stokes equations as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts solving nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. We propose a new framework based on matrix product states (MPSs) and matrix product operators (MPOs), in addition to the Krylov subspace methods. For example, the solution variables of the Navier-Stokes equations are represented by MPSs, and the linear and nonlinear terms are processed by MPOs. The time evolution of the operators is attained by a fast-forwarding algorithm using Krylov subspace methods. Furthermore, we discuss various techniques for efficient encoding of MPSs, measurement reduction for MPOs, and use of tensor operations to treat multi-variate, multi-physics characteristics of Navier-Stokes.

Gopalakrishnan Meena, Murali [ORNL] (ORCID:0000000↗
NASA NTRS1995

Unified approach for incompressible flows

A unified approach for solving incompressible flows has been investigated in this study. The numerical CTVD (Centered Total Variation Diminishing) scheme used in this study was successfully developed by Sanders and Li for compressible flows, especially for the high speed. The CTVD scheme possesses better mathematical properties to damp out the spurious oscillations while providing high-order accuracy for high speed flows. It leads us to believe that the CTVD scheme can equally well apply to solve incompressible flows. Because of the mathematical difference between the governing equations for incompressible and compressible flows, the scheme can not directly apply to the incompressible flows. However, if one can modify the continuity equation for incompressible flows by introducing pseudo-compressibility, the governing equations for incompressible flows would have the same mathematical characters as compressible flows. The application of the algorithm to incompressible flows thus becomes feasible. In this study, the governing equations for incompressible flows comprise continuity equation and momentum equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. Thus, the CTVD schemes can be implemented. In addition, the physical and numerical boundary conditions are properly implemented by the characteristic boundary conditions. Accordingly, a CFD code has been developed for this research and is currently under testing. Flow past a circular cylinder was chosen for numerical experiments to determine the accuracy and efficiency of the code. The code has shown some promising results.

Chang, Tyne-Hsien↗
NASA NTRS1993

Unified approach for incompressible flows

An unified approach for solving both compressible and incompressible flows was investigated in this study. The difference in CFD code development between incompressible and compressible flows is due to the mathematical characteristics. However, if one can modify the continuity equation for incompressible flows by introducing pseudocompressibility, the governing equations for incompressible flows would have the same mathematical characters as compressible flows. The application of a compressible flow code to solve incompressible flows becomes feasible. Among numerical algorithms developed for compressible flows, the Centered Total Variation Diminishing (CTVD) schemes possess better mathematical properties to damp out the spurious oscillations while providing high-order accuracy for high speed flows. It leads us to believe that CTVD schemes can equally well solve incompressible flows. In this study, the governing equations for incompressible flows include the continuity equation and momentum equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. Thus, the CTVD schemes can be implemented. In addition, the boundary conditions including physical and numerical boundary conditions must be properly specified to obtain accurate solution. The CFD code for this research is currently in progress. Flow past a circular cylinder will be used for numerical experiments to determine the accuracy and efficiency of the code before applying this code to more specific applications.

Chang, Tyne-Hsien↗
NASA NTRS1991

The equations of nearly incompressible fluids. I - Hydrodynamics, turbulence, and waves

An attempt is made to develop a more general theory of nearly incompressible fluids that can then be applied to many different fields. A perturbation expansion is developed for the fully compressible fluid equations which, in the limit of low Mach number (sound or Alfvenic Mach number), reduce to the appropriate incompressible fluid equations. The method developed derives modified systems of fluid equations in which the compressibility effects are admitted only weakly in terms of the incompressible hydrodynamic solutions ('nearly incompressible hydrodynamics'). Molecular viscosity is included self-consistently, and the role of thermal conduction in an ideal fluid is also considered. With heat conduction included, two distinct routes to incompressibility are found to be possible, distinguished according to the relative magnitudes of the temperature, density, and pressure fluctuations.

Zank, G. P.↗
NASA NTRS1992

Nearly incompressible fluid dynamics

An evaluation is made of general theoretical models pertinent to investigations of both compressible and incompressible MHD turbulence in the solar wind. It is noted that the underlying incompressible description for turbulence investigations in the solar wind should be 2D incompressible MHD, rather than the usual 3D description. 'Nearly incompressible' MHD can account for the observed density, temperature, and magnetic fluctuation spectra; the density fluctuations are found to be partially slaved to the magnetic field fluctuations, as well as to be generated by the incompressible flow field.

Zank, G. P.↗
NASA NTRS2004

Computational Challenges of Viscous Incompressible Flows

Over the past thirty years, numerical methods and simulation tools for incompressible flows have been advanced as a subset of the computational fluid dynamics (CFD) discipline. Although incompressible flows are encountered in many areas of engineering, simulation of compressible flow has been the major driver for developing computational algorithms and tools. This is probably due to the rather stringent requirements for predicting aerodynamic performance characteristics of flight vehicles, while flow devices involving low-speed or incompressible flow could be reasonably well designed without resorting to accurate numerical simulations. As flow devices are required to be more sophisticated and highly efficient CFD took become increasingly important in fluid engineering for incompressible and low-speed flow. This paper reviews some of the successes made possible by advances in computational technologies during the same period, and discusses some of the current challenges faced in computing incompressible flows.

Kwak, Dochan↗
NASA NTRS1990

A study of the topology of dissipating motions in direct numerical simulations of time-developing compressible and incompressible mixing layers

A preliminary investigation of the geometry of flow patterns in numerically simulated compressible and incompressible mixing layers was carried out using 3-D critical point methodology. Motions characterized by high rates of kinetic energy dissipation and/or high enstrophy were of particular interest. In the approach the partial derivatives of the velocity field are determined at every point in the flow. These are used to construct the invariants of the velocity gradient tensor and the rate-of-strain tensor (P, Q, R, and P(sub s), Q(sub s), R(sub s) respectively). For incompressible flow the first invariant is zero. For the conditions of the compressible simulation, the first invariant is found to be everywhere small, relative to the second and third invariants, and so in both cases the local topology at a point is mainly determined by the second and third invariants. The data at every grid point is used to construct scatter plots of Q versus R and Q(sub s) versus R(sub s). Most points map to a cluster near the origin in Q-R space. However, fine scale motions, that is motions which are characterized by velocity derivatives which scale with the square root of R(sub delta), tend to map to regions which lie far from the origin. Definite trends are observed for motions characterized by high enstrophy and/or high dissipation. The observed trends suggest that, for these motions, the second and third invariants of the velocity gradient and rate-of-strain tensors are strongly correlated. The second and third invariants of the rate-of-strain tensor are related by K(-Q(sub s))(exp 3/2), which is consistent with the above scaling of velocity derivatives. The quantity K appears to depend on Reynolds number with an upper limit K = 2(the square root of 3)/9 corresponding to locally axisymmetric flow. For both the compressible and incompressible mixing layer, regions corresponding to high rates of dissipation are found to be characterized by comparable magnitudes of R(sub ij)R(sub ij) and S(sub ij)S(sub ij). For the incompressible mixing layer, regions characterized by the highest values of enstrophy are found to have relatively low strain rates.

Chen, J. H.↗
NASA NTRS1993

Nearly incompressible fluids. II - Magnetohydrodynamics, turbulence, and waves

The theory of nearly incompressible (NI) fluid dynamics developed previously for hydrodynamics is extended to magnetohydrodynamics (MHD). Based on a singular expansion technique, modified systems of fluid equations are obtained for which the effects of compressibility are admitted only weakly in terms of the different possible incompressible solutions. NI MHD represents the interface between the compressible and incompressible magnetofluid descriptions in the subsonic regime. It is shown that three distinct NI descriptions exist corresponding to each of the three possible plasma beta regimes. The detailed theory of weakly compressible corrections to the various incompressible MHD descriptions is presented, and the implications for the solar wind are discussed.

Zank, G. P.↗
NASA NTRS1995

Progress in incompressible Navier-Stokes computations for propulsion flows and its dual-use applications

Development of an incompressible Navier-Stokes solution procedure was performed for the analysis of a liquid rocket engine pump components and for the mechanical heart assist devices. The solution procedure for the propulsion systems is applicable to incompressible Navier-Stokes flows in a steadily rotating frame of reference for any general complex configurations. The computer codes were tested on different complex configurations such as liquid rocket engine inducer and impellers. As a spin-off technology from the turbopump component simulations, the flow analysis for an axial heart pump was conducted. The baseline Left Ventricular Assist Device (LVAD) design was improved by adding an inducer geometry by adapting from the liquid rocket engine pump. The time-accurate mode of the incompressible Navier-Stokes code was validated with flapping foil experiment by using different domain decomposition methods. In the flapping foil experiment, two upstream NACA 0025 foils perform high-frequency synchronized motion and generate unsteady flow conditions for a downstream larger stationary foil. Fairly good agreement was obtained between unsteady experimental data and numerical results from two different moving boundary procedures. Incompressible Navier-Stokes code (INS3D) has been extended for heat transfer applications. The temperature equation was written for both forced and natural convection phenomena. Flow in a square duct case was used for the validation of the code in both natural and forced convection.

Kiris, Cetin↗
NASA NTRS2003

Successes and Challenges of Incompressible Flow Simulation

During the past thirty years, numerical methods and simulation tools for incompressible flows have been advanced as a subset of CFD discipline. Even though incompressible flows are encountered in many areas of engineering, simulation of compressible flow has been the major driver for developing computational algorithms and tools. This is probably due to rather stringent requirements for predicting aerodynamic performance characteristics of flight vehicles, while flow devices involving low speed or incompressible flow could be reasonably well designed without resorting to accurate numerical simulations. As flow devices are required to be more sophisticated and highly efficient, CFD tools become indispensable in fluid engineering for incompressible and low speed flow. This paper is intended to review some of the successes made possible by advances in computational technologies during the same period, and discuss some of the current challenges.

Kwak, Dochan↗
DOE OSTI2025

Bayesian inference of nuclear incompressibility from collective flow in mid-central Au+Au collisions at 400–1500 MeV/nucleon

The incompressibility K of symmetric nuclear matter (SNM) is determined through a Bayesian analysis of collective flow data from Au + Au collisions at beam energies $E = 400 -1500$ MeV/nucleon. This analysis utilizes a Gaussian process (GP) emulator applied to the isospin-dependent quantum molecular dynamics (IQMD) model for heavy-ion collisions, both with and without incorporating the momentum dependence of the single-nucleon potentials. Specifically, at the 68% confidence level, using rapidity and transverse velocity dependence of proton elliptic flow data with and without consideration of the momentum dependence, the inferred incompressibility values are $K=188.9^{+2.9}_{-4.5}$ MeV and $256.1^{+8.2}_{-8.7}$ MeV at $E = 400$ MeV/nucleon, respectively. When the transverse momentum dependence of proton-like directed flow data is included, the inferred incompressibility values become $K=222.3^{+9.0}_{-9.9}$ MeV and $K=285.5^{+6.7}_{-7.3}$ MeV, respectively. Furthermore, we found that the value of K derived from observables of proton elliptic flow increases with beam energy. Finally, this indicates that the equation of state (EoS) of nuclear matter hardens at higher densities and temperatures in reactions with higher beam energies.

Bayesian inference↗
DOE OSTI2023

An $\bar{F}$ Meshfree Treatment for Nearly Incompressible Materials

The computational modeling of nearly incompressible materials is a difficult task for many numerical methods, and even after several decades of investigation, it is still an active research area. This report seeks to address the treatment of incompressible materials in meshfree methods using a synergistic combination of two treatments. The first treatment is an $\bar{F}$ method, where the decomposed dilatational and deviatoric parts are calculated over different smoothing domains. The second treatment “activates” additional nodes throughout the domain to increase the flexibility of the model. We implement this synergistic combination in the context of the reproducing kernel particle method (RKPM) and present results for the Cook’s membrane benchmark problem. The results are compared with those using the composite tet10 finite element with a volume-averaged J formulation. We show that the combined treatment is an effective way to deal with nearly incompressible materials in a meshfree framework and compares well with other highly-effective treatments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A Scaling Study for Incompressible Multispecies Solver in Vertex-CFD

Multispecies incompressible flows occur widely in engineering and environmental applications, such as chemical reactors, fuel cells, ocean mixing, and biomedical systems. However, accurately resolving the complex transport and mixing phenomena associated with multiple interacting species remains computationally challenging, especially for large-scale problems. In this study, we present a robust, high-performance computing--enabled multispecies incompressible Navier–Stokes solver integrated within the Vertex-CFD framework. Our solver employs a fully coupled, implicit, finite element--based formulation that accurately captures the advection, diffusion, and interaction of multiple species in incompressible flows by leveraging the Kokkos library for parallel computing to achieve high computational efficiency. For pressure coupling, the entropically damped artificial compressibility method is utilized. We validated the solver against canonical test cases, including multispecies advection, diffusion, and Bateman systems; the results demonstrate second- and third-order spatial accuracy and consistent convergence. Additionally, we demonstrated the strong and weak scaling study results obtained on the leadership-class high-performance computing system, Frontier at Oak Ridge National Laboratory.

Oz, Furkan [ORNL] (ORCID:0000000265831724)↗
DOE OSTI2024

The inviscid incompressible limit of Kelvin–Helmholtz instability for plasmas

The Kelvin–Helmholtz Instability (KHI) is an interface instability that develops between two fluids or plasmas flowing with a common shear layer. KHI occurs in astrophysical jets, solar atmosphere, solar flows, cometary tails, planetary magnetospheres. Two applications of interest, encompassing both space and fusion applications, drive this study: KHI formation at the outer flanks of the Earth’s magnetosphere and KHI growth from non-uniform laser heating in magnetized direct-drive implosion experiments. Here, we study 2D KHI with or without a magnetic field parallel to the flow. We use both the GAMERA code, which solves the compressible Euler equations, and the STRATOSPEC code, which solves the Navier-Stokes equations under the Boussinesq approximation, coupled with the magnetic field dynamics. GAMERA is a global three-dimensional MHD code with high-order reconstruction in arbitrary nonorthogonal curvilinear coordinates, which is developed for a large range of astrophysical applications. STRATOSPEC is a three-dimensional pseudo-spectral code with an accuracy of infinite order (no numerical diffusion). Magnetized KHI is a canonical case for benchmarking hydrocode simulations with extended MHD options. An objective is to assess whether or not, and under which conditions, the incompressibility hypothesis allows to describe a dynamic compressible system. For comparing both codes, we reach the inviscid incompressible regime, by decreasing the Mach number in GAMERA, and viscosity and diffusion in STRATOSPEC. Here, we specifically investigate both single-mode and multi-mode initial perturbations, either with or without magnetic field parallel to the flow. The method relies on comparisons of the density fields, 1D profiles of physical quantities averaged along the flow direction, and scale-by-scale spectral densities. We also address the triggering, formation and damping of filamentary structures under varying Mach number or Atwood number, with or without a parallel magnetic field. Comparisons show very satisfactory results between the two codes. The vortices dynamics is well reproduced, along with the breaking or damping of small-scale structures. We end with the extraction of growth rates of magnetized KHI from the compressible regime to the incompressible limit in the linear regime assessing the effects of compressibility under increasing magnetic field. The observed differences between the two codes are explained either from diffusion or non-Boussinesq effects.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1984

A solution procedure for three-dimensional incompressible Navier-Stokes equation and its application

An implicit, finite-difference procedure is presented for numerically solving viscous incompressible flows. For convenience of applying the present method to three-dimensional problems, primitive variables, namely the pressure and velocities, are used. One of the major difficulties in solving incompressible flows that use primitive variables is caused by the pressure field solution method which is used as a mapping procedure to obtain a divergence-free velocity field. The present method is designed to accelerate the pressure-field solution procedure. This is achieved by the method of pseudocompressibility in which the time derivative pressure term is introduced into the mass conservation equation. The pressure wave propagation and the spreading of the viscous effect is investigated using simple test problems. The present study clarifies physical and numerical characteristics of the pseudo-compressible approach in simulating incompressible flows. Computed results for external and internal flows are presented to verify the present procedure. The present algorithm has been shown to be very robust and accurate if the selection of the pseudo-compressibility parameter has been made according to the guidelines given.

Kwak, D.↗
NASA NTRS1986

Incompressible boundary-layer stability analysis of LFC experimental data for sub-critical Mach numbers

An incompressible boundary-layer stability analysis of Laminar Flow Control (LFC) experimental data was completed and the results are presented. This analysis was undertaken for three reasons: to study laminar boundary-layer stability on a modern swept LFC airfoil; to calculate incompressible design limits of linear stability theory as applied to a modern airfoil at high subsonic speeds; and to verify the use of linear stability theory as a design tool. The experimental data were taken from the slotted LFC experiment recently completed in the NASA Langley 8-Foot Transonic Pressure Tunnel. Linear stability theory was applied and the results were compared with transition data to arrive at correlated n-factors. Results of the analysis showed that for the configuration and cases studied, Tollmien-Schlichting (TS) amplification was the dominating disturbance influencing transition. For these cases, incompressible linear stability theory correlated with an n-factor for TS waves of approximately 10 at transition. The n-factor method correlated rather consistently to this value despite a number of non-ideal conditions which indicates the method is useful as a design tool for advanced laminar flow airfoils.

Berry, S. A.↗
NASA NTRS1989

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.

Hafez, Mohammed↗
NASA NTRS1991

Faster Algorithm For Computation Of Incompressible Flow

Improved algorithm yields faster numerical solutions of Navier-Stokes equations of steady or unsteady three-dimensional flow of incompressible fluid. In artificial-compressibility method, unsteady flow treated as incompressible in advancing from one time step to next, but at each time step (or in steady state), fluid treated as having variable compressibility enabling propagation of flow field, and subiterations performed in increments of pseudotime until effects of compressibility subside. Directly couples pressure and velocity fields at same time step and converts elliptic incompressible Navier-Stokes equations to hyperbolic form more amenable to numerical integration.

Rogers, S. E.↗